\begin{align*} \sum _ { \substack { z \ \equiv \ l _ 1 \ ( \mathrm { m o d } \ \lceil 2 N ^ { \epsilon } \rceil ) \\ s \ \equiv \ l _ 2 \ ( \mathrm { m o d } \lceil N ^ { \epsilon } \rceil ) } } p ( x - z , t - s ) ^ 2 & \le \sum _ { z \in Z } \sum _ { s = 0 } ^ t p ( z , s ) ^ 2 , \end{align*}


\begin{align*} f _ X ( x , y ) = 0 , \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ { d } } u ( t , x ) \psi ( x ) d x = & \int _ { \mathbb { R } ^ { d } } \phi ( x ) \psi ( x ) d x + \frac { 1 } { 2 } \int _ { t } ^ { T } \int _ { \mathbb { R } ^ { d } } u ( s , x ) \Delta \psi ( x ) d x d s \\ & + \int _ { t } ^ { T } \int _ { \mathbb { R } ^ { d } } u ( s , x ) \psi ( x ) { W } ( d s , x ) d x , \end{align*}


\begin{align*} C = \begin{pmatrix} A _ { 1 3 } & 0 _ { d \times d } \\ 0 _ { d \times d } & - A _ { 2 1 } \end{pmatrix} \ \ \ \ \ \ L = \begin{pmatrix} I _ { d \times d } & I _ { d \times d } - A _ { 1 1 } \\ I _ { d \times d } & - A _ { 1 1 } \end{pmatrix} . \end{align*}


\begin{align*} ( \xi \mid \zeta ) = \sum _ { i < j } \xi _ { i j } \zeta _ { i j } \end{align*}


\begin{align*} f ( B _ t ) = f ( B _ s ) - \int _ { s } ^ { t } \nabla _ X f ( B _ r ) \cdot B _ r \dd r + \frac { 1 } { 2 } \int _ { s } ^ { t } \mathrm { t r } [ \gamma ( B _ r ) ^ T \nabla ^ 2 f ( B _ r ) \gamma ( B _ r ) ] \dd r + \int _ { s } ^ { t } \nabla _ X f ( B _ r ) \gamma ( B _ r ) \dd W _ r \ , , \end{align*}


\begin{align*} \rho _ s ( x ) = ( x / e ) ^ { \sigma } \left ( 1 + \sigma / x \right ) ^ { x + \sigma } \exp \left ( \tilde { F } _ { x + \sigma } ( t ) + i \psi ( t ) \right ) \left ( 1 + O \left ( x ^ { - 1 } \right ) \right ) \end{align*}


\begin{align*} - a + b \frac { \cos 2 \theta } { | \sin 2 \theta | } = 0 . \end{align*}


\begin{align*} \tau _ \varepsilon ^ i \cdot \nu = \nabla _ \Gamma g _ i \cdot \{ ( I _ n - \varepsilon g _ i W ) ^ { - 1 } \nu \} = \nabla _ \Gamma g _ i \cdot \nu = 0 \quad \overline { S _ T } . \end{align*}


\begin{align*} [ b _ x ^ \ast , b _ y ^ \ast ] & = 0 = [ b _ x , b _ y ] , \\ [ b _ x , \ , b ^ \ast _ y ] & = \delta ( x - y ) - \phi ( x ) \overline { \phi ( y ) } : = \widehat \delta ( x - y ) , x , y \in \mathbb { R } ^ 3 . \end{align*}


\begin{align*} \tilde { H } _ n ( x _ 1 \vee \cdots \vee x _ n ) : = - \sum _ { \sigma \in \mathrm { S h } ( 1 , n - 1 ) } \epsilon ( \sigma ) \ ; h x _ { \sigma ( 1 ) } \vee x _ { \sigma ( 2 ) } \vee \cdots \vee x _ { \sigma ( n ) } . \end{align*}


\begin{align*} \frac { \lambda ^ c _ x \left ( t ( \ell + m ) \right ) } { \lambda ^ c _ { x ^ u } \left ( \tau ( \ell + m ) \right ) } = \frac { \lambda ^ c _ x \left ( t ( \ell ) \right ) \lambda ^ c _ { f ^ { t ( \ell ) } ( x ) } \left ( t ( \ell + m ) - t ( \ell ) \right ) } { \lambda ^ c _ { x ^ u } \left ( \tau ( \ell ) \right ) \lambda ^ c _ { f ^ { \tau ( \ell ) } ( x ^ u ) } \left ( \tau ( \ell + m ) - \tau ( \ell ) \right ) } \end{align*}


\begin{align*} T = \mathbb F _ { p ^ r } [ x , y ] / \langle { x ^ { p ^ k } - 1 , f ( y ) } \rangle = \left ( \mathbb F _ { p ^ r } [ x ] / \langle { x ^ { p ^ k } } - 1 \rangle \right ) [ y ] / \langle f ( y ) \rangle , \end{align*}


\begin{align*} \hat \mu _ j ( y ) = ( p _ j , y ) , \ \ y \in H , \ \ j = 1 , 2 . \end{align*}


\begin{align*} \Phi _ { \# } ^ { \infty } : = 2 \alpha ^ { 1 / 2 } ( h _ { \alpha } - e _ { \alpha } + H _ f ) ^ { - 1 } P u _ { \alpha } \otimes A _ { \infty } ^ + \Omega , \Phi _ * ^ { \infty } : = 2 \alpha ^ { 1 / 2 } P u _ { \alpha } \otimes H _ f ^ { - 1 } A _ { \infty } ^ + \Omega , \end{align*}


\begin{align*} \overline { \rm E n t } [ \overline { \mathbf { P } } | \overline { \mathbf { \Pi } } ^ { \lambda } ] = \int _ { \Omega } { \rm E n t } [ \overline { \mathbf { P } } ^ { x } | \mathbf { \Pi } ^ { \lambda ( x ) } ] \ , d x . \end{align*}


\begin{align*} & \mu ^ 3 + \lambda ^ 3 = 1 . \end{align*}


\begin{align*} p _ 1 ( \lambda , \mu ) & = 1 + 2 \lambda + 3 \lambda + 4 \lambda ^ 2 + 5 \lambda \mu + 6 \mu ^ 2 + 7 \lambda ^ 3 + 8 \lambda ^ 2 \mu + 9 \lambda \mu ^ 2 + 1 0 \mu ^ 3 = 0 , \\ p _ 2 ( \lambda , \mu ) & = 1 0 + 9 \lambda + 8 \mu + 7 \lambda ^ 2 + 6 \lambda \mu + 5 \mu ^ 2 + 4 \lambda ^ 3 + 3 \lambda ^ 2 \mu + 2 \lambda \mu ^ 2 + \mu ^ 3 = 0 . \end{align*}


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


\begin{align*} s = \sqrt { \beta ^ 2 + x ^ 2 } \end{align*}


\begin{align*} b _ j = a _ n + b _ { n - 1 } - a _ { n - 1 } + f ( a _ n ) \end{align*}


\begin{align*} Q ( \exp ( \pi ) ) & = \sum _ { n = 0 } ^ \infty \frac { 1 } { n ! } \sum _ { k = 0 } ^ n \binom { n } { k } Q _ k ^ 1 ( \pi \vee \cdots \vee \pi ) \vee \pi \vee \cdots \vee \pi \\ & = Q ^ 1 ( \exp ( \pi ) ) \vee \exp ( \pi ) \end{align*}


\begin{align*} \Lambda ( s , \xi _ { \ell } ) = | D | ^ { s / 2 } ( 2 \pi ) ^ { - s } \Gamma ( s + | \ell | / 2 ) L ( s , \xi _ { \ell } ) \end{align*}


\begin{align*} Y _ 1 = 0 , \cdots , Y _ { n - r } = 0 . \end{align*}


\begin{align*} a _ { n + k _ { \beta } + j } { } = { } \left \{ \begin{array} { l l } \beta ^ { j } a _ { n + k _ { \beta } } & \textrm { i f $ j = 1 , \ldots , l $ } , \\ \displaystyle \sum _ { i = l } ^ { j - 1 } \beta ^ { j - i - 1 } a _ { n + i } + \beta ^ { j } a _ { n + k _ { \beta } } & \textrm { i f $ j = l + 1 , \ldots , t $ } . \end{array} \right . \end{align*}


\begin{align*} \breve { M } = B ( x , t , k ) \breve { M } ^ { r } ( x , t , k ) \begin{pmatrix} 1 & 0 \\ 0 & \frac { k - i \kappa } { k } \end{pmatrix} , k \in \mathbb { C } , \end{align*}


\begin{align*} m ( a ) = [ a ] \ ; . \end{align*}


\begin{align*} \tanh ( d _ H ( 0 , \gamma ( t ) ) ) = \frac { \tanh ( d ) } { \cos ( \beta ) } \leq \frac { \tanh ( d ) } { \cos ( \bar \beta ) } = \tanh ( d _ H ( 0 , \gamma ( \bar t ) ) ) . \end{align*}


\begin{align*} v = a ^ l \ , , \partial _ n v = \partial _ n a ^ l \quad \textrm { o n } \Gamma ^ l \ , , \quad \textrm { f o r e v e r y } l = 0 , 1 , \ldots , L \ , , \end{align*}


\begin{align*} L = \sup _ { | x | , | y | \leq R , x \neq y , t \neq s } \frac { U ^ { x , y } _ { s t } } { ( x - y ) ^ { \bar { \delta } } ( t - s ) ^ { \bar { \alpha } } } < \infty \end{align*}


\begin{align*} u _ i = n '^ { - 1 } a _ i \beta _ b ^ { - n ' } \prod _ { 1 \leq s \leq t , s \neq b } ( \beta _ b ^ { n ' } - \beta _ s ^ { n ' } ) ^ { - 1 } . \end{align*}


\begin{align*} u _ n ( t ) - u _ n ( s ) = \int _ { s } ^ { t } \Delta u _ n ( r ) d r + \int _ { s } ^ { t } P _ n \Big [ u _ n ( r ) \log \big ( u _ n ( r ) \big ) \Big ] d r + \int _ { s } ^ { t } P _ n \Big [ \sigma \big ( u _ n ( r ) \big ) \Big ] h ( r ) d r . \end{align*}


\begin{align*} { \rm i n t } [ \mathbf { P } ] : = \mathbf { E } _ { \mathbf { P } } [ | C | \square _ { 1 } ] . \end{align*}


\begin{align*} \tilde D : = \tilde T ^ { - 1 } V = \{ x \in V ^ 1 : T ^ { \# } x \in ( j ^ * \circ c ) V \} . \end{align*}


\begin{align*} u _ { t } + 2 u u _ { x } = 0 , u _ { t } - 2 v u _ { x x } = 0 \end{align*}


\begin{align*} U _ k d & = \begin{bmatrix} d _ n u _ { k - n + 1 } + \dots + d _ 0 u _ { k + 1 } \\ \vdots \\ d _ n u _ { k - n + L - 1 } + \dots + d _ 0 u _ { k + L - 1 } \end{bmatrix} \\ & = ( T \otimes I _ m ) u _ { [ k - n + 1 , k + L - 1 ] } \end{align*}


\begin{align*} \| u \| _ { \mathcal { H } } = \inf \left \{ \tau > 0 : \rho _ { \mathcal { H } } \left ( \frac { u } { \tau } \right ) \leq 1 \right \} . \end{align*}


\begin{align*} c _ 0 ^ 3 = c _ \infty ^ 3 + \frac { M ( \phi ) } { j _ M ^ * } . \end{align*}


\begin{align*} { \sf H } = { \sf U } _ { 2 n } { \rm o r } { \sf S p } _ { 2 n } \end{align*}


\begin{align*} M ^ * = \begin{cases} M \cup \{ a _ 1 , a _ 2 \} , & ; \\ ( M \cup \{ a _ 1 \} ) \smallsetminus \{ a _ 2 \} , & . \end{cases} \end{align*}


\begin{align*} & \# \left \{ \{ x _ h \in \mathbb { Z } _ p \mid s _ { h } x _ h \ne 0 \} \bigcap \{ x _ h \in \mathbb { Z } _ q \mid x _ h \ne x _ l , \ \ l < h \} \right \} \\ = \ , & q - \# \left \{ \{ x _ h \in \mathbb { Z } _ p \ | \ s _ { h } x _ h = 0 \} \bigcup \{ x _ 1 , \ldots , x _ { h - 1 } \} \right \} \\ = \ , & q - d _ h - h + 1 . \end{align*}


\begin{align*} \binom { 2 n - 1 } { 4 } - s + d . \end{align*}


\begin{align*} \pi ^ \sharp = \sum _ { ( T ^ * ) } \frac { \langle \pi , R ^ G _ { T , s } \rangle _ { G ^ F } } { \langle R ^ G _ { T , s } , R ^ G _ { T , s } \rangle _ { G ^ F } } R ^ G _ { T , s } , \end{align*}


\begin{align*} r = \frac { p ^ 2 ( p - n ) } { q ( p - n ) + p ^ 2 } , \quad \frac { r } { p - r } = \left ( \frac { q } { p } + \frac { n } { p - n } \right ) ^ { - 1 } . \end{align*}


\begin{align*} \Delta ( y _ 1 , y _ 2 ) = ( v _ 2 , w _ 2 ) . \end{align*}


\begin{align*} \xi _ 1 ( x _ 1 ) \cdots \xi _ d ( x _ d ) = \sum _ { k = 0 } ^ n \langle \delta _ { \gamma _ { x _ 1 \cdots x _ d v _ 0 } ( k ) } , \delta _ { \gamma _ { v _ 0 } ( n - k ) } \rangle = \varphi _ n ( x _ 1 \cdots x _ d ) \end{align*}


\begin{align*} E ^ * ( M ^ * ( t ) u ) = E ^ * ( M ^ * ( s ) u ) \end{align*}


\begin{align*} \delta _ { k - 1 } : = \tau _ { k - 1 } - \tau _ k \ , \delta _ 0 : = s - \tau _ 1 \ , . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ n m _ i \psi _ j ( \Phi ( P _ i ) ) = 0 , \ \ \ \ \ \ j \in \{ 1 , \ldots , d \} . \end{align*}


\begin{align*} b _ i \big ( a _ 0 \otimes \ldots \otimes a _ k \big ) = \begin{cases} a _ 0 \otimes \ldots \otimes a _ i a _ { i + 1 } \otimes \ldots \otimes a _ k & , \\ a _ k a _ 0 \otimes \ldots \otimes a _ { k - 1 } & , \end{cases} \end{align*}


\begin{align*} g < 0 , \ , \ , \partial _ \eta g = 0 , \ , \ , \partial _ \tau g \leq 0 , \ , \ , \partial _ \xi g \leq 0 , \ , \ , w ^ 2 \partial _ { \eta } ^ 2 g \geq 0 a t z _ { m i n } , \end{align*}


\begin{align*} { z } ^ { * } = \int _ { \mathbb { R } ^ { d } } \exp \left ( - V ( x ) \right ) \ , d x . \end{align*}


\begin{align*} \mathrm { T r } \left ( \alpha ^ { v + l } \left ( y _ { 0 } \alpha ^ { j _ { 0 } } + \cdots + y _ { t - 1 } \alpha ^ { j _ { t - 1 } } \right ) \right ) = 0 . \end{align*}


\begin{align*} \Omega _ t : = \bigcup ^ N _ { k = 1 } B ( z _ k , t ) , \ \ \ 0 < t < \delta _ 0 . \end{align*}


\begin{align*} \begin{multlined} M _ { \otimes 0 } ( A ) ( ( \vec { \alpha } _ i , \vec { \alpha } _ o ) , ( \vec { \beta } _ i , \vec { \beta } _ o ) ) \\ = ( - 1 ) ^ { | \vec \alpha _ o | | \vec \beta _ i | } A ( ( \vec \alpha _ i \sqcup \vec \beta _ i , \vec \alpha _ o \sqcup \vec \beta _ o ) ) \ ; , \end{multlined} \end{align*}


\begin{align*} U ( A ) ( x ) = A ( \bar { \Phi } _ \times ^ U ( x ) ) \ ; . \end{align*}


\begin{align*} \mathcal { F } ( \Delta ) = \big \{ [ n ] \setminus F _ { p } : p = 1 , \dots , d \big \} \cup \mathcal { F } . \end{align*}


\begin{align*} \mathbb { Q } ( S , 2 ) = \{ b \in \mathbb { Q } ^ { * } / \mathbb { Q } ^ { * 2 } : v _ { p } ( b ) \equiv 0 \pmod 2 p \notin S \} \end{align*}


\begin{align*} \log Z ^ { a , b } _ { m , n } = W _ { n , m } + N _ { n , m } = S _ { n , m } + E _ { n , m } \end{align*}


\begin{align*} e ^ { i t \Delta } u _ 0 ( x ) = \tfrac { A } { 1 + i t } \exp \bigl \{ - \tfrac { | x | ^ 2 } { 4 ( 1 + i t ) } \bigr \} . \end{align*}


\begin{align*} \Sigma ( \kappa ; \eta , P ) : = \sum _ { T < \gamma \le 2 T } Z ' ( \gamma ) \ , Z \Big ( \gamma + \frac { 2 \pi \kappa } { \log T } \Big ) \ , \Big | M \Big ( \frac 1 2 + i \gamma + \frac { 2 \pi i \eta } { \log T } , P \Big ) \Big | ^ 2 < 0 \end{align*}


\begin{align*} \frac { \partial } { \partial \overline { \tau } } \frac { v ^ 2 } { Q ( \overline { \tau } , 1 ) } = \frac { i v ^ 2 Q _ { \tau } } { Q ( \overline { \tau } , 1 ) ^ 2 } , 2 i v ^ 2 \frac { \partial } { \partial \overline { \tau } } Q _ { \tau } = Q ( \tau , 1 ) , i v ^ 2 \frac { \partial } { \partial \overline { \tau } } \frac { Q ( \overline { \tau } , 1 ) } { v ^ 2 } = Q _ { \tau } . \end{align*}


\begin{align*} S ^ { ( 2 ) } ( E _ { p } / \mathbb { Q } ) = \begin{cases} ( 1 , \pm 1 ) E _ { p } [ 2 ] \cup ( 1 , \pm p ) E _ { p } [ 2 ] \cong ( \mathbb { Z } / 2 \mathbb { Z } ) ^ { 4 } , & p \equiv 1 \mod 8 , \\ ( 1 , \pm 1 ) E _ { p } [ 2 ] \cong ( \mathbb { Z } / 2 \mathbb { Z } ) ^ { 3 } , & p \equiv 5 \mod 8 . \end{cases} \end{align*}


\begin{align*} \dim { W _ 1 \cap W _ 2 } \ , = \ , \dim W _ 1 + \dim W _ 2 - n . \end{align*}


\begin{align*} \delta _ 3 = & \phi ( \phi _ { s s } \phi _ { z z } - \phi _ { s z } ^ 2 ) + \phi _ s ( \phi _ s \phi _ { z z } - \phi _ z \phi _ { s z } ) + \phi _ z ( \phi _ { s s } \phi _ z - \phi _ s \phi _ { s z } ) \\ = & - \det \left ( \begin{array} { c c c } - \phi & \phi _ s & \phi _ z \\ \phi _ s & \phi _ { s s } & \phi _ { s z } \\ \phi _ z & \phi _ { s z } & \phi _ { z z } \end{array} \right ) , \end{align*}


\begin{align*} d s ^ { 2 } = \sum _ { \alpha , \beta = 1 } ^ { 3 N - 7 } N _ { \alpha \beta } d s ^ { \alpha } d s ^ { \beta } + \sum _ { i = 1 } ^ { N - 1 } \mid \pmb { r } _ { i } \mid ^ { 2 } ( d \lambda ) ^ { 2 } \end{align*}


\begin{align*} \mu ^ \lambda _ { s , t } = \tau _ { \lambda X } \mu _ { s , t } , \end{align*}


\begin{align*} \Gamma ' ( X Y ) = P Q = Q P + C \neq Q P = \Gamma ( X Y ) . \end{align*}


\begin{align*} [ x _ 0 , \ldots , x _ m ] f = \frac { f ^ { ( m ) } ( x _ 0 ) } { m ! } \hbox { i f $ x _ 0 = x _ 1 = \cdots = x _ m $ } . \end{align*}


\begin{align*} v ( t , x ) : = \ , \bar u \left ( X ( t , 0 , \cdot ) ^ { - 1 } ( x ) \right ) x \in \R ^ n \ , , \end{align*}


\begin{align*} \int _ { Q _ T } - G \omega ' ( p ) f ' ( u ) | \nabla p | ^ 2 \leq B \int _ { Q _ T } \omega ' ( p ) f ' ( u ) | \nabla p | ^ 2 = - B \int _ { Q _ T } \omega ( p ) \nabla \cdot ( f ' ( u ) \nabla p ) \end{align*}


\begin{align*} E \Big ( A _ N ^ { ( 1 ) } ; A _ N ^ { ( 2 ) } \Big ) = \O \Big ( N ^ { 3 } \exp { \big ( - ( \log { N } ) ^ { \frac { 1 } { 2 } + \delta } \big ) } \Big ) , \ , \mbox { f o r s o m e } \delta > 0 . \end{align*}


\begin{align*} \Sigma _ 0 : \begin{cases} x _ { k + 1 } & = A x _ k + B u _ k , \\ y _ k & = C x _ k + D u _ k \end{cases} \end{align*}


\begin{align*} u ( x , t ) + 6 u ^ 2 ( x , t ) u _ { x } ( x , t ) + u _ { x x x } ( x , t ) = 0 \end{align*}


\begin{align*} \left \{ \begin{alignedat} { 2 } - \Delta u & = a ( x ) u ^ { - \gamma } \ ; \ ; & & \mbox { i n } \ ; \ ; \Omega , \\ u & > 0 & & \mbox { i n } \ ; \ ; \Omega , \\ u & = 0 & & \mbox { o n } \ ; \ ; \partial \Omega . \end{alignedat} \right . \end{align*}


\begin{align*} \psi ^ { - 1 } \left ( \overline { J R _ { \operatorname { r e d } } } \left ( R _ { \operatorname { r e d } } / ( x ) R _ { \operatorname { r e d } } \right ) \right ) = \psi ^ { - 1 } \left ( \overline { J \left ( R _ { \operatorname { r e d } } / ( x ) R _ { \operatorname { r e d } } \right ) } \right ) . \end{align*}


\begin{align*} D f ( x ) v ^ * ( x ) = \lambda ^ * _ x v ^ * ( x _ 1 ) , \end{align*}


\begin{align*} L _ { H } \left [ p \right ] = { \displaystyle \int \limits _ { - \infty } ^ { \infty } } p \left ( x \right ) e ^ { - x ^ { 2 } } d x , p \in \mathbb { R } \left [ x \right ] . \end{align*}


\begin{align*} \| x ( t , s _ 0 , x _ 0 ) \| = \| x ( t ) \| < \varepsilon , \end{align*}


\begin{align*} B ^ 4 ( a ) = \left \{ z \in \C ^ 2 \ ; \big | \ ; \pi | z | ^ 2 \le a \right \} \end{align*}


\begin{align*} M = \{ x \in S : f ( x ) < 1 \} , \end{align*}


\begin{align*} Y _ s ^ { t , x } = \phi ( B _ { T - t } ^ { t , x } ) + \int _ s ^ T Y _ r ^ { t , x } W ( d r , B _ { r - t } ^ { t , x } ) - \int _ s ^ t Z ^ { t , x } _ r d B _ r \ , , \ s \in [ t , T ] . \end{align*}


\begin{align*} m ( \pi , \sigma ) = \langle \pi \otimes \omega _ \psi ^ \vee , \sigma \rangle _ { G ^ F } \leq 2 ^ n ( n ! ) ^ 2 . \end{align*}


\begin{align*} \begin{cases} M _ 1 ( k ) M _ 2 ( k ) & \\ M _ 1 ( k ) ( k + \frac { \bar h _ 1 ^ \vee } { 2 } + 1 ) & \end{cases} \end{align*}


\begin{align*} \cos ( \alpha + \mu \pi / n ) = \cos ( \alpha ) \cos ( \mu \pi / n ) - \sin ( \alpha ) \sin ( \mu \pi / n ) , \end{align*}


\begin{align*} \frac { \sum _ { \xi _ { k + 1 } = - } \tau _ \xi p ^ { \xi } _ i } { \sum _ { \xi _ { k + 1 } = - } \tau _ \xi q ^ { \xi } _ i } = \frac { p _ i } { q _ i } 1 \leq i \leq k . \end{align*}


\begin{align*} \overline { ( \phi ( a ) | \phi ( b ) ) } = ( a | b ) , \end{align*}


\begin{align*} \rho _ 0 = \Tilde { \rho } _ 1 \wedge e ^ { i \phi ^ * ( \omega _ R ) } . \end{align*}


\begin{align*} [ A ] _ V = A _ V / \sim / V _ { } \ ; , \end{align*}


\begin{align*} H \varphi = E \varphi , \ E < 0 . \end{align*}


\begin{align*} \zeta _ { j i } \left ( \frac { w } { z } \right ) \psi ^ \epsilon _ i ( z ) e _ j ( w ) = \zeta _ { i j } \left ( \frac { z } { w } \right ) e _ j ( w ) \psi ^ \epsilon _ i ( z ) \ , , \end{align*}


\begin{align*} d X ^ \epsilon ( t ) = \sigma ( X ^ \epsilon ( t ) ) d \zeta ^ \epsilon ( t ) \end{align*}


\begin{align*} R _ 2 = ( 2 H ( y , \hat \eta _ 0 ) ) ^ { - \frac { 1 } { 2 } } \Big ( 2 g _ 0 \hat \eta _ 1 , \frac { \partial \hat \eta _ 1 } { \partial \tau } \Big | _ { \tau = 0 } \Big ) = \alpha ( g _ 1 - g _ 0 ) , \end{align*}


\begin{align*} x = \tfrac { \sqrt { - 1 } } { 2 } ( X _ \theta - X _ { - \theta } ) , e = \tfrac { 1 } { 2 } ( X _ \theta + X _ { - \theta } + \sqrt { - 1 } h _ \theta ) , f = \tfrac { 1 } { 2 } ( X _ \theta + X _ { - \theta } - \sqrt { - 1 } h _ \theta ) . \end{align*}


\begin{align*} h _ \tau ( t ) : = \int _ { 0 } ^ t \left ( g _ \tau ^ 2 ( s ) - 1 \right ) \ d s , \end{align*}


\begin{align*} \cos G ( \theta ) = - \dfrac { 1 + \bar B \ , \tan \theta } { \sqrt { 1 + \bar C \ , \tan ^ 2 \theta + 2 \ , \bar B \ , \tan \theta } } \end{align*}


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


\begin{align*} e _ { i , j } ^ { + } ( u ) \mid 0 \rangle = 0 f o r \ a l l i < j . \end{align*}


\begin{align*} G _ i : = \left \{ \left ( m A r _ i , 0 , n r _ i ^ 2 \right ) : m , n \in \Z \right \} , \end{align*}


\begin{align*} c ( \sigma ^ { 2 } , \alpha ) = c \big ( \sigma , \sigma ( \alpha ) \big ) \cdot c ( \sigma , \alpha ) = c ( \sigma , \alpha ) ^ { 2 } , \end{align*}


\begin{align*} 0 = \partial _ \eta G _ { s u m } = \partial _ \eta g _ { s u m } a t z _ { m a x } . \end{align*}


\begin{align*} \phi _ { A , i , j } \ , = \ , \pi ^ { i } | _ { C _ i ^ j } : C _ i ^ j \rightarrow A \cap X \end{align*}


\begin{align*} q ^ \epsilon ( t ) = Q ^ \epsilon ( t ) - P ^ \epsilon ( t ) . \end{align*}


\begin{align*} \operatorname * { R e s } _ { \mathrm { i } \kappa _ { n } } \varphi \left ( k \right ) = \mathrm { i } c _ { n } ^ { 2 } \left ( t \right ) \psi \left ( \mathrm { i } \kappa _ { n } \right ) , \operatorname * { R e s } _ { - \mathrm { i } \kappa _ { n } } \varphi \left ( k \right ) = - \mathrm { i } c _ { n } ^ { 2 } \left ( t \right ) \psi \left ( \mathrm { i } \kappa _ { n } \right ) , \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*} \int _ { \mathbb { D } } \mathrm { R e } \ , h = 0 , \end{align*}


\begin{align*} \mathcal S = ( \Omega , { \mathcal F } , P ) \end{align*}


\begin{align*} \max _ { i , j = 1 , \dots , n } | a _ { i j } ^ \varepsilon ( x , t ) | \leq c _ 1 , \max _ { i = 1 , \dots , n } | b _ i ^ \varepsilon ( x , t ) | \leq c _ 1 , | c ^ \varepsilon ( x , t ) | \leq c _ 1 \end{align*}


\begin{align*} \left \{ \begin{aligned} \dot { x } ( t ) & \in - N _ { C ( t ) } ( x ( t ) ) \textrm { a . e . } t \in [ a , b ] , \\ x ( 0 ) & = x _ 0 \in C ( 0 ) . \end{aligned} \right . \end{align*}


\begin{align*} \pi ^ { - \frac { r s } { 2 } } \Gamma \left ( \frac { r s } { 2 } \right ) \zeta ( r s ) E ^ P _ { s , \varphi } ( g ) = \pi ^ { - \frac { r } { 2 } ( 1 - s ) } \Gamma \left ( \frac { r ( 1 - s ) } { 2 } \right ) \zeta \left ( r - r s \right ) { E ^ P _ { 1 - s , \varphi } \left ( ( g ^ \top ) ^ { - 1 } \right ) } . \end{align*}


\begin{align*} S _ \iota : = \{ s \in S : G _ s = G _ \iota \} , \end{align*}


\begin{align*} = \left \langle \ , A ( { \bf 0 } ) , \ , z _ n ( t ) \ , \right \rangle _ E + \int _ 0 ^ { t } \ , d \tau \ , \left \langle \ , A \left ( z _ n ( \tau ) \right ) - A ( { \bf 0 } ) , \ , \dot { z } _ n ( \tau ) \ , \right \rangle _ E \end{align*}


\begin{align*} \tilde { \nabla } _ E T _ { \theta } e _ 3 = \omega _ { 3 4 } ( E ) T _ { \theta } e _ 4 + \frac { \kappa _ 2 } { \kappa _ 1 } T _ { \theta } e _ 5 - \frac { i \mu _ 2 } { \kappa _ 1 } T _ { \theta } e _ 6 - \kappa _ 1 e ^ { i \theta } d g _ { \theta } ( \overline { E } ) , \end{align*}


\begin{align*} E _ n = \{ M \in _ { n \times n } ( k ) \ , | \ , M ^ 2 = M \} \end{align*}


\begin{align*} \tau = \frac { \log ( ( 1 - \alpha - \delta - \beta ) t + 1 ) } { 1 - \alpha - \delta - \beta } , \end{align*}


\begin{align*} & \int _ { - \gamma _ \epsilon } ^ 0 \phi _ \epsilon ( t ) \ , d t \ , + \ , \int _ { 0 } ^ { \gamma _ \epsilon } \phi _ \epsilon ( t ) \ , d t \ , = \ , 0 \end{align*}


\begin{align*} [ \psi ^ \epsilon ( z ) , \psi ^ { \epsilon ' } ( w ) ] = 0 \ , , \psi ^ \pm _ { \mp b ^ \pm } \cdot ( \psi ^ \pm _ { \mp b ^ \pm } ) ^ { - 1 } = ( \psi ^ \pm _ { \mp b ^ \pm } ) ^ { - 1 } \cdot \psi ^ \pm _ { \mp b ^ \pm } = 1 \ , , \end{align*}


\begin{align*} { \mathcal { F } } _ { \tilde { \mathcal { D } } } ( J , J ' _ { \sigma , k } ) = \frac { { \mathcal { A } } _ { j ' _ { \sigma ( k ) } } ^ { J \setminus J ' _ { \sigma , k } } } { x _ { 1 } - x _ { j ' _ { \sigma ( k ) } } + 2 } - \frac { \tilde { \mathcal { D } } _ { j ' _ { \sigma ( k ) } } ^ { J \setminus J ' _ { \sigma , k } } } { x _ { 1 } - x _ { j ' _ { \sigma ( k ) } } + 1 } \ , . \end{align*}


\begin{align*} 2 \lambda _ 1 \int _ { 0 } ^ { T } \int _ { D } \partial _ x ( u _ r \times g ( u _ r ) ) \cdot \partial _ x u _ r \dd x \dd r & = - 2 \lambda _ 1 \int _ { 0 } ^ { T } \int _ { D } ( u _ r \times g ( u _ r ) ) \cdot \partial _ x ^ 2 u _ r \dd x \dd r \\ & = - 2 \lambda _ 1 \int _ { 0 } ^ { T } \int _ { D } ( u _ r \times g ( u _ r ) ) \cdot ( u _ r \times \partial _ x ^ 2 u _ r ) \dd x \dd r \ , . \end{align*}


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


\begin{align*} u ( x , t ) = A + O \left ( t ^ { - \frac { 1 } { 2 } } e ^ { - 8 t \kappa _ { \delta } ( 3 \xi - \kappa _ { \delta } ^ 2 ) } \right ) . \end{align*}


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


\begin{align*} J _ 1 \ge \frac 1 2 \frac d { d t } \sum \limits _ { | \alpha | = 0 } ^ { 2 } \left \| \sqrt { B } \partial _ \tau ^ \alpha \partial _ y u \right \| _ { L ^ 2 } ^ 2 - C D ( t ) ^ { \frac 1 2 } E ( t ) . \end{align*}


\begin{align*} v = u _ { s } + u _ 0 + \sum _ { l \in \{ 1 , . . . , s - 1 \} \cap I _ A } \sigma _ { A , s } ( l ) u _ l \end{align*}


\begin{align*} \begin{gathered} \bullet \otimes \bullet = \bullet \ ; , \\ \bullet \otimes \star = \star \otimes \bullet = \star \otimes \star = \star \ ; . \end{gathered} \end{align*}


\begin{align*} \mathcal { H } _ \lambda & : = \{ f \in H : \sum _ { k \in \N } \lambda _ k | \hat { f } _ k | ^ 2 < \infty \} \end{align*}


\begin{align*} \begin{aligned} & | H _ t - F _ 0 ( D ^ 2 H , x , t ) - P _ f | \leq C | ( x , t ) | ^ { k } , \\ & H ( x ' , 0 , t ) \equiv P _ g ( x ' , 0 , t ) , \end{aligned} \end{align*}


\begin{align*} \nu _ { s , t } ( x - y ) : = \int _ { \R ^ d } \rho _ { s , t } ( x - z , y - z ) d z , \end{align*}


\begin{align*} F \left [ \begin{array} { c } \left \{ \textbf { a } , \textbf { m } \right \} \\ \left \{ \textbf { b } , \textbf { n } \right \} \end{array} \ ; \middle | \ ; \{ \textbf { x } \} \right ] = \sum _ { s _ 1 = 0 } ^ { \infty } \dots \sum _ { s _ r = 0 } ^ { \infty } \frac { ( \textbf { a } ) _ { \textbf { m } \cdot \textbf { s } } } { ( \textbf { b } ) _ { \textbf { n } \cdot \textbf { s } } } \frac { \textbf { x } ^ \textbf { s } } { \textbf { s } ! } \end{align*}


\begin{align*} \begin{aligned} \circ _ { s , m , t } : \hom ( s , m ) \times \hom ( m , t ) & \rightarrow \hom ( s , t ) \ ; , \\ ( g , f ) & \mapsto g \circ f \ ; . \end{aligned} \end{align*}


\begin{align*} \lVert { g _ z } \rVert _ q = m _ p ( z ) ^ { - 1 } = K _ p ( z ) ^ { 1 / p } = \lVert { L _ z } \rVert , \end{align*}


\begin{align*} \sum _ x ( g ( u ) ^ G _ i ) _ { x y } = \sum _ x \sum _ j u _ { x _ i , y _ j } = \sum _ j \sum _ x u _ { x _ i , y _ j } = \sum _ j \sum _ w u _ { v _ i , w _ j } = 1 , \end{align*}


\begin{align*} u _ t ( x ) = \frac { 1 } { 1 - \frac { e x } { t } } \quad v _ t ( x ) = t \log \frac { t } { e x } \end{align*}


\begin{align*} & c h \ , M ^ W ( \L ) = e ^ \nu q ^ { \ell } F ^ { N S } ( q ) , \end{align*}


\begin{align*} q \alpha ^ 2 + ( 3 p - 3 q + 1 ) \alpha + ( 2 q - 4 p - 1 ) = 0 \end{align*}


\begin{align*} T I ( G ) = \sum _ { i j \in E } F ( d _ i , d _ j ) , \end{align*}


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


\begin{align*} M u _ { [ k - n , k ] } = & \bar { M } _ r u _ { [ k - n , k - r ] } \end{align*}


\begin{align*} T ( \R ^ n \setminus E ) = \left [ T ( \R ^ n ) \setminus T ( E ) \right ] \cup \left [ T ( \R ^ n \setminus E ) \cap T ( E ) \right ] . \end{align*}


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


\begin{align*} v _ j ( t , x ) = \bar u _ j ( X ( 0 , t , x ) ) . \end{align*}


\begin{align*} \frac { \Delta t m _ { n + 1 } ^ \epsilon } { \epsilon } & = \Delta \beta _ n + \epsilon ( m _ n ^ { \epsilon , \Delta t } - m _ { n + 1 } ^ { \epsilon , \Delta t } ) \\ \zeta ^ \epsilon ( t _ { n + 1 } ) - \zeta ^ \epsilon ( t _ n ) & = \int _ { t _ n } ^ { t _ { n + 1 } } \frac { m ^ \epsilon ( t ) } { \epsilon } d t = \beta ( t _ { n + 1 } ) - \beta ( t _ n ) + \epsilon ( m ^ \epsilon ( t _ n ) - m ^ \epsilon ( t _ { n + 1 } ) ) \end{align*}


\begin{align*} \int _ { \R ^ N } | \nabla u | ^ { p - 2 } \nabla u \nabla \phi \ , { \rm d } x & = \int _ { \R ^ N } f ( \cdot , u , v , \nabla u , \nabla v ) \phi \ , { \rm d } x , \\ \int _ { \R ^ N } | \nabla v | ^ { q - 2 } \nabla v \nabla \psi \ , { \rm d } x & = \int _ { \R ^ N } g ( \cdot , u , v , \nabla u , \nabla v ) \psi \ , { \rm d } x ; \end{align*}


\begin{align*} \frac { \partial } { \partial w } \Omega _ { k + 1 , D } ( \tau , w ) = 0 . \end{align*}


\begin{align*} \delta : = \frac { 1 } { \mathtt { m } } \| \rho \| _ { s } + \frac { 1 } { \sqrt { \kappa } } \| \Pi _ 0 ^ { \bot } \phi \| _ { s } \ , , \sigma : = \Pi _ 0 \phi \ , . \end{align*}


\begin{align*} \dim ( \mathbb M ^ r _ X \otimes _ R K ) = ( r + 1 ) \frac { g ( g + 1 ) } { 2 } + r . \end{align*}


\begin{align*} \dot z ^ { \perp } = \Lambda z ^ { \perp } + \Pi ^ { \perp } X _ { Z _ 2 } ( z ^ { \leq } , z ^ { \perp } ) \ ; \end{align*}


\begin{align*} \Vert D f \Vert ( A ) : = \inf \{ \Vert D f \Vert ( W ) : \ , A \subset W , \ , W \subset X \} . \end{align*}


\begin{align*} \lim _ { r \to \infty } K _ { \mu } ( r ) \sqrt { r } e ^ r = \sqrt { \pi / 2 } , \end{align*}


\begin{align*} f ( a _ n ) = \begin{cases} [ \beta ] & n - 1 \in X ^ c \cap Y \\ [ \beta ] - 2 & n - 1 \in X \cap Y ^ c \\ [ \beta ] - 1 & \end{cases} \end{align*}


\begin{align*} \tau _ \varepsilon ^ i ( y , t ) = \{ I _ n - \varepsilon g _ i ( y , t ) W ( y , t ) \} ^ { - 1 } \nabla _ \Gamma g _ i ( y , t ) , ( y , t ) \in \overline { S _ T } , \ , i = 0 , 1 . \end{align*}


\begin{align*} m ( a ) = \bigsqcup _ { l \in L ^ { a _ X } } a _ M ( l ) \ ; . \end{align*}


\begin{align*} A ( x ) \langle M \rangle : = \big \{ A ( x ) \langle y \rangle ~ : ~ y \in M \big \} . \end{align*}


\begin{align*} R _ { a v g } ( i ) = \frac { 1 } { B } \sum _ { b } \hat { \mathbf { R } } _ { D , b } ( i , i ) , \\ R _ { m a x } ( i ) = \max _ b \hat { \mathbf { R } } _ { D , b } ( i , i ) , \\ R _ { m i n } ( i ) = \min _ b \hat { \mathbf { R } } _ { D , b } ( i , i ) \end{align*}


\begin{align*} \widetilde { h } ( x ) : = \limsup _ { i \to \infty } \sum _ { l = i } ^ { N _ i } a _ { i , l } h _ l ( x ) , x \in U . \end{align*}


\begin{align*} V ^ { \times d } & \longrightarrow V ^ { \times r } , \ ; \ ; \begin{bmatrix} v _ 1 \\ \vdots \\ v _ { d } \end{bmatrix} \longmapsto A \begin{bmatrix} v _ 1 \\ \vdots \\ v _ { d } \end{bmatrix} . \end{align*}


\begin{align*} \omega ^ 2 - Q _ 2 ^ 2 = x _ 3 Q _ 1 \end{align*}


\begin{align*} \sum _ { 1 \le u \le n } \frac { [ x ^ { n - \Sigma _ { \mathrm { k } } - u } ] S ( x ) } { [ x ^ n ] S ( x ) } [ x ^ u ] T _ 1 ( x ) = \mathcal { O } \bigg ( z _ n ^ { \Sigma _ { \mathrm { k } } } \sum _ { u \ge 1 } [ x ^ u ] T _ 1 ( x ) z _ n ^ u \bigg ) = \mathcal { O } \big ( z _ n ^ { \Sigma _ { \mathrm { k } } } T _ 1 ( z _ n ) \big ) = o \big ( z _ n ^ { \Sigma _ { \mathrm { k } } } \big ) . \end{align*}


\begin{align*} \frac { d \varphi _ { s , t } ( z ) } { d t } = G ( \varphi _ { s , t } ( z ) , \ , t ) , \varphi _ { s , s } ( z ) = z \end{align*}


\begin{align*} r _ 1 \theta _ 1 + r _ 2 \theta _ 2 + \ldots + r _ n \theta _ n = 0 \mod 1 , \end{align*}


\begin{align*} D _ m ( X , Y ) : = \min _ { \pi \in S _ n } \max _ { j = 1 } ^ n | x _ j - y _ { \pi ( j ) } | , \end{align*}


\begin{align*} S _ { q , r } ( m ) = p \left ( q - 1 \right ) q ^ { v - 1 } w + r \in \Omega _ p \end{align*}


\begin{align*} ( \beta + d ) ( 1 - \delta _ 0 ) = ( \alpha + d ) . \end{align*}


\begin{align*} \| z _ \alpha ( t ) \| _ { 0 } = \| z _ \alpha ( t _ 0 ) \| _ { 0 } , \forall t , t _ 0 \in [ - T _ R , T _ R ] \ , , \forall \alpha \ , . \end{align*}


\begin{align*} | f ( x ) - f ( y ) | \le \int _ { \gamma } g \ , d s : = \int _ 0 ^ { \ell _ { \gamma } } g ( \gamma ( s ) ) \ , d s , \end{align*}


\begin{align*} \lambda \left ( \sigma _ v = \cdot , \ v \in U \mid \sigma _ v = \eta _ v , v \in U ^ c \right ) = \lambda \left ( \sigma _ v = \cdot , \ v \in U \mid \sigma _ v = \eta _ v , v \in \partial _ V U \right ) , \end{align*}


\begin{align*} g _ { \theta } ( x ) = \min _ { z \in \R ^ d } g ( z ) + \frac { 1 } { 2 \theta } \norm { x - z } ^ 2 . \end{align*}


\begin{align*} \phi _ { \epsilon } \ast u ( x ) = P _ \Omega [ \phi _ { \epsilon } \ast u ] ( x ) , x \in \Omega . \end{align*}


\begin{align*} \omega ( B ) = \cap _ { D \ge 0 } [ \cup _ { d ( h , \partial \Sigma ) \ge D } S ( h ) B ] _ { \Phi } , \end{align*}


\begin{align*} & \| \varphi \circ \psi - \tilde \varphi \circ \psi \| _ { \C ^ k } \leq \sum _ { j = 0 } ^ k \omega ^ j \sup _ { | \alpha | = k - j } \| \partial ^ { \alpha } \varphi - \partial ^ { \alpha } \tilde \varphi \| _ { \C ^ 0 } \prod _ { i = j } ^ { k - 1 } \| ( D \psi ) ^ t \| _ { \C ^ i } \end{align*}


\begin{align*} \mathcal P _ { ( \ker L ) ^ \perp } \biggl ( \begin{array} { c } q \\ \int _ 0 ^ 1 u \ , d z \end{array} \biggr ) = \sum _ { \mathbf k \in 2 \pi \mathbb T ^ 2 \setminus \lbrace ( 0 , 0 ) \rbrace } \bigl ( b _ { \mathbf k } ^ + V _ { \mathbf k } ^ + + b _ { \mathbf k } ^ - V _ { \mathbf k } ^ - \bigr ) , \end{align*}


\begin{align*} \int _ { \mathbb { D } } f \circ { F _ a } \cdot { \eta } = \int _ { \mathbb { D } } f \cdot ( { \eta \circ { F _ a ^ { - 1 } } } ) | ( F _ a ^ { - 1 } ) ' | ^ 2 = 0 , \ \ \ \forall \ , f \in { A ^ 2 ( \mathbb { D } ) } , \end{align*}


\begin{align*} \vartheta \ln \left ( \gamma _ { n } \right ) = 2 \left ( \gamma _ { n - 1 } - \gamma _ { n + 1 } + 1 \right ) , \end{align*}


\begin{align*} d \mathbf { P } _ { N , \beta } ( X _ { N } ) = \frac { 1 } { Z _ { N , \beta } } \exp \left ( - \beta \mathcal { H } _ { N } ( X _ { N } ) \right ) d X _ { N } , \end{align*}


\begin{align*} ( x - t ) f ( x , t ) = \varphi ( x ) \psi ( t ) . \end{align*}


\begin{align*} \mathbf { v } \left ( \mathrm { i } s - 0 \right ) = \mathbf { v } \left ( \mathrm { i } s + 0 \right ) \left ( \begin{array} [ c ] { c c } 1 & - 2 \mathrm { i } \pi \chi _ { - } \left ( s \right ) \delta \left ( s \right ) \\ - 2 \mathrm { i } \pi \chi _ { + } \left ( s \right ) \delta \left ( s \right ) & 1 \end{array} \right ) , \end{align*}


\begin{align*} O ( x ) : = \{ \alpha ( t , x ) \ , : \ , t \in G \} \ , . \end{align*}


\begin{align*} ( { } ^ { \partial ( x ) } y ) x y ^ { - 1 } x ^ { - 1 } = 1 , \ x , y \in \Gamma . \end{align*}


\begin{align*} F ( u _ 0 , u _ 1 ; v _ 0 , v _ 1 ) = 0 \end{align*}


\begin{align*} { \bf E } ^ { k } = \sum _ { i = 1 } ^ { \lfloor k / 2 \rfloor } { \bf E } ^ { i } { \bf E } ^ { k - i } , \end{align*}


\begin{align*} [ 2 a ] _ n = \begin{cases} t & \\ 0 & \end{cases} \end{align*}


\begin{align*} \tilde { \psi } _ { t , n } = e ^ { j \left ( \pi - \angle ( \tilde { \vartheta } _ { t , n } ^ * \pm j \tilde { \vartheta } _ { r , n } ^ * ) \right ) } . \end{align*}


\begin{align*} ( n - 2 ) / 2 \leq \sum _ { i = 1 } ^ { n } x _ i \leq n / 2 \end{align*}


\begin{align*} H ( G , f ) = \min _ \mathcal { S } \left ( 1 + Z ^ + _ S ( f ) ' \Gamma _ S ^ { - 1 } Z ^ + _ S ( f ) \right ) ^ { - 1 } \end{align*}


\begin{align*} \varphi ( [ x , y ] ) & = \frac 1 2 \left ( [ \varphi ( x ) , y ] + [ x , \varphi ( y ) ] \right ) , \ x \in \S ( q ) _ 0 , \\ \varphi ( [ x , y ] ) & = \frac 1 2 \left ( [ \varphi ( x ) , y ] - [ x , \varphi ( y ) ] \right ) , \ x \in \S ( q ) _ 1 . \end{align*}


\begin{align*} N = ( 1 - o _ n ( 1 ) ) \frac { \binom { n } i } { \binom { r } i } \binom { r } { t } . \end{align*}


\begin{align*} m ( a ) = ( a , a ) \ ; . \end{align*}


\begin{align*} B _ t = B _ s - \int _ { s } ^ { t } B _ r \dd r + \int _ { s } ^ { t } B _ r \times \dd W _ r \ , \end{align*}


\begin{align*} T & = \frac { 1 } { 2 } ( \textbf { H } + 1 ) \partial _ \varphi - \partial _ \theta , \\ J T & = \frac { 1 } { 2 } ( \textbf { H } - 1 ) \partial _ \varphi - \partial _ \theta . \end{align*}


\begin{align*} f '' ( x ) - m \tanh x f ' ( x ) + \frac { f ( x ) } { \cosh ^ 2 ( x ) } \frac { m } { m + p - 2 } ( \hat { \lambda } + m - p ) = 0 . \end{align*}


\begin{align*} S _ { q } = \left \{ R \left ( k \right ) , \mathrm { d } \rho \left ( k \right ) : k \geq 0 \right \} , \end{align*}


\begin{align*} h _ s ( g ) = \sum _ { \gamma \in \Gamma _ \infty \backslash \Gamma } \tau ( \gamma g ) \cdot ( \gamma g ) ^ s . \end{align*}


\begin{align*} I = \{ ( g , x , y ) : g x = x _ 0 \} \subseteq G \times X \times Y \end{align*}


\begin{align*} \tilde { \nabla } _ E T _ { \theta } e _ 5 = \omega _ { 5 6 } ( E ) T _ { \theta } e _ 6 - \frac { \kappa _ 2 } { \kappa _ 1 } \left ( T _ { \theta } e _ 3 + i T _ { \theta } e _ 4 \right ) , \end{align*}


\begin{align*} c _ k ( Y , r \lambda ) = r c _ k ( Y , \lambda ) . \end{align*}


\begin{align*} \| \Theta [ n + 1 ] \| _ \mathrm { H S } & \le g N ^ { 3 \beta } \| \upsilon \| _ { L ^ \infty } \| \phi _ { n + 1 } \| _ { L ^ 2 } ^ 2 + \frac { 1 } { N } \| ( \Theta ^ \mathrm { p a i r } [ n + 1 ] ) \| _ \mathrm { H S } \\ & \le g \| \upsilon \| _ { L ^ \infty } \Big \{ N ^ { 3 \beta } + \frac { C ^ 2 } { 2 c } \Big \} , \end{align*}


\begin{align*} \rho ( M ) = ( \delta _ { 1 , 2 , \ldots , m } ( M ) : \ldots : \delta _ { n + 1 , n + 2 , \ldots , n + m } ( M ) ) , \end{align*}


\begin{align*} ( x - t ) ^ { m } _ { + } = \begin{cases} ( x - t ) ^ { m } & x \leq t _ { 0 } \\ 0 & x < t \end{cases} \end{align*}


\begin{align*} J : L ^ 2 ( E , \mathbb { C } ^ 2 ) \to L ^ 2 ( \mathbb { R } ^ 2 , \mathbb { C } ^ 2 ) , ( J \varphi ) ( x _ 1 , x _ 2 ) : = \begin{cases} \varphi ( x _ 1 , x _ 2 ) & \textup { i f } x _ 2 \geq 0 \\ ( \sigma _ 1 \varphi ) ( x _ 1 , - x _ 2 ) & \textup { i f } x _ 2 < 0 \end{cases} \ ; . \end{align*}


\begin{align*} \Phi _ 1 \Phi _ 2 = \Phi _ 3 \Phi _ 4 . \end{align*}


\begin{align*} [ 1 a ] _ n = \sum _ { k = 2 } ^ \infty [ k ] _ { n - 1 } . \end{align*}


\begin{align*} x ( b ) - x ( a ) = \int _ { a } ^ { b } { \rm d } [ A ( s ) ] x ( s ) + g ( b ) - g ( a ) . \end{align*}


\begin{align*} b _ N = a _ N + b _ k - a _ k + f ( a _ N ) \end{align*}


\begin{align*} \varepsilon _ 4 = f _ 6 ( s ) + f _ 7 ( z ) + s f _ 8 ( z ) . \end{align*}


\begin{align*} \sum _ { U \subset A _ s \atop | U | = \ell } \prod _ { u \in U } X _ d ( 1 / 2 + u ) \mathcal { B } ^ { ( d ) } ( A _ s - U + U ^ { - } ; \ell ) \ll D ^ { - k \sigma + \epsilon } ( | t | + 2 ) ^ { - k \sigma } . \end{align*}


\begin{align*} D ^ { 2 } z ( t ) + [ a - 2 b \cos ( 2 t ) ] z ( t ) = 0 \end{align*}


\begin{align*} \Bigl | \mathrm { d i v } \ , \bar { \nu } + \overline { H } \Bigr | = \Bigl | \mathrm { d i v } \ , \bar { \nu } - \overline { \mathrm { d i v } _ \Gamma \nu } \Bigr | \leq c | d | \Bigl | \overline { W } \Bigr | \leq c | d | \quad \overline { N } \end{align*}


\begin{align*} M = \begin{pmatrix} W & I & Q \\ O & 0 & 0 \\ V & 0 & S \end{pmatrix} \end{align*}


\begin{align*} \frac { 1 } { p _ \theta } : = \frac { \theta } { p _ 1 } + \frac { 1 - \theta } { p _ 2 } , \frac { 1 } { q _ \theta } : = \frac { \theta } { q _ 1 } + \frac { 1 - \theta } { q _ 2 } , \alpha _ \theta : = \theta \alpha _ 1 + ( 1 - \theta ) \alpha _ 2 , \end{align*}


\begin{align*} - \partial _ s w - \Delta w + \theta _ \infty h ( \bar x _ \infty , \bar t _ \infty ) | D w | ^ \gamma = 0 . \end{align*}


\begin{align*} 0 = [ \star , \star ] _ G = 2 [ \mu _ 0 , \hbar m _ \star ] + [ \hbar m _ \star , \hbar m _ \star ] . \end{align*}


\begin{align*} & U _ { + } = \begin{pmatrix} 0 & A \\ 0 & 0 \end{pmatrix} , U _ { - } = \begin{pmatrix} 0 & 0 \\ - \sigma A & 0 \end{pmatrix} , \\ & V _ { + } ( k ) = \begin{pmatrix} 0 & 4 k ^ 2 A \\ 0 & 0 \end{pmatrix} , V _ { - } ( k ) = \begin{pmatrix} 0 & 0 \\ - 4 \sigma k ^ 2 A & 0 \end{pmatrix} . \end{align*}


\begin{align*} M _ { \otimes 0 } ( A ) ( ( i , j ) ) = A ( \Phi _ \cdot ^ { a , b } ( i , j ) ) \ ; . \end{align*}


\begin{align*} \lambda _ i ( p ) = - \frac { d ( p ) ^ \top \tau ^ i ( p ) } { d ( p ) ^ \top \nu } \nu _ i , i = 1 , 2 , 3 , 4 . \end{align*}


\begin{align*} f ( x ) \coloneqq 1 - x _ 1 ^ 2 - \cdots - x _ d ^ 2 , ~ g _ { j , \pm } ( x ) \coloneqq ( 1 \pm x _ j ) ^ 2 , \ ; j = 1 , \dots , d . \end{align*}


\begin{align*} \mathcal { E } _ { V } ^ { \theta } ( { \mu } _ { \theta } ^ { \epsilon } ) = \mathcal { E } _ { V } ^ { \theta } ( \mu _ { \theta } ) + \epsilon C + \frac { 1 } { \theta } | X | \epsilon \log \epsilon + O ( \epsilon ^ { 2 } ) , \end{align*}


\begin{align*} \mathbf { v } _ { + } ( k ) = \mathbf { v } _ { - } ( k ) \mathbf { J } ( k , t ) , k \in \Sigma . \end{align*}


\begin{align*} \left \vert x \right \vert _ { r , q } = \sup \left \{ \sum _ { i = 1 } ^ { n } x _ { i } y _ { i } : \left \vert y \right \vert _ { r , q } ^ { \circ } \leq 1 \right \} \end{align*}


\begin{align*} G _ 1 = \frac { \theta ( P - q ) } { R } , G _ 2 = \frac { P - q } { a } , G _ 3 = \frac { \theta ^ 2 } { 2 q } \frac { P + q } { P - q } , \end{align*}


\begin{align*} c _ 1 ( L ) = - N \epsilon _ { 1 , 2 } \ , , \end{align*}


\begin{align*} | \tau _ \varepsilon ^ i ( y , t ) | \leq c , | \tau _ \varepsilon ^ i ( y , t ) - \nabla _ \Gamma g _ i ( y , t ) | \leq c \varepsilon , ( y , t ) \in \overline { S _ T } , \ , i = 0 , 1 \end{align*}


\begin{align*} \sum _ { i = 1 } ^ n b _ i Y _ i \leq C _ q \left ( \left \vert b \right \vert _ 1 + e ^ { t ^ 2 / q } \left \vert b \right \vert _ { q / 2 } \right ) \end{align*}


\begin{align*} & \lceil \log _ q \big ( q + q ( q - 1 ) ( \max \{ 3 \lceil \frac { \sqrt { 2 } } { 3 } q \rceil , 9 \} - 2 ) \big ) \rceil \\ & \geq \lceil \log _ q \big ( q + q ( q - 1 ) ( q + 4 - 2 ) \big ) \rceil = 4 . \end{align*}


\begin{align*} ( x , x ) = ( \alpha ( 0 , x ) , x ) \in V [ O , U ] \subseteq V \ , . \end{align*}


\begin{align*} a _ { 1 1 } = \lim _ { k \rightarrow \infty } k a _ 1 ( k ) = \frac { A } { 2 i } , a _ 2 ' ( 0 ) = \frac { 2 i } { A } . \end{align*}


\begin{align*} \bigl [ ( \bar { \nu } - \varepsilon \bar { \tau } _ \varepsilon ^ i ) \cdot \nabla \rho ^ \varepsilon \bigr ] ( x , t ) + k _ d ^ { - 1 } \left [ \Bigl ( \overline { V _ \Gamma } + \varepsilon \ , \overline { \partial ^ \circ g _ i } + \varepsilon ^ 2 \bar { g } _ i \bar { \tau } _ \varepsilon ^ i \cdot \overline { \nabla _ \Gamma V _ \Gamma } \Bigr ) \rho ^ \varepsilon \right ] ( x , t ) = 0 . \end{align*}


\begin{align*} N _ 1 ( c ) & = c ^ 2 \mathcal { Q } ( G _ 0 ) + c ^ 3 \mathcal C ( G _ 0 ) , \\ N _ 2 ( c , H ) & = \mathcal { Q } ( H ) + 2 c \mathcal { Q } ( G _ 0 , H ) + 3 c ^ 2 \mathcal { C } ( G _ 0 , G _ 0 , H ) + 3 c \mathcal { C } ( G _ 0 , H , H ) . \end{align*}


\begin{align*} \sum _ { i = 0 } ^ { q } ( - 1 ) ^ { i } t _ i b _ { j _ i } \leq 0 . \end{align*}


\begin{align*} \dot { y } ( t ) = L ( t ) y _ t + G ( t , y _ t ) , \\ \end{align*}


\begin{align*} \lim _ { k \rightarrow + \infty } \| u _ { t _ k } \times \partial ^ 2 _ x u _ { t _ k } \| ^ 2 _ { L ^ 2 } = 0 \ , . \end{align*}


\begin{align*} K \otimes _ R \R ^ j G ( F M ) \simeq \R ^ j G _ K ( F _ K K \otimes _ R M ) = 0 . \end{align*}


\begin{align*} u ( 0 ) = \cdots = | D ^ k u ( 0 ) | = | D g ( 0 ) | \cdots = | D ^ k g ( 0 ) | = 0 . \end{align*}


\begin{align*} [ e _ i ( z ) , f _ j ( w ) ] = \frac { \delta _ { i j } } { q _ i - q _ i ^ { - 1 } } \delta \left ( \frac { z } { w } \right ) \left ( \psi ^ + _ i ( z ) - \psi ^ - _ i ( z ) \right ) \ , , \end{align*}


\begin{align*} L \circ D H = D H \circ D f , \end{align*}


\begin{align*} W ( z , r ) : = \{ ( y , s ) \in \Omega , \ , | y - z | < r / 2 , \ , r / 2 \leq | s | \leq 2 r \} . \end{align*}


\begin{align*} \phi ( s ' \otimes a s \otimes m ) & = \phi ( s ' s \otimes a m ) = \phi ( a ( s ' s \otimes m ) ) = a \phi ( f ( r ' ) s \otimes m ) = a \phi ( r ' f ( 1 _ R ) s \otimes m ) \\ & = r ' a \phi ( 1 _ S s \otimes m ) = ( 1 _ S \otimes r ' a ) \phi ( s \otimes m ) = ( f ( 1 _ R r ' ) \otimes a ) \phi ( s \otimes m ) = s ' \otimes a \phi ( s \otimes m ) , \end{align*}


\begin{align*} x ^ 2 + y ^ 2 + ( t - \epsilon _ 1 ) ( t - \epsilon _ 2 ) \cdots ( t - \epsilon _ { 2 r } ) = 0 \end{align*}


\begin{align*} \mathbf { R } _ { I W , b } = d i a g \left ( \frac { 1 } { B } \sum _ b \mathbf { \hat { R } } _ { D , b } \right ) \end{align*}


\begin{align*} \| \mathfrak { D } - \mathfrak { D } _ 0 \| _ { C ^ m ( D \times D ; [ 0 , + \infty ) ) } \le c _ m t , m = 1 , 2 , \dots . \end{align*}


\begin{align*} \mathbb { X } ^ { N } _ t = 1 + \int _ 0 ^ t \mathbb { X } ^ { N } _ { s ^ - } \otimes d \mathbf { X } _ s + \sum _ { 0 < s \leq t } \mathbb { X } ^ { N } _ { s ^ - } \otimes \{ \exp ^ { ( N ) } ( \log ^ { ( [ p ] ) } ( \Delta \mathbf { X } _ s ) ) - \Delta \mathbf { X } _ s \} . \end{align*}


\begin{align*} f _ \ast ( F ) \subseteq G , H \in \mathsf { d o m } ( f _ \ast ) F \subseteq H G = f _ \ast ( H ) . \end{align*}


\begin{align*} \begin{aligned} p _ \circ a & = ( \gamma _ a ^ { p - 1 } + \dots + \gamma _ a + 1 ) ( a ) \\ & = ( p + \binom { p } { 2 } \delta _ a + \dots + \binom { p } { p - 1 } \delta _ a ^ { p - 2 } ) ( a ) + \delta _ a ^ { p - 1 } ( a ) \end{aligned} \end{align*}


\begin{align*} \mathrm { d } \mu _ { y } \left ( x \right ) = \left ( \int P _ { x + \mathrm { i } y } \left ( t \right ) \mathrm { d } \mu \left ( t \right ) \right ) \mathrm { d } x \rightarrow \mathrm { d } \mu \left ( x \right ) , y \rightarrow 0 , \end{align*}


\begin{align*} \kappa ( \gamma _ 1 \vee \cdots \vee \gamma _ n \otimes m ) = \sum _ { k = 0 } ^ n \sum _ { \sigma \in S h ( k , n - k ) } \epsilon ( \sigma ) \gamma _ { \sigma ( 1 ) } \vee \cdots \vee \gamma _ { \sigma ( k ) } \otimes \kappa _ { n - k } ( \gamma _ { \sigma ( k + 1 ) } \vee \cdots \vee \gamma _ { \sigma ( n ) } \otimes m ) . \end{align*}


\begin{align*} y ( t ) & = u _ \varphi ( t ) ( 0 ) \\ & = ( T _ 0 ( t - s ) \varphi ) ( 0 ) + j ^ { - 1 } \bigg ( \int _ s ^ t T _ 0 ^ { \odot \star } ( t - \tau ) [ L ( \tau ) u _ \varphi ( \tau ) + G ( \tau , u _ \varphi ( \tau ) ) ] r ^ { \odot \star } d \tau \bigg ) ( 0 ) \\ & = \varphi ( 0 ) + \int _ s ^ t L ( \tau ) u _ \varphi ( \tau ) + G ( \tau , u _ \varphi ( \tau ) ) d \tau . \end{align*}


\begin{align*} Q = \sum _ { 0 \leq i < j \leq 3 } a _ { i j } x _ i x _ j . \end{align*}


\begin{align*} \psi \left ( s , t , r , \theta \right ) = \left ( r e ^ { s / 2 + i \theta } , \sqrt { L / \pi } e ^ { s / 2 + 2 \pi i t / L } \right ) . \end{align*}


\begin{align*} L ( A ) & : = \{ x \in P \mid x \leq y y \in A \} , \\ U ( A ) & : = \{ x \in P \mid y \leq x y \in A \} , \end{align*}


\begin{align*} \int _ { \mathbb { D } } \frac { | g ( w ) | ^ p } { | w | ^ { p k _ p } } = \frac { 1 } { 2 - p k _ p } \int ^ 1 _ 0 M \left ( s ^ { \frac { 1 } { 2 - p k _ p } } \right ) d s . \end{align*}


\begin{align*} \Delta _ { m - 1 } = \left \{ ( x _ 1 , \ldots , x _ m ) \in [ 0 , 1 ] ^ m \colon x _ 1 + \cdots + x _ m = 1 \right \} . \end{align*}


\begin{align*} P _ D ( x , z ) = \int _ D G _ D ( x , z ) \nu ( z - y ) d z , x \in D , \ ; z \in \overline { D } ^ c . \end{align*}


\begin{align*} \beta ( \mu , \phi ) \ ; = \ ; \int - \phi | { \mathbf { H } } | ^ 2 \ , + \ , D \phi \cdot \bigl ( ( T _ x \mu ) ^ \bot \cdot { \mathbf { H } } \bigr ) \ , d \mu . \end{align*}


\begin{align*} R [ y , y ' , \ldots , y ^ { ( r ) } , \ldots ] : = \bigcup _ { r = 0 } ^ { \infty } S ^ r ; \end{align*}


\begin{align*} \gamma _ { A , \ell } & = \sum _ { i \in I } \frac { \dim W _ i } { \dim G _ { A , \ell } - \dim ( G _ { A , \ell } ) _ W } \leq \sum _ { i \in I } \frac { \dim W _ i } { \dim G _ { A _ i , \ell } - \dim ( G _ { A _ i , \ell } ) _ W } \leq \sum _ { i \in I } \gamma _ { A _ i , \ell } . \end{align*}


\begin{align*} \hat { J } ( u _ { 0 } ) = \mathbb { E } _ { v _ 0 \in \mathbb { B } ^ d } \left [ J ( u _ { 0 } + \epsilon v _ 0 ) \right ] . \end{align*}


\begin{align*} \| \varphi _ n \| _ { M _ d } \leq \| \varphi _ n \| _ { A } \leq \left \| \dfrac { \chi _ { F _ n } } { n ^ { 1 / 2 } } \right \| ^ 2 _ 2 = 1 , \end{align*}


\begin{align*} \rho ^ 2 ( x , b ) : = h ^ 2 ( x , b ) + h '^ 2 ( x , b ) . \end{align*}


\begin{align*} H _ { \infty } : = h _ { \alpha } + H _ f - 2 \alpha ^ { 1 / 2 } \mathrm { R e } P A _ { \infty } + \alpha \| \lambda _ { \infty } \| ^ 2 + 2 \alpha A _ { \infty } ^ { + } A _ { \infty } ^ - , \end{align*}


\begin{align*} ( \overline { \mathbf { \Pi } } ^ { \lambda } ) ^ { x } = \mathbf { \Pi } ^ { \lambda ( x ) } . \end{align*}


\begin{align*} \operatorname { s u p p } T \subset \bigcup _ { i = 1 } ^ s L _ { P _ i } . \end{align*}


\begin{align*} b _ k \left ( \coprod _ { i = 1 } ^ m ( X _ i , \omega _ i ) \right ) = \max _ { k _ 1 + \cdots + k _ m = k } \sum _ { i = 1 } ^ m b _ { k _ i } ( X _ i , \omega _ i ) . \end{align*}


\begin{align*} \mathcal { L } \omega ( y ) : = \mathcal { L } ( x , y , D \omega , D ^ { 2 } \omega ) , \end{align*}


\begin{align*} U _ T ( t , \tau ) = U _ { T - \tau } ( t - \tau , 0 ) = U _ { S } ( S - T + t , 0 ) U _ { T - \tau } ( T - S - \tau , 0 ) . \end{align*}


\begin{align*} A B + C D = 0 , \end{align*}


\begin{align*} M _ { \lambda _ 1 } = \left \{ ( s , y ) \in \R \times Y \mid 0 < s < f _ 1 ( y ) \right \} \end{align*}


\begin{align*} \begin{aligned} \| \rho ^ \varepsilon - \bar { \eta } \| _ { C ( \overline { Q _ { \varepsilon , T } } ) } & \leq c _ T \Bigl ( \| \rho _ 0 ^ \varepsilon - \bar { \eta } _ 0 \| _ { C ( \overline { \Omega _ \varepsilon ( 0 ) } ) } + \| f ^ \varepsilon - \bar { f } \| _ { C ( Q _ { \varepsilon , T } ) } \Bigr ) \\ & + \varepsilon c _ T \Bigl ( \| \eta _ 0 \| _ { C ( \Gamma ( 0 ) ) } + \| \eta \| _ { C ^ { 2 , 1 } ( S _ T ) } \Bigr ) \end{aligned} \end{align*}


\begin{align*} J _ { \mu A } = ( I + \mu A ) ^ { - 1 } , \mu > 0 . \end{align*}


\begin{align*} X _ L = \sum _ { \mathbf { l } : l _ 0 + l _ 1 + \ldots = L } E _ { \mathbf { l } } \ ; . \end{align*}


\begin{align*} \tilde { g } ( \theta , \theta ) = \rho ( z ) | \vartheta | ^ 2 , & \rho = \sum _ { k = 1 } ^ { n } | \partial _ { z } w _ k \circ w _ i ^ { - 1 } | ^ 2 , \\ \alpha ^ { * } \tilde { g } ' ( \theta , \theta ) = \tilde { g } ' ( \theta ' , \theta ' ) = \rho ' ( z ' ) | \vartheta ' | ^ 2 , & \rho ' = \sum _ { k = 1 } ^ { n } | \partial _ { z ' } w ' _ k \circ w _ i ^ { ' - 1 } | ^ 2 . \end{align*}


\begin{align*} I _ { 2 } = - \pi \mu ^ { \prime } \left ( \alpha \right ) . \end{align*}


\begin{align*} \mathrm { s c l } _ G ( g ) = \lim _ { n \to \infty } \frac { \mathrm { c l } _ G ( g ^ n ) } { n } , \end{align*}


\begin{align*} \overline { \mathrm { m o } } \ , ( Z \cap E _ { k _ 1 } ) \ , = \ , \overline { \mathrm { m o } } \ , ( Z ) . \end{align*}


\begin{align*} \sum _ { k = 0 } ^ \infty \binom { 2 s } { k } ( - 1 ) ^ { k } ( k + w ) ! \frac { \Gamma ( p + q - 2 m ) } { \Gamma ( p + q - 2 m + k + w + 1 ) } = \frac { \Gamma ( w + 1 ) \Gamma ( p + q - 2 m + 2 s ) } { \Gamma ( p + q - 2 m + 2 s + w + 1 ) } \ , . \end{align*}


\begin{align*} \partial _ t \Phi _ \varepsilon ^ i ( X , t ) = ( V _ \Gamma \nu ) ( y , t ) + \varepsilon \{ \partial ^ \circ g _ i ( y , t ) \nu ( y , t ) - g _ i ( y , t ) \nabla _ \Gamma V _ \Gamma ( y , t ) \} . \end{align*}


\begin{align*} \psi ( \gamma ) & = \sum _ { \alpha \in \Gamma , i \in I } \sum _ { \beta \in \Gamma , j \in I } a _ { \alpha , i } b _ { \beta , j } \langle \lambda _ \Gamma ( \gamma ) \delta _ \alpha , \delta _ \beta \rangle \langle v _ i , v _ j \rangle \\ & = \sum _ { i \in I } \langle \lambda _ \Gamma ( \gamma ) ( \sum _ { \alpha \in \Gamma } a _ { \alpha , i } \delta _ \alpha ) , ( \sum _ { \beta \in \Gamma } b _ { \beta , i } \delta _ \beta ) \rangle . \end{align*}


\begin{align*} \log \left | G \left ( \frac { z } { z _ k } ; 2 \right ) \right | = { \rm R e } \left ( \log G \left ( \frac { z } { z _ k } ; 2 \right ) \right ) \le \sum _ { j = 3 } ^ \infty \left | \frac { z } { z _ k } \right | ^ j \le 2 \left | \frac { z } { z _ k } \right | ^ 3 . \end{align*}


\begin{align*} \Lambda _ p : = \int _ \Omega \int _ I \max \left \{ \ell ^ { \frac { n } { n - p } } , \frac { ( { \rm d i s t } ( x , \partial \Omega ) ) ^ { \frac { n } { n - p } } } { ( \sup | b | ) ^ { \frac { n } { n - p } } } \right \} \exp \left ( \frac { \ell p ^ 2 } { p - n } \| D _ x b ( s , x ) \| \right ) \dd s \dd x < + \infty , \end{align*}


\begin{align*} \mathcal { C } ( \mathfrak { f } , g _ 0 , g _ 0 ) = - 2 \mathcal { C } ( \mathfrak { f } , g _ 0 , g _ 1 ) - \mathcal { C } ( \mathfrak { f } , g _ 1 , g _ 1 ) . \end{align*}


\begin{align*} \omega _ 0 ( \mathcal C ) = ( x \partial _ x + y \partial _ y + 1 ) ^ 2 = \omega _ 0 ( \mathcal C ' ) + 1 \ , . \end{align*}


\begin{align*} \overline g | _ { U ' } = g _ { \mathcal C , F } = d x ^ 2 + x ^ 2 g _ F , \end{align*}


\begin{align*} R _ k = J _ n + J _ m . \end{align*}


\begin{align*} \overline { \mathbf { F } } _ { R , N } ( C ) = \frac { 1 } { T ^ { d } } \int _ { \mathbb { T } ^ { d } } \delta _ { \left ( x , \left ( \theta _ { N ^ { \frac { 1 } { d } } x } C \right ) \bigg | _ { \square _ { R } } \right ) } \ , d x . \end{align*}


\begin{align*} & G ^ { ( 2 ) } _ { \mathcal R } ( x , y ) = \\ & \frac { i e ^ { i k _ { + } \vert x \vert } } { 4 \pi } \left \{ \int ^ { + \infty } _ { - \infty } \left [ g ( s _ b ) \sqrt { s - s _ b } + \frac { g ( s _ b ) - g ( 0 ) } { s _ b } ( s - s _ b ) ^ { \frac 3 2 } \right ] e ^ { - \vert x \vert s ^ 2 } d s + G ^ { ( 2 ) } _ { \mathcal R , 1 } ( x , y ) \right \} , \end{align*}


\begin{align*} z = \sum _ { \alpha } z _ \alpha , z _ \alpha : = \Pi _ \alpha u \ . \end{align*}


\begin{align*} B _ p ^ { X , k Y } = B _ p ^ X - \sum _ { l = 0 } ^ { k - 1 } B _ p ^ { \perp , \l } . \end{align*}


\begin{align*} \int _ t ^ \infty e ^ { - b \tau + \eta | \tau - s | } d \tau = \begin{dcases} \frac { e ^ { - b t } } { b - \eta } e ^ { \eta ( t - s ) } , & t \geq s \\ \frac { e ^ { - b t } } { b + \eta } e ^ { \eta ( s - t ) } - \frac { e ^ { - b s } } { b + \eta } + \frac { e ^ { - b s } } { b - \eta } , & t \leq s . \end{dcases} \end{align*}


\begin{align*} \chi ^ 0 _ { s t } = f ( s , \cdot ) \star \mu _ { s t } ^ \omega ( \theta _ s ^ 1 ) - f ( s , \cdot ) \star \mu _ { s t } ^ \omega ( \theta _ s ^ 2 ) . \end{align*}


\begin{align*} \partial _ t \bar { \eta } ( x , t ) = \overline { \partial ^ \circ \eta } ( x , t ) + d ( x , t ) R ( x , t ) \overline { \nabla _ \Gamma V _ \Gamma } ( x , t ) \cdot \overline { \nabla _ \Gamma \eta } ( x , t ) \end{align*}


\begin{align*} \mathcal { P } _ { A , L } ( q ) & = \prod _ { a \in A } \left [ m _ { | a | } \delta _ { * , L } \left ( \sum _ { l \in a } ( q _ l - q _ { l + 1 } ) \right ) \prod _ { l \in a } \widehat { B } _ \# ( q _ l - q _ { l + 1 } ) \right ] . \end{align*}


\begin{align*} J = \{ ( j , k ) \mid k = 0 , 1 , 2 , \ j = 1 , \dotsc , b ^ k \} . \end{align*}


\begin{align*} & W _ t ^ T X _ t = X _ t ^ T W _ t , \\ & Z _ t ^ T Y _ t = Y _ t ^ T Z _ t , \\ & Z ^ T _ t X _ t - Y _ t ^ T W _ t = I _ { d \times d } \end{align*}


\begin{align*} \overline f ( x , y ) : = \int _ 0 ^ 1 f ( x , y , z ' ) \ , d z ' , ~ ~ \widetilde f ( x , y , z ) : = f ( x , y , z ) - \overline f ( x , y ) , \end{align*}


\begin{align*} g = ( \partial _ \xi w + \partial _ \tau w ) - e ^ { - X } \frac { \delta } { 2 } w e ^ { \xi } . \end{align*}


\begin{align*} \limsup _ { n \to \infty } c ^ n n ! \prod _ { p \in R } \left | n ! \right | _ p ^ 2 = 0 , c = c ( t ; R ) : = 4 | t | \prod _ { p \in R } | t | _ p ^ 2 . \end{align*}


\begin{align*} - \Delta u + V ( x ) u = ( I _ { \alpha } \ast | u | ^ { p } ) | u | ^ { p - 2 } u , x \in \mathbb { R } ^ N , \end{align*}


\begin{align*} \sum _ { n = 1 } ^ { \left \lfloor T r \right \rfloor } \prod _ { i = 1 } ^ { m } ( \mathsf { D } _ { d } ^ { i - 1 } R _ { T , \left \lfloor T r \right \rfloor - n } ^ { ( 1 ) } ( d ) ) ^ { j _ { i } } T ^ { 1 / 2 - d } \pi _ { \left \lfloor T r \right \rfloor - n } ( d ) \xi _ { n } \Rightarrow \int _ { 0 } ^ { r } \prod _ { i = 1 } ^ { m } ( \mathsf { D } _ { d } ^ { i - 1 } R ^ { ( 1 ) } ( d ) ) ^ { j _ { i } } \mathsf { d } W . \end{align*}


\begin{align*} \mathcal { A } = \frac { h ^ 2 } { 2 } \Delta _ { \mathbb { S } ^ 2 } & - \left [ \lambda _ 1 v _ r \times g ' ( v _ r ) - \lambda _ 2 v _ r \times ( v _ r \times g ' ( v _ r ) ) \right ] \cdot \nabla \\ & \quad + \left [ \lambda _ 1 v _ r \times \Delta v _ r - \lambda _ 2 v _ r \times ( v _ r \times \Delta v _ r ) \right ] \cdot \nabla \ , . \end{align*}


\begin{align*} \eqref { i n v a r i a n t 2 } = B \big ( \left [ [ x , y ] , \alpha ( z ) \right ] , t \big ) + B \big ( \left [ [ y , z ] , \alpha ( x ) \right ] , t \big ) + B \big ( \left [ [ x , z ] , \alpha ( y ) \right ] , t \big ) . \end{align*}


\begin{align*} & d ^ { 2 ^ h + 1 } = x . \end{align*}


\begin{align*} u _ { z _ 0 } ( z ) = \begin{cases} \frac { 2 z } { \ell ^ 2 - \bar { z } _ 0 z } & z \in B _ { \ell } ( 0 ) , \\ \frac { 2 } { \bar z - \bar { z } _ 0 } & z \in \mathbb C \setminus B _ { \ell } ( 0 ) . \end{cases} \end{align*}


\begin{align*} \nabla \Phi _ { \epsilon , y } = a _ { \epsilon , A ( y ) } \nabla \phi _ { \epsilon } \ , \ , \mbox { a . e . i n } \Omega . \end{align*}


\begin{align*} [ C ( x ) , B ( y ) ] & = \mathbf { l } _ { \tilde { D } \tilde { D } } ( x , y ) \tilde { D } ( x ) \tilde { D } ( y ) + \mathbf { l } _ { A A } ( x , y ) A ( x ) A ( y ) + \mathbf { m } _ { A A } ( x , y ) A ( y ) A ( x ) \\ & + \mathbf { m } _ { A \tilde { D } } ( x , y ) A ( y ) \tilde { D } ( x ) + \mathbf { l } _ { A \tilde { D } } ( x , y ) A ( x ) \tilde { D } ( y ) + \mathbf { l } _ { \tilde { D } A } ( x , y ) \tilde { D } ( x ) A ( y ) \ , , \end{align*}


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


\begin{align*} \pm \dim \ker \xi _ { \Delta _ { g _ s } } ( \beta _ \mu ) = \pm \dim \ker ( \Delta _ { g _ F } + \mu ) . \end{align*}


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


\begin{align*} & G ^ { ( 3 ) } _ { \mathcal R } ( x , y ) = \\ & \frac { e ^ { i k _ { + } \vert x \vert } } { \sqrt { \vert x \vert } } \frac { e ^ { i \frac { \pi } 4 } } { \sqrt { 8 \pi k _ { + } } } \left ( \frac { 2 i \sin \theta _ { \hat x } \widetilde { \mathcal S } ( \cos \theta _ { \hat x } , n ) } { n ^ 2 - 1 } \right ) e ^ { - i k _ { + } \vert y \vert \cos ( \theta _ { \hat x } + \theta _ { \hat y } ) } + G ^ { ( 3 ) } _ { \mathcal R , R e s } ( x , y ) \end{align*}


\begin{align*} \frac { \dim H - \dim H _ { V _ I } } { \dim V _ I } = \frac { \dim G _ { A , \ell } ^ \circ - \dim ( G _ { A , \ell } ^ \circ ) _ { V _ \ell ( A _ I ) } } { \dim V _ \ell ( A _ I ) } = \frac { \dim G _ { A _ I , \ell } ^ \circ } { 2 \dim A _ I } , \end{align*}


\begin{align*} \Big ( \Big ( \frac { \partial } { \partial \overline { z } _ i } \big | _ { y _ 0 } \Big ) ^ { H } \tilde { g } \Big ) ( y _ 0 , Z _ N ) = \Big ( \frac { \partial } { \partial \overline { z } _ i } \big ( g \cdot z _ N ^ { \otimes k } \big ) \Big ) ( y _ 0 ) \exp \Big ( - \frac { \pi } { 2 } | Z _ N | ^ 2 \Big ) . \end{align*}


\begin{align*} K _ p ( \cdot , z ) = K _ { 2 , p , z } ( \cdot , z ) , \ \ \ \forall \ , z \in \Omega . \end{align*}


\begin{align*} \langle z , \bar x \rangle _ 1 = ( T ^ \# x ) ( z ) = ( ( j ^ * \circ c ) y ) ( z ) = ( c y ) ( j z ) = \langle j z , \bar y \rangle = \langle z , \tilde T ^ { - 1 } \bar y \rangle _ 1 . \end{align*}


\begin{align*} \frac { d } { d t } \int _ { \Omega _ { \varepsilon } ( t ) } \rho ^ \varepsilon \ , d x = \int _ { \Omega _ \varepsilon ( t ) } f ^ \varepsilon \ , d x , t \in ( 0 , T ) \end{align*}


\begin{align*} n _ { \pm } : = \sum _ { k = 1 } ^ { d } ( - q ) ^ { \mp k } m _ k , \end{align*}


\begin{align*} n ^ \mathrm { p a i r } ( x , y ) = \sum _ j { p _ j ( x ) \overline { p _ j ( y ) } } , m ^ \mathrm { p a i r } ( x , y ) = - \sum _ j { u _ j ( x ) \overline { p _ j ( y ) } } . \end{align*}


\begin{align*} a _ r = a _ { r + 2 t - 1 } + b _ r + \begin{cases} C & \ r = - t + m , \\ - C & \ r = - t - m , \\ 0 & \end{cases} \end{align*}


\begin{align*} F ^ { \alpha } _ { \Delta , S } ( u ) = F ( u ) + \alpha [ F ^ { \alpha } _ { \Delta , S } ( L _ n ^ { - 1 } ( u ) ) - S ( L _ n ^ { - 1 } ( u ) ) ] ~ ~ u \in I _ n , \end{align*}


\begin{align*} \phi ( x , \lambda ) = \lambda _ { \mathcal { E } } ( x ) + \lambda x _ m ^ r . \end{align*}


\begin{align*} \Vert \alpha ^ f _ { 2 } \Vert ^ 2 = 2 ( 1 - K ) . \end{align*}


\begin{align*} X ^ { \odot \star } = X ^ { \odot \star } _ { - } ( s ) \oplus X ^ { \odot \star } _ { 0 } ( s ) \oplus X ^ { \odot \star } _ { + } ( s ) , \forall s \in \mathbb { R } , \end{align*}


\begin{align*} \psi _ 1 ( x , t , k ) = \psi _ 2 ( x , t , k ) e ^ { - ( i k x + 4 i k ^ 3 t ) \sigma _ 3 } \sigma \Lambda S ^ { - 1 } ( k ) \Lambda e ^ { ( i k x + 4 i k ^ 3 t ) \sigma _ 3 } , k \in \mathbb { R } \backslash \{ 0 \} . \end{align*}


\begin{align*} \log _ { + } z : = \log ( 1 \vee z ) . \end{align*}


\begin{align*} M : = \varepsilon ^ * \Omega ^ 1 _ { S / \Z _ \ell } \cong \left ( \bigoplus _ { i = 1 } ^ n \Z _ \ell \dd X _ i \right ) / \langle \dd P _ j , \ , 1 \leq j \leq r \rangle . \end{align*}


\begin{align*} \tilde { \psi } _ { r , n } = j \tilde { \psi } _ { t , n } \tilde { \psi } _ { r , n } = - j \tilde { \psi } _ { t , n } . \end{align*}


\begin{align*} A = \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} , \ > B = \begin{bmatrix} 1 \\ 0 \end{bmatrix} , \ > C = I _ 2 , \ > D = 0 . \end{align*}


\begin{align*} \sup _ { a _ j : \ , a _ 1 = 1 } \left ( F ( - 1 ) \right ) = - \frac 1 4 \sec ^ 2 { \frac { \pi } { N + 2 } } . \end{align*}


\begin{align*} \limsup _ { \tau \to 0 , R \to \infty , N \to \infty } \sup _ { C \in \mathcal { A } } d _ { \mathcal { P } ( \Omega \times { \rm C o n f i g } ) } ( \overline { \mathbf { P } } _ { N } ( C ) , \overline { \mathbf { P } } _ { N } ( \mathcal { R } C ) ) - \frac { 1 } { | I | } \sum _ { i \in I } d _ { \rm C o n f i g } ( \theta _ { x _ { i } } \cdot C , \theta _ { x _ { i } } \cdot \mathcal { R } C ) = 0 , \end{align*}


\begin{align*} \begin{cases} \partial _ t ^ 2 \mathcal { E } ^ { ( j ) } ( x , t ) = - \Delta _ x \mathcal { E } ^ { ( j ) } ( x , t ) , & x _ 1 > 0 , \\ \mathcal { E } ^ { ( j ) } ( 0 , x _ 2 , x _ 3 , t ) = 0 , & j = 2 , 3 . \end{cases} \end{align*}


\begin{align*} K _ { p , \varphi _ p } ( z ) = \frac { 2 - p k _ p } { 2 \pi } + \frac { 4 - p k _ p } { 2 \pi } | z | ^ 2 + o ( | z | ^ 2 ) \ \ \ ( z \rightarrow 0 ) , \end{align*}


\begin{align*} S _ { q , r } ( m ) & = \frac { p \left ( q - 1 \right ) q ^ v w + \left ( q - 1 \right ) r + r } { q } \\ & = p \left ( q - 1 \right ) q ^ { v - 1 } w + r \\ & = \left ( 1 - \frac { 1 } { q } \right ) \left ( p \cdot q ^ v w \right ) + r \\ & = \left ( 1 - \frac { 1 } { q } \right ) \left ( m - r \right ) + r . \end{align*}


\begin{align*} \xi _ s = \frac { \tau - s } \tau y _ 0 . \end{align*}


\begin{align*} \begin{cases} \displaystyle \frac { 1 - \nu ^ 2 } { E } \Delta ^ 2 v = - \theta & \textrm { i n $ \Omega $ } \\ [ 2 m m ] \nabla ^ 2 v \ , t = 0 & \textrm { o n $ \partial \Omega $ \ , . } \end{cases} \end{align*}


\begin{align*} \vartheta = z \partial _ { z } . \end{align*}


\begin{align*} \int _ { 0 } ^ { t } \langle \epsilon _ { I } , \mathbb { X } _ { s ^ - } \rangle \ d X ^ { j } _ s = & \ \langle ( \epsilon _ { I } ; \epsilon _ { j } ) ^ { \thicksim } , \mathbb { X } _ t \rangle . \end{align*}


\begin{align*} F I O ( S p ( d , \R ) , q , v _ s ) = \bigcup _ { \chi \in S p ( d , \R ) } F I O ( \chi , q , v _ s ) \end{align*}


\begin{align*} u ( t ) = e ^ { - ( t - t _ 0 ) \nu _ 1 ( - \Delta ) ^ { \alpha } } u _ 0 - \int _ { t _ 0 } ^ t e ^ { - ( t - s ) \nu _ 1 ( - \Delta ) ^ { \alpha } } { \mathbb P } _ H \div ( u ( s ) \otimes u ( s ) - B ( s ) \otimes B ( s ) ) \d s , \\ B ( t ) = e ^ { - ( t - t _ 0 ) \nu _ 2 ( - \Delta ) ^ { \alpha } } B _ 0 - \int _ { t _ 0 } ^ t e ^ { - ( t - s ) \nu _ 2 ( - \Delta ) ^ { \alpha } } { \mathbb P } _ H \div ( B ( s ) \otimes u ( s ) - u ( s ) \otimes B ( s ) ) \d s , \end{align*}


\begin{align*} D v v ^ * = v B v ^ * = u ( p B p ) u ^ * . \end{align*}


\begin{align*} H _ N ( \alpha , \beta ; q ) = \sum _ { \lambda \vdash d } \frac { ( \dim \mathsf { V } ^ \lambda ) ^ 2 } { d ! } \omega _ \alpha ( \lambda ) \Omega _ { \frac { q } { N } } ^ { - 1 } ( \lambda ) \Omega _ { \frac { 1 } { N } } ^ { - 1 } ( \lambda ) \omega _ \beta ( \lambda ) = \left \langle \omega _ \alpha \Omega _ { \frac { q } { N } } ^ { - 1 } \Omega _ { \frac { 1 } { N } } ^ { - 1 } \omega _ \beta \right \rangle . \end{align*}


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


\begin{align*} \alpha + \beta = 2 \nu , \ , \beta / \gamma ^ 2 = \nu - 1 , \ , \beta ^ 3 / \gamma ^ 4 = ( \nu - 1 ) ^ 2 ( \nu - 2 ) . \end{align*}


\begin{align*} \partial _ \alpha f + \frac { 1 } { \alpha - 1 } f = - \frac { 1 } { ( \alpha - 1 ) t } \left [ \frac { \log ( 1 - t ) } { ( 1 - t ) ^ { 2 \alpha - 2 } } - \frac { \log ( 1 + t ) } { ( 1 + t ) ^ { 2 \alpha - 2 } } \right ] \ , \end{align*}


\begin{align*} H = \kappa _ \gamma + ( n - 2 ) \kappa ( C _ t ) . \end{align*}


\begin{align*} \begin{aligned} \| \mathfrak Q _ j \| _ { W ^ { \sigma - 1 , p } _ x } \lesssim \| \mathfrak q _ j \| _ { W ^ { \sigma - 1 , p } _ x } \end{aligned} \end{align*}


\begin{align*} Q ( x _ 1 \vee \cdots \vee x _ n ) & = \sum _ { k = 1 } ^ { n } \sum _ { \sigma \in S h ( k , n - k ) } \epsilon ( \sigma ) Q _ { k } ^ 1 ( x _ { \sigma ( 1 ) } \vee \cdots \vee x _ { \sigma ( k ) } ) \vee x _ { \sigma ( k + 1 ) } \vee \cdots \vee x _ { \sigma ( n ) } \end{align*}


\begin{align*} \phi _ { \tau _ 0 } ( \upsilon ) - \phi _ { \tau _ 0 } ( \upsilon _ 0 ) & = \left ( \phi _ { \tau _ 0 + \upsilon _ 0 } ( \upsilon - \upsilon _ 0 ) - \mathrm { i d } \right ) \phi _ { \tau _ 0 } ( \upsilon _ 0 ) , \end{align*}


\begin{align*} & \phi _ x + i k \sigma _ 3 \phi = U ( x , t ) \phi , \\ & \phi _ t + 4 i k ^ 3 \sigma _ 3 \phi = V ( x , t , k ) \phi , \end{align*}


\begin{align*} s _ k = \sqrt { - 1 } \frac { ( k + 1 ) } { \sqrt { 2 | k + h ^ \vee | } } . \end{align*}


\begin{align*} I _ A : = \{ 1 , \ldots , n \} \setminus J _ A . \end{align*}


\begin{align*} ( p _ 1 , u + v ) ( p _ 2 , u + \tilde \alpha v ) = ( p _ 1 , u - v ) ( p _ 2 , u - \tilde \alpha v ) , \ \ u , v \in H . \end{align*}


\begin{align*} \mathfrak { d _ 1 } : = \frac { \mathfrak { d } ( \mathfrak { S } _ { 1 , \delta = 0 } ) } { ( \mathfrak { d } ( \mathfrak { S } _ { 1 } ) } = \frac { 1 } { 2 } . \end{align*}


\begin{align*} m \left ( 1 - \exp \left \{ - 2 \pi i \frac { j + \theta _ 1 / 2 } { m } \right \} + \frac { z } { m } \right ) = 2 \pi i ( j + \theta _ 1 / 2 ) + z + o ( 1 / m ) . \end{align*}


\begin{align*} \begin{aligned} \langle \widetilde { \pi } ( f ) l ' , \widetilde { v } \rangle & = \int _ { G _ n } f ( g ) \langle \widetilde { \pi } ( g ) l ' , \widetilde { v } \rangle d g \\ & = \int _ { G _ n } f ( g ) \langle l ' , \widetilde { \pi } ( g ^ { - 1 } ) ( \widetilde { v } ) \rangle d g . \end{aligned} \end{align*}


\begin{align*} \sum _ { n = 0 } ^ { N - 1 } \dd _ { 2 p } \Bigl ( \Phi \bigl ( \zeta ^ \epsilon ( t _ { n + 1 } ) - \zeta ^ \epsilon ( t _ n ) , Y _ { n } ^ { \epsilon , \Delta t } \bigr ) , \varphi \bigl ( \zeta ^ \epsilon ( t _ { n + 1 } ) - \zeta ^ \epsilon ( t _ n ) , Y _ { n } ^ { \epsilon , \Delta t } \bigr ) \Bigr ) \le C _ p ( T ) \Delta t ^ { \frac 1 2 } . \end{align*}


\begin{align*} a \otimes b = a \sqcup b \ ; . \end{align*}


\begin{align*} ( X _ I ) ^ S = \{ w _ K \in W \mid K \subset \Delta , K \subset I \} . \end{align*}


\begin{align*} \mathcal { B } \subseteq \bigcup \limits _ { i = 0 } ^ { m _ { q + 1 } - 1 } [ t _ i - 2 \theta _ { q + 1 } , t _ i + 3 \theta _ { q + 1 } ] . \end{align*}


\begin{align*} c _ 1 \coloneqq c _ 1 ( \overline { \alpha } ) = 2 ^ { k } \left ( \max _ { 1 \leq j \leq k } \{ | \alpha _ j | \} \right ) ^ { k } \quad c _ 2 \coloneqq c _ 2 ( \overline { \alpha } ; R ) \coloneqq c _ 1 \prod _ { p \in R } \left ( \max _ { 1 \leq i \leq k } \{ \left | \alpha _ i \right | _ p \} \right ) ^ { k + 1 } . \end{align*}


\begin{align*} \Psi \in \Gamma _ s ( \mathfrak { h } ) , \Psi = ( \Psi ^ { ( 0 ) } , \Psi ^ { ( 1 ) } , \Psi ^ { ( 2 ) } , \ldots ) , \end{align*}


\begin{align*} R ( t , \varphi ) : = G ( t , \varphi ) r ^ { \odot \star } , \forall t \in \mathbb { R } , \ \varphi \in X , \end{align*}


\begin{align*} S ( n ) = \frac { 3 n + 1 } { 8 } \equiv - 1 \pmod { 4 } . \end{align*}


\begin{align*} \MoveEqLeft \triangle _ k ^ N ( f ( k ) g ( k ) h ( k ) ) \\ & = \sum _ { m _ 1 + m _ 2 + m _ 3 = N } C _ { m _ 1 , m _ 2 , m _ 3 } \triangle _ k ^ { m _ 1 } f ( k ) \triangle _ k ^ { m _ 2 } g ( k + m _ 1 ) \triangle _ k ^ { m _ 3 } h ( k + m _ 1 + m _ 2 ) , \end{align*}


\begin{align*} \langle v , v \rangle = \sum _ i v _ i v _ i = 0 . 3 5 5 \ ; , \end{align*}


\begin{align*} \begin{aligned} - \operatorname { d i v } \mathcal { A } ( x , u , \nabla u ) & = \mathcal { B } ( x , u , \nabla u ) & & \Omega , \\ u & = 0 & & \partial \Omega , \end{aligned} \end{align*}


\begin{align*} ( Q _ 1 ' ) ^ { - 1 } = \begin{bmatrix} x & - \lambda _ 2 & \dots & - \lambda _ n \\ 0 & 1 & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & 1 \end{bmatrix} \end{align*}


\begin{align*} \int _ 1 ^ \infty t ^ { 2 s } ( \Theta ( t g ) - 1 ) \frac { d t } { t } = s . \end{align*}


\begin{align*} \N = \left \{ n _ r = \left ( \begin{array} { c c c } 1 & r \\ 0 & 1 \end{array} \right ) , r \in \R \right \} \ , . \end{align*}


\begin{align*} \langle N _ { 1 , 2 } \psi , N _ { 2 , 2 } \psi \rangle _ { L ^ 2 ( \Gamma _ T ) } & = - \lambda \int _ { \Gamma _ T } \left ( s \xi + \frac { \tau } { 2 } \right ) \left ( \frac { \tau } { 2 } - s \alpha \right ) \partial _ \nu ^ A \eta ^ 0 ( \partial _ t \log \gamma ) \psi ^ 2 \ , \d S \ , \d t \end{align*}


\begin{align*} \partial _ t n _ { m + 1 } - \alpha \Delta n _ { m + 1 } = - n _ { m + 1 } \sum _ { i = 1 } ^ { \ell } \beta _ i \rho _ { i , m + 1 } \end{align*}


\begin{align*} \begin{cases} & \frac { \partial X } { \partial t } ( x , t ) = ( G ( \nu , X ) F ^ { \beta } ( \l _ i ) - 1 ) u \nu , \\ & X ( \cdot , 0 ) = X _ 0 , \end{cases} \end{align*}


\begin{align*} G _ { \mathcal R } ( x , y ) & = G ^ { ( 1 ) } _ { \mathcal R } ( x , y ) + G ^ { ( 3 ) } _ { \mathcal R } ( x , y ) + G ^ { ( 4 ) } _ { \mathcal R } ( x , y ) , \end{align*}


\begin{align*} A d _ { g } ( Y ) = \frac { d } { d t } \mid _ { t = 0 } ( g e ^ { t Y } g ^ { - 1 } ) \end{align*}


\begin{align*} \frac { \Gamma ( z , s ) } { \Gamma ( x , t + 1 ) } = 1 + O ( N ^ { - 1 / 4 + 3 \epsilon } ) . \end{align*}


\begin{align*} [ A ] _ i = \sum _ { 0 \leq j < b } \widetilde A _ { i , j , j } \ ; . \end{align*}


\begin{align*} \left ( \sum _ { r , s \ge 0 } - \sum _ { r , s < 0 } \right ) q ^ { r s } x ^ r y ^ s = \frac { ( q ) ^ 2 _ { \infty } ( x y , q / x y ; q ) _ \infty } { ( x , q / x , y , q / y ; q ) _ \infty } . \end{align*}


\begin{align*} \frac { \sum _ { i \in Q } \exp ( w _ { f _ i } + \sum _ { j \in Q \setminus \{ i \} } \nu _ { z _ j } ) } { \exp ( \nu _ z + \sum _ { j \in Q } \nu _ { z _ j } ) + \sum _ { i \in Q } \exp ( w _ { f _ i } + \sum _ { j \in Q \setminus \{ i \} } \nu _ { z _ j } ) } = \frac { \sum _ { i \in Q } \exp ( w _ { f _ i } - \nu _ { z _ i } - \nu _ z ) } { 1 + \sum _ { i \in Q } \exp ( w _ { f _ i } - \nu _ { z _ i } - \nu _ z ) } . \end{align*}


\begin{align*} f ( q ) = \underset { n \geq 0 } { \sum } a _ n q ^ n . \end{align*}


\begin{align*} \langle 0 \mid a _ { n } ^ { i } = \langle 0 \mid b _ { n } ^ { i j } = \langle 0 \mid c _ { n } ^ { i j } = 0 f o r \ a l l n < 0 . \end{align*}


\begin{align*} Q = \zeta ^ { N + 1 } + \zeta ^ N c _ { 1 } ^ { \vec \eta } ( \mathcal { E } ) \ldots + c _ { N + 1 } ^ { \vec \eta } ( \mathcal { E } ) \end{align*}


\begin{align*} U _ t = \rho ( B _ t ( D ) ) \end{align*}


\begin{align*} k = \frac { \zeta } { \sqrt { - 4 8 t k _ 0 } } - k _ 0 = \frac { \eta } { \sqrt { \tau } } \zeta - k _ 0 \end{align*}


\begin{align*} \partial _ t \bar { \eta } ( y , t ) = \partial ^ \circ \eta ( y , t ) , ( y , t ) \in \overline { S _ T } \end{align*}


\begin{align*} \sum \limits _ { j = 1 } ^ m \langle g _ { \theta _ j } , v _ j \rangle = 0 , \end{align*}


\begin{align*} 1 ( i ) = \{ \} \forall i \ ; . \end{align*}


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


\begin{align*} N = \bigoplus _ { i = 1 } ^ t N _ i , \end{align*}


\begin{align*} a _ 0 ( t ) : = \begin{bmatrix} e ^ { t / 3 } & & \\ & e ^ { - t / 6 } & \\ & & e ^ { - t / 6 } \end{bmatrix} \in G , \end{align*}


\begin{align*} \beta _ { x } ( s ) \lambda ^ u _ { \Phi ^ c _ x ( s ) } v ^ u ( f \circ \Phi ^ c _ x ( x ) ) = \lambda ^ u _ x \beta _ { x _ 1 } ( \lambda ^ c _ x s ) v ^ u \left ( \Phi ^ c _ { x _ 1 } ( \lambda ^ c _ x s ) \right ) . \end{align*}


\begin{align*} H _ m = \sqrt { - \Delta + m ^ 2 } - \frac { \kappa } { | x | } , 0 < \kappa < \kappa ^ { * } : = \frac { 2 } { \pi } , \end{align*}


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


\begin{align*} \mathcal { A } _ \kappa = \left \{ m \in \mathcal { A } : \mathcal { E } _ \kappa ( m ) - 8 \pi < 0 \right \} . \end{align*}


\begin{align*} & X ^ { s , \gamma , p } _ { ( 0 , t ) } = \begin{cases} L ^ \gamma ( 0 , t ; W ^ { s , p } _ x ) , \ \ \ \frac { 2 \alpha } { \gamma } + \frac 3 p = 2 \alpha - 1 + s , 1 \leq \gamma < \infty , 1 \leq p \leq \infty , s \geq 0 , \\ C ( [ 0 , t ] ; W ^ { s , p } _ x ) , \ \ , \ \ \gamma = \infty , 1 \leq p \leq \infty , s \geq 0 . \end{cases} \end{align*}


\begin{align*} X _ { n + 1 } ^ { 0 , \Delta t , { \rm E } } = X _ { n } ^ { 0 , \Delta t , { \rm E } } + \Delta \beta _ n \sigma ( X _ n ^ { 0 , \Delta t , { \rm E } } ) \end{align*}


\begin{align*} X = \Gamma \left ( x _ 0 + \int _ 0 ^ { \cdot } T ^ { X - \tilde { W } ^ H + \tilde { W } ^ H } f ( d r , x _ 0 ) + \tilde { W } ^ H \right ) = \Gamma \left ( x _ 0 + \int _ 0 ^ { \cdot } T ^ { \tilde { W } ^ H } f ( d r , x _ 0 + \theta _ r ) + \tilde { W } ^ H \right ) , \end{align*}


\begin{align*} s _ { r } = 2 ( s _ 0 + \tau ( \bar r - 3 ) ) \ , . \end{align*}


\begin{align*} y _ R ( t ) = & S ( t - \tau ) z _ R ( \tau ) - \int _ \tau ^ t S ( t - s ) ( i \nu z ( s ) + \epsilon ( \gamma z ( s ) - \mu \overline { z } ( s ) ) ) \d s \\ & - \tfrac { 1 } { 2 } \int _ \tau ^ t S ( t - s ) ( y _ R ( s ) F _ \Phi ) \d s - i \int _ \tau ^ t S ( t - s ) ( y _ R ( s ) \d W ( s ) ) , \end{align*}


\begin{align*} \psi ^ { \varepsilon } ( t , x , y ) : = \psi ( t , x ) + \varepsilon \chi _ { \delta } ( y ) , \end{align*}


\begin{align*} \begin{gathered} ( \vec { \alpha } \hat \sqcup \vec { \beta } ) ( ( 0 , i ) ) = ( 0 , \vec \alpha ( i ) ) \ ; , \\ ( \vec { \alpha } \hat \sqcup \vec { \beta } ) ( ( 1 , i ) ) = ( 1 , \vec \beta ( i ) ) \\ \end{gathered} \end{align*}


\begin{align*} f ( s ) : = - \sqrt { 2 / k _ { + } } e ^ { - i \frac { \pi } { 4 } } H _ { \theta _ c } ( s ) H _ { \pi - \theta _ c } ( s ) \sqrt { s - s _ b ^ { * } } F ( s ) \frac { d \zeta ( s ) } { d s } \frac { 2 i \sin \zeta ( s ) } { n ^ 2 - 1 } . \end{align*}


\begin{align*} \overline { L } ^ n = \pi _ { i _ 1 } ^ * \overline { L } _ { i _ 1 } \cdots \pi _ { i _ n } ^ * \overline { L } _ { i _ n } = 0 , \end{align*}


\begin{align*} \chi _ \upsilon ( s ) = \frac { 1 } { | W _ { G _ s } ( T _ \upsilon ) ^ F | } \sum _ { x \in W _ G ( T ) ^ F } { } ^ { \gamma x } \chi ( s ) , \end{align*}


\begin{align*} \mathcal { F } _ { ( w ) } ( B ; \ell ) : = \frac { \tilde { \mathcal { B } } _ { ( w ) } ^ { ( 2 ) } ( B ; \ell ) } { \mathcal { B } _ { ( w ) } ^ { ( 2 ) } ( B ; \ell ) } = \prod _ { p > 2 \atop p \nmid \ell } \frac { \Big ( 1 + \sum _ { n = 1 } ^ \infty \frac { \tau _ B ( p ^ { 2 n } ) } { p ^ n } \Big ( \frac { p ^ w } { p ^ w + 1 } \Big ) \Big ) } { \Big ( 1 + \sum _ { n = 1 } ^ \infty \frac { \tau _ B ( p ^ { 2 n } ) } { p ^ n } \Big ) } \prod _ { p | \ell } \frac { p ^ w } { p ^ w + 1 } . \end{align*}


\begin{align*} \langle \sigma _ A \rangle _ { \Lambda , \beta } = \langle \prod _ { x \in A } \sigma _ x \rangle _ { \Lambda , \beta } . \end{align*}


\begin{align*} { r } ^ { r } _ 1 ( k ) = \frac { k - i \kappa } { k } r _ 1 ( k ) , { r } ^ { r } _ 2 ( k ) = \frac { k } { k - i \kappa } r _ 2 ( k ) . \end{align*}


\begin{align*} \Delta _ X ^ { k + 1 } : = \big \{ ( x _ 0 , \ldots , x _ k ) \in X ^ { k + 1 } \mid x _ 0 = \ldots = x _ k \big \} \ . \end{align*}


\begin{align*} \lim _ { x \rightarrow \infty } \frac { \# \{ \ell < x : a _ { \ell } ( f ) \equiv 0 \pmod { p } \} } { \pi ( x ) } = \frac { p } { p ^ 2 - 1 } . \end{align*}


\begin{align*} S ( n ) = \frac { 3 n + 1 } { 2 } = 3 b 2 ^ v - 1 . \end{align*}


\begin{align*} R _ 2 = ( 2 H ( y , \hat \eta _ 0 ) ^ { - \frac { 1 } { 2 } } \Big ( 2 g _ 0 \hat \eta _ \tau , \frac { \partial \hat \eta _ \tau } { \partial \tau } \Big ) \Big | _ { \tau = 0 } \end{align*}


\begin{align*} \nabla \sigma _ \varepsilon ( x , t ) = ( - 1 ) ^ { i + 1 } \varepsilon \bar { g } ( x , t ) \{ \bar { \nu } ( x , t ) - \varepsilon \bar { \tau } _ \varepsilon ^ i ( x , t ) \} , x \in \Gamma _ \varepsilon ^ i ( t ) , \ , t \in [ 0 , T ] , \ , i = 0 , 1 , \end{align*}


\begin{align*} P ( \{ S _ \ell = y \} \cap A _ t ) \le P \big ( S _ \ell = y \big ) P \big ( \hat { S } ^ 1 _ { t - \ell } \ge \psi ( t ) - \sqrt { \ell } \big ) \ , . \end{align*}


\begin{align*} \alpha _ { n , 1 } \left ( x \right ) = - \frac { n \left ( n - 1 \right ) } { 2 \left ( 2 n - 1 \right ) } x ^ { n - 2 } , \end{align*}


\begin{align*} \widetilde { M } ( \gamma , \delta ) = M ( \Phi _ \sqcup ^ \alpha ( \gamma ) , \Phi _ \sqcup ^ \alpha ( \delta ) ) \ ; . \end{align*}


\begin{align*} [ k ] _ n = \begin{cases} t ^ n & k = n + 1 ; \\ 0 & k \neq n + 1 . \end{cases} \end{align*}


\begin{gather*} \beta _ \lambda ( r ) : = \frac { 1 } { \lambda } \bigl ( r - J _ \lambda ( r ) \bigr ) : = \frac { 1 } { \lambda } \bigl ( r - ( I + \lambda \beta ) ^ { - 1 } ( r ) \bigr ) , \\ \beta _ { \Gamma , \lambda } ( r ) : = \frac { 1 } { \lambda } \bigl ( r - J _ { \Gamma , \lambda } ( r ) \bigr ) : = \frac { 1 } { \lambda } \bigl ( r - ( I + \lambda \beta _ \Gamma ) ^ { - 1 } ( r ) \bigr ) { \rm f o r ~ } r \in \mathbb { R } . \end{gather*}


\begin{align*} l _ 3 & \coloneq \left [ \left ( 1 - \dfrac { g } { 2 } \right ) l _ 1 + \dfrac { g } { 2 } l _ 2 \right ] + \underbrace { \left [ ( g - 1 ) l _ 1 - ( g + 1 ) l _ 2 \right ] } _ { } \\ & = \dfrac { g } { 2 } l _ 1 - \dfrac { g + 2 } { 2 } l _ 2 , \end{align*}


\begin{align*} \phi _ { \tau _ 0 } ( \upsilon ) = \phi _ { \tau _ 0 + \tau } ( \upsilon - \tau ) \cdot \phi _ { \tau _ 0 } ( \tau ) . \end{align*}


\begin{align*} H ( x , t , \xi ) = h | \xi | ^ \gamma + b ( x , t ) \cdot \xi , \end{align*}


\begin{align*} \Big | E \sum _ { u = 1 } ^ p Z _ u - E \sum _ { u = 1 } ^ p \underline { Z } _ u \Big | \le O \Big ( \frac { p n } { ( \log n ) ^ { \alpha / 4 } } \Big ) = o ( \bar h _ 4 ( n ) ) \ , , \end{align*}


\begin{align*} f _ N ( t ) = c t + N ^ { - 1 / 2 } \sum _ { s = 1 } ^ t y ( s ) . \end{align*}


\begin{align*} 2 \mathrm { R e } \langle u _ { \alpha } \otimes R _ * \ , | \ , H _ y R _ y \rangle & = 2 \alpha \mathrm { R e } \langle u _ { \alpha } \otimes R _ * \ , | \ , V _ y R _ y \rangle \\ & = 2 \alpha \mathrm { R e } \langle u _ { \alpha } \otimes R _ * \ , | \ , V _ y \Phi _ { \# } ^ y \rangle + 2 \alpha \mathrm { R e } \langle u _ { \alpha } \otimes R _ * \ , | \ , V _ y R _ y ^ { \# } \rangle , \end{align*}


\begin{align*} \Delta ( \eta \circ \theta ) = 4 | \partial _ { \bar { z } } \theta | ^ 2 \eta '' \circ \theta , \end{align*}


\begin{align*} ( 2 A + \nabla t B ) \delta ^ { n + 1 / 2 } = ( 2 A - \nabla t B ) \delta ^ { n } \end{align*}


\begin{align*} X \circ \Pi ( - r , \pi - \theta ) & = \left ( \overline { h ( r e ^ { i \theta } ) } , a + b - t \circ \Pi ( r , \theta ) \right ) , \end{align*}


\begin{align*} V ^ { \varepsilon } ( T , x , y ) = g ( x , y ) . \end{align*}


\begin{align*} H ( k ) - \tilde H ( k ) = \sum _ { r \in E } b _ r e ^ { - \frac { r + 2 } { 2 } k } \end{align*}


\begin{align*} \lim _ { n \to \infty } d ( \phi ^ { - n } ( x ) , \phi ^ { - n } ( y ) ) = 0 . \end{align*}


\begin{align*} d ^ 0 _ { 0 , \frac q 2 } & = 0 , \ d ^ 1 _ { 0 , \frac q 2 } ( m , i ) = \begin{cases} 0 , & ( m , i ) \ne \left ( 0 , - \frac { 3 q } 2 \right ) , \\ 1 , & ( m , i ) = \left ( 0 , - \frac { 3 q } 2 \right ) . \end{cases} \end{align*}


\begin{align*} \beta _ { \pm , \gamma } ( k ) = ( 2 \pi ) ^ { 3 / 2 } \frac { \omega ^ { 1 / 2 } ( k ) } { 2 \pi } \alpha _ { \pm , \gamma } ( k ) . \end{align*}


\begin{align*} \max _ { k \in [ n - 1 ] } \mu ( n , k ) = \sup _ { \eta , k } \eta ( n , k ) \ ; , \end{align*}


\begin{align*} & \chi = \chi _ { j 1 } \otimes \cdots \otimes \chi _ { j \lambda _ j } \otimes \chi _ { j 1 } ' \otimes \cdots \otimes \chi _ { j \lambda _ j ' } ' , \\ & \eta = \eta _ { j 1 } \otimes \cdots \otimes \eta _ { j \mu _ j } \otimes \eta _ { j 1 } ' \otimes \cdots \otimes \eta _ { j \mu _ j ' } ' . \end{align*}


\begin{align*} \langle \Lambda _ { M _ 1 } , \Lambda _ { M _ 2 } \rangle = | M _ 1 \cap M _ 2 | \pmod 2 . \end{align*}


\begin{align*} \langle \alpha ^ g _ 3 ( X _ { \theta _ 0 } , X _ { \theta _ 0 } , X _ { \theta _ 0 } ) , \alpha ^ g _ 3 ( X _ { \theta _ 0 } , X _ { \theta _ 0 } , X _ { \theta _ 0 } ) \rangle = 0 , \end{align*}


\begin{align*} \partial _ x ^ \alpha g ( x ) = c \int _ { \R ^ d } \frac { g ( y ) - g ( x ) } { | y - x | ^ { d + \alpha } } \d y . \end{align*}


\begin{align*} b _ { x } = \sum _ j { \{ u _ j ( x ) \alpha _ j - \overline { p _ j ( x ) } \alpha _ j ^ \ast \} } , b ^ \ast _ { x } = \sum _ j { \{ \overline { u _ j ( x ) } \alpha _ j ^ \ast - p _ j ( x ) \alpha _ j \} } . \end{align*}


\begin{align*} \begin{gathered} [ A ] _ S ( i , j ) = A _ S ( ( 0 , i ) , ( 0 , j ) ) \ ; , \\ [ A ] _ \sigma ( i ) = A _ \sigma ( ( 0 , i ) ) \ ; . \\ \end{gathered} \end{align*}


\begin{align*} & G _ { \mathcal R , \zeta _ o , 2 } ( x , y ) = \frac { i } { 2 \pi } \int _ { \mathcal I _ { \zeta _ o , \frac { \pi } 2 - i \infty } } \frac { 2 i \sin \zeta \mathcal S _ { - } ( \cos \zeta , n ) } { n ^ 2 - 1 } { e ^ { i k _ { + } \left ( - \vert y \vert \cos ( \zeta + \theta _ { \hat y } ) + \vert x \vert \cos ( \zeta - \theta _ { \hat x } ) \right ) } } d \zeta , \end{align*}


\begin{align*} \xi _ { \Delta _ { g _ 0 } } ( \zeta ) = \zeta ^ 2 + ( n - 1 ) \zeta , \end{align*}


\begin{align*} \mu ( Z ) < \mu ( X _ 0 ) < \mu ( Y _ 0 ) < \cdots < \mu ( Y _ k ) = \mu ( Y ) . \end{align*}


\begin{align*} \sigma _ i \coloneqq \sigma _ i ( n , \overline { \alpha } ) = ( - 1 ) ^ i \sum _ { \substack { i _ 1 + \ldots + i _ k = i \\ 0 \leq i _ j \leq n } } \binom { n } { i _ i } \binom { n } { i _ 2 } \cdots \binom { n } { i _ k } \alpha _ 1 ^ { n - i _ i } \alpha _ 2 ^ { n - i _ 2 } \cdots \alpha _ k ^ { n - i _ k } . \end{align*}


\begin{align*} \displaystyle \lim _ { r \to \infty } { r ^ { \rho - \rho ( r ) } } \cdot { \displaystyle \sum _ { | z _ k | > r } \frac 1 { z _ k ^ \rho } } = 0 . \end{align*}


\begin{align*} \gamma _ k = & \frac { | \mathbf { h } _ k ^ H \mathbf { \Theta } _ i \mathbf { G } \mathbf { w } _ k | ^ 2 } { \sum _ { \ell \in \mathcal { K } \setminus k } | \mathbf { h } _ k ^ H \mathbf { \Theta } _ i \mathbf { G } \mathbf { w } _ \ell | ^ 2 + \sigma _ k ^ 2 } . \end{align*}


\begin{align*} g ( n ) = \begin{cases} 3 n + 1 & \\ n / 2 & \end{cases} \end{align*}


\begin{align*} \mathcal { C } _ { \mathbf { i } } = \{ x \in \mathbb { R } ^ N \mid [ \varsigma _ { \mathbf { i } , i } ] _ { t r o p } ( x ) \ge 0 i \in I . \} . \end{align*}


\begin{align*} \frac { d ^ 2 } { d s ^ 2 } \Big \lvert _ { s = 0 } g _ H ( T _ * N , N ) & = - \frac { d } { d s } \Big \lvert _ { s = 0 } \Bigl ( g _ H ( T _ * N , N ^ \perp ) \bigl ( - \kappa _ \eta + g _ H ( T _ * N , N ) \kappa _ 1 \bigr ) \Bigr ) \\ & = \bigl ( \kappa _ \eta - g _ H ( T _ * N , N ) \kappa _ 1 \bigr ) \frac { d } { d s } \Big \lvert _ { s = 0 } g _ H ( T _ * N , N ^ \perp ) , \end{align*}


\begin{align*} \sideset { } { '' } \sum _ { \substack { \boldsymbol { r } , \boldsymbol { t } \in \mathbb { Z } ^ d \setminus \{ \boldsymbol { 0 } \} \\ r _ { j _ \beta } v = r _ { j _ \beta } w } } | c _ { \boldsymbol { r } } c _ { \boldsymbol { t } } | \ll _ d \frac { s ^ { 2 d - q } } { N ^ { 2 - q / d } } \frac { ( v , w ) ^ q } { \sqrt { | v w | ^ q } } , \end{align*}


\begin{align*} \mathcal { M } ( A ) ( i ) = \mathcal { H } ( A ( i ) ) \end{align*}


\begin{align*} J ( h ) = - \partial _ \tau h - \eta \partial _ \xi h + \big ( ( w + \bar { w } ) \partial _ { \eta } ^ 2 \bar { w } \big ) h + w ^ 2 \partial _ { \eta } ^ 2 h . \end{align*}


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


\begin{align*} \left ( \begin{array} { c c } \tilde { v } _ { 1 } ( x , t ) \\ \tilde { v } _ { 2 } ( x , t ) \end{array} \right ) = \frac { 2 i \sigma } { A } \left ( \begin{array} { c c } v _ 1 ( x , t ) \\ v _ 2 ( x , t ) \end{array} \right ) . \end{align*}


\begin{align*} \begin{cases} - \partial _ t \zeta - \Delta \zeta = \tilde m ^ { q _ 0 ' - 1 } & \\ \zeta ( z , s ) = 0 & \\ \zeta ( z , \bar s ) = 0 & . \end{cases} \end{align*}


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


\begin{align*} \chi = \bigotimes _ j \chi _ { j 1 } \otimes \cdots \otimes \chi _ { j \lambda _ j } , \eta = \bigotimes _ j \eta _ { j 1 } \otimes \cdots \otimes \eta _ { j \mu _ j } , \end{align*}


\begin{align*} \alpha ^ { g _ \theta } ( X , Y ) = T _ \theta \circ \alpha ^ g ( J _ \theta X , Y ) , \ \ X , Y \in T M . \end{align*}


\begin{align*} B _ 1 \cap B _ 2 = \emptyset , \ \ \ B _ 1 \cup B _ 2 = B , \ \ \ \ P ^ * ( A _ 1 \triangle B _ 1 ) + P ^ * ( A _ 2 \triangle B _ 2 ) < \varepsilon . \end{align*}


\begin{align*} u _ { z _ 0 } ( z ) = f ( z ) + \overline { g ( z ) } z \in \overline { \Omega } , \end{align*}


\begin{align*} A ( t ) x ^ 2 + C ( t ) y ^ 2 = H ( t ) z ^ 2 \end{align*}


\begin{align*} 1 + \sum _ { d = 1 } ^ \infty \frac { z ^ { 2 d } } { d ! d ! } N ^ { 2 d } \sum _ { \substack { \lambda \vdash d \\ \ell ( \lambda ) \leq N } } s _ \lambda ( a _ 1 , \dots , a _ N ) s _ \lambda ( b _ 1 , \dots , b _ N ) \frac { ( \dim \mathsf { V } ^ \lambda ) ^ 2 } { ( \dim \mathsf { W } _ M ^ \lambda ) ( \dim \mathsf { W } _ N ^ \lambda ) } , \end{align*}


\begin{align*} h _ L '' ( x ) = ( m - p ) \tanh ( x ) h ' _ L ( x ) + ( 1 - m ) h _ L ( x ) . \end{align*}


\begin{align*} \int _ { \R + i } & \frac { 1 } { \lambda ^ 2 - z ^ 2 } f _ 1 ( \lambda ) \lambda \coth ( \frac { \pi \lambda } { 2 } ) \ , d \lambda \\ & = \frac { 1 } { 2 } \int _ { \R + i } \frac { 1 } { \lambda - z } f _ 1 ( \lambda ) \coth ( \frac { \pi \lambda } { 2 } ) \ , d \lambda + \frac { 1 } { 2 } \int _ { \R + i } \frac { 1 } { \lambda + z } f _ 1 ( \lambda ) \coth ( \frac { \pi \lambda } { 2 } ) \ , d \lambda \ , . \end{align*}


\begin{align*} F ^ { ( j ) } - F = \int _ { w _ j } ^ { w _ j ' } \frac { \beta _ { e _ j } e ^ t } { \alpha _ { e _ j } + \beta _ { e _ j } e ^ t } ~ t . \end{align*}


\begin{align*} F ( p , q ) = \log ( p ) - \log \Gamma ( 1 - p ) - \log \Gamma ( 1 + p ) + \log ( 2 \pi ) + \log | \sin ( \pi ( p + q i ) | - \log | \sin ( \pi p ) | - \pi q . \end{align*}


\begin{align*} B ( \C _ { e _ { \alpha } q ^ { - i + 1 } } ) = \begin{cases} \C _ { e _ { \alpha } q ^ { - i } } & 1 \le i < k _ { \alpha } \\ 0 & i = k _ { \alpha } \\ \end{cases} , z ( \C _ { e _ { \alpha } } ) = \C _ { e _ { \alpha } } , \end{align*}


\begin{align*} \| P _ \lambda ( s + h ) - P _ \lambda ( s ) \| & = \frac { 1 } { 2 \pi } \bigg | \bigg | \oint _ { \partial C _ \lambda } ( z I - U ( s + T + h , s + h ) ) ^ { - 1 } - ( z I - U ( s + T , s ) ) ^ { - 1 } d z \bigg | \bigg | , \end{align*}


\begin{align*} p = p ( t ) : = 1 - 6 t . \end{align*}


\begin{align*} \left \{ \begin{alignedat} { 3 } \partial ^ \circ ( g \eta ) - g V _ \Gamma H \eta - k _ d \ , \mathrm { d i v } _ \Gamma ( g \nabla _ \Gamma \eta ) & = g f & & & & S _ T , \\ \eta | _ { t = 0 } & = \eta _ 0 & & & & \Gamma ( 0 ) , \end{alignedat} \right . \end{align*}


\begin{align*} ( a \otimes b ) ( i ) = a ( i ) \sqcup b ( i ) \ ; . \end{align*}


\begin{align*} { \dot { W } } _ { \varepsilon , \eta } ( s , B _ s ) = & \int _ 0 ^ s \int _ { R ^ d } \varphi _ \eta ( s - r ) p _ \varepsilon ( B _ s - y ) W ( d r , y ) d y \ , , \end{align*}


\begin{align*} - \frac { h ^ { \prime \prime \prime } } { h ^ { \prime } } - \frac { h ^ { \prime \prime } } { h } + 8 \frac { { h ^ { \prime } } ^ 2 } { h ^ 2 } - \frac { 8 } { h ^ 2 } = 0 , \end{align*}


\begin{align*} \int _ { B ( 0 , r ) } \frac { 1 } { | z | ^ { n - 1 } } \dd z = r \sigma _ { n - 1 } . \end{align*}


\begin{align*} q _ { n } ( x ) & = q \left ( x \right ) \chi _ { \left ( - n , \infty \right ) } \left ( x \right ) \rightarrow q \left ( x \right ) \\ & \Rightarrow \begin{array} [ c ] { c c c c } S _ { n } & \rightarrow & S _ { q } & \\ q _ { n } \left ( x , t \right ) & \rightarrow & q \left ( x , t \right ) & t > 0 \end{array} \end{align*}


\begin{align*} g _ 1 = - 1 , g _ 2 = \frac { 3 + \cos \theta } { 3 + \cos \theta + 4 \alpha \cos \theta + 2 \beta } \end{align*}


\begin{align*} \partial _ { \bar { z } } r = i \partial _ { \bar { z } } \theta ( z ) w e ^ { - i \theta ( z ) } - i \partial _ { \bar { z } } \theta ( z ) \overline { w } e ^ { i \theta ( z ) } + \eta ' ( \theta ( z ) ) \partial _ { \bar { z } } \theta ( z ) = \eta ' ( \theta ( z ) ) \partial _ { \bar { z } } \theta ( z ) \neq 0 \end{align*}


\begin{align*} ( T _ t f ) \oplus ( T _ s f ) = T _ { s + t } f \ , . \end{align*}


\begin{align*} \Phi = \varphi _ 1 \land y \leq z \ , \& \dots \& \ , \varphi _ m \land y \leq z \Longrightarrow y \leq z . \end{align*}


\begin{align*} w \cdot e _ j = \pm e _ { w ( j ) } , \end{align*}


\begin{align*} \| \nabla \phi \| _ { L ^ p ( \mathbb C ^ n ) } \le 2 ( C _ p ^ \ast + 1 ) \sum _ { j = 1 } ^ n \| \partial \phi / \partial \bar { z } _ j \| _ { L ^ p ( \mathbb C ^ n ) } , \ \ \ \forall \ , \phi \in C ^ 1 _ 0 ( \mathbb C ^ n ) . \end{align*}


\begin{align*} \left \{ \begin{aligned} \frac { \partial } { \partial t } \omega ^ { T , n } - \mathcal { L } ( x , y , D \omega ^ { T , n } , D ^ { 2 } \omega ^ { T , n } ) = 0 , & ( 0 , T ] \times D _ { n } , \\ \omega ^ { T , n } ( 0 , y ) = g ( x , y ) , & D _ { n } , \\ \omega ^ { T , n } ( t , y ) = 0 , & [ 0 , T ] \times \partial D _ { n } , \end{aligned} \right . \end{align*}


\begin{align*} y _ n C ( x _ n ) & = y _ n C ( z _ n ) + y _ n z _ n C ' ( z _ n ) \delta _ n + y _ n z _ n ^ 2 C '' ( z _ n ) \delta _ n ^ 2 / 2 + o ( C '' ( z _ n ) \delta _ n ^ 2 ) \\ & = y _ n C ( z _ n ) + y _ n z _ n C ' ( z _ n ) \delta _ n + o ( y _ n C ' ( z _ n ) \delta _ n ) . \end{align*}


\begin{align*} \mathbb { Z } _ 2 ^ { [ \delta \times k ] } \coloneqq \left \{ A = ( A _ { i , j } ) \in \mathbb { Z } _ 2 ^ { N \times k } \left | \begin{array} { l } \sum \limits _ { j = 1 , \ldots , k } A _ { i , j } = \delta _ i \ , ( i \in [ N ] ) , \\ \sum \limits _ { i = 1 , \ldots , N } A _ { i , j } = 1 \ , \ , ( j \in [ k ] ) \end{array} \right . \right \} . \end{align*}


\begin{align*} | u ( y ) | \leq \max \left \{ H ( r ) ^ { 1 / ( p - 1 ) } , \dfrac { r } { \delta _ 0 } \right \} : = \bar H ( r ) \end{align*}


\begin{align*} \epsilon \| m ^ \epsilon ( t _ N ) - m _ N ^ { \epsilon , \Delta t } \| = { \rm O } ( \Delta t ^ { \frac 1 2 } ) \end{align*}


\begin{align*} \frac { \dd ^ N \sinh t } { \dd t ^ N } & = - i ^ { N + 1 } \sin \Big ( i t + \frac { \pi N } { 2 } \Big ) \end{align*}


\begin{align*} W _ j ( \omega ) = \left \{ \begin{array} { c } p ^ { i + 1 } ( V ) \ \ \mbox { i f } \ \ \omega \in \hat \Omega _ j = \Omega - \Omega _ j \\ ( p ^ { i + 1 , j } ) ^ { - 1 } ( x _ { i j } ( \omega ) ) \cap p ^ { i + 1 } ( V ) \ \ \mbox { i f } \ \ \omega \in \Omega _ j \end{array} \right . \end{align*}


\begin{align*} S _ { \ell } ( m ) = \frac { \left ( 2 ^ { \ell } - 1 \right ) m + 1 } { 2 ^ e } \end{align*}


\begin{align*} S ^ v ( n ) = 4 c + 1 \equiv 1 \pmod { 4 } . \end{align*}


\begin{align*} f ( \omega ) = \begin{cases} \left ( \frac { j _ M ^ 0 } { j _ M ^ * } \right ) ^ { 1 / 3 } \omega ^ { - 7 / 6 } \left ( 1 + O ( \epsilon \omega ^ { \delta } ) \right ) & \mbox { f o r } \omega \leq 1 , \\ \left ( \frac { j _ M ^ \infty } { j _ M ^ * } \right ) ^ { 1 / 3 } \omega ^ { - 7 / 6 } \left ( 1 + O ( \epsilon \omega ^ { - \delta } ) \right ) & \mbox { f o r } \omega > 1 , \end{cases} \end{align*}


\begin{align*} \mu _ { n + 1 } & \le \mu _ 0 + \frac { g } { 2 } \big ( E _ \mathrm { H } ( \phi _ 0 ) \big ) ^ 2 + \frac { g } { 2 } \big ( \tilde { \mathcal { E } } [ n + 1 ] ( \phi _ { n + 1 } ) \big ) ^ 2 \\ & + \frac { 2 g C } { c } \| \upsilon \| _ { L ^ \infty } + \frac { g C ^ 2 } { N c } \mathcal { E } _ \mathrm { H a r t r e e } ( \phi _ 0 ) , \end{align*}


\begin{align*} \partial _ r : = \sum _ { \alpha = d + 1 } ^ n \frac { t _ \alpha } { | t | } \partial _ \alpha . \end{align*}


\begin{align*} G = g - b \delta \end{align*}


\begin{align*} \begin{cases} x ^ 2 + \Delta ( t ) + y ( y + z ( b - a ) ) = 0 , \\ y = ( t - a ) z . \end{cases} \end{align*}


\begin{align*} A ( \varphi ) = \omega _ { n - 1 } \int \limits _ 0 ^ \varphi \sin ^ { n - 2 } \theta \ , d \theta \ , . \end{align*}


\begin{align*} ~ ~ ~ ~ ~ ~ \mu ' ( q , n ) = \begin{cases} 0 & W _ { n - 1 } \geq W _ q \ ; , \\ \mu ( q , n ) + \mu ( n - 1 , n ) & \end{cases} \end{align*}


\begin{align*} \dfrac { \partial D h ( x - T x ) } { \partial x } = \ ( \begin{matrix} \dfrac { \partial h _ { x _ 1 } ( x - T x ) } { \partial x _ 1 } & \cdots & \dfrac { \partial h _ { x _ 1 } ( x - T x ) } { \partial x _ n } \\ \vdots & \vdots & \vdots \\ \dfrac { \partial h _ { x _ n } ( x - T x ) } { \partial x _ 1 } & \cdots & \dfrac { \partial h _ { x _ n } ( x - T x ) } { \partial x _ n } \end{matrix} \ ) \end{align*}


\begin{align*} | \Omega \rangle = | \rangle \otimes \ldots \otimes | \rangle \ , , \end{align*}


\begin{align*} \phi _ 1 ( x , t , k ) = \phi _ 2 ( x , t , k ) S ( k ) , k \in \mathbb { R } \backslash \{ 0 \} \end{align*}


\begin{align*} \left \{ \begin{aligned} & ( u _ 2 ) _ s - 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*} \limsup _ { k \to \infty } \eta ( k ) = \lim _ { k \to \infty } \eta ( k ) = \inf _ k \eta ( k ) \end{align*}


\begin{align*} \lim _ { x \to 0 ^ + } u _ t ( x ) = 1 \quad \lim _ { x \to 0 ^ + } v _ t ( x ) = \infty , \end{align*}


\begin{align*} \left \{ \begin{alignedat} { 2 } \mathcal { L } _ 1 u = f ( x , u , v ) \ ; \ ; & \mbox { i n $ \Omega $ , } \ ; \ ; u > 0 \ ; \ ; & & \mbox { i n $ \Omega $ , } \ ; \ ; u \lfloor _ { \partial \Omega } = 0 , \\ \mathcal { L } _ 2 v = g ( x , u , v ) \ ; \ ; & \mbox { i n $ \Omega $ , } \ ; \ ; v > 0 \ ; \ ; & & \mbox { i n $ \Omega $ , } \ ; \ ; v \lfloor _ { \partial \Omega } = 0 , \end{alignedat} \right . \end{align*}


\begin{align*} F = \alpha \alpha _ 0 ^ { - 1 } \sigma _ 2 \alpha ^ { - 1 } \sigma \alpha _ 0 \alpha ^ { - 1 } \end{align*}


\begin{align*} \bar H = \max \{ H _ 0 , H _ 1 , \cdots , H _ d \} \ , \underbar H = \min \{ H _ 0 , H _ 1 , \cdots , H _ d \} \ , . \end{align*}


\begin{align*} K = 2 A _ { T , X } + 1 . \end{align*}


\begin{align*} v \in \left [ 0 \ , , \frac \pi 8 \right ] \quad u \in \bigcup _ { n = 0 } ^ \infty J _ n J _ n = \left [ 2 n \pi \ , , 2 n \pi + \frac \pi 8 \right ] . \end{align*}


\begin{align*} V : = U \backslash ( s - x + F ' ) \end{align*}


\begin{align*} - 2 \sum _ { x \in X _ n } \omega ( x ) \int _ { - \infty } ^ \infty K _ t ( x , y ) W ( y ) \mathrm { d } y & = - 2 t ^ { - 1 } \sum _ { x \in X _ n } \omega ( x ) W ( x ) \\ & = - 2 t ^ { - 1 } \int _ { - \infty } ^ \infty W ^ 2 ( x ) d x = - \frac { 2 } { \sqrt { 2 } t } . \end{align*}


\begin{align*} \alpha _ { n + 1 , k } \left ( x \right ) - x \alpha _ { n , k } \left ( x \right ) + \eta _ { n , k } x ^ { n - 1 } + { \displaystyle \sum \limits _ { j = 1 } ^ { k - 1 } } \alpha _ { n - 1 , j } \left ( x \right ) \eta _ { n , k - j } = 0 . \end{align*}


\begin{align*} \lVert \theta k \rVert = \theta k , k \le ( 2 \chi ) ^ { - 1 } , \end{align*}


\begin{align*} \hat { f } ( k ) = \int _ { \R ^ d } f ( x ) e ^ { - 2 \pi i k \cdot x } d x , k \in \R ^ d , \end{align*}


\begin{align*} \frac { \dd } { \dd v } | y ( v ) | = \langle \frac { y ( v ) } { | y ( v ) | } , \dot y ( v ) \rangle = \langle \frac { y ( v ) } { | y ( v ) | } , B ( v ) y ( v ) \rangle \le \| B ( v ) \| | y ( v ) | v \in ( s , t ) \ , . \end{align*}


\begin{align*} A ( k , \nu ) = \frac { ( \nu | \nu + 2 \rho ^ \natural ) } { 2 ( k + h ^ \vee ) } + \frac { ( \xi | \nu ) } { k + h ^ \vee } ( ( \xi | \nu ) - k - 1 ) . \end{align*}


\begin{align*} H g H = H g \sqcup H g h = H a b \sqcup H b . \end{align*}


\begin{align*} d _ \alpha ( n ) & = \ ! \begin{cases} 0 & n \in ( X \cap Y ) \ ! \cup \ ! ( X ^ c \cap Y ^ c ) \\ 1 & n \in X ^ c \cap Y \\ - 1 & n \in X \cap Y ^ c . \end{cases} \ ! \ ! \implies \Delta ^ 2 _ n \ ! = \ ! \begin{cases} [ \beta ] - 1 \\ [ \beta ] \\ [ \beta ] - 2 . \end{cases} \end{align*}


\begin{align*} f ( A , \theta , \lambda \omega , S ) = \det ( \lambda ) ^ { - w } \cdot f ( A , \theta , \omega , S ) . \end{align*}


\begin{align*} \hat { n } ( x ) \times \mathcal E ( x ) = 0 , \hat { n } ( x ) \cdot \mathcal B ( x ) = 0 , x \in \Sigma _ 0 , \end{align*}


\begin{align*} C = T ^ { - 1 } ( ( S ^ { - 1 } ( A [ \underline { x } ] ) ) [ \underline y ] ) \simeq T ^ { - 1 } ( \widetilde { S } ^ { - 1 } ( A [ \underline { x } , \underline y ] ) ) \simeq U ^ { - 1 } ( A [ \underline { x } , \underline y ] ) \end{align*}


\begin{align*} \mathcal { P } _ 1 ( \varepsilon ) = \Big \{ p \in \mathcal { P } : \inf _ { q \in \mathcal { P } _ 0 } \sup _ { A \subseteq [ \ell _ 1 ] \times [ \ell _ 2 ] \times [ n ] } \big | p ( A ) - q ( A ) \big | \geq \varepsilon \Big \} . \end{align*}


\begin{align*} q ^ \epsilon ( t _ n ) = Q ^ \epsilon ( t _ n ) - P ^ \epsilon ( t _ n ) ~ , q _ n ^ { \epsilon , \Delta t } = Q _ n ^ { \epsilon , \Delta t } - P _ n ^ { \epsilon , \Delta t } , \end{align*}


\begin{align*} \mathbf { R } _ { I W , b } = \mathbf { \hat { R } } _ { D , b } \end{align*}


\begin{align*} b ( t ) : = \frac 5 6 s ( 1 - s ) ^ 2 p ^ 5 r ^ 2 = a ( t ) . \end{align*}


\begin{align*} | x _ j - z | _ { \infty } = r _ j . \end{align*}


\begin{align*} I _ 1 ( q ) = \log \left ( \frac { \mathrm { s i n } ( u / 2 ) } { u / 2 } \right ) \cdot \frac { 1 } { d - 1 ! } \overline { \pi } _ * ( \mathrm { c } _ 1 ( S ) \otimes \dots \otimes 1 ) . \end{align*}


\begin{align*} k < j _ 1 \hbox { \rm \ \ a n d \ \ } j _ \ell + k < j _ { \ell + 1 } \hbox { \rm \ \ f o r \ \ } \ell = 1 , 2 , \dots , i - 1 \ , , \end{align*}


\begin{align*} A ^ p ( \mathbb { D } , \varphi _ p ) : = \left \{ g \in \mathcal { O } ( \mathbb { D } ) : \ \| g \| _ { p , \varphi _ p } ^ p = \int _ { \mathbb { D } } | g | ^ p e ^ { - \varphi _ p } < \infty \right \} , \end{align*}


\begin{align*} c _ { \boldsymbol { r } } = \begin{cases} \omega _ d \frac { s ^ d } { N } , & \mbox { i f } \boldsymbol { r } = \boldsymbol { 0 } , \\ \frac { s ^ { d / 2 } } { \sqrt { N } \| \boldsymbol { r } \| _ 2 ^ { d / 2 } } J _ { d / 2 } \big ( \frac { 2 \pi s } { N ^ { 1 / d } } \| \boldsymbol { r } \| _ 2 \big ) , & \mbox { i f } \boldsymbol { r } \in \mathbb { Z } ^ d \setminus \{ \boldsymbol { 0 } \} . \end{cases} \end{align*}


\begin{align*} \frac { \dim G _ { L } - \dim ( G _ L ) _ { V _ I ' } } { \dim V _ I ' } = \frac { \dim G - \dim G _ { V _ I } } { \dim V _ I } . \end{align*}


\begin{align*} \tilde \Psi ^ * e ^ { - t P _ 1 } ( x , y ) = e ^ { - t P _ 2 } ( x , y ) \ , ( t , x , y ) \in ( 0 , \infty ) \times U _ 2 \times U _ 2 . \end{align*}


\begin{align*} f _ { \delta , m } ( t ) : = \frac { e ^ { - 2 \pi i \delta t } } { m \left ( 1 - e ^ { - 2 \pi i t } + z / m \right ) } . \end{align*}


\begin{align*} u _ N ^ e = u ( \xi , t ) = \sum _ { i = m - 2 } ^ { m + 3 } \delta _ i ( t ) Q _ i ( \xi ) , \end{align*}


\begin{align*} \begin{aligned} \sum _ \gamma \Big ( \frac { \sin \frac 1 2 ( \gamma - \tau ) \log x } { \frac 1 2 ( \gamma - \tau ) \log x } \Big ) ^ { \ ! 2 } & = \frac { \log \frac { \tau } { 2 \pi } } { \log x } - \frac 2 { \log x } \sum _ { n \le x } \frac { \Lambda ( n ) } { n ^ { 1 / 2 } } \Big ( 1 - \frac { \log n } { \log x } \Big ) \cos ( \tau \log n ) \\ & + O \Big ( \frac 1 { \tau \log x } \Big ) + O \Big ( \frac { x ^ { 1 / 2 } } { ( \tau \log x ) ^ 2 } \Big ) . \end{aligned} \end{align*}


\begin{align*} \psi _ { \rho _ { N } } \left ( x , t , k \right ) = \psi \left ( x , t , k \right ) - { \displaystyle \sum \limits _ { n = 1 } ^ { N } } \rho \left ( I _ { n } \right ) e ^ { 8 \kappa _ { n } ^ { 3 } t } y _ { n } \left ( x , t \right ) \int _ { x } ^ { \infty } \psi \left ( s , t , k \right ) \psi \left ( s , t , \mathrm { i } \kappa _ { n } \right ) \mathrm { d } s , \end{align*}