\begin{align*} \tau = \mathcal { T } \circ \mathcal { S } \colon J ^ { - 1 } ( \mu ) & \to \pi ^ * \left ( ( T ^ 1 _ k ) ^ * ( Q / G ) \right ) \end{align*}


\begin{align*} b _ \alpha ( s ) = \begin{cases} b ( s ) & s \leq \bar \rho - \alpha , \\ b ( \bar \rho - \alpha ) & s > \bar \rho - \alpha \end{cases} \end{align*}


\begin{align*} \left \{ \begin{aligned} & \dot { \phi } = \frac { 1 } { 2 } \Delta \phi - V \phi - \beta | \phi | ^ { 2 } \phi + \Omega L _ z \phi + \mu _ { \phi } ( t ) \phi , \mathbf { x } \in \mathbb { R } ^ d , \ ; t \geq 0 , \\ & \phi ( \mathbf { x } , 0 ) = \phi _ { 0 } ( \mathbf { x } ) , \mathbf { x } \in \mathbb { R } ^ d , \end{aligned} \right . \end{align*}


\begin{align*} \delta = s ^ { 2 } ( 1 + r \pi ^ { d ( \delta ) } ) = s ^ { 2 } ( 1 + r \pi ^ { d ( c ) } ) , \quad \ ; r , s \in \mathcal { O } _ { F } ^ { \times } \ , \end{align*}


\begin{align*} M _ { d _ 0 } ( f , H ) : = { 1 \over \# X _ f ^ - ( H ) } \sum _ { \chi \in X _ f ^ - ( H ) } \vert L ( 1 , \chi ' ) \vert ^ 2 \end{align*}


\begin{align*} \lambda = - 3 \pm i \sqrt { 4 \pi ^ 2 ( l ^ 2 - l m + m ^ 2 ) - 9 } . \end{align*}


\begin{align*} \langle X f , f \rangle = \int _ \R x f ( x ) \overline { f ( x ) } \ , d x = \int _ \R f ( x ) \overline { ( x f ( x ) ) } \ , d x = \langle f , X f \rangle , \end{align*}


\begin{align*} C _ q = { C \over q } \varphi ( x _ q , t _ q ) ^ { 1 - q } , \end{align*}


\begin{align*} m ^ { ( 2 ) } \rightarrow \Delta _ { R } ( k ) , \vert J ^ { ( 2 ) } ( x , t , k ) - I \vert = \mathcal { O } \left ( e ^ { - \gamma t } \right ) , \end{align*}


\begin{align*} \widetilde { f } ( 0 + ) = \frac { 2 a } { 1 + a ^ 2 } \frac { 1 } { \sqrt { 2 \pi } } \int f ( x ) \psi _ + ( x , 0 ) \ , d x \mbox { a n d } \widetilde { f } ( 0 - ) = \frac { 1 } { a } \widetilde { f } ( 0 + ) . \end{align*}


\begin{align*} Y _ { s , t } ( x , v ) = X _ { s , t } ( x + \check \tau _ { t , x } v , v ) - ( x + ( s - \tau _ { x } ) v ) , \end{align*}


\begin{align*} \overline { \{ g _ { \alpha } : \alpha \in S _ 2 \} } = A ^ 2 ( D , e ^ { - \varphi } ) , \end{align*}


\begin{align*} M _ b T _ a e ^ { - \pi c t ^ 2 } = e ^ { - \pi c ( t - a ) ^ 2 } e ^ { 2 \pi i b t } . \end{align*}


\begin{align*} G : = \left \{ \begin{aligned} & \begin{pmatrix} 1 & 0 \\ - r e ^ { 2 i t \theta } & 1 \end{pmatrix} , & k \in U _ 1 , \\ & \begin{pmatrix} 1 & - r ^ * e ^ { - 2 i t \theta } \\ & 1 \end{pmatrix} , & k \in U _ 1 ^ * , \\ & I , \textnormal { e l s e w h e r e } \end{aligned} \right . \end{align*}


\begin{align*} \pi ( F ) = \int _ { \mathbf { H } } F ( \mathbf { h } ) \pi ( \mathbf { h } ) \ , d \mathbf { h } . \end{align*}


\begin{align*} \forall \gamma ^ 1 > _ \mathbb { R } 0 , { \gamma ' } ^ 1 > _ \mathbb { R } 0 , x ^ X , { x ' } ^ X \left ( x = _ X x ' \land \gamma = _ \mathbb { R } \gamma ' \rightarrow J ^ A _ \gamma x = _ X J ^ A _ { \gamma ' } x ' \right ) . \end{align*}


\begin{align*} I _ 6 \leq & c \int _ 0 ^ \tau \frac { 1 } { | \Omega _ R | } \int _ { \Omega _ R } p ( \rho ) \varepsilon ^ { - 2 } \int _ { \Omega _ R } \left ( P ( \rho ) - P ( r ) - P ' ( r ) ( \rho - r ) \right ) \\ \leq & c \int _ 0 ^ \tau \frac { 1 } { | \Omega _ R | } \int _ { \Omega _ R } p ( \rho ) \mathcal E ( \rho , u | r , U ) ( t ) \end{align*}


\begin{align*} t _ j ^ S = t _ { j - 1 } ^ S + \tau ^ { \rm R e a c } _ { m } R _ j + \tau ^ { \rm R e a c } _ { V _ j } + \tau _ { \rm f } . \end{align*}


\begin{align*} \frac { ( \delta ( v _ 1 ) + \cdots + \delta ( v _ n ) ) ^ 2 - \delta ( v _ 1 ) ^ 2 - \cdots - \delta ( v _ n ) ^ 2 } { 2 } = \frac { 1 - ( \delta ( v _ 1 ) ^ 2 + \cdots + \delta ( v _ n ) ^ 2 ) } { 2 } . \end{align*}


\begin{align*} S ^ T J S = J \end{align*}


\begin{align*} \left \| f \right \| _ { \mathcal { D } ( \mathcal { L } ) } : = \left \| ( \lambda - \mathcal { L } ) f \right \| _ { L ^ 2 ( M ) } . \end{align*}


\begin{align*} \partial _ t \| u ( \cdot , t ) \| ^ 2 _ { L ^ 2 _ { \varLambda ^ q } ( { \mathbb R } ^ n ) } + \mu \ , \sum _ { j = 1 } ^ n \| \partial _ j u ( \cdot , t ) \| ^ 2 _ { L ^ 2 _ { \varLambda ^ { q } } ( { \mathbb R } ^ n ) } = ( B _ q ( w , u ) ( \cdot , t ) , u ( \cdot , t ) ) ^ 2 _ { L ^ 2 _ { \varLambda ^ { q } } ( { \mathbb R } ^ n ) } . \end{align*}


\begin{align*} \sum _ { j = 0 } ^ { \mathcal { J } } \sum _ { p \in \mathcal { S } ( j ) } ( \log p ) \left | L ( \tfrac { 1 } { 2 } , f \otimes \chi _ { 8 p } ) \right | ^ 2 | \mathcal { M } ( p ) | ^ 2 \ll X . \end{align*}


\begin{align*} \sigma ( A ^ * ) : = \Big \{ \lambda ^ p _ { k } , \ , k \geq k _ 0 ; \ \ \lambda ^ h _ k , \ , | k | \geq k _ 0 \Big \} \cup \Big \{ \lambda _ 0 \Big \} \cup \Big \{ \widehat \lambda _ n , \ , 1 \leq n \leq n _ 0 \Big \} . \end{align*}


\begin{align*} \nabla _ v \mu ( v ) = - v \psi ( v ) , \end{align*}


\begin{align*} \begin{bmatrix} i \delta _ 1 & 0 \\ 0 & i \delta _ 2 \end{bmatrix} = \Omega _ 2 \left ( \begin{bmatrix} i \varepsilon & 0 \\ 0 & i \varepsilon \end{bmatrix} \right ) , \end{align*}


\begin{align*} \hat { V } _ A ( \tau ) = \left [ - A e ^ { \left [ \frac { \alpha } { 2 } - l + \frac { 2 \eta } { \alpha } \left ( \frac { \alpha } { 2 } - l \right ) \right ] \tau } , A e ^ { \left [ \frac { \alpha } { 2 } - l + \frac { 2 \eta } { \alpha } \left ( \frac { \alpha } { 2 } - l \right ) \right ] \tau } \right ] ^ { l - 1 } . \end{align*}


\begin{align*} \left ( H - b \beta \Lambda \right ) \phi _ { i , , \beta } = 2 \beta b \left ( \frac { \alpha } { 2 } - i + \tilde \lambda \right ) \phi _ { i , , \beta } , \end{align*}


\begin{align*} \mathcal { E } _ { \Sigma } ( f _ j , f _ j ) = \mathcal { E } _ { \Sigma } ( g _ j , g _ j ) + 2 \varepsilon \mathcal { E } _ { \Sigma } ( g _ j , f ) + \varepsilon ^ 2 \mathcal { E } _ { \Sigma } ( f , f ) . \end{align*}


\begin{align*} \int _ \mathcal { Q } | Z f ( x , \omega ) | \ , d x \leq \int _ \mathcal { Q } \sum _ { k \in \Z ^ d } | f ( x + k ) | \ , d x = \norm { f } _ 1 , \end{align*}


\begin{align*} \tau \left [ q _ { t } + v \cdot \nabla q - q \cdot \nabla v + ( \nabla \cdot v ) q \right ] + q = - \kappa \nabla \theta , \end{align*}


\begin{align*} | S _ p \cup \{ p \} | = | S \cup \{ 0 \} | = d + 1 . \end{align*}


\begin{align*} \phi _ { \delta } ^ { x _ 0 } ( t ) = e ^ { \mu t } , t \in [ 0 , T ] , \end{align*}


\begin{align*} \int _ 0 ^ T f \dd ( \eta - \S ' ( u ; h ) ) = 0 \forall f \in G [ 0 , T ] . \end{align*}


\begin{align*} \mathfrak { S } \bf x = \bf e , \end{align*}


\begin{align*} \pi f _ A ( t ) + \pi f _ A ( t ^ { - 1 } ) = 1 , \end{align*}


\begin{align*} \left \{ \begin{array} { r c l } \dot { \xi } _ { r , m } ( t ) & = & A \xi _ { r , m } ( t ) + \alpha ( \xi _ { r , m } ( t ) ) - \rho ( \xi _ { r , m } ( t ) ) + \sum _ { j = 1 } ^ r \sigma ^ j ( \xi _ { r , m } ( t ) ) \dot { B } _ m ^ j ( t ) \medskip \\ \xi _ { r , m } ( 0 ) & = & x . \end{array} \right . \end{align*}


\begin{align*} 0 = I _ 0 \subseteq I _ 1 \subseteq . . . \subseteq I _ n = B \end{align*}


\begin{align*} A ^ { 0 } ( \partial _ { x } ^ { \alpha } u ) _ { t } - B ^ { i j } \partial _ { i } \partial _ { j } ( \partial _ { x } ^ { \alpha } u ) & = A ^ { 0 } \partial _ { x } ^ { \alpha } \left ( ( A ^ { 0 } ) ^ { - 1 } [ f - A ^ { i } \partial _ { i } u - D u ] \right ) \\ & + A ^ { 0 } G _ { \alpha } ( ( A ^ { 0 } ) ^ { - 1 } B ^ { i j } , \partial _ { i } \partial _ { j } u ) . \end{align*}


\begin{align*} \underset { n \rightarrow \infty } { \varliminf } \vert x _ 1 ^ n - \mathbb { E } [ x ^ n _ 1 ] \vert = \underset { n \rightarrow \infty } { \varliminf } \vert x _ 1 - \mathbb { E } [ x ^ n _ 1 ] \vert . \end{align*}


\begin{align*} \check H _ { c t } ^ 1 ( X ; A ) = \frac { \mathcal C _ h ( X , S ^ 1 ) } { \mathcal C _ h ( X , \R ) } . \end{align*}


\begin{align*} A = n \ , \frac { 2 \gamma a } { e ^ { - 2 \gamma a } - 1 } B = n \ , \frac { 2 \gamma a } { e ^ { - 2 \gamma a } - 1 } \ , e ^ { - 2 \gamma a } , \gamma < 0 . \end{align*}


\begin{align*} \delta _ { \alpha , \beta } g & = \mathfrak { D } _ { { \rm l e f t } , \frac { ( \alpha + \beta + 2 ) x - ( \beta - \alpha ) } { 2 ( 1 - x ^ 2 ) } } ( ( 1 - x ^ 2 ) ^ { 1 / 2 } g ) \\ - \delta ^ * _ { \alpha , \beta } g & = \mathfrak { D } _ { { \rm r i g h t } , - \frac { ( \alpha + \beta ) x - ( \beta - \alpha ) } { 2 ( 1 - x ^ 2 ) } } ( ( 1 - x ^ 2 ) ^ { 1 / 2 } g ) . \end{align*}


\begin{align*} [ T u , T v ] = T \Big ( \rho ( T u ) v - \rho ( T v ) u \Big ) , \ ; \forall u , v \in V . \end{align*}


\begin{align*} = \int _ { U _ t ( \mathbb { A } ) P ^ t _ n ( \Q ) \backslash G ( \mathbb { A } ) / K _ 1 ( \mathfrak { n } ) K _ { \infty } } \int _ { U _ t ( \Q ) \backslash U _ t ( \mathbb { A } ) } \sum _ { \xi , \gamma } \phi ( \tilde { \tau } _ t ( \xi g \times \gamma h ) , s ) \mathbf { f } ( n h ) \mathbf { d } n \mathbf { d } h , \end{align*}


\begin{align*} u \rhd _ s v = v s v ^ { - 1 } u \in G . \end{align*}


\begin{align*} \underline \dim _ { { \rm l o c } } ( \mu , x ) = \liminf _ { r \rightarrow 0 ^ { + } } \frac { \log ( \mu ( B ( x , r ) ) ) } { \log ( r ) } \ \mbox { a n d } \ \overline \dim _ { { \rm l o c } } ( \mu , x ) = \limsup _ { r \rightarrow 0 ^ { + } } \frac { \log ( \mu ( B ( x , r ) ) ) } { \log ( r ) } . \end{align*}


\begin{align*} \lambda _ { \ell } ( t ) = C ( u _ 0 ) ( 1 + o ( 1 ) ) ( T - t ) ^ { \frac { 2 \ell } { \alpha } } t \to T . \end{align*}


\begin{align*} G = \{ x \rightarrow x y , \ ; \ ; y \rightarrow x ^ 2 \} \end{align*}


\begin{align*} f = S ^ { - 1 / 2 } S ( S ^ { - 1 / 2 } f ) = \sum _ { \gamma \in \Gamma } \langle f , S ^ { - 1 / 2 } e _ \gamma \rangle S ^ { - 1 / 2 } e _ \gamma , \end{align*}


\begin{align*} \varphi ( a b ) = \varphi ( b \sigma ^ { \varphi } ( a ) ) , \psi ( a b ) = \psi ( b \sigma ^ { \psi } ( a ) ) , \forall a , b \in A . \end{align*}


\begin{align*} \begin{cases} \lambda _ 3 \lambda _ 2 & = \frac { ( a - 2 ) \pm \sqrt { a ^ 2 - 4 a } } { 2 } \\ \lambda _ 2 \lambda _ 1 & = \frac { ( b - 2 ) \pm \sqrt { b ^ 2 - 4 b } } { 2 } \\ \lambda _ 1 \lambda _ 3 & = \frac { ( c - 2 ) \pm \sqrt { c ^ 2 - 4 c } } { 2 } \end{cases} \end{align*}


\begin{align*} 2 \varepsilon ( w _ { \widehat { \delta } } ) + \varepsilon ( \Delta _ { 4 k + 2 } ^ 2 ) = 0 . \end{align*}


\begin{align*} C = C ' \left ( 1 + \rho ^ { 2 s } \| c \| _ { L ^ \infty ( B _ \rho ( e _ 1 ) ) } \right ) \end{align*}


\begin{align*} D _ { p q } = N Y ^ { o p } _ p \times _ { Y _ 0 } Y _ 1 \times _ { Y _ 0 } N X _ q \end{align*}


\begin{align*} \begin{cases} \ \\ [ 5 p t ] \ \end{cases} \in \ \R _ { \ell } . \end{align*}


\begin{align*} B ( M _ { - \eta } T _ \xi g _ 0 ) ( z ) = e ^ { - \pi i \xi \cdot \eta } e ^ { - \frac { \pi } { 2 } | w | ^ 2 } K _ w ( z ) . \end{align*}


\begin{align*} \mu _ { \mathrm { d i s c } } ( \delta _ { 1 4 } ) - \mu _ { \mathrm { d i s c } } ( \delta _ { 1 3 } ) = \mu _ { \mathrm { d i s c } } ( \gamma _ { 2 } \cup { \gamma _ { 3 } } ) - \mu _ { \mathrm { d i s c } } ( \gamma _ { 1 } \cup { \gamma _ { 4 } } ) = 2 \end{align*}


\begin{align*} \frac 1 { \omega ( P ) ^ { \frac 2 p - 1 } } \sum \limits _ { Q \subseteq P } \mu ( Q ) | \psi _ Q ( q _ { j } ( \cdot , y ) ) ( x _ Q ) | ^ 2 & \lesssim \sup _ { x \in P } \frac 1 { \omega ( B ( x , r ^ { - j } ) ) ^ { \frac 2 p } } \sum \limits _ { k = k _ 0 } ^ { \infty } r ^ { - 2 | k - j | \varepsilon } \\ & \leqslant C _ j \sup _ { x \in P } \frac 1 { \omega ( B ( x , r ^ { - j } ) ) ^ { \frac 2 p } } . \end{align*}


\begin{align*} V _ { g _ 0 } h _ \alpha ( x , - \omega ) = e ^ { \pi i x \cdot \omega } B h _ \alpha ( z ) e ^ { - \frac { \pi } { 2 } | z | ^ 2 } = e ^ { \pi i x \cdot \omega } e ^ { - \frac { \pi } { 2 } | z | ^ 2 } e _ \alpha . \end{align*}


\begin{align*} v _ { \varepsilon } ( \mathbf { x } _ { \varepsilon } ) = \sigma _ 3 \sqrt { \tilde { \lambda } _ 0 \underline { m } } \ , | \mathbf { x } _ { \varepsilon } - \mathbf { y } _ { \varepsilon } | = \sigma _ 3 \sqrt { \tilde { \lambda } _ 0 \underline { m } } \ , ( | \mathbf { x } _ { \varepsilon } - \bar { \mathbf { x } } _ { \varepsilon } | + | \bar { \mathbf { x } } _ { \varepsilon } - \mathbf { y } _ { \varepsilon } | ) \ge \end{align*}


\begin{align*} \inf _ { ( \rho , m ) } \mathcal L ( \rho , m , S ) = \ < S ( 1 ) , \rho ^ b \ > - \ < S ( 0 ) , \rho ^ a \ > - \int _ 0 ^ 1 \max ( H ( \dot S , \nabla _ G S ) , 0 ) d t , \end{align*}


\begin{align*} \Big | \frac { 1 } { 2 \varepsilon n ^ 2 } \sum _ { x } \Phi _ n ( s , \tfrac { x } { n } ) \sum _ { r = 1 } ^ { \varepsilon n } \int _ { \Omega - \Omega _ 1 ( x ) } \overleftarrow { \eta } ^ { \ell } ( x ) [ \eta ( x + 1 ) - \eta ( x + 1 + r ) ] [ f ( \eta ) - f ( \eta ^ { x + 1 , x + 1 + r } ) ] d \nu _ { b } \Big | \end{align*}


\begin{align*} \int _ { - \infty } ^ \infty W \psi ( x , p ) \ , d p = | \psi ( x ) | ^ 2 , \\ \int _ { - \infty } ^ \infty W \psi ( x , p ) \ , d x = | \phi ( p ) | ^ 2 ; \end{align*}


\begin{align*} J \cup K \cup L & = ( \mathbb { N } \setminus \{ \bar { a } _ n \} ) \cup \{ \bar { a } _ n \} = \mathbb { N } , \end{align*}


\begin{align*} a : = - \frac { 6 \sigma _ 3 } { \sigma _ 1 } > 0 . \end{align*}


\begin{align*} 0 = \lim _ { n \to \infty } u ( k _ n , r _ n ) = u ( k _ 0 + d , R / 2 ) , \end{align*}


\begin{align*} \bar { \Delta } ^ { \rm P } \triangleq \lim \limits _ { N \to \infty } \frac { 1 } { N } \sum _ { j = 1 } ^ { N } \Delta \left ( t ^ S _ j - 1 \right ) . \end{align*}


\begin{align*} m _ 1 + 2 m _ 2 + \cdots + ( d k + e ) m _ { d k + e } = b _ r + d k - d + e . \end{align*}


\begin{align*} \cal { M } ( \phi _ { 1 , \Delta } ) & = P \circ \begin{bmatrix} \lambda _ 1 & 0 \\ \lambda _ 1 + 1 & - 1 \end{bmatrix} \circ P ^ { - 1 } , \\ \cal { M } ( \phi _ { 2 , \Delta } ) & = P \circ \begin{bmatrix} \lambda _ 2 & 0 \\ - ( \lambda _ 2 + 1 ) & - 1 \end{bmatrix} \circ P ^ { - 1 } , \\ \cal { M } ( \phi _ { 3 , \Delta } ) & = P \circ \begin{bmatrix} - 1 & 0 \\ 0 & \lambda _ 3 \end{bmatrix} \circ P ^ { - 1 } \end{align*}


\begin{align*} u + H ( x , D u ) = 0 \end{align*}


\begin{align*} \nu _ * = \alpha _ * + \frac { \gamma _ * } { p _ 0 } ‐ \frac { \gamma _ * } { q } . \end{align*}


\begin{align*} L ( s , f \otimes \chi _ d ) & = \sum _ { n = 1 } ^ { \infty } \frac { \lambda _ f ( n ) \chi _ d ( n ) } { n ^ s } = \prod _ { p \nmid d } \left ( 1 - \frac { \lambda _ f ( p ) \chi _ d ( p ) } { p ^ s } + \frac { 1 } { p ^ { 2 s } } \right ) ^ { - 1 } . \end{align*}


\begin{align*} \begin{aligned} \big ( D _ E ^ + \big ) = - \big ( D _ E ^ - \big ) & = - \int _ { N } a ( D _ E ^ - ) + \frac { h ( \nu B \nu ) + \eta ( \nu B \nu ) } { 2 } \\ & = \int _ { N } a ( D _ E ^ + ) + \frac { h ( B ) - \eta ( B ) } { 2 } , \end{aligned} \end{align*}


\begin{align*} x = \sum _ { i = 0 } ^ { 3 j + t } \alpha _ { i } e _ { i } , \quad \alpha _ { 3 j + t } \neq 0 . \end{align*}


\begin{align*} \begin{aligned} | \widetilde \mu ( x ) - \widetilde \mu ( y ) | = | \mu ( x ) - \mu ( y ) | \leq c \cdot ( 1 + | x | ^ { \ell _ \mu } + | y | ^ { \ell _ \mu } ) \cdot | x - y | . \end{aligned} \end{align*}


\begin{align*} p ( x ) = \phi ( g _ { j ( 1 ) } ( x ) , \ldots , g _ { j ( T ) ( T ) } ( x ) ) , \deg ( p ) \leq \delta , \end{align*}


\begin{align*} \forall t \in \R \colon S ( t ) : = \begin{cases} \{ - t ^ 2 \} & t \leq 0 , \\ \{ \sqrt t \} & t > 0 . \end{cases} \end{align*}


\begin{align*} \int _ { B ( x , r ) } | \nabla u | ^ { p - 2 } \nabla u \cdot \nabla ( u - v ) \ , d \mu = \int _ { B ( x , r ) } ( u - v ) f \ , d \nu . \end{align*}


\begin{align*} \mathcal { H } ^ { \zeta } ( E ) = \lim _ { t \to 0 ^ + } \mathcal { H } ^ { \zeta } _ t ( E ) . \end{align*}


\begin{align*} g | _ S = \begin{cases} g ( x ) & x \in S \\ - \infty & . \end{cases} \end{align*}


\begin{align*} \int _ a ^ b c \dd g = c ( g ( b ) - g ( a ) ) \int _ a ^ b f \dd c = 0 \end{align*}


\begin{align*} \ell : = \frac { | m h \pm n k | } { d } . \end{align*}


\begin{align*} \underset { x \in \mathbb { R } ^ n } { } \ c ^ \top x + \frac { 1 } { 2 } x ^ \top Q x + \| D x \| _ 1 + \delta _ { \mathcal { K } } ( x ) , \ A x = b , \end{align*}


\begin{align*} \binom { c _ n k n } l \frac { ( k ( n - s - 1 ) ) _ { c _ n k n - l } } { ( k n ) _ { c _ n k n } } = \frac { \binom { c _ n k n } l } { \prod _ { i = 1 } ^ l ( k n - c _ n k n + i ) } \prod _ { t = 0 } ^ { c _ n k n - l - 1 } \frac { k ( n - s - 1 ) - t } { k n - t } . \end{align*}


\begin{align*} H ^ 0 = C _ 0 , H ^ i = ( D _ i ) , V _ k ^ i = U ^ { k } _ i H ^ i . \end{align*}


\begin{align*} & \textnormal { d e t } \Phi _ { R , L } ( x , t , k ) = 1 , k \in \mathbb { R } \backslash [ - C , C ] . \end{align*}


\begin{align*} A \norm { c } _ { \ell ^ 2 } ^ 2 \leq \norm { \sum _ { \gamma \in \Gamma } c _ \gamma f _ \gamma } _ \mathcal { H } ^ 2 \leq B \norm { c } _ { \ell ^ 2 } ^ 2 , \forall c = ( c _ \gamma ) _ { \gamma \in \Gamma } \in \ell ^ 2 ( \Gamma ) . \end{align*}


\begin{align*} i \partial _ t u - \partial _ { x x } u + V ( x ) u \pm \left | u \right | ^ { 2 } u = 0 , u ( 0 , x ) = u _ { 0 } ( x ) , \end{align*}


\begin{align*} \rho = - \bar { p } ( 1 - \xi ) / \mathrm { l i } ( 1 - \xi ) , \ \ \mu _ i = - \vartheta _ i \ln ( 1 - \xi ) , \ \forall i , \end{align*}


\begin{align*} \left [ z ^ 1 , z ^ 2 \right ] = \left [ \cos s \ , e ^ { i \Phi _ 1 } , \sin s \ , e ^ { i \Phi _ 2 } \right ] . \end{align*}


\begin{align*} \left | \frac { \psi ( x ) - x } { x } \right | & \leq A ( x _ 0 , \sigma ) \cdot \left ( B _ 2 r \right ) ^ { 5 - 2 \sigma } \exp \left ( \frac { B _ 2 ( 5 - 8 \sigma ) } { 3 } r \right ) \\ & = A ( x _ 0 , \sigma ) \cdot \left ( B _ 2 \frac { \log ^ { 3 / 5 } x } { ( \log \log x ) ^ { 1 / 5 } } \right ) ^ { 5 - 2 \sigma } \exp \left ( \frac { B _ 2 ( 5 - 8 \sigma ) } { 3 } \frac { \log ^ { 3 / 5 } x } { ( \log \log x ) ^ { 1 / 5 } } \right ) . \end{align*}


\begin{align*} x _ q = \lambda y _ q + ( 1 - \lambda ) z _ q , w _ q ( x _ q , t _ q ) = \left ( \lambda u ( y _ q , t _ q ) ^ q + ( 1 - \lambda ) u ( z _ q , t _ q ) ^ q \right ) ^ { \frac { 1 } { q } } . \end{align*}


\begin{align*} S ( \alpha _ i ) B ( \alpha _ i ) = 1 , ( S B ) ^ { ( \ell ) } ( \alpha _ i ) = 0 \quad 1 \le \ell \le \mu _ i - 1 . \end{align*}


\begin{align*} { \rm s p } _ 0 ( \tau ) = \frac { 1 } { 2 } p ^ e S , S = { \bigcup } _ { u = 0 } ^ n { \bigcup } _ { z = 1 } ^ { e } { \bigcup } _ { \alpha \in \Xi _ { u , z } } U ( \alpha ) . \end{align*}


\begin{align*} \Delta f _ i + f _ i = 0 . \end{align*}


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


\begin{align*} M _ { n + 1 } ( x ) = x \sum _ { k = 0 } ^ n { n \choose k } M _ { k } ( x ) M _ { n - k } ( x ) . \end{align*}


\begin{align*} \norm { V _ g f } _ 2 = \norm { f } _ 2 \norm { g } _ 2 \ , . \end{align*}


\begin{align*} \log \Bigl ( \frac { 1 } { 1 - X } \Bigr ) = - \log ( 1 - X ) = \sum _ { n = 1 } ^ \infty \frac { X ^ n } { n } . \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ { d - 1 } } \psi \tilde \psi \ , d x & = 1 , \\ \int _ { \mathbb { R } ^ { d - 1 } } \beta ( | g _ j ^ Q | ^ \frac { 1 } { p } \psi ) \tilde \psi \ , d x & = ( - 1 ) ^ { j + 1 } | g _ j ^ Q | ^ \frac { 1 } { p } \\ \int _ { \mathbb { R } ^ { d - 1 } } \tilde \psi \ , d x & = 0 . \end{align*}


\begin{align*} \Upsilon _ { \omega } ^ { + } : = \left \{ Y _ { \alpha \omega } ^ { m i } , Y _ { \beta \omega } ^ { m j } \right \} , \quad \Upsilon _ { \omega } ^ { - } : = \left \{ Y _ { \alpha \omega } ^ { m j } , Y _ { \beta \omega } ^ { m i } \right \} , \quad \Upsilon _ { \omega } : = \Upsilon _ { \omega } ^ { + } \cup \Upsilon _ { \omega } ^ { - } \end{align*}


\begin{align*} J ( \sigma , X , u ) = \left ( \begin{matrix} J _ 1 ( \sigma , X , u ) \\ J _ 2 ( \sigma , X , u ) \end{matrix} \right ) = \left ( \begin{matrix} \Im \int e ^ { - i \sigma } u ( x + X ) Q ( x ) d x \\ \Re \int e ^ { - i \sigma } u ( x + X ) \nabla Q ( x ) d x \end{matrix} \right ) , \end{align*}


\begin{align*} L ( 1 , \chi ) = L ( 1 , \chi ' ) \prod _ { q \mid d _ 0 } \left ( 1 - \frac { \chi ( q ) } { q } \right ) ^ { - 1 } \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ d } \left ( \sum _ { j = 1 } ^ d \int _ 0 ^ 1 x _ j U _ j ^ k ( \tau _ { t x } \omega ) d t \right ) \partial _ i \eta ( x ) d x = \int _ 0 ^ 1 \sum _ { j = 1 } ^ d \int _ { \mathbb { R } ^ d } \frac { y _ j } { t ^ { d + 1 } } U _ j ^ k ( \tau _ y \omega ) \partial _ i \eta \left ( \frac { y } { t } \right ) d y d t \end{align*}


\begin{align*} e ^ { e _ 0 t } \alpha _ + ( t ) = \int _ t ^ \infty e ^ { e _ 0 s } B ( R , \mathcal { Y } _ - ) d s . \end{align*}


\begin{gather*} \beta ( v ) = \sum R '' \otimes R ' v \end{gather*}


\begin{align*} \sigma f = f ^ { ( - 1 ) } \sigma , f \in \overline { k } ( t ) . \end{align*}


\begin{align*} \sum _ { \rho \in R _ \epsilon } ( \rho ) = \frac { 1 } { 2 \pi i } \int _ { \partial { R _ \epsilon } } \log \left ( \frac { s } { 2 } \right ) \ d s \end{align*}


\begin{align*} T _ u ^ { \sigma } ( q ) = \frac { - 1 } { \log 2 } \left ( q \cdot \log \Big ( \frac { u } { 1 - u } \Big ) + \log \left ( \frac { 1 } { 2 } \sqrt { 1 + 4 \Big ( \frac { 1 - u } { u ^ 2 } \Big ) ^ q } - \frac { 1 } { 2 } \right ) \right ) . \end{align*}


\begin{align*} | B | + ( \lambda - 1 ) \max B & \ \leqslant \ k - | \{ 1 , 2 , \ldots , k \} \cap A | + ( \lambda - 1 ) k \\ & \ \leqslant \ \lambda m - | \{ 1 , 2 , \ldots , k \} \cap A | \\ & \ \leqslant \ | A | - | \{ 1 , 2 , \ldots , k \} \cap A | \ = \ | F | . \end{align*}


\begin{align*} \rho ( \alpha ) \left [ \begin{array} { c } \rho ( z ) \\ u _ { 0 2 } \end{array} \right ] \mu ( z ) \lambda ( \alpha , z ) ^ { - 1 } \mu ( \alpha z ) ^ { - 1 } = \left [ \begin{array} { c } \rho ( \alpha ) \rho ( z ) \\ u _ { 0 2 } \end{array} \right ] \lambda ( \rho ( \alpha ) , \rho ( z ) ) \mu ( z ) \lambda ( \alpha , z ) ^ { - 1 } \mu ( \alpha z ) ^ { - 1 } . \end{align*}


\begin{align*} a ^ { \dagger } a \ , e _ { \lambda } ^ { a ^ { \dagger } a - \lambda } ( t ) = e _ { \lambda } ^ { a ^ { \dagger } a - \lambda } ( t ) a ^ { \dagger } a = a ^ { \dagger } e _ { \lambda } ^ { a a ^ { \dagger } - \lambda } ( t ) a = a ^ { \dagger } e _ { \lambda } ^ { a ^ { \dagger } a + 1 - \lambda } ( t ) a . \end{align*}


\begin{align*} \ge \sigma _ 3 \sqrt { \tilde { \lambda } _ 0 \underline { m } } \ , | \mathbf { x } _ { \varepsilon } - \bar { \mathbf { x } } _ { \varepsilon } | = \sigma _ 3 \sqrt { \tilde { \lambda } _ 0 \underline { m } } \ , d ( \mathbf { x } _ { \varepsilon } , \partial \Omega ) ; . \end{align*}


\begin{align*} L ( \tfrac { 1 } { 2 } , f \otimes \chi _ { 8 p } ) = & 2 \sum ^ { \infty } _ { \substack { n = 1 } } \frac { \lambda _ f ( n ) \chi _ { 8 p } ( n ) } { \sqrt { n } } V \left ( \frac { n } { p } \right ) , \end{align*}


\begin{align*} \partial _ t u = \Delta u - 3 ( n - 4 ) u ^ 2 - ( n - 4 ) | x | ^ 2 u ^ 3 . \end{align*}


\begin{align*} \int ^ { T } _ { 0 } \int _ { \Omega } & \big ( \rho u \cdot \partial _ t \phi + \rho u \otimes u : \nabla _ x \phi + \rho ^ { \gamma } { \rm { d i v } } _ x \phi - \mathbb { S } ( \nabla _ x u ) : \nabla _ x \phi \\ & + ( j - n u ) \cdot \phi \big ) \ , d x d t + \int _ { \Omega } \rho _ 0 u _ 0 \cdot \phi ( 0 , x ) \ , d x = 0 , \end{align*}


\begin{align*} \log a = \log ( 1 + x v ) = \sum _ { m = 1 } ^ { \infty } ( - 1 ) ^ { m + 1 } \frac { ( x v ) ^ m } { m } . \end{align*}


\begin{align*} \left \langle X \right \rangle _ 0 = 0 , \ \ \ \ \ \ \ \ \left \langle X \right \rangle _ k = \sum _ { i = 1 } ^ k \mathbf { E } ( \xi _ i ^ 2 | \mathcal { F } _ { i - 1 } ) , \ \ \ \ k = 1 , . . . , n . \end{align*}


\begin{align*} M ( v , X ( k ) \setminus S ) = \sum \limits _ { y \in X ( k ) \setminus S } M ( v , y ) \end{align*}


\begin{align*} \mathcal { H } ^ { n - 1 } ( Z _ { \psi _ \lambda } \cap B ^ n _ 1 ) & \leq C \sum _ { m = 0 } ^ \infty e ^ { - c m } J _ { ( m + 1 ) \lfloor \sqrt { \lambda } \rfloor } . \end{align*}


\begin{align*} \forall x , y \in X \left ( x = y \rightarrow T x = T y \right ) \end{align*}


\begin{align*} \mathrm { G L } _ n ( \mathbb { B } ) = \{ g \in M _ n ( \mathbb { B } ) : \det ( g ) \neq 0 \} , \mathrm { S L } _ n ( \mathbb { B } ) = \{ g \in M _ n ( \mathbb { B } ) : \det ( g ) = 1 \} . \end{align*}


\begin{align*} A ( \phi ( x ) ) \ & = \ \frac { f ( \phi ( x ) ) - g ( \phi ( x ) ) } { h ( \phi ( x ) ) + 1 } \\ [ 1 1 p t ] & = \ \frac { ( f \circ \phi ) ( x ) - ( g \circ \phi ) ( x ) } { ( h \circ \phi ) ( x ) + 1 } \end{align*}


\begin{align*} | \Lambda ( M ) | = \sum _ { \Gamma \in H _ { 1 } ( Y ) } | \Lambda ( M , \Gamma ) | \end{align*}


\begin{align*} \tau ( a _ 1 \cdots a _ n ) & = 0 , \\ \ \ \ \tau ' ( a _ 1 \cdots a _ n ) & = \sum _ { j = 1 } ^ n \tau ( a _ 1 \cdots a _ { j - 1 } \tau ' ( a _ j ) a _ { j + 1 } \cdots a _ n ) . \end{align*}


\begin{align*} \tau ( a _ 1 \cdots a _ n ) = 0 . \end{align*}


\begin{align*} u ( t ) = \boldsymbol { S } ( t ) u _ { 0 } + \int \nolimits _ { 0 } ^ { t } \boldsymbol { S } ( t - \tau ) f ( \tau ) ) d \tau , \end{align*}


\begin{align*} \mathbb { E } \bigg [ \frac { \rho } { | \hat { h } _ i | ^ 2 } \Big | | \hat { h } _ i | ^ 2 \ge \mu _ i \bigg ] & = \frac { \rho } { \mathbb { P } ( | \hat { h } _ i | ^ 2 \ge \mu _ i ) } \int \limits _ { \mu _ i } ^ { \infty } \frac { e ^ { - x / \vartheta _ i } } { \vartheta _ i x } \mathrm { d } x = - \frac { \rho } { ( 1 - \xi ) \vartheta _ i } \mathrm { l i } ( 1 - \xi ) , \end{align*}


\begin{align*} \nabla _ { \nabla y ^ { k } } \nabla y ^ { l } = \widetilde { \bar { \nabla } _ { \bar { \nabla } z ^ { k } } \bar { \nabla } z ^ { l } } + A ( \nabla y ^ { k } , \nabla y ^ { l } ) , \end{align*}


\begin{align*} \widehat { A } + ( \widehat { A } ) ^ T = 2 I . \end{align*}


\begin{align*} \mu ^ { \# } ( k , \ell , m , n ) : = \int \overline { \mathcal { K } ^ { \# } ( x , k ) } \mathcal { K } ^ { \# } ( x , \ell ) \overline { \mathcal { K } ^ { \# } ( x , m ) } \mathcal { K } ( x , n ) \ , d x . \end{align*}


\begin{align*} \mathbf { h } = \sqrt { \frac { K } { K + 1 } } \mathbf { h } _ { L O S } + \sqrt { \frac { 1 } { K + 1 } } \mathbf { h } _ { N L O S } , \end{align*}


\begin{align*} \frac { d } { d x } f ( x ) = \frac { \sigma \sqrt { \log ( 1 / \delta ) } } { \sqrt { x } } - G _ 0 , \end{align*}


\begin{align*} \begin{aligned} | f _ n ( x , y ) | & \leq n ^ { - 1 / 2 } | | y | ^ { \ell _ \mu } - | x | ^ { \ell _ \mu } | \cdot | \sigma ( y ) | \le 4 | x - y | \cdot \frac { 1 + | y | ^ { \ell _ \sigma + 1 } } { 1 + | y | } \leq c \cdot ( 1 + | y | ^ { \ell _ \sigma } ) \cdot | x - y | . \end{aligned} \end{align*}


\begin{align*} \tilde { g } _ j ( x ) = g _ j ( x ) + \sum _ { k _ y = 1 } ^ { \infty } ( - 1 ) ^ { k _ y } \left ( g _ { 2 k _ y n _ y - j } ( x ) + g _ { 2 k _ y n _ y + j } ( x ) \right ) . \end{align*}


\begin{align*} S _ t = S _ t ( Y _ 0 , d W _ \cdot ) , \end{align*}


\begin{align*} \tau ( s , \mu + \eta ) = \bigoplus _ { 1 \leq i \leq n } \tilde { \omega } _ { r f } ^ { \sum _ { j ' = 0 } ^ { r f - 1 } \alpha ' _ { ( s , \mu ) , j ' , i } p ^ { j ' } } . \end{align*}


\begin{align*} 0 = \sum _ { z < p } \Psi ( z ) \Psi ( f ( z ) ) . \end{align*}


\begin{align*} \dot { \varrho } _ { L _ { \alpha _ j } } ( \pi ^ { \prime } ) = \varrho _ { \alpha } ( \pi ) . \end{align*}


\begin{align*} I ( \delta _ { k + 1 } \cup { \gamma } , \delta _ { k } \cup { \gamma } ) = I ( \delta _ { k + 1 } , \delta _ { k } ) + 2 \# ( \mathbb { R } \times \gamma \cap { v ^ { z } } ) = 2 + 2 \# ( \mathbb { R } \times \gamma \cap { v ^ { z } } ) . \end{align*}


\begin{align*} ( u , \eta , \tau ) \big | _ { t = 0 } = ( u _ 0 , \eta _ 0 , \tau _ 0 ) . \end{align*}


\begin{align*} \sum _ { n \in D } w ( n ) | y _ n | & \ = \ \sum _ { n \in D \cap ( A \cup B ) ^ c } w ( n ) | x _ n | + t \sum _ { n \in D \cap B } w ( n ) \\ & \ \leqslant \ \sum _ { n \in D \cap ( A \cup B ) ^ c } w ( n ) | x _ n | + t \sum _ { n \in E } 1 \\ & \ = \ \sum _ { n \in G } w ( n ) | z _ n | \ \leqslant \ \| z \| _ \omega . \end{align*}


\begin{align*} \phi : T \rightarrow k [ H ] , t _ i \mapsto Y ^ { a _ i } i = 1 , \ldots , m . \end{align*}


\begin{align*} \mathrm { W F } ( \overline { u } _ n ) = \left \{ \begin{array} { l @ { \quad : \quad } l } \{ 0 \} \times \mathbb { R } _ { < 0 } & \alpha = i \pi \\ \{ 0 \} \times \mathbb { R } _ { > 0 } & \alpha = - i \pi \\ \{ 0 \} \times \mathbb { R } \setminus \{ 0 \} & \end{array} \right . \end{align*}


\begin{align*} \det \Phi ( \xi ^ { ( - i ) } ) \neq 0 \end{align*}


\begin{align*} m ^ { ( 1 ) } ( x , t , k ) = m ( x , t , k ) e ^ { i t [ g ( k , \xi ) - \theta ( k , \xi ) ] \sigma _ 3 } . \end{align*}


\begin{align*} \mathcal { H } : = \Biggl \{ \prod _ { i = 1 } ^ d \bigl [ a _ { i \tau ( i , t _ i - 1 ) } , a _ { i \tau ( i , t _ i ) } \bigr ] \Biggm | t = ( t _ 1 , \ldots , t _ d ) \in \prod _ { i = 1 } ^ d \{ 1 , \ldots , l _ i \} \Biggr \} . \end{align*}


\begin{align*} \phi * \psi = \phi \succ \psi + \phi \prec \psi . \end{align*}


\begin{align*} \forall x \in \R \colon \Phi ( x ) : = \begin{cases} \R & x \leq 0 , \\ [ x ^ 2 , \infty ) & x > 0 \end{cases} \end{align*}


\begin{align*} B _ \ell ( n + 1 ) - \frac { \ell } { \ell + 1 } \frac { B _ { \ell + 1 } ( n + 1 ) } { n } = \frac { 1 } { \ell + 1 } n ^ { \ell } + O _ \ell \left ( n ^ { \ell - 1 } \right ) \end{align*}


\begin{align*} Z = \frac { 1 } { \lambda _ 1 } + \frac { \rho _ 1 } { \lambda _ 2 } + \frac { \rho _ 1 \rho _ 2 } { \lambda _ 3 } + \frac { \rho _ 1 \rho _ 2 \rho _ 3 } { \lambda _ 0 } . \end{align*}


\begin{align*} \Sigma = \bigsqcup _ { g \in ( \mathbb { Z } / l \mathbb { Z } ) } g ( \tilde { p } ) . \end{align*}


\begin{align*} \sigma ( z ) = z \prod _ { \gamma \in \Gamma \backslash 0 } \left ( 1 - \frac { z } { \gamma } \right ) e ^ { \frac { z } { \gamma } + \frac { z ^ 2 } { 2 \gamma ^ 2 } } . \end{align*}


\begin{align*} \Sigma _ { a } ^ { b } = 2 \sum _ { \substack { H _ a \le \gamma < T \\ a \le \beta < b } } \frac { x ^ { \beta - 1 } } { \gamma } . \end{align*}


\begin{align*} \begin{cases} \partial _ { z } \Phi & = \tau + R _ 1 ( T b - \kappa \tau ) - R _ 2 T n + \mathcal O ( R _ 1 ^ 2 , R _ 2 ^ 2 ) , \\ \partial _ { R _ 1 } \Phi & = n + \mathcal O ( R _ 1 , R _ 2 ) , \\ \partial _ { R _ 2 } \Phi & = b + \mathcal O ( R _ 1 , R _ 2 ) , \end{cases} \end{align*}


\begin{align*} \partial _ s h _ { V _ \ast } + ( v - V _ \ast ) \cdot \nabla _ x h _ { V _ \ast } - \nabla ( \phi * _ x \rho [ h _ { V _ \ast } ] ) \cdot \nabla _ v \mu & = - e _ 0 \nabla \Phi ( x ) \cdot \nabla _ v \mu , h _ { V _ \ast } ( 0 , \cdot ) = 0 . \end{align*}


\begin{align*} A ( z ) R ( z ) + B ( z ) S ( z ) = 1 \mbox { f o r a l l $ z \in \C $ . } \end{align*}


\begin{align*} \mu ( \{ \lambda : | \lambda - z _ 0 | \leq r \} ) = - r \frac { d } { d r } L ( U ^ \mu ; z _ 0 , r ) . \end{align*}


\begin{align*} - 4 \delta _ 0 ~ \leq ~ a _ n ( t , 0 + ) ~ = ~ - { \sigma _ n ( t ) \over 2 } ~ \leq ~ - 2 \delta _ 0 , 2 \delta _ 0 ~ \leq ~ a _ n ( t , 0 - ) ~ = ~ { \sigma _ n ( t ) \over 2 } ~ \leq ~ 4 \delta _ 0 . \end{align*}


\begin{align*} \sum _ { n \geq 0 } t ^ n \sum _ { \lambda \in \mathcal { P } ( n ) } L ( \lambda ) = \left ( \sum _ { n \geq 1 } \frac { a _ n t ^ { n } } { 1 - t ^ { n } } \right ) \prod _ { n \geq 1 } \frac { 1 } { ( 1 - t ^ n ) } . \end{align*}


\begin{align*} \boldsymbol { \Phi } = { \rm { d i a g } } ( e ^ { j \phi _ 1 } , e ^ { j \phi _ 2 } , \dots , e ^ { j \phi _ N } ) \end{align*}


\begin{align*} \int ( w ( \tau _ 0 ) - Q _ b ) \phi _ { 1 , b , \beta } \rho _ \beta d y = \varepsilon _ 1 ( \tau _ 0 ) \| \phi _ { 1 , b , \beta } \| ^ 2 . \end{align*}


\begin{align*} H _ n : = - \partial _ { x x } - n ( n + 1 ) ^ 2 ( x ) , n \in \mathbb { N } _ 0 , \end{align*}


\begin{align*} \ell _ \rho ( \gamma \vert _ { [ 0 , t ] } ) = \int _ 0 ^ t \rho ( y + s ) \ , d s = \begin{cases} \frac { 1 } { \beta - 1 } \left [ \frac { 1 } { y ^ { \beta - 1 } } - \frac { 1 } { ( y + t ) ^ { \beta - 1 } } \right ] & y \ge 1 , \\ 1 - y + \frac { 1 } { \beta - 1 } \left [ 1 - \frac { 1 } { ( t + y ) ^ { \beta - 1 } } \right ] & y < 1 \le y + t , \\ t & y + t < 1 . \end{cases} \end{align*}


\begin{align*} g ( x _ 0 ) - \underline { g } & \geq \sum _ { k = 0 } ^ K g ( x _ k ) - g ( x _ { k + 1 } ) \\ & \geq K \min \left ( c _ 1 \tau _ 1 \min ( \underline { \alpha _ 1 } , \underline { t _ 1 } ) \varepsilon _ 1 ^ 2 , c _ 2 \tau _ 2 ^ 2 \min \left ( \min \left ( \alpha _ { 0 2 } , \dfrac { 3 | 2 c _ 2 - 1 | \varepsilon _ 2 } { M _ g } \right ) , \underline { t _ 2 } \right ) ^ 2 \varepsilon _ 2 \right ) . \end{align*}


\begin{align*} H _ { i } ^ + = \left \lbrace x \in \mathbb { R } ^ n \middle | \left \langle \tilde a _ i , x \right \rangle \leqslant b _ i \right \rbrace . \end{align*}


\begin{align*} I _ H ( w ) = w + [ 0 , \lambda H ) \times \{ 0 \} . \end{align*}


\begin{align*} f ( t ) = \sum _ { k \in \Z } f ( k ) \frac { \sin ( \pi ( t - k ) ) } { \pi ( t - k ) } , t \in \R . \end{align*}


\begin{align*} \int _ { \mathbb S ^ 1 } B J _ { x x } B _ x \ , d x = - \int _ { \mathbb S ^ 1 } B _ x J _ { x } B _ x \ , d x - \int _ { \mathbb S ^ 1 } B J _ { x } B _ { x x } \ , d x , \end{align*}


\begin{align*} \forall \ , R > 0 \ ; : \ ; \lim _ { k \to + \infty } \sup _ { \mathbf { y } \in \R ^ N } \int _ { \mathbf { y } + B _ R ( \mathbf { y } ) } \rho _ { n _ k } = 0 \ ; , \end{align*}


\begin{align*} \begin{aligned} \partial _ t \rho _ 1 = & \partial _ x \left ( \partial _ x \rho _ 1 + \rho _ 1 \partial _ x \phi \right ) , \\ \partial _ t \rho _ 2 = & \partial _ x \left ( \partial _ x \rho _ 2 - 2 \rho _ 2 \partial _ x \phi \right ) , \\ - \partial _ x ^ 2 \phi = & 1 2 ( x - 0 . 5 ) ^ 2 + \rho _ 1 - 2 \rho _ 2 . \end{aligned} \end{align*}


\begin{align*} \lim _ { t \rightarrow 0 ^ + } \frac { t } { s ( r , t ) } = \frac { \lambda _ 1 ( \mu ) ^ 2 - r ^ 2 } { \lambda _ 1 ( \mu ) ^ 2 } . \end{align*}


\begin{align*} \Pi ^ \mathcal { F } _ \star ( X , Y ) = - \frac { 1 } { 2 c } \ , \mathbf { g } _ { t \star } ( X , Y ) \ , V _ \perp = - \frac { 1 } { 2 c } \ , \mathbf { g } _ { t \star } ( X , Y ) \star V _ \perp \end{align*}


\begin{align*} \mathcal { T } _ s ^ { - 1 } F ( x , t ) = F ( \tfrac { x + t } { 2 } , x - t ) . \end{align*}


\begin{align*} { \big \| e ^ { i t H } h \big \| } _ { L ^ \infty _ x } = \left \Vert \int \mathcal { K } ^ { \# } ( x , k ) e ^ { i k ^ 2 t } h ^ { \# } ( k ) \ , d k \right \Vert _ { L ^ \infty _ x } \lesssim \frac { 1 } { \sqrt { t } } { \big \| h ^ \# \big \| } _ { L ^ \infty _ k } + \frac { 1 } { t ^ { \frac { 3 } { 4 } } } { \big \| \partial _ k h ^ \# \big \| } _ { L ^ 2 _ k } . \end{align*}


\begin{align*} f _ \R ( k ) \ = \ \zeta ( k + 2 ) ( k = 0 , 1 , \ldots ) . \end{align*}


\begin{align*} \Phi ' _ i = h ' _ i \cdot \Psi ' _ i \cdot \phi ( h ' _ i ) ^ { - 1 } \end{align*}


\begin{align*} a ^ { n + 1 } = \gamma _ 1 a + \gamma _ 2 a ^ 2 + \dots + \gamma _ { n } a ^ { n } \end{align*}


\begin{align*} E & = O ( \epsilon ) \\ F & = O ( \epsilon ) \\ G & = 6 a _ { 0 } ^ { ( 1 ) } ( b _ { 0 } ( 0 ) ) + O ( \epsilon ) \\ H & = O ( \epsilon ) \end{align*}


\begin{align*} f ( w , x , y ) = \frac 1 2 \left ( \frac { 1 - \left | w \right | ^ 2 } { 1 + \left | w \right | ^ 2 } \right ) ^ 2 , \end{align*}


\begin{align*} \mathcal A ( \rho , m ) = \frac 1 2 \int _ 0 ^ 1 \Big ( \frac { m _ { 1 2 } ^ 2 } { \theta _ { 1 2 } ( \rho ) } + \frac { m _ { 2 3 } ^ 2 } { \theta _ { 2 3 } ( \rho ) } \Big ) d t , \end{align*}


\begin{align*} Y _ m ^ j \leq \sum _ { k = 1 } ^ m | \eta _ m ^ { j , k } | \end{align*}


\begin{align*} \Phi _ { \widehat \lambda _ n } : = ( \xi _ { \widehat \lambda _ n } , \eta _ { \widehat \lambda _ n } ) , \ 1 \leq n \leq n _ 0 . \end{align*}


\begin{align*} \begin{aligned} & \left \{ \varsigma _ { Q _ j , m } = 1 , \varsigma _ { Q _ j , i } = 1 \right \} \bigcap _ { r \in [ i - 1 ] } \left \{ \varsigma _ { Q _ j , r } = 0 \right \} \\ & = \left \{ \varsigma _ { Q _ j , i } = 1 , \varsigma _ { Q _ j , i - 1 } = 0 \right \} \\ & = \left \{ \varsigma _ { Q _ j , i } = 1 \right \} / \left \{ \varsigma _ { Q _ j , i - 1 } = 1 \right \} . \end{aligned} \end{align*}


\begin{align*} b \Phi ( a _ 0 , a _ 1 ) = \Phi ( a _ 0 a _ 1 ) - \Phi ( a _ 1 a _ 0 ) . \end{align*}


\begin{align*} \Bigg | \ln \frac { \mathbf { P } \left ( S _ n / ( \sqrt { n } \sigma ) > x \right ) } { 1 - \Phi \left ( x \right ) } \Bigg | = O \bigg ( \frac { x ^ 3 } { \sqrt { n } } \bigg ) , \ \ \ \ n \rightarrow \infty . \end{align*}


\begin{align*} | \Omega | & = \int _ \Omega u ^ { \frac { p } { p + 1 } } ( t , x ; s , u _ 0 ) u ^ { - \frac { p } { p + 1 } } ( t , x ; s , u _ 0 ) d x \\ & \le \big ( \int _ \Omega u ( t , x ; s , u _ 0 ) d x \big ) ^ { \frac { p } { p + 1 } } \big ( \int _ \Omega u ^ { - p } ( t , x ; s , u _ 0 ) d x \big ) ^ { \frac { 1 } { p + 1 } } \quad \forall \ , t > s . \end{align*}


\begin{align*} \lim _ { n \to \infty } \| u ( t , \cdot ; s , u _ n , a _ n , b _ n ) - u ( t , \cdot ; s , u _ 0 , a _ 0 , b _ 0 ) \| _ \infty = 0 \ , \ , { \rm u n i f o r m l y \ , \ , i n } \ , \ , t \in [ s , s + T ] \end{align*}


\begin{gather*} \dim Q / X ^ + = d - \ell ^ + - 2 = \ell ^ - , \\ \dim Q / X ^ - = d - \ell ^ - - 2 = \ell ^ + . \end{gather*}


\begin{align*} \alpha ( t , x ) = a ( x ) , ( t , x ) \in [ r , u ) \times X . \end{align*}


\begin{align*} \langle X , \nabla H \rangle = - \frac { 1 } { \alpha } \langle X , \nu \rangle ^ { - \frac { \alpha + 1 } { \alpha } } h _ { i j } \langle X , e _ { i } \rangle \langle X , e _ { j } \rangle \leq 0 , \end{align*}


\begin{align*} \begin{aligned} \| C \hat { V } \| _ F ^ 2 \geq \sum ^ { l } _ { t = 1 } \bar { \sigma } ^ 2 _ { t } - \frac { 2 \theta \| S \| ^ 2 _ F } { \sqrt { \alpha } } - \theta ( k + \sqrt { k } \xi ) \| C \| ^ 2 _ F . \end{aligned} \end{align*}


\begin{align*} \int _ a ^ b f \dd g = 0 \forall f \in G [ a , b ] . \end{align*}


\begin{align*} f = \sum \limits _ { k = - \infty } ^ { \infty } T _ { N } ^ { - 1 } D _ { k } ^ { N } D _ { k } ( f ) = \sum \limits _ { k = - \infty } ^ { \infty } D _ { k } D _ { k } ^ { N } T _ { N } ^ { - 1 } ( f ) \end{align*}


\begin{align*} d _ K ( h ) = \inf _ { g \in K } \| h - g \| = \eta . \end{align*}


\begin{align*} \overline { H } _ n ( \mathbf { s } ) = \overline { H } _ n ( s _ 1 , s _ 2 , \ldots , s _ r ) : = \sum \limits _ { 0 \leq k _ 1 < k _ 2 < \cdots < k _ r \leq n - 1 } \prod _ { j = 1 } ^ { r } \frac { 1 } { ( 2 k _ j + 1 ) ^ { s _ j } } , \end{align*}


\begin{align*} \frac { \dd } { \dd t } \widetilde { e } ^ { n , m } ( t ) + A ^ 2 \widetilde { e } ^ { n , m } ( t ) = - P _ n A P ( F ( X ^ m ( t ) ) - F ( X ^ { n } ( t ) ) ) . \end{align*}


\begin{align*} & \chi ^ 0 = \chi _ 0 : = \frac { \int _ { \mathbb { R } ^ d } \left ( \frac { 1 } { 2 } | \nabla \phi _ 0 | ^ 2 + V | \phi _ 0 | ^ 2 + \beta | \phi _ 0 | ^ 4 - \Omega \overline { \phi } _ 0 L _ z \phi _ 0 \right ) \mathrm { d } \mathbf { x } } { \| \phi _ 0 \| ^ 2 } + \sigma R ^ \prime ( \| \phi _ 0 \| ^ 2 - 1 ) . \end{align*}


\begin{align*} n ^ 0 _ \beta = \left ( - 1 \right ) ^ { \dim \left ( M _ \beta \right ) } e \left ( M _ \beta \right ) , \end{align*}


\begin{align*} & \frac { \tilde { \phi } ^ { n + 1 } - 2 \phi ^ n + \phi ^ { n - 1 } } { \tau ^ 2 } + \eta ^ n \frac { \tilde { \phi } ^ { n + 1 } - \phi ^ { n - 1 } } { 2 \tau } = - G ( \phi ^ n ) + \lambda ^ n \phi ^ n , \mathbf { x } \in \mathbb { R } ^ d , \\ & \phi ^ { n + 1 } = \tilde { \phi } ^ { n + 1 } / \| \tilde { \phi } ^ { n + 1 } \| , n \geq 1 . \end{align*}


\begin{align*} \partial _ \tau w = \partial _ y ^ 2 w + \frac { d + 1 } { y } \partial _ y w - \beta ( \tau ) \Lambda _ y w - 3 ( d - 2 ) w ^ 2 - ( d - 2 ) y ^ 2 w ^ 3 , \end{align*}


\begin{align*} F ( x , \omega ) = e ^ { 2 \pi i x \cdot \omega } V _ g f ( x , \omega ) \ , V _ g f ( - x , - \omega ) \end{align*}


\begin{align*} \dfrac { ( D ^ { \alpha - 1 } ) ^ * ( p _ \alpha f ) ( x ) } { p _ \alpha ( x ) } = ( D ^ { \alpha - 1 } ) ^ { * } ( f ) ( x ) + x f ( x ) + \frac { 1 } { p _ \alpha ( x ) } ( R ^ { \alpha } ) ^ * ( p _ \alpha , f ) ( x ) , \end{align*}


\begin{align*} \iint _ { \R ^ { 2 d } } | k ( x , t ) | ^ 2 \ , d ( x , t ) = | \Omega | | T | . \end{align*}


\begin{align*} \Psi _ { Q + i P , \nu , \tilde { \nu } } = { \bf L } _ { - \nu _ k } \dots { \bf L } _ { - \nu _ 1 } \tilde { \bf L } _ { - \tilde { \nu } _ { k ' } } \dots \tilde { \bf L } _ { - \tilde { \nu } _ 1 } \Psi _ { Q + i P } \end{align*}


\begin{align*} [ V _ 2 ] [ V _ 4 ] = \left ( \begin{matrix} 1 & 0 & t _ 1 & a + c t _ 1 + d t _ 3 & t _ 3 \\ 0 & 1 & t _ 2 & b + c t _ 2 + d t _ 4 & t _ 4 \end{matrix} \right ) . \end{align*}


\begin{align*} \frac { } { t } \left ( \frac { x ^ { - \nu _ 2 ( t ) } } { t } \right ) = \frac { x ^ { - \nu _ 2 ( t ) } \log x } { R _ 1 t ^ 2 } \left ( \frac { D ( 1 - 2 \log \log t ) + \log t } { \log ^ 3 t } - \frac { R _ 1 } { \log x } \right ) . \end{align*}


\begin{align*} f ( \lambda ^ { w _ 1 } x _ 1 , \ldots , \lambda ^ { w _ n } x _ n ) = \lambda ^ d f ( x _ 1 , \ldots , x _ n ) \end{align*}


\begin{align*} S ( \mathbf { P } _ { \ ! \sigma } ) = S ( \mathbf { p } _ { \sigma _ d } ; \ldots ; \mathbf { p } _ { \sigma _ 1 } ) = - \sum _ { n = 1 } ^ { d } C _ { n } ^ { ( d ) , \sigma } ( \mathbf { P } _ { \ ! \sigma } ) \cdot \int \ ! \log \boldsymbol { \mu } _ n ^ { \sigma } \ , \mathrm { d } \mathbf { p } _ { \sigma _ n } . \end{align*}


\begin{align*} f ( t ) = \iint _ { \R ^ { 2 d } } F ( x , \omega ) M _ \omega T _ x g ( t ) \ , d ( x , \omega ) \end{align*}


\begin{align*} \mathfrak { B } _ 1 \times \mathfrak { B } _ { 2 , \{ 1 , 2 \} } = [ 1 \bar { 3 } \ , \bar { 2 } , \bar 1 \ , \bar 2 \ , \bar 3 ] . \end{align*}


\begin{align*} f ^ j ( x ) & : = \sum _ { Q \in \mathcal { Q } } \psi _ Q ( x ) | g _ j ^ Q | ^ \frac { 1 } { p } , \\ h ^ j ( x ) & : = \sum _ { Q \in \mathcal { Q } } \mathbb { 1 } _ Q ( x ) \Theta ^ { Q , j } ( \lambda x ) . \end{align*}


\begin{align*} f ( x ) = \begin{cases} v ( x ) , & x + \delta > 0 , \\ u ( x ) , & x + \delta \leq 0 , \end{cases} \end{align*}


\begin{align*} - f '' + f + 1 - \rho \max ( f ^ * - f , 0 ) = 0 , \end{align*}


\begin{align*} \Box y = 2 \langle \Box \phi , \phi \rangle - 2 \left ( | \phi _ t | ^ 2 - | \phi _ x | ^ 2 \right ) = 2 \left ( | \phi _ t | ^ 2 - | \phi _ x | ^ 2 \right ) ( y - 1 ) . \end{align*}


\begin{align*} n _ 1 , \ldots , n _ L = n \gg 1 , \end{align*}


\begin{align*} \begin{aligned} d X ^ x _ t & = \mu ( X ^ x _ t ) \ , d t + \sigma ( X ^ x _ t ) \ , d W _ t , t \in [ 0 , \infty ) , \\ X ^ x _ 0 & = x , \end{aligned} \end{align*}


\begin{align*} \psi _ Z ( s ) = \prod _ { h = 1 } ^ m \prod _ { i = 1 } ^ { r _ { 2 h } } \left ( s - i + 2 h - 1 \right ) \end{align*}


\begin{align*} \sum \limits _ { \mathbf { p } _ k \in \mathcal { P } ( d , k ) } f ( \mathbf { p } _ k ) & = \sum \limits _ { 1 \le p _ 1 < \cdots < p _ k \le d } f ( \mathbf { p } _ k ) = \sum _ { p _ { k } = 1 } ^ { d } \sum _ { p _ { k - 1 } = 1 } ^ { p _ k - 1 } \cdots \sum _ { p _ 1 = 1 } ^ { p _ 2 - 1 } f ( \mathbf { p } _ k ) . \end{align*}


\begin{align*} \mathcal { T } ( R , L ) = \mathcal { T } ( R + \tau \ , * , L ) - \tau \ ; \mathcal { T } ( * , L ) \succeq 0 , \end{align*}


\begin{align*} \varepsilon \circ S = \varepsilon . \end{align*}


\begin{align*} \kappa ( f ) = \sum _ { ( d , e ) \in D _ { 1 } ( P G ) } | f ( d ) \cap f ( e ) | \end{align*}


\begin{align*} \Phi ' = J _ n , \Psi ' = \mathrm { d i a g } [ 1 , 1 , 1 , - \alpha , - \alpha , - \alpha ] , \end{align*}


\begin{align*} z = k ^ + \cdot ( g _ 0 ^ { - 1 } \cdot \phi ( g ^ + ) \cdot g _ 0 ) . \end{align*}


\begin{align*} \dot \rho = \nabla _ { S } \mathcal H ( \rho , S ) + \nabla _ { S } \mathcal H _ 1 ( \rho , S ) \dot W ^ { \delta } , \ ; \\ \dot S = - \nabla _ { \rho } \mathcal H ( \rho , S ) - \nabla _ { \rho } \mathcal H _ 1 ( \rho , S ) \dot W ^ { \delta } , \end{align*}


\begin{align*} \langle \mathcal { A } ( \tilde { f } ) ; v \rangle _ H = \langle f ; v \rangle _ H , \end{align*}


\begin{align*} f _ * = - \frac { \lambda _ 0 } { 2 ( p - 1 ) } Q + \sum _ { j = 1 } ^ N \mu _ j \left ( \partial _ { x _ j } Q \right ) \end{align*}


\begin{align*} h ( t ) = A _ 1 t \log ^ { - \alpha } t \exp \left ( - C u ( t ) \right ) , \end{align*}


\begin{align*} \sup _ { t \in ( 0 , \infty ) } t ^ { \frac { s } { \alpha } } \left \| e ^ { - \mu t \Lambda ^ { \alpha } } * f \right \| _ { H ^ s } \leq & \ C ( \alpha , s ) \| f \| _ { L ^ 2 } , \\ \lim _ { t \to 0 } t ^ { \frac { s } { \alpha } } \left \| e ^ { - \mu t \Lambda ^ { \alpha } } * f \right \| _ { H ^ s } = & \ 0 . \end{align*}


\begin{align*} C _ { \ell } ^ { a v } ( t ) = \frac { c _ { \ell } R _ \ell ( t ) } { t } , \end{align*}


\begin{align*} \dim _ { \mathrm H } F = \sup _ { \mathbf { p } _ { \sigma _ 2 } \in \mathcal { P } _ { 2 } ^ { \sigma } } \ ; t ( \mathbf { p } _ { \sigma _ 2 } ; \ , ( \mathbf { p } _ { \sigma _ 2 } ) _ 1 ^ { \sigma } ) , \end{align*}


\begin{align*} \langle X f , f \rangle = \int _ { \R } X f ( x ) \overline { f ( x ) } \ , d x = \int _ \R x | f ( x ) | ^ 2 \ , d x . \end{align*}


\begin{align*} \overline { M } = \sqrt { 1 + \| P \| } \overline { L } . \end{align*}


\begin{align*} \mathcal { A } ^ 1 _ m = \mathcal { A } ^ { 1 , i n i t } _ m + \mathcal { C } _ m + \mathcal { D } _ m , \end{align*}


\begin{align*} \varphi ( a _ n \cdots a _ 1 v b _ 1 \cdots b _ m ) = \tau ( a _ n b _ m ) \cdots \tau ( a _ 1 b _ 1 ) \varphi ( v ) \end{align*}


\begin{align*} \operatorname { T o r } ^ S _ * ( k [ K ] , k ) = { H } _ * \big ( \operatorname { K o s } ^ { { k [ K ] } } ( v _ 1 , \ldots , v _ m ) \big ) \end{align*}


\begin{align*} \phi \circ \theta _ T ( u , v ) = \theta _ { T ^ { ' } } ( \psi ( u ) , \psi ( v ) ) \circ \phi , \forall u , v \in V . \end{align*}


\begin{align*} \mathcal { A } ^ n = \mathcal { D } _ x ^ T \mathcal { M } ^ n \mathcal { D } _ x + \mathcal { D } _ z ^ T \mathcal { M } ^ n \mathcal { D } _ z , \end{align*}


\begin{align*} K _ i ( \mathcal { A } ^ \infty _ G ( X , E ) ) \cong K _ i ( C ^ * _ G ( X , E ) ) , \ i = 0 , 1 . \end{align*}


\begin{align*} 1 \geq \mu _ { \mathrm { d i s c } } ( \gamma _ { 1 } \cup { \gamma _ { 2 } } ) - \mu _ { \mathrm { d i s c } } ( \gamma _ { 3 } ) = 1 2 - \mu _ { \mathrm { d i s c } } ( \gamma _ { 3 } ) \end{align*}


\begin{align*} \Pi ( A , B ) ( x , y ) = \left ( \begin{pmatrix} a ( x , y ) \\ h ( x , y ) \end{pmatrix} , \begin{pmatrix} b ( x , y ) + c x + d x ^ { 2 } + e x ^ { 3 } + f x ^ { 4 } \\ x \end{pmatrix} \right ) \end{align*}


\begin{align*} \phi ( \mathfrak { g } , s ) = \chi _ v ( \det ( d _ p ) ) ^ { - 1 } | \det ( d _ p ) | _ { v } ^ { - s } j ( \mathfrak { g } , z _ 0 ) ^ { - k } | j ( \mathfrak { g } , z _ 0 | ^ { k - s } . \end{align*}


\begin{align*} \mu ( \phi ( x , b ) ) = \nu ' ( \phi ( x , b ) ) \mu ( c l _ B ( \phi ( x , b ) ) ) \end{align*}


\begin{align*} g ^ { - 1 } b \phi ( g ) = k ^ o _ 1 \omega k . \end{align*}


\begin{align*} f _ c ( y ) = c ^ { \frac { 1 } { p - 1 } } f ( c y ) . \end{align*}


\begin{align*} \int _ c ^ x H ( v ) d v & = \int _ c ^ x \int _ v ^ b \frac { \partial f } { \partial v } ( u , v ) d u d v \\ & = \int _ c ^ x \int _ c ^ u \frac { \partial f } { \partial v } ( u , v ) d v d u + \int _ x ^ b \int _ c ^ x \frac { \partial f } { \partial v } ( u , v ) d v d u . \end{align*}


\begin{align*} \sum _ { k = 0 } ^ { N - 1 } \cos ( 2 \pi k / N ) \sin ( 2 \pi k / N ) = 0 , \end{align*}


\begin{align*} \langle \mathcal { A } ( G ( u ) ) ; v \rangle _ H = \langle u ; v \rangle _ H . \end{align*}


\begin{align*} h ( \mathcal { I } ) : = \Delta _ { \mathbf { L } ( \mathbf { t } ) } ( \mathcal { I } ) \cdot \Delta _ { \mathbf { R } ( \mathbf { t } ) } ( \mathcal { I } ) , \mathcal { I } \in \mathfrak { G } ( \mathbf { L } ) \end{align*}


\begin{align*} \hat \sigma = s _ * \cdot \frac { \frac 1 q ‐ \frac { 1 } { p _ 0 } } { ‐ s _ * ‐ \frac { 1 } { p _ 0 } + \frac { 1 } { p _ 1 } + \frac { 2 s _ * } { q } } . \end{align*}


\begin{align*} \gamma _ { k } ^ { \langle n \rangle } = \frac { \left ( \sum _ { m = 1 } ^ M \sqrt { \eta _ { m k } } \alpha _ { m k } \right ) ^ 2 } { \sum _ { m = 1 } ^ M \beta _ { m k } \eta _ { m k } \alpha _ { m k } + \frac { 1 } { \gamma _ t } } . \end{align*}


\begin{align*} \mathcal { T } _ { 0 } : = \mathcal { J } , \quad \mathcal { T } _ { u } : = ( \mathcal { T } _ { u - 1 } ) _ { i _ { u } } ^ { \alpha _ { u } } \end{align*}


\begin{align*} \rho _ { t } = \frac { 1 } { { i } _ { t - 1 } } \left ( \rho _ { n + 1 } + i _ { t } \cdot \rho _ { t - 1 } \right ) t = 1 , \dots , n \end{align*}


\begin{align*} \left \langle \tilde a _ i , x \right \rangle = b _ i \ ; ( 1 \leqslant i \leqslant m ) . \end{align*}


\begin{align*} ( a + b ) \circ c = a \circ c - c + b \circ c \end{align*}


\begin{align*} \alpha : = \left ( a \gamma Q ^ { \gamma - 3 } _ 0 \right ) ^ { - 1 / 2 } , \ \ \ \beta : = \frac { Q _ 0 V ^ 2 _ 0 } { \nu } , \ \ \ \delta : = \frac { Q _ 0 V _ 0 } { \nu } , \end{align*}


\begin{align*} H ^ { ( 2 m ) } _ n \sim \frac { ( 1 - 4 \alpha ) c _ { 2 m } } { m ( 1 - 2 \alpha ) - 1 } \cdot n ^ { - 1 } ( n \to \infty ) . \end{align*}


\begin{align*} \frac { \phi ^ { n + 1 } - \phi ^ { n } } { \tau } & = \left ( \frac 1 2 \Delta - \vartheta ^ n \right ) \phi ^ { n + 1 } + \bigg ( \vartheta ^ n - V - \beta | \phi ^ n | ^ 2 + \Omega L _ z + \chi ^ n - \sigma R ^ \prime ( \| \phi ^ n \| ^ 2 - 1 ) \bigg ) \phi ^ n , \\ \frac { \chi ^ { n + 1 } - \chi ^ { n } } { \xi ^ n \tau } & = 1 - \| \phi ^ { n + 1 } \| ^ 2 , n = 0 , 1 , \ldots \end{align*}


\begin{align*} c ^ * > c _ 0 ^ * : = \frac { ( \lambda _ 1 + \gamma ) ^ 2 ( \lambda _ 2 - \lambda _ 1 ) ^ 2 } { 4 ( 2 \lambda _ 1 \lambda _ 2 - \lambda _ 1 ^ 2 + 4 \lambda _ 2 \gamma + 3 \gamma ^ 2 ) } \end{align*}


\begin{align*} \Phi \natural \Phi ^ { \xi , \eta } ( x , \omega ) = \iint _ { \R ^ { 2 d } } \langle \varphi , \rho ( x ' , \omega ' ) \varphi \rangle \langle \rho ( \xi , \eta ) \varphi , \rho ( x - x ' , \omega - \omega ' ) \varphi \rangle e ^ { \pi i ( x \cdot \omega ' - x ' \cdot \omega ) } \ , d ( x ' , \omega ' ) . \end{align*}


\begin{align*} \hat { \sigma } : \Delta _ { d + 1 } & \to \R ^ N ; \\ ( a _ 0 , a _ 1 , \dots , a _ d ) & \mapsto A \sigma ( a _ 1 / A , \dots , a _ d / A ) , A = \sum _ { i = 0 } ^ d a _ i . \end{align*}


\begin{align*} \mathcal { L i } _ { K , s } ( z ) ^ { ( - 1 ) } = \frac { z } { \theta - t } + \mathcal { L i } _ { K , s } ( z ) . \end{align*}


\begin{align*} \mathfrak { Z } = \mathfrak { Z } _ n = \mathfrak { Z } _ { m , r } = \{ z \in \C _ n ^ n : U ( z ) \in \Omega \} , U ( z ) = \left [ \begin{array} { c } z \\ u _ 0 \end{array} \right ] , u _ 0 = \left [ \begin{array} { c c } 0 & 1 _ r \\ 1 _ { 2 m } & 0 \end{array} \right ] . \end{align*}


\begin{align*} F = \begin{bmatrix} a & 1 \\ b & - a \end{bmatrix} , \tilde F = \begin{bmatrix} \tilde a & 1 \\ \tilde b & - \tilde a \end{bmatrix} , \end{align*}


\begin{align*} F ( z ) = \sum _ { \alpha \in \N _ 0 ^ d } c _ \alpha z ^ \alpha . \end{align*}


\begin{align*} X _ { s , t } ( x , \Psi _ { s , t } ( x , v ) ) = x - ( t - s ) v . \end{align*}


\begin{align*} U ^ \mu ( \lambda ) = \int _ \mathbb { C } \log \frac { 1 } { | \lambda - z | } d \mu ( z ) , \lambda \in \mathbb { C } , \end{align*}


\begin{align*} \mu ( \phi ( x , b ) ) = \frac { 1 } { | H | } \mu ( X ( b ) ) = \frac { | \Sigma | } { | G | } \mu ( X ( b ) ) = \nu ' ( \phi ( x , b ) ) \mu ( X ( b ) ) \end{align*}


\begin{align*} \tau _ t ( a _ n \cdots a _ 1 v b _ 1 \cdots b _ m ) = t \cdot \tau ( a _ n \cdots a _ 1 \tau ' ( v ) b _ 1 \cdots b _ m ) + o ( t ) \end{align*}


\begin{align*} X _ { n , k } ^ { \mathcal { S } } = M + \tau ( A ^ { ( n ) } _ { M , n , l } ) \ \ W _ { \mathcal { S } ( n , k ; M ) } \cap A ^ { ( n ) } _ { M , n , l } . \end{align*}


\begin{align*} - 2 f _ 3 ( t , x ) & = \int _ { t - d } ^ { t + d } \int _ { \lambda \in \R } e ^ { i s \lambda } \hat { g } ( \lambda ) d \lambda d s \\ & = \int _ { t - d } ^ { t + d } \int _ { | \lambda | \leq 1 } e ^ { i s \lambda } \hat { g } ( \lambda ) d \lambda d s + \int _ { t - d } ^ { t + d } \int _ { | \lambda | > 1 } e ^ { i s \lambda } \hat { g } ( \lambda ) d \lambda d s . \end{align*}


\begin{align*} f ( t ) = ( 1 + t ) \log _ { \lambda } ( 1 + t ) - t . \end{align*}


\begin{align*} | R _ j ( x , y ) | & \lesssim | y _ j - x _ j | \int _ 0 ^ \infty \frac 1 { V ( x , y , \sqrt t ) } \frac t { \| x - y \| ^ 2 } e ^ { - c d ( x , y ) ^ 2 / t } \frac { d t } { t \sqrt t } \\ & \leqslant \frac 1 { \| x - y \| } \bigg ( \int _ 0 ^ { d ( x , y ) ^ 2 } + \int _ { d ( x , y ) ^ 2 } ^ \infty \bigg ) \frac 1 { V ( x , y , \sqrt t ) } e ^ { - c d ( x , y ) ^ 2 / t } \frac { d t } { \sqrt t } \\ & = : I _ 1 + I _ 2 . \end{align*}


\begin{align*} \beta = \sum m _ { \mathbf { t } } \ell ^ { \mathbf { t } } + \sum n _ { i , j } \gamma _ { \mathbf { m _ 1 } , \mathbf { m _ 2 } } + \sum p _ { \mathbf { s } , \mathbf { t } } \sigma _ { \mathbf { s } , \mathbf { t } } \end{align*}


\begin{align*} \mathfrak { i } : \mathbb { B } \stackrel { \sim } \longrightarrow \{ x \in M _ 2 ( \mathbb { K } ) : \overline { x } I J = I J x \} . \end{align*}


\begin{align*} ( \nabla _ X A ) Y = ( \nabla _ Y A ) X + 2 \langle \xi , E ^ { \perp } _ 3 \rangle ( \langle Y , E ^ { \top } _ 3 \rangle X - \langle X , E ^ { \top } _ 3 \rangle Y ) . \end{align*}


\begin{align*} \langle a \rangle _ n : = \begin{cases} \mathbb { D } _ { n + a - 1 } ^ { q ^ { - ( a - 1 ) } } & , \\ 1 / \mathbb { L } _ { - a - n } ^ { q ^ n } & , \\ 0 & . \end{cases} \end{align*}


\begin{align*} U _ { J , z } \langle \alpha \rangle = \sum _ { \beta : \mathrm { a d m i s s i b l e \ , \ , o r b i t \ , \ , s e t \ , \ , w i t h \ , \ , } [ \beta ] = \Gamma } \# \{ \ , u \in \mathcal { M } _ { 2 } ^ { J } ( \alpha , \beta ) / \mathbb { R } ) \ , | \ , ( 0 , z ) \in u \ , \} \cdot \langle \beta \rangle . \end{align*}


\begin{align*} \check { \sigma } ^ { \check { \varphi } } ( \varphi ( - a ) ) = \varphi ( - S ^ 2 ( a ) \delta _ { \varphi } ^ { - 1 } ) , \check { \delta } _ { \check { \varphi } } ( a ) = \varepsilon ( \sigma ^ { \varphi } ( a ) ) , \check { \nu } = \nu ^ { - 1 } . \end{align*}


\begin{align*} \Phi _ { \lambda ^ p _ k } : = ( \xi _ { \lambda ^ p _ k } , \eta _ { \lambda ^ p _ k } ) \ \ \lambda ^ p _ k , \ \ \forall k \geq k _ 0 , \end{align*}


\begin{align*} \tau ^ * = \sup \{ \tau _ 1 \ge \tau _ 0 ( \varepsilon , b , \beta ) ( \tau ) \in V _ 1 [ A , \eta , \tilde \eta ] ( \tau ) , \forall \tau \in [ \tau _ 0 , \tau _ 1 ] \} . \end{align*}


\begin{align*} \phi _ v ( \tilde { \tau } _ m ( h _ v \times 1 ) ) = \chi _ { v } ( \det ( r ) ) | \det ( r ) | _ v ^ { - s } . \end{align*}


\begin{align*} m _ \pm ( x , k ) = 1 \pm \int _ x ^ { \pm \infty } D _ k ( \pm ( y - x ) ) V ( y ) m _ \pm ( y , k ) \ , d y , D _ k ( x ) = \frac { e ^ { 2 i k x } - 1 } { 2 i k } ; \end{align*}


\begin{align*} \begin{aligned} & u ' + 2 u ' u '' + 1 = 0 \\ \Longrightarrow & u '' = \frac { - 1 - u ' } { 2 u ' } = \frac { 1 } { - 2 u ' } - \frac { 1 } { 2 } \to + \infty \end{aligned} \end{align*}


\begin{align*} \tilde \psi ( \bar z ) = \psi _ { \max ( y _ j ) } \left ( \begin{pmatrix} x _ 0 \\ y _ 0 \end{pmatrix} , \ldots , \begin{pmatrix} x _ { q - 1 } \\ y _ { q - 1 } \end{pmatrix} \right ) + \sum _ { t = 1 } ^ { \max ( y _ j ) - 1 } d _ { q - 2 } \chi _ t \left ( \begin{pmatrix} x _ 0 \\ y _ 0 \end{pmatrix} , \ldots , \begin{pmatrix} x _ { q - 1 } \\ y _ { q - 1 } \end{pmatrix} \right ) \end{align*}


\begin{align*} A & = \left ( \sum _ { n \leq x } \sum _ { n < m \leq y } + \sum _ { n \leq x } \sum _ { y < m \leq M } + \sum _ { x < n \leq N } \sum _ { n < m \leq M } \right ) \frac { 1 } { m ^ { s _ 1 } n ^ { s _ 2 } ( m + n ) ^ { s _ 3 } } \\ & = \sum _ { n \leq x } \sum _ { n < m \leq y } \frac { 1 } { m ^ { s _ 1 } n ^ { s _ 2 } ( m + n ) ^ { s _ 3 } } + A _ 1 + A _ 2 , \end{align*}


\begin{align*} - t e _ { \lambda } ^ { - r } ( t ) & = - t \sum _ { n = 0 } ^ { \infty } \frac { ( - r ) _ { n , \lambda } } { n ! } t ^ { n } = t \sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n - 1 } \frac { \langle r \rangle _ { n , \lambda } } { n ! } t ^ { n } \\ & = \sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \langle r \rangle _ { n - 1 , \lambda } \frac { t ^ { n } } { ( n - 1 ) ! } = \sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \langle r \rangle _ { n - 1 , \lambda } n \frac { t ^ { n } } { n ! } , \end{align*}


\begin{align*} \underset { t \to \infty } { \lim } \Delta _ { \ell , \pi } ^ { a v } ( t ) = \lim _ { t \to \infty } \sum _ { i = 1 } ^ { R _ \ell ^ \pi ( t ) } \frac { \frac { ( T _ { \ell i } ^ \pi ) ^ 2 } { 2 } + T _ { \ell i } ^ \pi Z _ { \ell i } ^ \pi } { t } . \end{align*}


\begin{align*} \mathcal { K } ( x , k ) : = \frac { 1 } { \sqrt { 2 \pi } } \begin{cases} T ( k ) \psi _ + ( x , k ) & k \geq 0 \\ T ( - k ) \psi _ - ( x , - k ) & k < 0 , \end{cases} \end{align*}


\begin{align*} \alpha _ p ( U _ 0 \bot p U _ 0 , T ) = 2 ( 1 + p ^ { - 1 } ) ^ 2 p . \end{align*}


\begin{align*} \max _ { j \in \lbrace 1 , \ldots , n \rbrace } \frac { | e _ j ^ \intercal ( A x _ 0 - b ) | ^ 2 } { \norm { A ^ \intercal e _ j } _ 2 ^ 2 \norm { A x _ 0 - b } _ 2 ^ 2 } \geq \sum _ { j = 1 } ^ n \frac { \norm { A ^ \intercal e _ j } _ 2 ^ 2 } { \norm { A } _ F ^ 2 } \frac { | e _ j ^ \intercal ( A x _ 0 - b ) | ^ 2 } { \norm { A ^ \intercal e _ j } _ 2 ^ 2 \norm { A x _ 0 - b } _ 2 ^ 2 } = \frac { 1 } { \norm { A } _ F ^ 2 } . \end{align*}


\begin{align*} o ( t ) = \tau _ t ( a _ 1 \cdots a _ n ) + \sum _ { r = 1 } ^ n ( - 1 ) ^ r \sum _ { 1 \leq k _ 1 < \ldots < k _ r \leq n } \left ( \prod _ { i = 1 } ^ { r } \tau _ t ( a _ { k _ i } ) \right ) \tau _ t ( a _ 1 \cdots \hat { a } _ { k _ 1 } \cdots \hat { a } _ { k _ r } \cdots a _ n ) \end{align*}


\begin{align*} \begin{aligned} & \sum \limits _ { i = 1 } ^ { N } { \left ( { { n _ { i + 1 } } - { n _ i } } \right ) } , \\ & \sum \limits _ { i = 1 } ^ { N } { \left ( { { n _ { i + 1 } } - { n _ i } } \right ) \left ( { 2 { \tau _ { \rm c } } + 2 \mathcal { T } + { n _ { i + 1 } } + { n _ i } } \right ) } . \end{aligned} \end{align*}


\begin{align*} \partial _ t X _ t = \Big ( \dot { x } ( t ) , \dot { x } ( t ) \cdot \nabla u ( t , x ( t ) ) + \partial _ t u ( t , x ( t ) ) \Big ) . \end{align*}


\begin{align*} \frac { a _ 1 } { a _ 1 } = 1 \leq 1 + \log ( 1 + a _ 1 ) \end{align*}


\begin{align*} \check { \Delta } ( \omega ) ( \chi \otimes 1 ) = \varphi ( - b _ { ( 2 ) } ) \otimes \varphi ( - S ^ { - 1 } ( b _ { ( 1 ) } ) a ) . \end{align*}


\begin{align*} E _ { y } ( x ) = \sum _ { j = 0 } ^ { 2 \lceil y \rceil } \frac { x ^ { j } } { j ! } . \end{align*}


\begin{align*} E ( 0 ) = \alpha , E ( 1 ) < \alpha , \end{align*}


\begin{align*} \phi \prec \psi : = m _ A ( \phi \otimes \psi ) \Delta _ \prec , \phi \succ \psi : = m _ A ( \phi \otimes \psi ) \Delta _ \succ \end{align*}


\begin{align*} \phi _ { i , o u t , \beta } ( y ) = \phi _ { i , \infty , \beta } ( y ) + \tilde { \lambda } ( \tilde { \phi } _ { i , \beta } ( y ) + R _ { i , 1 } ( y ) ) + R _ { i , 2 } ( y ) , \end{align*}


\begin{align*} { \rm t r } _ g ^ Y ( \kappa ) : = \int _ { G / Z _ g } \int _ Y c _ Y ( h g h ^ { - 1 } y ) { \rm t r } ( h g h ^ { - 1 } \kappa ( h g ^ { - 1 } h ^ { - 1 } y , y ) ) d x \ , d ( h Z ) . \end{align*}


\begin{align*} ~ \alpha _ { i 0 } \preceq _ i \cdots \preceq _ i \alpha _ { i n } \ ; \ ; \ ; \upsilon _ { i 0 } = \alpha _ { i 0 } \upsilon _ { i j } \preceq _ i \upsilon _ { i n } \upsilon _ { i j } \preceq _ i \alpha _ { i j } \ ; \ ; j \in \{ 1 , \ldots , n \} , \end{align*}


\begin{align*} \P ( \cup _ { \emptyset \neq U ' \subset U , W ' \subset U _ { \mathsf k ' } } \{ \mathcal L ( W ' ) = \mathcal L ( U ' ) \} \mid \mathcal L ( U ) = \mathbf L ) \geq \chi _ { \mathbf L } \ , . \end{align*}


\begin{align*} [ x + u , y + v , z + v ] _ { \overline { T } } = [ u , v , w ] _ T + \theta _ T ( v , w ) x - \theta _ T ( u , w ) y + D _ T ( u , v ) z . \end{align*}


\begin{align*} ( M \rightarrow Q _ n ^ { [ M ] } ) : = I ( M ; Q _ n ) , \end{align*}


\begin{align*} & \phi = \phi _ S + \phi _ R , \phi _ S = \phi _ 0 + \phi _ + + \phi _ - , \\ & \phi _ \ast ( x ) : = \frac { 1 } { \sqrt { 2 \pi } } \chi _ \ast ( x ) \int \mathcal { K } _ \ast ^ \# ( x , k ) \ , ( \mathcal { F } ^ \# \phi ) ( k ) \ , d k , \ast \in \{ 0 , + , - \} \\ & \phi _ R ( x ) : = \frac { 1 } { \sqrt { 2 \pi } } \int \mathcal { K } _ R ^ \# ( x , k ) \ , ( \mathcal { F } ^ \# \phi ) ( k ) \ , d k . \end{align*}


\begin{align*} \lim _ { t \rightarrow 0 ^ + } \frac { i t } { \omega _ 2 ( i t ) } & = \lim _ { t \rightarrow 0 ^ + } \frac { s ( | \lambda | , t ) t - t ^ 2 } { | \lambda | ^ 2 } = \frac { | \lambda | ^ 2 - \lambda _ 2 ( \mu ) } { | \lambda | ^ 2 } . \end{align*}


\begin{align*} \partial _ t ^ 2 \phi - \partial _ x ^ 2 \phi + \sin \phi = 0 , \end{align*}


\begin{align*} ( r a - r ^ { - 1 } ) v = y , \end{align*}


\begin{align*} - \begin{bmatrix} H + \rho I & - A ^ T \\ A & \delta I \end{bmatrix} \begin{bmatrix} \mathbf { x } \\ \mathbf { y } \end{bmatrix} + \begin{bmatrix} - \mathbf { g } + \rho \mathbf { x } _ k \\ \mathbf { b } + \delta \mathbf { y } _ k \end{bmatrix} \in N _ D ( \mathbf { x } , \mathbf { y } ) . \end{align*}


\begin{align*} B _ m ( \theta ) = \textbf { a } _ m ^ T ( \theta ) \textbf { w } _ m = \sum _ { n = 1 } ^ { N _ s } e ^ { - j 2 \pi f _ 0 \tau _ { n m } ( \theta ) } e ^ { j \psi _ { n m } } . \end{align*}


\begin{align*} s _ c = \frac { N } { 2 } - \frac { \gamma + 2 } { 2 ( p - 1 ) } , \end{align*}


\begin{align*} \left ( \int _ \R | f ' ( x ) | ^ 2 \ , d x \right ) ^ { 1 / 2 } = \left ( \int _ \R | ( 2 \pi i \omega ) \widehat { f } ( \omega ) | ^ 2 \ , d \omega \right ) ^ { 1 / 2 } . \end{align*}


\begin{align*} x _ i [ y _ j - ( \alpha _ 1 g _ 1 + \cdots + \alpha _ r g _ r ) ] & = ( \beta _ 1 g _ 1 + \cdots + \beta _ r g _ r ) y _ j \\ y _ j [ x _ i - ( \beta _ 1 g _ 1 + \cdots + \beta _ r g _ r ) ] & = ( \alpha _ 1 g _ 1 + \cdots + \alpha _ r g _ r ) x _ i \end{align*}


\begin{align*} \partial _ t u = \partial _ r ^ 2 u + \frac { d + 1 } { r } \partial _ r u - 3 ( d - 2 ) u ^ 2 - ( d - 2 ) r ^ 2 u ^ 3 , ( r , t ) \in \R _ + \times \R _ + . \end{align*}


\begin{align*} \mathcal { P } ( v ) : = \partial _ \xi ^ 2 v + \frac { d + 1 } { \xi } \partial _ \xi v - 3 ( d - 2 ) ( 2 Q + \xi ^ 2 Q ^ 2 ) v + \bar B ( v ) + \theta ( \tau ) \Lambda _ \xi Q + \theta ( \tau ) \Lambda _ \xi v - b ( \tau ) \partial _ \tau v , \end{align*}


\begin{align*} \langle ( - \triangle _ { g } ) ^ { \alpha } u , v \rangle = \int _ { \mathbb { R } ^ { d } } \int _ { \mathbb { R } ^ { d } } g \bigg { ( } \frac { u ( x ) - u ( y ) } { | x - y | ^ { \alpha } } \bigg { ) } \frac { v ( x ) - v ( y ) } { | x - y | ^ { \alpha } } \frac { d x d y } { | x - y | ^ { d } } , \end{align*}


\begin{align*} \zeta _ 2 ( z ) \wp _ 2 ( z ) = - \frac 1 2 \big ( \wp _ 2 ' ( z ) + f _ 2 ' ( z ) \big ) \end{align*}


\begin{align*} \left \{ \begin{array} { r c l } & & \partial _ t A _ \mu ( x , t ) + \partial ^ \nu F _ { \mu \nu } ( x , t ) + \left [ A ^ \nu , F _ { \mu , \nu } \right ] ( x , t ) = 0 , ~ ~ t > 0 , \\ & & A _ \mu ( x , 0 ) = A _ { \mu , 0 } ( x ) . \end{array} \right . \end{align*}


\begin{align*} \varphi _ t ( x _ q , t _ q ) = { 1 \over q } \varphi ( x _ q , t _ q ) ^ { 1 - q } \psi _ t ( x _ q , t _ q ) = { 1 \over q } \varphi ( x _ q , t _ q ) ^ { 1 - q } ( \lambda h _ q + ( 1 - \lambda ) k _ q ) , \end{align*}


\begin{align*} B f _ 1 ( x ) & = \sum \limits _ { j = k } ^ { \infty } q ^ { - ( j + 1 ) } g _ j \ast f _ 1 ( x ) \\ & = 0 . \end{align*}


\begin{align*} a _ h ( w , v ) = \sum _ { E \in { \cal T } _ h } \sum _ { \iota \in I ^ E } [ \mu \nabla w \cdot \nabla v + \boldsymbol { \beta } \cdot \nabla w \ , v + \sigma w v ] ( \xi ^ E _ \iota ) \ , \omega ^ E _ \iota \ , , \end{align*}


\begin{align*} \norm { \eta _ { \lambda ^ h _ k } - \tilde { \psi } _ k } _ { L ^ 2 } = \norm { \eta _ { \lambda ^ h _ k } } _ { L ^ 2 } \leq C | k | ^ { - 1 } , \end{align*}


\begin{align*} \dot \rho = \nabla _ { S } \mathcal H ( \rho , S ) ( 1 + \dot W ^ { \delta } ) ^ 2 , \ ; \\ \dot S = - \nabla _ { \rho } \mathcal H ( \rho , S ) ( 1 + \dot W ^ { \delta } ) ^ 2 , \end{align*}


\begin{align*} \left [ \mathfrak { A } \chi _ 1 \left ( \mathfrak { A } ( \mathcal { M } _ 1 ) \right ) , \mathfrak { A } \chi _ 2 \left ( \mathfrak { A } ( \mathcal { M } _ 2 ) \right ) \right ] = 0 . \end{align*}


\begin{align*} ( \log ( | x | ) ) ' = \tfrac { 1 } { x } . \end{align*}


\begin{align*} \varphi _ { \gamma + Z _ { r , m } } ( t ) \leq \varphi _ { \gamma + Z _ { r , m } } ( T ) = \frac { 1 } { \gamma + Z _ { r , m } } \big ( e ^ { ( \gamma + Z _ { r , m } ) T } - 1 \big ) \leq \frac { e ^ { \gamma T } } { \gamma } e ^ { Z _ { r , m } T } , t \in [ 0 , T ] . \end{align*}


\begin{align*} \forall k \in \N \colon x _ k : = \frac 1 k , y _ k : = \frac { 1 } { k ^ 2 } , x _ k ^ * : = 1 , \lambda _ k : = \frac { k } { 2 } , \end{align*}


\begin{align*} \Delta _ l ( g _ j \ast f ) & = \Delta _ l ( f \ast g _ j ) \\ & = \mathcal { F } ^ { - 1 } \phi _ l \mathcal { F } ( f \ast g _ j ) \\ & = ( \mathcal { F } ^ { - 1 } \phi _ l \mathcal { F } f ) \ast g _ j , \end{align*}


\begin{align*} D _ { \infty } ( \xi ) : = \exp \left \{ - \frac { 1 } { 2 \pi i } \left [ \left ( \int _ { - \eta } ^ { - C _ L } + \int _ { C _ L } ^ { \eta } \right ) \frac { \log ( 1 - r ( s ) r ^ { * } ( s ) ) } { X _ { L } ( s ) } d s + \left ( \int _ { - C _ L } ^ { - C _ R } + \int _ { C _ R } ^ { C _ L } \right ) \frac { \log r _ { + } ( s ) } { X _ { L + } ( s ) } d s \right ] \right \} . \end{align*}


\begin{align*} m ^ { P C } _ { \eta , + } ( \zeta ) = m ^ { P C } _ { \eta , - } ( \zeta ) J ^ { P C } ( \zeta ) , \zeta \in \Sigma ^ { p c } , \end{align*}


\begin{align*} U ( \mathcal { T } ) : = \{ g \in \mathrm { G L } _ { 2 n } ( K ) : g ^ { \ast } \mathcal { T } g = \mathcal { T } \} . \end{align*}


\begin{align*} y ^ i ( x , \omega ) = \int _ 0 ^ 1 \langle \nabla y ^ i ( \gamma _ \omega ( t ) , \omega ) , \gamma _ \omega ' ( t ) \rangle d t = \sum _ { k = 1 } ^ d \int _ 0 ^ 1 D _ k y ^ i ( 0 , \tau _ { \gamma _ \omega ( t ) } \omega ) \gamma _ { \omega , k } ' ( t ) \ 1 { \gamma _ \omega ( t ) \in W ' ( \omega ) } d t , \end{align*}


\begin{align*} I ( t , K ) : = e ^ { - i t K ^ 2 } a ( K ) \ , \mathrm { p . v . } \int _ \R e ^ { i q ^ 2 t } G ( q ) \frac { \psi ( K - q ) } { K - q } \ , d q \end{align*}


\begin{align*} n = q , \theta = \hat \theta = 0 . \end{align*}


\begin{align*} \langle v , y ( x , \omega ) \rangle & = \sum _ { i = 1 } ^ d \sum _ { k = 1 } ^ d \int _ 0 ^ 1 v _ i D _ k y ^ i ( 0 , \tau _ { \gamma ( t ) } \omega ) \gamma _ { \omega , k } ' ( t ) \ 1 { \gamma _ \omega ( t ) \in W ' ( \omega ) } d t \\ & = \sum _ { k = 1 } ^ d \int _ 0 ^ 1 \langle v , D _ k y ( 0 , \tau _ { \gamma _ \omega ( t ) } \omega ) \rangle \ 1 { \gamma _ \omega ( t ) \in W ' } \gamma _ \omega ' ( t ) d t . \end{align*}


\begin{align*} P - \alpha Q = u ( R + \alpha S + \alpha ^ 2 U ) ^ 2 \end{align*}


\begin{align*} | \Delta _ j ( 1 \otimes f - f \otimes 1 ) & | _ { C ^ \alpha ( X \times X ) } = | \Delta _ j ( 1 \otimes f - f \otimes 1 ) | _ { C ^ \alpha ( U _ j ) } \leq \\ \leq & | \Delta _ j | _ { C ^ \alpha ( U _ j ) } \| 1 \otimes f - f \otimes 1 \| _ { C ^ 0 ( U _ j ) } + \| \Delta _ j \| _ { C ^ 0 ( U _ j ) } | 1 \otimes f - f \otimes 1 ) | _ { C ^ \alpha ( U _ j ) } \lesssim \\ \lesssim & \| f \| _ { C ^ \beta ( X ) } j ^ { - \gamma } , \end{align*}


\begin{align*} \sigma _ 1 \overline { \Phi _ { R , L } ( x , t , \overline { k } ) } \sigma _ 1 = \Phi _ { R , L } ( x , t , k ) , \overline { \Phi _ { R , L } ( x , t , - \overline { k } ) } = \Phi _ { R , L } ( x , t , k ) . \end{align*}


\begin{align*} Z ( S ) = \{ P \in \mathbb A _ { \mathbb C } ^ n \ , : \ , f ( P ) = 0 f \in S \} . \end{align*}


\begin{align*} \frac { \tilde { \lambda } _ 0 + B } { 1 - A } = \tilde { \lambda } _ 0 + ( \tilde { \lambda } _ 0 A + B ) \frac { 1 } { 1 - A } = \tilde { \lambda } _ 0 + ( \tilde { \lambda } _ 0 A + B ) \sum _ { k = 0 } ^ { + \infty } A ^ k \end{align*}


\begin{align*} \forall \gamma ^ 1 , x ^ X , p ^ X \left ( \gamma > _ \mathbb { R } 0 \land \tilde { \rho } > _ \mathbb { R } - \gamma \land p = _ X J ^ A _ { \gamma } x \rightarrow \gamma ^ { - 1 } ( x - _ X p ) \in A p \right ) . \end{align*}


\begin{align*} \sum z _ i ^ 2 = 0 \ , , \sum z _ i ^ 3 = 0 \footnote { O n e c a n c o n s i d e r t h e c o o r d i n a t e s o f a p o i n t o n t h e c u r v e a s t h e r o o t s o f a n e q u a t i o n o f t h e 5 t h d e g r e e , w h i c h h a s b e e n b r o u g h t i n t o B R I N G ' s n o r m a l f o r m b y T s c h i r n h a u s t r a n s f o r m a t i o n . } \ , . \end{align*}


\begin{align*} \Phi ^ P : = \bigl \{ ( u , \phi ) \in \R ^ { d \times ( n + 1 ) } \times \R \bigm | \phi = \phi ( u _ { 1 n } , \ldots , u _ { d n } ) , \ u \in P \bigr \} . \end{align*}


\begin{align*} g ( x _ 0 ) - \underline { g } & \geq \sum _ { k = 0 } ^ K g ( x _ k ) - g ( x _ { k + 1 } ) \\ & \geq K c _ 1 \tau _ 1 \min ( \underline { \alpha _ 1 } , t _ 1 ( x _ k ) ) \norm { \nabla g ( x _ k ) } ^ 2 \\ & \geq K c _ 1 \tau _ 1 \min ( \underline { \alpha _ 1 } , \underline { t _ 1 } ) \varepsilon _ 1 ^ 2 . \end{align*}


\begin{align*} P _ f ( x ) = \bigcup _ { j \in \omega } ( f ^ { - 1 } ) ^ { ( j ) } ( x ) . \end{align*}


\begin{align*} \pi ( x ) = \# \{ p \leq x : p \} , \end{align*}


\begin{align*} \lim _ { n \to \infty } \Lambda _ { n , 1 } = 0 , \lim _ { n \to \infty } \Lambda _ { n , 2 } = \frac 1 2 . \end{align*}


\begin{align*} w ^ { ( 1 ) } ( t , x ) ~ = ~ \begin{cases} \overline { w } ( x ) & \qquad \mathrm { i f } ~ ~ x \in ( t , 0 ) \cup ( 0 , \infty ) , \\ [ 2 m m ] \overline { w } ( x + \tau _ 0 - t ) & \qquad \mathrm { i f } ~ ~ x \in ( - \infty , t ) , \end{cases} \end{align*}


\begin{align*} \| f - { \mathsf { C } } _ { d _ x , d _ y } [ f ] \| _ 1 = \int _ D \left | f ( x , y ) - { \mathsf { C } } _ { d _ x , d _ y } [ f ] ( x , y ) \right | d x d y \le 4 \sum _ { i = d _ x + 1 } ^ { \infty } \sum _ { j = d _ y + 1 } ^ { \infty } \left | c _ { i , j } \right | , \end{align*}


\begin{align*} \frac { \phi _ 0 + \phi _ 1 } { 2 } - \beta _ a \epsilon ( a ) \frac { \phi _ 1 - \phi _ 0 } { h } = \phi ^ b ( a ) , \frac { \phi _ { N + 1 } + \phi _ N } { 2 } + \beta _ b \epsilon ( b ) \frac { \phi _ { N + 1 } - \phi _ N } { h } = \phi ^ b ( b ) . \end{align*}


\begin{align*} x _ { n + 1 } = J ^ S _ { \mu _ n } ( x _ n + \mu _ n T _ { \lambda _ n } x _ n ) \end{align*}


\begin{align*} x _ { k + 1 } = x _ k - A ^ \intercal e _ { \mathrm { r e m } ( \zeta _ { k - 1 } , n ) + 1 } \frac { e _ { \mathrm { r e m } ( \zeta _ { k - 1 } , n ) + 1 } ^ \intercal ( A x _ k - b ) } { \norm { A ^ \intercal e _ { \mathrm { r e m } ( \zeta _ { k - 1 } , n ) + 1 } } _ 2 ^ 2 } , \end{align*}


\begin{align*} \Vert \varphi \Vert _ { 0 , E , \omega } = \left ( \sum _ { \iota \in I ^ E } \varphi ^ 2 ( \xi ^ E _ \iota ) \ , \omega ^ E _ \iota \right ) ^ { 1 / 2 } \ , . \end{align*}


\begin{align*} { \rm A d } _ { \hat q } \hat v = \hat q \hat v \hat q ^ * . \end{align*}


\begin{align*} \prod _ { j = j _ 0 ( \delta ) + 1 } ^ { n - 1 } \left ( 1 + \dfrac { \delta } { j } \right ) = \prod _ { j = k + 1 } ^ { n - 1 } \dfrac { j - k } { j } = \dfrac { \Gamma ( n - k ) \times k ! } { \Gamma ( n ) } . \end{align*}


\begin{align*} x _ 1 ^ \prime : x _ 2 ^ \prime = x _ 1 : x _ 2 \ , , y _ 1 ^ \prime : y _ 2 ^ \prime = y _ 1 : y _ 2 \ , , \end{align*}


\begin{align*} \mathcal { E } _ 2 : = \left \{ \exists \ , y ^ { \prime } \in \left ( w + [ - 4 \lambda H , 5 \lambda H ) \times \{ H ^ \prime \} \right ) \cap \mathbb { L } : X ^ { y ^ { \prime } } _ { H - H ^ \prime } - \pi _ 1 ( y ^ \prime ) \geq \left ( v - v ^ { \prime } \right ) H \right \} . \end{align*}


\begin{align*} \rho ( f ) = \cfrac { 1 } { r _ 0 + \cfrac { 1 } { r _ 1 + \cfrac { 1 } { r _ 2 + \dotsb } } } . \end{align*}


\begin{align*} \Box \phi = \left ( | \phi _ t | ^ 2 - | \phi _ x | ^ 2 \right ) \phi + a ( x ) \phi _ t , \ ; \phi [ 0 ] \in \mathbb { S } ^ k \times T \mathbb { S } ^ k . \end{align*}


\begin{align*} \widehat { V } _ Q ^ { - 1 } = \widehat { V } _ { - Q } \widehat { D } _ { L , m } ^ { - 1 } = \widehat { D } _ { L ^ { - 1 } , - m } . \end{align*}


\begin{align*} s _ { i j } = \min \bigl \{ m _ { i j t _ i ( H ) } , s _ { i n } \bigr \} i \in \{ 1 , \ldots , d \} \ ; j \in \{ 0 , \ldots , n \} . \end{align*}


\begin{align*} u _ { t } + u u _ { x } - u _ { x x } = 0 , \ \ x \in \R , \ t > 0 . \end{align*}


\begin{align*} p _ { K '' | K } ( k '' ) & = \frac { \binom { K } { k '' } } { \binom { Q } { K } } \ ! \ ! \ ! \ ! \sum _ { \substack { \forall \mathcal { I } , \mathcal { J } \subset \mathcal { Q } : \\ \mathcal { I } \cap \mathcal { J } = \emptyset , \\ | \mathcal { I } | = k '' , | \mathcal { J } | = K - k '' } } \ ! \ ! \ ! \ ! \prod _ { i \in \mathcal { I } } \xi _ i \prod _ { j \in \mathcal { J } } ( 1 - \xi _ j ) \end{align*}


\begin{align*} K ( \omega , z ) = ( \omega - z ) \left ( \omega - z - \frac { 1 } { G _ { { \mu } } ( \omega ) } \right ) , z , \omega \in \mathbb { C } ^ + . \end{align*}


\begin{align*} \begin{aligned} \max _ { \boldsymbol { \Theta } , \mathbf { w } _ m } & \biggl | \sum _ { n = 1 } ^ N g _ n c _ n h _ n B _ r ( \theta _ I ) + h _ d B _ d ( \theta _ d ) \biggr | ^ 2 \\ \textrm { s . t . } & \| \mathbf { w } _ m \| ^ 2 \leqslant 1 , \ : m = 1 , 2 , \\ & \phi _ n \in [ 0 , 2 \pi ) , \ : \forall n = 1 , 2 , \ldots , N . \end{aligned} \end{align*}


\begin{align*} \mathcal { H } ^ n & = \left ( - 1 + \frac { \tau } { 2 } \eta ^ n - \frac { \tau ^ 2 } { 2 } \vartheta ^ n \right ) \phi ^ { n - 1 } + \frac { \tau ^ 2 } { 4 } \Delta \phi ^ { n - 1 } \\ & \quad \ ; + 2 \phi ^ n + \tau ^ 2 \left ( \Big ( \vartheta ^ n + \chi ^ n - \sigma R ^ \prime ( \| \phi ^ n \| ^ 2 - 1 ) \Big ) \phi ^ n - \frac { 1 } { 2 } \Delta \phi ^ n - G ( \phi ^ n ) \right ) \end{align*}


\begin{align*} D _ u x ^ n _ t = \int _ u ^ t \tilde { f } ' _ n ( s , x _ s ^ n ) D _ u x _ s ^ n d s + K _ H ( t , u ) \end{align*}


\begin{align*} \psi _ c ( x ) = \int _ { - \infty } ^ x g _ c ( x ' ) d x ' , x \in \R . \end{align*}


\begin{align*} G = \{ x \rightarrow x y , \ ; \ ; y \rightarrow x y \} . \end{align*}


\begin{align*} \frac { d } { d t } \xi ^ N ( k , t ) = & \ i \sum _ { \substack { m + n = k , \\ | m | , | n | , | k | \leq N } } m | n | e ^ { \frac 1 2 \mu ( | k | ^ \alpha - | m | ^ \alpha - | n | ^ \alpha ) t } \xi ^ N ( m ) \xi ^ N B ( n ) \\ & - \frac 1 2 \mu | k | ^ \alpha \xi ^ N ( k ) . \end{align*}


\begin{align*} 1 = \sum _ { I \in \mathcal { K } _ k } \vert I \vert ^ { \frac { \log 2 } { \log 3 } } . \end{align*}


\begin{align*} x _ 0 \in \mathrm { d o m } A , \ ; x _ { n + 1 } = J ^ A _ { \gamma _ n } x _ n \end{align*}


\begin{align*} \sigma _ { \mathcal { X } } = \left ( \mathcal { E } _ p , \mathcal { E } _ { p _ 1 } , \mathcal { E } _ { p _ 2 } , \mathcal { E } _ { p _ 3 } , \mathcal { E } _ { p _ 4 } , \Phi ( \sigma _ { \widetilde { X } } ) \right ) , \end{align*}


\begin{align*} \begin{aligned} | N _ { G } ( v ) \cap ( V ( H ) \cup \{ u _ { 1 } , \cdots , u _ { l - 1 } \} ) | & \geq d _ { G } ( v ) - | N _ { G } ( v ) \cap V ( B ) | \\ & - | N _ { G } ( v ) \cap ( V ( T ' ) \setminus \{ u _ { 1 } , \cdots , u _ { l - 1 } \} ) | \\ & \geq ( t + 3 ) - 2 - ( t - \lfloor \frac { l - 1 } { 2 } \rfloor ) \\ & \geq \lfloor \frac { l - 1 } { 2 } \rfloor + 1 . \end{aligned} \end{align*}


\begin{align*} \begin{pmatrix} A & B \\ C & D \end{pmatrix} \end{align*}


\begin{align*} \begin{aligned} & \sum _ { j \notin L _ { i k } } \lambda _ { i j } \leq \delta _ { i k } , \ \sum _ { j \notin R _ { i k } } \lambda _ { i j } \leq 1 - \delta _ { i k } k \in K _ i , \\ & \lambda _ i \in \Lambda _ i , \ \delta _ i \in \{ 0 , 1 \} ^ { \lceil \log _ 2 l _ i \rceil } , \end{aligned} \end{align*}


\begin{align*} X _ \varepsilon = _ { \cal L } X _ \mu + Z _ \varepsilon , \end{align*}


\begin{align*} g = g ^ T \oplus g ^ \Sigma , \end{align*}


\begin{align*} S _ { k } ( G ) = \bigcup _ { ( d , e ) \in D _ { k } ( G ) } d \times e \cup e \times d . \end{align*}


\begin{align*} f = \lim _ { N \to \infty } \sum _ { k = 1 } ^ N e _ { \pi ( k ) } . \end{align*}


\begin{align*} \tanh { x } = 8 x \sum _ { k = 0 } ^ { \infty } \frac { 1 } { 4 x ^ 2 + \pi ^ 2 \left ( 2 k + 1 \right ) ^ 2 } , x \in \mathbb { C } . \end{align*}


\begin{align*} | \det { \mathfrak { S } } | & = | A ( \beta _ 1 ) \cdots A ( \beta _ { k } ) | \geqslant c _ j ^ k \prod _ { i = 1 } ^ { k } | \beta _ i - \alpha _ j | ^ { m _ j } \\ & = c _ j ^ k | B ( \alpha _ j ) | ^ { m _ j } \geqslant c _ j ^ k \delta ^ { m _ j } \geqslant c _ { j } ^ { K } \delta ^ { m _ j } . \end{align*}


\begin{align*} \varphi ( x + x ^ { \prime } ) = \varphi ( x ) x ^ { \prime } \in B _ { l ( x ) } ^ { n } . \end{align*}


\begin{align*} 1 \geq \mu _ { \mathrm { d i s c } } ( \gamma _ { 0 } \cup { \gamma _ { 1 } } ) - \mu _ { \mathrm { d i s c } } ( \gamma _ { 2 } ) = n + 3 - \mu _ { \mathrm { d i s c } } ( \gamma _ { 2 } ) . \end{align*}


\begin{align*} \begin{cases} \theta ( z _ 1 , [ \mathfrak { X } , z _ 2 ] ) D ( \mathfrak { X } ) + \theta ( [ \mathfrak { X } , z _ 1 ] , z _ 2 ) D ( \mathfrak { X } ) + \theta ( [ \mathfrak { X } , z _ 1 ] , [ \mathfrak { X } , z _ 2 ] ) = 0 , \\ \theta ( [ \mathfrak { X } , z _ 1 ] , [ \mathfrak { X } , z _ 2 ] ) D ( \mathfrak { X } ) = 0 . \end{cases} \end{align*}


\begin{align*} \begin{array} { l } f _ 0 = 1 , \mbox { a n d } f _ k = z \sum \limits _ { \ell = 0 } ^ { k - 1 } f _ \ell + z \sum \limits _ { \ell = 0 } ^ { k - 1 } g _ \ell + z \sum \limits _ { \ell = 0 } ^ { k - 1 } h _ \ell , k \geq 1 , \\ g _ k = z f _ { k + 1 } + z g _ { k + 1 } + z h _ { k + 1 } , k \geq 0 , \\ h _ k = z f _ k + z g _ k + z h _ k , k \geq 0 . \\ \end{array} \end{align*}


\begin{align*} \mathbf { v } _ i = \frac { 1 } { \delta } ( B _ 1 ( \alpha _ i ) , B _ 2 ( \alpha _ i ) , \dots , B _ L ( \alpha _ i ) ) \end{align*}


\begin{align*} \Phi \big ( ( x , v ) , ( x ' , v ' ) \big ) : = \big ( ( | x - x ' | + | v - v ' | ) \wedge 1 \big ) \big ( \mathcal W ( x , v ) + \mathcal W ( x ' , v ' ) \big ) \end{align*}


\begin{align*} B ^ \perp = \{ \alpha \in X ( A ) \mid \alpha ( b ) = 1 b \in B \} . \end{align*}


\begin{align*} R = \log \left ( 1 + \frac { P _ t \Bigl | \bigl ( \mathbf { g } ^ T \boldsymbol { \Theta } \mathbf { H } + \mathbf { h } _ d ^ T \bigr ) \mathbf { w } \Bigr | ^ 2 } { \sigma _ n ^ 2 } \right ) \end{align*}


\begin{align*} A ( f , g ) ( x , \omega ) & = e ^ { \pi i x \cdot \omega } V _ g f ( x , \omega ) = e ^ { \pi i x \cdot \omega } e ^ { - 2 \pi i x \cdot \omega } V _ { \widehat { g } } \widehat { f } ( \omega , - x ) \\ & = e ^ { - \pi i x \cdot \omega } V _ { \widehat { g } } \widehat { f } ( \omega , - x ) = A ( \widehat { f } , \widehat { g } ) ( \omega , - x ) . \end{align*}


\begin{align*} b ( \tau ) \sim e ^ { \frac { 2 } { \alpha } ( \frac { \alpha } { 2 } - l ) \tau } . \end{align*}


\begin{align*} \Delta _ { \alpha \alpha } ^ { ( \ell ) } = \frac { C _ W } { n _ { \ell } } \sum _ { j = 1 } ^ { n _ \ell } X _ j . \end{align*}


\begin{align*} \Xi _ { k + 1 } & = \mathcal { A } \Xi _ k + \mathcal { B } \mathcal { G } ( \Psi ( \mathbf { u } _ k , { \tilde { \Xi } _ k } ) + E ( \mathbf { u } _ k , { \tilde { \Xi } _ k } ) + D ( \omega _ k ) ) , \\ { \mathbf { \tilde { y } } _ k } & { = \mathcal { C } \Xi _ k + \mathbf { w } _ k } , \end{align*}


\begin{align*} P _ d ( x ) = \sum _ { | J | = d } p _ J x ^ J , \ Q _ d ( x ) = \sum _ { | J | = d } q _ J x ^ J , \ x ^ J = x _ 0 ^ { j _ 0 } \cdot \ldots \cdot x _ n ^ { j _ n } . \end{align*}


\begin{align*} \pi : x = ( x _ n ) _ { n \in \mathbb { N } } \mapsto \lim _ { n \to + \infty } f _ { ( x _ 1 , . . . , x _ n ) } ( 0 ) . \end{align*}


\begin{align*} n ' = | T | + | ( U \cup V ) \setminus ( U _ 4 \cup V _ 4 ) | & \leq 3 0 h ^ 2 \varepsilon n + 1 6 0 h ^ 3 \varepsilon n ^ { \frac { 3 } { 2 } } + 3 \cdot ( 6 h ) ^ { s + 3 } \varepsilon ^ { \frac { s + 1 } { 2 } } n ^ { \frac { s + 2 } { 2 } } . \end{align*}


\begin{align*} \begin{aligned} ( \rho ^ { n + 1 } , \phi ^ { n + 1 } ) = & \arg \min _ { ( \rho , \phi ) \in \mathcal { A } , m } \left \{ \frac { 1 } { 2 \tau } \sum _ { i = 1 } ^ s \int _ { \Omega } F ( \rho _ i , m _ i ) D ^ { - 1 } _ i d x + E ( \rho , \phi ) \right \} , \\ & \rho _ i - \rho _ i ^ n + \nabla \cdot ( m _ i ) = 0 , m _ i \cdot \mathbf { n } = 0 , x \in \partial \Omega . \end{aligned} \end{align*}


\begin{align*} d \omega _ { t } = \theta \wedge d \theta _ { t } ^ { c } = \theta \wedge \omega _ { t } . \end{align*}


\begin{align*} D = \{ ( u , v ) : | u | \leq 1 8 \pi , | v | \leq 1 8 \pi \} . \end{align*}


\begin{align*} \left | \dfrac { \sqrt { s _ n ^ 2 } } { \sqrt { \sigma _ n ^ 2 } } - 1 \right | = \dfrac { | s _ n ^ 2 - \sigma _ n ^ 2 | } { \sqrt { \sigma _ n ^ 2 } \cdot ( \sqrt { s _ n ^ 2 } + \sqrt { \sigma _ n ^ 2 } ) } \leq \dfrac { | s _ n ^ 2 - \sigma _ n ^ 2 | } { \sigma _ n ^ 2 } , \end{align*}


\begin{align*} \lim _ { n \to \infty } \frac { | \hat { I } _ { = 2 , < \epsilon } ( k _ { n } ) | } { k _ { n } } = 1 . \end{align*}


\begin{align*} o ( t ) = & \tau _ t \Big ( ( a _ 1 - \tau _ t ( a _ 1 ) ) \cdots ( a _ n - \tau _ t ( a _ n ) ) \Big ) \\ = & \tau _ t ( a _ 1 \cdots a _ n ) + \sum _ { r = 1 } ^ n ( - 1 ) ^ r \sum _ { 1 \leq k _ 1 < \ldots < k _ r \leq n } \left ( \prod _ { i = 1 } ^ { r } \tau _ t ( a _ { k _ i } ) \right ) \tau _ t ( a _ 1 \cdots \hat { a } _ { k _ 1 } \cdots \hat { a } _ { k _ r } \cdots a _ n ) \end{align*}


\begin{align*} \rho _ c ( x ) = \rho ( x ) - a x ^ 4 < P ( x ) , \textrm { f o r } | x | \le c / 2 . \end{align*}


\begin{align*} e _ { - i } & = \lfloor \frac { 2 n - 1 } { n - 1 } i \rfloor - i + 1 = \lfloor \frac { n } { n - 1 } i \rfloor + 1 \\ a _ i & \ge ( n - 1 ) i + 1 . \end{align*}


\begin{align*} \left ( 1 + \frac { \tau } { 2 } \eta ^ n + \frac { \tau ^ 2 } { 2 } \vartheta ^ n \right ) \tilde { \phi } ^ { n + 1 } - \frac { \tau ^ 2 } { 4 } \Delta \tilde { \phi } ^ { n + 1 } = \mathcal { H } ^ n . \end{align*}


\begin{align*} ( \alpha + \beta ) \circ \gamma = ( \alpha \circ \gamma ) + ( \beta \circ \gamma ) , \\ ( \alpha \cdot \beta ) \circ \gamma = ( \alpha \circ \gamma ) \cdot ( \beta \circ \gamma ) \end{align*}


\begin{align*} \Psi _ V = & \sum _ { i = 1 } ^ n \big [ \dot { \mu } _ i \sigma _ i \Lambda \R _ i - ( \dot { x } _ i - \mu _ i ) \sigma _ i \partial _ y \R _ i \big ] + I + J + K + L , \end{align*}


\begin{align*} \Psi _ V = & E _ V - \sum _ { i = 1 } ^ n ( \dot { x } _ i - \mu _ i ) \sigma _ i \partial _ y \widetilde { R } _ i + \sum _ { i = 1 } ^ n \bigg ( \dot { \mu } _ i + \sum ^ n _ { \substack { j = 1 , \\ j \not = i } } \frac { a _ { i j } } { x ^ 3 _ { i j } } + \sum ^ n _ { \substack { k , j = 1 , \\ j \not = i } } \frac { b _ { i j k } \mu _ k } { x ^ 3 _ { i j } } \bigg ) \frac { \sigma _ i \Lambda \widetilde { R } _ i } { 1 + \mu _ i } , \end{align*}


\begin{align*} A ^ * = \begin{pmatrix} \partial _ x & \partial _ x \\ [ 4 p t ] \partial _ x & \partial _ { x x } + \partial _ x \end{pmatrix} , \end{align*}


\begin{align*} b \circ a \circ \overline { b } = b \circ a \circ \overline { b } \circ \overline { a } + \lambda _ { b \circ a \circ \overline { b } \circ \overline { a } } ( a ) \in T _ + ( A ) . \end{align*}


\begin{align*} m _ { 0 , \lambda } ( V ) = 0 , \mbox { i f $ \lambda \neq ( n ) , 1 ^ n $ } . \end{align*}


\begin{align*} \lambda _ h = \lambda _ 0 - i \frac { \eta _ 0 } { 4 \pi } { \lambda _ 0 } ^ \frac { 5 } { 2 } { U _ 0 } ^ 2 h + \mathcal { O } ( h ^ 2 ) . \end{align*}


\begin{align*} \mathrm { R } _ t { } ^ \delta _ { \alpha \beta \gamma } = \frac { \mathbf { g } _ { \alpha \gamma } \delta ^ \delta _ \beta - \mathbf { g } _ { \beta \gamma } \delta ^ \delta _ \alpha } { 2 c } , ~ ~ ~ ~ ~ \mathrm { R i c } _ t { } _ { \beta \gamma } = \mathrm { R } _ t { } ^ \alpha _ { \alpha \beta \gamma } = - \frac { \mathbf { g } _ { \beta \gamma } } { 2 c } , ~ ~ ~ ~ ~ \mathfrak { R } _ t = \mathrm { R i c } _ t { } _ { \beta \beta } = - \frac { 1 } { c } . \end{align*}


\begin{align*} & \overline { \ < P s i > _ { m , n } ( t , \tilde { y } ) - \ < P s i > _ { m , n } ( s , \tilde { y } ) } \\ & = i \int _ s ^ t \int _ { \mathbb { R } ^ d } \overline { a _ { m , n } ( t - t ' , \tilde { y } - x ' ) } W ( d t ' , d x ' ) + i \int _ 0 ^ s \int _ { \mathbb { R } ^ d } \Big [ \overline { a _ { m , n } ( t - t ' , \tilde { y } - x ' ) - a _ { m , n } ( s - t ' , \tilde { y } - x ' ) } \Big ] W ( d t ' , d x ' ) . \end{align*}


\begin{align*} & 2 ( x - y ) \cdot ( \widetilde \mu ( x ) - \widetilde \mu ( y ) ) + ( p _ 1 - 1 ) \cdot ( \widetilde \sigma ( x ) - \widetilde \sigma ( y ) ) ^ 2 \\ & \qquad = 2 ( y - x ) \cdot ( \widetilde \mu ( y ) - \widetilde \mu ( x ) ) + ( p _ 1 - 1 ) \cdot ( \widetilde \sigma ( y ) - \widetilde \sigma ( x ) ) ^ 2 , \end{align*}


\begin{align*} \sum _ { l \in \Z } \widehat { f } ( \omega + l ) = \sum _ { k \in \Z } f ( k ) e ^ { - 2 \pi i k \omega } . \end{align*}


\begin{gather*} e _ { P _ j } ( K / k ) = l ^ { \beta _ j } , 1 \leq \beta _ j \leq n , 1 \leq j \leq r , \quad e _ { \infty } ( K / k ) = l ^ { t ' } , 0 \leq t ' \leq n \end{gather*}


\begin{align*} d x ( t ) = ( \alpha _ { - 1 } x ( t ) ^ { - 1 } - \alpha _ { 0 } + \alpha _ { 1 } x ( t ) - \alpha _ { 2 } x ( t ) ^ { \rho } ) d t + \sigma x ( t ) ^ { \theta } d ^ { \circ } B _ t ^ H \end{align*}


\begin{align*} ( x , \omega , \tau ) \bullet ( x ' , \omega ' , \tau ' ) = ( x + x ' , \omega + \omega ' , \tau + \tau ' + \tfrac { 1 } { 2 } ( x ' \cdot \omega - x \cdot \omega ' ) ) \end{align*}


\begin{align*} U \pi ( x , \omega , \tau ) f & = U \Big ( \sum _ { k = 1 } ^ n c _ k e ^ { 2 \pi i \tau } \pi ( x , \omega , 0 ) \pi ( x _ k , \omega _ k , 0 ) g \Big ) \\ & = \sum _ { k = 1 } ^ n c _ k e ^ { 2 \pi i \tau } \rho ( x , \omega ) \rho ( x _ k , \omega _ k ) \varphi \\ & = \rho ( x , \omega , \tau ) U f . \end{align*}


\begin{align*} \left | \log \frac { x } { y } \right | = \int _ { \min ( x , y ) } ^ { \max ( x , y ) } \frac { d u } { u } \geq \frac { | x - y | } { \max ( x , y ) } , y > 0 . \end{align*}


\begin{align*} [ a \cdot v _ 0 , \varphi ( a ^ 2 \cdot v _ 0 ) ] = [ [ \zeta '' , a ] , a ] \cdot ( a \cdot v _ 0 ) = { \rm I I } ( [ [ \zeta '' , a ] , a ] , a ) \cdot v _ 0 . \end{align*}


\begin{align*} w _ { 1 2 } = \lambda _ 1 \lambda _ 2 , w _ { 1 3 } = \lambda _ 1 \lambda _ 3 , w _ { 2 3 } = \lambda _ 2 \lambda _ 3 , w _ { 1 2 3 } = \lambda _ 1 \lambda _ 2 \lambda _ 3 . \end{align*}


\begin{align*} \mathbb { Z } _ p ^ { v } \cong \frac { A ( \tau ) / N } { p ( A ( \tau ) / N ) } = \frac { A ( \tau ) / N } { ( p A ( \tau ) + N ) / N } \cong \frac { A ( \tau ) } { p A ( \tau ) + N } . \end{align*}


\begin{align*} \hat { \sigma } ( \xi ) = \int _ 0 ^ \xi ( \xi - \zeta ) \partial _ \xi ^ 2 \hat { \sigma } ( \zeta ) d \zeta . \end{align*}


\begin{align*} { { \rm d } ^ n \over { \rm d } x ^ n } \sec ( x ) = Q _ n ( \tan ( x ) ) \sec ( x ) . \end{align*}


\begin{align*} X _ { s , t } ( x , v ) = x - ( t - s ) \zeta _ { s , t , x } ( v ) . \end{align*}


\begin{align*} B _ 1 ' = \left ( \frac { 2 } { 3 c } \right ) ^ { 3 / 5 } \left ( \frac { 1 } { \beta } \right ) ^ { 1 / 5 } , \end{align*}


\begin{align*} \Xi _ \perp : = \{ X \in \Xi ~ | ~ \mathbf { g } ( X , \Xi _ t ) = 0 \} \Omega _ t : = \{ \omega \in \Omega ~ | ~ \mathbf { g } ^ { - 1 } ( \omega , \Omega _ \perp ) = 0 \} , \end{align*}


\begin{align*} & \bigg ( \frac { \theta d } { 6 4 } \bigg ) ^ { - n - 2 } \int _ { B _ { \theta d / 6 4 } ^ + ( a ) } x _ 1 w ( x ) \dd x \\ \leqslant & \ , C \bigg ( \partial _ 1 w ( 0 ) + ( r \theta ) ^ { - n - 2 } + \tau \theta ^ { 2 s } d ^ { - n + 2 s - 2 } \bigg ) \\ = & \ , C \tau d ^ { - n - 2 } \bigg ( \bigg ( 1 - \frac \theta 2 \bigg ) ^ { - n - 2 } - 1 \bigg ) \partial _ 1 \zeta ( 0 ) + C ( r \theta ) ^ { - n - 2 } + C \tau \theta ^ { 2 s } d ^ { - n + 2 s - 2 } . \end{align*}


\begin{align*} T ( t ) f ( x ) = \begin{cases} f ( x - t ) & \mbox { f o r } x > t \\ 0 & \mbox { f o r } x \leq t . \end{cases} \end{align*}


\begin{align*} \| \phi ( \cdot , t ) \| ^ 2 \equiv \| \phi _ 0 \| ^ 2 = 1 , \frac { \mathrm { d } } { \mathrm { d } t } \mathcal { F } ( t ) = - 2 \eta ( t ) \| \dot { \phi } ( \cdot , t ) \| ^ 2 , \forall \ , t > 0 . \end{align*}


\begin{align*} \begin{cases} \forall \gamma ^ 1 , p ^ X , x ^ X \left ( \gamma > _ \mathbb { R } 0 \land \gamma ^ { - 1 } ( x - _ X p ) \in A p \rightarrow p = _ X J ^ A _ { \gamma } x \right ) , \\ \forall \gamma ^ 1 , p ^ X , x ^ X \left ( \gamma > _ \mathbb { R } 0 \land p = _ X J ^ A _ { \gamma } x \rightarrow \gamma ^ { - 1 } ( x - _ X p ) \in A p ) \right ) , \end{cases} \end{align*}


\begin{gather*} \lim _ { \eta \downarrow 0 } \left [ \frac { \eta } { 1 - ( 1 - \eta ) ^ { \alpha } } - \frac { 1 } { \alpha } \right ] \eta ^ { - 1 } = \frac { \alpha - 1 } { 2 \alpha } \quad \lim _ { \eta \downarrow 0 } \frac { \eta ^ { 2 / \alpha } } { 1 - ( 1 - \eta ) ^ 2 } \ , \eta ^ { - 2 / \alpha + 1 } = \frac 1 2 . \end{gather*}


\begin{align*} \int _ \Omega u ( t , x ; s , u _ 0 ) d x \le \max \left \{ \int _ \Omega u _ 0 ( x ) d x , M _ 0 ^ * \right \} = M _ 0 ^ * \forall \ , t \ge s . \end{align*}


\begin{align*} \mathcal { R } : = \int _ { \R ^ 3 } \int _ { \R ^ 3 } | v - v _ * | ^ \gamma g _ * f ^ 2 d v _ * d v \lesssim \Vert g \Vert _ { L ^ 2 _ { | \gamma | + 2 } } \Vert f \Vert _ { H ^ s _ { \gamma / 2 } } ^ 2 . \end{align*}


\begin{align*} \Gamma ( \pi ) & = \lim _ { t \rightarrow \infty } \frac { 1 } { N } \sum _ { \ell = 1 } ^ { N } ( C _ { \ell , \pi } ^ { a v } ( t ) + \rho _ { \ell } \Delta _ { \ell , \pi } ^ { a v } ( t ) ) , \end{align*}


\begin{align*} \begin{cases} \ f ( x ) \ = \ \max \{ A ( n ) - E ( n ) : n \leq x \} \\ [ 1 1 p t ] \ g ( x ) \ = \ \min \{ A ( n ) + E ( n ) : n \leq x \} \end{cases} . \end{align*}


\begin{align*} \mathcal L = p ( t , S ) \frac { \partial ^ 2 } { \partial S ^ 2 } + w ( t , S ) \frac { \partial } { \partial S } + z ( t , S ) , \end{align*}


\begin{align*} \mathrm { s g n } _ X ( g ) = \prod _ { \{ u , v \} \in E ( X ) } \frac { u ^ g - v ^ g } { u - v } . \end{align*}


\begin{align*} \begin{cases} \forall \gamma ^ 1 , r ^ 1 , x ^ X , y ^ X \Big ( \gamma > _ \mathbb { R } 0 \land r > _ \mathbb { R } 0 _ \mathbb { R } \rightarrow \norm { J ^ A _ \gamma x - _ X J ^ A _ \gamma y } _ X \\ \qquad \qquad \qquad \leq _ \mathbb { R } \norm { r ( x - _ X y ) + _ X ( 1 - r ) ( J ^ A _ \gamma x - _ X J ^ A _ \gamma y ) } _ X \Big ) , \\ \forall \gamma ^ 1 , p ^ X , x ^ X \left ( \gamma > _ \mathbb { R } 0 \land \gamma ^ { - 1 } ( x - _ X p ) \in A p \rightarrow p = _ X J ^ A _ { \gamma } x \right ) , \end{cases} \end{align*}


\begin{align*} \mathcal P ' ( x ) + \mathcal P ( x ) \tilde F ( x ) = F ( x ) \mathcal P ( x ) . \end{align*}


\begin{align*} H = 2 ( \partial _ + \star y ^ + - 1 - y ^ - \star \partial _ - ) , ~ ~ ~ ~ ~ ~ E ^ \pm = \displaystyle \frac { 1 } { \sqrt { a } } \partial _ 0 \star y ^ \pm - 2 \sqrt { a } y ^ 0 \star \partial _ \mp . \end{align*}


\begin{align*} R > \int _ { y _ 0 } ^ y \rho d y = 1 - y _ 0 + \frac { 1 } { \beta - 1 } ( 1 - y ^ { 1 - \beta } ) . \end{align*}


\begin{align*} \Big | \partial _ r ^ k \left ( ( f * g ) ( x ) \right ) \Big | \lesssim \sum _ { j = 1 } ^ k \frac { \left ( | D ^ j f | * | g | \right ) ( x ) } { | x | ^ { k - j } } , \forall x \in \mathbb { R } ^ N . \end{align*}


\begin{align*} \dot X _ { 0 , t } = b ( X _ { 0 , t } ) . \end{align*}


\begin{align*} c ^ K _ r ( \widetilde { W } , \widetilde { W ' } ) = c ^ K _ { - r } \big ( \pi ( w _ n ) W , \pi ' ( w _ { n - 1 } ) W ' \big ) . \end{align*}


\begin{align*} \int n ( y ) \Phi _ 1 ( y ) \ , d y & = 2 \int _ { 2 \sqrt { a } v } ^ \infty \Phi _ 1 ( y ) \ , d y \\ & \le 2 \int _ 0 ^ \infty \Phi _ 1 ( y ) \ , d y \\ & = 1 . \end{align*}


\begin{align*} \hat q = q + q _ d \epsilon , \end{align*}


\begin{align*} \Delta _ G & = \left \{ \widehat { x } _ i - \widehat { x } _ { i + 1 } \ , , 1 \leq i \leq p + q - 1 \ , , \ ; 2 \ , \widehat { x } _ { p + q } \right \} \ , , \\ \intertext { a n d } \Delta _ K & = \left \{ \begin{cases} \widehat { x } _ i - \widehat { x } _ { i + 1 } \ , , 1 \leq i \leq p - 1 \ , , 2 \ , \widehat { x } _ { p } \ , , \\ \widehat { x } _ i - \widehat { x } _ { i + 1 } \ , , p + 1 \leq i \leq p + q - 1 \ , , 2 \ , \widehat { x } _ { p + q } \end{cases} \right \} \ , , \end{align*}


\begin{align*} h _ q ( S _ q ) = \begin{cases} - 1 & q = p 2 , \\ 1 & . \end{cases} \end{align*}


\begin{align*} T B ( f ) = 7 T B _ { 5 } ( f ) . \end{align*}


\begin{align*} \vartheta _ 3 ( \tau ; z ) = \sum _ { k \in \Z } e ^ { \pi i \tau k ^ 2 } e ^ { 2 \pi i k z } , ( \tau , z ) \in \mathbb { H } \times \C . \end{align*}


\begin{align*} \Bigg | \ln \frac { \mathbf { P } \left ( S _ n / ( \sqrt { n } \sigma ) > x \right ) } { 1 - \Phi \left ( x \right ) } \Bigg | = O \bigg ( ( 1 + x ) \frac { \ln n } { \sqrt { n } } \bigg ) \end{align*}


\begin{align*} \Phi ( X _ \tau ) = 1 - \lambda _ { i } ( M ) \pm O _ { M , k , \delta } ( \gamma ) . \end{align*}


\begin{align*} 1 / \underline { \alpha _ 2 } & = \max \left ( 1 / \alpha _ { 0 2 } , \dfrac { M _ g } { 3 | 2 c _ 2 - 1 | \varepsilon _ 2 } \right ) \\ & \leq \varepsilon _ 2 ^ { - 1 } \max \left ( 1 / \alpha _ { 0 2 } , \dfrac { M _ g } { 3 | 2 c _ 2 - 1 | } \right ) \\ & \leq b _ 3 \beta \varepsilon _ 2 ^ { - 1 } . \end{align*}


\begin{align*} Q ( x ) = C x + O ( x ^ { 1 1 / 1 2 + \varepsilon } ) \end{align*}


\begin{align*} M _ { d _ 0 } ( p , H ) = \frac { \pi ^ 2 } { 2 } \left \{ \prod _ { q \mid d _ 0 } \left ( 1 - \frac { 1 } { q ^ 2 } \right ) \right \} \left ( 1 + \frac { N _ { d _ 0 } ' ( p , H ) } { p } \right ) , \end{align*}


\begin{align*} S _ { m , n } = \left \{ ( s , t , s ^ { m + 2 } + \varepsilon t ^ { n + 2 } ) , \ \ ( s , t ) \in \R ^ 2 \right \} \ , . \end{align*}


\begin{align*} P = F P F ^ T + G K _ { Z } G ^ T . \end{align*}


\begin{align*} & \sum _ { \overset { { i + j + k = s } } { i , j , k \geq 0 } } [ T _ i ( u ) , T _ j ( v ) , T _ k ( w ) ] \\ & = \sum _ { \overset { { i + j + k = s } } { i , j , k \geq 0 } } T _ i \Big ( D ( T _ j ( u ) , T _ k ( v ) ) w + \theta ( T _ j ( v ) , T _ k ( w ) ) u - \theta ( T _ j ( u ) , T _ k ( w ) ) v \Big ) , \end{align*}


\begin{align*} D ( s , \mathbf { f } , \chi ) = \sum _ { \xi \in K _ 1 ( \mathfrak { n } ) \backslash \mathfrak { X } / K _ 1 ( \mathfrak { n } ) } \lambda _ { \mathbf { f } } ( \xi ) \chi ^ { \ast } ( l ( \xi ) ) l ( \xi ) ^ { - s } , \mathrm { R e } ( s ) \gg 0 . \end{align*}


\begin{align*} u ( r , t ) = \lambda ^ { - 1 } ( t ) Q \left ( \frac { r } { \sqrt { \lambda ( t ) } } \right ) + \tilde u ( r , t ) , t \in [ 0 , T ) , \end{align*}


\begin{align*} \kappa _ G ( s , y , y ^ \prime ) = \int _ { { \rm I m } \lambda = r } s ^ { i \lambda } G ( \lambda ) ( y , y ^ \prime ) d \lambda \end{align*}


\begin{align*} \bigg \| \int _ 0 ^ t S _ { t - s } \alpha ( \xi ( s ; x ) ) d s \bigg \| & \leq \int _ 0 ^ t \| S _ { t - s } \alpha ( \xi ( s ; x ) ) \| d s \leq M \int _ 0 ^ t e ^ { \beta ( t - s ) } \| \alpha ( \xi ( s ; x ) ) \| d s \\ & \leq M B \int _ 0 ^ t e ^ { \beta ( t - s ) } d s = M B \varphi _ { \beta } ( t ) \end{align*}


\begin{align*} \left ( 1 + \frac { \tau } { 2 } \eta ^ n + \frac { \tau ^ 2 } { 2 } \vartheta ^ n + \frac { \tau ^ 2 } { 4 } \Big ( ( \varrho ^ { x } _ { p } ) ^ 2 + ( \varrho ^ { y } _ { q } ) ^ 2 \Big ) \right ) \widehat { ( \tilde { \phi } ^ { n + 1 } ) } _ { p q } = \widehat { ( \mathcal { H } ^ n ) } _ { p q } , p , q = - \frac { M } { 2 } , \ldots , \frac { M } { 2 } - 1 . \end{align*}


\begin{align*} ~ x \notin \cup _ i W _ i ~ ~ B f _ 1 ( x ) = 0 , \end{align*}


\begin{align*} \tilde { f } = \int _ { 0 } ^ { + \infty } \mathcal { P } _ t ( f ) d t = ( - \mathcal { L } ) ^ { - 1 } ( f ) . \end{align*}


\begin{align*} \forall x ^ X , y ^ X \left ( \norm { x + _ X y } ^ 2 _ X + \norm { x - _ X y } ^ 2 _ X = _ \mathbb { R } 2 \left ( \norm { x } ^ 2 _ X + \norm { y } ^ 2 _ X \right ) \right ) . \end{align*}


\begin{align*} f ( z ) = C \tan ( a z + b ) + D z + E , a , b , C , D , E \in \R . \end{align*}


\begin{align*} \frac { I _ t ( \mu , R _ n ) \mu ( B _ n ) } { \mu ( R _ n ) ^ 2 } \le \beta _ 2 r _ n ^ { - a _ j t + a _ j \sum _ { i = 1 } ^ j s _ i + \sum _ { i = 1 } ^ d s _ i - \sum _ { i = 1 } ^ j a _ i s _ i } \le \beta _ 2 . \end{align*}


\begin{align*} \bar { X } _ W = \{ V \in X ( k ) : V \subset W \} . \end{align*}


\begin{align*} \min \{ v _ p ( u _ i ) , k \} = \min \{ v _ p ( u _ n ) + ( n - i ) a , k \} \end{align*}


\begin{align*} a ( 0 ) = 0 \mbox { o r } G ( 0 ) = 0 . \end{align*}


\begin{align*} X ^ r : = \underbrace { X \times _ B \ldots \times _ B X } _ { } , \end{align*}


\begin{align*} \nabla f _ { \mu } ( x ) = \mathbb { E } _ U \left [ \frac { f \left ( x + \mu U \right ) - f ( x ) } { \mu } U \right ] = \mathbb { E } _ { U , \xi } \left [ \frac { F \left ( x + \mu U , \xi \right ) - F ( x , \xi ) } { \mu } U \right ] , \end{align*}


\begin{align*} \mathcal { N } ^ { ( 1 ) } _ { R , 1 } ( t , k ) & = e ^ { - i t k ^ 2 } \int \overline { \mathcal { K } _ { R } ( x , k ) } u _ { M _ 1 } ( t , x ) \overline { u _ { M _ 2 } } ( t , x ) u _ { M _ 3 } ( t , x ) \ , d x . \end{align*}


\begin{align*} \bar f ( x ) - f ( x ) = ( - \Delta ) ^ s ( \bar u - u ) ( x ) & = C \int _ { \R ^ n _ + \setminus B _ R } \bigg ( \frac 1 { \vert x - y \vert ^ { n + 2 s } } - \frac 1 { \vert x _ \ast - y \vert ^ { n + 2 s } } \bigg ) ( u - \bar u ) ( y ) \dd y . \end{align*}


\begin{align*} e _ { 1 } = \frac { 1 } { \sqrt { 2 } } ( \nabla r - \sqrt { - 1 } J \nabla r ) , \end{align*}


\begin{align*} B _ r \cdot f ( A _ r ) = B \cdot f ( A _ r ^ 2 ) = B \cdot f ( A _ r ) = B _ r . \end{align*}


\begin{align*} J _ { 0 } ( \alpha , \beta , Z ) + J _ { 0 } ( \beta , \gamma , Z ' ) = J _ { 0 } ( \alpha , \gamma , Z + Z ' ) . \end{align*}


\begin{align*} \mathcal { J } - j \leq \frac { \log ( 1 / \alpha _ { j } ) } { \log 2 0 } , \mbox { a n d } \sum _ { X ^ { \alpha _ { j } } < p \leq X ^ { \alpha _ { j + 1 } } } \frac { 1 } { p } = \log \alpha _ { j + 1 } - \log \alpha _ { j } + o ( 1 ) = \log 2 0 + o ( 1 ) \leq 1 0 . \end{align*}


\begin{align*} \Phi _ 1 ( f _ * ) = \frac { 1 } { 2 } \int ( L _ + f _ * ) f _ * d x = - \frac { \lambda _ 0 ^ 2 } { 4 ( p - 1 ) } \int \left ( | \cdot | ^ { - ( N - \gamma ) } * Q ^ { p } \right ) Q ^ p d x = 0 , \end{align*}


\begin{align*} i \partial _ t U _ k ^ A + \Delta U _ k ^ A + \left ( | \cdot | ^ { - ( N - \gamma ) } * \big | U _ k ^ A \big | ^ p \right ) \big | U _ k ^ A \big | ^ { p - 2 } U _ k ^ A = O ( e ^ { - ( k + 1 ) e _ 0 t } ) \mathcal { S } ( \mathbb { R } ^ N ) . \end{align*}


\begin{align*} m ( x , t , k ) : = \left \{ \begin{aligned} & \left ( \frac { [ \Phi _ L ( x , t , k ) ] _ 1 } { a ( k ) } , [ \Phi _ R ( x , t , k ) ] _ 2 \right ) e ^ { i t \theta ( k , \xi ) \sigma _ 3 } , k \in \mathbb { C } _ { + } \\ & \left ( [ \Phi _ R ( x , t , k ) ] _ 1 , \frac { [ \Phi _ L ( x , t , k ) ] _ 2 } { a ^ { * } ( k ) } \right ) e ^ { i t \theta ( k , \xi ) \sigma _ 3 } , k \in \mathbb { C } _ { - } , \end{aligned} \right . \end{align*}


\begin{align*} D \left ( { 1 + x ^ 2 \over a ^ 2 } \right ) = 0 . \end{align*}