\begin{align*} \Delta _ { \mu } ^ p \mathbf { E } _ { \mu } ( g , \mu ) = C \cdot \mathbf { E } _ l ( g , \mu ) , \ , \ , \ , \ , \ , C \in \overline { \Q } ^ { \times } . \end{align*}


\begin{align*} ( x ) _ { n } = \sum _ { k = 0 } ^ { n } S _ { 1 , \lambda } ( n , k ) ( x ) _ { k , \lambda } , ( n \ge 0 ) . \end{align*}


\begin{align*} H ^ * [ P , Q , \varepsilon ] : = \forall p \in P \exists q \in Q \left ( \norm { p - q } \leq \varepsilon \right ) \end{align*}


\begin{align*} s _ n ^ 2 & : = \sum _ { i = 1 } ^ n \dfrac { \Gamma ( i ) ^ 2 } { \Gamma ( i + \alpha ) ^ 2 } , \intertext { a n d } \sigma _ n ^ 2 & : = \begin{cases} \dfrac { \Gamma ( n ) ^ 2 } { \Gamma ( n + \alpha ) ^ 2 } \cdot \dfrac { n } { 1 - 2 \alpha } & ( - 1 < \alpha < 1 / 2 ) , \\ [ 4 m m ] \dfrac { \Gamma ( n ) ^ 2 } { \Gamma ( n + \alpha ) ^ 2 } \cdot n \log n & ( \alpha = 1 / 2 ) . \end{cases} \end{align*}


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


\begin{align*} \sum _ { i = 1 } ^ { n } \frac { \partial f } { \partial \kappa _ { i } } = \frac { 1 } { k } H _ { k } ^ { \frac { 1 - k } { k } } \sum _ { i = 1 } ^ { n } \frac { \partial H _ { k } } { \partial \kappa _ { i } } = H _ { k } ^ { \frac { 1 - k } { k } } H _ { k - 1 } \geq 1 , \end{align*}


\begin{align*} \sum _ { \gamma \in \Gamma } A e _ \gamma = A \sum _ { \gamma \in \Gamma } e _ \gamma = A f . \end{align*}


\begin{align*} x _ 1 ^ \prime = - x _ 1 \ , , x _ i ^ \prime = x _ i ( i = 2 , 3 , 4 , 5 ) \end{align*}


\begin{align*} u _ { n + 1 } & = \varepsilon _ n u _ n ^ 2 , \\ v _ { n + 1 } & = \varepsilon _ n u _ n v _ n + 1 . \\ \end{align*}


\begin{align*} \Delta _ { g _ M } f + h e ^ f = c , \end{align*}


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


\begin{align*} \int _ \R | f ( x ) | ^ 2 \ , d x = - \int _ \R x f ( x ) \overline { f ' ( x ) } \ , d x - \int _ \R x \overline { f ( x ) } f ' ( x ) \ , d x = - 2 \Re \left ( \int _ \R x f ( x ) \overline { f ' ( x ) } \ , d x \right ) , \end{align*}


\begin{align*} \mathbf { M } _ { 1 2 \overline { 3 } } = \mathbf { F } _ { 1 2 \overline { 3 } } - \mathbf { F } _ { 2 1 \overline { 3 } } - \mathbf { F } _ { \overline { 1 } 2 \overline { 3 } } + \mathbf { F } _ { \overline { 1 } \ , \overline { 2 } \ , \overline { 3 } } \quad \mathbf { M } _ { 1 \bar { 3 } 2 } = \mathbf { F } _ { 1 \bar { 3 } 2 } - \mathbf { F } _ { 2 \overline { 3 } 1 } - \mathbf { F } _ { \overline { 1 } \ , \overline { 3 } 2 } + \mathbf { F } _ { \overline { 1 } \ , \overline { 3 } \ , \overline { 2 } } . \end{align*}


\begin{align*} & S = \int _ { \Bbb R ^ N } f _ t ( x , u ) g _ s ( u , y ) d \omega ( u ) = \int _ { \Bbb R ^ N } \Big ( f _ t ( x , u ) - f _ t ( x , y ) \Big ) g _ s ( u , y ) d \omega ( u ) . \end{align*}


\begin{align*} \varphi _ S ( a ^ * a ) = \varphi ( a ^ * a \delta _ { \varphi } ) = \varphi ( \delta _ { \varphi } ^ { 1 / 2 } a ^ * a \delta _ { \varphi } ^ { 1 / 2 } ) \geq 0 , \end{align*}


\begin{align*} \frac { d } { d x } \sum _ { k = 1 } ^ { p + 1 } S _ { 2 , \lambda } ( p + 1 , k ) \frac { x ^ { k } } { k } = \frac { d } { d x } \bigg ( e ^ { - x } \sum _ { n = 0 } ^ { \infty } \big ( ( 1 - \lambda ) _ { p , \lambda } + ( 2 - \lambda ) _ { p , \lambda } + \cdots + ( n - \lambda ) _ { p , \lambda } \big ) \frac { x ^ { n } } { n ! } \bigg ) . \end{align*}


\begin{align*} & Q ( y ) = \frac { \kappa _ 0 } { y ^ 2 } + \frac { g ( y ) } { y ^ 4 } + O \bigg ( \frac { 1 } { y ^ 6 } \bigg ) , \ ; \ ; | y | \rightarrow + \infty , \\ & Q ' ( y ) = - \frac { 2 \kappa _ 0 } { y ^ 3 } + O \bigg ( \frac { 1 } { | y | ^ 5 } \bigg ) , \ ; \ ; | y | \rightarrow + \infty . \end{align*}


\begin{align*} \partial _ t h + \mathcal { L } h = R , ( x , t ) \in \mathbb { R } ^ N \times ( t _ 0 , + \infty ) , \end{align*}


\begin{align*} \sup _ { x \in \mathbb { R } , \ , k \in \mathbb { R } } \left | \left \langle x \right \rangle ^ { \beta } \partial _ { k } ^ { j } \mathcal { Q } ( x , k ) \right | < \infty , j = 0 , 1 , \end{align*}


\begin{align*} | C \cap F | = O _ d \left ( A _ { F } ^ { \frac { d - 2 } { d - 1 } } \log A _ F \right ) . \end{align*}


\begin{align*} & \forall \gamma ^ 1 , x ^ X \big ( \gamma > _ \mathbb { R } 0 \land \tilde { \rho } > _ \mathbb { R } - \gamma \\ & \qquad \qquad \qquad \land \exists y ^ X , z ^ X ( z \in A y \land x = _ X y + _ X \gamma z ) \rightarrow \gamma ^ { - 1 } ( x - _ X J ^ A _ { \gamma } x ) \in A ( J ^ A _ { \gamma } x ) \big ) , \end{align*}


\begin{align*} \left \{ \begin{array} { r c l } d X ( t ) & = & \big ( \frac { d } { d x } X ( t ) + \alpha _ { \rm H J M } ( X ( t ) ) \big ) d t + \sigma ( X ( t ) ) d W _ t \medskip \\ X ( 0 ) & = & h , \end{array} \right . \end{align*}


\begin{align*} \theta ( r ) & \ge \sqrt { \psi ^ \prime ( \varphi ' ( \mu _ { \min } ) ) } \int _ 0 ^ r \sqrt { s } \ , d s \\ & = \frac { 2 } { 3 } \sqrt { \psi ^ \prime ( \varphi ' ( \mu _ { \min } ) ) } r ^ { 3 / 2 } . \end{align*}


\begin{align*} [ X , \mathrm { A d } ( g ^ { - 1 } ) \cdot E _ { \theta } ] & = \mathrm { A d } ( g ^ { - 1 } ) \cdot [ \mathrm { A d } ( g ) \cdot X , E _ { \theta } ] \\ & = \mathrm { A d } ( g ^ { - 1 } ) \cdot [ w ^ { - 1 } \cdot X , E _ { \theta } ] \\ & = \theta ( w ^ { - 1 } \cdot X ) \ , \mathrm { A d } ( g ^ { - 1 } ) \cdot E _ { \theta } \\ & = ( w \cdot \theta ) ( X ) \ , \mathrm { A d } ( g ^ { - 1 } ) \cdot E _ { \theta } \ , . \end{align*}


\begin{align*} L f ( \eta ) = \displaystyle \sum _ { x \in \mathbb { Z } } g ( \eta ( x ) ) \sum _ { y \in \mathbb { Z } } p ( x , y ) [ f ( \eta ^ { x y } ) - f ( \eta ) ] , \end{align*}


\begin{align*} N _ { n , i } = N _ { n , i _ 1 } ^ 1 \otimes \cdots \otimes N _ { n , i _ d } ^ d , \end{align*}


\begin{align*} g _ j \ast f _ 1 ( x ) & = \int \limits _ { K } a ( y ) \Phi _ { A _ j } ( y ) f _ 1 ( x - y ) d y \\ & = \int \limits _ { \mathfrak { D } ^ * } a ( y ) f _ 1 ( x - y ) d y , \end{align*}


\begin{align*} \lim _ { N \to \infty } \limsup _ { n \to \infty } k \sum _ { l = 1 } ^ k \binom { c _ n k n } l ( k ) _ l \sum _ { j = 1 } ^ { n - N } \frac { ( k ( j - 1 ) ) _ { c _ n k n - l } } { ( k n ) _ { c _ n k n } } \frac 1 { k ( n - j + 1 ) - l } = 0 . \end{align*}


\begin{align*} \left \| x - \sum _ { n = 1 } ^ { \lceil \lambda m \rceil } e ^ * _ { \rho _ x ( n ) } ( x ) e _ { \rho _ x ( n ) } \right \| & \ \leqslant \ \mathbf C _ \ell \left \| x - \sum _ { n = 1 } ^ { m } e ^ * _ { \rho _ x ( n ) } ( x ) e _ { \rho _ x ( n ) } \right \| \\ & \ \leqslant \ \mathbf C _ \ell C \sigma ^ { \mathcal { U } _ X , L } _ m ( x ) . \end{align*}


\begin{align*} | \mathcal { F } _ 2 | & = \sum _ { i = 0 } ^ { ( s - 2 ) / 2 } \binom { ( s + d - 3 ) / 2 } { \left \lfloor i / 2 \right \rfloor } \binom { ( s + d - 3 ) / 2 - \left \lfloor i / 2 \right \rfloor } { ( s - 2 ) / 2 - i } , \\ | \mathcal { F } _ 1 | + | \mathcal { F } _ 3 | & = \sum _ { i = 0 } ^ { ( s - 2 ) / 2 } \binom { ( s + d - 5 ) / 2 } { \left \lfloor i / 2 \right \rfloor } \binom { ( s + d - 1 ) / 2 - \left \lfloor i / 2 \right \rfloor } { ( s - 2 ) / 2 - i } , \end{align*}


\begin{align*} A x = b , A \in \mathbb { R } ^ { m \times n } , b \in \mathbb { R } ^ { m } , \end{align*}


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


\begin{align*} & a ' = \tfrac { 1 } { a + 1 } , & a '' & = \tfrac { a } { a + 1 } , \\ & b ' = \tfrac { \sin \alpha } { \sin \gamma } \tfrac { 1 } { b + 1 } , & b '' & = \tfrac { \sin \alpha } { \sin \gamma } \tfrac { b } { b + 1 } , \\ & c ' = \tfrac { \sin \beta } { \sin \gamma } \tfrac { 1 } { c + 1 } , & c '' & = \tfrac { \sin \beta } { \sin \gamma } \tfrac { c } { c + 1 } . \\ \end{align*}


\begin{align*} K _ { \ell , v , j } & = L _ { v , j } ( j \neq \nu _ v ) , \\ \mathrm { r k } ( K _ { \ell , v , \nu _ v } ) & = n _ { v , \nu _ v } - 1 . \end{align*}


\begin{align*} \begin{cases} & \partial _ { t } \rho ( t , u ) = [ - ( - \Delta ) ^ { \gamma / 2 } \rho ^ m ] ( t , u ) , \ ; \ ; u \in \mathbb { R } , \ ; t \in [ 0 , T ] , \\ & \rho ( 0 , u ) = g ( u ) , \ ; \ ; u \in \mathbb { R } , \end{cases} \end{align*}


\begin{align*} \begin{cases} & \partial _ { t } \rho ( t , u ) = [ - ( - \Delta ) ^ { \gamma / 2 } \rho ^ 2 ] ( t , u ) , \ ; \ ; u \in \mathbb { R } , \ ; t \in [ 0 , T ] , \\ & \rho ( 0 , u ) = g ( u ) , \ ; \ ; u \in \mathbb { R } . \end{cases} \end{align*}


\begin{align*} X _ { ( j + 1 ) K } ^ y - X _ { j K } ^ y \geq \Big ( v _ + + \frac { \theta } { 2 r } \Big ) K \end{align*}


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


\begin{align*} D _ i v ( \omega ) = - \int _ { \mathbb { R } ^ d } f ( \tau _ x \omega ) \partial _ i \phi ( x ) d x . \end{align*}


\begin{align*} \liminf _ { t \downarrow 0 } \frac { 1 } { t } d _ K \big ( S _ t h + t ( \alpha ( h ) - \rho ( h ) + \sigma ( h ) u ) \big ) = 0 , h \in K u \in U _ 0 . \end{align*}


\begin{align*} [ { \bf L } _ n , \chi _ T ] = [ { \bf L } ^ 0 _ n , \chi _ T ] = i \chi _ T ' { \bf A } _ n , \end{align*}


\begin{align*} l _ { \pi _ { \lambda } } ( z ) = \int _ { \mathbb { R } } \mathrm { e } ^ { z x } \ , \mathrm { d } \pi _ { \lambda } ( x ) = \mathrm { e } ^ { - \lambda } \sum _ { k = 0 } ^ { \infty } \mathrm { e } ^ { z k } \frac { \lambda ^ { k } } { k ! } = \exp \left ( \lambda ( \mathrm { e } ^ { z } - 1 ) \right ) . \end{align*}


\begin{align*} \frac { c | m h \pm n k | } { g x Q } W \left ( \frac { c | m h \pm n k | } { g x Q } \right ) = \frac { 1 } { 2 \delta } \int _ { - \delta } ^ { \delta } \frac { c | m h \pm e ^ { \xi } n k | } { g x Q } W \left ( \frac { c | m h \pm e ^ { \xi } n k | } { g x Q } \right ) \ , d \xi + O \left ( \frac { c n k \delta } { g x Q } \right ) . \end{align*}


\begin{align*} ( \mathcal { F } \otimes 1 ) ( \Delta \otimes \mathrm { i d } ) ( \mathcal { F } ) = & ( 1 \otimes \mathcal { F } ) ( \mathrm { i d } \otimes \Delta ) ( \mathcal { F } ) , \\ ( \epsilon \otimes \mathrm { i d } ) ( \mathcal { F } ) = & 1 = ( \mathrm { i d } \otimes \epsilon ) ( \mathcal { F } ) . \end{align*}


\begin{align*} H _ { 0 } ( \xi ) = \Lambda _ \xi Q _ \sigma ( \xi ) , \end{align*}


\begin{align*} ( s \otimes _ \star t ) ^ * = ( \overline { \mathcal { R } } _ 1 \rhd s ^ * ) \otimes _ \star ( \overline { \mathcal { R } } _ 2 \rhd t ^ * ) , ( s \otimes _ \star t ) ^ { * _ \star } = ( \overline { \mathcal { R } } _ 1 \rhd s ^ { * _ \star } ) \otimes _ \star ( \overline { \mathcal { R } } _ 2 \rhd t ^ { * _ \star } ) . \end{align*}


\begin{align*} \psi ( x ) : = \mu ^ { \frac { d - 1 } { p } } \varphi ( \mu x ) \\ \tilde { \psi } ( x ) : = \mu ^ { \frac { d - 1 } { p ' } } \tilde { \varphi } ( \mu x ) \end{align*}


\begin{align*} | | \psi _ i | | _ H & \leq \int _ 0 ^ 1 \left ( \max _ { x \in M } H _ i ( x ) - \min _ { x \in M } H _ i ( x ) \right ) d t \\ & = | \phi ( x _ i ) | \left ( \max _ { | v | \leq \delta } H ( v ) - \min _ { | v | \leq \delta } H ( v ) \right ) \\ & \leq 2 \delta | \phi ( x _ i ) | \end{align*}


\begin{align*} \begin{aligned} e ^ { - t } u ( x ) \leq & \int _ t ^ { t + b } e ^ { - s } \left ( C _ p \left | \dot { \xi } ( s ) - \frac { x - y } { b } \right | ^ q + f \left ( \xi ( s ) + \frac { b + t - s } { b } ( x - y ) \right ) \right ) d s \\ & + e ^ { - ( b + t ) } u ( \xi ( b + t ) ) . \end{aligned} \end{align*}


\begin{align*} \Omega _ 1 \times \Omega _ 2 \to \Omega _ N , U _ 1 \times U _ 2 \mapsto R ^ { - 1 } \left [ \begin{array} { c c } U _ 1 & 0 \\ 0 & \mathfrak { J } \overline { U _ 2 } \end{array} \right ] , \mathfrak { J } = \left [ \begin{array} { c c c c } J _ { m } ' & 0 & 0 & 0 \\ 0 & 0 & - 1 _ r & 0 \\ 0 & 1 _ r & 0 & 0 \\ 0 & 0 & 0 & J _ { m } ' \end{array} \right ] . \end{align*}


\begin{align*} D = - \sum _ { \substack { i , j = 1 , \\ i \not = j } } ^ n \partial _ y \Bigg [ \bigg ( \frac { \tau _ i ( \mathcal { L } B _ { i j , 0 } ) } { x _ { i j } ^ 3 } \bigg ) ( \varphi _ { i j } - 1 ) \Bigg ] - \sum _ { \substack { i , j = 1 , \\ i \not = j } } ^ n \partial _ y \bigg ( \frac { \tau _ i ( [ | D | , \varphi _ { i j } ] B _ { i j , 0 } ) } { x _ { i j } ^ 3 } \bigg ) . \end{align*}


\begin{align*} \mathcal { M } ' _ K : = \Bigl \{ ( \lambda , y ) \in \{ 0 , 1 \} ^ { | V | + | E | } \Bigm | \lambda \in \Lambda , \ y _ e = \prod _ { v \in e } \lambda _ v , \ e \in E \Bigr \} . \end{align*}


\begin{align*} \lim _ { T \to \infty } \ , \sup _ { \tau \in [ 0 , 1 ] } S \left ( \omega _ { V ( 0 ) } \circ \alpha _ T ^ { 0 \to \tau } \bigg | \omega _ { V ( \tau ) } \right ) = 0 . \end{align*}


\begin{align*} \mathcal { S } ( h , k ) \sim \sum _ { \ell = 0 } ^ { \min \{ | A | , | B | \} } \mathcal { I } _ \ell ( h , k ) , Q \to \infty . \end{align*}


\begin{align*} g ( x ) = { \rm d i s t } ^ { ‐ \beta } ( x , \ , \Gamma ) , \ ; \ ; w ( x ) = { \rm d i s t } ^ { ‐ \sigma } ( x , \ , \Gamma ) , \ ; \ ; v ( x ) = { \rm d i s t } ^ { ‐ \lambda } ( x , \ , \Gamma ) , \end{align*}


\begin{align*} z ^ 2 ( z - 1 ) ( z - K ) y '' ( z ) = K n ( n + 1 ) y ( z ) . \end{align*}


\begin{gather*} G ^ { \prime } ( t ) = \int \limits _ { \mathbb { Q } _ { p } } u _ { t } ( x , t ) w ( x ) d x = - \gamma \int \limits _ { \mathbb { Q } _ { p } } ( \boldsymbol { D } _ { x } ^ { \alpha } u ) ( x , t ) w ( x ) d x \\ + \int \limits _ { \mathbb { Q } _ { p } } F ( u ( x , t ) ) w ( x ) d x + \int \limits _ { \mathbb { Q } _ { p } } ( \boldsymbol { D } _ { x } ^ { \alpha _ { 1 } } u ^ { 3 } ) ( x , t ) w ( x ) d x . \end{gather*}


\begin{align*} D _ \alpha ^ { J } : = d _ 1 ^ { j _ 1 } \cdots d _ m ^ { j _ { p } } \bigg | _ { x = x _ \alpha } \end{align*}


\begin{align*} | I _ m \cap I _ n | = 0 , \end{align*}


\begin{align*} \sum _ { \substack { p > x ^ { 1 / 2 } \\ p | d } } \frac { \lambda ^ 2 _ f ( p ) - 2 } { p } \ll \frac 1 { x ^ { 1 / 2 } } \sum _ { \substack { p > x ^ { 1 / 2 } \\ p | d } } 1 \ll \frac { \log X } { x ^ { 1 / 2 } \log x } . \end{align*}


\begin{align*} \operatorname { I n d _ c } ( D ) : = [ P _ { Q } ] - [ e _ 1 ] \in K _ 0 ( \mathcal { A } _ G ^ c ( X , E ) ) \ ; \ ; \ ; \ ; e _ 1 : = \left ( \begin{array} { c c } 0 & 0 \\ 0 & 1 \end{array} \right ) . \end{align*}


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


\begin{align*} \sum _ { r \in \mathbb { Z } } c ^ F _ r \big ( r _ l ( \widetilde { V } ) , r _ l ( \widetilde { \mathcal { S } } ) \big ) q _ F ^ { - \frac { r } { 2 } } X ^ { - e f r } = r _ l \big ( \gamma ( X , \pi _ E , \widehat { \sigma _ E } , \psi _ E ) \big ) \sum _ { r \in \mathbb { Z } } c ^ F _ r \big ( r _ l ( V ) , r _ l ( \mathcal { S } ) \big ) q _ F ^ { \frac { r } { 2 } } X ^ { e f r } \end{align*}


\begin{align*} \hat t d ( x , x _ 0 ) = \hat t d ( \gamma _ x ( 1 ) , \gamma _ x ( 0 ) ) = d ( \gamma _ x ( \hat t ) , \gamma _ x ( 0 ) ) \le S \end{align*}


\begin{align*} \widehat { \psi _ { p ^ { 2 k } } } ( \mathbf { u } ) = \frac { 1 } { p ^ { 2 n k } } \sum _ { \mathbf { c } \in ( \mathbb { Z } / p ^ { 2 k } \mathbb { Z } ) ^ n } \psi _ { p ^ { 2 k } } ( f _ \mathbf { c } ) \exp \left ( \frac { 2 \pi i ( c _ 1 + n ) u _ 1 } { p ^ { 2 k } } \right ) . \end{align*}


\begin{align*} \partial _ t \widetilde { h } + \mathcal { L } \widetilde { h } = R \end{align*}


\begin{align*} K ( x ) : = - \log | \tanh x | \end{align*}


\begin{align*} \hbox { $ s ( b , c , d ) = s ( a b , a c , d ) $ f o r a n y $ a \in { \mathbb Z } $ w i t h $ \gcd ( a , d ) = 1 $ . } \end{align*}


\begin{align*} \langle f , \varphi \rangle = \frac { 1 } { \overline { \langle g , \widetilde { g } \rangle } } \iint _ { \R ^ { 2 d } } \langle f , M _ \omega T _ x g \rangle \overline { V _ { \widetilde { g } } \varphi ( x , \omega ) } \ , d ( x , \omega ) = \langle \widetilde { f } , \varphi \rangle , \end{align*}


\begin{align*} T = 1 \mbox { a n d } J = 1 , \end{align*}


\begin{align*} \mathcal { A } ^ c _ G ( X , E ) : = \big ( C ^ \infty _ c ( G ) \hat { \otimes } \Psi ^ { - \infty } ( S , E | _ S ) \big ) ^ { K \times K } , \end{align*}


\begin{align*} 0 = \langle ( P X - X P ) g , g \rangle = \frac { 1 } { 2 \pi i } \norm { g } _ 2 ^ 2 . \end{align*}


\begin{align*} \dot { y } _ R ( t ) = 2 \Im \int _ { \mathbb { R } ^ N } \nabla \varphi _ R ( x ) \cdot \nabla u \bar { u } d x = 2 R \Im \int _ { \mathbb { R } ^ N } \nabla \varphi \left ( \frac { x } { R } \right ) \cdot \nabla u \bar { u } d x , \end{align*}


\begin{align*} \sum _ { a \leqslant k \leqslant n } k ^ { \ell - 1 } = \frac { 1 } { \ell } B _ \ell ( n + 1 ) - \frac { 1 } { \ell } B _ \ell ( a ) . \end{align*}


\begin{align*} ( \tau _ j f ) ( { \bf x } ) = f ( { \bf x } - { \bf e } _ { j } ) , \end{align*}


\begin{align*} 0 = E _ 0 \subset E _ 1 \subset \ldots \subset E _ l = E \end{align*}


\begin{align*} \left \{ \begin{array} { l l } h _ n ( X _ t ) - h _ n ( X _ 0 ) = M _ t ^ { h _ n } + N _ t ^ { h _ n } , \\ N _ t ^ { h _ n } = - \int _ 0 ^ t h _ n ( X _ s ) { \rm d } A _ s ^ { \overline { \mu } ^ n } \end{array} \right . \end{align*}


\begin{align*} H _ 0 = \big \{ \{ k \} \colon k \in S \big \} H _ 0 ' = \big \{ \{ k \} \colon k \in S ' \big \} \end{align*}


\begin{align*} T B _ { k } ( f ) = \sum _ { \gamma \in \Gamma _ { k } ( G ) } t b ( f ( \gamma ) ) . \end{align*}


\begin{align*} x ^ 2 = x \cdot x = x ^ T x . \end{align*}


\begin{align*} \int _ 0 ^ 1 \sum _ { j \neq i } \partial _ j \left ( \frac { y _ j } { t ^ d } \eta \left ( \frac { y } { t } \right ) \right ) + \frac { y _ i } { t ^ { d + 1 } } \partial _ i \eta \left ( \frac { y } { t } \right ) d t = - \int _ 0 ^ 1 \frac { 1 } { t ^ { d - 1 } } \frac { d } { d t } \left ( \eta \left ( \frac { y } { t } \right ) \right ) d t = - \eta ( y ) \end{align*}


\begin{align*} \Phi _ 1 ( h _ 1 ) = \frac { 1 } { 2 } \int \left ( L _ + h _ 1 \right ) h _ 1 , \Phi _ 2 ( h _ 2 ) = \frac { 1 } { 2 } \int \left ( L _ - h _ 2 \right ) h _ 2 . \end{align*}


\begin{align*} Z _ i = Z _ { i ; | | } + Z _ { i ; \perp } , Z _ { i ; | | } \in \ker ( \kappa ^ { ( \ell ) } ) , \ , Z _ { i ; \perp } \in \ker ( \kappa ^ { ( \ell ) } ) ^ \perp \end{align*}


\begin{align*} \mathfrak { P } _ k ( z _ 0 \times z _ 0 ) ^ { - 1 } \overline { \mathbf { f } ( g _ { \mathbf { h } } \times h _ { \mathbf { h } } ) } = \sum _ { j = 1 } ^ e \mathfrak { P } _ k ( z _ 0 ) ^ { - 1 } \overline { \mathbf { g } _ j ( g _ { \mathbf { h } } ) } \mathfrak { P } _ k ( z _ 0 ) ^ { - 1 } h _ j ( \mathbf { h } ) . \end{align*}


\begin{align*} \mathbf { P } \Big ( \frac { 1 } { \sigma } ( \hat { \theta } _ n - \theta ) \sqrt { \Sigma _ { k = 1 } ^ n X _ { k - 1 } ^ 2 } < - \Phi ^ { - 1 } ( 1 - \kappa _ n / 2 ) \Big ) \sim \kappa _ n / 2 , \ \ \ n \rightarrow \infty . \end{align*}


\begin{align*} \forall x ^ X , y ^ X , { x ' } ^ X , { y ' } ^ { X } \left ( x = _ X x ' \land y = _ X y ' \rightarrow \chi _ A x y = _ 0 \chi _ A x ' y ' \right ) \end{align*}


\begin{align*} a \otimes b = \sum _ { i = 1 } ^ n \Delta ( p _ i ) ( 1 \otimes q _ i ) , \end{align*}


\begin{align*} \bar { u } = A w \qquad . \end{align*}


\begin{align*} z ^ 3 y ^ 3 + a z y ^ 5 + x ^ 6 + b y ^ 6 = 0 \ , . \end{align*}


\begin{align*} & d _ { v } ^ { n } : = \frac { \left \| \phi ^ { n } - \phi ^ { n - 1 } \right \| _ { \infty } } { \tau } < \varepsilon _ { v } \end{align*}


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


\begin{align*} D \cdot C _ n = E _ Z \cdot C _ n - C _ n ^ 2 = - ( K _ Z + C _ n ) \cdot C _ n = - \deg ( K _ { C _ n } ) . \end{align*}


\begin{align*} l _ { \pi _ { \lambda , \beta } } ( z ) = \int _ { \mathbb { R } } \mathrm { e } ^ { z x } \ , \mathrm { d } \pi _ { \lambda , \beta } ( x ) = \sum _ { k = 0 } ^ { \infty } \frac { \big ( \mathrm { e } ^ { z } \lambda \big ) ^ { k } } { k ! } E _ { \beta } ^ { ( k ) } \big ( - \lambda \big ) = E _ { \beta } \big ( \lambda ( \mathrm { e } ^ { z } - 1 ) \big ) . \end{align*}


\begin{align*} \left ( \begin{array} { c c c c } 0 & a _ { 1 , 2 } & \cdots & a _ { 1 , n } \\ a _ { 2 , 1 } & 0 & \cdots & a _ { 2 , n } \\ \vdots & \vdots & \ddots & \vdots \\ a _ { n , 1 } & a _ { n , 2 } & \cdots & 0 \end{array} \right ) \left ( \begin{array} { c c c c } c _ 1 \\ c _ 2 \\ \vdots \\ c _ n \end{array} \right ) = \left ( \begin{array} { c c c c } x _ 1 \\ x _ 2 \\ \vdots \\ x _ n \end{array} \right ) . \end{align*}


\begin{align*} \alpha \ = \ \sum \limits _ { n = 0 } ^ \infty \ \frac { f ( n ) } { 2 ^ n } ( f ( n ) \in \{ 0 , 1 \} ) . \end{align*}


\begin{align*} L _ 0 = C ( A + \| \beta _ 0 \| _ \infty + 3 + T ) \end{align*}


\begin{align*} \varphi ( \xi _ 1 , \xi _ 2 ) = ( \varphi _ X ( \xi _ 1 ) \varphi _ X ( \xi _ 2 ) ) ^ { 1 - z } \varphi _ X ( \xi _ 1 + \xi _ 2 ) ^ { z } . \end{align*}


\begin{align*} \int _ { \mathcal { Q } } Z f ( x , \omega ) \ , d \omega & = \int _ \mathcal { Q } \left ( \sum _ { k \in \Z ^ d } f ( x + k ) e ^ { - 2 \pi i k \cdot \omega } \right ) \ , d \omega \\ & = \sum _ { k \in \Z ^ d } \int _ \mathcal { Q } f ( x + k ) e ^ { - 2 \pi i k \cdot \omega } \ , d \omega = \sum _ { k \in \Z ^ d } f ( x + k ) \int _ \mathcal { Q } e ^ { - 2 \pi i k \cdot \omega } \ , d \omega \\ & = f ( x ) , \end{align*}


\begin{align*} \langle ( P ^ { \nu _ \alpha } _ t ) ^ * ( f ) - f ; g \rangle _ { L ^ 2 ( \mu _ \alpha ) } & = \int _ { 0 } ^ t \left \langle ( D ^ { \alpha - 1 } ) ^ { * } ( f ) + \frac { 1 } { p _ \alpha } ( R ^ { \alpha } ) ^ * ( p _ \alpha , f ) ; \nabla P _ s ^ { \nu _ \alpha } ( g ) \right \rangle _ { L ^ 2 ( \mu _ \alpha ) } d s . \end{align*}


\begin{align*} \frac { e ^ { \delta z } - e ^ { - \delta z } } { 2 \delta z } = 1 + O ( \delta | z | ) . \end{align*}


\begin{align*} P ' Q & = - \left ( \sum _ { i = 1 } ^ n P _ i \right ) \left ( \sum _ { j = 1 } ^ n - \lambda _ j ^ 2 P _ j \right ) \\ & = - \sum _ { j = 1 } ^ n \lambda _ j ^ 2 P _ j ^ 2 - \sum _ { \substack { 1 \leq i , j \leq n \\ i \neq j } } \lambda _ j ^ 2 P _ i P _ j \end{align*}


\begin{align*} \big ( \sup _ { n \in \N } S _ n T _ n \big ) f \geq \big ( \sup _ { n \in \N } S _ n T _ m \big ) f = \sigma \lim _ n ( S _ n T _ m ) f = \sigma \lim _ n S _ n ( T _ m f ) = S T _ m f , \end{align*}


\begin{align*} \norm { K } _ { H . S . } = \left ( \sum _ { n \in \N } \norm { K e _ n } _ \mathcal { H } ^ 2 \right ) ^ { 1 / 2 } . \end{align*}


\begin{align*} \mathbf { E } ( \mathfrak { g } , s ) : = E _ k ^ { \tilde { \tau } _ m } ( \mathfrak { g } , s ; \chi , \mathfrak { n } ) : = \sum _ { \gamma \in P _ N \backslash G _ N } \phi ( \gamma \mathfrak { g } , s ) , \mathfrak { g } \in G _ N \end{align*}


\begin{align*} \ddot x _ 1 ( t ) = - \theta _ 1 x _ 1 ( t ) - \theta _ 2 \dot x _ 1 ( t ) + \theta _ 3 u ( t ) \end{align*}


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


\begin{align*} d [ c a _ { i , j } ] : = \min \{ d ( c a _ { i , j } ) , \alpha _ { i - 1 } , \alpha _ { j } \} \ , , c \in F ^ { \times } \ , . \end{align*}


\begin{align*} S _ { 2 , \lambda } ( n , k ) = S _ { 2 , \lambda } ( n - 1 , k - 1 ) + ( k - ( n - 1 ) \lambda ) S _ { 2 , \lambda } ( n - 1 , k ) , ( \mathrm { s e e } \ [ 8 ] ) , \end{align*}


\begin{align*} \xi ( s ; r , x ) & = S _ { s - r } x + \int _ r ^ s S _ { s - u } \alpha ( u , \xi ( u ; r , x ) ) d u , s \in [ r , \infty ) . \end{align*}


\begin{align*} e ^ { - t } C _ p q \left | \dot { \xi } ( t ) \right | ^ { q - 2 } \dot { \xi } ( t ) = \int _ 0 ^ t e ^ { - s } D f \left ( \xi ( s ) \right ) d s + \tilde { c } \end{align*}


\begin{align*} H ( z , \chi ) = L ( z , \chi ) L ( 2 z , \chi ^ { 2 } ) G ( z , \chi ) , \ ( \Re z > 1 ) \end{align*}


\begin{align*} a = \left ( \int _ 0 ^ \infty \frac { 1 } { u } d \mu ( u ) \right ) ^ { - 1 } , b = \int _ 0 ^ \infty u \ , d \mu ( u ) . \end{align*}


\begin{align*} S ^ T J S = \begin{pmatrix} A ^ T C - C ^ T A & A ^ T D - C ^ T B \\ B ^ T C - D ^ T A & B ^ T D - D ^ T B \end{pmatrix} = \begin{pmatrix} 0 & I \\ - I & 0 \end{pmatrix} . \end{align*}


\begin{align*} ( A \setminus \{ \bar { a } _ n \} ) \cup ( B \setminus \{ \bar { a } _ n \} ) \cup ( C \setminus \{ \bar { a } _ n \} ) & = ( A \cup B \cup C ) \setminus \{ \bar { a } _ n \} \\ & = \mathbb { N } \setminus \{ \bar { a } _ n \} \end{align*}


\begin{align*} L _ n ( x , y ) = D ^ n ( x ) . \end{align*}


\begin{align*} \left \{ \begin{aligned} & \mathrm { d } X _ { t } = \theta f ( X _ { t } ) \mathrm { d } t + \sigma \mathrm { d } W _ { t } + \mathrm { d } L _ { t } - \mathrm { d } R _ { t } , \\ & X _ { 0 } = x \in [ l , u ] , \end{aligned} \right . \end{align*}


\begin{align*} \widetilde { A } ( \xi ; U ) : = \sum _ { i = 1 } ^ { 3 } \widetilde { A } ^ { i } ( U ) \xi _ { i } \quad \mbox { a n d } \widetilde { B } ( \xi ; U ) : = \sum _ { i , j = 1 } ^ { 3 } \widetilde { B } ^ { i j } ( U ) \xi _ { i } \xi _ { j } , \end{align*}


\begin{align*} \frac { 2 v ( h e _ 1 ) - v ( h e _ 1 + y ) - v ( h e _ 1 - y ) } h & = 2 \partial ^ h _ 1 v ( 0 ) - \partial ^ h _ 1 v ( y ) - \partial ^ h _ 1 v ( - y ) - \frac { v ( y ) + v ( - y ) } h \end{align*}


\begin{align*} \norm { A - \mathbb { S } A } & = \norm { ( I - \mathbb { S } ) A } = \norm { ( I - \mathbb { S } ) ( I - Q _ { Z _ k } Q _ { Z _ k } ^ T ) A } \\ & \le \norm { ( I - \mathbb { S } ) } \ , \norm { ( I - Q _ { Z _ k } Q _ { Z _ k } ^ T ) A } . \end{align*}


\begin{align*} \frac { \tilde { \phi } ^ { n + 1 } - \phi ^ n } { \tau ^ 2 } + \eta ^ n \frac { \tilde { \phi } ^ { n + 1 } - \phi ^ n } { 2 \tau } = 0 . \end{align*}


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


\begin{align*} \Gamma _ m ( s ) = \pi ^ { m ( m - 1 ) } \prod _ { i = 0 } ^ { m - 1 } \Gamma ( s - 2 i ) , m \in \Z , \end{align*}


\begin{align*} \eta : = \frac { \delta } { 2 \phi _ { \delta } ^ { x _ 0 } ( T ) } , \end{align*}


\begin{align*} v _ { k _ \bullet } = v _ + - 2 \varepsilon _ \bullet , v _ { k + 1 } = v _ k + \frac { \varepsilon _ \bullet } { k ^ 2 } , \ ; \ ; k \geq k _ \bullet . \end{align*}


\begin{align*} r ( \mathbf { x } , \mathbf { y } ) : = \begin{bmatrix} \mathbf { x } \\ \mathbf { y } \end{bmatrix} - \Pi _ { D } ( \begin{bmatrix} \mathbf { x } \\ \mathbf { y } \end{bmatrix} - \begin{bmatrix} H \mathbf { x } + \mathbf { g } - A ^ T \mathbf { y } \\ A \mathbf { x } - \mathbf { b } \end{bmatrix} ) , \end{align*}


\begin{align*} \partial _ x \pi _ 2 \tilde A _ { k n } & = O ( 1 ) , \\ \partial _ y \pi _ 2 \tilde A _ { k n } & = \epsilon ^ { 2 ^ { k n } } \tilde b ^ { ( 1 + O ( \alpha ^ n ) ) | \bar \upsilon _ { k n } | } , \end{align*}


\begin{align*} \mathcal { O } _ { k , N } ( t ) : = \int _ 0 ^ t \tilde { E } _ { k , N } ( t - s ) \textmd { d } W ( s ) . \end{align*}


\begin{align*} P _ n ( n + 1 , x ) = \frac { \Gamma ( x ) } { \Gamma ( x - n - 1 ) } \left ( \psi ( x ) - \psi ( x - n - 1 ) \right ) . \end{align*}


\begin{align*} \rho ( t , x ) & = G * _ { t , x } S + S \end{align*}


\begin{align*} I ' ( \omega ) & = \int _ { \R } ( - 2 \pi i x ) e ^ { - \pi M x ^ 2 } e ^ { - 2 \pi i \omega x } \ , d x = \frac { i } { M } \int _ { \R } \dfrac { d } { d x } ( e ^ { - \pi M x ^ 2 } ) e ^ { - 2 \pi i \omega x } \ , d x \\ & \stackrel { } { = } - \frac { i } { M } \int _ { \R } e ^ { - \pi M x ^ 2 } \dfrac { d } { d x } ( e ^ { - 2 \pi i \omega x } ) \ , d x = - \frac { 2 \pi \omega } { M } I ( \omega ) . \end{align*}


\begin{align*} \mathbf { f } ( \alpha g k _ { \infty } k ) = j ( k _ { \infty } , z _ 0 ) ^ { - k } \mathbf { f } ( g ) , \end{align*}


\begin{align*} L ( g ) : = | | X | | \end{align*}


\begin{align*} L ^ { h _ \alpha } _ t & = \exp \left ( M ^ { ( \alpha ) } _ t - \frac { \ , 1 \ , } { 2 } \ < M ^ { ( \alpha ) , c } \ > _ t \right ) \prod _ { 0 < s \leq t } \frac { h _ { \alpha } ( X _ s ) } { h _ { \alpha } ( X _ { s - } ) } \exp \left ( 1 - \frac { h _ { \alpha } ( X _ s ) } { h _ { \alpha } ( X _ { s - } ) } \right ) \\ & = \frac { h _ { \alpha } ( X _ t ) } { h _ { \alpha } ( X _ 0 ) } \exp \left ( A _ t ^ { \overline { \mu } } + \alpha \int _ 0 ^ t \left ( h _ { \alpha } ^ { - 1 } - 1 \right ) ( X _ s ) { \rm d } s \right ) . \end{align*}


\begin{align*} f = \sum _ { \gamma \in \Gamma } c _ \gamma e _ \gamma \end{align*}


\begin{align*} \left ( 1 - \frac { 1 } { p ^ w } \right ) \left ( 1 + \frac { p ^ { w - 1 } - 1 } { p ( p - 1 ) } \right ) = 1 + \frac { 1 } { p ^ { 3 - w } } + O \left ( \frac { 1 } { p ^ { 1 + \varepsilon } } \right ) . \end{align*}


\begin{align*} \min \left \{ I _ b [ \gamma ] ; \gamma \in \mathrm { A C } \left ( [ 0 , b ] ; \Omega \right ) , \gamma ( 0 ) = x , \gamma ( b ) = \xi ( b ) \right \} . \end{align*}


\begin{align*} \begin{aligned} & \Phi ^ P _ { Y , g } ( A _ 0 , A _ 1 , \dots , A _ m ) \\ : = & \int _ { h \in M / Z _ M ( g ) } \int _ { K N } \int _ { G ^ { \times m } } C \big ( H ( g _ 1 . . . g _ m k ) , H ( g _ 2 . . . g _ m k ) , \dots , H ( g _ m k ) \big ) \\ & \operatorname { T r } \Big ( A _ 0 \big ( k h g h ^ { - 1 } n k ^ { - 1 } ( g _ 1 \dots g _ m ) ^ { - 1 } \big ) \circ A _ 1 ( g _ 1 ) \dots \circ A _ m ( g _ m ) \Big ) d g _ 1 \cdots d g _ m d k d n d h . \end{aligned} \end{align*}


\begin{align*} \dot { \hat q } \equiv { d { \hat q } \over d t } = { 1 \over 2 } \hat q \hat \xi ^ b , \end{align*}


\begin{align*} R \left [ \begin{array} { c } z _ 1 \\ u _ { 0 1 } \end{array} \right ] = \left [ \begin{array} { c } z _ 2 \\ u _ { 0 2 } \end{array} \right ] \mu ( z _ 1 ) , \end{align*}


\begin{align*} \langle \nabla ( f _ h ) ( x ) ; u \rangle = - \int _ 0 ^ { + \infty } e ^ { - t } P _ t ^ { \Sigma } ( \langle u ; \nabla ( h ) \rangle ) ( x ) d t . \end{align*}


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


\begin{align*} \int _ { T - \tau } ^ { T } \mod { \widetilde { U } ^ p ( t ) } ^ 2 d t = \int _ { 0 } ^ { \tau } \mod { \widetilde { U } ^ p ( T - t ) } ^ 2 d t & \geq C \sum _ { k \geq N } \mod { a _ k } ^ 2 | 1 - e ^ { \tilde { \lambda } _ k } | ^ 2 e ^ { 2 \Re ( \tilde { \lambda } _ k ) \tau } \\ & \geq C \sum _ { k \geq N } \mod { a _ k } ^ 2 e ^ { 2 \Re ( \tilde { \lambda } _ k ) \tau } , \end{align*}


\begin{align*} L ( U ^ \mu ; z _ 0 , r ) = \frac { 1 } { 2 \pi } \int _ { - \pi } ^ \pi U ^ \mu ( z _ 0 + r e ^ { i \theta } ) d \theta . \end{align*}


\begin{align*} \left ( \begin{array} { c } u _ 1 ( - t ) \\ u _ 2 ( - t ) \\ u _ 3 ( - t ) \end{array} \right ) = M ( - 1 ) \left ( \begin{array} { c } u _ 1 ( t ) \\ u _ 2 ( t ) \\ u _ 3 ( t ) \end{array} \right ) \end{align*}


\begin{align*} S ' _ n : = \sum _ { j = n _ 0 } ^ n f ( j ) \mbox { a n d } I ' _ n : = \int _ { n _ 0 } ^ n f ( t ) \ , d t . \end{align*}


\begin{align*} H _ 2 : C _ A = \Pi _ 2 A \in I _ d ( 2 ) , \end{align*}


\begin{align*} \begin{aligned} & | \sigma _ n ^ 2 ( z ) - \sigma _ n ^ 2 ( y ) | \\ & \le ( | \sigma _ n ( z ) | + | \sigma _ n ( y ) | ) \cdot | \sigma _ n ( z ) - \sigma _ n ( y ) | \le ( | \sigma ( z ) | + | \sigma ( y ) | ) \cdot | \sigma _ n ( z ) - \sigma _ n ( y ) | \\ & \le c \cdot \bigl ( ( 1 + | z | ^ { 2 \ell _ \sigma + 1 } + | y | ^ { 2 \ell _ \sigma + 1 } ) \cdot \bigl ( | z - y | + n ^ { - 1 / 2 } \bigr ) . \end{aligned} \end{align*}


\begin{align*} \tau _ x & = { \mathcal T } _ { t , x , 0 } , \check \tau _ { t , x } = \check { \mathcal T } _ { t , x , 0 } , \\ { \mathcal T } _ { t , x , v } & = \tau _ { x - \check { \mathcal T } _ { t , x , v } v } \check { \mathcal T } _ { t , x , v } > 0 . \end{align*}


\begin{align*} { } \left \{ \begin{aligned} \mathrm { d } X _ { t } & = b ( X _ { t } ) \mathrm { d } t + \sigma \mathrm { d } B ^ { H } _ { t } , \\ X _ { 0 } & = x , \end{aligned} \right . \end{align*}


\begin{align*} - ( \triangle ) ^ \alpha u + V ( x ) u = K ( x ) f ( u ) , \ \ \ \mathbb { R } ^ { d } , \end{align*}


\begin{align*} \varphi ( r , t , \xi ) \ , = \ , - \frac { 1 } { t } \cdot \log \left ( r - \xi \right ) \end{align*}


\begin{align*} g _ { v } ( u , w ) = \frac { 1 } { 2 } \left . \frac { \partial ^ 2 } { \partial s \partial t } L ( v + t u + s w ) \right | _ { t = s = 0 } \end{align*}


\begin{align*} q \colon [ 0 , 1 ] \times \Delta _ d & \to \Delta _ { d + 1 } \\ ( t , b _ 1 , \dots , b _ d ) & \mapsto ( ( 1 - t ) ( 1 - \sum _ { i = 1 } ^ d b _ i ) , ( 1 - t ) b _ 1 , \dots , ( 1 - t ) b _ d ) . \end{align*}


\begin{align*} H = \partial _ \xi ^ 2 + \frac { d + 1 } { \xi } \partial _ \xi - 3 ( d - 2 ) \left ( 2 Q ( \xi ) + \xi ^ 2 Q ^ 2 ( \xi ) \right ) . \end{align*}


\begin{align*} \left \{ \begin{array} { r c l } d Y ( t ) & = & L ( Y ( t ) ) d t + \sigma ( Y ( t ) ) d W ( t ) \medskip \\ Y ( 0 ) & = & x , \end{array} \right . \end{align*}


\begin{align*} X ( v \otimes a ) = \sum _ s \delta _ s ( v ) ( 1 \otimes a ) , v \in V , a \in A . \end{align*}


\begin{align*} \frac { \mathbf { P } \Big ( \frac { 1 } { \sigma } ( \hat { \theta } _ n - \theta ) \sqrt { \Sigma _ { k = 1 } ^ n X _ { k - 1 } ^ 2 } < - x \Big ) } { 1 - \Phi \left ( x \right ) } = \exp \bigg \{ \theta _ 2 C \frac { \ln n + x ^ { 3 } } { \sqrt { n } } \bigg \} , \end{align*}


\begin{align*} \lim _ { M \to \infty } \frac { | \bigcup _ { p _ { i } \in S _ { \theta } , \ , \ , p _ { i } < p _ { l } } \Lambda _ { ( p _ { i } , \infty ) } ( M , \Gamma ) | } { | \Lambda ( M , \Gamma ) | } = 0 . \end{align*}


\begin{align*} [ N x , N y , N z ] = & N \Big ( [ N x , N y , z ] + [ x , N y , N z ] + [ N x , y , N z ] \\ - & N \Big ( [ N x , y , z ] + [ x , N y , z ] + [ x , y , N z ] - N [ x , y , z ] \Big ) \Big ) . \end{align*}


\begin{align*} \hat { \theta } _ n - \theta = \frac { \sum _ { k = 1 } ^ n X _ { k - 1 } \varepsilon _ k } { \sum _ { k = 1 } ^ n X _ { k - 1 } ^ 2 } . \end{align*}


\begin{align*} k _ w ( t ) = \frac { t ^ 2 } { 2 } \big ( k _ w '' ( 0 ) + \sqrt { t } R \big ) . \end{align*}


\begin{align*} \widehat { v } ( t , x ) = H _ 1 \ast \widehat { \mu } _ t ( x ) + \frac 1 m \sum _ { i = 1 } ^ m K _ 1 ( \widehat { Y } ^ i ( t ) - x ) . \end{align*}


\begin{align*} R ( n , \alpha , \varepsilon , Z , T , f ) = \inf \{ \sum _ { i } e ^ { - \alpha n + f _ { n } ( x _ i ) } : Z \subset \cup _ { i } B _ { n } ( x _ i , \varepsilon ) \} , \end{align*}


\begin{align*} A g _ 0 ( 0 , \omega ) = e ^ { - \frac { \pi } { 2 } \omega ^ 2 } , \end{align*}


\begin{align*} \sum p ( 5 n + 4 ) X ^ { 5 n + 4 } = \frac { \alpha ( X ^ { 2 5 } ) } { \alpha ( X ^ 5 ) ^ 6 } ( \alpha _ 0 ^ 2 \alpha _ 2 ^ 2 - 3 \alpha _ 0 \alpha _ 1 ^ 2 \alpha _ 2 + \alpha _ 1 ^ 4 ) \overset { \eqref { s h o r t } } { = } 5 \frac { \alpha ( X ^ { 2 5 } ) } { \alpha ( X ^ 5 ) ^ 6 } \alpha _ 1 ^ 4 \overset { \eqref { a l p h a 1 } } { = } 5 X ^ 4 \frac { \alpha ( X ^ { 2 5 } ) ^ 5 } { \alpha ( X ^ 5 ) ^ 6 } . \end{align*}


\begin{align*} \begin{aligned} | N _ { G } ( h ) \backslash V ( B \cup T ' ) | & = d _ { G } ( h ) - | N _ { G } ( h ) \cap V ( B ) | \\ & - | N _ { G } ( h ) \cap ( V ( T ' ) \backslash \{ u _ { l + 1 } , \cdots , u _ { r } \} ) | \\ & \geq t + 1 - ( t - \lfloor \frac { r - l } { 2 } \rfloor ) \\ & \geq 1 + \lfloor \frac { r - l } { 2 } \rfloor , \end{aligned} \end{align*}


\begin{align*} D ( \varphi ) : = \frac 1 2 \sum _ { i = 1 } ^ n \int _ \R ( K _ \alpha * \varphi _ i ) \varphi _ i = \frac 1 { 2 n ^ 2 } \sum _ { i = 1 } ^ n \frac 1 { \ell _ i ^ 2 } \iint _ { ( 0 , \ell _ i ) ^ 2 } K _ \alpha ( x - y ) \ , d y d x . \end{align*}


\begin{align*} & \partial _ t \rho _ i = \nabla \cdot \left [ D _ i ( x ) \left ( \nabla \rho _ i + \frac { z _ i e } { k _ B T } \rho _ i \nabla \phi \right ) \right ] , i = 1 , \cdots , s , \\ & - \nabla \cdot ( \epsilon ( x ) \nabla \phi ) = 4 \pi \left ( f ( x ) + \sum _ { i = 1 } ^ { s } z _ i e \rho _ i \right ) . \end{align*}


\begin{align*} H _ k ( x ) = \prod _ { \nu \in M _ k } \max \{ 1 , | x | _ \nu \} h _ k ( x ) = \log H _ k ( x ) = \sum _ { \nu \in M _ k } \log | x | _ \nu . \end{align*}


\begin{align*} \| K _ \alpha * \rho \| _ \infty \leq \| \rho \| _ \infty \int _ \R K _ \alpha = \int _ \R K . \end{align*}


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


\begin{align*} \lim _ { n \to \infty } \norm { V _ { g _ 0 } f _ n - F } _ { L ^ { p , q } } = 0 . \end{align*}


\begin{align*} \frac { 1 } { \sqrt { n } } \le | \langle \mathbf { v } _ i , \mathbf { y } \rangle _ { \C ^ { L } } | = \frac { 1 } { \delta } \Big | \sum _ { j = 1 } ^ L B _ j ( \alpha _ i ) \overline y _ j \Big | \mbox { f o r a l l $ 1 \leq i \leq n $ . } \end{align*}


\begin{align*} \gamma _ { n , s } : = \frac { \sin ( \pi s ) \Gamma ( n / 2 ) } { \pi ^ { \frac n 2 + 1 } } . \end{align*}


\begin{align*} \langle z | ( a ^ { \dagger } a ) _ { k , \lambda } | z \rangle = \sum _ { l = 0 } ^ { k } S _ { 2 , \lambda } ( k , l ) | z | ^ { 2 l } = \phi _ { k , \lambda } ( | z | ^ { 2 } ) . \end{align*}


\begin{align*} L = \frac { f ' } f , F ( z ) = z - \frac { f ( z ) } { f ' ( z ) } , F ' = \frac { f f '' } { ( f ' ) ^ 2 } . \end{align*}


\begin{align*} C ( \Sigma ) : = \{ r x : x \in \Sigma , \ , r > 0 \} \end{align*}


\begin{align*} - \Delta u = { } & 1 , \textrm { i n } \Omega , \\ u = { } & 0 , \textrm { o n } \partial \Omega . \end{align*}


\begin{align*} l ( x , y , \omega ) & = \inf \{ f ( x , u , \omega ) - u \cdot y \} \\ & = \begin{cases} + \infty & , \\ f _ 0 ( x , \omega ) + y \cdot H ( x , \omega ) & , \\ - \infty & \end{cases} \end{align*}


\begin{align*} \nabla _ t ^ \mathcal { F } : = \mathrm { p r } _ { t \star } \circ \nabla ^ \mathcal { F } | _ { \Xi _ { t \star } \otimes _ { \mathbb { K } [ [ \nu ] ] } \Xi _ { t \star } } \colon \Xi _ { t \star } \otimes _ { \mathbb { K } [ [ \nu ] ] } \Xi _ { t \star } \rightarrow \Xi _ { t \star } \end{align*}


\begin{align*} H _ R = \left ( \frac { 1 } { ( \beta - 1 ) R } \right ) ^ { 1 / ( \beta - 1 ) } . \end{align*}


\begin{align*} S _ r \Sigma : = \Sigma \circ \pi _ r , \Sigma \in L _ 2 ^ 0 ( H ) . \end{align*}


\begin{align*} 1 - \Phi \left ( \overline { \lambda } \right ) = \Big ( 1 - \Phi ( x ) \Big ) \exp \bigg \{ \theta _ { 1 } c _ { 6 } \ , ( 1 + x ) \Big ( x ^ 2 ( \epsilon _ n + \delta _ n ) + x \delta _ n \sqrt { | \ln \delta _ n | } \Big ) \bigg \} , \end{align*}


\begin{align*} \frac { d } { d t } \int | B | \ , d x = \frac { d } { d t } \int B \ , d x = \int B _ x J \ , d x - \mu \int \Lambda ^ \alpha B \ , d x \end{align*}


\begin{align*} \mu f ( x _ 1 , \ldots , x _ k , x _ { k + 1 } ) \ = \ x _ { k + 1 } \end{align*}


\begin{align*} ( n + 1 ) \widetilde { L } _ { n + 1 } + ( \alpha - 1 - x - 2 n ) \widetilde { L } _ n + ( n - \alpha ) \widetilde { L } _ { n - 1 } = 0 , \end{align*}


\begin{align*} L ( \rho , \phi , m , v , \xi ) : = & \frac { 1 } { 2 \tau } \sum _ { i = 1 } ^ s \int _ { \Omega } F ( \rho _ i , m _ i ) D ^ { - 1 } _ i d x + E ( \rho , \phi ) + \int _ { \partial \Omega } \xi ( \alpha \phi + \beta \partial _ n \phi - \phi ^ b ) d s \\ & + \sum _ { i = 1 } ^ s \int _ { \Omega } v _ i ( \rho _ i - \rho _ i ^ n + \nabla \cdot m _ i ) d x + \int _ { \Omega } v _ { s + 1 } ( f + \sum _ { i = 1 } ^ s z _ i \rho _ i + \nabla \cdot ( \epsilon ( x ) \nabla \phi ) ) d x . \end{align*}


\begin{align*} \sin ( c _ k + i d _ k ) = O ( k ^ { - 1 } ) , . \end{align*}


\begin{align*} P _ { i J ( i ) } : = \bigl \{ u _ i \in P _ i \bigm | u _ { i j } = a _ { i j } j \leq \min ( J _ i ) , \ u _ { i j } = u _ { i n } j \geq \max ( J _ i ) \bigr \} . \end{align*}


\begin{align*} \Xi _ \pm ^ { n } = \left \{ ( s _ 1 , \ldots , s _ n ; \xi _ 1 , \ldots , \xi _ n ) \in T ^ { * } \Sigma ^ n \ , | \ , \pm \xi _ i \ge 0 , 0 \leq i \leq 1 \right \} , \end{align*}


\begin{align*} \lim _ { \alpha \to 0 ^ + } \frac { \S ( u + \alpha h ) ( t ) - \S ( u ) ( t ) } { \alpha } = \S ' ( u ; h ) ( t ) \forall t \in [ 0 , T ] . \end{align*}


\begin{align*} \Delta ( \delta _ { \varphi } ) = \Delta ( 1 ) ( \delta _ { \varphi } \otimes \delta _ { \varphi } ) \Delta ( 1 ) , S ( \delta _ { \varphi } ) = \delta _ { \varphi } ^ { - 1 } , \varepsilon ( \delta _ { \varphi } a ) = \varepsilon ( a \delta _ { \varphi } ) = \varepsilon ( a ) , \forall a \in A , \end{align*}


\begin{align*} u _ t + ( u \cdot \nabla ) u - ( B \cdot \nabla ) B + \nabla P = & \ \nu \Delta u , \\ B _ t + ( u \cdot \nabla ) B - ( B \cdot \nabla ) u + \nabla \times ( ( \nabla \times B ) \times B ) = & \ \mu \Delta B , \\ \nabla \cdot u = 0 , \ \ \nabla \cdot B = & \ 0 , \end{align*}


\begin{align*} \tilde B ^ { k + 1 } _ t + B ^ k \tilde J ^ { k + 1 } _ x + \tilde B ^ k J ^ k _ x - J ^ k \tilde B ^ { k + 1 } _ x - \tilde J ^ k B _ x ^ k + \mu \Lambda ^ \alpha \tilde B ^ { k + 1 } = 0 \end{align*}


\begin{align*} v ( z ) = - \sum _ { n = - 1 } ^ \infty v _ n z ^ { n + 1 } . \end{align*}


\begin{align*} \widetilde { f } = \frac { 1 } { \langle \widetilde { g } , g \rangle } \iint _ { \R ^ { 2 d } } F ( x , \omega ) M _ \omega T _ x \widetilde { g } \ , d ( x , \omega ) . \end{align*}


\begin{align*} f _ 2 = \left ( \tfrac { c _ 0 } { 2 c _ 1 } \right ) \cdot \frac { { \bar f _ 2 } + f _ 1 ^ 2 } { X ^ { p ^ s - 2 p ^ { s - 1 } } } . \end{align*}


\begin{align*} \big \{ ( x , y ) \in \R ^ 2 \ y = e ^ x \big \} \end{align*}


\begin{align*} \chi _ \eta ( k ) = \left ( \frac { k - \eta } { k + \eta } \right ) ^ { \frac { 1 } { 4 } } = \left ( \frac { k - \sqrt { - 2 \xi } } { k + \sqrt { - 2 \xi } } \right ) ^ { \frac { 1 } { 4 } } . \end{align*}


\begin{align*} & \theta _ { i j } ( \rho ) [ v _ { i j } - ( \lambda _ i - \lambda _ j ) ( 1 + \dot W ^ { \delta } ) ] = 0 , \ ; \forall ( i , j ) \in E , \\ & \ < \dot \lambda , \rho \ > + \frac 1 4 \sum _ { i j } v _ { i j } ^ 2 \theta _ { i j } ( \rho ) + \frac 1 2 \sum _ { i j } v _ { i j } ( \lambda _ j - \lambda _ i ) \theta _ { i j } ( \rho ) \\ & + \frac 1 2 \sum _ { i j } v _ { i j } ( \lambda _ j - \lambda _ i ) \theta _ { i j } ( \rho ) d W ^ { \delta } ( t ) = 0 , \ ; \mathcal L ^ 1 \ ; \end{align*}


\begin{align*} V ( h ) & = p \left ( | \cdot | ^ { - ( N - \gamma ) } * ( Q ^ { p - 1 } \Re h ) \right ) Q ^ { p - 1 } \\ & + \left ( | \cdot | ^ { - ( N - \gamma ) } * Q ^ p \right ) \left ( \left ( \frac { p } { 2 } - 1 \right ) Q ^ { p - 2 } \bar { h } + \frac { p } { 2 } Q ^ { p - 2 } h \right ) \end{align*}


\begin{align*} \mathcal { E } \left ( G _ { 0 ^ + } ( u ) , v \right ) = \langle u ; v \rangle _ H . \end{align*}


\begin{align*} \begin{cases} \ f ( x ) = s _ x \\ [ 3 p t ] \ g ( x ) = s _ x + 2 ^ { - x } \end{cases} . \end{align*}


\begin{align*} \exp ( \alpha + \beta ) & = \sum \frac { ( \alpha + \beta ) ^ n } { n ! } = \sum _ { n = 0 } ^ \infty \sum _ { k = 0 } ^ n \binom { n } { k } \frac { \alpha ^ k \beta ^ { n - k } } { n ! } \\ & = \sum _ { n = 0 } ^ \infty \sum _ { k = 0 } ^ n \frac { \alpha ^ k \beta ^ { n - k } } { k ! ( n - k ) ! } = \sum \frac { \alpha ^ n } { n ! } \cdot \sum \frac { \beta ^ n } { n ! } = \exp ( \alpha ) \exp ( \beta ) . \end{align*}


\begin{align*} \norm { ( I - B ^ { - 1 } S ) } _ { o p } = \sup _ { \norm { f } _ \mathcal { H } = 1 } | \langle ( I - B ^ { - 1 } S ) f , f \rangle | \leq \frac { B - A } { B } = 1 - \frac { A } { B } < 1 . \end{align*}


\begin{align*} V ( \R ^ d | \R ^ d ) = \left \lbrace v \in L ^ 2 ( \R ^ d ) : ( v ( x ) - v ( y ) ) K _ s ^ { 1 / 2 } ( x , y ) \in L ^ 2 ( \R ^ d \times \R ^ d ) \right \rbrace \end{align*}


\begin{align*} \omega _ 1 \colon & t _ i \leftrightarrow s _ i , \\ \omega _ 2 \colon & \begin{cases} s _ i \mapsto s _ i , t _ i \mapsto t _ i , & ( 1 \leq i \leq n - 1 ) \\ t _ n \mapsto - t _ n , s _ n \mapsto - s _ n . & \end{cases} \end{align*}


\begin{align*} F ( x ) = \begin{pmatrix} f _ { 1 1 } ( x ) & \cdots & f _ { 1 r _ 2 } ( x ) \\ \vdots & \ddots & \vdots \\ f _ { r _ 1 1 } ( x ) & \cdots & f _ { r _ 1 r _ 2 } ( x ) \end{pmatrix} G ( x ) = \begin{pmatrix} g _ { 1 1 } ( x ) & \cdots & g _ { 1 q _ 2 } ( x ) \\ \vdots & \ddots & \vdots \\ g _ { q _ 1 1 } ( x ) & \cdots & g _ { q _ 1 q _ 2 } ( x ) \end{pmatrix} , \end{align*}


\begin{align*} \underline { b } : = \inf _ { x \in [ 0 , 1 ] } b ( x ) \mbox { a n d } \bar { b } : = \sup _ { x \in [ 0 , 1 ] } b ( x ) , \end{align*}


\begin{align*} \left | \frac { z - \beta _ { k } } { \beta _ 1 - \beta _ { k } } \right | \le \frac { 1 } { M } + 2 = \frac { 2 M + 1 } { M } \mbox { f o r a l l $ | z | = 1 $ } . \end{align*}


\begin{align*} \big | I _ 0 ( t , K ) \big | = \Big | \ , \mathrm { p . v . } \int _ \R \big [ H ( q ; t ) - H ( K ; t ) \big ] \frac { \psi ( K - q ) } { K - q } \chi _ 1 \big ( ( K - q ) t ^ 3 \big ) \ , d q \Big | \\ \lesssim t \ , \int _ \R \frac { 1 } { | K - q | ^ { 1 / 2 } } \chi _ 1 \big ( ( K - q ) t ^ 3 \big ) \ , d q \lesssim | t | ^ { - 1 / 2 } . \end{align*}


\begin{align*} F _ { u } ( z , \chi ) \ll \prod _ { p \mid u } ( 1 + a ( p ^ l ) p ^ { - 1 / 2 } ) \ll d ( u ) , \ \Re z \geq 1 / 2 , \ \ ( u = \prod _ { p } p ^ l ) . \end{align*}


\begin{align*} \begin{aligned} & \ \ O ( p { \rm l o g } \ , m + p { \rm l o g } \ , n + p ^ 3 + n p l + n d ^ 2 + d ^ 3 ) \\ & = O \left ( \frac { k ^ { 8 } } { \epsilon ^ { 4 } \delta } { \rm l o g } \ , m + \frac { k ^ { 1 6 } } { \epsilon ^ { 8 } \delta ^ 2 } { \rm l o g } \ , n + \frac { k ^ { 2 4 } } { \epsilon ^ { 1 2 } \delta ^ 3 } + \frac { k ^ { 8 } } { \epsilon ^ { 4 } \delta } n l + n d ^ 2 + d ^ 3 \right ) . \end{aligned} \end{align*}


\begin{align*} f _ { \varphi } ( x _ 0 , \cdots , x _ k ) & : = \int _ { G ^ { \times ( k + 1 ) } } c _ X ( g _ 0 ^ { - 1 } x _ 0 ) \cdots c _ X ( g _ k ^ { - 1 } x _ k ) \varphi ( g _ 0 , g _ 1 , \cdots , g _ k ) { \rm d } g _ 0 \cdots { \rm d } g _ k , \\ \Phi ( \varphi ) & : = ( d _ { x _ 1 } d _ { x _ 2 } \cdots d _ { x _ k } f _ { \varphi } ) ( x , \cdots , x ) . \end{align*}


\begin{align*} L ( x , y , t ) = \frac { x y \sqrt { y ^ 2 - x ^ 2 } } { \sqrt { y ^ 2 - x ^ 2 } \cosh { ( t \sqrt { y ^ 2 - x } ) } - y \sinh { t \sqrt { y ^ 2 - x ^ 2 } } } . \end{align*}


\begin{align*} \tilde { L } ( q , v ) = \frac { 1 } { 2 } \mathcal { G } ( v , v ) = \frac { 1 } { 2 } g _ { i j } ( q ) v ^ i v ^ j . \end{align*}


\begin{align*} \mu ( \phi ( x , b ) ) = \frac { \mu ( Y ( b ) ) } { | H | } . \end{align*}


\begin{align*} \frac { 1 } { t } d _ K ( S _ t x + t v ) & = \frac { 1 } { t } d _ K \bigg ( x + t ( A x + v ) - t \bigg ( A x - \frac { S _ t x - x } { t } \bigg ) \bigg ) \\ & \leq \frac { 1 } { t } d _ K \big ( x + t ( A x + v ) \big ) + \bigg \| A x - \frac { S _ t x - x } { t } \bigg \| , \end{align*}


\begin{align*} \tau _ E ( m ) : = \sum _ { m _ 1 \cdots m _ j = m } m _ 1 ^ { - \xi _ 1 } \cdots m _ j ^ { - \xi _ j } , \end{align*}


\begin{align*} \xi ( t ) = \left \{ \begin{aligned} & b , & t \leq t _ s , \\ & b / ( t - t _ s + 1 ) , & t > t _ s . \end{aligned} \right . \end{align*}


\begin{align*} \mathbf { I } _ \zeta ( z ) = \frac { \left ( \frac { 1 } { 2 } z \right ) ^ \zeta } { \Gamma ( \zeta + \frac { 1 } { 2 } ) \Gamma ( \frac { 1 } { 2 } ) } \int _ 0 ^ \pi \cosh \left ( z \cos \theta \right ) \sin ^ { 2 \zeta } ( \theta ) d \theta , \end{align*}


\begin{align*} \langle f , S ^ { - 1 } f \rangle = \sum _ { \gamma \in \Gamma } c _ \gamma \langle e _ \gamma , S ^ { - 1 } f \rangle = \sum _ { \gamma \in \Gamma } c _ \gamma \overline { a _ \gamma } = \langle c , a \rangle _ { \ell ^ 2 } \end{align*}


\begin{align*} \left | \int _ { \mathbb { R } ^ { d - 1 } } \Psi _ { a , \zeta , \mu , \sigma } \tilde { \Psi } _ { a , \zeta , \mu , \sigma } \ , d x - 1 \right | & < \zeta , \\ \left | \int _ { \mathbb { R } ^ { d - 1 } } \beta ( | a | ^ \frac { 1 } { p } \Psi _ { a , \zeta , \mu , \sigma } ) \tilde { \Psi } _ { a , \zeta , \mu , \sigma } \ , d x - \sigma | a | ^ \frac { 1 } { p } \right | & < \zeta \\ \int _ { \mathbb { R } ^ { d - 1 } } \tilde { \Psi } _ { a , \zeta , \mu , \sigma } \ , d x & = 0 , \end{align*}


\begin{align*} \overline { V _ 1 ^ c } \cap \overline { V _ 2 ^ c } = ( [ 0 , 0 . 2 5 ] + \Z ) \cap ( [ 0 . 5 , 0 . 7 5 ] + \Z ) = \emptyset \end{align*}


\begin{align*} \dot { \varrho } _ { m a j _ { \alpha } ( \pi ) } ( \pi ^ { \prime } ) = \begin{cases} m a j _ { \alpha } ( \pi ) & \pi _ { \mu } ^ { \circ } \neq 1 1 \leq \alpha \leq \mu - 2 \\ 0 & \alpha = \mu - 1 \pi _ { \mu } ^ { \circ } = 1 . \end{cases} \end{align*}


\begin{align*} f ( t ) = \int _ \R \widehat { f } ( \omega ) e ^ { 2 \pi i t \omega } \ , d \omega , \end{align*}


\begin{align*} \begin{aligned} & \lim _ { n \to \infty } k \sum _ { l = 1 } ^ k \binom { c _ n k n } l ( k ) _ l \sum _ { j = n - N + 1 } ^ { n - 1 } \frac { ( k ( j - 1 ) ) _ { c _ n k n - l } } { ( k n ) _ { c _ n k n } } \frac 1 { k ( n - j + 1 ) - l } = \\ & k \sum _ { l = 1 } ^ k \binom k l ( \frac c { 1 - c } ) ^ l \sum _ { s = 1 } ^ { N - 1 } ( 1 - c ) ^ { k ( s + 1 ) } \frac 1 { k ( s + 1 ) - l } . \end{aligned} \end{align*}


\begin{align*} ( E _ r ^ { s , t } ) _ \lambda = 0 . \end{align*}


\begin{align*} [ \mathfrak { X } , T D ( \mathfrak { X } ) ( u ) - [ \mathfrak { X } , T ( u ) ] ] = 0 . \end{align*}


\begin{align*} y = \sum _ { i = 0 } ^ { 3 j + 1 } \beta _ { i } e _ { i } , \quad ~ \beta _ { 3 j } , \beta _ { 3 j + 1 } \neq 0 . \end{align*}


\begin{align*} \omega _ { p } ^ { p } ( A ) & = \underset { \theta \in \mathbb { R } } { \sup } \norm { R e ( e ^ { i \theta } A ) } _ { p } ^ { p } \\ & \leq \frac { 1 } { 2 ^ { p - 2 } } \underset { i = j } { \sum } \omega _ { p } ^ { p } ( a _ { i j } ) \end{align*}


\begin{align*} \dot { \mathfrak { p } } _ 1 = ( \partial _ t \epsilon , R _ 1 ) - \dot { \mathfrak { q } _ 1 } ( \epsilon , \partial _ y R _ 1 ) + \frac { 1 } { 2 } \int \partial _ t \psi _ 1 \epsilon ^ 2 + \int \psi _ 1 \partial _ t \epsilon \epsilon . \end{align*}


\begin{align*} \mathcal K = \left ( \begin{array} { c c } H & \sigma \\ \tau & K \end{array} \right ) , \end{align*}


\begin{align*} I ( \hat { \alpha } \cup { ( \gamma , N ) \cup { ( \delta _ { 1 } , 1 ) } } , \hat { \alpha } \cup { ( \gamma , N - p _ { i } ) } \cup { ( \delta _ { 2 } , 1 ) } ) = 2 . \end{align*}


\begin{align*} \alpha = \sum _ { k _ 1 , \ldots , k _ n \ge 0 } c _ { k _ 1 , \ldots , k _ n } \beta _ 1 ^ { k _ 1 } \ldots \beta _ n ^ { k _ n } \end{align*}


\begin{align*} ( s - t ) ^ 2 - \frac { s - t } { h ( s ) } + | \lambda | ^ 2 = 0 . \end{align*}


\begin{align*} \mathcal { L } ( h , k ) : = \frac { 1 } { 2 } \sum _ { \substack { 1 \leq q < \infty \\ ( q , h k ) = 1 } } W \left ( \frac { q } { Q } \right ) \sum _ { \substack { 1 \leq m , n < \infty \\ ( m n , q ) = 1 } } \frac { \tau _ A ( m ) \tau _ B ( n ) } { \sqrt { m n } } V \left ( \frac { m } { X } \right ) V \left ( \frac { n } { X } \right ) \sum _ { \substack { c > C , d \geq 1 \\ c d = q \\ d | m h \pm n k } } \phi ( d ) \mu ( c ) , \end{align*}


\begin{align*} Z [ u ] = \frac { \| u \| _ 2 ^ { ( N + \gamma ) - ( N - 2 ) p } \| \nabla u \| _ 2 ^ { N p - ( N + \gamma ) } } { \int \left ( | \cdot | ^ { - ( N - \gamma ) } * | u | ^ p \right ) | u | ^ p d x } . \end{align*}


\begin{align*} \widetilde { U } ( t ) = \widetilde { U } ^ p ( t ) + \widetilde { U } ^ h ( t ) = U ( t ) - U ( t - 1 ) . \end{align*}


\begin{align*} \phi _ { \varepsilon } = C _ { \varepsilon } \ , \tilde u _ { \varepsilon } + \hat { \phi } _ { \varepsilon } \end{align*}


\begin{align*} \int _ { \Omega _ { \varepsilon _ { n _ k } } } | 2 w - e ^ { - \beta _ { \varepsilon _ { n _ k } } \Psi _ { \varepsilon _ { n _ k } } ( \mathbf { q } ) } V _ { \varepsilon _ { n _ k } } | | V _ { \varepsilon _ { n _ k } } | = o \left ( e ^ { D \sigma _ 1 \beta _ { \varepsilon _ { n _ k } } \Psi _ { \varepsilon _ { n _ k } } ( \mathbf { q } ) } \right ) \end{align*}


\begin{align*} \int e ^ { - \frac { z ^ 2 } { 4 } } ( \sqrt { r } y - z \sqrt { 1 - r } ) ^ { \omega + 3 } d z = - \frac { 1 } { ( \omega + 4 ) \sqrt { 1 - r } } \int \frac { z } { 2 } e ^ { - \frac { z ^ 2 } { 4 } } ( \sqrt { r } y - z \sqrt { 1 - r } ) ^ { \omega + 4 } d z \end{align*}


\begin{align*} w _ F = \frac { 1 } { 2 } \Big { ( } \nabla ^ F ( \vec H _ F ) + \ , \langle ( A _ F ) , \vec H _ { F } \rangle _ { g _ { \textnormal { e u c } } } ^ { \sharp _ { g _ F } } + \ , \langle ( A _ F ^ 0 ) , \vec H _ { F } \rangle _ { g _ { \textnormal { e u c } } } ^ { \sharp _ { g _ F } } \Big { ) } , \end{align*}


\begin{align*} \int _ \mathbb { R } [ ( D _ { \rm l e f t } ) ^ { \alpha } u ( x ) ] v ( x ) \ , d x = \int _ \mathbb { R } u ( x ) [ ( D _ { \rm r i g h t } ) ^ { \alpha } v ( x ) ] \ , d x . \end{align*}


\begin{align*} 1 = \mathrm { i n d } ( v _ { + } ^ { \infty } ) = \mu _ { \mathrm { d i s c } } ( \gamma _ { 2 } ) - 6 = n - 4 \end{align*}


\begin{align*} \sum _ { k = 1 } ^ { p + 1 } S _ { 2 , \lambda } ( p + 1 , k ) \frac { x ^ { k } } { k } = e ^ { - x } \sum _ { n = 0 } ^ { \infty } \big ( ( 1 - \lambda ) _ { p , \lambda } + ( 2 - \lambda ) _ { p , \lambda } + \cdots + ( n - \lambda ) _ { p , \lambda } \big ) \frac { x ^ { n } } { n ! } . \end{align*}


\begin{align*} \delta ( D ) : = \liminf _ { n \to \infty } \frac { | D \cap [ - n , n ] | } { 2 n + 1 } . \end{align*}


\begin{align*} \Omega _ { M _ c \star } = \Omega _ { \mathcal { C } ^ c \star } / \Omega _ { \mathcal { C } \mathcal { C } ^ c \star } = \{ [ \omega ] = \omega + \Omega _ { \mathcal { C } \mathcal { C } ^ c \star } ~ | ~ \omega \in \Omega _ { \mathcal { C } ^ c \star } \} . \end{align*}


\begin{align*} \partial _ { x _ { j ; \alpha } } z _ { i ; \alpha } ^ { ( \ell ) } , j = 0 , \ldots , n _ 0 , \end{align*}


\begin{align*} \partial _ t U _ t F ( \varphi ) = \Delta U _ t F ( \varphi ) - \left ( A _ { \epsilon } \varphi , \nabla U _ t F ( \varphi ) \right ) . \end{align*}


\begin{align*} \varphi : \Z & \to \R \\ z & \mapsto \sum _ { n = 1 } ^ { | z | } \frac 1 n . \end{align*}


\begin{align*} c ( w ) = \min \left ( \left \{ k \in \omega : { h } ^ { k + 1 } ( { t } _ { w } ) = \ ; \uparrow \right \} \right ) . \end{align*}


\begin{align*} \norm { J ^ A _ \gamma x - J ^ A _ { \gamma ' } x } & \leq \norm { J ^ A _ \gamma x - J ^ A _ { \gamma ' } x + \vert \gamma \vert ( \gamma ^ { - 1 } ( x - J ^ A _ \gamma x ) - \gamma '^ { - 1 } ( x - J ^ A _ { \gamma ' } x ) ) } \\ & = \norm { \left ( 1 - \frac { \gamma } { \gamma ' } \right ) x + \left ( \frac { \gamma } { \gamma ' } - 1 \right ) J ^ A _ { \gamma ' } x } \\ & = \left \vert 1 - \frac { \gamma } { \gamma ' } \right \vert \norm { x - J ^ A _ { \gamma ' } x } \end{align*}


\begin{align*} H ^ i ( \mathcal { Y } _ K , \mathcal { A } _ { \mathcal { Y } _ K } ^ { \otimes m } ( - \mathcal { E } _ K ) ) = 0 \end{align*}


\begin{align*} c \otimes a = \sum _ i ( p _ i \otimes 1 ) \Delta ( q _ i ) , d \otimes b = \sum _ j \Delta ( z _ j ) ( w _ j \otimes 1 ) . \end{align*}


\begin{align*} \langle \varepsilon , \partial _ \tau \phi _ i \rangle _ { L ^ 2 _ \rho } = \sum _ { j = 0 } ^ { M } \varepsilon _ j \langle \phi _ j , \partial _ \tau \phi _ i \rangle _ { L ^ 2 _ \rho } + \langle \varepsilon _ { - } , \partial _ \tau \phi _ i \rangle _ { L ^ 2 _ \rho } \end{align*}


\begin{align*} \tilde { \phi } _ { x x } ( x ) & = \int _ { \R } \phi _ { x x } ( t , x ) \psi ( t ) \ , d t \\ & = \int _ { \R } \left ( - | \phi _ x | ^ 2 \phi + \phi _ { t t } + | \phi _ t | ^ 2 \phi + a ( x ) \phi _ t \right ) ( t , x ) \psi ( t ) \ , d t \\ & = \int _ { \R } - | \phi _ x | ^ 2 \phi \psi ( t ) - \phi _ { t } \psi _ t ( t ) + ( | \phi _ t | ^ 2 \phi + a ( x ) \phi _ t ) \psi ( t ) \ , d t . \end{align*}


\begin{align*} - \phi _ { u v } = ( \phi _ u \cdot \phi _ v ) \phi + \frac { 1 } { 4 } a \phi _ t = : F ( t , x ) = F ( u , v ) . \end{align*}


\begin{align*} T ^ T T = A ^ { - 1 } . \end{align*}


\begin{align*} & Z _ { \lambda } = \Theta _ { \lambda } + \frac { 1 } { 2 } P _ { \sigma \nu } \omega ^ { \sigma } \wedge \omega ^ { \nu } + Q _ { \sigma , \nu } ^ { j } \omega ^ { \sigma } \wedge \omega _ { j } ^ { \nu } + \frac { 1 } { 2 } R _ { \sigma , \nu } ^ { i , j } \omega _ { i } ^ { \sigma } \wedge \omega _ { j } ^ { \nu } , \end{align*}


\begin{align*} c _ { 1 } : = \mathrm { s i g n } \left [ ( i - j ) ( \alpha - \beta ) ( i - \beta ) ( \alpha - j ) \right ] , \ , c _ { 2 } : = \mathrm { s i g n } \left [ ( i - j ) ( \alpha - \beta ) ( i - \alpha ) ( j - \beta ) \right ] . \end{align*}


\begin{align*} d f _ p = \lambda g _ { v } ( v , \cdot ) = g _ { \lambda v } ( \lambda v , \cdot ) \end{align*}


\begin{align*} \mathcal { F } _ { k } ^ j = \sigma ( \zeta _ { j - k } , x _ { j - k + 1 } , W _ { j - k + 1 } , \ldots , W _ { j - 1 } , \zeta _ { j - 1 } , x _ j ) , \end{align*}


\begin{align*} i \partial _ t h + \Delta h - h + V ( h ) + R ( h ) = 0 , \end{align*}


\begin{align*} h ( \beta \otimes \alpha ) p ( x , w ) = p ( \beta x \alpha , w ) = p ( x \beta \alpha , w ) = p ( x \alpha \beta , w ) . \end{align*}


\begin{align*} T M = Q \oplus \Sigma = Q \oplus \langle U \rangle \oplus \langle V \rangle , \end{align*}


\begin{align*} \bar { R } _ { \nu j l i } = - \left ( \frac { \lambda '' } { \lambda } + \frac { c - \lambda '^ { 2 } } { \lambda ^ { 2 } } \right ) ( \delta _ { i j } r _ { l } - \delta _ { j l } r _ { i } ) r _ { \nu } , \end{align*}


\begin{align*} ( \sigma , R ) ( \tau , S ) = ( \sigma \tau , R ^ { \tau } S ) \end{align*}


\begin{align*} \check x _ { t , x , v } : = x - \check { \mathcal T } _ { t , x , v } v . \end{align*}


\begin{align*} \left \{ R _ j = a \right \} = \left \{ \varsigma _ { Q _ { j } , m } = 1 \right \} \bigcap _ { i \in [ a ] } \left \{ \varsigma _ { i + Q _ { j - 1 } , m } = 0 \right \} . \end{align*}


\begin{align*} r _ k = \begin{cases} \frac { k \theta + 1 } { 2 } , & \mbox { i f } \ \ k = 1 , . . . , 4 \\ 5 / 2 + \frac { a \theta } { 2 } , & \mbox { i f } \ \ k = 5 \\ 5 / 2 + a \theta , & \mbox { i f } \ \ k = 6 , \end{cases} \end{align*}


\begin{align*} \sum _ { r \in \mathbb { Z } } c ^ F _ { - r } \big ( r _ l ( \pi _ F ) ^ { ( l ) } ( w _ n ) W , \sigma _ F ^ { ( l ) } ( w _ { n - 1 } ) W ' \big ) q _ F ^ { - \frac { r } { 2 } } X ^ { - r } = \gamma \big ( X , r _ l ( \pi _ F ) ^ { ( l ) } , \sigma _ F ^ { ( l ) } , \overline { \psi } _ F ^ l \big ) \sum _ { r \in \mathbb { Z } } c ^ F _ r ( W , W ' ) q _ F ^ { \frac { r } { 2 } } X ^ { r } . \end{align*}


\begin{align*} 2 ^ { ( \mu _ * ( 1 ‐ \tilde \lambda ) ‐ \alpha _ * \tilde \lambda ) k _ * t } \cdot n ^ { ‐ \frac 1 2 + \frac 1 q } \stackrel { \eqref { t i l d e _ n u _ d e f } } { = } n ^ { ‐ \tilde \nu } . \end{align*}


\begin{align*} \langle f ( t ) , g ( s ) \rangle : = - \mathrm { R e s } _ t ( f ( t ) , g ( t ) ) , \end{align*}


\begin{align*} ( ( f , g ) ) _ k : = ( f , g ) _ { L ^ 2 _ k } + \eta \int _ 0 ^ { + \infty } ( \mathcal { S } _ L ( \tau ) f , \mathcal { S } _ L ( \tau ) g ) _ { L ^ 2 _ v } d \tau . \end{align*}


\begin{align*} \int _ 0 ^ { \infty } e ^ { - s } \left ( C _ p \left | \dot { \tilde { \xi } } ( s ) \right | ^ q + f \left ( \tilde { \xi } ( s ) \right ) \right ) d s = f ( \xi ( 0 ) ) = u ( \xi ( 0 ) ) = u ( x ) . \end{align*}


\begin{align*} \alpha _ { n + 2 } ( x ) = \frac { - \varepsilon } { A ^ n _ 1 } D \alpha _ 0 ( x ) = \frac { - \varepsilon } { A ^ n _ 1 } \sum _ { j = 2 } ^ \infty j ( j - 1 ) a _ j x ^ { j - ( m + 2 ) } , \end{align*}


\begin{align*} b ( t , h ) : = a ( h ) + \sum _ { j = 1 } ^ r \sigma ^ j ( h ) \dot { B } _ m ^ j ( t ) . \end{align*}


\begin{align*} - \frac { N - 2 } { 2 } \| \nabla Q \| _ 2 ^ 2 - \frac { N } { 2 } \| Q \| _ 2 ^ 2 + \frac { N + \gamma } { 2 p } \int \left ( | \cdot | ^ { - ( N - \gamma ) } * Q ^ p \right ) Q ^ { p } d x = 0 . \end{align*}


\begin{align*} \mathrm { s d } ( u ) = \inf \left \{ \delta \in \mathbb { R } \ , | \ , \lim _ { \lambda \rightarrow 0 } \lambda ^ { \delta } u ( \lambda x ) = 0 \right \} . \end{align*}


\begin{align*} \mathbf { C } _ i ( \psi ) ( x , t ) : = \int _ D \psi ( x , q , t ) U ' _ i ( \frac { 1 } { 2 } | q _ i | ^ 2 ) q _ i q _ i ^ T \ ; \mathrm { d } q , \ ; \ ; i = 1 , \ldots , K . \end{align*}


\begin{align*} f ( t , x , v ) = - \int _ 0 ^ t E ( s , X _ { s , t } ( x , v ) ) \cdot \nabla _ v \mu ( V _ { s , t } ( x , v ) ) - \int _ 0 ^ t e _ 0 \nabla \Phi ( X _ { s , t } ( x , v ) - X ( s ) ) \cdot \nabla _ v \mu ( V _ { s , t } ( x , v ) ) . \end{align*}


\begin{align*} \sum _ { \ell = 0 } ^ { k _ c } \phi _ { t _ { r , i } + \ell } \phi ^ \top _ { t _ { r , i } + \ell } \ge C _ d I _ p . \end{align*}


\begin{align*} \Psi _ k ( \lambda ) = \sum _ { i = 1 } ^ k \ln \mathbf { E } \big ( e ^ { \lambda \xi _ i } \big | \mathcal { F } _ { i - 1 } \big ) . \end{align*}


\begin{align*} L ' _ { v d } \wedge L ' _ { e } = L ' _ { i v d } . \end{align*}


\begin{align*} ( \gamma \cdot f ) ( x ) & : = \langle \frac { d } { d t } \Big | _ { t = 0 } \gamma ( \exp ^ g _ y ( t \cdot f ( y ) \nu ( y ) ) ) , \nu ( \gamma ( y ) ) \rangle _ g \Big | _ { y = \gamma ^ { - 1 } ( x ) } \\ & = f ( \gamma ^ { - 1 } ( x ) ) \cdot \langle \gamma _ * \nu ( \gamma ^ { - 1 } ( x ) ) , \nu ( x ) \rangle _ g , \end{align*}


\begin{align*} g _ j ( y ) = a ( y ) \Phi _ { A _ j } ( y ) , \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 _ 1 ( x ) } \overleftarrow { \eta } ^ { \ell } ( x ) [ \eta ( x + 1 ) - \eta ( x + 1 + r ) ] [ f ( \eta ) - f ( \eta _ { 1 , x , r } ) ] d \nu _ { b } \Big | , \end{align*}


\begin{align*} T \log \left ( \frac { a ^ 2 + T ^ 2 } { 4 } \right ) = 2 T \log \left ( \frac { T } { 2 } \right ) + \mathcal { O } \left ( \frac { 1 } { T } \right ) \end{align*}


\begin{align*} ( w - v ) ( x ) & = \begin{cases} 0 , & x \in \{ v > 0 \} \cap \R ^ n _ + \\ u ( x ) - ( 1 - \theta / 2 ) ^ { - n - 2 } u ( a ) \zeta ( x ) , & x \in \{ v \leqslant 0 \} \cap \R ^ n _ + . \end{cases} \end{align*}


\begin{align*} \lim _ { | x | , | \omega | \to 0 } \norm { D ^ \alpha X ^ \beta ( M _ \omega T _ x g - g ) } _ \infty = 0 , \end{align*}


\begin{align*} Y _ { \alpha \beta } ^ { i j } = - Y _ { \alpha \omega } ^ { i j } \cdot Y _ { \omega \beta } ^ { i j } = - Y _ { \alpha \omega } ^ { i m } \cdot Y _ { \omega \delta } ^ { m j } \cdot Y _ { \omega \beta } ^ { i m } \cdot Y _ { \omega \beta } ^ { i m } . \end{align*}


\begin{align*} ( n + 1 ) Q ( x ) - ( 1 - x ) Q ' ( x ) = \mu _ { n , m } x ^ m . \end{align*}


\begin{align*} \pi ( M _ { p } ) = \begin{cases} M _ { y } , & y \in Q \ \ p = g ( y ) , \\ 0 , & o t h e r w i s e . \end{cases} \end{align*}


\begin{align*} \sum _ { n = 0 } ^ \infty A _ n ( x , y ) { t ^ n \over n ! } = { y - x \over 1 - x y ^ { - 1 } e ^ { ( y - x ) t } } . \end{align*}


\begin{align*} K _ 1 \cap \dotsb \cap K _ j \cap \dotsb \cap K _ t = G _ m \cap G _ { s ( K _ 1 ) } \cap \dotsb \cap G _ { s ( K _ t ) } = G _ { s ( K _ 1 ) } \cap \dotsb \cap G _ { s ( K _ t ) } \end{align*}


\begin{align*} \left ( \begin{array} { c } u _ 1 ( \bar { T } ) \\ u _ 2 ( \bar { T } ) \\ u _ 3 ( \bar { T } ) \end{array} \right ) = M ( 2 p - 1 ) \left ( \begin{array} { c } u _ 1 ( \bar { T } ) \\ u _ 2 ( \bar { T } ) \\ u _ 3 ( \bar { T } ) \end{array} \right ) , \end{align*}


\begin{align*} H _ k : C _ A = \Pi _ { k } A \in I _ d ( k ) , \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ { d } } K ( x ) \frac { f ( x , u ) } { \vert u \vert ^ { g ^ + - 1 } } \vert u \vert ^ { g ^ + - 1 } u d x & = \int _ { \mathbb { R } ^ { d } } \int _ { \mathbb { R } ^ { d } } g \left ( \frac { u ( x ) - u ( y ) } { \vert x - y \vert ^ { \alpha } } \right ) \frac { u ( x ) - u ( y ) } { \vert x - y \vert ^ { \alpha + d } } d x d y \\ & + \int _ { \mathbb { R } ^ { d } } g ( u ) u d x . \end{align*}


\begin{align*} - u _ { x x } + u _ x + \rho _ x + \lambda u = g , \end{align*}


\begin{align*} \frac { N ^ K _ j ( t \log K ) } { N ^ K _ { i } ( t \log K ) } = \exp ( \log K ( \beta _ { j } ( t ) - \beta _ { i } ( t ) ) \le e ^ { L \ , \rho ( j \delta _ K , i \delta _ K ) \ , \log K } . \end{align*}


\begin{align*} \delta = \frac { \int _ 0 ^ \infty e ^ { - \lambda \tau } P _ { s + \tau } f ( x ) d \tau } { \sup _ { t > 0 } P _ t f ( x ) } \end{align*}


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


\begin{align*} C \mathcal { H } ^ { \frac { \log 2 } { \log 3 } } _ { \infty } ( \Omega _ { k } ) \leq \mathcal { H } ^ { \frac { \log 2 } { \log 3 } } _ { \infty } ( K _ { 1 / 3 } ) \leq \mathcal { H } ^ { \frac { \log 2 } { \log 3 } } _ { \infty } \Big ( \bigcup _ { I \in \mathcal { K } _ k } I \Big ) = \mathcal { H } ^ { \frac { \log 2 } { \log 3 } } _ { \infty } ( \Omega _ { k } ) \le 1 , \end{align*}


\begin{align*} L f ( \eta ) = \sum _ { x , y \in \Z } p ( x , y ) \left [ f ( \eta ^ { x y } ) - f ( \eta ) \right ] , \end{align*}


\begin{align*} x \in \mathrm { r a n } ( I d + \gamma A ) & \exists y \left ( x \in y + \gamma A y \right ) \\ & \exists y , z \left ( z \in A y \land x = y + \gamma z \right ) \\ & \exists y , z \left ( z \in A y \land z = \frac { 1 } { \gamma } ( x - y ) \right ) . \end{align*}


\begin{align*} \log _ { q _ s } \abs { \psi ( t ) } _ s = \langle \psi , \omega ( t ) \rangle \end{align*}


\begin{align*} \lim _ { x \to 0 } \norm { T _ x f - f } _ 2 = 0 \end{align*}


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


\begin{align*} - \widetilde { P } _ h ( z ) ^ * : = - h P - i Q - ( - \bar { z } ) , \end{align*}


\begin{align*} ( n + 1 ) ^ { - \# \mathcal { I } } e ^ { n H ( \mathbf { p } ) } \leq \# \{ ( i _ 1 , \ldots , i _ n ) \in \mathcal { I } ^ n : \tau ( i _ 1 , \ldots , i _ n ) = \mathbf { p } \} \leq e ^ { n H ( \mathbf { p } ) } . \end{align*}


\begin{align*} | B | + N \ = \ k - s + N \ \leqslant \ ( N + 1 ) m - s \ = \ | A | - s \ = \ | F | . \end{align*}


\begin{align*} L ( s , \operatorname { s y m } ^ 2 f ) = & \zeta ( 2 s ) \sum _ { n \geq 1 } \frac { \lambda ( n ^ 2 ) } { n ^ s } = \prod _ { p } \left ( 1 - \frac { \lambda ( p ^ 2 ) } { p ^ s } + \frac { \lambda ( p ^ 2 ) } { p ^ { 2 s } } - \frac { 1 } { p ^ { 3 s } } \right ) ^ { - 1 } . \end{align*}


\begin{align*} A ^ \varepsilon = A ^ \varepsilon ( n ) = \{ ( t , \ , m ) : \ ; 0 \le t \le \hat t ( n ) , \ ; m \ge m ^ * _ t \} \end{align*}


\begin{align*} \sum _ { n = 0 } ^ { \infty } \frac { x ^ { n } } { n ! } z ^ { n } & = \sum _ { m = 0 } ^ { \infty } \frac { z ^ { m } } { m ! } A _ { m } ^ { \lambda , \beta } ( x ) \cdot \sum _ { k = 0 } ^ { \infty } \frac { z ^ { k } } { k ! } M _ { \lambda , \beta } ( n ) \\ & = \sum _ { n = 0 } ^ { \infty } \frac { z ^ { n } } { n ! } \left ( \sum _ { k = 0 } ^ { n } \binom { n } { k } A _ { k } ^ { \lambda , \beta } ( x ) M _ { \lambda , \beta } ( n - k ) \right ) . \end{align*}


\begin{align*} \ker \mathcal { L } = \{ i Q , \partial _ { x _ 1 } Q , . . . , \partial _ { x _ N } Q \} . \end{align*}


\begin{align*} \# \{ \{ z , z ' \} \mid z \in A , \ ; z ' \in \mathcal { C } ^ \vee ( Q ) - A , \ ; | z - z ' | = \delta \} \geq c _ 1 n ^ { - 1 } \# A \end{align*}


\begin{align*} D ( [ \C ^ 2 / G ] ) = \left \langle \phi ( D ( Y ) ) ^ \perp , \phi ( D ( Y ) ) \right \rangle . \end{align*}


\begin{align*} M _ n = \dfrac { S _ n } { a _ n } \to M _ { \infty } \mbox { a . s . a n d i n $ L ^ 2 $ a s $ n \to \infty $ . } \end{align*}


\begin{align*} i \partial _ t w + \Delta w - w + S ( \mathcal { V } _ { l ( k ) } ^ A + w ) - S ( \mathcal { V } _ { l ( k ) } ^ A ) = - \varepsilon _ { l ( k ) } = O ( e ^ { - ( l ( k ) + 1 ) e _ 0 t } ) , \end{align*}


\begin{align*} J _ L ( g , g \cdot \xi _ 1 , \dots , g \cdot \xi _ k ) = \big ( { \rm A d } ^ * _ { g ^ { - 1 } } \big ) ^ k \pmb { F } \ell ( \xi _ 1 , \dots , \xi _ k ) . \end{align*}


\begin{align*} & \partial _ t u ^ { \epsilon } \rightarrow \partial _ t u \ ; \ ; \ ; \ ; L ^ \infty ( 0 , T ; H ^ 1 ) , \\ & \frac { 1 } { \epsilon } P ' ( \rho ^ \epsilon ) \nabla \phi ^ \epsilon \rightarrow v \ ; \ ; \ ; \ ; L ^ { \infty } ( 0 , T ; H ^ 1 ) , \ ; \ ; \ ; \ ; T > 0 , \end{align*}


\begin{align*} w _ t - \partial _ y ( | D | w + w - | w | ^ { p - 1 } w ) = 0 . \end{align*}


\begin{align*} h _ { i , j } ( \lambda ) = \lambda _ i + \lambda ' _ j - i - j + 1 , \end{align*}


\begin{align*} h _ { x y } ( a ) = \sum _ { b \in \Lambda _ { x y } ( V ' ) ( a ) } g ( b ) . \end{align*}


\begin{align*} ( \mathbf { x } _ { k + 1 } , \mathbf { y } _ { k + 1 } ) = \mathcal { P } ( \mathbf { x } _ { k } , \mathbf { y } _ { k } ) , \hbox { w h e r e } \mathcal { P } = ( I + \Sigma ^ { - 1 } T _ { \mathcal { L } } ) ^ { - 1 } \hbox { a n d } \Sigma : = b l o c k d i a g ( \rho I _ d , \delta I _ m ) . \end{align*}


\begin{align*} q ( z ) = \pi A ' z ^ 2 + 2 \pi b ' \cdot z + c ' . \end{align*}


\begin{align*} \bigg ( \int _ { \Omega \cap B _ { \gamma / 2 } } | w | ^ j d x \bigg ) ^ { 1 / j } \leq c ( d , \chi ) j \forall j = 1 , 2 , \ldots \end{align*}


\begin{align*} ( - \Delta ) ^ s ( \tilde u - \tau \varphi ^ { ( 2 ) } ) ( a ) & \leqslant - C a _ 1 \int _ { B _ { 1 / 2 } ^ + } \frac { y _ 1 ( \tilde u - \tau \varphi ^ { ( 2 ) } ) ( y ) } { \vert a _ \ast - y \vert ^ { n + 2 s + 2 } } \dd y \\ & \leqslant - C a _ 1 \bigg ( \int _ { B _ { 1 / 2 } ^ + } y _ 1 \tilde u ( y ) \dd y - \tau \bigg ) \\ & = - C a _ 1 \bigg ( \frac 1 { \rho ^ { n + 1 } } \int _ { B _ { \rho / 2 } ^ + } y _ 1 u ( y ) \dd y - \tau \bigg ) . \end{align*}


\begin{align*} \operatorname { C o v } _ { \mu } ( g _ j , f ) = \mathcal { E } _ { \Sigma } ( g _ j , f ) . \end{align*}


\begin{align*} \begin{aligned} \mathbb { P } \left ( R _ j = a \right ) & = \mathbb { P } \left ( \varsigma _ { Q _ { j } , m } = 1 \right ) \cdot \prod _ { i \in [ a ] } \mathbb { P } \left ( \varsigma _ { i + Q _ { j - 1 } , m } = 0 \right ) \\ & = \left ( 1 - \epsilon _ m \right ) \epsilon ^ a _ m . \end{aligned} \end{align*}


\begin{align*} \Delta ^ q _ R : = \{ ( x _ 0 , \ldots , x _ q ) \mid d ( x _ i , x _ j ) \le R \forall i , j \} . \end{align*}


\begin{align*} u ^ { i - 1 } \beta _ i ( x y ) - u ^ { i - 1 } x \beta _ i ( y ) - u ^ { i - 2 } \beta _ m ( x ) \beta _ { i - m } ( y ) - \dots - u ^ { i - 1 } \beta _ i ( x ) y = 0 \end{align*}


\begin{align*} \frac { c } { Q } \int _ { 1 } ^ { \infty } W \left ( \frac { c x } { Q } \right ) \ , d x = \int _ { 0 } ^ { \infty } W ( x ) \ , d x - \int _ { 0 } ^ { c / Q } W ( x ) \ , d x = \int _ { 0 } ^ { \infty } W ( x ) \ , d x + O \left ( \frac { c } { Q } \right ) . \end{align*}


\begin{align*} \lim _ { X \rightarrow + \infty } \frac { N ( \cup _ { p > M } \mathcal { W } _ p ( V ) , X ) } { X ^ { \dim V } } = O \left ( \frac { 1 } { \log M } \right ) , \end{align*}


\begin{align*} \| \operatorname { H e s s } \hat { f } ^ j \| _ { C ^ { 0 , \alpha } ( B _ { R } ( \hat x , \hat h _ \delta ) ) } \leq \Psi ( \delta \mid n , r , v , \rho , \alpha , Q ) , \ \ j = 1 , \dots , n , \end{align*}


\begin{align*} \mathcal { D } ^ \prime & = \left \{ ( s _ 1 , s _ 2 , s _ 3 ) \mid \sigma _ 1 \geq 0 , \sigma _ 2 \geq 0 , \sigma _ 3 > 0 , t _ 1 \geq 0 , t _ 2 \geq 0 , 2 \leq t _ 3 \leq T , \right . \\ & \left . \sigma _ 1 + \sigma _ 3 \leq 1 , \sigma _ 2 + \sigma _ 3 \leq 1 , \sigma _ 1 + \sigma _ 2 + \sigma _ 3 > 3 / 2 , \right . \\ & \left . s _ 1 + s _ 3 \neq 1 , s _ 2 + s _ 3 \neq 1 , s _ 1 + s _ 2 + s _ 3 \neq 2 \right \} . \end{align*}


\begin{align*} \int _ { G _ { n } ( \Q ) \backslash G _ { n } ( \mathbb { A } ) / K _ 1 ( \mathfrak { n } ) K _ { \infty } } \mathbf { E } ( g \times h , s ) \mathbf { f } ( h ) \mathbf { d } h = c _ k ( s ) D ( s , \mathbf { f } , \chi ) \mathbf { f } ( g ) , \end{align*}


\begin{gather*} \Box \phi = \left ( | \phi _ { t } | ^ 2 - | \phi _ { x } | ^ 2 \right ) \phi + \mathbf { 1 } _ { \omega } f ^ { \phi ^ { \perp } } , \ ; \phi [ 0 ] = u [ 0 ] , \end{gather*}


\begin{align*} i \partial _ t u - \partial _ { x x } u + V ( x ) u \pm b _ 1 ( x ) \left | u \right | ^ { 2 } u + b _ 2 ( x ) u ^ 3 + b _ 3 ( x ) | u | u ^ 2 + b _ 4 ( x ) | u | ^ 3 = 0 \end{align*}


\begin{align*} \frac { R _ { \pm } ( k ) } { T ( k ) } = \frac { 1 } { 2 i k } \int e ^ { \mp 2 i k x } V ( x ) m _ { \mp } ( x , k ) \ , d x , \end{align*}


\begin{align*} \big \{ ( x , y ) \in \R ^ 2 \ \exists \ n \in \N , \ y = n x \big \} \end{align*}


\begin{align*} C = & \partial _ y \Bigg [ \eta _ i \int _ 0 ^ 1 \frac { p ( p - 1 ) } { 2 } \bigg | 1 + s \sum _ { \substack { j = 1 , \\ j \not = i } } ^ n \frac { \sigma _ i \sigma _ j \R _ j } { \R _ i } \bigg | ^ { p - 3 } \bigg ( 1 + s \sum _ { \substack { j = 1 , \\ j \not = i } } ^ n \frac { \sigma _ i \sigma _ j \R _ j } { \R _ i } \bigg ) \times \sigma _ i \R _ i ^ p \bigg ( \sum _ { \substack { j = 1 , \\ j \not = i } } ^ n \frac { \sigma _ i \sigma _ j \R _ j } { \R _ i } \bigg ) ^ 2 \ , \dd s \Bigg ] . \end{align*}


\begin{align*} \int n ( y ) \Phi _ 1 ( y ) \ , d y = \int \Phi _ 1 ( y ) \ , d y = 1 . \end{align*}


\begin{align*} d _ F \circ \Delta _ { a _ 1 } \circ \cdots \circ \Delta _ { a _ r } \bigl ( ~ _ r F _ s ( z ) \bigr ) = \Delta _ { b _ 1 - 1 } \circ \cdots \circ \Delta _ { b _ s - 1 } \bigl ( ~ _ r F _ s ( z ) \bigr ) . \end{align*}


\begin{align*} P = W _ { - 1 } \overset { W ( - 1 , 0 ] } { \leadsto } W _ { 0 } \overset { W ( 0 , 1 ] } { \leadsto } \cdots \overset { W ( d - 2 , d - 1 ] } { \leadsto } W _ { d - 1 } \overset { W ( d - 1 , d ] } { \leadsto } W _ { d } = Q \end{align*}


\begin{align*} \Gamma _ { g _ 1 , k } ( f ) = \Gamma _ { g _ 2 , k } ( f ) \end{align*}


\begin{align*} \epsilon ( X , \pi , \pi ' , \psi _ K ) = ( q _ K ^ { \frac { 1 } { 2 } } X ) ^ { n ( \pi , \pi ' , \psi _ K ) } \epsilon ( \pi , \pi ' , \psi _ K ) . \end{align*}


\begin{align*} \widetilde { M } ( A \otimes ^ I B ) = \widetilde { M } ( A \otimes B ) \cap M ( A \otimes ^ I B ) , \end{align*}


\begin{align*} \sum _ { j = 1 } ^ \infty | b _ j ( z ) | ^ 2 \le \frac { C } { \delta ^ { 2 \max m _ j } } \Big ( 1 + \frac { 1 } { \delta ^ 2 } \log \frac { 1 } { \delta } \Big ) ^ 2 . \end{align*}


\begin{align*} \begin{array} { l c l } K ^ { \frac { \theta } { 2 } , \nu } = 0 \ m = 2 & & K ^ { \theta , \mu } = \delta _ { 0 , \mu } ( 1 + z ^ { m - 2 } ) . \end{array} \end{align*}


\begin{align*} \left \{ \begin{array} { r c l } \dot { \xi } ( t ) & = & A \xi ( t ) + \alpha ( \xi ( t ) ) \\ \xi ( 0 ) & = & x , \end{array} \right . \end{align*}


\begin{align*} J _ { 0 } ( \alpha , \beta , Z ) - J _ { 0 } ( \alpha , \beta , Z ' ) = < - c _ { 1 } ( \xi ) + 2 \mathrm { P D } ( \Gamma ) , Z - Z ' > . \end{align*}


\begin{align*} \begin{aligned} - f '' + f + 1 & \geq 0 , \\ f - f ^ * & \geq 0 , \\ ( - f '' + f + 1 = 0 ) \vee ( f - f ^ * & = 0 ) , \end{aligned} \end{align*}


\begin{align*} \ddot { y } _ R ( t ) = 8 s _ c ( p - 1 ) \delta ( t ) + A _ R ( u ( t ) ) \ge 4 s _ c ( p - 1 ) \delta ( t ) , \forall t \in [ \delta , \tau ] , \end{align*}


\begin{align*} f ( x ) - f ( x ^ \star ) & = \int _ 0 ^ 1 \tfrac { 1 } { \lambda } \langle \nabla f ( \lambda x + ( 1 - \lambda ) x ^ \star ) , \lambda x + ( 1 - \lambda ) x ^ \star - x ^ \star \rangle d \lambda \\ & \geq \int _ 0 ^ 1 \tfrac { \gamma } { \lambda } \left ( f ( \lambda x + ( 1 - \lambda ) x ^ \star ) - f ( x ^ \star ) + \tfrac { \mu _ s \lambda ^ 2 } { 2 } \| x - x ^ \star \| ^ 2 \right ) d \lambda \\ & \geq \tfrac { \gamma \mu _ s } { 4 } \| x - x ^ \star \| ^ 2 , \end{align*}


\begin{align*} \overline { j ( \alpha , z _ 1 ) } \delta ( \alpha z _ 1 , \alpha z _ 2 ) j ( \alpha , z _ 2 ) = \delta ( z _ 1 , z _ 2 ) . \end{align*}


\begin{align*} { \rho } _ n = \left ( n + \frac { 1 } { 2 } \right ) \pi + \xi _ n , n \in \mathbf { N } . \end{align*}


\begin{align*} \forall y ^ \sigma ( A _ 0 ( y ) \rightarrow r [ s / x ^ \rho ] = _ \tau r [ t / x ^ \rho ] ) \equiv \exists y ^ \sigma A _ 0 ( y ) \rightarrow r [ s / x ^ \rho ] = _ \tau r [ t / x ^ \rho ] , \end{align*}


\begin{align*} & \widetilde { d } _ T ( f ) ( u _ 1 , \cdots , u _ { p + 1 } ) \\ & = \sum _ { 1 \leq i < j \leq p + 1 } ( - 1 ) ^ { i + j } f ( \rho ( T u _ i ) u _ j - \rho ( T u _ j ) u _ i , u _ 1 , \cdots , \widehat { u _ i } , \cdots , \widehat { u _ j } , \cdots , u _ { p + 1 } ) \\ & + \sum _ { i = 1 } ^ { p + 1 } ( - 1 ) ^ { i + 1 } \Big ( [ T u _ i , f ( u _ 1 , \cdots , \widehat { u _ i } , \cdots , u _ { p + 1 } ) ] + T \rho ( f ( u _ 1 , \cdots , \widehat { u _ i } , \cdots , u _ { p + 1 } ) ) ( u _ i ) \Big ) . \end{align*}


\begin{align*} \mathbf { d } z = \delta ( z ) ^ { - n + 1 } \prod _ { h \leq k } [ ( i / 2 ) d z _ { h k } \wedge d \overline { z } _ { h k } ] , \end{align*}


\begin{align*} I _ 1 = R _ 0 + O \bigg ( \frac { ( X C h k ) ^ { \varepsilon } Q ^ { 2 } ( h , k ) } { C \sqrt { h k } } \bigg ) + O \Big ( X ^ { \varepsilon } Q ^ { \frac { 3 } { 2 } } h ^ { \varepsilon } k ^ { \varepsilon } \Big ) , \end{align*}


\begin{align*} \sum _ a X _ a ( J ^ a _ \xi ) = 0 \end{align*}


\begin{align*} \lim _ { x \to \infty } E _ { \psi } ( x ) = 0 . \end{align*}


\begin{align*} [ u ] _ { ( \alpha , G ) } = \inf \bigg { \{ } \lambda > 0 : \ \overline { \rho } ( \alpha ; \frac { u } { \lambda } ) \leq 1 \bigg { \} } . \end{align*}


\begin{align*} \bar { \Delta } ^ { \rm P } _ { \rm R e a c t i v e } = - 1 - \tau _ { \rm f } + \mathbb { E } T ^ { \rm R e a c } _ j + \mathbb { E } \tau ^ { \rm R e a c } _ { V _ { j - 1 } } . \end{align*}


\begin{align*} f ( x , y ) \sim \sum _ { j = 0 } ^ { \infty } { \vphantom { \sum } } ' g _ j ( x ) T _ j ( y ) , \mbox { w h e r e } ~ ~ ~ g _ j ( x ) = \sum _ { i = 0 } ^ { \infty } { \vphantom { \sum } } ' c _ { i , j } T _ i ( x ) . \end{align*}


\begin{align*} \lim _ { \omega \to 0 } \norm { M _ \omega f - f } _ 2 = \lim _ { \omega \to 0 } \norm { T _ \omega \widehat { f } - \widehat { f } } _ 2 = 0 . \end{align*}


\begin{align*} \mathfrak { B } ^ J = \{ w \in \mathfrak { B } _ n \ , | \ , 0 < w _ 1 < \cdots < w _ { p _ 1 } , w _ { p _ 1 + 1 } < \cdots < w _ { p _ 2 } , \ldots , w _ { p _ k + 1 } < \cdots < w _ { n } \} . \end{align*}


\begin{align*} | A _ R ( u ( t ) ) | & = \Big | A _ R ( u ) - A _ R \left ( e ^ { i ( t + \theta ( t ) ) } Q ( x - X ( t ) ) \right ) \Big | \\ & \lesssim e ^ { - C R _ 0 } \delta ( t ) + \delta ( t ) ^ 2 + \delta ( t ) ^ { 2 p } , R \ge R _ 0 + | X ( t ) | . \end{align*}


\begin{align*} \int _ M V \ , d V _ g = 0 . \end{align*}


\begin{align*} \int _ 0 ^ { \infty } e ^ { - s } \left ( C _ p \left | \dot { \tilde { \xi } } ( s ) \right | ^ q + f \left ( \tilde { \xi } ( s ) \right ) \right ) d s = \int _ 0 ^ { \infty } e ^ { - s } f ( \xi ( 0 ) ) d s = f ( \xi ( 0 ) ) . \end{align*}


\begin{align*} 0 = J ^ { ' } ( \widehat { u } ) + \lambda _ 0 \varphi ^ * _ { \widehat { u } } \ \ \ ( W ^ { \alpha , G } ( \mathbb { R } ^ d ) ) ^ * , \ \ \ \varphi ^ * _ { \widehat { u } } \in \partial \varphi ( \widehat { u } ) . \end{align*}


\begin{align*} q _ k = { n + k \choose k } ( 0 \leq k \leq m ) . \end{align*}


\begin{align*} E _ { 1 } = - \frac { 1 } { 2 \pi i } \int _ { \Gamma _ E } \left ( I + \mu ( s ) \right ) ( J ^ E ( s ) - I ) d s : = I _ 1 + I _ 2 + I _ 3 . \end{align*}


\begin{align*} \mathcal { U } ^ 1 ( h , k ) = - \mathcal { L } ^ 0 ( h , k ) + O \big ( ( X h k ) ^ \varepsilon X C \big ) + O \bigg ( Q \frac { ( X C H K ) ^ { \varepsilon } } { \sqrt { H K } } \bigg ) . \end{align*}


\begin{align*} n = q , \quad \hat \theta = \frac { | 1 + J ' ( \alpha , \alpha ) | } { \sqrt { 2 q ( q - 1 ) } } , \end{align*}


\begin{align*} P _ n ( 1 ) = 2 ^ n E _ { n } , \end{align*}


\begin{align*} \int _ 0 ^ 1 \langle T _ { \gamma ( t ) } G , \gamma ' ( t ) \rangle \ 1 { \gamma ( t ) \in W ' } d t = 0 , \end{align*}


\begin{align*} F ( \mu ) = 0 , \end{align*}


\begin{align*} \big ( D _ E ^ + \big ) = \tfrac 1 4 \big ( - \sigma ( N ) + 2 \chi ( M ) + 3 \sigma ( M ) + 2 b _ 0 ( \partial M ) + 2 b _ 2 ( \partial M ) \big ) , \end{align*}


\begin{align*} \forall b \in [ B ] , \abs { \pi _ b } = 2 \quad i _ { \pi _ b ( 1 ) } = i _ { \pi _ b ( 2 ) } \end{align*}


\begin{align*} D _ A = \begin{bmatrix} D _ 1 \\ 0 _ { m - n , n } \end{bmatrix} D _ B = \begin{bmatrix} D _ 2 & 0 _ { n , m - n } & 0 _ { n , \ell - m } \\ 0 _ { m - n , n } & I _ { m - n } & 0 _ { m - n , \ell - m } \end{bmatrix} . \end{align*}


\begin{align*} B f ( w ) & = e ^ { - \pi i \xi \cdot \eta } e ^ { \frac { \pi } { 2 } | w | ^ 2 } V _ { g _ 0 } f ( \xi , - \eta ) \\ & = e ^ { - \pi i \xi \cdot \eta } e ^ { \frac { \pi } { 2 } | w | ^ 2 } e ^ { - \pi i \xi \cdot ( - \eta ) } \langle f , \rho ( \xi , - \eta , 0 ) g _ 0 \rangle \\ & = e ^ { \frac { \pi } { 2 } | w | ^ 2 } \langle f , \rho ( \overline { w } , 0 ) g _ 0 \rangle . \end{align*}


\begin{align*} \Lambda _ { ( n , m ) } ( M , \Gamma ) : = \{ \ , \ , \alpha \in \Lambda ( M , \Gamma ) \ , | \ , ( E ( \alpha ) , H ( \alpha ) ) = ( n , m ) \ , \} \end{align*}


\begin{align*} \norm { A f - \sum _ { \gamma \in F } A e _ \gamma } _ { \mathcal { B } _ 2 } = \norm { A ( f - \sum _ { \gamma \in F } e _ \gamma ) } _ { \mathcal { B } _ 2 } \leq \norm { A } _ { o p } \norm { f - \sum _ { \gamma \in F } e _ \gamma } _ { \mathcal { B } _ 1 } < \varepsilon , \end{align*}


\begin{align*} \dim _ H \Big ( \limsup _ { B \in \mathcal { Q } } B ^ { \delta } \cap K _ { 1 / 3 } ^ { ( 0 ) } \Big ) = \min \left \{ \frac { \log 2 } { \log 3 } , \frac { 1 } { \delta } \right \} . \end{align*}


\begin{align*} X = \begin{cases} \alpha , & \ 0 . 5 , \\ \epsilon & \ 0 . 5 , \end{cases} \end{align*}


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


\begin{align*} ( 1 - x ^ 2 ) Y '' + [ - \alpha + \beta - ( \alpha + \beta + 2 ) x ] \ , Y ' + ( n + 1 ) ( n - \alpha - \beta ) Y = 0 , \end{align*}


\begin{align*} \forall ( t , u ) \in [ 0 , T ] \times \mathbb { R } , G ( t , u ) = \sum _ { j = 0 } ^ { k } t ^ { j } G _ j ( u ) . \end{align*}


\begin{align*} \begin{aligned} ( \lfloor \frac { a - 1 } { 2 } \rfloor + \lfloor \frac { j - l } { 2 } \rfloor + 1 & - \lceil \frac { p - 1 } { 2 } \rceil ) + \frac { a } { 2 } + l - a + p \\ & \geq \lfloor \frac { j - l } { 2 } \rfloor + l + \lceil \frac { p } { 2 } \rceil \\ & \geq \lfloor \frac { j - l } { 2 } \rfloor + l + \lceil \frac { j - l + 1 } { 2 } \rceil \\ & \geq j , \\ \end{aligned} \end{align*}


\begin{align*} \delta ^ * = O ( \delta + \delta \delta ^ * ) , \forall t \in D _ { \delta _ 0 } , \end{align*}


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


\begin{align*} \sum _ { t = 1 } ^ T \eta _ t \norm { \nabla f ( \bar x _ t ) } ^ 2 & \leq f ( x _ 1 ) - f ( x _ { T + 1 } ) + 2 L \sum _ { t = 1 } ^ T { \eta _ t ^ 2 } \norm { \nabla f ( \bar x _ t ) } ^ 2 \\ \end{align*}


\begin{align*} \sup _ { f \in M } \| f \| _ { Y _ q ( \tilde \Omega _ t ) } \underset { \mathfrak { Z } _ 0 } { \lesssim } \sum \limits _ { l = t } ^ \infty \sum \limits _ { m = 0 } ^ \infty d _ 0 ( W _ { l , m } , \ , l _ q ^ { \nu _ { l , m } } ) . \end{align*}


\begin{align*} \widehat { g } ( \omega ) = C \ , e ^ { - \gamma \omega ^ 2 } e ^ { - 2 \pi i \delta \omega } \prod _ { k = 1 } ^ \infty \frac { e ^ { 2 \pi i \delta _ k \omega } } { 1 + 2 \pi i \delta _ k \omega } , \end{align*}


\begin{align*} \lim _ { L \to \infty } \sup _ { u , v \in C _ { R + 1 } } | K _ { x , L } ( u , v ) - K _ x ( u - v ) | = 0 , \end{align*}


\begin{align*} P _ d ( x ) = \sum _ { | J | = d } \sqrt { \binom { d } { J } } a _ { J } x ^ J , \end{align*}


\begin{align*} R _ \mu ( n , \alpha , \varepsilon , \delta , T , f ) = \inf \{ \sum _ { i } e ^ { - \alpha n + f _ { n } ( x _ i ) } : \mu \big ( \cup _ { i } B _ { n } ( x _ i , \varepsilon ) \big ) \geq 1 - \delta \} , \end{align*}


\begin{align*} u \left ( t , x \right ) = \frac { e ^ { i \frac { x ^ { 2 } } { 4 t } } } { \sqrt { - 2 i t } } ( \mathcal { F } f ) \left ( t , - \frac { x } { 2 t } \right ) + \mathcal { O } \big ( { \big \| \partial _ { k } ( \mathcal { F } f ) ( t ) \big \| } _ { L _ k ^ 2 } \ , t ^ { - \frac { 3 } { 4 } } \big ) , t \geq 1 , \end{align*}


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


\begin{align*} \langle \ ! \langle C _ { n } ^ { \lambda , \beta } , Q _ { m } ^ { \pi _ { \lambda , \beta } } \rangle \ ! \rangle _ { \pi _ { \lambda , \beta } } & = \int _ { \mathbb { R } } m ! \sum _ { k = m } ^ { \infty } \frac { S ( k , m ) } { k ! } \left ( \nabla ^ { k } C _ { n } ^ { \lambda , \beta } ( x ) \right ) \ , \mathrm { d } \pi _ { \lambda , \beta } ( x ) \\ & = m ! \sum _ { k = m } ^ { \infty } \frac { S ( k , m ) } { k ! } k ! s ( n , k ) \\ & = m ! \sum _ { k = m } ^ { n } s ( n , k ) S ( k , m ) \\ & = n ! \delta _ { n , m } . \end{align*}


\begin{align*} \mathcal { B } = \{ z \in M _ n ( \C ) : 1 - z ^ { \ast } z > 0 \} , \end{align*}


\begin{align*} f _ { Y _ i } ( y ) & = \frac { f _ { | \hat { h } _ i | } ( y ) } { 1 - F _ { | \hat { h } _ i | } ( \sqrt { \mu _ i / \vartheta } ) } = 2 y e ^ { \mu _ i / \vartheta _ i - y ^ 2 } \end{align*}


\begin{align*} & \int _ c ^ x ( f ( u , u ) - f ( u , c ) ) d u + \int _ c ^ b ( f ( u , x ) - f ( u , c ) ) d u \\ & = \int _ c ^ x f ( u , u ) d u + \int _ c ^ b f ( u , x ) d u - \int _ c ^ b f ( u , c ) d u \\ & = \int _ c ^ x f ( u , u ) d u + F _ 2 ( x ) - F _ 2 ( c ) . \end{align*}


\begin{align*} \lim _ { n \to \infty } ( h _ n ^ j h _ n ^ k ) ^ { - \frac { 1 } { 2 } } \Big | B _ n ^ j ( R ) \cap B _ n ^ k ( R ) \Big | = 0 , \end{align*}


\begin{align*} C ^ q ( U , A ) = \varinjlim _ { A _ 1 \sqcup \cdots \sqcup A _ n = U } C ^ q _ { A _ 1 , \ldots , A _ n } ( U , A ) \end{align*}


\begin{align*} \norm { w } = \frac { \norm { x - z } } { \vert \gamma _ n \vert } \leq \frac { \norm { x } + \norm { z } } { \vert \gamma _ n \vert } \leq \frac { 2 L } { \vert \gamma _ n \vert } . \end{align*}


\begin{align*} \varphi : U & \to A \\ x & \mapsto \begin{cases} \varphi _ 1 ( x ) & x \in U _ 1 \\ \varphi _ 2 ( x ) & x \in U _ 2 \setminus U _ 1 \\ \vdots & \vdots \\ \varphi _ n ( x ) & x \in U _ n \cap U _ 1 ^ c \cap \cdots \cap U _ { n - 1 } ^ c \\ 0 & x \in U _ 1 ^ c \cap \cdots \cap U _ n ^ c \end{cases} \end{align*}


\begin{align*} \left \langle m _ { \Lambda } ^ { * } E ^ { i j } _ 0 , f \otimes g \right \rangle \equiv \Lambda ^ { - 2 } \left \langle E ^ { i j } _ 0 , ( { m _ { \Lambda } } _ * f ) \otimes ( { m _ { \Lambda } } _ * g ) \right \rangle = \left \langle E ^ { i j } _ 0 , f \otimes g \right \rangle \end{align*}


\begin{align*} \mathbf { F } = \{ v _ k = ( v _ { k , 1 } , \ldots , v _ { k , d } ) \mid k = 1 , \ldots , K \} , \end{align*}


\begin{align*} \left ( \frac { d \psi } { d \zeta } + \frac { i \zeta } { 2 } \sigma _ 3 \psi \right ) \psi ^ { - 1 } = \left [ \frac { d m ^ { P C } _ { \eta } } { d \zeta } + m ^ { P C } _ { \eta } \frac { i \nu } { \zeta } \sigma _ 3 \right ] \left ( m ^ { P C } _ { \eta } \right ) ^ { - 1 } + \frac { i \zeta } { 2 } \left [ \sigma _ 3 , m ^ { P C } _ { \eta } \right ] \left ( m ^ { P C } _ { \eta } \right ) ^ { - 1 } , \end{align*}


\begin{align*} g = \exp ( t ^ { s } \xi _ { s } ) \dots \exp ( t ^ { 1 } \xi _ 1 ) . \end{align*}


\begin{align*} a _ { \inf } > \begin{cases} \frac { \mu \chi ^ 2 } { 4 } { \rm i f } 0 < \chi \leq 2 \cr \mu ( \chi - 1 ) { \rm i f } \chi > 2 . \end{cases} \end{align*}


\begin{align*} A ^ T C = C ^ T A , \ B ^ T D = D ^ T B A ^ T D - C ^ T B = I , \end{align*}


\begin{align*} ( T _ t ^ \alpha ) ^ * ( f ) = P ^ { \nu _ \alpha } _ t ( f ) . \end{align*}


\begin{align*} u = - v ( r , t ) x , \end{align*}


\begin{align*} ( - \Delta ) ^ s u + c u & = 0 \Omega ^ + . \end{align*}


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


\begin{align*} d ( D ) = \sqrt { \lambda D / N } , \end{align*}


\begin{align*} [ q _ 1 , q _ 2 ] : = \{ x \in \R \mid Q ( x ) = 0 \} \end{align*}


\begin{align*} \tilde { \mathbf { y } } _ k [ t ] = \mathbf { D } \mathbf { y } _ k [ t ] . \end{align*}


\begin{align*} \norm { K } _ { H . S . } = \norm { k } _ 2 . \end{align*}


\begin{align*} \partial _ \tau \epsilon _ 0 \| \phi _ 0 \| ^ 2 _ \rho + \epsilon _ 0 ( \tau ) \partial _ \tau \| \phi _ 0 \| ^ 2 _ { \rho } = \langle \partial _ \tau \epsilon , \phi _ 0 \rangle _ { L ^ 2 _ \rho } + \langle \epsilon , \partial _ \tau \phi _ 0 \rangle _ { L ^ 2 _ \rho } . \end{align*}


\begin{align*} \big \{ ( x , y , z ) \ x = y = z \big \} . ] \end{align*}


\begin{align*} & \mathcal { A } ^ c _ G ( X , E ) \\ \cong & \big \{ \Phi : G \to \Psi ^ { - \infty } ( S , E | _ S ) , K \times K \big \} . \end{align*}


\begin{align*} \Delta u + k ^ 2 ( 1 + \eta ) u = 0 \end{align*}


\begin{align*} \lim _ { R \to \infty } \sum _ { 1 \leq i _ 1 , i _ 2 \leq m } \frac { 1 } { | B _ R | } \int _ { B _ R } ( k _ { i _ 1 i _ 2 } ( x ) ) ^ 2 d x = 0 , \end{align*}


\begin{align*} \pi ( M _ p ) = \sum _ { p \leq x } \mu _ { P } ( p , x ) F _ { f ( x ) } = \sum _ { q \in Q } \Big ( \sum _ { p \leq x , \ f ( x ) = q } \mu _ { P } ( p , x ) \Big ) F _ q . \end{align*}


\begin{align*} B _ 2 : = 2 + \delta , B _ 3 : = 2 \left [ \frac { k } { m } + \delta \left ( \frac { k } { m } - \frac { 1 } { 2 } \right ) - \frac { \delta ^ 2 } { 4 } \right ] . \end{align*}


\begin{align*} T ( x ; k , r ) : = \sum _ { m \leq x , \ , m \equiv k \ , ( m o d \ , r ) } a ( m ) \end{align*}


\begin{align*} \Psi _ { ( \Sigma , U ) } ( f ) & = \Psi _ { ( \Sigma , U ) } \left ( \mathfrak { D } ^ { ( \mu ) } ( i , i _ U ) ( f | _ { \Sigma } ) \right ) \\ & = \mathfrak { P } _ { \ell } ( i , i _ U ) \left ( \Psi _ { ( \Sigma _ f , U _ f ) } ( f | _ { \Sigma } ) \right ) . \end{align*}


\begin{align*} \tau _ t ( a _ 1 \cdots a _ k ) = \tau ( a _ 1 \ldots a _ k ) + t \cdot \tau ' ( a _ 1 \ldots a _ k ) + o ( t ) . \end{align*}


\begin{align*} \phantom { \quad x , y , z , w \in V . } \langle R _ { x , y } ( z ) , w \rangle = \langle R ( x \wedge y ) , w \wedge z \rangle , \quad x , y , z , w \in V . \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 _ 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*} \sum _ { \l ^ \circ \in \L ^ \circ } A g ( \l ^ \circ ) = - \sum _ { \l ^ \circ \in \L ^ \circ } A g ( \l ^ \circ ) = 0 . \end{align*}


\begin{align*} ( x , y ) _ { F , p ^ n } : = \frac { \rho _ { F } ( x ) ( \sqrt [ p ^ n ] { y } ) } { \sqrt [ p ^ n ] { y } } \in \mu _ { p ^ n } \ \ ( x , y \in F ^ { \times } ) , \end{align*}


\begin{align*} T B _ { 1 4 } ( f ) = 2 T B _ { 6 } ( f ) . \end{align*}


\begin{align*} M = \bigoplus _ { j = 0 } ^ 1 M _ j ( \pi ^ j ) , \end{align*}


\begin{align*} B ^ { \left [ n + 1 \right ] } = \left \langle \bigcup _ { i = 1 } ^ n B ^ { [ i ] } * B ^ { [ n - i ] } \right \rangle _ + \end{align*}


\begin{align*} \lim _ { n \to \infty } \frac { E _ { n , k } ( \tau ( A ^ { ( n ) } _ { c _ n k n , n - s , l } ) | \mathcal { W A } ^ { ( n , k ) } _ { M _ n , n - s , l } ) } { n k } = \frac { 1 - c } { k + 1 } . \end{align*}


\begin{align*} \psi ( x ) = \frac { 1 } { 2 \pi i } \int _ { c - i T } ^ { c + i T } \left ( - \frac { \zeta ' ( s ) } { \zeta ( s ) } \right ) \frac { x ^ s } { s } \ , d s + O ^ * \left ( \sum _ { n = 1 } ^ \infty \left ( \frac { x } { n } \right ) ^ c \Lambda ( n ) \min \left ( 0 . 5 0 1 , \frac { 1 } { \pi T \left | \log \frac { x } { n } \right | } \right ) \right ) . \end{align*}


\begin{align*} \frac { \partial z ^ { k } } { \partial x ^ j } = \left ( \left \{ \frac { \partial x ^ s } { \partial z ^ t } \right \} \right ) / \operatorname { d e t } ( B ) , \end{align*}


\begin{align*} a _ i = ( - 1 ) ^ i q ^ { i ^ 2 } { k \choose k - i } _ q S _ k ^ i ( V , V ' ) . \end{align*}


\begin{align*} & - \nabla \cdot ( \epsilon ( x ) \psi ) = \tilde { f } ( x ) + \sum _ { i = 1 } ^ s z _ i \rho _ i , \quad , x \in \Omega , \\ & \psi = 0 , x \in \partial \Omega , \end{align*}


\begin{align*} \sum _ { \substack { 0 < \gamma < T \\ \beta \le 1 / 2 } } \frac { 1 } { \gamma } = \sum _ { \substack { 0 < \gamma < T } } \frac { 1 } { \gamma } - \sum _ { \substack { H _ 0 \le \gamma < T \\ \beta > 1 / 2 } } \frac { 1 } { \gamma } \le \sum _ { \substack { 0 < \gamma < T } } \frac { 1 } { \gamma } - \sum _ { \substack { H _ 0 \le \gamma < T \\ 1 / 2 < \beta < \sigma } } \frac { 1 } { \gamma } . \end{align*}


\begin{align*} r = \sqrt { P _ t } \Bigl ( \mathbf { g } ^ T \boldsymbol { \Theta } \mathbf { H } + \mathbf { h } _ d ^ T \Bigr ) \mathbf { w } s + n . \end{align*}


\begin{align*} U ^ 1 ( x ) = e ^ { i \theta _ 2 } Q ( x + x _ 2 ) \ ; \Rightarrow \ ; u _ n ( x + x _ n ^ 1 ) = e ^ { i \theta _ 2 } Q ( x + x _ 2 ) + r _ n ^ 1 ( x + x _ n ^ 1 ) . \end{align*}


\begin{align*} u ( r , \theta ) = ( r ^ { - \alpha } - r ^ { \alpha } ) \mathrm { s i n } ( \alpha \theta ) \end{align*}


\begin{align*} 0 & = \norm { x - J ^ A _ { \gamma } ( x + \gamma u ) + \gamma ( u - \gamma ^ { - 1 } ( ( x + \gamma u ) - J ^ A _ { \gamma } ( x + \gamma u ) ) ) } \\ & \geq \norm { x - J ^ A _ { \gamma } ( x + \gamma u ) } . \end{align*}


\begin{align*} - M \left ( \int _ { \Omega } g ( \vert \nabla u \vert ) d x \right ) \Delta _ g u = f ( u ) , \ \ \Omega , \end{align*}


\begin{align*} \begin{aligned} ( \widetilde \sigma ( x ) - \widetilde \sigma ( y ) ) ^ 2 & = ( \widetilde \sigma ( x ) - \widetilde \sigma ( z ) + \widetilde \sigma ( z ) - \widetilde \sigma ( y ) ) ^ 2 \\ & \leq ( 1 + \delta _ { x , y } ) \cdot ( \widetilde \sigma ( x ) - \widetilde \sigma ( z ) ) ^ 2 + \Bigl ( 1 + \frac { 1 } { \delta _ { x , y } } \Bigr ) \cdot ( \widetilde \sigma ( z ) - \widetilde \sigma ( y ) ) ^ 2 , \end{aligned} \end{align*}


\begin{align*} \chi ( \mathcal { I } \mid _ { \alpha \beta } ^ { i j } ) : = \left \{ h ( \mathcal { I } ) \cdot h ( \mathcal { I } _ { \alpha \beta } ^ { i j } ) , h ( \mathcal { I } _ { \alpha } ^ { i } ) \cdot h ( \mathcal { I } _ { \beta } ^ { j } ) , h ( \mathcal { I } _ { \beta } ^ { i } ) \cdot h ( \mathcal { I } _ { \alpha } ^ { j } ) \right \} \end{align*}


\begin{align*} \tau _ { E _ s } ( m ) = m ^ { - s } \tau _ E ( m ) . \end{align*}


\begin{align*} \xi ^ \alpha \cdot [ x _ 0 : \dots : x _ n ] = \left [ \xi ^ { \alpha m / q _ 0 } x _ 0 : \dots : \xi ^ { \alpha m / q _ n } x _ n \right ] , \end{align*}


\begin{align*} \max _ { x \in L _ 0 } h _ i ( x ) - \min _ { y \in L _ 0 } h _ i ( y ) & = \left | \int _ 0 ^ { d _ { L _ 0 } ( x _ i , y _ i ) } ( d h _ i ) _ { \gamma _ i ( t ) } ( \dot { \gamma } _ i ( t ) ) d t \right | \\ & \leq | | d h _ i | | \int _ 0 ^ { d _ { L _ 0 } ( x _ i , y _ i ) } | \dot { \gamma } _ i ( t ) | d t \\ & \leq \mathrm { D i a m } ( L _ 0 ) | | d h _ i | | . \end{align*}


\begin{align*} ( l - 1 ) b _ 2 + ( t _ \sigma + 1 ) b _ 1 = m b _ 1 s _ \sigma + ( q - l p ) \pi \ + r _ { p , q } , \end{align*}


\begin{align*} f ( x ) = \mathcal { O } ( e ^ { - a \pi x ^ 2 } ) \widehat { f } ( \omega ) = \mathcal { O } ( e ^ { - b \pi \omega ^ 2 } ) , \end{align*}


\begin{align*} P _ U = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} , L _ U = \begin{bmatrix} 1 & 0 \\ h & 1 \end{bmatrix} , h \in \mathbb C . \end{align*}


\begin{align*} b ( x ) = 1 \textrm { o n } \left [ \frac { S _ 0 } { 8 } , \frac { 3 S _ 0 } { 8 } \right ] , \ ; \ ; b ( x ) = 0 \textrm { o n } \S \setminus \left [ 0 , \frac { S _ 0 } { 2 } \right ] , \end{align*}


\begin{align*} \Xi ^ { M _ c } : = \Xi _ { \mathcal { C } ^ c } / \Xi _ { \mathcal { C } \mathcal { C } ^ c } : = \{ [ X ] : = X + \Xi _ { \mathcal { C } \mathcal { C } ^ c } ~ | ~ X \in \Xi _ { \mathcal { C } ^ c } \} \end{align*}


\begin{align*} M : = \Bigl \{ ( \gamma _ i , \delta _ i ) \Bigm | \gamma _ { i t } = \prod _ { k \in \eta _ i ( t ) } \delta _ { i k } \prod _ { k \notin \eta _ i ( t ) } ( 1 - \delta _ { i k } ) \ ; t \in \{ 1 , \ldots , 2 ^ { \lceil \log _ 2 l _ i \rceil } \} , \ \delta _ i \in [ 0 , 1 ] ^ { \lceil \log _ 2 l _ i \rceil } \Bigr \} . \end{align*}


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


\begin{align*} \int _ { \R ^ N } f ( x ) d \omega ( x ) = 0 . \end{align*}


\begin{align*} V _ { g _ 0 } f ( 0 , - \omega ) = B f ( i \omega ) e ^ { - \frac { \pi } { 2 } \omega ^ 2 } . \end{align*}


\begin{align*} \Gamma ( \pi ) \ge \frac { 1 } { N } \sum _ { \ell = 1 } ^ N \left ( \frac { \rho _ \ell \beta _ \ell ^ \pi } { 2 T _ { \ell , \pi } ^ { a v } } + \frac { \rho _ \ell T _ { \ell , \pi } ^ { a v } } { 2 } + \frac { c _ \ell } { T _ { \ell , \pi } ^ { a v } } \right ) , \end{align*}


\begin{align*} \zeta ( k ) = t ^ { 1 / 2 } \sqrt { 2 g '' ( \eta ) } ( k - \eta ) , \end{align*}


\begin{align*} \left \| f _ * \left ( \Phi _ 1 ( e ^ { i \theta } ) , \dots , \Phi _ { n } ( e ^ { i \theta } ) \right ) \right \| = \max _ { \theta } \{ | f _ * ( \lambda _ 1 , \dots , \lambda _ { n } ) | : ( \lambda _ 1 , \dots , \lambda _ { n } ) \in \sigma _ T ( \Phi ( e ^ { i \theta } ) , \dots , \Phi _ { n } ( e ^ { i \theta } ) ) \} . \end{align*}


\begin{align*} \mathbf A = \left ( \frac { 2 5 } { 1 2 } \mathbf I - k \mathbf L \right ) , \tilde { \mathbf y ^ { n + 4 , ( \ell ) } } = k \tilde { \mathbf b } ^ { n + 4 } + 4 \tilde { \mathbf V } ^ { n + 3 , ( \ell ) } - 3 \tilde { \mathbf V } ^ { n + 2 , ( \ell ) } + \frac { 4 } { 3 } \tilde { \mathbf V } ^ { n + 1 , ( \ell ) } - \frac { 1 } { 4 } \tilde { \mathbf V } ^ { n , ( \ell ) } . \end{align*}


\begin{align*} \chi \circ d ( x ) & = ( - \bar { \rho } \circ \pi _ { \ell } ^ { \overline { \Sigma } } \circ d ( x ) , \bar { \rho } \circ \pi _ r ^ { \overline { \Sigma } } \circ d ( x ) ) \\ & = ( - d _ { \Sigma _ 0 } \circ \bar { \rho } \circ \pi _ { \ell } ^ { \overline { \Sigma } } ( x ) , d _ { \Sigma _ 0 } \circ \bar { \rho } \circ \pi _ { r } ^ { \overline { \Sigma } } ( x ) ) \\ & = d _ 0 \circ \chi ( x ) , \end{align*}


\begin{align*} P ( X ) \ = \ \alpha _ 0 X ^ k + \alpha _ 1 X ^ { k - 1 } + \cdots + \alpha _ { k - 1 } X + \alpha _ k \end{align*}


\begin{align*} \widehat { K } _ { \mathrm { u - c p t } } ^ \mathbb { R } & = \big ( | \mathbf { s } ^ H \mathbf { y } | ^ 2 / N ^ 2 - \sigma ^ 2 / N \big ) / \bar { p } . \end{align*}