\begin{align*} M = \begin{pmatrix} v _ { 1 , 1 } & v _ { 2 , 1 } & & v _ { d , 1 } \\ \vdots & \vdots & \cdots & \vdots \\ v _ { 1 , d } & v _ { 2 , d } & & v _ { d , d } \end{pmatrix} \end{align*}


\begin{align*} \norm { A \oplus A } _ { p } = 2 ^ { \frac { 1 } { p } } \norm { A } _ { p } . \end{align*}


\begin{align*} \rho _ { i , j } ^ * \geq \delta + r _ p ( h ) , j _ p < j < j _ { p + 1 } , p = 1 , \cdots , k , \end{align*}


\begin{align*} d \Theta ^ { ( d ) } _ { u n i f } = \frac { d \theta _ 1 } { 2 \pi } \cdots \frac { d \theta _ d } { 2 \pi } . \end{align*}


\begin{align*} \abs { \frac { 1 } { A ( x ) } - \frac { 1 } { \alpha } } \ = \ \frac { \abs { \alpha - A ( x ) } } { \abs { A ( x ) } \abs { \alpha } } \ \leq \ \frac { a } { x + 1 } , \end{align*}


\begin{align*} & \phi _ 1 ^ + ( y ) = \mathbf { 1 } _ { [ \alpha _ 1 , + \infty ) } ( y ) , \quad \phi _ n ^ - ( y ) = \mathbf { 1 } _ { ( - \infty , \alpha _ n ] } ( y ) , \\ & \phi _ i ^ - ( y ) = \psi \bigg ( \frac { 3 ( y - \alpha _ i ) } { \alpha _ i - \alpha _ { i + 1 } } \bigg ) \mathbf { 1 } _ { [ \alpha _ { i + 1 } , \alpha _ i ] } ( y ) , \forall i = 1 , \ldots , n - 1 , \\ & \phi _ i ^ + ( y ) = [ 1 - \phi _ { i - 1 } ^ - ( y ) ] \mathbf { 1 } _ { [ \alpha _ i , \alpha _ { i - 1 } ] } ( y ) , \forall i = 2 , \ldots , n . \end{align*}


\begin{align*} ( \lambda , y ) = \sum _ { e \in E } \chi ( e ) y _ e \end{align*}


\begin{align*} \alpha _ t = \frac { 1 } { 2 } \min \left \{ \frac { 1 } { \rho } , \sqrt { \frac { \Delta } { ( n ^ 2 + 2 n ) \rho L _ { f , 0 } ^ 2 ( T + 1 ) } } \right \} , \end{align*}


\begin{align*} L ' \otimes \Z _ v & = \begin{cases} \displaystyle { \bigoplus _ { j = 0 } ^ 1 } L ' _ { v , j } ( \pi ^ j ) & ( v = 2 , p _ 1 , \cdots , p _ k ) , \\ L ' _ { v , 0 } & ( \mathrm { o t h e r w i s e } ) , \\ \end{cases} \\ K _ { \ell } \otimes \Z _ v & = \begin{cases} \displaystyle { \bigoplus _ { j = 0 } ^ 1 } K _ { \ell , v , j } ( \pi ^ j ) & ( v = 2 , p _ { k ' + 1 } , \cdots , p _ k ) , \\ K _ { \ell , v , 0 } & ( \mathrm { o t h e r w i s e } ) , \end{cases} \end{align*}


\begin{align*} \tilde { \tau } ( \psi ) ( x , t ) & : = \frac { 1 } { } \tau _ 1 ( \psi ) ( x , t ) - \frac { 1 } { ^ 2 } \Big ( \int _ { D \times D } \tilde { \gamma } ( q , q ' ) \psi ( x , q , t ) \psi ( x , q ' , t ) \ ; \mathrm { d } q \mathrm { d } q ' \Big ) \mathbb { I } \\ & = \frac { 1 } { } \tau _ 1 ( \psi ) ( x , t ) - \frac { \tilde { \mathfrak { z } } } { ^ 2 } \eta ^ 2 ( x , t ) \mathbb { I } , \end{align*}


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


\begin{align*} d ' ( \lambda _ 1 , \lambda _ 2 ) = \max \{ d _ G ( \lambda _ 1 , \lambda _ 2 ) , d _ H ( w _ 1 , w _ 2 ) \} . \end{align*}


\begin{align*} D _ E ( \phi \otimes \varepsilon ) = \sum _ { i = 1 } ^ { 2 n } ( e _ i \nabla ^ S _ { e _ i } \phi ) \otimes \varepsilon + ( e _ i \phi ) \otimes \nabla ^ E _ { e _ i } \varepsilon . \end{align*}


\begin{align*} ( m + 2 ) ( m + 1 ) s ^ m w _ { t t } + \varepsilon ( n + 2 ) ( n + 1 ) t ^ n w _ { s s } = 0 \ , , \end{align*}


\begin{align*} \gamma ^ t ( x ) : \S & \rightarrow \mathcal { M } , \\ x & \mapsto \gamma ^ t ( x ) : = \phi ( t \cdot T , x ) . \end{align*}


\begin{align*} s ( x ) \ = \ \min \bigg \{ n : E ( n ) \leq \frac { 1 } { x + 1 } \bigg \} . \end{align*}


\begin{align*} \left \{ \begin{aligned} \Delta \Phi & = \varepsilon ^ { - 2 } ( 1 - \left ( x ^ { 2 } + y ^ { 2 } \right ) ) & \Omega , \\ \Phi & = g & \partial \Omega . \end{aligned} \right . \end{align*}


\begin{align*} \mathcal { V } _ Q \ , \rho ( x , \omega ) \ , \mathcal { V } _ Q ^ { - 1 } f ( t ) & = e ^ { \pi i Q t \cdot t } e ^ { \pi i x \cdot \omega } e ^ { 2 \pi i \omega \cdot ( t - x ) } e ^ { - \pi i Q ( t - x ) \cdot ( t - x ) } f ( t - x ) \\ & = e ^ { \pi i ( \omega + Q x ) \cdot x } e ^ { 2 \pi i ( \omega + Q x ) \cdot ( t - x ) } f ( t - x ) \\ & = \rho ( x , Q x + \omega , \tau ) f ( t ) = \rho ( V _ Q ( x , \omega ) , \tau ) f ( t ) , \end{align*}


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


\begin{align*} \gamma ^ \prime & = r n + 1 2 r - 1 4 n - 6 m - 8 \Theta - 4 \Delta \ , ; \\ t & = \frac { 1 } { 6 } \left ( r ^ 3 - 3 r ^ 2 - 5 8 r - 3 r \left ( n + 3 m + 3 \Theta \right ) + 4 2 n + 7 8 m + 7 8 \Theta \right ) \ , ; \\ t ^ \prime & = \frac { 1 } { 6 } \left ( r ^ 3 - 3 r ^ 2 - 5 8 r - 3 r \left ( m + 3 n + 3 \Theta \right ) + 4 2 m + 7 8 n + 7 8 \Theta \right ) \ , , \end{align*}


\begin{align*} B _ - ( y , u ) : = \{ t \in [ 0 , T ] \colon y ( t ) = - r \exists \varepsilon > 0 y - u = \mathrm { c o n s t } [ t , t + \varepsilon ) \} , \end{align*}


\begin{align*} \Big ( \sum _ j u _ j ( t ) \Big ) ^ 2 = \big ( u _ { j _ 1 } ( t ) + \cdots + u _ { j _ m } ( t ) \big ) ^ 2 \lesssim \big ( \max _ \ell u _ { j _ \ell } ( t ) \big ) ^ 2 \leq \sum _ { \ell } u _ { j _ \ell } ( t ) ^ 2 \leq \sum _ j u _ j ( t ) ^ 2 . \end{align*}


\begin{align*} \mathfrak { i } : M _ n ( \mathbb { H } ) \stackrel { \sim } \longrightarrow \{ x \in M _ { 2 n } ( \C ) : \overline { x } J _ n ' = J _ n ' x \} . \end{align*}


\begin{align*} g _ j \ast f _ 1 ( x ) & = \int \limits _ { y \in ( x - W _ i ) \cap \mathfrak { D } ^ * } a ( y ) f _ 1 ( x - y ) d y . \end{align*}


\begin{align*} \begin{aligned} & h _ q + G _ \beta ^ \ast ( v ( y _ q , t _ q ) , 0 , Y _ q , K ) \geq 0 , \\ & k _ q + G _ \beta ^ \ast ( v ( z _ q , t _ q ) , 0 , Z _ q , K ) \geq 0 . \end{aligned} \end{align*}


\begin{align*} S ( H _ { 1 5 } , 1 5 f ) = \frac { 1 4 f - \left ( 1 2 + 3 \varepsilon _ d \right ) d + 1 } { 3 } \hbox { a n d } N _ { 1 5 } ' ( f , H ) = - \frac { 3 2 + 1 5 \varepsilon _ d } { 4 8 } d + \frac { 1 - 2 \varepsilon _ d } { 4 8 } . \end{align*}


\begin{align*} 0 = \langle J ^ { ' } ( \widehat { u } ) , \widehat { u } \rangle + \lambda _ 0 \langle \varphi ^ { * } _ { \widehat { u } } , \widehat { u } \rangle = \lambda _ 0 \langle \varphi ^ { * } _ { \widehat { u } } , \widehat { u } \rangle , \ \ \ \varphi ^ * _ { \widehat { u } } \in \partial \varphi ( \widehat { u } ) \end{align*}


\begin{align*} \frac { B } { A } = e ^ { 2 a \gamma } . \end{align*}


\begin{align*} \nu ^ { \epsilon , L } ( d \varphi ) = e ^ { - V _ 0 ^ { \epsilon , L } ( \varphi ) } \gamma ^ { \epsilon , L } ( d \varphi ) , \end{align*}


\begin{align*} J _ { 1 1 } = C + O ( E _ y ) , \end{align*}


\begin{align*} \tilde { \Gamma } ( f , g ) ( x ) = \frac { 1 } { \alpha } \left ( \Gamma ( f , g ) ( x ) + \Gamma ^ * ( f , g ) ( x ) \right ) . \end{align*}


\begin{align*} \Delta _ 0 = \sum _ { 1 \le i < j \le d } D _ { i , j } ^ 2 . \end{align*}


\begin{align*} \mathcal { H } ^ { \zeta } _ t ( E ) = \inf \left \{ \sum _ { n \in \mathbb { N } } \zeta ( \vert B _ n \vert ) : \ ( B _ n ) _ { n \in \N } \mbox { c l o s e d b a l l s , } \vert B _ n \vert \leq t E \subset \bigcup _ { n \in \mathbb { N } } B _ n \right \} . \end{align*}


\begin{align*} \mathcal { K } ^ { \# } ( x , k ) = ( k ) \mathcal { K } ( x , k ) . \end{align*}


\begin{align*} G ( \phi ^ { n } ) = \frac { \delta E } { \delta \overline { \phi } } ( \phi ^ { n } ) = - \frac 1 2 \Delta { \phi } ^ { n } + V \phi ^ n + \beta | \phi ^ n | ^ 2 \phi ^ n - \Omega L _ z \phi ^ n , n = 0 , 1 , \ldots . \end{align*}


\begin{align*} T _ { \ell i } ^ m = \sum _ { j = 1 } ^ { K _ { \ell i } ^ { m } } X _ { \ell i j } , \end{align*}


\begin{align*} \lim _ { t \rightarrow 0 ^ + } s ( r , t ) = s ( r , 0 ) , r > 0 . \end{align*}


\begin{align*} \lambda = 2 \Lambda \end{align*}


\begin{align*} - ( \omega ( z ) - z ) ^ 2 + \frac { \omega ( z ) - z } { G _ { { \mu } } ( \omega ( z ) ) } + | \lambda | ^ 2 = 0 . \end{align*}


\begin{align*} C ( x _ 1 , \cdots , x _ n ) : = \left | \begin{array} { l l l } x _ 1 ^ 1 & \cdots & x _ 1 ^ n \\ & \cdots & \\ x _ n ^ 1 & \cdots & x _ n ^ n \end{array} \right | . \end{align*}


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


\begin{align*} \left \{ \begin{array} { l l } \frac { 4 | S _ { \ell } | } { | N | } = \frac { 2 } { | T _ { \ell ' } | } & \mbox { i f } a = 1 , \mbox { a n d } \\ \frac { 2 | S _ { \ell } | } { | N | } \left ( \alpha ( a ) + \alpha ( a ^ { - 1 } ) \right ) = \frac { 1 } { | T _ { \ell ' } | } \left ( \alpha ( a ) + \alpha ( a ^ { - 1 } ) \right ) & \mbox { i f } 1 \neq a \in ( \mu _ { q - 1 } ) _ { \ell ' } . \end{array} \right . \end{align*}


\begin{align*} ( t , x ) \mapsto ( T ( t ) f ) ( x ) = \lim _ { n \to \infty } \big ( T _ n ( t ) f \big ) ( x ) \end{align*}


\begin{align*} u = x y , \ ; \ ; 2 v = x + y . \end{align*}


\begin{align*} N _ t ^ { R \nu } = - A _ t ^ { \nu } , t \in [ 0 , \zeta [ \end{align*}


\begin{align*} \lambda _ { n - k + 1 } = \min _ { U } \left \{ \max _ { f \in U } \left \{ \frac { \langle f , A f \rangle } { \langle f , f \rangle } \right \} ~ \middle | ~ ( U ) = k \right \} . \end{align*}


\begin{align*} x _ i = a _ i x _ 1 + b _ i x _ 2 ~ ~ ~ \mbox { f o r } 3 \leq i \leq 2 k . \end{align*}


\begin{align*} r _ { h , i } ( w ) = F _ h ( \varphi _ i ) - a _ h ( w , \varphi _ i ) \ , , i \in I _ h \ , , \end{align*}


\begin{align*} F ^ { i j } \bar { R } _ { \nu j l i } \bar { g } ( \lambda \partial _ { r } , e _ { l } ) & = - \lambda \left ( \frac { \lambda '' } { \lambda } + \frac { c - \lambda '^ { 2 } } { \lambda ^ { 2 } } \right ) F ^ { i j } ( \delta _ { i j } r _ { l } - \delta _ { j l } r _ { i } ) r _ { \nu } r _ { l } \\ & = - u \left ( \frac { \lambda '' } { \lambda } + \frac { c - \lambda '^ { 2 } } { \lambda ^ { 2 } } \right ) \Big ( ( \sum _ { l } r _ { l } ^ { 2 } ) ( \sum _ { i } F ^ { i i } ) - F ^ { i j } r _ { i } r _ { j } \Big ) . \end{align*}


\begin{align*} A _ \gamma : = \frac { 1 } { \gamma } ( I d - J ^ A _ \gamma ) . \end{align*}


\begin{align*} \lambda _ { i , \infty , \beta } = 2 \beta \left ( \frac { \alpha } { 2 } - i \right ) , \end{align*}


\begin{align*} & N ^ K _ { j } = N ^ K _ { j - \lfloor j \delta _ K \rfloor / \delta _ K } \quad p ( j \delta _ K ) = p ( ( j - \lfloor j \delta _ K \rfloor / \delta _ K ) \delta _ K ) , \mbox { w h e n } j \geq 1 / \delta _ K , \\ & N ^ K _ { j } = N ^ K _ { j + \lceil | j | \delta _ K \rceil / \delta _ K } \quad p ( j \delta _ K ) = p ( ( j + \lceil | j | \delta _ K \rceil / \delta _ K ) \delta _ K ) , \mbox { w h e n } j < 0 . \end{align*}


\begin{align*} T B _ { 6 } ( f ) = T B _ { 5 } ( f ) , \end{align*}


\begin{align*} x _ { \rm T T L S } - x _ { \rm Q i T T L S } = - ( \hat { \mathbf { V } } _ { 1 1 } ^ T ) ^ { \dag } \check { E } ^ T x _ { \rm T T L S } - x _ { \rm T T L S } + { ( \hat { \mathbf { V } } _ { 1 1 } ^ T ) } ^ { \dag } \hat { \mathbf { V } } _ { 1 1 } ^ T x _ { \rm T T L S } - ( \hat { \mathbf { V } } _ { 1 1 } ^ T ) ^ { \dag } e . \end{align*}


\begin{align*} u ^ i ( t , \mu ) = h ^ i ( t ) g ^ i ( \mu ) \end{align*}


\begin{align*} \widetilde { \mathcal { G } } _ { \alpha , \beta } \widetilde { P } _ n ^ { \alpha , \beta } = - n ( n - \alpha - \beta + 1 ) \widetilde { P } _ n ^ { \alpha , \beta } . \end{align*}


\begin{align*} \tilde { \lambda } _ { \varepsilon } \le \frac { \int _ { \tilde { \Omega } _ { \varepsilon } } | \nabla P _ { \tilde { \Omega } _ { \varepsilon } } w | ^ 2 } { \int _ { \tilde { \Omega } _ { \varepsilon } } \tilde { m } _ 0 \ , | P _ { \tilde { \Omega } _ { \varepsilon } } w | ^ 2 } = \frac { \tilde { \lambda } _ 0 + B } { 1 - A } \end{align*}


\begin{align*} \left \{ \begin{aligned} u ( x ) + \frac { 1 } { 2 } | u ^ \prime ( x ) | ^ 2 - f ( x ) & \leq 0 \ , ( - 1 , 1 ) , \\ u ( x ) + \frac { 1 } { 2 } | u ^ \prime ( x ) | ^ 2 - f ( x ) & \geq 0 [ - 1 , 1 ] , \end{aligned} \right . \end{align*}


\begin{align*} m _ { i , \lambda } ( \pi _ 1 ^ { - 1 } ( \omega ) ) = 0 . \end{align*}


\begin{align*} p ( Z ) & = \left ( ( B \circ A ^ { r _ 0 } ) ^ { r _ 1 } \circ A , B \circ A ^ { r _ 0 } \right ) . \end{align*}


\begin{align*} | X ( K _ q ) - \mu _ d ( X ) q ^ d | = O ( q ^ { d - 1 / 2 } ) . \end{align*}


\begin{align*} u ( x ) & = \gamma _ { n , s } \int _ { \R ^ n _ + \setminus B _ r ^ + } \bigg ( \frac { r ^ 2 - \vert x \vert ^ 2 } { \vert y \vert ^ 2 - r ^ 2 } \bigg ) ^ s \bigg ( \frac 1 { \vert x - y \vert ^ n } - \frac 1 { \vert x _ \ast - y \vert ^ n } \bigg ) u ( y ) \dd y \end{align*}


\begin{align*} \gamma = \frac { P _ t \Bigl | \sum _ { n = 1 } ^ N g _ n c _ n h _ n B _ r ( \theta _ I ) + h _ d B _ d ( \theta _ d ) \Bigr | ^ 2 } { \sigma _ n ^ 2 } . \end{align*}


\begin{align*} \tilde T _ n ( k ) : = \sum \limits _ { \substack { A \subset \{ 1 , \dots , d \} \\ | A | = k } } \tilde S _ { n , A } ^ { \rm M } , \tilde S _ { n , A } ^ { \rm M } : = \frac 1 n \sum _ { i , j = 1 } ^ n \prod _ { p \in A } \tilde I _ { i , j } ^ { ( p ) } , \end{align*}


\begin{align*} \mathcal { P } ( M _ { - \omega } f ) ( x ) = \sum _ { k \in \Z ^ d } f ( k + x ) e ^ { - 2 \pi i \omega \cdot ( k + x ) } = \sum _ { k \in \Z ^ d } T _ { - x } M _ { - \omega } f ( k ) , \end{align*}


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


\begin{align*} & \widehat { R } _ j ^ i ( s ) = \left | \int _ 0 ^ s \left ( h _ j ^ i ( z ) - h ^ i ( z ) \right ) g ^ i \left ( \frac { 1 } { N } \sum _ { n = 1 } ^ { N } \delta _ { X ^ n ( z ) } \right ) \ , d z \right | \\ & + M \int _ 0 ^ T \left | g ^ i _ j \left ( \frac { 1 } { N } \sum _ { n = 1 } ^ { N } \delta _ { X ^ n ( z ) } \right ) - g ^ i \left ( \frac { 1 } { N } \sum _ { n = 1 } ^ { N } \delta _ { X ^ n ( z ) } \right ) \right | \ , d z . \end{align*}


\begin{align*} E _ 2 ( x ) = \sum _ { x \leq p \leq 2 x } \frac { 1 } { p } \sum _ { \substack { 0 < a < p \\ g c d ( n , p - 1 ) = 1 } } e ^ { \frac { i 2 \pi a ( \tau ^ { n } - u ) } { p } } \cdot \frac { 1 } { p } \sum _ { \substack { 0 < b < p \\ \gcd ( m , p - 1 ) = 1 } } e ^ { \frac { i 2 \pi b ( \tau ^ { e m } - v ) } { p } } \ll x ^ { 1 - 2 \varepsilon } . \end{align*}


\begin{align*} \int _ \Omega g ( u ( x ) ) d x = \int _ { \Omega \setminus \Omega _ 0 } g ( u ( x ) ) d x = \int _ { \Omega \setminus \Omega _ 0 } \lim _ { n \rightarrow \infty } g ( u _ n ( x ) ) d x \leq \lim _ { n \rightarrow \infty } \inf \int _ { \Omega \setminus \Omega _ 0 } g ( u _ n ( x ) ) d x . \end{align*}


\begin{align*} \frac { \partial \phi _ k ( x , z , t _ i ) } { \partial t } \approx \sum _ { n = \alpha } ^ { \beta } a _ n \phi ( x , z , t _ i + n \Delta t ) , i = 0 , 1 , . . . N , \end{align*}


\begin{align*} V _ { 1 , 1 } = \int _ { D } \left | f _ { x y } ( x , y ) \right | \omega ( x , y ) d x d y , \end{align*}


\begin{align*} \partial _ t f + v \cdot \nabla _ x f + { \rm { d i v } } _ v \big ( ( \alpha u _ N \chi _ { \{ \alpha | u _ N | \leq N \} } - v ) f \big ) - \Delta _ v f + \alpha { \rm { d i v } } _ v ( f L [ f ] ) = 0 . \end{align*}


\begin{align*} k ( s , t ) = ( s - t ) \bigg ( \frac { 1 } { h ( s ) } - s + t \bigg ) , s > 0 , \ , t \in \R . \end{align*}


\begin{align*} & \Sigma ^ \prime _ i = ( \Sigma ^ \prime _ { i - 1 } ) ^ \ast ( v _ { i - 1 } + v ) i = 1 , \dots , k \\ \end{align*}


\begin{align*} s - t > | \lambda | ^ 2 h ( s ) = | \lambda | ^ 2 \int _ \mathbb { R } \frac { t } { t ^ 2 + u ^ 2 } d \mu \boxplus \nu ( t ) > \frac { | \lambda | ^ 2 t } { t ^ 2 + M } \end{align*}


\begin{align*} \langle g _ 0 , M _ \omega T _ x g _ 0 \rangle = e ^ { - \pi i x \cdot \omega } e ^ { - \frac { \pi } { 2 } ( x ^ 2 + \omega ^ 2 ) } \end{align*}


\begin{align*} a _ { k , l } ( \rho ) : = ( - 1 ) ^ { k + l } \frac { \det [ U _ j ( y _ r ) ] _ { j = \overline { 1 , k - 1 } , \ , r = \overline { 1 , k } \setminus l } } { \det [ U _ j ( y _ r ) ] _ { j , r = \overline { 1 , k } } } . \end{align*}


\begin{align*} g _ j ( X _ n ) = p _ j ( X _ n ) + O ( n ^ { - k + 1 } ) , \end{align*}


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


\begin{align*} & \lim _ { n \longrightarrow \infty } ( A _ n , B _ n , C _ n , N _ n , K _ { W _ n } ) = ( A , B , C , N , K _ { W } ) , \ : K _ W \succ 0 , \\ & \lim _ { n \longrightarrow \infty } ( F _ n , G _ n , \Gamma _ n , D _ n , K _ { Z _ n } ) = ( F , G , \Gamma , D , K _ { Z } ) , \ : K _ Z \succeq 0 \end{align*}


\begin{align*} R _ - ( \kappa _ x ) = \frac { E _ { \text r } ^ - ( \kappa _ x ) } { E _ { \text i } ^ + ( \kappa _ x ) } \qquad T _ + ( \kappa _ x ) = \frac { E _ { \text t } ^ + ( \kappa _ x ) } { E _ { \text i } ^ + ( \kappa _ x ) } , \end{align*}


\begin{align*} \pi ( 0 , 0 , \tau ) \pi ( x , \omega , \tau ' ) = \pi ( x , \omega , \tau + \tau ' ) = \pi ( x , \omega , \tau ' ) \pi ( 0 , 0 , \tau ) , \end{align*}


\begin{align*} I _ 1 ( t , K ) = a ( K ) \int _ \R e ^ { - i 2 y K t + i y ^ 2 t } G ( y + K ) \frac { \psi ( y ) } { y } \chi _ 2 \big ( y t ^ \gamma \big ) \chi _ 1 ( ( y + K ) t ^ { 3 \alpha } ) \ , d y . \end{align*}


\begin{align*} \check { \mathcal T } _ { t , x , v } : = \check { \mathcal T } _ { t , x _ 1 , v _ 1 } : = [ t - \mathcal T _ { t , x _ 1 , v _ 1 } ] _ + . \end{align*}


\begin{align*} h ^ { - \frac { 1 } { 2 } + \alpha } k ^ { - \frac { 1 } { 2 } + \beta } \mathcal { G } ( 2 - \alpha - \beta , \alpha , \beta ; A , B , h , k ) = \mathcal { K } ( 0 , 0 , 2 - \alpha - \beta ; A , B , \alpha , \beta , h , k ) . \end{align*}


\begin{align*} \hat { b } _ { t } = \frac { 1 } { h } \int _ { 0 } ^ { T } K \left ( \frac { \tau - t } { h } \right ) \mathrm { d } X _ { \tau } , \end{align*}


\begin{align*} \partial _ y \Phi _ 1 ( t , y ) & = - \frac { 2 \mu _ i } { \sqrt { t } } [ \phi _ i ^ - ] ' ( y ) - \frac { 2 \mu _ { i + 1 } } { \sqrt { t } } [ \phi _ { i + 1 } ^ + ] ' ( y ) \\ & = - \frac { 2 \mu _ i } { \alpha _ i - \alpha _ { i + 1 } } \psi ' \bigg ( \frac { 3 ( y / \sqrt { t } - \alpha _ i ) } { \alpha _ i - \alpha _ { i + 1 } } \bigg ) + \frac { 2 \mu _ { i + 1 } } { \alpha _ i - \alpha _ { i + 1 } } \psi ' \bigg ( \frac { 3 ( y / \sqrt { t } - \alpha _ i ) } { \alpha _ i - \alpha _ { i + 1 } } \bigg ) \end{align*}


\begin{align*} A _ { n } x = b _ { n } . \end{align*}


\begin{align*} t v c = t u b = s x b = s y \tilde { b } . \end{align*}


\begin{align*} g : = f - \tilde f . \end{align*}


\begin{align*} \mathfrak { F } _ { ( \alpha , \beta ) } ( g ) = \{ \alpha \Z \times \beta \Z \mid \alpha \beta < 1 \} . \end{align*}


\begin{align*} { \mathcal L } ( \varphi ) = \sum _ { ( d , e ) \in D _ { 1 } ( H G ) \cup D _ { 2 } ( H G ) } \varepsilon ( d , e ) \ell _ { D } ( d , e ) . \end{align*}


\begin{align*} & \Phi _ { \gamma , \delta } ( \Phi _ { \gamma , \delta } ( d , e , s - r ) , e , t - s ) = e ^ { \gamma ( t - s ) } \Phi _ { \gamma , \delta } ( d , e , s - r ) + \varphi _ { \gamma } ( t - s ) ( e + \delta ) \\ & = e ^ { \gamma ( t - s ) } \big ( e ^ { \gamma ( s - r ) } d + \varphi _ { \gamma } ( s - r ) ( e + \delta ) \big ) + \varphi _ { \gamma } ( t - s ) ( e + \delta ) \\ & = e ^ { \gamma ( t - r ) } d + \varphi _ { \gamma } ( t - r ) ( e + \delta ) = \Phi _ { \gamma , \delta } ( d , e , t - r ) , \end{align*}


\begin{align*} \alpha ( t , x ) = a _ n ( x ) , ( t , x ) \in [ t _ n , t _ { n + 1 } ) \times X . \end{align*}


\begin{align*} | \Omega | ^ 2 = \left ( \int _ { \Omega } u _ 0 ^ { - p / 2 } ( x ) u _ 0 ^ { p / 2 } ( x ) d x \right ) ^ 2 \le \int _ \Omega u _ 0 ^ { - p } ( x ) d x \cdot \int _ \Omega u _ 0 ^ p ( x ) d x \le M _ 1 ^ * \int _ \Omega u _ 0 ^ p ( x ) d x . \end{align*}


\begin{align*} H X _ b ^ 0 ( X , A ) = \begin{cases} A & e ( X ) = 0 \\ 0 & e ( X ) > 0 ; \end{cases} \end{align*}


\begin{align*} \Box \phi = \left ( | \phi _ { t } | ^ 2 - | \phi _ { x } | ^ 2 \right ) \phi + \mathbf { 1 } _ { \omega } f ^ { \phi ^ { \perp } } \end{align*}


\begin{align*} L i _ { C , \mathfrak { s } } ( { \bf z } ) : = \sum _ { i _ 1 > \cdots > i _ r \geq 0 } \frac { z _ 1 ^ { q ^ { i _ 1 } } \cdots z _ r ^ { q ^ { i _ r } } } { L _ { i _ 1 } ^ { s _ 1 } \cdots L _ { i _ r } ^ { s _ r } } \in \mathbb { C } _ { \infty } . \end{align*}


\begin{align*} x _ i x _ k = \rho U _ { i k } \ , , \end{align*}


\begin{align*} P ^ N \tilde { \tau } _ { m } ( G _ { n } \times G _ { n } ) = \bigcup _ { \xi \in G _ { n } } P ^ N \tilde { \tau } _ { m } ( \xi \times 1 _ { n } ) = \bigcup _ { \xi \in G _ { n } } P ^ N \tilde { \tau } _ { m } ( 1 _ n \times \xi ) . \end{align*}


\begin{align*} U ^ k _ i f ( x _ k ) = \frac { 1 } { \prod \limits _ { j = i + 1 } ^ k R ( j ) } \sum \limits _ { x _ { k - 1 } < x _ k } \ldots \sum \limits _ { x _ { i } < x _ { i + 1 } } f ( x _ i ) \end{align*}


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


\begin{align*} \kappa ( y ) = - \psi \left ( \frac { 1 } { 2 } \right ) + \psi \left ( \frac { 1 + \lvert y \rvert } { 2 } \right ) . \end{align*}


\begin{align*} \Delta _ \succ ( a _ 1 \dotsm a _ n ) : = \ 1 \otimes a _ 1 \dotsm a _ n + \sum _ { \substack { 1 \not \in S \subsetneq [ n ] \\ S \neq \varnothing } } w _ S \otimes w _ { J ^ S _ 1 } \vert \dotsm \vert w _ { J ^ S _ m } . \end{align*}


\begin{align*} { \rm s p e c } _ { L ^ 2 } ( D _ { \partial } ) \cap [ - \alpha , \alpha ] = \emptyset . \end{align*}


\begin{align*} I _ f ( u ) = \int _ { \overline \Omega } g _ u ^ p \ , d \mu - p \int _ { \partial \Omega } u f \ , d \nu & \le \int _ { \overline \Omega } g ^ p \ , d \mu - p \int _ { \partial \Omega } u f \ , d \nu \\ & = \lim _ { k \to \infty } \left ( \int _ { \overline \Omega } g _ k ^ p \ , d \mu - p \int _ { \partial \Omega } v _ k f \ , d \nu \right ) . \end{align*}


\begin{align*} \nabla _ { e _ i } R ( X , Y ) = - 2 \mu _ i R ( X , Y ) , \end{align*}


\begin{align*} a : V \times V \to \mathbb { R } \ , , a ( w , v ) = \int _ \Omega \mu \nabla w \cdot \nabla v + \boldsymbol { \beta } \cdot \nabla w \ , v + \sigma w \ , v \ , , \end{align*}


\begin{align*} \chi _ { f , G } ^ I ( \sigma , 1 ) = ( - 1 ) ^ n \chi _ { \widetilde { f } , \widetilde { G } } ^ { \overline { I } } ( \sigma , 1 ) \end{align*}


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


\begin{align*} \deg ( \pi ) = 2 . \end{align*}


\begin{align*} e = R _ { s s } \cdot N , \ \ \ \ \ \ f = R _ { s t } \cdot N , \ \ \ \ \ g = R _ { t t } \cdot N . \end{align*}


\begin{align*} ( D ^ 2 , S ^ 1 ) ^ \sigma = \left \{ ( z _ 1 , \ldots , z _ m ) \in ( D ^ 2 ) ^ m \ , : \ , z _ i \in S ^ 1 i \notin \sigma \right \} . \end{align*}


\begin{align*} P ( x ) : = \int _ 0 ^ x Q ( y ) \ , d y x \in \R \end{align*}


\begin{align*} \langle f , M _ \omega T _ x g \rangle = 0 , \ , \forall ( x , \omega ) \in \R ^ { 2 d } \Longrightarrow f = 0 . \end{align*}


\begin{align*} \sum _ { \alpha \in Y } \alpha ( a ) = \begin{cases} | Y | & Y \subseteq a ^ \perp \\ 0 & \end{cases} \end{align*}


\begin{align*} \boldsymbol { \varphi } = \{ \varphi _ n ^ { \sigma } : \ , \mathcal { I } _ n ^ { \sigma } \to \mathbb { R } \ , | \ , \sigma \in \mathcal { A } , \ , 1 \leq n \leq d \} \end{align*}


\begin{align*} U ^ 2 \subseteq V , \ ; U V \subseteq U , \ ; V ^ 2 \subseteq U , \ ; U V ^ 2 = 0 . \end{align*}


\begin{align*} \mathbf { E } _ l ^ { \ast } \left ( \left [ \begin{array} { c c } q & \sigma \hat { q } \\ 0 & \hat { q } \end{array} \right ] , s \right ) = \sum _ { h \in S } c ( h , q , s ) e _ { \mathbb { A } } ( \lambda ( h \sigma ) ) , \end{align*}


\begin{align*} w = { } & \frac { 1 } { 2 ( a ^ 2 + c ^ 2 ) } \bigg ( 1 - a ^ 2 \left ( x + \frac { b } { a \sqrt { D } } \right ) ^ 2 \\ { } & - b \left ( x + \frac { b } { a \sqrt { D } } \right ) \left ( y - \frac { 2 a } { \sqrt { D } } \right ) - c ^ 2 \left ( y - \frac { 2 a } { \sqrt { D } } \right ) ^ 2 \bigg ) . \end{align*}


\begin{align*} E ( \rho , \phi ) = \int _ { \Omega } \bigg ( \sum _ { i = 1 } ^ s \rho _ i \log \rho _ i + \frac { 1 } { 2 } \epsilon ( x ) | \nabla \phi | ^ 2 \bigg ) d x - \int _ { \Gamma _ D } \epsilon ( x ) \phi _ D ^ b \partial _ n \phi d s + \frac { 1 } { 2 \beta _ R } \int _ { \Gamma _ R } | \phi | ^ 2 d s . \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 } + \rho _ \ell \gamma _ \ell + \frac { c _ \ell } { T _ { \ell , \pi } ^ { a v } } \right ) , \end{align*}


\begin{align*} \sum _ { \substack { \ell = 1 \\ ( \ell , v ) = 1 } } ^ { \infty } \frac { 1 } { \phi ( u \ell ) \ell ^ s } = \frac { 1 } { \phi ( u ) } \zeta ( 1 + s ) R ( s ; u , v ) , \end{align*}


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


\begin{align*} \langle \cdot , \cdot \rangle _ \star : = \langle \cdot , \cdot \rangle \circ \overline { \mathcal { F } } \rhd \colon \Xi _ \star \otimes _ { \mathcal { X } _ \star } \Omega _ \star \rightarrow \mathcal { X } _ \star . \end{align*}


\begin{align*} Q ^ { \pm \infty } = \begin{pmatrix} 0 & C \\ C & 0 \end{pmatrix} , V ^ { \pm \infty } = 4 k ^ 2 Q ^ { \pm \infty } - 2 i k \sigma _ 3 ( Q ^ { \pm \infty } ) ^ 2 + 2 ( Q ^ { \pm \infty } ) ^ { 3 } . \end{align*}


\begin{align*} \| \operatorname { H e s s } y ^ j \| _ { C ^ { 0 , \alpha } ( B _ { r } ( p ) ) } \leq \tau , j = 1 , \dots , m . \end{align*}


\begin{align*} B _ l = \left \{ Q : \omega ( Q \cap \Omega _ l ) > { 1 \over 2 } \omega ( Q ) \omega ( Q \cap \Omega _ { l + 1 } ) \leqslant { 1 \over 2 } \omega ( Q ) \right \} \end{align*}


\begin{align*} S J ^ { - 1 } = \begin{pmatrix} B & - A \\ D & - C \end{pmatrix} , \end{align*}


\begin{align*} \widehat { \Phi } ( x ) = 1 + \widehat { \kappa } \big ( x \ , \widehat { \Phi } ( x ) \big ) . \end{align*}


\begin{align*} \bar { \Xi } _ L & \stackrel { \eqref { X i } } { = } \begin{bmatrix} \bar { y } _ { 1 , [ L , L + d _ 1 - 1 ] } \\ \vdots \\ \bar { y } _ { m , [ L , L + d _ m - 1 ] } \end{bmatrix} \stackrel { \eqref { l a s t _ d i _ o u t p u t s } } { = } \begin{bmatrix} H _ { d _ 1 } ( y _ { 1 , [ L , N + d _ 1 - 1 ] } ) \\ \vdots \\ H _ { d _ m } ( y _ { m , [ L , N + d _ m - 1 ] } ) \end{bmatrix} \alpha \\ & = H _ 1 ( \Xi _ { [ L , N ] } ) \alpha . \end{align*}


\begin{align*} \hat { I } _ { ( \infty , m ) } ( k ) : = \bigcup _ { n = 0 } ^ { \infty } \hat { I } _ { ( n , m ) } ( k ) \end{align*}


\begin{align*} \L ^ \circ = J \L ^ \perp . \end{align*}


\begin{align*} D ^ * \Phi ( ( x , 0 ) ; ( u , 0 ) ) ( \lambda , \mu ) \subset \begin{cases} \bigl ( D ^ * S ( x ; u ) ( \mu ) - \lambda , - \mu \bigr ) & - \lambda \in \mathcal N _ \Omega ( x ^ 1 ; u ^ 1 ) , \\ \varnothing & \end{cases} \end{align*}


\begin{align*} w ( x , t ) : = u ( x , t ) - \chi ( x , t ) - v ( x , t ) = \psi ( x , t ) - v ( x , t ) . \end{align*}


\begin{align*} B _ { k n } = ( \Phi _ { Z } ^ { k } ) ^ { - 1 } \circ p B _ { k n } \circ \Phi _ { Z } ^ { k } \end{align*}


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


\begin{align*} Z f ( x + l , \omega ) = e ^ { 2 \pi i l \cdot \omega } Z f ( x , \omega ) . \end{align*}


\begin{align*} ( u ( t , x ; s , u _ 0 ) , v ( t , x ; s , u _ 0 ) ) = ( u ( t , x ; \tilde s , \tilde u _ 0 ) , v ( t , x ; \tilde s , \tilde u _ 0 ) ) \forall \ , t \ge \tilde s , \end{align*}


\begin{align*} \mathcal P ( 0 ) = U ^ { - 1 } \tilde U . \end{align*}


\begin{align*} ( v - y _ i ( t ) ) ( y _ i ' ( t ) - u _ i ' ( t ) ) \geq 0 \forall v \in Z ~ t \in ( 0 , T ) i = 1 , 2 . \end{align*}


\begin{align*} | \langle e , M _ \omega T _ x f \rangle | = | V _ f f ( x - \Delta t , \omega - \Delta \omega ) | = | A f ( x - \Delta t , \omega - \Delta \omega ) | . \end{align*}


\begin{align*} \epsilon ( x ) = h _ + e ^ { \lambda _ 1 x } + h _ - e ^ { \lambda _ 2 x } - \frac { 1 } { \lambda _ 1 - \lambda _ 2 } \int _ { x } ^ \infty \left ( e ^ { \lambda _ 1 ( x - x ' ) } - e ^ { \lambda _ 2 ( x - x ' ) } \right ) g ( \epsilon ( x ' ) ) d x ' , \end{align*}


\begin{align*} \left [ ( \psi _ 1 \circ f ) ^ * \lambda \right ] = f ^ * \left [ \lambda + \mathrm { F l u x } ( \{ \psi _ t \} ) \right ] = [ \sigma ' ] - f ^ * \pi ^ * [ \sigma ] = 0 , \end{align*}


\begin{align*} H _ m ( G _ m ( t ) ) = t ^ 2 . \end{align*}


\begin{align*} \widehat D ^ * \Phi ( x , y ) ( \lambda , \mu ) = \begin{cases} \bigl ( \widehat D ^ * S ( x ^ 1 , x ^ 2 + y ^ 2 ) ( \mu ) - \lambda , - \mu \bigr ) & - \lambda \in \widehat { \mathcal N } _ \Omega ( x ^ 1 + y ^ 1 ) , \\ \varnothing & \end{cases} \end{align*}


\begin{align*} w ( \alpha ) + ( w i n d ( \eta _ { \kappa _ { - } } ) - 1 ) - w ( \beta ) = \# ( \gamma _ { 0 } \cap { \pi ( v \cap { [ - s _ { * } , s _ { * } ] \times S ^ { 3 } } ) } ) = \# ( \mathbb { R } \times \gamma _ { 0 } \cap v ) . \end{align*}


\begin{align*} \lim _ { \epsilon \to 0 } P _ 0 ^ \omega \left ( \sup _ { 0 \leq t \leq T } | \epsilon \chi ( X _ { t / \epsilon ^ 2 } ^ \omega , \omega ) | > \delta \right ) = 0 . \end{align*}


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


\begin{align*} \mathrm { R } ^ \mathcal { F } _ { t \star } ( X , Y , Z ) = \frac { ( \overline { \mathcal { R } } _ 1 \rhd Y ) \star \mathbf { g } _ { t \star } ( \overline { \mathcal { R } } _ 2 \rhd X , Z ) - X \star \mathbf { g } _ { t \star } ( Y , Z ) } { 2 c } , \mathrm { R i c } ^ \mathcal { F } _ { t \star } ( Y , Z ) = - \frac { \mathbf { g } _ { t \star } ( Y , Z ) } { 2 c } \end{align*}


\begin{align*} \frac { B } { A } = e ^ { 2 a \gamma } \coth \left ( \frac { \gamma n a } { 2 } \right ) ^ 2 . \end{align*}


\begin{align*} \sum _ { n = 0 } P _ n ( x ) { t ^ { n + 1 } \over ( n + 1 ) ! } = \log \left ( \sum _ { n = 0 } ^ \infty Q _ n ( x ) { t ^ n \over n ! } \right ) = \log \left ( { 1 \over \cos ( t ) - x \sin ( t ) } \right ) , \end{align*}


\begin{align*} \langle N _ { i } ^ { { \bf m } * } , N _ \ell \rangle = 0 , \ell \in { \bf m _ 1 } . \end{align*}


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


\begin{align*} \eta _ n = \lambda ^ { - 1 } _ n \phi ( \lambda _ n ) \cdot h ^ { - 1 } , \lambda _ { n + 1 } = \lambda _ n \cdot \eta _ n , \end{align*}


\begin{align*} | 2 w - e ^ { - \beta _ { \varepsilon } \Psi _ { \varepsilon } ( \mathbf { q } ) } V _ { \varepsilon } | = w + P _ { \Omega _ { \varepsilon } } w \le 2 | w | . \end{align*}


\begin{align*} \zeta ^ { ( h ) } _ n = \frac { 4 } { n ^ 2 \delta ^ 2 _ n ( k ) } \cdot | \mathcal P ( d , k ) | \cdot | \mathcal I _ h | \cdot O ( n ^ { k ( 2 - h ) } ) = O ( n ^ { ( k - 1 ) ( 2 - h ) } ) = o ( 1 ) , \end{align*}


\begin{align*} \Gamma _ t & = \prod _ { i = 1 } ^ { t } ( 1 - \alpha _ i ) = \prod _ { i = 2 } ^ { t } \frac { k - 1 } { k } = \frac { 1 } { k } \\ \end{align*}


\begin{align*} \tilde \theta = s _ * \frac { \alpha _ * + \gamma _ * / p _ 0 ‐ \gamma _ * / q } { \mu _ * + \alpha _ * + \gamma _ * ( s _ * + 1 / p _ 0 ‐ 1 / p _ 1 ) } , \end{align*}


\begin{align*} \alpha * \beta & = ( s _ 1 \circ a _ { j _ 1 } \circ \cdots a _ { j _ f } ) * ( s _ 2 + y _ { l _ 1 } + \cdots + y _ { l _ g } ) \\ & = ( a _ { j _ 1 } \circ \cdots a _ { j _ f } ) * ( y _ { l _ 1 } + \cdots + y _ { l _ g } ) \end{align*}


\begin{align*} \arg \Gamma \left ( \frac { \sigma + i T } { 2 } \right ) & = \frac { T } { 4 } \log \left ( \frac { \sigma ^ 2 + T ^ 2 } { 4 } \right ) + \left ( \frac { \sigma - 1 } { 2 } \right ) \arg \left ( \frac { \sigma + i T } { 2 } \right ) - \frac { T } { 2 } + o ( 1 ) \end{align*}


\begin{align*} \mathcal G _ 0 ( X + \xi , Y + \eta ) : = \frac { 1 } { 2 } ( \xi ( Y ) + \eta ( X ) ) . \end{align*}


\begin{align*} V + a ^ j Y _ j = W + b ^ k \overline { Y } _ k , \end{align*}


\begin{align*} \begin{array} { l } \alpha _ { 1 1 } ( r , \theta ) = { \left ( z _ { t t } - z _ { s s } \right ) \sin \theta \cos \theta + 2 z _ { s t } \cos ^ { 2 } \theta } , \\ \alpha _ { 1 2 } ( r , \theta ) = { - z _ { s s } } , \\ \alpha _ { 2 1 } ( r , \theta ) = { z _ { t t } } , \\ \alpha _ { 2 2 } ( r , \theta ) = { \left ( z _ { t t } - z _ { s s } \right ) \sin \theta \cos \theta - 2 z _ { s t } \sin ^ { 2 } \theta } , \end{array} \end{align*}


\begin{align*} \Phi _ { a , b } ( t ) : = t ^ { a } A ^ b S ( t ) Q ^ \frac { 1 } { 2 } , t \in ( 0 , \infty ) . \end{align*}


\begin{align*} g ( x ) - y = g ( z + t ) - Q _ z ( t ) = \int _ z ^ { z + t } ( g '' ( u ) + A ) ( z + t - u ) \ , d u \in \ge C M | t | ^ 2 . \end{align*}


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


\begin{align*} B _ { i j } ( t , y ) = B _ { i j , 0 } ( y ) + \sum _ { k = 1 } ^ n \mu _ k ( t ) \mathfrak { B } _ { i j k } ( y ) . \end{align*}


\begin{align*} g ( k ) = g ( | k | ) , \end{align*}


\begin{align*} C _ b = 0 , C _ W = \sigma _ 1 ^ { - 2 } , \end{align*}


\begin{align*} \tilde M _ 2 ( p , q , \tau , s , u _ 0 ) = & C _ 6 \Big ( \int _ \Omega u ^ { - p } ( \tau , x ) d x + M _ 1 ( p ) + \tilde M _ 1 ( p , \tau , s , u _ 0 ) \Big ) ^ { \frac { 2 ( q + 1 ) ( q + 1 - \varepsilon _ 0 / 2 ) } { \varepsilon _ 0 p } } \\ & - C _ 6 \Big ( 2 M _ 1 ( p ) \Big ) ^ { \frac { 2 ( q + 1 ) ( q + 1 - \varepsilon _ 0 / 2 ) } { \varepsilon _ 0 p } } . \end{align*}


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


\begin{align*} [ X _ a , X _ b ] & = { \rm H o r } ( [ X _ a , X _ b ] ) + { \rm V e r } ( [ X _ a , X _ b ] ) = [ \overline { X } _ a , \overline { X } _ b ] ^ h + { \rm V e r } ( [ X _ a , X _ b ] ) \\ & = { \rm V e r } ( [ X _ a , X _ b ] ) . \end{align*}


\begin{align*} G ( \theta ) = \int _ { a } ^ { b } \big ( \partial _ { \theta } f ( x , \theta ) \big ) ^ { 2 } \pi _ { a , b } ( x ) \mathrm { d } x . \end{align*}


\begin{align*} \mathbb { E } _ { i _ 1 , i _ 2 , \ldots , i _ { j } } \langle y , Q _ j x \rangle = \mathbb { E } _ { i _ 1 , i _ 2 , \ldots , i _ { j - 1 } } \langle \mathcal { L } y , Q _ { j - 1 } x \rangle = \mathbb { E } _ { i _ 1 , i _ 2 , \ldots , i _ { j - 2 } } \langle \mathcal { L } ^ 2 y , Q _ { j - 2 } x \rangle = \cdots = \langle \mathcal { L } ^ j y , x \rangle . \end{align*}


\begin{align*} ( ( \sigma ^ { \varphi } ) ^ { n / 2 } S ^ { n } ( b ) ) ^ * \delta _ { \varphi } ^ n ( ( \sigma ^ { \varphi } ) ^ { n / 2 } S ^ { n } ( b ) ) = 0 . \end{align*}


\begin{align*} \tilde { \lambda } _ { \varepsilon _ { n _ k } } = \tilde { \lambda } _ 0 + \Phi \ , e ^ { - \beta _ { \varepsilon _ { n _ k } } \tilde { \Psi } _ { \varepsilon _ { n _ k } } ( \mathbf { x } _ { \varepsilon _ { n _ k } } ) } + o \biggl ( e ^ { - \beta _ { \varepsilon _ { n _ k } } \tilde { \Psi } _ { \varepsilon _ { n _ k } } ( \mathbf { x } _ { \varepsilon _ { n _ k } } ) } \biggr ) k \to + \infty \ ; , \end{align*}


\begin{align*} \lim _ { n \to \infty } E \left [ \left ( \dfrac { S _ n } { \sqrt { n \log n } } \right ) ^ k \right ] = \mu _ k ( k \in \mathbb { N } ) . \end{align*}


\begin{align*} P _ n ^ { k } X _ { n + 1 } = X _ n , n \geq 1 . \end{align*}


\begin{align*} \mathcal { P } _ t = ( P ^ { \nu _ \alpha } _ { \frac { t } { \alpha } } ) ^ * \circ P ^ { \nu _ \alpha } _ { \frac { t } { \alpha } } = P ^ { \nu _ \alpha } _ { \frac { t } { \alpha } } \circ ( P ^ { \nu _ \alpha } _ { \frac { t } { \alpha } } ) ^ * . \end{align*}


\begin{align*} \mathcal { N } _ \mu ( \mathcal { B } , \mathcal { U } , s ) & = \left \{ n \in \mathbb { N } \ : s _ \mu ( B _ n , U _ n ) \geq s \right \} . \end{align*}


\begin{align*} \begin{cases} e ^ { - 2 Q } \delta _ { s t } \leq \bar { h } _ { s t } \leq e ^ { 2 Q } \delta _ { s t } ; \\ ( r / 2 ) ^ { 1 + \alpha } \left \| \frac { \partial } { \partial x ^ { j } } \bar { h } _ { s t } \right \| _ { C ^ { 0 , \alpha } ( B _ { r / 2 } ( \bar { p } ) ) } \leq e ^ { 2 Q } , \end{cases} \end{align*}


\begin{align*} { A ( r ) = \left \{ x \in A \ : \ \forall \tilde { r } \leq r , \ m ( B ( x , \tilde { r } ) \cap A ) \geq \frac { 3 } { 4 } m ( B ( x , \tilde { r } ) ) \right \} } \end{align*}


\begin{align*} \mathcal { H } ^ { s } _ { t } ( E ) = \inf \left \{ \sum _ { n \in \mathbb { N } } \vert B _ n \vert ^ s : \ , \vert B _ n \vert \leq t , \ B _ n E \subset \bigcup _ { n \in \mathbb { N } } B _ n \right \} . \end{align*}


\begin{align*} ( \phi \vee \psi ) ( x _ 0 , \ldots , x _ { p + q } ) = \phi ( x _ 0 , \ldots , x _ p ) \psi ( x _ p , \ldots , x _ { p + q } ) . \end{align*}


\begin{align*} a ( u , v ) = F ( v ) \forall v \in V \ , . \end{align*}


\begin{align*} \begin{dcases} - \int _ { 0 } ^ { T } \overline { e ^ { \lambda ^ p _ k ( T - t ) } } p ( t ) \ , d t & = m _ { 1 , k } , \forall k \geq k _ 0 , \\ - \int _ { 0 } ^ { T } \overline { e ^ { \widehat \lambda _ { n _ j } ( T - t ) } } p ( t ) \ , d t & = m _ { 1 , j } , \forall \ , 1 \leq j \leq j _ 0 , \end{dcases} \end{align*}


\begin{align*} V = E _ 0 \oplus E _ 1 \oplus \ldots \oplus E _ r , \end{align*}


\begin{align*} \lambda ( \alpha , z ) ^ { \ast } \eta ( \alpha z ) \lambda ( \alpha , z ) = \eta ( z ) , \ , \ , \ , \delta ( \alpha z ) = | j ( \alpha , z ) | ^ { - 2 } \delta ( z ) . \end{align*}


\begin{align*} \lim _ { T \to \infty } \omega _ { V ( 0 ) } \circ \alpha _ T ^ { 0 \to \tau } ( \dot V ( \tau ) ) = \omega _ { V ( \tau ) } ( \dot V ( \tau ) ) . \end{align*}


\begin{align*} \langle \mathfrak { x } _ 1 , \mathfrak { y } _ 1 , \ldots , \mathfrak { x } _ b , \mathfrak { y } _ b , \mathfrak { z } _ 1 , \ldots , \mathfrak { z } _ { s } \rangle & = G , \\ \mathfrak { z } _ 1 + \cdots + \mathfrak { z } _ { s } & = 0 , \\ b - 1 + \sum \limits _ { i = 1 } ^ { s } \Big ( 1 - \frac { 1 } { | \mathfrak { z } _ { i } | } \Big ) & = \frac { g - 1 } { | G | } . \end{align*}


\begin{align*} \left ( \frac { d \psi } { d \zeta } + \frac { i \zeta } { 2 } \sigma _ 3 \psi \right ) _ { + } = \left ( \frac { d \psi } { d \zeta } + \frac { i \zeta } { 2 } \sigma _ 3 \psi \right ) _ { - } J ^ { \psi } . \end{align*}


\begin{align*} \sum _ { i = 0 } ^ { a - 1 } \binom { a + b - 2 } { \lfloor i / 2 \rfloor } \left ( \binom { a + b - 2 - \lfloor i / 2 \rfloor } { a - i - 1 } + \binom { a + b - 2 - \lfloor i / 2 \rfloor } { a - i - 2 } \right ) , \end{align*}


\begin{align*} a : = \psi _ + ( - \infty , 0 ) \in \R \setminus \{ 0 \} . \end{align*}


\begin{align*} \eta ^ 1 _ { A _ Y } ( V ) : A _ Y ( V ) & \to \alpha _ * \alpha ^ { - 1 } A _ Y ( V ) = A _ Y ( \alpha \circ \alpha ^ { - 1 } ( V ) ) \\ \varphi & \mapsto \varphi | _ { \alpha \circ \alpha ^ { - 1 } ( W ) } \end{align*}


\begin{align*} A ( M _ \eta T _ \xi f , M _ \eta T _ \xi g ) ( x , \omega ) = e ^ { 2 \pi i ( x \cdot \eta - \omega \cdot \xi ) } A ( f , g ) ( x , \omega ) . \end{align*}


\begin{align*} u ( a ) = \tau d ^ { - n - 2 } , \end{align*}


\begin{align*} \int _ M \nabla _ g u ( x ) \nabla _ g w ( x ) { \rm d } v _ g - \mu \int _ M \frac { u ( x ) w ( x ) } { d ^ 2 _ g ( x _ 0 , x ) } { \rm d } v _ g + \int _ M u ( x ) w ( x ) { \rm d } v _ g = \lambda \int _ M \alpha ( x ) \xi _ x w ( x ) { \rm d } v _ g . \end{align*}


\begin{align*} \| u - Q \| _ { H ^ 1 } < \varepsilon _ 0 \Rightarrow \exists ! \ ; ( \sigma , X ) , \ ; | \sigma | + | X | \le \eta _ 0 J ( \sigma , X , u ) = 0 . \end{align*}


\begin{align*} \tilde { f } _ { \delta } = \int _ 0 ^ { + \infty } P ^ { \delta } _ t ( g ) d t , \end{align*}


\begin{align*} w ^ \mu = \lambda _ \mu ^ { - 1 } ( v ^ \mu - u ^ \mu ) , k ^ \mu = \lambda _ \mu ^ { - 1 } ( H _ u ^ \mu - H _ v ^ \mu ) , \tau ^ \mu = \lambda _ \mu ^ { - 1 } ( \xi ^ \mu - \eta ^ \mu ) . \end{align*}


\begin{align*} ( u , v ) _ { H ^ 2 _ * } = \int _ U u _ { x x } v _ { x x } d x . \end{align*}


\begin{align*} \mathbf { P } \bigg ( \Big | \frac 1 n \sum _ { i = 1 } ^ n \mathbf { E } ( \eta _ i ^ 2 | \mathcal { F } _ { i - 1 } ) - \sigma ^ 2 \Big | \geq x \bigg ) \leq \frac { 1 } { H } \exp \bigg \{ - H x ^ 2 n \bigg \} . \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*} [ T u , T v , T w ] = T \Big ( D ( T u , T v ) w + \theta ( T v , T w ) u - \theta ( T u , T w ) v \Big ) , \forall u , v , w \in V . \end{align*}


\begin{align*} A b _ 0 ( x , \omega ) = \int _ \R b _ 0 ( t + \tfrac { x } { 2 } ) b _ 0 ( t - \tfrac { x } { 2 } ) e ^ { - 2 \pi i \omega \cdot t } \ , d t . \end{align*}


\begin{align*} \lim _ { T \to \infty } s ( \nu _ 0 \circ \alpha _ { T \Psi } ^ { 0 \to \tau } | \nu _ \tau ) = 0 . \end{align*}


\begin{align*} \check { \psi } ( \omega ) = \varepsilon ( a ) , \omega = \psi ( a - ) , a \in A . \end{align*}


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


\begin{align*} h ^ 0 ( X , \iota _ * F ) \leq \dfrac { \chi ( X , \iota _ * F ) } { 2 } + \dfrac { 1 } { 2 } \sum _ { i = 1 } ^ { n } \lVert \overline { p _ i p _ { i - 1 } } \rVert \end{align*}


\begin{align*} \left \{ \begin{aligned} \Delta u & = \varepsilon ^ { - 2 } \left ( e ^ { u } - \left ( x ^ { 2 } + y ^ { 2 } \right ) e ^ { - u } \right ) & B _ 1 ( 0 ) , \\ u & = 0 & \partial B _ 1 ( 0 ) . \end{aligned} \right . \end{align*}


\begin{align*} H o m ( X \times \R , Y ) = H o m ( X , \Omega Y ) . \end{align*}


\begin{align*} \frac { f ' } { f } = P \psi , \end{align*}


\begin{align*} \Sigma = \dot \bigcup _ { F \in \mathcal { F } _ P } N _ F \end{align*}


\begin{align*} e _ { \lambda } ^ { x } ( t ) = \sum _ { k = 0 } ^ { \infty } ( x ) _ { n , \lambda } \frac { t ^ { k } } { k ! } , ( \mathrm { s e e } \ [ 6 , 7 ] ) , \end{align*}


\begin{align*} \frac { \tilde { \phi } ^ { n + 1 } - \phi ^ n } { \tau } = \left ( \frac { 1 } { 2 } \Delta - V - \beta | \phi ^ n | ^ 2 + \Omega L _ z + \mu ^ n \right ) \phi ^ n , \end{align*}


\begin{align*} H ' \cap \{ 0 , \dots , k - 1 \} = \left ( H ' \cap \{ 0 , \dots , j - 1 \} \right ) \cup \{ j \} . \end{align*}


\begin{align*} \tilde \psi _ { i , \beta } ( y ) = - \phi _ { i , \infty , \beta } \int _ 1 ^ y \frac { ( y ' ) ^ { - ( d + 1 ) } e ^ { 2 \beta \frac { ( y ' ) ^ 2 } { 4 } } } { \phi ^ 2 _ { i , \infty , \beta } ( y ' ) } d y ' . \end{align*}


\begin{align*} | \widehat B ( \alpha _ i ) | = \frac { | \beta _ k | | B ( \alpha _ i ) | } { 2 D | \alpha _ i - \beta _ k | } \geqslant \frac { 1 } { 3 D } | B ( \alpha _ i ) | \geqslant \frac { \delta } { 3 D } . \end{align*}


\begin{align*} ( 2 \pi ) ^ 2 \mu ( k , \ell , m , n ) = \mu _ S ( k , \ell , m , n ) + \mu _ R ( k , \ell , m , n ) ; \end{align*}


\begin{align*} T f _ 1 ( x ) & = \langle K ( x , y ) u ( y ) f _ 1 ( y ) \rangle \\ & = \int _ { \R ^ N } K ( x , y ) u ( y ) [ f _ 1 ( y ) - f _ 1 ( x ) ] d \omega ( y ) + f _ 1 ( x ) \langle K ( x , \cdot ) , u ( \cdot ) \rangle \\ & = : I + I \ ! I . \end{align*}


\begin{align*} \lambda w _ i + ( 1 - \lambda ) h _ i = \frac { 1 } { 2 } \pm b i \end{align*}


\begin{align*} \bigg \{ \sqrt { \binom { d } { J } } x ^ J \bigg \} _ { | J | = d } \end{align*}


\begin{align*} \int _ { \R ^ n _ + } \frac { \dd z } { \vert e _ 1 + z \vert ^ { n + 2 s } } & = \frac { \pi ^ { \frac { n - 1 } 2 } \Gamma \big ( \frac { 1 + 2 s } 2 \big ) } { \Gamma \big ( \frac { n + 2 s } 2 \big ) } \int _ 0 ^ \infty \frac { \dd z _ 1 } { ( z _ 1 + 1 ) ^ { 1 + 2 s } } = \frac { \pi ^ { \frac { n - 1 } 2 } \Gamma \big ( \frac { 1 + 2 s } 2 \big ) } { 2 s \Gamma \big ( \frac { n + 2 s } 2 \big ) } . \end{align*}


\begin{align*} g _ \gamma ( t ) = \sqrt { \gamma } \ , e ^ { - \gamma | t | } , t \in \R , \end{align*}


\begin{align*} K _ 1 \ast K _ 2 = \{ \sigma \cup \tau \ , : \ , \sigma \in K _ 1 , \ , \tau \in K _ 2 \} . \end{align*}


\begin{align*} \sum _ { i = \lfloor N - \eta ^ { - 1 } \rfloor + 1 } ^ { N - 1 } [ ( N - i ) \eta ] ^ { - 1 / \alpha } \leq \frac { \alpha } { \alpha - 1 } \ , \eta ^ { - 1 } . \end{align*}


\begin{align*} W ( t ) = K ( t ) \ast W _ { 0 } + \int _ { 0 } ^ { t } { K ( t - \tau ) \ast f ( \tau ) \ , d \tau } , \end{align*}


\begin{align*} \partial _ \xi F ^ f ( x , z , \xi ) & = F ^ f ( x , z , \xi ) \cdot g ^ 0 ( x ) G ( x , z , \xi ) ^ { - 1 } \xi , \\ \partial _ z F ^ f ( x , z , \xi ) & = F ^ f ( x , z , \xi ) \cdot \left ( \frac { n \partial _ z f } { 2 ( 1 + f ) } \Big | _ { ( x , z ) } + t r ( G ^ { - 1 } \partial _ z G ) | _ { ( x , z , \xi ) } \right ) \ , , \end{align*}


\begin{align*} \rho _ S ( z , \tau ) = \rho ( S z , \tau ) . \end{align*}


\begin{align*} | G _ k ( f ) ( x ) | & = \Big | \int _ { \R ^ N } D _ k ( x , y ) D ^ M _ K ( f ) ( y ) d \omega ( y ) \Big | = \Big | \int _ { \R ^ N } D _ k ( x , y ) [ D ^ M _ K ( f ) ( y ) - D ^ M _ K ( f ) ( x ) ] d \omega ( y ) \Big | \\ & \lesssim \| f \| _ \eta \int _ { \R ^ N } | D _ k ( x , y ) | \| x - y \| ^ \eta d \omega ( y ) \lesssim r ^ { - k \eta } \| f \| _ \eta . \end{align*}


\begin{align*} \lambda _ { \varepsilon } = \lambda ^ 1 ( B ^ { \varepsilon } ( \mathbf { x } _ { \varepsilon } ) ) , \end{align*}


\begin{align*} f _ { \alpha } ^ { ( \gamma ) } ( o ) = f ^ { ( \gamma ) } ( o ) \Leftrightarrow f ^ { ( \gamma ) } ( o ) = \sum _ { \beta < \gamma } a _ { \beta } g _ { \beta } ^ { ( \gamma ) } ( o ) + a _ { \gamma } g _ { \gamma } ^ { ( \gamma ) } ( o ) . \end{align*}


\begin{align*} V \left ( \frac { g _ 2 g _ 3 u } { H X } \right ) = \frac { 1 } { 2 \pi i } \Psi \Big ( \frac { u } { X Q ^ { \vartheta } } \Big ) \int _ { ( \varepsilon ) } \left ( \frac { X H } { g _ 2 g _ 3 u } \right ) ^ { s _ 3 } \widetilde { V } ( s _ 3 ) \ , d s _ 3 . \end{align*}


\begin{align*} \int _ x ^ { + \infty } p _ \delta ( y ) d y = \frac { 1 } { \delta } \int _ { x ^ { \delta } } ^ { + \infty } z ^ { \frac { 1 } { \delta } - 1 } e ^ { - z } d z = \frac { 1 } { \delta } \Gamma \left ( \frac { 1 } { \delta } , x ^ { \delta } \right ) , \end{align*}


\begin{align*} n ! = \sqrt { 2 \pi n } \left ( \dfrac { n } { e } \right ) ^ n \left ( 1 + \dfrac { 1 } { 1 2 n } + \frac { 1 } { 2 8 8 n ^ { 2 } } + \cdots \right ) \end{align*}


\begin{align*} \Omega = \{ ( z _ 1 , \dots , z _ n ) \in \mathbb D ^ n \ , : \ , f _ i \circ \pi _ n ( z _ 1 , \dots , z _ n ) = 0 \ , , \ , 1 \leq i \leq n - 1 \} . \end{align*}


\begin{align*} u \mapsto \| u \| _ { C ^ { \gamma } } = \| u \| _ { L ^ \infty } + [ u ] _ { C ^ { \gamma } } . \end{align*}


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


\begin{align*} N ( T ) = \cup _ { T \le t < 0 } \tilde \Gamma ( t ) . \end{align*}


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


\begin{align*} \left \{ \begin{aligned} \Delta u & = \sigma \varepsilon ^ { - 2 } \left ( e ^ { u } - \left ( x ^ { 2 } + y ^ { 2 } \right ) e ^ { - u } \right ) & \Omega , \\ u & = \sigma g & \partial \Omega , \end{aligned} \right . \end{align*}


\begin{align*} \int _ \Omega | \nabla u | ^ { p - 2 } \nabla u \cdot \nabla \phi \ , d \mu = \int _ { \partial \Omega } \phi f d \nu \int _ \Omega | \nabla v | ^ { p - 2 } \nabla v \cdot \nabla \phi \ , d \mu = \int _ { \partial \Omega } \phi g d \nu . \end{align*}


\begin{align*} N ( R ; F ) = \sum _ { H \in H ( n - m ) } N ( R ; F , H ) . \end{align*}


\begin{align*} c = - b \ , . \end{align*}


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


\begin{align*} \frac { \partial u } { \partial Z _ { j } } = \sum _ { i = 1 } ^ { n } \frac { \partial g _ { i } } { \partial Z _ { j } } h _ { i } + \sum _ { i = 1 } ^ { n } g _ { i } \frac { \partial h _ { i } } { \partial Z _ { j } } , j = 1 , \cdots , n . \end{align*}


\begin{align*} \mathcal { O } _ { k , N } ( t _ m ) = & \frac { \sin \pi \alpha } { \pi } \int _ 0 ^ { t _ m } \int _ \sigma ^ { t _ m } \tilde { E } _ { k , N } ( t _ m - \sigma ) ( t _ m - s ) ^ { \alpha - 1 } ( s - \sigma ) ^ { - \alpha } \ , \dd s \ , \dd W ( \sigma ) \\ = & \frac { \sin \pi \alpha } { \pi } \int _ 0 ^ { t _ m } \int _ 0 ^ s \tilde { E } _ { k , N } ( t _ m - \sigma ) ( t _ m - s ) ^ { \alpha - 1 } ( s - \sigma ) ^ { - \alpha } \ , \dd W ( \sigma ) \ , \dd s . \end{align*}


\begin{align*} w \xi ^ { - 1 } = \left ( ( u ^ { ( 1 ) } \times \cdots \times u ^ { ( k - 1 ) } ) \zeta '^ { - 1 } \times u ^ { ( k ) } \right ) \eta ^ { - 1 } . \end{align*}


\begin{align*} w ( t ) = M ( w ) ( t ) , \end{align*}


\begin{align*} \langle J ^ { ' } ( t _ { \widehat { w } ^ + } \widehat { w } ^ { + } + s _ { \widehat { w } ^ - } \widehat { w } ^ { - } ) , \widehat { w } ^ + \rangle = 0 \ \ \langle J ^ { ' } ( t _ { \widehat { w } ^ + } \widehat { w } ^ { + } + s _ { \widehat { w } ^ - } \widehat { w } ^ { - } ) , \widehat { w } ^ - \rangle = 0 . \end{align*}


\begin{align*} \mathcal { L } ( \mathcal { A } ) = 0 , \mathcal { A } \coloneqq \{ \mathbf { x } \in \R ^ N : u _ { \ast } > 0 , \nabla u _ { \ast } = 0 \} \ ; . \end{align*}


\begin{align*} \norm { x ^ * - J ^ S _ { \mu _ 0 } ( x ^ * + \mu _ 0 T ^ \circ x ^ * ) } = 0 \end{align*}


\begin{align*} I _ \ell ( \tau ) = e ^ { \left ( \frac { 2 \ell } { \alpha } - 1 \right ) \tau } . \end{align*}


\begin{align*} J ^ E ( x , t , k ) = \left \{ \begin{aligned} & m ^ { G P } ( x , t , k ) J ^ { ( 3 ) } ( x , t , k ) \left ( m ^ { G P } \left ( x , t , k \right ) \right ) ^ { - 1 } , & k \in \Gamma ^ { ( 3 ) } \backslash U _ { \xi } , \\ & m ^ { G P } ( x , t , k ) m ^ { m o d } ( x , t , k ) \left ( m ^ { G P } \left ( x , t , k \right ) \right ) ^ { - 1 } , & k \in \partial U _ { \xi } , \end{aligned} \right . \end{align*}


\begin{align*} \| u \| _ { S \left ( \dot { H } ^ { s _ c } \right ) } = \sup \left \lbrace \| u \| _ { L _ t ^ q L _ x ^ r } : ( q , r ) ( \ref { S t r i c h a r t z p a i r } ) ( \ref { r e s t r i c t i o n o n S t r i c h a r t z p a i r } ) \right \rbrace . \end{align*}


\begin{align*} \limsup _ { L \to \infty } \max _ { 0 \leq | \alpha | \leq 3 } \sup _ { x \in Q } L ^ { - 2 | \alpha | } | D _ x ^ { \alpha } D _ y ^ { \alpha } ( K _ L ) _ { i _ 1 i _ 2 } ( x , y ) | _ { y = x } | \leq M , \ i _ 1 , i _ 2 = 1 , 2 , \dots , m . \end{align*}


\begin{align*} W g _ 0 ( x , \omega ) = 2 ^ d e ^ { - 2 \pi ( x ^ 2 + \omega ^ 2 ) } , \end{align*}


\begin{align*} ( x , b ) + ( x ' , b ' ) = ( \rho ( x + \varphi ( b , x ' ) + \gamma ( b , b ' ) , b + b ' ) , b + b ' ) \end{align*}


\begin{align*} f ( x , y ) \ = \ \frac { x y } { ( x ^ 2 + y ^ 2 ) ^ 2 } \end{align*}


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


\begin{align*} \frac { \tilde { \phi } ^ { n + 1 } - \phi ^ n } { \tau } = \left ( \frac { 1 } { 2 } \Delta - \vartheta ^ n \right ) \tilde { \phi } ^ { n + 1 } + \Big ( \vartheta ^ n - V - \beta | \phi ^ n | ^ { 2 } + \Omega L _ z + \mu ^ n \Big ) \phi ^ n , \end{align*}


\begin{align*} ( \mathbf { v } \mathrel { \trianglelefteq } ( D , E , F ) \ , \& \ , \mid E \cap \mathbf { v } _ 2 \mid = \omega ) \Rightarrow \mid H \cap \mathbf { v } _ 2 \mid = \omega \end{align*}


\begin{align*} \mathbb { E } \left [ \nabla f ( X _ { t } ^ { x ; l ^ \epsilon } ) \nabla _ { x } X _ { t } ^ { x ; l ^ \epsilon } G \right ] = \mathbb { E } \left [ D _ { s } f ( X _ { t } ^ { x ; l ^ \epsilon } ) \nabla _ { x } X _ { \gamma ^ { \epsilon } _ { s } } ^ { x ; l ^ \epsilon } G \right ] . \end{align*}


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


\begin{align*} M _ { u } : = \sum _ { u \leq v } \mu _ { \mathfrak { S } _ n } ( u , v ) F _ { v } , \end{align*}


\begin{align*} \sum _ { t = 1 } ^ { T } Z _ t \leq \sigma ^ 2 \log ( 1 / \delta ) \end{align*}


\begin{align*} X = \partial Y _ 0 \cong G \times _ K S . \end{align*}


\begin{gather*} \partial _ y ( \mathcal { L } Y ^ - ) = - e _ 0 Y ^ - , \partial _ y ( \mathcal { L } \textbf { } Y ^ + ) = e _ 0 Y ^ + , \\ Y ^ { + } ( y ) = Y ^ - ( - y ) , \quad \int _ { \mathbb { R } } Y ^ \pm = 0 , \\ ( Y ^ - , \mathcal { L } Y ^ - ) = ( Y ^ + , \mathcal { L } Y ^ + ) = 0 , \\ ( Y ^ - , \mathcal { L } Y ^ + ) = ( Y ^ + , \mathcal { L } Y ^ - ) \not = 0 . \end{gather*}


\begin{align*} \begin{array} { l l } e & = r ^ { m _ { j } - 2 } \left ( e _ 1 ( { \theta } ) + r e _ { 2 } ( r , \theta ) \right ) , \\ f & = r ^ { m _ { j } - 2 } \left ( f _ 1 ( { \theta } ) + r f _ { 2 } ( r , \theta ) \right ) , \\ g & = r ^ { m _ { j } - 2 } \left ( g _ 1 ( { \theta } ) + r g _ { 2 } ( r , \theta ) \right ) . \end{array} \end{align*}


\begin{align*} \varepsilon _ { \delta _ { 1 } \delta _ { 2 } \delta _ { 3 } } ^ { a _ { 1 } a _ { 2 } a _ { 3 } } : = - \mathrm { s i g n } \left [ \prod _ { u < w } ( a _ { u } - a _ { w } ) \cdot \prod _ { x < z } ( \delta _ { x } - \delta _ { z } ) \cdot \prod _ { r \neq s } ( a _ { r } - \delta _ { s } ) \right ] . \end{align*}


\begin{align*} \nu _ m ( d u ) = \frac { 2 } { \| u \| ^ d } \left ( \int _ 0 ^ { + \infty } g _ m ( 2 w ) L _ { \frac { d } { 2 } } \left ( \sqrt { 2 w } \| u \| \right ) d w \right ) d u , \end{align*}


\begin{align*} \mu ^ \# _ { R , 1 } ( { \bf k } ) = \sum _ { ( A _ 1 , A _ 2 , A _ 3 , A _ 4 ) \in \mathcal { X } _ R } \int _ { \R } \prod _ \ast \mathcal { K } ^ \# _ { A _ j } ( k _ j , x ) \ , d x , \ , \mathcal { X } _ R : = \{ S , R \} ^ 4 \smallsetminus \{ ( S , S , S , S ) \} , \end{align*}


\begin{align*} P _ { n , m } f ( x ) = P _ { n , 0 } f ( x ) + \sum _ { \mu = 1 } ^ m \langle f , f _ { n , \mu } \rangle f _ { n , \mu } . \end{align*}


\begin{align*} M _ { 3 , 2 } = \begin{bmatrix} 1 & - \lambda _ 3 ( \lambda _ 2 + 1 ) \\ 0 & \lambda _ 3 \lambda _ 2 \end{bmatrix} . \end{align*}


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


\begin{align*} \psi _ { \bar w } ^ Z \vert _ { \Gamma _ i } = Z ^ { \bar w } \circ L _ Z , \bar w \prec \bar t _ { n } ^ i . \end{align*}


\begin{align*} \left \{ \begin{aligned} & \mathrm { d } Y _ { t } = ( R _ { t } + \theta _ { 1 } ) \mathrm { d } t + \sigma \mathrm { d } W ^ { 1 } _ { t } + \mathrm { d } L ^ { 1 } _ { t } + \mathrm { d } R _ { t } , \\ & \mathrm { d } R _ { t } = \theta _ { 2 } ( 1 - R _ { t } ) \mathrm { d } t + \sigma \mathrm { d } W ^ { 2 } _ { t } + \mathrm { d } L ^ { 2 } _ { t } , \\ & Y _ { 0 } = y \in [ a , b ] , R _ { 0 } = r > 0 , \end{aligned} \right . \end{align*}


\begin{align*} \varphi = \varphi | _ { U _ 1 } + \cdots + \varphi | _ { U _ n } + \varphi | _ V \end{align*}


\begin{align*} \begin{aligned} y ^ * & \in \mathcal N _ { \mathbf T ( u ) } ( w _ s ( u , v ) ) \cap \ker \nabla g ( \bar x ) ^ * , \\ z ^ * & \in \mathcal T _ { \mathcal N _ { \mathbf T ( u ) } ( w _ s ( u , v ) ) } ( y ^ * ) , \end{aligned} \end{align*}


\begin{align*} V _ g f ( x , \omega ) = \langle f , \pi ( \l ) g \rangle = \int _ \R f ( t ) \overline { g ( t - x ) } e ^ { - 2 \pi i \omega t } \ , d t . \end{align*}


\begin{align*} \vert J ^ E - I \vert = \vert m ^ { G P } ( m ^ { m o d } - I ) ( m ^ { G P } ) ^ { - 1 } \vert \in \mathcal { O } ( t ^ { - 1 / 2 } ) . \end{align*}


\begin{align*} & \mathcal { L } \mathfrak { A } _ { i j k , 1 } = \mu _ k E _ { i j k } , \\ & \mathcal { L } \mathfrak { A } _ { i j , 2 } = F _ { i j } + \sum _ { k = 1 } ^ n \mu _ k ( p - 1 ) \Lambda Q Q ^ { p - 2 } \mathfrak { A } _ { i j k , 1 } . \end{align*}


\begin{align*} S ( a , b , 3 \delta f ) = s ( a - \delta f , b , 3 \delta f ) + s ( a + \delta f , b , 3 \delta f ) . \end{align*}


\begin{align*} \tfrac { 1 } { 4 \pi } \norm { f } _ 2 ^ 2 = \tfrac { 1 } { 2 } | \langle [ X , P ] f , f \rangle | \leq \norm { ( X - a ) f } _ 2 ^ 2 \norm { ( P - b ) f } _ 2 ^ 2 . \end{align*}


\begin{align*} T B _ { 8 } ( f ) = T B _ { 6 } ( f ) , \end{align*}


\begin{align*} C B ^ T - D A ^ T = - I \ ; \Leftrightarrow \ ; ( D A ^ T - C B ^ T ) ^ T = I ^ T \ ; \Leftrightarrow \ ; A D ^ T - B C ^ T = I . \end{align*}


\begin{align*} \pi _ { q , i } : = \frac { ( q - 1 ) _ \ell \cdot \ell ^ { - i } - 1 } { 2 } \ , , \end{align*}


\begin{align*} M ( w ) ( t ) = - i \int _ t ^ { + \infty } e ^ { i ( t - s ) ( \Delta - 1 ) } \left [ S ( \mathcal { V } _ { l ( k ) } ^ A + w ) - S ( \mathcal { V } _ { l ( k ) } ^ A ) + \varepsilon _ { l ( k ) } \right ] ( s ) d s . \end{align*}


\begin{align*} \lhd \ , T ^ + y , x \ , \rhd = \lhd \ , y , T x \rhd . \end{align*}


\begin{align*} S = \{ ( X , g ) : g X \} . \end{align*}


\begin{align*} \langle P M _ l T _ k g , g \rangle & = \langle X T _ l M _ { - k } \widehat { g } , \widehat { g } \rangle \\ & = l \langle T _ l M _ { - k } \widehat { g } , \widehat { g } \rangle + \langle T _ l M _ { - k } X \widehat { g } , \widehat { g } \rangle \\ & = \langle M _ l T _ k P g , g \rangle . \end{align*}


\begin{align*} x y \cdot \rho y = x \end{align*}


\begin{align*} \mathbf { y } _ k ^ { c p } [ t ] = \sum _ { m = 1 } ^ M \mathbf { g } _ { m k } [ t ] \ast \textbf { x } _ m ^ { c p } [ t ] + \mathbf { z } _ k [ t ] , \end{align*}


\begin{align*} 2 ^ { \mu _ * k _ * t } \cdot 2 ^ { ‐ \tilde m _ t ( s _ * + 1 / q ‐ 1 / p _ 1 ) } n ^ { 1 / q ‐ 1 / p _ 1 } = 2 ^ { ‐ \alpha _ * k _ * t } \cdot 2 ^ { ‐ \tilde m _ t ( 1 / q ‐ 1 / p _ 0 ) } n ^ { 1 / q ‐ 1 / p _ 0 } \stackrel { \eqref { t i l d e _ t h e t a _ d e f } } { = } n ^ { ‐ \tilde \theta } ; \end{align*}


\begin{align*} \sum _ { n \geq 0 } A _ n ( q ) \frac { t ^ n } { n ! } = \frac { 1 - q } { 1 - q e ^ { t ( 1 - q ) } } . \end{align*}


\begin{align*} C L h ^ { j } = A h ^ { j } - B \overline { h } ^ { j } + C \left [ g ^ { 2 } F ^ { j } + g \lambda G ^ { j } + \lambda ^ { 2 } H ^ { j } \right ] , \end{align*}


\begin{align*} Z ( M _ l T _ k g ) ( x , \omega ) = e ^ { 2 \pi i ( l \cdot x - k \cdot \omega ) } Z g ( x , \omega ) . \end{align*}


\begin{align*} \lim _ { n \to \infty } h _ n ^ j \xi _ n ^ j = \infty \end{align*}


\begin{align*} u _ { i j } ( x ; J ) = a _ { i j } j \leq \min ( J _ i ) u _ { i j } ( x ; J ) = f _ i ( x ) j \geq \max ( J _ i ) u _ { i j } ( x ; J ) = u _ { i j } ( x ) . \end{align*}


\begin{align*} P ( \rho ) - P ( r ) - P ' ( r ) ( \rho - r ) \geq \begin{cases} c ( \rho - r ) ^ 2 , & \rho \in ( \alpha _ 1 , \bar \rho - \alpha _ 1 ) , \\ \frac { p ( r ) } { 2 } , & \rho \in [ 0 , \alpha _ 1 ] , \\ \frac { P ( \rho ) } { 2 } > 1 , & \rho \in [ \bar \rho - \alpha _ 1 , \bar \rho ) . \end{cases} \end{align*}


\begin{align*} a ^ n - \gamma _ { n - 1 } a ^ { n - 1 } - \dots - \gamma _ { 1 } a = 0 , \end{align*}


\begin{align*} \deg ( \pi ) \ = \ 1 ] \end{align*}


\begin{align*} \begin{aligned} & f _ i = a _ { i 0 } z _ { i 0 } + \sum _ { j = 1 } ^ { n _ i } ( a _ { i j } - a _ { i j - 1 } ) z _ { i j } , z _ i \in \Delta _ i , \\ & z _ { i j } \geq \delta _ { i j } \geq z _ { i j + 1 } j \in \{ 1 , \ldots , n _ i - 1 \} , \\ & \delta _ { i } \in \{ 0 , 1 \} ^ { n _ i - 1 } , \end{aligned} \end{align*}


\begin{align*} P ( z , \partial _ z , \lambda ) f ^ { \lambda + 1 } = b ( \lambda ) f ^ \lambda \end{align*}


\begin{align*} P _ r = x _ r z \sum _ { j \geq 0 } ( P _ 1 + P _ 2 + \cdots + P _ { k + 1 - r } ) ^ j = \frac { x _ r z } { 1 - P _ 1 - P _ 2 - \cdots - P _ { k + 1 - r } } \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*} A \alpha _ \mathbf { p } ^ 4 + B \alpha _ \mathbf { p } ^ 3 + C \alpha _ \mathbf { p } ^ 2 + 1 = 0 \end{align*}


\begin{align*} \pi _ c ( x ) = x - \frac { \left \langle c , x - z \right \rangle } { \| c \| ^ 2 } c . \end{align*}


\begin{gather*} \omega ( \dot { x } _ i ^ \pm ( u ) ) = \dot { x } _ { i } ^ \mp ( u ) , \omega ( \dot { h } _ i ( u ) ) = \dot { h } _ i ( u ) , \\ \varsigma ( \dot { x } _ i ^ \pm ( u ) ) = \dot { x } _ i ^ \pm ( - u ) , \varsigma ( \dot { h } _ i ( u ) ) = \dot { h } _ i ( - u ) . \end{gather*}


\begin{align*} \sigma _ { } ( \mathcal { L } ) = \{ i \xi : \xi \in \mathbb { R } , | \xi | \ge 1 \} , \sigma ( \mathcal { L } ) \cap \mathbb { R } = \{ - e _ 0 , 0 , e _ 0 \} e _ 0 > 0 . \end{align*}


\begin{align*} \partial _ \xi ^ k T _ i ( \xi ) = \partial _ \xi ^ k \left ( C _ i \xi ^ { - \gamma + 2 i } \right ) + O \left ( \xi ^ { - \gamma + 2 i - 2 - k } \ln \xi \right ) , \xi \to + \infty . \end{align*}


\begin{align*} \kappa = ( \sigma ^ { \varphi } ) ^ { - 1 } \circ S ^ 2 , \rho = \sigma ^ { \varphi _ S } \circ S ^ 2 . \end{align*}


\begin{align*} \left | f _ \delta ( x ) \right | \leq \frac { 1 } { p _ \delta ( x ) } \left ( \int _ x ^ { + \infty } p _ \delta ( y ) d y \right ) ^ { \frac { 1 } { q } } \| g \| _ { L ^ p ( \mu _ \delta ) } = G _ \delta ( x ) \| g \| _ { L ^ p ( \mu _ \delta ) } , \end{align*}


\begin{align*} U ^ s \nu ( x ) : = \int _ M \varrho ( x , y ) ^ { - s } \nu ( d y ) , x \in M , \end{align*}


\begin{align*} \Psi ( \hat { f } _ 0 , \cdots , \hat { f } _ n ) = \int _ { \mathbb { R } ^ n } \hat { f } _ 0 d \hat { f } _ 1 \cdots d \hat { f } _ n . \end{align*}


\begin{align*} d ^ 2 ( \rho ^ 0 , \rho ^ 1 ) : = \min _ { ( \rho , u ) \in K } \sum _ { i = 1 } ^ 2 \int _ 0 ^ 1 \int _ { \Omega } | u _ i | ^ 2 \rho _ i d x d t . \end{align*}


\begin{align*} f ( t ) = \sum _ { k \in \Z ^ d } c _ k e ^ { 2 \pi i k \cdot t } , \end{align*}


\begin{align*} & B ^ { k } \triangleq \left \lbrace w \in X ^ k : \| w \| _ { X ^ k } \le 1 \right \rbrace , \\ & X ^ k \triangleq \Big \{ w : \left \langle \nabla \right \rangle w \in S ( [ t _ k , + \infty ) , L ^ 2 ) : \| w \| _ { X ^ k } = \sup _ { t \ge t _ k } \| \left \langle \nabla \right \rangle w \| _ { S ( [ t , + \infty ) , L ^ 2 ) } e ^ { ( k + \frac { 1 } { 2 } ) e _ 0 t } < \infty \Big \} , \end{align*}


\begin{align*} \underset { x \in \mathbb { R } ^ n , w \in \mathbb { R } ^ n } { } \ c ^ \top x + \frac { 1 } { 2 } x ^ \top Q x + \| D w \| _ 1 + \delta _ { \mathcal { K } } ( w ) , \ A x = b , w - x = 0 . \end{align*}


\begin{align*} \mathcal { C } : = \big \{ f : { \bf k } : = ( k _ 1 , k _ 2 , k _ 3 , k _ 4 ) \in \R ^ 4 \rightarrow \C , \ , \ , \mbox { s . t . } \ , f ( { \bf k } ) = \prod _ \ast a _ j ( k _ j ) , \ , \ , a _ j \in A \big \} , \end{align*}


\begin{align*} \begin{aligned} & \liminf _ { \alpha \to 0 } \left ( - \frac { | \tilde { x } - \tilde { y } | ^ 4 } { \varepsilon ^ 4 } + 2 \alpha \max _ { x \in K } | x | ^ 2 - \alpha ( | \tilde { x } | ^ 2 + | \tilde { y } | ^ 2 ) \right ) \\ \leq & \liminf _ { \alpha \to 0 } \left ( - \frac { | \tilde { x } - \tilde { y } | ^ 4 } { \varepsilon ^ 4 } + 2 \alpha \max _ { x \in K } | x | ^ 2 \right ) < 0 , \end{aligned} \end{align*}


\begin{align*} \frac { d } { d t } \int | B | \ , d x = 0 . \end{align*}


\begin{align*} x _ { k + 1 } = x _ k + E _ i ( E _ i ^ \intercal A ^ \intercal A E _ i ) ^ \dagger E _ i ^ \intercal A ^ \intercal ( b - A x _ k ) . \end{align*}


\begin{gather*} ( \phi \prec \psi ) \prec \rho = \phi \prec ( \psi * \rho ) , \\ ( \phi \succ \psi ) \prec \rho = \phi \succ ( \psi \prec \rho ) , \\ \phi \succ ( \psi \succ \rho ) = ( \phi * \psi ) \succ \rho . \end{gather*}


\begin{align*} \sum _ { i = 1 } ^ { k } | \{ \lambda \in \Lambda \mid \lambda \supset e \cup e _ { i } \} | = 2 \sum _ { 1 \leq i < j \leq k } | \{ \lambda \in \Lambda \mid \lambda \supset e \cup e _ { i } \cup e _ { j } \} | \equiv 0 \pmod 2 . \end{align*}


\begin{align*} z _ { k + 1 } = z _ k - B ^ { - 1 } \bar { A } ^ \intercal S _ { i _ k } ( S _ { i _ k } ^ \intercal \bar A B ^ { - 1 } \bar { A } ^ \intercal S _ { i _ k } ) ^ \dagger S _ { i _ k } ^ \intercal ( \bar A z _ k - b ) , \end{align*}


\begin{align*} \tfrac { \partial } { \partial x } \big ( x ^ 3 e ^ { - x } \big ) = ( 3 - x ) \ , x ^ 2 e ^ { - x } , \end{align*}


\begin{align*} G \cap H & = ( ( D \setminus \{ \bar { b } _ n \} ) \cap ( E \setminus \{ \bar { b } _ n \} ) ) \cup \\ & \qquad \qquad \cup ( ( D \setminus \{ \bar { b } _ n \} ) \cap [ \bar { a } _ n \mapsto \bar { b } _ n ] ( K ) ) \\ & \qquad \qquad \cup ( ( E \setminus \{ \bar { b } _ n \} ) \cap [ \bar { a } _ n \mapsto \bar { b } _ n ] ( J ) ) \\ & \qquad \qquad \cup ( [ \bar { a } _ n \mapsto \bar { b } _ n ] ( J ) \cap [ \bar { a } _ n \mapsto \bar { b } _ n ] ( K ) ) \end{align*}


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


\begin{align*} x _ { k + 1 } = x _ k - A ^ \intercal W _ k ( W _ k ^ \intercal A A ^ \intercal W _ k ) ^ \dagger W _ k ^ \intercal ( A x _ k - b ) , \end{align*}


\begin{align*} M _ { ( k _ 1 , \ell _ 1 , \ldots , k _ r , \ell _ r ) } = \left ( \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix} \right ) ^ { k _ 1 } \left ( \begin{matrix} 1 & 0 \\ 1 & 1 \end{matrix} \right ) ^ { \ell _ 1 } \dots \left ( \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix} \right ) ^ { k _ r } \left ( \begin{matrix} 1 & 0 \\ 1 & 1 \end{matrix} \right ) ^ { \ell _ r } . \end{align*}


\begin{align*} \tan ^ 2 ( 2 r \pi ) = \frac { 4 \tan ^ 2 ( r \pi ) } { ( 1 - \tan ^ 2 ( r \pi ) ) ^ 2 } \end{align*}


\begin{align*} h ( x ) = R ^ { Y } ( h \overline { \mu } ) ( x ) + 1 , x \in E . \end{align*}


\begin{align*} m ^ { G P } = \Delta _ \eta ( k ) : = \frac { 1 } { 2 } \left ( \begin{array} { c c } \chi _ \eta ( k ) + \chi ^ { - 1 } _ \eta ( k ) & i \left ( \chi _ \eta ( k ) - \chi _ \eta ^ { - 1 } ( k ) \right ) \\ - i \left ( \chi _ \eta ( k ) - \chi _ { \eta } ^ { - 1 } ( k ) \right ) & \chi _ \eta ( k ) + \chi _ \eta ^ { - 1 } ( k ) \end{array} \right ) , \end{align*}


\begin{align*} f ( x ) = g ( x ) \end{align*}


\begin{align*} w ( S ^ j _ k ) & = \frac { 1 } { q ^ { j ^ 2 } { k \choose k - j } _ q } \sum \limits _ { i = 0 } ^ j q ^ { { j - i \choose 2 } } { j \choose i } _ q { k + i \choose i } _ q \\ & \leq \frac { q ^ { j k } } { q ^ { j ^ 2 } { k \choose k - j } _ q } \sum \limits _ { i = 0 } ^ j q ^ { - i } \\ & \leq \left ( \frac { q } { q - 1 } \right ) ^ { \min ( j , k - j ) + 1 } \end{align*}


\begin{align*} \frac { 1 } { 2 } \frac { d } { d t } \| B \| ^ 2 _ { L ^ 2 } + \mu \| \Lambda ^ { \frac { \alpha } { 2 } } B \| ^ 2 _ { L ^ 2 } = - \int _ { \mathbb S ^ 1 } B J _ x B \ , d x + \int _ { \mathbb S ^ 1 } B _ x J B \ , d x . \end{align*}


\begin{align*} \Omega \simeq \bigcup _ { n = 1 } ^ { N _ { e l m } } E ^ n , \phi _ k ( x , z ) = \bigoplus _ { n = 1 } ^ { N _ { e l m } } \phi _ k ^ n ( x ^ n , z ^ n ) , \left ( x ^ n , z ^ n \right ) \in E ^ n , \end{align*}


\begin{align*} \nu _ 0 ( A ) = \iint _ A \ \frac { 1 } { d ( x , y ) ^ { \theta p + Q - 1 } } \ , d \nu \times \nu ( x , y ) , \end{align*}


\begin{align*} I \ ! I _ 1 = \int _ { \| u - y \| > t } \frac 1 { V ( x , u , t + d ( x , u ) ) } \Big ( \frac { t } { t + d ( x , u ) } \Big ) ^ { \varepsilon _ 0 } \frac 1 { V ( u , y , s + d ( u , y ) ) } \Big ( \frac { s } { s + \| u - y \| } \Big ) ^ { \varepsilon _ 0 } d \omega ( u ) \end{align*}


\begin{align*} \psi ( x ) = \frac { 1 } { 2 \pi i } \int _ { c - i T } ^ { c + i T } f + O ^ * ( A ( x ) + B ( x ) + 6 \log x ) . \end{align*}


\begin{align*} \omega _ \theta ^ S ( t , \eta ^ L _ r ( t ) ) = e ^ t \omega ^ S _ { 0 , \theta } ( e ^ { - t } r _ 0 ) \quad \omega ^ S _ { 0 , \theta } = \omega ^ S _ \theta ( t ) | _ { t = 0 } . \end{align*}


\begin{align*} ( - \Delta ) ^ s \partial _ 1 v ( 0 ) & = - c _ { n , s } \int _ { \R ^ n } \frac { \partial _ 1 v ( y ) } { \vert y \vert ^ { n + 2 s } } \dd y \\ & = c _ { n , s } \int _ { \R ^ n } v ( y ) \partial _ 1 \vert y \vert ^ { - n - 2 s } \dd y \\ & = - c _ { n , s } ( n + 2 s ) \int _ { \R ^ n } \frac { y _ 1 v ( y ) } { \vert y \vert ^ { n + 2 s + 2 } } \dd y \\ & = - 2 c _ { n , s } ( n + 2 s ) \int _ { \R ^ n _ + } \frac { y _ 1 v ( y ) } { \vert y \vert ^ { n + 2 s + 2 } } \dd y \end{align*}


\begin{align*} \mu ( k , \ell , m , n ) = \sqrt { 2 \pi } \delta _ 0 ( k - \ell + m - n ) + \mu ^ { \# } _ L ( k , \ell , m , n ) + \mu ^ { \# } _ R ( k , \ell , m , n ) . \end{align*}


\begin{align*} \ln \hat q = { 1 \over 2 } ( \theta x + \epsilon p ^ b ) . \end{align*}


\begin{align*} \mathcal { I } _ 1 ^ * ( h , k ) = J _ 1 + J _ 2 + J _ 3 + O \big ( X ^ { - \frac { 1 } { 2 } + \varepsilon } Q ^ { \frac { 5 } { 2 } } ( h k ) ^ { \varepsilon } \big ) + O \big ( X ^ { \varepsilon } Q ^ { \frac { 3 } { 2 } + \varepsilon } ( h k ) ^ { \varepsilon } \big ) . \end{align*}


\begin{align*} \lambda _ 1 x \cdot \lambda _ 2 ( x \cdot y ) = \lambda _ 3 y \end{align*}


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


\begin{align*} T ( x , t ) : = \mathcal { F } ^ { - 1 } \left [ \frac { 1 } { \sqrt { 2 \pi } } \exp \left ( \frac { - t \xi ^ { 2 } + i \gamma t \xi ^ { 3 } } { 1 + \xi ^ { 2 } } \right ) \right ] ( x ) . \end{align*}


\begin{align*} D W _ j & = \prod _ { i < k , i \neq j , k \neq j } ( \lambda _ i - \lambda _ k ) ^ 2 \cdot \prod _ { k \neq j } ( \lambda _ j - \lambda _ k ) \cdot \sum _ { k \neq j } ( \lambda _ j - \lambda _ k ) ^ { - 1 } \end{align*}


\begin{align*} \langle f , D ( c _ \gamma ) \rangle _ \mathcal { H } = \langle ( \langle f , e _ \gamma \rangle _ \mathcal { H } ) , ( c _ \gamma ) \rangle _ { \ell ^ 2 ( \Gamma ) } . \end{align*}


\begin{align*} f ( x ) & = \int _ 0 ^ \infty \psi _ t \ast q _ t \ast f ( x ) \frac { d t } { t } \\ & = - \sum \limits _ { j = - \infty } ^ \infty \int _ { r ^ { - j } } ^ { r ^ { - j + 1 } } \psi _ { j } \ast q _ { j } \ast f ( x ) \frac { d t } { t } + \sum \limits _ { j = - \infty } ^ \infty \int _ { r ^ { - j } } ^ { r ^ { - j + 1 } } \Big [ \psi _ { j } \ast q _ { j } \ast f ( x ) - \psi _ t \ast q _ t \ast f ( x ) \Big ] \frac { d t } { t } \\ & = - \ln r \sum \limits _ { j = - \infty } ^ \infty \psi _ { j } \ast q _ { j } \ast f ( x ) + R _ 1 ( f ) , \end{align*}


\begin{align*} ( - \Delta _ p ) ^ { \theta } u = f \in L ^ { p ' } ( Z ) . \end{align*}


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


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


\begin{align*} \| f ( x ) \| _ p ^ p \leq \sum \limits _ { l = - \infty } ^ \infty \| \sum \limits _ { Q \in B _ l } \omega ( Q ) D _ k ( x , x _ Q ) { \widetilde D } _ k ( f ) ( x _ Q ) \| _ p ^ p . \end{align*}


\begin{align*} \hat { G } ( t , \xi ) & = \hat { \phi } ( \xi ) | \xi | \hat { \psi } _ \xi ( t | \xi | ) . \end{align*}


\begin{align*} B _ { i } ^ j ( X ) = \varepsilon _ j g ( \nabla _ { X } e _ i , e _ j ) = - \varepsilon _ j g ( e _ i , \nabla _ { X } e _ j ) = - \varepsilon _ j \varepsilon _ i B ^ { i } _ j ( X ) . \end{align*}


\begin{align*} \varphi _ { \mathbf { L } } : \mathbb { C } [ y _ { \mathcal { I } } : \ , \mathcal { I } \in \mathfrak { G } ( \mathbf { L } ) ] \longrightarrow \mathbb { C } [ x _ { \alpha } : \ , \alpha \in [ n ] ] , \varphi _ { \mathbf { L } } ( y _ { \mathcal { I } } ) : = \prod _ { \alpha \in \mathcal { I } } x _ { \alpha } . \end{align*}


\begin{align*} \Bigg | \partial _ r ^ k \left ( \frac { 1 } { r } \partial _ r Q \right ) \Bigg | \lesssim \sum _ { j = 0 } ^ k r ^ { - ( k - j + 1 ) } | \partial _ r ^ { j + 1 } Q | \lesssim Q ( r ) , \forall \ ; r \ge 1 \end{align*}


\begin{align*} ( f \bullet g ) ( x ) : = q g ( x ) f ( x g ( x ) ) . \end{align*}


\begin{align*} T _ m M = V _ 0 ( m ) + V _ 1 ( m ) + \ldots + V _ s ( m ) + V _ 0 ' ( m ) \end{align*}


\begin{align*} W ^ J : = \{ w \in W \ , | \ , \ell ( w ) < \ell ( w s ) \ \ s \in J \} \end{align*}


\begin{align*} ( \rho , u ) : = ( \tilde \rho , \tilde u ) + ( \check \rho , \check u ) , \end{align*}


\begin{align*} u ( x ) = \inf \left \lbrace \int _ 0 ^ \infty e ^ { - s } \left ( L \left ( \gamma ( s ) , - \dot { \gamma } ( s ) \right ) \right ) d s : \gamma \in \mathrm { A C } ( [ 0 , \infty ) ; \overline { \Omega } ) , \gamma ( 0 ) = x \right \rbrace , \end{align*}


\begin{align*} \langle ( F L ) ^ a ( \pmb { v } ) , w _ q \rangle = \left . \frac { d } { d s } \right | _ { s = 0 } L ( v _ { 1 q } , \dots , v _ { a q } + s w _ q , \dots , v _ { k q } ) , w _ q \in T Q . \end{align*}


\begin{align*} b ( \tau ) \sim e ^ { \frac { 1 } { \alpha } ( \frac { \alpha } { 2 } - 1 ) \tau } . \end{align*}


\begin{align*} \frac { u ( h e _ 1 ) } h & = \frac 1 { h } \gamma _ { n , s } \int _ { \R ^ n _ + \setminus B _ r ^ + } \bigg ( \frac { r ^ 2 - h ^ 2 } { \vert y \vert ^ 2 - r ^ 2 } \bigg ) ^ s \bigg ( \frac 1 { \vert h e _ 1 - y \vert ^ n } - \frac 1 { \vert h e _ 1 + y \vert ^ n } \bigg ) u ( y ) \dd y \end{align*}


\begin{align*} \forall \delta \in \mathfrak { C } _ { c } , \ , a \in \mathfrak { C } _ { r } : h ( \mathcal { A } _ { \delta } ^ { a } ) \neq 0 \Leftrightarrow a = \varrho ( \delta ) . \end{align*}


\begin{align*} \Im \int x \cdot \nabla u ( x , t ) \bar { u } ( x , t ) d x = \frac { 1 } { 4 } \dot { y } ( t ) > 0 \end{align*}


\begin{align*} F ( t , x , v ) = \mu ( v ) + f ( t , x , v ) . \end{align*}


\begin{align*} x ^ \prime = \alpha _ 1 \alpha _ 2 \beta _ 1 \beta _ 2 x \ , , y ^ \prime = \alpha _ 1 ^ 2 \beta _ 1 ^ 2 y \ , , z ^ \prime = \alpha _ 1 \beta _ 1 \alpha _ 2 \beta _ 2 z \ , , w ^ \prime = \alpha _ 2 ^ 2 \beta _ 2 ^ 2 w \ , , \end{align*}


\begin{align*} & \varphi ( b _ 0 a _ 1 b _ 1 a _ 2 b _ 2 \cdots a _ n b _ n ) \\ = & \varphi ( b _ 0 a _ 1 ( c _ 1 + \tau ( b _ 1 ) ) a _ 2 \cdots ( c _ { n - 1 } + \tau ( b _ { n - 1 } ) ) a _ n b _ n ) \\ = & \varphi ( b _ 0 a _ 1 c _ 1 a _ 2 \cdots c _ { n - 1 } a _ n b _ n ) \\ & + \sum _ { r = 1 } ^ n \sum _ { 1 \leq k _ 1 < \cdots < k _ r \leq n - 1 } \varphi ( b _ 0 a _ 1 c _ 1 a _ 2 \cdots \hat { c } _ { k _ 1 } \cdots \hat { c } _ { k _ r } \cdots c _ { n - 1 } a _ n b _ n ) \tau ( b _ { k _ 1 } ) \cdots \tau ( b _ { k _ r } ) \end{align*}


\begin{align*} | \varphi ( z ) | & \leq C _ { \varepsilon } \mathrm { e } ^ { \varepsilon | z | } \left ( \sum _ { n = 0 } ^ { \infty } ( n ! ) ^ { 2 } 2 ^ { n q } | \varphi _ { n } | ^ { 2 } \right ) ^ { 1 / 2 } \left ( \sum _ { n = 0 } ^ { \infty } 2 ^ { - n q } \sigma _ { \varepsilon } ^ { - 2 n } \right ) ^ { 1 / 2 } \\ & = C _ { \varepsilon } ( 1 - 2 ^ { - q } \sigma _ { \varepsilon } ^ { - 2 } ) ^ { - 1 / 2 } \| \varphi \| _ { q , 1 , \pi _ { \lambda , \beta } } \mathrm { e } ^ { \varepsilon | z | } , \end{align*}


\begin{align*} u ( 0 , x ) ~ = ~ \overline { w } ( x ) + \left ( c _ 1 \cdot \chi _ { \strut ] - \infty , 0 [ } + c _ 2 \cdot \chi _ { \strut ] 0 , + \infty [ } \right ) \cdot \phi ( x , 0 ) . \end{align*}


\begin{align*} G _ R ( y ) = ( - R ) \vee \left ( y \wedge R \right ) . \end{align*}


\begin{align*} \lambda ^ p _ k & = - k ^ 2 \pi ^ 2 - 2 c _ k k \pi - 2 i d _ k k \pi + O ( 1 ) , k \geq k _ 0 , \\ \lambda ^ h _ k & = - 1 - \alpha _ { 1 , k } - i ( 2 k \pi + \alpha _ { 2 , k } ) , | k | \geq k _ 0 , \end{align*}


\begin{align*} W _ 1 ( \mu _ n , \gamma _ { \Sigma } ) \leq \dfrac { \| \Sigma ^ { - \frac { 1 } { 2 } } \| _ { o p } \| \Sigma \| _ { H S } } { n } \left ( \sum _ { k = 1 } ^ n ( U _ { \Sigma , \mu _ k } - 1 ) \right ) ^ { \frac { 1 } { 2 } } , \end{align*}


\begin{align*} x _ 1 ^ \prime = \varepsilon ^ 2 y _ 1 \ , , y _ 1 ^ \prime = - \varepsilon ^ 4 x _ 2 \ , , x _ 2 ^ \prime = \varepsilon ^ 3 y _ 2 \ , , y _ 2 ^ \prime = \varepsilon x _ 1 \ , , \end{align*}


\begin{align*} \operatorname { H e s s } z ^ { j } ( \bar { \nabla } z ^ { k } , \bar { \nabla } z ^ { l } ) = \operatorname { H e s s } y ^ { j } ( \nabla y ^ { k } , \nabla y ^ { l } ) \end{align*}


\begin{align*} C _ F ( \rho ^ a , \rho ^ b ) & = \Bigg \{ \rho \in H ^ { 1 } ( [ 0 , 1 ] ; \mathcal P ( G ) ) , m \in L ^ 2 ( [ 0 , 1 ] ; \mathcal S ^ { N \times N } ) \Bigg | ( \rho ( 0 ) , \rho ( 1 ) ) = ( \rho ^ a , \rho ^ b ) , \\ & d \rho _ i ( t ) + \underset { j \in N ( i ) } { \sum } m _ { i j } d t + \underset { j \in N ( i ) } { \sum } ( \Sigma _ j - \Sigma _ i ) \theta _ { i j } ( \rho ) d W ^ { \delta } ( t ) = 0 . \Bigg \} \end{align*}


\begin{align*} S _ { n + k ^ { \prime } } \phi ( x ) = S _ { n } \phi ( x ) \times \frac { n } { n + k ^ { \prime } } . \end{align*}


\begin{align*} - g \begin{pmatrix} \varphi ^ { j } \\ \psi ^ { j } \end{pmatrix} _ { s } - \begin{pmatrix} 0 & - e \\ g & - 2 f \end{pmatrix} \begin{pmatrix} \varphi ^ { j } \\ \psi ^ { j } \end{pmatrix} _ { t } = \Lambda \begin{pmatrix} \varphi ^ { j } \\ \psi ^ { j } \end{pmatrix} + \begin{pmatrix} e H ^ { j } - g F ^ { j } \\ 2 f H ^ { j } - g G ^ { j } \end{pmatrix} \end{align*}


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


\begin{align*} \mathrm { v a r } [ \widehat { K } _ { \mathrm { a - c p t - d } } ^ \mathbb { R } ] & = \frac { \psi } { 2 } \Big ( \frac { 1 } { N _ 2 \bar { \gamma } _ c ' } \ ! - \ ! \frac { \mathrm { l i } ( 1 - \xi ) K ' } { ( 1 + N _ 1 \bar { \gamma } ' ) ( 1 - \xi ) } \Big ) . \end{align*}


\begin{align*} x \geq 0 , ~ ~ ~ \omega = \mathcal { A } x ^ { m - 1 } + q \geq 0 , ~ ~ ~ \mbox { a n d } ~ ~ x ^ { T } ( \mathcal { A } x ^ { m - 1 } + q ) = 0 \end{align*}


\begin{align*} W _ n = \prod _ { i = 1 } ^ { k _ n } I _ 2 ( m _ { n , i } ) \end{align*}


\begin{align*} g ( x ) & = \sum _ { k = 1 } ^ { m } p _ k \theta _ k e ^ { - \theta _ k x } \\ \bar { G } ( x ) & = \sum _ { k = 1 } ^ { m } p _ k e ^ { - \theta _ k x } \end{align*}


\begin{gather*} P _ { \alpha , \beta } ^ l ( y ) : = \{ ( t , x ) : x \in [ y , y + l ] , t \in [ \alpha + x - y , \beta - x + y ] \} , \end{gather*}


\begin{align*} \lim _ { s \rightarrow 0 ^ + } s ^ 2 \Big ( 1 - s h ( s ) \Big ) = \lim _ { s \rightarrow 0 ^ + } \int _ 0 ^ \infty \frac { s ^ 2 u ^ 2 } { s ^ 2 + u ^ 2 } d \mu ( u ) = \lambda _ 2 ( \mu ) ^ 2 . \end{align*}


\begin{align*} d \mathbf { P } _ \lambda = Z _ n ( \lambda ) d \mathbf { P } . \end{align*}


\begin{align*} \check H ^ q _ { c t } ( X , A ) \cong \check H _ { c t } ^ q ( U _ 1 \cap U _ 2 , A ) \cong \check H _ { c t } ^ { q + 1 } ( U _ 1 \cup U _ 2 , A ) = \check H _ { c t } ^ { q + 1 } ( X \times \R , A ) \end{align*}


\begin{align*} f _ t - \frac { 1 } { 2 \sqrt { t } } f '' + \frac { 1 } { 2 \sqrt { t } } + \rho \max ( f ^ * - f , 0 ) = 0 , \end{align*}


\begin{align*} \{ u , v , w \} = \theta ( T v , T w ) u , \forall u , v , w \in V . \end{align*}


\begin{align*} 1 = \int _ 0 ^ 1 Q _ \alpha ( x ) \ , d x \leq \int _ 0 ^ 1 Q _ \alpha ( 1 ) x \ , d x = \frac 1 2 Q _ \alpha ( 1 ) . \end{align*}


\begin{align*} i k \mathcal { K } ^ { \# } _ { S } ( x , k ) = \partial _ x \mathcal { K } _ { S , \# } ' ( x , k ) - \mathcal { K } _ { S , \# } '' ( x , k ) \end{align*}


\begin{align*} u _ t ( x , t ) - | \nabla u ( x , t ) | m ( \{ u ( \cdot , t ) < u ( x , t ) \} ) = 0 \end{align*}


\begin{align*} F _ { \mu } ^ { [ \alpha , \beta ] , \rho , \varepsilon } = \left \{ x \in \mathbb { R } ^ d : \overline { \dim } ( \mu , x ) \in [ \alpha , \beta ] \forall r < \rho , \ \mu ( B ( x , r ) ) \geq r ^ { \overline { \dim } ( \mu , x ) + \varepsilon } \right \} . \end{align*}


\begin{align*} z = r ^ m P ( \theta ) , \end{align*}


\begin{align*} E = E | _ { U _ 1 ^ 2 } \cup \cdots \cup E | _ { U _ n ^ 2 } \cup E | _ { B ^ 2 } \end{align*}


\begin{align*} \hat { \omega } ( \pi ) : = ( N - \lambda _ 1 ) \cdot \lambda _ { \# ( \pi ) } \cdot \prod _ { i = 1 } ^ { \# ( \pi ) - 1 } ( \lambda _ { i } - \lambda _ { i + 1 } - 1 ) \ \ \ \ \hat { \omega } ( \emptyset ) = N + 1 . \end{align*}


\begin{align*} \forall u \in \mathbb S _ { \mathbb X } \colon \ker D ^ * \Phi ( ( \bar x , \bar y ) ; ( u , 0 ) ) = \{ 0 \} \end{align*}


\begin{align*} \sum _ { t = 1 } ^ { T } ( \norm { \xi _ t } ^ 2 - \sigma ^ 2 ) \leq \sigma ^ 2 \log ( 1 / \delta ) \end{align*}


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


\begin{align*} R ( z ) = - \frac { 1 } { \delta } \mbox { a n d } S ( z ) = \frac { 1 } { \delta } . \end{align*}


\begin{align*} \mathbb P ( S _ n ^ c ) = O ( n ^ { - \infty } ) . \end{align*}


\begin{align*} h ' ( s ) = 1 + A ( 1 + \delta ) s ^ { \delta } , \ \ \ \ h '' ( s ) = A \delta ( 1 + \delta ) s ^ { \delta - 1 } . \end{align*}


\begin{align*} \lambda x \cdot x y = y \end{align*}


\begin{align*} \nabla _ \nu \xi = \textstyle \sum \limits _ { i = 2 } ^ 4 \nu e _ i \nabla _ { e _ i } \xi = - B \xi - A \xi . \end{align*}


\begin{align*} \langle P _ d , Q _ d \rangle = \sum _ { | J | = d } \binom { d } { J } ^ { - 1 } p _ J q _ J , \end{align*}


\begin{align*} { _ { 0 } \mathfrak { I } ^ { \beta } f ( t ) } = \frac { 1 } { \Gamma ( \beta ) } \int _ o ^ t ( t - \eta ) ^ { \beta - 1 } f ( \eta ) ~ \mathrm { d } \eta = \frac { t ^ \beta } { \Gamma { ( \beta + 1 ) } } . \end{align*}


\begin{align*} B _ T ( e , y ^ * ) ( x ^ * ) = \langle ( T ( e ) ) ( x ^ * ) , y ^ * \rangle \ , \ , ( e , y ^ * ) \in E \times Y ^ * x ^ * \in X ^ * , \end{align*}


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


\begin{align*} \lambda _ { Z } = \eta ^ { r _ 0 } \circ \xi ( 0 ) . \end{align*}


\begin{align*} J _ 4 = \sum _ { \substack { i , j = 1 , \\ i \not = j } } ^ n \frac { \partial _ y ( \tau _ i F _ { i j , 3 } ) } { x _ { i j } ^ 2 } + O ( \Gamma ) , \end{align*}


\begin{align*} e ( \vect { r } ) = \int _ { V _ S } j ( \vect { s } ) \ , h ( \vect { r } , \vect { s } ) \ , d \vect { s } \end{align*}


\begin{align*} E _ { \psi } ( x ) = \Big | \frac { \psi ( x ) - x } { x } \Big | . \end{align*}


\begin{align*} \left \{ T ( h ) , \Psi ( f ) \right \} _ \ell ^ \Sigma [ \varphi ] & = - \int _ \mathcal { I } T ( h ) ^ { ( 1 ) } [ \varphi ] ( s ) \frac { d \Psi ( f ) ^ { ( 1 ) } [ \phi ] ( s ) } { d s } \ , \mathrm { d } s \\ & = - \int _ \mathcal { I } h ( s ) \varphi ( s ) \sqrt { \gamma ' ( s ) } \frac { d f } { d s } ( s ) \ , \mathrm { d } s \\ & = - \Psi \left ( h f ' \right ) [ \varphi ] . \end{align*}


\begin{align*} J _ 1 ^ n ( t , u ) & : = \int _ u ^ t \exp \Big ( \int _ s ^ t \tilde { f } ' ( r , x _ r ) d r \Big ) \Big ( \tilde { f } ' _ n ( s , x _ s ^ n ) - K _ s \Big ) K _ H ( s , u ) d s \end{align*}


\begin{align*} \mu _ { R } ( k , \ell , m , n ) = \mu _ { R , 1 } ( k , \ell , m , n ) + \mu _ { R , 2 } ( k , \ell , m , n ) \end{align*}


\begin{align*} \norm { U ^ { - 1 } - \sum _ { k = 1 } ^ n ( I - U ) ^ k } _ { o p } \to 0 , n \to \infty . \end{align*}


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


\begin{align*} W ^ { \alpha , G } ( \mathbb { R } ^ { d } ) = \bigg { \{ } u \in L ^ { G } ( \mathbb { R } ^ { d } ) : \ \overline { \rho } ( \alpha ; u ) < \infty \bigg { \} } , \end{align*}


\begin{align*} \mathbf { D } = \left [ \begin{aligned} \Omega _ N ^ { 0 0 } & & \cdots & & \Omega _ N ^ { 0 ( N - 1 ) } \\ \vdots & & \ddots & & \vdots \\ \Omega _ N ^ { ( N - 1 ) 0 } & & \cdots & & \Omega _ N ^ { ( N - 1 ) ( N - 1 ) } \end{aligned} \right ] \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ d } K ( x ) f ( x , w ^ + ) w ^ + d x & = A ^ + ( w ) \end{align*}


\begin{align*} 0 \leq \varphi ( t ) \leq 1 ~ \forall t \in \R , \varphi ( t ) = 1 ~ \forall t \in ( - \infty , - 1 ] , \varphi ( t ) = 0 ~ \forall t \in [ 0 , \infty ) . \end{align*}


\begin{align*} ( M _ { z } ^ * ) ^ j k _ { \overline y } = y ^ j k _ { \overline y } \rightarrow 0 \ ; \textup { a s } j \rightarrow \infty , \end{align*}


\begin{align*} L ( v _ 1 , \dots , v _ k ) = \frac { 1 } { 2 } \mathcal { G } ( v _ 1 , v _ 1 ) + \dots + \frac { 1 } { 2 } \mathcal { G } ( v _ k , v _ k ) , \end{align*}


\begin{align*} P = \left \{ \sum _ { i = 1 } ^ { d - 1 } \lambda _ i b _ i : \lambda _ i \in [ \alpha _ i , \beta _ i ] \cap \Z \right \} . \end{align*}


\begin{align*} \P ( \exists v \in \Lambda _ n : v \mbox { i s u n d e t e r m i n e d } ) & \leq \sum _ { v \in \Lambda _ n } \sum _ { 1 \leq \ell \leq r } \sum _ { I _ 1 , \ldots , I _ \ell } \sum _ { B ' _ { I _ 1 } , \ldots , B ' _ { I _ \ell } } \P ( \sigma | _ { B _ { I _ j } ( v ) } = \sigma | _ { B ' _ { I _ j } } \mbox { f o r } 1 \leq j \leq \ell ) \\ & \leq n ^ d r \binom { r - 1 } { \ell - 1 } n ^ { d \ell } q ^ { - ( r - 1 ) ^ { d } ( 1 + \ell ) } = o ( 1 ) \ , , \end{align*}


\begin{align*} \Lambda ( s , f \otimes \chi _ d ) = \left ( \frac { | d | } { 2 \pi } \right ) ^ s \Gamma ( s + \tfrac { \kappa - 1 } { 2 } ) L ( s , f \otimes \chi _ d ) = i ^ \kappa \epsilon ( d ) \Lambda ( 1 - s , f \otimes \chi _ d ) . \end{align*}


\begin{align*} [ e _ x , e _ y ] = - 2 e _ z , [ e _ x , e _ \theta ] = e _ y , [ e _ y , e _ \theta ] = - e _ x . \end{align*}


\begin{align*} \epsilon ^ { ( \ell + 1 ) } & \leq \frac { C } { \ell ^ 3 } + \epsilon ^ { ( \ell ) } \left [ 1 - \frac { 2 - \delta } { \ell } \right ] + C ( \epsilon ^ { ( \ell ) } ) ^ 3 . \end{align*}


\begin{align*} \prod _ { p \nmid h k } \sum _ { \substack { 0 \le q , \ell < \infty \\ \min ( \ell , q ) = 0 } } p ^ { - q w } \phi ^ { \star } ( p ^ q ) \frac { \tau _ A ( p ^ { \ell } ) \tau _ B ( p ^ { \ell } ) } { p ^ { \ell ( 1 + s _ 1 + s _ 2 ) } } = \prod _ { p \nmid h k } \left ( 1 + \sum _ { q = 1 } ^ { \infty } p ^ { - q w } \phi ^ { \star } ( p ^ q ) + \sum _ { \ell = 1 } ^ { \infty } \frac { \tau _ A ( p ^ { \ell } ) \tau _ B ( p ^ { \ell } ) } { p ^ { \ell ( 1 + s _ 1 + s _ 2 ) } } \right ) \end{align*}


\begin{align*} & [ T _ t ( u ) , T _ t ( v ) , T _ t ( w ) ] = T _ t \Big ( D ( T _ t ( u ) , T _ t ( v ) ) w + \theta ( T _ t ( v ) , T _ t ( w ) ) u - \theta ( T _ t ( u ) , T _ t ( w ) ) v \Big ) , \end{align*}


\begin{align*} X ^ 2 \ll & \left ( \sum _ { 2 < p \leq X } ( \log p ) L ( \tfrac { 1 } { 2 } , f \otimes \chi _ { 8 p } ) \mathcal { M } ( p ) \right ) ^ 2 \leq \ \mathcal { N } \sum _ { 2 < p \leq X } ( \log p ) \left | L ( \tfrac { 1 } { 2 } , f \otimes \chi _ { 8 p } ) \right | ^ 2 | \mathcal { M } ( p ) | ^ 2 \ll \mathcal { N } X . \end{align*}


\begin{align*} & \alpha _ i = \sum _ { j = i } ^ k \theta _ j , \forall i = 1 , \ldots , k , \\ & \alpha _ { k + 1 } = 0 , \quad \alpha _ { n + 1 - i } = - \alpha _ i , \forall i = 1 , \ldots , k . \end{align*}


\begin{align*} \nabla P = \nabla ( a \rho ^ \gamma ) = \frac { a \gamma } { \gamma - 1 } \rho \nabla ( \rho ^ { \gamma - 1 } ) = \frac { a \gamma } { \gamma - 1 } \rho \nabla P _ 1 \end{align*}


\begin{align*} \nu _ { t , m } = \lceil 2 \cdot 2 ^ { \gamma _ * k _ * t } \cdot 2 ^ m \rceil , \end{align*}


\begin{align*} \alpha = 1 + \alpha _ m t ^ m + \alpha _ { m + 1 } t ^ { m + 1 } + \cdots + \alpha _ { n - 1 } t ^ { n - 1 } \end{align*}


\begin{align*} S ( k ) = \begin{pmatrix} a ( k ) & b ^ * ( k ) \\ b ( k ) & a ^ * ( k ) \end{pmatrix} . \end{align*}


\begin{align*} J _ q ( n , k , t ) = S _ k ^ { k - t } = \frac { 1 } { q ^ { ( k - t ) ^ 2 } { k \choose t } _ q } \sum \limits _ { i = 0 } ^ { k - t } ( - 1 ) ^ { k - t - i } q ^ { { k - t - i \choose 2 } } { k - t \choose i } _ q { k + i \choose i } _ q N _ k ^ { i } \end{align*}


\begin{align*} x = \lambda y _ j + ( 1 - \lambda ) z _ j , \big ( \lambda u ( y _ j , t ) ^ q + ( 1 - \lambda ) u ( z _ j , t ) ^ q \big ) ^ { 1 \over q } \le u _ { q , \lambda } ( x , t ) + \frac { 1 } { j } . \end{align*}


\begin{align*} I _ 1 & = - \frac { 1 } { 2 \pi i } \oint _ { \partial U _ { \delta } ( \pm \eta ) } \left ( \sum _ { j = 1 } ^ { \infty } \frac { T _ { \pm \eta , j } ( \xi , k ) } { t ^ j } \right ) d s = t ^ { - 1 } \textnormal { R e s } _ { k = \eta } T _ { \eta , 1 } + t ^ { - 1 } \textnormal { R e s } _ { k = - \eta } T _ { - \eta , 1 } + \mathcal { O } ( t ^ { - 2 } ) , \end{align*}


\begin{align*} \overline { H } _ n ( s ) = \sum \limits _ { k = 1 } ^ n ( - 1 ) ^ { k - 1 } \binom { n } { k } { _ { s + 1 } F _ s } \left ( \left \{ \frac { 1 } { 2 } \right \} ^ { s } , 1 - k ; \left \{ \frac { 3 } { 2 } \right \} ^ { s } ; 1 \right ) . \end{align*}


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


\begin{align*} \left ( \frac { 1 } { \tau ^ 2 } + \frac { \eta ^ n } { 2 \tau } + \frac { \vartheta ^ n } { 2 } - \frac 1 4 \Delta \right ) \Big ( { \phi } ^ { n + 1 } - \phi ^ n \Big ) = 0 , \end{align*}


\begin{align*} J _ 5 = D - \sum _ { \substack { i , j , k = 1 , \\ i \not = j } } ^ n \frac { \mu _ k \partial _ y [ \tau _ i ( \mathcal { L } \mathfrak { B } _ { i j k } ) ( \varphi _ { i j } - 1 ) ] } { x ^ 3 _ { i j } } - \sum _ { \substack { i , j , k = 1 , \\ i \not = j } } ^ n \partial _ y \bigg ( \frac { \mu _ k \tau _ i ( [ | D | , \varphi _ { i j } ] \mathfrak { B } _ { i j k } ) } { x _ { i j } ^ 3 } \bigg ) , \end{align*}


\begin{align*} \arg \pi ^ { - \frac { \sigma + i T } { 2 } } = \arg \left ( \pi ^ { - \frac { \sigma } { 2 } } e ^ { - \left ( \frac { i \log \pi } { 2 } \right ) T } \right ) \end{align*}


\begin{align*} { \bf E } \left ( { \bf A } , { \mathbb V } \right ) = \left \{ { \bf A } _ { \theta } : \theta \left ( x , y \right ) = \sum _ { i = 1 } ^ { s } \theta _ { i } \left ( x , y \right ) e _ { i } \ \ \ \ \left \langle \left [ \theta _ { 1 } \right ] , \left [ \theta _ { 2 } \right ] , \dots , \left [ \theta _ { s } \right ] \right \rangle \in { \bf T } _ { s } ( { \bf A } ) \right \} . \end{align*}


\begin{align*} f _ r \left ( s \right ) = - 1 + e ^ { - \frac { i \beta \eta } { 2 } \omega ^ 2 _ r s } , f _ { r 1 } \left ( s \right ) = \frac { \omega ^ 3 _ r } { 4 } \left ( \frac { \beta ^ 2 \eta ^ 2 } { 2 } + \gamma - \eta ^ 2 \right ) s e ^ { - \frac { i \beta \eta } { 2 } \omega ^ 2 _ r s } \end{align*}


\begin{align*} \int _ { v \in G _ 0 \cdot \Lambda \cdot s _ i ( L _ i ) } f ( v ) m _ { \infty } ( v ) \ , d v = | W _ 0 | \int _ { b \in \Lambda \cdot L _ i } \int _ { g \in G _ 0 } f ( g \cdot s _ i ( b ) ) \ , d g \ , d b , \end{align*}


\begin{align*} \rho = \frac { d _ { \Gamma ( x , t ) } } { \varepsilon } - h _ 1 ( S ( x , t ) , t ) - \varepsilon h _ { 2 , \varepsilon } ( S ( x , t ) , t ) , \end{align*}


\begin{align*} \widehat { \Phi } ( x ) = 1 + \widehat { \beta } ( x ) \widehat { \Phi } ( x ) . \end{align*}


\begin{align*} \frac { ( \alpha + \beta ) ^ 2 } { | \alpha \beta | } \geq \begin{cases} \frac { \sigma } { 2 | \beta | } , & \ \alpha + \beta > \alpha / 2 ; \\ \frac { \sigma ^ 2 } { 2 | \beta | ^ 2 } , & \ \alpha + \beta \leq \alpha / 2 . \end{cases} \end{align*}


\begin{align*} T _ 2 & \lesssim \int _ { d ( y , z ) \geqslant \ 1 2 d ( x , y ) } \frac 1 { V ( x , z , t + d ( x , z ) ) } \Big ( \frac { t } { t + d ( x , z ) } \Big ) ^ { \varepsilon _ 1 } \frac 1 { \omega ( B ( y , d ( x , y ) ) ) } d \omega ( z ) \\ & = \frac 1 { \omega ( B ( y , d ( x , y ) ) ) } \int _ { \R ^ N } \frac 1 { V ( x , z , t + d ( x , z ) ) } \Big ( \frac { t } { t + d ( x , z ) } \Big ) ^ { \varepsilon _ 1 } d \omega ( z ) . \end{align*}


\begin{align*} \forall x , y \in X \left ( x = y \rightarrow \forall k \in \mathbb { N } \left ( H ^ * \left [ T x , T y , \frac { 1 } { k + 1 } \right ] \right ) \right ) . \end{align*}


\begin{align*} d _ \triangle ( a , b ) \ = \ \sum _ { i \in [ n ] } ( b _ i - a _ i ) - n \min _ { i \in [ n ] } ( b _ i - a _ i ) \ = \ \sum _ { i \in [ n ] } ( b _ i - a _ i ) + n \max _ { i \in [ n ] } ( a _ i - b _ i ) \enspace , \end{align*}


\begin{align*} \delta ^ { 1 } ( \alpha ) ( x , y , z ) = \partial _ { \rho } ( \alpha ) ( [ x , y ] , z ) - \rho ( z ) \partial _ { \rho } ( \alpha ) ( x , y ) . \end{align*}


\begin{align*} c _ l = \int _ 0 ^ { 1 / b } C _ { b , \gamma } ^ 2 \ , e ^ { - 2 \gamma t } e ^ { - 2 \pi i \frac { l } { a } t } \ , d t = C _ { b , \gamma } ^ 2 \ , \frac { a \left ( 1 - e ^ { - \frac { 2 \gamma } { b } } \right ) } { 2 a \gamma + 2 \pi i l } . \end{align*}


\begin{align*} i \partial _ t h + \Delta h + i \dot { \alpha } Q - i \dot { X } \cdot \nabla Q - \dot { \theta } Q = O ( \delta + \delta \delta ^ * ) L ^ 2 . \end{align*}


\begin{align*} \lim _ { t \to 0 } \| t ^ { \frac { \beta } { \alpha } } B \| _ { H ^ { \frac 5 2 - \alpha + \beta } } = 0 , \ \ \ \beta > 0 . \end{align*}


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


\begin{align*} \lim \limits _ { t \rightarrow + \infty } \frac { F ( x , t u ) } { \vert t u \vert ^ { g ^ + } } \vert u \vert ^ { g ^ + } = + \infty , \ \ x \in A . \end{align*}


\begin{align*} [ \bar { b } _ n \mapsto \bar { a } _ n ] ( G ) \cup [ \bar { b } _ n \mapsto \bar { a } _ n ] ( H ) \cup [ \bar { b } _ n \mapsto \bar { a } _ n ] ( I ) & = [ \bar { b } _ n \mapsto \bar { a } _ n ] ( G \cup H \cup I ) \\ & = [ \bar { b } _ n \mapsto \bar { a } _ n ] ( \mathbb { N } ) = \{ \bar { a } _ n \} \end{align*}


\begin{align*} \frac { \partial z _ { q ; \alpha } ^ { ( L + 1 ) } } { \partial W _ { i j } ^ { ( 1 ) } } = x _ { j ; \alpha } \sigma ' ( z _ { i ; \alpha } ^ { ( 1 ) } ) \frac { \partial z _ { q ; \alpha } ^ { ( L + 1 ) } } { \partial \sigma ( z _ { i ; \alpha } ^ { ( 1 ) } ) } . \end{align*}


\begin{align*} A & = \{ ( a , 0 ) \in G \mid a \neq 0 \} \\ B & = \{ ( 0 , b ) \in G \mid b \neq 0 \} \\ C & = \{ ( a , b ) \in G \mid a , b \neq 0 \} \\ \end{align*}


\begin{align*} \det ( q _ { \infty } ) ^ { - l } c ( h , q , l ) = C \cdot e _ { \infty } ( i \lambda ( q ^ { \ast } h q ) ) . \end{align*}


\begin{align*} \langle J ^ { ' } ( t _ 2 u ) , u \rangle = 0 . \end{align*}


\begin{align*} \Tilde { \xi } ( s ) : = \left \{ \begin{aligned} & \xi ( s ) + \left ( 1 - \frac { s } { T } \right ) h , 0 \leq s \leq T , \\ & \xi ( s ) , s \geq T , \end{aligned} \right . \end{align*}


\begin{align*} \Upsilon _ { \pm } ( w ; m h , n k ) = \Upsilon _ { \pm } ( w ; m h , n k ; c , Q ) : = \int _ { 0 } ^ { \infty } \frac { c | m h \pm n k | } { g x Q } W \left ( \frac { c | m h \pm n k | } { g x Q } \right ) x ^ { w - 1 } \ , d x . \end{align*}


\begin{align*} \tilde { f } ' ( s , y ) \le K _ 1 + s ^ { 2 H - 1 } K _ 2 \le s ^ { 2 H - 1 } K = K _ s \end{align*}


\begin{align*} \forall q \geq 1 ~ \exists C = C ( q ) I _ C = O ( n ^ { - q } ) . \end{align*}


\begin{align*} \div u = \div ( \rho u ) = 0 , \end{align*}


\begin{align*} \mathbf L \tilde { \mathbf V } + \mathbf b + \mathbf q ( \tilde { \mathbf V } ) = 0 . \end{align*}


\begin{align*} \epsilon | \hat { I } _ { \geq \epsilon } ( k _ { n } ) | < A ( \alpha _ { k _ { n } } ) - A ( \alpha _ { 0 } ) = s n R - A ( \alpha _ { 0 } ) < C \sqrt { k _ { n } } \end{align*}


\begin{align*} & \omega _ 2 ( x , y , z ) - \omega _ 1 ( x , y , z ) = ( \varphi _ 2 - \varphi _ 1 ) ( [ x , y ] , z ) - \rho ( z ) ( \varphi _ 2 - \varphi _ 1 ) ( x , y ) \\ & \quad \ ; \ ; \ ; = \partial _ { \rho } ( \alpha ) ( [ x , y ] , z ) - \rho ( z ) \partial _ { \rho } ( x , y ) \\ & \quad \ ; \ ; \ ; = \delta ^ { 1 } ( \alpha ) ( x , y , z ) , \ ; \alpha \in C ^ { 1 } _ { L i e } ( L , V ) , \end{align*}


\begin{align*} \begin{pmatrix} \varphi ^ { j } \\ \psi ^ { j } \\ \varphi _ { s } ^ { j } - F ^ { j } \end{pmatrix} = \begin{pmatrix} x _ { s } & y _ { s } & z _ { s } \\ x _ { t } & y _ { t } & z _ { t } \\ x _ { s s } & y _ { s s } & z _ { s s } \end{pmatrix} \begin{pmatrix} u ^ { j } \\ v ^ { j } \\ w ^ { j } \end{pmatrix} = M ( s , t ) \begin{pmatrix} u ^ { j } \\ v ^ { j } \\ w ^ { j } \end{pmatrix} \ , . \end{align*}


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


\begin{align*} = R ^ { - 1 } \left [ \begin{array} { c c } g _ 1 & 0 \\ 0 & g _ 2 \end{array} \right ] R R ^ { - 1 } \left [ \begin{array} { c c } U ( z _ 1 ) & 0 \\ 0 & \mathfrak { J } \overline { U ( z _ 2 ) } \end{array} \right ] \left [ \begin{array} { c c } \lambda ( g _ 1 , z _ 1 ) & 0 \\ 0 & \overline { \lambda ( g _ 2 , z _ 2 ) } \end{array} \right ] ^ { - 1 } B ( g _ 1 z _ 1 , g _ 2 z _ 2 ) ^ { - 1 } S \end{align*}


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


\begin{align*} \lambda _ { q _ n } ( w ) * a = \lambda _ { q _ { n + 1 } } ( w ) * a . \end{align*}


\begin{align*} V _ g f ( S ( x , \omega ) ) = e ^ { \pi i ( x \cdot \omega - x ' \cdot \omega ' ) } V _ { \widehat { S } ^ { - 1 } g } ( \widehat { S } ^ { - 1 } f ) ( x , \omega ) , \end{align*}


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


\begin{align*} \lim _ { n \rightarrow \infty } \sup _ { t \in \lbrack 0 , T ] } \left \Vert u _ { n } ( t ) - u ( t ) \right \Vert _ { s } = 0 . \end{align*}


\begin{align*} u _ t ( \varphi ) : = \log U _ t e ^ { - V _ 0 ( \varphi ) } , \end{align*}