\begin{align*} & \mathbb { P } \left \{ v ^ T \cdot C _ t \cdot v = 0 , \ \ v \neq 0 \right \} \\ & = \mathbb { P } \left \{ v ^ T D ^ j Y _ t = 0 , \ 1 \leq i \leq d , \ \ v \neq 0 \right \} \\ & = \mathbb { P } \left \{ \int _ { 0 } ^ { t } v ^ T J _ { t \leftarrow s } V _ j ( Y _ s ) d D ^ j X ^ j _ s = 0 , \ 1 \leq i \leq d , \ \ v \neq 0 \right \} . \end{align*}


\begin{align*} \psi ( x , y ) : = \Big ( k \circ { ( \delta - k \circ \overline { k } ) ^ { - 1 / 2 } } \Big ) ( x , y ) . \end{align*}


\begin{align*} \mathcal S _ A ^ 0 ( D ; \ell ) = \frac { 1 } { 2 \pi i } \int _ { ( a ) } \breve W ( s ) N ^ s \sum _ { \atop ( d , \ell ) = 1 } \mu ^ 2 ( 2 d ) \Psi \left ( \frac d D \right ) \mathcal B ^ { ( 2 d ) } ( A _ s ; \ell ) ~ d s + O ( D ^ { 1 / 2 } \log Y + D Y ^ { - 1 } ) \end{align*}


\begin{align*} O ( X ) \subseteq V _ 1 \cup \ldots \cup V _ n \mbox { a n d } \bigcup _ { j = 1 } ^ n ( V _ j \times V _ j ) \subseteq V \ , . \end{align*}


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


\begin{align*} \Omega _ { M C } = \begin{pmatrix} i \rho _ 1 + j \overline { \omega _ 3 } & - \frac { \overline { \omega _ 1 } } { \sqrt { 2 } } + j \frac { \omega _ 2 } { \sqrt { 2 } } \\ \frac { \omega _ 1 } { \sqrt { 2 } } + j \frac { \omega _ 2 } { \sqrt { 2 } } & i \rho _ 2 + j \tau \end{pmatrix} . \end{align*}


\begin{align*} ( \beta + 2 d ) ( 1 - \delta _ 0 ) = 2 ( \alpha + d ) , \end{align*}


\begin{align*} q ( \bar { t } , \bar { x } , \bar { y } ) : = \frac 1 2 h ^ 2 ( \bar { t } , \bar { x } , \bar { y } ) , \end{align*}


\begin{align*} \partial _ a ( a _ 1 \ldots a _ d ) = \underset { i = 1 } { \overset { d } { \sum } } \delta _ { a , a _ i } a _ { i + 1 } \ldots a _ d a _ 1 \ldots a _ { i - 1 } , \end{align*}


\begin{align*} f \sigma _ 1 = \rho _ 1 - \rho _ 2 , \tau = f \frac { \sigma _ 2 + i \sigma _ 3 } { \sqrt { 2 } } . \end{align*}


\begin{align*} \lambda _ { \tilde \mu } : = \frac { | \tilde \mu ( z ) | ^ 2 } { ( - 2 { \rm R e } z ) } d x d y \end{align*}


\begin{align*} \| \tilde { M } \| \leq \| M \| \cdot \| R \| = \| M \| , | \det \tilde { M } | = | \det M | . \end{align*}


\begin{align*} \tilde H ' _ 1 ( s ) & = h '' g _ H ( T _ * N , \dot \eta ) g _ H ( \nu , T _ * N ) + h ' \frac { d } { d s } g _ H ( \nu , T _ * N ) \\ & = h '' g _ H ( T _ * N , \dot \eta ) g _ H ( \nu , T _ * N ) - h ' \Bigl ( g _ H ( \nabla _ { \dot \eta } \dot \eta ^ \perp , T _ * N ) + g _ H ( \dot \eta ^ \perp , \nabla _ { \dot \eta } T _ * N ) \Bigr ) \\ & = h '' g _ H ( T _ * N , \dot \eta ) g _ H ( \nu , T _ * N ) - h ' \Bigl ( - g _ H ( T _ * N , \dot \eta ) \kappa _ \eta + g _ H ( T _ * N , \dot \eta ) g _ H ( T _ * N ^ \perp , \dot \eta ) \kappa _ 1 \Bigr ) , \end{align*}


\begin{align*} B \left ( \sum _ { i j } c _ { i j } v _ i \otimes _ { H S } \lambda _ j \right ) = \sum _ { i j } c _ { i j } \beta ( v _ i , \lambda _ j ) = \sum _ i c _ { i i } . \end{align*}


\begin{align*} D ^ \epsilon _ \Lambda = \Lambda ' _ 1 + D ^ + _ Z = \{ \ , \Lambda ' _ 1 + \Lambda ' \mid \Lambda ' \in D ^ + _ Z \ , \} . \end{align*}


\begin{align*} \alpha ( t , x ) = \Psi ( \iota _ { \mathsf { b } } ( t ) ) \end{align*}


\begin{align*} h _ i \cdot \varphi ^ P _ s ( g ) & = \begin{cases} s ( r - \ell ) \cdot \varphi _ s ^ P ( g ) & 1 \leq i \leq \ell \\ - s \ell \cdot \varphi ^ P _ s ( g ) & \ell < i \leq r , \end{cases} \end{align*}


\begin{align*} \sum _ { j = 1 } ^ N \chi _ V ( x _ j ) \ , , \end{align*}


\begin{align*} \begin{array} { c c } ( B R u ) _ { j } ( x , t ) = - \displaystyle \sum _ { k \neq j } \sum _ { l = 1 } ^ { n } \int _ { 0 } ^ { 1 } \int _ { x _ j } ^ { x } d _ { j } ( \xi , x , t ) b _ { j k } ( \xi , \omega _ j ( \xi ) ) c _ { k } ( x _ k , \xi , \omega _ j ( \xi ) ) & \\ \displaystyle \times r _ { k l } ( \eta , \omega _ k ( x _ k , \xi , \omega _ j ( \xi ) ) ) u _ { l } ( \eta , \omega _ k ( x _ k , \xi , \omega _ j ( \xi ) ) ) \ , d \xi d \eta , \ ; \ ; \ ; j \le n . & \end{array} \end{align*}


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


\begin{align*} ( u , 0 ) \in L \ \Rightarrow \ u = 0 ~ ~ \mbox { f o r a l l } ~ ~ L \in \mathcal { S } F ( \bar { x } , \bar { y } ) . \end{align*}


\begin{align*} \alpha _ { 1 , 0 } = r , \alpha _ { 1 , 2 } = d . \end{align*}


\begin{align*} M = \begin{pmatrix} M \rvert _ { 0 \cdot } \\ M \rvert _ { 1 \cdot } \end{pmatrix} \ ; , \end{align*}


\begin{align*} d _ { c } ^ { 2 } ( f ) ( x , y , z ) & = \theta \left ( x , [ \alpha ( y ) , z ] \right ) + \theta \left ( y , \delta ( z , \alpha ( x ) \right ) + \beta \Big ( \theta \left ( z , [ x , y ] \right ) \Big ) \\ + & \rho \left ( x \right ) \theta ( \alpha ( y ) , z ) + \rho \left ( y \right ) \theta ( z , \alpha ( y ) ) + \beta \Big ( \rho \left ( z \right ) \theta ( x , y ) \Big ) . \end{align*}


\begin{align*} I = \Bigl \{ i = 1 , \dotsc , r - 1 \Bigm | \frac { d _ i + \cdots + d _ { r - 1 } } { r - i } = \frac { d } { r } \Bigr \} , \end{align*}


\begin{gather*} \gamma ( x , y , \cdot ) = \gamma ' ( x , y , \cdot ) - B _ { \mathfrak a } \left ( ( \theta ' + \frac { 1 } { 2 } d ( - \tau ) ) \wedge ( - \tau ) \right ) ( x , y , \cdot ) . \end{gather*}


\begin{align*} \Omega _ { \frac { q } { N } } ^ { - 1 } ( \lambda ) = \prod _ { \Box \in \lambda } \frac { 1 } { 1 + \frac { q c ( \Box ) } { N } } = \sum _ { r = 0 } ^ \infty \left ( - \frac { 1 } { N } \right ) ^ r q ^ r f _ r ( \lambda ) , \end{align*}


\begin{align*} \frac { e ^ { - ( \lambda + \theta _ 1 \pi i + T w ) } } { 1 - e ^ { - ( \lambda + \theta _ 1 \pi i + T w ) } } = - 1 + \frac { 1 } { 1 - e ^ { - ( \lambda + \theta _ 1 \pi i + T w ) } } , \end{align*}


\begin{align*} I _ n : = \mathbb { P } \left ( \sum _ { m = 1 } ^ n \max _ { 0 \le k < 2 ^ { n - m } } \left | \sum _ { i = k 2 ^ m + 1 } ^ { ( k + 1 ) 2 ^ m } Y _ { i , m } \right | \ge C _ 1 ( b ) \varepsilon _ 1 b _ { 2 ^ { n } } \right ) \to 0 n \to \infty , \end{align*}


\begin{align*} \varphi ^ P _ s \left ( \Psi ( v ) g \right ) = | e _ r \cdot \Psi ( v ) g | ^ { - r s } = | v \cdot g | ^ { - r s } . \end{align*}


\begin{align*} - \partial _ t u - \Delta u + h ( x , t ) | D u | ^ \gamma = f ( x , t ) \end{align*}


\begin{align*} b _ { 3 } ( x , y , z , t ) = - \int _ 0 ^ z \nabla _ H \cdot \tilde { b } ( x , y , \xi , t ) \ , d \xi . \end{align*}


\begin{align*} \lim _ { t \to \infty } \frac { f ( t ) } { t } \log P \bigg ( | \mathrm { N b d } ( B [ 0 , t ] , a ) | \le \bigg ( \frac { \pi } { \sqrt { 2 } } \bigg ) ^ 3 f ( t ) ^ { 3 / 2 } \omega _ 3 \bigg ) = - 1 , \end{align*}


\begin{align*} \underset { n \rightarrow + \infty } { \lim } \int _ { G } ( R _ { \alpha } \ast | u _ { \lambda _ n } | ^ p ) | u _ { \lambda _ n } | ^ p \ , d \mu = \int _ { G } ( R _ { \alpha } \ast | u | ^ p ) | u | ^ p \ , d \mu . \end{align*}


\begin{align*} \psi ( u ) = e \Big ( \sum _ { j = 1 } ^ n \psi _ j u _ { j j } \Big ) , \psi ' _ p ( u ) = e \Big ( \sum _ { j = 1 } ^ n \psi ' _ j u _ { j j } \Big ) \end{align*}


\begin{align*} \alpha _ x ( t ) = \alpha ( t , x ) \end{align*}


\begin{align*} \left ( e ^ { - 2 \theta } \nu \left ( u _ 0 + \sum _ { l = 1 } ^ { j - 1 } [ M _ A v ] _ l \right ) - E - i \eta \right ) \end{align*}


\begin{align*} \int _ { 0 } ^ { t } \mathbb { E } \left [ \int _ { D } h ^ 2 | \partial _ x u _ r | ^ 2 \dd x \right ] \dd r = \int _ { 0 } ^ { t } \mathbb { E } \left [ \int _ { D } h ^ 2 | u _ r \times \partial _ x u _ r | ^ 2 \dd x \right ] \dd r \leq \int _ { 0 } ^ { t } \| h \| _ { L ^ \infty } ^ 2 \mathbb { E } \left [ \| u _ r \times \partial _ x u _ r \| _ { L ^ 2 } ^ 2 \right ] \dd r \ , . \end{align*}


\begin{align*} \dim Z = \dim ( \textbf { d } ) \cdot M = \dim ( \textbf { d } ) - \dim ( M ) < \dim ( \textbf { d } ) . \end{align*}


\begin{align*} R = \left ( \begin{matrix} U ^ T \ \ & 0 \\ 0 \ \ & I _ { ( p - \ell ) \times ( p - \ell ) } \end{matrix} \right ) . \end{align*}


\begin{align*} \tau \sum _ { \ell = 1 } ^ { \infty } \frac { 1 } { ( 1 + \tau ) ^ { 2 \ell } } = \frac { 1 } { 2 + \tau } \le \frac 1 2 \end{align*}


\begin{align*} \hat D _ i . L _ 1 = \hat D _ i . L _ { i j } \iff \hat D _ j . L _ 1 = \hat D _ j . L _ { i j } . \end{align*}


\begin{align*} \left \| P \right \| _ R : = \sum _ { l = r _ 1 } ^ { r _ 2 } \left \| P _ l \right \| _ R \ , . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ n \mathbb { P } ( | X _ i | > \varepsilon n ^ { 1 / p } ) & \ge \sum _ { i = \lfloor n / 2 \rfloor } ^ n \mathbb { P } ( | X _ i | > \varepsilon n ^ { 1 / p } ) \\ & \ge \sum _ { i = \lfloor n / 2 \rfloor } ^ n \frac { 1 } { n } \ge \frac { 1 } { 2 } . \end{align*}


\begin{align*} ( \sigma _ { \widetilde i } ) ^ r = ( L _ { \widetilde i } \ , \pi ) ^ r \stackrel { \eqref { p i L x p i } } { = } \prod _ { k = 0 } ^ { r - 1 } ( L _ { \widetilde k } ^ { - 1 } L _ { \widetilde { i + k } } ) \pi ^ r = \prod _ { k = 0 } ^ { r - 1 } L _ { \widetilde k } ^ { - 1 } \prod _ { k = 0 } ^ { r - 1 } L _ { \widetilde { i + k } } = \mathrm { i d } \end{align*}


\begin{align*} \chi _ { \varepsilon , \lambda } ( h _ a ) = | a | ^ { i \lambda } \left ( \frac { a } { | a | } \right ) ^ \varepsilon ( a \in \R ^ \times ) \ , . \end{align*}


\begin{align*} X _ { - } ( s ) : = \bigcap _ { \lambda \in \Lambda _ 0 \cup \Lambda _ + } R _ { \lambda } ( s ) , \end{align*}


\begin{align*} \det \limits _ { i \notin X , j \notin Y } M _ { i , j } = ( - 1 ) ^ { \sum _ { k = 1 } ^ p ( x ^ k + y ^ k ) } \det ( M ) \det _ { i \in Y , j \in X } M ^ { - 1 } _ { i , j } . \end{align*}


\begin{align*} ( M \circ N ) _ { i j } = \sum _ k M _ { i k } N _ { k j } \ ; . \end{align*}


\begin{align*} \dim X = \dim ( X \times _ R R ( t _ 1 , . . . , t _ n ) ) . \end{align*}


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


\begin{align*} \zeta = ( x + t y ) f ( x , y , t ) = s ( x , y ) u ( x , y , t ) \neq 0 \end{align*}


\begin{align*} c _ k ( Y , \lambda ) = \sup _ { R > 0 } b _ k ( [ - R , 0 ] \times Y ) . \end{align*}


\begin{align*} \hat { E } _ { n } ^ { \epsilon , \Delta t } = E _ n ^ { \epsilon , \Delta t } - \frac { C \epsilon ^ 2 } { ( n + 1 ) ^ 2 } | p _ 0 ^ \epsilon | ^ 2 , \end{align*}


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


\begin{align*} | \hat { f } _ \# ( k ) | & = \left | \int _ { \Lambda _ L } e ^ { - i 2 \pi k \cdot x } f ( x ) d x \right | \leq \int _ { \Lambda _ L } | f ( x ) | d x \leq \| f \| _ 1 . \end{align*}


\begin{align*} M : = \begin{bmatrix} x & 0 & - e _ 3 ( y + z ) \\ z - y & x - z & e _ 2 y - e _ 1 z \\ 0 & - y & x - e _ 1 y + z \\ 0 & 0 & z \end{bmatrix} . \end{align*}


\begin{align*} B ( \Phi ( x ) , \Phi ( y ) ) & = B \left ( x - \tau ( x ) - \frac { 1 } { 2 } B _ { \mathfrak a } \big ( \tau ( x ) , \tau ( \cdot ) \big ) , y - \tau ( y ) - \frac { 1 } { 2 } B _ { \mathfrak a } \big ( \tau ( y ) , \tau ( \cdot ) \big ) \right ) \\ & = B _ { \mathfrak a } \left ( \tau ( x ) , \tau ( y ) \right ) - \frac { 1 } { 2 } B _ { \mathfrak a } \big ( \tau ( y ) , \tau ( x ) \big ) - \frac { 1 } { 2 } B _ { \mathfrak a } \big ( \tau ( x ) , \tau ( y ) \big ) \\ & = 0 = B ( x , y ) \end{align*}


\begin{align*} g _ { n } \left ( z \right ) = \frac { n } { 2 } - \gamma _ { n } \left ( z \right ) . \end{align*}


\begin{align*} q _ { \delta , j , m } = O ( 1 / j ^ 2 ) . \end{align*}


\begin{align*} c _ { - j } : = - \frac { 1 } { 2 } \int _ { - 1 / 2 } ^ { 1 / 2 } \left \{ \log \left ( 1 + \left ( \frac { y / 2 \pi + \theta _ 1 / 2 + j } { x / 2 \pi } \right ) ^ 2 \right ) - \log \left ( 1 + \left ( \frac { y / 2 \pi + \theta _ 1 / 2 + j + \phi } { x / 2 \pi } \right ) ^ 2 \right ) \right \} \mathrm { d } \phi , \end{align*}


\begin{align*} \frac { 1 } { N \theta } \log \left ( \int _ { \mathbb { R } ^ { d \times N } } \exp \left ( - \beta \mathcal { H } ^ { * } _ { N } \right ) d \pi ^ { \otimes N } \right ) = \frac { 1 } { N \theta } \log \left ( \int _ { \mathbb { R } ^ { d \times N } } \exp \left ( - \beta \mathcal { H } _ { N } \right ) d X _ { N } \right ) - \frac { \log z ^ { * } } { \theta } . \end{align*}


\begin{align*} T = \sum _ { i = 1 } ^ s \lambda _ i T _ { L _ { P _ i } } . \end{align*}


\begin{align*} \gcd ( p ^ r + 1 , p ^ s - 1 ) = \left \{ \begin{array} { r l } 1 , & { \rm { i f } } \ \frac { s } { \gcd ( r , s ) } \ { \rm { i s \ o d d \ a n d } } \ p \ { \rm i s \ e v e n , } \\ 2 , & { \rm { i f } } \ \frac { s } { \gcd ( r , s ) } \ { \rm { i s \ o d d \ a n d } } \ p \ { \rm i s \ o d d , } \\ p ^ { \gcd ( r , s ) } + 1 , & { \rm { i f } } \ \frac { s } { \gcd ( r , s ) } \ { \rm { i s \ e v e n . } } \\ \end{array} \right . \end{align*}


\begin{align*} | M ^ * \cap M _ 1 | + | M ^ * \cap M _ 2 | = | M \cap M _ 1 | + | M \cap M _ 2 | + 1 \end{align*}


\begin{align*} H \circ k + k \circ H ^ T + ( \upsilon _ N m ) + k \circ ( \upsilon _ N \overline { m } ) \circ k = 0 , \| k \| _ \mathrm { o p } < 1 , \end{align*}


\begin{align*} \begin{pmatrix} W & X \\ Y & Z \end{pmatrix} \end{align*}


\begin{align*} u _ m ( t ) = U ( t , s ) \varphi + j ^ { - 1 } \int _ s ^ t U ^ { \odot \star } ( t , \tau ) f _ m ( \tau ) d \tau , \forall m \in \mathbb { N } , \ t \in I , \end{align*}


\begin{align*} H _ b & : = \big ( p - b { A } ( x ) \big ) \cdot \sigma , \end{align*}


\begin{align*} \left \{ \begin{aligned} & \gamma _ 0 \psi _ { 2 , 2 2 } ( 0 , 0 , i \kappa ) = \psi _ { 2 , 1 2 } ( 0 , 0 , i \kappa ) \\ & \gamma _ 0 \psi _ { 2 , 1 2 } ( 0 , 0 , i \kappa ) = \psi _ { 2 , 2 2 } ( 0 , 0 , i \kappa ) , \end{aligned} \right . \end{align*}


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


\begin{align*} \mathbb { E } \left [ 2 \int _ { 0 } ^ { t } \int _ { D } \partial _ x h ^ 2 u _ r \cdot u _ r \dd x \dd r \right ] = 2 t \| \partial _ x h \| _ { L ^ 2 } ^ 2 \ , . \end{align*}


\begin{align*} \left \{ \begin{array} { l } - i \gamma ^ \mu \partial _ \mu \psi + m \psi = [ V _ b * ( \psi ^ \dagger \psi ) ] \gamma ^ 0 \psi , \\ \psi | _ { t = 0 } = \psi _ 0 , \end{array} \right . \end{align*}


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


\begin{align*} \kappa = \kappa ( s ) = 1 0 0 0 0 s ^ { - 1 } ( 1 - s ) ^ { - 4 } \textrm { a n d } \qquad \omega = 1 0 0 ( \kappa + 1 ) \delta . \end{align*}


\begin{align*} G = P _ { V } ( x ) + K \subseteq P _ U ( x ) + ( \overline { O } + K ) \ , . \end{align*}


\begin{align*} \hat { f } _ \theta ( k ) = e ^ { - d \theta / 2 } \hat { f } ( e ^ { - \theta } k ) = e ^ { - d \theta / 2 } Q _ \lambda ( e ^ { - \theta } ( k - a ) ) e ^ { - \frac { \pi } { \lambda } e ^ { - 2 \theta } | k - a | ^ 2 } e ^ { - 2 \pi i e ^ { - \theta } ( k - a ) \cdot x _ 0 } . \end{align*}


\begin{align*} \int _ { u ( y ) } ^ { \bar { u } ( y ) } \frac { d \eta } { \bar { w } } = \int _ 0 ^ { u ( y ) } \frac { \bar { w } - w } { \bar { w } w } d \eta . \end{align*}


\begin{align*} \mathcal { T } ^ * ( x , t ) : = t ^ { p _ * ( x ) } + \mu ( x ) ^ { \frac { q _ * ( x ) } { q ( x ) } } t ^ { q _ * ( x ) } ( x , t ) \in \overline { \Omega } \times [ 0 , \infty ) \end{align*}


\begin{align*} \textbf { A } _ { \textbf { S T } } = \begin{pmatrix} I _ { d \times d } & - I _ { d \times d } & 0 _ { d \times d } & 0 _ { d \times d } \\ 0 _ { d \times d } & 0 _ { d \times d } & I _ { d \times d } & I _ { d \times d } \\ 0 _ { d \times d } & 0 _ { d \times d } & 0 _ { d \times d } & - I _ { d \times d } \\ - I _ { d \times d } & 0 _ { d \times d } & 0 _ { d \times d } & 0 _ { d \times d } \end{pmatrix} \end{align*}


\begin{align*} \chi ( s + a ) \ , \chi ( 1 - s + b ) = \Big ( \frac { t } { 2 \pi } \Big ) ^ { - a - b } \bigg ( 1 + O \Big ( \frac { 1 } { 1 + | t | } \Big ) \bigg ) \end{align*}


\begin{align*} \sum _ { i = 1 } ^ n b _ i W _ i \leq C _ q \left \vert b \right \vert _ 1 + C _ q t ^ 2 e ^ { t ^ 2 / q } \sum _ { i = 1 } ^ n i ^ { - 1 + 2 / q } b _ { [ i ] } \end{align*}


\begin{align*} V = \{ x _ 4 ^ 2 Q + R = 0 \} . \end{align*}


\begin{align*} S = w - \bar { w } , \end{align*}


\begin{align*} u ( x , t ) = \Sigma _ m \xi _ m ( t ) B _ m ( x ) \end{align*}


\begin{align*} ( q _ 1 z - w ) ( q _ 2 z - w ) ( q _ 3 z - w ) \psi ^ \epsilon ( z ) f ( w ) = ( z - q _ 1 w ) ( z - q _ 2 w ) ( z - q _ 3 w ) f ( w ) \psi ^ \epsilon ( z ) \ , , \end{align*}


\begin{align*} \lim _ { T \rightarrow + \infty } \mathbb { E } \left [ \sup _ { t \geq T } \left | \int _ { D } [ ( u _ t - B _ t ) \cdot ( u _ t - B _ t ) - \alpha ] \dd x \right | \right ] = 0 \ , . \end{align*}


\begin{align*} \omega = \omega _ M + d \theta \wedge d \varphi . \end{align*}


\begin{align*} E \Delta _ { n , k } = \frac { \pi ^ 2 } { 8 } \bar h _ 4 ( n ) ( 1 + o ( 1 ) ) \ , . \end{align*}


\begin{align*} \sum _ { j = 0 } ^ { \ell } ( - 1 ) ^ j \binom { d - 1 } { j } a _ { \ell - j } ^ n = \delta _ { 0 , \ell } , \ell = 0 , 1 , \ldots , d - 1 . \end{align*}


\begin{align*} M ( ( A , B ) ) = A \otimes B \ ; . \end{align*}


\begin{align*} \left \lbrace \begin{aligned} Q ^ \epsilon ( t ) & = q ^ \epsilon ( t ) + \epsilon p ^ \epsilon ( t ) \\ P ^ \epsilon ( t ) & = \epsilon p ^ \epsilon ( t ) . \end{aligned} \right . \end{align*}


\begin{align*} 0 < r _ 1 \hbox { \rm \ \ a n d \ \ } r _ \ell < r _ { \ell + 1 } \hbox { \rm \ \ f o r \ \ } \ell = 1 , 2 , \dots , i - 1 \ , . \end{align*}


\begin{align*} \Lambda ( { Z _ \mu } ) ( \tau ) & = \Lambda ( \mathbf { b } ) ( \tau ) K N , \\ | \Lambda ( { Z _ \mu } ) ( \tau ) | & \le ( P - \tau ) K N . \end{align*}


\begin{align*} \rho _ 0 = ( z _ 1 + d z ^ { 1 2 } ) \wedge e ^ { i \omega _ 0 } , \end{align*}


\begin{align*} \Big | \sin \Big ( i t + \frac { \pi N } { 2 } \Big ) \Big | \lesssim \begin{cases} 1 , & 0 < t < 1 , \\ e ^ t , & t > 1 , \end{cases} \end{align*}


\begin{align*} T ( \Gamma ) \cap B _ j = \emptyset , \end{align*}


\begin{align*} n = b 2 ^ { v + 1 } - 1 \end{align*}


\begin{align*} \left | B _ { n + 1 , \mu , 0 } ( 1 ) \Lambda _ p \right | _ p \leq \left | \sum _ { j = 1 } ^ k \lambda _ j S _ { n + 1 , \mu , j } ( 1 ) \right | _ p \end{align*}


\begin{align*} L _ i = \left [ h _ i ( \pmb { \theta } ) \frac { 1 } { x _ i } - g _ i ( \pmb { \theta } ) \right ] \end{align*}


\begin{align*} \left . \partial _ \nu u _ { z _ 0 } \right | _ \Omega - \left . \partial _ \nu u _ { z _ 0 } \right | _ { \Omega ^ c } = - 2 i \tau f ' \end{align*}


\begin{align*} \beta _ 2 ( \check V ) = \beta _ 2 ( \hat V ) = n / 2 , \end{align*}


\begin{align*} A = \bigoplus _ { i } M _ { n _ i } \otimes X _ i \ ; . \end{align*}


\begin{align*} \lim _ { \epsilon \to 0 } \omega ( \epsilon ) = 0 . \end{align*}


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


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


\begin{align*} \langle 0 \mid l _ { i + 1 , i } ^ { ( - 1 ) } = 0 a n d \langle 0 \mid l _ { i , i } ^ { ( - 1 ) } = - \frac { 1 } { h } ( d _ { i } - d _ { i - 1 } ) \langle 0 \mid \end{align*}


\begin{align*} S ( m ) = \frac { 3 m + 1 } { 2 ^ e } = m \end{align*}


\begin{align*} W ^ { 1 , \mathcal { H } _ i } ( \Omega ) = \left \{ u \in L ^ { \mathcal { H } _ i } ( \Omega ) \ , : \ , | \nabla u | \in L ^ { \mathcal { H } _ i } ( \Omega ) \right \} \end{align*}


\begin{align*} 0 = \mathrm { T r } \left ( \gamma b \right ) = \mathrm { T r } \left ( \left ( \omega b ^ { - 1 } \right ) b \right ) = \mathrm { T r } \left ( \omega \right ) . \end{align*}


\begin{align*} \langle \mathcal { A } ( u ) , v \rangle _ { \mathcal { W } } = \langle \mathcal { F } ( u ) , v \rangle _ { \mathcal { W } } + \langle \mathcal { G } ( u ) , v \rangle _ { \mathcal { W } } v \in \mathcal { W } . \end{align*}


\begin{align*} \left ( \frac { d m _ 0 } { d \zeta } + \frac { i \zeta } { 2 } \sigma _ 3 m _ 0 \right ) _ { + } = \left ( \frac { d m _ 0 } { d \zeta } + \frac { i \zeta } { 2 } \sigma _ 3 m _ 0 \right ) _ { - } J _ 0 . \end{align*}


\begin{align*} \begin{aligned} \Lambda ( \mathbf { c , c ^ \prime } ) ( P j + k ) = & \Lambda ( \mathbf { a , a ^ \prime } ) ( j ) \Lambda ( \mathbf { b } , \mathbf { b ' } ) ( k ) \\ & + \Lambda ( \mathbf { a , a ^ \prime } ) ( j + 1 ) \Lambda ( \mathbf { b } , \mathbf { b ' } ) ( - P + k ) , \\ & 0 \le k < P - N < j < N . \end{aligned} \end{align*}


\begin{align*} \Xi _ { \beta } : = \left \{ \zeta \in \mathbb C ; \zeta ^ 2 - a \zeta - \mu = 0 , \mu \in { \rm S p e c } ( \Delta _ { g _ F } ) \right \} \cap \Gamma _ { \beta } . \end{align*}


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


\begin{align*} X _ { i , m } & = h ^ { ( i , m ) } \left ( ( \phi _ { i , m } ( W _ j ) ) _ { j \in J } \right ) \end{align*}


\begin{align*} \Delta _ g u = 0 \end{align*}


\begin{align*} d _ 2 ( \mu _ { k + 1 } ^ { \tau } , \mu _ { k } ^ { \tau } ) ^ 2 = \int _ M \lvert \nabla \phi _ { k , k + 1 } ^ { c } \rvert ^ 2 d \mu _ { k + 1 } ^ { \tau } . \end{align*}


\begin{align*} X ( x + 1 , t ) - X ( x - 1 , t ) = \sum _ { z \in \Z } \sum _ { s = 1 } ^ t \Delta ( x - z , t - s ) \xi ( z , s ) \Gamma ( z , s ) . \end{align*}


\begin{align*} \mathbb { E } ( f ( \overline { x _ k } ) ) - m i n ( f ) = \mathcal { O } ( h ^ { 1 / 2 } ) + \mathcal { O } \left ( \frac { 1 } { k h } \right ) + \frac { \sigma _ * ^ 2 d } { 2 } . \end{align*}


\begin{align*} \Phi \sigma _ x = \sigma ' _ { \Phi ( x ) } \Phi . \end{align*}


\begin{align*} R _ l : = H _ l \ltimes N _ l . \end{align*}


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


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


\begin{align*} G \left ( k ^ { 2 } , x \right ) = \frac { f _ { + } \left ( x , k \right ) f _ { - } \left ( x , k \right ) } { W \left \{ f _ { + } \left ( x , k \right ) , f _ { - } \left ( x , k \right ) \right \} } , \end{align*}


\begin{align*} 2 L \left [ x P _ { n } P _ { n + 1 } \right ] & = 2 h _ { n + 1 } , L \left [ \left ( x ^ { 2 } - z ^ { 2 } \right ) P _ { n + 1 } \partial _ { x } P _ { n } \right ] = n h _ { n + 1 } , \\ L \left [ - z ^ { 2 } P _ { n } \partial _ { x } P _ { n + 1 } \right ] & = - z ^ { 2 } \left ( n + 1 \right ) h _ { n } . \end{align*}


\begin{align*} \mathcal { M } : = \left \{ \mathrm { l e v e l } ( f ) \ | \ f \in \mathcal { H } ( g ) \right \} . \end{align*}


\begin{align*} | [ n ] \times [ m ] | = | [ n ] | | [ m ] | = m n \ ; . \end{align*}


\begin{align*} V ( z ) = \int _ 0 ^ \infty e ^ { - z k } e ^ { - \frac 7 4 k } w ( k ) \ , d k = W ( z + \tfrac 7 4 ) \end{align*}


\begin{align*} \underset { k = 0 } { \rm R e s } M ^ { ( 2 ) } ( x , t , k ) = \frac { A } { 2 i } M ^ { ( 1 ) } ( x , t , 0 ) . \end{align*}


\begin{align*} \mathrm { P f } ( M ) : = \frac { 1 } { 2 ^ n n ! } \sum _ { \pi \in S _ { 2 n } } \mathrm { s g n } ( \pi ) \ , \prod _ { j = 1 } ^ n m _ { \pi ( 2 j - 1 ) \ , \pi ( 2 j ) } . \end{align*}


\begin{align*} \prod _ { k \in \mathbb { Z } } \left ( 1 + \left ( \frac { q } { p + k } \right ) ^ 2 \right ) = \frac { 1 } { 1 + ( q / p ) ^ 2 } \left | \frac { \Gamma ( p ) } { \Gamma ( p + q i ) } \right | ^ 2 \left | \frac { \Gamma ( - p ) } { \Gamma ( - p + q i ) } \right | ^ 2 \end{align*}


\begin{align*} V = X + W + X ^ \vee \end{align*}


\begin{align*} f ( t ) = \sum _ { | I | \leq p } f ^ { I } \langle \epsilon _ { I } , \mathbb { X } _ { t ^ - } \rangle , \end{align*}


\begin{align*} \liminf \limits _ { t \to \infty } \frac { X _ { \underline { n } ( t ) } } { t } = \liminf \limits _ { t \to \infty } \frac { X _ { \underline { n } ( t ) } } { \underline { n } ( t ) } \frac { \underline { n } ( t ) } { t } \stackrel { } \ge c _ 1 > 0 \end{align*}


\begin{align*} \dot { h } _ t ( z ) : = \frac { d h _ t ( z ) } { d t } = - h ' ( z + t ) \frac { 1 - 2 t ^ 2 S h ( z + t ) } { \left ( 1 + t P h ( z + t ) \right ) ^ 2 } , \end{align*}


\begin{align*} \lim _ { i \to \infty } u _ i = - 1 N \setminus \Sigma . \end{align*}


\begin{align*} \lim _ { x \to \infty } \frac { V ( x ) } { 2 } + g ( x ) = \infty . \end{align*}


\begin{align*} K \left ( k / \mathrm { i } , s ; x , t \right ) : = \int _ { x } ^ { \infty } \psi \left ( z , t ; k \right ) \psi \left ( z , t ; \mathrm { i } s \right ) \mathrm { d } z , \ \ \operatorname { I m } k \geq 0 , s \geq 0 . \end{align*}


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


\begin{align*} \frac { d \hat x _ \tau } { d \tau } ( T , \hat y , \hat \eta _ \tau ) + \frac { \partial \hat x _ \tau } { \partial \eta } \frac { d \hat \eta _ \tau } { d \tau } = 0 . \end{align*}


\begin{align*} \begin{aligned} & \underset { x _ { 1 } , \hdots , x _ { m } } { } & & \sum _ { i = 1 } ^ { m } f _ { i } ( x _ { i } ) \\ & & & \sum _ { i = 1 } ^ { m } x _ { i } = \sum _ { i = 1 } ^ { m } R _ { i } \end{aligned} \end{align*}


\begin{align*} U _ { \ell , k } : = P _ { \ell , k } \times Q _ k , ( \ell , k ) \in J . \end{align*}


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


\begin{align*} L ^ { - T } = \begin{pmatrix} A _ { 1 1 } ^ T & - A _ { 4 1 } ^ T \\ A _ { 1 2 } ^ T & - A _ { 4 2 } ^ T \end{pmatrix} , \end{align*}


\begin{align*} \| K \| ^ 2 _ { \gamma ( H ; E ) } = \tilde { \mathbb { E } } \left \| \sum _ { k \in \mathbb { N } } \gamma _ k K e _ k \right \| ^ 2 _ E . \end{align*}


\begin{align*} \dfrac { d x } { d \tau } = D [ A ( t ) x + g ( t ) ] . \end{align*}


\begin{align*} \mathcal { G } : = \bigcup _ { q \geq 0 } I _ q ^ c \setminus \{ 0 , T \} , \ \ \mathcal { B } : = [ 0 , T ] \setminus \mathcal { G } . \end{align*}


\begin{align*} U _ j = \{ y \in X \ , : \ , ( y , \alpha ( t _ j , x ) ) \in W \} \ , . \end{align*}


\begin{align*} \widehat { Q } ^ 1 _ 1 F = Q '^ 1 _ 1 \circ F - ( - 1 ) ^ { \abs { F } } F \circ Q \end{align*}


\begin{align*} K _ r ^ { \perp } = i \left ( H _ { 2 r + 1 } { \overline { H } _ { 2 r + 2 } } - { \overline { H } _ { 2 r + 1 } } H _ { 2 r + 2 } \right ) . \end{align*}


\begin{align*} ( u , \partial _ { y } \tilde { h } ) | _ { y = 0 } = \bf { 0 } , \end{align*}


\begin{align*} - 2 \Lambda d \varphi + d \left ( ( 2 - m - n ) \Lambda + | \nabla _ { B } \varphi | ^ { 2 } - \Delta _ { B } \varphi - \frac { m } { f } \nabla _ { B } \varphi ( f ) \right ) = 0 , \end{align*}


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


\begin{align*} \chi _ { \psi } ^ n = \left \{ ( e ^ { j ( \pi - \angle \varphi _ n ^ + ) } , e ^ { j ( \frac { 3 } { 2 } \pi - \angle \varphi _ n ^ + ) } ) , ( e ^ { j ( \pi - \angle \varphi _ n ^ - ) } , e ^ { j ( \frac { 1 } { 2 } \pi - \angle \varphi _ n ^ - ) } ) \right \} . \end{align*}


\begin{align*} \rho _ s ( x ) = & \left ( 1 + O \left ( x ^ { - 1 } \right ) \right ) ( x / e ) ^ { \sigma } \left ( 1 + \sigma / x \right ) ^ { x + \sigma } \\ & \times \exp \left ( \tfrac 1 2 ( x + \sigma ) \log \big ( 1 + \left ( \tfrac { t } { x + \sigma } \right ) ^ 2 \big ) - t \arctan \left ( \tfrac { t } { x + \sigma } \right ) \right ) \\ & \times \exp \left ( i \phi _ 1 ( t ) + i \phi _ 2 ( t ) + i t ( - 1 + \log x ) \right ) \end{align*}


\begin{align*} h = \partial _ y \psi , g = - \partial _ x \psi , \psi | _ { y = 0 } = 0 . \end{align*}


\begin{align*} x = \begin{cases} u v \ & \ x \in H \cap H ^ { g _ 1 } \\ s ^ 2 t , v , u ^ 2 v , u ^ 2 , s ^ 2 t v , s ^ 2 t u ^ 2 v s ^ 2 t u ^ 2 \ & \ x \in H \cap H ^ { g _ 2 } . \end{cases} \end{align*}


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


\begin{align*} S ( m ) = \frac { 3 m + 1 } { 2 ^ e } \end{align*}


\begin{align*} \varphi _ 1 = \bigwedge _ { i \leq p } \gamma _ j \land \bigwedge _ { j \leq q } \lnot \psi _ i , \end{align*}


\begin{align*} P _ 2 \left ( n + 1 \right ) = P _ 2 ( n ) + P _ 2 ( n - 1 ) + \cdots + P _ 2 ( n - r + 1 ) + 1 \end{align*}


\begin{align*} \begin{aligned} T _ { 3 } \leq \int _ { B _ { R } ^ { + } \backslash B _ { \rho } ^ { + } } | \bar { \boldsymbol { w } } | | \nabla \eta ^ { 6 } | \omega ^ { 2 } d x d y \leq \frac { C ( \log R ) ^ { \frac { 1 } { 2 } } } { R - \rho } . \end{aligned} \end{align*}


\begin{align*} m = 2 ^ v w - 1 . \end{align*}


\begin{align*} \iint _ { \Omega } \mathcal { A } \nabla u \cdot \nabla \varphi \frac { d t } { | t | ^ { n - d - 1 } } d x = 0 . \end{align*}


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


\begin{align*} | p ( r , \theta ) | = o ( \log r ) , \end{align*}


\begin{align*} \langle \Omega | W _ { 2 m , 0 } ( x ) | \Omega \rangle & = - \sum _ { k _ 1 , \ldots , k _ N = 0 } ^ \infty \prod _ { j = 1 } ^ N \binom { 2 s } { k _ j } ( - 1 ) ^ { k _ { t o t } } k _ { t o t } ! \frac { \Gamma ( p + q - 2 m ) } { \Gamma ( p + q - 2 m + k _ { t o t } + 1 ) } \ , . \end{align*}


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


\begin{align*} \limsup _ { s \to \infty } \frac { P _ { k - 1 } ( s ) L _ k ( s ) ^ { 1 + \alpha } \Theta ' ( s ) } { \Theta ( s ) } = \infty \alpha > 0 k \ge 1 \ , . \end{align*}


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


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


\begin{align*} R _ { 2 j } = \sum _ { n - 1 \le \delta _ 1 \le \ldots \le \delta _ { j - 2 } \le n } R _ { 2 j } ( \delta ) , \end{align*}


\begin{align*} \overline { E } \Big ( \sum \limits _ { j = 1 } ^ m e ^ { i \theta _ j } \langle T _ { \theta _ j } e _ 6 , v _ j \rangle \Big ) = - \omega _ { 5 6 } ( \overline { E } ) \sum \limits _ { j = 1 } ^ m e ^ { i \theta _ j } \langle T _ { \theta _ j } e _ 5 , v _ j \rangle . \end{align*}


\begin{align*} \partial _ { t } u - 6 u ^ { \prime } + u ^ { \prime \prime \prime } = 0 , \ \ \ x , t \in \mathbb { R } , \end{align*}


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


\begin{align*} \left \{ \begin{aligned} \partial _ r ^ 2 \eta _ 2 ( r ) & = \frac { 1 } { g } \{ \nabla _ \Gamma g \cdot \nabla _ \Gamma \eta _ 0 - k _ d ^ { - 1 } ( \partial ^ \circ g ) \eta _ 0 + k _ d ^ { - 2 } g V _ \Gamma ^ 2 \eta _ 0 \} , r \in ( g _ 0 , g _ 1 ) , \\ \partial _ r \eta _ 2 ( g _ i ) & = \nabla _ \Gamma g _ i \cdot \nabla _ \Gamma \eta _ 0 - k _ d ^ { - 1 } ( \partial ^ \circ g _ i ) \eta _ 0 + k _ d ^ { - 2 } g _ i V _ \Gamma ^ 2 \eta _ 0 , i = 0 , 1 . \end{aligned} \right . \end{align*}


\begin{align*} d _ Y ^ 2 ( \eta \otimes f ) = [ Y ^ 2 , \eta \otimes f ] \end{align*}


\begin{align*} \frac { 1 } { 2 \pi } \int _ { \mathbb { R } } \phi ^ 2 ( x ) \d x = 1 . \end{align*}


\begin{align*} B ^ E ( T ^ n \omega ) \begin{bmatrix} u ( n ) \\ u ( n - 1 ) \end{bmatrix} = p ( T ^ n \omega ) \begin{bmatrix} u ( n + 1 ) \\ u ( n ) \end{bmatrix} \end{align*}


\begin{align*} I \times \Omega _ \eta : = \bigcup _ { t \in I } \{ t \} \times \Omega _ { \eta ( t ) } \end{align*}


\begin{align*} J _ \omega = \bigoplus _ r J _ r , \end{align*}


\begin{align*} I + I I = & \dfrac { 2 \delta } { p - 1 } + \dfrac { ( 1 - \delta ) ^ p } { p ( p - 1 ) } - \dfrac { 1 } { p ( p - 1 ) } - \dfrac { \delta ^ p } { p ( p - 1 ) } \\ & = \dfrac { 2 \delta } { p - 1 } + \dfrac { p \xi ^ { p - 1 } ( - \delta ) } { p ( p - 1 ) } - \dfrac { \delta ^ p } { p ( p - 1 ) } , \\ & \geq \dfrac { 2 \delta } { p - 1 } - \dfrac { \delta } { p - 1 } - \dfrac { \delta ^ p } { p ( p - 1 ) } = \dfrac { \delta } { p - 1 } - \dfrac { \delta ^ p } { p ( p - 1 ) } \end{align*}


\begin{align*} d _ n = d ( \bar x _ n , \partial \Omega ) , M _ n = | u _ n ( x _ n , \bar t _ n ) | , r _ n = | x _ n - \bar x _ n | , \end{align*}


\begin{align*} \det ( A _ { \omega , q } ( \varkappa ) - E ) = \Delta _ { \omega , q } ( E ) + 2 ( - 1 ) ^ { q - 1 } \cos ( q \varkappa ) \end{align*}


\begin{align*} \iint _ { \Omega } \mathcal { A } \nabla u \cdot \nabla ( u - u _ j ) \frac { d t d x } { | t | ^ { n - d - 1 } } = \iint _ { \Omega } \mathcal { A } ^ j \nabla u _ j \cdot \nabla ( u - u _ j ) \frac { d t d x } { | t | ^ { n - d - 1 } } = 0 . \end{align*}


\begin{align*} f ( 0 ) = \frac { \int _ { \mathbb { D } } { f } e ^ { - \varphi } } { \int _ { \mathbb { D } } { e ^ { - \varphi } } } , \ \ \ \forall \ , f \in { A ^ 2 ( \mathbb D ) } . \end{align*}


\begin{align*} \mathcal { H } ( x , t ) = t ^ { p ( x ) } + \mu ( x ) t ^ { q ( x ) } \quad ( x , t ) \in \Omega \times [ 0 , \infty ) , \end{align*}


\begin{align*} \{ w \in \mathbb { C } : | w - C | < r _ { S G } \} \subset \Delta _ { S G } , \ , \ , \ , \ , r _ { S G } = \left ( \frac { e - 1 } { e + 1 } \right ) - | C - 1 | . \end{align*}


\begin{align*} & F ^ { ( 1 ) } _ { \theta } ( s ) : = 1 + P ( s ) \cos \left ( \frac { \theta - \theta _ { \hat x } } { 2 } \right ) + Q ( s ) \sin \left ( \frac { \theta - \theta _ { \hat x } } { 2 } \right ) , \\ & F ^ { ( 2 ) } _ { \theta } ( s ) : = P ( s ) \sin \left ( \frac { \theta + \theta _ { \hat x } } { 2 } \right ) + Q ( s ) \cos \left ( \frac { \theta + \theta _ { \hat x } } { 2 } \right ) , \\ & F ^ { ( 3 ) } _ { \theta } ( s ) : = \cos \left ( \frac { \theta - \theta _ { \hat x } } { 2 } \right ) + P ( s ) . \end{align*}


\begin{align*} ( T _ \nu f _ 1 , \ldots , T _ \nu f _ g , T _ \nu e _ 1 , \ldots , T _ \nu e _ g ) = ( f _ 1 , \ldots , f _ g , e _ 1 , \ldots , e _ g ) \ , { } ^ t m _ \nu . \end{align*}


\begin{align*} g = - c _ 0 e ^ { - \beta \xi } ( 1 - \eta ) e ^ { \frac { 8 } { 3 } \eta - \frac { 8 } { 3 } } + \varepsilon \tau + w i n \overline { D _ { T ^ * } } . \end{align*}


\begin{align*} ( F ( A ) ) _ { i j } = \begin{cases} 1 & j = A ( i ) \\ 0 & \end{cases} \ ; . \end{align*}


\begin{align*} J ( x , t , k ) = \left ( \begin{array} { c c } 1 + r _ 1 ( k ) r _ 2 ( k ) & r _ 2 ( k ) e ^ { - 2 i k x - 8 i k ^ 3 t } \\ r _ 1 ( k ) e ^ { 2 i k x + 8 i k ^ 3 t } & 1 \end{array} \right ) , k \in \mathbb { R } \backslash \{ 0 \} . \end{align*}


\begin{align*} K _ 1 ^ { \ast } = \frac { 1 } { 2 } - K . \end{align*}


\begin{align*} \phi \colon G & \to \langle a , b , c \mid [ b , c ] = 1 , b ^ q c ^ p = a ^ p = 1 \rangle \\ & = \langle a \mid a ^ p = 1 \rangle * \langle b , c \mid [ b , c ] = 1 , b ^ q c ^ p = 1 \rangle \\ & = \mathbb { Z } _ p * \mathbb { Z } . \end{align*}


\begin{align*} J _ \Psi ( x ) = \det ( D \Psi ( x ) ) \ , . \end{align*}


\begin{align*} L _ { \widetilde i } = \prod _ { k = 1 + i } ^ { 0 } D _ { \widetilde { k } } ^ { - 1 } . \end{align*}


\begin{align*} \mathbf { 1 } ( 0 ) = 1 \ ; . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ s n _ i ( B _ { P _ i } - A _ { P _ i } ) \simeq 0 , \end{align*}


\begin{align*} \dim ( \textbf { d } ) \cdot M = \dim ( \textbf { d } ) - \dim _ { ( \textbf { d } ) } ( M ) = \dim ( \textbf { d } ) - \dim ( M ) . \end{align*}


\begin{align*} \dim ( \textbf { d } ) = \dim ( \textbf { d } ) \cdot M + \dim _ { ( \textbf { d } ) } ( M ) = \dim ( \textbf { d } ) \cdot M + m . \end{align*}


\begin{align*} G = { \mathcal { L } } ^ { - 1 } e ^ { x / 3 } \left [ \Phi + { \mathcal { Q } } ( G ) + { \mathcal { C } } ( G ) \right ] . \end{align*}


\begin{align*} \partial _ { \epsilon } g ( x ) : = \left \{ x ^ \ast \in X ^ \ast \colon \langle x ^ \ast , y - x \rangle \leq g ( y ) - g ( x ) + \epsilon , y \in X \right \} . \end{align*}


\begin{align*} \frac d { d t } \sum \limits _ { \beta = 0 } ^ { 1 } \sum \limits _ { | \alpha | = 0 } ^ { 3 - \beta } ( \| \sqrt { B } \partial _ \tau ^ \alpha \partial _ y ^ \beta u \| _ { L ^ 2 } ^ 2 & + \| \sqrt { A } \partial _ \tau ^ \alpha \partial _ y ^ \beta \tilde { h } \| _ { L ^ 2 } ^ 2 ) + C _ 0 D ( t ) \\ & \le C D ( t ) ^ { \frac 1 4 } E ( t ) ^ { \frac 5 4 } + C D ( t ) ^ { \frac 1 2 } E ( t ) + C D ( t ) E ( t ) ^ { \frac 1 4 } . \end{align*}


\begin{align*} Q _ { \varepsilon , T } = \bigcup _ { t \in ( 0 , T ] } \Omega _ \varepsilon ( t ) \times \{ t \} , \partial _ \ell Q _ { \varepsilon , T } = \bigcup _ { t \in ( 0 , T ] } \partial \Omega _ \varepsilon ( t ) \times \{ t \} , \end{align*}


\begin{align*} G _ S = \prod _ { v \in S } G _ v \times \prod _ { v \notin S } K _ v , \end{align*}


\begin{align*} \cos F ( \delta ) = g ( \tan \delta ) . \end{align*}


\begin{align*} \frac 1 q \leq \frac 1 { q _ 1 } + \frac 1 { q _ 2 } , 1 + \frac 1 p = \frac 1 { p _ 1 } + \frac 1 { p _ 2 } . \end{align*}


\begin{align*} \varphi ( s _ 1 \cdots s _ d ) = \xi _ 1 ( s _ 1 ) \cdots \xi _ d ( s _ d ) s _ 1 , . . . , s _ d \in \Lambda . \end{align*}


\begin{align*} ( X _ { i , m } ) _ { ( i , m ) \in L } = \alpha ( \vec { W } ) \end{align*}


\begin{align*} \lambda _ { n } \left ( z \right ) = \left [ 2 \left ( \gamma _ { n } + \gamma _ { n + 1 } + \gamma _ { n + 2 } - z ^ { 2 } - 1 \right ) - n \right ] \gamma _ { n + 1 } , \end{align*}


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


\begin{align*} f \star g ( x ) = \int _ { \mathbb { R } ^ N } f ( x - z ) g ( z ) d z , \ , \ , \mbox { f o r a n y } x \in \Omega , \end{align*}


\begin{align*} ( v ^ * , 0 ) \in L ^ * \ \Rightarrow \ v ^ * = 0 . \end{align*}


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


\begin{align*} 2 ^ { a + b } = 2 ^ a 2 ^ b \ ; . \end{align*}


\begin{align*} \mathbb H ^ r _ { g , \mathbb K } = \mathbb K [ \mathcal T _ 0 ] \oplus \left ( \bigoplus _ { l = D ( g , r ) + 1 } ^ { D _ p ( g , r , 1 ) + \varrho ^ - _ { p } ( g , r ) } \mathbb K [ \mathcal T _ 0 ] h _ { l } \right ) , \end{align*}


\begin{align*} L W = ( s - 1 ) W = \left \{ \begin{pmatrix} x \\ - x \end{pmatrix} ; x \in M _ { 1 , 2 } ( \R ) \right \} \ , . \end{align*}


\begin{align*} ( n - 2 ) \ , d / 2 + 1 < \sum _ { i = 1 } ^ n k _ i \leq n d / 2 . \end{align*}


\begin{align*} \lim _ { m \to \infty } A _ m ( x + i y ) = \max \{ x , 0 \} \end{align*}


\begin{align*} y _ { 1 1 } = & \phi ^ 2 \left [ ( \phi \Omega ) _ z ^ 2 + \phi \phi _ { z z } ( z ( \phi \Omega ) _ z + s ( \phi \Omega ) _ s ) - ( r ^ 2 - s ^ 2 ) ( \phi ^ 2 ( \phi _ { s s } \phi _ { z z } - \phi _ { s z } ^ 2 ) - \Omega \delta _ 2 ) \right ] \\ y _ { 1 2 } = & y _ { 2 1 } = \phi ^ 3 \left [ \phi _ { s z } ( \phi \Omega ) _ z - \phi _ { z z } ( \phi \Omega ) _ s \right ] \\ y _ { 2 2 } = & - \phi ^ 4 ( \phi _ { s s } \phi _ { z z } - \phi _ { s z } ^ 2 ) . \end{align*}


\begin{align*} r ( X , Y ) & = \left ( \rho ^ \nabla \right ) ^ { J , + } ( X , Y ) + \sum _ { i , k = 1 } ^ { 2 n } g \left ( J X , N ( e _ i , e _ k ) \right ) g \left ( Y , N ( e _ i , e _ k ) \right ) \\ & + \sum _ { i , k = 1 } ^ { 2 n } g \left ( N ( X , e _ k ) , e _ i \right ) g \left ( N ( Y , e _ k ) , J e _ i \right ) , \end{align*}


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


\begin{align*} X = \frac { u + C v + B } { \gamma } . \end{align*}


\begin{align*} m ( t ) = \frac { 1 } { 1 - \beta } \inf _ { s \leq t } \left ( w ( s ) + \frac { \alpha } { 1 - \alpha } \sup _ { u \leq s } \left ( w ( u ) + \beta m ( u ) \right ) \right ) = : \phi ^ - ( w , m ) ( t ) . \end{align*}


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


\begin{align*} \int _ { x = - \infty } ^ { \infty } | \mathcal K ( x ) | \ , e ^ { b x } \ , \dd x < \infty . \end{align*}


\begin{align*} \breve { J } ^ r ( x , t , k ) = \overline { \breve { J } ^ r ( x , t , - \bar { k } ) } \end{align*}


\begin{align*} b H _ l c + H _ l b c = c \end{align*}


\begin{align*} \Pi ( z ) = \prod _ { n = 1 } ^ \infty E _ { p _ n } ( \frac { z } { a _ n } ) . \end{align*}


\begin{align*} M _ G ( \lambda ) : = E [ e ^ { \lambda \gamma _ G ( [ 0 , 1 ] ^ 2 ) } ] \ , , \end{align*}


\begin{align*} X \circ \Pi ( - r , \pi - \theta ) = \left ( ( R _ { \Gamma } \circ h \circ \Pi ) ( r , \theta ) , a + b - t \circ \Pi ( r , \theta ) \right ) , \end{align*}


\begin{align*} D g _ { \ell } ( t , s ) = \frac { D \lambda ^ u _ { f ^ { - \ell - 1 } ( \Gamma _ x ( t , s ) ) } D f ^ { - \ell - 1 } ( \Gamma _ x ( t , s ) ) D \Gamma _ x ( t , s ) } { \lambda ^ u _ { f ^ { - \ell - 1 } ( \Gamma _ x ( t , s ) ) } } . \end{align*}


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


\begin{align*} S ( n ) = \frac { 3 n + 1 } { 1 6 } \equiv 1 \pmod { 4 } . \end{align*}


\begin{align*} A & = \int _ { \R ^ n } \int _ 0 ^ 1 \left [ \int _ 0 ^ 1 D ^ 2 h \ ( x - T x - s t \xi + \tau ( 2 s t \xi ) \ ) ( s \xi ) \ , d \tau \right ] \cdot \xi \ , d s \ , \phi ( x ) \ , d x \\ B & = \int _ { \R ^ n } \left [ \int _ 0 ^ 1 D h ( x - T x - s t \xi ) \cdot \xi \ , d s \right ] \ , \dfrac { \phi ( x - t \xi ) - \phi ( x ) } { t } \ , d x . \end{align*}


\begin{align*} \left \{ \begin{aligned} & \partial _ t u + u \partial _ x u + v \partial _ { y } u - \partial _ { y } ^ 2 u = 0 , i n \Omega , \\ & \partial _ x u + \partial _ y v = 0 , i n \Omega , \\ & u | _ { y = 0 } = v | _ { y = 0 } = 0 \quad \mbox { a n d } \displaystyle \lim _ { y \to + \infty } u ( t , x , y ) = 1 , \\ & u | _ { t = 0 } = u _ 0 , u | _ { x = 0 } = u _ 1 , \end{aligned} \right . \end{align*}


\begin{align*} a _ { i j } = \delta _ { i j } - \frac 1 2 \ ( \dfrac { \partial ( x _ i - T _ i x ) } { \partial x _ j } + \dfrac { \partial ( x _ j - T _ j x ) } { \partial x _ i } \ ) = \delta _ { i j } - \frac 1 2 \ ( \delta _ { i j } - \dfrac { \partial T _ i x } { \partial x _ j } + \delta _ { i j } - \dfrac { \partial T _ j x } { \partial x _ i } \ ) = \frac 1 2 \ ( \dfrac { \partial T _ i x } { \partial x _ j } + \dfrac { \partial T _ j x } { \partial x _ i } \ ) \end{align*}


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


\begin{align*} X = ( \R \times \Q _ p \times \Q _ q ) / \Gamma \end{align*}


\begin{align*} D _ i = \{ \omega - Q = 0 , x _ i = 0 \} \end{align*}


\begin{align*} y ^ { \tau ( x ) } x ^ { - 1 } y ^ { - 1 } x , \ \ y = r _ i ^ a \in \varPhi , \ \ x = r _ j ^ b \in \varPhi , \ \ a , b \in \mathfrak F , \ i , j = 1 , \ldots , l . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ s m _ i \alpha ( P _ i , t ) \simeq 0 \end{align*}


\begin{align*} u _ j : \R ^ { 2 n + 1 } \to \R , \quad ( t _ 0 , . . . , t _ { 2 n } ) \mapsto t _ j j = 0 , \ldots , 2 n . \end{align*}


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


\begin{align*} \delta \chi _ { s u t } = \bigl ( f ( s , \cdot ) - f ( u , \cdot ) \bigr ) \star \mu ^ \omega _ { u t } ( \theta _ s ) + f ( u , \cdot ) \star \mu ^ \omega _ { u t } ( \theta _ s ) - f ( u , \cdot ) \star \mu ^ \omega _ { u t } ( \theta _ u ) . \end{align*}


\begin{align*} \phi & = X ^ \flat \wedge T ^ \flat - ( J X ) ^ \flat \wedge ( J T ) ^ \flat , \\ & = { e ^ u } \left ( d x \wedge \left ( \frac { 1 } { 2 } ( \textbf { H } - 1 ) d \theta + d \varphi \right ) + d y \wedge \left ( \frac { 1 } { 2 } ( \textbf { H } + 1 ) d \theta + d \varphi \right ) \right ) , \end{align*}


\begin{align*} \sum _ { n = 0 } ^ { N - 1 } \dd _ { 2 p } ( R _ { n , 2 } ^ { \epsilon , \Delta t } , 0 ) \le C _ p ( T ) \Delta t ^ { \frac 1 2 } . \end{align*}


\begin{align*} \begin{array} { l } P ^ * \{ u \ne \pi _ n x \ \& \ v \ne \pi _ n y \} = P ^ * \{ ( \pi _ { n } x , \pi _ { n } y ) \notin W ^ n \} \\ \le P ^ * \{ \pi _ { n } x \notin Q \} + P ^ * \{ \pi _ { n } y \notin Q \} + P ^ * \{ \| \pi _ { n } x - \pi _ { n } y \| _ X \ge \rho \} \le 2 P ^ * \{ x \notin Q _ 0 \} + \delta \\ < 2 \varepsilon + \delta < 3 \varepsilon \ \ \ ( n \ge n _ 0 ) . \end{array} \end{align*}


\begin{align*} h ( x ) = \int \limits _ { \mathbb { S } ^ { n - 1 } } P ( x , \eta ) h _ b ( \eta ) d \sigma ( \eta ) \end{align*}


\begin{align*} \frac { \partial Q _ \lambda } { \partial x _ i } = d w _ i ( x _ i ^ { d - 1 } - \lambda x _ 1 ^ { w _ 1 } \cdots x _ i ^ { w _ i - 1 } \cdots x _ n ^ { w _ n } ) , \end{align*}


\begin{align*} H _ y ^ { } : = h _ { \alpha } + H _ f ^ + - 2 \alpha ^ { 1 / 2 } \mathrm { R e } ( P A _ { y } ( x ) ) + \alpha A ^ 2 _ { y } ( x ) + \alpha V _ y ( x ) , \end{align*}


\begin{align*} c \land a = 0 \ , \ , \Longleftrightarrow \ , \ , c \leq \lnot a ; \end{align*}


\begin{align*} \nabla f ( x _ 0 ) = \nabla ( x _ { 1 } ) - \nabla f ( x _ { 0 } ) , \end{align*}


\begin{align*} \left \Vert \alpha _ { 3 } \right \Vert ^ 2 = 1 - K . \end{align*}


\begin{align*} \begin{gathered} ( a \otimes b ) _ X = a _ X \sqcup b _ X \ ; , \\ ( a \otimes b ) _ M ( l _ 1 \sqcup l _ 2 ) = a _ M ( l _ 1 ) \times b _ M ( l _ 2 ) \ ; . \end{gathered} \end{align*}


\begin{align*} \nabla \eta ^ 0 = ( \partial _ \nu \eta ^ 0 ) \nu \Gamma . \end{align*}


\begin{align*} \lim \limits _ { r \to \infty } \frac { r ^ { \frac { 3 } { 4 } } } { ( \log r ) ^ { \frac { 1 } { 8 } } } | \omega ( r , \theta ) | = 0 , \end{align*}


\begin{align*} ( U _ 1 ^ { p _ 1 } \circ \ldots \circ U _ n ^ { p _ n } ) ( x ) = x \end{align*}


\begin{align*} \rho s _ i \rho ^ { - 1 } = s _ { i + 1 } . \end{align*}


\begin{align*} V : = c \biggl [ \sum _ { x \in \Z } \sum _ { t = 0 } ^ \infty \Delta ( x , t ) ^ 4 ( \mu _ 4 - \mu _ 2 ^ 2 ) + \biggl ( \sum _ { x \in \Z } \sum _ { t = 0 } ^ \infty \Delta ( x , t ) ^ 2 \mu _ 2 \biggr ) ^ 2 \biggr ] , \end{align*}


\begin{align*} [ C ( x ) , B ( y ) ] & = \frac { 1 } { 1 + x + y } \left ( A ( y ) A ( x ) - D ( x ) D ( y ) \right ) - \frac { x + y } { ( x - y ) ( x + y + 1 ) } \left ( A ( x ) D ( y ) - A ( y ) D ( x ) \right ) \ , . \end{align*}


\begin{align*} \left \{ \begin{alignedat} { 2 } - \Delta u = \lambda u ^ { q _ 1 } - \frac { u ^ { p _ 1 } } { v ^ { \beta _ 1 } } \ ; \ ; & \mbox { i n $ \Omega $ , } \ ; \ ; u > 0 \ ; \ ; & & \mbox { i n $ \Omega $ , } \ ; \ ; u \lfloor _ { \partial \Omega } = 0 , \\ - \Delta v = \mu v ^ { q _ 2 } - \frac { u ^ { p _ 2 } } { v ^ { \beta _ 2 } } \ ; \ ; & \mbox { i n $ \Omega $ , } \ ; \ ; v > 0 \ ; \ ; & & \mbox { i n $ \Omega $ , } \ ; \ ; v \lfloor _ { \partial \Omega } = 0 . \end{alignedat} \right . \end{align*}


\begin{align*} E _ { \gamma } : = ( D ^ - \times V ) \cup _ { \phi _ { \gamma } } ( D ^ + \times V ) \end{align*}


\begin{align*} \| w ' \| _ { L ^ p ( \Omega ) } & \leq C _ p \left ( \| \nabla w ' \| _ { L ^ 2 ( \Omega ) } + \| w ' \| _ { L ^ 2 ( \Omega ) } \right ) . \end{align*}


\begin{align*} ( F ' ( t ) , F ' ( s ) ) = ( \alpha ( t , \alpha ( s - t , x ) ) , \alpha ( t , \alpha ( 0 , x ) ) ) \in U \ , . \end{align*}


\begin{align*} P _ { g , v , m } ( \nu _ 1 , \ldots , \nu _ g , T ^ { ( 2 ) } ) = P _ { g , v , m } ( \nu _ 1 , \ldots , \nu _ g , D _ 2 ) \ ; \ ; \textup { f o r a l l } \ ; \ ; 0 \leq m < v \leq g \end{align*}


\begin{align*} \frac { 1 } { 4 \epsilon ^ { 2 } } \mathrm { t r a c e } ( \mathrm { O P } _ { \hbar } ^ { T } ( f ( q ) ) \mathrm { O P } _ { \hbar } ^ { T } ( ( 2 \pi \hbar ) ^ { d } \nu ) ) = \frac { 1 } { 4 \epsilon ^ { 2 } } \underset { \mathbb { R } ^ { 2 } \times \mathbb { R } ^ { 2 } } { \int } | q | ^ { 2 } \nu ( d q d p ) + \frac { \hbar } { 4 \epsilon ^ { 2 } } . \end{align*}


\begin{align*} ( f * g ) ( [ s , u ] ) = \sum _ { s \leq t \leq u } f ( [ s , t ] ) g ( [ t , u ] ) ( [ s , u ] \in ( P ) ) . \end{align*}


\begin{align*} \sum _ { n \le x } a _ n \log \frac x n = \frac 1 { 2 \pi i } \int _ { a - i \infty } ^ { a + i \infty } \alpha ( s ) \frac { x ^ s } { s ^ 2 } \ , d s \end{align*}


\begin{align*} \mathrm { d } U _ t ^ x = ( \mu - F ( U _ t ^ x ) ) \mathrm { d } t + \sigma \mathrm { d } W _ t , U _ 0 ^ x = x . \end{align*}


\begin{align*} \hat { \R } _ { q ^ n } : = \{ S _ i \} _ { i \in \Lambda _ n } \subseteq \R _ { q ^ n } \ , , \end{align*}


\begin{align*} b _ k ( X , r \omega ) = r \cdot b _ k ( X , \omega ) . \end{align*}


\begin{align*} \begin{cases} & ( - \Delta ) ^ { s _ 1 } u ( x ) + ( - \Delta ) ^ { s _ 2 } u ( x ) + \lambda u ( x ) + V ( x ) u ( x ) = g ( u ( x ) ) , \ x \in \mathbb { R } ^ { d } , \\ & \int _ { \mathbb { R } ^ d } | u ( x ) | ^ 2 d x = a , \end{cases} \end{align*}


\begin{align*} G ( z , t ) = ( \tau ( t ) - z ) ( 1 - \overline { \tau ( t ) } z ) \ , p ( z , t ) \end{align*}


\begin{align*} D _ x b ( x ) = \begin{cases} 0 & x < 0 x > e , \\ \log \frac { 1 } { x } & 0 < x < e . \end{cases} \end{align*}


\begin{align*} 0 & = q n \cdot d ^ 1 _ { 0 , \frac q 2 } ( n , j ) , \\ 0 & = \frac { q n } 2 \cdot d ^ 1 _ { 0 , \frac q 2 } ( - n , i ) - \frac { q n } 2 \cdot d ^ 1 _ { 0 , \frac q 2 } ( n , j ) . \end{align*}


\begin{align*} \sum _ { j = n _ 1 } ^ { n _ 2 } \beta _ j \leq ( n _ 2 - n _ 1 + 2 ) \beta . \end{align*}


\begin{align*} C _ 0 = c _ 0 ^ { - 1 } \end{align*}


\begin{align*} R ( k ) = - \left ( \frac { h } { \sqrt { k ^ { 2 } } + \sqrt { k ^ { 2 } + h ^ { 2 } } } \right ) ^ { 2 } , \ \ \mathrm { d } \rho \left ( k \right ) = \frac { 2 k } { \pi h ^ { 2 } } \sqrt { h ^ { 2 } - k ^ { 2 } } \mathrm { d } k . \end{align*}


\begin{align*} E _ { \overline { k } } = \exp \sum _ { g = 0 } ^ k F _ g \quad E _ { \underline { k + 1 } } = \exp \sum _ { g = k + 1 } ^ \infty F _ g . \end{align*}


\begin{align*} \sum _ { x \in V _ x } u ^ \rho _ { x y } & = \sum _ { r = 1 } ^ k \sum _ { x \in V _ X } \delta _ { x \sigma ^ r ( y ) } p _ r + \sum _ { s = 1 } ^ k \sum _ { x \in V _ X } \delta _ { x \tau ^ s ( y ) } q _ s - 1 = \sum _ { r = 1 } ^ k p _ r + \sum _ { s = 1 } ^ k q _ s - 1 = 1 , \end{align*}


\begin{align*} \lim \limits _ { \abs { ( x , y ) } { } \rightarrow \infty } \abs { v _ \varepsilon } { } = 0 , \end{align*}


\begin{align*} f ( d ) \in H _ 1 \cap H _ 2 = H , \end{align*}


\begin{align*} M - D = M _ 0 \sqcup M _ 1 \end{align*}


\begin{align*} - 1 + f + 2 f ^ 2 = 0 , d \gamma _ 1 = \frac { 1 } { 2 } ( 5 - f ) \sigma _ 2 \wedge \sigma _ 3 . \end{align*}


\begin{align*} \phi \star \psi = \mu \circ ( \phi \otimes \psi ) \circ \Delta . \end{align*}


\begin{align*} A ' & : = \{ x ' \mid x \in A \} , \\ A _ 0 & : = \{ x \in P \mid x ' \in A \} . \end{align*}


\begin{align*} X _ k = h _ k ( ( Y _ j ) _ { j \in \tilde { J } _ k } ) \forall k \in K \end{align*}


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


\begin{align*} \dim { \left ( \overline { R } / q \right ) _ { m } } & = \dim { \left ( R / ( q \cap R ) \right ) _ { m \cap R } } + \operatorname { t r d e g } _ { R / ( q \cap R ) } \overline { R } / q - \operatorname { t r d e g } _ { \kappa \left ( ( m \cap R ) / ( q \cap R ) \right ) } \kappa ( m / q ) \\ & = \dim { R } + 0 - 0 = \dim { \overline { R } } , \end{align*}


\begin{align*} s c o r e = \# l o c k s \cdot e ^ { m a g n i t u d e } \end{align*}


\begin{align*} N _ { \sigma } ( D ) & = \sum _ { \Gamma \in S } \sigma _ { \Gamma } ( D ) \cdot \Gamma \\ & = \sum _ { \Gamma \in S } \lim _ { \epsilon \to 0 ^ + } \sigma _ { \Gamma } ( D + \epsilon A ) \cdot \Gamma \\ & = \lim _ { \epsilon \to 0 ^ + } ( \sum _ { \Gamma \in S } \sigma _ { \Gamma } ( D + \epsilon A ) \cdot \Gamma \ ) . \end{align*}


\begin{align*} \eta _ k ( u _ k ) = \begin{cases} q _ k ^ + & \| u _ k \| _ { \mathcal { H } _ k } < 1 , \\ p _ k ^ - & \| u _ k \| _ { \mathcal { H } _ k } \geq 1 , \\ \end{cases} \end{align*}


\begin{align*} \mathcal { F } ( \overline { \mathbf { P } } ) = \overline { { \rm E n t } } [ \overline { \mathbf { P } } | \overline { \mathbf { \Pi } } ^ { \mu _ { \theta } } ] . \end{align*}


\begin{align*} \aleph _ { \alpha , L } = \frac { 8 \alpha ^ { \eta } } { 9 \pi } \Big \langle \arctan \left ( \frac { 3 \alpha ^ { - \eta } } { 1 6 ( h _ 1 - e _ 1 ) } \right ) \Big \rangle _ { x u _ 1 } = \frac { 1 6 } { 3 } \alpha ^ { \eta } + O ( \alpha ^ { 2 \eta } ) , \end{align*}


\begin{align*} V _ 0 ( x ) = I _ F ( x ) - I _ F ( 0 ) \varphi _ F ( x ) , x \geq 0 . \end{align*}


\begin{align*} \Phi ( t , x ) - \varphi ( t , x ) = { \rm O } ( | t | ^ 3 ) \end{align*}


\begin{align*} \partial _ t p - | \nabla p | ^ 2 - \gamma p ( \Delta p + G ) = 0 . \end{align*}


\begin{align*} v v ^ * ( D ' \cap q P q ) v v ^ * = u ( p A p ) u ^ * . \end{align*}


\begin{align*} - \int _ { \mathbb { T } ^ 1 } u \times ( u \times \partial ^ 2 _ x u ) \cdot \partial ^ 2 _ x u \dd x = \int _ { \mathbb { T } ^ 1 } ( \partial _ x ^ 2 u + u | \partial _ x u | ^ 2 ) \cdot \partial ^ 2 _ x u \dd x \ , , \end{align*}


\begin{align*} \sum _ { w \in \widehat W ^ \natural } d e t ( w ) c h \ , M ^ W ( w . \widehat \nu _ h ) = \left ( \sum _ { w \in \widehat W ^ \natural } d e t ( w ) c h \ , M ^ W ( w . \widehat \nu _ 0 ) \right ) e ^ { ( 0 , \tfrac { 2 h ^ 2 + ( h ^ \vee - 1 ) h } { 2 ( k + h ^ \vee ) } - h ) } . \end{align*}


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


\begin{align*} \begin{multlined} \sigma _ 0 \sigma _ 0 ( A ) ( ( i , j ) ) \\ = ( - 1 ) ^ { | i | | j | } ( - 1 ) ^ { | j | | i | } A ( ( i , j ) ) = A ( ( i , j ) ) \ ; . \end{multlined} \end{align*}


\begin{align*} F ( \zeta _ { N } ^ { - 1 } ) = U ( - 1 ; \zeta _ { N } ) . \end{align*}


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


\begin{align*} A ( x ) B ( y ) = \mathbf { f } ( x , y ) B ( y ) A ( x ) + \mathbf { g } _ { A } ( x , y ) B ( x ) A ( y ) + \mathbf { g } _ { \tilde { D } } ( x , y ) B ( x ) \tilde { D } ( y ) \ , , \end{align*}


\begin{align*} X _ { s } = - u _ s \nabla _ { x } \Phi ( Y _ { s } , X _ { s } ) \end{align*}


\begin{align*} \lambda _ { m , n } = \varepsilon _ { 1 } 2 ^ { b m } 2 ^ { a n } L ( 2 ^ { n } ) . \end{align*}


\begin{align*} e _ j ( \tilde { z } ) = ( - 1 ) ^ j h _ j ( z ) \mod \tilde { J } \ / ; e _ j ( \hat { z } ) = ( - 1 ) ^ j G _ j ( z ) \mod \hat { J } \ / . \end{align*}


\begin{align*} \widetilde { \varphi } ( g ) = \int _ { \Omega } \varphi ( \gamma ( g w ) ) d w g \in G . \end{align*}


\begin{align*} \Psi _ y = u _ { \alpha } \otimes \Phi _ y + R _ y , \end{align*}


\begin{align*} \mathcal { I } ( a ) = \log a \end{align*}


\begin{align*} \frac { \partial z _ \epsilon } { \partial t } \ , - \ , \triangle z _ \epsilon \ , + \ , \frac { 2 q } { \epsilon } ( | D z _ \epsilon | ^ 2 \ , - \ , 1 ) \ ; = \ ; 0 . \end{align*}


\begin{align*} g = \frac { \partial _ { \tau , \xi } w } { v } \end{align*}


\begin{align*} X ( r e ^ { i \theta } ) = \left ( r \cos { \theta } , r \sin { \theta } , \frac { \theta } { \pi } \right ) , w = r e ^ { i \theta } \in \mathbb { H } . \end{align*}


\begin{align*} w \to u _ w ^ { } = \frac { 1 } { 4 \pi i } \int _ { ( \frac { 1 } { 2 } ) } \frac { \langle f , E _ s \rangle \cdot E _ s } { \lambda _ s - \lambda _ w } d s \end{align*}


\begin{align*} \pi ( x ) = \frac { x } { \log x } + \frac { x } { \log ^ 2 x } + \frac { 2 x } { \log ^ 3 x } + \frac { 6 x } { \log ^ 4 x } + \ldots + \frac { ( n - 1 ) ! x } { \log ^ n x } + O \left ( \frac { x } { \log ^ { n + 1 } x } \right ) \end{align*}


\begin{align*} ( q + \xi ) ^ 2 - E - i \eta & = q ^ 2 + 2 \xi q + \xi ^ 2 - i \eta \end{align*}


\begin{align*} \| \mu \| _ { K } : = \sup _ { t \in G } \left | \mu \right | ( t + K ) \ , . \end{align*}


\begin{align*} ( m ( a ) \sup _ { 0 \leq k \leq \lfloor T c ( a ) \rfloor } \ , ( S _ k - a u k + \zeta _ { k + 1 } ) ) _ { u > 0 } ~ \overset { { \rm f . d . } } { \longrightarrow } ~ ( X _ l ( u , T ) ) _ { u > 0 } , a \to 0 + , l = 1 , 2 , 3 , \end{align*}


\begin{align*} I I & = \dfrac { n } { r ^ n } \int _ 0 ^ r \rho ^ { n - 1 } \ , I ( \rho , r , y ) \ , d \rho \end{align*}


\begin{align*} \nabla J ( u _ 0 ) = \frac { d } { \epsilon } \cdot \mathbb { E } _ { v _ 0 \in \mathbb { S } ^ { d - 1 } } \left [ \epsilon \langle \nabla J ( u _ 0 ) , v _ 0 \rangle v _ 0 \right ] . \end{align*}


\begin{align*} \Omega \cdot \varphi ^ P _ s = \sum _ { i } h _ i ^ 2 \cdot \varphi ^ P _ s + \sum _ { i < j } ( - h _ i + h _ j ) \cdot \varphi ^ P _ s . \end{align*}


\begin{align*} m \otimes n = m + n \ ; . \end{align*}


\begin{align*} \Upsilon : = \exp \bigg \{ \int _ t ^ T \int _ t ^ T \alpha _ { H _ 0 } | u - v | ^ { 2 H _ 0 - 2 } \big [ q ( B ^ 1 _ u , B ^ 1 _ v ) + 2 q ( B ^ 1 _ u , B ^ 2 _ v ) + q ( B ^ 2 _ u , B ^ 2 _ v ) \big ] d u d v \bigg \} \ , . \end{align*}


\begin{align*} \| u \| _ { 1 , \mathcal { H } } = \| u \| _ { \mathcal { H } } + \| \nabla u \| _ { \mathcal { H } } , \end{align*}


\begin{align*} a _ n = \left \{ ( x _ 1 , \ldots , x _ n ) \in [ 0 , 1 ] ^ n : ( n - 2 ) / 2 \leq \sum _ { i = 1 } ^ { n } x _ i \leq n / 2 \right \} \end{align*}


\begin{align*} c _ { i i } = 2 \ , , c _ { i , i \pm 1 } = - 1 \ , , m _ { i , i \pm 1 } = \mp 1 \ , , \mathrm { a n d } c _ { i j } = 0 = m _ { i j } \mathrm { o t h e r w i s e } \ , . \end{align*}


\begin{align*} S _ { n } ^ { ( k ) } = \sum _ { i = 0 } ^ { n } \frac { C _ i \alpha _ i ^ n } { f ' ( \alpha _ i ) } \end{align*}


\begin{align*} W ^ 2 \gamma & = W ( \nabla _ \alpha - ( \lambda - \alpha ) Z ) [ \gamma ] = \nabla ^ 2 _ \alpha \gamma - ( \lambda - \alpha ) Z \nabla _ \alpha \gamma - W [ \lambda - \alpha ] \cdot Z \gamma - ( \lambda - \alpha ) W Z \gamma . \end{align*}


\begin{align*} \begin{gathered} a _ { i j } ^ \varepsilon ( x , t ) = a _ { j i } ^ \varepsilon ( x , t ) \quad ( x , t ) \in Q _ { \varepsilon , T } , \ , i , j = 1 , \dots , n , \\ \sum _ { i , j = 1 } ^ n a _ { i j } ^ \varepsilon ( x , t ) \xi _ i \xi _ j \geq 0 \quad ( x , t ) \in Q _ { \varepsilon , T } , \ , \xi = ( \xi _ 1 , \dots , \xi _ n ) ^ T \in \mathbb { R } ^ n \end{gathered} \end{align*}


\begin{align*} \tilde { U } ( { \cdots } ) = \begin{cases} \dfrac { ( - 1 ) ^ m } { m ! } , & j = 0 , \\ 0 , & . \end{cases} \end{align*}


\begin{align*} \Delta ^ \wedge & = \partial ^ 2 _ t + \frac 1 { t + R } \partial _ t + \frac 1 { ( t + R ) ^ 2 } \partial _ \theta ^ 2 \\ & = \partial ^ 2 _ t + \frac 1 R \sum _ { m \ge 0 } \left ( \frac t R \right ) ^ m \partial _ t + \frac 1 { R ^ 2 } \left ( \sum _ { m \ge 0 } \left ( \frac t R \right ) ^ m \right ) ^ 2 \partial _ \theta ^ 2 . \end{align*}


\begin{align*} Z _ r \left ( g g ^ \top , \frac { r s } { 2 } \right ) = \sum _ { 0 \neq v \in \Z ^ r } { | v \cdot g | ^ { - 2 ( r s / 2 ) } } = \zeta ( r s ) \sum _ { v \in \Z ^ r } { | v \cdot g | ^ { - r s } } , \end{align*}


\begin{align*} \eta _ q ^ { \epsilon , { \bf h } } ( t ) & = h _ q + \frac { 1 } { \epsilon } \int _ 0 ^ { t } \eta _ p ^ { \epsilon , { \bf h } } ( s ) d s \\ & = h _ q + \int _ 0 ^ t D f ( q ^ \epsilon ( s ) ) . \eta _ q ^ { \epsilon , { \bf h } } ( s ) d s + \int _ 0 ^ t D \sigma ( q ^ { \epsilon , { \bf h } } ( s ) ) . \eta _ q ^ { \epsilon , { \bf h } } ( s ) d \beta ( s ) + \epsilon \bigl ( \eta _ p ^ { \epsilon , { \bf h } } ( 0 ) - \eta _ p ^ { \epsilon , { \bf h } } ( t ) \bigr ) . \end{align*}


\begin{align*} u _ w ^ { } = u _ { 1 - w } ^ { } - \frac { \langle f , E _ w \rangle \cdot E _ w } { 2 w - 1 } . \end{align*}


\begin{align*} \varphi & : = z \phi _ { x ^ 0 } + \frac { s } { r } \phi _ r + \phi _ s , \\ W & : = \frac { 1 } { \phi } \left \{ \frac { \varphi } { 2 } - s \phi U - \frac { \phi _ z \Omega } { 2 \Lambda } ( \varphi _ z - 2 \phi _ { x ^ 0 } ) - ( r ^ 2 - s ^ 2 ) \left [ { \phi _ s } U - { \phi _ z } V \right ] \right \} , \\ U & : = \frac { 1 } { 2 \Lambda } \left \{ \left ( \varphi _ s - \frac { 2 } { r } \phi _ r \right ) \phi _ { z z } - \left ( \varphi _ z - 2 \phi _ { x ^ 0 } \right ) \phi _ { s z } \right \} , \end{align*}


\begin{align*} : & x ' _ i = \begin{cases} ( q ' , b ) & i = i _ 0 \\ x _ j & \end{cases} \\ : & x ' _ i = \begin{cases} a & i = i _ 0 \\ ( q ' , x _ { i } ) & i = i _ 0 + \delta \\ x _ i & \end{cases} \end{align*}


\begin{align*} \mathbb { P } \left \{ \left \vert \sum _ { j = 1 } ^ k a _ j ( Z _ j - 1 ) \right \vert > r \right \} \leq 2 \exp \left ( - c \min \left \{ \left ( \frac { r } { \left \vert a \right \vert } \right ) ^ 2 , \frac { r } { \left \vert a \right \vert _ \infty } \right \} \right ) \end{align*}


\begin{align*} c _ 0 ^ { - 1 } \leq 1 - r \kappa _ \alpha ( y ) \leq c _ 0 , y \in \Gamma , \ , r \in [ - \delta , \delta ] , \ , \alpha = 1 , \dots , n - 1 \end{align*}


\begin{align*} \delta \Phi ^ { \rm E } ( t _ 1 , t _ 2 , x ) & = x + ( t _ 1 + t _ 2 ) \sigma ( x ) - \bigl ( \Phi ^ { \rm E } ( t _ 2 , x ) + t _ 1 \sigma ( \Phi ^ { \rm E } ( t _ 2 , x ) ) \bigr ) \\ & = t _ 1 \bigl ( \sigma ( x ) - \sigma ( x + t _ 2 \sigma ( x ) ) \bigr ) \end{align*}


\begin{align*} \pi _ t ( p ) = \begin{cases} F _ t \circ \rho ^ { - 1 } ( x ) , & p \in U \\ p , & p \in M - U _ { 2 t } = M _ { 2 t } \end{cases} . \end{align*}


\begin{align*} G \left ( k ^ { 2 } , x \right ) = - \frac { \varphi \left ( x , k \right ) \psi \left ( x , k \right ) } { 2 \mathrm { i } k } . \end{align*}


\begin{align*} \Delta ^ 2 \widetilde { w } ^ \alpha _ 0 = 0 \qquad \textrm { i n } \Omega \end{align*}


\begin{align*} \tau \Big ( ( a _ 1 - \tau ( a _ 1 ) ) ( a _ 2 - \tau ( a _ 2 ) ) \cdots ( a _ k - \tau ( a _ k ) ) \Big ) = 0 . \end{align*}


\begin{align*} \forall \ , x , y , \ \alpha ^ s ( x , y ) = 0 \Longleftrightarrow \forall \ , n \in \Z , \ \alpha ^ s ( f ^ n ( x ) , f ^ n ( y ) ) = 0 . \end{align*}


\begin{align*} \eta _ \delta ( x ) : = \begin{cases} 1 & | x - a _ 0 | \leq \delta , \\ \frac { 2 \log | x - a _ 0 | } { \log \delta } - 1 & \delta < | x - a _ 0 | \leq \delta ^ \frac { 1 } { 2 } , \\ 0 & \end{cases} \end{align*}


\begin{align*} A ( t ) x ^ 2 + B ( t ) x y + C ( t ) y ^ 2 + ( t - a ) ( t - b ) z ^ 2 = 0 \end{align*}


\begin{align*} R _ y = \kappa \Phi _ { \# } ^ y + R ^ { \# } _ y , \end{align*}


\begin{align*} f \otimes g = f \hat \sqcup g \ ; . \end{align*}


\begin{align*} C _ { n } \left ( x ; z \right ) = \phi \left ( x ; z \right ) + \gamma _ { n } \left ( z \right ) + \gamma _ { n + 1 } \left ( z \right ) - n - \frac { 1 } { 2 } . \end{align*}


\begin{align*} g _ n : = f _ n - f , \end{align*}


\begin{align*} N _ { i , p ( u ) } = \begin{cases} 1 , & u _ { i + 1 } > u \geq u _ { i } \\ 0 , & \end{cases} \end{align*}


\begin{align*} \Delta ( \tilde E _ s - h _ s ) & = ( \Delta - \lambda _ s ) ( \tilde E _ s - h _ s ) + \lambda _ s ( \tilde E _ s - h _ s ) \\ & = ( \Delta - \lambda _ s ) ( \tilde \Delta - \lambda _ s ) ^ { - 1 } ( \Delta - \lambda _ s ) h _ s + \lambda _ s ( \tilde E _ s - h _ s ) \\ & = ( \Delta - \lambda _ s ) h _ s + \lambda _ s ( \tilde E _ s - h _ s ) \in L ^ 2 , \end{align*}


\begin{align*} \int _ 0 ^ { \infty } \norm { \nabla f ( X ( s ) ) } ^ 2 d s \geq \sum _ { k \geq k ' } \int _ { t _ k } ^ { t _ k + \frac { \varepsilon } { 2 L } } \norm { \nabla f ( X ( s ) ) } ^ 2 d s \geq \sum _ { k \geq k ' } \frac { \delta ^ 2 \varepsilon } { 8 L } = \infty , \end{align*}


\begin{align*} U : = \{ g \in G : g - 1 _ V \textrm { i s i n v e r t i b l e } \} . \end{align*}


\begin{align*} u ( t ) = e ^ { i t \Delta } u _ 0 - i \int _ 0 ^ t e ^ { i ( t - s ) \Delta } F ( u ( s ) ) \ , d s , \end{align*}


\begin{align*} 0 < \delta \ll 1 , \mbox { a n d } \mu : = 1 - 2 \delta \ . \end{align*}


\begin{align*} F _ { s i } = \mathbf { D } ^ { \ell } F _ { \ell i } \end{align*}


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


\begin{align*} - \mathfrak { s } \kappa \int _ 0 ^ t \sum _ { k \in \mathbb { N } } \operatorname { R e } \langle z \Phi e _ k | z | ^ 2 , z \Phi e _ k \rangle _ { L _ x ^ 2 } \d s + \mathfrak { s } \kappa \int _ 0 ^ t \operatorname { R e } \langle z | z | ^ 2 , z F _ { \Phi } \rangle _ { L _ x ^ 2 } \d s = 0 . \end{align*}


\begin{align*} | a ( q ) + a ( \theta ( q ) ) | & = O _ \lambda \left ( r ^ 2 ( \| \nabla _ a ^ 2 a \| _ { L _ \infty ( B ) } + \| \partial _ z a \| _ { L _ \infty ( B ) } ) \right ) , \end{align*}


\begin{align*} \begin{gathered} \Phi _ { + 0 } ^ { a , b } : \{ 0 , \ldots , a - 1 \} \rightarrow \{ 0 , \ldots , a + b - 1 \} \ ; , \\ \Phi _ { + 0 } ^ { a , b } ( i ) = i \ ; . \end{gathered} \end{align*}


\begin{align*} \Omega _ { n , \le s } = \Omega _ { n } \setminus \bigcup _ { k > s } \Omega _ { n , k } . \end{align*}


\begin{align*} ( - i u ) ^ { \gamma \cdot \mathrm c _ 1 ( S ) } \cdot \left \langle \gamma _ 1 , \dots , \gamma _ N \right \rangle ^ { - \infty } _ { g , \gamma } = ( e ^ { i u / 2 } - e ^ { - i u / 2 } ) ^ { \gamma \cdot \mathrm c _ 1 ( S ) } \cdot \left \langle \gamma _ 1 , \dots , \gamma _ N \right \rangle ^ { 0 } _ { g , \gamma } . \end{align*}


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


\begin{align*} \frac { \omega - q } { \prod _ { i \in I } \ell _ i } = \frac { \prod _ { i \notin I } \ell _ i } { \omega + q } , \end{align*}


\begin{align*} & T _ { 0 , L } [ E + i \eta ; \varphi _ p , \varphi _ p ] = \frac { 1 } { \nu ( p ) - E - i \eta } . \end{align*}


\begin{align*} w _ n : = v _ n - u . \end{align*}


\begin{align*} P _ D ( \lambda ) = ( \lambda + 2 ) ^ { n - k } \left [ \prod _ { i = 1 } ^ k ( \lambda - n _ i + 2 ) - \sum _ { i = 1 } ^ k n _ i \prod _ { j = 1 , j \neq i } ^ k ( \lambda - n _ j + 2 ) \right ] . \end{align*}


\begin{align*} \nabla _ { a X } \ , n = a \nabla _ X \ , n , \nabla _ X ( a n ) = X ( a ) \ , n + a \ , \nabla _ X \ , n , \end{align*}


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


\begin{align*} \mu _ { f } = \frac { f _ { \overline { z } } } { f _ { z } } . \end{align*}


\begin{align*} Q _ N g _ n = - B ^ N ( P _ N w _ n , P _ N w _ n ) - B ^ N ( P _ N w _ n , Q _ N w _ n ) - B ^ N ( Q _ N w _ n , P _ N w _ n ) - D B ^ N ( u ) P _ N w _ n + Q _ N g _ { n - 1 } , \end{align*}


\begin{align*} \lim _ { { \rm R e } z \to 0 ^ + } ( 2 { \rm R e } z ) \left | P h ( z ) \right | = 0 . \end{align*}


\begin{align*} L _ { R } \left [ p \right ] = { \displaystyle \int \limits _ { I } } p \left ( x \right ) e ^ { - a x ^ { 2 } } d x , p \in \mathbb { R } \left [ x \right ] , a > 0 . \end{align*}


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


\begin{align*} m _ { \jmath _ 0 } = e _ { T _ { \rm a } , S } = - 1 . \end{align*}


\begin{align*} R _ { i , j } ( s , t ) : = \Phi ( \psi _ i ) _ s ( v _ { i , j } Z ^ t ) \end{align*}


\begin{align*} l _ { i + 1 , i } ^ { ( 0 ) } \mid 0 \rangle = 0 a n d l _ { i i } ^ { ( 0 ) } \mid 0 \rangle = - \frac { 1 } { h } ( c _ { i } - c _ { i - 1 } ) \mid 0 \rangle , \end{align*}


\begin{align*} D _ { p } ( g , r , 1 ) = \binom { g + r } { r } \end{align*}


\begin{align*} V = \bigoplus _ { i = 1 } ^ n V _ i . \end{align*}


\begin{align*} [ 2 b ] _ n = \begin{cases} 0 & \\ q ^ { n - 1 } t \left ( \frac { t ^ { n - 3 } - 1 } { t - 1 } \right ) & \end{cases} \end{align*}


\begin{align*} P _ { V } ( x ) + K = G \ , . \end{align*}


\begin{align*} L _ k ( s ) : = E _ k ^ { - 1 } ( s ) s \in E _ k ( \R ) = ( s _ k , + \infty ) \ , . \end{align*}


\begin{align*} \acute { \mu } ( \alpha , e ) = \begin{cases} ( d / r , \ , e / r ) , & r > 0 , \\ + \infty , & r = 0 . \end{cases} \end{align*}


\begin{align*} & u _ { 0 A } ( x ) : = \left \{ \begin{aligned} & 0 , \ \ x < 0 , \\ & A , \ \ x > 0 , \end{aligned} \right . \end{align*}


\begin{align*} \beta _ 2 ( \overline V ) & = 1 + d + s _ 1 \\ \beta _ 4 ( \overline V ) & = 1 + d + s _ 1 \\ \beta _ 3 ( \overline V ) & = \beta _ 3 ( \hat V ) . \end{align*}


\begin{align*} 2 \mathrm { R e } \langle u _ { \alpha } \otimes \Phi _ y \ , | \ , H _ y R _ y \rangle = & - 2 \mathrm { R e } ( \overline { \Phi } ^ { ( 0 ) } _ y \kappa ) \| \Phi _ { \# } ^ y \| ^ 2 _ { \# } - C \alpha ( \varepsilon _ 7 + \alpha ) \| R _ y ^ { \# } \| ^ 2 \\ & - C \alpha ( \varepsilon _ 8 + \alpha ) \| R _ * \| ^ 2 - C \alpha \varepsilon _ 7 \| P R ^ { \# } _ y \| ^ 2 + O ( \alpha ^ 4 ) . \end{align*}


\begin{align*} [ v ^ \sharp , f X ] \ , = \ , v ^ \sharp ( f ) \ , X \ , + \ , f \ , [ v ^ \sharp , X ] = 0 . \end{align*}


\begin{align*} { \mathcal { L } } & = a - \mathcal { K } * , \\ { \mathcal { Q } } ( G ) ( x ) & = \mathcal { Q } ( g ) ( \omega ) , \\ { \mathcal { C } } ( G ) ( x ) & = \mathcal { C } ( g ) ( \omega ) . \end{align*}


\begin{align*} e \leq f \iff e = f e = e f \end{align*}


\begin{align*} P ( A _ n ^ c ) : = P \Big ( \min ( R _ { n } , R _ { n , 2 n } ) \le \frac { n } { 2 \log n } \Big ) \le 8 C _ 1 ( \log n ) ^ { - 2 } . \end{align*}


\begin{align*} \begin{cases} ( - \Delta ) ^ s u ( x ) = v ^ p ( x ) , & x \in \Omega , \\ ( - \Delta ) ^ t v ( x ) = u ^ q ( x ) , & x \in \Omega , \\ u ( x ) = - \Delta u ( x ) = \cdots = ( - \Delta ) ^ { s - 1 } u ( x ) = 0 , & x \in \partial \Omega , \\ v ( x ) = - \Delta v ( x ) = \cdots = ( - \Delta ) ^ { t - 1 } v ( x ) = 0 , & x \in \partial \Omega . \end{cases} \end{align*}


\begin{align*} \ddot \kappa _ \gamma ( 0 ) = - ( n - 2 ) \ddot \kappa ( C _ 0 ) - H _ 1 '' . \end{align*}


\begin{align*} B ( k , \nu ) = \frac { ( \nu | \nu + 2 \rho ^ \natural ) } { 2 ( k + h ^ \vee ) } - \frac { ( k + 1 ) ^ 2 } { 4 ( k + h ^ \vee ) } . \end{align*}


\begin{align*} \Psi ( Z ) & = q ^ A \zeta ^ B s ^ C \prod _ { ( n , r , m ) > 0 } ( 1 - q ^ n \zeta ^ { r } s ^ { t m } ) ^ { \mathrm { m u l t } ( n , r , m ) } \\ & = q ^ A \zeta ^ B s ^ C \prod _ { d | N } \prod _ { ( n , r , m ) > 0 } ( 1 - q ^ { d n } \zeta ^ { d r } s ^ { d t m } ) ^ { \mathrm { m u l t } _ d ( n m , r ) } \end{align*}


\begin{align*} \langle \alpha _ j ^ \ast \alpha _ k \rangle = N _ 0 ( E _ j ) \delta _ { j k } \ , , N _ 0 ( E _ j ) : = \Big ( \frac { 1 } { e ^ { \beta ( T ) E _ j } - 1 } \Big ) , \beta ( T ) : = 1 / k _ B T , T \ge 0 . \end{align*}


\begin{align*} d _ { C C } ( \hat { \mathbb { X } } ^ N _ { 1 } , \hat { \mathbb { Y } } ^ N _ { 1 } ) = d _ { C C } ( \hat { \mathbb { X } } ^ N _ { 0 , 1 } , \hat { \mathbb { Y } } ^ N _ { 0 , 1 } ) \lesssim \lim _ { \delta \to 0 } \inf _ { \lambda \in \Lambda } \{ | \lambda | \vee d _ p ( \hat { \mathbf { X } } ^ { \phi , \delta } _ \lambda , \hat { \mathbf { Y } } ^ { \phi , \delta } ) \} = : \beta _ { p - v a r } ( \hat { \mathbf { X } } , \hat { \mathbf { Y } } ) . \end{align*}


\begin{align*} ( z ^ * ) ^ { \otimes d } \circ \mu _ { X ' } = ( z ^ * ) ^ { \otimes d } \circ \mu _ A \circ \mu _ X = ( z ^ * ) ^ { \otimes d } \circ \mu _ X \end{align*}


\begin{align*} M ( A ) = \mathbf { 1 } ^ { \mathcal { H } } \ ; . \end{align*}


\begin{align*} \frac { \partial ^ n } { \partial x ^ n } V ( y , x ) = \delta _ n ^ 0 + O _ n \Big ( x ^ { - n / 2 } ( y / x ) ^ { \sqrt { x } } \Big ) \frac { \partial ^ n } { \partial x ^ n } V ( y , x ) \ll _ n x ^ { - n / 2 } ( x / y ) ^ { \sqrt { x } } , \end{align*}


\begin{align*} \langle M _ { 1 , 2 } \psi , M _ { 2 , 2 } \psi \rangle _ { L ^ 2 ( \Omega _ T ) } & = - \int _ { \Gamma _ T } \partial _ t \psi \partial _ \nu ^ A \psi \ , \d S \ , \d t . \end{align*}


\begin{align*} E _ 1 ^ { p q } = ( K ^ q ( T _ p ) \otimes \Q ) _ A \Rightarrow K ^ { p + q } ( T ) \otimes \Q , \end{align*}


\begin{align*} \omega _ { 5 6 } ( \overline { E } ) = \frac { i } { \kappa _ 2 ^ 2 - \mu _ 2 ^ 2 } \left ( \kappa _ 2 \overline { E } ( \mu _ 2 ) - \mu _ 2 \overline { E } ( \kappa _ 2 ) \right ) . \end{align*}


\begin{align*} l ( T ^ F , s ) : = \left | \{ ( j , k ) : 1 \leq k \leq \lambda _ j ' , \ s ' _ { j k } \neq 1 \} \right | . \end{align*}


\begin{align*} n ^ * : = \limsup _ { T \to \infty } \frac { 1 } { N ( T ) } \sum _ { 0 < \gamma _ d \le T } m ( \gamma _ d ) ^ 2 \end{align*}


\begin{align*} W _ { i j } = - \underline { D } _ i \nu _ j , H = - \mathrm { d i v } _ \Gamma \nu \quad \Gamma , i , j = 1 , \dots , n \end{align*}


\begin{align*} C _ 1 ( b ) & = \frac { I _ F ' ( b ) \varphi _ F ( b ) - I _ F ( b ) \varphi _ F ' ( b ) } { \psi ' ( b ) \varphi _ F ( b ) - \psi ( b ) \varphi _ F ' ( b ) } , \\ C _ 2 ( b ) & = \frac { I _ F ' ( b ) \psi ( b ) - I _ F ( b ) \psi ' ( b ) } { \psi ' ( b ) \varphi _ F ( b ) - \psi ( b ) \varphi _ F ' ( b ) } . \end{align*}


\begin{align*} Z _ n ( \lambda , T ) : = \sum _ { N \geq 0 } \frac { T ^ N } { N ! } \int _ { \mathbb { T } _ n ^ N } g _ N ^ \lambda ( x _ 1 , \ldots , x _ N ) \mathrm { d } x _ 1 \ldots \mathrm { d } x _ N = \sum _ { k \geq 0 } \sum _ { 0 \leq \ell \leq n } T ^ { n k } e ^ { - \lambda \ell } \mathrm { V o l } ^ { ( n ) } _ { k , \ell } . \end{align*}


\begin{align*} \phi _ i ( v ) \ = \ \sum _ { j = 1 } ^ n \phi _ i ( \lambda _ j \cdot v _ j ) \ = \ \sum _ { j = 1 } ^ n \phi _ i ( \lambda _ j ) \cdot \phi _ i ( v _ j ) , \end{align*}


\begin{align*} \hat \mu _ 1 ( u + v ) \hat \mu _ 2 ( u + v ) = \hat \mu _ 1 ( u - v ) \hat \mu _ 2 ( u - v ) , \ \ u , v \in L . \end{align*}


\begin{align*} \Delta ^ 2 \bar v _ h = - \theta _ h \qquad \textrm { i n } \R ^ 2 \ , , \end{align*}


\begin{align*} J _ { \gamma } ( v ) : = \int _ { \Omega } | \nabla v | + x _ n ^ { 2 \gamma } \chi _ { \{ u > 0 \} } d x . \end{align*}


\begin{align*} ( F _ t f ) ( y , Z _ N ) : = f \big ( y , t Z _ N \big ) , ( y , Z _ N ) \in B _ { \frac { \epsilon } { t } } ( N ) . \end{align*}


\begin{align*} h ^ 0 ( \mathbb P ^ { r - 1 } , \mathcal I _ { \Sigma , \ , \mathbb P ^ { r - 1 } } ( i ) ) \leq h ^ 1 ( \mathbb P ^ { r } , \mathcal I _ { S } ( i - 1 ) ) \leq i + \binom { i } { 2 } s - \binom { r + i - 1 } { i - 1 } . \end{align*}


\begin{align*} \gamma _ { \mathfrak n } ( x , v , w ) & = B \left ( d ' ( x , v ) , w \right ) = B _ { \mathfrak a } ( \rho ( x ) w , v ) \\ \gamma _ { \mathfrak n } ( v , w , x ) & = B \left ( d ' ( v , w ) , x \right ) = B _ { \mathfrak a } \left ( \rho ( x ) v , w \right ) \end{align*}


\begin{align*} B = \{ \phi ^ 2 = p q \psi \} . \end{align*}


\begin{align*} - \Delta _ p u = a ( x ) u ^ { - \gamma } + \lambda f ( u ) \ ; \ ; \mbox { i n } \ ; \ ; \Omega , u > 0 \ ; \ ; \mbox { i n } \ ; \ ; \Omega , u = 0 \ ; \ ; \mbox { o n } \ ; \ ; \partial \Omega \end{align*}


\begin{align*} \eta _ a f = c _ P f ( i a ) = \int _ 0 ^ 1 f ( x + i a ) d x \end{align*}


\begin{align*} \beta _ { a } ( I _ { u _ \ell } ) = \beta _ { a , b } ( I _ { u _ \ell } ) & = \big | \big \{ w \in \mathcal { L } _ t ^ f ( u ) : \min ( u ) = 1 \big \} \big | - \big | \big \{ w \in \mathcal { L } _ t ^ f ( v ) \setminus \{ v \} : \max ( w ) = n \big \} \big | \\ & = \big | \big \{ u _ \ell > u _ { \ell + 1 } > \dots > u _ { d - 1 } \} \big | - | \{ v _ { \ell + 1 } > \dots > v _ d \} | = 0 , \end{align*}


\begin{align*} d ' ( p ) : = s - \big ( f _ i ( P _ j ) p \big ) . \end{align*}


\begin{align*} \mu ^ { * } : = \frac { \exp \left ( - \frac { ( 1 - \lambda ) \theta } { 2 } V \right ) } { \int _ { M } \exp \left ( - \frac { ( 1 - \lambda ) \theta } { 2 } V \right ) } , \end{align*}


\begin{align*} x _ { k + 1 } = x _ k - h \nabla f ( x _ k ) + \sigma ( k , x _ k ) \triangle W _ k , \end{align*}


\begin{align*} L _ 0 G _ { s u m } = L _ 0 g _ { s u m } = - ( g _ { s u m } \frac { L v } { v } + 2 \frac { w ^ 2 } { v } \partial _ { \eta } v \partial _ { \eta } g _ { s u m } ) = - g _ { s u m } \frac { L v } { v } > 0 a t z _ { m a x } , \end{align*}


\begin{align*} \frac { 4 - 4 { h ^ \prime } ^ 2 - 2 h h ^ { \prime \prime } } { h ^ 2 } = - \frac { 5 h ^ \prime h ^ { \prime \prime } + h h ^ { \prime \prime \prime } } { 2 h h ^ \prime } = \frac { 1 } { 8 } { \textbf { H } ^ \prime } ^ 2 \geq 0 . \end{align*}


\begin{align*} \gamma _ { A , \ell } = \frac { \dim W } { \dim G _ { A , \ell } - \dim ( G _ { A , \ell } ) _ W } \end{align*}


\begin{align*} \sigma _ 1 + a \sigma _ 2 = 0 , \ \ \sigma ' _ 1 + a \sigma ' _ 2 = 0 \end{align*}


\begin{align*} \Delta \log ( 1 - K ) = 6 K . \end{align*}


\begin{align*} ( i d \times r ) ( r \times i d ) ( i d \times r ) = ( r \times i d ) ( i d \times r ) ( r \times i d ) . \end{align*}


\begin{align*} x P _ { n } \left ( x ; z \right ) = P _ { n + 1 } \left ( x ; z \right ) + \gamma _ { n } ( z ) P _ { n - 1 } \left ( x ; z \right ) , n \geq 0 , \end{align*}


\begin{align*} \sigma _ m ( \mathcal B , x ) _ { \mathbb { X } } \ = \ \sigma _ m ( x ) \ : = \ \inf \left \{ \left \| x - \sum _ { n \in A } b _ n e _ n \right \| : | A | = m , b _ n \in \mathbb { F } \right \} . \end{align*}


\begin{align*} \frac { \partial \overline { H } _ 6 } { \partial \overline { z } } = 3 i \overline { H } _ 6 \omega _ { 1 2 } ( \overline { \partial } ) - \overline { H } _ 5 \omega _ { 5 6 } ( \overline { \partial } ) . \end{align*}


\begin{align*} \phi _ g ( \tau , z ) = - \frac { T _ 2 \varphi _ g ( \tau , z ) } { \varphi _ g ( \tau , z ) } - f _ g ( \tau , z ) , \end{align*}


\begin{align*} \begin{gathered} A ( ( l , c _ 1 , c _ 2 , \ldots ) ) \\ = \begin{cases} 1 & n ( c _ 1 ) = l c _ 1 = c _ 2 = \ldots \\ 0 & \end{cases} \ ; , \end{gathered} \end{align*}


\begin{align*} \mathcal { E } ^ { * } ( \mu ) + \frac { 1 } { \theta } { \rm e n t } [ \mu | \pi ] = \mathcal { E } _ { V } ( \mu ) + \frac { 1 } { \theta } \left ( { \rm e n t } [ \mu ] - \log z ^ { * } \right ) . \end{align*}


\begin{align*} S _ g = ( n + 1 ) \Lambda - \left ( \Delta _ B \varphi + \frac { \nabla _ B \varphi ( f ) } { f } \right ) - ( m - 1 ) \left ( \frac { \Delta _ B f } { f } + ( m - 1 ) \frac { | \nabla _ B f | ^ 2 } { f ^ 2 } \right ) . \end{align*}


\begin{align*} j _ { M , \epsilon } ^ \infty = 1 - \epsilon . \end{align*}


\begin{align*} \mathcal { U } ( x ) = \mathcal { M } ( x ) \hat { \mathcal { K } } ( x ) \hat { \mathcal { M } } ( x ) \ , , \end{align*}


\begin{align*} \mathcal { H } ^ { H , T } = \{ h : h ( t ) = \int _ 0 ^ t K ^ H ( t , s ) f ( s ) d s , \mbox { f o r s o m e } f \in L ^ 2 ( [ 0 , T ] ) \} \end{align*}


\begin{align*} \| x \| _ { \mathcal { C G } _ q ^ \omega } \ \ge \ \left ( \sum _ { k = 1 } ^ { n _ s } ( \omega ( k ) \vartheta _ k ( x _ s ) ) ^ q \frac { 1 } { k } \right ) ^ { 1 / q } \ \stackrel { \eqref { f o u r } } { \gtrsim } \ ( \widetilde { \zeta } ( n _ s ) ) ^ { 1 / q } \| 1 _ { V _ s } \| \ \stackrel { \eqref { z e t a } } { \gtrsim } \ \omega ( n _ s ) \| 1 _ { V _ s } \| . \end{align*}


\begin{align*} V _ { I } : = 1 _ { A _ t } \sum _ { \ell \in I } G ( 0 , S _ \ell ) \ , , \ ; I \subseteq [ 0 , t ] \cap \Z \ , , \end{align*}


\begin{align*} R _ l : = H _ l \ltimes N _ l , \end{align*}


\begin{align*} g _ n & = \left ( \frac { d } { d t } P _ N u _ n + \nu A P _ N u _ n + B _ N ( u _ n , u _ n ) \right ) - \left ( \frac { d } { d t } P _ N u + \nu A P _ N u + B _ N ( u , u ) \right ) \\ & = B _ N ( P _ N u + Q _ N v _ n , P _ N u + Q _ N v _ n ) - B _ N ( P _ N u + Q _ N u , P _ N u + Q _ N u ) \\ & = B _ N ( Q _ N w _ n , Q _ N w _ N ) + D B _ N ( u ) Q _ N w _ n , \end{align*}


\begin{align*} \partial _ \eta w = 0 o n \eta = 0 . \end{align*}


\begin{align*} \begin{pmatrix} \scriptstyle ( W - X Z ^ { - 1 } Y ) ^ { - 1 } & \scriptstyle - ( W - X Z ^ { - 1 } Y ) ^ { - 1 } X Z ^ { - 1 } \\ \scriptstyle - Z ^ { - 1 } Y ( W - X Z ^ { - 1 } Y ) ^ { - 1 } & \scriptstyle Z ^ { - 1 } + Z ^ { - 1 } Y ( W - X Z ^ { - 1 } Y ) ^ { - 1 } X Z ^ { - 1 } \end{pmatrix} \ ; . \end{align*}


\begin{align*} \gamma _ { o i r 2 } ( G \Box H ) \leq \omega ( h ) = \alpha ( G ) \big { ( } | V ( H ) | - \alpha ( H ) \big { ) } + \big { ( } | V ( G ) | - \alpha ( G ) \big { ) } | V ( H ) | - \beta ( G ) . \end{align*}


\begin{align*} \begin{gathered} A ( \mathbf { 1 } ) = \sum _ i \left ( R ( ( i , \mathbf { 1 } ) ) R ( ( i , \mathbf { 1 } ) ) + R ( ( i , \mathbf { i } ) ) R ( ( i , \mathbf { i } ) ) \right ) \geq 0 \ ; , \\ A ( \mathbf { i } ) = \sum _ i \left ( R ( ( i , \mathbf { 1 } ) ) R ( ( i , \mathbf { i } ) ) - R ( ( i , \mathbf { i } ) ) R ( ( i , \mathbf { 1 } ) ) \right ) = 0 \ ; . \end{gathered} \end{align*}


\begin{align*} ( \gamma . L _ { P _ j } ) = m _ j . \end{align*}


\begin{align*} \frac { \log ( 1 - x ) } { \exp \bigl ( \log ( 1 - x ) \bigr ) - 1 } = - \frac { \log ( 1 - x ) } { x } . \end{align*}


\begin{align*} \rho = \frac { n + i - k - l } { n + i } \geq \frac { 1 } { 2 } . \end{align*}


\begin{align*} F ^ { \mathrm { s e l f } } ( \alpha ) \coloneqq \sum _ { j = 1 } ^ J \mathcal { G } ( W ^ j _ { 0 } ; \Omega _ { D } ( \alpha ) ) + \frac { E } { 1 - \nu ^ 2 } \sum _ { j = 1 } ^ J \frac { { | b _ j | ^ 2 } } { 8 \pi } \log D \ , , \end{align*}


\begin{align*} & \widetilde { \mathcal S } ( \cos \zeta , n ) = \mathcal S ( \cos \zeta , n ) , & & \zeta \in \mathcal D _ { - \frac \pi 2 + \theta _ { \hat { x } } + i \infty , \zeta _ o } , \\ & \widetilde { \mathcal S } ( \cos \zeta , n ) = - \mathcal S ( \cos \zeta , n ) , & & \zeta \in \mathcal \mathcal D _ { \zeta _ o , \frac \pi 2 + \theta _ { \hat { x } } - i \infty } . \end{align*}


\begin{align*} U _ r : = \{ ( x , y ) \in X \times X \ , : \ , d ( x , y ) < r \} \ , . \end{align*}


\begin{align*} | \mbox { s m } ( X , \prec ) | = | X | . \end{align*}


\begin{align*} \alpha ( \| \lambda _ y \| ^ 2 - \| \lambda _ { \infty } \| ^ 2 ) = \alpha \int _ { \mathbb { R } ^ 3 } \mathrm { d } k \ , f _ y ( k ) , \end{align*}


\begin{align*} \| \psi \| _ { L ^ 3 ( \mathbb { R } ^ 3 \times \mathbb { R } ^ 3 ) } & \le C \| \psi \| _ { H ^ 1 ( \mathbb { R } ^ 3 \times \mathbb { R } ^ 3 ) } , \\ \| \psi \| _ { L ^ 2 ( \mathbb { R } ^ 3 \times \mathbb { R } ^ 3 ) } & \le C \| \psi \| _ { H ^ 1 _ \mathrm { t r a p } ( \mathbb { R } ^ 3 \times \mathbb { R } ^ 3 ) } . \end{align*}


\begin{align*} \mathcal { H } ( x , t ) : = t ^ { p ( x ) } + \mu ( x ) t ^ { q ( x ) } \quad ( x , t ) \in \Omega \times [ 0 , \infty ) . \end{align*}


\begin{align*} K _ q ^ \Psi ( x ) : = \begin{cases} \displaystyle { \frac { \| D \Psi ( x ) \| ^ q } { J _ \Psi ( x ) } } & J _ \Psi ( x ) \neq 0 , \\ 1 & . \end{cases} \end{align*}


\begin{align*} \mathcal { A } [ X ] ( r , \mu ) = X _ 0 ( \mu ) + \int \limits _ 0 ^ r A ( X ( r ' , \mu ) ) d r ' , \\ \mathcal { B } [ Y ] ( r , \mu ) = Y _ 0 ( \mu ) + \int \limits _ 0 ^ r B ( Y ( r ' , \mu ) ) d r ' . \end{align*}


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


\begin{align*} \tilde { \boldsymbol { \theta } } _ i = \mathrm { d i a g } ( \tilde { \boldsymbol { \beta } } _ i ) \tilde { \boldsymbol { \psi } } _ i = \mathrm { d i a g } ( \tilde { \boldsymbol { \psi } } _ i ) \tilde { \boldsymbol { \beta } } _ i , \forall i \in \{ t , r \} . \end{align*}


\begin{align*} \Gamma _ s ( \mathfrak { h } _ + ) = \bigoplus _ { n = 0 } ^ { \infty } \mathfrak { h } _ + ^ { \otimes _ s n } , \mathfrak { h } _ + : = L ^ 2 ( \mathbb { R } ^ + \times \mathbb { R } ^ 2 ; \mathbb { C } ^ 2 ; \mathrm { d } k ) , \end{align*}


\begin{align*} \phi _ { i j } = \pm \phi _ { i k } + q _ i \psi _ i . \end{align*}


\begin{align*} w \partial _ \eta ^ 2 w \leq 0 \Rightarrow \partial _ \tau w \leq 0 \ , \ , a n d \ , \ , \partial _ \xi w \leq 0 \ , \ , \Rightarrow w ^ 2 \partial _ { \eta } ^ 2 w = \partial _ \tau w + \eta \partial _ \xi w \leq 0 . \end{align*}


\begin{gather*} d ^ 2 _ { Q } ( \theta , \gamma ) = d ^ 2 _ { Q } ( \theta ' , \gamma ' ) \iff \left \{ \begin{array} { l l } \theta ' = \theta + d ^ 1 \tau \\ \gamma ' = \gamma + d ^ 2 \sigma - B \left ( \tau \wedge ( \theta + \frac { 1 } { 2 } d ^ 1 \tau ) \right ) \\ \end{array} \right . \end{gather*}


\begin{align*} \left | \sum _ { l = 2 } ^ n \sigma _ l \omega _ { j _ l } + \sigma _ 1 \omega _ { j _ 1 } \right | \ . \end{align*}


\begin{align*} \partial \Omega _ { \eta ( t ) } = \{ y + \eta ( t , y ) \nu ( y ) : \ , y \in \partial \Omega \} , \end{align*}


\begin{align*} \frak { d } ( \mathfrak { S } _ 3 ) = \frac { \pi ( x ) } { p - 1 } . \end{align*}


\begin{align*} ( R _ { T , \left \lfloor T r \right \rfloor - n } ^ { ( 1 ) } ) ^ { j } = ( ( - \log T + \sum _ { k = 0 } ^ { T } ( k + d ) ^ { - 1 } ) - \sum _ { k = \left \lfloor T r \right \rfloor - n } ^ { T } ( k + d ) ^ { - 1 } ) ^ { j } = ( - \psi ( d ) - \sum _ { k = \left \lfloor T r \right \rfloor - n } ^ { T } ( k + d ) ^ { - 1 } ) ^ { j } + o ( 1 ) . \end{align*}


\begin{align*} \begin{cases} d \tilde Y _ t = - { \dot { W } } _ { \varepsilon , \eta } ( t , B _ t ) \tilde Y _ t d t - Y _ t ^ { \varepsilon , \eta } \nabla _ x { \dot { W } } _ { \varepsilon , \eta } ( t , B _ t ) I _ { [ 0 , t ] } ( r ) d t + \tilde Z _ t d B _ t , \ \ r \leq t \le T \\ \tilde Y _ T = D _ r ^ B \xi \ , . \end{cases} \end{align*}


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


\begin{align*} M = N \prod _ { \ell } \ell ^ { \alpha ( \ell ) } \quad , \end{align*}


\begin{align*} & \int _ M \langle \nabla _ M f ( x , t ) , \frac { \nabla \phi _ { k , k + 1 } ^ c ( x ) } { \tau } \rangle _ x d \mu _ { k + 1 } ^ { \tau } ( x ) = \int _ { M } \langle \nabla _ M f ( x , t ) , \nabla ( W \ast \mu _ { k + 1 } ^ { \tau } ) \rangle _ x d \mu _ { k + 1 } ^ { \tau } ( x ) . \end{align*}


\begin{align*} \sigma _ a ( b ) = ( \lambda _ a ( 1 ) ) ^ { - } \circ b = ( 7 ^ a ) ^ { - } \circ b = \begin{cases} 1 + 7 b \quad { \rm i f \ ; a \ ; i s \ ; e v e n , } \\ 7 + 7 b \quad { \rm i f \ ; a \ ; i s \ ; o d d . } \end{cases} \end{align*}


\begin{align*} p ( s , t , x , y ) = q _ { t - s } ( x - y ) + \int _ s ^ t \partial _ y q _ { t - r } \star \bigl ( b ( r , \cdot ) p ( s , r , x , \cdot ) \bigr ) \ , d r , \end{align*}


\begin{align*} x \circ y = x + \gamma _ x ( y ) , \end{align*}


\begin{align*} W _ { 2 m , 0 } ( x ) | \Omega \rangle = \frac { 1 } { 2 m - p - q - 2 N s } | \Omega \rangle \ , , \end{align*}


\begin{align*} \sigma _ { ( a , i ) } ^ { - 1 } ( ( b , j ) ) = ( \alpha ^ h ( b ) + \alpha ^ i ( g ) , j + h ) . \end{align*}


\begin{align*} P _ A \left ( \sigma ( x ^ 1 ) = y ^ 1 , \ldots , \sigma ( x ^ p ) = y ^ p \right ) & : = \frac { 1 } { Y _ { m _ 1 , m _ 2 } ( A ) } \sum _ { \sigma } \prod _ { k = 1 } ^ p \mathbf { 1 } \{ \sigma ( x ^ i ) = y ^ i \} \prod _ { x \in \mathbb { T } _ { m _ 1 , m _ 2 } } w _ A ( x ^ i , y ^ i ) \\ & = \frac { \prod _ { i = 1 } ^ k w _ A ( x ^ i , y ^ i ) } { Y _ { m _ 1 , m _ 2 } ( A ) } \frac { \partial } { \partial W _ { x ^ 1 , y ^ 1 } } \ldots \frac { \partial } { \partial W _ { x ^ p , y ^ p } } Y _ { m _ 1 , m _ 2 } ( A ) . \end{align*}


\begin{align*} \begin{multlined} A ( i ) = \sum _ j R ( ( j , i ) ) R ( ( j , i ) ) \\ = \sum _ j ( R ( j , i ) ) ^ 2 \geq 0 \forall a \ ; . \end{multlined} \end{align*}


\begin{align*} \int \limits _ { { \Bbb B } ^ 2 } \ , J ( z , F ) \ , d m ( z ) \leqslant m ( F ( { \Bbb B } ^ 2 ) ) = \pi \ , . \end{align*}


\begin{align*} \| g \| = \| g \| _ \infty + v ( g ) \end{align*}


\begin{align*} \mathcal { L } ( x , y , q , N ) : = b ( x , y ) \cdot q + ( \varrho ( x , y ) \varrho ^ { \top } ( x , y ) N ) \end{align*}


\begin{align*} H _ { \omega } : \ell ^ 2 ( \mathbb { Z } ) \rightarrow \ell ^ 2 ( \mathbb { Z } ) , H _ { \omega } = \Delta + V _ { \omega } \end{align*}


\begin{align*} \omega _ \psi ^ { ( \nu ) } ( s ) = ( - 1 ) ^ { l ( Z _ \jmath ^ { F ^ \nu } , s ) } \vartheta _ { Z _ \jmath } ^ { ( \nu ) } ( s ) q ^ { \frac { \nu } { 2 } \dim V ^ s } . \end{align*}


\begin{align*} Z = m _ 1 Z _ 1 \dot { + } \ldots \dot { + } m _ l Z _ l . \end{align*}


\begin{align*} \phi _ 0 ( 1 \otimes \phi _ 1 ( \gamma \otimes m ) ) + \phi _ 1 ( Q _ 1 ( \gamma ) \otimes m ) + \phi _ 2 ( Q _ 0 ( 1 ) \vee \gamma \otimes m ) + ( - 1 ) ^ { \abs { \gamma } } \phi _ 1 ( \gamma \otimes \phi _ 0 ( 1 \otimes m ) ) = 0 \end{align*}


\begin{align*} \sum _ { n = 1 } ^ { N } \frac 1 n \le C ( T ) | \log ( \Delta t ) | ~ , \sum _ { n = 1 } ^ { N } \frac { 1 } { n ^ 2 } \le C \end{align*}


\begin{align*} - k _ d \partial _ r ^ 2 \eta _ 0 ( r ) & = 0 , \\ - V _ \Gamma \partial _ r \eta _ 0 ( r ) + k _ d H \partial _ r \eta _ 0 ( r ) - k _ d \partial _ r ^ 2 \eta _ 1 ( r ) & = 0 , \end{align*}


\begin{align*} x _ \sigma ' \ ; : = \ ; \prod _ { \substack { \rho \in \Sigma [ 1 ] \smallsetminus \sigma [ 1 ] } } { x _ \rho ' } \ ; = \ ; \prod _ { \substack { \rho \in \Sigma [ 1 ] \\ \varsigma _ \sigma \not \subset \tau _ \rho } } { x _ \rho ' } \end{align*}


\begin{align*} M ( A ) _ M ( ( i , \alpha ) , ( j , \beta ) ) = \mathcal { H } _ { \alpha \beta } ( A _ M ( i , j ) ) \ ; . \end{align*}


\begin{align*} \mathcal { M } ( x ) = \mathcal { L } ^ { [ 1 ] } ( x ) \mathcal { L } ^ { [ 2 ] } ( x ) \cdots \mathcal { L } ^ { [ N ] } ( x ) \ , , \hat { \mathcal { M } } ( x ) = \mathcal { L } ^ { [ N ] } ( x ) \mathcal { L } ^ { [ N - 1 ] } ( x ) \cdots \mathcal { L } ^ { [ 1 ] } ( x ) \ , , \end{align*}


\begin{align*} \phi \left ( t ; z \right ) \partial _ { z } S = 2 t e ^ { - z ^ { 2 } } . \end{align*}


\begin{align*} 2 \min f - \max f = 2 ( [ \beta ] - 2 ) - [ \beta ] = [ \beta ] - 4 \geq 1 . \end{align*}


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


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


\begin{align*} \begin{cases} f ( x ) + f ( y ) = f ( x + l ) + f ( z ) \\ g ( x ) + g ( y ) = g ( x + l ) + g ( z ) , \end{cases} \ , 1 \leq x < x + l \leq z < y \leq N . \end{align*}


\begin{align*} \lim _ { k \rightarrow \infty } k \left ( \breve { M } ^ r ( x , t , k ) - I \right ) = \lim _ { k \rightarrow \infty } k ( E ( x , t , k ) - I ) = \mathfrak { B } ^ r ( \xi , t ) - \overline { \mathfrak { B } ^ r ( \xi , t ) } + R ( \xi , t ) . \end{align*}


\begin{align*} \mathbb M ^ r _ X = \bigoplus _ { w \in W ( r ) } \mathbb M ^ r _ X ( w ) : = \mathbb S ^ r _ { X , L } = \bigoplus _ { w \in W ( r ) } \mathbb S ^ r _ { X , L } ( w ) , \ \ \ \mathbb M _ X : = \mathbb S _ { X , L } . \end{align*}


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


\begin{align*} \{ z = \xi _ 5 ^ i u , \ , w = \xi _ 5 ^ i v \} ( i = 0 , \ldots , 4 ) \end{align*}


\begin{align*} \sum _ { i + j = \ell } e _ i ( z ) e _ j ( \hat { z } ) \ : = \ : \hat { g } _ { \ell } ( z , \zeta , q ) \ / . \end{align*}


\begin{align*} d Y _ t = V ( Y _ t ) \diamond d \mathbf { X } _ t , \end{align*}


\begin{align*} \theta ' \left ( s ( x ) , s ( y ) \right ) & = f \left ( s ' ( \theta ( x , y ) ) \right ) + f \left ( s ( [ x , y ] ) \right ) - \rho \left ( s ( x ) \right ) f ( s ( y ) ) - \rho \left ( s ( y ) \right ) f ( s ( x ) ) \\ & = f \left ( \left [ s ( x ) , s ( y ) \right ] \right ) - \rho \left ( s ( x ) \right ) f ( s ( y ) ) - \rho \left ( s ( y ) \right ) f ( s ( x ) ) . \end{align*}


\begin{align*} Y _ t = \xi + \int _ t ^ T Y _ s W ( d s , B _ s ) - \int _ t ^ T Z _ s d B _ s , t \in [ 0 , T ] \ , , \end{align*}


\begin{align*} \frac { 1 } { \Delta t ^ { \frac 1 2 } } \mathcal { R } ( \sqrt { \Delta t } , \Delta t ) = \frac { 1 } { \Delta t } \int _ { 0 } ^ { \Delta t } ( 1 - e ^ { - \frac { t } { \Delta t } } ) d t = \int _ 0 ^ 1 ( 1 - e ^ { - s } ) d s = e ^ { - 1 } \end{align*}


\begin{align*} \overline { \alpha _ { \mathfrak J } } \circ i & = i \circ \beta \\ \alpha \circ \pi & = \pi \circ \overline { \alpha _ { \mathfrak J } } \\ i ( \mathfrak { a } ) & = \ker \pi \\ B ( i ( v ) , i ( w ) ) & = B ( v , w ) \end{align*}


\begin{align*} \delta _ k = \tfrac 1 4 - \theta _ k ^ 2 , \qquad & k = 0 , \ldots , n - 1 , \\ \mathcal E _ k = t _ k \ , \frac { \partial \mathcal W } { \partial t _ k } , \qquad & k = 2 , \ldots , n - 2 . \end{align*}


\begin{align*} & \pi _ { \varepsilon , \lambda } ( h _ a ) = \left ( \frac { a } { | a | } \right ) ^ \varepsilon \begin{pmatrix} | a | ^ { i \lambda } & 0 \\ 0 & | a | ^ { - i \lambda } \end{pmatrix} ( a \in \R ^ \times ) \ , \\ & \pi _ { \varepsilon , \lambda } ( s ) = s \ , . \end{align*}


\begin{align*} | f _ \eta ^ \varepsilon | & \leq c \varepsilon \sum _ { \xi = \eta , \zeta _ 0 , \zeta _ 1 } \left ( | \bar { \xi } | + \Bigl | \overline { \partial ^ \circ \xi } \Bigr | + \Bigl | \overline { \nabla _ \Gamma \xi } \Bigr | + \Bigl | \overline { \nabla _ \Gamma ^ 2 \xi } \Bigr | \right ) \end{align*}


\begin{align*} J _ \nu ( t ) = \frac { 1 } { 2 \pi } \int _ 0 ^ { 2 \pi } \cos ( t \sin { \theta } - \nu \theta ) d \theta . \end{align*}


\begin{align*} \lim \limits _ { y \to \infty } ( u , \tilde { h } ) = ( 0 , 0 ) . \end{align*}


\begin{align*} \int _ { s } ^ { t } u _ r \times W _ r \cdot \partial _ x u _ r \partial _ x \dot { h } _ r \dd r = \int _ { s } ^ { t } u _ r \times \partial _ x u _ r \cdot W _ r \partial _ x \dot { h } _ r \dd r \ , , \end{align*}


\begin{align*} D _ { \bullet } ^ { - 1 } B _ { \bullet } = ( B _ { ( ( p ) ) } ) ^ { N _ { n , \bullet } } , \end{align*}


\begin{align*} f ( \mathbf { x } ) = \sum _ { p \in ( n ) _ m } \left ( \sum _ { p _ 0 \in [ n ] \setminus \{ p _ 1 \} } \mathbf { x } _ { p _ 0 , p _ 1 } \right ) \left ( \prod _ { i = 1 } ^ { m - 1 } \mathbf { x } _ { p _ i , p _ { i + 1 } } \right ) \left ( \sum _ { p _ { m + 1 } \in [ n ] \setminus \{ p _ m \} } \mathbf { x } _ { p _ m , p _ { m + 1 } } \right ) \ ; . \end{align*}


\begin{align*} \mathcal { G } ^ { \neq } \left ( \delta _ { x } - \delta _ { x } ^ { \epsilon } , { \rm e m p } _ { N } \right ) = \int _ { y : 0 < | y - x | \leq r } h ^ { \delta _ { x } - \delta _ { x } ^ { \epsilon } } d { \rm e m p } _ { N } ( y ) + \int _ { y : | y - x | \geq r } h ^ { \delta _ { x } - \delta _ { x } ^ { \epsilon } } d { \rm e m p } _ { N } ( y ) , \end{align*}


\begin{align*} \lim _ { T \to \infty } \sup _ { k : \ , t _ k ^ { ( \lambda , \ , 2 ) } \geq T } \big ( \sigma B ( t _ k ^ { ( \lambda , \ , 2 ) } ) - u t _ k ^ { ( \lambda , \ , 2 ) } + j _ k ^ { ( \lambda , \ , 2 ) } \big ) = - \infty . \end{align*}


\begin{align*} 2 T ^ { m } + U ^ { k } = V ^ { k } \end{align*}


\begin{align*} V ( x , i ) = \sup _ { D , R } \mathrm { E } _ { x , i } \left [ \int _ { 0 } ^ { e _ { \lambda _ { i } } } e ^ { - \int _ { 0 } ^ { t } \delta _ { Y _ s } \mathrm { d } s } \mathrm { d } D _ { t } - \phi \int _ { 0 } ^ { e _ { \lambda _ { i } } } e ^ { - \int _ { 0 } ^ { t } \delta _ { Y _ s } \mathrm { d } s } \mathrm { d } R _ { t } + e ^ { - \int _ 0 ^ { e _ { \lambda _ { i } } } \delta _ { Y _ s } d s } V ( U _ { e _ { \lambda _ { i } } } , Y _ { e _ { \lambda _ { i } } } ) \right ] , \end{align*}


\begin{align*} \left \vert \langle \alpha _ { r + 1 } ^ { ( r + 1 , 0 ) } , \alpha _ { r + 1 } ^ { ( r + 1 , 0 ) } \rangle \right \vert ^ 2 = \frac { F ^ { 2 r + 2 } } { 2 ^ { 2 r + 4 } } \left ( \Vert \alpha ^ f _ { r + 1 } \Vert ^ 4 - 4 ^ r ( K _ r ^ { \perp } ) ^ 2 \right ) . \end{align*}


\begin{align*} L ^ { \pm } ( u ) = ( l _ { i j } ^ { \pm } ( u ) ) _ { 1 \leq i , j \leq n } . \end{align*}


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


\begin{align*} \partial _ \eta w = 0 o n \overline { D } \cap \{ \eta = 0 \} \end{align*}


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


\begin{align*} \sigma _ { \widetilde { i - 1 } } & = L _ { \widetilde { i - 1 } } \pi \stackrel { \eqref { p i L x } } { = } \pi ^ { - 1 } L _ { \widetilde 1 } ^ { - 1 } L _ { \widetilde i } \pi ^ 2 = \sigma _ { \widetilde 1 } ^ { - 1 } \sigma _ { \widetilde i } \sigma _ { \widetilde 0 } = \sigma _ { \widetilde 1 } ^ { - 1 } \sigma ^ i _ { \widetilde 1 } \sigma _ { \widetilde 0 } ^ { 1 - i } \sigma _ { \widetilde 0 } = \sigma ^ { i - 1 } _ { \widetilde 1 } \sigma _ { \widetilde 0 } ^ { 2 - i } . \end{align*}


\begin{align*} \square _ { R } ( x ) = \left [ x - \frac { R } { 2 } , x + \frac { R } { 2 } \right ] ^ { d } . \end{align*}


\begin{align*} H [ \phi , k ; \mu ] \circ k + k \circ H ^ T [ \phi , k ; \mu ] + \Theta [ \phi , k ] + k \circ \overline { \Theta [ \phi , k ; \mu ] } \circ k = 0 , \end{align*}


\begin{align*} V = \{ x _ 4 ^ 2 Q + x _ 4 K + R = 0 \} , \end{align*}


\begin{align*} ( \exp ( s - t ) - 1 ) q > - 1 , \quad \begin{cases} q < \frac { 1 } { 1 - \exp ( s - t ) } & s < t , \\ q \ge 1 & s > t . \end{cases} \end{align*}


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