\begin{align*} ( - L y _ s , y _ s ) _ V & = \frac { 1 } { 2 } \sum \limits _ { e \in E } \frac { \delta _ E ( e ) } { | e | ^ 2 } \sum \limits _ { u , v \in e } \left ( y _ s ( u ) - y _ s ( v ) \right ) ^ 2 \\ & = \frac { 1 } { 2 } \sum \limits _ { e \in \partial S } \frac { \delta _ E ( e ) } { | e | ^ 2 } 2 | S \cap e | | e - S | \frac { 1 } { | S | } \\ & \le \frac { 1 } { 4 } \sum \limits _ { e \in \partial S } \delta _ E ( e ) \frac { 1 } { | S | } \\ & \le \frac { 1 } { 4 } \delta _ E ( \max ) \frac { | \partial S | } { | S | } . \end{align*}


\begin{align*} L _ { n s } ( \dot { x } , x , t ) = \frac { 1 } { C _ 3 \dot { x } } \ , \end{align*}


\begin{align*} \lim _ { n _ 1 , \dots , n _ k \to \infty } \mu ( A _ 0 \cap T ^ { - n _ 1 } A _ 1 \cap T ^ { - n _ 1 - n _ 2 } A _ 2 \cap \cdots \cap T ^ { - n _ 1 - n _ 2 - \cdots - n _ k } A _ k ) = \mu ( A _ 1 ) \mu ( A _ 2 ) \cdots \mu ( A _ k ) . \end{align*}


\begin{align*} \bar c _ i ( \alpha , g ) = \bar c _ i ( \alpha , h ) . \end{align*}


\begin{align*} \prod \limits _ { k = 1 } ^ { \infty } \Bigg ( \frac { z - \mu _ { k } ^ { 2 } } { z - \lambda _ { k } ^ { 2 } } \Bigg ) = 1 - \sum \limits _ { k = 1 } ^ { \infty } \frac { a _ k } { \lambda _ { k } ^ { 2 } - z } \end{align*}


\begin{align*} \left ( \sum _ { k = 1 } ^ { \infty } \int _ 0 ^ 1 \int _ { 0 } ^ { 1 } [ ( y + k ) ^ \alpha - ( y + k - z ) ^ \alpha ] [ ( k + y ) ^ \alpha - k ^ \alpha ] d z d y \right ) \int _ 0 ^ \tau \gamma _ s ( \sigma ' \sigma ) ^ 2 ( X _ s ) d s . \end{align*}


\begin{align*} t r ( A ( t , \delta ) ^ { 9 } ) = t ^ { 9 } - 9 t ^ { 7 } \delta + 2 7 t ^ { 5 } \delta ^ 2 - 3 0 t ^ { 3 } \delta ^ 3 + 9 t \delta ^ 4 \in S ^ 9 , \ ; \forall t , \delta \in R . \end{align*}


\begin{align*} f ( x ) = x ^ 2 - 2 x = x ( x - \lambda ) \end{align*}


\begin{align*} \lambda _ i \psi ( v _ i , v _ j ) = \psi ( \lambda _ i v _ i , v _ j ) = \psi ( \Phi v _ i , v _ j ) = - \psi ( v _ i , \Phi v _ j ) = - \lambda _ j \psi ( v _ i , v _ j ) . \end{align*}


\begin{align*} \int _ 0 ^ L \abs { \omega _ x } ^ 2 d x = o ( 1 ) \int _ 0 ^ { \infty } \int _ 0 ^ L \sigma ( s ) \abs { \eta _ x } ^ 2 d x d s = o ( 1 ) , \end{align*}


\begin{align*} E \doteq \bigcap \limits _ { A \in \mathcal { U } } \{ \rho \in \mathcal { U } ^ { \ast } : \rho ( \mathfrak { 1 } ) = 1 , \ ; \rho ( | A | ^ { 2 } ) \geq 0 \} \ . \end{align*}


\begin{align*} \begin{aligned} x _ i { } & = { } \frac { U } { L } \sum \limits _ { j = i } ^ { N - 1 } h _ j \\ g _ i { } & = { } U \\ f _ i { } & = { } \Delta - \frac { U ^ 2 } { 2 L } \sum \limits _ { j = 0 } ^ { i - 1 } h _ j \Big ( 2 - h _ j \frac { - \kappa } { 1 - \kappa } \Big ) \end{aligned} \end{align*}


\begin{align*} \begin{aligned} h _ * = \arg \max _ { 0 < h { } \leq { } \bar { h } ( \kappa ) } 2 h - h ^ 2 \frac { - \kappa } { - ( 1 + \kappa ) + 2 \min \left ( 1 , \frac { 1 } { h } \right ) } \end{aligned} \end{align*}


\begin{align*} \xi ( s ) - \mu = e ^ { s - t } ( \xi ( t ) - \mu ) . \end{align*}


\begin{align*} \sup \limits _ { m \in \N } \sum \limits _ { i = 0 } ^ { m - 1 } \P \{ \ell ^ { ( A ) } _ i > m - i \} < \infty \end{align*}


\begin{align*} \big | \left < \rho _ { S , \mathcal { O } } ( \beta ^ * * \beta ) \phi _ { x , r } , \psi _ i \right > \big | = \big | \left < \phi _ { x , r } , \rho _ { S , \mathcal { O } } ( \beta ^ * * \beta ) \psi _ i \right > \big | = \lambda _ i \big | \left < \phi _ { x , r } , F _ \pi \right > \big | . \end{align*}


\begin{align*} \mathcal { U } ^ { + } \doteq \{ A \in \mathcal { U } : A = \mathrm { g } _ { \pi } ( A ) \} \subseteq \mathcal { U } . \end{align*}


\begin{align*} \langle a , u _ i \rangle \geq n \mbox { f o r } i = 1 , \ldots , m \mbox { a n d } \langle a , \gamma _ j \rangle \leq r d _ j - 1 \mbox { f o r s o m e } 1 \leq j \leq p . \end{align*}


\begin{align*} S \left ( W ^ C _ { \bullet , \bullet } ; p \right ) = \frac { 2 } { ( - ( K _ S + \Omega ) ) ^ 2 } \int _ 0 ^ \tau h ( u ) d u . \end{align*}


\begin{align*} H ^ { 0 } \left ( X , K _ X \right ) = H ^ { 0 } \left ( \mathbb { P } ^ 1 \times C , K _ { \mathbb { P } ^ 1 \times C } \right ) \oplus \bigoplus _ { \chi \ne \chi _ { 0 0 0 } } { H ^ { 0 } \left ( \mathbb { P } ^ 1 \times C , K _ { \mathbb { P } ^ 1 \times C } + L _ { \chi } \right ) } , \end{align*}


\begin{align*} T ( x ) = \begin{cases} x ( 1 + 2 ^ \gamma x ^ \gamma ) & x \in [ 0 , \frac { 1 } { 2 } ) , \\ 2 x - 1 & x \in [ \frac { 1 } { 2 } , 1 ] , \end{cases} \end{align*}


\begin{align*} \Vert J _ { 3 } \Vert _ { p } = \mathcal { O } \left ( \left ( v ( | z _ { 1 } | ) + v ( | z _ { 2 } | ) \right ) \left ( \frac { \omega ( h _ { 1 } ) } { v ( h _ { 1 } ) } + \frac { \omega ( h _ { 2 } ) } { v ( h _ { 2 } ) } \right ) \right ) . \end{align*}


\begin{align*} \sum _ { k \ge 1 } \left ( \frac { \mu _ k ^ 2 } { \lambda _ { k + 1 } ^ 2 } - 1 \right ) = \infty . \end{align*}


\begin{align*} \langle \sigma _ 1 ^ { r } \sigma _ { 1 2 } \rangle ^ { \mathbf { s } , o } = \langle \sigma _ 2 ^ { s } \sigma _ { 1 2 } \rangle ^ { \mathbf { s } , o } = - 1 . \end{align*}


\begin{align*} u = y = 0 \Gamma \times ( 0 , \infty ) , \end{align*}


\begin{align*} T _ m ( d ) = \frac 1 2 \sum _ { \substack { Q \in \Gamma _ \infty \backslash \mathcal Q _ { d ^ 2 } \\ Q = [ a , b , c ] , a \neq 0 } } \chi _ d ( Q ) \int _ { C _ Q } e ( m x ) \phi _ { 2 , m } ( y , s ) \ , d z . \end{align*}


\begin{align*} h | _ { D ^ \vee _ 1 } = h _ 2 : D ^ \vee _ 1 \rightarrow \mathbb C \end{align*}


\begin{align*} f _ t \left ( \int _ { \Omega } ^ { } \alpha ( \omega ) d \omega \right ) = 0 \leq \int _ { \Omega } ^ { } f _ { T ( \omega ) } \circ \alpha ( \omega ) d \omega . \end{align*}


\begin{align*} ( f * \phi _ \sigma ) ( x ) = \int _ { \R ^ d } \frac { \phi _ { \sigma } ( x - y ) } { \phi _ { \sigma } ( y ) } f ( y ) \phi _ { \sigma } ( y ) d y . \end{align*}


\begin{align*} \left \{ \begin{array} [ c ] { l c c } \frac { \partial } { \partial t } \omega = - R i c ^ { T } ( \omega ) & \mathrm { o n } & ( 0 , T _ { 0 } ) \times M _ { \operatorname { r e g } } , \\ \omega ( 0 ) = \omega _ { 0 } & \mathrm { o n } & M . \end{array} \right . \end{align*}


\begin{align*} | g ( z ^ m ) | + \sum _ { k = N } ^ { \infty } | b _ k | | z | ^ k \leq d ( 0 , \partial { \Omega } ) \end{align*}


\begin{align*} [ P ^ E _ t ] - \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \end{align*}


\begin{align*} { \bf T } _ j ( \alpha ^ * ) = [ T _ j ^ 1 ( \alpha ^ * ) , \ldots , T _ j ^ m ( \alpha ^ * ) ] ^ T = \displaystyle \sum _ { k = 1 } ^ { \mu _ j } Q _ j ^ k ( \alpha ^ * ) { \bf r } _ j ^ k j = 1 , \ldots , L \end{align*}


\begin{align*} \sum _ { i = 1 } ^ n g ^ n _ i \frac { 1 } { n } \rightarrow \int _ 0 ^ 1 g ( x ) d x = U , \end{align*}


\begin{align*} k _ 1 + k _ 2 = g = h + 1 = h _ 0 + h _ 1 + 1 \mathrlap , \end{align*}


\begin{align*} \mathbb S ( \Gamma ) = \{ f \in \ell ^ \infty \Gamma \mid f - R _ t ( f ) \in c _ 0 ( \Gamma ) , \ t \in \Gamma \} \subset \ell ^ \infty \Gamma . \end{align*}


\begin{align*} 0 < C \leq \prod \limits _ { k = N } ^ { \infty } \Bigg ( \frac { \lambda _ { N } ^ { 2 } + \mu _ { k } ^ { 2 } } { \lambda _ { N } ^ { 2 } + \lambda _ { k } ^ { 2 } } \Bigg ) \leq 1 \end{align*}


\begin{align*} H _ { ( \mathbb P ^ 2 , H + H ) } ( y ) = e ^ { \frac { H \log y } { 2 \pi i } } \sum _ { d \geq 0 } y ^ d \frac { 1 } { \Gamma ( 1 + \frac { H } { 2 \pi i } + d ) } [ \textbf { 1 } ] _ { d , d } , \end{align*}


\begin{align*} \mathcal { F } ( \tilde A , \tilde B , \tilde C , \tilde D ) = & { \| A - \tilde A \| } _ F ^ 2 + { \| B - \tilde B \| } _ F ^ 2 \\ & + { \| C - \tilde C \| } _ F ^ 2 + { \| D - \tilde D \| } _ F ^ 2 . \end{align*}


\begin{align*} \delta ^ \ast = \frac { 2 - n } { 2 } , \end{align*}


\begin{align*} ( \alpha ( \xi ) | \alpha ( \zeta ) ) & = L ( \alpha ( \xi ) \otimes \Omega _ \phi ) ^ \ast L ( \alpha ( \zeta ) \otimes \Omega _ \phi ) \\ & = L ( U ( \xi \otimes \Omega _ \phi ) ) ^ \ast L ( U ( \zeta \otimes \Omega _ \phi ) ) \\ & = L ( \xi \otimes \Omega _ \phi ) ^ \ast U ^ \ast U L ( \zeta \otimes \Omega _ \phi ) \\ & = ( \xi | \zeta ) . \end{align*}


\begin{align*} ( { } ^ J G _ { v , { \bf w } , > 0 } ) ^ \sharp \dot { s } ^ \sharp _ 0 \dot { x } ^ \sharp = { } ^ { { I ^ ! } ^ \flat } \ ! \ ! G _ { v ^ \sharp ( s _ 0 x ) ^ \sharp , { \bf w ^ \sharp s ^ \sharp _ 0 x ^ \sharp } , > 0 } . \end{align*}


\begin{align*} \begin{array} [ c ] { c } \frac { \det ( g _ { \alpha \overline { \beta } } ^ { T } + \varphi _ { \alpha \overline { \beta } } ) } { \det ( g _ { \alpha \overline { \beta } } ^ { T } ) } = e ^ { - \kappa \varphi + F } . \end{array} \end{align*}


\begin{align*} F _ p = L ( h _ \theta ) + \sum _ { i \in I } \partial _ i L ( h _ \theta ) Q _ \theta ^ i \end{align*}


\begin{align*} f \colon \mathbb { P } ^ { n - 1 } \hookrightarrow \mathbb { P } = \mathbb { P } \left ( \mathsf { M a t } _ { n + m , s + 1 } ( \mathbb C ) \right ) , [ \alpha ] \mapsto [ A _ { \alpha } ] . \end{align*}


\begin{align*} u ( x , t ) = - { \sf U } _ \infty ( x ) - \theta ( x ) - \Theta _ J ( x ) \end{align*}


\begin{align*} j ( a ) ( \gamma \beta ^ { - 1 } ) = \lim _ { n } j ( f _ { n } ) ( \gamma \beta ^ { - 1 } ) . \end{align*}


\begin{align*} \dfrac { \partial \lambda _ i } { \partial \eta } = - \dfrac { f _ \eta ( \lambda _ i ) } { f _ \lambda ( \lambda _ i ) } , \ \ \frac { \partial ^ 2 \lambda _ i } { \partial \eta ^ 2 } = - \frac { f _ { \eta \eta } ( \lambda _ i ) + 2 f _ { \lambda \eta } ( \lambda _ i ) \lambda _ { i , \eta } + f _ { \lambda \lambda } ( \lambda _ i ) \lambda _ { i , \eta } ^ 2 } { f _ \lambda ( \lambda _ i ) } . \end{align*}


\begin{align*} \begin{aligned} \mathrm { C o n t } ( Q ^ 0 ) | _ { b _ l = 0 } & = - ( h r + 1 - b _ { I \setminus \{ a _ l \} } ) \mathrm { C o n t } ( Q ) \\ \mathrm { C o n t } ( Q ^ j ) | _ { b _ l = 0 } & = ( r - b _ { I _ j } + | Q _ j | ) \mathrm { C o n t } ( Q ) . \end{aligned} \end{align*}


\begin{align*} A _ { \mu } ^ p = \{ f : R \cdot \mathbb { D } \rightarrow \mathbb { C } : f \mbox { h o l o m o r p h i c w i t h } \Vert f \Vert _ p < \infty \} . \end{align*}


\begin{align*} U = \left ( \begin{array} { c c } A & B \\ C & D \end{array} \right ) . \end{align*}


\begin{align*} \partial _ s \left ( e ^ { f _ o } Q \right ) - \Delta ^ { f _ o } _ { g _ o } \left ( e ^ { f _ o } Q \right ) = 0 , \end{align*}


\begin{align*} j ( i ) = \begin{cases} ( i + 1 ) / 2 & \hbox { i f $ i $ i s o d d ; } \\ i / 2 & \hbox { i f $ i $ i s e v e n , } \end{cases} \end{align*}


\begin{align*} \mathbb { V } ( \overline { S } _ n ) = \frac { 1 } { n ^ 2 } \sum _ { 1 \leq i \leq n } \mathbb { E } ( { X _ i } ^ 2 ) + \frac { 2 } { n ^ 2 } \sum _ { 1 \leq i \textless j \leq n } \mathbb { E } ( X _ i X _ j ) - \frac { 1 } { n ^ 2 } \mathbb { E } ^ 2 \Big ( \sum _ { 1 \leq i \leq n } { X _ i } \Big ) . \end{align*}


\begin{align*} \rho _ { \mu _ t g } ( \gamma \cup \delta ) = \rho _ { \mu _ t g } ( \gamma ) \star \rho _ { \mu _ t g } ( \delta ) , \end{align*}


\begin{align*} \lim _ { t \rightarrow ( T ^ * ) ^ { - } } N ( t ) = + \infty . \end{align*}


\begin{align*} \frac { \partial ^ 2 u } { \partial t ^ 2 } + \frac { \partial ^ 2 } { \partial x ^ 2 } \Biggl ( a \frac { \partial ^ 3 u } { \partial t \partial x ^ 2 } \Biggr ) + \frac { \partial ^ 2 } { \partial x ^ 2 } \Biggl ( a \frac { \partial ^ 2 u } { \partial x ^ 2 } \Biggr ) = 0 , \end{align*}


\begin{align*} g ( x ) = c ( x , y ) - g ^ c ( y ) . \end{align*}


\begin{align*} \varphi _ k ^ \lambda ( z ) = ( 2 \pi ) ^ { n / 2 } \lambda ^ { n / 2 } \sum _ { | \nu | _ 1 = k } \Phi ^ \lambda _ { \nu , \nu } ( z ) . \end{align*}


\begin{align*} \lim _ { R \rightarrow \infty } \lim _ { i \rightarrow \infty } \Psi _ { i } ( x , R ) = c _ { n , \sigma p } \lim _ { R \rightarrow \infty } \lim _ { i \rightarrow \infty } \int _ { B ^ { c } _ { R } ( 0 ) } \frac { u ^ { p - 1 } _ { i } ( y ) } { | y | ^ { n + \sigma p } } d y \geq 0 . \end{align*}


\begin{align*} \mathbb E \big [ \sup _ { t } \| O ( t ) \| _ { l ^ \infty _ n } ^ p \big ] = \mathbb E \big [ \sup _ { t , x } \left | o ^ n ( t , x ) - o ^ n ( 0 , 0 ) \right | ^ { p } \big ] \le C ( p , T ) . \end{align*}


\begin{align*} & P _ { \ell } ( z ) = \left [ \dfrac { 1 } { ( n - 1 ) ! ^ r } \right ] \circ { \rm { E v a l } } _ z \circ \mathcal { T } _ { \bold { c } } \bigcirc _ { j = 1 } ^ { n - 1 } B ( \theta _ t + j ) \left ( t ^ { \ell } \prod _ { i = 1 } ^ m ( t - \alpha _ i ) ^ { r n } \right ) \enspace , \\ & P _ { \ell , i , s } ( z ) = \psi _ { i , s } \left ( \dfrac { P _ { \ell } ( z ) - P _ { \ell } ( t ) } { z - t } \right ) \ \ 1 \le i \le m , 0 \le s \le r - 1 \enspace , \end{align*}


\begin{align*} \norm { P } _ { m , s } \vcentcolon = \sum _ { | \alpha | \leq m } \| a _ \alpha \| _ { s - | \alpha | , - s } = \sum _ { | \alpha | \leq m } \| a _ \alpha \| _ { s , | \alpha | - s } \end{align*}


\begin{align*} \rho ( g ) : \mathcal B ( \mathcal H ) \to \mathcal B ( \mathcal H ) \ , \ A \mapsto A ^ g = \rho ( g ) A \rho ( g ) ^ { - 1 } \ , \ \forall g \in S U ( 3 ) \ . \end{align*}


\begin{align*} & \hat { \eta } _ { x , p ; \delta } ( \xi ) = e ^ { - i x \xi } \hat { \eta } _ \delta ( \xi - p ) \end{align*}


\begin{align*} & ( - 1 ) ^ { k } \left ( C _ m ( n ) - \sum _ { j = 1 - k } ^ k ( - 1 ) ^ j u _ m \big ( n - j ( 3 j - 1 ) / 2 \big ) \right ) \geqslant 0 , \end{align*}


\begin{align*} \int _ { \Xi ^ r _ j } x ^ k e ^ { x ^ r / \hbar } d x = \delta _ { j k } \end{align*}


\begin{align*} \| u \| _ 1 : = \sup _ { ( x , t ) \in \R \times [ 0 , T ] } | u ( x , t ) | . \end{align*}


\begin{align*} \sum _ { k = N } ^ M a _ k \bigg ( \sum _ { i = k } ^ M a _ i \bigg ) ^ s b _ k \approx \sum _ { k = N + 1 } ^ M ( b _ k - b _ { k - 1 } ) \bigg ( \sum _ { i = k } ^ M a _ i \bigg ) ^ { s + 1 } + \bigg ( \sum _ { k = N } ^ M a _ k \bigg ) ^ { s + 1 } b _ N . \end{align*}


\begin{align*} \mathcal G _ { \rho } ( z ) = - \frac { \big ( \partial \bar { \partial } \log \rho \big ) ( z ) } { \rho ( z ) ^ 2 } , z \in \Omega . \end{align*}


\begin{align*} H ( p , G ) = \tilde { Y } _ O ( P , G ) ^ { 2 / p ' } \leq C ( P ) ^ { 2 / p ' } = B ( p ) \end{align*}


\begin{align*} \lim _ { N \to \infty } | E ( N ) - E _ { 2 - \mathrm { m o d e } } - E ^ \mathrm { B o g } | = 0 . \end{align*}


\begin{align*} \lim _ { \epsilon \downarrow 0 } \frac { \mathsf { W } _ 2 \big ( ( 1 + \epsilon h ) \mu , \mu \big ) } { \epsilon } = \| h \mu \| _ { \dot { H } ^ { - 1 , 2 } ( \mu ) } , \end{align*}


\begin{align*} X _ { 2 } ( h ) = 0 \end{align*}


\begin{align*} \mathit { \Omega } _ { \mathfrak { m } } \left ( c _ { + } \right ) \doteq \left \{ \omega \in \mathit { \Omega } _ { \mathfrak { m } \left ( c _ { + } \right ) } ^ { \sharp } : e _ { ( \cdot ) } \left ( \omega \right ) = c _ { + } \mathfrak { a } _ { + } \right \} \subseteq E _ { 1 } \ , \end{align*}


\begin{align*} \mathbf V = \begin{pmatrix} F _ 1 \\ F _ 2 \\ F _ 3 \end{pmatrix} \cdot \begin{pmatrix} G _ 1 & G _ 2 & G _ 3 \end{pmatrix} \end{align*}


\begin{align*} T _ f = M _ f + \sum _ { g \prec f } t _ { g f } ( q ) M _ g , L _ f = M _ f + \sum _ { g \prec f } \ell _ { g f } ( q ) M _ g , \end{align*}


\begin{align*} h : = ( h _ 1 , \ldots , h _ n ) : X ^ \vee \rightarrow \mathbb C ^ n , \end{align*}


\begin{align*} \Phi \mapsto ( \phi _ 1 , \dots , \phi _ n ) \phi _ k = \Phi ^ { - 1 } ( \{ n - k + 1 , \dots , n \} ) . \end{align*}


\begin{align*} \mathcal { B } ( ( u , y ) , ( \phi ^ 1 , \phi ^ 2 ) ) = \mathcal { L } ( \phi _ 1 , \phi _ 2 ) , \forall \ , ( \phi ^ 1 , \phi ^ 2 ) \in H _ L ^ 1 ( 0 , L ) \times H _ L ^ 1 ( 0 , L ) , \end{align*}


\begin{align*} \partial _ s \left ( e ^ { f _ o } v \right ) = \Delta _ { g _ o } ^ { f _ o } \left ( e ^ { f _ o } v \right ) \end{align*}


\begin{align*} z = a + i b . \end{align*}


\begin{align*} E _ c ( \Psi _ t ) ( 0 , h ) = v ' \Longleftrightarrow \begin{pmatrix} 0 \\ h \end{pmatrix} = P _ c ( \Psi _ t ) \cdot v ' , \end{align*}


\begin{align*} \mathcal { D } _ { [ n , j ] } = \bigcup _ { k = i } ^ { n - j + i } \widetilde { \mathcal { D } } _ { [ n , j , k ] } , \end{align*}


\begin{align*} \begin{aligned} \frac { d b ( t ) } { d t } & = g _ 0 + { g _ 1 } N ( b ( t ) , c ( t ) ) - b ( t ) , \\ \frac { d c ( t ) } { d t } & = 2 a _ 0 + 2 { a _ 1 } N ( b ( t ) , c ( t ) ) - 2 c ( t ) . \end{aligned} \end{align*}


\begin{align*} h ( X _ { H , V } ) \geq \lim \limits _ { n \to + \infty } \dfrac { \log _ 2 \left ( \max \limits _ { v \in \{ 0 , \dots , p - 1 \} ^ n } \Pi _ { j = 0 } ^ { p - 1 } N _ { v ^ j } \right ) } { p n } \end{align*}


\begin{align*} Q ^ { n , 8 } _ \tau = n ^ { 2 \alpha + 1 } \int ^ { \tau } _ 0 \gamma _ { \eta _ n ( s ) } ( s - \eta _ n ( s ) ) ^ { 2 \alpha + 1 } ( \sigma ' \sigma ) ^ 2 ( X _ { \eta _ n ( s ) } ) d s . \end{align*}


\begin{align*} N _ { \pi _ { \bar { Y } _ { \mathcal { T } } | \bar { X } } } ( x ^ n , b _ { \mathcal { T } } ) \coloneqq \abs { \left \{ \ y ^ { n } _ { \mathcal { T } } : \ \mathcal { B } _ i ( y ^ { n } _ { i } ) = b _ i \ f o r \ i \in \mathcal { T } \ \wedge \ y ^ { n } _ { \mathcal { T } } \in \mathcal { T } ^ { n } _ { \pi _ { \bar { Y } _ { \mathcal { T } } | \bar { X } } } ( x ^ n ) \right \} } , \end{align*}


\begin{align*} \frac { r _ { M _ { 1 } } } { a _ { M _ { 1 } } } = [ b _ { 1 } , \cdot \cdot \cdot , ( b _ { i } - 1 ) , \cdot \cdot \cdot , b _ { l } ] . \end{align*}


\begin{align*} \int _ { \{ \Psi < - t \} } \left \vert \frac { \tilde f _ 1 + \tilde f _ 2 } { 2 } \right \vert ^ 2 + \int _ { \{ \Psi < - t \} } \left \vert \frac { \tilde f _ 1 - \tilde f _ 2 } { 2 } \right \vert ^ 2 \\ = \frac { \int _ { \{ \Psi < - t \} } | \tilde f _ 1 | ^ 2 + \int _ { \{ \Psi < - t \} } | \tilde f _ 2 | ^ 2 } { 2 } = G ( t ) , \end{align*}


\begin{align*} \chi ' ( b ) = a ^ { - 1 } \chi ( b ) \kappa ( b ) a \kappa ( b ) ^ { - 1 } \ , \ , \ , \ , b \in B . \end{align*}


\begin{align*} \psi _ { A _ { k } } = A _ { k } \left [ \frac { \cos ( k x ) } { \cos ( k a / 2 ) } - \frac { \cosh ( k ' x ) } { \cosh ( k ' a / 2 ) } \right ] \end{align*}


\begin{align*} e ^ { \max } = \max _ { j = 1 , \dots , N } E ^ 2 ( t _ j ) \end{align*}


\begin{align*} \delta ^ { ( s ) } w = \alpha w ^ { \phi ^ s } , \end{align*}


\begin{align*} \sum \limits _ { i = 0 } ^ { \infty } \mu \left ( \left [ a _ i , \infty \right ) \right ) < \infty . \end{align*}


\begin{align*} Q ^ { n , 2 } _ \tau = Q ^ { n , 5 } _ \tau + Q ^ { n , 6 } _ \tau , \end{align*}


\begin{align*} v _ i = - g ^ { j k } u _ j h _ { k i } - \frac { f ' ( u ) } { f ( u ) } v u _ i . \end{align*}


\begin{align*} p ( t , v , g ) = \rho ( t , v ) \mathcal { G } ( g ; g _ { } ( t ) , a ( t ) ) . \end{align*}


\begin{align*} \frac { d ^ 2 x ^ i ( t ) } { d t ^ 2 } + { G } ^ i \left ( \gamma , \frac { d \gamma } { d t } \right ) = 0 , 1 \le i \le n . \end{align*}


\begin{align*} P ( T , x ) & = \dfrac { \displaystyle \prod _ { j = 1 } ^ { l ( T ) } W _ j ^ { n ( T , l ( T ) + 1 - j ) } } { W _ 0 ^ { n ( T , l ( T ) ) } \displaystyle \prod _ { j = 1 } ^ { l ( T ) - 1 } W _ j ^ { n ( T , l ( T ) - j ) } } = \dfrac { \displaystyle \prod _ { j = 1 } ^ { l ( T ) } W _ j ^ { n ( T , l ( T ) + 1 - j ) } } { \displaystyle \prod _ { j = 1 } ^ { l ( T ) - 1 } W _ j ^ { n ( T , l ( T ) - j ) } } . \end{align*}


\begin{align*} F _ { m } ( z , s ) = \sum _ { \gamma \in \Gamma _ \infty \backslash \Gamma } f _ m ( \gamma z , s ) , \end{align*}


\begin{align*} \pi _ k = a _ 1 \cdot \mathtt { w } ^ + ( \phi _ 1 ) + \ldots + a _ s \cdot \mathtt { w } ^ + ( \phi _ s ) \geqslant c _ k \end{align*}


\begin{align*} H _ { k , j ; n , l , r } ^ { \mu , m , i ; l i n } ( t , x , \zeta ) : = \int _ { 0 } ^ t \int _ { \R ^ 3 } \int _ { \R ^ 3 } \int _ { \mathbb { S } ^ 2 } \varphi _ { m ; - 1 0 M _ t } ( t - s ) \big [ ( t - s ) \big ( \mathfrak { K } ^ { \mu , E } _ { k ; n } ( y , \omega , v , \zeta ) + \hat { \zeta } \times \mathfrak { K } ^ { \mu , B } _ { k ; n } ( y , \omega , v , \zeta ) \big ) + \mathfrak { K } ^ { e r r ; \mu , E } _ { k ; n } ( y , v , \zeta ) \end{align*}


\begin{align*} X = \left \{ x \in \gg _ { 1 } \mid \left [ x , x \right ] = 0 \right \} . \end{align*}


\begin{align*} G / B = N ' \rtimes P ' \end{align*}


\begin{align*} ( M _ \varphi ) _ { i , j } = | \varphi ( j ) | _ i i , j \in \{ 0 , 1 \} , \end{align*}


\begin{align*} \langle f , I ^ { \circ } ( w _ 1 , z ) w _ 2 \rangle = 0 , \mbox { f o r a n y } \ f \in ( M ^ 3 ( 0 ) ) ^ * , \ w _ 1 \in M ^ 1 , \ w _ 2 \in M ^ 2 ( 0 ) . \end{align*}


\begin{align*} \sum _ { n = 0 } ^ \infty { \mathcal R } _ 1 ( e _ n ) ( z ) { { \mathcal R } _ 1 ( e _ n ) ( w ) } ^ * = \mathbb K ^ { ( \alpha , \beta ) } ( z , w ) , \ ; z , w \in \Omega _ 0 . \end{align*}


\begin{align*} [ u ] _ { s , p , \R ^ n } \vcentcolon = \left ( \frac { C ( n , s , p ) } { 2 } \int _ { \R ^ n } \int _ { \R ^ n } \frac { \abs { u ( x ) - u ( y ) } ^ p } { \abs { x - y } ^ { n + s p } } d x d y \right ) ^ { 1 / p } \end{align*}


\begin{align*} { \rm P r } \left ( \left . Z > \lambda \right | \mathcal { H } _ 1 \right ) = Q \left ( \frac { \lambda - 1 - \varrho \bar { \gamma } } { \sqrt { ( 1 + 2 \varrho \bar { \gamma } ) / M \bar { N } } } \right ) . \end{align*}


\begin{align*} d V _ { \max } = \sigma _ { \max } ( x ) d x , \end{align*}


\begin{align*} \begin{aligned} Y _ { \lambda } ( x , p ) & = \sum _ { n = 1 } ^ { \infty } \sum _ { k = 1 } ^ { n } ( k ) _ { p , \lambda } \frac { x ^ { n } } { n ! } \\ & = \sum _ { n = 1 } ^ { \infty } \Big ( ( 1 ) _ { p , \lambda } + ( 2 ) _ { p , \lambda } + \cdots + ( n ) _ { p , \lambda } \Big ) \frac { x ^ { n } } { n ! } \end{aligned} \end{align*}