\begin{align*} S _ 2 = \sum _ { u , v , z } b _ { u \vee z } b _ { v \vee z } \int _ E w _ u ( x ) w _ v ( x ) d x , \end{align*}


\begin{align*} \lambda _ n u _ n ( x _ n ) - \rho _ n u _ n ^ { p - 1 } ( x _ n ) = u _ n '' ( x _ n ) \le 0 , \end{align*}


\begin{align*} & \big ( \big ( K _ 1 ^ 2 \partial _ i \bar { \partial } _ j \log K _ 1 \big ) ( z , w ) \big ) _ { i , j = 1 } ^ m \\ & \quad \quad = \big ( \big ( K _ 2 ^ 2 \partial _ i \bar { \partial } _ j \log K _ 2 \big ) ( z , w ) \big ) _ { i , j = 1 } ^ m + \big ( \big ( K _ 3 ^ 2 \partial _ i \bar { \partial } _ j \log K _ 3 \big ) ( z , w ) \big ) _ { i , j = 1 } ^ m + \big ( \left \langle \gamma _ j ( w ) , \gamma _ i ( z ) \right \rangle \big ) _ { i , j = 1 } ^ m . \end{align*}


\begin{align*} ( \alpha , \beta ) _ { L , n } = T _ { L | K } ( \lambda _ { \rho } ( \alpha ) \psi _ { L , n } ( \beta ) ) \cdot _ { \rho } v _ n \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \abs { \frac { H _ n ( 1 - s ) } { H _ n ( s ) } } = 1 . \end{align*}


\begin{align*} \sum _ { 1 = K _ s \subsetneq K _ { s - 1 } \subsetneq \dots \subsetneq K _ 0 = F } ( - 1 ) ^ s \beta _ { K _ 0 , \dots , K _ s } = ( - 1 ) ^ { \sum _ i e _ i + \sum _ i f _ i } \prod _ i Q _ i ( e _ i ) \end{align*}


\begin{align*} C ( J f ( x ) ) = C ^ * ( J f ( x ) ) . \end{align*}


\begin{align*} \widetilde { M } _ f : = \frac { 1 } { [ | W _ f | ] } M _ f S _ 0 = \sum _ { \tau \in W ^ f } q ^ { \ell ( ^ f w _ 0 ) - \ell ( \tau ) } M _ { f \cdot \tau } \end{align*}


\begin{align*} \eta _ { x t } + \eta _ { x s } - w _ x = 0 . \end{align*}


\begin{align*} \left | \Phi _ { i } ( x , R ) \right | \leq C ( p , n , \sigma , \alpha , \mathcal { M } ) \begin{cases} \varepsilon ^ { \min \{ ( 1 - \sigma ) p , ( \sigma p + \alpha ) ( p - 2 ) + \alpha \} } , & , \\ \varepsilon ^ { \min \{ ( 1 - \sigma ) p , p + \alpha - 2 \} } , & , \end{cases} \end{align*}


\begin{align*} w ( \eta ) = \sup \limits _ \xi \left ( \eta \xi - J ( \xi ) \right ) \end{align*}


\begin{align*} \sum \limits _ { i = 1 } ^ \infty \mu [ a _ { i + \kappa _ n } , \infty ) = \sum \limits _ { i = \kappa _ n + 1 } ^ \infty \mu [ a _ i , \infty ) < \frac { 1 } { 2 ^ n } . \end{align*}


\begin{align*} J _ 1 ( t ) & = \langle E ( t ) , F _ n ( U ( t ) ) - F _ n ( \tilde U ( t ) ) \rangle + \langle E ( t ) , F _ n ( \tilde U ( t ) ) - F _ n ( \mathbb U ( t ) ) \rangle \\ & \le \frac { 3 } { 2 } \| E ( t ) \| ^ 2 + \frac { 1 } { 2 } \| F _ n ( \mathbb U ( t ) ) - F _ n ( \tilde U ( t ) ) \| ^ 2 , \end{align*}


\begin{align*} \Omega ( \alpha , A , z + a f _ k ) = \{ z + a f _ k \} \cup \bigcup _ { n = 1 } ^ \infty \Omega ( \alpha - 1 , A _ n , x ^ k _ n + a _ n f _ { p ( n ) } ) . \end{align*}


\begin{align*} \big \langle \Phi , ( \mathcal { N } _ \perp ^ 2 + 1 ) \Phi \big \rangle \le \ ; & C \\ \langle \Phi , \mathfrak { D } ^ 2 \Phi \rangle \le \ ; & C N \\ \big \langle U _ N ^ * \Phi , \ , \mathcal { N } _ - \ , U _ N ^ * \Phi \big \rangle \le \ ; & C \min \{ N , \frac { 1 } { T } \} \end{align*}


\begin{align*} \sigma _ { p , \alpha } ^ 2 = \Big ( \frac { p } { 1 + \alpha } \Big ) ^ { 2 / p } \frac { \Gamma ( 3 / p ) } { \Gamma ( 1 / p ) } . \end{align*}


\begin{align*} ( - \Delta ) ^ s u _ f + \sum _ { | \alpha | \leq m } a _ \alpha D ^ \alpha u _ f & = 0 \mbox { i n } \quad \Omega , u _ f = f \quad \mbox { i n } \quad \Omega _ e \end{align*}


\begin{align*} M ' = \oplus _ { n \in \mathbb { N } } M ( \frac { n } { T } ) ^ * , \end{align*}


\begin{align*} \beta \circ f | _ K = \beta | _ { E _ 1 \times \cdots \times E _ { j - 1 } \times M } \circ f | _ K \end{align*}


\begin{align*} \sum _ { Q \in \Gamma \backslash \mathcal Q _ D } \chi _ d ( Q ) \int _ { C _ Q } \varphi ( z ) y ^ { s - 1 } \ , | d z | = \pi ^ { - s - \frac 1 2 } \sqrt { d } \ , \Gamma ( \tfrac s 2 + \tfrac { i r } 2 + \tfrac 1 4 ) \Gamma ( \tfrac s 2 - \tfrac { i r } 2 + \tfrac 1 4 ) L ( s + \tfrac 1 2 , \varphi \times \chi _ d ) . \end{align*}


\begin{align*} ( \mathrm { c u r l } \ \boldsymbol { u } , \mathrm { c u r l } \ \boldsymbol { v } ) + ( \boldsymbol { u } , \boldsymbol { v } ) = ( \boldsymbol { f } , \boldsymbol { v } ) , \forall \boldsymbol { v } \in \boldsymbol { H } _ 0 ( \mathrm { c u r l } ; \Omega ) . \end{align*}


\begin{align*} \lim \limits _ { \ell \rightarrow \infty } \dfrac { \mathbb { E } [ \ell , k ] } { \ell } = \dfrac { b } { a } \end{align*}


\begin{align*} H _ { \mu } ^ 1 ( \mathcal { G } ) : = \left \{ u \in H ^ 1 ( \mathcal { G } ) : \int _ { \mathcal { G } } | u | ^ 2 \ , d x = \mu \right \} . \end{align*}


\begin{align*} B _ x & : = \inf \{ t \ge 0 : x \in \eta _ n ( t ) \} , \\ D _ x & : = \sup \{ t \ge 0 : x \in \eta _ n ( t ) \} , \\ L _ x & : = D _ x - B _ x . \end{align*}


\begin{align*} \tau ^ \mathrm { a n } ( M / S , F ) & = \sum _ k \bigg \{ \frac { 2 ^ { 2 k } \big ( k ! \big ) ^ 2 } { ( 2 k + 1 ) ! } \tau ^ \mathrm { B L } ( M / S , F ) \bigg \} ^ { [ 2 k ] } \ ; , \\ \tau ^ \mathrm { t o p } ( M / S , F ) & = \sum _ k \bigg \{ \ ! - \frac { k ! } { ( 2 \pi ) ^ k } \tau ^ \mathrm { I K } ( M / S , F ) + \frac { \zeta ' ( - k ) \mathrm { r k } F } { 2 } \int _ Z \mathrm { e } ( T Z ) \mathrm { c h } ( T Z ) \bigg \} ^ { [ 2 k ] } \ ; , \end{align*}


\begin{align*} \phi _ w ( g _ \infty , g _ p ) : = \sum _ { \gamma \in \Gamma _ { p , \ell } } w ( g _ \infty \gamma ) \chi ( g _ p \gamma ) \hbox { f o r $ ( g _ \infty , g _ p ) \in G _ \infty \times G _ p . $ } \end{align*}


\begin{align*} \dot { K } _ { \delta } ( \hat \tau ^ * ) = \dot { \hat { \tau } } ^ * \frac { k } { 2 } \textrm { s e c h } ^ 2 \left ( k ( \hat \tau ^ * - \delta ) \right ) \end{align*}


\begin{align*} ( 1 \otimes x _ 2 \cdot 1 \otimes x _ 3 ) _ { \lambda } = 1 \otimes x _ 2 x _ 3 + \sum _ i m _ i \otimes y _ i \end{align*}


\begin{align*} \norm { F ^ { ( \iota ) } ( \sqrt L ) f _ j } _ p ^ p \lesssim \sum _ { j = 0 } ^ \infty \Vert F ^ { ( \iota ) } ( \sqrt L ) f _ j \Vert _ p ^ p . \end{align*}


\begin{align*} 0 \longrightarrow \mathbb { L } : = ( \hat { \beta } ) \longrightarrow \mathbb { Z } ^ { m + 1 } \stackrel { \hat { \beta } } { \longrightarrow } N \oplus \mathbb Z \longrightarrow 0 , \end{align*}


\begin{align*} \Vert P _ n f \Vert _ p \leq c _ p \Vert f \Vert _ p \mbox { f o r a l l } f \in A _ { \mu } ^ p \mbox { a n d a l l } n \mbox { a n d } \lim _ { n \rightarrow \infty } \Vert f - P _ n f \Vert _ p = 0 . \end{align*}


\begin{align*} \overline B ' : = \sup _ { s > 0 } \left ( \int _ s ^ \infty V ( t ) ^ { - \frac 1 { p - 1 } } \ , d t \right ) ^ { p - 1 } \left ( \int _ 0 ^ s W ( t ) \ , d t \right ) \end{align*}


\begin{align*} G ( V , V ) G ( \mbox { g r a d } b , P _ U U ) = G ( U , U ) G ( \mbox { g r a d } b , P _ V V ) , \end{align*}


\begin{align*} F = ( f , f \circ S ^ { - m } , f \circ S ^ { - 2 m } , \dotsc , f \circ S ^ { - ( n - 1 ) m } ) , \end{align*}


\begin{align*} \Gamma ( \mathbf v , d ) & = \{ \mathbf a \in \mathbb Z ^ 3 \mid \mathbf a \times \mathbf v / d \in M ( \mathbf v , d ) \} \\ & = \{ \mathbf a \in \mathbb Z ^ 3 \mid \mathbf a \times \mathbf v ( \mathbf a \times \mathbf v ) \times \mathbf v \} . \end{align*}


\begin{align*} k = q ( r ^ B _ A ( k ) ) ^ { - 1 } r ^ B _ A ( k ) . \end{align*}


\begin{align*} I m ( s _ 1 ^ { \pm 1 } - I d ) = \mathbb { R } e _ 2 \ \ \ \ \ \ \ I m ( s _ 2 ^ { \pm 1 } - I d ) = \mathbb { R } e _ 1 \end{align*}


\begin{align*} X _ 1 = \partial _ x , X _ 2 = \partial _ y , & X _ 3 = x \partial _ x , X _ 4 = y \partial _ y , \\ X _ 5 = y \partial _ x , X _ 6 = x \partial _ y , X _ 7 & = x ^ 2 \partial _ x + x y \partial _ y , X _ 8 = x y \partial _ x + y ^ 2 \partial _ y . \end{align*}


\begin{align*} B = \lim _ { Q \in \Psi _ n } \nu ( Q ) \mbox { a n d } \overline B = \lim _ { Q \in \Psi _ n } \nu _ Q ( F ) . \end{align*}


\begin{align*} \| N _ t u _ 0 \| _ { H ^ { 1 + \varepsilon } ( \mathbb { T } ) } & = \left \| \left ( L _ { t } [ S _ t u _ 0 ] S _ t - W _ t ^ \gamma \right ) u _ 0 \right \| _ { H _ x ^ { 1 + \varepsilon } ( \mathbb { T } ) } \\ & \leq \tilde { C } ( \| f \| _ { H ^ 1 _ x ( \mathbb { T } ) } , \gamma , \varepsilon ) , \end{align*}


\begin{align*} \sup _ n \left | \sum _ { k = 1 } ^ \infty t _ { n , k } a _ k w _ { m _ k } ( x ) \right | < \infty \end{align*}


\begin{align*} \bold { q } ( t ) = - \int _ 0 ^ { \infty } g ( s ) \theta _ x ( t - s ) d s , \end{align*}


\begin{align*} \bold { q } _ t ( t ) + k \bold { q } = - \theta _ x ( t ) . \end{align*}


\begin{align*} \tilde y _ i = \left \{ \begin{array} { c c } y _ i y _ \mathrm { r } ^ { c _ i } & 1 \leq i \leq \mathrm { r } - 1 \\ y _ \mathrm { r } ^ { - c } & i = \mathrm { r } \end{array} \right . \end{align*}


\begin{align*} r _ n [ B , M ] ^ * = \{ r _ n ( N ) : N \in [ B , M ] ^ * \} , \end{align*}


\begin{align*} \begin{array} [ c ] { c } \frac { \partial \varphi } { \partial t } = \varphi ^ { \prime } ( t ) \leq C ( 1 + t ) e ^ { - t } , \end{array} \end{align*}


\begin{align*} Y _ { \lambda } ( x , p ) = \sum _ { k = 1 } ^ { \infty } ( k ) _ { p , \lambda } \sum _ { n = k } ^ { \infty } \frac { x ^ { n } } { n ! } = \sum _ { k = 1 } ^ { \infty } ( k ) _ { p , \lambda } \bigg ( e ^ { x } - \sum _ { l = 0 } ^ { k - 1 } \frac { x ^ { l } } { l ! } \bigg ) . \end{align*}


\begin{align*} { \mathcal Q } = { \mathcal P } ^ t { \mathcal P } \end{align*}


\begin{align*} { \bf U } ^ k ( \alpha _ i ) \cong \displaystyle \sum _ { j = 1 } ^ { L } T _ j ^ k ( \alpha _ i ) { \boldsymbol \xi } _ j k = 1 , \ldots , m , \ i = 1 , \ldots , p , \end{align*}


\begin{align*} \begin{cases} & \frac { \partial X } { \partial t } ( x , t ) = u ^ \alpha \rho ^ \delta f ^ { - \beta } ( \kappa ) \nu ( x , t ) , \\ & X ( \cdot , 0 ) = X _ 0 . \end{cases} \end{align*}


\begin{align*} \| ( A + \lambda I ) ^ { - 1 } \| { = \frac 1 { \operatorname { d i s t } ( \lambda , \sigma ( A ) ) } } \leq \frac { 1 } { \sin \theta | \lambda | } | \arg \lambda | < \pi - \theta \end{align*}


\begin{align*} R _ { \alpha \beta 3 \alpha } = & | \nabla u | ^ { - 1 } ( \nabla _ \alpha \nabla _ \beta - \nabla _ \beta \nabla _ \alpha ) \nabla _ \alpha u \\ = & - | \nabla u | ^ { - 1 } \nabla _ \alpha ( k _ { \alpha \beta } | \nabla u | ) + | \nabla u | ^ { - 1 } \nabla _ \beta ( k _ { \alpha \alpha } | \nabla u | ) \\ = & - \nabla _ \alpha k _ { \alpha \beta } + \nabla _ { \beta } k _ { \alpha \beta } + k _ { \alpha 3 } k _ { \alpha \beta } - k _ { \beta 3 } k _ { \alpha \alpha } , \end{align*}


\begin{align*} \left \{ \begin{array} { r c l } \dd X ^ { i , N } _ t & = & V ^ { i , N } _ t \dd t \\ \dd V ^ { i , N } _ t & = & F \big ( X ^ { i , N } _ t , V ^ { i , N } _ t , \mu _ { \mathcal { X } ^ N _ t } \big ) \dd t + \sigma \big ( X ^ i _ t , V ^ { i , N } _ t , \mu _ { \mathcal { X } ^ N _ t } \big ) \dd B ^ { i } _ t , \end{array} \right . \end{align*}


\begin{align*} ( - \Delta ) ^ s u _ 2 ^ * + \sum _ { | \alpha | \leq m } ( - 1 ) ^ { | \alpha | } D ^ \alpha ( a _ { 2 , \alpha } u _ 2 ^ * ) & = 0 \mbox { i n } \ ; \ ; \Omega , u _ 2 ^ * - f _ 2 \in \widetilde H ^ s ( \Omega ) , \end{align*}


\begin{align*} \overline { G } _ { n } ( u ) = \int _ { u } ^ { \infty } { g } _ { n } ( v ) \mathrm { d } v \end{align*}


\begin{align*} \mathrm { d } \Gamma _ \perp ( A ) : = \sum _ { m , n \ge 3 } \langle u _ m , A u _ n \rangle a ^ * _ m a _ n \end{align*}


\begin{align*} \| P _ \delta ( E ) \| _ { \mathrm { o p } } = \| P ^ \parallel _ \delta ( E ^ \parallel ) \| _ { \mathrm { o p } } \| P ^ \perp _ \delta ( E ^ \perp ) \| _ { \mathrm { o p } } \end{align*}


\begin{align*} \langle f , I ^ { \circ } ( w _ 1 , z ) u _ { p + \frac { j _ 2 } { T } } w _ 2 \rangle = 0 \end{align*}


\begin{align*} \gamma _ 2 : = \inf \{ k : k \in \R _ + , | V ( x , v , \tau _ { \ast } , 0 ) | | \leq 2 ^ { k M _ t } \} . \end{align*}


\begin{align*} \Lambda _ 1 = ( 0 , 1 , 2 , 3 ) \end{align*}


\begin{align*} s + 1 = r _ 1 + \cdots + r _ { w - 1 } + s _ w \enspace . \end{align*}


\begin{align*} \sum \limits _ { n = 1 } ^ N \frac { a _ n ^ N } { \lambda _ n ^ 2 - z } \to \sum \limits _ { n \ge 1 } \frac { a _ n } { \lambda _ n ^ 2 - z } N \to \infty \end{align*}


\begin{align*} ( D ^ { \nu } _ i ) _ { j k } = ( D ^ { \nu } _ j ) _ { i k } \end{align*}


\begin{align*} \pi _ { i _ 1 } \pi _ J = \pi _ { i _ 1 } \cdot \pi _ { J _ 1 } \cdots \pi _ { J _ m } \end{align*}


\begin{align*} a _ s ( y ) & = a ( x _ 0 + s y ) \\ f _ s ( y ) & = s f ( x _ 0 + s y ) . \end{align*}


\begin{gather*} T _ { 2 , 2 } ( m ) = \left \lfloor \frac m 3 \left \lfloor \frac { m - 1 } 2 \right \rfloor \right \rfloor - [ m \equiv 5 \bmod 6 ] \\ T _ { 2 , b } ( m ) = \begin{cases} \left \lfloor \frac { b - 2 } 3 \binom m 2 \right \rfloor + T _ { 2 , 2 } ( m ) & 2 \mid m \land 2 \mid b \\ \left \lfloor \frac { b - 1 } 3 \binom m 2 \right \rfloor - [ m \equiv 5 \bmod 6 \lor b \equiv 5 \bmod 6 ] & \end{cases} \end{gather*}


\begin{align*} { e ^ { W \left [ \eta _ T \left ( t \right ) \right ] } } = { e ^ { W \left [ \eta \left ( t / T \right ) \right ] } } \sim e ^ { T \int w ( \eta ( \tau ) ) d \tau } \end{align*}


\begin{align*} \int \exp _ 2 ^ * ( s _ p ( q ) ) \ p \ d \mu = \int \frac q p \log \frac q p p \ d \mu - \int \frac q p \ d \mu + 1 = - \int \log \frac p q \ q \ d \mu \ . \end{align*}


\begin{align*} \rho _ 2 = s \Big ( - i \omega _ 2 + \big ( \frac { \xi } { 2 s } + \sqrt { 1 - \frac { \vert \xi \vert ^ 2 } { 4 s ^ 2 } } \omega _ 1 \big ) \Big ) : = s \omega ^ { * } _ 2 ( s ) . \end{align*}


\begin{align*} \sum _ { k = 1 } ^ \infty \frac { 1 } { 1 \vee \log \left ( \phi ^ { - 1 } ( k ) \right ) } = \infty . \end{align*}


\begin{align*} \lambda _ i = \frac { \kappa _ i - \kappa _ 1 } { \kappa _ 2 - \kappa _ 1 } \end{align*}


\begin{align*} \sum _ { i = 3 , 4 } \big | \mathbf { P } \big ( T _ { k , j ; n } ^ { \mu , i } ( \mathfrak { m } , E ) ( t , x , \zeta ) + \hat { \zeta } \times T _ { k , j ; n } ^ { \mu , i } ( \mathfrak { m } , B ) ( t , x , \zeta ) \big ) \big | + 2 ^ { ( \gamma _ 1 - \gamma _ 2 ) M _ { t ^ { \star } } } \big | T _ { k , j ; n } ^ { \mu , i } ( \mathfrak { m } , E ) ( t , x , \zeta ) + \hat { \zeta } \times T _ { k , j ; n } ^ { \mu , i } ( \mathfrak { m } , B ) ( t , x , \zeta ) \big | \end{align*}


\begin{align*} [ x , y z ] = [ x , z ] [ x , y ] ^ z \mbox { a n d } [ x y , z ] = [ x , z ] ^ y [ y , z ] . \end{align*}


\begin{align*} I _ { X _ + } ( y , z ) = z e ^ { t _ + / z } \sum _ { d \in \mathbb K _ { + } } y ^ { d } \left ( \prod _ { i = 0 } ^ { m } \frac { \prod _ { a \leq 0 , \langle a \rangle = \langle D _ i \cdot d \rangle } ( \bar { D } _ i + a z ) } { \prod _ { a \leq D _ i \cdot d , \langle a \rangle = \langle D _ i \cdot d \rangle } ( \bar { D } _ i + a z ) } \right ) \textbf { 1 } _ { [ - d ] } , \end{align*}


\begin{align*} f _ { \ell k + 1 } \circ S ^ \ell = f _ { ( \ell - 1 ) k + 1 } \circ S ^ { \ell - 1 } = \dotsb f _ { k + 1 } \circ S ^ { - 1 } = f _ 1 , \end{align*}


\begin{align*} \langle A _ { i j } ( t ) g [ \mathbf { A } ] \rangle = \sum _ { n = 0 } ^ { \infty } \frac { 1 } { n ! } w ^ { ( n + 1 ) } _ { i j i _ 1 j _ 1 . . . i _ n j _ n } \biggl \langle \frac { \delta ^ n \ , g [ \mathbf { A } ] } { \delta A _ { i _ 1 j _ 1 } ( t _ 1 ) \ , \dots \ , \delta A _ { i _ n j _ n } ( t _ n ) } \biggr \rangle \end{align*}


\begin{align*} X ( t , x ) = \sum _ { \alpha = 1 } ^ r b _ \alpha ( t ) X _ \alpha ( x ) , t \in \mathbb { R } , x \in N , \end{align*}


\begin{align*} \nu \left ( L ( h _ \rho ) - L _ p ( h _ \rho ) \right ) > \beta _ b + b \gamma _ \rho = \nu ( L ( h _ \rho ) ) . \end{align*}


\begin{align*} c _ { n - 1 , \xi _ { \boldsymbol { s } k } } ^ { \ast } = \frac { \Gamma \left ( ( n - 1 ) / 2 \right ) } { ( 2 \pi ) ^ { ( n - 1 ) / 2 } } \left [ \int _ { 0 } ^ { \infty } x ^ { ( n - 3 ) / 2 } \overline { G } _ { n } \left ( \frac { 1 } { 2 } \xi _ { \boldsymbol { s } k } ^ { 2 } + x \right ) \mathrm { d } x \right ] ^ { - 1 } , \end{align*}


\begin{align*} n ( \Gamma ) = \sum _ { v \in \mathrm { V e r t } ( \Lambda ) } n ( v ) - ( r + s ) | E | . \end{align*}


\begin{align*} L _ { \mathcal { L } } = \| \tfrac { 1 } { N } A ^ T B A \| \leq \tfrac { 1 } { N } \| A ^ T A \| . \end{align*}


\begin{align*} \left \{ \begin{array} { l l } T & = \lambda ^ 3 + \mu ^ 3 + \nu ^ 3 ; \\ L & = \lambda ^ 3 + \mu ^ 3 + \nu ^ 3 - \left ( \lambda ^ 2 \mu + \lambda \mu ^ 2 + \lambda \nu ^ 2 + \lambda ^ 2 \nu + \mu ^ 2 \nu + \mu \nu ^ 2 \right ) + 3 \lambda \mu \nu . \end{array} \right . \end{align*}


\begin{align*} m _ v \coloneqq m - \sum _ { 0 \leq i < k } m _ i = \lim _ { n \to \infty } \int _ { B _ { r _ n } \setminus \cup _ { 0 \leq i < k _ n } B ^ i _ n } u _ n . \end{align*}


\begin{align*} \lambda _ { b , k } = \int _ 0 ^ 1 x ^ { k - 2 } ( 1 - x ) ^ { b - k } \Lambda ( d x ) \end{align*}


\begin{align*} y ^ * = \lim _ { n \rightarrow \infty } \left ( z ^ * + \sum _ { m = 1 } ^ n c _ m \widetilde { u } _ m \right ) \end{align*}


\begin{align*} \nabla ( G _ 0 - \Gamma ) \in L ^ { \bar q } ( \Omega ) , \bar q = \frac { N ( p - 1 ) } { N - 1 } , \end{align*}


\begin{align*} v = W _ t u _ 0 \pm T [ W _ t u _ 0 , v , \cdots , v ] + z . \end{align*}


\begin{align*} S ( x , y ) = \frac { \partial G ^ m } { \partial y ^ m } - y ^ m \frac { \partial \left ( \ln \sigma _ F \right ) } { \partial x ^ m } , \end{align*}


\begin{align*} \mathcal { F } ( \tilde A , \tilde B ) = { \| A - ( J - R ) Q \| } _ F ^ 2 + { \| B - ( R - J ) C ^ T \| } _ F ^ 2 , \end{align*}


\begin{align*} M _ 2 = f _ k ( i ) f _ k ( i + 1 ) + O _ \alpha \Big ( \frac { \tau _ 3 ( i ) \tau _ 3 ( i + 1 ) } { \sqrt { i } } \Big ) . \end{align*}


\begin{align*} \theta _ 0 ( t ) = \sum _ { n = r + 1 } ^ \infty \psi _ 1 ^ { [ n * ] } ( t ) , \end{align*}


\begin{align*} \delta ( x _ 1 ^ { a _ 1 } \cdots x _ { i - 1 } ^ { a _ { i - 1 } } x _ { i + 1 } ^ { a _ { i + 1 } } \cdots x _ n ^ { a _ n } d x _ 1 \wedge \cdots \wedge \widehat { d x _ i } \wedge \cdots \wedge d x _ n ) = ( - 1 ) ^ { i - 1 } \hbar ^ { - 1 } r _ i x _ 1 ^ { a _ 1 } \cdots x _ { i - 1 } ^ { a _ { i - 1 } } x _ i ^ { r _ i - 1 } x _ { i + 1 } ^ { a _ { i + 1 } } \cdots x _ n ^ { a _ n } \Omega . \end{align*}


\begin{align*} E = \left ( n _ 1 + n _ 2 + n _ 3 + \dfrac { 3 } { 2 } \right ) \omega \end{align*}


\begin{align*} \P \biggl ( \sup _ { x \in Q } Y ( t , x ) > R \mathrel { \Big | } A \biggr ) & \geq C \P \biggl ( \sup _ { x \in Q } g ( t - \tau , x - \eta ) > 4 R \mathrel { \Big | } A \biggr ) \\ & = C \P ( ( 2 \pi ( t - \tau ) ) ^ { - \frac d 2 } > 4 R \mid A ) . \end{align*}


\begin{align*} T _ { e , i } = \{ 4 k d + i , 4 k d + e + i , 4 k d + 2 e + i , \ldots , 4 k d + ( 4 k - 1 ) e + i \} . \end{align*}


\begin{align*} H _ { 2 - \mathrm { m o d e } } : = P ^ { \otimes N } H _ N P ^ { \otimes N } \end{align*}


\begin{align*} \left | \sum _ { 0 < | c | \le h ^ \delta } \sum _ { k = 1 } ^ \infty k ^ { - 4 } S _ { k } ( c ; \xi ) I _ { k , \ell } ( { } ^ t g c ) \right | \ll _ { w , \ell , \theta , \epsilon } \| g \| _ E h ^ { 7 / 2 + 3 \delta + \epsilon } . \end{align*}


\begin{align*} \sum \limits _ { i } ^ { N } \alpha _ i = 0 \end{align*}


\begin{align*} z _ Q = - \frac { b } { 2 a } + i \frac { | d | } { 2 a } \end{align*}


\begin{align*} K _ X ^ 2 = 1 6 n , p _ g \left ( X \right ) = 2 n , \chi \left ( \mathcal { O } _ X \right ) = 2 n , q \left ( X \right ) = 1 . \end{align*}


\begin{align*} \sum \nolimits _ { k , \beta } a _ { k , \beta } \ , \zeta _ { k , \beta } = 0 . \end{align*}


\begin{align*} \nu ( f - l ( h _ \sigma ) ) \geq \min \{ \nu ( f ) , \nu ( l ( h _ \sigma ) ) \} > \nu _ \sigma ( f ) = \beta _ b + b \gamma _ \sigma , \end{align*}


\begin{align*} w _ m ^ * = \frac { \varrho | g _ m | ^ 2 } { \sum _ { k = 1 } ^ { M } \varrho | g _ k | ^ 2 } , m = 1 , 2 , \ldots , M \end{align*}


\begin{align*} \begin{cases} - V '' + V = V ^ { p - 1 } , U _ 0 > 0 & , \\ V ( 0 ) = \max _ \R V , \\ V ( x ) \to 0 & . \end{cases} \end{align*}


\begin{align*} \left ( \Lambda _ { r , s } \right ) _ { r ' , s ' } = \Lambda \end{align*}


\begin{align*} \begin{cases} r _ + = 2 a + 3 k \ , \ \ r _ - = k \\ \vdots \\ r _ + = a + 3 k \ , \ \ r _ - = a + k \end{cases} \ , \end{align*}


\begin{align*} \frac { 1 } { 2 } g '' ( r ) + \frac { \alpha } { 2 } g ' ( r ) - \lambda g ( r ) = 0 \end{align*}


\begin{align*} \kappa \sqrt { \ss } = \tfrac 1 2 C _ { \widehat { r } } = ( \pi - \phi ( \widehat { r } ) ) \sinh ( \beta \widehat { r } ) . \end{align*}


\begin{align*} D _ { q ^ { - 1 } } v ( x ) - q p ( x ) v ( x ) = 1 . \end{align*}


\begin{align*} & \mathrm { D T V } _ { ( p , q ) } ( X ) = - \mathrm { D T E } _ { ( p , q ) } ^ { 2 } ( X ) + \mu ^ { 2 } + 2 \mu \sigma \mathrm { D T E } _ { ( p , q ) } ( Y ) + \sigma ^ { 2 } \left ( L _ { 1 } + 2 L _ { 2 } \right ) , \end{align*}


\begin{align*} \chi _ t : = \max _ { x \in \Lambda } \sum _ { y \in \Lambda } S _ t ( x , y ) . \end{align*}


\begin{align*} \Im H ( x + i y ) \begin{cases} > 0 , & y > f ( x ) , \\ < 0 , & y < f ( x ) . \end{cases} \end{align*}


\begin{align*} & \int f ( x ) h ( x / q ) \left ( \frac { 1 } { q } u ( x ) u ( x / q ) + \frac { 1 } { q } x r ( x ) ( q - 1 ) u ( x / q ) \right ) y ( x ) d _ q x \\ & = f ( x / q ) h ( x / q ) \left ( y ( x / q ) u ( x / q ) - D _ { q ^ { - 1 } } y ( x ) \right ) . \end{align*}


\begin{align*} N _ f : = \frac { [ | W _ \zeta | ] } { [ | W _ f | ] } \widetilde { N } _ f . \end{align*}


\begin{align*} \mathbb { E } [ \ell , k ] = \left . \left [ \dfrac { \ell } { 2 } \dbinom { \ell } { k } + \sum \limits _ { i = 0 } ^ { ( k - 1 ) / 2 } \dbinom { \ell } { k - 2 i - 1 } \right ] \middle / \dbinom { \ell } { k } \right . \end{align*}


\begin{align*} \bigcup _ { i = 1 } ^ { k + 1 } X _ i & = \biggl ( \bigcup _ { i = 1 } ^ { k } X _ i \biggr ) \bigcup X _ { k + 1 } \\ & = \biggl ( \bigsqcup _ { i = 1 } ^ { k } X _ i ' \biggr ) \bigcup X _ { k + 1 } \\ & = \biggl ( \bigsqcup _ { i = 1 } ^ { k } X _ i ' \biggr ) \bigsqcup \Biggl ( X _ { k + 1 } \backslash \biggl ( \bigsqcup _ { i = 1 } ^ { k } X _ i ' \biggr ) \Biggr ) \end{align*}


\begin{align*} \widetilde { \mathcal { I } } _ { [ m , j ] } : = \{ ( i _ { 1 } , \ldots , i _ { m } ) \in \mathbb { Z } ^ { m } : - p \leq i _ { 1 } \leq ( m - 1 ) p + j \ , \ , \mbox { a n d } \ , \ , i _ { k - 1 } - p \leq i _ { k } \leq ( m - k ) p + j , \ , \ , 2 \leq k \leq m \} . \end{align*}


\begin{align*} H _ { ( Z _ + , D _ { Z , + } ) } ( y ) = e ^ { \frac { t _ + } { 2 \pi i } } \sum _ { d \in \mathbb K _ { + } } y ^ d \left ( \frac { \prod _ { i = 1 } ^ 2 \Gamma ( 1 + \frac { v _ i } { z } + v _ i \cdot d ) } { \prod _ { i \in M _ 0 } \Gamma ( 1 + \frac { \bar D _ i } { z } + D _ i \cdot d ) } \right ) \textbf { 1 } _ { [ d ] } [ \textbf { 1 } ] _ { ( D _ i \cdot d ) _ { i \in I _ + } , v _ 2 \cdot d } . \end{align*}


\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 } ( y _ S ( u ) - y _ S ( v ) ) ^ 2 \\ & \le \frac { 1 } { 2 } \sum \limits _ { e \in \partial S } \frac { \delta _ E ( e ) } { | e | ^ 2 } 2 | S \cap e | | e - S | 4 \\ & \le | \partial S | \delta _ E ( \max ) = b ( G ) \delta _ E ( \max ) . \end{align*}


\begin{align*} K ( x , y , P ) : = \frac { g _ y ( R _ y ( u ) , u ) } { g _ y ( y , y ) g _ y ( u , u ) - g _ y ( y , u ) g _ y ( y , u ) } , \end{align*}


\begin{align*} | \Gamma | ^ 2 - | \Gamma _ 1 | ^ 2 = u u ^ * . \end{align*}


\begin{align*} D \psi _ s \left ( \frac { \partial } { \partial { x ^ i } } \right ) = \frac { \partial \widehat \psi _ s ^ a } { \partial x ^ i } \frac { \partial } { \partial { y ^ a } } . \end{align*}


\begin{align*} \frac { \partial c ( v ) } { \partial t } = - a ( v ) c ( v ) + \sum _ { v ' > v } \bar b _ { v | v ' } a ( v ' ) c ( v ' ) . \end{align*}


\begin{align*} \pi _ J \coloneqq \prod _ { k = 1 } ^ m e _ { | J _ k | } ( y _ 1 , y _ 2 , \ldots , y _ { \max J _ k } ) \in M , \end{align*}


\begin{align*} h _ 2 ( j ) = j + 1 . \end{align*}


\begin{align*} \nabla _ X U = A _ X U + \nu \nabla _ X U , \end{align*}


\begin{align*} \norm { \rho _ t ( \beta ) f _ t } ^ 2 & = \sum _ { \pi : \lambda _ { \pi _ \infty } \in t \Omega } \norm { \pi _ { p } ( \beta ) f _ \pi } ^ 2 = \sum _ { \pi : \lambda _ { \pi _ \infty } \in t \Omega } \abs { \beta ( \omega _ { \pi _ { p } } ) } ^ 2 \\ & = \sum _ { \pi : \lambda _ { \pi _ \infty } \in t \Omega } ( \beta ^ \ast \ast \beta ) ( \omega _ { \pi _ { p } } ) = \sum _ { \pi : \lambda _ { \pi _ \infty } \in t \Omega } \chi _ { \pi _ p } ( \phi _ { \beta ^ \ast \ast \beta } ) . \end{align*}


\begin{align*} \left \{ \begin{array} [ c ] { l l l } \frac { \partial } { \partial t } \varphi ( x , t ) & = & \log \frac { ( \widehat { \omega } _ { t } + \sqrt { - 1 } \partial _ { B } \overline { \partial } _ { B } \varphi ) ^ { n } \wedge \eta _ { 0 } } { \Omega \wedge \eta _ { 0 } } , \\ \widehat { \omega } _ { t } + \sqrt { - 1 } \partial _ { B } \overline { \partial } _ { B } \varphi & > & 0 , \\ \varphi ( 0 ) & = & \varphi _ { 0 } . \end{array} \right . \end{align*}


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


\begin{align*} \beta ( s ) = \sum _ { n \geq 1 } \frac { \chi ( n ) } { n ^ s } = \prod _ { p \geq 3 } \left ( 1 - \frac { \chi ( p ) } { p ^ s } \right ) ^ { - 1 } , \end{align*}


\begin{align*} s = - \frac { H } { 2 \pi i } - \frac { m } { d } - \bar { f } , m \geq 1 \end{align*}


\begin{align*} m ^ \tau _ { U , \ , \eta } \ = \ \sup _ { \alpha \subset U , \ , \alpha } m _ \alpha \ , . \end{align*}


\begin{align*} \beta ( s ) = \begin{cases} \displaystyle r _ n \pi ^ { s } & , \\ \displaystyle r _ n \pi ^ { s } - u _ n & , \end{cases} \end{align*}


\begin{align*} \Phi ( h _ + ) = \xi _ - - h _ - , \Phi ( \xi _ + ) = \xi _ - \end{align*}


\begin{align*} z G _ { \mu } ( z ) = \frac { 1 } { 1 - \eta _ { \mu } \left ( \frac { 1 } { z } \right ) } , z \in \mathbb { C } \setminus \mathbb { R } _ { + } , \end{align*}


\begin{align*} A \cup \{ k \} = \{ k \} \cup \bigcup _ { j = 1 } ^ \infty ( A _ j \cup \{ p ( j ) \} ) , \end{align*}


\begin{align*} I _ { ( X _ + , D _ + ) } ( y , z ) = z e ^ { t _ + / z } \sum _ { d \in \mathbb K _ { + } } y ^ { d } \left ( \prod _ { i \in M _ 0 } \frac { \prod _ { a \leq 0 , \langle a \rangle = \langle D _ i \cdot d \rangle } ( \bar { D } _ i + a z ) } { \prod _ { a \leq D _ i \cdot d , \langle a \rangle = \langle D _ i \cdot d \rangle } ( \bar { D } _ i + a z ) } \right ) \textbf { 1 } _ { [ - d ] } I _ { D _ + , d } , \end{align*}


\begin{align*} \tau ^ * ( n ) : = \sum \limits _ { \substack { n = n _ 1 n _ 2 \\ \gcd ( n _ 1 , n _ 2 ) = 1 } } 1 . \end{align*}


\begin{align*} c _ { i j } ^ { \mathcal { S } } ( t ) : = \begin{cases} c _ { i j } ( t ) & \mathrm { i f } \ , \ , i , j \in \mathcal { J } \ , \ , \mathrm { a n d } \ , \ , | i - j | \le \ell \\ 0 & \mathrm { o t h e r w i s e } , \end{cases} \end{align*}


\begin{align*} \dim \left [ \gg , x \right ] = \sum _ { \mu } \dim \left [ \gg ^ { \mu } , x \right ] , \end{align*}


\begin{align*} \sum _ { Q \in \Gamma \backslash \mathcal Q _ D } \chi _ d ( Q ) \int _ { C _ Q } i \partial _ z \varphi ( z ) y ^ { s } \ , d z = - 2 \pi i \sum _ { n \neq 0 } n a _ \varphi ( n ) G ( n , d ) \int _ 0 ^ \infty y ^ { s + \frac 1 2 } K _ { i r } ( 2 \pi | n | y ) \ , d y \end{align*}


\begin{align*} \sum _ { j = 0 } ^ { \infty } c _ j ^ 2 = \left \| e ^ { f _ o } v ( \cdot , 0 ) \right \| ^ 2 _ { L ^ 2 ( d \nu _ * ) } \leq \overline C , \end{align*}


\begin{align*} E r r H ^ 2 _ { i , i _ 1 } ( t _ 1 , t _ 2 ) = \int _ { t _ 1 } ^ { t _ 2 } \int _ { \R ^ 3 } \int _ { \R ^ 3 } \int _ { \R ^ 3 } e ^ { i X ( s ) \cdot \xi + i s \mu _ 1 | \xi - \eta | } \mathcal { F } \big [ ( \hat { v } - \hat { V } ( s ) ) \times B f \big ] \cdot \nabla _ v \big [ ( \hat { V } ( s ) \cdot \xi - \hat { v } \cdot \eta + \mu _ 1 | \xi - \eta | ) ^ { - 1 } \end{align*}


\begin{align*} S _ { \log , k } ( x ) : = \sum _ { n _ 1 \cdots n _ k \le x } \log ( n _ 1 , \ldots , n _ k ) = \sum _ { d ^ k \delta \le x } \Lambda ( d ) \tau _ k ( \delta ) = \sum _ { p ^ { \nu k } \delta \le x } ( \log p ) \tau _ k ( \delta ) . \end{align*}


\begin{align*} \mathrm { D T E } _ { ( p , q ) } ( X ) = \mu + \sigma \frac { c _ { 1 } \left [ \overline { G } _ { 1 } \left ( \frac { 1 } { 2 } \xi _ { p } ^ { 2 } \right ) - \overline { G } _ { 1 } \left ( \frac { 1 } { 2 } \xi _ { q } ^ { 2 } \right ) \right ] } { F _ { Y } ( \xi _ { p } , \xi _ { q } ) } , \end{align*}


\begin{align*} \inf _ { 2 \leq l \leq s } \inf _ { ( x , y ) \in \bar B } \sum _ { i = 1 } ^ { n - 2 } ( \varphi _ i ^ l ( x , y ) - \Phi _ i ( x , y ) ) ^ 2 > \frac { 4 \varepsilon } { 9 } . \end{align*}


\begin{align*} \frac { f _ { \lambda \lambda } ( \lambda _ i ) } { f _ \lambda ( \lambda _ i ) } = 2 \sum _ { j ( \neq i ) } \frac { 1 } { \lambda _ i - \lambda _ j } . \end{align*}


\begin{align*} ( W ^ k f ) _ { p _ 1 \dots p _ { m - k } q _ 1 \dots q _ { m - k } i _ 1 \dots i _ k } = \sigma ( q _ 1 \dots q _ { m - k } i _ 1 \dots i _ k ) ( W f ^ { i _ 1 \dots i _ k } ) _ { p _ 1 \dots p _ { m - k } q _ 1 \dots q _ { m - k } } . \end{align*}


\begin{align*} \lambda ( k ) = ( - 1 ) ^ { \Big [ \frac { 2 A p k } { q } \Big ] } \end{align*}


\begin{align*} C = \{ r e ^ { i h _ { \mu } ( r ) } : r \in ( 0 , + \infty ) \} , \end{align*}


\begin{align*} f ( q ^ n x ) = ( - 1 ) ^ n q ^ n f ( x ) ( n \in \mathbb { N } _ 0 ) . \end{align*}


\begin{align*} { \rm C a p } _ p ( B _ { \hat { d } } ( \mathcal O , 1 ) , B _ { \hat { d } } ( \mathcal O , 2 ^ i ) ) : = \inf \int _ X g _ u ^ p d \mu \end{align*}


\begin{align*} \int _ 0 ^ t \dot \kappa ^ { \epsilon , L } _ s \ , d s \geq \log ( t + 1 ) - c _ d ( \lambda , \mu , m ^ 2 = 1 ) . \end{align*}


\begin{align*} \tilde { \bf u } _ m ( \alpha ) = { \bf u } _ m ( \alpha ) + \tilde { \boldsymbol { \eta } } _ m ( \alpha ) . \end{align*}


\begin{align*} \omega _ 1 ^ { * } \cdot \nabla \varphi _ 1 = \omega _ 1 ^ { * } \cdot A _ 1 , \omega _ 2 ^ { * } \cdot \nabla \varphi _ 2 = - \omega _ 2 ^ { * } \cdot A _ 2 . \end{align*}


\begin{align*} 0 = d \omega _ { i A } = { \bar \Omega } _ { i A } , 1 \leq i \leq n , 1 \le A \le n + 2 . \end{align*}


\begin{align*} \int _ { \R ^ 3 } \hat { v } \times \nabla _ x f ( t , x , v ) d v = 0 , \end{align*}


\begin{align*} R ^ { n , 3 } _ \tau = n ^ { 2 \alpha + 1 } \int _ 0 ^ \tau \gamma _ s ( \sigma ' \sigma ) ^ 2 ( X _ s ) \int _ 0 ^ s \psi ^ 2 _ { n , 1 } ( u , s ) d u d s . \end{align*}


\begin{align*} V _ { \epsilon } \triangleq \begin{cases} V _ { \min } & \epsilon < \frac 1 2 \\ V _ { \max } & \epsilon \geq \frac 1 2 . \end{cases} \end{align*}


\begin{align*} \varphi _ { \mu } ( z ) = \gamma + \int _ { \mathbb { R } } \frac { 1 + s z } { z - s } \ , d \sigma ( s ) , z \in \mathbb { C } ^ { + } , \end{align*}


\begin{align*} \Big ( x \frac { d } { d x } \Big ) _ { p , \lambda } \frac { 1 } { 1 - x } = \sum _ { n = 0 } ^ { \infty } \Big ( x \frac { d } { d x } \Big ) _ { p , \lambda } x ^ { n } = \sum _ { n = 0 } ^ { \infty } ( n ) _ { p , \lambda } x ^ { n } = \frac { 1 } { 1 - x } F _ { p , \lambda } \Big ( \frac { x } { 1 - x } \Big ) . \end{align*}


\begin{align*} \mathcal { D } _ { [ n , j ] } = \bigcup _ { k = 0 } ^ { n - j } \mathcal { D } _ { [ n , j , k ] } , \end{align*}


\begin{align*} \log _ q N _ e > \log _ q ( q ^ e - 2 q ^ { e / 2 } g ) = e / 2 + \log _ q ( q ^ { e / 2 } - 2 g ) \geq e / 2 \geq \log _ q N \ . \end{align*}


\begin{align*} E ^ 2 ( t ) = | u ( t ) - \hat U ( t ) | \end{align*}


\begin{align*} \limsup _ { N \to \infty } \frac { \sum _ { k = 1 } ^ N f ( S _ k p / q ) - E ( f ) N } { \sqrt { 2 N \log \log N } } = \sigma \mathrm { a . s . } \end{align*}


\begin{align*} 0 = V _ 0 \subset V _ 1 \subset \cdots \subset V _ n = H _ * ( X ; \Q ) \end{align*}


\begin{align*} U ( t ) = \exp \left ( { \rm i } f _ 1 ( t ) \widehat { H } _ 1 \right ) \times \ldots \times \exp \left ( { \rm i } f _ r ( t ) \widehat { H } _ r \right ) { , } \end{align*}


\begin{align*} \Phi ( \infty ) : = \lim _ { z \to \infty } \Phi ( z ) = 1 \end{align*}


\begin{align*} \psi _ m : = \pi ^ m _ 1 \circ \varphi ^ m _ { p ( m , 1 ) , 1 } . \end{align*}


\begin{align*} \{ [ v ' ] \mid v ' \in { \downarrow _ 1 } ( v _ 1 ) \} = \{ [ v ' ] \mid v ' \in { \downarrow _ 1 } ( v _ 2 ) \} . \end{align*}


\begin{align*} \dim \gg ^ { \mu } = 2 \dim \gg _ { 1 } ^ { \mu } . \end{align*}


\begin{align*} \kappa _ 3 = \sum _ { k = 1 } ^ \infty \int _ 0 ^ 1 [ ( x + k ) ^ \alpha - k ^ \alpha ] ^ 2 d x . \end{align*}


\begin{align*} D _ + = \sum _ { i \in ( M _ + \cup \{ j \} ) } \bar D _ i \subset X _ + D _ - = \sum _ { i \in M _ + } \bar D _ i \subset X _ - . \end{align*}


\begin{align*} n = \lambda _ 1 + \lambda _ 2 + \cdots + \lambda _ k . \end{align*}


\begin{align*} \exp _ 2 ^ * ( a y ) = \int _ 0 ^ { a y } \frac { a y - s } { 1 + s } \ d s = a \int _ 0 ^ y \frac { a ( y - t ) } { 1 + a t } \ d t \leq \max ( a , a ^ 2 ) \int _ 0 ^ y \frac { y - t } { 1 + t } \ d t \ , \end{align*}


\begin{align*} D ^ n _ x ( 1 ) = \left \lbrace ( y ^ 1 , y ^ 2 , . . . , y ^ n ) \in T _ x M ^ n : F ( x , y ) < 1 \right \rbrace , y = y ^ { \epsilon } z ^ i _ { \epsilon } \textnormal { a n d } z = \left ( z ^ i _ { \epsilon } \right ) = \left ( \frac { \partial \varphi ^ i } { \partial x ^ { \epsilon } } \right ) . \end{align*}


\begin{align*} \Sigma _ \omega = \{ \sigma _ I : \bar { I } \in \mathcal A _ \omega \} . \end{align*}


\begin{align*} w _ m \left ( \alpha \oplus \frac { \varepsilon _ 1 } { 2 ^ { k _ 1 } } \oplus \ldots \oplus \frac { \varepsilon _ l } { 2 ^ { k _ l } } \right ) = \prod _ { j = 1 } ^ l r _ { k _ j } \left ( \alpha \oplus \frac { \varepsilon _ j } { 2 ^ { k _ j } } \right ) \end{align*}


\begin{align*} \phi ( N \times \{ 0 \} ) = S ^ { 1 } \cdot x . \end{align*}


\begin{align*} \| \phi _ 1 * \phi _ 2 * \cdots * \phi _ N \| _ { q ( P ) } \leq Y \prod _ { k = 1 } ^ N \| \phi _ k \| _ { p _ k } \end{align*}


\begin{align*} \frac { 1 } { r ! } \Big ( \frac { d } { d x } \Big ) ^ { r } \bigg [ \frac { x ^ { r } } { ( 1 - x ) ^ { 2 } } F _ { p , \lambda } \Big ( \frac { x } { 1 - x } \Big ) \bigg ] & = \frac { 1 } { r ! } \Big ( \frac { d } { d x } \Big ) ^ { r } \bigg [ \sum _ { n = 1 } ^ { \infty } \Big ( ( 1 ) _ { p , \lambda } + ( 2 ) _ { p , \lambda } + \cdots + ( n ) _ { p , \lambda } \Big ) \bigg ] x ^ { n + r } \\ & = \sum _ { n = 1 } ^ { \infty } \Big ( ( 1 ) _ { p , \lambda } + ( 2 ) _ { p , \lambda } + \cdots + ( n ) _ { p , \lambda } \Big ) \binom { n + r } { r } x ^ { n } . \end{align*}


\begin{align*} \psi _ { j } ( z ) : = \langle ( z I - \mathcal { W } ) ^ { - 1 } \hat { e } _ { j } , \hat { e } _ { 0 } \rangle = \sum _ { n = 0 } ^ { \infty } \frac { \langle \mathcal { W } ^ { n } \ , \hat { e } _ { j } , \hat { e } _ { 0 } \rangle } { z ^ { n + 1 } } , 0 \leq j \leq p , \end{align*}


\begin{align*} \limsup _ { N \to \infty } \frac { \sum _ { k = 1 } ^ N f ( S _ k \alpha ) - E ( f ) N } { \sqrt { 2 N \log \log N } } = \sigma \textrm { a . s . } \end{align*}


\begin{align*} \Big ( \sum _ { g \in \mathrm { G a l } ( K / \Q ) } g . \varphi _ k \Big ) _ { k = 0 , \cdots , p ^ r - 1 } \ ; . \end{align*}


\begin{align*} w = \frac { h ( w ) } { h ( w ) ^ { p + 1 } + 1 } . \end{align*}


\begin{align*} G _ Z ( \alpha ) = e ^ { \lambda _ Z ( e ^ { \alpha } - 1 ) } \ , . \ , \end{align*}


\begin{align*} \mathbb { G } ( X _ n , X _ 0 ) = \sum _ { m = 0 } ^ \infty p _ m ( X _ n , X _ 0 ) \leq \gamma ( N + 1 ) ^ 3 ( n + 1 ) ^ 2 p _ n ( X _ n , X _ 0 ) + \sum _ { m = N + 1 } ^ \infty p _ m ( X _ n , X _ 0 ) . \end{align*}


\begin{align*} 1 - F ( z ) = \prod _ { k \ge 1 } \left ( \frac { z - \mu _ k ^ 2 } { z - \lambda _ k ^ 2 } \right ) \ , . \end{align*}


\begin{align*} \log \ , \left | \dfrac { k ! } { ( \beta ) _ k } \right | _ p \le \begin{cases} C _ { p , k } \log ( p ) & \ \ \ \ p \mid N _ k \\ 0 & \ \ \enspace , \end{cases} \end{align*}


\begin{align*} \varPsi ( u ) = \displaystyle \int _ { \Omega } \int _ { \Omega } \varPhi _ { x , y } \left ( \dfrac { | u ( x ) - u ( y ) | } { | x - y | ^ s } \right ) \dfrac { d x d y } { | x - y | ^ N } + \int _ { \Omega } \widehat { \varPhi } _ { x } \left ( | u ( x ) | \right ) d x . \end{align*}


\begin{align*} P _ c ( \Psi _ t ) ( x ) = [ \nabla _ i \Psi _ t ( x ) \nabla _ i \nabla _ j \Psi _ t ( x ) - \nabla _ l \Psi _ t \Gamma _ { i j } ^ l ( x ) \nabla _ k \nabla _ k \Psi _ t ( x ) - \nabla _ l \Psi _ t \Gamma _ { k k } ^ l ( x ) - \frac { \sum _ k ( \nabla _ k \nabla _ k \Psi _ t - \nabla _ l \Psi _ t \Gamma _ { k k } ^ l ) } { n } ( x ) ] ^ T \end{align*}


\begin{align*} \langle { x ^ \eta ( T ) } \rangle = \int e ^ { T \eta a } \rho _ { \bar { \xi } } ( a ) d a \sim e ^ { T \ , w ( \eta ) } \end{align*}


\begin{align*} \lim _ { \delta \rightarrow 0 } \int _ { x _ 0 - \delta } ^ { x _ 0 + \delta } ( a u '' ) ' v ' d x = 0 . \end{align*}


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


\begin{align*} v ( \Box \phi , w ) & = \inf \{ v ( \phi , w ' ) : w R w ' \} , & v ( \lozenge \phi , w ) & = \sup \{ v ( \phi , w ' ) : w R w ' \} . \end{align*}


\begin{align*} d ( L ) = \prod _ { i = 1 } ^ { n } b _ { i i } . \end{align*}


\begin{align*} \begin{aligned} \relax [ \mu , L ] = [ 0 , L _ { \mathcal { L } } ] + \beta [ - \tfrac { 1 } { \sigma } , \tfrac { 1 } { \lambda - \sigma } ] \approx [ - \tfrac { \beta } { \sigma } , \tfrac { 1 } { N } \| A ^ T A \| + \tfrac { \beta } { \lambda - \sigma } ] \end{aligned} \end{align*}


\begin{align*} \mu ( B _ Q ( R ) \cap B _ O ( R ) ) = n C _ { \alpha } e ^ { - \frac { r } { 2 } } ( 1 + O ( e ^ { - ( \alpha - \frac 1 2 ) r } + e ^ { - r } ) ) . \end{align*}


\begin{align*} \prod _ { b \in \lambda } \dfrac { 1 - t ^ { n + c ( b ) } } { 1 - t ^ { h ( b ) } } = \prod _ { 1 \leq i < j \leq n } \dfrac { 1 - t ^ { \lambda _ i - \lambda _ j + j - i } } { 1 - t ^ { j - i } } . \end{align*}


\begin{align*} K _ { \log , k } = \sum _ p \frac { \log p } { p ^ k - 1 } , K _ { \omega , k } = \sum _ p \frac 1 { p ^ k } , K _ { \Omega , k } = \sum _ p \frac 1 { p ^ k - 1 } . \end{align*}


\begin{align*} F ^ n _ \tau = n ^ { \alpha + \frac 1 2 } \int ^ { \tau } _ 0 ( t - s ) ^ { \alpha } \sigma ' ( X _ s ) \ , \left ( \int ^ { \eta _ n ( s ) } _ 0 \psi _ { n , 2 } ( u , s ) \sigma ( X _ { \eta _ n ( u ) } ) \ , d W _ u \right ) d s , \end{align*}


\begin{align*} p _ { k } : = \xi _ { k } , \overline { y } _ { k + 1 } : = y _ { k } - \frac { 1 } { \sigma _ { k } } \left ( g ( x _ { k } ) + \nabla g ( x _ { k } ) ^ { \top } \xi _ { k } \right ) , \overline { Z } _ { k + 1 } : = \Sigma _ { k } , \end{align*}


\begin{align*} & y _ { i } \leq \langle ( y _ 1 , \ldots y _ s ) , \gamma _ j \rangle \leq d _ j - y _ { s + 3 } \leq d _ j , \ i = 1 , \ldots , s \\ & y _ { s + 3 } \leq y _ { s + 1 } \leq \langle ( y _ 1 , \ldots , y _ s ) , u \rangle \leq \langle ( y _ 1 , \ldots y _ s ) , \gamma _ j \rangle \leq d _ j - y _ { s + 3 } \leq d _ j , \end{align*}


\begin{align*} \varphi ( x ^ 1 , x ^ 2 ) = ( f ( x ^ 1 ) \cos x ^ 2 , f ( x ^ 1 ) \sin x ^ 2 , x ^ 1 ) , \end{align*}


\begin{align*} L ^ \sigma [ \psi ] ( x ) : = \textup { t r } \big ( \sigma \sigma ^ T D ^ 2 \psi ( x ) \big ) \end{align*}


\begin{align*} \sum _ { k = \overline { k } } ^ { R } e ^ { - 2 \beta k } I _ k \geq \sum _ { k = \overline { k } } ^ { R } e ^ { - 2 \beta k } \widetilde { c } \ss e ^ { - \alpha ( R - k ) } Z _ k = \widetilde { c } \ss e ^ { - 2 \beta R } \sum _ { k = \overline { k } } ^ R Z _ k > \widetilde { c } \ss e ^ { - 2 \beta R } \frac { \widetilde { \eta } } { 2 } ( R - \overline { k } + 1 ) . \end{align*}


\begin{align*} \dot { x } _ t = f ( x _ t ) + \epsilon ( L _ G ( g ( x _ t ) ) ) , \end{align*}


\begin{align*} g ( P _ X Y , Z ) = g ( Y , P _ X Z ) , \end{align*}


\begin{align*} \begin{cases} w _ t = \Delta _ x w - q { \sf U } _ \infty ^ { q - 1 } w + ( F _ 1 [ \tilde v _ 1 , \tilde w _ 1 ] - F _ 2 [ \tilde w _ 1 ] ) ( 1 - \chi _ { { \sf s q } } ) \\ + k [ v ] + ( g + N [ \epsilon , \tilde v _ 1 , \tilde w _ 1 ] ) { \bf 1 } _ { | \xi | > 1 } & x \in \R ^ n , \ t \in ( 0 , T ) , \\ w = w _ 0 & x \in \R ^ n , \ t = 0 . \end{cases} \end{align*}


\begin{align*} \lim \limits _ { i \to \infty } \textstyle \frac { \log _ q n _ i } { \mu _ q ( n _ i ) } = 0 . \end{align*}


\begin{align*} \mathcal { W } _ { q } ( \mu , \nu ) & : = { \bigg ( \inf _ { \pi \in \Pi ( \mu , \nu ) } \int _ \mathcal { X } { d ( x , y ) } ^ q \mathrm { d } \pi ( x , y ) \bigg ) } ^ { 1 / q } \\ & = \inf \{ [ \mathbf { E } { d ( X , Y ) } ^ q ] ^ { 1 / q } ; \hbox { l a w } ( X ) = \mu , \hbox { l a w } ( Y ) = \nu \} , \end{align*}


\begin{align*} \mathcal { F } _ 1 ( v , x ) & = ( x - \beta ( v ) ) \displaystyle \prod _ { w \in c ( v ) } \mathcal { F } _ 1 ( w , x ) - \displaystyle \sum _ { w \in c ( v ) } \left ( \mathcal { F } _ 2 ( w , x ) \displaystyle \prod _ { t \in c ( v ) , t \neq w } \mathcal { F } _ 1 ( t , x ) \right ) , \\ \mathcal { F } _ 2 ( v , x ) & = \displaystyle \prod _ { w \in c ( v ) } \mathcal { F } _ 1 ( w , x ) , \end{align*}


\begin{align*} w ^ 3 + x ^ 3 + y ^ 3 + z ^ 3 = 0 , \end{align*}


\begin{align*} \frac { 1 } { 2 T } \mu ( \Lambda _ { S ^ * _ \Gamma \mathbb { T } ^ 2 \backslash S ^ * \Gamma } ) = 0 . \end{align*}


\begin{align*} \sigma _ 1 \leqslant a _ { m + 2 } = \left ( a _ { m + 1 } ^ 2 - \left | \operatorname { d e t } \left ( a _ { m + 1 } ^ 2 - A ^ { H } A \right ) \right | \left ( \frac { n - 1 } { ( n + 1 ) a _ { m + 1 } ^ 2 - \| A \| _ { F } ^ { 2 } } \right ) ^ { n - 1 } \right ) ^ { 1 / 2 } \leqslant a _ { m + 1 } \end{align*}


\begin{align*} & \zeta ( \Delta _ n , s ) = V _ \alpha ( s ) \Bigg ( a ( s ) \ , n ^ { \alpha - 2 s } + \sum _ { m = 0 } ^ { M - 1 } b _ m ( s ) \ , n ^ { - 2 m } \Bigg ) + O ( n ^ { - 2 M - 2 s + 2 } ) , \\ & \zeta ( \widetilde { \Delta } _ n , s ) = V _ \alpha ( s ) \Bigg ( \widetilde { a } ( s ) \ , n ^ { \alpha - 2 s } + \sum _ { m = 0 } ^ { M - 1 } \widetilde { b } _ m ( s ) \ , n ^ { - 2 m } \Bigg ) + O ( n ^ { - 2 M - 2 s + 2 } ) . \end{align*}


\begin{align*} \abs * { \sum _ { t = 1 } ^ { T } ( f _ t ( M ) - F ( M ) ) } \le C \sqrt { 2 T H d \log \frac { 3 T ^ 2 } { \delta } } . \end{align*}


\begin{align*} d t _ 1 \left ( C _ 1 ( [ x , y ] _ \mathfrak g , z ) - [ C _ 1 ( x , y ) , \varrho ( z ) ^ 2 ] + \circlearrowleft ( x , y , z ) \right ) = 0 . \end{align*}


\begin{align*} \nabla \Psi _ { t , \eta _ i } ^ { q ( t ) } \cdot \nabla v _ k - \frac { \mathrm { t r } _ g ( \nabla \Psi _ { t , \eta _ i } ^ { q ( t ) } \cdot \nabla v _ k ) } { n } g + \nabla v _ k \cdot \nabla \Psi _ { t , \eta _ i } ^ { q ( t ) } - \frac { \mathrm { t r } _ g ( \nabla v _ k \cdot \nabla \Psi _ { t , \eta _ i } ^ { q ( t ) } ) } { n } g + \nabla v _ k \cdot \nabla v _ k - \frac { \mathrm { t r } _ g ( \nabla v _ k \cdot \nabla v _ k ) } { n } g = h . \end{align*}


\begin{align*} \bigcup _ { i = 1 } ^ N B ( x _ i , \frac { 1 } { 2 } \sin \theta ) \subseteq B ( 0 , 1 + \frac { 1 } { 2 } \sin \theta ) . \end{align*}


\begin{align*} \mu \int _ { B _ s } | \nabla u | ^ 2 \ , d x & \le \frac { 1 } { \mu } \L { 2 } { \nabla u } { B _ t \setminus B _ s } \L { 2 } { \nabla \eta } { B _ 1 } + \Big | \int _ { B _ t } f ( x ) ( \eta u ) \ , d x \Big | \\ & \le \frac { C ( \mu ) } { 2 \tau ^ { 2 } } \L { 2 } { \nabla u } { B _ t \setminus B _ s } ^ { 2 } + 2 \tau ^ { 2 } \L { 2 } { \nabla \eta } { 2 } ^ 2 + C ( N ) . \end{align*}


\begin{align*} \frac { d ^ 2 y } { d x ^ 2 } + p ( x ) \frac { d y } { d x } + r ( x ) y ( x ) = 0 , \end{align*}


\begin{align*} \int _ \Omega f = \int _ { \Omega ^ * } f ^ * . \end{align*}


\begin{align*} \hat { \tau } ^ * = - \frac { \xi _ x \nu _ x + \xi _ y \nu _ y } { \nu _ x ^ 2 + \nu _ y ^ 2 + \varepsilon } , \end{align*}


\begin{align*} \label [ I e q ] { e q : m e a n f i e l d j u m p s d e } & X ^ i _ t = X ^ i _ 0 + \int _ 0 ^ t a ( X ^ i _ s ) \dd s \\ & + \int _ { 0 } ^ t \int _ { 0 } ^ { + \infty } \int _ \Theta { \left \{ \psi { \left ( X ^ i _ { s ^ - } , \mu _ { \mathcal { X } ^ N _ { s ^ - } } , \theta \right ) } - X ^ { i } _ { s ^ - } \right \} } \ 1 _ { \big ( 0 , \lambda { \big ( X ^ i _ { s ^ - } , \mu _ { \mathcal { X } ^ N _ { s ^ - } } \big ) } \big ] } ( u ) \ , \ , \mathcal { N } ^ i ( \dd s , \dd u , \dd \theta ) . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ n \partial _ i F _ i = \Omega ( \frac { \partial } { \partial y _ 1 } , \ldots , \frac { \partial } { \partial y _ n } ) . \end{align*}


\begin{align*} \sum _ { i \le k } c _ i d _ i \le \sum _ { i \le k - 1 } ( c _ i - c _ { i + 1 } ) D _ i \ + \ c _ k D _ k = \sum _ { i \le k } c _ i ( D _ i - D _ { i - 1 } ) . \end{align*}


\begin{align*} - \lim _ { y \rightarrow \infty } y \Im \varphi _ \mu ( i y ) = \int _ \mathbb { R } 1 + s ^ 2 \ , d \sigma _ \mu ( s ) = + \infty . \end{align*}


\begin{align*} P _ 1 = M _ f ( \Lambda _ 1 ) & \\ P _ 2 = M _ f ( \Lambda _ 2 ) & \\ P _ 3 = M _ f ( \Lambda _ 3 ) & , \end{align*}


\begin{align*} \tilde \varphi ( \cdot ) = \lim _ { i \to \omega } \varphi ( A _ i ^ { 1 / 2 } \cdot A _ i ^ { 1 / 2 } ) , \end{align*}


\begin{align*} C _ { n , s } = \left ( \int _ { \R ^ n } \frac { 1 - \cos ( x _ 1 ) } { | x | ^ { n + 2 s } } \ , d x \right ) ^ { - 1 } < \infty , \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } E [ \Lambda _ n ] & = \exp \left ( - \frac { \lambda ^ 2 } 2 \kappa _ 1 \right ) \lim _ { n \rightarrow \infty } E \left [ \exp \left ( i \mu Y ^ { n , 1 } _ t \right ) \right ] \\ & = E \left [ \exp \left ( i \lambda Z _ t \right ) \right ] E \left [ \exp \left ( i \mu Y ^ { \infty , 1 } _ t \right ) \right ] , \end{align*}


\begin{align*} U _ { z } = \int _ { \mathrm { T } _ { d } } x _ { 1 } ^ { z _ { 1 } } \cdots x _ { d } ^ { z _ { d } } \mathrm { d } \mathrm { P } \left ( x \right ) = \int _ { \Theta _ { d } } \mathrm { e } ^ { i \theta \cdot z } \mathrm { d } \left ( \mathrm { F } _ { \ast } \mathrm { P } \right ) \left ( \theta \right ) \ , z \in \mathbb { Z } ^ { d } \ , \end{align*}


\begin{align*} B [ \Phi , \Phi _ 1 ] : = \int _ 0 ^ r \left [ \phi ' \phi ' _ 1 - \frac { z } { 2 } \phi ' \phi _ 1 + \frac { z } { 2 } e _ \sigma \psi ' \phi _ 1 \right ] + \int _ 0 ^ r \left [ \psi ' \psi ' _ 1 - \frac { z } { 2 } \psi ' \psi _ 1 + \frac { z } { 2 } e _ \sigma \phi ' \psi _ 1 \right ] ~ \end{align*}


\begin{align*} f ( y , v _ 1 , \ldots , v _ k ) = \int _ 0 ^ 1 d f ( y , t v _ 1 , \ldots , t v _ k , 0 , v _ 1 , \ldots , v _ k ) \ , d t \ , . \end{align*}


\begin{align*} ( f _ 1 + f _ 2 ) ( x ) = f _ 1 ( x ) + f _ 2 ( x ) , ( \lambda f ) ( x ) = \lambda f ( x ) . \end{align*}


\begin{align*} \mbox { S p a n } \Bigl \{ \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} e ^ { 2 \pi i n \cdot x } , \frac { 1 } { \sqrt 2 } \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} e ^ { 2 \pi i n \cdot x } , \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} e ^ { 2 \pi i n \cdot x } \Bigr \} _ { | n | \leq k } \end{align*}


\begin{align*} D = a W + b L _ { x y } + b _ 0 R _ 0 + b _ 1 R _ 1 + \Delta , \end{align*}


\begin{align*} c ( t ) e ^ { \frac { k \alpha ( t ) } { 2 } } = e ^ { - \alpha ( t ) } l \Longrightarrow c ( t ) e ^ { \frac { \alpha ( t ) ( k + 2 ) } { 2 } } = l , \end{align*}


\begin{align*} \small \frac { 1 - 1 4 \delta } { 2 + 2 \gamma } = \frac { 1 } { 4 + 4 \gamma } , \varsigma \in \left ( 0 , \frac { 1 - 1 4 \delta } { 2 + 2 \gamma } \right ) , \quad \varsigma + \varsigma \gamma + 7 \delta - \frac 1 2 = \frac { 1 } { 8 } + \frac { 1 } { 4 } - \frac 1 2 = - \frac { 1 } { 8 } , \end{align*}


\begin{align*} \mathsf { D } ( r ) : = \{ a D ^ \times \mid a \in \mathcal { A } ( D ) a \mid _ D r \} . \end{align*}


\begin{align*} I _ { \alpha } ( n ) = \frac { \sqrt { 2 \pi } u } { \sqrt { ( 1 + u ) n } } \left ( \frac { u } { e ^ { 1 / u } } \right ) ^ n \left ( 1 + \sum _ { r = 1 } ^ { R - 1 } \frac { a _ r ( u ) } { n ^ r } + O \left ( \frac { ( \log n ) ^ R } { n ^ R } \right ) \right ) \end{align*}


\begin{align*} j _ \delta ' ( u _ \delta ) h = ( p _ \delta , h ) + \alpha ( u , h ) + ( u - \overline { u } , h ) , \forall h \in U . \end{align*}


\begin{align*} \big | d e t \big ( \begin{bmatrix} \sin \theta \cos \phi - \sin \tilde { \theta } \cos \tilde { \phi } & \sin \theta \sin \phi - \sin \tilde { \theta } \sin \tilde { \phi } & \cos \theta - \cos \tilde { \theta } \\ - s \sin \theta \sin \phi & s \sin \theta \cos \phi & 0 \\ s \cos \theta \cos \phi & s \cos \theta \sin \phi & - s \sin \theta \\ \end{bmatrix} \big ) \big | \end{align*}


\begin{align*} G ( o , x ) & = \int _ { 0 } ^ { \varepsilon D ^ 2 } h _ t ( o , x ) d t + \int _ { \varepsilon D ^ 2 } ^ { b D ^ 2 } h _ t ( o , x ) d t + \int _ { b D ^ 2 } ^ { \infty } h _ t ( o , x ) d t . \end{align*}


\begin{align*} \sum \limits _ { 1 \leq n \leq N } f _ k ( n ) f _ k ( n + 1 ) = \sum _ { \substack { w _ 1 \leq N , w _ 2 \leq N + 1 \\ ( w _ 1 , w _ 2 ) = 1 } } \frac { g _ k ( w _ 1 ) g _ k ( w _ 2 ) } { w _ 1 w _ 2 } \Big ( \frac { N } { w _ 1 w _ 2 } + O ( 1 ) \Big ) , \end{align*}


\begin{align*} v : = - \overline { g } ( T , \mu ) . \end{align*}


\begin{align*} \vartheta ( x , y , z ) = [ Q , \Phi _ 2 ( x , y , z ) ] . \end{align*}


\begin{align*} \begin{array} { l l } \frac { 1 } { 2 } ( L _ Z g ) ( X , Y ) + \frac { 1 } { \lambda ^ 2 } R i c ^ N ( \tilde { X } , \tilde { Y } ) + \frac { \mu } { \lambda ^ 2 } h ( \tilde { X } , \tilde { Y } ) = 0 . \end{array} \end{align*}


\begin{align*} G / N = ( G / K ) / ( N / K ) = H / f ( [ G , G ] ) \end{align*}


\begin{align*} S _ n = \{ f \in K [ x ] \mid \nu _ Q ( f ) < \nu ( f ) \mbox { f o r e v e r y } Q \in \Psi _ n \} . \end{align*}


\begin{align*} D _ + = D _ { + , 1 } + \cdots + D _ { + , n } + E , \end{align*}


\begin{align*} X _ t ^ \varepsilon ( x _ 0 , v _ 0 ) = X _ { \varepsilon t } \left ( x _ 0 , v _ 0 / \varepsilon \right ) \mbox { a n d } V _ t ^ \varepsilon ( x _ 0 , v _ 0 ) = \varepsilon V _ { \varepsilon t } \left ( x _ 0 , v _ 0 / \varepsilon \right ) . \end{align*}


\begin{align*} f ^ { - 1 } \circ X \circ f = \lambda \frac { z - a } { 1 - \bar { a } z } . \end{align*}


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


\begin{align*} \mathcal R _ { p } ( \mathcal O ) < \infty \implies \begin{cases} 1 \leq p < Q & 1 < Q < \infty , \\ p = 1 & Q = 1 . \end{cases} \implies R _ { p } ( h , \mathcal O ) < \infty \end{align*}


\begin{align*} ( ( - \Delta ) ^ s + q ) v & = 0 \quad \quad \Omega , \\ v & = f \quad \quad \Omega _ e . \end{align*}


\begin{align*} \tau ^ \mathrm { a n } ( M ' / S , \mathbb { 1 } ) = \tau ^ \mathrm { t o p } ( M ' / S , \mathbb { 1 } ) \ ; . \end{align*}


\begin{align*} P ^ { n , 1 } _ \tau = n ^ { \alpha + \frac 1 2 } \int ^ { \tau } _ 0 ( t - s ) ^ { \alpha } ( \sigma ' \sigma ) ( X _ { \eta _ n ( s ) } ) \Xi ^ { n , 2 } _ s d s \end{align*}


\begin{align*} \begin{aligned} - \kappa \left ( 1 + \kappa \right ) h ^ 3 + \big [ 3 \kappa + \left ( 1 + \kappa \right ) ^ 2 \big ] h ^ 2 - 4 \left ( 1 + \kappa \right ) h + 4 = 0 \end{aligned} \end{align*}


\begin{align*} - \Delta v _ e ^ 1 = 0 , v _ e ^ 1 ( x , 0 ) = - v _ p ^ 0 ( x , 0 ) , v _ e ^ 1 ( x , \infty ) = 0 , u _ e ^ 1 = \int _ x ^ \infty v _ { e Y } ^ 1 ( \theta , Y ) { \rm d } \theta , \end{align*}


\begin{align*} w ( \eta _ 1 , . . . , \eta _ d ) \equiv w _ { \xi } ( \eta _ 1 , . . . , \eta _ d ) = \lim \limits _ { T \to \infty } \frac 1 T \ln \left \langle e ^ { \int \limits _ 0 ^ T ( \xi _ 1 \eta _ 1 + . . . + \xi _ d \eta _ d ) d t } \right \rangle \end{align*}


\begin{align*} \Lambda _ n : = \exp \left ( i \lambda N ^ { n , 1 } _ { \eta _ n ( t ) } + i \mu Y ^ { n , 1 } _ { t } \right ) . \end{align*}


\begin{align*} \langle \Lambda _ { \gamma } f , g \rangle & = B _ { \gamma } ( u _ f , g ) = B _ q ( \gamma ^ { 1 / 2 } u _ f , \gamma ^ { 1 / 2 } g ) = B _ q ( v _ f , g ) = \langle \Lambda _ { q } f , g \rangle . \end{align*}


\begin{align*} \lim _ { t \rightarrow \infty } \mathbb E \| u ( t , s , \zeta _ { s } ) - u ( t ) \| ^ { 2 } = 0 . \end{align*}


\begin{align*} \lesssim 2 ^ { 4 \epsilon M _ t } \| \mathfrak { m } ( \cdot , \zeta ) \| _ { \mathcal { S } ^ \infty } \big ( \min \{ 2 ^ { n + 4 j / 3 } , 2 ^ { k + n } \} \mathbf { 1 } _ { l > - j } + \min \{ 1 + 2 ^ { n + j } , 2 ^ { k / 2 + n } \} \mathbf { 1 } _ { l = - j } \big ) \end{align*}


\begin{align*} \Big \{ x \in \R ^ d \mid & \ x _ { i j } = 0 \ \forall i \notin M _ P j \in N _ i , \sum _ { i \in \bar M } x _ { i n _ i } = | \bar M | - 1 , \\ & \sum _ { i \in M _ P \cap M _ 0 } a _ { i 1 } x _ { i 1 } + \sum _ { j \in N _ { i ' } - n _ { i ' } } a _ { i ' j } x _ { i ' j } = b - \sum _ { i \in \bar M - i ' } a _ { i n _ i } \Big \} , \end{align*}


\begin{align*} \left | \frac { \d A _ { \alpha } } { \d X _ { \gamma } } \right | = 1 + \left ( \frac { \d M _ { \beta } } { \d X _ { \gamma } } \right ) \delta _ { \beta \gamma } \Delta t + O ( \Delta t ^ 2 ) \end{align*}


\begin{align*} & ( b _ i ^ 2 - b _ i ) = 0 k < i < \ell \\ & b _ i b _ j = 0 k < i < \ell , j > i . \end{align*}


\begin{align*} \sup _ { t > 0 } \frac { c \eta _ t } { m _ t } \leq { \bf 1 } _ { d = 2 } c ( m ^ 2 ) + c _ 0 { \bf 1 } _ { d = 3 } \lim _ { m \rightarrow \infty } c ( m ^ 2 ) = 0 , \end{align*}


\begin{align*} \tilde { \mathbf S } = \dfrac { 1 } { 2 } \begin{pmatrix} 7 & - 2 \\ 2 & - 7 \end{pmatrix} \ , \end{align*}


\begin{align*} F _ 0 ( z , s ) = y ^ s + \frac { \Lambda ( 2 s - 1 ) } { \Lambda ( 2 s ) } y ^ { 1 - s } + 2 \sqrt y \sum _ { n \neq 0 } \frac { | n | ^ { s - \frac 1 2 } \sigma _ { 1 - 2 s } ( | n | ) } { \Lambda ( 2 s ) } K _ { s - \frac 1 2 } ( 2 \pi | n | y ) e ( n x ) , \end{align*}


\begin{align*} H _ 0 = G _ 0 - \Gamma \in L ^ \infty ( \Omega ) . \end{align*}


\begin{align*} w _ t & = \Delta _ x w - q { \sf U } _ \infty ^ { q - 1 } w + ( F _ 1 [ \tilde v _ 1 , \tilde w _ 1 ] - F _ 2 [ \tilde w _ 1 ] ) ( 1 - \chi _ { { \sf s q } } ) \\ & + k [ v ] + ( g + N [ \epsilon , \tilde v _ 1 , \tilde w _ 1 ] ) { \bf 1 } _ { | \xi | > 1 } x \in \R ^ n , \ t \in ( 0 , T ) . \end{align*}


\begin{align*} \mathfrak { g } _ { A _ I , k } ^ { r , s } = \bigoplus _ { \substack { k _ 1 , k _ 2 \ge 0 \\ 0 < k _ 1 + k _ 2 \le k } } \mathfrak { g } ^ { r , s } _ { ( k _ 1 , k _ 2 ) } . \end{align*}


\begin{align*} S _ { j } ( z ) = S _ { 0 } ( z ) ^ { j + 1 } , 1 \leq j \leq p . \end{align*}


\begin{align*} u ( x , t ) \ge M _ n \left \{ \frac { ( t - x - R ) ^ { 1 - b } } { 1 + t - x } \right \} ^ { a _ n } \left \{ \frac { ( t - x - R ) ^ { 1 - a } } { 1 + t - x } \right \} ^ { b _ n } \quad \mbox { i n } \ D \end{align*}


\begin{align*} \mathfrak { D } : = U _ N ( \mathcal { N } _ 1 - \mathcal { N } _ 2 ) U _ N ^ * \qquad ( \mathfrak { D } \Phi ) _ { s , d } = d \Phi _ { s , d } . \end{align*}


\begin{align*} S ( r ) < \dfrac { e ^ { \frac { r - 1 } { 2 } } } { 2 ^ r } \sum _ { k = 2 r } ^ { \infty } \dfrac { 1 } { \pi ^ k } = \dfrac { \pi } { \sqrt { e } ( \pi - 1 ) } \Bigl ( \dfrac { \sqrt { e } } { 2 \ r ^ 2 } \Bigr ) ^ r < \dfrac { 1 } { 1 0 ^ r } . \end{align*}


\begin{align*} \begin{aligned} b _ h ( u ^ { \Delta t } _ { h } , u ^ { \Delta t } _ { h } , P _ h ( u ^ { \Delta t } _ { h } \phi ) ) & = b _ h ( u ^ { \Delta t } _ { h } , u ^ { \Delta t } _ { h } , u ^ { \Delta t } _ { h } \phi ) + R _ { n l } . \end{aligned} \end{align*}


\begin{align*} - \Delta _ p ( \mathit \Gamma + u ) + \Delta _ p \mathit \Gamma = f \Omega \setminus \{ 0 \} \end{align*}


\begin{align*} \overline { p } ( n ) = \frac { 1 } { 2 \pi } \underset { 2 \nmid k } { \sum _ { k = 1 } ^ { \infty } } \sqrt { k } \underset { ( h , k ) = 1 } { \sum _ { h = 0 } ^ { k - 1 } } \dfrac { \omega ( h , k ) ^ 2 } { \omega ( 2 h , k ) } e ^ { - \frac { 2 \pi i n h } { k } } \dfrac { d } { d n } \biggl ( \dfrac { \sinh \frac { \pi \sqrt { n } } { k } } { \sqrt { n } } \biggr ) , \end{align*}


\begin{align*} \nabla ^ s u ( x , y ) = \sqrt { \frac { C _ { n , s } } { 2 } } \frac { u ( x ) - u ( y ) } { | x - y | ^ { n / 2 + s + 1 } } ( x - y ) , \end{align*}


\begin{align*} \int _ { \mathbb { D } \setminus \mathcal { H } } \frac { 1 } { \abs { f ' ( z ) } ^ q } d x d y = \sum _ { \gamma \in \Gamma } \int _ { \gamma \Phi ^ * } \frac { 1 } { \abs { f ' ( z ) } ^ q } d x d y \approx _ { \Gamma , q } \sum _ { \gamma \in \Gamma } \frac { \abs { \gamma ' ( 0 ) } ^ { q + 2 } } { \abs { \rho ( \gamma ) ' ( 0 ) } ^ q } . \end{align*}


\begin{align*} O _ j ^ h \subseteq ( H \cap P _ i ^ { g _ j } ) ^ h = H ^ h \cap P _ i ^ { p g _ k } = H \cap P _ i ^ { g _ k } \end{align*}


\begin{align*} C _ 3 ^ { x y z } = C _ 3 ^ { y z x } = C _ 3 ^ { z x y } \ . \end{align*}


\begin{align*} ( A _ i ) _ { x y } = \begin{cases} 1 , \\ 0 . \end{cases} \end{align*}


\begin{align*} \langle \Theta \nabla ^ s u , \nabla ^ s \phi \rangle _ { L ^ 2 ( \R ^ { 2 n } ) } = \langle ( - \Delta ) ^ { s / 2 } ( \gamma ^ { 1 / 2 } u ) , ( - \Delta ) ^ { s / 2 } ( \gamma ^ { 1 / 2 } \phi ) \rangle _ { L ^ 2 ( \R ^ n ) } + \langle q \gamma ^ { 1 / 2 } u , \gamma ^ { 1 / 2 } \phi \rangle _ { L ^ 2 ( \R ^ n ) } \end{align*}


\begin{align*} \sum _ { k = 1 } ^ \infty \frac { 1 } { 1 \vee \log \max \{ \phi ^ { - 1 } ( k ) , k ^ 2 \} } = \infty \sum _ { k = 1 } ^ \infty \frac { 2 ^ k } { 1 \vee \max \{ \log \phi ^ { - 1 } ( 2 ^ k ) , k \log 4 \} } = \infty . \end{align*}


\begin{align*} \prod _ { i = 1 } ^ d s _ i ^ { t _ i } \ , ( t _ i ! ) . \end{align*}


\begin{align*} \widetilde { B _ 0 } S _ n ( t ) \in \mathcal { C } _ { r _ n } r _ n = \frac { p q } { n p + q } , n = 0 , 1 , \ldots , l . \end{align*}


\begin{align*} \frac k { \gcd ( k , j _ 1 , j _ 2 , \dots , j _ m ) } \binom { k } { j _ 1 , j _ 2 , \dots , j _ m } . \end{align*}


\begin{align*} f _ 0 ( z ) = z \exp { \int _ { 0 } ^ { z } \frac { \psi ( t ) - 1 } { t } d t } . \end{align*}


\begin{align*} ( F ( L , \mathbf U ) f ) ^ \mu ( x ) = F ( L ^ \mu , \mu ) f ^ \mu ( x ) \end{align*}


\begin{align*} \Delta _ { a n g } = \frac { 1 } { 2 \sinh ^ 2 ( \beta r ) } \frac { \partial ^ 2 } { \partial \theta ^ 2 } \end{align*}


\begin{align*} Q = Q ^ T > 0 , \ \ \ J = - J ^ T , \ \ \ R = R ^ T \geq 0 . \end{align*}


\begin{align*} \frac { 1 } { 2 } \frac { d } { d t } \left ( \int _ 0 ^ L ( \rho \abs { u _ t } ^ 2 + \alpha _ 1 \abs { u _ x } ^ 2 + \mu \abs { y _ t } ^ 2 + \beta \abs { \gamma u _ x - y _ x } ^ 2 ) d x \right ) + \delta \int _ 0 ^ L \omega _ x \overline { u _ t } d x = 0 . \end{align*}


\begin{align*} \dim H ^ 1 ( G , V ) = \dim H ^ 1 ( H , V ) . \end{align*}


\begin{align*} A \phi ( x ) = a \cdot \nabla \phi ( x ) + \frac { 1 } { 2 } \nabla \cdot ( Q \nabla \phi ) ( x ) + \int _ { \R ^ d } \left ( \phi ( x + y ) - \phi ( x ) - \nabla \phi ( x ) y 1 _ { \{ | y | < 1 \} } \right ) \nu ( d y ) \end{align*}


\begin{align*} \frac { \sin ( k a ) } { k a } = \mp \frac { \sinh ( k ' a ) } { k ' a } , \end{align*}


\begin{align*} \langle \mathcal { H } _ { q } ^ { n } \ , e _ { r } , e _ { 0 } \rangle = \langle e _ { r } , e _ { 0 } \rangle = \delta _ { r , 0 } = \sum _ { \gamma \in \mathcal { S } _ { 1 } ( 0 , r ) } w ( \gamma ) . \end{align*}


\begin{align*} \mathcal { G } ( O _ { n + 1 } ) & = \mathcal { G } ( h _ 1 ) \oplus \dots \oplus \mathcal { G } ( h _ i ) \oplus \dots \oplus \mathcal { G } ( h _ p ) & \\ & = \mathcal { G } ( h _ 1 ) \oplus \dots \oplus [ \mathcal { G } ( h _ i - 1 ) \oplus 1 ] \oplus \dots \oplus \mathcal { G } ( h _ p ) & \eqref { o b 1 } \\ & = \mathcal { G } ( O ' _ n ) \oplus 1 & \end{align*}


\begin{align*} h ( F _ \ast X , F _ \ast Y ) = \lambda ^ 2 ( p ) g ( X , Y ) , ~ \forall X , Y \in \Gamma ( K e r F _ \ast ) ^ \bot ~ ~ p \in M . \end{align*}


\begin{align*} \binom { \sum _ { i = 1 } ^ { n } c _ i N _ i } { 2 } & = \sum _ { i = 1 } ^ { n } \binom { c _ i N _ i } { 2 } + \sum _ { 1 \leq l < k \leq n } c _ l c _ k N _ l N _ k \\ & = \sum _ { i = 1 } ^ { n } \left ( \sum _ { k _ 1 + \cdots + k _ { N _ i } = 2 } \binom { c _ i } { k _ 1 } \cdots \binom { c _ i } { k _ { N _ i } } \right ) + \sum _ { 1 \leq l < k \leq n } c _ l c _ k N _ l N _ k \\ & = \sum _ { i = 1 } ^ { n } N _ i \binom { c _ i } { 2 } + \sum _ { i = 1 } ^ { n } c _ i ^ 2 \binom { N _ i } { 2 } + \sum _ { 1 \leq l < k \leq n } c _ l c _ k N _ l N _ k . \end{align*}


\begin{align*} 0 = V _ 0 \subset \cdots \subset V _ r = V \end{align*}


\begin{align*} \log \delta _ n = - \frac { Q ^ { - 1 } ( \epsilon _ n ) ^ 2 } { 2 } - \frac { 1 } { 2 } \log n . \end{align*}


\begin{align*} \Lambda _ { i } ^ { ( s ) t } A _ i ^ { ( s ) } \Lambda _ { i } ^ { ( s ) } = B ^ { ( s ) } . \end{align*}


\begin{align*} \Gamma [ z ] & = \chi ( t / \tilde { T } ) W _ t T [ u _ 0 ] \pm i \chi ( t / \tilde { T } ) \int _ 0 ^ t W _ { t - s } \left ( \mathfrak { F } ( v ) + \mathcal { H L } _ B [ z , v , \cdots , v ] \right ) \ , d s . \end{align*}


\begin{align*} M _ { n , k } ( m ) = \dbinom { m + n } { n - ( m - k ) + 1 } - \dbinom { m + n } { n - ( m - k ) - 1 } . \end{align*}


\begin{align*} D _ \pm = \sum _ { i \in ( M _ + \cup M _ - ) \setminus S _ \pm } \bar { D } _ i . \end{align*}


\begin{gather*} { \varrho _ m ( \gamma , \upsilon ) } \hat h | _ { k - \frac 1 2 } ( \gamma , \upsilon ) = \hat h \end{gather*}


\begin{align*} f ( x , u , \xi ) = \bigl ( 1 + u ^ { p ( \frac { 1 } { p ^ \star } - 1 ) } \bigr ) \abs { \xi } ^ { p ( x ) } , \end{align*}


\begin{align*} Q ' : = \dd ^ { } + Q + \sum _ { k \geq 1 , i _ 1 , \ldots , i _ k = 1 , \ldots , \mathrm { d i m } ( \mathfrak { g } ) } \frac { 1 } { k ! } \xi ^ { i _ 1 } \odot \cdots \odot \xi ^ { i _ k } \Phi _ { k - 1 } ( \xi _ { i _ 1 } , \ldots , \xi _ { i _ { k } } ) , \end{align*}


\begin{align*} \chi _ 4 ( \hat { \beta } ) = N - k \end{align*}


\begin{align*} | I | - 3 = { \sum _ { i \in I } a _ i - ( r - 2 ) \over r } + { \sum _ { i \in I } b _ i - ( s - 2 ) \over s } + \sum _ { i \in I } d _ i , \end{align*}


\begin{align*} Q _ \mathfrak { g } ( x _ 1 \wedge \cdots \wedge x _ k ) : = \sum _ { 1 \leq i < j \leq k } ( - 1 ) ^ { i + j - 1 } [ x _ i , x _ j ] _ \mathfrak { g } \wedge x _ 1 \wedge \cdots \widehat { x } _ i \cdots \widehat { x } _ j \cdots \wedge x _ k , \end{align*}


\begin{align*} s _ { i _ { k } } \cdots s _ { i _ \ell } \lambda = \lambda - \alpha _ { i _ \ell } - \alpha _ { i _ { \ell - 1 } } - \cdots - \alpha _ { i _ k } \end{align*}


\begin{align*} \langle \phi , \mu \rangle : = \int _ { \R ^ d } \phi ( x ) \mu ( d x ) = : \mu ( \phi ) . \end{align*}


\begin{align*} & \limsup \limits _ { \delta \rightarrow 0 } \limsup _ { n \rightarrow \infty } \limsup _ { t \rightarrow \infty } \| P _ \delta ( W _ { n , m ; \textrm { f a r } } ) e ^ { - i t H _ 0 } \varphi \| \\ & \leq \limsup \limits _ { \delta \rightarrow 0 } \limsup _ { n \rightarrow \infty } \limsup _ { t \rightarrow \infty } \| P _ \delta ( W _ { n , m ; \textrm { f a r } } ) e ^ { - i t H } \psi \| = 0 \end{align*}


\begin{align*} | Z _ 2 | = 0 ~ ~ E _ { u _ n } ( x , y ) \longrightarrow E _ u ( x , y ) ~ ~ ( x , y ) \in \R ^ { 2 N } \setminus Z _ 2 . \end{align*}


\begin{align*} 2 A _ X Y = \nu [ X , Y ] - \lambda ^ 2 g ( X , Y ) \left ( \nabla _ \nu \frac { 1 } { \lambda ^ 2 } \right ) , \end{align*}


\begin{align*} R \rightarrow W ( R ) = R ^ { \mathbb N } \end{align*}


\begin{align*} \int _ 0 ^ \infty \ 1 _ { \{ \overline f > \tau \} } ( r ) g ( r ) \ , d r & = \int _ 0 ^ { b _ \tau } g ( r ) \ , d r \leq b _ \tau ^ \beta \sup _ { s > 0 } s ^ { - \beta } \int _ 0 ^ s g ( r ) \ , d r \\ & = \beta \int _ 0 ^ \infty \ 1 _ { \{ \overline f > \tau \} } ( r ) \ , \frac { d r } { r ^ { 1 - \beta } } \ \underline \nu _ \beta ( g ) \ , . \end{align*}


\begin{align*} u ( x , t ) \ge M _ n \left \{ \log \left ( \frac { 1 + t - x } { 1 + R } \right ) \right \} ^ { a _ n } \quad \mbox { i n } \ D \end{align*}


\begin{align*} A ( x , v ) = a x - b v , \mbox { a n d } B ( x , v ) = x + v ( v - 1 ) ( v - c ) , \end{align*}


\begin{align*} & A _ { \tilde { i } , \tilde { j } | a } : = \Big \{ \overline { a } \in A _ { \tilde { i } , \tilde { j } } : \forall p = 1 , . . . , i + j , \overline { a } _ p = a _ p \Big \} . \end{align*}


\begin{align*} \det ( C ( \alpha , \beta ) ) = \sum _ { \gamma } \det ( A ( \alpha , \gamma ) ) \det ( B ( \gamma , \beta ) ) , \end{align*}


\begin{align*} H ' = - \frac { \lambda ^ 2 } { 2 } \left ( \nabla _ \nu \frac { 1 } { \lambda ^ 2 } \right ) . \end{align*}


\begin{align*} A _ { n } ^ { ( 1 ) } = \sum _ { \gamma \in \mathcal { M } _ { n } ^ { ( 1 ) } } w ( \gamma ) , \end{align*}


\begin{align*} L \tilde u = \left \{ \begin{array} { c c } - \div \kappa \nabla \tilde u & \Omega ^ - \\ - \Delta \tilde u & \Omega ^ + \end{array} \right . \end{align*}


\begin{align*} \mathcal { W } \ , \hat { e } _ { n } = \hat { e } _ { n - 1 } + \sum _ { k = 0 } ^ { p } a _ { n } ^ { ( k ) } \hat { e } _ { n + k } , n \in \mathbb { Z } , \end{align*}


\begin{align*} f = a _ { \rho 0 } ( f ) + a _ { \rho 1 } ( f ) Q _ \rho + \ldots + a _ { \rho r } ( f ) Q _ \rho ^ r \end{align*}


\begin{align*} \lim _ { N \to \infty } \prod \limits _ { k = 1 } ^ { N - 1 } \Bigg ( \frac { \lambda _ N ^ 2 + \mu _ { k } ^ { 2 } } { \lambda _ N ^ 2 + \lambda _ { k + 1 } ^ { 2 } } \Bigg ) = \infty , \end{align*}


\begin{align*} \lim _ { \delta \rightarrow 0 } \int _ 0 ^ { x _ 0 - \delta } a u '' v '' d x = \int _ 0 ^ { x _ 0 } a u '' v '' d x , \lim _ { \delta \rightarrow 0 } \int _ { x _ 0 + \delta } ^ 1 a u '' v '' d x = \int _ { x _ 0 } ^ 1 a u '' v '' d x \end{align*}


\begin{align*} \int _ \Sigma u ^ * \omega = \int _ { \partial \Sigma } \lambda = \int _ { - \infty } ^ \infty d f _ L - \int _ { - \infty } ^ \infty d f _ { L ' } = ( f _ L - f _ { L ' } ) ( q ) - ( f _ L - f _ { L ' } ) ( p ) . \end{align*}


\begin{align*} \int _ 0 ^ L \abs { \omega } ^ 2 d x = o ( 1 ) \int _ 0 ^ L \abs { \Lambda _ x } ^ 2 d x = o ( 1 ) . \end{align*}


\begin{align*} h _ { m n } : = \ ; & \big \langle u _ m , ( - \Delta + V _ \mathrm { D W } ) u _ n \big \rangle \\ w _ { m n p q } : = \ ; & \big \langle u _ m \otimes u _ n , w , u _ p \otimes u _ q \big \rangle . \end{align*}


\begin{align*} \mathcal S _ { \kappa , \gamma } = \left \{ \left [ n _ 0 , n _ 1 , \ldots , n _ { 2 ^ { \gamma } - 1 } \right ] \colon n _ i \geq 0 , \ ; \sum _ { i = 0 } ^ { 2 ^ \gamma - 1 } n _ i = \kappa \right \} . \end{align*}


\begin{align*} \mathbb T ^ { m | n } _ \zeta : = \mathbb T ^ { m | n } S _ \zeta . \end{align*}


\begin{align*} \chi ^ \Lambda ( g , \nu ) = \sum _ { x \in \Lambda } \avg { \varphi _ 0 \varphi _ x } _ { g , \nu } . \end{align*}


\begin{align*} \phi ( x ) = A e ^ { i k x } + B e ^ { - i k x } + C e ^ { k ' x } + D e ^ { - k ' x } . \end{align*}


\begin{align*} W _ n = \bigcup _ { w \in \Sigma _ { \beta } ^ n } \bigcup _ { 0 \leq k \leq \left \lfloor \frac { \beta ^ n } { \varphi ( n ) } \right \rfloor } \left \{ ( x , y ) \in I _ { n , \beta } ( w ) \times J _ { n , \beta } ( k ) : | T _ { \beta } ^ n x - f ( x , y ) | < \varphi ( n ) \right \} . \end{align*}


\begin{align*} d ^ j f ( x , y _ 1 , \ldots , y _ j ) ( v ) = d ^ j ( f ^ \wedge ) ( ( x , v ) , ( y _ 1 , 0 ) , \ldots , ( y _ j , 0 ) ) \end{align*}


\begin{align*} \psi ( x ) = x ( 1 + { ( b + 1 ) C \over b - a } x ^ a y ^ b + \cdots ) . \end{align*}


\begin{align*} I = \bigcap _ { 1 \leq j _ 1 < \cdots < j _ { s - d + 1 } \leq s } ( t _ { j _ 1 } , \ldots , t _ { j _ { s - d + 1 } } ) . \end{align*}


\begin{align*} \Phi _ 4 ( \xi , \eta , \sigma , \kappa , \zeta ) : = \hat { \zeta } \cdot ( \xi + \eta + \sigma + \kappa ) + \mu _ 3 | \kappa | + \mu _ 2 | \sigma | + \mu | \xi | + \mu _ 1 | \eta | . \end{align*}


\begin{align*} K e r ( M ) \cap K e r ( \tau ) = K e r ( \tau ^ M ) . \end{align*}


\begin{align*} H _ { \lambda } ( q ) : = \prod _ { x \in \lambda } ( 1 - q ^ { h ( x ) } ) , \end{align*}


\begin{align*} \Phi = \sum _ { k \geq 0 } \Phi ^ { ( k ) } \end{align*}


\begin{align*} \frac { 1 } { 2 } \frac { d } { d t } \left ( \rho \int _ 0 ^ L \abs { u _ t } ^ 2 d x + \alpha \int _ 0 ^ L \abs { u _ x } ^ 2 d x \right ) - \gamma \beta \Re \left ( \int _ 0 ^ L y _ x \overline { u _ { x t } } \right ) + \delta \Re \left ( \int _ 0 ^ L \omega _ x \overline { u _ t } d x \right ) = 0 \end{align*}


\begin{align*} h _ q ( \delta ) = \delta \log _ q ( q - 1 ) - \delta \log _ q \delta - ( 1 - \delta ) \log _ q ( 1 - \delta ) , \delta \in [ 0 , 1 - q ^ { - 1 } ] , \end{align*}


\begin{align*} \mathcal { C } \colon = C ^ * . \end{align*}


\begin{align*} \langle e _ 1 \rangle \subset \langle e _ 1 , e _ 2 \rangle \subset \cdots \subset \langle e _ 1 , e _ 2 , \ldots , e _ n \rangle = \C ^ n \end{align*}


\begin{align*} N _ n ( t , u ) = t ^ n N _ n ( 1 / t , u / t ) . \end{align*}


\begin{align*} \liminf \limits _ { t \rightarrow \infty } \| \chi _ { S _ { v t } } e ^ { - i t H } \psi \| = 0 \end{align*}


\begin{align*} \mathrm { L i } _ { k , \lambda } ( t ) = \sum _ { n = 1 } ^ { \infty } \frac { ( - \lambda ) ^ { n - 1 } ( 1 ) _ { n , 1 / \lambda } } { ( n - 1 ) ! n ^ { k } } t ^ { n } , ( \mathrm { s e e } \ [ 8 ] ) . \end{align*}


\begin{align*} \liminf _ { z \rightarrow \alpha } \int _ \mathbb { R } \frac { d \nu ( s ) } { | z - s | ^ 2 } = \int _ \mathbb { R } \frac { d \nu ( s ) } { ( \alpha - s ) ^ 2 } = \lim _ { z \rightarrow _ { \sphericalangle } \alpha } \int _ \mathbb { R } \frac { d \nu ( s ) } { | z - s | ^ 2 } , \end{align*}


\begin{align*} \begin{cases} \dot { x } = - y + d x + l \ , x ^ 2 + m \ , x \ , y + n \ , y ^ 2 , \\ \dot { y } = x + b x \ , y , \end{cases} \end{align*}


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


\begin{align*} g ( x ) = \int _ { \mathbb { R } } \frac { 1 + s ^ { 2 } } { ( x - s ) ^ { 2 } } \ , d \rho ( s ) , x \in \mathbb { R } . \end{align*}


\begin{align*} \langle \alpha _ { i _ k } ^ \vee , s _ { i _ { k + 1 } } \cdots s _ { i _ { \ell - 1 } } s _ { i _ \ell } \lambda \rangle = 1 \end{align*}


\begin{align*} \kappa _ 3 = \sum _ { k = 1 } ^ \infty \int _ { 0 } ^ 1 \left [ ( x + k ) ^ { \alpha } - k ^ { \alpha } \right ] ^ 2 d x , \end{align*}


\begin{align*} \psi _ n ( x ) = \lambda _ n ^ { \frac 1 { p - 2 } } e ^ { - \gamma \ , \lambda _ n ^ { \frac { 1 } { 2 } } | x - { \rm v } _ 1 | } . \end{align*}


\begin{align*} \label [ I e q ] { e q : L i n f t y T i n f t y } \frac { \dd } { \dd t } T _ { \infty , t } \Phi = \mathcal { L } _ \infty [ T _ { \infty , t } \Phi ] = T _ { \infty , t } \mathcal { L } _ \infty \Phi . \end{align*}


\begin{align*} \frac { 1 } { \sigma _ { k - 1 } } \sum ^ { m } _ { j = 1 } \lambda _ { j } ( \left [ - X ( x _ k ) \right ] _ { + } ) ^ 2 = \frac { 1 } { \sigma _ { k - 1 } } \norm { \left [ - X ( x _ k ) \right ] _ { + } } _ { \mathrm { F } } ^ 2 \to 0 \end{align*}


\begin{align*} \mathbb { P } _ x \bigl ( X _ n = o d ( o , X _ m ) \leq r \bigr ) & \leq \mathbb { P } _ x \bigl ( \tau _ \dagger > n d ( o , X _ m ) \leq r \bigr ) \\ & \leq \left ( 1 - \frac { c _ 1 ( \log r ) ^ \gamma } { r ^ 2 } \right ) ^ n \leq \exp \left [ - \frac { c _ 1 ( \log r ) ^ \gamma n } { r ^ 2 } \right ] , \end{align*}


\begin{align*} k ( x ) = \left \{ \begin{array} { l c } 1 + \frac { x } { c } , & \\ \\ x , & \end{array} \right . \end{align*}


\begin{align*} c _ j = \frac { f ^ { ( j ) } ( \lambda ) } { j ! } \end{align*}


\begin{align*} S ( h + ( t _ 1 \ldots , t _ n ) ) = \left \langle ( ( h _ 1 + t _ 1 ) [ D _ 1 ] + \ldots + ( h _ m + t _ m ) [ D _ m ] ) ^ m , [ X _ { \Sigma , \Lambda } ] \right \rangle = t _ { i _ 1 } \ldots t _ { i _ n } \cdot n ! \cdot \left \langle D _ { i _ 1 } \ldots D _ { i _ n } , [ X _ { \Sigma , \Lambda } ] \right \rangle + \ldots \end{align*}


\begin{align*} \begin{bmatrix} P A + A ^ { T } P & P B - A ^ { T } C ^ { T } \\ B ^ { T } P - C A & - ( C B + B ^ { T } C ^ { T } ) \end{bmatrix} \leq 0 . \end{align*}


\begin{align*} \int _ { b _ 1 } ^ { b _ 2 } \left | v _ { x } \right | ^ 2 d x = o \left ( \lambda ^ { - \frac { 1 } { 2 } } \right ) \int _ { d _ 1 } ^ { d _ 2 } \left | z _ { x } \right | ^ 2 d x = o \left ( \lambda ^ { - \frac { 1 } { 2 } } \right ) . \end{align*}


\begin{align*} ( C _ { R } ( Q ) ) ^ h = C _ { R [ t ] } ( Q ^ h ) = \langle x _ 1 , \ldots , x _ c \rangle . \end{align*}


\begin{align*} \Phi _ 0 ( x ) [ f ] = \varrho ( x ) [ f ] , \ ; \ , \ ; [ Q , \Phi _ 0 ( x ) ] = 0 , \forall x \in \mathfrak { g } [ 1 ] , f \in \mathcal { O } . \end{align*}


\begin{align*} \Theta : \ell ^ 2 ( \mathcal { F } _ \perp ) \to \ell ^ 2 ( \mathcal { F } _ \perp ) \qquad ( \Theta \Phi ) _ { s , d } = \Phi _ { s , d - 1 } . \end{align*}


\begin{align*} \left ( 1 - q \right ) ^ { - f ( u , v ) } = \prod _ { i , j } \left ( 1 - u ^ i v ^ j q \right ) ^ { - p _ { i j } } \end{align*}


\begin{align*} S _ 0 H _ \sigma = q ^ { - \ell ( \sigma ) } S _ 0 = H _ \sigma S _ 0 . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } E \left ( \exp \left ( i \lambda N ^ { n , 1 } _ { \eta _ n ( t ) } + i \mu Y ^ { n , 1 } _ { t } \right ) \right ) = E \left ( \exp \left ( i \lambda Z _ { t } \right ) \right ) E \left ( \exp \left ( i \mu Y ^ { \infty , 1 } _ { t } \right ) \right ) , \end{align*}


\begin{align*} \beta _ 0 \cdot D w ( x _ 0 ) = e ^ { - \kappa \phi ( x _ 0 ) } [ \beta _ 0 \cdot D ( u - u _ 0 ) ( x _ 0 ) - \kappa \beta _ 0 \cdot \gamma ( x _ 0 ) ( u - u _ 0 ) ( x _ 0 ) ] \ge 0 , \end{align*}


\begin{align*} H ^ * ( X _ { D } ) = \mathbb { Z } [ x _ { a _ 1 b _ 1 } , x _ { a _ 2 b _ 2 } , \dots , x _ { a _ k b _ k } ] / \langle p _ 1 , \dots , p _ k \rangle , \end{align*}


\begin{align*} T = \inf \Big \{ t \in \Phi : \widehat { \mathsf { F D P } } ( t ) \leq q \Big \} , \end{align*}


\begin{align*} \hat { \pi } ^ i _ t ( m ^ i ) \triangleq \P ^ { g } ( M ^ i = m ^ i | x ^ i _ { 1 : t } , y _ { 1 : t } ) , i = 1 , 2 . \end{align*}


\begin{align*} \frac { \P ( S _ { N + 1 } \in \cdot \setminus \{ 0 , 1 , \ldots , N \} \ | \ S _ N ) } { \P ( S _ { N + 1 } \notin \{ 0 , 1 , \ldots , N \} \ | \ S _ N ) } = \delta _ { N + 1 } ( \cdot ) \end{align*}


\begin{align*} P ( u , v ) = \frac { c ( u , v ) } { c ( u ) + K ( u ) } P ( u , \dagger ) = \frac { K ( u ) } { c ( u ) + K ( u ) } , P ( \dagger , \dagger ) = 1 , \end{align*}


\begin{align*} \omega f ( x ) = f ( \omega ^ { - 1 } x ) , \omega \in W . \end{align*}


\begin{align*} \tilde { D } _ i = ( D _ i , 0 ) , i \in \{ 1 , \ldots , m \} ; \end{align*}


\begin{align*} s ( g ) = \min \{ j \in \N \ : \ I ( g ) _ j \ne 0 \} \leq d - 1 . \end{align*}


\begin{align*} \boldsymbol { x } _ { A } ^ { p o s t } = \boldsymbol { V } _ { A } ^ { p o s t } \left ( ( \boldsymbol { V } _ { A } ^ { p r i } ) ^ { - 1 } \boldsymbol { x } _ { A } ^ { p r i } + \frac { \mathbf { F } ^ { H } \boldsymbol { y } } { \sigma _ { n } ^ { 2 } } \right ) \end{align*}


\begin{align*} \frac { \partial C } { \partial z ^ i _ { \epsilon } } v ^ i = 0 , \end{align*}


\begin{align*} w _ { \xi } ( \eta _ { 1 } , . . . , \eta _ { d } ) = \left ( { b + c } \right ) \mathop \sum \limits _ { k = 1 } ^ d \left ( { \frac { { d + 1 } } { 2 } - k } \right ) { \eta _ k } + \frac { 1 } { 2 } \mathop \sum \limits _ { k , p = 1 } ^ d \left ( { \left ( { b + c } \right ) { \delta _ { k p } } - a } \right ) { \eta _ k } { \eta _ p } \ , \end{align*}


\begin{align*} \bar { \tau } ^ { M , 0 } = \tau ^ { M , 0 } \otimes I _ 3 \in \R ^ { 3 n \times 3 n } . \end{align*}


\begin{align*} \mathcal { X } _ R ( x ) : = \mathcal { X } \left ( \frac { x } { R } \right ) \qquad R > 0 , \end{align*}


\begin{align*} B _ q ( v , \phi ) = F ( \phi ) \quad v - f \in \Tilde { H } ^ s ( \Omega ) \end{align*}


\begin{align*} \mathcal { P } _ { 1 } : & \underset { \left \{ \mathbf { V } _ { 2 , p } \right \} _ { p = 1 } ^ { P _ { 2 } } } { \max } \lambda \\ & \textrm { t r } \left ( \mathbf { V } _ { 2 , p } \right ) \leq P _ { t } , \\ & \mathbf { J } _ { \textrm { e f f } } \left ( \left \{ \mathbf { V } _ { 2 , p } \right \} _ { p = 1 } ^ { P _ { 2 } } \right ) \succeq \lambda \mathbf { I } , \\ & \textrm { r a n k } \left ( \mathbf { V } _ { 2 , p } \right ) = 1 , p = 1 , 2 \ldots , P _ { 2 } . \end{align*}


\begin{align*} A _ { \bar { S } } ( k ) e = e - A _ { \bar { S } S } ( k ) e \end{align*}


\begin{align*} K _ g ^ 2 : = | E \cdot \omega | ^ 2 + | B \cdot \omega | ^ 2 + | E - \omega \times B | ^ 2 + | B + \omega \times E | ^ 2 . \end{align*}


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


\begin{align*} \sum _ { k = 1 } ^ \infty \frac { 1 } { 1 \vee \log \phi ^ { - 1 } ( a ( k ) ) } = \infty . \end{align*}


\begin{align*} \label [ I e q ] { e q : m c k e a n v l a s o v - p d e } \partial _ t f _ t ( x ) = - \nabla _ x \cdot \{ b ( x , f _ t ) f _ t \} + \frac { 1 } { 2 } \sum _ { i , j = 1 } ^ d \partial _ { x _ i } \partial _ { x _ j } \{ a _ { i j } ( x , f _ t ) f _ t \} \end{align*}


\begin{align*} \widetilde { N _ H } ( n ) = C a r d ( \{ v : | v | = n , u \not \sqsubset v w _ 1 v u \in \mathcal { L } _ H \} ) . \end{align*}


\begin{align*} ( A _ { 0 } , A _ { 1 } , \ldots , A _ { p - 1 } ) = \cfrac { \mathbf { 1 } } { \mathbf { d } _ 1 + \cfrac { \mathbf { 1 } } { \mathbf { d } _ 2 + \cfrac { \mathbf { 1 } } { \mathbf { d } _ 3 + \mathbf { v } _ 3 } } } . \end{align*}


\begin{align*} \sum _ { n _ 1 , \ldots , n _ k \le x } f ( [ n _ 1 , \ldots , n _ k ] ) = C _ { f , k } \frac { x ^ { k ( r + 1 ) } } { ( r + 1 ) ^ k } + O \big ( x ^ { k ( r + 1 ) - \frac 1 { 2 } + \varepsilon } \big ) , \end{align*}


\begin{align*} \Delta \left ( a \right ) & = \int _ 0 ^ { \infty } \frac { \ln \left ( \left | a \right | - i x \right ) } { \cosh \left ( \pi x \right ) } \ , \mathrm { d } x + \int _ 0 ^ { \infty } \frac { \ln \left ( \left | a \right | + i x \right ) } { \cosh \left ( \pi x \right ) } \ , \mathrm { d } x \end{align*}


\begin{align*} \frac { d \beta } { d t } ( t ) = \beta ^ 2 ( t ) + K ( t ) , \frac { d \alpha } { d t } ( t ) + 2 \beta ( t ) = 0 . \end{align*}


\begin{align*} & \{ f , \{ g , h \} \} + \{ g , \{ h , f \} \} + \{ h , \{ f , g \} \} = \\ & ( X , \delta _ X D f \wedge \delta _ X ( D g , \wedge \delta _ X D h ) ) + \circlearrowright \\ + & ( X , \delta _ X D f \wedge \delta _ X ( O _ { X , D g } D h - O _ { X , D h } D g ) ) + \circlearrowright \\ + & ( X , \delta _ X f \wedge \delta _ X ( D ^ 2 g ( \nu _ { X } D f , \cdot ) - D ^ 2 f ( \nu _ { X } D g , \cdot ) ) ) + \circlearrowright , \end{align*}


\begin{align*} \mu _ { V _ { n , k } X ^ { ( n ) } } ( A ) : = \mathbb { P } ( V _ { n , k } X ^ { ( n ) } \in A ) , \end{align*}


\begin{align*} N ^ { n , 1 } _ \tau = N ^ { n , 3 } _ \tau + N ^ { n , 4 } _ \tau , \end{align*}


\begin{align*} \psi ( x ' ) = \int _ { 0 } ^ { \infty } \int _ { 0 } ^ { \infty } \alpha _ { t _ 1 , t _ 2 } ( x ' ) d t _ 1 d t _ 2 \end{align*}


\begin{align*} ( j + 1 ) _ m = \sum _ { k = 0 } ^ m ( - 1 ) ^ k P _ k \left ( m , x \right ) \left ( j + x \right ) ^ { m - k } , \end{align*}


\begin{align*} ( 1 - \lambda ) _ { n - k , \lambda } = ( 1 ) _ { n - k , \lambda } \big ( 1 - ( n - k ) \lambda \big ) . \end{align*}


\begin{align*} L _ K ( G ) : = \mathop { \oplus } \limits _ { n = 1 } ^ \infty ( \gamma _ n ( G ) / \gamma _ { n + 1 } ( G ) ) \otimes _ { \Z } K . \end{align*}


\begin{align*} \chi ^ { \epsilon , L } _ t ( \lambda , \mu , m ^ 2 ) : = \epsilon ^ d \sum _ { x \in \Lambda _ { \epsilon , L } } \big < \varphi _ 0 \varphi _ x \big > ^ { \epsilon , L } _ { \lambda , \mu + 1 / t , m ^ 2 } . \end{align*}


\begin{align*} & \psi _ { { i , s } } = \psi _ { i , 0 } \circ ( \theta _ t + \gamma _ 1 ) \circ \cdots \circ ( \theta _ t + \gamma _ s ) \enspace , \\ & \psi _ { i , 0 } \circ \mathcal { T } _ { \bold { c } } = [ \alpha _ i ] \circ { \rm { E v a l } } _ { \alpha _ i } \ \ 1 \le i \le m \enspace . \end{align*}


\begin{align*} [ x , y , z ] _ { \mathfrak { g } _ 0 } & = l _ 3 ( x , y , z ) , \\ [ \alpha , \beta , \gamma ] _ { \mathfrak { g } _ 1 } & = l _ 3 ( d ( \alpha ) , d ( \beta ) , \gamma ) = l _ 3 ( d ( \alpha ) , \beta , d ( \gamma ) ) = l _ 3 ( \alpha , d ( \beta ) , d ( \gamma ) ) . \end{align*}


\begin{align*} \lim _ { i \rightarrow \infty } \Phi _ { i } ( x , R ) = 0 . \end{align*}


\begin{align*} \label [ I e q ] { e q : m c k e a n v l a s o v c o n t i n u i t y } \partial _ t f ^ N _ t = - \nabla \cdot \left ( \left ( \mathbf { b } ^ N - \frac { \sigma ^ 2 } { 2 } \nabla \log f ^ N _ t \right ) f ^ N _ t \right ) . \end{align*}


\begin{align*} ( C _ i \beta _ i ) \cap ( C _ i \beta _ i ) ^ { \bot _ { A _ i } } = \begin{cases} C _ i \beta _ i , & \mbox { i f } C _ i \beta _ i \overline { C _ i \beta _ i } = 0 ; \\ 0 , & \mbox { o t h e r w i s e . } \end{cases} \end{align*}


\begin{align*} \mathcal { E } ( E _ { 1 } ) = \left \{ \hat { \rho } \in E _ { 1 } : \hat { \rho } \right \} = \bigcap \limits _ { A \in \mathcal { U } } \left \{ \hat { \rho } \in E _ { 1 } : \lim \limits _ { L \rightarrow \infty } \hat { \rho } ( \left \vert A _ { L } \right \vert ^ { 2 } ) = | \hat { \rho } ( A ) | ^ { 2 } \right \} \ . \end{align*}


\begin{align*} \frac { d \Omega } { d t } = \operatorname { d e x p } ^ { - 1 } _ { \Omega ( t ) } ( A ( t ) ) , \qquad \Omega ( 0 ) = { \bf 0 } \ , , \end{align*}


\begin{align*} L _ f S _ \zeta & = M _ f S _ \zeta + \sum _ { g \prec f } \ell _ { g f } ( q ) M _ g S _ \zeta \\ & = \widetilde { N } _ f + \sum _ { g \prec f ; g \in \Z ^ { m | n } } \ell _ { g f } ( q ) q ^ { - \ell ( \tau _ g ) } \widetilde { N } _ { g \cdot \tau ( g ) } , \end{align*}


\begin{align*} \tanh ( k ' a / 2 ) = 0 , \end{align*}


\begin{align*} \hat { H } = - \frac { \hslash ^ { 2 } } { 2 m } \partial _ { x } ^ { 2 } + \frac { \beta \hslash ^ { 4 } } { 3 m } \partial _ { x } ^ { 4 } \end{align*}


\begin{align*} M _ 3 = f _ k ( j ) f _ k ( j + 1 ) + O _ { \alpha } \Big ( \frac { \tau _ 3 ( j ) \tau _ 3 ( j + 1 ) } { \sqrt { j - i - 1 } } \Big ) . \end{align*}


\begin{align*} ( \alpha , \beta ) _ { M , n } & = \Phi _ M ( \beta ) ( \xi ) - \xi \\ & = t _ { M | L } ( \Phi _ L ( \beta ) ) ( \xi ) - \xi \\ & = \prod _ i h _ i ( \xi ) - \xi \\ & = \sum _ i ( h _ i ( \xi ) - \xi ) \end{align*}


\begin{align*} \Gamma _ P : = \bigsqcup _ { j = 1 } ^ h \Gamma _ { P , j } : = \bigsqcup _ { j = 1 } ^ h \Gamma _ { 0 , k _ 1 ( j ) , k _ 2 ( j ) , 1 , \{ ( a _ i , b _ i ) \} _ { i \in I _ j } } . \end{align*}


\begin{align*} \left < \hat { H } , H \right > & = \left < \sum _ { A \in \left [ b \right ] ^ d } \hat { w } _ A h _ { d , b , A } , \frac { 1 } { n } \sum _ { i = 1 } ^ n \sum _ { B \in \left [ b \right ] ^ d } h _ { d , b , B } \ 1 \left ( X _ i \in \Lambda _ { d , b , B } \right ) \right > \\ & = \frac { 1 } { n } \sum _ { i = 1 } ^ n \sum _ { A \in \left [ b \right ] ^ d } \hat { w } _ A \ 1 \left ( X _ i \in \Lambda _ B \right ) b ^ d = \frac { 1 } { n } \sum _ { i = 1 } ^ n \hat { H } ( X _ i ) . \end{align*}


\begin{align*} L _ + = \R ^ n , L _ - = ( e ^ { i \phi _ 1 } , \ldots e ^ { i \phi _ n } ) \R ^ n , 0 < \phi _ i < \pi , T X = \R ^ n \otimes \C , \end{align*}


\begin{align*} S = \{ f \in D [ X ] \mid ( A _ f ) _ v = D \} . \end{align*}


\begin{gather*} \bigwedge _ { i = 1 } ^ { n - 2 } \neg c _ i \lor c _ { i + 1 } \\ \bigwedge _ { i = 1 } ^ n c _ { i - 1 } \lor \neg a _ i \lor b _ i \qquad \bigwedge _ { i = 1 } ^ n c _ { i - 1 } \lor a _ i \lor b _ i \lor \neg c _ i \\ \bigwedge _ { i = 1 } ^ n c _ { i - 1 } \lor \neg a _ i \lor \neg b _ i \lor \neg c _ i \qquad \bigwedge _ { i = 1 } ^ n c _ { i - 1 } \lor a _ i \lor \neg b _ i \lor c _ i \end{gather*}


\begin{align*} \sigma _ { k } ^ { i i } = \frac { C _ n ^ k \times k } { C _ n ^ { k - 1 } } \sigma _ { k - 1 } \ge C ( n , k ) \sigma _ { k } ^ { \frac { k - 1 } { k } } . \end{align*}


\begin{align*} \phi ( \beta ( g ^ { - 1 } , \psi ^ { - 1 } ( \alpha ( g , x ) , v ) ) ) = ( x , A ( g , x , v ) ) \mbox { f o r a l l $ \ , g \in U $ , $ x \in V $ , a n d $ v \in F $ , } \end{align*}


\begin{align*} h _ { \psi } ( r ^ m ) + R ^ N ( r ) + h _ { \psi } ( - 1 ) = 0 , \end{align*}


\begin{align*} \begin{cases} u _ t + ( - \Delta ) ^ { d } u = | x | ^ { \alpha } | u | ^ { p } + \zeta ( t ) { \mathbf w } ( x ) \ \quad \mbox { f o r } ( x , t ) \in \mathbb { R } ^ { N } \times ( 0 , \infty ) , \\ u ( x , 0 ) = u _ 0 ( x ) , x \in \mathbb { R } ^ { N } , \end{cases} \end{align*}


\begin{align*} f ( A ) & = f ( 2 ^ K ) + \sum _ { p \in A } f ( p ) + \sum _ { p \notin A } f ( A _ { p } ) \\ & \le f ( 2 ^ K ) + \sum _ { \substack { p > 2 \\ p \in A } } f ( p ) + \sum _ { \substack { p > 2 \\ p \notin A } } b _ p f ( p ) + \sum _ { \substack { p > 2 \\ p \notin A } } \sum _ { k = 1 } ^ { K - 1 } f ( ( A ^ k ) _ { 2 ^ k p } ) , \end{align*}


\begin{align*} \lim _ { \kappa \to 0 } \kappa ^ p \nu _ p ( \{ ( a , r ) \in \omega \times \R _ + : \ c _ d ' r | \nabla f ( a ) | \pm C A r ^ 2 > \kappa \} = c _ { d , p } \int _ { \omega } | \nabla f ( a ) | ^ p \ , d a \ , . \end{align*}


\begin{align*} ( g , n ) \cdot ( g ' , n ' ) = ( g g ' , n + n ' + \sigma ( g , g ' ) ) . \end{align*}


\begin{align*} \mathcal { N } = \phi ^ { - 1 } \big ( \mathbb { A } \big ( \rho ( \tfrac 1 3 \sigma ^ * ) \big ) \big ) , \end{align*}


\begin{align*} \int _ { N _ d ( K ) } H ( x , y , t ) d \mathrm { v o l } ( y ) \lesssim \sum _ { \rho = 1 } ^ \infty ( 1 + \rho ^ n ) e ^ { - \frac { \rho ^ 2 } { 4 t } - \frac { ( n - 1 ) ^ 2 } { 4 } t - \frac { n - 1 } { 2 } \rho } \mathrm { v o l } ( \mathrm { A n } ( \rho ) \cap N _ d ( K ) ) , \end{align*}


\begin{align*} f ' ( A ( z ) ) = B ' ( g ( z ) ) g ' ( z ) \left ( A ^ { - 1 } \right ) ' ( A ( z ) ) . \end{align*}


\begin{align*} \rho _ { { v } ^ { - 1 } } \circ \theta ( \omega ' ) = \rho _ { { v } ^ { - 1 } } ( \omega ) = \Phi _ K ( w ) ( \omega ) & = \Phi _ K ( u ) ( \theta ( \omega ' ) ) \\ & = \theta ( \Phi _ K ( u ) ( \omega ' ) ) \\ & = \theta \circ [ u ^ { - 1 } ] _ f ( \omega ' ) . \end{align*}


\begin{align*} \sum _ { n = 1 } ^ \infty a _ n \phi _ n ( x ) . \end{align*}


\begin{align*} [ e _ \lambda , e _ \lambda ^ \vee ] = h , \ [ h , e _ \lambda ^ \vee ] = - 2 e _ \lambda ^ \vee . \end{align*}


\begin{align*} \mathcal { E } ( \mathcal { B } _ { 2 , k } ) & = \begin{cases} 2 \left ( \cot \left ( \dfrac { \pi } { 2 k + 2 } \right ) - 1 \right ) , & 2 \nmid k , \\ 2 \left ( \csc \left ( \dfrac { \pi } { 2 k + 2 } \right ) - 1 \right ) , & 2 \mid k . \end{cases} \end{align*}


\begin{align*} \Phi ( \lambda ) + \lambda ( 1 - \Phi ( \lambda ) ) \frac { \Phi ( \lambda ) } { { \phi ( \lambda ) } } & \geq \Phi ( \lambda ) + \lambda ( 1 - \Phi ( \lambda ) ) \frac { \lambda } { 1 + \lambda ^ 2 } \\ & = \frac { \lambda ^ 2 } { 1 + \lambda ^ 2 } + \Phi ( \lambda ) \frac { 1 } { \lambda ^ 2 + 1 } \geq \frac { \lambda ^ 2 } { 1 + \lambda ^ 2 } \geq \frac { 1 } { 2 } . \end{align*}


\begin{align*} d ( L ) = | \operatorname { d e t } ( B ) | . \end{align*}


\begin{align*} ( M ^ { \prime } , \alpha ^ { \prime } ) = ( M - \overset { \circ } { \mathcal { N } } ( { \mathsf { S _ { \mathbf { p } } ^ { 1 } } } ) , \alpha ) \underset { \Psi } { \cup } ( { \mathbf { C } } \times { \mathbf { S } ^ { 3 } } , \delta _ { a } ) . \end{align*}


\begin{align*} P _ { \rm F A } ^ { } ( \lambda ) = & { \rm P r } \left ( \left . \frac { \sum _ { m = 1 } ^ { M } ( T _ { { m } } - \alpha ) T _ { { m } } } { \sum _ { k = 1 } ^ { M } ( T _ { k } - \alpha ) } > \lambda \right | \mathcal { H } _ 0 \right ) \end{align*}


\begin{align*} \textstyle r \leq \epsilon _ 0 : = e ^ { - \frac { 1 } { 2 \delta } } \frac { R } { 2 } , \end{align*}


\begin{align*} X = p \partial _ { 1 } + q \partial _ { 2 } . \end{align*}


\begin{align*} a q ( r ^ B _ A ( k ) ) ^ { - 1 } r ^ B _ A ( k ) \stackrel { ( \ref { s p l } ) } = a k = a k ' \stackrel { ( \ref { s p l } ) } = a q ( r ^ B _ A ( k ' ) ) ^ { - 1 } r ^ B _ A ( k ' ) . \end{align*}


\begin{align*} ( L ( x ) , x ) _ V = - \sum _ { e \in E } \frac { 1 } { 2 } \frac { \delta _ E ( e ) } { | e | ^ 2 } \sum _ { u , v \in e } ( x ( u ) - x ( v ) ) ^ 2 \le 0 . \end{align*}


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


\begin{align*} \widetilde { T } _ { k , j ; n } ^ { b i l ; \mu , i } ( \mathfrak { m } , U ) ( t , x , \zeta ) = \int _ 0 ^ t \int _ { \R ^ 3 } \int _ { \R ^ 3 } e ^ { i x \cdot \xi + i \mu ( t - s ) | \xi | } | \xi | ^ { - 1 } \mathcal { F } \big ( ( E + \hat { v } \times B ) f \big ) ( s , \xi , v ) \cdot \nabla _ v \big [ \big ( \frac { m ^ 0 _ U ( \xi , v ) } { i ( \mu | \xi | + \hat { v } \cdot \xi ) } \end{align*}


\begin{align*} \| ( c ( x ) , s ( x ) ) \| _ Z = 1 \end{align*}


\begin{align*} \Big ( B ( - \theta _ z ) z - A ( - \theta _ z ) \Big ) f ( z ) = B ( 0 ) \enspace . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \sup _ { \tau \in [ 0 , T ] } E [ | R ^ { n , M , 1 } _ \tau | ^ 2 ] = 0 . \end{align*}


\begin{align*} \begin{aligned} h _ * = \arg \max _ { 1 { } \leq { } h < 2 } 2 h + \frac { \kappa h ^ 3 } { 2 - h \left ( 1 + \kappa \right ) } \end{aligned} \end{align*}


\begin{align*} L ^ n = \{ \beta \in L ^ { \times } ; ~ ( \alpha , \beta ) _ { L , n } = 0 ~ \forall \alpha \in W _ L \} . \end{align*}


\begin{align*} ( ~ , ~ ) _ { \rho , L , n } : \mathfrak { p } _ L \times L ^ { \times } & \rightarrow W _ { \rho } ^ n \\ ( \alpha , \beta ) & \mapsto \Phi _ L ( \beta ) ( \xi ) - \xi , \end{align*}


\begin{align*} \widetilde { \Omega } = \widetilde { \Omega } _ \xi : = \Big \{ \sup _ { 0 \le t \le T } \max _ { i \in [ N ] } N ^ { 2 / 3 } \widehat { i } ^ { 1 / 3 } | \lambda _ i ( t ) - \gamma _ i ( t ) | \le N ^ \xi \Big \} \end{align*}


\begin{align*} H _ { k , j ; n , l , r ; p , q ; \star } ^ { \mu , m , i ; n o n , a } ( t , x , \zeta ) = \int _ { 0 } ^ t \int _ { \R ^ 3 } \int _ { \R ^ 3 } \int _ { \mathbb { S } ^ 2 } E B ^ i ( t , s , x - y + ( t - s ) \omega , \omega , v ) \cdot \nabla _ v \big ( ( t - s ) \mathcal { H } ^ { \star ; \mu , E , i } _ { k , j ; n , l , r } ( y , \omega , v , \zeta ) + \mathcal { H } ^ { \star ; e r r ; \mu , E , i } _ { k , j ; n , l , r } ( y , v , \omega , \zeta ) \big ) \end{align*}


\begin{align*} \label [ I e q ] { e q : o p t i m a l c o u p l i n g j u m p p a r a m } \overline { X } { } ^ i _ { T ^ i _ n } = \psi \big ( \overline { X } { } ^ i _ { T ^ { i , - } _ n } , f _ { T ^ i _ n } , \theta \big ) , X ^ i _ { T ^ i _ n } = \psi \Big ( X ^ { i } _ { T ^ { i , - } _ n } , \mu _ { X ^ i _ { T ^ { i , - } _ n } } , \theta \Big ) , \end{align*}


\begin{align*} = \int _ { t _ 1 } ^ { t _ 2 } \int _ { ( \R ^ 3 ) ^ 4 } e ^ { i X ( s ) \cdot ( \xi + \eta + \sigma + \kappa ) + i s ( \mu _ 2 | \sigma | + \mu | \xi | + \mu _ 1 | \eta | ) } \clubsuit K _ { k _ 3 , j _ 3 ; n _ 3 } ^ { \mu _ 3 , i _ 3 } ( s , \kappa , V ( s ) ) \cdot { } ^ 3 \clubsuit K ( s , \xi , \eta , \sigma , X ( s ) , V ( s ) ) d \xi d \eta d \sigma d \kappa d s \end{align*}


\begin{align*} ( L y _ i ) ( v ) & = \begin{cases} - \frac { \delta _ E ( e _ 0 ) } { c } \frac { 1 } { | e _ 0 | } y _ i ( v ) & v \in e _ u , \\ 0 & . \end{cases} \end{align*}


\begin{align*} \begin{array} { l l } R i c ( U , X ) & = ( m - n ) g ( \nabla _ U H , X ) - \sum \limits _ { i = n + 1 } ^ { m } g ( ( \nabla _ { U _ i } T ) _ U U _ i , X ) \\ & + \sum \limits _ { j = 1 } ^ { n } g ( ( \nabla _ X A ) _ { X _ j } X _ j , U ) - \sum \limits _ { j = 1 } ^ { n } g ( ( \nabla _ { X _ j } A ) _ X X _ j , U ) \\ & - \sum \limits _ { j = 1 } ^ { n } g ( T _ U X _ j , \nu [ X , X _ j ] ) , \end{array} \end{align*}


\begin{align*} \beta ^ { \prime } ( 0 ) = - 2 \ln \left ( \frac { 2 \Gamma \left ( \frac { 3 } { 4 } \right ) } { \Gamma \left ( \frac { 1 } { 4 } \right ) } \right ) . \end{align*}


\begin{align*} H _ u ( f ) = P _ + ( u \bar f ) , f \in H ^ 2 ; \end{align*}


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


\begin{align*} \int \frac { f ( x ) } { f ( x / q ) } \left ( v ( x / q ) - \frac { 1 } { q } x ( 1 - q ) \right ) r ( x ) y ( x ) d _ q x = y ( x / q ) - v ( x / q ) D _ { q ^ { - 1 } } y ( x ) , \end{align*}


\begin{align*} \lim \limits _ { t \rightarrow \infty } \| e ^ { - i t H _ 0 } \psi - e ^ { - i t H } \varphi \| = 0 \end{align*}


\begin{align*} N E ( X _ \omega ) = C _ \omega ^ { \prime , \vee } = \left \{ d \in H _ 2 ( X _ \omega ; \mathbb R ) : \eta \cdot d \geq 0 \eta \in C _ \omega ^ \prime \right \} . \end{align*}


\begin{align*} ( D ^ 2 u ) = f \quad \ \mathbb { R } ^ n , \end{align*}


\begin{align*} & \langle \Lambda _ { \gamma _ 1 } f , g \rangle = \langle \Lambda _ { \gamma _ 2 } f , g \rangle \Leftrightarrow \langle \Lambda _ { q _ 1 } f , g \rangle = \langle \Lambda _ { q _ 2 } f , g \rangle \end{align*}


\begin{align*} \Lambda _ { A } : = \bigcup _ { T } ^ { \infty } \Lambda _ { A , T } \ , , \qquad \Lambda _ { A , T } : = \bigcup _ { t = - T } ^ { T } \phi _ t ( A ) \end{align*}


\begin{align*} \mathcal T _ { \kappa , 2 } = \left \{ [ n _ 0 , n _ 1 , n _ 2 , n _ 3 ] \in \mathcal S _ { \kappa , 2 } \colon n _ 1 = n _ 2 \right \} . \end{align*}


\begin{align*} \theta ( t ) = \sum _ { n = r + 1 } ^ \infty \phi * \psi _ 1 ^ { [ n * ] } ( t ) < \infty , \end{align*}


\begin{align*} K \phi \varphi _ { i } ( \gamma ) + K \phi \gamma \varphi ^ { \prime } _ { i } ( \gamma ) = \theta . \end{align*}


\begin{align*} E ( G / A ) = E ( N _ G ( A ) / A ) = \pm | W ( K ) | \end{align*}


\begin{align*} \mu _ { L , L ' } ( p ) = \frac { 1 } { \pi } ( \sum \phi _ i + \theta _ L ( p ) - \theta _ { L ' } ( p ) ) \end{align*}


\begin{align*} r \cdot w _ 1 \cdots w _ s = h _ s \cdots h _ 1 \cdot r ' . \end{align*}


\begin{align*} d V _ { B H } = \frac { \int \limits _ { 0 } ^ { \pi } \sin ^ { n - 2 } ( t ) d t } { \int \limits _ { 0 } ^ { \pi } \frac { \sin ^ { n - 2 } ( t ) } { \phi ( b \cos ( t ) ) ^ n } d t } d V _ { \alpha } , \end{align*}


\begin{align*} \mathcal { G } ( n + 1 ) \oplus \mathcal { G } ( n + 1 ) \oplus \mathcal { G } ( 2 a c + 1 ) = \mathcal { G } ( n ) \oplus \mathcal { G } ( n ) \oplus \mathcal { G } ( 2 a c + 3 ) , \ ; a \neq 2 \bmod 6 \end{align*}


\begin{align*} I - S S ^ * = e _ 0 e _ 0 ^ * , I - \Sigma \Sigma ^ * = q q ^ * , \end{align*}


\begin{align*} \sum _ { j = m _ 0 } ^ \infty | y _ n ^ * ( x _ j ^ k ) | < \epsilon / 8 . \end{align*}


\begin{align*} D _ I : = \cap _ { i \in I } D _ i . \end{align*}


\begin{align*} \mathrm { P } _ { ( \Phi , - \hat { f } \left ( 0 \right ) \delta _ { \Psi } ) } ^ { \sharp } = \mathrm { P } _ { ( \Phi , - \hat { f } \left ( 0 \right ) \delta _ { \Psi } ) } ^ { \flat } \leq \inf _ { \gamma \in \left ( 0 , 1 \right ) } \mathrm { P } _ { \Phi + \mathcal { K } _ { \gamma } \left ( \Psi , - f \right ) } ^ { \sharp } \ , \Psi \in \mathbb { S } , \ \Phi \in \mathcal { W } _ { 1 } , \ f \in \mathfrak { D } _ { 0 , + } \ , \end{align*}


\begin{align*} F ( z ) = \int _ \R \frac { d \rho ( s ) } { s - z } \forall z \in \C \setminus \R \end{align*}


\begin{align*} \frac { d } { d t } \langle X \rangle _ t = - \frac { 1 } { 2 } \sum \limits _ { j } \langle [ C _ { j } , [ C _ { j } , X ] ] ) \rangle _ t . \end{align*}


\begin{align*} u ( x , t ) \ge M _ n \left \{ \frac { t - x - R } { ( 1 + t - x ) ^ { 1 + b } } \right \} ^ { a _ n } \left \{ \log \left ( \frac { 1 + t + x } { 1 + t - x } \right ) \right \} ^ { b _ n } \quad \mbox { i n } \ D \end{align*}


\begin{align*} ( \phi \cdot \psi ) ( F _ * ^ { e + d } r ) = \phi \left ( F _ * ^ e ( \psi ( F _ * ^ d r ) ) \right ) . \end{align*}


\begin{align*} s _ { \theta } = ( 1 - \theta ) s _ 0 + \theta s _ 1 = \frac { t - r } { t - s } s + \frac { r - s } { t - s } t = r . \end{align*}


\begin{align*} a _ i = W _ c ( \mu , \nu _ { i / I } ) - W _ c ( \mu , \nu _ { ( i - 1 ) / I } ) \forall i \in \mathcal I . \end{align*}


\begin{align*} f ( x , y , n , z ) : = \frac { n ^ 2 } { \pi ^ 2 } \sin ^ 2 \Big ( \frac { \pi x } { n } \Big ) + \frac { n ^ 2 } { \pi ^ 2 } \sin ^ 2 \Big ( \frac { \pi y } { n } \Big ) + z ^ 2 . \end{align*}


\begin{align*} u ( x , t ) = \lambda ^ { - \frac { n - 2 } { 2 } } { \sf Q } ( y ) + \lambda ^ { - \frac { n - 2 } { 2 } } \sigma { \sf A } _ 1 ( 1 + o ) | y | \to \infty . \end{align*}


\begin{align*} \nu = \mathcal { D } \Big ( \sum _ { j = 1 } ^ \infty A _ { \bullet , j } Z _ j + \sigma ( I _ k - A A ^ T ) ^ { 1 / 2 } N _ k \Big ) \end{align*}


\begin{align*} \begin{gathered} a _ { 1 1 } = 1 , a _ { 1 2 } = - \lambda _ 7 , a _ { 1 3 } = - \lambda _ 8 , a _ { 2 1 } = \lambda _ 1 , \\ a _ { 2 2 } = ( 1 + \lambda _ 5 \lambda _ 6 ) e ^ { \lambda _ 3 } - \lambda _ 1 \lambda _ 7 , a _ { 2 3 } = \lambda _ 5 e ^ { \lambda _ 3 } - \lambda _ 1 \lambda _ 8 , a _ { 3 1 } = \lambda _ 2 , \\ a _ { 3 2 } = \lambda _ 6 e ^ { \lambda _ 4 } - \lambda _ 2 \lambda _ 7 , a _ { 3 3 } = e ^ { \lambda _ 4 } - \lambda _ 2 \lambda _ 8 . \end{gathered} \end{align*}


\begin{align*} T _ { A , B } : = \left ( \begin{array} { c | r } A & B \\ \hline B & A \end{array} \right ) , \end{align*}


\begin{align*} \widetilde { T } _ { k , j ; n , l , r } ^ { S ; \mu , m , i } ( \mathfrak { m } , U ) ( t , x , \zeta ) = \int _ 0 ^ t \int _ { \R ^ 3 } \int _ { \R ^ 3 } \int _ { \mathbb { S } ^ 2 } \big [ ( t - s ) K _ { k ; n } ^ { \mu } ( \mathfrak { m } ) ( y , \zeta , \omega ) + \widetilde { K } _ { k ; n } ^ { \mu } ( \mathfrak { m } ) ( y , \zeta ) \big ] f ( s , x - y + ( t - s ) \omega , v ) \big ( E ( s , x - y + ( t - s ) \omega ) \end{align*}


\begin{align*} \int _ { \mathbb { R } } ( x - s ) ^ { - 2 } \ , d \nu ( s ) = + \infty , x \in \mathbb { R } , \end{align*}


\begin{align*} \Theta = \dfrac { \prod _ { i = 1 } ^ m \alpha ^ r _ i \prod _ { s = 0 } ^ { r - 1 } a _ { 0 , s } ^ m } { ( n - 1 ) ! ^ { r ^ 2 m } } \cdot { \rm { d e t } } \left ( { \rm { E v a l } } _ { \alpha _ i } \bigcirc _ { { { w } } = 0 } ^ s ( \theta _ t + \gamma _ { r - s + { { w } } } ) ^ { - 1 } ( t ^ n H _ { \ell } ( t ) ) \right ) _ { \substack { 0 \le \ell \le r m - 1 \\ 1 \le i \le m , 0 \le s \le r - 1 } } . \end{align*}


\begin{align*} w = \sum _ { p \in J } S ( p ) ^ { d } w \geq \sum _ { p \in J } \prod _ { j = 1 } ^ d \frac { \tilde S ( p , \pi _ j ( p ) , \alpha , \lambda ) } { { \lambda + k _ j \choose k _ j } } \geq \frac { 1 } { \prod _ { j = 1 } ^ d { \lambda + k _ j \choose k _ j } } { \lambda + n \choose n } \gtrsim _ { n } 1 , \end{align*}


\begin{align*} \Delta \cdot L _ { x y } = \left ( D - a W - b L _ { x y } - b _ 0 R _ 0 - b _ 1 R _ 1 \right ) \cdot L _ { x y } \leq \left ( D - b L _ { x y } \right ) \cdot L _ { x y } = \frac { 3 + 2 b ( 4 n - 1 ) } { 4 n ( 4 n + 1 ) } < \frac { 1 } { \lambda } . \end{align*}