\begin{align*} P _ T = P _ { \rm P B S } \left ( \left | { \mathcal { K } } \right | - 1 \right ) + P _ { \rm M B S } + P _ { \rm V L C } \left ( \left | { \mathcal { V } } \right | \right ) + \sum \limits _ { \forall j } P _ j , \end{align*}


\begin{align*} \hat { A } ( y | x ) = A ( x ) \ , \ \forall \ ( y | x ) \in \hat { \Omega } \ . \end{align*}


\begin{align*} { \bf h } | _ { \partial \bf N } & = \pm 2 \bar u v d t ^ 2 + h ^ \intercal \\ A ' | _ { \bar { \bf g } } ( \bf h ) & = \pm \big ( v \nu ( \bar u ) + \bar u ( \nu ( u ) ) ' \big ) d t ^ 2 + A ' | _ g ( h ) . \end{align*}


\begin{align*} 0 & = \int _ \Sigma \Big \langle Q ( h , v ) , \big ( k ^ \intercal , H ' ( k ) \big ) \Big \rangle d \sigma \\ & = \int _ \Sigma \Big \langle \big ( v A + \bar u A ' ( h ) - \nu ( v ) g ^ \intercal , 2 v \big ) , \big ( - 2 \zeta A ' ( h ) , \zeta A \cdot A ' ( h ) \big ) \Big \rangle \ , d \sigma \\ & = - \int _ \Sigma 2 \bar u \zeta | A ' ( h ) | ^ 2 \ , d \sigma \end{align*}


\begin{align*} \Big ( f ( u _ 2 ) - f ( u _ 1 ) - f ' ( u _ 2 ) \omega _ + \Big ) \omega _ + = 0 \end{align*}


\begin{align*} \sum _ { \substack { n \\ r _ { Q } ( n ) = 0 } } n \ll \max \left \{ \frac { p ^ { 3 + \epsilon } } { ( \min Q ^ { * } ) ^ { 2 } } , p ^ { 5 / 2 + \epsilon } \right \} . \end{align*}


\begin{align*} \kappa _ { i [ j k ] } = \mathbf { A } _ { [ j k ] , i } = \frac { 1 } { 2 } \left ( A _ { j k , i } - A _ { k j , i } \right ) = \frac { 1 } { 2 } \left ( - \epsilon _ { j k l } \vartheta _ { l , i } + \epsilon _ { k j l } \vartheta _ { l , i } \right ) = \frac { 1 } { 2 } \left ( \epsilon _ { k j l } + \epsilon _ { k j l } \right ) \vartheta _ { l , i } = \epsilon _ { k j l } \ , \vartheta _ { l , i } . \end{align*}


\begin{align*} f ( x ) = \sum _ { | I | = 0 , 1 , \dots , } \frac { 1 } { | I | ! } \partial ^ I f ( p ) ( x - p ) ^ I \mbox { f o r a l l } x \in U \end{align*}


\begin{align*} F ( \eta ) : = T - \frac { 1 } { \eta } I , \eta \in \Omega . \end{align*}


\begin{gather*} \lim _ { N \to \infty } \dfrac { | \Phi _ N g \cap \Phi _ N | } { | \Phi _ N | } = 1 \\ \lim _ { N \to \infty } \dfrac { | \Phi _ N \cap g \Phi _ N | } { | \Phi _ N | } = 1 \end{gather*}


\begin{align*} f _ i ( r ) : = \Big \lfloor \frac { r } { 1 + ( 1 + \tilde \varepsilon ) ^ { - i } } \Big \rfloor , r \in \N . \end{align*}


\begin{align*} & \log \left ( \prod _ { \mathrm { N } ( v ) \leq q ^ { m } } \det ( 1 - \rho ( \mathrm { F r o b } _ { v } ) \mathrm { N } ( v ) ^ { - s } ) ^ { - 1 } \right ) \\ & = \sum _ { 1 \leq k \deg ( v ) \leq m } \frac { \mathrm { t r } ( \rho ( \mathrm { F r o b } _ { v } ) ^ { k } ) } { k } q ^ { - k \deg ( v ) s } + \sum _ { k = 2 } ^ { \infty } \frac { 1 } { k } \sum _ { \frac { m } { k } < \deg ( v ) \leq m } \mathrm { t r } ( \rho ( \mathrm { F r o b } _ { v } ) ^ { k } ) q ^ { - k \deg ( v ) s } . \end{align*}


\begin{align*} \nabla _ T D _ i = [ D _ i , C _ T ] = 0 \quad \forall T \in \Gamma \forall i . \end{align*}


\begin{align*} G r _ w ( E ) = G r _ w ( F ) . \end{align*}


\begin{align*} | ( k + 1 ) ^ { 1 - s } - k ^ { 1 - s } | = | 1 - s | \left \lvert \int _ k ^ { k + 1 } y ^ { - s } \dd y \right \rvert \asymp | 1 - s | k ^ { - \Re ( s ) } \end{align*}


\begin{align*} f ( t ) = ( \kappa \ , * \ , \phi ) ( t ) \ , = \ , \int _ 0 ^ t \kappa ( t - \tau ) \phi ( \tau ) \ , d \tau , \ t > 0 \end{align*}


\begin{align*} D \sigma _ T ( x ) = \frac { 1 } { \pi } \Big ( \log \Big ( \frac { \sin ( \pi x _ 1 ) } { \sin ( \pi ( x _ 1 + x _ 2 ) ) } , \log \Big ( \frac { \sin ( \pi x _ 2 ) } { \sin ( \pi ( x _ 1 + x _ 2 ) ) } \Big ) \Big ) . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \int _ { \mathbb { T } ^ d } \langle b ( u _ n ( x ) ) , \nabla u _ n ( x ) - \nabla u ( x ) \rangle d x = 0 . \end{align*}


\begin{align*} \Pi _ V J _ { z / | z | } ^ \perp x = \lambda \Pi _ V x \forall x \in \C ^ n . \end{align*}


\begin{align*} \tilde { u } ( x , t ) = \begin{cases} u ( x , t ) , & t \in [ 0 , T _ ] \\ u _ 1 ( x , t ) , & t \in [ T _ , T _ + T _ 1 ] , \end{cases} \end{align*}


\begin{align*} g x _ i = a _ { i 1 } x _ 1 + \cdots + a _ { i n } x _ n \end{align*}


\begin{align*} G ^ \prime = \begin{pmatrix} 1 & 1 & \cdots & 1 \\ g _ { 2 1 } ^ \prime & g _ { 2 2 } ^ \prime & \cdots & g _ { 2 n } ^ \prime \\ \vdots & \vdots & & \vdots \\ g _ { k 1 } ^ \prime & g _ { k 2 } ^ \prime & \cdots & g _ { k n } ^ \prime \end{pmatrix} = \begin{pmatrix} G _ 1 ^ \prime & G _ 2 ^ \prime & \cdots & G _ n ^ \prime \end{pmatrix} . \end{align*}


\begin{align*} \| \varphi _ { 2 k j _ 0 } - \rho \| _ \infty & = \max \Big { \{ } \sup \{ | \varphi _ { 2 k j } ( z ) - \rho ( z ) | ; \ z \in \mathbb { B } _ n , \ | \varphi _ { k j } ( z ) | > r \} , \\ & \sup \{ | \varphi _ { 2 k j } ( z ) - \rho ( z ) | ; \ z \in \mathbb { B } _ n , \ | \varphi _ { k j } ( z ) | \leq r \} \Big { \} } . \end{align*}


\begin{align*} \sup _ { z \in \mathbb { D } } \bigg | \frac { 1 - \overline { b ( a ) } b ( z ) } { 1 - \overline { a } z } \bigg | = o \left ( \frac { 1 } { 1 - | a | ^ 2 } \right ) \ , | b ( a ) | \to 1 , \end{align*}


\begin{align*} x _ { \rm t } [ n ] = \delta \ , x _ { \rm c } [ n ] + P _ { \rm b i a s } , \end{align*}


\begin{align*} \sum _ { m = 0 } ^ { n } \left ( - 1 \right ) ^ { m } \sum _ { k = 0 } ^ { m } \binom { m } { k } \binom { n - m } { k } \rho ^ { k } = \left \{ \begin{array} { c c c } 0 & & n \\ ( 1 - \rho ) ^ { n / 2 } & & n \end{array} \right . . \end{align*}


\begin{align*} \left ( I - \mathcal { P } ^ - \right ) s = 0 \end{align*}


\begin{gather*} m ( f _ 1 ) + m ( f _ 2 ) = \lim _ { \epsilon \rightarrow 0 } \ , \Bigl \{ \int _ \epsilon ^ 1 f _ 1 ( x ) \ , d x + \int _ 0 ^ { 1 - \epsilon } f _ 2 ( x ) \ , d x \Bigr \} \\ = \lim _ { \epsilon \rightarrow 0 } \int _ 0 ^ { 1 - \epsilon } \bigl \{ f _ 1 ( x + \epsilon ) + f _ 2 ( x ) \bigr \} \ , d x \le \lim _ { \epsilon \rightarrow 0 } \int _ 0 ^ { 1 - \epsilon } c ( x + \epsilon , x ) \ , d x = 0 . \end{gather*}


\begin{align*} k _ i ^ N ( t , \nu ^ { N , 1 } _ { \boldsymbol { x } } , \ldots , \nu ^ { N , N } _ { \boldsymbol { x } } , \lambda ^ { N , 1 } _ { \boldsymbol { \varphi } } , \ldots , \lambda ^ { N , N } _ { \boldsymbol { \varphi } } ) : = \rho _ i ^ N \big ( - R ^ { - 1 } B [ P _ t \nu ^ { N , 1 } _ { \boldsymbol { x } } + \lambda ^ { N , 1 } _ { \boldsymbol { \varphi } } ] , \ldots , - R ^ { - 1 } B [ P _ t \nu ^ { N , N } _ { \boldsymbol { x } } + \lambda ^ { N , N } _ { \boldsymbol { \varphi } } ] \big ) . \end{align*}


\begin{align*} v = 0 , ( \nu ' ) ( h ) ( \bar u ) + \nu ( v ) = 0 , A ' ( h ) = 0 \mbox { o n } \hat \Sigma . \end{align*}


\begin{align*} \Delta _ { \mathbb S ^ d } f = - \lambda f , \end{align*}


\begin{align*} D ^ q _ * ( f ) ( z ) = \sum _ { n = 1 } ^ { \infty } a _ n \frac { \Gamma ( q n + 1 ) } { \Gamma ( q ( n - 1 ) + 1 ) } z ^ { q ( n - 1 ) } . \end{align*}


\begin{align*} & \mathcal { P } ^ - ( f ( z ) e ^ { 2 i z x } ) = \frac { 1 } { 2 \pi i } \lim _ { \varepsilon \rightarrow 0 } \int _ { R } \frac { f ( s ) e ^ { 2 i s x } } { s - ( z - i \varepsilon ) } \mathrm { d } s . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \langle A ( u _ n ) , u _ n \rangle = \langle A ( u ) , u \rangle . \end{align*}


\begin{align*} \mathcal A _ { \mathcal X } ^ { \mathcal D } = \left \{ ( s , \alpha ) | s \in S , \alpha \in A _ { X _ s } ^ { D _ s } \right \} \end{align*}


\begin{align*} & \int _ { B _ { 1 / 2 } } | \nabla v | ^ { 3 - 3 \theta } \d x \leq C \end{align*}


\begin{align*} \begin{aligned} & ( I - F ) \left [ z \left ( \Psi ^ { - } _ 1 - e ^ { - i c _ - } e _ 1 \right ) - \left ( \widehat { \Psi } ^ - _ { 1 1 } e _ 1 + \widehat { \Psi } ^ - _ { 2 1 } e _ { 2 } \right ) \right ] \\ = & z d e _ { 2 } - ( I - F ) \left ( \widehat { \Psi } ^ - _ { 1 1 } e _ 1 + \widehat { \Psi } ^ - _ { 2 1 } e _ 2 \right ) , \end{aligned} \end{align*}


\begin{align*} \langle e _ { i } , A e _ { j } \rangle = \langle A e _ { i } , A e _ { j } \rangle = \langle e _ { i } , e _ { j } \rangle = 0 \end{align*}


\begin{align*} \alpha ( A ) & = \inf \{ t > 0 : A \mbox { i s i n c l u d e d i n s o m e b a l l o f r a d i u s $ t $ } \} , \\ \beta ( A ) & = \inf \{ t > 0 : A \mbox { h a s a d i a m e t e r l e s s t h a n $ t $ } \} , \end{align*}


\begin{align*} 0 = \frac { \dd } { \dd \tau } E [ b + \tau \tilde { b } + s ( \tau ) b _ 0 ] \Bigg | _ { \tau = 0 } & = \int _ { \mathbb { R } ^ { 3 } } b \cdot ( \tilde { b } + \dot { s } ( 0 ) b _ 0 ) \dd x \\ & = \int _ { \mathbb { R } ^ { 3 } } b \cdot \tilde { b } \dd x - \frac { 1 } { \partial _ { s } j ( 0 , 0 ) } \left ( \int _ { \mathbb { R } ^ { 3 } } b \cdot b _ 0 \dd x \right ) \partial _ { \tau } j ( 0 , 0 ) , \\ & = \int _ { \mathbb { R } ^ { 3 } } b \cdot \tilde { b } \dd x - \frac { \mu } { 2 } \partial _ { \tau } j ( 0 , 0 ) . \end{align*}


\begin{gather*} \int f \ , d P = \sup _ k \int f \wedge k \ , d P . \end{gather*}


\begin{align*} ( \delta _ n ^ k ) ' = \sum _ { J = \{ i _ 0 < \hdots < \widehat { i _ k } < \hdots < i _ n \} \subset K = \{ i _ 0 < \hdots < i _ n \} } ( \nu _ { K } ^ J ) _ ! \end{align*}


\begin{align*} \mathcal { Q } ^ { r } ( t ) : = \mathcal { M } ^ { r } ( \delta ) + ( r - 1 ) \theta _ { 1 } \varrho _ { 0 } ^ { 1 - \sigma } \int _ { \delta } ^ { t } ( \mathcal { M } ^ { r } ( s ) ) ^ { \sigma + 1 } \ d s . \end{align*}


\begin{align*} v _ i = 1 \{ \theta _ i ^ \star = 0 \} - 1 \{ \theta _ i ^ \star = \pi \} , i = 1 , . . . , N . \end{align*}


\begin{align*} { \bf g } | _ { \partial \bf N } & = \pm u ^ 2 d t ^ 2 + g ^ \intercal \\ A _ { \bf g } & = \pm u \nu ( u ) d t ^ 2 + A _ g . \end{align*}


\begin{align*} Z _ { V } ( [ \alpha ] ) = - \int _ { V } \alpha + \sqrt { - 1 } \int _ { V } \omega , \end{align*}


\begin{align*} \max _ { - N \leq j \leq N } \Vert M _ { H _ j } : S _ { 1 } ^ { 2 N + 1 } \rightarrow S _ { 1 } ^ { 2 N + 1 } \Vert \leq \Vert M _ { \widetilde { H } \vert _ { \mathbb { Z } _ N \times \mathbb { Z } _ N } } ^ { ( N ) } : S _ { p _ 1 } ^ { 2 N + 1 } \times S _ { p _ 2 } ^ { 2 N + 1 } \rightarrow S _ { 1 } ^ { 2 N + 1 } \Vert . \end{align*}


\begin{align*} \begin{aligned} \left ( C _ h - e ^ { - \frac { 1 } { 2 } \zeta h j ^ s } \right ) \vert \eta ( \mathbf 0 ) v _ j ( \mathbf 0 ) \vert & \lesssim j ^ { - \beta + s } + j ^ { \gamma - ( \alpha + 1 ) s } + j ^ { - \beta + \frac { 1 } { 2 } s } + j ^ { - \beta + ( - \alpha + \frac { 1 } { 2 } ) s } \\ & + j ^ { \gamma - \alpha s } + j ^ { - \frac { 1 3 } { 2 4 } s } + j ^ { - \frac { 1 7 } { 1 2 } s } , \end{aligned} \end{align*}


\begin{align*} \sum _ { n = 0 } ^ \infty \frac { q ^ { n ^ 2 } X ^ n } { \prod _ { i = 0 } ^ { n - 1 } ( q ^ n - q ^ i ) } = \left ( \sum _ { n = 0 } ^ \infty \frac { q ^ { n ^ 2 - n } X ^ n } { \prod _ { i = 0 } ^ { n - 1 } ( q ^ n - q ^ i ) } \right ) \cdot \left ( \sum _ { n = 0 } ^ \infty X ^ n \right ) . \end{align*}


\begin{align*} \mathcal { W } ( f V W \Delta ^ { - n - 1 } ) = \mathcal { g } ^ { \Delta } ( f V , W ) = \mathcal { g } ^ { \Delta } ( V , f W ) , \end{align*}


\begin{align*} \alpha = \frac { c } { 4 \pi k \sqrt { k ^ 2 + r _ * ^ 2 } } , \ \ \beta = \frac { \alpha } { 2 } ( 3 r _ * ^ 2 + 4 k ^ 2 ) . \end{align*}


\begin{align*} \dot { x } ( t ) & = - A ^ { \circ } x ( t ) \quad ( t > 0 ) , \\ y ( t ) & = B ^ { \circ } x ( t ) \quad ( t \geq 0 ) , \\ x ( 0 ) & = x _ 0 \in X ^ \circ . \end{align*}


\begin{align*} 0 = \partial ^ 2 _ { x _ i x _ j } \phi - \partial ^ 2 _ { x _ i x _ j } \psi - \partial ^ 2 _ { x _ i y } \psi \partial _ { x _ j } T - \partial ^ 2 _ { x _ j y } \psi \partial _ { x _ i } T - \partial ^ 2 _ y \psi \partial _ { x _ j } T \partial _ { x _ i } T - \partial _ y \psi \partial ^ 2 _ { x _ j x _ i } T \\ + { ( \partial _ y T ) ^ { - 2 } } ( \partial _ y T \partial ^ 3 _ { x _ j x _ i y } T - \partial ^ 2 _ { x _ i y } T \partial ^ 2 _ { x _ j y } T ) \end{align*}


\begin{align*} d \in \sigma _ l ( A ) , \ , v = ( x , 1 ) ^ T \in V _ l ( d ) , \forall x \in S _ L ( d - a , b ) . \end{align*}


\begin{align*} \lim _ { r \to \infty } \int _ { | x | = r } \sum _ { i , j = 1 } ^ n \left ( \frac { \partial h _ { i j } } { \partial x _ i } - \frac { \partial h _ { i j } } { \partial x _ j } \right ) \frac { x _ j } { | x | } \ , d \sigma _ { \bar g } = 0 . \end{align*}


\begin{align*} g _ { i j } = \left \langle e _ i , e _ j \right \rangle , h _ { i j } = \left \langle \overline { D } _ { e _ i } e _ j , \mathbf { n } \right \rangle , \end{align*}


\begin{align*} z ^ { - 1 } k ^ { - 1 } b ( k ) = - \frac { 1 } { 2 } \widehat { \bar { u } _ x K \psi ^ - _ { 1 1 } } ( z ) + i \widehat { \bar { u } } ( z ) . \end{align*}


\begin{align*} D _ 1 = \displaystyle \sum _ { \substack { X \in S ( 0 ) \\ t = 2 } } \Theta _ { \theta } \begin{bmatrix} m _ 1 & X \\ 0 & m _ 2 \end{bmatrix} \overline { \psi _ A ( { m _ 1 } ^ { - 1 } X } ) = - q ^ 8 ( 1 - q ) . \end{align*}


\begin{align*} \Phi \ , : \ , \left ( \C ^ { Z _ m \times Z _ m } \right ) ^ { G _ m } \to \bigoplus _ { i = 1 } ^ k \C ^ { m _ i \times m _ i } , \end{align*}


\begin{align*} \mathcal { H } _ m ( u ) : = \mathcal { H } _ m \left ( \frac { \partial ^ 2 u } { \partial \overline { q _ l } \partial { q _ k } } \right ) . \end{align*}


\begin{align*} \begin{aligned} \Theta _ \gamma ( v ) = 1 \textrm { i f } 0 \leq \gamma \leq 1 \ , , \textrm { a n d } \Theta _ \gamma ( v ) = \nu ^ { - 1 } ( v ) \textrm { i f } - 3 < \gamma < 0 \ , . \end{aligned} \end{align*}


\begin{align*} \int _ { D ( 0 , 2 ) \backslash D ( 0 , 1 ) } | \phi | ^ { 2 } \frac { 1 } { r } \dd z \dd r = 1 . \end{align*}


\begin{align*} \mathfrak { W } _ 2 ^ { [ k , k ] } ( 1 + \varepsilon , 1 - \varepsilon ) & = 1 - \frac { k - 1 } { 2 ( 2 k - 1 ) } \varepsilon ^ 2 + O ( \varepsilon ^ 3 ) , \\ \mathfrak { M } _ 2 ^ { [ r ] } ( 1 + \varepsilon , 1 - \varepsilon ) & = 1 - \frac { 1 - r } { 2 } \varepsilon ^ 2 + O ( \varepsilon ^ 3 ) \end{align*}


\begin{align*} m _ { n } ( x ) = \frac { \nu _ n ( A _ { h , j } ) } { \mu _ n ( A _ { h , j } ) } \end{align*}


\begin{align*} Q _ { j , - } - \mathcal { P } ^ - \left ( Q _ { j , - } J \right ) = \mathcal { P } ^ - ( D _ j ) , Q _ { j , + } - \mathcal { P } ^ - \left ( Q _ { j , - } J \right ) = \mathcal { P } ^ + ( D _ j ) . \end{align*}


\begin{align*} D & : = P + Q \intertext { a n d } \overline { D } & : = \overline { P } + \overline { Q } \end{align*}


\begin{align*} 1 + \sum _ { v ^ * } \ ! \Bigg ( \prod _ { s \ge 1 } \frac { q ^ { \langle v ^ s \ ! , \ , v ^ s \rangle } } { q ^ { \langle \beta ^ s \ ! , \ , \beta ^ s \rangle } } \frac { c ( v ^ s , q ) } { \mathit { g l } ( v ^ s , q ) } X ^ { s v ^ s } \Bigg ) = \ , \mathrm { E x p } \Bigg ( \sum _ { v \in \mathbb { N } ^ n \backslash \{ 0 \} } \ ! \frac { a ( v , q ) } { q - 1 } X ^ v \Bigg ) , \end{align*}


\begin{align*} & \partial _ { t } \left ( \frac { \rho ( u , t ) ^ { 2 } } { 2 } \right ) - \Theta \sigma ^ { i j } _ { k } \nabla _ { i j } \left ( \frac { \rho ( u , t ) ^ { 2 } } { 2 } \right ) \\ & = ( k + 1 ) h \Theta \sigma _ { k } - \rho ^ { 2 } + \sigma _ { k } \nabla _ { i } h \nabla _ { j } \Theta - \Theta \sigma ^ { i j } _ { k } w _ { m i } w _ { m j } . \end{align*}


\begin{align*} \int _ { \mathfrak X } | \omega | & = { \pi _ s } _ ! \int _ { \mathfrak U } | \omega | = { \pi _ s } _ ! \Big ( { \psi _ s } _ ! \int _ { \mathfrak Y } | \omega | \Big ) = { \phi _ s } _ ! \int _ { \mathfrak Y } | \omega | , \end{align*}


\begin{align*} & P \otimes _ { G } Q \rightarrow P \otimes _ { G } Q \\ & p \otimes q \mapsto p \kappa ^ { - 1 } \otimes \kappa q = p \otimes q \end{align*}


\begin{align*} g \rightarrow g ^ { \sigma } : = \sigma ^ { - 1 } ( g ^ { - 1 } ) ^ { \vee } \sigma \end{align*}


\begin{align*} \int _ { \mathfrak U } | \omega | & = { \pi ' _ s } _ ! \int _ { \mathfrak U ' } | \omega | = { \pi ' _ s } _ ! \Big ( { i ' _ s } ^ * \int _ { \mathfrak X ' } | \omega | \Big ) \\ & = ( \mathfrak i _ 0 ^ * \circ { \pi _ s } _ ! ) \int _ { \mathfrak X ' } | \omega | \\ & = \mathfrak i _ 0 ^ * \int _ { \mathfrak X } | \omega | , \end{align*}


\begin{align*} | k _ i ^ N ( t , \nu ^ { N , 1 } _ { \boldsymbol { x } } , \ldots , \nu ^ { N , N } _ { \boldsymbol { x } } , \lambda ^ { N , 1 } _ { \boldsymbol { \phi } ( t , \boldsymbol { x } ) } , \ldots , \lambda ^ { N , N } _ { \boldsymbol { \phi } ( t , \boldsymbol { x } ) } ) - k \big ( t , \nu ^ { N , i } _ { \boldsymbol { x } } , \Phi ( t , \nu ^ { N , i } _ { \boldsymbol { x } } ) \big ) | = O \Big ( \frac { 1 } { N } \Big ( 1 + \frac { 1 } { N } \underset { j \neq i } \sum | x ^ i - x ^ j | \Big ) \Big ) ; \end{align*}


\begin{align*} d _ 0 | q | ^ 2 \wedge d _ 1 | q | ^ 2 = 4 \sum _ { l = 0 } ^ { n - 1 } | q _ l | ^ 2 \omega ^ l \wedge \omega ^ { n + l } + \sum _ { | j - k | \not = n } a _ { j k } \omega ^ j \wedge \omega ^ k \end{align*}


\begin{align*} ( a \wedge b ) \cdot c & = ( a \cdot b ) \ : \cdot \ : ( a \cdot c ) \\ a \cdot ( b \wedge c ) & = ( a \cdot b ) \wedge ( a \cdot c ) . \end{align*}


\begin{align*} \int _ \Omega \Delta u _ 1 \wedge \dots \wedge \Delta u _ m \wedge \beta _ n ^ { n - m } = \int _ \Omega \Delta v _ 1 \wedge \dots \wedge \Delta v _ m \wedge \beta _ n ^ { n - m } . \end{align*}


\begin{align*} \rho _ k ^ { - 1 } L _ k \overset { \eqref { c o n i c - L } } { = } C + \rho _ k ^ { - 1 } B + \rho _ k ^ { - 1 } L _ { \nabla g } \sum _ { i = 0 } ^ { k - 1 } \rho _ i \eta _ i + \rho _ k ^ { - 2 } \overset { \eqref { d e f 1 } } { \leq } C + \rho _ 0 ^ { - 1 } B + \rho _ 0 ^ { - 1 } L _ { \nabla g } \theta + \rho _ 0 ^ { - 2 } \overset { \eqref { d e f 2 } } { = } L . \end{align*}


\begin{align*} \sum _ { l = 1 } ^ M \sum \limits _ { \substack { j = 1 \\ j \neq l } } ^ M \sum _ { q = 1 } ^ M a _ q ^ 2 \hat { \phi } ^ { \left ( l , q \right ) ^ * } \hat { \phi } ^ { \left ( j , q \right ) } = 2 \sum _ { l = 1 } ^ { M - 1 } \sum _ { j = i + 1 } ^ { M } \sum _ { q = 1 } ^ M a _ q ^ 2 \Re \left [ \hat { \phi } ^ { \left ( l , q \right ) ^ * } \hat { \phi } ^ { \left ( j , q \right ) } \right ] . \end{align*}


\begin{align*} Q _ { \delta _ N } L _ { \delta _ N } u _ N = L _ { \delta _ N } u _ N - \sum _ { j = 1 } ^ m \sum _ { h = 1 } ^ 2 C _ { j , h , N } \frac { \partial } { \partial z _ { j , h } } ( - \delta _ N ^ 2 ( K ( z _ { N , j } ) \nabla V _ { \delta _ N , Z _ N , j } ) ) . \end{align*}


\begin{align*} Q ( \mathcal { O } ^ { ( p ) } ) = d ( \mathcal { O } ^ { ( p - 1 ) } ) , \end{align*}


\begin{align*} 1 + \gamma _ { w , k , m } = \frac { 1 + \Delta _ { w , k + 1 , m } } { 1 + \Delta _ { w , k , m } } , ~ | \Delta _ { w , i , m } | \le \frac { w - m } { m } \Big | \frac { z _ { w } } { z _ { w - m } } \Big | ^ i , ~ i = k , k + 1 , \end{align*}


\begin{align*} N = \begin{cases} ( d + 1 ) ( t + T ) - 1 , & s = 1 \\ d s t + d T + t s - 1 + T + 1 , & s > 1 \end{cases} \end{align*}


\begin{align*} s _ { \min } ( A ) = s _ n ( A ) = \min \limits _ { x : \ , \lVert x \rVert _ 2 = 1 } \lVert A x \rVert _ 2 , s _ { \max } ( A ) = s _ 1 ( A ) = \max \limits _ { x : \ , \lVert x \rVert _ 2 = 1 } \lVert A x \rVert _ 2 . \end{align*}


\begin{align*} \nabla _ { V } \hat X & = \omega ( V ) \\ \nabla _ V \omega & = - R ( \hat X , V ) + T _ h ( V ) . \end{align*}


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


\begin{align*} G ( x , y ) = \begin{cases} C _ 0 \left ( \ln \frac { 2 } { | x - y | } + O ( 1 ) \right ) , \quad & \mathrm { i f } \ \ | x - y | \le 2 , \\ O \Big ( e ^ { - | x - y | / 2 } \Big ) , & \mathrm { i f } \ \ | x - y | > 2 , \\ \end{cases} \end{align*}


\begin{align*} 0 \rightarrow E \rightarrow \bigoplus _ { j = 1 } ^ n E _ j \rightarrow \mathcal { T } _ E \rightarrow 0 , \end{align*}


\begin{align*} H ^ 2 ( \mathbb { D } ) = \{ f = \sum _ { n = 0 } ^ { \infty } a _ n z ^ n \mid ( a _ n ) _ { n \in \mathbb { N } } \in \ell ^ 2 \} . \end{align*}


\begin{align*} H ^ 2 ( \mathbb { D } ) & = \{ \sum _ { n = 0 } ^ { \infty } a _ n z ^ n \mid ( a _ n ) \in \ell ^ 2 \} \\ Y & = \{ \sum _ { n = 0 } ^ { \infty } a _ n f _ n \mid ( a _ n ) \in \ell ^ 2 \} \end{align*}


\begin{align*} E [ M _ v ( t ) ^ 2 ] = E \big [ [ M _ v ] ( t ) \big ] . \end{align*}


\begin{align*} F _ m ( n ) = 1 + H ( n , m ) - r _ m ( n ) \sum _ { k = 1 } ^ m \frac { \mu ( k ) } { k } , \end{align*}


\begin{align*} I _ { R _ 1 } = \int _ { \Gamma ^ - \setminus \Gamma ^ - _ h } e ^ { - \tau ( \mathbf d + \mathrm i \mathbf d ^ \perp ) } \mathrm d \sigma , I _ { R _ 2 } = \int _ { \Gamma ^ + \setminus \Gamma ^ + _ h } e ^ { - \tau ( \mathbf d + \mathrm i \mathbf d ^ \perp ) } \mathrm d \sigma . \end{align*}


\begin{align*} A _ i f ( x ) : = \sum _ { \substack { y \sim x \\ \tau ( y ) = \tau ( x ) + i } } f ( y ) , \end{align*}


\begin{align*} [ \kappa ] _ n ^ 2 \stackrel { d e f } { = } \kappa _ 1 ^ 2 + \sum _ { i = 2 } ^ n ( \kappa _ i - \kappa _ { i - 1 } ) ^ 2 , \ n \ge 2 . \end{align*}


\begin{align*} \left \{ \begin{aligned} & \partial _ t F = Q ( F , F ) , \\ & F | _ { t = 0 } = F _ 0 , \end{aligned} \right . \end{align*}


\begin{align*} ( \theta _ 1 , \theta _ 2 ) = ( a ^ x \alpha ^ y , b ^ x \beta ^ y ) \end{align*}


\begin{align*} S _ i = { { w } _ i } \rho c _ v \frac { \Delta t } { 2 } \partial _ t ^ 2 T , \end{align*}


\begin{align*} f _ { k } ( x , y ) : = \begin{cases} f ( x , - k ) , & y < - k , \\ f ( x , y ) , & | y | \le k , \\ f ( x , k ) , & y > k . \end{cases} \end{align*}


\begin{align*} k ^ 2 \int _ { \mathcal C _ h } ( q - 1 ) v u _ 0 \mathrm d \mathbf x = - \sum _ { m = 1 } ^ 3 I _ m , \int _ { \Gamma _ h } \eta u _ 0 v \mathrm d \sigma = I - \sum _ { m = 4 } ^ 9 I _ m , \end{align*}


\begin{align*} & M i n i m i z e _ { x , y \in \mathbb { R } ^ { n } } \ \ \alpha _ J ^ T y + 2 b _ J ^ T x \\ & s . t . \ \ \alpha _ k ^ T y + 2 b _ k ^ T x + c _ k \le 0 \ \ ( \forall \ , k \in \{ 1 , . . . , m \} ) \\ & s . t . \ \ x _ i ^ 2 - y _ i \le 0 , \ \ \forall \ , i \end{align*}


\begin{align*} \begin{aligned} \vert J _ 1 \vert \lesssim ( 1 + \tau ) ( 1 + \tau ^ { - \frac { 2 } { 5 } } ) e ^ { - \zeta h \tau } \end{aligned} \end{align*}


\begin{align*} ( g _ t , u _ t ) = \psi _ t ^ * \big ( \left . \big ( \bar g , \bar u \big ) \right | _ { M \setminus \Omega _ t } \big ) . \end{align*}


\begin{align*} \mathcal { K } = \{ \mathbf { x } \in \mathbb { R } ^ { 2 } ~ | ~ \theta _ { m } < \arg ( x _ { 1 } + \mathrm i x _ { 2 } ) < \theta _ { M } \} , \end{align*}


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


\begin{align*} \begin{array} { r l } \min & \sum _ { j = 1 } ^ n z _ j ^ { p } \\ & z \ge 0 \ , , \ , M z + q \ge 0 \ , , \ , z ^ \top ( M z + q ) = 0 \ , , \end{array} \end{align*}


\begin{align*} A _ n ( \ell ) a _ { \ell } = Q _ { n , 1 } ( \ell ) \beta _ { n , 1 } ^ { \ell } + \cdots + Q _ { n , s ( n ) } ( \ell ) \beta _ { n , s ( n ) } ^ { \ell } . \end{align*}


\begin{align*} \Delta u _ 1 \wedge \dots \wedge \Delta u _ n = n ! \det ( A _ 1 , A _ 2 , \dots , A _ n ) \Omega _ { 2 n } . \end{align*}


\begin{align*} H _ { \mathrm { C V } } ( s , i , p , q , l ) & \coloneqq ( s + i ) l w + \beta ( 1 - \theta l ) ^ 2 s i \left ( q - p \right ) + i \phi ( i ) \left ( \frac { w } { r } + \chi \right ) - ( \gamma + \phi ( i ) ) i q \\ & = \beta \theta ^ 2 s i ( q - p ) l ^ 2 + \left [ ( s + i ) w - 2 \beta \theta s i ( q - p ) \right ] l \\ & + \beta s i ( q - p ) + i \phi ( i ) \left ( \frac { w } { r } + \chi \right ) - ( \gamma + \phi ( i ) ) i q \ , , \end{align*}


\begin{align*} \begin{aligned} \det ( s I + A ) & = \prod \limits _ { k = 1 } ^ n ( s + \lambda _ k ( A ) ) = \sum \limits _ { m = 0 } ^ n \sum \limits _ { 1 \leq j _ 1 < \dots < j _ m \leq n } \lambda _ { j _ 1 } ( A ) \cdots \lambda _ { j _ m } ( A ) s ^ { n - m } = \sum \limits _ { m = 0 } ^ n \mathcal { H } _ m ( A ) s ^ { n - m } \end{aligned} \end{align*}


\begin{align*} \left \| ( I - F ) ^ { - 1 } \right \| _ { L _ x ^ { \infty } L _ z ^ 2 \rightarrow L _ x ^ { \infty } L _ z ^ 2 } \leq \sum _ { n = 0 } ^ { \infty } \frac { 1 } { n ! } \left \| \widetilde { Q } ( u ) \right \| _ { L ^ 1 } ^ n = \mathrm { e } ^ { \left \| \widetilde { Q } ( u ) \right \| _ { L ^ 1 } } . \end{align*}


\begin{align*} \frac { \partial } { \partial s } f ( p , s ) = \log a - \frac { p / 2 + 2 } { s } \geq \log a - \frac { p _ k / 2 + 2 } { s _ k } \end{align*}


\begin{align*} \begin{cases} \Delta _ \Psi v \geq 0 & \O \\ \partial _ { \vec \nu } v = - 2 \mathrm { I I } ( X , X ) \leq 0 & \partial \O . \end{cases} \end{align*}


\begin{align*} \theta _ { S , \mu } \left ( \gamma ( \tau , z ) \right ) ( c \tau + d ) ^ { - \frac { g } { 2 } } e \left ( - c S [ z ] ( c \tau + d ) ^ { - 1 } \right ) = \sum _ { \eta \in ( 2 S ) ^ { - 1 } \mathbb Z ^ g / \mathbb Z ^ g } \varepsilon _ { S } ( \eta , \mu ; \gamma ) \theta _ { S , \eta } ( \tau , z ) , \end{align*}


\begin{align*} \lim _ { t \to 0 } j ( t ) = \int _ { \Omega } ( K _ H ( x ) \nabla u | \nabla u ) d x > 0 . \end{align*}


\begin{align*} \sigma _ { X } \coloneqq \alpha _ { \underline { X } } ( X _ { \mathbb { C } } ) ( \mathrm { i d } _ { \mathbb { C } } ) = \mathrm { i d } _ { X } \times \tau \in \mathrm { H o m } _ { \mathbb { C } } ( \overline { X _ { \mathbb { C } } } , X _ { \mathbb { C } } ) , \end{align*}


\begin{align*} \Sigma _ 0 = \Sigma \setminus ( S _ p ( F ) \cup S _ { \infty } ( F ) ) \supset S . \end{align*}


\begin{align*} W ( \xi _ 0 , k , x , \mathsf { M } ^ k ) = W ( 1 - \xi _ 0 , k , x , \mathsf { M } ^ k ) , ~ \forall \xi _ 0 \in [ 0 , 1 ] , \forall x \in \mathbb { R } . \end{align*}


\begin{align*} \gamma ^ k = \max _ { x _ n ^ D \in \mathcal { N } _ D } 5 0 0 \cdot \frac { | \Gamma _ D | \mathrm { N } _ I \lambda _ d ^ 2 } { | \Omega | \mathrm { N } _ D \lambda _ 1 } \frac { | \nabla w ( x _ n ^ D ) | ^ 2 } { | \nabla w ( y _ n ) | ^ 2 } . \end{align*}


\begin{align*} & < v _ t , \xi > + \int _ { \mathbb { R } ^ { 3 } } ( ( v + u _ { \infty } ) \cdot \nabla v - ( b + B _ { \infty } ) \cdot \nabla b ) \cdot \xi \dd x + \nu \int _ { \mathbb { R } ^ { 3 } } \nabla v : \nabla \xi \dd x = 0 , \\ & < b _ t , \zeta > + \int _ { \Omega } ( b \times v + B _ { \infty } \times v + b \times u _ { \infty } ) \cdot \nabla \times \zeta \dd x + \mu \int _ { \Omega } \nabla \times b \cdot \nabla \times \zeta \dd x = 0 , \end{align*}


\begin{align*} \epsilon ( x _ a , t ) : = \int _ 0 ^ t h ( x _ a + \delta u ( \tau ) ) u ( \tau ) - \dfrac { b _ { 1 , \delta } ( x _ a ) } { 2 } \ , d \tau \end{align*}


\begin{align*} - \Delta u = f & D , \\ [ 1 m m ] u = 0 & \partial D . \end{align*}


\begin{align*} \Delta _ { \Psi } u = \Delta u - g ( \nabla \Psi , \nabla u ) \end{align*}


\begin{align*} | \mathfrak { I } _ { q _ 1 , \ldots , q _ 4 , \kappa } ( \ell ) | \leq \frac { C } { \ell ^ { 3 } } , \end{align*}


\begin{align*} I _ { \varepsilon } ( u ) = \frac { 1 } { 2 } \int _ { \Omega } ( K _ H ( x ) \nabla u | \nabla u ) d x - \frac { 1 } { ( p + 1 ) \varepsilon ^ 2 } \int _ { \Omega } \left ( u - q \ln \frac { 1 } { \varepsilon } \right ) ^ { p + 1 } _ + d x , \ \ \forall u \in \mathcal { H } ( \Omega ) . \end{align*}


\begin{align*} F _ n ( \eta ) : = T _ n - \frac { 1 } { \eta } I _ n . \end{align*}


\begin{align*} | \phi ^ { N , i } ( 0 , \boldsymbol { \xi } ) - \Psi ^ { N , i } ( 0 , \boldsymbol { \xi } ) | \leq \frac { C } { N } \Big ( 1 + \frac { 1 } { N } \underset { j = 1 } { \overset { N } \sum } | \xi ^ { i } - \xi ^ { j } | ^ 2 + \frac { 1 } { N ^ 2 } \underset { i , j = 1 } { \overset { N } { \sum } } | \xi ^ { i } - \xi ^ { j } | ^ 2 \Big ) ^ { \frac { 1 } { 2 } } , \end{align*}


\begin{align*} \sum _ { r = 1 } ^ n { T _ { i j r } T _ { k \ell r } } = 0 \forall i , j , k , \ell \in \{ 1 , \ldots , n \} . \end{align*}


\begin{align*} \frac { d ^ 2 } { d t ^ 2 } X ( \gamma ( t ) ) = B ( t ) X ( \gamma ( t ) ) + D ( t ) \end{align*}


\begin{align*} ( \nu S \textbf { \textit { u } } ) ^ i = - ( \nabla ^ i u _ n + \nabla _ n u ^ i ) . \end{align*}


\begin{align*} \langle z , w \rangle _ { \mathbb C } : = \frac { 1 } { 2 } \left ( \overline z w + \overline w z \right ) = \frac { 1 } { 2 } \left ( z \overline w + w \overline z \right ) . \end{align*}


\begin{align*} \begin{cases} \mathcal { L } _ H q = 0 , \\ \nabla ^ { \perp } q \cdot \nu | _ { \partial \Omega } = - v _ n | _ { \partial \Omega } . \\ \end{cases} \end{align*}


\begin{align*} M _ d ^ h = M _ d ^ { h ' } . \end{align*}


\begin{align*} \psi _ { \tau } ( p ) \stackrel { d e f } { = } \left [ \ p \int _ 0 ^ { \infty } \ t ^ { p - 1 } \ T _ { \tau } ( t ) \ d t \ \right ] ^ { 1 / p } = | | \tau | | L _ p ( \Omega ) , \end{align*}


\begin{align*} b ( x _ 1 , x _ 2 ) = \nabla \times ( \phi ( x _ 1 , x _ 2 ) \nabla z ) + G ( x _ 1 , x _ 2 ) \nabla z . \end{align*}


\begin{align*} \dot { s } \varepsilon ( z ) \dot { s } \varepsilon ( 1 ) \eta = \dot { s } ( \varepsilon ( z - 1 ) - \varepsilon ( z ) ) \eta + \dot { s } \varepsilon ( z ) \varpi . \end{align*}


\begin{align*} \int _ \R ( | T _ i ( x _ i ) | + | x _ i | ) Q _ i ( x _ i ) d x _ i = \int _ \R | x _ i | Q ^ * _ i ( x _ i ) d x _ i + \int _ \R | x _ i | Q _ i ( x _ i ) d x _ i < \infty . \end{align*}


\begin{align*} X = \{ z = ( z _ k ) : | | z | | < \infty \} , \end{align*}


\begin{align*} F _ n - n ^ p \ \ = \ \ \mathcal { C } _ n ^ { ( p ) } \ - \ 2 \sum _ { j = 0 } ^ { p / 2 } \binom { p } { 2 j + 1 } \mathcal { C } _ n ^ { ( p - 2 j - 1 ) } \ + 2 \binom { p } { 2 \lfloor p / 2 \rfloor + 1 } F _ n \mbox { f o r $ p $ o d d } . \end{align*}


\begin{align*} \frac { 1 } { p } + \frac { 1 } { 1 + \epsilon _ { 0 } } = \frac { 1 } { \tilde { p } } \mbox { a n d } \frac { 1 } { n + 1 } + \frac { 1 } { p } \leq \frac { 1 } { \tilde { p } } < \frac { 1 } { p } + \min \left \{ \frac { 1 } { p } , \frac { 1 } { n } \right \} . \end{align*}


\begin{align*} C _ E = \sum _ { n \geq 1 } \frac { \mu ( n ) } { [ \Q ( E [ n ] ) : \Q ] } , \end{align*}


\begin{align*} \int _ X f \ , d \mu = \int _ X f \circ \varphi _ \alpha \ , d \mu , \end{align*}


\begin{align*} a _ 2 ( x ) = \frac { \Gamma ( n - 2 ) \operatorname { v o l } ( \mathbb { S } ^ { n - 2 } ) } { ( 2 \pi ) ^ { n - 1 } } \bigg ( a _ { 2 , 1 } H ^ 2 + a _ { 2 , 2 } \sum _ { \alpha } \kappa _ \alpha ^ 2 + a _ { 2 , 3 } R + a _ { 2 , 4 } R _ { n n } \bigg ) \end{align*}


\begin{align*} \int _ \Omega \left ( { \mathcal D } _ { \psi } f ( w ) + ( \frac { \partial p _ 0 } { \partial x } + \frac { \partial q _ 0 } { \partial y } ) f + ( \frac { \partial p _ 1 } { \partial x } + \frac { \partial q _ 1 } { \partial y } ) { i } f \right ) d x d y = \int _ { \partial \Omega } f d \sigma _ { \psi } ( w ) , \end{align*}


\begin{align*} \begin{aligned} \sigma ( a , r ) = \int _ { B ( a , r ) } \Delta u \wedge \beta _ n ^ { n - 1 } . \end{aligned} \end{align*}


\begin{align*} X X ^ \top = l \ , V ^ \top V \mbox { a n d } X ^ \top X = k \ , W ^ \top W . \end{align*}


\begin{align*} S ^ { p q , r s } _ k = \frac { \partial ^ 2 S _ k [ D ^ 2 u ] } { \partial r _ { p q } \partial r _ { r s } } . \end{align*}


\begin{align*} \lim _ { \substack { n \rightarrow \infty \\ n \in \Lambda } } X _ n ( t , \omega ) = X ( t , \omega ) , \end{align*}


\begin{align*} \begin{aligned} C _ m ( K ) = \inf \left \{ \int _ { \Omega } ( \Delta u ) ^ m \wedge \beta _ n ^ { n - m } : u \in \mathcal { U } ^ * ( K , \Omega ) \right \} \end{aligned} \end{align*}


\begin{align*} \mathcal { B } _ q ( \psi _ m ) ( p ) = \displaystyle \frac { p ^ m } { \sqrt { \Gamma ( q m + 1 ) } } , \end{align*}


\begin{align*} \gamma _ i + \bar { \gamma _ j } = 1 \ , \ , { \rm a n d } \ , \ , \gamma _ 3 + \bar { \gamma } _ 4 = 1 , \end{align*}


\begin{align*} \Phi _ { n , e } ^ { ( i ) } ( x , y ) = \prod _ { \substack { m | n \\ ( m , e ) = 1 \\ \overline m m \equiv i \bmod e } } ( x ^ { n / m } - \zeta ^ { \overline m } _ { e } y ^ { n / m } ) ^ { \mu ( m ) } . \end{align*}


\begin{align*} \int _ { \partial \Omega } \nabla c \cdot \nu c \psi ^ { 2 } = 0 . \end{align*}


\begin{align*} W ^ 2 ( r ) + f ( r ) \frac { d W ( r ) } { d r } = W ^ { \prime 2 } ( r ) - f ( r ) \frac { d W ' ( r ) } { d r } + E _ 1 - E _ 0 , \end{align*}


\begin{align*} F _ n - n ^ 3 \ = \ \sum _ { i = 1 } ^ { n } \left ( i ^ 3 - 2 \left ( \binom { 3 } { 2 } i ^ 2 + \binom { 3 } { 0 } i ^ 0 \right ) \right ) \cdot F _ { n - i } . \end{align*}


\begin{align*} \hat { A } ^ { \pm } = \mp \sqrt { f ( r ) } \frac { d } { d r } \sqrt { f ( r ) } + W ( r ) . \end{align*}


\begin{align*} \phi _ { f ' } ( x ) = \alpha ( v x + w ) ^ { q + 1 } ( \phi _ f \circ \mu ( x ) ) , \end{align*}


\begin{align*} \sum _ { j = 0 } ^ { n } ( - 1 ) ^ { j } \binom { n } { j } \frac { \left ( \beta \right ) ^ { \left ( j + m \right ) } } { \left ( \beta \right ) ^ { \left ( j \right ) } } = ( - 1 ) ^ { n } n ! \binom { m } { n } \frac { \left ( \beta \right ) ^ { \left ( m \right ) } } { \left ( \beta \right ) ^ { \left ( n \right ) } } . \end{align*}


\begin{align*} \nabla _ i \frac { A _ { j k } } { \mathcal { H } ^ 2 } & = \frac { 1 } { \mathcal { H } ^ 2 } \nabla _ i A _ { j k } - \frac { 1 } { \left ( \mathcal { H } ^ 2 \right ) ^ 2 } A _ { j k } \nabla _ i \mathcal { H } ^ 2 \\ & = \frac { 1 } { \left ( \mathcal { H } ^ 2 \right ) ^ 2 } ( \mathcal { H } ^ 2 \nabla _ i A _ { j k } - A _ { j k } \nabla _ i \mathcal { H } ^ 2 ) , \end{align*}


\begin{align*} < n \times u , v > _ { \partial \Omega } = \int _ { \partial \Omega } u \times v \cdot n \dd H = \int _ { \partial \Omega } \alpha \wedge \beta = \int _ { \partial \Omega } d ( \phi \wedge \beta ) = 0 . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \mathbb { E } \sup _ { t \leq T \wedge \sigma _ n ^ M } \Big [ \varphi _ n ( t ) \Vert X ( t , x _ n ) - X ( t , x ) \Vert _ H ^ 2 \Big ] = 0 . \end{align*}


\begin{align*} h ( x ) & = \frac { \left [ \Phi \left ( \frac { x - \mu } { \sigma } \right ) \right ] ^ { \alpha - 1 } \ , \left [ 1 - \Phi \left ( \frac { x - \mu } { \sigma } \right ) \right ] ^ { \beta - 1 } \ , \phi \left ( \frac { x - \mu } { \sigma } \right ) } { \sigma \ , \mathrm { B } ( \alpha , \beta ) \ , \left [ 1 - I _ { \Phi ( \frac { x - \mu } { \sigma } ) } ( \alpha , \beta ) \right ] } , \end{align*}


\begin{align*} M _ I = \bigoplus _ { J \subset I } R _ J . \end{align*}


\begin{align*} L _ { \mu } ^ h f ( x ) = \frac { 1 } { h ^ 2 } \int [ f ( x + h y ) - f ( x ) ] p _ { \mu } ( x , d y ) , \end{align*}


\begin{align*} f ^ \star = \min f ( W _ 1 ) = \min f ( W _ 2 ) \ , . \end{align*}


\begin{align*} X _ { M _ \mathbf i } ( x _ 1 , \dots , x _ { i _ 1 } , \dots , x _ n ) = X _ { M _ { \mathbf i ' } } ( x _ 1 ' , \dots , x _ { i _ 1 } ' , \dots , x _ n ' ) \end{align*}


\begin{align*} & \int _ { A _ { n , j } \cap A _ n } \int _ 0 ^ { \tau _ { n + 1 } ( \omega ) - \tau _ n ( \omega ) } D ^ p ( x _ { \tau _ n + t } ( \omega ) ) d t d \P \\ & = \int _ { A _ { n , j } \cap A _ n } \int _ 0 ^ { \eta _ j ( K ) } D ^ p ( x _ { \tau _ n + t } ( \omega ) ) d t d \P + \int _ { A _ { n , j } \cap A _ n } \int _ { \eta _ j ( K ) } ^ { \tau _ { n + 1 } ( \omega ) - \tau _ n ( \omega ) } D ^ p ( x _ { \tau _ n + t } ( \omega ) ) d t d \P . \end{align*}


\begin{align*} \begin{aligned} b ( \omega , v _ 1 - v ) = \beta ( \theta ) | v _ 1 - v | ^ \gamma \ , , \ \cos \theta = \frac { ( v _ 1 - v ) \cdot \omega } { | v _ 1 - v | } \ , , \ - 3 < \gamma \leq 1 \ , , \end{aligned} \end{align*}


\begin{align*} \begin{aligned} d u _ m & = i 2 ^ m \left ( \overline { u _ { m + 1 } } u _ { m + 2 } - \frac { \delta } { 2 } \overline { u _ { m - 1 } } u _ { m + 1 } - \frac { \delta - 1 } { 4 } u _ { m - 2 } u _ { m - 1 } \right ) \\ & - \delta 2 ^ { 2 m } u _ m + q _ m d W _ t ^ { ( m ; R ) } + i p _ m d W _ t ^ { ( m ; I ) } , \end{aligned} \end{align*}


\begin{align*} r _ { i , j } ( w ) . : = | \{ k ; k \leq i , w ( k ) \leq j \} | . \end{align*}


\begin{align*} \partial _ { \vec \nu } | X | ^ { 2 } & = - 2 \mathrm { I I } ( X , X ) , \partial \O . \end{align*}


\begin{align*} a \wedge b : = ( a \cdot b ) a = ( b \cdot a ) b . \end{align*}


\begin{align*} \sum _ { \substack { k = 0 \\ n - k \equiv 0 \ , ( \mathrm { m o d } \ , 2 ) } } ^ n \binom { n } { k } \big ( 2 ^ k L _ { j k } + 2 L _ j ^ k \big ) ( \sqrt { 5 } F _ j ) ^ { n - k } \frac { 2 ^ { n - k + 2 } - 1 } { n - k + 2 } B _ { n - k + 2 } = L _ j ^ n , \end{align*}


\begin{align*} \begin{aligned} \min & \ \frac { 1 } { 2 } x ^ T A x + b ^ T x \\ \mbox { s . t . } & \ \frac { 1 } { 2 } x ^ T B _ i x + c _ i ^ T x + d _ i \leq 0 , i = 1 , \dots , m , \\ & \ - 1 \leq x _ i \leq 1 , i = 1 , \dots , n , \end{aligned} \end{align*}


\begin{align*} \frac { ( p ^ 3 - p ^ 2 ) } { ( p ^ 2 - p ) } = p \end{align*}


\begin{align*} ( \sigma _ 1 , \dots , \sigma _ { n - 1 } ) \subset \bigoplus _ { i = 1 } ^ { n - 1 } ( E _ { i , q _ i } , E _ { i + 1 , q _ i ' } ) . \end{align*}


\begin{align*} \begin{aligned} | v _ 1 | ^ 2 - | v | ^ 2 = | v _ 1 - v | ^ 2 + 2 v \cdot ( v _ 1 - v ) \ , . \end{aligned} \end{align*}


\begin{align*} B _ 1 + \cdots + B _ k \subset \{ n \in \N : T ^ n x _ { \vec 0 } \in E \} = A \end{align*}


\begin{align*} \div _ { \Psi } X = e ^ { \Psi } \div ( e ^ { - \Psi } X ) \end{align*}


\begin{align*} u v = T _ u v + T _ v u + R ( u , v ) , \end{align*}


\begin{align*} ( x \wedge y ) \cdot z = ( x \cdot y ) x \cdot z = ( x \cdot y ) \cdot ( x \cdot z ) , \end{align*}


\begin{align*} { \mathcal { I } } _ h = \inf \left \{ \frac { 1 } { 2 } \int _ { \Omega } | B | ^ { 2 } \dd x \ \middle | \ B \in L ^ { 2 } _ { \sigma } ( \Omega ) , \ \int _ { \Omega } \textrm { c u r l } ^ { - 1 } B \cdot B \dd x = h \right \} . \end{align*}


\begin{align*} y \left [ i \right ] = \sum \limits _ { \left ( { l , l ' } \right ) \in { \cal S } } { { \bf { h } } _ l ^ H { { \bf { F } } _ { l ' } } { \bf { d } } \left [ { i - { n _ l } - { \kappa _ { l ' } } } \right ] } + z \left [ { i } \right ] . \end{align*}


\begin{align*} \gamma ^ { q ^ n } = \gamma ^ { q ^ { n - i } } . \end{align*}


\begin{align*} E _ n = \frac { - \varepsilon _ 1 } { \sqrt { \varepsilon _ 1 ^ 2 + | \hat { x } | ^ 2 } } e _ 1 + \frac { | \hat { x } | } { \sqrt { \varepsilon _ 1 ^ 2 + | \hat { x } | ^ 2 } } e _ 2 . \end{align*}


\begin{align*} T = 2 n \exp ( \log 2 n / 1 0 3 . 9 5 \log \log 2 n ) , \end{align*}


\begin{align*} \eta ( 1 ) = \gamma ^ { - 1 } \mbox { a n d } \eta ( g ) = 1 . \end{align*}


\begin{align*} \iota ( \upsilon ) = \frac { - 1 } { 2 \pi } \sin ( 2 \pi \upsilon ) + \frac { - \sin ( 2 \pi \upsilon ) + 4 \pi \upsilon - \sin ( 4 \pi \upsilon ) + \frac { 1 } { 3 } \sin ( 6 \pi \upsilon ) } { 8 \left ( - \sin ( 2 \pi \upsilon ) + \frac 1 3 \sin ( 6 \pi \upsilon ) - ( 2 \pi \upsilon + \frac 1 2 \sin ( 4 \pi \upsilon ) - 2 \sin ( 2 \pi \upsilon ) ) \right ) } \left ( - 4 \upsilon + \frac { 1 } { \pi } \sin ( 4 \pi \upsilon ) \right ) \end{align*}


\begin{align*} \beta _ { q } ( n ) = \lim _ { v \to \infty } \frac { \# \{ \vec { x } \in ( \Z / p ^ { v } \Z ) ^ { 4 } | Q ( \vec { x } ) \equiv n \pmod { p ^ { v } } \} } { p ^ { 3 v } } , \end{align*}


\begin{align*} & a ( k ) = e ^ { - i c } + \mathcal { O } \left ( k ^ { - 1 } \right ) , \ \ b ( k ) = \mathcal { O } \left ( k ^ { - 1 } \right ) , k \rightarrow \infty , \\ & a ( k ) = e ^ { - i c } \left ( 1 + \mathcal { O } \left ( k ^ 2 \right ) \right ) , \ \ b ( k ) = \mathcal { O } \left ( k ^ 3 \right ) , k \rightarrow 0 , \end{align*}


\begin{align*} H _ 1 = \sum _ { X \in S ( 0 ) } \Theta \begin{pmatrix} m _ 1 & X \\ 0 & m _ 2 \end{pmatrix} \overline { \psi _ A ( { m _ 1 } ^ { - 1 } X } ) = - \theta ( a ) q ^ 8 ( 1 - q ) ^ 2 ( 1 - q ^ 2 ) . \end{align*}


\begin{align*} f = \sum _ { \mathcal { J } } \sum _ { \beta } f _ { \mathcal { J } , \beta } ( \epsilon ( x ^ { \mu } ) ) \prod _ { j \in \mathcal { J } } \theta _ j ^ { \beta ^ j } . \end{align*}


\begin{align*} \| A _ j x \| ^ 2 - \big | \langle x | A _ j x \rangle \big | ^ 2 & = \| ( A _ j - z _ j I ) x \| ^ 2 - \big | \langle x | ( A _ j - z _ j I ) x \rangle \big | ^ 2 \\ & \leq \| ( A _ j - z _ j I ) x \| ^ 2 . \end{align*}


\begin{align*} & \sup _ { x \in \mathbb { R } } \left \| \mathcal { P } ^ + \left ( z ^ { - i } \bar { r } _ 1 ( z ) e ^ { 2 i z x } \right ) \right \| _ { L _ z ^ \infty } \leq \frac { 1 } { \sqrt { 2 } } \left \| z ^ { - i } r _ 1 ( z ) \right \| _ { H ^ 1 } , i = 0 , 1 , \\ & \sup _ { x \in \mathbb { R } } \left \| \mathcal { P } ^ - \left ( r _ 2 ( z ) \mathrm { e } ^ { - 2 i z x } \right ) \right \| _ { L _ { z } ^ { \infty } } \leq \frac { 1 } { \sqrt { 2 } } \left \| r _ 2 ( z ) \right \| _ { H ^ { 1 } } . \end{align*}


\begin{align*} \| ( \lambda x _ v + ( 1 - \lambda ) x _ w ) \| ^ 2 = \lambda \| x _ v \| ^ 2 + ( 1 - \lambda ) \| x _ w \| ^ 2 - \lambda ( 1 - \lambda ) \| x _ v - x _ w \| ^ 2 . \end{align*}


\begin{align*} \phi _ { i s } ( \partial _ r ) = 0 \quad \forall r \neq i \neq s \neq r , \end{align*}


\begin{align*} L ( x , \gamma ) : = J ( x ) + \sum \limits _ { k = 1 } ^ m \gamma _ k f _ k ( x ) = x ^ T A ( \gamma ) x + 2 b ( \gamma ) ^ T x + c ( \gamma ) \end{align*}


\begin{align*} ( M - X ^ m ( X ^ k - 1 ) I _ p ) \begin{pmatrix} X _ 1 X _ 2 \cdots X _ { p - 1 } \\ X _ 2 \cdots X _ { p - 1 } \\ \vdots \\ X _ { p - 1 } \\ 1 \end{pmatrix} = \begin{pmatrix} 0 \\ \vdots \\ 0 \end{pmatrix} \end{align*}


\begin{align*} M ' ( S ) = \varinjlim _ i N ( T _ i ) = \varinjlim _ i \varprojlim _ { n \in \Delta } N ( T _ { i , n } ) = \varprojlim _ { n \in \Delta } \varinjlim _ i N ( T _ { i , n } ) = \varprojlim _ { n \in \Delta } M ' ( S _ n ) , \end{align*}


\begin{align*} \int _ { X _ j } | u _ { \epsilon , j } | ^ 2 _ { h } e ^ { v _ \epsilon ( \Psi ) - \delta \max \{ \psi + T , 2 \log | F | \} } c ( - v _ \epsilon ( \Psi ) ) \le \int _ { X _ j } v '' _ { \epsilon } ( \Psi ) | f F | ^ 2 _ h e ^ { - u ( - v _ { \epsilon } ( \Psi ) ) } < + \infty . \end{align*}


\begin{align*} \mathrm { H } ( X ) = & \ln \left \{ \sqrt { 2 \pi } \sigma \mathrm { B } ( \alpha , \beta ) \right \} + \frac { 1 } { 2 \sigma ^ 2 } \left [ \mathrm { E } ( X ^ 2 ) - 2 \mu \mathrm { E } ( X ) + \mu ^ 2 \right ] \\ & + \frac { 1 } { \mathrm { B } ( \alpha , \beta ) } \sum _ { n = 1 } ^ { \infty } \frac { 1 } { n } \left [ ( \alpha - 1 ) \mathrm { B } ( \alpha , n + \beta ) + ( \beta - 1 ) \mathrm { B } ( n + \alpha , \beta ) \right ] . \end{align*}


\begin{align*} F ( \eta ) : = T - \frac { 1 } { \eta } I , \eta \in \Omega . \end{align*}


\begin{align*} \phi ( \sigma , \tau _ 1 \tau _ 2 ) = \widetilde \sigma \widetilde { \tau _ 1 } \widetilde { \tau _ 2 } { \widetilde { \sigma } } ^ { - 1 } { \widetilde { \tau _ 2 } } ^ { - 1 } { \widetilde { \tau _ 1 } } ^ { - 1 } , \phi ( \sigma , \tau _ 1 ) \phi ( \sigma , \tau _ 2 ) = \widetilde \sigma \widetilde { \tau _ 1 } { \widetilde \sigma } ^ { - 1 } { \widetilde { \tau _ 1 } } ^ { - 1 } [ \widetilde \sigma , \widetilde { \tau _ 2 } ] \end{align*}


\begin{align*} \left ( \sqrt { p ' _ j } - \sqrt { q ' _ j } \right ) ^ 2 & = q ' _ j \left ( \sqrt { 1 + \frac { p ' _ \gamma - q ' _ j } { q ' _ j } } - 1 \right ) ^ 2 \leq \frac { ( p ' _ i - q ' _ j ) ^ 2 } { q ' _ j } . \end{align*}


\begin{align*} l ( y _ p ) \leq \bigvee _ { 1 \leq i \leq n } l ( \Psi ( v _ i ) ) = \bigvee _ { 1 \leq i \leq n } \Phi ( l ( v _ i ) ) = \bigvee _ { 1 \leq i \leq n } \Phi ( p _ i ) = \Phi \left ( \bigvee _ { 1 \leq i \leq n } p _ i \right ) = \Phi ( p ) . \end{align*}


\begin{align*} \int _ { \Omega } ( K _ H ( x ) \nabla u | \nabla u ) d x = \int _ { \Omega } q ^ 2 \left ( K _ H ( x ) \nabla \left ( \frac { u } { q } \right ) | \nabla \left ( \frac { u } { q } \right ) \right ) d x . \end{align*}


\begin{align*} \Delta \Delta \Psi _ \delta ( x ) = p ( p + 2 ) \left ( | x | ^ 2 + \delta \right ) ^ { - \frac { p } { 2 } - 4 } ( A | x | ^ 4 + B | x | ^ 2 + C ) , \end{align*}


\begin{align*} \int _ { \{ z \in M : - \Psi ( z ) \in N \} } | \tilde { F } | ^ 2 _ h = 0 \end{align*}


\begin{align*} ( I - Q ) F ^ q ( B _ \phi u + \psi ( B _ \phi u , p ) , p ) = 0 . \end{align*}


\begin{align*} D '' u _ { m , m ' , \epsilon , j } + P _ { m , m ' } \big ( \sqrt { 2 \pi b N _ 1 \delta \tilde \delta _ m + N _ 1 \tilde { \lambda } _ { m ' } } h _ { m , m ' , \epsilon , j } \big ) = \lambda \end{align*}


\begin{align*} \alpha = \alpha _ { ( 0 ) } + \alpha _ { ( 1 ) } \end{align*}


\begin{align*} \delta ^ 2 ( 1 - \epsilon ^ 2 ) b _ { \alpha } u _ n ( x ) \leq u ( y ) - u ( x ) \leq C b _ { \alpha } ^ 2 , \mbox { w i t h } y = ( x ' , \delta ^ 2 b _ \alpha ) \in \partial _ 2 \omega \end{align*}


\begin{align*} A _ { 2 } = \sum _ { \beta \in F ^ { \times } } \sum _ { X \in S ( \beta ) } \Theta _ { \theta } \begin{bmatrix} m _ { 1 } & X \\ 0 & m _ { 2 } \end{bmatrix} \overline { \psi _ A ( { m _ { 1 } } ^ { - 1 } X } ) = q ^ { 8 } ( q - 1 ) ^ { 2 } . \end{align*}


\begin{align*} \partial _ z ^ * = M _ z , \end{align*}


\begin{align*} \kappa _ { V , I } : = \sum _ { J \subset I } ( - 1 ) ^ { | J | - | I | } K _ { V , J } . \end{align*}


\begin{align*} \delta ^ { \frac { 2 } { p - 1 } } s _ \delta ^ { - \frac { 2 } { p - 1 } } \phi ' ( 1 ) = a / \ln \frac { s _ \delta } { R } . \end{align*}


\begin{align*} J ( x , \delta ) = \int _ 1 ^ { x } | \theta ( y + \delta y ) - \theta ( y ) - \delta y | ^ 2 d y \end{align*}


\begin{align*} ( \mu _ T ( y , z ) | \mu _ T ( x , w ) ) = ( \mu _ T ( x , y ) | \mu _ T ( z , w ) ) . \end{align*}


\begin{align*} K _ { j } \nu ( x ) = \int _ { X } k _ j ( x , y ) d \nu ( y ) , K ^ { * } \nu ( x ) = \sup _ { j } \vert K _ j \nu ( x ) \vert . \end{align*}


\begin{align*} L ^ { * } : = \left \lbrace y \in E \mid h \left ( L , y \right ) \subset { \mathcal O } _ K \right \rbrace . \end{align*}


\begin{align*} \Psi ^ q + v + \psi ( v , p ( s ) ) = \Psi ^ q + v + \frac 1 2 ( v , p ( s ) - p ( 0 ) ) H \begin{pmatrix} v \\ p ( s ) - p ( 0 ) \end{pmatrix} + O ( \norm { ( v , s ) } ^ 3 ) , \end{align*}


\begin{align*} \begin{pmatrix} A & B \\ C & D \end{pmatrix} \begin{pmatrix} x \\ F ( x ) \end{pmatrix} + \begin{pmatrix} u \\ v \end{pmatrix} = \begin{pmatrix} \pi ( x ) \\ F ' ( \pi ( x ) ) \end{pmatrix} . \end{align*}


\begin{align*} g _ 1 ^ { \prime + } ( 0 ) - g _ 2 ^ { \prime + } ( 0 ) & = \sum _ { i = 1 } ^ n \int _ \R \big ( \hat { f } ' _ i ( x _ i ) Q _ i ( x _ i ) - Q _ i ' ( x _ i ) \big ) \big ( T _ i ( x _ i ) - x _ i \big ) d x _ i . \end{align*}


\begin{align*} | \theta ^ j _ { \alpha \beta } ( x ' , t ; y ' ) | & = t | \partial _ j K ^ L _ { \alpha \beta } ( x ' - y ' , t ) | \lesssim \frac { t } { | ( x ' - y ' , t ) | ^ n } , \\ | \nabla _ { y ' } \theta ^ j _ { \alpha \beta } ( x ' , t ; y ' ) | & \le t | \nabla ^ 2 K ^ L _ { \alpha \beta } ( x ' - y ' , t ) | \lesssim \frac { t } { | ( x ' - y ' , t ) | ^ { n + 1 } } . \end{align*}


\begin{align*} \delta ^ 2 \int _ { \Omega } \left ( ( K ( x ) - K ( z _ j ) ) \nabla V _ { \delta , Z , j } | \nabla V _ { \delta , Z , i } \right ) = O \left ( \frac { \delta ^ 2 } { | \ln \varepsilon | ^ 2 } \right ) . \end{align*}


\begin{align*} E _ { \xi ^ { \bar { \theta } } _ 1 ( u ) } \biggl [ \varphi \biggl ( x + u + \sum _ { i = 2 } ^ { n } Z ^ { { \bar { \theta } } [ u ] } _ i \biggr ) \biggr ] = & E _ { \xi ^ { \bar { \theta } } _ 1 ( u ) } \biggl [ \varphi \biggl ( x + u + \sum _ { i = 1 } ^ { n - 1 } Z ^ { \theta ^ { [ n - 1 ] } } _ i \biggr ) \biggr ] \\ = & V \bigg ( \frac { \xi _ 0 f _ 1 ( u ) } { \xi _ 0 f _ 1 ( u ) + ( 1 - \xi _ 0 ) f _ 2 ( u ) } , n - 1 , x + u \bigg ) . \end{align*}


\begin{align*} H _ { N } ( t ) : = H ( f _ { N , t } ; \nu _ { N , t } ) , \forall \ , t \in [ 0 , T ] . \end{align*}


\begin{align*} & [ C _ I , C _ J ] = 0 I \cap J = \emptyset \ \ \ \ I \subseteq J \ , , \\ & C _ I = \frac { 1 } { 2 } \sum _ { i , j \in I \atop i \neq j } C _ { i j } - ( | I | - 2 ) \sum _ { i \in I } C _ i \ , , \end{align*}


\begin{align*} \left \{ \begin{array} { l l } - \Delta ( c ( t ) - c ( s ) ) + n ( t ) ( c ( t ) - c ( s ) ) = - c ( s ) ( n ( t ) - n ( s ) ) , & x \in \Omega , \\ \nabla ( c ( t ) - c ( s ) ) \cdot \nu + ( c ( t ) - c ( s ) ) = 0 , & x \in \partial \Omega , \end{array} \right . \end{align*}


\begin{align*} p & : = \left ( 1 / 2 - 2 \epsilon - \epsilon ^ { 1 + \alpha } + \epsilon ^ { 1 + \beta } - \epsilon ^ { 1 + \delta } , 1 / 2 + 2 \epsilon , \epsilon ^ { 1 + \alpha } - \epsilon ^ { 1 + \beta } , \epsilon ^ { 1 + \delta } \right ) , \\ q & : = \left ( 1 / 2 , 1 / 2 - \epsilon ^ { 1 + \alpha } , \epsilon ^ { 1 + \alpha } , 0 \right ) , \end{align*}


\begin{align*} ( \mathbf { x } , - \mathbf { y } ) M _ { I J } = 0 . \end{align*}


\begin{align*} h ( a , x ) : = \min _ k \mu _ k ( a , x ) > 0 . \end{align*}


\begin{align*} \tilde H ( p , s ) = \tilde G ( p , s ) - F ( p , s ) . \end{align*}


\begin{align*} d ( J ^ * d \theta _ i ) = 0 , \mbox { l o c a l l y } J ^ * d \theta _ i = d y _ i . \end{align*}


\begin{align*} 0 \le V _ { \mathrm { o r i g } } - V _ { \mathrm { d e t } } \le \frac { n T ^ 2 } { 2 } \sum _ { i , j = 1 } ^ n \int _ { \R ^ n } | \partial _ { i j } g | ^ 2 \ , d \gamma _ { y ^ * , T } , \end{align*}


\begin{align*} S t _ { D N D C } \lim _ { n \rightarrow \infty } | | \mathcal { Y } _ { n } ( 1 , z ) - 1 | | _ { \infty } = 0 , \end{align*}


\begin{align*} g ' _ 2 = - \frac { 1 } { \nu } I \widehat { g _ 1 ' } I + \frac { 2 i } { \mu \nu } h _ z J g _ 1 ' \end{align*}


\begin{align*} \int _ { \Omega } - \delta _ N ^ 2 \left ( \frac { \partial K ( z _ { N , i } ) } { \partial z _ { i , h } } \nabla V _ { \delta _ N , Z _ N , i } \right ) u _ N = O \left ( \frac { \varepsilon _ N ^ 2 } { | \ln \varepsilon _ N | ^ { p - 1 } } \right ) . \end{align*}


\begin{align*} { \rm D } ^ 2 \widetilde { W } ( { \rm D } \phi ) . ( \xi \otimes \eta , \xi \otimes \eta ) = ( b _ 1 + b _ 2 ) \| \xi \| ^ 2 \| \eta \| ^ 2 \sin ^ 2 \theta + ( b _ 1 + b _ 3 ) \| \xi \| ^ 2 \| \eta \| ^ 2 \cos ^ 2 \theta , \end{align*}


\begin{align*} \underline { h } _ { \mu } ( f _ { 1 , \infty } ) : = \int \underline { h } _ { \mu } ( f _ { 1 , \infty } , x ) \ d \mu ( x ) . \end{align*}


\begin{align*} \sum _ { x y \ne 1 } f _ { x , y } = A + B , \sum _ { z \in \bar { \mathbb { F } } ^ * _ q } g _ { z } = C , \sum _ { u \in \bar { \mathbb { F } } _ q } h _ { u } = 0 . \end{align*}


\begin{align*} \ \begin{cases} \partial _ t v _ n + \partial _ x v _ n \cdot B + \frac { 1 } { 2 } \mathrm { T r } \big [ \partial _ { x x } v _ n \Sigma \Sigma ^ \top \big ] + f _ n ( t , \partial _ { x } v _ n \Sigma , x ) [ 0 , 1 ) \times \R ^ \ell = 0 , \\ v _ n ( 1 , x ) = F ( x ) x \in \R ^ \ell \end{cases} \end{align*}


\begin{align*} \omega : = \R \setminus B _ R ( 0 ) = \{ x \in \R : \Vert x \Vert \ge R \} \end{align*}


\begin{align*} \Psi ^ \pm _ 2 ( x ; z ) = \frac { 1 } { 2 i k } \begin{pmatrix} \psi ^ \pm _ { 1 2 } \\ - \bar { u } _ x \psi ^ \pm _ { 1 2 } + 2 i k \psi ^ \pm _ { 2 2 } \end{pmatrix} , \end{align*}


\begin{align*} n _ { i } & = 3 n _ { i - 1 } - 3 n _ { i - 2 } + n _ { i - 3 } + \kappa \\ \left ( n + a \right ) _ { i } & = 3 \left ( n + a \right ) _ { i - 1 } - 3 \left ( n + a \right ) _ { i - 2 } + \left ( n + a \right ) _ { i - 3 } + \lambda \\ \left ( n + b \right ) _ { i } & = 3 \left ( n + b \right ) _ { i - 1 } - 3 \left ( n + b \right ) _ { i - 2 } + \left ( n + b \right ) _ { i - 3 } - \lambda \end{align*}


\begin{align*} u ^ \infty _ 1 ( \hat { \mathbf x } ; u ^ i ) = u ^ \infty _ 2 ( \hat { \mathbf x } ; u ^ i ) \end{align*}


\begin{align*} \kappa _ { p , d } = 2 ^ { 1 - p } \frac { \Gamma \left ( \frac { d - p } { 2 } \right ) } { \Gamma \left ( \frac { d } { 2 } \right ) \Gamma \left ( \frac { p } { 2 } \right ) } . \end{align*}


\begin{align*} B _ q ( \phi ) ( z ) : = \displaystyle \int _ { \mathbb { R } } \overline { A _ q ( z , x ) } \phi ( x ) d x . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ M \sum \limits _ { \substack { j = 1 \\ j \neq l } } ^ M \sum _ { l = 1 } ^ M a _ l ^ 2 \phi ^ { \left ( i , l \right ) ^ * } \phi ^ { \left ( j , l \right ) } = 2 \sum _ { i = 1 } ^ { M - 1 } \sum _ { q = i + 1 } ^ { M } \sum _ { r = 1 } ^ M a _ r ^ 2 \Re \left [ \phi ^ { \left ( i , r \right ) ^ * } \phi ^ { \left ( q , r \right ) } \right ] . \end{align*}


\begin{align*} \lambda _ n ( A ) = \frac { \lambda ( A \cap A _ n ) } { \lambda ( A _ n ) } , \end{align*}


\begin{align*} \Lambda = \bigcup _ { n = 0 } ^ { \infty } \Lambda _ n = \sum _ { k = 0 } ^ { \infty } ( R ^ { t } ) ^ { k } L , \end{align*}


\begin{align*} u _ n & = \sum _ { j = 0 } ^ \infty \Xi _ j ( \vec u _ { 0 , n } ) = \sum _ { j = 0 } ^ \infty \Xi _ j ( \vec u _ { 0 } + \vec \phi _ n ) , \end{align*}


\begin{align*} B _ n = ( \mathbb Z / 2 \mathbb Z ) ^ n \rtimes S _ n \end{align*}


\begin{align*} \ln \left \{ 1 - \Phi \left ( \frac { x - \mu } { \sigma } \right ) \right \} = \sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ n } { n } \ , \Phi \left ( \frac { x - \mu } { \sigma } \right ) ^ n . \end{align*}


\begin{align*} ( d / d t ) v + A v = 0 , v \in \R ^ D , \end{align*}


\begin{align*} & \nabla _ V \left ( \nabla _ Y \hat X - \omega ( Y ) \right ) \\ & = \nabla _ Y \nabla _ V \hat X + R ( V , Y ) \hat X - ( \nabla _ V \omega ) ( Y ) - \omega ( \nabla _ V Y ) \\ & = \nabla _ Y ( \omega ( V ) ) + R ( V , Y ) \hat X + R ( \hat X , V ) Y - T _ h ( V , Y ) - \omega ( \nabla _ V Y ) \mbox { ( b y \eqref { e q : O D E } ) } \\ & = ( \nabla _ Y \omega ) ( V ) - R ( Y , \hat X ) V - T _ h ( V , Y ) \mbox { ( b y B i a n c h i i d e n t i t y ) } \end{align*}


\begin{align*} 2 a = ( a + b ) + ( a - b ) \end{align*}


\begin{align*} \hat { b } _ { 1 , \delta } ( x ) = \left \{ \begin{array} { c c } b _ { 1 , \delta } ( x ) & x \in \bar \Delta \\ 0 & x \not \in \hat \Delta \ , . \end{array} \right . \end{align*}


\begin{align*} & ( i ) \ \ \ \nabla ( \gamma _ 0 J + \sum \limits _ { k = 1 } ^ m \gamma _ k f _ k ) ( x ^ * ) = 0 \\ & ( i i ) \ \ \gamma _ k f _ k ( x ^ * ) = 0 \ \ k \in \{ 1 , . . . , m \} \\ & ( i i i ) \ \gamma _ { 0 } A _ J + \sum \limits _ { k = 1 } ^ m \gamma _ k A _ k \succeq 0 \end{align*}


\begin{align*} \lambda _ n = ( \lambda { m } ) ^ n . \end{align*}


\begin{align*} R = \frac { 1 } { 2 J ( X _ { J } ) } \left [ 2 \delta ( X _ { J } ) g \wedge S + 2 \omega ( X _ { J } ) g \wedge g + E \wedge g + F \wedge S \right ] , \end{align*}


\begin{align*} x \cdot ( x \wedge y ) = x \cdot y \in I . \end{align*}


\begin{align*} B _ \gamma ( z ) = A ( z ) \bigg ( \gamma + \sum _ n \Big ( \frac { 1 } { t _ n - z } - \frac { 1 } { t _ n } \Big ) \mu _ n \bigg ) . \end{align*}


\begin{align*} M N A P P ( m , n ) = & \biggl \{ A \in N A P P ( m , n ) : A \mbox { i s t h e m i n i m a l e l e m e n t i n t h e s e t } \\ & \{ A , \rho ( A ) , \sigma ( A ) , ( \sigma \rho ) ( A ) \} \mbox { u n d e r t h e l e x i c o g r a p h i c o r d e r } \biggr \} \end{align*}


\begin{align*} \begin{aligned} \int _ { \Lambda _ { h } } ( w - v ) \partial _ { \nu } u _ { 0 } - u _ { 0 } \partial _ { \nu } ( w - v ) \mathrm { d } \sigma - \int _ { \Gamma _ h ^ { \pm } } \eta u _ 0 v \mathrm d \sigma & = k ^ { 2 } \int _ { S _ { h } } ( q - 1 ) v u _ { 0 } \mathrm { d } \mathbf x . \end{aligned} \end{align*}


\begin{align*} \partial _ i g ( x ) = \frac { 1 } { n } D _ m G \bigg ( \frac { 1 } { n } \sum _ { k = 1 } ^ n \delta _ { x _ k } , x _ i \bigg ) , \partial _ { i j } g ( x ) = \frac { 1 } { n ^ 2 } D _ m ^ 2 G \bigg ( \frac { 1 } { n } \sum _ { k = 1 } ^ n \delta _ { x _ k } , x _ i , x _ j \bigg ) , \ \ i \neq j . \end{align*}


\begin{align*} F ' _ m ( n ) = 1 + H ' ( n , m ) - r _ m ( n ) \sum _ { \begin{smallmatrix} k = 1 \\ k | m \end{smallmatrix} } ^ m \frac { \mu ( k ) } { k } , \end{align*}


\begin{align*} \sigma ( P ) ( x , \xi ) = \mathfrak a _ { 2 } + \mathfrak a _ { 1 } + \mathfrak a _ { 0 } . \end{align*}


\begin{align*} \widetilde { \sigma } _ { j i } & = ( 2 \ , \mu ^ * + \varkappa ) ( u _ { i , j } + u _ { j , i } ) + \varkappa \ , \epsilon _ { j i m } \Big ( \underbrace { \frac { 1 } { 2 } \epsilon _ { m k s } u _ { s , k } } _ { r _ m } - \vartheta _ m \Big ) + \lambda \ , u _ { k , k } \delta _ { j i } , \\ * m _ { j i } & = \beta \ , \vartheta _ { j , i } + \gamma \ , \vartheta _ { i , j } + \alpha \ , \vartheta _ { k , k } \delta _ { j i } \ , . \end{align*}


\begin{align*} \pi _ 1 : G \to M , \pi _ 1 ( s ( q ) z ) = \overline { \pi } ( q ) \chi _ 1 ( z ) \forall q \in Q , z \in Z . \end{align*}


\begin{align*} d _ i ( x ) = \left \{ \begin{array} { l l l l l } d _ i x \in X _ { k - 1 } \subseteq ( \Sigma X ) _ { m - 1 } & \mbox { i f } 0 \leq i \leq k , \\ x \in X _ { k } \subseteq ( \Sigma X ) _ { m } & \mbox { i f } k + 1 \leq i \leq m , \end{array} \right . \end{align*}


\begin{align*} 0 ^ p \cdot F _ 2 + 1 ^ p \cdot F _ 1 + 2 ^ p \cdot F _ 0 \ & = \ A _ p F _ 2 + B _ p F _ 3 - \sum _ { k = 0 } ^ p \binom { p } { k } B _ k 2 ^ { p - k } , \intertext { a n d i f w e s i m p l i f y t h i s a n d p u l l o u t t h e $ k = p $ t e r m f r o m t h e s u m , w e g e t } 1 \ & = \ A _ p + 2 B _ p - B _ p - \sum _ { k = 0 } ^ { p - 1 } \binom { p } { k } B _ k 2 ^ { p - k } . \end{align*}


\begin{align*} \begin{cases} \mathcal { L } _ H \varphi ( x ) = - \cdot ( K _ H ( x ) \nabla \varphi ( x ) ) = \frac { 1 } { \varepsilon ^ 2 } f ( \varphi ( x ) - \mu ) , \ & x \in \Omega , \\ \varphi ( x ) = 0 , \ & x \in \partial \Omega . \end{cases} \end{align*}


\begin{align*} d \tilde { x } _ t = \lambda _ A \tilde { x } _ t d t + e ^ { \lambda _ A t } B ( x _ t , x _ t ) d t - A \tilde { x } _ t d t + e ^ { \lambda _ A t } \sigma d W _ t . \end{align*}


\begin{align*} L ^ 2 ( z ) = 4 e ^ { L _ j z } \cosh ^ 2 \Big ( \frac { \sqrt { 5 } F _ j } { 2 } z \Big ) . \end{align*}


\begin{align*} V _ 1 ^ { - 1 } ( k ) R ( x ; z ) V _ 1 ( k ) = V _ 2 ^ { - 1 } ( k ) R ( x ; z ) V _ 2 ( k ) = J ( x ; k ) , z \in \mathbb { R } , k \in \mathbb { R } \cup i \mathbb { R } . \end{align*}


\begin{align*} \begin{aligned} & S _ h \subset T _ h \subset \left \{ b \in L ^ { 2 } _ { \sigma , \textrm { a x i } } ( \mathbb { R } ^ { 3 } ) \ \middle | \ H [ b ] = h \right \} , \\ & I _ h = \inf \left \{ E [ b ] \ \middle | \ b \in T _ h \right \} . \end{aligned} \end{align*}


\begin{align*} Q _ \delta L _ \delta \omega = Q _ \delta l _ { 1 , \delta } + Q _ \delta l _ { 2 , \delta } + Q _ \delta R _ \delta ( \omega ) , \end{align*}


\begin{align*} \langle \mathcal { G } ^ { \mathrm { e n d } } \cdot , \cdot \rangle = \mathcal { G } \end{align*}


\begin{align*} a _ { n - i } ( \gamma ^ { ( q ^ { n - i } - q ^ n ) } - 1 ) = 0 \end{align*}


\begin{align*} \begin{aligned} & z '' _ { b e n } + ( M ^ S _ { b e n } - P ) z _ { b e n } = 0 \\ & z '' _ { n b } + ( M ^ S _ { n b } - P ) z _ { n b } = 0 , \end{aligned} \end{align*}


\begin{align*} \int _ { 0 } ^ { \lambda } \mu [ \xi ( \mu , t ) - \xi ( \mu , \tau ) ] \ d \mu = - \int _ { \tau } ^ { t } \int _ { \lambda } ^ { \infty } \int _ { 0 } ^ { \lambda } \nu \Lambda ( \mu , \nu ) \xi ( \mu , s ) \xi ( \nu , s ) \ d \nu d \mu d s . \end{align*} % \end{align*}


\begin{align*} \begin{cases} \Delta _ { \Psi } v = f ' ( u ) \ , v & \O \\ v = 0 & \partial \O . \end{cases} \end{align*}


\begin{align*} A _ k = \sum _ { j = 0 } ^ { k - 1 } \binom { k } { j } B _ j . \end{align*}


\begin{align*} 0 = \int _ \Sigma \Delta x _ { n + 1 } \ , d A = \int _ \Sigma \left ( \eta x _ { n + 1 } ^ m + \lambda \right ) \nu _ { n + 1 } \ , d A = \int _ \Sigma \eta x _ { n + 1 } ^ m \nu _ { n + 1 } \ , d A , \end{align*}


\begin{align*} e _ 2 ( C ) = 2 - I ( C ) - s ( C ) ; \end{align*}


\begin{align*} \alpha _ i + \bar { \alpha } _ j = 1 . \end{align*}


\begin{align*} \partial _ j \xi _ \tau ( 0 , \omega ) = 2 \pi i \ ; y _ j ( \xi ) \ ; \xi ( \omega ) . \end{align*}


\begin{align*} E _ { \xi _ 0 } \biggl [ \varphi \biggl ( x + \sum _ { i = 1 } ^ { n } Z _ i ^ \theta \biggr ) \biggr ] = \xi _ 0 E _ 1 \biggl [ \varphi \biggl ( x + \sum _ { i = 1 } ^ { n } Z _ i ^ \theta \biggr ) \biggr ] + ( 1 - \xi _ 0 ) E _ 2 \biggl [ \varphi \biggl ( x + \sum _ { i = 1 } ^ { n } Z _ i ^ \theta \biggr ) \biggr ] . \end{align*}


\begin{align*} X _ i \left ( t , x _ { 0 } , t _ { 0 } \right ) : = \lambda _ i ( t - t _ { 0 } ) + x _ { 0 } , t \in [ 0 , T ] . \end{align*}


\begin{align*} f ( x ) = a x ^ { q + 1 } + b x ^ q + c x + d \end{align*}


\begin{align*} ( D _ { a ^ + } ^ { \alpha } f ) ( x ) : = \frac { d } { d x } \left [ ( { \bf I } _ { a ^ + } ^ { 1 - \alpha } f ) ( x ) \right ] \end{align*}


\begin{align*} \frac { \partial f } { \partial y } ( t z ) = - \frac { \psi _ 0 } { \psi _ 1 } \frac { \partial f } { \partial x } ( t z ) . \end{align*}


\begin{align*} \kappa ( t ) = h _ { \alpha } ( t ) \cdot \ , \kappa _ 1 ( t ) , \ \kappa _ 1 ( t ) = \sum _ { k = 0 } ^ { + \infty } \ , a _ k t ^ k , \ a _ 0 \not = 0 , \ 0 < \alpha < 1 , \end{align*}


\begin{align*} \psi ( p ) = \psi _ { p _ 0 } [ \eta ] ( p ) \stackrel { d e f } { = } \inf _ { r \in R ( p ) } g ( p _ 0 , r , | \eta | _ r ) , \ p \le p _ 0 . \end{align*}


\begin{align*} \rho ( t ) = \begin{cases} 0 , & t = 0 , \\ \exp ( - 1 / t ) , & 0 < t < T , \end{cases} \end{align*}


\begin{align*} x ^ { t + 1 } = ( I + \gamma _ t B ) ^ { - 1 } \left ( x ^ t + \alpha _ t ( x ^ t - x ^ { t - 1 } ) - \gamma _ t ( F ( x ^ t ) + \beta _ t ( F ( x ^ t ) - F ( x ^ { t - 1 } ) ) ) \right ) , \end{align*}


\begin{gather*} f \in L , \ , \ , f \le c \quad Q ( f ) = \sup _ { g \in L , \ , g \le c } Q ( g ) \quad Q \in M ; \end{gather*}


\begin{align*} \dot { s } \varepsilon ( z ) \dot { s } \varepsilon ( 1 ) \xi _ 2 = \dot { s } ( \varepsilon ( z - 1 ) - \varepsilon ( z ) ) \xi _ 2 + \dot { s } \varepsilon ( z ) \varpi . \end{align*}


\begin{align*} P _ { e , w } ( q ) = \sum _ { 1 \leq i \leq k } q ^ { \frac { \ell ( w ) - \ell ( w _ i ) } { 2 } } . \end{align*}


\begin{align*} \left \{ \begin{aligned} a x = \lambda x , \\ c x + d = \lambda . \end{aligned} \right . \end{align*}


\begin{align*} \mathfrak h = \frac { _ { n - 1 } ( \Sigma ) } { \min \{ _ n ( A ) , _ n ( B ) \} } . \end{align*}


\begin{align*} S _ r = \min _ { k } \mathbb { E } \left [ \bar { R _ { c , k } } \right ] + \sum _ { k = 1 } ^ K \mathbb { E } \left [ \bar { R } _ k \right ] , \end{align*}


\begin{align*} F ( x ) = \frac { 1 } { \operatorname { B } ( a , b ) } \int ^ { x } _ { 0 } \frac { 2 \theta } { y ^ { 3 } } \exp \left ( - \frac { a \theta } { y ^ { 2 } } \right ) \left \{ 1 - \exp \left ( - \frac { \theta } { y ^ { 2 } } \right ) \right \} ^ { b - 1 } \mathrm { d } y . \end{align*}


\begin{align*} \mathbf { w } = \frac { w } { k } \overrightarrow { \zeta } , \end{align*}


\begin{align*} \sup _ { x \in \mathbb { R } , t > 0 } | \mathcal { I } ( x , t ) | \leq \bar { \tau } = \max \left ( \sum _ { i = 1 } ^ p \frac { 2 R } { \lambda _ i } , \sum _ { i = p + 1 } ^ n \frac { 2 R } { \lambda _ i } \right ) . \end{align*}


\begin{align*} \tau _ { n } ^ M : = & T \wedge \inf \Big \{ t \geq 0 : \Vert Y _ n ( t ) \Vert _ H ^ 2 > M \Big \} \\ & \wedge \inf \Big \{ t \geq 0 : \int _ 0 ^ t \Vert Y _ n ( s ) \Vert _ { V } ^ { \alpha } d s > M \Big \} . \end{align*}


\begin{align*} h _ 0 \ , * \ h _ 1 = h _ 1 . \end{align*}


\begin{align*} p ( x ) = p _ 1 ( x ) + p _ 2 ( x ) \geq 0 + g _ a ( x ) = g _ a ( x ) . \end{align*}


\begin{align*} u ^ k = \arg \min _ { w \in H ^ 1 ( \Omega ) } \bigg ( \frac { | \Omega | } { 2 \mathrm { N } _ I } \sum _ { n = 1 } ^ { \mathrm { N } _ I } ( w - u ^ { k - 1 } ) ^ 2 ( x _ n ^ I ) + \tau _ k \tilde { N } ( w ; w ) \bigg ) . \end{align*}


\begin{align*} \L ^ n _ { \overline { A } } f \to \int f \ d \mu \ , \ \ \ \L ^ n _ { \overline { A } ^ * } f \to \int f \circ \theta ^ { - 1 } \ d \mu ^ * = \int f \ d \left ( \theta ^ { - 1 } _ * \mu ^ * \right ) \ , \end{align*}


\begin{align*} \psi _ { 2 n } = \begin{pmatrix} \psi _ { 2 } \\ & \psi _ { 2 } \\ & & \ddots \\ & & & \psi _ { 2 } \end{pmatrix} = \bigoplus _ { 1 } ^ { n } \psi _ { 2 } , \psi _ { 2 } = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} . \end{align*}


\begin{align*} \sigma _ { W _ { \rm s } } ^ 2 = \frac { 1 } { K } \sum _ { n = 0 } ^ { K - 1 } \sigma _ { \rm a } ^ 2 ( x [ n ] ) \simeq \mathbb { E } \{ \sigma _ { \rm a } ^ 2 ( x [ n ] ) \} , \end{align*}


\begin{align*} P _ { n , k } ( z ) = \sum _ { i = 0 } ^ { k + d _ n } \alpha _ { n , k , i } \binom { z } { i } \end{align*}


\begin{align*} \mathsf { M } ^ n = & \{ \mathsf { m } ^ n _ 1 , \cdots , \mathsf { m } ^ n _ n \} \\ = & \left \{ g \bigl ( \xi _ 0 \bigr ) , g \bigl ( \xi _ 1 ^ { \mathsf { M } ^ n } \bigr ) , \cdots , g \bigl ( \xi _ { n - 1 } ^ { \mathsf { M } ^ n } \bigr ) \right \} \in \Theta _ n , \end{align*}


\begin{align*} \forall t > 0 , \ H _ { m i n } ( E [ t ] ) = t H _ { m i n } ( E ) . \end{align*}


\begin{align*} C u r \mathcal { G } = \mathbb { C } [ \partial ] \otimes \mathcal { G } , [ a _ \lambda b ] = [ a , b ] , \end{align*}


\begin{align*} r ( a b ) = ( r a ) b = ( - 1 ) ^ { p ( r ) p ( a ) } a ( r b ) \end{align*}


\begin{align*} ( \partial - \nabla ) _ X Y = - \sum _ { i = 1 , 2 } \sqrt { c _ i } \ \langle X _ i , Y \rangle \nu _ i + B ( X , Y _ T ) - B ^ * ( X , Y _ N ) \end{align*}


\begin{align*} \sigma _ { k } ( \lambda ) = \sum _ { i _ { 1 } < \ldots < i _ { k } } \lambda _ { i _ { 1 } } \ldots \lambda _ { i _ { k } } , \ \ k = 1 , \ldots , n , \end{align*}


\begin{align*} \chi _ 2 \geq ( 1 - \sum \limits _ { j = 3 } ^ n w _ j ) \chi - \chi _ 1 - \sum \limits _ { j = 3 } ^ n k _ j + ( n - 2 ) r \end{align*}


\begin{align*} ( \nu ( u ) ) ' = \nu ' ( h ) ( \bar u ) + \nu ( v ) \end{align*}


\begin{align*} \int _ { \Omega } - \delta _ N ^ 2 \left ( ( K ( x ) - K ( z _ { N , i } ) ) \nabla \frac { \partial V _ { \delta _ N , Z _ N , i } } { \partial z _ { i , h } } \right ) u _ N = O \left ( \frac { \varepsilon _ N ^ 2 } { | \ln \varepsilon _ N | ^ { p - 1 } } \right ) . \end{align*}


\begin{align*} g _ { i j } = \left \langle e _ i , e _ j \right \rangle , h _ { i j } = \left \langle \overline { D } _ { e _ i } e _ j , \mathbf { n } \right \rangle , \end{align*}


\begin{align*} \frac { d } { d \tau } \mathcal { U } _ { \lambda } ( \tau ) = C _ { 1 } \Theta _ { \lambda } ( \tau ) \geq C _ { 1 } \mathcal { U } _ { \lambda } ( \tau ) . \end{align*}


\begin{align*} \psi _ 1 ^ - ( x , t ; k ) = a ( k ) \psi _ 1 ^ + ( x , t ; k ) + b ( k ) e ^ { 2 i k ^ 2 x + 2 i \eta ^ 2 t } \psi _ 2 ^ + ( x , t ; k ) , k \in \mathbb { R } \cup i \mathbb { R } , \end{align*}


\begin{align*} \frac \varpi \pi x ' = \varphi _ M ^ { - 1 } ( \frac { \varpi ^ p } \pi x ' ) . \end{align*}


\begin{align*} D ( p ) = \frac { \Gamma ( 3 - p ) } { \left [ \Gamma \left ( 2 - \frac { p } { 2 } \right ) \right ] ^ 2 \Gamma \left ( 3 - \frac { p } { 2 } \right ) } , \end{align*}


\begin{align*} a \cdot b : = \bigvee \{ c \in A \ : | \ : c \wedge a \le b \} . \end{align*}


\begin{align*} Y _ n = \{ x y - t ^ { 2 n + 1 } z ^ 2 = 0 \} \subset \mathbb { A } ^ 1 _ \mathbb { C } \times _ \C \mathbb { P } ^ 2 _ { \mathbb { C } } = \mathrm { P r o j } _ { \mathbb { C } [ t ] } ( \mathbb { C } [ t ] [ x , y , z ] ) \end{align*}


\begin{align*} f ( u , y ) = \mp \sum _ { u < v < y } f ( u , v ) f ( v , y ) . \end{align*}


\begin{align*} \sum _ { J } \frac { ( - i ) ^ { | J | } } { J ! } \partial _ { \xi } ^ { J } q \ , \partial _ { x ^ \prime } ^ { J } q - \sum _ { J } \frac { ( - i ) ^ { | J | } } { J ! } \partial _ { \xi } ^ { J } b \ , \partial _ { x ^ \prime } ^ { J } q - \frac { \partial q } { \partial x _ n } + c = 0 , \end{align*}


\begin{align*} ( \partial \alpha ) ( u , v , w ) = \sum _ { \mathfrak { S } } \langle \alpha _ u v , w \rangle u , v , w \in E , \end{align*}


\begin{align*} \det ( D ^ 2 ( a _ k \hat C _ { q _ k , N } ) ) = c _ k \delta _ { q _ k } . \end{align*}


\begin{align*} g _ { a b } = \delta _ { a b } - \frac { 1 } { 3 } R _ { a c b d } x ^ { c } x ^ { d } + o ( { \mathbf { x ^ { 2 } } } ) , \end{align*}


\begin{align*} \frac { d } { d t } \mathbb { E } [ u _ t ] \bigg | _ { t = 0 } = \frac { d } { d t } \int _ \Omega W ( x , u _ t ( x ) , \nabla u _ t ( x ) ) \ , d x \bigg | _ { t = 0 } = 0 . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ { d } [ a _ i ( t , x , u , z ) - a _ i ( t , x , u , \tilde { z } ) ] ( z _ i - \tilde { z } _ i ) > 0 . \end{align*}


\begin{align*} & = \sup _ { ( x ' , y ' ) \in X \times \mathbb R } \{ \psi ( x ' , y ) - \frac { 1 } { 2 \epsilon } \left ( \| x - x ' \| ^ 2 - \| x \| ^ 2 \right ) - \frac { 1 } { 2 } \left ( | y - y ' | ^ 2 - \| y \| ^ 2 \right ) \} \\ & = \sup _ { ( x ' , y ' ) \in X \times \mathbb R } \{ \psi ( x ' , y ) + \frac { 1 } { \epsilon } x \cdot x ' + \frac { 1 } { \epsilon } y \cdot y ' - \frac { 1 } { 2 \epsilon } \| x ' \| ^ 2 - \frac { 1 } { 2 \epsilon } \| y ' \| ^ 2 \} \end{align*}


\begin{align*} a = \frac { u _ 1 - u _ 0 \beta } { \alpha - \beta } , b = \frac { u _ 0 \alpha - u _ 1 } { \alpha - \beta } \end{align*}


\begin{align*} h ( p ) = \sum _ k \lambda _ k a _ k ^ p \end{align*}


\begin{align*} \| \psi ( \mathbf x ) \| _ { L ^ { 4 } ( \Lambda _ h ) } \leq C \| \psi ( \mathbf x ) \| _ { H ^ { \frac { 7 } { 8 } , 8 } ( \Lambda _ h ) } \leq C \| \psi ( \mathbf x ) \| _ { H ^ { 1 , 8 } ( \mathcal C ^ h ) } = O ( \tau ^ { - \frac { 2 } { 5 } } ) , \end{align*}


\begin{align*} s _ t = \frac { \varphi ( m ) - 1 } { m - p _ t } ( r _ { p _ t } - r _ m ) . \end{align*}


\begin{align*} u ( x ) \geq l ( y ' ) + \frac { 1 } { 4 } \sum _ { \beta = 1 } ^ { n - 1 } b _ \beta y _ \beta ^ 2 - C b _ \alpha | \hat { y } | ^ 2 \end{align*}


\begin{align*} L _ p ( s , \vee ^ 2 ) = & \prod _ { 1 \leq i \leq j \leq M } ( 1 - \alpha _ { \pi } ( p , i ) \alpha _ { \pi } ( p , j ) p ^ { - s } ) ^ { - 1 } , \mbox { a n d } \\ L _ p ( s , \wedge ^ 2 ) = & \prod _ { 1 \leq i < j \leq M } ( 1 - \alpha _ { \pi } ( p , i ) \alpha _ { \pi } ( p , j ) p ^ { - s } ) ^ { - 1 } . \end{align*}


\begin{align*} q _ i = \begin{cases} 0 . 5 \tilde { q } _ i , & i \in [ m ] \\ 0 . 5 \tilde { q } _ { i - m } , & i \in [ k ] \setminus [ m ] . \end{cases} \end{align*}


\begin{align*} & \lambda _ p < \lambda _ q \equiv 0 < \lambda _ r , \\ & p = 1 , \dots , l , q = l + 1 , \dots , m , r = m + 1 , \dots , n , \end{align*}


\begin{align*} F = \bigcup _ { n = 0 } ^ \infty K _ n , \end{align*}


\begin{align*} \sigma _ { \rm a } ^ 2 ( x ) = \left ( \psi _ { 1 } x + \psi _ { 2 } \right ) T _ { \rm s } \ , e ^ { - \frac { \left ( \psi _ 1 x + \psi _ 2 \right ) \tau _ { \rm d } } { N _ { \rm a } } } - \frac { \left ( \psi _ { 1 } x + \psi _ { 2 } \right ) ^ 2 T _ { \rm s } \tau _ { \rm d } } { N _ { \rm a } } \left ( 2 - \frac { \tau _ { \rm d } } { T _ { \rm s } } \right ) e ^ { - \frac { 2 \left ( \psi _ 1 x + \psi _ 2 \right ) \tau _ { \rm d } } { N _ { \rm a } } } . \end{align*}


\begin{align*} \dot \phi _ i = \frac 1 M \sum _ { j = 1 } ^ M a _ { i - j } \sin ( \phi _ j - \phi _ i ) , i = 1 , \dots , M , \end{align*}


\begin{align*} \big | M ^ * _ v ( t ) - M ^ * _ v ( \delta n ) \big | & \le e ^ { - ( \lambda ^ * - \lambda ) t } \big | M _ v ( t ) - M _ v ( \delta n ) \big | + \big ( e ^ { - ( \lambda ^ * - \lambda ) \delta n } - e ^ { - ( \lambda ^ * - \lambda ) t } \big ) | M _ v ( \delta n ) | \\ & \le \delta e ^ { - ( \lambda ^ * - \lambda ) ( t - \delta n ) } + \delta \big ( 1 - e ^ { - ( \lambda ^ * - \lambda ) ( t - \delta n ) } \big ) = \delta , \end{align*}


\begin{align*} \begin{cases} \Delta _ \Psi u = f ( u ) & A ( r _ { 1 } , r _ { 2 } ) \\ u \equiv c _ { 1 } & \{ r _ { 1 } \} \times N \\ u \equiv c _ { 2 } & \{ r _ { 2 } \} \times N . \end{cases} \end{align*}


\begin{align*} \int _ { \mathfrak U } | \omega | = { \phi _ s } _ ! \int _ { \mathfrak Y } | \omega | , \end{align*}


\begin{align*} \sigma ( F ) = \Omega \setminus \rho ( F ) . \end{align*}


\begin{align*} c _ k = 0 \ \ c _ k = \left ( \frac { 1 } { p ^ { d / \tau } } \right ) ^ { p ^ { v _ p ( k ) } } \ \end{align*}


\begin{align*} \sigma ( \alpha + \beta ) = \sigma ( \alpha ) + \sigma ( \beta ) \end{align*}


\begin{align*} z = \prod _ { c = 0 } ^ { \ell - 1 } X _ c . \end{align*}


\begin{align*} Z ^ 1 ( v , s ) = \Psi ^ q + v \end{align*}


\begin{align*} ( \widehat { h } ^ i , \widehat { h } ^ i { } _ j , \widehat { h } ^ i { } _ { j k } ) & = ( \phi ( h ^ a ) ^ i , D \phi ( h ^ a ) ^ i { } _ \alpha h ^ \alpha { } _ j , H \phi ( h ^ a ) ^ i { } _ { \alpha \beta } h ^ \alpha { } _ j h ^ \beta { } _ k + D \phi ( h ^ a ) ^ i { } _ \alpha h ^ \alpha { } _ { j k } ) \\ * & = ( \phi ( h ^ a ) ^ i , ( D \phi ( h ^ a ) ^ i { } _ j , H \phi ( h ^ a ) ^ i { } _ { j k } ) ( h ^ i { } _ j , h ^ i { } _ { j k } ) ) , \end{align*}


\begin{align*} S _ { \mu , \nu , \lambda } : = \mathop { \mathrm { s u p } } \limits _ { \omega \in \mathcal { A } _ { \mu , \nu } } \mathcal { E } _ { \lambda } \left ( \omega \right ) , \end{align*}


\begin{align*} \langle - \Delta z , p \rangle = - \langle F ' ( \overline { y } ; z ) , p \rangle + \langle h , p \rangle . \end{align*}


\begin{align*} [ - T , T ] = \bigcup _ { j = - [ T / \tau ] - 1 } ^ { [ T \tau ] } [ j \tau , ( j + 1 ) \tau ] \cap [ - T , T ] . \end{align*}


\begin{align*} q , r , \tilde q , \tilde r \geq 2 \quad \tfrac { 2 } { q } + \tfrac { 1 } { r } = \tfrac { 1 } { 2 } = \tfrac { 2 } { \tilde q } + \tfrac { 1 } { \tilde r } . \end{align*}


\begin{align*} D _ 1 = \displaystyle \sum _ { \substack { X \in S ( 0 ) \\ t = 2 } } \Theta \begin{bmatrix} m _ 1 & X \\ 0 & m _ 2 \end{bmatrix} \overline { \psi _ A ( { m _ 1 } ^ { - 1 } X } ) = - \theta ( a ) q ^ 8 ( 1 - q ) . \end{align*}


\begin{align*} q _ i = \begin{cases} 0 . 5 \tilde { q } _ i , & i \in [ m ] \\ 0 . 5 \tilde { q } _ { i - m } , & i \in [ k ] \setminus [ m ] . \end{cases} \end{align*}


\begin{align*} \phi ( \alpha , \tau ) = \widetilde \alpha \widetilde \tau { \widetilde \alpha } ^ { - 1 } \cdot { \widetilde \tau } ^ { - 1 } = \alpha ( x _ { i _ 1 } \cdots x _ { i _ k } ) \cdot ( x _ { i _ 1 } \cdots x _ { i _ k } ) ^ { - 1 } = x _ { \alpha ( i _ 1 ) } \cdots x _ { \alpha ( i _ k ) } \cdot ( x _ { i _ 1 } \cdots x _ { i _ k } ) ^ { - 1 } = 1 . \end{align*}


\begin{align*} c ( x ) = I _ a x ^ 4 + 2 P _ { a b } x ^ 3 + ( 2 P _ { a c } + I _ b ) x ^ 2 + 2 P _ { b c } x + I _ c \not \equiv 0 . \end{align*}


\begin{align*} v = 0 , \nu ' ( h ) ( \bar u ) + \nu ( v ) = 0 , A ' ( h ) = 0 \mbox { o n } \hat \Sigma . \end{align*}


\begin{align*} \alpha ( g , ( z _ 1 + 1 , z _ 2 , \ldots ) ) = ( \xi + ( z _ 1 + 1 ) \psi _ 1 ) \alpha ( g , Z ) - \sum _ { \overline { \Gamma } \in { \rm B i c } ( { g } , { Z } ) _ 1 } m ( \overline { \Gamma } ) \alpha ( \overline { \Gamma } ) . \end{align*}


\begin{align*} \tau ( \widetilde K _ n ) \ge [ \Lambda ( g , 2 A , M ) ] ^ { - 1 } ( 1 + \varepsilon ) & \geq [ \Lambda ( g , A , M ) ] ^ { - 2 } ( 1 + \varepsilon ) \\ & > \left [ \frac { 1 + \varepsilon } { 1 + \frac { \varepsilon } { 2 } } \right ] ^ { - 1 } ( 1 + \varepsilon ) \\ & = 1 + \frac { \varepsilon } { 2 } , \end{align*}


\begin{align*} \beta _ m & = \min \left \{ \sum _ { \omega \in \Omega _ m ' } | \omega | x _ { \omega } q _ { \omega } \ , \ , : \ , \ , x _ { \omega } \geq 0 \ , \ , \forall \omega \in \Omega _ m ' , \ , \sum _ { \omega \in \Omega _ m ' } | \omega | x _ { \omega } = 1 , \ , \sum _ { \omega \in \Omega _ m ' } x _ { \omega } A _ { \omega } \succeq 0 \right \} \end{align*}


\begin{align*} & P _ 1 = e ^ { \frac { i } { 2 } \int _ { - \infty } ^ x | u _ y ( y , t ) | ^ 2 d y \widehat { \sigma } _ { 3 } } ( k P _ x + \frac { i } { 2 } | u _ x | ^ 2 \sigma _ { 3 } ) . \end{align*}


\begin{align*} \lim _ { n \to \infty } \| p _ n x \| _ { X _ n } = \| x \| _ X , \ , x \in X \lim _ { n \to \infty } \| q _ n y \| _ { Y _ n } = \| y \| _ Y , \ , y \in Y . \end{align*}


\begin{align*} Q ^ 2 - B Q - \Bigl [ I _ { n + 1 } \frac { \partial } { \partial x _ n } , Q \Bigr ] + C = 0 , \end{align*}


\begin{align*} \begin{aligned} \sup _ { x \in \left ( - \infty , x _ 0 \right ) } \| \langle x \rangle m ( x ; z ) \| _ { L _ z ^ 2 ( \mathbb { R } ) } & \leq \sqrt { \pi } \left ( \left \| u _ { x x } \right \| _ { L ^ { 2 , 1 } } + \frac { 1 } { 2 } \left \| u _ x ^ 3 \right \| _ { L ^ { 2 , 1 } } \right ) \\ & \leq c \left ( \| u \| _ { H ^ { 2 , 1 } } + \| u \| _ { H ^ { 2 , 1 } } ^ 3 \right ) , \end{aligned} \end{align*}


\begin{align*} X - W = c _ i W ^ { ( i ) } + O ^ { 3 , \alpha } ( | x | ^ { - q } ) . \end{align*}


\begin{align*} \begin{aligned} k ^ { - 1 } \psi ^ { \pm } _ { 2 1 } ( x ; k ) & = - \int _ { \pm \infty } ^ { x } e ^ { 2 i z ( x - y ) } \bar { u } _ y e ^ { - i c _ \pm } d y \\ & - \int _ { \pm \infty } ^ { x } e ^ { 2 i z ( x - y ) } \bar { u } _ y \left ( \psi ^ { \pm } _ { 1 1 } ( y ; k ) - e ^ { - i c _ \pm } \right ) d y . \end{aligned} \end{align*}


\begin{align*} N = \begin{pmatrix} Q & 0 _ { k \times k } \\ 0 _ { k \times k } & Q \end{pmatrix} . \end{align*}


\begin{align*} y _ n ^ 2 = \left ( \bar { y } _ n + \sum _ { \beta = 1 } ^ { n - 1 } \rho _ \beta ( x ' _ 0 ) y _ \beta \right ) ^ 2 \leq 2 \bar { y } _ n ^ 2 + C \epsilon ^ 2 \delta ^ 2 b _ \alpha | y ' | ^ 2 . \end{align*}


\begin{align*} A _ q ( z , x ) = A _ q ^ z ( x ) : = \displaystyle \sum _ { n = 0 } ^ \infty \frac { \overline { z } ^ n } { \sqrt { \Gamma ( q n + 1 ) } } \psi _ n ( x ) , \end{align*}


\begin{align*} \alpha ( H ) \leq \alpha \left ( \bigcup _ { i = 1 } ^ k H _ i \right ) \leq ( k - 1 ) \frac { n } { p } \end{align*}


\begin{align*} \begin{aligned} \left ( \sup _ { 0 \le s \le \tau } \int _ 0 ^ \tau \mathbf { 1 } _ { s \le t } \| S _ { Z } ( t , s ) \| ^ 2 d t \right ) ^ { 1 / 2 } \le e ^ { C _ 0 } \left ( \int _ 0 ^ { \tau } e ^ { ( t - \tau _ 1 ) \epsilon | x _ 0 | } d t \right ) ^ { 1 / 2 } \le \frac { e ^ { C _ 0 } } { \sqrt { \epsilon } } | x _ 0 | ^ { \epsilon - 1 / 2 } \end{aligned} \end{align*}


\begin{align*} r = | x | u _ r = \dfrac { x } { | x | } \cdot \nabla u . \end{align*}


\begin{align*} p + \alpha \overline { u } + \mu = 0 , \\ u _ { a } \le \overline { u } \le u _ { b } , \ ; \ ; \mu _ { a } \ge 0 , \ ; \ ; \mu _ { b } \ge 0 , \\ \mu _ { a } ( u _ { a } - \overline { u } ) = \mu _ { b } ( \overline { u } - u _ { b } ) = 0 , \end{align*}


\begin{align*} \rho ( F ) = \{ \eta \in \Omega : F ( \eta ) ^ { - 1 } \} \end{align*}


\begin{align*} \Big \| \sum \limits _ { k = 2 } ^ { n } \mu ( k ) ( I - S ) h _ k - ( 1 - z ) \Big \| _ { H ^ 2 ( \mathbb { D } ) } \longrightarrow 0 n \to \infty \end{align*}


\begin{align*} \begin{array} { l l l } \mathsf { g } _ { \mathbb { D } , \tilde { \mathbb { F } } ( \mathsf { S } ) } ( \tilde { F } ( L ) ) & = & \tilde { F } ( \mathsf { g } _ { \mathbb { C } , \mathsf { S } } ( L ) ) \circ \Phi _ { S , S } \\ & = & \tilde { F } ( \mathsf { d } _ { \mathbb { C } , \mathsf { S } } ( R ) ) \circ \Phi _ { S , S } \\ & = & \mathsf { d } _ { \mathbb { D } , \tilde { \mathbb { F } } ( \mathsf { S } ) } ( \tilde { F } ( R ) ) . \end{array} \end{align*}


\begin{align*} K _ Y = ( K _ X + A _ X ) \vert _ Y = ( 2 - n ) A _ X \vert _ Y \end{align*}


\begin{align*} \alpha _ \beta ( z ) = z \frac { \Phi ( z ) } { \phi ( z ) } - \frac { ( 2 - \beta ) \Phi ( z ) - 1 } { 1 - \Phi ( z ) } . \end{align*}


\begin{align*} F ( \lambda ) x = 0 . \end{align*}


\begin{align*} W _ n ( t ) = Q _ n W ( t ) : = \sum _ { i = 1 } ^ n \langle W ( t ) , h _ i \rangle h _ i , \end{align*}


\begin{align*} \{ ( \xi , \rho ) \} = \mathcal { S } ( \beta ) \cap \mathcal { S } ( \beta ) ^ \perp . \end{align*}


\begin{align*} D _ n ( x , t _ X , t _ Y ) = \begin{cases} ( t _ X + t _ Y ) \Delta _ n ( x , \frac { t _ X } { t _ X + t _ Y } ) & \\ 0 & \end{cases} \end{align*}


\begin{align*} \Theta ( \beta _ 2 ( t ) ) & = f ( y _ 2 ( t ) ) - f ^ \star \\ & = f ( y _ 1 ( \alpha ( t ) ) ) - f ^ \star = \Theta ( \beta _ 1 ( \alpha ( t ) ) ) \end{align*}


\begin{align*} { } ^ { \psi } \mathcal D \circ { } ^ { \psi } \mathcal T [ \mathfrak { f } ] = \mathfrak { f } , \ \ \forall \mathfrak { f } \in L _ 2 ( \Omega , \mathbb H ) \cup C ( \Omega , \mathbb H ) . \end{align*}


\begin{align*} 0 = M ^ { ( 0 ) } \subset M ^ { ( 1 ) } \subset \cdots \subset M ^ { ( n ) } = M , 0 = M _ { ( 0 ) } \subset M _ { ( 1 ) } \subset \cdots \subset M _ { ( n ) } = M . \end{align*}


\begin{align*} \begin{cases} d x _ t = B ( x _ t , x _ t ) d t - A x _ t d t + \sigma d W _ t \\ x _ t | _ { t = 0 } = x _ 0 \in \R ^ n . \end{cases} \end{align*}


\begin{align*} F ^ { q , \dagger } _ v ( v ( \tau ) , p ( s ( \tau ) ) ) ( u _ \ell + \omega ( \tau ) ) = \nu ( \tau ) ( u _ \ell + \omega ( \tau ) ) , \end{align*}


\begin{align*} ( A + F ) y = ( b + f ) . \end{align*}


\begin{align*} S t _ { D N P } \lim _ { n \rightarrow \infty } | | \mathcal { Y } _ { n } ( z ^ { 2 } , z ) - z ^ { 2 } | | _ { \infty } = 0 . \\ \end{align*}


\begin{align*} D _ { a , k } : = \begin{pmatrix} D _ { a } \\ & \ddots \\ & & D _ { a } \\ & & & 1 _ { 2 n - 2 k } \end{pmatrix} = \left ( \bigoplus _ { 1 } ^ { k } D _ { a } \right ) \bigoplus 1 _ { 2 ( n - k ) } , D _ { a } : = \begin{pmatrix} a & 0 \\ 0 & a ^ { - 1 } \end{pmatrix} . \end{align*}


\begin{align*} \lim _ { n \to \infty } \int _ { t _ 1 } ^ { t _ 2 } \sum _ { i = m } ^ { n } \sum _ { j = n + 1 } ^ { \infty } g _ i V _ { i , j } \psi _ i ( s ) \psi _ j ( s ) d s = 0 \end{align*}


\begin{align*} \lambda _ { \phi _ { \psi } } ( w ) = \bigoplus _ { i = 0 } ^ { a } \bigoplus _ { j = 0 } ^ { d } | w | ^ { \frac { a + d } { 2 } - ( i + j ) } \end{align*}


\begin{align*} L ( s , \vee ^ 2 ) = \prod _ p L _ p ( s , \vee ^ 2 ) , \mbox { a n d } L ( s , \wedge ^ 2 ) = \prod _ p L _ p ( s , \wedge ^ 2 ) , \end{align*}


\begin{align*} \dim ( K _ i \cap K _ j ) = 1 \quad \quad K _ 1 \cap K _ 2 \cap K _ 3 = 0 \quad . \end{align*}


\begin{align*} O _ { n , n } ( R ) \subseteq O _ { n + 1 , n + 1 } ( R ) : A \mapsto \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & A \end{pmatrix} . \end{align*}


\begin{align*} \partial _ { \theta _ i } \partial _ { \theta _ j } = - \partial _ { \theta _ j } \partial _ { \theta _ i } \hbox { i f } 1 \leq i \neq j \leq n \qquad \hbox { a n d } \partial _ { \theta _ i } ^ 2 = 0 \hbox { f o r } 1 \leq i \leq n \end{align*}


\begin{align*} x = Q _ { \mathrm { B } } ( u ) = I _ u ^ { - 1 } ( \alpha , \beta ) = \sum _ { i = 1 } ^ { \infty } d _ i \ , u ^ { i / \alpha } . \end{align*}


\begin{align*} \begin{aligned} u ( x ) \geq u ( x _ 0 ) - C b _ \alpha | x - x _ 0 | = \ , & u ( x _ 0 ) - C b _ \alpha ( x _ n - \rho ( 0 , \tilde { x } ) ) \\ \geq \ , & u ( x _ 0 ) - C \epsilon _ 0 \delta ^ 2 b _ \alpha ^ 2 . \end{aligned} \end{align*}


\begin{align*} \begin{pmatrix} \bar { x } _ { t , n } \\ \bar { x } _ { t , 1 } \\ \end{pmatrix} = \int _ 0 ^ t S ( t - s ) \begin{pmatrix} F ( s ) \\ G ( s ) \\ \end{pmatrix} d s - \int _ 0 ^ t S ( t - s ) \begin{pmatrix} 0 \\ \sigma _ 1 d W _ s ^ { ( 1 ) } \\ \end{pmatrix} . \end{align*}


\begin{align*} \begin{aligned} \mathcal { P } ^ { + } \left ( z ^ { - 1 } \bar { r } _ 1 ( z ) e ^ { - 2 i z x } \right ) & = \int _ { 2 x } ^ { \infty } \widehat { ( z ^ { - 1 } \bar { r } _ 1 ( z ) ) } ( \xi ) e ^ { i z ( y - 2 x ) } d \xi . \end{aligned} \end{align*}


\begin{align*} n ^ { 3 } + \left ( n + 1 \right ) ^ { 3 } - \left ( \frac { \beta } { 3 } \left ( n + 2 \right ) \right ) ^ { 3 } + \left ( \frac { \beta } { 3 } \left ( n - 1 \right ) \right ) ^ { 3 } = 0 \end{align*}


\begin{align*} \Phi ( [ a \cos ( 2 \pi \ell \phi ) + b \sin ( 2 \pi \ell \phi ) ] u _ \ell + [ - a \sin ( 2 \pi \ell \phi ) + b \cos ( 2 \pi \ell \phi ) ] w _ \ell , p ) \\ = [ c \cos ( 2 \pi \ell \phi ) + d \sin ( 2 \pi \ell \phi ) ] u _ \ell + [ - c \sin ( 2 \pi \ell \phi ) + d \cos ( 2 \pi \ell \phi ) ] w _ \ell . \end{align*}


\begin{align*} \frac { d \mu _ 1 ^ { n } ( t ) } { d t } = 0 , \end{align*}


\begin{gather*} \mathcal { K } _ { \psi } ( \Omega ) : = \{ w \in W ^ { 1 , p } ( \Omega ) : w \geq \psi \ \ \Omega \} \end{gather*}


\begin{align*} e _ 2 ( C ) = 2 - I ( C ) - s ( C ) ; \end{align*}


\begin{align*} B ( x , y ) = \begin{pmatrix} x _ 1 y _ 3 \\ - x _ 2 y _ 3 \\ ( x _ 2 - x _ 1 ) ( y _ 2 + y _ 1 ) \end{pmatrix} . \end{align*}


\begin{align*} \nabla _ X \psi = \frac { 1 } { 2 } X _ 1 \cdot V \cdot \psi - \frac { 1 } { 2 } S ( X ) \cdot N \cdot \psi . \end{align*}


\begin{align*} \lambda _ { - , 1 } = - 2 \sqrt { \delta ( 1 - \delta ) } , E _ { - , 1 } = \left \{ \begin{pmatrix} \bar { a } _ 1 \sqrt { \frac { \delta } { \delta - 1 } } \\ - \bar { b } _ 1 \sqrt { \frac { \delta } { \delta - 1 } } \\ 0 \\ 1 \end{pmatrix} , \begin{pmatrix} - \bar { b } _ 1 \sqrt { \frac { \delta } { \delta - 1 } } \\ - \bar { a } _ 1 \sqrt { \frac { \delta } { \delta - 1 } } \\ 1 \\ 0 \end{pmatrix} \right \} . \end{align*}


\begin{align*} \phi = \left \{ \begin{array} { l l } u , & { \rm o n } \Omega \setminus D , \\ \max \{ u , v \} , & { \rm o n } D , \end{array} \right . \end{align*}


\begin{align*} \mathcal { A } _ q ( p , x ) = \mathcal { A } _ q ^ p ( x ) : = \displaystyle \sum _ { n = 0 } ^ \infty \frac { \overline { p } ^ n } { \sqrt { \Gamma ( q n + 1 ) } } \psi _ n ( x ) , p \in \mathbb { H } , x \in \mathbb { R } \end{align*}


\begin{align*} I _ h \leq E [ b ] \leq \liminf _ { n \to \infty } E [ b _ n ] = I _ h . \end{align*}


\begin{align*} \log \det u _ { , i j } = - v _ p x ^ p + u _ { , q } \xi ^ q + c . \end{align*}


\begin{align*} \sigma \geq v + \sum _ { k = 1 } ^ l a _ k \hat C _ { q _ k , N } . \end{align*}


\begin{align*} M = \left ( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right ) \in \mathrm { G L } _ 2 ( A [ t ^ { \pm 1 } ] ) \cap ( \mathrm { M a t } _ { 2 , 2 } ( A [ t ] ) \setminus \mathrm { M a t } _ { 2 , 2 } ( t A [ t ] ) ) . \end{align*}


\begin{align*} I _ h \leq E [ b ] \leq \liminf _ { n \to \infty } E [ b _ n ] = I _ h , \end{align*}


\begin{align*} \frac { \partial u } { \partial t } = \mathcal { L } ( u ) + f ( t , x ) , \mathcal { B } ( u ) = g ( t , x ) , \end{align*}


\begin{align*} \delta _ { + } ( z ) \delta _ { - } ^ { - 1 } ( z ) = \mathrm { e } ^ { \log \left ( 1 + \bar { r } _ 1 ( z ) r _ 2 ( z ) \right ) } = 1 + \bar { r } _ 1 ( z ) r _ 2 ( z ) , z \in \mathbb { R } . \end{align*}


\begin{align*} D ( X ) = \sum _ { 0 \le i \ne j \le n - 1 } B _ { i j } X ^ { 2 ^ i + 2 ^ j } . \end{align*}


\begin{align*} u ( x ' , t ) : = P _ t ^ L * f ( x ' ) , ( x ' , t ) \in \R ^ n _ + , \end{align*}


\begin{align*} L _ X g = h . \end{align*}


\begin{align*} - F v = \norm { v } _ { V } F \left ( \frac { - v } { \norm { v } _ { V } } \right ) = \norm { v } _ { V } ( G \widetilde { w } + z ) = G ( \norm { v } _ { V } \widetilde { w } ) + \norm { v } _ { V } z , \end{align*}


\begin{align*} x _ { \lambda } x _ { \mu } x _ { \nu } = ( x _ { \lambda } x _ { \mu } ) x _ { \nu } ( \lambda \mu ) \nu \end{align*}


\begin{align*} \sum _ { j = 1 } ^ d F _ j = F _ 1 + \cdots + F _ d = \left \{ \sum _ { j = 1 } ^ d a _ j : a _ j \in F _ j \ \forall 1 \le j \le d \right \} . \end{align*}


\begin{align*} { \mathcal C } = { \mathcal C } _ { \mathbf { x } _ 0 , \theta _ 0 } : = \left \{ \mathbf { y } \in \mathbb R ^ 3 ~ | ~ 0 \leqslant \angle ( \mathbf y - \mathbf { x } _ 0 , \mathbf { v } _ 0 ) \leqslant \theta _ 0 \right \} \ ( \theta _ 0 \in ( 0 , \pi / 2 ) ) \end{align*}


\begin{align*} \frac { p } { p + 1 } \sum _ { n = 1 } ^ { \infty } \frac { 2 ^ { \omega ( \gcd ( n , p ) ) } \cdot 2 r _ { Q ^ { \ast } } ( n ) ^ { 2 } } { n } \sum _ { d = 1 } ^ { \infty } \psi \left ( d \sqrt { \frac { n } { p } } \right ) \end{align*}


\begin{align*} f ^ \lambda \ast _ \lambda \varphi _ { k , \lambda } ^ { n - 1 } ( z ) = ( 2 \pi ) ^ { - p - q } \lambda ^ { p + q } \ , P ( z ) \ , g ^ \lambda \ast _ \lambda \varphi _ { k - p , \lambda } ^ { n + p + q - 1 } ( z ) \end{align*}


\begin{align*} \sigma ( \delta ) = ( \det \sigma ) \ \delta . \end{align*}


\begin{align*} F ( x ^ { t + 1 } ) - F ( x ) + \frac { \alpha _ t ^ 2 } { 2 \gamma _ t } \| x - z ^ { t + 1 } \| ^ 2 \overset { \eqref { F - i n e q } } { \leq } \ & ( 1 - \alpha _ t ) \left ( F ( x ^ t ) - F ( x ) \right ) + \frac { \alpha _ { t } ^ 2 ( 1 - \beta _ t ) } { 2 \gamma _ { t } } \| x - z ^ t \| ^ 2 \\ \overset { \eqref { a l p h a - e q n } } { = } \ & ( 1 - \alpha _ t ) \left ( F ( x ^ t ) - F ( x ) + \frac { \alpha _ { t - 1 } ^ 2 } { 2 \gamma _ { t - 1 } } \| x - z ^ t \| ^ 2 \right ) . \end{align*}


\begin{gather*} \mathcal { A } _ { j m } ( x , \xi ) = \int _ { B _ 1 ( 0 ) } \phi ( y ) \mathcal { A } _ j ( x + m y , \xi ) d y , \end{gather*}


\begin{align*} \begin{aligned} 0 & \to \left ( T _ { \mu _ g ( \kappa ) ^ { - 1 } ( X , D , \alpha , m ) } \right ) _ { ( X , D , \alpha , m ) } \to \left ( T _ { \mathcal B _ g ( \kappa ) } \right ) _ { ( X , D , \alpha , p , m ) } \to \left ( T _ { \mathcal A _ g ( \kappa ) } \right ) _ { ( X , D , \alpha , m ) } \\ & \xrightarrow { \left ( d \mathfrak c _ g ( \kappa ) \right ) _ { ( X , D , \alpha , m ) } } H ^ 1 \left ( X , K _ X ^ { \otimes 2 } ( - D ) \right ) \to 0 \end{aligned} \end{align*}


\begin{align*} D = i \gamma ^ { j } \nabla ^ { ( s ) } _ { e _ { j } } , \end{align*}


\begin{align*} \frac { B _ \gamma ( z ) } { A ( z ) } = \gamma + \frac { \pi } { i } \frac { e ^ { 2 \pi i z } - 1 } { e ^ { 2 \pi i z } + 1 } - \pi \frac { e ^ { 2 \pi z } - 1 } { e ^ { 2 \pi z } + 1 } . \end{align*}


\begin{align*} \begin{cases} a _ { \alpha + n + 1 } - a _ { n + 1 } = d _ { \alpha + n } - d _ { \alpha + n + 1 } \\ b _ n = a _ { \alpha + 1 } + ( d _ { \alpha + 1 } - d _ \alpha ) = b \\ d _ { \alpha + n } = d _ \alpha + ( b _ 1 - a _ { \alpha + 1 } ) + \sum _ { j = 2 } ^ n ( a _ j - a _ { \alpha + j } ) \\ d = d _ \alpha + b - a _ { \alpha + 1 } + \sum _ { j = 2 } ^ \infty ( a _ j - a _ { \alpha + j } ) \end{cases} \ \end{align*}


\begin{align*} A _ t + B \times u + \nabla Q = - \mu \nabla \times B . \end{align*}


\begin{align*} \begin{aligned} | x ( t ) - z & ( t ) | \le | x ( 0 ) - z ( 0 ) | + \int _ 0 ^ t \left | \dot { x } ( \tau ) - \dot { z } ( \tau ) \right | \ , d \tau \\ \le & \ , | x ( 0 ) - z ( 0 ) | + \gamma L _ r \int _ 0 ^ t | x ( \tau ) - z ( \tau ) | \ , d \tau \\ & + \gamma ^ 2 \bar k ( L _ r , M _ r , \delta ) t \end{aligned} \end{align*}


\begin{align*} E ' _ 0 = \kappa \left ( \frac { B _ 3 } { 2 \sqrt { \kappa B _ 4 } } + L + 5 \right ) ^ 2 - \left ( \frac { Q } { 2 ( L + 2 ) } - \sqrt { \kappa B _ 4 } \right ) ^ 2 , \end{align*}


\begin{align*} \lambda _ n ( A ) = \frac { \lambda ( A \cap A _ n ) } { \lambda ( A _ n ) } , \end{align*}


\begin{align*} \mathcal { L } _ H \varPhi = W = f _ \varepsilon \left ( \varPhi - \frac { \alpha } { 2 } | x ' | ^ 2 | \ln \varepsilon | \right ) \ \ \ B _ { R ^ * } ( 0 ) , \end{align*}


\begin{align*} L ( x ) = \sum _ { i = 0 } ^ d b _ i x ^ { q ^ i } \end{align*}


\begin{align*} f ( t ) = \frac { 1 } { \Gamma ( \alpha ) } \ , \int _ 0 ^ t ( t - \tau ) ^ { \alpha - 1 } \phi ( \tau ) \ , d \tau , \ t > 0 , \ 0 < \alpha < 1 . \end{align*}


\begin{align*} B _ q ( \phi ) ( z ) & = \int _ { \mathbb { C } } f ( w ) | w | ^ { \frac { 2 } { q } - 2 } e ^ { - \frac { | w | ^ 2 } { q } } \overline { K _ q ( w , z ) } d A ( w ) \end{align*}


\begin{align*} \sigma _ { n + m } \left ( \Delta \cdot \Delta ' \right ) = \sigma _ n ( \Delta ) \sigma _ m ( \Delta ' ) \end{align*}


\begin{align*} g _ n \big ( t , q , x \big ) : = \sup _ { z \in \mathbb { R } ^ d } \Big ( q \cdot z - f _ n ( t , z , x ) \Big ) . \end{align*}


\begin{align*} \begin{aligned} h ( x ) = & \ , h _ 0 + A \sin ( 1 0 x ) \\ & \ , + \left \{ \begin{array} { c c } ( x - \pi ) ^ 2 - 1 & x < \pi \\ \cos ( x - \pi ) - 2 & x \in [ \pi , \ , 2 \pi ) \\ ( x - 2 \pi ) ^ 2 - 3 & x \ge 2 \pi \end{array} \right . \end{aligned} \end{align*}


\begin{align*} \begin{aligned} u ( x _ 0 ) & = \varphi ( x _ 0 ) = \frac { 1 } { 2 } \sum _ { \beta = \alpha + 1 } ^ { n - 1 } b _ \beta x _ \beta ^ 2 + \frac { 1 } { 6 } \sum _ { \xi , \beta , \gamma \geq \alpha + 1 } \varphi _ { \xi \beta \gamma } ( 0 ) x _ \xi x _ \beta x _ \gamma + O ( | \tilde { x } | ^ 4 ) \\ & \geq \frac { 1 } { 2 } \delta ^ 2 b _ \alpha ^ 2 - C \delta ^ 3 b _ \alpha ^ 2 \geq \frac { 7 } { 1 6 } \delta ^ 2 b _ \alpha ^ 2 \end{aligned} \end{align*}


\begin{align*} \nabla _ X \psi _ 1 = \frac { 1 } { 2 } X _ 1 \cdot \psi _ 1 - \frac { 1 } { 2 } S ( X ) \cdot N \cdot \psi _ 1 \end{align*}


\begin{align*} D _ { \uparrow } V ( u ) ( h ) = \liminf _ { \substack { t \downarrow 0 \\ h ' \to h } } \frac { V ( u + t h ' ) - V ( u ) } { t } . \end{align*}


\begin{align*} a \cos ( x ) + b \sin ( x ) = r \cos ( x + \theta ) . \end{align*}


\begin{align*} b _ { \sigma \otimes \tau } = b _ { \sigma } \otimes b _ { \tau } . \end{align*}


\begin{align*} \sum _ i F ^ { i i } h _ { i i 1 1 } = - \sum _ { p , q , r , s } F ^ { p q , r s } h _ { p q 1 } h _ { r s 1 } = - \sum _ { p \neq q } F ^ { p p , q q } h _ { p p 1 } h _ { q q 1 } + \sum _ { p \neq q } F ^ { p p , q q } h _ { p q 1 } ^ 2 . \end{align*}


\begin{align*} E ^ L _ Y ( u ) = \varphi ^ L _ { \sigma , s } ( E ^ L _ X ( v ) ) \end{align*}


\begin{align*} \frac { f ' ( x ) } { f ( x ) } ~ { \rm f o r } ~ f ( x ) = \frac { p ( x ) } { g ( x ) } , ~ g ( x ) = \prod _ { j = 1 } ^ k ( x - z _ j ) , \end{align*}


\begin{align*} \begin{cases} - \varepsilon ^ 2 ( K ( x ) \nabla u ) = ( u - q | \ln \varepsilon | ) ^ { p } _ + , \ \ & x \in \Omega , \\ u = 0 , \ \ & x \in \partial \Omega , \end{cases} \end{align*}


\begin{align*} \begin{aligned} 0 = \ , & ( A \Psi \pm T _ \alpha ( u - \varphi ) ) _ n \\ = \ , & - A + 2 A K x _ n \\ & \pm \left [ ( u - \varphi ) _ { \alpha n } + \sum _ { \beta < n } B _ { \alpha \beta } ( x _ { \beta } ( u - \varphi ) _ { n n } - x _ n ( u - \varphi ) _ { \beta n } ) - \sum _ { \beta < n } B _ { \alpha \beta } ( u - \varphi ) _ { \beta } \right ] \\ < \ , & 0 , \end{aligned} \end{align*}


\begin{align*} P [ \Delta _ { M _ n } < s _ n ] = O ( r _ n ^ { 1 + \kappa } ) \end{align*}


\begin{align*} U \cdot V = U _ { i _ 1 \cdots i _ n } V _ { i _ 1 \cdots i _ n } , \quad | U | ^ 2 = U \cdot U . \end{align*}


\begin{align*} \lambda _ 1 = q \sqrt { q } \sigma _ 1 ( \textbf { z } ) , \lambda _ 2 = q ^ 2 \sigma _ 2 ( \textbf { z } ) , \lambda _ 3 = q \sqrt { q } \sigma _ 3 ( \textbf { z } ) . \end{align*}


\begin{align*} \sum _ { \gamma _ { \chi } } \frac { \frac { 1 } { 2 } } { \frac { 1 } { 4 } + \gamma _ { \chi } ^ 2 } & \leq \left ( \dfrac { 1 } { 2 } + \dfrac { 2 } { \log q - 2 } \right ) \log { \dfrac { q } { \pi } } + 2 \left ( 1 - \dfrac { 2 } { \log q } \right ) ^ { - 2 } \log \log q \leq \dfrac { 1 } { 2 } \log { \dfrac { q } { \pi } } + 2 . 6 \log \log q . \end{align*}


\begin{align*} g _ 1 ' : = \frac { 1 } { 2 } ( 1 + i I ) v ' \end{align*}


\begin{align*} & \mathcal { I } _ 1 \leq C T ^ { \frac { - \gamma - 1 } { p - 1 } } R ^ N , \\ & \mathcal { I } _ 2 \leq C T ^ { \frac { - \gamma - 1 } { p - 1 } } R ^ { N - \frac { 2 p } { p - 1 } } , \\ & \mathcal { I } _ 3 \leq C T ^ { 1 - \frac { \gamma } { p - 1 } } R ^ { N - \frac { 2 p } { p - 1 } } . \end{align*}


\begin{align*} a _ { ( n ) } \cdot v = 0 , n \geq N , \end{align*}


\begin{align*} \left | \Pi _ { \mathrm { k e r } A ^ \perp } \frac { d ^ \ell } { d t ^ \ell } X _ t | _ { t = 0 } \right | \ge \frac { \gamma } { C _ J } K ^ { \ell + 1 } , \end{align*}


\begin{align*} \exists \ C = C [ \psi ] \ \in ( 0 , \infty ) \ \Rightarrow \ | | \zeta | | G \psi \le C ( \psi ) . \end{align*}


\begin{align*} \left [ g , h ^ n \right ] = \left [ g , h ^ { n - 1 } \right ] \left [ g , h \right ] ^ { h ^ { n - 1 } } = \left [ g , h ^ { n - 1 } \right ] t ^ { h ^ { n - 1 } } = \left [ g , h ^ { n - 1 } \right ] t \end{align*}


\begin{align*} & \frac { 1 } { \varepsilon ^ 2 } \int _ \Omega \left ( v ^ \tau _ \varepsilon - q \ln \frac { 1 } { \varepsilon } \right ) ^ { p + 1 } _ + d x = \int _ { \hat { B } _ \tau ( 0 ) } q ^ { p + 1 } ( \bar { x } + \varepsilon y ) ( \hat { U } ( y ) + \ln \tau ) ^ { p + 1 } _ + d y \\ \rightrightarrows & q ^ { p + 1 } ( \bar { x } ) \int _ { \hat { B } _ \tau ( 0 ) } ( \hat { U } ( y ) + \ln \tau ) ^ { p + 1 } _ + d y , \end{align*}


\begin{align*} x _ i ^ 2 = \epsilon ^ 2 = 1 , 1 \leq i \leq n , \end{align*}


\begin{align*} 0 = & p s _ { \delta _ N , i } \left ( \frac { \delta _ N } { s _ { \delta _ N , i } } \right ) ^ { \frac { 2 p } { p - 1 } } \int _ { \mathbb { R } ^ 2 } \phi ( T _ { z _ { N , i } } y ) ^ { p - 1 } _ + \phi ' ( T _ { z _ { N , i } } y ) \frac { ( T _ { z _ { N , i } } ) _ h ^ t \cdot T _ { z _ { N , i } } y } { | T _ { z _ { N , i } } y | } \tilde { u } _ { N , i } ( y ) d y + O \left ( \frac { \varepsilon _ N ^ 2 } { | \ln \varepsilon _ N | ^ { p - 1 } } \right ) . \end{align*}


\begin{align*} { } _ a D _ g ^ { \alpha } \circ { } _ { a } I _ g ^ { \alpha } = I , \end{align*}


\begin{align*} \int _ { B _ { \rho } } r ^ { 2 - n } | \nabla u | ^ 2 \d x & \leq \theta \int _ { B _ { 8 \rho } } r ^ { 2 - n } | \nabla u | ^ 2 \d x \rho \leq 1 / 8 \varepsilon \leq \varepsilon _ 0 , \end{align*}


\begin{align*} v _ { \alpha } ( x ) = \begin{cases} x _ n ^ { \alpha } ( | x ' | ^ 2 - C _ \alpha ) \quad C _ { \alpha } = \frac { 1 + 2 D ^ 2 } { \alpha ( 1 - \alpha ) } & \alpha \in [ \frac { 2 } { n } , 1 ) , \\ ( 1 + 2 D ^ 2 ) x _ n \log ( x _ n / D ) + x _ n ( | x ' | ^ 2 - D ^ 2 ) & \alpha = 1 . \end{cases} \end{align*}


\begin{align*} & \int _ { \{ z \in M : - \Psi ( z ) \in S _ { i , n } \backslash U _ k \} } | F _ { t _ 0 , \tilde { c } } | ^ 2 _ h c ( - \Psi ) \\ = & \int _ { \{ z \in M : - \Psi ( z ) \in S _ { i , n } \backslash U _ k \} } | \tilde F | ^ 2 _ h c ( - \Psi ) + \int _ { \{ z \in M : - \Psi ( z ) \in S _ { i , n } \backslash U _ k \} } | F _ { t _ 0 , \tilde { c } } - \tilde F | ^ 2 _ h c ( - \Psi ) . \end{align*}


\begin{align*} \Lambda = \Lambda ( i _ 1 , \cdots , i _ m ) = \prod _ { a = 1 } ^ { m } \Lambda _ a ^ { i _ a } , \medspace R = R ( j _ 1 , \cdots , j _ n ) = \prod _ { b = 1 } ^ { n } R _ b ( j _ b ) \cdot C . \end{align*}


\begin{align*} A ^ { q - 1 } = \frac { r } { ( r + 1 ) ^ q } , \end{align*}


\begin{align*} \int ^ { \infty } _ { 0 } \frac { 1 } { x ^ 2 } f ( x ) \mathrm { d } x = & \frac { 2 \theta } { \operatorname { B } ( a , b ) } \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ n \Gamma ( b ) } { \Gamma ( b - n ) n ! } \int ^ { \infty } _ { 0 } \frac { \exp ( - t ) } { [ 2 ( a + n ) \theta ] ^ 2 } \mathrm { d } t \\ = & \frac { 1 } { \theta \operatorname { B } ( a , b ) } \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ n \Gamma ( b ) } { \Gamma ( b - n ) n ! } \frac { 1 } { ( a + n ) ^ 2 } . \end{align*}


\begin{align*} M _ { \pm } ( x ; z ) = \left ( M _ { \pm , 1 } ( x ; z ) , M _ { \pm , 2 } ( x ; z ) \right ) . \end{align*}


\begin{align*} q _ d ^ * = - ( d - 1 ) + O ( d ) \left ( - \frac { 1 - \log 2 } { 2 } d \right ) , d \to \infty . \end{align*}


\begin{align*} D / D = \{ 1 \} ^ { [ k ] } \cup ( G \setminus \{ 1 \} ) ^ { [ \lambda ] } . \end{align*}


\begin{align*} g ( \Gamma ) = h ^ 1 ( \Gamma ) + \sum _ { v \in V } g ( v ) , h ^ 1 ( \Gamma ) = | E | - | V | + 1 . \end{align*}


\begin{align*} \beta _ 1 ( u ) = 0 & \Longrightarrow \beta _ 1 ( u _ { \varepsilon } ) = 0 , \\ \beta _ 1 ( u ) = \beta ' _ 3 ( u ; Q ) = 0 & \Longrightarrow \beta ' _ 3 ( u _ { \varepsilon } ; Q ) = 0 . \end{align*}


\begin{align*} \overline { \pi } ( q _ 1 ) \overline { \pi } ( q _ 2 ) = c _ \chi ( q _ 1 , q _ 2 ) \overline { \pi } ( q _ 1 q _ 2 ) \forall q _ 1 , q _ 2 \in Q . \end{align*}


\begin{align*} y [ n ] = \mu _ { \rm a } ( x [ n ] ) + w _ { \rm s } [ n ] , \end{align*}


\begin{align*} \int f ( x ) \ , \overline { g ( x ) } \ , d x = h \sum _ n f ( n h ) \ , \overline { g ( n h ) } . \end{align*}