\begin{align*} \varphi _ { | N } = \frac { 1 } { n } ( \psi _ 1 + \cdots + \psi _ n ) \end{align*}


\begin{align*} \dot \theta _ k = \frac { - 1 } M \sum _ { j = 1 } ^ M b _ { k - j } \sin ( \theta _ j - \theta _ k ) + \frac { 1 } M \sum _ { j = 1 } ^ M b _ { 1 - j } \sin ( \theta _ j ) . \end{align*}


\begin{align*} \tau ( K ) : = \bigcap _ { K \subset U } \ker ( r : S H _ { M } ( M ; \Lambda ) \rightarrow S H _ { M } ( M - U ; \Lambda ) ) \end{align*}


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


\begin{align*} ( \nu _ { * } F ) _ { \mathbb { C } } ( T ) = F ( \nu ^ { * } \nu _ { * } T ) = F ( T \sqcup \overline { T } ) , \end{align*}


\begin{align*} \xi _ { t } = h ( \xi ) \end{align*}


\begin{align*} \exp \left ( a _ { \alpha + 1 } + d _ { \alpha + 1 } - d _ \alpha \right ) = 1 - \exp \left ( c _ { n + 1 } + b _ { n + 1 } - b _ n \right ) \ . \end{align*}


\begin{align*} & \frac { 1 } { | \mathcal F _ { N , k } | } \sum _ { f \in \mathcal F _ { N , k } } R _ 2 ( g , \rho ) ( f ) = \frac { T ( g , \rho ) } { 4 L } \\ & + \O \left ( \frac { 1 } { L } \right ) + \O \left ( \frac { L ( \log \log x ) ^ 2 } { \pi _ N ( x ) } \right ) + \O \left ( \frac { x ^ { \pi _ N ( x ) c } 8 ^ { \nu ( N ) } } { k N } \right ) , \end{align*}


\begin{align*} \alpha _ t ^ { \xi } = - R ^ { - 1 } B y _ t ^ { \xi } - h ( \mu _ t ) . \end{align*}


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


\begin{align*} \varphi _ a ( z ) = \dfrac { a - P _ a ( z ) - s _ a Q _ a ( z ) } { 1 - \langle z , a \rangle } , z \in \mathbb { B } _ n , \end{align*}


\begin{align*} \mathrm { d i v } ( \widetilde { \mathcal { R } } q ) : = \pi _ { \widehat { Q } _ h ^ \perp \cap \widetilde { P } _ { k - 1 } ^ { \mathrm { d i s c } } ( \mathcal { T } ) } q = \begin{cases} \pi _ { \widehat { Q } _ h ^ \perp } q & k \geq d , \\ \pi _ { \widetilde { P } _ { k - 1 } ^ { \mathrm { d i s c } } ( \mathcal { T } ) } q & k < d , \end{cases} \quad q \in Q , \end{align*}


\begin{align*} u _ n = a ( \theta _ 1 ^ n - \zeta \theta _ 2 ^ n ) \end{align*}


\begin{align*} \mathcal { C } _ n ^ { ( p ) } \ = \ A _ p F _ n + B _ p F _ { n + 1 } - \sum _ { k = 0 } ^ p \binom { p } { k } B _ k n ^ { p - k } . \end{align*}


\begin{align*} S _ q = B _ q \circ \mathcal F \circ B _ q ^ { - 1 } . \end{align*}


\begin{align*} [ \Delta , X ] = 0 . \end{align*}


\begin{align*} \omega _ { u v } = \begin{cases} x _ u y _ v & t ( u , \bullet ) = t ( \bullet , v ) \\ 0 & \end{cases} \end{align*}


\begin{align*} \Phi : = \sup \{ \zeta \in U S C ( \overline { \Omega } \times \mathbb R ^ m ) : \zeta | _ { \Omega \times \mathbb R ^ m } F \star \mathcal P \zeta | _ { \partial \Omega \times \mathbb R ^ m } \le \phi \} \end{align*}


\begin{align*} \nu ( H ' | _ g ( h ) ) & = - \tfrac { 1 } { 2 } R ' ( h ) + \tfrac { 1 } { 2 } R '^ \Sigma | _ { g ^ \intercal } ( h ^ \intercal ) + A _ g \cdot ( A _ g \circ h ) \\ & - A _ g \cdot A ' ( h ) - H _ g H ' ( h ) \\ & + \tfrac { 1 } { 4 } \big ( - R _ g + R ^ \Sigma _ g - | A _ g | ^ 2 - H _ { g } ^ 2 \big ) h ( \nu , \nu ) \\ & - \tfrac { 1 } { 2 } \Delta _ \Sigma h ( \nu , \nu ) + g ^ { a b } \omega ( e _ a ) e _ b ( H _ g ) . \end{align*}


\begin{align*} \rho c _ v \partial _ { t _ 1 } T = { \bar F } ^ { ( 1 ) } , \end{align*}


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


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


\begin{align*} A ( G ) : = L _ 2 ( G ) \ast L _ 2 ( G ) = \{ s \mapsto \langle \l _ s \xi , \eta \rangle \mid \xi , \eta \in L _ 2 ( G ) \} . \end{align*}


\begin{align*} | | B _ q ( \psi _ m ) | | _ { M L _ q ( \mathbb { C } ) } = 1 = | | \psi _ m | | _ { L ^ 2 ( \mathbb { R } ) } . \end{align*}


\begin{align*} k ( t , x , y ) : = \rho ( - R ^ { - 1 } B P _ t x - R ^ { - 1 } B y ) \quad \end{align*}


\begin{align*} \frac { \mathcal { H } ^ 2 _ { m a x } } { \mathcal { H } ^ 2 _ { m i n } } = \frac { \operatorname { R } _ { m a x } } { \operatorname { R } _ { m i n } } & \to 1 , \\ \operatorname { d i a m } ( \Sigma _ t ) & \to 0 . \end{align*}


\begin{align*} \begin{cases} \lambda + 2 < p , & \alpha \leq 2 , \\ \lambda + 2 < \alpha + p - 2 , & \alpha > 2 . \end{cases} \end{align*}


\begin{align*} \sup _ { p \ge 1 } \left \{ \ \frac { | | \tau | | _ { p , \Omega } } { \psi _ { m , L } ( p ) } \ \right \} = C ( m , L ) < \infty \end{align*}


\begin{align*} \left | \dfrac { \gamma \alpha \omega - \gamma \omega } { - \gamma \alpha \omega } \right | = \dfrac { | \alpha \omega - \omega | } { | - \gamma \alpha \omega | | c \omega + d | | c \alpha \omega + d | } = \dfrac { 2 ^ { - 1 } } { | a \alpha \omega + b | | c \omega + d | } . \end{align*}


\begin{align*} l ( y ' ) : = u ( x _ 0 ) + \sum _ { \beta = 1 } ^ { n - 1 } \nabla _ \beta u ( x _ 0 ) y _ \beta . \end{align*}


\begin{align*} \begin{cases} d _ n = \log ( 1 - e ^ { a _ { n + 1 } } ) \\ b _ n = \log ( 1 - e ^ { c _ { n + 1 } } ) \end{cases} \ \forall \ n \in \N \ , \end{align*}


\begin{align*} ( B _ \phi f ) ( x ) = f ( x + \phi ) - f ( \phi ) . \end{align*}


\begin{align*} \det ( Y ) ^ { k / 2 } | F ( Z ) | \ll _ { n , \epsilon } \begin{cases} k ^ { \frac { 3 n ( n + 1 ) } { 8 } } \det ( Y ) ^ { \frac { n + 1 } { 4 } + \epsilon } \\ k ^ { \frac { n ( n + 1 ) } { 4 } } \det ( Y ) ^ { \frac { 3 ( n + 1 ) } { 8 } + \epsilon } . \end{cases} \end{align*}


\begin{align*} \ \begin{aligned} y \left [ n \right ] = & \left ( { \sum \limits _ { l = 1 } ^ L { { \bf { h } } _ l ^ H { { \bf { f } } _ l } } } \right ) s \left [ { n - { n _ { \max } } } \right ] + \\ & \sum \limits _ { l = 1 } ^ L { \sum \limits _ { l ' \ne l } ^ L { { \bf { h } } _ l ^ H { { \bf { f } } _ { l ' } } s \left [ { n - { n _ { \max } } + { n _ { l ' } } - { n _ l } } \right ] } } + z \left [ n \right ] . \end{aligned} \end{align*}


\begin{align*} \sigma _ { p q } ( t ) = \sigma _ { x y } ( ( 1 - t ) a + t b ) . \end{align*}


\begin{align*} & \hphantom { { } = { } } \phi \circ ( \varphi \circ f ) ( x ) \\ * & = \widehat { \varphi } ( f ( o ) ) + D \phi ( p ) ^ i { } _ \alpha a ^ \alpha { } _ \beta x ^ \beta \\ * & \hphantom { { } = { } } + \frac 1 2 \left ( H \phi ( p ) ^ i { } _ { \alpha \beta } a ^ \alpha { } _ j a ^ \beta { } _ k + D \phi ( p ) ^ i { } _ \alpha b ^ \alpha { } _ { j k } \right ) x ^ j x ^ k + ( ) . \end{align*}


\begin{align*} \Phi ^ \dagger _ { v p } ( v , p ) [ v _ 1 , p _ 1 ] & = Q ^ \dagger F ^ { q , \dagger } _ { v v } ( v + \psi ^ \dagger ( v , p ) , p ) [ v _ 1 + \psi _ v ^ \dagger ( v , p ) v _ 1 ] \psi ^ \dagger _ p ( v , p ) p _ 1 \\ & + Q ^ \dagger F ^ { q , \dagger } _ { v p } ( v + \psi ^ \dagger ( v , p ) , p ) [ v _ 1 + \psi _ v ^ \dagger ( v , p ) v _ 1 , p _ 1 ] \\ & + Q ^ \dagger F ^ { q , \dagger } _ { v } ( v + \psi ^ \dagger ( v , p ) , p ) \psi ^ \dagger _ { v p } ( v , p ) [ v _ 1 , p _ 1 ] \end{align*}


\begin{align*} \omega : = \{ x \in \Omega : | x _ \beta | < \delta \frac { b _ \alpha } { \sqrt { b _ \beta } } , \beta = \alpha + 1 , \ldots , n - 1 , x _ n < \delta ^ 2 b _ \alpha \} , \end{align*}


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


\begin{align*} \begin{aligned} | ( \langle \boldsymbol { \nu } , G \rangle - \bar G ) \cdot ( \langle \boldsymbol { \nu } , R \rangle - \bar R ) | & \le c _ 3 \langle \boldsymbol { \nu } , \eta ( \lambda | \bar U ; x , t ) \rangle = c _ 3 \mathcal { H } ( \boldsymbol { \nu } , U , \bar { U } ; x , t ) \ ; . \end{aligned} \end{align*}


\begin{align*} u _ 1 ( \mathbf x ) = u _ 2 ( \mathbf x ) , \ \forall \mathbf x \in \mathbb R ^ 3 \setminus ( \overline { \Omega } _ 1 \cup \overline { \Omega } _ 2 ) . \end{align*}


\begin{align*} v _ \epsilon { ' } ( t ) = \int _ { - \infty } ^ { t } ( \frac { 1 } { B - 4 \epsilon } \mathbb { I } _ { ( - t _ 0 - B + 2 \epsilon , - t _ 0 - 2 \epsilon ) } * \rho _ { \frac { 1 } { 4 } \epsilon } ) ( s ) d s . \end{align*}


\begin{align*} \langle z _ 1 ^ * , v \rangle & = 2 \int _ \S \sin ( 2 \pi \ell x ) v ( x ) \ \d x , \\ \langle z _ 2 ^ * , v \rangle & = - 2 \int _ \S \cos ( 2 \pi \ell x ) v ( x ) \ \d x , \end{align*}


\begin{align*} u _ n = r u _ { n - 1 } + s u _ { n - 2 } , \end{align*}


\begin{align*} \mathcal { J } ' ( \overline { u } ; h ) = \langle \overline { y } - g , S ' ( \overline { u } ; h ) \rangle + \alpha \langle \overline { u } , h \rangle \ge 0 , h \in T _ { \mathcal { C } _ { a d } } ( \overline { u } ) . \end{align*}


\begin{align*} | \{ j \le 2 : j \not = i , m _ j > 0 \} | = \begin{cases} 0 \\ 1 \end{cases} \end{align*}


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


\begin{align*} { \bar F _ i } = - { { w } _ i } { \bar F } , \end{align*}


\begin{align*} ( T _ 1 - f _ 1 ) \cdot ( T _ 2 + f _ 2 ) \cdot \Psi _ 1 = i \Psi _ 1 \end{align*}


\begin{align*} \sum _ { k = 0 } ^ { m } \gamma _ { k } y _ { k } \ge 0 \ \ \forall \ y \in \Omega _ { 0 } \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \mathbb { E } \int _ 0 ^ T \Vert X _ n ( t ) - X ( t ) \Vert _ H ^ { \kappa } d t = 0 , \forall \ \kappa \in [ 1 , \alpha ) . \end{align*}


\begin{align*} \begin{cases} \ \ \Delta w + k ^ { 2 } ( 1 + V ) w = 0 & \mbox { $ \mathrm { i n } \ \ \Omega $ } , \\ \ \ \Delta v + k ^ { 2 } v = 0 & \mbox { $ \mathrm { i n } \ \ \Omega $ } , \\ \ \ w = v , \ \partial _ { \nu } w = \partial _ { \nu } v + \eta v & \mbox { $ \mathrm { o n } \ \ \Gamma $ } , \end{cases} \end{align*}


\begin{align*} \begin{cases} \dim H ^ 1 ( S , T _ S ) = 6 8 ; \\ \dim H ^ 2 ( S , T _ S ) = 6 . \end{cases} \end{align*}


\begin{align*} U ( t + T , s + T ) = U ( t , s ) , t \geq s . \end{align*}


\begin{align*} V ' = \{ v \in V \mid ( v , e _ 1 ) \in E ( v , f _ 1 ) \in E \} . \end{align*}


\begin{align*} Z ( \theta ) = \sum \limits _ { i \in R } \prod \limits _ { F \in \mathrm { f a c e t } ( \Gamma ) } \theta ^ { F } _ { i _ { F } } . \end{align*}


\begin{align*} { \textit { ( i ) } } ~ | | \xi _ 1 | | _ { \mathfrak { Y } } & = \sup \bigg \{ | | S ( \xi _ 1 , \widehat { D } ) | | ~ : ~ \widehat { D } \in \mathfrak { Y } \bigg \} \\ & = \sup \bigg \{ | | \sum _ { i = 1 } ^ { n } \xi _ 1 ( d _ i ) \overline { \mu } ( D _ i ) | | ~ : ~ \widehat { D } = ( D _ i , d _ i ) , ~ i = 1 , 2 , . . \bigg \} \\ & \geq 0 \end{align*}


\begin{align*} x [ n ] = \frac { 1 } { \sqrt { K } } \sum _ { k = 0 } ^ { K - 1 } X [ k ] e ^ { \frac { 2 \pi n k j } { K } } . \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*} a _ { ( n ) } \cdot v = 0 , n \geq N , \end{align*}


\begin{align*} \mathbf { h } ( H _ { \rho } ( x ) ) = R _ \rho \mathbf { h } ( x ) \end{align*}


\begin{align*} \zeta ' & \ = \sum _ { z \in \mathbb { F } ^ * _ { q ^ n } } \overline { U _ { q ^ n } } \Lambda ( z ) { \bf 1 } _ { - } - ( q ^ n - 1 ) \overline { U _ { q ^ n } } { \bf 1 } _ { - } \\ & \ = \sum _ { x , y \in \mathbb { F } ^ * _ { q ^ n } , x y \ne 1 } \varepsilon ( x ) \Lambda ( y ) { \bf 1 } _ { - } - ( q ^ n - 1 ) \overline { U _ { q ^ n } } { \bf 1 } _ { - } \end{align*}


\begin{align*} \frac { 1 } { f ( x ) } \sigma _ { k } ( x ) \varphi ( h ) G ( \nabla h ) = 1 \end{align*}


\begin{align*} \{ \lambda _ { t } ^ { N , j } \} _ { 1 \leq j \leq N } \subset \Big \{ \lambda \in \mathbb C \ , : \ , \left | \lambda - 1 \right | \leq \frac { 1 } { N - 1 } \sum _ { j \not = i } h ' ( \mu ^ { N , j } _ { \boldsymbol { \alpha } ^ * _ t } ) \Big \} . \end{align*}


\begin{align*} G = i \ \overline { v } \ \widehat { v } . \end{align*}


\begin{align*} \mathcal { S } _ 1 = \frac { \psi _ 1 T _ { \rm s } } { \sqrt { 2 \pi } } \ , \mathcal { W } _ 2 + \psi _ { 2 } \ , T _ { \rm s } \ , e ^ { - \frac { \psi _ { 2 } \tau _ { \rm d } } { N _ { \rm a } } + \frac { \psi _ 1 ^ 2 \tau _ { \rm d } ^ 2 } { 2 N _ { \rm a } ^ 2 } } \left [ Q \left ( \kappa _ { \rm b } + \frac { \psi _ { 1 } \tau _ { \rm d } } { N _ { \rm a } } \right ) - Q \left ( \kappa _ { \rm t } + \frac { \psi _ { 1 } \tau _ { \rm d } } { N _ { \rm a } } \right ) \right ] . \end{align*}


\begin{align*} d _ \textrm { H S } ( A , B ) = \| A - B \| _ \textrm { H S } \forall A , B \in \mathrm { U } ( n ) . \end{align*}


\begin{align*} \deg _ G ( v ) = \frac { \sum _ { v \in S \in V ^ { ( k - 1 ) } } \deg _ G ( S ) } { k - 1 } \geq \frac { \deg _ H ( v ) \sqrt { d } } { k - 1 } > d , \end{align*}


\begin{align*} D '' u _ { m ' , \epsilon , j } + P _ { m ' } \big ( \sqrt { N _ 1 \tilde { \lambda } _ { m ' } } h _ { m ' , \epsilon , j } \big ) = D '' \left ( ( 1 - v ' _ { \epsilon } ( \Psi ) ) f F ^ { 1 + \delta } \right ) . \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ { 2 } } | b | ^ { 2 } \dd x ' = \int _ { \mathbb { R } ^ { 2 } } \left ( | \nabla \phi | ^ { 2 } + | G | ^ { 2 } \right ) \dd x ' . \end{align*}


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


\begin{align*} \Omega _ w = \{ ( F _ \bullet , V _ \bullet ) ; \dim V _ i \cap F _ j \geq r _ { i , j } ( w ) \} . \end{align*}


\begin{align*} f \circ \gamma ( t ) = \gamma ( t + b _ \gamma ( f ) ) . \end{align*}


\begin{align*} \left \| \div \big ( | \nabla u | \nabla u \big ) \right \| _ { L ^ 1 ( B _ { 3 / 4 } ) } \leq C \| \nabla u \| _ { L ^ 2 ( B _ { 1 } ) } ^ 2 . \end{align*}


\begin{align*} \lim _ { \norm { \eta _ n } _ { H ^ 1 } \to 0 } \frac { 1 } { \norm { \eta _ n } _ { H ^ 1 } } \norm { A ^ { \tilde \Psi + \eta _ n } [ \eta _ 1 , \dots , \eta _ { n - 1 } ] - A ^ { \tilde \Psi } [ \eta _ 1 , \dots , \eta _ { n - 1 } ] - A ^ { \tilde \Psi } [ \eta _ 1 , \dots , \eta _ n ] } _ { H ^ 1 } = 0 . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ n | f _ i ( s ) | ^ 2 = \mu \log K ( G ( s ) , \overline { G ( s ) } ) ~ ~ s \in U . \end{align*}


\begin{align*} H _ r ( E ^ { \vee } ) = H _ r ( E ) ^ { - 1 } \end{align*}


\begin{align*} \mu _ { i + 1 } = ( I - \gamma g _ { \mu _ i } ) \# \mu _ i . \end{align*}


\begin{align*} - \epsilon ^ 2 \delta ^ 2 b _ { \alpha } u _ n ( x ) \leq u ( 0 ) - u ( x ) + \sum _ { \beta = 1 } ^ { n - 1 } x _ \beta u _ \beta ( x ) \leq C b _ \alpha ^ 2 \end{align*}


\begin{align*} L = a _ n x ^ { q ^ n } + a _ { n - 1 } x ^ { q ^ { n - 1 } } + \cdots + a _ 0 x , \end{align*}


\begin{align*} k t ' = k ( t - p ) > \sum _ { i \in [ \chi ] } x _ i . \end{align*}


\begin{align*} J _ \nu ( t ) = \sum _ { k = 0 } ^ \infty \frac { ( - 1 ) ^ k } { k ! \Gamma ( k + \nu + 1 ) } \left ( \frac { t } { 2 } \right ) ^ { 2 k + \nu } \end{align*}


\begin{align*} f ( z , t ) = \int _ \Omega \int _ { \C ^ n } \widehat { f } ( a , w ) e _ a ( ( - w , 0 ) ( z , t ) ) d w \ , d \nu ( a ) . \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*} \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*} T _ { W , \psi } ( \mu _ \infty ) - I ( \mu _ \infty ) & \ge \limsup _ { n \to \infty } \big ( T _ { W _ { n J } , \psi } ( \mu _ n ( Q ^ n ) ) - I ( \mu _ n ( Q ^ n ) ) \big ) = \limsup _ { n \to \infty } M _ n ^ \psi ( Q ^ n ) / n . \end{align*}


\begin{align*} \left | \dfrac { - \gamma \omega } { - \gamma \alpha \omega } \right | & = \left | 1 + \dfrac { \gamma \alpha \omega - \gamma \omega } { - \gamma \alpha \omega } \right | \leq \max \left ( 1 , \left | \dfrac { \gamma \alpha \omega - \gamma \omega } { - \gamma \alpha \omega } \right | \right ) , \end{align*}


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


\begin{align*} \begin{pmatrix} 1 / d & \sqrt { 1 - 1 / d ^ 2 } \\ \sqrt { 1 - 1 / d ^ 2 } & - 1 / d \end{pmatrix} , \end{align*}


\begin{align*} f ( z ) = a _ 0 + \left ( x - \frac { \psi _ 0 } { \psi _ 1 } y \right ) a _ 1 + \left ( x - \frac { \psi _ 0 } { \psi _ 1 } y \right ) ^ 2 h _ 2 ( z ) , \\ \end{align*}


\begin{align*} & \int _ 0 ^ T \int _ { \mathbb { R } ^ { N } } \omega \varphi d x d t \leq C ( p ) \biggr ( \mathcal { J } _ 1 + k \mathcal { J } _ 2 + \mathcal { J } _ 3 \biggr ) , \end{align*}


\begin{align*} X \coloneqq \{ B \mid A \subset B \subset C , \ \dim B = b , \ B \cap D = E \} \end{align*}


\begin{align*} \left \{ \begin{array} { l l } - \Delta \delta c + \tilde { n } _ { 1 } \delta c = - c _ { 2 } \delta \tilde { n } , & x \in \Omega , \\ \nabla \delta c \cdot \nu + \delta c = 0 , & x \in \partial \Omega , \end{array} \right . \end{align*}


\begin{align*} A ( y _ 1 x ) - A ( y _ 1 x ' ) = A ( 1 ^ \infty ) - A ( 1 0 ^ \infty ) = c - d \ . \end{align*}


\begin{align*} F ' _ m ( n ) = F _ { p _ t } ( n ) = 0 , n \leq p _ t . \end{align*}


\begin{align*} \mathcal V = \mathcal V ( e , L , M ) \coloneqq \{ N \mid L \subset N \subset M , \ \} \end{align*}


\begin{align*} 2 E \left ( \omega _ { n } \right ) & = \int _ { \Pi } \int _ { \Pi } \omega _ { n } ( x ) G _ { \Pi } ( x , y ) \omega _ { n } ( y ) \mathrm { d } x \mathrm { d } y \\ & \geq \int _ { \Pi \cap B _ { L } ( 0 ) } \int _ { \Pi \cap B _ { L } ( 0 ) } \omega _ { n } ( x ) G _ { \Pi } ( x , y ) \omega _ { n } ( y ) \mathrm { d } x \mathrm { d } y , \end{align*}


\begin{align*} \sum _ { r = 1 } ^ p \sum _ { s = 1 } ^ q \alpha _ { r s } ( ( X ^ * X - \| X \| ^ 2 ) e _ r \otimes f _ s ) = 0 . \end{align*}


\begin{align*} H _ { m i n } ( E \otimes ( F _ 1 [ t ] \times F _ 1 [ t ] ^ { \vee } ) ) & = H _ { m i n } ( E \otimes F _ 1 [ t ] \times E \otimes F _ 1 [ t ] ^ { \vee } ) \\ & = \min H _ { m i n } ( E \otimes F _ 1 [ t ] , E \otimes F _ 1 [ t ] ^ { \vee } ) \\ & \leq H _ { m i n } ( E \otimes F _ 1 [ t ] ) = t H _ { m i n } ( E \otimes F _ 1 ) \leq t H _ { m i n } ( E ) H _ { m i n } ( F _ 1 ) \end{align*}


\begin{align*} f _ { + } ( z , t , s ) & : = \exp \left ( - \sum _ { n = 1 } ^ { \infty } \frac { z ^ { n } } { n } \int _ { \{ S _ n > 0 \} } e ^ { i t S _ n } e ^ { i { s } \cdot { C } _ n } d \mathbb { P } \right ) , \\ f _ { - } ( z , t , s ) & : = \exp \left ( + \sum _ { n = 1 } ^ { \infty } \frac { z ^ { n } } { n } \int _ { \{ S _ n \le 0 \} } e ^ { i t S _ n } e ^ { i { s } \cdot { C } _ n } d \mathbb { P } \right ) . \end{align*}


\begin{align*} \Phi ^ \dagger ( v , p ) : = Q ^ \dagger F ^ { q , \dagger } ( v + \psi ^ \dagger ( v , p ) , p ) , \end{align*}


\begin{align*} \widetilde { \nu } _ t ^ { * , \xi } = \eta _ t \nu _ t ^ { * , \xi } , \widetilde { \varphi } _ t ^ { * , \xi } = \eta ^ { - 1 } _ t \varphi _ t ^ { * , \xi } , \quad \widetilde { \Lambda } _ t ^ { 0 , * , \xi } = \eta ^ { - 1 } _ t \Lambda _ t ^ { 0 , * , \xi } , t \in [ 0 , T ] . \end{align*}


\begin{align*} \C = \left \{ ( \lambda + \mu \theta ^ { i _ 1 } , \lambda + \mu \theta ^ { i _ 2 } , \cdots , \lambda + \mu \theta ^ { i _ n } ) \ , | \ , \lambda , \mu \in \mathbb { F } _ { p ^ e } \right \} \end{align*}


\begin{align*} 0 < \Psi \leq \frac { 3 } { 2 } ( a _ 3 = 0 ) ( \beta - \gamma = 0 , \ ) . \end{align*}


\begin{align*} B _ q ( \psi _ m ) ( z ) & = \int \limits _ { \mathbb R } \overline { A _ q ( z , x ) } \psi _ m ( x ) d x \\ & = \int \limits _ { \mathbb R } \left ( \sum _ { j = 0 } ^ { \infty } \frac { z ^ j } { \sqrt { \Gamma ( q j + 1 ) } } \psi _ j ( x ) \right ) \psi _ m ( x ) d x , \\ & = \sum _ { j = 0 } ^ { \infty } \frac { z ^ j } { \sqrt { \Gamma ( q j + 1 ) } } \left ( \int \limits _ { \mathbb R } \psi _ j ( x ) \psi _ m ( x ) d x \right ) . \\ \end{align*}


\begin{align*} & g \big ( \lfloor ( 1 - ( 1 + \tilde \varepsilon ) ^ { - j } ) r \rfloor \big ) \geq 1 6 \ , g \big ( \lfloor ( 1 - ( 1 + \tilde \varepsilon ) ^ { - j - 1 } ) r \rfloor \big ) , j = 1 , 2 , \dots , i - 1 ; \\ & g \big ( \lfloor ( 1 - ( 1 + \tilde \varepsilon ) ^ { - i } ) r \rfloor \big ) \leq \sqrt { x } . \end{align*}


\begin{align*} \lim _ { M \rightarrow \infty } \lim _ { n \rightarrow \infty } I I _ 2 \leq C \bigg \{ \lim _ { n \rightarrow \infty } \mathbb { E } \int _ 0 ^ { T } \Vert X _ n ( s ) - X ( s ) \Vert _ H d s \bigg \} ^ { \frac { \alpha - \theta } { \alpha } } = 0 . \end{align*}


\begin{align*} \sum _ { k = 2 } ^ n \frac { \mu ( k ) } { k } ( I - S ) h _ k ( z ) = \sum _ { k = 1 } ^ n \frac { \mu ( k ) } { k } \left [ \log ( 1 - z ^ k ) - \log ( 1 - z ) - \log k \right ] \end{align*}


\begin{align*} z ^ k & \mapsto f _ k ( s ) = - \frac { 1 } { s } \left ( ( k + 1 ) ^ { 1 - s } - k ^ { 1 - s } \right ) & & ( k \in \mathbb { N ^ * } ) \\ 1 & \mapsto - \frac { 1 } { s } \end{align*}


\begin{align*} \begin{pmatrix} v _ 1 \\ v _ 2 \end{pmatrix} = - \frac { 1 } { k ^ 2 + x _ 1 ^ 2 + x _ 2 ^ 2 } \begin{pmatrix} x _ 1 x _ 2 & - k ^ 2 - x _ 1 ^ 2 \\ k ^ 2 + x _ 2 ^ 2 & - x _ 1 x _ 2 \end{pmatrix} \begin{pmatrix} \partial _ { x _ 1 } \varphi \\ \partial _ { x _ 2 } \varphi \end{pmatrix} . \end{align*}


\begin{align*} | Z _ 3 | & = \left | \frac { 1 } { \log x } \sum _ { \rho } \frac { x ^ { \rho - s } - x ^ { 2 ( \rho - s ) } } { ( s - \rho ) ^ 2 } \right | \\ & \leq \frac { x ^ { \frac { 1 } { 2 } - \sigma } + x ^ { 1 - 2 \sigma } } { \log x } \sum _ { \gamma } \frac { 1 } { ( \sigma - \beta ) ^ 2 + ( t - \gamma ) ^ 2 } . \end{align*}


\begin{align*} \Tilde { x } \in \mathcal { S } ^ n : = \{ x \in \mathbb { R } ^ n \ | \ \| x \| ^ 2 = \lambda \| x _ v \| ^ 2 + ( 1 - \lambda ) \| x _ w \| ^ 2 \} . \end{align*}


\begin{align*} \S = \{ v \in C ^ { 2 } ( \overline \O ) : v | _ { ( \partial \O ) _ { j } } = \phi _ { j } \} . \end{align*}


\begin{align*} \epsilon _ s P ( x _ 1 , \ldots , x _ n ) = P ( v _ 1 , \ldots , v _ n ) = 0 . \end{align*}


\begin{align*} Z _ n = M _ n + r _ n N _ n , \end{align*}


\begin{align*} G ( \beta , 1 ) = H ( \beta , 1 ) \circ [ g _ { \beta } ] _ { 1 } = \beta \circ \widetilde { \beta ( 1 ) } \simeq i d _ { X _ { U ' } ^ { I } } \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } E \left ( \omega _ { n } \right ) = E ( \omega ) , \end{align*}


\begin{align*} f _ a ( x ) : = \dfrac { 1 } { T } \int _ { 0 } ^ { T } f ( x , t ) \ , d t . \end{align*}


\begin{align*} \begin{aligned} & \dfrac { b _ { 1 , \delta } ( x ) } { 2 } = \ , \int _ 0 ^ 1 h ( x + \delta \sin ( 2 \pi t ) ) \sin ( 2 \pi t ) \ , d t \\ = & \ , \int _ 0 ^ 1 [ \tilde { h } ( x + \delta \sin ( 2 \pi t ) ) + m ( x + \delta \sin ( 2 \pi t ) ) ] \sin ( 2 \pi t ) \ , d t \\ = & \ , \int _ 0 ^ 1 \tilde { h } ( x + \delta \sin ( 2 \pi t ) ) \sin ( 2 \pi t ) \ , d t \\ & \ , + \int _ 0 ^ 1 [ m ( x + \delta \sin ( 2 \pi t ) ) - m ( x ) ] \sin ( 2 \pi t ) \ , d t . \end{aligned} \end{align*}


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


\begin{align*} S _ { s } ( x ) = - s \sum _ { \zeta ( \rho ) = 0 } \frac { x ^ { \rho - s } } { \rho ( \rho - s ) } . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \sup _ { y \in \Pi } \int _ { B _ { R } ( y ) \cap \Pi } x _ { 2 } \omega _ { n } \mathrm { ~ d } x = 0 . \end{align*}


\begin{align*} \mathcal U ( s , i ) = [ 0 , \bar L ] , ( s , i ) \in C . \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*} \displaystyle T ( f ) ( x ) = \int _ { \mathbb { C } } { A _ q ( z , x ) } f ( z ) | z | ^ { \frac { 2 } { q } - 2 } e ^ { - \frac { | z | ^ 2 } { q } } d A ( z ) , \end{align*}


\begin{align*} | \rho ( x ) | = ( x , \Sigma ) : = \inf \{ d ( x , y ) \mid y \in \Sigma \ , \} . \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*} \begin{cases} - \varepsilon ^ 2 ( K _ H ( x ) \nabla u ) = \left ( u - \left ( \frac { \alpha | x | ^ 2 } { 2 } + \beta \right ) \ln \frac { 1 } { \varepsilon } \right ) ^ { p } _ + , \ \ & x \in B _ { R ^ * } ( 0 ) , \\ u = 0 , \ \ & x \in \partial B _ { R ^ * } ( 0 ) , \end{cases} \end{align*}


\begin{align*} { \rm a r g m a x } _ { j \in [ n ] \backslash I } | \langle v _ { k + 1 } ( B ) , ( B _ { j , [ m ] } ) ^ \top \rangle | & \le | \langle v _ { k + 1 } ( B _ { I , [ m ] } ) , ( B _ { i _ { k + 1 } ( B _ { I , [ m ] } ) , [ m ] } ) ^ \top \rangle | \\ & = { \rm a r g m a x } _ { i \in I \backslash I _ { k } ( B ) } | \langle e _ 1 , ( B _ { i , [ m ] } ) ^ \top \rangle | , \end{align*}


\begin{align*} \tilde { b } = \nabla \times ( \phi \nabla \theta ) + \frac { 1 } { \tau } G \nabla \theta , \end{align*}


\begin{align*} v _ \beta = y ^ { 2 ( r _ 0 + \cdots + r _ { a ( \beta ) } - \beta + 1 ) } . \end{align*}


\begin{align*} \omega ^ * : = \R \setminus [ - R , 0 ] = ( - \infty , - R ) . \end{align*}


\begin{align*} \langle \alpha _ i , \alpha _ j \rangle _ H = \begin{cases} c _ i & , \\ c _ i c _ { i j } & , \\ 0 & . \end{cases} \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*} | G _ { z _ i } ( T _ { z _ i } z _ i , T _ { z _ i } z _ j ) | + | \nabla G _ { z _ i } ( T _ { z _ i } z _ i , T _ { z _ i } z _ j ) | \le C , \ \ 1 \leq i \neq j \leq m , \end{align*}


\begin{align*} ( \ell ^ { \infty } , \beta _ { 0 } ) ' = ( \ell ^ { \infty } , \mu ( \ell ^ { \infty } , \ell ^ { 1 } ) ) ' = \ell ^ { 1 } . \end{align*}


\begin{align*} \begin{aligned} \int _ { \R ^ 3 } | v | ^ k e ^ { - a | v | ^ 2 } \d v = \Big \{ \int _ { | v | \leq 1 } + \int _ { | v | > 1 } \Big \} | v | ^ k e ^ { - a | v | ^ 2 } \d v \ , . \end{aligned} \end{align*}


\begin{align*} u _ n = a \alpha ^ n + b \beta ^ n , \end{align*}


\begin{align*} q - h \ge \log _ { \theta } \Big ( 1 + \frac { d \rho } { \epsilon _ 0 } \Big ) ~ { \rm f o r } ~ h = 0 , 1 , \dots , q - 1 , \end{align*}


\begin{align*} \nabla \mathcal { N } _ { \epsilon } ( x ) = W _ { L } \cdot \sigma _ { \epsilon } ' ( N _ { \epsilon } ^ { ( L - 1 ) } ( x ) ) \cdot W _ { L - 1 } \cdot \ldots \cdot \sigma _ { \epsilon } ' ( N _ { \epsilon } ^ { ( 1 ) } ( x ) ) \cdot W _ { 1 } . \end{align*}


\begin{align*} P ^ g ( v _ 1 , \ldots , v _ n ) = P ( w _ 1 , \ldots , w _ n ) . \end{align*}


\begin{align*} \mathrm { H o m } ( V , W ) = \bigoplus _ { \alpha \in K ( \mathcal { I } ) } \mathrm { H o m } ( V , W ) _ { \alpha } , \mathrm { H o m } ( V , W ) _ { \alpha } = \{ f \in \mathrm { H o m } ( V , W ) | f ( V _ { i } ) \subset W _ { j } , [ ( j , i ) ] = \alpha \} . \end{align*}


\begin{align*} & E _ { * } ( g _ 1 , w ) = \\ & \int _ { - \infty } ^ \infty \int _ 0 ^ 1 \int _ 0 ^ \infty \int _ 0 ^ \infty \Psi _ * \left ( \lambda , \theta , t , u + \left ( \frac { t _ 1 } { t } \right ) ^ 2 u _ 1 \right ) e ^ { i \Phi \left ( \lambda , \theta , t , u + \left ( \frac { t _ 1 } { t } \right ) ^ 2 u _ 1 \right ) } W ( \lambda , t , u ) d \lambda d \theta d t d u . \end{align*}


\begin{gather*} \phi ^ + ( x ) : = \lim _ { n \rightarrow \infty } \frac 1 n \sum ^ { n - 1 } _ { k = 0 } \phi ( f ^ k ( x ) ) , \\ \phi ^ - ( x ) : = \lim _ { n \rightarrow \infty } \frac 1 n \sum ^ { n - 1 } _ { k = 0 } \phi ( f ^ { - k } ( x ) ) , \\ \bar \phi ( x ) : = \lim _ { n \rightarrow \infty } \frac 1 { 2 n - 1 } \sum ^ { n - 1 } _ { k = - ( n - 1 ) } \phi ( f ^ k ( x ) ) . \end{gather*}


\begin{align*} ( I - F ) \Psi ^ - _ 1 = 0 \Longleftrightarrow F \Psi ^ - _ 1 = \Psi ^ - _ 1 . \end{align*}


\begin{align*} M / ( x _ 1 ^ n , \dots , x _ k ^ n ) = ( ( M / x _ 1 ^ n ) / x _ 2 ^ n ) / \dots / x _ k ^ n \end{align*}


\begin{align*} l \sum h _ { k } \nabla _ { k } \chi & = l \sum h _ { k } \left [ \frac { - f _ { k } } { f } + \frac { h _ { k } } { h } + \frac { \varphi ^ { ' } } { \varphi } h _ { k } + \frac { ( \nabla G \cdot e _ { i } ) \cdot w _ { k i } } { G } \right ] \\ & \leq C _ { 1 } l + l \frac { \nabla G \cdot e _ { k } } { G } h _ { k } w _ { k k } = C _ { 1 } l + l \frac { \nabla G \cdot e _ { k } } { G } \rho \rho _ { k } \leq C _ { 2 } l , \end{align*}


\begin{align*} ( \mathcal { F } _ N ) = \begin{pmatrix} N + 2 \\ 2 \end{pmatrix} \ , . \end{align*}


\begin{align*} M ( - t ) = \frac { 1 } { \sqrt 2 \pi } \sum _ { r , j = 0 } ^ \infty \ , \pi _ r \ , \ , c _ { r , j } \ , \int _ { - \infty } ^ { \infty } \ , x ^ { j } \ , \exp \left ( - t x - \frac { x ^ 2 } { 2 } \right ) \mathrm { d } x . \end{align*}


\begin{align*} \bar { x } _ n - x _ n ^ 0 = \frac { - \varepsilon _ 1 \bar { \xi } _ 1 + | \hat { x } | \bar { \xi } _ 2 } { \sqrt { \varepsilon _ 1 ^ 2 + | \hat { x } | ^ 2 } } \leq | D \rho ( 0 , \tilde { x } ) | | \bar { \xi } _ 1 | + | \bar { \xi } _ 2 | \leq C \epsilon _ 0 \delta ^ 2 b _ \alpha \end{align*}


\begin{align*} a _ { n , k } & = \begin{cases} O ( 1 ) \quad & H < H ( k ) \\ O ( \log n ) \quad & H = H ( k ) \\ O ( n ^ { ( 2 H - 1 ) k - 1 } ) \quad & H > H ( k ) \end{cases} \end{align*}


\begin{align*} \lim _ { s \to \infty } b _ \eta ( \xi ( s ) ) = - \infty , \lim _ { s \to - \infty } b _ \eta ( \xi ( s ) ) = \infty , \end{align*}


\begin{align*} h ( s ) = h _ { a , \Delta } ( s ) = \left ( \dfrac { a } { a ^ 2 + s ^ 2 } \right ) \left ( \dfrac { e ^ { 2 \pi a \Delta } + e ^ { - 2 \pi a \Delta } - 2 \cos ( 2 \pi \Delta s ) } { \left ( e ^ { \pi a \Delta } - e ^ { - \pi a \Delta } \right ) ^ 2 } \right ) \end{align*}


\begin{align*} u = ( Q _ { \delta } ( - \delta ^ 2 ( K ( x ) \nabla ) ) ) ^ { - 1 } p Q _ \delta ( V _ { \delta , Z } - q ) ^ { p - 1 } _ + u + ( Q _ { \delta } ( - \delta ^ 2 ( K ( x ) \nabla ) ) ) ^ { - 1 } h , \ \ u \in E _ { \delta , Z } . \end{align*}


\begin{align*} \frac { 2 \pi } { N } = \int _ { \alpha _ k } ^ { \beta _ k } \frac { s b ' ( s ) } { b ( s ) } d m ( s ) \geq | b ' ( s _ k ) | | \beta _ k - \alpha _ k | \ , k = 1 , \ldots , N . \end{align*}


\begin{align*} - \Delta u + V ( x ) u + \frac { \gamma } { 2 \pi } \left ( \log ( | \cdot | ) \ast u ^ 2 \right ) u = b | u | ^ { p - 2 } u \qquad \ \mathbb { R } ^ 2 . \end{align*}


\begin{align*} \mathbb { E } \left [ y _ i ^ * y _ i \right ] = a _ c ^ 2 \lvert \mathbf { h } _ { i , * } \mathbf { p } _ c \rvert ^ 2 + \sum _ { l = 1 } ^ M a _ l ^ 2 \lvert \mathbf { h } _ { i , * } \mathbf { p } _ l \rvert ^ 2 + \sigma _ n ^ 2 . \end{align*}


\begin{align*} \begin{pmatrix} \theta ^ 2 & \theta \cdot \sigma \\ \theta \cdot \sigma & \sigma ^ 2 \end{pmatrix} = \begin{pmatrix} 2 & 0 \\ 0 & - 2 \end{pmatrix} . \end{align*}


\begin{align*} \hat { f } ( \chi ) = \int _ G f ( g ) \chi ( g ) d g \end{align*}


\begin{align*} c = \mathop { \inf } \limits _ { \gamma \in \Gamma } \mathop { \max } \limits _ { t \in [ 0 , 1 ] } I ( \gamma ( t ) ) , \end{align*}


\begin{align*} \varphi ^ * x ^ { \mu } = y ^ { \mu } , \varphi ^ * \theta _ { i , a } = \eta _ { i , a } , \end{align*}


\begin{align*} \mathcal { L } _ \Phi : = { \big \{ F ( z , \omega ) = f ( \Phi ^ { - 1 } ( z , \omega ) , \omega ) ; f \in L ^ 2 _ { \rm l o c } ( \mathbb { R } ^ n ; L ^ 2 ( \Omega ) ) \ ; \ ; \big \} } \end{align*}


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


\begin{gather*} \mathcal { A } _ j ( x , \xi ) = D _ { \xi } F _ j ( x , \xi ) , \end{gather*}


\begin{align*} b ^ \Lambda _ \lambda = \eta ( \tau ) ^ \ell c ^ \Lambda _ \lambda \end{align*}


\begin{align*} \bigcup _ { y = 1 } ^ z I _ y = \{ 1 , \dots , n \} \end{align*}


\begin{align*} \pi _ 1 ( \theta _ { 1 } \theta _ { 2 } ) = \theta _ 1 \theta _ 2 = - \pi _ 1 ( \theta _ { 1 } ) \pi _ 1 ( \theta _ { 2 } ) ~ . \end{align*}


\begin{align*} J ( x ; k ) = \begin{cases} \begin{pmatrix} | r ( k ) | ^ 2 & \overline { r ( k ) } \mathrm { e } ^ { - 2 i k ^ 2 x } \\ r ( k ) e ^ { 2 i k ^ 2 x } & 0 \end{pmatrix} , k \in \mathbb { R } , \\ \\ \begin{pmatrix} - | r ( k ) | ^ 2 & - \overline { r ( k ) } e ^ { - 2 i k ^ 2 x } \\ r ( k ) e ^ { 2 i k ^ 2 x } & 0 \end{pmatrix} , k \in i \mathbb { R } . \end{cases} \end{align*}


\begin{align*} \lim _ { n \to \infty } \frac { 1 } { n } \log \int _ { \R ^ n } e ^ { f ( x ) } d x = \sup _ { Q \in \P ( \R ) } \left ( \int _ { \R } V \ , d Q + \frac 1 2 \int _ { \R } \int _ { \R } K ( x - y ) Q ( d x ) Q ( d y ) - H ( Q ) \right ) . \end{align*}


\begin{align*} \begin{aligned} & I _ { h , W , \gamma } = \\ & \inf \left \{ \frac { 1 } { 2 } \int _ { \mathbb { R } ^ { 3 } } | b | ^ { 2 } \dd x \ \middle | \ b = \nabla \times ( \phi \nabla \theta ) + G \nabla \theta \in L ^ { 2 } _ { \sigma , \textrm { a x i } } ( \mathbb { R } ^ { 3 } ) , 2 \int _ { \mathbb { R } ^ { 3 } } ( \phi - \phi _ { \infty } ) _ { + } \frac { G } { r ^ { 2 } } \dd x = h \ \right \} , \end{aligned} \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 } - \tfrac { 3 } { 2 } < \gamma < 0 \ , . \end{aligned} \end{align*}


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


\begin{align*} \mu ( X ( s _ 1 ) \in B _ 1 , \cdots , X ( s _ r ) \in B _ r ) = \frac { 1 } { ( 2 \pi ) ^ { \frac { r } { 2 } } . | C | } \int _ { B _ 1 \times \cdots \times B _ r } \exp \big ( - \frac { 1 } { 2 } . { } ^ { t } { x } . C ^ { - 1 } . x \big ) d x , \end{align*}


\begin{align*} \left | \int _ M f ( x ) \right | = \left | \int _ M ( f ( x ) - \tilde f ( x ) ) \right | \leq { \rm V o l } ( \tilde \Sigma ^ r ) \cdot { \rm V o l } ( M ) . \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*} \mathcal { S } ^ { k - 1 } : = \{ \underline { x } \in [ 0 , 1 ] ^ { k } : | | \underline { x } | | = 1 \} , ~ \mathcal { D } ^ { k } ( A ) : = \{ \rho ( \underline { a } ) : \underline { a } \in A ^ { k } \} ~ \mbox { a n d } ~ \mathcal { D } ^ { \underline { k } } ( A ) : = \{ \rho ( \underline { a } ) : \underline { a } \in A ^ { \underline { k } } \} , \end{align*}


\begin{align*} \frac { \partial \tilde x } { \partial x } = h ^ { \frac { 1 } { 3 } } + x \cdot \frac { 1 } { 3 } h ^ { - \frac { 2 } { 3 } } \cdot h _ x ' , \end{align*}


\begin{align*} \alpha ( p ^ i ) = \frac { p ^ { i + 1 } - p ^ { i } } { p ^ { i + 1 } - 1 } , \end{align*}


\begin{align*} - \epsilon _ 1 b _ { n - 1 } u _ n ( x ) \leq u ( 0 ) - u ( x ) + \sum _ { \alpha = 1 } ^ { n - 1 } x _ \alpha u _ \alpha ( x ) \leq C b _ { n - 1 } ^ 2 . \end{align*}


\begin{align*} f ( x ) = \sum _ { m = 0 } ^ \infty \frac { x ^ m } { \Gamma \bigl ( 1 + \frac { m } { 2 } \bigr ) m ! } , \end{align*}


\begin{align*} \varphi ( [ m , n ] ) = \int _ { \widehat { N } } \chi ( [ m , n ] ) \ , \mathrm { d } \mu _ \varphi ( \chi ) = \int _ { \widehat { N } } \chi ^ m ( n ) \chi ^ { - 1 } ( n ) \ , \mathrm { d } \mu _ \varphi ( \chi ) = 1 . \end{align*}


\begin{align*} f \left ( x , \frac { 1 } { 2 ^ n } \right ) = \frac { h _ n ( x ) } { 2 ^ { n + 1 } } f ( x , 0 ) = 0 . \end{align*}


\begin{align*} \begin{aligned} \int _ { \mathcal C } \vert e ^ { \rho \cdot \mathbf x } \vert \vert \psi ( \mathbf x ) \vert \mathrm d \mathbf x \leq \frac { 1 } { \tau ^ 3 } \| e ^ { - \mathbf d \cdot \mathbf y } \| _ { L ^ { \frac { 8 } { 7 } } ( \mathcal C ) } \left \| \psi \left ( \frac { \mathbf y } { \tau } \right ) \right \| _ { L ^ 8 ( \mathcal C ) } \lesssim \tau ^ { - \frac { 1 2 1 } { 4 0 } } , \end{aligned} \end{align*}


\begin{align*} | \nabla f | = \alpha ( f ) \Delta f = \beta ( f ) . \end{align*}


\begin{align*} | F _ y ( x , y ) - p _ y ( x , y ) | = | F _ y ( x , y ) - h ^ y ( x , y ) | < 2 \varepsilon . \end{align*}


\begin{align*} \nabla G ( U , x , t ) ^ T \nabla A ( U , x , t ) & = \nabla A ( U , x , t ) ^ T \nabla G ( U , x , t ) \\ \nabla f _ \alpha ( U , x , t ) ^ T \nabla A ( U , x , t ) & = \nabla f _ \alpha ( U , x , t ) ^ T \nabla G ( U , x , t ) \qquad \alpha = 1 , \dots , d , \end{align*}


\begin{align*} \begin{aligned} \left | \int _ { \Omega } d _ 0 u \wedge d _ 1 v \wedge \alpha \right | ^ 2 \leq \int _ { \Omega } d _ 0 u \wedge d _ 1 u \wedge \alpha \cdot \int _ { \Omega } d _ 0 v \wedge d _ 1 v \wedge \alpha . \end{aligned} \end{align*}


\begin{align*} E ^ { 1 } _ { p , q } = H _ { q } ( G , C _ { p } ) \Rightarrow H _ { p + q } ( G , C _ { * } ) \end{align*}


\begin{align*} ( R ( X , T ) S ) ^ h = 0 \quad \forall T , S \in \Delta , \ , X \in T M . \end{align*}


\begin{align*} \Tilde { L } ( x ) : = \gamma _ 0 f _ 0 ( x ) + \sum \limits _ { k = 1 } ^ m \gamma _ k f _ k ( x ) \ge 0 \ \ x \in \mathbb { R } ^ n . \end{align*}


\begin{align*} E ^ L _ Y ( u ) \supset E ^ L _ Y ( s \sigma ( v ' ) w ) = \varphi _ { \sigma , s } ^ L ( E ^ L _ X ( v ' ) ) = \varphi _ { \sigma , s } ^ L ( E ^ L _ X ( v ) ) . \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*} \dfrac { \varepsilon } { 2 } \leq \beta ( \varphi _ { 2 k } ( a _ 1 ) , \rho ( a _ 1 ) ) = \beta ( \varphi _ { 2 k } ( a _ 1 ) , \rho \circ \varphi _ { 2 k } ( a _ 1 ) ) \leq \beta ( \varphi _ i ( a _ 1 ) , \rho \circ \varphi _ i ( a _ 1 ) ) . \end{align*}


\begin{align*} F _ { j } ( s ) = \prod _ { p } ^ { } \left ( 1 + \frac { \lambda ^ { 2 } _ { s y m ^ { j } f } ( p ) l ( p ) } { p ^ { s } } + \cdots + \frac { \lambda ^ { 2 } _ { s y m ^ { j } f } ( p ^ { m } ) l ( p ^ { m } ) } { p ^ { m s } } + \cdots \right ) . \end{align*}


\begin{align*} [ L _ \lambda L ] = ( \partial + 2 \lambda ) L , [ L _ \lambda H ] = ( \partial + \lambda ) H , [ H _ \lambda L ] = \lambda H , [ H _ \lambda H ] = 0 . \end{align*}


\begin{align*} \exists \ \theta > 1 \ \Rightarrow \ K _ p ( \theta ) : = | | \kappa _ p ( \xi ) | | _ { \theta } < \infty , \end{align*}


\begin{align*} H \bigg ( \mathrm { L a w } \bigg ( \sum _ { i = 1 } ^ n t _ i X _ i \bigg ) \bigg ) \le \sum _ { i = 1 } ^ n t _ i H ( Q _ i ) . \end{align*}


\begin{align*} f _ { N } ( x , y | \rho ) = f _ { N } ( x ) f _ { N } ( x ) \sum _ { n \geq 0 } \rho ^ { n } h _ { n } ( x ) h _ { n } ( y ) , \end{align*}


\begin{align*} \hat h \left ( A _ \phi \begin{pmatrix} a \\ b \end{pmatrix} , s \right ) = \hat h \left ( \begin{pmatrix} a \\ b \end{pmatrix} , s \right ) \end{align*}


\begin{align*} \varphi _ { r , h } ( t , x ) : = g _ r ( t ) f _ h ( \phi ( x ) ) . \end{align*}


\begin{align*} a ^ i _ S ( T ) = \begin{cases} 1 , & T \cap F _ i = S \\ 0 , & \end{cases} \end{align*}


\begin{align*} \Lambda ( s , \pi \otimes \chi _ { 8 d } ) : = & \pi ^ { - M s / 2 } \prod ^ M _ { j = 1 } \Gamma \Big ( \frac { s + \mu _ { \pi \otimes \chi _ { 8 d } } ( j ) } { 2 } \Big ) L ( s , \pi \otimes \chi _ { 8 d } ) = \epsilon ( s , \pi \otimes \chi _ { 8 d } ) \Lambda ( 1 - s , \pi \otimes \chi _ { 8 d } ) , \end{align*}


\begin{align*} \Omega = \left \{ ( \xi , z ) \in \mathbb { C } ^ { d _ { 0 } } \times D : | | \xi | | ^ { 2 } < \varphi ( z ) \right \} \end{align*}


\begin{align*} \{ l _ 1 , l _ 2 , \dots , l _ r , l _ 1 ' , l _ 2 ' , \dots , l _ r ' \} = \cup _ { u = 1 } ^ t \mathcal I ( s _ u ) , \end{align*}


\begin{align*} \varphi ^ L _ { \sigma , s ' } ( D ) = \{ a , b \} \varphi ^ L _ { \sigma , s ' } ( E ) = \varphi ^ L _ { \sigma , s ' } ( C _ i ) \setminus \{ a \} = C \setminus \{ a \} . \end{align*}


\begin{align*} \Delta \rho _ 1 ( x ) \leq \psi _ { K , - H } ( \rho _ 1 ( x ) ) = - \psi _ { K , H } ( \rho ( x ) ) . \end{align*}


\begin{align*} L = \bigoplus _ { i = 1 } ^ \ell \mathfrak a _ i b _ i . \end{align*}


\begin{align*} \begin{aligned} \int _ { \Omega } A \cdot B \dd x & \leq \frac { 1 } { f _ 1 ^ { + } } \int _ { \Omega } | B | ^ { 2 } \dd x , \\ - \int _ { \Omega } A \cdot B \dd x & \leq \frac { 1 } { - f _ 1 ^ { - } } \int _ { \Omega } | B | ^ { 2 } \dd x , \end{aligned} \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*} a ( z ) = 1 + k \int _ { \mathbb { R } } u _ y \psi ^ - _ { 2 1 } ( x ; k ) \mathrm { d } x . \end{align*}


\begin{align*} \sum _ { i \geq 1 \atop { d \geq 0 } } a _ { i , d } ( \underline { b } ) = a ( \underline { b } ) . \end{align*}


\begin{align*} \begin{aligned} \Delta u _ 1 \wedge \dots \wedge \Delta u _ k \wedge \beta _ n ^ { n - m } \wedge \omega \geq 0 . \end{aligned} \end{align*}


\begin{align*} L ( x ) : = \sum _ { \beta = 1 } ^ { n - 1 } \rho _ \beta ( y ' ) ( x _ \beta - y _ \beta ) . \end{align*}


\begin{align*} u = \int ^ x \frac { \dd x } { \sqrt { f _ 5 ( x ) } } + \int ^ { x ' } \frac { \dd x ' } { \sqrt { f _ 5 ( x ' ) } } , v = \int ^ x \frac { x \dd x } { \sqrt { f _ 5 ( x ) } } + \int ^ { x ' } \frac { x ' \dd x ' } { \sqrt { f _ 5 ( x ' ) } } , \end{align*}


\begin{align*} \delta z = - \frac { 1 } { 8 } { F _ z } . \end{align*}


\begin{align*} X _ { n + 1 } = A X _ { n } + B u _ n + \xi _ { n + 1 } \end{align*}


\begin{align*} \begin{aligned} \Psi ^ - _ { 2 1 } ( x ; z ) & = \int _ { - \infty } ^ { x - \delta } e ^ { 2 i z ( x - y ) } \nu ( y ; z ) d y + \nu ( x ; z ) \int _ { x - \delta } ^ { x } e ^ { 2 i z ( x - y ) } \mathrm { d } y \\ & + \int _ { x - \delta } ^ { x } e ^ { 2 i z ( x - y ) } ( \nu ( y ; z ) - \nu ( x ; z ) ) \mathrm { d } y \equiv I _ 1 + I _ 2 + I _ 3 , \end{aligned} \end{align*}


\begin{align*} \begin{cases} d X _ { t , n } = X _ { t , 1 } X _ { t , n - 1 } d t + \sigma _ { n } d W _ t ^ { ( n ) } \\ d X _ { t , n - 1 } = \sigma _ { n - 1 } d W _ t ^ { ( n - 1 ) } \\ d X _ { t , j } = 0 & j \not \in \{ n , n - 1 \} \end{cases} \end{align*}


\begin{align*} i _ { \Gamma } ^ * A : = ( I , \Gamma ) ^ T A ( I , \Gamma ) = B + C \Gamma + \Gamma ^ T C ^ T + \Gamma D \Gamma ^ T . \end{align*}


\begin{align*} \nabla _ X \varphi = \frac { 1 } { 2 } ( \sqrt { c _ 1 } \ X _ 1 \cdot \nu _ 1 + \sqrt { c _ 2 } \ X _ 2 \cdot \nu _ 2 ) \cdot \varphi - \frac { 1 } { 2 } B ( X ) \cdot \varphi . \end{align*}


\begin{align*} \begin{aligned} u ( \bar { x } ) \leq \ , & u ( x ^ * ) + C \epsilon _ 0 \delta ^ 2 b _ { \alpha } ^ 2 + C b _ { \alpha } \bar { x } _ n + C \delta ^ 3 b _ \alpha ^ 2 \\ \leq \ , & u ( x ^ * ) + C \left ( \epsilon _ 0 + \delta \right ) \delta ^ 2 b _ \alpha ^ 2 \leq \left ( 1 + \frac { c _ 0 } { 3 } \right ) u ( x ^ * ) \end{aligned} \end{align*}


\begin{align*} P V _ { \delta , \hat { x } , \hat { q } , z } ( x ) = V _ { \delta , \hat { x } , \hat { q } , z } ( x ) - \frac { \hat { q } } { \ln \frac { R } { s _ \delta } } g _ { \hat { x } } ( T _ { \hat { x } } x , T _ { \hat { x } } z ) , \ \ \forall x \in \Omega , \end{align*}


\begin{align*} \pi _ 0 ( e ) = \pi _ 0 ( e ^ 2 ) = \pi _ 0 ( e ) ^ 2 + \sum _ { \omega \in W _ f ^ + } \pi _ { - \omega } ( e ) \pi _ { \omega } ( e ) . \end{align*}


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


\begin{align*} M ( - t ) = \frac { 1 } { \sqrt 2 \pi } \sum _ { r = 0 } ^ \infty \pi _ r \ , \ , \int _ { - \infty } ^ { \infty } \Phi ( x ) ^ r \ , \exp \left ( - t \ , x - \frac { x ^ 2 } { 2 } \right ) \mathrm { d } x . \end{align*}


\begin{align*} \mathcal { F } _ { \geq p } t e l ( \mathcal { C } ) : = \lbrace x \in t e l ( \mathcal { C } ) \mid \mathcal { A } ( x ) \geq p \rbrace . \end{align*}


\begin{align*} \frac { d \mu _ g ^ n } { d t } = \sum _ { i = 1 } ^ { n - 1 } \sum _ { j = i } ^ { n - 1 } ( i g _ { j + 1 } - i g _ j - g _ i ) V _ { i , j } \psi _ i \psi _ j \end{align*}


\begin{align*} \nabla ( c \varphi ) \cdot \nu = \nabla c \cdot \nu \varphi + c \nabla \varphi \cdot \nu = ( \gamma - c ) \varphi + c \nabla \varphi \cdot \nu \quad \mbox { o n } \ , \ , \partial \Omega , \end{align*}


\begin{align*} \frac { \partial \tilde x } { \partial y } = x \cdot \frac { 1 } { 3 } h ^ { - \frac { 2 } { 3 } } \cdot h _ y ' , \end{align*}


\begin{align*} \chi = \begin{cases} \max \{ 1 + \beta , 1 + \lambda , 1 + \gamma + \frac { 2 \theta } { \alpha } \} , & \alpha \leq 2 , \\ \max \{ 1 + \beta , 3 + \lambda - \alpha , 3 + \gamma + \theta - \alpha \} , & \alpha > 2 . \end{cases} \end{align*}


\begin{align*} & \limsup _ { n \to + \infty } \sum _ { i = 0 } ^ { n - 1 } \frac { 1 } { \inf _ { I _ { i , n } \backslash U _ k } c ( t ) } \int _ { I _ { i , n } \backslash U _ k } c ( s ) e ^ { - s } d s \\ \le & \limsup _ { n \to + \infty } \sum _ { i = 0 } ^ { n - 1 } \frac { \sup _ { I _ { i , n } \backslash U _ k } c ( t ) } { \inf _ { I _ { i , n } \backslash U _ k } c ( t ) } \int _ { I _ { i , n } \backslash U _ k } e ^ { - s } d s \\ = & \int _ { ( t ' _ 2 , t ' _ 1 ] \backslash U _ k } e ^ { - s } d s . \end{align*}


\begin{align*} | \Gamma _ { \ell , m , n } | = ( q - 1 ) ^ 3 q ^ { 3 \ell + m - n + 6 } \end{align*}


\begin{align*} \dot x = x Q ( x ) \Longleftrightarrow \dot x _ i = \sum _ k x _ k Q _ { k i } ( x ) \ , \ , \ , \ , i , \end{align*}


\begin{align*} \hat \Phi ^ \dagger ( a , s ) & = \left [ \gamma _ 1 a ^ 3 + \mathcal O ( a ^ 4 ) \right ] + s \left [ \gamma _ 2 a + \mathcal O ( a ^ 2 ) \right ] + \mathcal O ( s ^ 2 ) \\ & = a ( \gamma _ 1 a ^ 2 + \gamma _ 2 s ) + \mathcal O ( a ^ 4 + \abs { s } a ^ 2 + s ^ 2 ) \end{align*}


\begin{align*} \sigma ( \sqrt { a } ) = - \sqrt { a } , \ & \sigma ( \sqrt { d } ) = \sqrt { d } \\ \tau ( \sqrt { a } ) = \sqrt { a } , \ & \tau ( \sqrt { d } ) = - \sqrt { d } . \end{align*}


\begin{align*} \hat { u } _ i = c _ 1 \frac { \partial \phi } { \partial x _ 1 } + c _ 2 \frac { \partial \phi } { \partial x _ 2 } . \end{align*}


\begin{align*} \mathcal { Q } = \bigoplus _ { i = 1 } ^ n \mathcal { Q } _ i \mathcal { R } = \bigoplus _ { i = 1 } ^ { n - 1 } J _ { p _ i * } ( J _ { p _ i } ^ * ( \mathcal { Q } _ { i + 1 } ) ) . \end{align*}


\begin{align*} k ( t ) = t ^ { \beta - 1 } \ , E _ { \alpha , \beta } ( - t ^ \alpha ) , \end{align*}


\begin{align*} \begin{cases} \dot { u } ( t ) = A ( t ) u ( t ) , & t \geq s , \\ u ( s ) = x \in X . \end{cases} \end{align*}


\begin{align*} h ( \mu ) = \frac { 2 ^ { 2 - \alpha - \beta } } { \sqrt { 2 \pi } \ , \sigma \ , \mathrm { B } ( \alpha , \beta ) \left [ 1 - I _ { \frac { 1 } { 2 } } ( \alpha , \beta ) \right ] } . \end{align*}


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


\begin{align*} \begin{aligned} ( \Pi _ V \tilde { Y } _ { \tau / 2 } ) & = \int _ 0 ^ { \tau / 2 } \left \| \Pi _ V e ^ { | z | J ^ \perp ( t - s ) } P ^ { - 1 } \Pi _ { \mathrm { k e r } A ^ \perp } \sigma \right \| _ F ^ 2 d s \ge c _ 1 \tau \end{aligned} \end{align*}


\begin{align*} \widetilde { P } _ { k } ^ { \mathrm { d i s c } } ( \mathcal { T } ) : = \left \{ q _ h \in { P } _ { k } ^ { \mathrm { d i s c } } ( \mathcal { T } ) : ( q _ h , 1 ) _ T = 0 T \in \mathcal { T } \right \} . \end{align*}


\begin{align*} N = n ^ { 3 } + \left ( n + 1 \right ) ^ { 3 } = \left ( n + a \right ) ^ { 3 } + \left ( n + b \right ) ^ { 3 } \end{align*}


\begin{align*} f _ { G } ( x , y | \rho ) = f _ { g } ( x | \beta ) f _ { g } ( y | \beta ) \sum _ { j \geq 0 } \rho ^ { n } l _ { n } ( x | \beta ) l _ { n } ( y | \beta ) , \end{align*}


\begin{align*} \Delta [ M _ v ] ( t ) = ( \Delta M _ v ( t ) ) ^ 2 = e ^ { - 2 \lambda t } ( v A ^ * _ k ) ^ 2 = e ^ { - 2 \lambda t } \lambda ^ 2 v _ { k } ^ 2 , \end{align*}


\begin{align*} i _ { \Gamma } ^ * A : = ( I , \Gamma ) ^ T A ( I , \Gamma ) = B + C \Gamma + \Gamma ^ T C ^ T + \Gamma D \Gamma ^ T . \end{align*}


\begin{align*} s _ { m } = - \sum _ { \emptyset \neq I \subset \{ 1 , \dots , m \} } ( - 1 ) ^ { | I | } \langle a _ { I } \rangle \in \mathbb { Z } [ R ^ { * } ] , \end{align*}


\begin{align*} \beta ' = \beta \pi _ { 2 3 } , \delta ' = \delta \pi _ { 2 3 } , \gamma ' = \gamma \pi _ { 1 2 } , \gamma '' = \pi _ { 2 3 } \gamma \pi _ { 2 3 } . \end{align*}


\begin{align*} { \mathcal R } ( f ) : = \mathcal { W } \ , ( f \Delta ^ { - m + 1 } ) = \frac { n - 2 } { 1 2 } v _ { n - 1 } \int _ { M } f R ( g ) v o l _ { g } . \end{align*}


\begin{align*} R _ { q _ 1 , \ldots , q _ 4 , \kappa } = \# \{ k _ { i j } : k _ { i j } \neq 0 , i < j \} \end{align*}


\begin{align*} \Delta _ n ( x , \xi _ 0 ) : = & W ( \xi _ 0 , n , x , \mathsf { L } ^ n ) - W ( \xi _ 0 , n , x , \mathsf { R } ^ n ) \\ = & E _ { \xi _ 0 } \biggl [ \varphi \biggl ( x + \sum _ { i = 1 } ^ { n } Z _ i ^ { \mathsf { L } ^ n } \biggr ) \biggr ] - E _ { \xi _ 0 } \biggl [ \varphi \biggl ( x + \sum _ { i = 1 } ^ { n } Z _ i ^ { \mathsf { R } ^ n } \biggr ) \biggr ] . \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*} \begin{aligned} \mathbf { S } _ { \infty m } ( q , 1 ) = \big \{ ( q _ { i j } ) _ { \substack { i \geq 1 \\ 0 \leq j \leq m } } \big | \ , q < q _ { i j } < 1 \ , , q _ { i j } < q _ { i , j - 1 } ( \forall i \geq 1 , 0 \leq j \leq m ) \ , , q _ { i + 1 , 0 } < q _ { i m } ( \forall i \geq 1 ) \big \} \ , . \end{aligned} \end{align*}


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


\begin{align*} \liminf _ { n \to \infty } M _ n ^ { \psi } / n \ge \lim _ { n \to \infty } M _ n ^ { \psi } ( Q ^ n ) / n = T _ { W , \psi } ( \mu ) - I ( \mu ) . \end{align*}


\begin{align*} \mu ^ * + \varkappa & = \mu ^ { \rm N } + \varkappa ^ { \rm N } , \mu ^ * = \mu ^ { \rm N } - \varkappa ^ { \rm N } , \lambda = \lambda ^ { \rm N } , \\ \beta & = \gamma ^ { \rm N } - \beta ^ { \rm N } , \ \ \gamma = \gamma ^ { \rm N } + \beta ^ { \rm N } , \qquad \alpha = \alpha ^ { \rm N } \ , , \end{align*}


\begin{align*} [ \omega ] = 2 [ H ] - [ E ] , [ \alpha ] = 6 [ H ] + [ E ] . \end{align*}


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


\begin{align*} v = 0 , \nu ( v ) = 0 \mbox { o n } \hat \Sigma . \end{align*}


\begin{align*} \lambda _ 0 \alpha ^ m + \lambda _ 1 \alpha ^ { m - 1 } \beta + \cdots + \lambda _ m \beta ^ m = 0 . \end{align*}


\begin{align*} S t ( e _ { 1 } ) = \left \{ \begin{pmatrix} 1 & c & ^ { t } u \\ 0 & 1 & 0 \\ 0 & x & B \end{pmatrix} \Bigg | \ , \ , u + \ , ^ { t } B \psi x = 0 ; \ , \ , 2 c + \langle x , x \rangle = 0 ; \ , \ , B \in O _ { n - 1 , n - 1 } . \right \} . \end{align*}


\begin{align*} E X ^ { m } Y ^ { n - m } = \sum _ { j = 0 } ^ { \min ( m , n - m ) } \rho ^ { j } H _ { m , j } H _ { n - m , j } . \end{align*}


\begin{align*} \xi _ { k , i } ( x , t ) = \left \{ \begin{array} { l l } 2 \sqrt { C _ { d } \kappa } \ , \theta _ { k } ( t ) { \rm R e } ( \sigma _ { k , i } ( x ) ) , & k \in \Z _ { + } ^ d , \\ 2 \sqrt { C _ { d } \kappa } \ , \theta _ { k } ( t ) { \rm I m } ( \sigma _ { k , i } ( x ) ) , & k \in \Z _ { - } ^ d , \end{array} \right . \end{align*}


\begin{align*} \frac { \partial \xi ( \mu , t ) } { \partial t } = & - \frac { \partial } { \partial \mu } \bigg ( \xi ( \mu , t ) \int _ 0 ^ { \mu } \nu \Lambda ( \mu , \nu ) \xi ( \nu , t ) \ d \nu \bigg ) \\ & - \int _ { \mu } ^ { \infty } \Lambda ( \mu , \nu ) \xi ( \mu , t ) \xi ( \nu , t ) \ d \nu , \ ( \mu , t ) \in \mathbb { R } _ { > 0 } ^ 2 , \ \end{align*}


\begin{align*} \int _ { \mathfrak Y ( n ) } | \mathfrak h ( n ) _ { \eta } ^ * \omega ( n ) | = ( \phi _ 0 ) _ ! \int _ { S m ( \widetilde { \mathfrak Y ( n ) } ) } | \phi _ { \eta } ^ * \mathfrak h ( n ) _ { \eta } ^ * \omega ( n ) | . \end{align*}


\begin{align*} \sigma ^ L ( G ^ L ( X ) ) = G ( \{ \varphi ^ L _ { \sigma , s } ( E ^ L _ X ( v ) ) \mid ( s , v ) \in D \} ) . \end{align*}


\begin{align*} \begin{aligned} \biggr { | } \int _ { \partial \Omega \cap B _ { \eta } } \nabla c \cdot \nu c \psi ^ { 2 } \biggr { | } & = \biggr { | } \int _ { \partial \Omega \cap B _ { \eta } } ( \gamma - c ) c \psi ^ { 2 } \biggr { | } \\ & \le \gamma ^ { 2 } \int _ { \partial \Omega \cap B _ { \eta } } \psi ^ { 2 } \\ & \le \gamma ^ { 2 } \int _ { \partial \Omega } \psi ^ { 2 } . \end{aligned} \end{align*}


\begin{align*} \left \{ \begin{aligned} \dot { X } & = \frac { 1 } { \tau } ( Z - X ) , \\ \dot { Z } & = - \nabla f ( X ) \end{aligned} \right . \end{align*}


\begin{align*} \sum _ { \ell \in L _ { i } } c _ { \ell } ( 0 ) \cdot u _ { i } ^ { \gamma _ { \ell } } = 0 \end{align*}


\begin{align*} \sigma _ { n - 1 } ( \kappa ) = \sigma , \partial \Sigma = \Gamma , \end{align*}


\begin{align*} \tilde h _ d ( \alpha _ d ) & \leq O ( 1 ) + \log \frac { d \cdot \Gamma ( \frac { d } { 2 } ) } { \alpha _ d d ^ { \frac { d - 1 } { 2 } } } + d \log 2 \\ & = O ( 1 ) + \log \frac { \Gamma ( \frac { d } { 2 } ) } { C e ^ { - c d } d ^ { \frac { d - 1 } { 2 } } } + d \log 2 . \end{align*}


\begin{align*} \lim _ { t \to \infty } \frac 1 { t ^ n | A | } \int _ { A _ t } f ( x ) \ , d x = M ( f ) , \end{align*}


\begin{align*} \rho _ n = \left ( | \nabla \phi _ n | ^ { 2 } + | G _ n | ^ { 2 } \right ) \frac { 2 \pi } { r } . \end{align*}


\begin{align*} \nabla u ( x _ { 0 } ) \in \mathrm { s p a n } { \nabla d ( x _ { 0 } ) } = ( T _ { x } \Sigma _ { d ( x _ { 0 } ) } ) ^ { \perp } . \end{align*}


\begin{align*} \tilde { F } _ k ( x , y ) = \begin{pmatrix} \max _ { i = 1 , \ldots , 2 ^ k } x _ i \\ \max _ { i = 1 , \ldots , 2 ^ k } y _ i \end{pmatrix} . \end{align*}


\begin{align*} | \beta _ k - \alpha _ k | \leq C _ 1 \frac { 2 \pi } { N } \left ( s _ k , \{ z \in \mathbb { D } : | b ( z ) | \leq C \} \right ) \ , k = 1 , \ldots , N . \end{align*}


\begin{align*} \mathcal { L } _ g : = \begin{pmatrix} \mu \Delta _ { B } + ( \lambda + \mu ) \operatorname { g r a d } \operatorname { d i v } + \mu \operatorname { R i c } + \rho \omega ^ 2 & - \beta \operatorname { g r a d } \\ i \omega \theta _ { 0 } \beta \operatorname { d i v } & \alpha \Delta _ { g } + i \omega \gamma \end{pmatrix} , \end{align*}


\begin{align*} ( \tilde h , \tilde v ) : = ( \hat h , \hat v ) + \big ( L _ Y \bar g , Y ( \bar u ) \big ) \end{align*}


\begin{align*} \Omega _ X ^ { [ p ] } : = j _ * \Omega ^ p _ { X ^ { s m } } = ( \Omega ^ p _ X ) ^ { \vee \vee } \end{align*}


\begin{align*} x _ { i , j } + x _ { i + 1 , j } + x _ { i + 1 , j } + x _ { i + 1 , j + 1 } = S , \mbox { \ \ f o r a l l } 1 \leq i \leq m \mbox { a n d } 1 \leq j \leq n _ 1 . \end{align*}


\begin{align*} & \int _ { J _ U ^ V } \left ( A _ { \vartheta \varphi } { } ^ { ( \vartheta , \varphi ) } \mathcal I _ { a } ^ { \vec { \alpha } } [ F ] ( W , Z ) + B _ { \vartheta \varphi } { \bf i } { } ^ { ( \vartheta , \varphi ) } \mathcal I _ { a } ^ { \vec { \alpha } } [ F ] ( W , Z ) \right ) d Z \wedge d Z ^ { * } \\ = & \int _ { \partial J _ U ^ V } { } ^ { ( \vartheta , \varphi ) } \mathcal I _ { a } ^ { \vec { \alpha } } [ F ] ( W , Z ) d \sigma _ { ( \vartheta , \varphi ) } ( Z ) , \end{align*}


\begin{align*} \Lambda = \cup _ { m \in \mathbb { N } ^ 2 } \Lambda ^ m . \end{align*}


\begin{align*} [ F ( \gamma , \beta ) : F ] = [ F ( \gamma , \beta ) : F ( \beta ) ] [ F ( \beta ) : F ] = [ F ( \gamma , \beta ) : F ( \beta ) ] ( q ^ n - 1 ) . \ \end{align*}


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


\begin{align*} \int _ { \R ^ n } \partial _ i f ( x ) \big ( T _ i ( x _ i ) - x _ i \big ) Q ( x ) d x & = \int _ { \R } \hat { f } ' _ i ( x _ i ) \big ( T _ i ( x _ i ) - x _ i \big ) Q _ i ( x _ i ) d x _ i , \end{align*}


\begin{align*} \sigma _ { W _ { \rm s , i d } } ^ 2 = \left ( \psi _ { 1 } \kappa _ { \rm b } + \psi _ { 2 } \right ) T _ { \rm s } - \psi _ { 1 } \kappa _ { \rm b } Q ( \kappa _ { \rm b } ) T _ { \rm s } + \psi _ { 1 } \kappa _ { \rm t } Q ( \kappa _ { \rm t } ) T _ { \rm s } + \psi _ { 1 } T _ { \rm s } f ( \kappa _ { \rm b } ) - \psi _ { 1 } T _ { \rm s } f ( \kappa _ { \rm t } ) . \end{align*}


\begin{align*} M _ { \hat I } = \varprojlim _ n M / ( x _ 1 ^ n , \dots , x _ k ^ n ) . \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*} a ( u , v ) = ( f , v ) ~ v \in H ^ 1 _ 0 ( D ) , \end{align*}


\begin{align*} X _ { \alpha } v _ k = \sum _ { i = 1 } ^ { k } X _ { m - k + 1 } \cdots X _ { m - k + i - 1 } [ X _ { \alpha } , X _ { m - k + i } ] X _ { m - k + i + 1 } \cdots X _ m v _ 0 \end{align*}


\begin{align*} \begin{cases} \frac { d } { d t } X _ t = B ( X _ t , X _ t ) \\ X _ t | _ { t = 0 } = x _ 0 \end{cases} \end{align*}


\begin{align*} \begin{aligned} W ( \xi _ 0 , k + 1 , x , \mathsf { L } ^ { k + 1 } ) = & E _ { \xi _ 0 } \biggl [ \varphi \biggl ( x + \sum _ { i = 1 } ^ { k + 1 } Z _ i ^ { \mathsf { L } ^ { k + 1 } } \biggr ) \biggr ] \\ = & E _ { 1 - \xi _ 0 } \biggl [ \varphi \biggl ( x + \sum _ { i = 1 } ^ { k + 1 } Z _ i ^ { \mathsf { R } ^ { k + 1 } } \biggr ) \biggr ] \\ = & W ( 1 - \xi _ 0 , k + 1 , x , \mathsf { R } ^ { k + 1 } ) . \end{aligned} \end{align*}


\begin{align*} \begin{aligned} \epsilon ( \eta _ a , t ) = & \sum _ { n = 1 } ^ N \int _ { n - 1 } ^ { n } F ( \eta _ a , \tau ) - F _ a ( \eta _ a ) \ , d \tau \\ & + \int _ { N } ^ { t } F ( \eta _ a , \tau ) - F _ a ( \eta _ a ) \ , d \tau \\ = & \int _ { N } ^ { t } F ( \eta _ a , \tau ) - F _ a ( \eta _ a ) \ , d \tau . \end{aligned} \end{align*}


\begin{align*} R _ { i j } = g ^ { k l } R _ { i k l j } , \end{align*}


\begin{align*} p _ i = \begin{cases} 0 . 5 \tilde { q } _ i ( 1 + \tilde { \delta } _ i ) , & i \in [ m ] \\ 0 . 5 \tilde { q } _ { i - m } ( 1 - \tilde { \delta } _ { i - m } ) , & i \in [ k ] \setminus [ m ] . \end{cases} \end{align*}


\begin{align*} \widetilde { \lambda } ( m ) \ r _ n ( m ) = E _ { n } \ \widetilde { r } _ { n + 1 } ( m ) + F _ n \ \widetilde { r } _ { n } ( m ) + G _ n \ \widetilde { r } _ { n - 1 } ( m ) \ , , \end{align*}


\begin{align*} \sup _ { z \in \mathbb { B } _ n } \beta ( \varphi _ { k j } ( z ) , \rho ( z ) ) & \leq \sup _ { z \in \mathbb { B } _ n } k _ { V ^ { - 1 } \mathbb { B } _ n } ( V ^ { - 1 } \varphi _ { k j } ( z ) , V ^ { - 1 } \rho ( z ) ) \\ & = \sup _ { z \in V ^ { - 1 } \mathbb { B } _ n } k _ { V ^ { - 1 } \mathbb { B } _ n } ( V ^ { - 1 } \varphi _ { k j } V ( z ) , V ^ { - 1 } \rho V ( z ) ) . \end{align*}


\begin{align*} \{ 1 , \ldots , \ell \} \simeq \bigcup _ { i = 1 } ^ L \{ m _ 1 + \cdots + m _ { i - 1 } + 1 , \ldots , m _ 1 + \cdots + m _ i \} \end{align*}


\begin{align*} X ( t ) = X ( 0 ) + \int _ 0 ^ t F ( X ( s ) ) d s + \int _ 0 ^ t G ( X ( s ) ) d W ( s ) , \end{align*}


\begin{align*} \begin{aligned} \Phi _ 2 ( \gamma ) : = \{ b _ 0 | \ , - 2 ( 2 - b _ 0 - \gamma ) > - 3 \ , , 2 \gamma > - 3 \} . \end{aligned} \end{align*}


\begin{align*} f ( z , t ) = ( 2 \pi ) ^ { - n - 1 } \ , \int _ { - \infty } ^ \infty e ^ { i \lambda t } \ , \big ( \sum _ { k = 0 } ^ \infty \rho _ k ^ \lambda ( f ) e ^ { n - 1 } _ { k , \lambda } ( z , 0 ) \big ) | \lambda | ^ n d \lambda . \end{align*}


\begin{align*} { f _ i } \left ( { { \bf { x } } + { { \bf { c } } _ i } \Delta t , t + \Delta t } \right ) - { f _ i } \left ( { { \bf { x } } , t } \right ) = { \left ( { { { { \bf { \hat M } } } ^ { - 1 } } { \bf { \hat \Lambda \hat M } } } \right ) _ { i j } } \left [ { { f _ i } \left ( { { \bf { x } } , t } \right ) - f _ i ^ { \left ( { e q } \right ) } \left ( { { \bf { x } } , t } \right ) } \right ] + { { \bf { F } } _ i } \left ( { { \bf { x } } , t } \right ) , \end{align*}


\begin{align*} G \cdot R - \bar G \cdot \bar R - \bar R \cdot \nabla \bar G ( \nabla & \bar A ) ^ { - 1 } ( A - \bar A ) - \bar G \cdot ( P - \bar P ) = ( G - \bar G ) \cdot ( R - \bar R ) \\ & + \bar R \cdot G ( U | \bar U ; x , t ) + \bar G \cdot \left [ A _ t - \bar A _ t + f _ { \alpha , x _ \alpha - } \bar f _ { \alpha , x _ \alpha } \right ] \ , . \end{align*}


\begin{align*} \hat W _ r ( q + k ) - \hat W _ r ( 2 q ) & \le \frac { 2 } { \pi } \left ( \frac { 1 } { q + k } + \frac { 1 } { 2 q } \right ) \\ & \le \frac { 4 } { \pi q } \\ & \le 4 r - \frac { 2 } { \pi } \sin ( 2 \pi r ) \\ & = \hat W _ r ( 0 ) - \hat W _ r ( 1 ) . \end{align*}


\begin{align*} \sum _ { k = 0 } ^ { n } { n \choose k } F _ { j k } F _ { j ( n - k ) } = \frac { 2 ^ n L _ { j n } - 2 L _ j ^ n } { 5 } , \end{align*}


\begin{align*} e _ { a _ j } \left [ t + 1 \right ] = & e _ { a _ j } \left [ t \right ] + 2 \mu \Re { \left \{ \hat { \phi } ^ { \left ( j , j \right ) } \right \} } - 4 \mu \left ( \sum _ { l = 1 } ^ M \lvert \hat { \phi } ^ { \left ( l , j \right ) } \rvert ^ 2 + M \sigma _ { e , i } ^ 2 \lVert \mathbf { p } _ j \rVert ^ 2 \right . \left . + \sum _ { q = 1 } ^ { M - 1 } \sum _ { r = q + 1 } ^ M f _ { q , r } ^ { \left ( j \right ) } \right ) a _ j \left [ t \right ] . \end{align*}


\begin{align*} j ^ 1 _ q ( F _ \varphi \circ f _ \varphi { } ^ { - 1 } ) & = ( q , F _ \varphi ( o ) , D F _ \varphi ( o ) ( D f _ \varphi ( o ) ) ^ { - 1 } ) \\ * & = ( q , F _ \varphi ( o ) , D F _ \varphi ( o ) ^ i { } _ { j \alpha } ( ( F _ \varphi ( o ) ) ^ { - 1 } ) ^ \alpha { } _ k ) \end{align*}


\begin{align*} \lim _ { M \rightarrow \infty } \mathbb { P } ( \tau _ u ^ M < T ) = 0 . \end{align*}


\begin{align*} v = D '' u _ 0 + P _ { \omega , h } ( \sqrt { \lambda } \tau _ 0 ) . \end{align*}


\begin{align*} \alpha ^ h ( \partial _ j , \partial _ k ) = 0 \quad \forall j \neq k . \end{align*}


\begin{align*} \psi _ n ( k h ) = \psi ( h ) \forall k \in K _ n , \forall h \in H . \end{align*}


\begin{align*} \mathbf { Q } _ { 0 , n } f & = \frac { q t } { q t - 1 } h _ n [ ( 1 - q t ) X / ( q t ) ] \cdot f \\ \mathbf { Q } _ { 1 , n } f & = \mathbf { D } _ n f . \end{align*}


\begin{align*} \begin{cases} - ( K _ H ( x ) \nabla v ) = \frac { 1 } { \varepsilon ^ 2 } \left ( v - q _ { \hat { x } } \ln \frac { 1 } { \varepsilon } \right ) ^ { p } _ + , \ \ \ \ & \ \Omega _ { \hat { x } } . \\ v = 0 , \ \ \ \ & \ \partial \Omega _ { \hat { x } } . \end{cases} \end{align*}


\begin{align*} 0 = \nabla _ { i } Q = \frac { \nabla _ { i } P } { P } - A \nabla _ { i } \left ( \frac { \rho ^ { 2 } } { 2 } \right ) , \end{align*}


\begin{align*} d = 2 n = 4 l + 2 \equiv 2 \mod 3 \end{align*}


\begin{align*} \begin{cases} \partial _ t w + \nabla w \cdot \nabla ^ \perp \varphi = 0 , \\ w = \mathcal { L } _ H \varphi , \\ \varphi | _ { \partial \Omega } = 0 . \end{cases} \end{align*}


\begin{align*} \tau ( x ) \omega : = \omega + \Lambda x - \left \lfloor \omega + \Lambda x \right \rfloor \end{align*}


\begin{align*} T ( t ) x = \tau \lim _ { n \to \infty } \left ( \frac { n } { t } R \left ( \frac { n } { t } , A \right ) \right ) ^ { n } x \end{align*}


\begin{align*} g _ i ^ { \left ( { n e q } \right ) } \left ( { { { \bf { x } } _ b } , t } \right ) = { g _ i } \left ( { { { \bf { x } } _ f } , t } \right ) - g _ i ^ { \left ( { e q } \right ) } \left ( { { { \bf { x } } _ f } , t } \right ) . \end{align*}


\begin{align*} \left \{ \begin{aligned} - \Delta w _ { n } & = f ( \cdot , y ) - \mathcal { N } _ { n } ( \cdot , y _ { n } ) , \ ; \ ; \Omega , \\ w _ n & = 0 , \ , \ ; \ ; \qquad \qquad \qquad \partial \Omega . \end{aligned} \right . \end{align*}


\begin{align*} \mathbb { x } _ c ^ { \mathcal { X } ' , \mathbb { F } } \circ \alpha ^ { m } ( \mathbb { C } ) _ c = \alpha ^ m ( \mathbb { D } ) _ { F c } \circ \mathbb { x } ^ { \mathcal { X } , \mathbb { F } } _ c . \end{align*}


\begin{align*} & ( I - Q ^ \dagger ) F ^ { q , \dagger } _ { v v } ( v + \psi ^ \dagger ( v , p ) , p ) [ v _ 1 + \psi ^ \dagger _ v ( v , p ) v _ 1 , v _ 2 + \psi ^ \dagger _ v ( v , p ) v _ 2 ] \\ & \quad + ( I - Q ^ \dagger ) F ^ { q , \dagger } _ v ( v + \psi ^ \dagger ( v , p ) , p ) \psi ^ \dagger _ { v v } ( v , p ) [ v _ 1 , v _ 2 ] = 0 . \end{align*}


\begin{align*} P _ { n , k } ( z ) = z ^ k A _ n ( z ) . \end{align*}


\begin{align*} \frac { 1 } { \sqrt { \lambda _ t } } - 1 \geq \frac { 1 } { 2 } \alpha _ 0 \sum _ { i = 1 } ^ t \sqrt { \gamma _ i / \gamma _ 0 } \quad \Rightarrow \lambda _ t \leq 4 \left ( 2 + \alpha _ 0 \sum _ { i = 1 } ^ t \sqrt { \gamma _ i / \gamma _ 0 } \right ) ^ { - 2 } . \end{align*}


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


\begin{align*} \alpha _ x = \mu _ x , \end{align*}


\begin{align*} \begin{aligned} B \times u \in L ^ { 4 / 3 } ( 0 , T ; L ^ { 2 } ( \mathbb { R } ^ { 3 } ) ) , \\ \nabla \times B \in L ^ { 2 } ( 0 , T ; L ^ { 2 } ( \mathbb { R } ^ { 3 } ) ) . \end{aligned} \end{align*}


\begin{align*} \begin{cases} \nabla w \cdot \nabla ^ \perp \varphi = 0 , \\ w = \mathcal { L } _ H \varphi , \\ \nabla ^ \perp \varphi \cdot \nu | _ { \partial \Omega } = v _ n | _ { \partial \Omega } \ln \frac { 1 } { \varepsilon } . \end{cases} \end{align*}


\begin{align*} a \approx a ^ \mathrm { a p p } ( s ) : = \sqrt { \frac { - \gamma _ 2 s } { \gamma _ 1 } } . \end{align*}


\begin{align*} v _ 3 \left ( h _ 2 ( t _ 0 ) \right ) = 1 , v _ 5 \left ( h _ 2 ( t _ 0 ) \right ) = 0 , v _ 7 \left ( h _ 2 ( t _ 0 ) \right ) = 1 . \end{align*}


\begin{align*} 0 . 2 5 I _ f ( p , q ) & \leq \sum _ { i \in A _ { 1 + \kappa , \infty } } p _ i f \left ( \frac { q _ i } { p _ i } \right ) \leq \sum _ { i \in A _ { 1 + \kappa , \infty } } p _ i f ( \nu ) = p ' f ( \nu ) , \end{align*}


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


\begin{align*} \frac { 1 } { \ln \frac { R } { s _ { \delta , j } } } = \frac { 1 } { \ln \frac { R } { \varepsilon } } + O \left ( \frac { \ln | \ln \varepsilon | } { | \ln \varepsilon | ^ 2 } \right ) . \end{align*}


\begin{align*} B _ n ( x ) - B _ n ( y ) = n y ^ { n - 1 } , \end{align*}


\begin{align*} b ( v , w ) = - ( - ) ^ { \deg v \deg w } b ( w , v ) , \end{align*}


\begin{align*} [ F ( \gamma ) : F ] = m , \end{align*}


\begin{align*} \beta = 2 i - 1 \end{align*}


\begin{align*} \int _ \Omega \left ( \eta x _ { n + 1 } ^ m + \lambda \right ) d V = \int _ \Sigma \langle Y , \nu \rangle \ , d A . \end{align*}


\begin{align*} \prod _ { i = 1 } ^ { l } ( 1 - Z _ i ) \ge \exp \big ( - 2 \sum _ { i = 1 } ^ { l } Z _ i \big ) = & \exp \bigg ( - 2 \sum _ { s = 1 } ^ { \lceil \log _ 2 ( l / 2 ) \rceil } 2 ^ s \log \big ( \tau ^ { - 1 / 2 ^ s } l / 2 ^ s \big ) \frac { 3 } { l } \bigg ) . \end{align*}


\begin{align*} m : = \inf _ { \partial \Omega } \frac { G ( ( D ^ 2 u ) ' ) } { \widetilde { f } } , \end{align*}


\begin{align*} K ( c _ 1 , \dots , c _ n , 0 ) = & p _ i \circ H ( c _ 1 , \dots , c _ n , m ) \\ = & p _ i \circ \iota _ { U _ k } ( c 1 , \dots , c _ n ) = p _ i ( c _ 1 , \dots , c _ n ) , \end{align*}


\begin{align*} \| P _ { i + 1 } ' v \| _ 2 & \geq 1 - \frac { 4 \tau } { c ' \tilde \varepsilon \sqrt { x } \ , h } - \sum _ { j = 1 } ^ i \frac { 2 ^ { i + 4 - j } \sqrt { \ell _ j / x } \ , \ , \tau } { c ' \tilde \varepsilon h ^ 2 \ , g \big ( \lfloor ( 1 - ( 1 + \tilde \varepsilon ) ^ { - j } ) r \rfloor \big ) } \\ & \geq 1 - 4 h ^ 2 - h \ , \sum _ { j = 1 } ^ i \frac { 2 ^ { i + 4 - j } \sqrt { 2 \tilde \varepsilon ( 1 + \tilde \varepsilon ) ^ { i - j } } } { 4 ^ { i - j } } > 1 - 1 - 4 h ^ 2 - 6 4 h \geq 1 / 2 , \end{align*}


\begin{align*} \sum _ { 1 \le i < j \le n } \partial _ { \theta _ j } \partial _ { \theta _ i } \Delta _ C & = \sum _ { 1 \le r < s \le k } \sum _ { i \in C _ { r } \atop j \in C _ { s } } \pm \partial _ { \theta _ j } \partial _ { \theta _ i } \Delta _ C ~ . \end{align*}


\begin{align*} \hat { u } _ i = c _ 1 \frac { \partial \phi } { \partial x _ 1 } + c _ 2 \frac { \partial \phi } { \partial x _ 2 } . \end{align*}


\begin{align*} \nabla _ X \Psi _ 2 = - \frac { 1 } { 2 } \left ( X - \langle X , T _ 1 \rangle ( T _ 1 + f _ 1 ) - \langle X , T _ 2 \rangle ( T _ 2 + f _ 2 ) \right ) \cdot \Psi _ 1 + \frac { 1 } { 2 } S ( X ) \cdot \Psi _ 2 , \end{align*}


\begin{align*} | \phi ^ { N , i } ( t _ 0 , \boldsymbol { \xi } ) - \Psi ^ { N , i } ( t _ 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*} \left \{ \begin{aligned} a = \lambda , \\ c + d x = \lambda x . \end{aligned} \right . \end{align*}


\begin{align*} \nu ( H ' | _ g ( h ) ) & = - \tfrac { 1 } { 2 } R ' ( h ) + \tfrac { 1 } { 2 } R '^ \Sigma | _ { g ^ \intercal } ( h ^ \intercal ) + A _ g \cdot ( A _ g \circ h ) \\ & - A _ g \cdot A ' ( h ) - H _ g H ' ( h ) \\ & + \tfrac { 1 } { 4 } \big ( - R _ g + R ^ \Sigma _ g - | A _ g | ^ 2 - H _ { g } ^ 2 \big ) h ( \nu , \nu ) \\ & - \tfrac { 1 } { 2 } \Delta _ \Sigma h ( \nu , \nu ) + g ^ { a b } \omega ( e _ a ) e _ b ( H _ g ) . \end{align*}


\begin{align*} & M = \begin{pmatrix} D & * \\ 0 & * \end{pmatrix} , D = d i a g ( d _ 1 , d _ 2 , \ldots , d _ { l _ 0 - 1 } ) \\ & \deg ( \det ( D ) ) \geq \deg ( \det ( D _ 0 ) ) , \deg ( \det ( M ) ) = \deg ( \det ( M _ 0 ) ) \end{align*}


\begin{align*} \sum _ { u \le y _ i } \binom n u \le 2 \binom n { y _ i } \le \frac { 2 n ^ { y _ i } } { y _ i ! } \le \exp \Big ( \frac n { k ^ { 2 ^ { i - 1 } } } ( 2 ^ { i - 1 } \log k + 1 ) \Big ) , \end{align*}


\begin{align*} \widetilde { X } = \widehat { X } = \overline { X } , \widetilde { \mathbb { P } } \otimes d t . \end{align*}


\begin{align*} H _ { m i n } ( E ) = \min \limits _ { 0 \neq F \subset E } H _ r ( F ) . \end{align*}


\begin{align*} \langle x \rangle \vee \langle y \rangle = \langle x \wedge y \rangle . \end{align*}


\begin{align*} \nabla \cdot ( \nabla n \otimes \nabla n ) = \frac { 1 } { 2 } \nabla ( | \nabla n | ^ 2 ) + \nabla n \cdot \Delta n . \end{align*}


\begin{align*} \dfrac { b _ { 1 , \delta } ( x ) } { 2 } = \int _ 0 ^ 1 [ h ( x + \delta \sin ( 2 \pi t ) ) - h ( x ) ] \sin ( 2 \pi t ) \ , d t . \end{align*}


\begin{align*} \begin{array} { l l l } L _ f \circ g & = & \mu \circ ( f \otimes i d _ S ) \circ \lambda _ S ^ { - 1 } \circ g \\ & = & \mu \circ ( f \otimes i d _ S ) \circ ( i d _ I \otimes g ) \circ \lambda _ I ^ { - 1 } \\ & = & f \bullet _ \mu g \end{array} \end{align*}


\begin{align*} \begin{aligned} \partial _ x ^ 2 \left ( e ^ { i c _ + ( x ) } u _ x ( x ) \right ) & = I _ 3 ' ( x ) + I _ 4 ' ( x ) , \end{aligned} \end{align*}


\begin{align*} & \rho _ L ( \theta ) = \sum _ { | l | \leq L } \widehat { \rho _ L } ( l ) e ( l \theta ) = \widehat { \rho _ L } ( 0 ) + \sum _ { 1 \leq l \leq L } \widehat { \rho _ L } ( l ) 2 \cos ( 2 \pi l \theta ) , \end{align*}


\begin{align*} \hat { \mu } _ { A ^ * } = \hat { \mu } _ A \ . \end{align*}


\begin{align*} { \rm V o l } ( { \tilde \Sigma } ^ { 3 r } ) \leq C ( n ) \sum _ { i = 1 } ^ s { \rm V o l } ( A \cap B _ { p _ i } ( r ) ) \leq C ( n ) \ , r \sum _ { i = 1 } ^ s { \rm V o l } ( \Sigma \cap B _ { p _ i } ( 3 r ) ) . \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*} N = n + \frac { n ( n - 1 ) } { 2 } , \end{align*}


\begin{align*} \sum _ { i = 1 } ^ n \hat f _ i ( s _ 0 , X ) \bar f _ i ( t _ 0 ) \not \equiv . \end{align*}


\begin{align*} \partial _ \ell \Q ^ t \partial _ \ell \Q \ , x = \sum _ { i = 1 } ^ d ( \partial _ \ell f _ i ) ^ 2 w ^ i , \Q ^ t \partial ^ 2 _ { \ell k } \Q \ , x = \sum _ { i = 1 } ^ d \left [ \partial ^ 2 _ { \ell k } f _ { i } [ w ^ i ] ^ \perp - \partial _ \ell f _ { i } \partial _ k f _ { i } w ^ i \right ] , \end{align*}


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


\begin{align*} \Delta ( \gamma ) : = \{ \pmb { \theta } : \ , \max _ { i , j } | \theta _ i - \theta _ j | \le \gamma \} . \end{align*}


\begin{align*} \mu = \frac { 2 } { \partial _ { s } j ( 0 , 0 ) } \left ( \int _ { \mathbb { R } ^ { 3 } } b \cdot b _ 0 \dd x \right ) . \end{align*}


\begin{align*} \int _ { D ( 0 , 2 ) \backslash D ( 0 , 1 ) } | \phi _ { n _ { k } } - \phi | ^ { 2 } \frac { 2 \pi ^ { 2 } } { r } \dd z \dd r = \int _ { B ( 0 , 2 ) \backslash B ( 0 , 1 ) } | \varphi _ { n _ { k } } - \varphi | ^ { 2 } \dd y \to 0 . \end{align*}


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


\begin{align*} a _ \lambda v = \sum _ { j \in \mathbb { Z _ { + } } } ( a _ { ( j ) } v ) \frac { \lambda ^ { j } } { j ! } . \end{align*}


\begin{align*} \log L ( s , \rho ) & = \log \left ( \prod _ { v } \det ( 1 - \rho ( \mathrm { F r o b } _ { v } ) \mathrm { N } ( v ) ^ { - s } ) ^ { - 1 } \right ) \\ & = \sum _ { \ell = 1 } ^ { \infty } \frac { 1 } { \ell } \left ( \sum _ { \deg ( v ) \mid \ell } \deg ( v ) \mathrm { t r } \left ( \rho ( \mathrm { F r o b } _ { v } ) ^ { \frac { \ell } { \deg ( v ) } } \right ) \right ) q ^ { - \ell s } . \end{align*}


\begin{align*} \begin{aligned} | K _ 2 ^ \chi f ( v ) | \leq W _ 1 ( v ) + W _ 2 ( v ) \ , , \end{aligned} \end{align*}


\begin{align*} V _ { \alpha , \beta } = \mathbb { C } [ \partial ] v , L _ \lambda v = ( \partial + \alpha \lambda + \beta ) v . \end{align*}


\begin{align*} \| A \| _ \textrm { H S } = \sqrt { \frac { 1 } { n } \mathrm { t r } ( A ^ * A ) } \forall A \in \mathrm { U } ( n ) . \end{align*}


\begin{align*} u ( y ) - u ( x ) & \leq h ( y ) = \sup _ { l \in \partial _ { L _ x } f ( x ) } l ( y ) \\ & \leq \sup _ { l \in \partial _ { L _ x } f ( x ) } f ( y ) - f ( x ) + l ( x ) \\ & = f ( y ) - f ( x ) \end{align*}


\begin{align*} L ( s , \Delta ) \coloneqq \sum _ { n = 1 } ^ { \infty } \frac { \tau ( n ) } { n ^ { s } } = \prod _ { p } \left ( 1 - \frac { \tau ( p ) } { p ^ { s } } - \frac { 1 } { p ^ { 2 s - 1 1 } } \right ) ^ { - 1 } , \Re ( s ) > \frac { 1 3 } { 2 } . \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*} \gamma _ i + \bar { \gamma _ j } = 1 \ , \ , { \rm a n d } \ , \ , \gamma _ 3 + \bar { \gamma } _ 4 = 1 , \end{align*}


\begin{align*} T _ h ^ { ( N ) } : L _ { p _ 1 } ( \mathbb { R } , S _ { q _ 1 } ^ N ) \times L _ { p _ 2 } ( \mathbb { R } , S _ { q _ 2 } ^ N ) \rightarrow L _ { p } ( \mathbb { R } , S _ { q } ^ N ) \end{align*}


\begin{align*} G _ { 1 ^ s 0 ^ { n - s } } = F _ { 1 ^ s } \end{align*}


\begin{align*} \xi _ y ( x ) = e ^ { 2 \pi i \ , y \cdot x } , \end{align*}


\begin{align*} D _ t ^ { ( 1 ) } u _ k = b _ { k 1 } + 2 b _ { k 2 } t + \cdots , \end{align*}


\begin{align*} H _ 0 ^ 2 ( D ) : = \left \{ v \in H ^ 2 ( D ) : v = \frac { \partial v } { \partial \nu } = 0 \partial D \right \} , \end{align*}


\begin{align*} \mathcal { H } _ \Phi : = { \big \{ F ( z , \omega ) = f ( \Phi ^ { - 1 } ( z , \omega ) , \omega ) ; \ ; f \in H ^ 1 _ { \rm l o c } ( \mathbb { R } ^ n ; L ^ 2 ( \Omega ) ) \ ; \ ; \big \} } \end{align*}


\begin{align*} & \gamma \beta ( \gamma ) = \frac { \gamma _ { t - 1 } ( 1 - \kappa _ t ) } { ( 1 - \kappa _ { t - 1 } ) } \left ( 1 + \frac { 2 \mu \gamma _ { t - 1 } } { 1 - \kappa _ { t - 1 } } \right ) ^ { - 1 } \leq \frac { 3 } { 2 } \gamma _ 0 , \\ & \alpha ( \gamma ) = \frac { \kappa _ { t - 1 } \gamma \beta ( \gamma ) } { \gamma _ { t - 1 } } = \frac { \kappa _ { t - 1 } ( 1 - \kappa _ { t } ) } { 1 - \kappa _ { t - 1 } } \left ( 1 + \frac { 2 \mu \gamma _ { t - 1 } } { 1 - \kappa _ { t - 1 } } \right ) ^ { - 1 } \leq \frac { 1 } { 2 } . \end{align*}


\begin{align*} | F _ y ( x , y ) - p _ y ( x , y ) | = | F _ y ( x , y ) - h ^ y ( x , y ) | < 2 \varepsilon . \end{align*}


\begin{align*} M _ I : = M _ { \varrho ( I ) } \textrm { a n d } R _ I : = R _ { \varrho ( I ) } . \end{align*}


\begin{align*} ( a / b ) ^ k = ( \alpha / \beta ) ^ l . \end{align*}


\begin{align*} \mathbb { M } ( \theta ) _ J = \sum _ { w \in X _ J } \Bbbk { \bf U } \dot { w } \eta ( \theta ) _ J = \sum _ { w \in X _ J } \Bbbk { \bf U } _ { w _ J w ^ { - 1 } } \dot { w } \eta ( \theta ) _ J . \end{align*}


\begin{align*} \widehat { C } _ { \psi } = C _ { \widehat { \psi } } , \end{align*}


\begin{align*} \left | \int _ { \mathbb { R } ^ { 3 } } \nabla \times B \cdot B 1 _ { ( 0 , \infty ) } ( \Phi ) \dd x \right | = \left | \int _ { \mathbb { R } ^ { 3 } } \nabla \times b \cdot B 1 _ { ( 0 , \infty ) } ( \Phi ) \dd x \right | \lesssim | | \nabla \times b | | _ { L ^ { 2 } } | | b | | _ { L ^ { 2 } } . \end{align*}


\begin{align*} V ( L , K [ n - 1 ] ) = \frac { 1 } { n } \int _ { S ^ { n - 1 } } h _ L ( u ) d S _ K ( u ) , \end{align*}


\begin{align*} \int _ X \alpha ^ { 2 n } = c _ X \cdot q _ X ( \alpha ) ^ n . \end{align*}


\begin{align*} v = X ( u ) = g ( \nabla u , 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*} A _ n : = B _ n + C _ n \longrightarrow B + C = : A . \end{align*}


\begin{align*} \left \langle T _ a g _ t , T _ a g _ j \right \rangle _ { \alpha } = \frac { ( 1 + o ( 1 ) ) ( a + 1 ) ( \alpha - \frac { 1 } { 2 } ) } { ( a - \alpha + \frac { 3 } { 2 } ) } ( j + 1 ) ^ { a + \frac { 1 - \alpha } { 2 } } ( t + 1 ) ^ { - a + \frac { \alpha - 3 } { 2 } } . \end{align*}


\begin{align*} \| w _ n \| _ { L ^ { \infty } } = \left \| F ^ n e _ 1 \right \| _ { L ^ { \infty } } \leq \frac { 1 } { n ! } \left \| \widetilde { Q } ( u ) \right \| _ { L ^ 1 } ^ n . \end{align*}


\begin{align*} \mathcal { C } ( h ) ( z ) : = \frac { 1 } { 2 \pi i } \int _ { \mathbb { R } } \frac { h ( s ) } { s - z } d s , z \in \mathbb { C } \backslash \mathbb { R } . \end{align*}


\begin{align*} & \Psi ^ - _ 1 ( x ; z ) : = e _ 1 + \int _ { - \infty } ^ x \left ( 1 , e ^ { 2 i z ( x - y ) } \right ) \widetilde { Q } \Psi ^ - _ 1 ( y ; z ) d y , \\ & \Psi ^ - _ 2 ( x ; z ) : = e _ 2 + \int _ { - \infty } ^ x \left ( e ^ { - 2 i z ( x - y ) } , 1 \right ) \widetilde { Q } \Psi ^ - _ 2 ( y ; z ) d y . \end{align*}


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


\begin{align*} | F ^ { i j } ( \nabla _ \alpha \varphi ) _ { i j } | \leq C \sqrt { b _ { n - 1 } } \sum _ { i = 1 } ^ n F ^ { i i } \mbox { i n } \omega _ 1 . \end{align*}


\begin{align*} \Tilde { x } = \sum \limits _ { i = 1 } ^ p \alpha _ i u _ i + \lambda x _ v + ( 1 - \lambda ) x _ w \in \mathcal { S } ^ n , \end{align*}


\begin{align*} ( f _ 1 ( \mathbf { x } ) , f _ 2 ( \mathbf { x } ) , \cdots , f _ { n - 1 } ( \mathbf { x } ) ) = ( f _ 2 ( \mathbf { y } ) , f _ 3 ( \mathbf { y } ) , \cdots , f _ n ( \mathbf { y } ) ) = ( f _ 2 ( \mathbf { x } ) , f _ 3 ( \mathbf { x } ) , \cdots , f _ n ( \mathbf { x } ) ) . \end{align*}


\begin{align*} ( \Delta u ) ^ { m } \wedge \beta _ n ^ { n - m } = m ! ( n - m ) ! \mathcal { H } _ m ( u ) \Omega _ { 2 n } . \end{align*}


\begin{align*} \Upsilon : = K _ M \bigcap \bigcap _ { k = 1 } ^ { \infty } \Gamma _ k \end{align*}


\begin{align*} ( \nabla ^ { \ell + 2 } _ \nu h ) ( e _ a , e _ b ) = 0 , \mbox { o r i n t h e l o c a l f r a m e } h _ { a b ; \tiny \underbrace { 0 \cdots 0 } _ { \ell + 2 - \mbox { t i m e s } } } = 0 . \end{align*}


\begin{align*} \beta ( z , w ) = \dfrac { 1 } { 2 } \log \dfrac { 1 + | \varphi _ z ( w ) | } { 1 - | \varphi _ z ( w ) | } , z , w \in \mathbb { B } _ n . \end{align*}


\begin{align*} \left ( \lambda + \rho \right ) ^ 2 = \left ( \mu + \nu \right ) ^ 2 . \end{align*}


\begin{align*} & f _ n \big ( t , ( \partial _ x v _ n \Sigma ) ( t , x ) , x \big ) = ( \partial _ x v _ n \Sigma \cdot q ) ( t , x ) - g _ n \big ( t , q ( t , x ) , x \big ) . \end{align*}


\begin{align*} \overline { \mathrm { s p a n } } _ { H ^ 2 ( \mathbb { D } ) } \{ ( I - S ) h _ k \mid k \geq 2 \} = H ^ 2 ( \mathbb { D } ) . \end{align*}


\begin{align*} { \rm O } ( \epsilon ^ 0 ) : g _ i ^ { ( 0 ) } = g _ i ^ { ( e q ) } , \end{align*}


\begin{align*} \tilde { \gamma } _ { \pi \otimes \chi _ { 8 d } , j } = \frac { M \gamma _ { \pi \otimes \chi _ { 8 d } , j } } { 2 \pi } \log X . \end{align*}


\begin{align*} A _ { l , u } & = \Big \{ i \in [ k ] : \frac { p _ i } { q _ i } \in [ l , u ) \Big \} , \\ A _ { l , \infty } & = \Big \{ i \in [ k ] : \frac { p _ i } { q _ i } \in [ l , \infty ] \Big \} . \end{align*}


\begin{align*} \theta _ { \Psi } ( r , \xi ) = \theta _ { \Psi } ( r ) \end{align*}


\begin{align*} \begin{aligned} F ( \eta , t ) & : = \left ( \begin{array} { c } - \varepsilon _ \delta ( { x } , \bar { y } , t ) u ( t ) \\ \varepsilon _ \delta ( { x } , \bar { y } , t ) \end{array} \right ) \end{aligned} \end{align*}


\begin{align*} \operatorname { E } ( X ^ r ) = \frac { \theta ^ { r / 2 } } { \operatorname { B } ( a , b ) } \ , S _ r ( a , b ) , r < 2 . \end{align*}


\begin{align*} I = \varinjlim _ { H \subset G } I ^ H . \end{align*}


\begin{align*} \Phi ( v , p ) = Q F ^ q ( v + \psi ( v , p ) , p ) , \end{align*}


\begin{align*} y \left [ n \right ] = { { \bf { h } } ^ H } \left [ n \right ] * { \bf { x } } \left [ n \right ] + z \left [ n \right ] = \sum \limits _ { l = 1 } ^ L { { \bf { h } } _ l ^ H { \bf { x } } \left [ { n - { n _ l } } \right ] } + z \left [ n \right ] , \end{align*}


\begin{align*} \begin{cases} \mathcal { L } _ g \textbf { \textit { U } } = 0 & \ \Omega , \\ \textbf { \textit { U } } = \textbf { \textit { V } } & \ \partial \Omega , \end{cases} \end{align*}


\begin{align*} \left \{ \begin{aligned} - \Delta y + \mathcal { N } ( \cdot , y ) & = u , \ ; \ ; \Omega , \\ y & = 0 , \ ; \ ; \partial \Omega . \end{aligned} \right . \end{align*}


\begin{align*} D = \{ c \} \cup \{ d \in C _ i \mid \varphi _ { \sigma , s } ^ L ( d ) = a \} \end{align*}


\begin{align*} \langle \lambda , \lambda ' \rangle : = \sum _ { \alpha = 1 } ^ N \lambda _ { \alpha } \lambda ' _ { \alpha } , \quad \ , \lambda = ( \lambda _ { \alpha } ) _ { 1 \le \alpha \le N } , \ , \ , \ , \lambda ' = ( \lambda ' _ { \alpha } ) _ { 1 \le \alpha \le N } \in \mathbb { C } ^ N . \end{align*}


\begin{align*} \phi ( \pi \cdot e _ { \sigma } ) = \phi ( e _ { \pi \sigma \pi ^ { - 1 } } ) = \phi ( e _ { ( \pi \sigma _ 1 \ , \pi \sigma _ 2 \ , \ldots \ , \pi \sigma _ m ) } ) = ( \overline { \pi r } , 1 ) = \pi \cdot \phi ( e _ { \sigma } ) , \end{align*}


\begin{align*} \theta _ 0 ( x ) - x = - \frac { 1 } { 2 \pi } \int _ { - \infty } ^ { \infty } \frac { x ^ { s } } { s } \bigg \{ \frac { \zeta ' ( s ) } { \zeta ( s ) } + g ( s ) \bigg \} d t . \end{align*}


\begin{align*} x _ 2 v _ 1 - x _ 1 v _ 2 + k v _ 3 = 0 . \end{align*}


\begin{align*} \sum _ { \substack { U \subset F _ s : \\ U \cap F _ { \ell + 1 } = T } } \alpha ^ s _ U . \end{align*}


\begin{align*} d ( h _ z ) = \frac { 2 } { \mu } \mu _ z h _ z d z + \frac { \sqrt { \tau _ 0 } } { 2 } \left ( \frac { 1 } { 2 } d \overline { z } + ( \alpha - I \beta ) d z \right ) \end{align*}


\begin{align*} S ( X ; \pi , \hat { \phi } , W ) = o ( X \log X ) , \end{align*}


\begin{align*} \lim _ { n \to \infty } | g _ { n + 1 } | \sum _ { i = n + 1 } ^ { \infty } \psi _ i ( t _ p ) = 0 . \end{align*}


\begin{align*} | \{ p \leq X : B ( p ) = B \} | = c ( B ) \int _ 2 ^ X \frac { d t } { \log t } + O _ K \left ( X e ^ { - c _ K \sqrt { \log X } } \right ) , \end{align*}


\begin{align*} \mathcal { C } _ n ^ { ( k ) } \ = \ A _ k F _ n + B _ k F _ { n + 1 } - \sum _ { j = 0 } ^ k \binom { k } { j } B _ j n ^ { k - j } . \end{align*}


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


\begin{align*} \bigg | \frac { \partial G _ { z _ i } ( T _ { z _ i } z _ i , T _ { z _ i } z _ j ) } { \partial z _ { i , \hbar } } \bigg | \leq C , \ \ 1 \leq i \neq j \leq m , \hbar = 1 , 2 . \end{align*}


\begin{align*} \deg ( ( g _ { m , ( m ' ; l ' ) } ^ \epsilon ) _ { i j } ) = ( - 1 ) ^ j i l + i j ( d - 2 ) + \frac { ( - 1 ) ^ i } { 2 } \left ( 1 - \left ( - 1 \right ) ^ j \epsilon \right ) \end{align*}


\begin{align*} C _ { \mathrm { o b s } } = \frac { C _ 1 } { T ^ { 1 / r } } \exp \left ( \frac { C _ 2 } { T ^ { \frac { \gamma _ 1 \gamma _ 3 } { \gamma _ 2 - \gamma _ 1 } } } + C _ 3 T \right ) , \end{align*}


\begin{align*} \det D ^ { 2 } u = \mu , \end{align*}


\begin{align*} R _ { D } ^ \pm = R _ { D ^ 0 } ^ \pm + d ^ { D ^ 0 } S | _ { E _ \pm \otimes E _ { \mp } \otimes E _ \pm } , \end{align*}


\begin{align*} \phi ^ { \zeta \theta } ( e _ { v w } ) : = ( \phi ^ \zeta ) ^ \theta ( e _ { v w } ) & = \tau ^ \zeta ( v , e ) \theta ( v ) + \phi ^ \zeta ( e _ { v w } ) + \tau ^ \zeta ( w , e ) \theta ( w ) \\ & = \zeta ( v ) \tau ( v , e ) \theta ( v ) + \phi ( e _ { v w } ) + \zeta ( w ) \tau ( w , e ) \theta ( w ) \\ & = \tau ( v , e ) \theta ^ \zeta ( v ) + \phi ( e _ { v w } ) + \tau ( w , e ) \theta ^ \zeta ( w ) \\ & = \phi ^ { \theta ^ \zeta } ( e _ { v w } ) = \phi ^ { \theta ^ \zeta \zeta } ( e _ { v w } ) \end{align*}


\begin{align*} g _ n ( t , \omega ) : = & \langle A ( t , X _ n ( t , \omega ) ) , X _ n ( t , \omega ) - X ( t , \omega ) \rangle , \\ F _ n ( t , \omega ) : = & f ( t ) ( 2 + \Vert X _ n ( t , \omega ) \Vert _ H ^ 2 ) + C \Vert X ( t , \omega ) \Vert _ V ^ { \alpha } \\ & + C \Vert X _ n ( t , \omega ) \Vert _ H ^ { \beta ( \alpha - 1 ) } \Vert X ( t , \omega ) \Vert _ V ^ { \alpha } . \end{align*}


\begin{align*} \frac { g ' ( x + \gamma ) } { g ' ( x ) } = e ^ { a \gamma } > 1 , \end{align*}


\begin{align*} S _ { s } ( x ) = - s \sum _ { \zeta ( \rho ) = 0 } \frac { x ^ { \rho - s } } { \rho ( \rho - s ) } . \end{align*}


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


\begin{align*} g ( x - k \gamma ) & = g ( x ) - \gamma \sum _ { j = 1 } ^ k \frac { g ( x - ( j - 1 ) \gamma ) - g ( x - j \gamma ) } { \gamma } \\ & < g ( x ) - \gamma \sum _ { j = 1 } ^ k ( \Lambda + \varepsilon ) ^ { - j } \frac { \Lambda - 1 + \delta } { \gamma } g ( x ) \\ & = g ( x ) \left ( 1 - ( \Lambda - 1 + \delta ) \sum _ { j = 1 } ^ k ( \Lambda + \varepsilon ) ^ { - j } \right ) \\ & < 0 , \end{align*}


\begin{align*} \begin{aligned} x _ n - y _ n = \ , & \sum _ { \beta = 1 } ^ { n - 1 } \rho _ \beta ( y ' ) ( x _ \beta - y _ \beta ) + O ( | x ' - y ' | ^ 2 ) \\ \geq \ , & \sum _ { \beta = 1 } ^ { n - 1 } \rho _ \beta ( y ' ) ( x _ \beta - y _ \beta ) - \kappa | x ' - y ' | ^ 2 \end{aligned} \end{align*}


\begin{align*} \| \psi _ { n + m } - \psi _ n \| _ 1 \leq \sum _ { i = 1 } ^ m \frac { 1 } { 2 \pi } \int _ 0 ^ { 2 \pi } \left | \arg \frac { \hat { b } _ { n + i } \left ( e ^ { i \theta } \right ) } { e ^ { i \theta } } \right | \ , d \theta . \end{align*}


\begin{align*} \frac { d _ 1 } { u ^ { q + 1 } } = \frac { h ^ q } { h } \end{align*}


\begin{align*} a _ { k - j , 0 } & = \frac { \Gamma \bigl ( - w _ 2 + k \bigr ) \Gamma \bigl ( 0 \bigr ) \Gamma \bigl ( w _ 2 + \frac { n } { 2 } \bigr ) } { \Gamma \bigl ( - w _ 2 + j \bigr ) \Gamma \bigl ( - j \bigr ) \Gamma \bigl ( w _ 2 + \frac { n - 2 k } { 2 } \bigr ) } \\ & = ( - 1 ) ^ j j ! \frac { \Gamma \bigl ( - w _ 2 + k \bigr ) \Gamma \bigl ( w _ 2 + \frac { n } { 2 } \bigr ) } { \Gamma \bigl ( - w _ 2 + j \bigr ) \Gamma \bigl ( w _ 2 + \frac { n - 2 k } { 2 } \bigr ) } . \end{align*}


\begin{align*} H ( \xi ) : = \begin{cases} 1 / b ' ( \xi ) & \ \xi \notin ( b ) \\ 0 & \ \xi \in ( b ) \end{cases} \end{align*}


\begin{align*} \mathbb { H } : = H \times [ H ^ 1 ( \mathcal { O } ) ^ 3 ] , \mathbb { V } : = V \times \Big \{ n \in H ^ 2 ( \mathcal { O } ) ^ 3 : \frac { \partial n } { \partial \nu } = 0 \Big \} , \end{align*}


\begin{align*} \varphi ^ L _ { \sigma , s } ( x ) = a < b = \varphi ^ L _ { \sigma , s } ( y ) . \end{align*}


\begin{align*} \overline { \mathfrak { D } _ { q ^ n } } \ \overline { U _ { q ^ n } } \zeta = ( A + B ) \overline { U _ { q ^ n } } \dot { s } \overline { U ^ * _ { q ^ n } } { \bf 1 } _ { - } + ( q ^ n - 1 ) C \overline { U _ { q ^ n } } { \bf 1 } _ { - } \in \Bbbk { \bf G } \zeta . \end{align*}


\begin{align*} c _ 1 ( q , 2 q , p ^ 0 ( 0 ) ) & = \frac 1 4 \left ( \frac { 1 } { - q } f ( - q r ) + \frac { 1 } { 3 q } f ( 3 q r ) - \frac { 2 } { q } f ( q r ) - H ( q , r ) \frac 1 q f ( q r ) \right ) \\ & = \frac 1 q g ( q r ) , \end{align*}


\begin{align*} u _ m - \beta ^ r u _ { m - r } = a ' \alpha ^ { m - r } t _ r . \end{align*}


\begin{align*} \forall \varepsilon > 0 , \lim _ { n \to \infty } \sum _ { m = 1 } ^ n \mathbb { E } _ { n , m } [ \| \chi _ { n , m } \| _ { L ^ 2 ( \mu ) } ^ 2 : \| \chi _ { n , m } \| _ { L ^ 2 ( \mu ) } > \varepsilon ] = 0 \end{align*}


\begin{align*} T _ w : = T _ { s _ 1 } T _ { s _ 2 } \ldots T _ { s _ { \ell ( w ) } } , \end{align*}


\begin{align*} \gamma ( s , \tau ) = \left ( r ^ * \cos \left ( \frac { s } { r ^ * } \right ) , r ^ * \sin \left ( \frac { s } { r ^ * } \right ) , \frac { c } { 4 \pi r ^ * } \tau \right ) ^ t , \end{align*}


\begin{align*} \int _ { 0 } ^ { \lambda } \mu \big [ \xi ( \mu , t ) - \xi ( \mu , 0 ) ] \\ b i g d \mu = - & \int _ { 0 } ^ { t } \int _ { \lambda } ^ { \infty } \int _ { 0 } ^ { \lambda } \nu \Lambda ( \mu , \nu ) \zeta ( \mu , s ) \xi ( \nu , s ) \ d \nu d \mu d s . \end{align*}


\begin{align*} D _ 2 = \displaystyle \sum _ { \beta \in F ^ { \times } } \sum _ { \substack { X \in S ( \beta ) \\ t = 1 } } \Theta _ { \theta } \begin{bmatrix} m _ 1 & X \\ 0 & m _ 2 \end{bmatrix} \overline { \psi _ A ( { m _ 1 } ^ { - 1 } X } ) = q ^ 8 . \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*} F _ n ( \eta ) : = T _ n - \frac { 1 } { \eta } I _ n . \end{align*}


\begin{align*} \mbox { $ N _ { P _ 2 ^ * } ( c _ 6 ) = \emptyset $ , a n d $ T $ i s n o t a j u m p a c r o s s $ c _ 1 c _ 7 $ . } \end{align*}


\begin{align*} D _ n ( x ) = \prod _ { k = 0 } ^ { n } m _ \tau ( ( R ^ { t } ) ^ { k } x ) . \end{align*}


\begin{align*} T ( t ) x = \tau \lim _ { n \to \infty } \left ( \frac { n } { t } R \left ( \frac { n } { t } , A \right ) \right ) ^ { n } x \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*} \Psi ^ { \pm } ( x ; z ) = I + \int _ { \pm \infty } ^ x e ^ { - i z ( x - y ) \widehat { \sigma } _ 3 } \widetilde { Q } \Psi ^ { \pm } ( y ; z ) d y . \end{align*}


\begin{align*} \sum _ { k = 1 } ^ n \frac { \mu ( k ) } { k } \log ( 1 - z ^ k ) \to - z \ \ell ^ q \end{align*}


\begin{align*} \int _ 0 ^ T \eta ^ { - \frac { 1 } { p - 1 } } \left | \eta _ t \right | ^ { \frac { p } { p - 1 } } d t = C T ^ { - \frac { N } { 2 } + 1 } . \end{align*}


\begin{align*} w = u _ 1 + u _ 2 + \cdots + u _ l . \end{align*}


\begin{align*} \{ | v _ 1 | > | v _ 2 | > \dots > | v _ N | \} \cap \{ | z v _ i | < | v _ { i + 1 } | | i = 1 , \dots N - 1 \} \subset \mathbb C ^ N \times \mathbb H \end{align*}


\begin{align*} x ( [ \mathbf { a } ] \otimes [ \mathbf { b } ] ) = 0 . \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*} \Delta v + \big ( \Delta ' ( h ) \big ) \bar u = 0 \mbox { i n } M \setminus \Omega . \end{align*}


\begin{align*} \left ( \begin{array} { c c } A _ 1 - \lambda _ 1 E _ n & A _ 2 - \lambda _ 2 E _ n \\ \overline { A _ 2 - \lambda _ 2 E _ n } & \overline { A _ 1 - \lambda _ 1 E _ n } \\ \end{array} \right ) \left ( \begin{array} { c } z _ 1 \\ \overline { z _ 2 } \\ \end{array} \right ) = \left ( \begin{array} { c } 0 \\ 0 \\ \end{array} \right ) , \end{align*}


\begin{align*} { f _ { \bar i } } \left ( { { { \bf { x } } _ b } , t + \Delta t } \right ) = f _ i ^ * \left ( { { { \bf { x } } _ b } , t } \right ) , \end{align*}


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


\begin{align*} \hat W _ r ( k ) = \frac { 2 } { \pi k } \sin ( 2 \pi k r ) = \frac { 1 } { k } f ( k r ) , \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*} \nabla _ V ( \nabla X ) = - R ( X , V ) + T _ h ( V ) \end{align*}


\begin{align*} g _ n ( t , q , x , \mu ) : = g \big ( t , \frac { q } { \sqrt { n } } , x , \mu \big ) \end{align*}


\begin{align*} \left \lvert 1 - 4 \mu \left ( \sum _ { l = 1 } ^ { M } \lvert \hat { \phi } ^ { \left ( l , j \right ) } \rvert ^ 2 + \sigma _ { e , i } ^ 2 \lVert \mathbf { p } _ j \rVert ^ 2 + \sum _ { q = 1 } ^ { M - 1 } \sum _ { r = q + 1 } ^ { M } f _ { q , r } ^ { \left ( j \right ) } \right ) \right \rvert < 1 \end{align*}


\begin{align*} A _ k \ & = \ \sum _ { j = 0 } ^ { k } \ , j ! \ , F _ { j + 1 } \begin{Bmatrix} k \\ j \end{Bmatrix} , \\ B _ k \ & = \ \sum _ { j = 0 } ^ { k } \ , j ! \ , F _ { j + 2 } \begin{Bmatrix} k \\ j \end{Bmatrix} . \end{align*}


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


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


\begin{align*} Z ^ { ( i ) } = \partial _ i \mbox { o r } Z ^ { ( i , j ) } = x _ i \partial _ j - x _ j \partial _ i \mbox { f o r } i , j = 1 , \dots , n . \end{align*}


\begin{align*} | G ( n , m ) | \leq \left ( m ^ s + \frac { n ^ s } { \sqrt { \log ( n ) } } + \left ( \frac { n } { m } \right ) ^ { \frac { s } { 1 - s } } \right ) \cdot \omega \left ( m \left | \sum _ { k = 1 } ^ m \frac { \mu ( k ) } { k } \right | \right ) . \end{align*}


\begin{align*} \lim _ { y \to \pm \infty } e ^ { - \psi ( x _ 0 , y ) } d _ x T ( x _ 0 , y ) = 0 . \end{align*}


\begin{align*} 0 = u c \lim _ i \| \varpi \otimes \varpi ( \cdot ) \xi _ i - \xi _ i \| \end{align*}


\begin{align*} \begin{aligned} \Delta u _ 1 \wedge \Delta u _ 2 \wedge \dots \wedge \Delta u _ n & = d _ 0 ( d _ 1 u _ 1 \wedge \Delta u _ 2 \wedge \dots \wedge \Delta u _ n ) = - d _ 1 ( d _ 0 u _ 1 \wedge \Delta u _ 2 \wedge \dots \wedge \Delta u _ n ) \\ & = d _ 0 d _ 1 ( u _ 1 \Delta u _ 2 \wedge \dots \wedge \Delta u _ n ) = \Delta ( u _ 1 \Delta u _ 2 \wedge \dots \wedge \Delta u _ n ) . \end{aligned} \end{align*}


\begin{align*} \Lambda _ { g } = A Q - D \end{align*}


\begin{align*} B _ n ( x ) - ( - 1 ) ^ n B _ n ( y ) = 0 , \end{align*}


\begin{align*} A _ 2 ^ 2 \le \frac { 1 } { n } \sum _ { i = 1 } ^ n \mathcal { W } _ 1 ^ 2 ( P _ i , Q ^ * _ i ) \le \frac { 1 } { n } \sum _ { i = 1 } ^ n \mathcal { W } _ 2 ^ 2 ( P _ i , Q ^ * _ i ) \le \frac { 2 R _ f } { \kappa n } . \end{align*}


\begin{align*} \frac { \partial \mathbb { E } \left [ \varepsilon | \hat { \mathbf { H } } \right ] } { \partial a _ i } = & 4 a _ i \sum _ { l = 1 } ^ { M - 1 } \sum _ { q = l + 1 } ^ { M } \Re \left [ \hat { \phi } ^ { \left ( l , i \right ) ^ * } \hat { \phi } ^ { \left ( q , i \right ) } \right ] - 2 \Re { \left \{ \hat { \phi } ^ { \left ( i , i \right ) } \right \} } + 4 a _ i \left ( \sum _ { l = 1 } ^ M \lvert \hat { \phi } ^ { \left ( l , i \right ) } \rvert ^ 2 + M \sigma _ { e , i } ^ 2 \lVert \mathbf { p } _ i \rVert ^ 2 \right ) . \end{align*}


\begin{align*} L ( E , \epsilon ) = \max \{ \log | \lambda | + | \epsilon | , 0 \} , \forall \ E \in \Sigma _ { 2 \lambda \cos , \alpha } . \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*} \gamma ( \| \cdot \| _ { \infty } , \tau _ { \operatorname { c o } } ) = \beta _ { 0 } = \mu ( \mathrm { C } _ { \operatorname { b } } ( \Omega ) , \mathrm { M } _ { \operatorname { t } } ( \Omega ) ) \end{align*}


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


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


\begin{align*} F ' _ m ( n ) = - 1 + F ' _ m ( n - 1 ) , \end{align*}


\begin{align*} \Gamma ( G , M ( S ) ) = [ H ^ 0 ( G , r ( M ) ( S ) ) \to H ^ 0 ( G , r ^ 2 ( M ) ( S ) ) \to \dots ] , \end{align*}


\begin{align*} \beta _ 1 + \bar { \beta _ j } = 1 \ , \ , { \rm a n d } \ , \ , \beta _ u + \bar { \beta } _ v = 1 . \end{align*}