\begin{align*} X _ i = 0 , \omega _ { i j } V _ i V _ j = \tfrac { 1 } { 2 } h _ { i j } V _ i V _ j , \omega _ { i a } V _ i = h _ { i a } V _ i , \omega _ { a b } = 0 \end{align*}


\begin{align*} \frac { 1 } { m } \sum _ { i = 1 } ^ m U _ i < \left ( \frac { \pi } { 4 } + \frac { 1 9 9 \pi } { 3 2 0 0 k } \right ) \| \mathbf { x } \| . \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*} \underline { h } _ { \mu } ( f _ { 1 , \infty } ) : = \int \underline { h } _ { \mu } ( f _ { 1 , \infty } , x ) \ d \mu ( x ) . \end{align*}


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


\begin{align*} \| \tilde { X } ( k ) \| ^ 2 \leq \frac { \sum \limits _ { i = 1 } ^ { p } \left \| \tilde { Y } _ i ( k ) \right \| ^ 2 + \sum \limits _ { j = 1 } ^ { q } \left \| \tilde { Z } _ j ( k ) \right \| ^ 2 } { p + q } . \end{align*}


\begin{align*} \Theta _ { { \mathrm { E u c l } } } ^ { \Sigma _ \infty } ( 0 , 1 0 \lambda K ) - \Theta _ { { \mathrm { E u c l } } } ^ { \Sigma _ \infty } ( 0 , 1 / ( 3 K ) ) = 0 . \end{align*}


\begin{align*} \frac { x _ i } { g } ( x _ 1 - x _ k ) = \begin{cases} x _ i & ( i = k ) \\ 0 & ( i \neq k ) . \end{cases} \end{align*}


\begin{align*} S _ k ^ { i j } H _ { i j } = - \sum _ i S _ k ^ { p q , r s } u _ { p q i } u _ { r s i } + B ( n - k + 1 ) S _ { k - 1 } + \Delta f . \end{align*}


\begin{align*} \Lambda _ { g } ( \textbf { \textit { V } } ) : = \begin{pmatrix} \lambda \nu \operatorname { d i v } + \mu \nu S & - \beta \nu \\ 0 & \alpha \partial _ \nu \end{pmatrix} \textbf { \textit { U } } \ \partial \Omega , \end{align*}


\begin{align*} - B _ p n ^ { p - p } \binom { p } { p } \sum _ { K = 0 } ^ { 0 } ( - 1 ) ^ { K + p } \binom { p - p } { K } , \end{align*}


\begin{align*} M _ { \delta , + } ( x ; z ) = M _ { \delta , - } ( x ; z ) ( I + \hat { R } _ { \delta } ( x ; z ) ) , z \in \mathbb { R } . \end{align*}


\begin{align*} k ( M ) = \inf \frac { { \rm V o l } _ { n - 1 } ( \Sigma ) } { { \rm V o l } _ n ( A ) | \log ( { \rm V o l } _ n ( A ) ) | } , \end{align*}


\begin{align*} \begin{gathered} \min _ { \mathbf { a } } \mathbb { E } \left [ \lVert \mathbf { s } ^ { \left ( \right ) } - \mathbf { y } ' \rVert ^ 2 \right ] \\ ~ ~ \left ( \mathbf { P } ^ { \left ( \right ) } \left ( \mathbf { a } ^ { \left ( \right ) } \odot \mathbf { a } ^ { \left ( \right ) } \right ) \mathbf { P } ^ { \left ( \right ) ^ { H } } \right ) = E _ { t r } , \end{gathered} \end{align*}


\begin{align*} M = \left ( \begin{array} { c c } \varepsilon _ 1 \mu & \varepsilon _ 1 \nu \\ \varepsilon _ 3 \mu & \varepsilon _ { 3 } \nu \end{array} \right ) \end{align*}


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


\begin{align*} \mathbb { E } \left [ \mathbf { y ' } ^ H \mathbf { y } ' \right ] = \sum _ { l = 1 } ^ M \sum \limits _ { \substack { j = 1 \\ j \neq l } } ^ M \mathbb { E } \left [ y _ l ^ * y _ j \right ] + 2 \sum _ { i = 1 } ^ { M } \mathbb { E } \left [ y _ i ^ * y _ i \right ] , \end{align*}


\begin{align*} \begin{cases} \nabla w \cdot \nabla ^ \perp ( u - q \ln \frac { 1 } { \varepsilon } ) = 0 , \\ w = \mathcal { L } _ H u , \\ \nabla ^ \perp u \cdot \nu | _ { \partial \Omega } = 0 . \end{cases} \end{align*}


\begin{align*} \mu ^ * & = \mu ^ { \rm N } - \varkappa ^ { \rm N } , \qquad \ , \varkappa = 2 \varkappa ^ { \rm N } , \qquad \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*} H _ k : = \begin{pmatrix} A _ k & b _ k \\ b _ k ^ T & c _ k \end{pmatrix} \ \ k = 1 , . . . , m H _ J : = \begin{pmatrix} A _ J & b _ k \\ b _ J ^ T & c _ J \end{pmatrix} \end{align*}


\begin{align*} m ^ a \ell ^ { t _ { \widetilde K } } = m ^ a ( \ell ' v ) ^ { t _ { \widetilde K } } = m ^ a { \ell ' } ^ { t _ { \widetilde K } } \cdot ( v + \ell ' ( v ) + \dots + \ell '^ { t _ { \widetilde K } - 1 } ( v ) ) . \end{align*}


\begin{align*} \Omega _ 1 \Delta \Omega _ 2 : = ( \Omega _ 1 \setminus \Omega _ 2 ) \cup ( \Omega _ 2 \setminus \Omega _ 1 ) \end{align*}


\begin{align*} \frac { \sum \limits _ { i = 1 } ^ { j + 1 } \chi _ i - r ( j + 1 ) } { \chi } - \sum \limits _ { i = 1 } ^ j w _ i > w _ { j + 1 } > \frac { \sum \limits _ { i = 1 } ^ { j + 1 } \chi _ i - r j } { \chi } - \sum \limits _ { i = 1 } ^ j w _ i . \end{align*}


\begin{align*} r b = \nabla \phi \times r \nabla \theta + G r \nabla \theta . \end{align*}


\begin{align*} L ( E ) = \max \bigg \{ \log \Big | \frac { E } { 2 } + \frac { \sqrt { E ^ { 2 } - 4 } } { 2 } \Big | , \log | \lambda | \bigg \} , \forall \ E \in \mathbb { C } . \end{align*}


\begin{align*} F ( x ) = I _ { \Phi ( x ) } ( \alpha , \beta ) = \frac { 1 } { \mathrm { B } ( \alpha , \beta ) } \int _ 0 ^ { \Phi ( x ) } \omega ^ { \alpha - 1 } \ , ( 1 - \omega ) ^ { \beta - 1 } \mathrm { d } \omega . \end{align*}


\begin{align*} \lim _ { | z | \rightarrow \infty } \Psi ^ - _ { 1 1 } ( x ; z ) = 1 + \frac { 1 } { 2 i } \int _ { - \infty } ^ { x } | u _ y ( y ) | ^ 2 \lim _ { | z | \rightarrow \infty } \Psi ^ - _ { 1 1 } ( y ; z ) \mathbf { d } y , \end{align*}


\begin{align*} A R ( G ) = \begin{cases} A \rtimes \{ I \} & d \\ A \rtimes \langle - I \rangle & d \end{cases} \end{align*}


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


\begin{align*} \sum _ { i = 1 } ^ n f _ i ( s ) \bar { f _ i } ( t ) = \mu \log K ( G ( s ) , \bar { G } ( t ) ) ~ ~ ( s , t ) \in U \times \hbox { c o n j } ( { U } ) , \end{align*}


\begin{align*} \mu _ { ( \boldsymbol { x } , \boldsymbol { \phi } ( t , \boldsymbol { x } ) ) } ^ { N , i } + \frac { 1 } { N - 1 } \underset { j \neq i } { \sum } h \Big ( \mu _ { ( \boldsymbol { x } , \boldsymbol { \phi } ( t , \boldsymbol { x } ) ) } ^ { N , j } \Big ) = - R ^ { - 1 } B \Big [ P _ t \nu ^ { N , i } _ { \boldsymbol { x } } + \frac { 1 } { N - 1 } \underset { j \neq i } { \sum } \phi ^ { N , j } ( t , \boldsymbol { x } ) \Big ] . \end{align*}


\begin{align*} \int _ { \R ^ { n - 1 } } \langle f ( x ' ) , h ( x ' ) \rangle d x ' = \lim _ { \varepsilon \to 0 ^ + } \int _ { \R ^ { n - 1 } } \langle f ( x ' ) , \widetilde { \Psi } _ { \varepsilon , \varepsilon ^ { - 1 } } * h ( x ' ) \rangle d x ' . \end{align*}


\begin{align*} S ^ { i j } _ k = \frac { \partial S _ k [ D ^ 2 u ] } { \partial r _ { i j } } . \end{align*}


\begin{align*} \mathbb { E } \Big [ \underset { t _ 0 \leq t \leq T } { \sup } | x _ t ^ i - x _ t ^ { * , i } | ^ 2 \Big ] \leq \frac { C } { N ^ 2 } \Big ( 1 + \frac { 1 } { N } \underset { j = 1 } { \overset { N } \sum } \mathbb { E } [ | \xi ^ i - \xi ^ j | ^ 2 ] + \frac { 1 } { N ^ 2 } \underset { i , j = 1 } { \overset { N } { \sum } } \mathbb { E } [ | \xi ^ { i } - \xi ^ { j } | ^ 2 ] \Big ) , \end{align*}


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


\begin{align*} \frac { \partial \hat { q } _ { \delta , i } } { \partial z _ { i , h } } = O ( 1 ) , \end{align*}


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


\begin{align*} & \lim _ { j \rightarrow \infty } \sup _ { z \in V ^ { - 1 } \mathbb { B } _ n } \Big { | } ( ( V ^ { - 1 } \varphi _ { k j } V ) ^ 1 , . . . , ( V ^ { - 1 } \varphi _ { k j } V ) ^ s ) ( z ) \Big { | } \\ & \qquad = \lim _ { j \rightarrow \infty } \| V ^ { - 1 } ( \varphi _ { k j } - \rho ) \| _ \infty \\ & \leq C \lim _ { j \rightarrow \infty } \| \varphi _ { k j } - \rho \| _ \infty = 0 . \end{align*}


\begin{align*} \sigma : = | u _ { , i j k } | ^ 2 \end{align*}


\begin{align*} K _ q ( p , s ) = E _ q ^ * ( p \overline { s } ) : = \displaystyle \sum _ { n = 0 } ^ \infty \frac { p ^ n \overline { s } ^ n } { \Gamma ( q n + 1 ) } , \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*} | G _ { \gamma , \omega } ( s ) - 1 | = \dfrac { | \alpha \omega - \omega | } { | b | | c \omega + d | } \leq \dfrac { 3 ^ { - 2 } } { | b | } < \varepsilon , & & \mathrm { i f } \ | b | > \frac { 3 ^ { - 2 } } { \varepsilon } . \end{align*}


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


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


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


\begin{align*} \Phi _ N = \{ L _ N , L _ N + 1 , \dots , M _ N - 1 \} \end{align*}


\begin{align*} \frac { 1 } { n } \sum _ { i = 1 } ^ n \delta _ { X _ i } \to Q , . \end{align*}


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


\begin{align*} \begin{aligned} C _ m ( U ) & = \sup \left \{ \int _ { U } ( \Delta u ) ^ m \wedge \beta _ n ^ { n - m } : u \in Q S H _ m ( \Omega ) \cap C ( \Omega ) , - 1 \leq u < 0 \right \} \\ & = \sup \left \{ \int _ { U } ( \Delta u ) ^ m \wedge \beta _ n ^ { n - m } : u \in Q S H _ m ( \Omega ) \cap C ^ { \infty } ( \Omega ) , - 1 \leq u < 0 \right \} \end{aligned} \end{align*}


\begin{align*} Q ( r , t ) : = \int _ { B _ { r } ( 0 ) } n ( x , t ) \ , d x = \sigma _ { d } \int _ { 0 } ^ { r } \rho ^ { d - 1 } n ( \rho , t ) \ , d \rho \end{align*}


\begin{align*} \sum _ { d = 0 } ^ j x _ { i , d } = x _ i , \ , \sum _ { i = 1 } ^ r x _ i = t . \end{align*}


\begin{align*} \sum _ { u = 1 } ^ t d _ u = 2 j , \end{align*}


\begin{align*} I _ \delta ( u ) = \frac { \delta ^ 2 } { 2 } \int _ { \Omega } \left ( K ( x ) \nabla u | \nabla u \right ) - \frac { 1 } { p + 1 } \int _ { \Omega } ( u - q ) ^ { p + 1 } _ + . \end{align*}


\begin{align*} \Gamma _ { \lambda } ( t ) : = \int _ { 0 } ^ { \lambda } \mu \xi ( \mu , t ) \ d \mu \mbox { a n d } \Theta _ { \lambda } ( t ) : = \int _ { \lambda } ^ { \infty } \mu \xi ( \mu , t ) \ d \mu , \end{align*}


\begin{align*} \sum _ { i = 1 } ^ m \mathcal { W } _ 2 ^ 2 ( P _ i , Q _ i ) \le \mathcal { W } _ 2 ^ 2 ( P , Q ) . \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*} Y = m ( X ) + \eta \end{align*}


\begin{align*} \xi ^ { 2 ^ k } = \left \{ \begin{array} { l l } \xi + 1 & \textrm { i f $ k $ i s o d d , } \\ \xi & \textrm { i f $ k $ i s e v e n } . \end{array} \right . \end{align*}


\begin{align*} g ( \Phi ) _ { t } + u \cdot \nabla g ( \Phi ) = 0 , \end{align*}


\begin{align*} \hat W _ r ( k ) = \frac 1 k f ( r k ) \end{align*}


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


\begin{align*} \abs { f } = \inf _ { x \in X } d _ f ( x ) . \end{align*}


\begin{align*} M _ { f _ { 1 , \infty } } ( \alpha , \epsilon , Z , \psi ) : = \lim _ { n \to \infty } M _ { f _ { 1 , \infty } } ( n , \alpha , \epsilon , Z , \psi ) . \end{align*}


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


\begin{align*} & \ \eta = \sum _ { x \in \bar { \mathbb { F } } _ q } a _ x \dot { s } \varepsilon ( u + x ) { \bf 1 } _ { \theta } + \sum _ { y , z \in \bar { \mathbb { F } } _ q } b ' _ { y , z } \varepsilon ( - ( u + y ) ^ { - 1 } ) \dot { s } \varepsilon ( ( u + y ) ^ 2 z - ( u + y ) ) { { \bf 1 } } _ { \theta } , \end{align*}


\begin{align*} \mathcal { T } ( x , \rho _ { 0 } ) = C ( \rho _ { 0 } ) + o ( 1 ) \end{align*}


\begin{align*} \nabla ^ { \bot } \left ( \Psi - W x _ 2 \right ) \cdot \nabla \omega = 0 . \end{align*}


\begin{align*} \phi ( \sigma _ 1 \sigma _ 2 , \tau ) = \phi ( \tau , \sigma _ 1 \sigma _ 2 ) = \phi ( \tau , \sigma _ 1 ) \phi ( \tau , \sigma _ 2 ) = \phi ( \sigma _ 1 , \tau ) \phi ( \sigma _ 2 , \tau ) . \end{align*}


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


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


\begin{align*} Z ^ i ( v ^ \mathrm { a p p } ( s ) , s ) - Z ^ i ( v ( s ) , s ) = \mathcal O ( s ^ { \min ( \frac 3 2 , \frac { i + 1 } 2 ) } ) . \end{align*}


\begin{align*} V \cdot \psi _ 1 = \psi _ 2 . \end{align*}


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


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


\begin{align*} U \cdot \nabla f = 0 , \end{align*}


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


\begin{align*} \frac { 1 } { r } \left ( \Phi _ t + u \cdot \nabla \Phi - \mu \left ( \Delta - \frac { 2 } { r } \partial _ r \right ) \Phi \right ) = 0 \textrm { o n } \ L ^ { 2 } ( \mathbb { R } ^ { 3 } ) . \end{align*}


\begin{align*} \begin{cases} - \delta ^ 2 ( K ( \hat { x } ) \nabla v ) = ( V _ { \delta , \hat { x } , \hat { q } , z } - \hat { q } ) ^ { p } _ + , \ \ & \ \Omega , \\ v = 0 , \ \ & \ \partial \Omega . \end{cases} \end{align*}


\begin{align*} \lim _ { n \to \infty } \int _ { D } ( \phi _ n - \phi _ \infty ) _ + G _ n \frac { 1 } { r } \dd z \dd r = \int _ { D } ( \phi - \phi _ \infty ) _ + G \frac { 1 } { r } \dd z \dd r . \end{align*}


\begin{align*} h \cdot ( g \cdot \gamma ) & = \theta _ n \circ ( h \times g \cdot \gamma ) \circ i , \\ ( h g ) \cdot \gamma & = \theta _ n \circ ( h g \times \gamma ) \circ i . \end{align*}


\begin{align*} 0 = ( \nabla _ T C _ S ( X _ i ) - C _ S C _ T ( X _ i ) - C _ { \nabla _ T S } ( X _ i ) ) - ( \nabla _ S C _ T ( X _ i ) - C _ T C _ S ( X _ i ) - C _ { \nabla _ S T } ( X _ i ) ) = d \lambda _ i ( T , S ) X _ i . \end{align*}


\begin{align*} \| u \| _ { \mathcal { C } ( \R ^ n _ + ) } : = \sup _ { Q \subset \R ^ { n - 1 } } \bigg ( \frac { 1 } { | Q | } \iint _ { T _ Q } | \nabla u ( x ) | ^ 2 \ , t \ , d x ' d t \bigg ) ^ { \frac 1 2 } , \end{align*}


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


\begin{align*} \gamma - m _ { 0 } < \gamma - c ( R ) = c _ { r } ( R ) < \frac { M _ { 0 } m _ { 0 } } { \sigma _ { d } } R . \end{align*}


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


\begin{align*} \begin{cases} \mathcal { L } _ H u = - ( K _ H ( x ) \nabla u ) = \frac { 1 } { \varepsilon ^ 2 } \left ( u - q \ln \frac { 1 } { \varepsilon } \right ) ^ { p } _ + , \ \ & x \in \Omega , \\ u = 0 , \ \ & x \in \partial \Omega , \end{cases} \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*} j ' ( t ) = & - \frac { 1 } { t ^ 2 \varepsilon ^ 2 } \int _ { \Omega } \left ( p ( t u - q \ln \frac { 1 } { \varepsilon } \right ) _ + ^ { p - 1 } t u ^ 2 - \left ( t u - q \ln \frac { 1 } { \varepsilon } \right ) _ + ^ { p } u ) d x \\ = & \frac { 1 } { t ^ 2 \varepsilon ^ 2 } \int _ { \Omega } \left ( t u - q \ln \frac { 1 } { \varepsilon } \right ) _ + ^ { p - 1 } u [ ( 1 - p ) ( t u - q \ln \frac { 1 } { \varepsilon } ) - p q \ln \frac { 1 } { \varepsilon } ] d x \\ \leq & 0 . \end{align*}


\begin{align*} X _ t \leq \int _ 0 ^ t \alpha _ s \sup _ { r \in [ 0 , s ] } X _ r \ , d s + M _ s + H _ s , \end{align*}


\begin{align*} f _ { i : n } ( x ) = \frac { f ( x ) } { \operatorname { B } ( i , n - i + 1 ) } \sum _ { l = 0 } ^ { n - i } \binom { n - i } { l } ( - 1 ) ^ l F ( x ) ^ { i + l - 1 } . \end{align*}


\begin{align*} \kappa ( t ) = ( \sqrt { t } ) ^ { \alpha - 1 } J _ { \alpha - 1 } ( 2 \sqrt { t } ) , \ k ( t ) = ( \sqrt { t } ) ^ { - \alpha } I _ { - \alpha } ( 2 \sqrt { t } ) , \ 0 < \alpha < 1 , \end{align*}


\begin{align*} b ( 0 , x ) \equiv b _ 0 , \xi ( 0 , x ) = \xi _ 0 , S ( 0 , x ) = x \cdot \xi _ 0 . \end{align*}


\begin{align*} | \nabla r | = 1 \end{align*}


\begin{align*} A \cdot B = \sum _ { k = 1 } ^ 3 \frac { 1 } { | V _ k | ^ 2 } ( A \cdot V _ k ) ( B \cdot V _ k ) + \sum _ { k = 1 } ^ 6 \frac { 1 } { | W _ k | ^ 2 } ( A \cdot W _ k ) ( B \cdot W _ k ) . \end{align*}


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


\begin{align*} X _ \ell = \sum _ { q \ge 0 } X _ \ell [ q ] , X _ \ell [ q ] : = \frac { b _ q } { q ! } \int _ { \mathbb S ^ d } H _ q ( T _ \ell ( x ) ) \ , d x , \end{align*}


\begin{align*} [ a _ { ( m ) } , b _ { ( n ) } ] = \sum _ { j \in \mathbb { Z _ { + } } } \left ( \begin{array} { c c c } m \\ j \end{array} \right ) ( a _ { ( j ) } b ) _ { ( m + n - j ) } . \end{align*}


\begin{align*} \widetilde { \mu } ( n ) \ r _ n ( m ) = H _ { m } \ \widetilde { r } _ { n } ( m + 1 ) + I _ m \ \widetilde { r } _ { n } ( m ) + J _ m \ \widetilde { r } _ { n } ( m - 1 ) \ , , \end{align*}


\begin{align*} | a _ { C } ( n ) | \leq \left ( \sum _ { i = 1 } ^ s | c _ { i } | \right ) \tau ( n ) \sqrt { n } \end{align*}


\begin{align*} i _ { \chi \theta } ( f ) ( I _ \bullet ) = \sum _ { J _ \bullet \in \partial _ { \chi \theta } ^ { - 1 } ( I _ \bullet ) } f ( J _ \bullet ) \end{align*}


\begin{align*} d ^ * ( \lceil x \rceil ) & = q ( c ( \lceil x \rceil ) ) \\ & = q ( c _ a c _ a c _ b c _ b c _ b c _ b c _ b c _ b c _ b c _ b ) \\ & = a ^ 2 b ^ 8 \end{align*}


\begin{align*} \mathcal { W } = \int _ { \kappa _ { \rm b } } ^ { \kappa _ { \rm t } } x \mu _ { \rm a } ( x ) f ( x ) \ , \mathrm { d } x . \end{align*}


\begin{align*} \Phi _ t ( x ' ) = t ^ { 1 - n } ( \partial _ n K ^ L ) ( x ' / t , 1 ) = t \ , ( \partial _ n K ^ L ) ( x ' , t ) = t \partial _ t K ^ L ( x ' , t ) = t \partial _ t P _ t ^ L ( x ' ) , \end{align*}


\begin{align*} \begin{aligned} \Big | \langle \boldsymbol { \nu } , A _ t \rangle - \bar A _ t + \langle \boldsymbol { \nu } , f _ { \alpha , x _ \alpha } \rangle - \bar f _ { \alpha , x _ \alpha } ) \Big | & \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*} \lambda _ 1 = q \sqrt { q } \sigma _ 1 ( \textbf { z } ) , \lambda _ 2 = q ^ 2 \sigma _ 2 ( \textbf { z } ) , \lambda _ 3 = q \sqrt { q } \sigma _ 3 ( \textbf { z } ) . \end{align*}


\begin{align*} G _ { \hat { x } } ( x , y ) = \frac { 1 } { 2 \pi } \ln \frac { 1 } { | x - y | } - h _ { \hat { x } } ( x , y ) , \ \ \forall x , y \in T _ { \hat { x } } ( \Omega ) . \end{align*}


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


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


\begin{align*} \left \{ \begin{array} { l r } \frac { \partial X ( x , t ) } { \partial t } = \frac { 1 } { f ( v ) } \sigma _ { k } ( x , t ) \varphi ( X \cdot v ) ( X \cdot v ) G ( X ) v - X ; \\ X ( x , 0 ) = X _ { 0 } ( x ) , \end{array} \right . \end{align*}


\begin{align*} { { \bf { F } } _ { { \mathop { \rm i n t } } } } = - G \psi \left ( { \bf { x } } \right ) \sum \limits _ { i = 1 } ^ { 1 8 } { \varpi \left ( { { { \left | { { { \bf { c } } _ i } } \right | } ^ 2 } } \right ) } \psi \left ( { { \bf { x } } + { { \bf { c } } _ i } \Delta t } \right ) { { \bf { c } } _ i } , \end{align*}


\begin{align*} L ( x ) = x ^ { 2 ^ r } + x ^ 2 + t x \end{align*}


\begin{align*} r ( k ) : = \frac { b ( k ) } { a ( k ) } , k \in \mathbb { R } \cup i \mathbb { R } , \end{align*}


\begin{align*} \Theta & = d \theta ^ 0 + \theta ^ 1 \wedge \theta ^ 0 , \\ * \Omega & = d \theta ^ 1 + \theta ^ 1 \wedge \theta ^ 1 , \end{align*}


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


\begin{align*} T ( x , y , w , z ) = ( x + \alpha , y + x , w + \alpha , z + w ) \end{align*}


\begin{align*} \hat { q } _ i = q ( z _ i ) + \frac { \hat { q } _ i } { \ln \frac { R } { \varepsilon } } g _ { z _ i } ( T _ { z _ i } z _ i , T _ { z _ i } z _ i ) - \Sigma _ { j \neq i } \frac { \hat { q } _ j } { \ln \frac { R } { \varepsilon } } \bar { G } _ { z _ j } ( T _ { z _ j } z _ i , T _ { z _ j } z _ j ) , \end{align*}


\begin{align*} \P ( X \geq \delta _ i ) & = \P ( Y \geq \delta _ i ' ) = \P ( Y \geq 2 ^ { - j } ) = \sum _ { i \in [ j ] } \P \left \{ Y = 2 ^ { - i } \right \} = \sum _ { i \in [ j ] } r 2 ^ i = 2 r \left ( 2 ^ { j } - 1 \right ) \leq 2 r 2 ^ j . \end{align*}


\begin{align*} \psi ^ \dagger _ p ( 0 , p _ 0 ) = 0 . \end{align*}


\begin{align*} \langle B ^ { \circ } x ^ { \circ } , u \rangle = \langle x ^ { \circ } , B u \rangle = \lim _ { n \to \infty } \langle x ^ { \circ } , B u _ { n } \rangle , \end{align*}


\begin{align*} \begin{aligned} p _ n ( \varsigma ) = 1 + \varsigma + \cdots + \varsigma ^ n \ , , \ \varsigma \in \R \end{aligned} \end{align*}


\begin{align*} \frac { \alpha } { 2 \beta } \min \{ \kappa - r , \lambda \} \leq \frac { \alpha } { 2 \beta } \min \{ \kappa - r , r \} \leq \frac { 1 } { 2 } \frac { \alpha \kappa } { 2 \beta } = \frac { 1 } { 2 } \min \{ \alpha + \gamma - \frac { 1 } { 2 } , \alpha \} \leq \frac { \alpha } { 2 } \leq \min \{ \alpha - \varepsilon , \frac { 1 } { 2 } \} . \end{align*}


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


\begin{align*} d v ' = \left \{ ( \log \sqrt { \tau _ 0 } ) _ z \ d z + \frac { i } { \sqrt { \tau _ 0 } } \left ( Q _ 0 d z + \frac { \tau _ 0 } { 4 } d \overline { z } \right ) J \right \} v ' . \end{align*}


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


\begin{align*} \left \{ \begin{aligned} \mathcal P _ X ^ D ( \alpha _ 1 , \dots , \alpha _ r ) _ x & = \mathcal P _ { X , x } & \\ \mathcal P _ X ^ D ( \alpha _ 1 , \dots , \alpha _ r ) _ x & = \alpha _ i & \end{aligned} \right . \end{align*}


\begin{align*} \int _ { \mathbb { R } } s V s ^ { H } d z = 0 \Longrightarrow \int _ { \mathbb { R } } { \rm R e } ( s V s ^ { H } ) \mathrm { d } z = 0 , \end{align*}


\begin{align*} \int _ { \mathfrak U } | \omega | = \pi ^ * _ s \int _ { \mathfrak X } | \pi _ { \eta } ^ * \omega | \end{align*}


\begin{align*} \sup \limits _ { X } e ^ { - u ( - v ( \Psi ) ) } \le \sup \limits _ { t \in [ T , t _ 0 + B ] } e ^ { - u ( t ) } = \frac { 1 } { \delta } c ( T ) e ^ { - T } + \int ^ { t _ 0 + B } _ T c ( t _ 1 ) e ^ { - t _ 1 } d t _ 1 . \end{align*}


\begin{align*} \delta _ \rho G ( y , z ) = - \delta _ \rho h ( y , z ) = \int _ { \partial \Omega } \frac { \partial G ( y , x ) } { \partial \mathbf { n } _ x } \frac { \partial G ( z , x ) } { \partial \mathbf { n } _ x } \rho ( x ) d \sigma _ x , \end{align*}


\begin{align*} \phi _ \mu ( g ) = \frac { J _ \lambda ^ { ( 2 ) } ( a _ 1 , \ldots , a _ N ) } { J _ \lambda ^ { ( 2 ) } ( 1 , \ldots , 1 ) } , \mu = 2 \lambda \end{align*}


\begin{align*} & X _ { i ; j k } - X _ { i ; k j } + X _ { j ; k i } - X _ { j ; i k } - X _ { k ; i j } + X _ { k ; j i } \\ & = ( R _ { k j \ell i } + R _ { i k \ell j } - R _ { j i \ell k } ) X ^ \ell \\ & = 2 R _ { i j \ell k } X ^ \ell = - 2 R _ { \ell k j i } X ^ \ell \end{align*}


\begin{align*} E : y ^ { 2 } = x ( x - 4 ^ { m } p ) ( x + p ) \end{align*}


\begin{align*} \int _ { \R ^ { n - 1 } } \sum _ { 1 \le \beta \le N } ( P ^ L _ { \alpha \beta } ) _ t ( x ' - y ' ) C _ { \beta } \ , d y ' = C _ { \alpha } , \quad \forall ( x ' , t ) \in \R ^ n _ + . \end{align*}


\begin{align*} \left \| \sum \limits _ { j = 1 } ^ d A _ { j } ^ * A _ { j } \right \| = \sum \limits _ { j = 1 } ^ d \left \| X _ { j } ^ * X _ { j } \right \| . \end{align*}


\begin{align*} \operatorname { r a n k } E ( M _ { k , \mathrm { i n } } ( N _ { ( k , - ) } ) ) + \operatorname { r a n k } { F ( ( M _ { k , \mathrm { o u t } } ' ) ^ { - 1 } ( N _ { ( - , k ) } ) ) } = \operatorname { r a n k } N _ { ( k , - ) } = \sum _ { j \in \Omega ( k , - ) } b _ { k j } e _ j . \end{align*}


\begin{align*} \dot { x } ( t ) & = - A x ( t ) + B u ( t ) \quad ( t > 0 ) , \\ x ( 0 ) & = x _ 0 \in X , \end{align*}


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


\begin{align*} \mathcal { V } ( \mu , \nu ; \alpha ^ { * , \xi } ) = \underset { \alpha \in \mathcal { U } _ { a d } ( 0 , T ) } { \inf } \mathcal { J ( } \mu , \nu ; \alpha ) . \end{align*}


\begin{align*} t ^ n | K | \big ( \sum _ { m = k } ^ n & { { n } \choose { m } } V ( K [ m ] , ( U + V ) [ n - m ] ) t ^ m \big ) \\ & \le c _ n \big ( \sum _ { m = i } ^ n { { n } \choose { m } } V ( K [ m ] , U [ n - m ] ) t ^ m \big ) \big ( \sum _ { m = j } ^ n { { n } \choose { m } } V ( K [ m ] , V [ n - m ] ) t ^ m \big ) . \end{align*}


\begin{align*} \mathcal C ^ { h } : = \mathcal C _ { \theta _ 0 } \cap B _ { h } , \Gamma _ h = \partial \mathcal C \cap B _ h , \Lambda _ h = \mathcal C \cap \partial B _ h , \end{align*}


\begin{align*} F _ 1 = \frac { 1 } { \sqrt { | c _ 1 | } } \ \langle \langle \nu _ 1 \cdot \varphi , \varphi \rangle \rangle . \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*} \xi ( F ) = u _ \S ( F ) + 2 . \end{align*}


\begin{align*} ( N _ j ( \varphi ) - P ) & = S ( I - T _ 1 + P ) ( N _ j ( \varphi ) - P ) \\ & = S ( I - C _ \varphi + P ) ( M _ j ( \varphi ) - P ) \\ & = \dfrac { 1 } { j } S ( C _ \varphi - C _ { \varphi _ { j + 1 } } ) \rightarrow 0 , \end{align*}


\begin{align*} \mathcal { D } _ { G , \kappa } ^ { \mathrm { c h } } \simeq \bigoplus _ { \lambda \in P ^ + } \mathbb { V } ^ { \kappa } _ \lambda \otimes \mathbb { V } ^ { \kappa ^ * } _ { \lambda ^ * } , \frac { 1 } { \kappa + h ^ \vee } + \frac { 1 } { \kappa ^ * + h ^ \vee } = 0 . \end{align*}


\begin{align*} & ( \nexists \ , x \in \mathbb { R } ^ n ) \ \ s . t . \\ & f ( x ) < 0 \\ & g ( x ) \le 0 \end{align*}


\begin{align*} \varphi ( - t ) ^ 2 H ( - t ) + \varphi ( t ) ^ 2 H ( t ) = 2 \varphi ( t ^ 4 ) ^ 2 G ( t ^ 4 ) . \end{align*}


\begin{align*} u _ n = P _ 1 ( n ) \alpha _ 1 ^ n + \cdots + P _ s ( n ) \alpha _ s ^ n \end{align*}


\begin{align*} \Delta _ { \Psi } d = - \frac { d } { d r } \log ( \theta _ { \Psi } ) = H _ { \Psi } . \end{align*}


\begin{align*} \beta _ x = \min \left \{ \frac { 1 } { 2 L _ y } , \frac { 1 } { 2 \eta _ x L _ { x y } ^ 2 } \right \} . \end{align*}


\begin{align*} \oint s _ { 1 } ( z ) s _ 2 ( z ) d z = 0 . \end{align*}


\begin{align*} X ^ v \circ X ^ w = q ^ { - \langle v , w \rangle } X ^ { v + w } \ , v , w \in \mathbb { N } ^ n . \end{align*}


\begin{align*} \int e ^ { W ( y | x ) } d \nu _ { A } ( y ) = \varphi _ A ( x ) . \end{align*}


\begin{align*} & X _ { i ; j k } - X _ { i ; k j } + X _ { j ; k i } - X _ { j ; i k } - X _ { k ; i j } + X _ { k ; j i } \\ & = ( R _ { k j \ell i } + R _ { i k \ell j } - R _ { j i \ell k } ) X ^ \ell \\ & = 2 R _ { i j \ell k } X ^ \ell = - 2 R _ { \ell k j i } X ^ \ell \end{align*}


\begin{align*} \begin{aligned} { \Gamma } _ m = \{ \mathcal { H } _ 1 ( A ) > 0 \} \cap \cdots \{ \mathcal { H } _ m ( A ) > 0 \} . \end{aligned} \end{align*}


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


\begin{align*} \int _ { 0 } ^ { \infty } \omega ( \mu ) \big [ \xi ( \mu , t ) - \xi ( \mu , 0 ) \big ] \ d \mu = \int _ { 0 } ^ { t } \int _ { 0 } ^ { \infty } \int _ { 0 } ^ { \mu } \varpi _ 1 ( \mu , \nu ) \xi ( \mu , s ) \xi ( \nu , s ) \Lambda ( \mu , \nu ) \ d \nu d \mu d s , \end{align*}


\begin{align*} \gcd ( d _ 1 m _ E , d _ 2 ) = 1 \ ; \Longrightarrow \ ; f ( d _ 1 d _ 2 ) = f ( d _ 1 ) f ( d _ 2 ) , \end{align*}


\begin{align*} \| \mu _ 1 - \mu _ 2 \| _ { L , M } ^ * : = \sup _ { f \in C _ b ( M ) , \ , | f | _ L \leqslant 1 } \Big | \langle f , \mu _ 1 \rangle - \langle f , \mu _ 2 \rangle \Big | \le 2 , \end{align*}


\begin{align*} I _ j = W _ j \setminus \bigcup _ { e \in G [ W _ j ] } e . \end{align*}


\begin{align*} \frac { \partial s ( z ) } { \partial z } = - \Phi ( z ) \ , [ 1 - \Phi ( z ) ] - z \ , \phi ( z ) \ , \left [ \alpha - ( \alpha + \beta ) \ , \Phi ( z ) \right ] + ( 2 - \alpha - \beta ) \ , \phi ^ 2 ( z ) . \end{align*}


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


\begin{align*} \begin{array} { l } { \rm S I N R } _ { k , j } ^ m = 2 y _ { k , j } ^ m \sqrt { { p _ { k , j } ^ m \left | G _ { k , j } ^ m \right | ^ 2 } } - y _ { k , j } ^ m \left ( { { \sum \limits _ { j ' \ne j } \sum \limits _ { k ' \ne { { k } } } { p _ { k ' , j ' } ^ { { m } } { { { \left | G _ { k ' , j } ^ m \right | ^ 2 } } } } } + N _ { \rm R F } B _ { \rm R F } } \right ) , \end{array} \end{align*}


\begin{align*} \beta _ { \bar { \mathbf { g } } } \mathbf { g } = \beta _ { \bar g } g + \bar u ^ { - 2 } u d u - \bar u ^ { - 1 } g ( \nabla _ { \bar g } \bar u , \cdot ) . \end{align*}


\begin{align*} \rho c _ v \left [ g _ i + \Delta t D _ i g _ i + \frac { \Delta t ^ 2 } { 2 } D _ i ^ 2 g _ i \right ] - g _ i + ( 1 - \rho c _ v ) \left [ g _ i + \Delta t d _ i g _ i + \frac { \Delta t ^ 2 } { 2 } d _ i ^ 2 g _ i \right ] = - { \left ( { { { \bf { M } } ^ { - 1 } } \Lambda { \bf { M } } } \right ) _ { i j } } \left [ { g _ i - g _ i ^ { ( e q ) } } \right ] + \Delta t { \bar F _ i } + \vartheta \Delta t { S _ i } , \end{align*}


\begin{align*} \kappa ( s ) = \alpha \ , \frac { \nu _ { n + 1 } ( s ) } { x _ { n + 1 } ( s ) } + \varpi . \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ { N } } u _ 0 ( \varphi ( 0 , x ) - k \Delta \varphi ( 0 , x ) ) d x = \nu ( 0 ) \int _ { \mathbb { R } ^ { N } } u _ 0 ( \phi ( x ) - k \Delta \phi ( x ) ) d x = 0 . \end{align*}


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


\begin{align*} I _ \varepsilon ( u _ \varepsilon ) = c _ \varepsilon , \ \ I ' _ \varepsilon ( u _ \varepsilon ) = 0 . \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*} M ( x ; z ) = I + \mathcal { C } \left ( M _ { - } ( x ; \cdot ) R ( x ; \cdot ) \right ) , z \in \mathbb { C } ^ { \pm } . \end{align*}


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


\begin{align*} \omega ( \vec { x } ) = \frac { c } { \sqrt { 1 + \norm { \vec { a } } ^ 2 } + \vec { a } \cdot \vec { x } } \end{align*}


\begin{align*} \begin{aligned} \Big [ | \langle \boldsymbol { \nu } , A \rangle - \bar A | + | \langle \boldsymbol { \nu } , f _ \alpha \rangle - \bar f _ \alpha | \Big ] & \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*} c _ \varepsilon = I _ \varepsilon ( u _ \varepsilon ) = \pi \min \limits _ { \overline { \Omega } } q ^ 2 \sqrt { d e t ( K _ H ) } \ln \frac { 1 } { \varepsilon } + O ( 1 ) . \end{align*}


\begin{align*} [ \Delta ^ M _ \Psi , \Delta ^ N _ \Gamma ] = 0 . \end{align*}


\begin{align*} \sum _ { \substack { \tau \in D ( \ell ) \\ \mathrm { s i g n } ( \tau ) = 1 } } \prod _ { j = 1 } ^ { \ell } \frac { 1 } { x _ { \tau ( j ) } - x _ j } = \sum _ { \substack { X = \bigsqcup _ { i = 1 } ^ { s } X _ i \\ | X _ i | \geqslant 2 \\ s \ } } \left ( \prod _ { j = 1 } ^ s f ( X _ i ) \right ) . \end{align*}


\begin{align*} \lambda _ k = \frac 1 4 \hat W _ r ( q ) \Big [ \hat W _ r ( q + k ) + \hat W _ r ( q - k ) - 2 \hat W _ r ( q ) \Big ] \end{align*}


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


\begin{align*} \begin{aligned} { \bf { d } } \left [ { i - { \kappa _ l } } \right ] & = { { { \bf { \bar x } } } _ { \rm { t } } } \Big [ { \left \lfloor { \frac { { i - { \kappa _ l } + { N _ { \rm { C P } } } } } { { K + { N _ { \rm { C P } } } } } } \right \rfloor , } \Big . \\ & \ \ \ \ \Big . { { \rm { m o d } } \left ( { i - { \kappa _ l } + { N _ { \rm { C P } } } , K + { N _ { \rm { C P } } } } \right ) - { N _ { \rm { C P } } } } \Big ] . \end{aligned} \end{align*}


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


\begin{align*} D = \max \left \{ D _ { x , 1 } \sqrt { \delta ' } + D _ { x , 2 } , D _ { y , 1 } \sqrt { \delta ' } + D _ { y , 2 } \right \} , \end{align*}


\begin{align*} y \left [ i \right ] = \left ( { \sum \limits _ { l = 1 } ^ L { { \bf { h } } _ l ^ H { \bf { H } } _ l ^ \bot { \bf { X } } _ l } } \right ) { \bf { d } } \left [ { i - { n _ { \max } } } \right ] + z \left [ i \right ] . \end{align*}


\begin{align*} & \lim _ { n \to + \infty } \sum _ { i = 0 } ^ { n - 1 } \int _ { \{ z \in M : - \Psi ( z ) \in I _ { i , n } \backslash U _ k \} } | \tilde F | ^ 2 _ h \\ \le & \frac { G ( T _ 1 ) } { \int _ { T _ 1 } ^ { + \infty } c ( s ) e ^ { - s } d s } \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 . \end{align*}


\begin{align*} \varepsilon ( z ) \Lambda ( z ^ { - 1 } ) = - \Lambda ( - z ^ { - 1 } ) , \dot { s } \Lambda ( z ) = - \Lambda ( - z ^ { - 1 } ) , \quad \\ \dot { s } \varepsilon ( x ) \Lambda ( y ) = \varepsilon ( - x ^ { - 1 } ) \Lambda ( x ( x y - 1 ) ) + \Lambda ( x ) - \Lambda ( y ^ { - 1 } ( x y - 1 ) ) , \end{align*}


\begin{align*} \begin{aligned} { { \bf { x } } _ { \rm t } } \left [ { m , n } \right ] & = \frac { 1 } { { \sqrt K } } \sum \limits _ { k = 0 } ^ { K - 1 } { { { \bf { x } } _ { \rm f } } \left [ { m , k } \right ] { e ^ { j \frac { { 2 \pi } } { K } k n } } } \\ & = \frac { 1 } { { \sqrt K } } \sum \limits _ { k = 0 } ^ { K - 1 } { { { \bf { u } } _ k } s \left [ { m , k } \right ] { e ^ { j \frac { { 2 \pi } } { K } k n } } } , \ n = 0 , \cdots , K - 1 . \end{aligned} \end{align*}


\begin{align*} \varphi '' + \Big ( ( m - 1 ) \frac { \sigma ' } { \sigma } - \Phi ' \Big ) \varphi ' - f ' ( u ) \varphi \leq \varphi '' + \Big ( ( m - 1 ) \frac { \sigma ' } { \sigma } - \Phi ' \Big ) \varphi ' + B = 0 . \end{align*}


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


\begin{align*} M _ x ^ 2 & = \frac { \omega } { 3 L _ { x y } } \left ( \Psi ^ 0 + \frac { 4 } { ( 1 - \theta ) ^ 2 } ( \delta _ x + \delta _ y ) + \frac { \Sigma ^ 2 } { 2 ( 1 - \theta ) } \right ) , \\ M _ y ^ 2 & = \frac { 1 } { 4 L _ { x y } \omega } \left ( \Psi ^ 0 + \frac { 4 } { ( 1 - \theta ) ^ 2 } ( \delta _ x + \delta _ y ) + \frac { \Sigma ^ 2 } { 2 ( 1 - \theta ) } \right ) , \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*} a \cos ( x ) + b \sin ( x ) = r \cos ( x + \theta ) . \end{align*}


\begin{align*} \begin{aligned} \| ( \gamma - c ) \varphi \| _ { W ^ { \frac { 1 } { 3 } , \frac { 3 } { 2 } } ( \partial \Omega ) } + & \| c \nabla \varphi \cdot \nu \| _ { W ^ { \frac { 1 } { 3 } , \frac { 3 } { 2 } } ( \partial \Omega ) } \\ & \le C ( \gamma \| \varphi \| _ { W ^ { 2 , \frac { 3 } { 2 } } ( \Omega ) } + \| \nabla c \| _ { L ^ { 2 } ( \Omega ) } \| \varphi \| _ { W ^ { 1 , 6 } ( \Omega ) } ) . \end{aligned} \end{align*}


\begin{align*} U = U [ C , \beta , \Delta , p ] = \stackrel { d e f } { = } \sup _ { p > \Delta } \ \sup _ { \xi \in L _ p } \ \sup _ { X \in L _ p } Y [ C , \beta , \Delta ] ( \xi , X , p ) , \end{align*}


\begin{align*} \int _ { \partial \Omega } \operatorname { T r } \textbf { \textit { K } } ( t , x ^ { \prime } , x ^ { \prime } ) \ , d S = \sum _ { k = 1 } ^ { \infty } e ^ { - t \tau _ k } \end{align*}


\begin{align*} h = 0 , \nabla h = 0 \mbox { o n } \hat \Sigma \end{align*}


\begin{align*} \gamma ^ i ( x ) = \frac 1 N \sum _ { j = 1 } ^ N K ( x ^ i - x ^ j ) \end{align*}


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


\begin{align*} h = 0 , \nabla h = 0 , \nabla ^ { k + 2 } h = 0 \mbox { o n } \hat \Sigma . \end{align*}


\begin{align*} I _ n : = \left [ n \dfrac { \bar { t } } { \gamma } , \ , ( n + 1 ) \dfrac { \bar { t } } { \gamma } \right ) n \in { \mathbb { N } } \end{align*}


\begin{align*} d = k ^ 2 + 2 n k = 4 m k + 2 k - k ^ 2 \equiv 2 k - k ^ 2 \mod 4 k \end{align*}


\begin{align*} \left \{ \begin{aligned} \mathcal ( \delta f ) ^ { - 1 } \left ( j ^ 2 _ { f ( x ) } \phi \right ) & = j ^ 2 _ x ( \phi \circ f ) & \\ \mathcal ( \delta f ) ^ { - 1 } \left ( j ^ 2 _ { f ( x ) } \phi \right ) & = j ^ { 2 ( n _ i + 1 ) } _ x ( \phi \circ f ) & \end{aligned} \right . \end{align*}


\begin{align*} \sup \limits _ { X } e ^ { - u ( - v ( \Psi ) ) } \le \sup \limits _ { t \in [ T , t _ 0 + B ] } e ^ { - u ( t ) } = \frac { 1 } { \delta } c ( T ) e ^ { - T } + \int ^ { t _ 0 + B } _ T c ( t _ 1 ) e ^ { - t _ 1 } d t _ 1 . \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*} \alpha _ m = \max \{ t \ , | \ , Q - t J - Y \succeq 0 , Y \in \R ^ { Z _ m \times Z _ m } _ { \geq 0 } \} . \end{align*}


\begin{align*} \langle x | A _ j x \rangle & = \langle U x | U A _ j U ^ * ( U x ) \rangle \\ & = \left \langle \left ( \sqrt { s _ 1 } y _ 1 , \dots , \sqrt { s _ m } y _ m , 0 , \dots , 0 \right ) | \left ( \sqrt { s _ 1 } A _ { j , 1 } y _ 1 , \dots , \sqrt { s _ m } A _ { j , m } y _ m , 0 , \dots , 0 \right ) \right \rangle \\ & = \sum \limits _ { k = 1 } ^ m s _ k \langle y _ k | A _ { j , k } y _ k \rangle . \end{align*}


\begin{align*} \Gamma _ { ( g , u ) } ( X ) = - \Delta _ g X - u ^ { - 1 } \nabla X ( \nabla _ g u , \cdot ) + u ^ { - 2 } X ( u ) d u . \end{align*}


\begin{align*} \widehat { \iota } ( D ) = \frac { \beta m + \gamma } { m + b _ 1 + 1 } . \end{align*}


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


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


\begin{align*} A = \mathcal { N } ( A ) - \mathcal { R } ( A ) . \end{align*}


\begin{align*} h _ { n } = \rho _ p ^ { n - j _ p } h _ { j _ p } + \frac { \rho _ p ( \rho _ p ^ { n - j _ p } - 1 ) } { 2 } . \end{align*}


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


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


\begin{align*} \eta ( U | \bar U ; x , t ) \ge \begin{cases} c _ 1 ' \big | U - \bar U \big | ^ 2 & \ ; | U | \le r _ 2 , \\ c _ 2 ' \ , \big | U - \bar U \big | ^ p & \ ; | U | \ge r _ 2 , \end{cases} \end{align*}


\begin{align*} I _ 1 = y \cdot f ( y - 1 ) + ( y - 1 ) \left ( f ( ( a + 1 ) y + b ) - f ( a y + b ) \right ) \end{align*}


\begin{align*} A _ { 1 } = \sum _ { X \in S ( 0 ) } \Theta _ { \theta } \begin{bmatrix} m _ { 1 } & X \\ 0 & m _ { 2 } \end{bmatrix} \overline { \psi _ A ( { m _ { 1 } } ^ { - 1 } X } ) = q ^ { 8 } ( q - 1 ) ^ { 3 } . \end{align*}


\begin{align*} E = \big \{ ( u , v ) \in V ^ 2 : \ ; \pi ( u , v ) > 0 \big \} . \end{align*}


\begin{align*} h _ k \dot { \alpha } ( t _ { k - 1 } ) \le \frac { r } { c } ( k = k _ 0 + 1 , k _ 0 + 2 , \ldots ) . \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ { 3 } } ( a _ { t } ^ { \varepsilon } + ( B \times u ) ^ { \varepsilon } + \mu \nabla \times b ^ { \varepsilon } ) \cdot \nabla \times \zeta \dd x = 0 . \end{align*}


\begin{align*} \begin{array} { r c l l } \xi ^ { \mathrm { i n } } & = & 0 & a . e . ~ \mbox { i n } ( R _ { 0 } , \infty ) , \\ \mathcal { I } _ { 1 } & = & 0 & a . e . ~ \mbox { i n } ( 0 , t _ { 0 } ) \times ( R _ { 0 } , \infty ) , \\ \mathcal { I } _ { 2 } & = & 0 & a . e . ~ \mbox { i n } ~ ( 0 , t _ { 0 } ) \times ( R _ { 0 } , \infty ) . \end{array} \end{align*}


\begin{align*} \Delta _ { h } = h ^ { - 1 } \Delta \ , h ^ { - 1 } , \end{align*}


\begin{align*} & \sum _ { A = 0 } ^ { 2 n - 1 } \left [ d _ 1 \left ( u - v \right ) \wedge ( \triangle v ) ^ { p - 1 } \wedge ( \triangle u ) ^ { m - p } \wedge \beta _ n ^ { n - m } \right ] _ A \cdot \nabla _ { A 0 ' } \varrho ( q ) \ , \Omega _ { 2 n } \\ = & d _ 0 \varrho ( q ) \wedge d _ 1 \left ( u - v \right ) \wedge ( \triangle v ) ^ { p - 1 } \wedge ( \triangle u ) ^ { m - p } \wedge \beta _ n ^ { n - m } , \end{align*}


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


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


\begin{align*} X _ t ^ i = X _ 0 ^ i + \int _ 0 ^ t \bigg ( b ( X _ s ^ i ) + \frac 1 { N - 1 } \sum _ { j \neq i } K ( X _ s ^ i - X _ s ^ j ) \bigg ) \d s + \sigma W _ t ^ i , ~ ~ ~ ~ i = 1 , \cdots , N , \end{align*}


\begin{align*} B _ n & \rightharpoonup B \textrm { i n } \ L ^ { 2 } ( \Omega ) , \\ A _ n = \textrm { c u r l } ^ { - 1 } B _ n & \to A = \textrm { c u r l } ^ { - 1 } B \textrm { i n } \ L ^ { 2 } ( \Omega ) . \end{align*}


\begin{align*} H ( f ; \mu ) : = \int _ { \Omega _ N ^ d } f \log f d \mu = \sum _ { \eta \in \Omega _ N ^ d } f ( \eta ) \log f ( \eta ) \mu ( \eta ) . \end{align*}


\begin{align*} \prescript { } { l } { u } - \prescript { } { l } { v } _ 1 = Q _ 1 \prescript { } { l } { f } , \end{align*}


\begin{align*} \left \{ \begin{aligned} & \partial _ t c ( t , \nu ) + \partial _ { \nu } c ( t , \nu ) A \nu + \frac { 1 } { 2 } \partial _ { \nu \nu } c ( t , \nu ) \sigma _ 0 ^ 2 = 0 , \\ & c ( T , \nu ) = 0 . \end{aligned} \right . \end{align*}


\begin{align*} u ^ k = \arg \min _ { w \in H ^ 1 ( \Omega ) } \Big ( \frac { 1 } { 2 } \| w - u ^ { k - 1 } \| _ { L _ 2 ( \Omega ) } ^ 2 + \tau _ k N ( w ) \Big ) , \end{align*}


\begin{align*} \chi _ P ( P _ a , P _ b ) & = \left | \sum _ { i = 1 } ^ k k P _ a ( i ) P _ b ( i ) - 1 \right | \\ & = \left | \sum _ { i = 1 } ^ k k \frac { 1 } { k } \left ( 1 + \epsilon v _ { a , i } \right ) \frac { 1 } { k } \left ( 1 + \epsilon v _ { b , i } \right ) - 1 \right | \\ & = \left | \sum _ { i = 1 } ^ k \frac { 1 } { k } \left ( 1 + \epsilon v _ { a , i } + \epsilon v _ { b , i } + \epsilon ^ 2 v _ { a , i } v _ { b , i } \right ) - 1 \right | \\ & = 0 , \end{align*}


\begin{align*} \phi ( \sigma , \tau ) = \epsilon ^ { \sum _ { r = 1 } ^ j \lambda _ r ( d _ r - 1 ) } . \end{align*}


\begin{align*} \Delta _ { \Psi } d = - \frac { d } { d r } \log ( \theta _ { \Psi } ) = H _ { \Psi } . \end{align*}


\begin{align*} \{ u _ { \alpha \beta } ( 0 ) \} _ { 1 \leq \alpha , \beta \leq n - 1 } = \mathrm { d i a g } \{ b _ 1 , \ldots , b _ { n - 1 } \} . \end{align*}


\begin{align*} \epsilon = \sum _ { i \in I } \bar { \nu } _ i ^ ! & \ ; & ( \delta ^ n _ k ) ^ ! = \sum _ { J = \{ i _ 0 < \hdots < \widehat { i _ k } < \hdots < i _ n \} \subset K = \{ i _ 0 < \hdots < i _ n \} } ( \nu _ { K } ^ J ) ^ ! \end{align*}


\begin{align*} x _ { i , 1 } + x _ { i , 2 } + \cdots + x _ { i , n } & = 1 \textrm { \ , f o r a l l $ i $ \ ; \textrm ( \emph { r o w s u m s e q u a l o n e } ) } , \\ x _ { 1 , j } + x _ { 2 , j } + \cdots + x _ { n , j } & = 1 \textrm { \ , f o r a l l $ j $ \ ; ( \emph { c o l u m n s u m s e q u a l o n e } ) } . \end{align*}


\begin{align*} D _ j ( U ^ { i j } D _ i w ) = f \quad \ , \ , \ , \ , B _ 1 \end{align*}


\begin{align*} \left | v _ { i , j } \right | ^ { 2 } \prod _ { k = 1 ; k \neq i } ^ { n } \left ( \lambda _ { i } ( A ) - \lambda _ { k } ( A ) \right ) = \prod _ { k = 1 } ^ { n - 1 } \left ( \lambda _ { i } ( A ) - \lambda _ { k } \left ( M _ { j } \right ) \right ) . \end{align*}


\begin{align*} \gamma \times T ^ n = \gamma ( n \bmod \Z ) \ , T ^ n \end{align*}


\begin{align*} ( \sum \limits _ { i = 1 } ^ j w _ i ) \chi - \sum \limits _ { i = 1 } ^ { j - 1 } \chi _ i + r ( j - 1 ) < \chi _ j < ( \sum \limits _ { i = 1 } ^ j w _ i ) \chi - \sum \limits _ { i = 1 } ^ { j - 1 } \chi _ i + r j , \end{align*}


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


\begin{align*} a ( u _ n , v _ n ) = \lambda _ n ( u _ n , v _ n ) v _ n \in X _ n , \end{align*}


\begin{gather*} \Lambda ( z ; \alpha ) : = \alpha ( S - z ^ { - 1 } ) ^ { - 1 } , \Gamma ( z ; \alpha ) : = \beta ( z - S ) ^ { - 1 } , \end{gather*}


\begin{align*} G _ { \alpha \beta } ^ T H ^ { \beta } _ i \overline { G _ { \alpha \beta } } & = H ^ { \alpha } _ i \\ & = C ^ { \alpha } _ { i } H ^ { \alpha } _ { 1 } ( \overline { C ^ { \alpha } _ { i } } ^ { T } ) = C ^ { \alpha } _ { i } G _ { \alpha \beta } ^ T H ^ { \beta } _ 1 \overline { G _ { \alpha \beta } } ( \overline { C ^ { \alpha } _ { i } } ^ { T } ) . \end{align*}


\begin{align*} D _ { x , 2 } & = \frac { 1 } { 2 L _ { x y } } \left ( \sqrt { n \sigma _ { G , r } ^ 2 } + L _ { y } \sqrt { n } M _ y + \| \nabla G ( Y ^ * ) \| \right ) \\ & + \frac { 1 } { 2 \mu _ { x } } \left ( \sqrt { n \sigma _ { F , r } ^ 2 } + L _ { x } \sqrt { n } M _ x + \| \nabla F ( X ^ * ) \| \right ) , \end{align*}


\begin{align*} \max _ { v \in \Lambda } e ( \alpha ^ { ( v ) } , G ^ { ( v ) } ) = e ( 0 , G ) . \end{align*}


\begin{align*} \kappa \stackrel { d e f } { = } \lim _ { n \to \infty } \kappa _ n . \end{align*}


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


\begin{align*} \frac { \tau ^ 2 _ h ( \varphi v ) ( x ) } { | h | ^ \alpha } & = \frac { \tau _ h ( \varphi ( x + h ) \tau _ h v ( x ) + v ( x ) \tau _ h \varphi ( x ) ) } { | h | ^ \alpha } \\ & = \varphi ( x + 2 h ) \frac { \tau ^ 2 _ h v ( x ) } { | h | ^ \alpha } + 2 \frac { \tau _ h v ( x ) \tau _ h \varphi ( x + h ) } { | h | ^ \alpha } + v ( x ) \frac { \tau ^ 2 _ { h } \varphi ( x ) } { | h | ^ \alpha } \end{align*}


\begin{align*} \theta = \alpha _ 2 \omega _ 2 + \cdots + \alpha _ n \omega _ n . \end{align*}


\begin{gather*} T ( A ) = 1 \quad A \in \mathcal { R } A \supset \{ \pi _ 1 > \pi _ 2 \} . \end{gather*}


\begin{align*} & \psi ^ { \pm } ( x , t ; k ) = \sigma _ 2 \overline { \psi ^ { \pm } ( x , t ; \bar { k } ) } \sigma _ 2 , \ \ \ \psi ^ { \pm } ( x , t ; k ) = \sigma _ 3 \psi ^ { \pm } ( x , t ; - k ) \sigma _ 3 , \end{align*}


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


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


\begin{align*} \psi _ { K , H } ( t ) = ( n - 1 ) \sqrt K \cdot \frac { ( n - 1 ) \sqrt K \sinh \sqrt K t + H \cosh \sqrt K t } { ( n - 1 ) \sqrt K \cosh \sqrt K t + H \sinh \sqrt K t } . \end{align*}


\begin{align*} L = a _ 0 x + a _ 1 x ^ q + a _ 2 x ^ { q ^ 2 } + \cdots + a _ n x ^ { q ^ n } . \end{align*}


\begin{align*} \delta _ { j k } X _ { i j \ell } + \delta _ { i \ell } X _ { i j k } = \delta _ { i j } X _ { i k \ell } + \delta _ { k \ell } X _ { i j k } . \end{align*}


\begin{align*} \phi _ { i s } ( \partial _ s ) = \frac { \Gamma _ { s i } ^ i } { \varphi _ s } , \quad \forall s \neq i , \end{align*}


\begin{align*} \frac { \partial z } { \partial \alpha } & = \frac { \phi ( z ) [ 1 - \Phi ( z ) ] } { - \frac { \partial s ( z ) } { \partial z } } \mathrm { s i g n } \left ( \frac { \partial z } { \partial \alpha } \right ) = - \mathrm { s i g n } \left ( \frac { \partial s ( z ) } { \partial z } \right ) , \end{align*}


\begin{align*} \Theta f ( x ' , t ) : = \int _ { \R ^ { n - 1 } } \theta ( x ' , t ; y ' ) f ( y ' ) \ , d y ' , ( x ' , t ) \in \R ^ n _ + . \end{align*}


\begin{align*} d ( z ) & = \sum _ { \alpha \in S _ + } : X ^ \alpha \varphi ^ \alpha : - \frac { 1 } { 2 } \sum _ { \alpha , \beta , \gamma \in S _ + } { f ^ { \alpha \beta } } _ \gamma : \varphi _ \gamma \varphi ^ \alpha \varphi ^ \beta : + \sum _ { \alpha \in S _ + } ( f | q ^ \alpha ) \varphi ^ \alpha + \sum _ { \alpha \in S _ { \frac { 1 } { 2 } } } : \varphi ^ \alpha \Phi _ \alpha : . \end{align*}


\begin{align*} ( I - F ) \Psi ^ - = I , \end{align*}


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


\begin{align*} x ( t ) - z ( t ) = x ( 0 ) - z ( 0 ) + \int _ 0 ^ t \dot { x } ( \tau ) - \dot { z } ( \tau ) \ , d \tau \end{align*}


\begin{align*} - x \cdot \nabla ( L u ) = f ' ( u ) \left ( x \cdot \nabla u \right ) , \end{align*}


\begin{align*} M _ { f _ { 1 , \infty } } ( \alpha , \epsilon , Z , \psi ) : = \lim _ { n \to \infty } M _ { f _ { 1 , \infty } } ( n , \alpha , \epsilon , Z , \psi ) . \end{align*}


\begin{align*} \begin{aligned} \mathcal { g } ^ { \Delta ^ { ( s ) } } ( V , W ) & : = \mathcal { W } ( \nabla ^ { ( s ) } _ { V } \nabla ^ { ( s ) } _ { W } ( \Delta ^ { ( s ) } ) ^ { - n - 2 } ) = 2 ^ { m } \mathcal { g } ^ { \Delta } ( V , W ) , \\ \mathcal { G } ^ { \Delta ^ { ( s ) } } ( V , W ) & : = \mathcal { W } ( \nabla ^ { ( s ) } _ { V } \nabla ^ { ( s ) } _ { W } ( \Delta ^ { ( s ) } ) ^ { - n } ) = 2 ^ { m } \mathcal { G } ^ { \Delta } ( V , W ) . \end{aligned} \end{align*}


\begin{align*} \frac { k l } { m } , k \ , \sqrt { \frac { l - m } { m ( l - 1 ) } } , l \ , \sqrt { \frac { k - m } { m ( k - 1 ) } } . \end{align*}


\begin{align*} \partial _ t \omega + u \cdot \nabla \omega = 0 \end{align*}


\begin{align*} \langle \vartheta _ l , \varphi _ l \rangle _ { \C } = 0 \ \ \textrm { o n } \ \C \ \textrm { f o r } \ l = 1 , 2 . \end{align*}


\begin{align*} | \Pi _ { \mathrm { k e r } A ^ \perp } X _ t | & = \left | \Pi _ { \mathrm { k e r } A ^ \perp } X _ 0 + t \Pi _ { \mathrm { k e r } A ^ \perp } B ( X _ 0 , X _ 0 ) + \Pi _ { \mathrm { k e r } A ^ \perp } \int _ 0 ^ t ( t - s ) \frac { d } { d s } B ( X _ s , X _ s ) d s \right | \\ & \ge c t | \Pi _ { V _ 1 } X _ 0 | | \Pi _ { V _ 2 } X _ 0 | - C \sqrt { \delta } K ^ { r } - C K \tilde { \eta } \int _ 0 ^ t | B ( X _ s , X _ s ) | d s , \end{align*}


\begin{align*} v ( u ) \leq d _ u ( \pi _ N ( x ) + L ) + ( i _ u - d _ u ) L = d _ u ( \pi _ N ( x ) ) + i _ u L < 2 i _ u \pi _ N ( x ) , L < \pi _ N ( x ) . \end{align*}


\begin{align*} \xi _ 2 = \kappa _ 0 \xi _ 1 ^ 2 + O ( | \xi _ 1 | ^ 3 ) \mbox { a s } \xi _ 1 \rightarrow 0 \end{align*}


\begin{align*} ( \sum \limits _ { i = 1 } ^ j w _ i ) \chi - \sum \limits _ { i = 1 } ^ { j - 1 } \chi _ i + r ( j - 1 ) < \chi _ j < ( \sum \limits _ { i = 1 } ^ j w _ i ) \chi - \sum \limits _ { i = 1 } ^ { j - 1 } \chi _ i + r j , \end{align*}


\begin{align*} \chi _ l \leqslant w _ l \chi + k _ l , ( l = 2 , \dots , n ) \end{align*}


\begin{align*} \kappa ( s ) = \alpha \ , \frac { \nu _ { n + 1 } ( s ) } { x _ { n + 1 } ( s , t ) } + \varpi . \end{align*}


\begin{align*} - \Gamma ^ k + \bar u ^ { - 1 } \bar g ^ { i k } \frac { \partial \bar u } { \partial x _ i } = 0 . \end{align*}


\begin{align*} \mathcal { T } ( x , \textbf { 1 } ) = \log \log x + C ( \textbf { 1 } ) + o ( 1 ) , \end{align*}


\begin{align*} \widetilde { q } _ { 2 d } ( 2 k ) = e ^ { \frac { i \pi ( 2 k ) ^ 2 } { 2 d } } = q _ d ( k ) . \end{align*}


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


\begin{align*} \mathcal { Q } = \int _ { 0 } ^ t v _ 1 ( s , x - \lambda t + \lambda s ) \dd s - \int _ { t _ { 1 , e n } ( x , t ) } ^ { t _ { 1 , e x } ( x , t ) } v _ 1 ( s , x - \lambda t + \lambda s ) \dd s \end{align*}


\begin{align*} \begin{aligned} | \bar { y } - \bar { y } _ a | = & \ , | \bar { y } - \bar { y } _ a \pm z _ 2 | \le | \bar { y } - z _ 2 | + | z _ 2 - \bar { y } _ a | \\ \le & \ , | \bar { y } - z _ 2 | + \gamma | \epsilon _ 2 ( \eta _ a , t ) | \\ \le & \ , | \bar { y } - z _ 2 | + \gamma L _ r 2 \delta \end{aligned} \end{align*}


\begin{align*} \frac { h _ { j t } + h _ { j } } { h _ { t } + h } & = w ^ { i k } \nabla _ { j } w _ { i k } + \nabla _ { j } \chi \\ & = w ^ { i i } ( h _ { j i i } + h _ { i } \delta _ { i j } ) + \nabla _ { j } \chi , \end{align*}


\begin{align*} \zeta ( x ) : = \begin{cases} \frac { \xi ( x ) } { p ( x ) } , & \{ p ( x ) \ne 0 \} , \\ f _ { - } ' ( \overline { y } ( x ) ) , & \{ p ( x ) = 0 \} , \end{cases} \end{align*}


\begin{align*} \left \{ \begin{array} { l } \large \displaystyle u _ t - k \Delta u _ t - \Delta u = I ^ \gamma _ { 0 + } ( | u | ^ { p } ) + \omega ( x ) , \ , \ ( t , x ) \in ( 0 , \infty ) \times \mathbb { R } ^ N , \\ u ( 0 , x ) = u _ 0 ( x ) , \ , x \in \mathbb { R } ^ N , \end{array} \right . \end{align*}


\begin{align*} b _ { i j } = \begin{cases} c _ { i j } & , \\ - c _ { i j } & , \\ 0 & . \end{cases} \end{align*}


\begin{align*} \mathcal K _ { \mathbf x _ 0 } ^ { h } = \mathcal K _ { \mathbf { x _ 0 } ; { \mathbf e _ 1 } , \ldots { \mathbf e _ \ell } } \cap B _ { h } . \end{align*}


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


\begin{align*} \begin{aligned} \int _ B u _ { p + N } \Delta u _ p \wedge \alpha & = B _ { p , N } + \int _ B u _ p \Delta u _ { p + N } \wedge \alpha \geq B _ { p , N } + \int _ B u _ { p + N } \Delta u _ { p + N } \wedge \alpha , \end{aligned} \end{align*}


\begin{align*} s _ i ( A ) = \min \limits _ { E : \ ; \dim ( E ) = n - i + 1 } \max \limits _ { x \in E , \ , \lVert x \rVert _ 2 = 1 } \lVert A x \rVert _ 2 , 1 \leq i \leq n , \end{align*}


\begin{align*} \chi _ i \leqslant w _ i \chi + k _ i , ( i = 1 , \dots , n - 1 ) \end{align*}


\begin{align*} \gamma _ i ( s , \tau ) = \bar { Q } _ { \frac { 2 \pi ( i - 1 ) } { m } } \gamma ( s , \tau ) . \end{align*}


\begin{align*} A _ { p , N } : = \sum _ { A = 0 } ^ { 2 n - 1 } \int _ S \left [ v _ p ( d _ 1 v _ { p + N } \wedge \alpha ) _ A \tau ( \mathbf { { n } } ) _ { A 0 } + v _ { p + N } ( d _ 0 v _ p \wedge \alpha ) _ A \tau ( \mathbf { { n } } ) _ { A 1 } \right ] d S , \end{align*}


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


\begin{align*} M _ + ( x ; z ) = M _ - ( x ; z ) ( I + R ( x ; z ) ) , z \in \mathbb { R } , \end{align*}


\begin{align*} \sum _ { k = 1 } ^ n \binom n k \frac { F _ { j k } } { L _ j ^ k } B _ { n - k } = 0 , \qquad \mbox { $ n $ e v e n } , \end{align*}


\begin{align*} \begin{cases} \partial _ t \mathbf { v } + ( \mathbf { v } \cdot \nabla ) \mathbf { v } = - \nabla P , \ \ & D \times ( 0 , T ) , \\ \nabla \cdot \mathbf { v } = 0 , \ \ & D \times ( 0 , T ) , \\ \mathbf { v } ( \cdot , 0 ) = \mathbf { v } _ 0 ( \cdot ) , \ \ & D , \end{cases} \end{align*}


\begin{align*} A ^ * ( y ) = \left [ \hat { A } \circ \hat { \sigma } ^ { - 1 } + W \circ \hat { \sigma } ^ { - 1 } - W \right ] ( y | x ) , \end{align*}


\begin{align*} \alpha _ i ( t , x _ 1 , \ldots , x _ n ) = \widehat \alpha _ i ( t , x _ i ) , \end{align*}


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


\begin{align*} K _ { V , I } = \sum _ { J \subset I } \kappa _ { V , J } . \end{align*}


\begin{align*} G _ { \Pi } \left ( x , y \right ) = G \left ( x , y \right ) - G \left ( \bar { x } , y \right ) , \forall \ , x , y \in \Pi , \end{align*}


\begin{align*} E _ { - m } = b _ 0 q _ { - m } + \frac { \partial q _ { - m } } { \partial x _ n } - i \sum _ \alpha \frac { \partial b _ 1 } { \partial \xi _ \alpha } \frac { \partial q _ { - m } } { \partial x _ \alpha } - \sum _ { \substack { - m \leqslant j , k \leqslant 1 \\ | J | = j + k + m } } \frac { ( - i ) ^ { | J | } } { J ! } \partial _ { \xi } ^ { J } q _ j \ , \partial _ { x ^ \prime } ^ { J } q _ k , m \geqslant 0 . \end{align*}


\begin{align*} \begin{aligned} \dot { \zeta } = & \ , A \zeta + B z \\ x = & \ , C \zeta \end{aligned} \end{align*}


\begin{align*} \begin{aligned} \mathcal { U } ^ * ( K , \Omega ) = \left \{ u \in Q S H _ m ( \Omega ) \cap C ( \Omega ) , u \vert _ { K } \leq - 1 , \varliminf _ { q \to \partial \Omega } u ( q ) \geq 0 \right \} . \end{aligned} \end{align*}


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


\begin{align*} \sup \limits _ i \int _ K | \tilde { F } _ i - ( 1 - b _ { t _ 0 , B } ( \Psi ) ) f F ^ { 1 + \delta } | ^ 2 _ h < + \infty . \end{align*}


\begin{align*} y [ n ] = \alpha x [ n ] + w _ { \rm d } [ n ] + w _ { \rm s } [ n ] . \end{align*}


\begin{align*} \left \{ \begin{array} { l l } \partial _ t u - \Delta u = f , \\ u ( 0 , x ) = u _ 0 ( x ) , \end{array} \right . \end{align*}


\begin{align*} { \bf E } \left ( { \bf A } , { \mathbb V } \right ) = \left \{ { \bf A } _ { \theta } : \theta \left ( x , y \right ) = \sum _ { i = 1 } ^ { s } \theta _ { i } \left ( x , y \right ) e _ { i } \ \ \ \ \left \langle \left [ \theta _ { 1 } \right ] , \left [ \theta _ { 2 } \right ] , \dots , \left [ \theta _ { s } \right ] \right \rangle \in { \bf T } _ { s } ( { \bf A } ) \right \} . \end{align*}


\begin{align*} g ( \kappa ) = \sqrt { 2 \pi } \left ( \kappa - \frac { \psi _ { 1 } \tau _ { \rm d } } { N _ { \rm a } } \right ) f \left ( \kappa + \frac { \psi _ { 1 } \tau _ { \rm d } } { N _ { \rm a } } \right ) , \end{align*}


\begin{align*} H ^ { s , p } : = \{ f \in L ^ { p } ( \mathbb { R } ^ { n } ) ; \mathcal { F } ^ { - 1 } [ ( 1 + \vert \xi \vert ^ { 2 } ) ^ { \frac { s } { 2 } } \mathcal { F } f ] \in L ^ { p } ( \mathbb { R } ^ { n } ) \} , \end{align*}


\begin{align*} \Re \bigg \{ \left ( \frac { L ' } { L } \right ) ' ( s , \chi ) \bigg \} = \sum _ { \gamma _ \chi } g _ { a } ( t - \gamma _ \chi ) + \sum _ { \gamma } g _ { a } ( \gamma ) + O ( 1 ) , \end{align*}


\begin{align*} \xi = - L - 1 , \eta = \frac { Q } { 2 ( L + 1 ) } , \zeta = \frac { \kappa ( L + 1 ) B _ 1 } { Q } . \end{align*}


\begin{align*} \operatorname { g r a d } f = \nabla ^ i f \frac { \partial } { \partial x _ i } = g ^ { i j } \frac { \partial f } { \partial x _ i } \frac { \partial } { \partial x _ j } , f \in C ^ { \infty } ( \Omega ) , \end{align*}


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


\begin{align*} \left \{ \begin{aligned} d x _ t ^ { \xi , \alpha } = & ~ [ A x _ t ^ { \xi , \alpha } + B \alpha _ t + f ( \nu _ t ) + b ( \mu _ t ) ] d t + \sigma d W _ t + \sigma _ 0 d W ^ 0 _ t , \\ x _ 0 ^ { \xi , \alpha } = & ~ \xi , \end{aligned} \right . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \inf _ { k \in { \mathbb Z } } \left ( \frac { \mu ( f ^ { k + n } ( W ) ) } { \mu ( f ^ { k } ( W ) ) } \right ) = \infty \tag * { $ \mathcal { U E } 1 $ } \end{align*}


\begin{align*} G _ - ( x ; k ) = \mathcal { P } ^ { - } \left ( G _ { - } J \right ) ( z ) + \mathcal { P } ^ { - } \left ( J \right ) , \end{align*}


\begin{align*} & T _ { 3 , r } ( \underline { p , q } ) = 2 ^ r \sum _ { n _ 1 , n _ 2 , \dots , n _ r \atop { 1 \leq n _ i \leq \pi _ N ( x ) } } G ( n _ 1 ) G ( n _ 2 ) \dots G ( n _ r ) \\ & \sum _ { l _ 1 , l _ 2 , \dots , l _ r \atop { 0 \leq l _ i \leq L } } U ( l _ 1 ) U ( l _ 2 ) \dots U ( l _ r ) \sum _ { l _ 1 ' , l _ 2 ' , \dots , l _ r ' \atop { 0 \leq l _ i ' \leq L } } U ( l _ 1 ' ) U ( l _ 2 ' ) \dots U ( l _ r ' ) \prod _ { i = 1 } ^ r I ( p _ i , q _ i , n _ i , l _ i , l _ i ' ) , \end{align*}


\begin{align*} \sup _ { n \in \mathbb { N } } \sup _ { j \in [ n ] ^ V } \sup _ { s _ 1 , . . . , s _ i \in [ 0 , T ] } \norm { D ^ i _ { s _ 1 , . . . , s _ i } A _ n ( j ) } _ { L ^ p ( P ) } < \infty \end{align*}


\begin{align*} \begin{array} { l l l } \mu \circ ( \eta \otimes ( f \circ \eta ) ) & = & \mu \circ ( \eta \otimes i d _ M ) \circ ( i d _ I \otimes ( f \circ \eta ) ) \\ & = & \lambda _ M \circ ( i d _ I \otimes ( f \circ \eta ) ) \\ & = & ( f \circ \eta ) \circ \lambda _ I . \\ & & \mbox { ( b y n a t u r a l i t y o f $ \lambda $ ) } \end{array} \end{align*}


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


\begin{align*} s < s _ 0 : = \frac { \vert \omega _ 1 \land \omega _ 2 \land \omega _ 3 \vert } { 1 2 8 \pi } \end{align*}


\begin{align*} \int _ { G ^ \wedge } \left ( 1 + \gamma ( \xi ) ^ 2 \right ) \ , { \vert [ T ( f ) ] ( \xi ) \vert } ^ 2 d \nu ( \xi ) = \int _ { G ^ \wedge } \vert f ( \xi ) \vert ^ 2 d \nu ( \xi ) . \end{align*}


\begin{align*} \vert S \cap [ 1 , n ] \vert \leq \sum _ { \ell = 1 } ^ U \vert S _ { \ell } \vert = O ( U ^ 2 ) = O ( ( \log n ) ^ 2 ) = o ( n / \log n ) \end{align*}


\begin{align*} f ( \kappa ) = \sigma , \partial \Sigma = \Gamma , \end{align*}


\begin{align*} Q ( h , v ) = \Big ( v A + \bar u A ' ( h ) - ( \nu ( u ) ) ' \bar g ^ \intercal , \ , 2 v \Big ) \mbox { o n } \hat \Sigma . \end{align*}


\begin{align*} \limsup _ { n \to \infty } T _ { W _ { n J } , \psi } ( \mu _ n ( Q ^ n ) ) & \le \limsup _ { n \to \infty } T _ { W _ { n J } , \psi _ m } ( \mu _ n ( Q ^ n ) ) \le \limsup _ { n \to \infty } T _ { W , \psi _ m } ( \mu _ n ( Q ^ n ) ) = T _ { W , \psi _ m } ( \mu _ \infty ) , \end{align*}


\begin{align*} u _ { n n } \sigma _ 1 ( b ) - \sum _ { \alpha \leq n - 1 } u _ { n \alpha } ^ 2 + \sigma _ 2 ( b ) = f . \end{align*}


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


\begin{align*} \mathcal { E } _ { m } = \displaystyle \sum _ { n = x _ { m } + 1 } ^ { y _ { m } } e _ { n } \ ; \ ; \ ; \mbox { a n d } \ ; \ ; \ ; \mathcal { F } _ { m } = \displaystyle \sum _ { n = x _ { m } + 1 } ^ { y _ { m } } g _ { n } . \\ \end{align*}


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


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


\begin{align*} \begin{aligned} \binom { n } { m } \det ( A _ 1 , \dots , A _ m , I , \dots , I ) \geq \mathcal { H } _ m ( A _ 1 ) ^ { \frac { 1 } { m } } \dots \mathcal { H } _ m ( A _ m ) ^ { \frac { 1 } { m } } . \end{aligned} \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*} d u = \Delta _ p u \ , d t + \sum _ { n = 0 } ^ { + \infty } \xi _ { n } \cdot \nabla u \circ d B _ t ^ { n } , \end{align*}


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


\begin{align*} [ \Delta _ { \Psi } , X ] = 0 , \O \end{align*}


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


\begin{align*} Z ( A ) = V ( I ( A ) ) \ , . \end{align*}


\begin{align*} \zeta _ \mathrm { F S O } = h _ f h _ g = h _ f \left [ \mathrm { e r f } \left ( \frac { \sqrt { \pi } \omega } { 2 \sqrt { 2 } \theta L } \right ) \right ] ^ 2 , \end{align*}


\begin{align*} \int _ { X } ( \alpha + \sqrt { - 1 } \omega ) ^ { 2 } = \int _ { X } ( \alpha ^ { 2 } - \omega ^ { 2 } ) + 2 \sqrt { - 1 } \int _ { X } \alpha \wedge \omega = 3 2 + 2 6 \sqrt { - 1 } . \end{align*}


\begin{align*} L ( x ) = a _ 0 x + a _ 1 x ^ { q ^ s } + a _ 2 x ^ { q ^ { 2 s } } + \cdots + a _ m x ^ { q ^ { m s } } \in F [ x ] , \end{align*}


\begin{align*} & F ( W ) c _ { ( \vartheta , \varphi ) } = \int _ { \gamma } F ( Z ) E _ { ( \vartheta , \varphi ) } ( Z , W ) d \sigma _ { ( \vartheta , \varphi ) } ( Z ) - \int _ { \Gamma } E _ { ( \vartheta , \varphi ) } ( Z , W ) \frac { \partial F } { \partial Z _ { ( \vartheta , \varphi ) } } d Z \wedge d Z ^ { * } , \end{align*}


\begin{align*} ( n - ( j + 2 ) ) r < \sum \limits _ { i = j + 2 } ^ n \chi _ i , \end{align*}


\begin{align*} \bar \beta _ n ( G ) = O ( n ^ { e ( G ) } ) , \end{align*}


\begin{align*} 1 = \vert \theta \vert \leq \sum _ { j = 2 } ^ { n } \alpha _ j \vert \omega \vert = \sum _ { j = 2 } ^ n \alpha _ j . \end{align*}


\begin{align*} \sum _ { p \leq x } \frac { \chi _ { - 4 } ( p ) ^ { 2 } } { 2 p } = \sum _ { p \leq x } \frac { 1 } { 2 p } = \frac { 1 } { 2 } \log \log x + c + o ( 1 ) \end{align*}


\begin{align*} \phi _ { \lambda } ( [ a _ { \mu } b ] ) = [ \phi _ { \lambda } ( a ) _ { \lambda + \mu } b ] + [ a _ { \mu } \phi _ { \lambda } ( b ) ] , \forall \ a , b \in \mathcal { A } . \end{align*}


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


\begin{align*} & \psi \equiv 1 \ ; \ ; o n Q _ 0 a n d s u p p ( \psi ) \subset Q , \\ [ 2 m m ] & \phi \equiv 1 \ ; \ ; o n \ ; \ ; Q _ 1 a n d s u p p ( \phi ) \subset Q _ 0 \subset Q . \end{align*}


\begin{align*} m : = b _ { n - 1 } \cdots b _ { \alpha + 1 } b _ \alpha ^ { k + \alpha - n } . \end{align*}


\begin{align*} \| e ^ { - \mathbf d \cdot \mathbf y } \| _ { L ^ { \frac { 8 } { 7 } } ( \Gamma ^ \pm ) } \leq \left ( \int _ { \Gamma ^ \pm } e ^ { - \frac { 8 } { 7 } \zeta \vert \mathbf y \vert } \mathrm d \sigma \right ) ^ \frac { 7 } { 8 } = 2 \left ( \frac { 7 } { 8 } \frac { 1 } { \zeta } \right ) ^ { \frac { 7 } { 8 } } , \end{align*}


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


\begin{align*} f ( x + k , \omega ) = f ( x , \tau ( k ) \omega ) . \end{align*}


\begin{align*} \vert I \vert \leq \sum _ { m = 1 } ^ 9 \vert I _ m \vert + \vert J _ 1 \vert + \vert J _ 2 \vert . \end{align*}


\begin{align*} V \oplus W = \bigoplus _ { i \in \mathcal { I } } ( V _ i \oplus W _ i ) , V \otimes W = \bigoplus _ { k \in \mathcal { I } } \left ( \bigoplus _ { i + j = k } V _ i \otimes W _ j \right ) . \end{align*}


\begin{align*} \begin{array} { l l l } \mu \circ ( ( R \circ \eta ) \otimes \eta ) & = & \mu \circ ( R \otimes i d _ M ) \circ ( \eta \otimes \eta ) \\ & = & \mu \circ ( i d _ M \otimes L ) \circ ( \eta \otimes \eta ) \\ & = & \mu \circ ( \eta \otimes i d _ M ) \circ ( i d _ I \otimes ( L \circ \eta ) ) \\ & = & \lambda _ M \circ ( i d _ I \otimes ( L \circ \eta ) ) \\ & = & L \circ \eta \circ \lambda _ I \end{array} \end{align*}


\begin{align*} F ^ { q , \dagger } _ v ( 0 , p ) [ u _ k ] = c _ 1 ( q , k , p ) u _ k . \end{align*}


\begin{align*} \widetilde { x } ( [ 0 , \ldots , \widehat { k } , k + 1 ] \otimes [ 0 , \ldots , \ell ] ) = 0 \end{align*}


\begin{align*} F ' ( y ; h ) ( x ) = f _ { x } ' ( y ( x ) ; h ( x ) ) . \end{align*}


\begin{align*} L ( p ) = \log F ( p , 8 / 3 ) , R ( p ) = \log \left ( e ^ { - p / 6 } \tilde G ( p , 2 ) \right ) \end{align*}


\begin{align*} \int _ { 0 } ^ \infty \int _ { \Omega } \omega ( t , x ) g _ r ( t ) f _ h ' ( \phi ( x ) ) \ , u ( t , x ) \cdot \psi ( x ) d x d t + \int _ { 0 } ^ \infty \int _ { \Omega } \omega ( t , x ) g _ r ' ( t ) f _ h ( \phi ( x ) ) d x d t = 0 . \end{align*}


\begin{align*} X ^ \Delta ( t _ { n + 1 } ) = X ^ \Delta ( t _ { n } ) + F ( X ^ \Delta ( t _ { n } ) ) \Delta + \Delta ^ { 1 / 2 } G ( X ^ \Delta ( t _ { n } ) ) \xi _ { n + 1 } . \end{align*}


\begin{align*} F ^ { i j } = \frac { \partial F ( D ^ 2 u ) } { \partial u _ { i j } } . \end{align*}


\begin{align*} W ( r ) = \frac { \xi } { r } f + \eta + \zeta \frac { r } { f } , \xi \le 0 , \zeta > 0 , \end{align*}


\begin{align*} f ^ \star : = \inf \limits _ { x \in S ( g ) } f ( x ) \ , , \end{align*}


\begin{align*} a = \Big ( \frac { 1 - q } { 2 } \Big ) ^ { k / p _ k } , b = \Big ( \frac { q } { 2 } \Big ) ^ { k / p _ k } . \end{align*}


\begin{align*} x = \sum _ { m = 0 } ^ \infty \sum _ { i = 0 } ^ { p - 1 } \frac { a _ { m p + i } } { ( \beta _ 0 \cdots \beta _ { p - 1 } ) ^ m \beta _ 0 \cdots \beta _ i } . \end{align*}


\begin{align*} 2 H = \eta z ^ m + \lambda . \end{align*}


\begin{align*} \beta _ { p , d } = \frac { \sqrt { \pi } \Gamma \left ( \frac { d - p } { 2 } \right ) } { \Gamma \left ( \frac { 1 - p } { 2 } \right ) \Gamma \left ( \frac { d } { 2 } \right ) } . \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*} \begin{aligned} \Delta ^ { ( s ) } = - \partial _ { i } \partial _ { i } + \frac { 1 } { 3 } R _ { i j k \ell } \ , x ^ { j } x ^ { k } \partial _ { i } \partial _ { \ell } + o ( \mathbf { x ^ { 2 } ) } \\ + \frac { 2 } { 3 } R _ { i j } \ , x ^ { i } \partial _ { j } + \frac { 1 } { 4 } R _ { i \ell j k } \ , x ^ { \ell } \gamma ^ { j } \gamma ^ { k } \partial _ { i } + o ( \mathbf { x ) } \\ + o ( \mathbf { 1 ) } , \end{aligned} \end{align*}


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


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


\begin{align*} \lambda ^ { - m } \ , \int _ { S ^ { n - 1 } } \widehat { f } ( \lambda , \omega ) \ , Y _ m ( \omega ) \ , d \omega = c _ { n , m } \int _ 0 ^ \infty r ^ { - m } f _ m ( r ) \ , \frac { J _ { n / 2 + m - 1 } ( \lambda r ) } { ( \lambda r ) ^ { n / 2 + m - 1 } } r ^ { n + 2 m - 1 } d r \end{align*}


\begin{align*} \partial _ { t } h = P - h > 0 { \rm f o r \ a l l } \ t \geq 0 . \end{align*}


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


\begin{align*} \hat W _ r ( 0 ) = k \hat W _ \frac { r } { k } ( 0 ) . \end{align*}


\begin{align*} \lim _ { | z | \rightarrow \infty } z ( \Psi ^ \pm _ 1 ( x ; z ) - e ^ { - i c _ \pm ( x ) } e _ 1 ) = \widehat { \Psi } ^ \pm _ { 1 1 } ( x ) e _ 1 + \widehat { \Psi } ^ \pm _ { 2 1 } ( x ) e _ 2 . \end{align*}


\begin{align*} W '' = e ^ { - \lambda ^ * T } \big ( W ' + W ^ D \big ) . \end{align*}


\begin{align*} d v v ^ { - 1 } = \frac { 1 } { 2 } a I + \frac { 1 } { \sqrt { \tau _ 0 } } \left ( \frac { \tau _ 0 } { 4 } d \overline { z } + Q _ 0 d z \right ) i J . \end{align*}


\begin{align*} \begin{aligned} \int _ { \Omega } | \nabla v _ { l } | ^ { 2 } + \int _ { \partial \Omega } | v _ { l } | ^ { 2 } + \int _ { \Omega } u _ { l } v _ { l } ^ { 2 } & = \int _ { \Omega } f v _ { l } + \int _ { \partial \Omega } \eta v _ { l } \\ & \le C ( \| f \| _ { L ^ { p } ( \Omega ) } \| v _ { l } \| _ { H ^ { 1 } ( \Omega ) } ) + \int _ { \partial \Omega } \eta v _ { l } . \end{aligned} \end{align*}


\begin{align*} h ( x , y ) \ge \frac { 1 } { 2 \pi } \ln \frac { 1 } { | x - y | + 2 \max \{ d i s t ( x , \partial \Omega ) , d i s t ( y , \partial \Omega ) \} } . \end{align*}


\begin{align*} \nu _ i = 1 & \mbox { a n d } j = i + 1 , \mbox { o r } \\ \nu _ i = 0 & \mbox { a n d } j \neq i + 1 . \end{align*}


\begin{align*} x y = y x = x \wedge y \end{align*}


\begin{align*} T _ { \mathrm { g e l } } : = \inf \left \{ t \geq 0 \ \ \mbox { s u c h t h a t } \ \ \mathcal { M } ^ { 1 } ( t ) < \mathcal { M } ^ { 1 } ( 0 ) = \varrho _ { 0 } \right \} . \end{align*}


\begin{align*} \frac { L ' } { L } ( s , \chi ) = - \sum _ { n \leq x y } \frac { \Lambda _ { x , y } ( n ) \chi ( n ) } { n ^ s } + \frac { 1 } { \log { y } } \sum _ { \rho _ { \chi } } \frac { x ^ { \rho _ { \chi } - s } - ( x y ) ^ { \rho _ { \chi } - s } } { \left ( \rho _ { \chi } - s \right ) ^ { 2 } } + \frac { 1 } { \log { y } } \sum _ { n = 0 } ^ { \infty } \frac { x ^ { - 2 n - \mathfrak { a } - s } - ( x y ) ^ { - 2 n - \mathfrak { a } - s } } { \left ( 2 n + \mathfrak { a } + s \right ) ^ { 2 } } \end{align*}


\begin{align*} ( \lambda _ { S } ) _ { i j } : = \delta _ { i , 0 } L _ { ( P _ S ) _ i } \mbox { a n d } ( \rho _ { S } ) _ { i j } : = \delta _ { j , 0 } R _ { ( P _ S ) _ j } , \end{align*}


\begin{align*} \begin{aligned} \dfrac { b _ { 1 , \delta } ( x ) } { 2 } & \geq \bar b & \forall \ , x \geq x ^ \star + \delta \\ \dfrac { b _ { 1 , \delta } ( x ) } { 2 } & \leq - \bar b & \forall \ , x \leq x ^ \star - \delta \ , . \end{aligned} \end{align*}


\begin{align*} \mathbb { E } \{ \sigma _ { \rm a } ^ 2 ( x ) \} = \int _ { - \infty } ^ { + \infty } f ( x ) \sigma _ { \rm a } ^ 2 ( x ) \ , \mathrm { d } x , \end{align*}


\begin{align*} \min _ { \xi , \eta } p \left ( \xi , \eta \right ) = \min \left \{ \min _ { \xi } p \left ( \xi , 1 \right ) , \min _ { \xi } p \left ( \xi , \frac { 9 9 } { 1 0 0 } \right ) \right \} . \end{align*}


\begin{align*} \left \| \psi \left ( \frac { \mathbf y } { \tau } \right ) \right \| _ { L ^ 8 ( \partial \mathcal C ) } \lesssim \tau ^ { \frac { 1 } { 4 } } \| \psi ( \mathbf x ) \| _ { H ^ { \frac { 7 } { 8 } , 8 } ( \partial \mathcal C ) } \lesssim \tau ^ { \frac { 1 } { 4 } } \| \psi ( \mathbf x ) \| _ { H ^ { 1 , 8 } ( \mathcal C ) } \lesssim \tau ^ { - \frac { 3 } { 2 0 } } , \end{align*}


\begin{align*} \rho { u _ x } = \sum \limits _ { i = 0 } ^ { 1 8 } { { c _ { i x } } { f _ i } + \frac { { \Delta t } } { 2 } { F _ x } } , \end{align*}


\begin{align*} \Phi ( v , p ) = 0 , \end{align*}


\begin{align*} M _ { I J } = \begin{pmatrix} G _ { I _ 1 } & G _ { I _ 2 } & \cdots & G _ { I _ { 2 k } } \\ G _ { J _ 1 } & G _ { J _ 2 } & \cdots & G _ { J _ { 2 k } } \end{pmatrix} , \end{align*}


\begin{align*} & \frac { d } { d t } \int _ { V } \Re ( \alpha _ { t } + \sqrt { - 1 } \omega ) ^ { p } - \cot \Theta _ { 0 } \cdot \Im ( \alpha _ { t } + \sqrt { - 1 } \omega ) ^ { p } \\ = { } & p \int _ { V } \left ( \Re ( \alpha _ { t } + \sqrt { - 1 } \omega ) ^ { p - 1 } - \cot \Theta _ { 0 } \cdot \Im ( \alpha _ { t } + \sqrt { - 1 } \omega ) ^ { p - 1 } \right ) \wedge \omega > 0 . \end{align*}


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


\begin{align*} f ( \lambda _ i ( A ) ) = f ( n + 1 - 2 i ) = \frac { 1 } { 2 n } \prod _ { k = 1 ; k \neq i } ^ { n } \left ( k - i \right ) , \ 1 \leq i \leq n . \end{align*}


\begin{align*} \partial _ t \Phi _ { + } ^ { 2 } + u \cdot \nabla \Phi _ { + } ^ { 2 } = \mu \left ( \Delta - \frac { 2 } { r } \partial _ r \right ) \Phi _ { + } ^ { 2 } - 2 \mu | \nabla \Phi _ { + } | ^ { 2 } \textrm { o n } \ L ^ { 1 } _ { \textrm { l o c } } ( \mathbb { R } ^ { 3 } ) . \end{align*}


\begin{align*} & \o ^ \beta ( t , x ) = \o _ 0 ^ \beta ( X ^ \beta ( 0 ; t , x ) ) , \\ & \frac { d } { d s } X ^ \beta ( s ; t , x ) = u ^ \beta ( s , X ^ \beta ( s ; t , x ) ) , \ \ X ^ \beta ( s ; t , x ) | _ { s = t } = x . \end{align*}


\begin{align*} k ( t ) = h _ { 1 - \alpha , \rho } ( t ) \ , + \ , \rho \ , \int _ 0 ^ t h _ { 1 - \alpha , \rho } ( \tau ) \ , d \tau , \end{align*}


\begin{align*} e ^ { - t \Lambda _ g } \textbf { \textit { V } } ( x ^ { \prime } ) = \int _ { \partial \Omega } \textbf { \textit { K } } ( t , x ^ { \prime } , y ^ { \prime } ) \textbf { \textit { V } } ( x ^ { \prime } ) \ , d S , \textbf { \textit { V } } \in ( L ^ { 2 } ( \partial \Omega ) ) ^ { n + 1 } . \end{align*}


\begin{align*} & \Phi ( \tau ) ( \Lambda _ g - \tau I ) = I \mod O P S ^ { - \infty } , \\ & ( \Lambda _ g - \tau I ) \Phi ( \tau ) = I \mod O P S ^ { - \infty } . \end{align*}


\begin{align*} \lambda = ( \lambda _ 1 , \ldots , \lambda _ k ) , \ \lambda _ 1 \geq \ldots \geq \lambda _ k > 0 , \ \sum \lambda _ i = m . \end{align*}


\begin{align*} l r ( e p o c h s ) = \begin{cases} 1 0 ^ { - 3 } , \textit { 1 $ \leq $ e p o c h s $ < $ 2 , 0 0 0 } \\ 1 0 ^ { - 4 } , \textit { e p o c h s $ \geq $ 2 , 0 0 0 } . \end{cases} \end{align*}


\begin{align*} U ( 1 , 1 , \Delta ) = 1 . \end{align*}


\begin{align*} [ \mathbb C _ x ] = \prod _ { i = 0 } ^ r e ^ { x _ i \omega _ i } \end{align*}


\begin{align*} h ( T ' ) & = y q \log ( y q ) + r \Delta ( q + 1 ) + f ( r ) \\ & = q y \log ( q ) + q y \log ( y ) + r \log ( q ) + r + f ( r ) + o ( 1 ) \\ & = m \log ( m - r ) - r \log ( y ) + r + r \log ( r ) + o ( 1 ) \\ & = m \log ( m ) - r \log \left ( \frac y r \right ) + o ( 1 ) , \end{align*}


\begin{align*} \mbox { e i t h e r } T _ { \rm m a x } = \infty \mbox { o r } \limsup _ { t \nearrow T _ { \rm m a x } } \| n ( \cdot , t ) \| _ { L ^ { \infty } ( \Omega ) } = \infty . \end{align*}


\begin{align*} \epsilon & \colon 1 _ { \mathcal { F _ { \mathbb { R } } } } \to \nu _ { * } \nu ^ { * } , \ , H ( S ) \stackrel { H ( \mathrm { p r } _ { 1 } ) } { \longrightarrow } ( \nu _ { * } \nu ^ { * } H ) ( S ) = H ( \nu _ { * } S _ { \mathbb { C } } ) \\ \eta & \colon \nu ^ { * } \nu _ { * } \to 1 _ { \mathcal { F } _ { \mathbb { C } } } , \ , ( \nu ^ { * } \nu _ { * } F ) ( T ) = F ( T \sqcup \overline { T } ) \stackrel { F ( j _ { T } ) } { \longrightarrow } F ( T ) \end{align*}


\begin{align*} E ( X - Y ) ^ { n } = \left \{ \begin{array} { c c c } 0 & & n \\ \frac { n ! } { ( n / 2 ) ! } ( 1 - \rho ) ^ { n / 2 } & & n \end{array} \right . . \end{align*}


\begin{align*} \prod _ { i = 1 } ^ { \pi ( a ) } q _ i = \prod _ { i = 1 } ^ { \pi ( a ) } p _ i ^ { \alpha _ i } . \end{align*}


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


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


\begin{align*} a ( u , v ) = ( f , v ) ~ v \in H ^ 1 _ 0 ( D ) , \end{align*}


\begin{align*} & g _ { j k } ( x _ 0 ) = g ^ { j k } ( x _ 0 ) = \delta _ { j k } , 1 \leqslant j , k \leqslant n , \\ & \frac { \partial g _ { j k } } { \partial x _ \alpha } ( x _ 0 ) = \frac { \partial g ^ { j k } } { \partial x _ \alpha } ( x _ 0 ) = 0 , 1 \leqslant \alpha \leqslant n - 1 , \end{align*}


\begin{align*} B _ 1 + \cdots + B _ k = \{ b _ 1 + \cdots + b _ k : b _ 1 \in B _ 1 , \dots , b _ k \in B _ k \} \end{align*}


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


\begin{align*} ( \sigma , \sigma ( 1 ) , \ldots , \sigma ( r - 1 ) ) = [ \sigma , \sigma ( r - 1 ) ] = \Delta \end{align*}


\begin{align*} \psi _ { 0 , H } ( t ) = \frac { ( n - 1 ) H } { ( n - 1 ) + t H } . \end{align*}


\begin{align*} \varphi ( - t ) ^ 2 G ( - t ) - \varphi ( t ) ^ 2 G ( t ) = 2 t \varphi ( t ^ 4 ) ^ 2 H ( t ^ 4 ) \end{align*}


\begin{align*} p = 1 \oplus 0 \oplus 1 \oplus 0 \oplus \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} , q = 0 \oplus 1 \oplus 1 \oplus 0 \oplus \begin{pmatrix} a ^ 2 & a b \\ a b & b ^ 2 \end{pmatrix} \end{align*}


\begin{align*} a _ n ( u _ n , v _ n ) = ( f _ n , v _ n ) v _ n \in X _ n , \end{align*}


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


\begin{align*} a ( T f , v ) = ( f , v ) v \in H _ 0 ^ 2 ( D ) . \end{align*}


\begin{align*} \nu = \int _ Y \nu _ y \ , \mathrm { d } \mu . \end{align*}


\begin{align*} X _ { \gamma } ' = \{ x ' \in X ' \ ; | \ ; x ' \ ; \tau \| \cdot \| \} \eqqcolon X ^ { \circ } \end{align*}


\begin{align*} \tilde { I _ j } = \{ i \in [ n / \ell ] : \ : V _ i \subseteq I _ j \} . \end{align*}


\begin{align*} \frac { \partial ^ 2 \varphi } { \partial x _ \beta ^ 2 } ( x ' , \rho ( x ' ) ) \geq b _ \beta - C \delta \tilde { b } _ \beta , \ \beta = 1 , \ldots , n - 1 \end{align*}


\begin{align*} U _ { \bf n } = \rm { d i a g } ( \ \underbrace { 1 , \ldots 1 } _ { n _ 0 } , \underbrace { \omega , \ldots , \omega } _ { n _ 1 } , \ldots , \underbrace { \omega ^ { \ell - 1 } , \ldots , \omega ^ { \ell - 1 } } _ { n _ { \ell - 1 } } \ ) . \end{align*}


\begin{align*} \begin{aligned} & \lim _ { | k | \rightarrow \infty } ( z \Psi ^ - ) _ { 2 1 } ( x ; k ) = - \frac { 1 } { 2 i } \lim _ { | z | \rightarrow \infty } \nu ( x ; z ) \\ & = - \frac { 1 } { 4 } \left ( 2 i \bar { u } _ { x x } ( x ) + | u _ x ( x ) | ^ { 2 } \bar { u } _ x ( x ) \right ) e ^ { i c _ \pm } ( x ) = \widehat { \Psi } ^ - _ { 2 1 } ( x ) , \end{aligned} \end{align*}


\begin{align*} \phi _ t + u \cdot \nabla \phi = 0 . \end{align*}


\begin{align*} \begin{cases} \frac { \partial } { \partial t } u _ j ( t , x ) = c _ j ( t ) \frac { \partial } { \partial x } u _ j ( t , x ) + q _ j ( t , x ) u _ j ( t , x ) , & x \in \left ( 0 , 1 \right ) , \ t \geq 0 , \\ u _ j ( t , 0 ) = f _ j ( x ) , & x \in \left ( 0 , 1 \right ) , \\ \phi ^ - _ { i j } c _ j ( t ) u _ j ( t , 1 ) = \omega _ { i j } \sum _ { k \in \mathbb { N } } { \phi ^ + _ { i k } c _ k ( t ) u _ k ( t , 1 ) } , & t \geq 0 . \end{cases} \end{align*}


\begin{align*} R ( \lambda , M _ { q } ) f = \frac { 1 } { \lambda - q } f , f \in \mathrm { C } _ { \operatorname { b } } ( \Omega ) , \end{align*}


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


\begin{align*} \phi _ x ^ \ell : = \frac 1 \ell \sum _ { y = 0 } ^ { \ell - 1 } \phi _ { x + y } , \bar \phi _ x ^ \ell : = E _ { \tilde \nu _ { N , t } } \left [ \phi _ x ^ \ell ~ \Big | ~ \sum _ { y = 0 } ^ { \ell - 1 } r _ { x + y } \right ] . \end{align*}


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


\begin{align*} E Q H _ n : & = \left \{ q \in R _ n : \left < p , q \right > = 0 \hbox { f o r a l l } p \in I _ n \right \} \\ & = \left \{ q \in R _ n : p ( \partial _ { \theta _ 1 } , \partial _ { \theta _ 2 } , \ldots , \partial _ { \theta _ n } ) q = 0 \hbox { f o r a l l } p \in I _ n \right \} ~ . \end{align*}


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


\begin{align*} f _ k ( x ) : = x ^ T A _ k x + 2 b _ k ^ T x + c _ k \end{align*}


\begin{align*} L ( s , \pi \otimes \pi ) = L ( s , \vee ^ 2 ) L ( s , \wedge ^ 2 ) , \end{align*}


\begin{align*} D _ { 2 k ; w _ 1 , w _ 2 } ^ { e ^ { 2 \Upsilon } g } \bigl ( u \otimes v \bigr ) = e ^ { ( w _ 1 + w _ 2 - 2 k ) \Upsilon } D _ { 2 k ; w _ 1 , w _ 2 } ^ { g } \bigl ( e ^ { - w _ 1 \Upsilon } u \otimes e ^ { - w _ 2 \Upsilon } v \bigr ) \end{align*}


\begin{align*} X = \{ \sum _ { n = 0 } ^ { \infty } a _ n z ^ n \mid \left ( a _ n / w _ n \right ) _ { n \in \mathbb { N } } \in \ell ^ 2 \} \end{align*}


\begin{align*} g ^ - ( x ^ { \mu } ) : = \sum _ { i _ 1 = 0 } ^ { \infty } \cdots \sum _ { i _ n = 0 } ^ { \infty } \frac { 1 } { i _ 1 ! \cdots i _ n ! } \partial _ 1 ^ { i _ 1 } \cdots \partial _ n ^ { i _ n } g ( \epsilon ( x ^ { \mu } ) ) ( \epsilon ( x ^ 1 ) - x ^ 1 ) ^ { i _ 1 } \cdots ( \epsilon ( x ^ n ) - x ^ n ) ^ { i _ n } . \end{align*}


\begin{align*} n \sum _ { \substack { a , c < k \\ \gcd ( a , c ) = 1 } } \left ( \frac 1 { a c } - \max \Big \{ \frac 1 a , \frac 1 c \Big \} \right ) - \frac n 2 \prod _ { p < k } \Big ( 1 - \frac 1 p \Big ) + O ( 1 ) . \end{align*}


\begin{align*} \langle \mathcal { A } _ q ^ s , \mathcal { A } _ q ^ p \rangle _ { L ^ 2 ( \mathbb { R , \mathbb { H } } ) } = K _ q ( p , s ) . \end{align*}


\begin{align*} \lambda ( Y ) : = \sup \lbrace \lambda ( Y , X ) : Y \subset X \rbrace . \end{align*}


\begin{align*} \Bigl [ I _ { n + 1 } \frac { \partial } { \partial x _ n } , Q \Bigr ] f = I _ { n + 1 } \frac { \partial } { \partial x _ n } ( Q f ) - Q \Bigl ( I _ { n + 1 } \frac { \partial } { \partial x _ n } \Bigr ) f = \frac { \partial Q } { \partial x _ n } f , f \in C ^ { \infty } ( \Omega ) . \end{align*}


\begin{align*} \Phi _ k : = \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 _ { \tau , k } \} \end{align*}


\begin{align*} \vert f _ { j } ( \mathbf 0 ) \vert = 0 , \end{align*}


\begin{align*} R = \max _ { q \in D } { \rm d i s t } ( q , p ) \end{align*}


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


\begin{align*} \begin{aligned} \dfrac { b _ { 1 , \delta } ( x ) } { 2 } & \geq - A + \underline { \delta } ^ \star ( \delta ) & \forall \ , x \geq x ^ \star + \delta \\ \dfrac { b _ { 1 , \delta } ( x ) } { 2 } & \leq A - \underline { \delta } ^ \star ( \delta ) & \forall \ , x \leq x ^ \star - \delta \ , , \end{aligned} \end{align*}


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


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


\begin{align*} \mathcal { M } _ { f _ { 1 , \infty } } ( N , \alpha + \delta , 6 \epsilon , \bigcup _ { n = N } ^ { \infty } Z _ { n , n ^ { - 2 } t } , \psi ) \leq \sum _ { n = N } ^ { \infty } \mathcal { M } _ { f _ { 1 , \infty } } ( N , \alpha + \delta , 6 \epsilon , Z _ { n , n ^ { - 2 } t } , \psi ) . \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*} \frac { \partial \tilde y } { \partial x } = y \cdot \frac { 1 } { 4 } g ^ { - \frac { 3 } { 4 } } \cdot \biggl ( \frac { 1 } { 6 } \int \limits _ { 0 } ^ 1 f _ { x y ^ { 4 } } ^ { ( 5 ) } ( x , s y ) ( 1 - s ) ^ 3 \ , d s \biggr ) , \end{align*}


\begin{align*} \operatorname { E } ( X ^ { - 1 } ) = \frac { \sqrt { \pi / \theta } } { 2 \operatorname { B } ( a , b ) } \sum _ { n = 0 } ^ b ( - 1 ) ^ n \binom { b - 1 } { n } \frac { 1 } { \sqrt { ( a + n ) ^ 3 } } . \end{align*}


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


\begin{align*} { \hat H } & = - \left ( ( 1 + \lambda r ^ 2 ) \hat { \Delta } + \lambda r \frac { \partial } { \partial r } + \lambda { \hat J } ^ 2 \right ) + { \cal V } ( r ) \\ & = - \left ( ( 1 + \lambda r ^ 2 ) \frac { \partial ^ 2 } { \partial r ^ 2 } + ( d - 1 + d \lambda r ^ 2 ) \frac { \partial } { \partial r } - \frac { { \hat J } ^ 2 } { r ^ 2 } \right ) + { \cal V } ( r ) , . \end{align*}


\begin{align*} \psi _ { 0 , H } ( t ) = \frac { ( n - 1 ) H } { ( n - 1 ) + t H } . \end{align*}


\begin{gather*} \int Q \bigl ( \oplus _ { i = 1 } ^ n f _ i \bigr ) \ , P _ \mathcal { U } ( d Q ) = \sum _ { i = 1 } ^ n \mu _ i ( f _ i ) \quad f _ i \in B ( \Omega _ i , \mathcal { A } _ i ) i \in I ; \end{gather*}


\begin{align*} \lim _ { \delta \rightarrow 0 + } \sup _ { n \in \mathbb { N } } \mathbb { P } \left ( \int _ 0 ^ { T - \delta } \Vert X _ n ( t + \delta ) - X _ n ( t ) \Vert _ Y ^ p d t > \epsilon \right ) = 0 . \end{align*}


\begin{align*} u _ x ( x , t ) = 2 i m ( x , t ) e ^ { 4 i \int _ { - \infty } ^ x | m ( s , t ) | ^ 2 d s } , \end{align*}


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


\begin{align*} { \bf { \bar q } } \left [ i \right ] = \sum \limits _ { l = 1 } ^ { L ' } { { { \bf { \bar H } } _ { l } ^ \bot { { { \bf { \bar X } } } _ { l } } } { \bf { d } } \left [ { i - { \kappa _ l } } \right ] } , \end{align*}


\begin{align*} \sum _ { k = 0 } ^ n \binom n k \frac { L _ { j k - 1 } } { L _ j ^ k } B _ { n - k } = \sqrt 5 \ , B _ n \Big ( \frac { \alpha ^ j } { L _ j } \Big ) , \qquad \mbox { $ n $ o d d } , \end{align*}


\begin{align*} \inf _ { x \in E , p _ n x = x _ n } \| x \| _ E \le b \| x _ n \| _ { E _ n } ( n \in \mathbb N , \forall x _ n \in E _ n ) \end{align*}


\begin{align*} \tau = \tau ' v , v = \sum _ { r = 1 } ^ j \lambda _ r ( e _ { i _ { r , 1 } } + \dots + e _ { i _ { r , d _ r } } ) \end{align*}


\begin{align*} \max _ { v \in \Lambda } e ( \alpha ^ { ( v ) } , G ^ { ( v ) } ) = e ( 0 , G ) . \end{align*}


\begin{align*} \prod _ { j \le A } ( \tau ( j ) n / j ) & \le \left ( \frac 1 A \sum _ { j \le A } \tau ( j ) \right ) ^ A \frac { n ^ { A } } { A ! } \\ & \le ( \log n ) ^ { A } \exp \Big ( A \log n - A \log A + A \Big ) \\ & \le \exp \Big ( \frac 3 2 A \log \log n + A \Big ) . \end{align*}


\begin{align*} \sum _ { r _ { Q } ( n ) = 0 } n \ll p ^ { 1 + \epsilon } m p ^ { - 1 / 4 + \epsilon } \ll p ^ { 5 / 2 + \epsilon } . \end{align*}


\begin{align*} h ( \upsilon ) = \frac 1 8 \left ( - f ( \upsilon ) + 8 \upsilon - f ( 2 \upsilon ) + \frac 1 3 f ( 3 \upsilon ) \right ) . \end{align*}


\begin{align*} \beta _ 1 ( u ) = \beta _ 2 ( u ) = \beta _ 3 ( u ) = 0 \Longrightarrow \beta _ 3 ( u _ { \varepsilon } ) = 0 . \end{align*}


\begin{align*} \frac { d } { d t } \theta _ u = \omega _ u + \frac { K } { | V | } \sum _ { v \colon \{ u , v \} \in E } \sin ( \theta _ v - \theta _ u ) , u \in V \end{align*}


\begin{align*} f ( x ) = \frac { 1 } { \mathrm { B } ( \alpha , \beta ) } \ , \phi ( x ) \ , \Phi ( x ) ^ { \alpha - 1 } \ , [ 1 - \Phi ( x ) ] ^ { \beta - 1 } , \end{align*}


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


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


\begin{align*} - \sum _ { j = 0 } ^ { p } B _ j n ^ { p - j } \sum _ { k = j } ^ p ( - 1 ) ^ k \binom { p } { k } \binom { k } { j } . \end{align*}


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


\begin{align*} \nabla _ Y \mathcal { P } ( X ) = - \mathcal { P } ( B ^ * ( Y , X ) ) - B ( Y , s ( X ) ) + B ^ * ( Y , t ( X ) ) . \end{align*}


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


\begin{align*} a _ { n , k } = \lambda ^ m a _ { n - 1 , k - 1 } + ( \frac { \lambda ^ m - 1 } { \lambda - 1 } ) ^ n a _ { n - 1 , k } . \end{align*}


\begin{align*} y ^ * = t \int _ { \R ^ n } \nabla f \ , d \gamma _ { y ^ * , t } , \end{align*}


\begin{align*} f ^ { - 1 } ( \overline { V } _ \bullet ) = \{ V _ \bullet ; V _ j = \overline { V } _ j j \in [ n ] \setminus \{ l \} , V _ { l - 1 } + X V _ i \subset V _ l \subset V _ { l + 1 } \} \end{align*}


\begin{align*} \begin{aligned} \langle \partial _ { t , x } ^ \beta f , \phi \rangle = \partial _ { t , x } ^ \beta \langle f , \phi \rangle - \langle f , \partial _ { t , x } ^ \beta \phi \rangle = - \langle f , \partial _ { t , x } ^ \beta \phi \rangle \ , , \end{aligned} \end{align*}


\begin{align*} ( \tilde { \kappa } _ 1 ) _ { i i } = \tilde { h } _ { 1 1 i i } + 2 \sum _ { p \neq 1 } \frac { \tilde { h } _ { 1 p i } ^ 2 } { \tilde { \kappa } _ 1 - \tilde { \kappa } _ p } = h _ { 1 1 i i } + 2 \sum _ { p \neq 1 } \frac { h _ { 1 p i } ^ 2 } { \kappa _ 1 - \tilde { \kappa } _ p } . \end{align*}


\begin{align*} f _ \xi ( z ) = \begin{cases} \frac { f ( z ) } { \xi - z } & , \\ 0 & . \end{cases} \end{align*}


\begin{align*} D ( x ) : = x \cdot A x . \end{align*}


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


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


\begin{align*} \hat { f } ( \chi ) = \int _ G f ( g ) \chi ( g ) d g \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*} V ^ { - 1 } ( \rho \circ \varphi ) V = V ^ { - 1 } \circ \rho \circ V ( V ^ { - 1 } \circ \varphi \circ V ) = 0 _ s \oplus \begin{bmatrix} ( V ^ { - 1 } \varphi V ) ^ { s + 1 } ( z ) \\ \vdots \\ ( V ^ { - 1 } \varphi V ) ^ n ( z ) \\ \end{bmatrix} . \end{align*}


\begin{align*} \tau _ y = \tau ' _ y v _ y , v _ y = \sum _ { \substack { 1 \leq r \leq j \\ C _ r \in O _ y } } \lambda _ r ( e _ { i _ { r , 1 } } + \dots + e _ { i _ { r , d _ r } } ) \end{align*}


\begin{align*} H = \Delta u + \frac { B } { 2 } | x | ^ 2 , \end{align*}


\begin{align*} R _ f \circ g = g \bullet _ \mu f . \end{align*}


\begin{align*} h = H [ b ] = 2 \int _ { \mathbb { R } ^ { 3 } } ( \phi - \phi _ { \infty } ) _ { + } \frac { G } { r ^ { 2 } } \dd x = 0 . \end{align*}


\begin{align*} a ( z ) = 1 + k \int _ { \mathbb { R } } u _ y ( y ) \psi ^ - _ { 2 1 } ( x ; k ) \mathrm { d } y . \end{align*}


\begin{align*} \begin{array} { r l } & a _ { n + 1 } \in f ( V ( h ) ) \\ \Leftrightarrow & \exists a \in V ( h ) \ , : \ , a _ { n + 1 } = f ( a ) \\ \Leftrightarrow & \exists a \in \R ^ { n } \ , : \ , ( a , a _ { n + 1 } ) \in V ( h , x _ { n + 1 } - f ) \\ \Leftrightarrow & a _ { n + 1 } \in \hat \pi ( V ( h , x _ { n + 1 } - f ) ) \ , . \\ \end{array} \end{align*}


\begin{align*} \phi _ p ( x ) = ( 1 + x ) ^ { - p / 2 } \end{align*}


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