\begin{align*} E [ X _ { \ell , \Delta } ^ 2 ] - E [ X _ { \ell , \Delta } ] = \Delta ^ 2 \nu ^ 2 E [ Z _ 0 ( \ell \Delta ) ^ 2 ] ( 1 + O ( \Delta ) ) . \end{align*}


\begin{align*} S \coloneqq \Lambda ^ { 1 . 5 + \delta } = ( I _ 2 - \Delta _ 2 ) ^ { 1 / 2 } ; \end{align*}


\begin{align*} \left \lbrace \begin{array} { l c r } ( d d ^ c u ) ^ n = H ^ n ( z , u ) \ , \omega ^ n & \textnormal { i n } & \in \Omega \\ \\ u = 0 & \textnormal { o n } & \partial \Omega , \end{array} \right . \end{align*}


\begin{align*} \forall \ , p \leq r < q , \ ; \ ; \ ; \ ; & \sharp \{ p \leq s \leq r \ , | \ , D _ { \lambda } ( s ) = \times \} > \sharp \{ p \leq s \leq r \ , | \ , D _ { \lambda } ( s ) = \circ \} \\ & \sharp \{ p \leq s \leq q \ , | \ , D _ { \lambda } ( s ) = \times \} = \sharp \{ p \leq s \leq q \ , | \ , D _ { \lambda } ( s ) = \circ \} . \\ \end{align*}


\begin{align*} L _ S [ f ] = \sum _ { T \subseteq S } ( - 1 ) ^ { | T | } E _ T [ f ] . \end{align*}


\begin{align*} M _ i - ( \times _ \dagger - \overline { \lambda } _ i - 1 ) & = M _ i - ( k _ i - \overline { \lambda } _ i - 1 ) + k _ i - \times _ \dagger \le M _ { i ' } - ( k _ { i ' } - \overline { \lambda } _ { i ' } - 1 ) + k _ { i ' } - \times _ \dagger \\ & = M _ { i ' } - ( \times _ \dagger - \overline { \lambda } _ { i ' } - 1 ) \end{align*}


\begin{align*} u _ { B , 1 } ( e ^ { i x } ) & = \exp ( ( x - \pi ) i B ) , 0 < x < 2 \pi , \end{align*}


\begin{align*} \mathcal { R } = \mathrm { E n d } _ { \kappa } ( \overline { A } _ 0 ) = \mathrm { M a t } _ n ( R ) , \mathcal { B } = \mathcal { R } \otimes _ { \mathbb { Z } } \mathbb { Q } = \mathrm { M a t } _ n ( B ) . \end{align*}


\begin{align*} \prod _ { \substack { ( a , b , c ) = 1 \\ a , b , c \geq 1 } } \left ( \frac { 1 } { 1 - x ^ a y ^ b z ^ c } \right ) ^ { \frac { c ^ 5 } { a ^ 3 b ^ 3 } } = \exp \left \{ L i _ 3 ( x ) L i _ 3 ( y ) L i _ { - 5 } ( z ) \right \} \end{align*}


\begin{align*} \l _ { { \rm m i n } } \ , \mathrm { t r } ( M ) \leq \mathrm { t r } ( X M ) = \mathrm { t r } ( M X ) \leq \l _ { { \rm m a x } } \ , \mathrm { t r } ( M ) \ , , \end{align*}


\begin{align*} P _ { i + 1 } = P _ i + \begin{cases} ( 1 , 0 ) , & w i t h \ p r o b a b i l i t y \ \alpha , \\ ( 0 , 1 ) , & w i t h \ p r o b a b i l i t y \ 1 - \alpha , \end{cases} \end{align*}


\begin{align*} \overline { \Pi } _ { \psi } [ \omega ] f ( [ \epsilon , x ] ) = e ^ { - \tfrac { \pi i } { 4 } } \nu ( \epsilon , \omega ) \int _ { \R } e ^ { 2 \pi i \epsilon x y } f ( [ \epsilon , - y ] ) d y ; \end{align*}


\begin{align*} b _ { n , j } b _ { \tau ( n ) , j } = b _ { n , j } b _ { n , j } \ge 0 . \end{align*}


\begin{align*} \mathcal { Q } ( f ) = \mathcal { A } ( f , f ) = T ( f ^ { 2 } ) - 2 f T ( f ) - 2 B ( A ( f ) , A ( f ) ) = 0 \end{align*}


\begin{align*} \left . \begin{array} { l l } \displaystyle u _ { \lambda } ( x ) = \lambda ^ { \frac { N } { 2 } } u ( \lambda x ) , v _ { \lambda } ( x ) = \lambda ^ { \frac { N } { 2 } } v ( \lambda x ) . \end{array} \right . \end{align*}


\begin{align*} f ( z ) = \sum _ { n = 1 } ^ { \infty } a _ n z ^ n , \ ; \mbox { f o r } \ ; z \in \mathbb { D } . \end{align*}


\begin{align*} { \Pr } _ S \left [ | C _ S ( v ) | \ge x \mid V _ j \cap S = W \right ] \le { \Pr } _ S \left [ | S ' ( v ) | \ge \frac { x } { \Delta + 1 } \mid V _ j \cap S = W \right ] \le { \Pr } _ { S } \left [ | C _ { S _ 2 } ( v ) | \ge \frac { x } { \Delta + 1 } \right ] . \end{align*}


\begin{align*} \overrightarrow { V _ \varepsilon } ^ { ( 0 ) } ( x ) = \nabla _ { \xi } p ( \xi ) \big | _ { \xi = \frac { x } { \varepsilon } } = \varepsilon \nabla _ x \big ( p ( \tfrac { x } { \varepsilon } ) \big ) , x \in \Omega ^ { ( 0 ) } _ \varepsilon , \end{align*}


\begin{align*} A ( t ) : = \left ( \begin{array} { c c } 0 & 1 \\ a ( t ) & 0 \end{array} \right ) , Q ( t ) : = \left ( \begin{array} { c c } 0 & 0 \\ q ( t ) - a ( t ) & 0 \end{array} \right ) F ( t , \xi ) : = \left ( \begin{array} { c } 0 \\ \widehat { f } ( t , \xi ) \end{array} \right ) . \end{align*}


\begin{align*} \Vert x _ 1 \bar { \otimes } x _ 2 \Vert _ p = \Vert x _ 1 \Vert _ p \ , \Vert x _ 2 \Vert _ p \end{align*}


\begin{align*} \sum _ { i = 1 } ^ n \hat f ( \{ i \} ) ^ 2 \leq 7 5 0 \ , \alpha ^ 2 \log ( 1 / \alpha ) . \end{align*}


\begin{align*} x _ i ' = x _ { \sigma ( i ) } b _ { i , j } ' = b _ { \sigma ( i ) , \sigma ( j ) } 1 \le i \le m , 1 \le j \le n , \end{align*}


\begin{align*} \partial _ { \bar { \nu } _ { \bar { \xi } _ i } } u ^ { ( i ) } _ { p \alpha + k } = \big ( w ^ { ( i ) } _ { p \alpha + k - 1 } + u ^ { ( i ) } _ { p \alpha + k - 1 } \big ) \ , \overline { V } ^ { ( i ) } \boldsymbol { \boldsymbol { \cdot } } \bar { \nu } _ { \bar { \xi } _ i } \ , - \ , \delta _ { p \alpha + k - 1 , \ , \alpha - 1 } \ , \varphi ^ { ( i ) } \big ( \bar { \xi } _ i , x _ i , t \big ) , \bar { \xi } _ i \in \partial \Upsilon _ i , \end{align*}


\begin{align*} \Psi _ { \omega } ( x ) : = \omega ^ { \frac { 1 } { 2 } } \Psi ( \omega ^ { \frac { 1 } { 2 } } x ) . \end{align*}


\begin{align*} \delta _ { x K } ( \sigma ) : = \begin{cases} 1 & \mbox { i f } \sigma \in x K \\ 0 & \mbox { o t h e r w i s e } . \end{cases} \end{align*}


\begin{align*} S _ { \mathrm { m a i n } } = \sum _ { d _ 1 , d _ 2 < D } \frac { 1 } { \sqrt { d _ 1 d _ 2 } } \sum _ { I d _ 1 , J d _ 2 < D } \alpha _ { I d _ 1 } \bar { \alpha } _ { J d _ 2 } \sum _ { L , M \geq 1 } \frac { G _ k ( L M d _ 1 d _ 2 \slash p ) } { \sqrt { L M } } \ , \ \delta _ { I L \ , J M } \end{align*}


\begin{align*} \begin{cases} w \in H ^ { s _ M } ( \mathbb { R } ^ n ) ( P ( ( - \Delta ) ^ s ) + q ) w = 0 \Omega w | _ { \Omega ^ c } = 0 , \\ w \equiv 0 . \end{cases} \end{align*}


\begin{align*} \left ( \mathord { \operatorname { p r o j } } _ { K _ N \cap Q } \circ { \operatorname { p r o j } _ { K _ N } } ^ { - 1 } \right ) \# \mu _ N = \left ( \mathord { \operatorname { p r o j } } _ { K _ N \cap Q } \circ { \mathrm { p r o j } _ { Q } } ^ { - 1 } \right ) \# \gamma , \end{align*}


\begin{align*} \widetilde { M } _ 1 \bar { \otimes } \widetilde { M } _ 2 = ( M _ 1 \bar { \otimes } M _ 2 ) \bar { \rtimes } _ \sigma G \supset ( M _ 1 \bar { \otimes } M _ 2 ) \bar { \rtimes } _ \sigma H = \widetilde { M _ 1 \bar { \otimes } M _ 2 } , \end{align*}


\begin{align*} M _ q ( G ) \geq \eta ( B ) \geq T + \max _ { 1 \le i _ 1 < \ldots < i _ q \le s } \sum _ { j = 1 } ^ q a _ { i _ j } , \end{align*}


\begin{align*} \eta _ 1 ^ { n } = \inf \{ E ( w ) \ , ; \ , w \in \mathcal E ^ 1 ( \Omega ) , \ , I _ g ( w ) = 1 \} . \end{align*}


\begin{align*} \mathcal { N } ( A , \| \cdot \| , \delta ) : = \min \Big \{ K \in \mathbb { N } _ { > 0 } \ , \ , \Big | \ , \ , \exists z _ 1 , \ldots z _ K \in \R ^ d \ , \ , \colon \ , \ , A \subset \bigcup _ { k = 1 } ^ K B ^ { \rm c l } ( z _ k , \delta ) \Big \} . \end{align*}


\begin{align*} \nabla \tilde { u } _ m ^ i ( y ) = ( \mu _ m ^ i ) ^ { \frac { n } { 2 } } \bigg [ \Big ( \nabla u _ m ( x ) \Big ) \chi _ m ^ i ( x ) + u _ m ( x ) \Big ( \nabla \chi _ m ^ i ( x ) \Big ) \bigg ] , \end{align*}


\begin{align*} S _ k ^ { l } = \tilde S _ k ^ { l } = 0 , l = 1 , \dots , p ; k \ne l - 1 + ( p + 1 ) k ' , \ ; k ' \in \mathbb { Z } _ + . \end{align*}


\begin{align*} \limsup _ { j \to + \infty } \big \| \exp ( A ) ^ j \exp ( B ) ^ { - \lfloor \varepsilon j \rfloor } \big \| = \infty , \mbox { o r } \limsup _ { j \to - \infty } \big \| \exp ( A ) ^ j \exp ( B ) ^ { - \lfloor \varepsilon j \rfloor } \big \| = \infty , \end{align*}


\begin{align*} \begin{aligned} \Pr ( X _ n > k ) & = \frac { 1 } { 2 ^ n } \sum _ { j = k + 1 } ^ n \binom { n } { j } & \\ & = \frac { 1 } { 2 } \sum _ { j = k } ^ { n - 1 } \frac { 1 } { 2 ^ j } \binom { j } { k } , & \end{aligned} \end{align*}


\begin{align*} a u = b v , a x = b y , c x = d y , \end{align*}


\begin{align*} \left \langle \boldsymbol { \nu } ; { \boldsymbol { M } } ( \boldsymbol { a } , - \log ( 1 - x ) ) \right \rangle & = \frac { 1 } { 1 - q } H ( A ( - \log ( 1 - x ) ) ) F ( A ( - \log ( 1 - x ) ) ) \\ & = \frac { 1 } { 1 - q } H ( I ^ { - 1 } _ { \boldsymbol { a } } ( q x ) / q ) ) F ( I ^ { - 1 } _ { \boldsymbol { a } } ( q x ) / q ) ) . \end{align*}


\begin{align*} p ' _ k ( x ) & = \frac { - \beta _ k e ^ { \alpha _ k h _ k + \beta _ k x + \gamma _ k } } { ( 1 + e ^ { \alpha _ k h _ k + \beta _ k x + \gamma _ k } ) ^ 2 } = - \beta _ k \frac { 1 } { ( 1 + e ^ { \alpha _ k h _ k + \beta _ k x + \gamma _ k } ) } \frac { e ^ { \alpha _ k h _ k + \beta _ k x + \gamma _ k } } { ( 1 + e ^ { \alpha _ k h _ k + \beta _ k x + \gamma _ k } ) } = - \beta _ k p _ k ( x ) ( 1 - p _ k ( x ) ) . \end{align*}


\begin{align*} \phi _ 2 \phi _ 1 = \psi _ 2 \tau \widehat { \tau } \widehat { \psi _ 1 } = \psi _ 2 \widehat { \psi _ 1 } [ \# G ] \end{align*}


\begin{align*} \left [ C \right ] _ { i , j } ^ { } = \begin{cases} 1 & \textnormal { i f } j = i = 0 ; \\ 1 - p _ j ^ { } & \textnormal { i f } j = i \neq 0 ; \\ p _ j ^ { } & \textnormal { i f } j = i + 1 ; \\ 0 & \textnormal { o t h e r w i s e } ; \end{cases} \end{align*}


\begin{align*} H _ { J } : = ( X _ { k } , k \in J ) \subset \mathbb { R } ^ { N } . \end{align*}


\begin{align*} \langle F , g \rangle = 1 . \end{align*}


\begin{align*} \lim _ { n \to \infty } \sup _ { x \in e M _ { + } e , ~ x \neq 0 } \abs { \frac { ( S _ n ( \mu ) - \bar { \mu } ) ( x ) } { \tau ( x ) } } = 0 \end{align*}


\begin{align*} \left \{ \begin{aligned} d \mathcal { Z } _ t ^ u & = e ^ { - \frac { \beta } { 2 } t } \mathcal { Z } _ t h ( x _ t ^ u , u _ t ) d \xi _ t , \ t \in [ 0 , \infty ) , \\ \mathcal { Z } _ 0 ^ u & = 1 . \end{aligned} \right . \end{align*}


\begin{align*} ( \rho \otimes _ 0 \varphi ) ^ { \sigma ' , s ' } _ { \sigma , s } ( a ' g a ) = \rho _ { a ' g a \sigma } ^ { \sigma ' } \cdot \varphi _ s ^ { s ' } ( a ' g a ) = & \rho _ { g \sigma } ^ { ( a ' ) ^ { - 1 } \sigma ' } \cdot \varphi _ s ^ { s ' } ( g ) \\ = & \rho _ { g \sigma } ^ { \sigma ' } \cdot \varphi _ s ^ { s ' } ( g ) = ( \rho \otimes _ 0 \varphi ) ^ { \sigma ' , s ' } _ { \sigma , s } ( g ) , \end{align*}


\begin{align*} \nu _ i ( a _ j ) = \nu _ i ( f _ j Q _ i ' ) < \nu _ i ( b _ j Q _ i ) . \end{align*}


\begin{align*} | \langle v ( t ) \rangle _ { K , \sigma } | ^ 2 & = \frac { \sigma ^ 4 \varepsilon ^ 2 } { ( \sigma ^ 2 + h ^ 2 ) ^ 2 } | \eta ( K ) | + \frac { 2 \sigma ^ 2 \varepsilon ^ 4 } { \sigma ^ 2 + h ^ 2 } \Im \big ( \overline { \eta ( K ) } V _ K ^ 1 ( t ) e ^ { - i t \theta _ K } \big ) \\ & \quad + \varepsilon ^ 6 \Big ( | V _ K ^ 1 ( t ) | ^ 2 + \frac { 2 \sigma ^ 2 } { \sigma ^ 2 + h ^ 2 } \Im \big ( \overline { \eta ( K ) } V _ K ^ 2 ( t ) e ^ { - i t \theta _ K } \big ) \Big ) + \mathcal { O } ( \varepsilon ^ 8 ) . \end{align*}


\begin{align*} { \rm W F } _ h ' ( \chi e ^ { - i t _ 0 h ^ { - 1 } \tilde { P } _ h ( z ) } \tilde { R } _ h ( z ) \chi ) \cap S ^ * ( \mathcal { M } \times \mathcal { M } ) & \subset \\ \kappa \big ( \{ ( x , \xi , y , \eta ) \ , : \ , ( e ^ { - t _ 0 H _ p } ( x , \xi ) , y , \eta ) \in \Delta ( T ^ * \mathcal { U } ) \cup & \ ; \Omega _ + \cup E ^ * _ + \times E ^ * _ - \setminus \{ 0 \} \xi = 0 , \eta \neq 0 \} \big ) . \end{align*}


\begin{align*} [ { f } , { f } ] ( X , Y ) = ( { f } \nabla _ Y { f } - \nabla _ { { f } Y } { f } ) X - ( { f } \nabla _ X { f } - \nabla _ { { f } X } { f } ) Y ; \end{align*}


\begin{align*} \lim _ { r \downarrow 0 } \frac { 1 } { r } { \rm d i s t } \ , ( ( x + \pi ) \cap \overline { B } _ r ( x ) , ( x + \pi ) \cap F \cap \overline { B } _ r ) = 0 \ , . \end{align*}


\begin{align*} t \cdot A = ( \mu _ 1 , \ldots , \mu _ n , \mu _ { n ^ * } , \ldots , \mu _ { 1 ^ * } ) \cdot A . \end{align*}


\begin{align*} K = \mathrm { E n d } _ { \overline { K } _ 0 } ( A _ 0 ) \otimes _ { \mathbb { Z } } \mathbb { Q } \hookrightarrow \mathcal { B } = \mathrm { E n d } _ { \overline { \mathbb { F } } _ \ell } ( \overline { A } _ 0 ) \otimes _ { \mathbb { Z } } \mathbb { Q } . \end{align*}


\begin{align*} & \beta _ { ( n - 1 ) P + 1 } = . . . = \beta _ { n P } , \ n \in \mathcal { N } , \\ & \beta _ i \in [ 0 , 1 ] , i \in \mathcal { I } , \\ & \alpha _ n = \frac { \sum _ { p \in \mathcal { P } } \beta _ { ( n - 1 ) P + p } } { P } , n \in \mathcal { N } . \end{align*}


\begin{align*} D _ n ( a , b ) : & = \left [ \binom { 2 i + 2 a } { i - j + a - b } - \binom { 2 i + 2 a } { i - j + a - b - 1 } \right ] _ { i , j = 0 } ^ { n - 1 } \end{align*}


\begin{align*} \R _ F ( x ) = \frac 1 3 , \ \ W _ F ( x ) = \frac 1 3 , \end{align*}


\begin{align*} \times \exp \left \{ \sum _ { n = 1 } ^ { \infty } \left ( \frac { - ( n + 1 ) ^ 3 y ^ { n + 1 } + y ^ 3 + 4 y ^ 2 + y } { ( 1 - y ) ^ 4 } \right ) \frac { z ^ n } { n ^ 4 } \right \} \end{align*}


\begin{align*} \det ( u _ { j \bar k } ( 0 ) ) = a u _ { n \bar n } ( 0 ) + b , \end{align*}


\begin{align*} J = \lambda \sum \limits _ { i = 1 } ^ N { { G _ { { \alpha _ 1 } , { \beta _ 1 } } } \left ( { { e _ i } } \right ) } + \left ( { 1 - \lambda } \right ) \sum \limits _ { i = 1 } ^ N { \sum \limits _ { j = 1 } ^ N { { G _ { { \alpha _ 2 } , { \beta _ 2 } } } \left ( { { e _ i } - { e _ j } } \right ) } } , \end{align*}


\begin{align*} \sum _ { y < p \le x ^ { 1 / 3 } } \frac { 1 } { p \log p } \omega \left ( \frac { \log x } { \log p } - 1 \right ) & = \frac { 1 } { \log x } \int _ { 3 ^ - } ^ { u } v \omega ( v - 1 ) \ , d G ( v ) \\ & = \frac { 1 } { \log x } \left ( \int _ { 3 } ^ { u } \omega ( v - 1 ) \ , d v + \int _ { 3 ^ - } ^ { u } v \omega ( v - 1 ) \ , d \left ( G ( v ) - \log \frac { v } { 2 } \right ) \right ) . \end{align*}


\begin{align*} \mathbf { c } _ 0 \cdot \mathbf { P } ( \Phi ) = - 2 E ( \Phi ) \end{align*}


\begin{align*} d t - \left ( \textstyle \sum _ { i = 1 } ^ { n - 2 } y _ i \ , d x _ i \right ) - y ' \ , d x ' + \phi ( x ' ) \ , d y ' - y '' \ , d x '' , \end{align*}


\begin{align*} F _ H ( w ) = \bigcap _ { y \in I _ H ( w ) } \left \{ X ^ y _ { [ 0 , \tau _ { H , w } ] } \subseteq B _ H ( w ) \right \} . \end{align*}


\begin{align*} \vec { H } ( \lambda _ t ) = \vec { h } _ { \bar u ( t ) } ( \lambda _ t ) , \qquad \forall \ , t \in [ 0 , 1 ] . \end{align*}


\begin{align*} \Psi _ 2 \left ( \rho _ 1 ( y _ 1 , y _ 1 ^ \prime ) ^ { p } , \boldsymbol { d } _ { \mathcal { S } _ 2 } ( s _ 2 , s _ 2 ^ \prime ) ^ p \right ) = \Psi _ 2 \circ \psi _ 2 \left ( \boldsymbol { d } _ { \mathcal { Y } _ 1 } ( y _ 1 , y _ 1 ^ \prime ) ^ { p } , \boldsymbol { d } _ { \mathcal { Y } _ 2 } ( y _ 2 , y _ 2 ^ \prime ) ^ p , \boldsymbol { d } _ { \mathcal { X } } ( x , x ^ \prime ) ^ p \right ) . \end{align*}


\begin{align*} k x _ p + l y _ p = n x _ p \leq m y _ p \end{align*}


\begin{align*} \mathbf { G } ^ 1 [ \ell , \sigma ] : = \epsilon ( \sigma ^ { - 1 } ) \mathbf { G } ^ 1 [ \ell ] \epsilon ( \sigma ) \end{align*}


\begin{align*} F ( n _ 1 , n _ 2 ) = \left ( \frac { \partial } { \partial t _ 1 } \right ) ^ { n _ 1 } \left ( \frac { \partial } { \partial t _ 2 } \right ) ^ { n _ 2 } \det \left ( g _ { i + j + 1 } ( x , t _ 1 , t _ 2 ) \right ) _ { i , j = 0 , \ldots , k - 1 } \Bigg | _ { t _ 1 = t _ 2 = 0 } . \end{align*}


\begin{align*} \alpha = \alpha \left ( \left \{ \left . \alpha _ i \ \right | \ Q _ i \in I _ { n _ 0 } \right \} \right ) . \end{align*}


\begin{align*} \varphi ' ( t ) - N c = - N ^ 2 \ , . \end{align*}


\begin{align*} \frac { y } { 1 - p y } - \frac { x } { 1 - p x } = \frac { y - x } { ( 1 - p y ) ( 1 - p x ) } , \end{align*}


\begin{align*} J _ 5 \geq & - C s ^ 2 \lambda ^ 4 \iint _ Q \xi \left | A \nabla \eta \cdot \nabla \eta \right | ^ 2 | u | ^ 2 d x d y d t - C \lambda ^ 2 \iint _ { Q } \xi \left | A \nabla u \cdot \nabla \eta \right | ^ 2 d x d y d t \\ & - C s ^ 2 \lambda ^ 3 \int _ 0 ^ T \int _ { \omega _ 0 } \xi | u | ^ 2 d x d y d t - C \lambda \iint _ Q \xi A \nabla u \cdot \nabla u d x d y d t . \end{align*}


\begin{align*} R _ { c } = \log _ { 2 } \left ( \det \left [ \left ( \rho _ { f } \hat { \textbf { G } } _ { c c } ^ T \textbf { P } _ { c } \textbf { P } _ { c } ^ { H } \hat { \textbf { G } } _ { c c } ^ * \right ) \textbf { R } _ { c } ^ { - 1 } + \textbf { I } _ { K _ c } \right ] \right ) \end{align*}


\begin{align*} \Phi ( \pi ) = \tilde { \delta } \mu _ k \tilde { \alpha } _ k \mu _ { k - 1 } \tilde { \alpha } _ { k - 1 } \cdots \mu _ 1 \tilde { \alpha } _ 1 \mu _ 0 . \end{align*}


\begin{align*} y ( q x ) = \sin ( x ) y ( x ) . \end{align*}


\begin{align*} & \sum _ { i \neq l } ^ k \int _ { \mathcal { A } _ l } U _ { \mu _ i , \xi _ i } \bigg | \ln \ln \Big ( e + ( - 1 ) ^ l U _ { \mu _ l , \xi _ l } + \sum _ { i \neq l } ^ k ( - 1 ) ^ i U _ { \mu _ i , \xi _ i } \Big ) \bigg | ^ { \frac { 2 n } { n + 2 } } d x = O \Big ( \frac { \epsilon } { | \ln \epsilon | ^ 2 } \Big ) . \end{align*}


\begin{align*} \limsup _ { \ell \to \infty } h _ { \mu } ( Z , \lambda _ { \ell , 0 } ) \le h _ { \mu } ( Z , \lambda _ { \kappa } ) = \frac { 1 } { 2 } \sum _ { i = 1 } ^ { 2 } h _ { \mu } \left ( Z , \lambda _ { \delta _ { A _ { i } } } \right ) . \end{align*}


\begin{align*} & \varGamma _ L \cap N _ { U _ r } \\ & = \left \{ [ \mathrm { i d } _ { U _ r } , \mathrm { i d } _ { ( L _ { r + 1 } ) _ \R } , \mathbf { x } , \psi ] \mid \mathbf { x } \in ( L _ { r + 1 } ) ^ { r + 1 } \hbox { a n d } b _ { i j } + \frac { ( x _ i , x _ j ) } { 2 } \in \Z \hbox { f o r } 0 \le i , j \le r \right \} . \end{align*}


\begin{align*} & ( \pounds _ { \xi _ i } { f } ) X \overset { \eqref { 3 . 3 B } } = \nabla _ { \xi _ i } ( { f } X ) - \nabla _ { { f } X } \ , \xi _ i - { f } ( \nabla _ { \xi _ i } X - \nabla _ { X } \ , \xi _ i ) \\ & \ = ( \nabla _ { \xi _ i } { f } ) X - \nabla _ { { f } X } \ , \xi _ i + { f } \nabla _ X \ , \xi _ i . \end{align*}


\begin{align*} \varphi ( x ) = \frac { m _ i ( x ) } { f ( x ) ^ { k } } \rlap { . } \end{align*}


\begin{align*} Z : = \frac { P } { R T \rho } = A + B \rho + C \rho ^ { 2 } + \cdots \end{align*}


\begin{align*} \sup _ { y \in B _ 1 } \abs { p _ h ( y ) - p _ 0 } \leq \sup _ { y \in B _ 1 } \left \{ \abs { p _ h ( y ) - p _ h ^ 0 } + \abs { p _ h ^ 0 - p _ 0 } \right \} = \omega ( \sigma _ h ) + { \rm o } ( 1 ) , \end{align*}


\begin{align*} R _ 0 = R _ 0 ( d , \nu ) : = \left ( \frac { d } { \nu \omega _ d } \right ) ^ { 1 / d } \end{align*}


\begin{align*} \lambda _ s ( b ) \lambda _ t ( b ' ) \xi ( r ) & = b \lambda _ t ( b ' ) \xi ( s ^ { - 1 } r ) = b b ' \xi ( t ^ { - 1 } s ^ { - 1 } r ) = b b ' \xi ( ( s t ) ^ { - 1 } r ) \\ & = \lambda _ { s t } ( b b ' ) \xi ( r ) , \end{align*}


\begin{align*} ( L _ a - c ) \phi _ 1 & = L _ a \phi _ 1 - c \phi _ 1 \\ & = - ( \gamma _ 1 + c ) \phi _ 1 \\ & \leq 0 . \end{align*}


\begin{align*} \prod _ { \substack { ( a , b , c ) = 1 \\ a , b , c \geq 1 } } \left ( \frac { 1 } { 1 - x ^ a y ^ b z ^ c } \right ) ^ { \frac { 1 } { a ^ s b ^ t c ^ u } } = \exp \left \{ \left ( \sum _ { i = 1 } ^ { \infty } \frac { x ^ i } { i ^ s } \right ) \left ( \sum _ { j = 1 } ^ { \infty } \frac { y ^ j } { j ^ t } \right ) \left ( \sum _ { k = 1 } ^ { \infty } \frac { z ^ k } { k ^ u } \right ) \right \} . \end{align*}


\begin{align*} X ( K ) \cap \tilde { \Gamma } = \bigcup _ { i = 1 } ^ \ell \left ( S _ i + \Gamma _ i \right ) , \end{align*}


\begin{align*} u ( t ) = A \sin ( t + B ) , \end{align*}


\begin{align*} \begin{aligned} & - M w _ i \leq \gamma _ i \leq M w _ i \\ & - M ( 1 - w _ i ) \leq \sum _ { j \neq i } v _ { i j } - x _ { 0 , i } \leq M ( 1 - w _ i ) \end{aligned} \end{align*}


\begin{align*} \sum _ { n = 1 } ^ { \infty } \frac { p ^ { 2 } + \lambda _ { n } ^ { 2 } } { p \left ( p + \frac { 1 } { \pi } \right ) + \lambda _ { n } ^ { 2 } } \ , \frac { e ^ { 2 \pi \lambda _ { n } } } { \left ( \sigma ( \lambda _ { n } ) e ^ { 2 \pi \lambda _ { n } } - 1 \right ) ^ { 2 } } \cdot \left \{ \pi \ , \sigma \left ( \lambda _ { n } \right ) + \frac { p } { \left ( p - \lambda _ { n } \right ) ^ { 2 } } \right \} = \frac { \pi } { 2 4 } \cdot \frac { 1 + \frac { 3 } { \pi p } ( 1 + \frac { 1 } { \pi p } ) } { \left ( 1 + \frac { 1 } { \pi p } \right ) ^ { 3 } } - \frac { 1 } { 8 } . \end{align*}


\begin{align*} ( M \vec x ) _ i = \max _ { j \in J _ i } \{ M _ { i j } + x _ j \} . \end{align*}


\begin{align*} T ( u _ { B , \tau } ) & = ( S \otimes I ) T ( u _ { J , \tau } ) ( S ^ { - 1 } \otimes I ) , \\ T _ n ( u _ { B , \tau } ) & = ( S \otimes I _ n ) T _ n ( u _ { J , \tau } ) ( S ^ { - 1 } \otimes I _ n ) . \end{align*}


\begin{align*} S _ { \mathrm { o f f } } ^ { c \geq p ^ 2 } : = \sum _ { d _ 1 , \ , d _ 2 < D } \ , \ , \ , \ & \sum _ { I d _ 1 , \ , J d _ 2 < D } x _ { I d _ 1 } \bar { x } _ { J d _ 2 } \sqrt { I J } \sum _ { L , M \geq 1 } \frac { G _ k ( L M d _ 1 d _ 2 \slash p ) } { \sqrt { L M } } \\ & \times \left ( 2 \pi i ^ { 2 k } \sum _ { \substack { p | c \\ c \geq p ^ 2 } } \frac { \mathcal { S } ( I L , J M , c ) } { c } \ , J _ { 2 k - 1 } \left ( \frac { 4 \pi \sqrt { I L J M } } { c } \right ) \right ) . \end{align*}


\begin{align*} 0 = \sum _ { d _ 1 , d _ 2 < D } \frac { 1 } { \sqrt { d _ 1 d _ 2 } } \sum _ { I d _ 1 , J d _ 2 < D } \alpha _ { I d _ 1 } \bar { \alpha } _ { J d _ 2 } & \sum _ { L , M \geq 1 } \frac { G _ k ( L M d _ 1 d _ 2 \slash p ) } { \sqrt { L M } } \\ & \times \sum _ { f \in \mathcal { B } _ { 2 k } ( p ) } \omega _ f \lambda _ f ( I d _ 1 ) \lambda _ f ( J d _ 2 ) \lambda _ f ( L d _ 1 ) \lambda _ f ( M d _ 2 ) . \end{align*}


\begin{align*} S & = \left \{ ( X _ i ^ \pm , \underline { X } _ i ^ \pm ) : 0 \leq i \leq 1 \right \} \cup \left \{ ( X _ i ^ + , \underline { X } _ i ^ + ) : \ 2 \leq i \leq k \right \} \cup \left \{ ( X _ i ^ + , \underline { X } _ i ^ - ) : \ 2 \leq i \leq k \right \} , \end{align*}


\begin{align*} \hat { \Phi } ( y ) : = \int _ { - \infty } ^ { \infty } \Phi ( t ) e ^ { - i t y } d t = \sqrt { 2 \pi } \Phi ( y ) . \end{align*}


\begin{align*} t = \tfrac { 1 } { \sqrt { N } } \ , , s = \tfrac { 1 } { \sqrt { L } } = \sqrt { \tfrac { M } { N } } \ , . \end{align*}


\begin{align*} \{ b \in B : b p = 0 \} \end{align*}


\begin{align*} & \underline m ( \tau _ i , \varepsilon ) : = \max \{ n \in \mathbb N : n \varepsilon \leq \tau _ i \} , \\ & \underline m ( \tau _ i , \varepsilon ) \varepsilon = , \\ & \overline m ( \tau _ i , \varepsilon ) : = \min \{ n \in \mathbb N : n \varepsilon \geq \tau _ i \} , \\ & \overline m ( \tau _ i , \varepsilon ) \varepsilon = . \end{align*}


\begin{align*} T ( f ) ( z ) : = - \frac { 1 } { \pi } \iint _ D \frac { f ( \zeta ) } { \zeta - z } \ , d \xi \ , d \eta , \end{align*}


\begin{align*} \sigma _ { R } ^ 2 ( t ) = \sum _ { n \geq 1 } \frac { 1 } { n ! } \int _ { - R } ^ { R } \int _ { - R } ^ R \gamma _ n ( t , x - y ) d x d y . \end{align*}


\begin{align*} \Phi ( x , y ) = \mu _ y ( u ) e ^ { \gamma } x \log y \prod _ { p \leq y } \left ( 1 - \frac { 1 } { p } \right ) + O \left ( \frac { x R ( y ) } { \log y } \right ) . \end{align*}


\begin{align*} \widehat { P } ^ { 2 n / 3 - 2 / 3 - \delta ( 1 - n / 8 ) + \varepsilon } \widehat { Y } \ll \widehat { P } ^ { 2 n / 3 + 1 / 3 - \eta - \delta ( 1 - n / 8 ) + \varepsilon } = \widehat { P } ^ { n - 5 - \eta + \varepsilon } , \end{align*}


\begin{align*} \xi ( \boldsymbol { p } ) = \min _ { 1 \leq i \neq j \leq \ell } | p _ i - p _ j | . \end{align*}


\begin{align*} \left \{ \begin{array} { @ { \ , } l l l } p ( q + a ) = 0 \\ q ^ 2 = a ^ 2 , \end{array} \right . \end{align*}


\begin{align*} 0 \leq a \leq \sum _ { j = 1 } ^ k w _ j a _ j , \ 0 \leq a _ j , w _ j \ \forall j \implies a ^ p \leq ( \sum _ j w _ j ) ^ p \sum _ { j = 1 } ^ k a _ j ^ p \ \ \forall p \in ( 0 , \infty ) . \end{align*}


\begin{align*} \sigma _ { k - 3 } ( \ell ) = \sigma _ { \frac { p - 1 } { 2 } } ( \ell ) \equiv 0 \pmod { p } \chi _ { - p } ( \ell ) = - 1 . \end{align*}


\begin{align*} \nu ( x ) = \frac { D u ( x ) - x } { | D u ( x ) - x | } \forall \ x \in \mathcal F . \end{align*}


\begin{align*} & E _ { \mu , \xi } = \mbox { s p a n } \left \{ P \psi ^ h _ { \mu , \xi } : h = 0 , 1 , \cdots , n , \ i = 1 , \cdots , k \right \} , \\ & E _ { \mu , \xi } ^ \perp = \left \{ \phi \in H ^ 1 _ 0 ( \Omega ) : \langle \phi , P \psi ^ h _ { \mu , \xi } \rangle = 0 : h = 0 , 1 , \cdots , n , \ i = 1 , \cdots , k \right \} . \end{align*}


\begin{align*} \int _ { \R ^ d } \left ( \exp \left | \frac { e ^ { - t H ^ { \beta } } g } { \lambda } \right | ^ p - 1 \right ) \ , d x & \leq \sum _ { k = 1 } ^ { \infty } \frac { { C ^ { p k } } t ^ { - \frac { d } { 2 \beta } ( \frac 1 q - \frac { 1 } { p k } ) { p k } } \| g \| _ { L ^ q } ^ { p k } } { k ! \lambda ^ { p k } } \\ & = t ^ { \frac { d } { 2 \beta } } \left ( \exp \left ( \frac { { C } t ^ { - \frac { d } { 2 \beta q } } \| f \| _ { L ^ q } } { \lambda } \right ) ^ p - 1 \right ) . \end{align*}


\begin{align*} B ^ { k + 2 } = B ^ { k + 1 } \cup \{ s \} \setminus \{ t \} = B ^ { k } \cup \{ p , s \} \setminus \{ q , t \} . \end{align*}


\begin{align*} 0 = \int _ 0 ^ { \rho } r \phi ^ n ( r ) \mathrm { d } r \int _ { \mathbb { S } ^ n } x _ l \mathrm { d } A + \rho ^ 2 \phi ^ n ( \rho ) \int _ { \mathbb { S } ^ n } u x _ l \mathrm { d } A + O ( \varepsilon ) \| u \| _ { L ^ 2 ( \mathbb { S } ^ n ) } . \end{align*}


\begin{align*} | ( \partial u ) _ { B _ { j _ 1 + 1 } } | \leq | ( \partial u ) _ { B _ { j _ 0 } } | + \left ( \frac { \sigma } { 2 } \right ) ^ { - \frac { d } { p } } \sum _ { j = j _ 0 } ^ { j _ 1 } \mathfrak { E } _ p ( \partial u , B _ j ) \leq \frac { 1 } { 1 6 } \lambda . \end{align*}


\begin{align*} \partial _ { v _ 1 \cdots v _ d } : = \partial _ { v _ 1 } \cdots \partial _ { v _ d } v _ 1 , \ldots , v _ d \in W . \end{align*}


\begin{align*} \mathcal { F } _ { \overline { c } } ^ { o p } \left ( b \right ) : = \mathcal { F } ^ { o p } \left ( \overline { c } , b \right ) . \end{align*}


\begin{align*} b ( u \circ v ) + b ( v \circ u ) = 0 , \forall u , v \in T , \end{align*}


\begin{align*} \left \{ \begin{array} { l l } & - \dfrac { 1 } { 2 m _ 1 } \Delta u ( x ) + \omega _ 1 u ( x ) = - a _ 1 v ( x ) u ( x ) , ( 1 . 5 a ) \\ \\ & - \dfrac { 1 } { 2 m _ 2 } \Delta v ( x ) + \omega _ 2 v ( x ) = - a _ 2 u ^ 2 ( x ) , ~ ( 1 . 5 b ) \end{array} \right . \end{align*}


\begin{align*} \tilde { \xi } _ { \ast } \left ( \bar { m } _ { K } \times \left ( p _ { \ast } \left ( m _ { K \cap Q } \right ) \times \lambda \right ) \right ) = \bar { m } _ { K } \times \xi _ { \ast } \left ( p _ { \ast } \left ( m _ { K \cap Q } \right ) \times \lambda \right ) = \bar { m } _ { K } \times \eta . \end{align*}


\begin{align*} W = h \tfrac { 1 } { 2 } \nabla _ \xi H ^ 2 ( \nabla w ) \end{align*}


\begin{align*} E [ X _ { \ell , \Delta } ^ 2 | { \cal F } _ { ( \ell + 1 ) \Delta } ] - E [ X _ { \ell , \Delta } | { \cal F } _ { ( \ell + 1 ) \Delta } ] = E [ X _ { \ell , \Delta } | { \cal F } _ { ( \ell + 1 ) \Delta } ] ^ 2 , \end{align*}


\begin{align*} \left \langle F ^ { \prime } ( u ) , u \right \rangle = 0 , \enspace u \in \overline { \operatorname { c o n v } } \mathcal { G } \Longrightarrow F ( u ) = 0 , \end{align*}


\begin{align*} T ( z ) = { \rm t r } ^ \flat ( \chi e ^ { - i t _ 0 h ^ { - 1 } \tilde { P } _ h ( z ) } \tilde { R } _ h ( z ) \chi ) \end{align*}


\begin{align*} x = ( 1 - \varepsilon / 2 ) \Biggl ( \frac { \log _ 2 T _ k \log _ 4 T _ k } { \sum _ { T _ { k - 1 } < p \leq T _ k } \frac { 1 } { 2 p ^ { 1 + 2 \sigma _ k } } \bigl ( \frac { 2 \log T _ k } { \log p } \bigr ) ^ 2 \sin ^ 2 \bigl ( \frac { \log p } { 2 \log T _ k } \bigr ) + \frac { 1 } { 8 p ^ { 2 + 4 \sigma _ k } } } \Biggr ) ^ { 1 / 2 } \ , , \end{align*}


\begin{align*} P _ { \widehat { \mathcal { H } } _ \chi ^ { \rm s t i f f } } \left ( \mathcal { A } _ { \chi , \varepsilon } ^ { \rm a p p } - z I \right ) ^ { - 1 } | _ { \widehat { \mathcal { H } } _ \chi ^ { \rm s t i f f } } = \left ( \frac { 1 } { \varepsilon ^ 2 } \mathcal { A } _ \chi - \mathcal { B } _ \chi ( z ) \right ) ^ { - 1 } . \end{align*}


\begin{align*} F _ { | g _ i | ^ 2 } ( x ) = \gamma ( M , M x ) / \Gamma ( M ) = 1 - \sum _ { i = 0 } ^ M \frac { ( M x ) ^ i } { i ! } e ^ { - M x } , \end{align*}


\begin{align*} ( \Delta + \lambda - q ) w = 0 \ ; \mathrm { i n } \ ; \mathrm { M } _ 1 , w _ { | \Gamma } = \partial _ \nu w _ { | \Gamma } = 0 . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ { k } \lambda _ { i } ^ { \gamma } ( A + \alpha B ) \leq \sum _ { i = 1 } ^ { k } \lambda _ i ^ { \gamma } ( A ) + \sum _ { i = 1 } ^ { k } \lambda _ i ^ { \gamma } ( \alpha B ) \hbox { f o r a l l $ 0 < \alpha < 1 $ } \end{align*}


\begin{align*} { \rm t r } ^ \flat B = \int _ \mathcal { M } \iota ^ * K _ B ( x ) d x = \langle \iota ^ * K _ B , 1 \rangle . \end{align*}


\begin{align*} W ^ { \theta , q } _ 0 \coloneqq \lbrace g \in W ^ { \theta , q } : g ( 0 ) = 0 \rbrace . \end{align*}


\begin{align*} N D _ { \lambda } ^ { d } = \left \{ \mathcal { K } \times I \in \mathcal { A } \ , : E _ { p , \tau } \left ( u \ , ; \ , { K } ^ { d } \times I \right ) > \frac { \lambda } { 1 6 } \right \} . \end{align*}


\begin{align*} \partial _ t v ^ { ( \mathsf { s } , \mathsf { c } ) } ( t , x ) \ , = \ , \frac { \mathsf { s } } { 2 } \ , \Delta _ x v ^ { ( \mathsf { s } , \mathsf { c } ) } ( t , x ) \ , + \ , \mathsf { c } \ , \dot { W } ( t , x ) \ , , \end{align*}


\begin{align*} g ( { f } X , { f } Y ) = - g ( X , Q \ , Y ) + \sum \nolimits _ { i } \eta ^ i ( X ) \ , \eta ^ i ( Y ) , X , Y \in \mathfrak { X } _ M , \end{align*}


\begin{align*} E _ r : Y ^ 2 = X ^ 3 + 2 0 r ^ 2 X . \end{align*}


\begin{align*} \left ( \frac { \pi } { \sqrt { c } } \right ) ^ { - s } \Gamma ( s ) \ , \tilde { \zeta } _ { p , p ^ { \prime } } \left ( s , c \right ) = \left ( \frac { \pi } { \sqrt { c } } \right ) ^ { - ( 1 - s ) } \Gamma \left ( 1 - s \right ) \ , \tilde { \zeta } _ { p ^ { \prime } , p } \left ( 1 - s , c \right ) . \end{align*}


\begin{align*} h = Z _ \mathbb { Q } ^ { - 1 } Y _ \mathbb { Q } X Y _ f Z _ f ^ { - 1 } . \end{align*}


\begin{align*} \mathcal { G } _ k ( h ) : = \sum _ { n _ 0 \leq j \leq k } \mathcal { N } _ j \mathcal { G } _ k ( h ' ) : = \sum _ { n _ 0 \leq j \leq k } \mathcal { N } _ j ' , \end{align*}


\begin{align*} & \sum _ { k = 0 } ^ { \infty } \frac { \binom { 2 k } { k } \binom { 4 k } { 2 k } } { x ^ k } ( 2 H _ { 4 k } - H _ { 2 k } ) = \frac { 1 } { 2 } \bigg ( \log \frac { x } { x - 6 4 } \bigg ) \sum _ { k = 0 } ^ { \infty } \frac { \binom { 2 k } { k } \binom { 4 k } { 2 k } } { x ^ k } , \end{align*}


\begin{align*} \Psi : = \sum _ { \phi \in \Lambda ( G , H ) } \Phi _ \phi . \end{align*}


\begin{align*} \| H ( f ) \| _ { L _ { | x | _ h } ^ p L _ { \theta } ^ { \bar { p } _ 1 } ( \mathbb H ^ n ) \rightarrow L _ { | x | _ h } ^ p L _ { \theta } ^ { \bar { p } _ 2 } ( \mathbb H ^ n ) } = G \| f \| _ { L _ { | x | _ h } ^ p L _ { \theta } ^ { \bar { p } _ 2 } ( \mathbb H ^ n ) } \end{align*}


\begin{align*} \sigma _ 1 \circ \sigma _ n = w \sigma _ { n + 1 } + ( 1 - w ) \sigma _ { n - 1 } , ~ n \in \N , \end{align*}


\begin{align*} \sum ^ n _ { k = 1 } \frac { d _ k } { w _ j - z _ k } = 0 , j = 1 , \dots \deg ( p _ + ( z ) ) , \end{align*}


\begin{align*} & \lim _ { N \to \infty } N ^ { - 1 } M ( \tau _ N ) = \begin{cases} \dfrac { \nu } { \lambda } , & p = 0 , \\ - \dfrac { \nu q \log ( q ) } { \lambda p } , & 0 < p < 1 . \end{cases} \end{align*}


\begin{align*} \rho ( a ) ^ * \rho ( a ) = \rho ( a ^ * a ) = \rho ( a a ^ * ) \end{align*}


\begin{align*} \| f \| _ { \Lambda ^ \beta _ d } : = \sup _ { d ( x , y ) \ne 0 } \frac { | f ( x ) - f ( y ) | } { d ( x , y ) ^ \beta } < \infty . \end{align*}


\begin{align*} | \phi ' ( t ) | & = \Big | \frac 1 n \sum _ { j = 1 } ^ n \alpha ( y _ j ) q _ t ( x , y _ j ) \Big | \leqslant \| \alpha \| _ { L ^ 2 ( \hat \nu ) } \Big ( \frac 1 n \sum _ { j = 1 } ^ n q _ t ( x , y _ j ) ^ 2 \Big ) ^ { 1 / 2 } . \end{align*}


\begin{align*} \int V f _ 1 \cdot J g \ , d \mu = \lim \int T ^ { k _ j } f _ 1 \cdot J g \ , d \mu & = \lim \int T ^ { k _ j } f _ 1 \otimes g \ , d \eta \\ & = \lim \int f _ 1 \otimes S ^ { - k _ j } g \ , d \eta = \lim \int J ^ { \ast } f _ 1 \cdot S ^ { - k _ j } g \ , d \nu . \end{align*}


\begin{align*} I _ { \tilde { Y } } = \langle x _ 0 \vartheta + y _ 0 \zeta + \alpha x _ 0 y _ 0 , ( x _ 0 , y _ 0 ) ^ { k + 2 } \rangle . \end{align*}


\begin{align*} \left \{ \begin{array} { l l } \partial ^ { 2 } _ { t } w ( t , k ) + a ( t ) \mathcal { H } _ { \hbar , V } w ( t , k ) + q ( t ) w ( t , k ) = a ( t ) \left ( \mathcal { H } _ { V } - \mathcal { H } _ { \hbar , V } \right ) v ( t , k ) , ~ ~ k \in \hbar \mathbb { Z } ^ { n } , \\ w ( 0 , k ) = 0 , k \in \hbar \mathbb { Z } ^ { n } , \\ \partial _ { t } w ( 0 , k ) = 0 , k \in \hbar \mathbb { Z } ^ { n } . \end{array} \right . \end{align*}


\begin{align*} D \left ( \alpha _ { x , e } \parallel \alpha _ { x , g } \right ) = \sup _ { n \in \mathbb { N } } \left \{ H _ { q _ { x , e } ^ { n } \parallel q _ { x , g } ^ { n } } \left ( \mathcal { P } _ { x , n } \right ) - \Delta _ { x , g } ( n ) \right \} . \end{align*}


\begin{align*} g _ { \lambda , A } ( v ) = \sup _ { v ' \in \mathcal { V } } \left \{ g ( v ' ) - \sum _ { \ell = 1 } ^ { L } \lambda _ \ell \tilde { c } _ { \ell } \left \{ s _ { \ell } , s _ { \ell } ' \right \} \right \} \end{align*}


\begin{align*} T _ \gamma ( x ) = \left \{ \begin{array} { l l } x ( 1 + 2 ^ \gamma x ^ \gamma ) & i f \ , x \in [ 0 , 1 / 2 [ \\ 2 x - 1 & i f \ , x \in [ 1 / 2 , 1 ] \end{array} \right . . \end{align*}


\begin{align*} 0 = \int _ a ^ t \| X ' \| ^ 2 + \langle ( R - \lambda I ) X , X \rangle d s \geq \int _ a ^ t \| X ' \| ^ 2 + \mu \langle X , X \rangle d s \geq 0 , \end{align*}


\begin{align*} \sup _ { \lambda > 0 } \lambda \int _ { \{ ( b - a ) / 2 < x < b / 3 \ , : \ , ( b - a ) / b > \lambda \} } x ^ { - 1 } \ , d x \lesssim \int _ a ^ b x ^ { - 1 } \ , d x = \log \frac { b } a \simeq \frac { b - a } a \simeq \frac { b - a } b , \end{align*}


\begin{align*} \nu _ 0 = { \rm I d } , \nu _ { j } = \nu _ { j } [ A , \varphi ] ( f ) : = f - \varphi ( A \nu _ { j - 1 } ( f ) ) , \ , j = 1 , 2 , 3 , \ldots \end{align*}


\begin{align*} \tfrac { 1 } { \beta ! } \big ( \partial ^ \beta \big ) | _ { z = 0 } u _ x = \left \{ \begin{array} { l l } ( \cdot - x ) ^ \mathbf { n } & \mbox { i f } \beta = e _ \mathbf { n } , \\ 0 & \mbox { o t h e r w i s e . } \end{array} \right . \ ; \end{align*}


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


\begin{align*} - \Delta _ p v = 0 \ ; \ ; B _ r ( x _ 0 ) . \end{align*}


\begin{align*} H ( x ) = \frac { q - I _ { \boldsymbol { a } } ( q x ) } { q \Pi _ { \boldsymbol { a } } ( q x ) } ; \end{align*}


\begin{align*} c ^ { s + \frac { \l - 1 } { 2 } } \L _ { f } \left ( s , \frac { a } { c } \right ) = \int _ { 0 } ^ { \infty } f \left ( \frac { a } { c } - \frac { t } { i c } \right ) t ^ { s + \frac { \l - 1 } { 2 } } \ , d t . \end{align*}


\begin{align*} p _ { k } ^ { } = \frac { n - k } { n } p . \end{align*}


\begin{align*} P ( x , y ) = \frac { y } { x ^ 2 + y ^ 2 } \end{align*}


\begin{align*} M _ { [ 1 ] } \diamond \cdots \diamond M _ { [ j ] } & = S _ { [ 1 ] } \diamond \cdots \diamond S _ { [ j ] } + \mathcal { O } ( \| H \| ^ 2 ) . \end{align*}


\begin{align*} C ( I d , \gamma ) ( [ U ] \mid U L ( p ) ) = C ( U L ( p ) , U L ( p ) \gamma ) ( [ U ] ) . \end{align*}


\begin{gather*} \psi : \left ( - \frac { W } { 2 } , \frac { W } { 2 } \right ) \times \left ( - \frac { H } { 2 } , \frac { H } { 2 } \right ) \subset \mathbb { R } ^ 2 \to \mathbb { R } , \\ \psi ( ( x _ 1 , x _ 2 ) ) = \min \{ W , H \} \cdot \max \left \{ \left ( \frac { x _ 1 } { W } \right ) ^ 2 , \left ( \frac { x _ 2 } { H } \right ) ^ 2 \right \} . \end{gather*}


\begin{align*} I _ n = I _ \alpha . \end{align*}


\begin{align*} \left ( { D } ^ { \alpha } _ { N } U ^ { n - \sigma } , U ^ { n , \sigma } \right ) \ , + \ , a \big ( l ( U ^ { n , \sigma } ) \big ) \ , ( \nabla U ^ { n , \sigma } , \nabla U ^ { n , \sigma } ) \ , = & \ , ( f ^ { n - \sigma } , U ^ { n , \sigma } ) , \ ; \ , \forall v _ h \in V _ h . \\ \end{align*}


\begin{align*} \eta _ 1 ( m ) : = - \frac { A } { m } \frac { E ( m ) } { K ( m ) } , \eta _ 2 ( m ) : = \frac { A } { m } \Big ( 1 - m - \frac { E ( m ) } { K ( m ) } \Big ) , \quad \eta _ 3 ( m ) : = \frac { A } { m } \Big ( 1 - \frac { E ( m ) } { K ( m ) } \Big ) . \end{align*}


\begin{align*} \| T _ { \rho } f \| _ q ^ q = \| T _ { \rho ' \frac { 1 } { \sqrt { q } } } f ' \| _ q ^ q \le \sum _ { S \subseteq [ n ] } \left ( \frac { \beta } { \sqrt { q } } \right ) ^ { q | S | } \mathbb { E } _ { x \sim \Omega ^ S } \| D _ { S , x } [ f ' ] \| _ 2 ^ q . \end{align*}


\begin{align*} \begin{cases} \nu ( x ) = \Psi ( \nu , \mu ) ( x ) \\ \mu ( y ) = \Phi ( \nu , \mu ) ( y ) . \end{cases} \end{align*}


\begin{align*} \omega _ 0 = \theta \wedge J \theta - d J \theta . \end{align*}


\begin{align*} M = \sigma _ { j _ r | j _ r } \circ \dots \circ \sigma _ { j _ 1 | j _ 1 } ( N ) , \end{align*}


\begin{align*} \Delta _ { 0 } = 0 , \Delta _ { 1 , 2 } = \Delta _ { 1 , 3 } = 0 \implies \Delta _ { 1 } = \Delta _ { 1 , 1 } . \end{align*}


\begin{align*} f _ { 1 , m , \sharp } ( x , y ) = \sum _ { n = 1 } ^ { \mathcal { N } _ 1 } h _ { 1 , n , m } ( x , y ) e ^ { i \alpha _ { n , m } ( y ) x } \end{align*}


\begin{align*} \begin{cases} A ( ( - \Delta ) ^ \alpha ) u = F & \Omega , \\ u = f & \Omega ^ c \end{cases} \end{align*}


\begin{align*} \rho ( j , w ) = | w | ^ 2 - R _ { j } ^ 2 \left ( \frac { w } { | w | } \right ) \end{align*}


\begin{align*} f ( x ) = e ^ { - z | x | ^ 2 + \xi \cdot x } , \end{align*}


\begin{align*} \frac { T _ i ^ { \infty } } { T _ i ^ { \infty } - T _ i ( u ) } \frac { e ^ { \rho _ i ( u ) } } { \theta _ i } = e ^ * _ i ( u ) . \end{align*}


\begin{align*} w \mapsto ( ( D F ) ( r , \lambda ' , 0 ) ) ( 0 , 0 , w ) = F ( r , \lambda ' , w ) \end{align*}


\begin{align*} \rho _ { \overline { w } } ( 0 , 0 ) = - i \lambda e ^ { i \pi \beta _ 1 } \end{align*}


\begin{align*} H _ 0 ( \nabla _ \xi H ( \xi ) ) = 1 \end{align*}


\begin{align*} C \in \mathsf { C } \mathbb { P } \qquad \Rightarrow \qquad \bigcup C = X . \end{align*}


\begin{align*} \left \{ \begin{array} { l l } & \dfrac { 1 } { 2 c ^ 2 m _ 1 } \partial _ t ^ 2 \phi - \dfrac { 1 } { 2 m _ 1 } \Delta \phi + \dfrac { m _ 1 c ^ 2 } { 2 } \phi = - a _ 1 \psi \overline { \phi } , \\ \\ & \dfrac { 1 } { 2 c ^ 2 m _ 2 } \partial _ t ^ 2 \psi - \dfrac { 1 } { 2 m _ 2 } \Delta \psi + \dfrac { m _ 2 c ^ 2 } { 2 } \psi = - a _ 2 \phi ^ 2 . \end{array} \right . \end{align*}


\begin{align*} b _ { \delta , T } ( \ell ) = b _ { \delta , T } ( \ell _ { \sf p } ) + b _ { \delta , T } ( \ell _ { \sf q } ) \end{align*}


\begin{align*} \mathrm { d } \mu _ g = \sqrt { \mathrm { d e t } ( g _ { i j } ) } \mathrm { d } A = \phi ^ { n - 1 } D \mathrm { d } A , \\ \end{align*}


\begin{align*} \rho ( t , w ) = A t + { \rm I m } ( e ^ { i \pi ( 1 - \beta _ 1 ) } w ) + B t ^ 2 + C | w | ^ 2 + \end{align*}


\begin{align*} \begin{cases} \rho _ t + ( \rho u ) _ x = 0 , \\ \rho u _ t + \rho u u _ x + p _ x = S _ x , \\ E _ t + ( u E + p u + q - S u ) _ x = 0 . \end{cases} \end{align*}


\begin{align*} \begin{cases} \tilde { P } ( ( - \Delta ) ^ { \tilde { s } } ) u = 0 & \mathbb { R } ^ n , \\ u = 0 & \Omega ^ c , \end{cases} \end{align*}


\begin{align*} U _ { \varepsilon } ( t , \xi ) : = \left ( \begin{array} { c } i \langle \xi \rangle \widehat { u } _ { \varepsilon } ( t , \xi ) \\ \partial _ { t } \widehat { u } _ { \varepsilon } ( t , \xi ) \end{array} \right ) , U _ { 0 } ( \xi ) : = \left ( \begin{array} { c } i \langle \xi \rangle \widehat { u } _ { 0 } ( \xi ) \\ \widehat { u } _ { 1 } ( \xi ) \end{array} \right ) , \end{align*}


\begin{align*} \Pr ( \Phi ( \mathcal { A } ) > 0 ) = 1 - \exp \left ( - 2 \pi \lambda R _ { m i n } R _ S \right ) . \end{align*}


\begin{align*} d u _ n + A u _ n \ , d t = f _ n ( u _ n ) \ , d t + \sigma ( u _ n ) B \ , d W , u _ n ( 0 ) = u _ 0 , \end{align*}


\begin{align*} 1 < l _ 1 \leq \frac { F ' _ 1 ( s ) s } { F _ 1 ( s ) } \leq m _ 1 : = \sup _ { 0 < s < \delta } \left ( 2 + \frac { 1 } { \log s } \right ) \leq 2 , \end{align*}


\begin{align*} \bar { u } ( x ) : = \sup \{ L ( x ) : L \ , \ L \leq u \ \ U , \ D L \in \Omega ^ * \} . \end{align*}


\begin{align*} \nu _ i ( f ) = \min _ { 1 \leq j \leq r } \left \{ v ( b _ j ) + \sum _ { k \in I } \lambda _ j ( Q _ k ) \nu ( Q _ k ) \right \} . \end{align*}


\begin{align*} Y ' ( t ) = A ( t , \lambda ) Y ( t ) , A ( t , \lambda ) = \begin{bmatrix} 0 & I \\ R ( t ) - \lambda I & 0 \end{bmatrix} \end{align*}


\begin{align*} \nabla \vect u | _ { \partial \Omega } = \partial _ { \vec { \vect n } } \vect u | _ { \partial \Omega } \otimes \vec { \vect n } . \end{align*}


\begin{align*} \begin{aligned} P _ b = & \Pr ( - \mu _ D / \sigma _ D ) = Q \left ( \sqrt { \frac { P _ s \zeta ^ 2 | \hat \eta - \eta | ^ 2 } { 2 ( N _ 0 + P _ s ( 1 - \zeta ^ 2 ) \sigma _ e ^ 2 L ) } } \right ) . \end{aligned} \end{align*}


\begin{align*} \gamma = \begin{bmatrix} a \beta & x \\ c \beta & y \end{bmatrix} \begin{bmatrix} ( \alpha \beta ) ^ { - 1 } & 0 \\ 0 & D \beta \end{bmatrix} \begin{bmatrix} 1 & \alpha ( b y - x d ) \\ 0 & 1 \end{bmatrix} \begin{bmatrix} \alpha & 0 \\ 0 & 1 \end{bmatrix} . \end{align*}


\begin{align*} \Phi ( \sigma ) = \psi _ { S } ( \Phi ( \pi ) ) = ( \psi _ { x _ { k } } \circ \cdots \circ \psi _ { x _ { 2 } } \circ \psi _ { x _ { 1 } } ) ( \Phi ( \pi ) ) . \end{align*}


\begin{align*} - ( - \Delta ) ^ s \psi ( r ) = - \check C F ( a , b , c , - r ^ 2 ) \textrm { f o r a l l } \ ; \ ; r > 1 \ , , \end{align*}


\begin{align*} \Lambda _ { \alpha } = { \rm d i a g } \big ( \lambda _ i ^ { ( k , \alpha ) } \big ) _ { i = 1 } ^ { n } \textrm { w i t h } \lambda _ i ^ { ( k , \alpha ) } = \lambda _ k - 2 \alpha ^ { \frac { 1 } { n } } \theta _ n ^ { i } + \lambda _ k \alpha ^ { \frac { 2 } { n } } \theta _ n ^ { 2 i } \textrm { a n d } \theta _ n = \exp \Big ( \frac { 2 \pi { \bf i } } { n } \Big ) . \end{align*}


\begin{align*} \sum _ { j = 0 } ^ { n - 1 } \alpha _ j y ^ j + y ^ { n } = 0 . \end{align*}


\begin{align*} S _ { \mathrm { m a i n } } \gg \log p \sum _ { L = 1 } ^ { \infty } \phi ( L ) | y _ L | ^ 2 , \end{align*}


\begin{align*} \sum \limits _ { k = 0 } ^ { + \infty } \sum \limits _ { p = 0 } ^ { 1 } \varepsilon ^ { p \alpha + k - 1 } \ , \mathfrak { b } _ { p \alpha + k - 1 } \end{align*}


\begin{align*} \Big \| \sum _ { j = 1 } ^ m \alpha _ j \nabla u _ j \Big \| _ { p , q , \mu } ^ q & \leq p \sum _ { j = 1 } ^ m \int _ 0 ^ \infty t ^ { q - 1 } ( \nabla u _ j ) _ { * \mu } \Big ( \frac { t } { | \alpha _ j | } \Big ) ^ \frac { q } { p } \dd t \\ & = p \sum _ { j = 1 } ^ m | \alpha _ j | ^ q \int _ 0 ^ \infty t ^ { q - 1 } ( \nabla u _ j ) _ { * \mu } ( t ) ^ \frac { q } { p } \dd t = \sum _ { j = 1 } ^ m | \alpha _ j | ^ q \| \nabla u _ j \| _ { p , q , \mu } ^ q \end{align*}


\begin{align*} m ( t ; \lambda ) = \frac { 4 \pi } { 3 } \cdot \begin{cases*} ( \frac { 1 + t } { 1 + \lambda } ) ^ 3 & i f $ 0 < \lambda < \frac { 1 } { t } $ , \\ ( \frac { 1 - t } { \lambda - 1 } ) ^ 3 & i f $ \frac { 1 } { t } \leq \lambda < 2 $ , \\ 0 & o t h e r w i s e . \end{cases*} \end{align*}


\begin{align*} \chi ^ \perp ( p , p - k ) 2 \mathrm { R e } \ , \int _ 0 ^ t \int _ 0 ^ { t _ 1 } G _ k ( t _ 2 ) e ^ { i t _ 2 ( E _ p - E _ { p - k } ) } \d t _ 2 \d t _ 1 = 2 \pi t \ , \alpha ^ P _ t ( p , k ) \end{align*}


\begin{align*} c _ { \ell } ( s _ \ell , s _ \ell ' ) & = | y _ \ell - y _ \ell ' | + \| x _ \ell - x _ \ell ' \| _ 2 . \end{align*}


\begin{align*} I _ 1 & \le \Bigg ( \int _ { I ( x _ 0 , C r ) } \bigg \{ \sum _ { k = k _ 0 + 1 } ^ \infty { \mathfrak R } ^ { - k \beta } \big | D _ k ( a ) ( x ) \big | \bigg \} ^ { p ' } d \omega ( x ) \Bigg ) ^ { 1 / p ' } \\ & \lesssim \Bigg ( \int _ { I ( x _ 0 , C r ) } \bigg \{ \sum _ { k = k _ 0 + 1 } ^ \infty { \mathfrak R } ^ { - k \beta } \bigg \} ^ { p ' } d \omega ( x ) \Bigg ) ^ { 1 / p ' } \\ & \leq C r ^ \beta \omega ( I ) ^ { 1 / p ' } . \end{align*}


\begin{align*} Q _ { \Sigma } [ \phi ] = \int _ { \Sigma } | \nabla \phi | ^ 2 - \left ( | A | ^ 2 + \mathrm { R i c } ( \nu , \nu ) \right ) \phi ^ 2 - \int _ { \partial \Sigma } h ( \nu , \nu ) \phi ^ 2 , \end{align*}


\begin{align*} \prod _ { \substack { ( a , b , c ) = 1 \\ a , b , c \geq 1 } } \left ( \frac { 1 } { 1 - x ^ a y ^ b z ^ c } \right ) ^ { \frac { c ^ 3 } { a b ^ 3 } } = \exp \left \{ L i _ 1 ( x ) L i _ 3 ( y ) L i _ { - 3 } ( z ) \right \} \end{align*}


\begin{align*} \| f ^ { = d } \| _ q \leq ( 1 / \rho ) ^ d \gamma ' = \gamma \left ( 9 9 \cdot 2 \ , r \sqrt { q } \sqrt { \frac { \log ( 1 / \gamma ) } { d } } \right ) ^ d \leq \gamma \left ( 2 0 0 r \sqrt { q } \cdot \max \left \{ \sqrt { q } , \sqrt { \log ( 1 / \gamma ) } \right \} \right ) ^ d , \end{align*}


\begin{align*} \Big \| \sum _ { n = 1 } ^ \infty g _ n f \Big \| _ { L ^ 1 ( \Omega ) } \leq \sum _ { n = 1 } ^ \infty \| g _ n f \| _ { L ^ 1 ( \Omega ) } \leq \| f \| _ X \sum _ { n = 1 } ^ \infty \| g _ n \| _ { X ' } . \end{align*}


\begin{align*} \int _ { \tilde { X } } f d \tilde { \boldsymbol { \beta } } _ { \tilde { \nu } } ( w ) = \int _ { G / \Gamma } \int _ { X } f _ { z } d \boldsymbol { \beta } _ { \nu } \left ( \tau ( z ) ^ { - 1 } . w \right ) d m _ { G / \Gamma } ( z ) , \mbox { w h e r e } w \in G / P , f \in L ^ { \infty } \left ( \tilde { X } \right ) , \end{align*}


\begin{align*} \delta _ t ( x + \lambda y ) = \ , \frac { 2 t } { \pi } \delta _ { x , 0 } \ , + \mathcal { O } ( 1 / t x ^ 2 ) + \mathcal { O } ( t ^ 3 \lambda ^ 2 | y | ^ 3 ) \ . \end{align*}


\begin{align*} S = \begin{bmatrix} 0 & - 1 \\ 1 & 0 \end{bmatrix} . \end{align*}


\begin{align*} \left ( u _ { \xi } , u _ { \eta } \right ) : = \left \{ \begin{array} { c c } 1 , & \xi = \eta , \\ 0 , & \xi \neq \eta , \end{array} \right . \end{align*}


\begin{align*} \left \{ \sum _ { i = 1 } ^ { m } a ^ { ( i - 1 ) } _ n \cdot R _ i \colon n \ge 0 \right \} . \end{align*}


\begin{align*} S & \ll \widehat { Y } ^ { ( n + 3 ) / 2 + \varepsilon } \sum _ { d = d _ 1 d _ 2 } | d _ 1 | \sum _ { d _ 2 = e f } | e | | f | \sum _ { | t _ 2 | \leq \widehat { Y } } \sum _ { | t _ 1 | \leq \frac { \widehat { Y } } { | t _ 2 | } } | t _ 1 | ^ { - 1 / 2 } \left ( \widehat { V } ^ n + \widehat { Y } ^ { n / 3 } \right ) \\ & \ll \widehat { Y } ^ { ( n + 3 ) / 2 + \varepsilon } | d | \left ( \widehat { V } ^ n + \widehat { Y } ^ { n / 3 } \right ) , \end{align*}


\begin{align*} \Pr [ \hat { M } = M ] \leq \inf _ { 0 \leq \gamma \leq 1 } \sum _ k \binom { n } { k } \frac { s ! } { ( k + s ) ! } ( 1 - Q ( ( 1 - \gamma ) \mu ) ) ^ { n - k } ( 1 - Q ( \gamma \mu ) ) ^ { k ( k + s ) } e ^ { k ( 2 \gamma - 1 ) \mu ^ 2 / 2 } \end{align*}


\begin{align*} \tilde { \zeta } _ { \infty , p ^ { \prime } } ( s , c ) : = \lim _ { p \rightarrow \infty } \tilde { \zeta } _ { p , p ^ { \prime } } ( s , c ) = \sum _ { m , n \neq 0 } \frac { p ^ { \prime 2 } + \lambda _ { n } ^ { \prime 2 } } { p ^ { \prime } \left ( p ^ { \prime } + \frac { 1 } { \pi } \right ) + \lambda _ { n } ^ { \prime 2 } } \frac { 1 } { \left ( m ^ { 2 } + c \lambda _ { n } ^ { \prime 2 } \right ) ^ { s } } = \zeta _ { \infty , p ^ { \prime } } ( s , c ) , \ , \ , \ , \ , \ , ( s ) > 1 . \end{align*}


\begin{align*} L \log L ( \Omega ) : = \left \{ f \in L _ { l o c } ^ 1 ( \Omega ) : \abs { f } _ { L \log L ( \Omega ) } < \infty \right \} , \end{align*}


\begin{align*} ( f * g ) ( \gamma _ 0 ) = \int _ { G _ { t ( \gamma _ 0 ) } } f ( \gamma ^ { - 1 } ) g ( \gamma \gamma _ 0 ) \ d \lambda _ { t ( \gamma _ 0 ) } = \int _ { G _ { s ( \gamma _ 0 ) } } f ( \gamma _ 0 \gamma ^ { - 1 } ) g ( \gamma ) \ d \lambda _ { s ( \gamma _ 0 ) } . \end{align*}


\begin{align*} \phi ( w ) = \begin{cases} 1 & \mbox { i f } ( \prod _ { i = 1 } ^ m a ( i ) \ast g _ w ( t ( i ) ) ) \ast a ( m + 1 ) \in A _ 1 \\ 2 & \mbox { o t h e r w i s e . } \end{cases} \end{align*}


\begin{align*} \lim _ { a \to \infty } \sup _ { x \in e M _ { + } e , ~ x \neq 0 } \abs { \frac { ( M _ a ( \mu ) - \bar { \mu } ) ( x ) } { \tau ( x ) } } = 0 . \end{align*}


\begin{align*} \phi ( E _ { i i } \otimes Y E _ { j j } Y ^ * ) = U _ Y ( E _ { i i } \otimes E _ { j j } ) V _ Y ^ * \end{align*}


\begin{align*} \langle T ( z ^ n \otimes \lbrace 0 , \infty \rbrace ) , f \rangle = \langle z ^ n \otimes \lbrace 0 , \infty \rbrace , T f \rangle = 0 . \end{align*}


\begin{align*} m ( \lambda ) = \int _ 0 ^ \infty m ( t ; \lambda ) \ , d t . \end{align*}


\begin{align*} \norm [ \big ] { ( M _ T ^ * ) ^ { 1 / q } } _ { L ^ p ( \Omega ) } = \norm [ \big ] { ( M _ T ^ * ) } ^ { 1 / q } _ { L ^ { p / q } ( \Omega ) } \lesssim q ^ { 1 / q } \norm [ \big ] { \overline { v } _ \lambda - \overline { v } _ \mu } _ { L ^ p ( \Omega ; L ^ { 2 q } ( 0 , t ; L ^ q _ x H ) ) } , \end{align*}


\begin{align*} K _ { \omega , \mathbf { c } } ( U _ 0 ) & = 2 L ( U _ 0 ) + 3 N ( U _ 0 ) + 2 \omega Q ( U _ 0 ) + \mathbf { c } \cdot \mathbf { P } ( U _ 0 ) \\ & = 2 \omega Q ( U _ 0 ) + \sqrt { \omega } \ \mathbf { c } _ 0 \cdot \mathbf { P } ( U _ 0 ) + 2 L ( U _ 0 ) + 3 N ( U _ 0 ) \end{align*}


\begin{align*} H _ { \xi } \cap H _ { \xi ' } \subseteq & \big ( G _ \xi \cap G _ { \xi ' } \cap [ \beta , \gamma ) \big ) \times X \\ \subseteq & \big ( R \cap [ \beta , \gamma ) \big ) \times X = \emptyset . \end{align*}


\begin{align*} \theta _ { Y ^ { \ast } , L } ( f ) ( y ) = \int _ { Y ^ { \ast } / Y ^ { \ast } \cap L } f ( [ \dot { y } ^ { \ast } , 0 ] + [ y , 0 ] ) d \dot { y } ^ { \ast } , \end{align*}


\begin{align*} \rho ( j , w ) = | w | ^ 2 - R _ { j } ^ 2 \left ( \frac { w } { | w | } \right ) \end{align*}


\begin{align*} V ( D ^ 2 u ( x _ 0 ) ) \equiv V ( \lambda _ 1 , \dots , \lambda _ n ) = \Bigg [ ( 1 + \lambda _ 1 ^ 2 ) . . . ( 1 + \lambda _ n ^ 2 ) \bigg ] ^ { \frac { 1 } { 2 } } . \end{align*}


\begin{align*} \overline { F } ( k , t , \mathbf { p } _ { n + 1 } ) \overset { } { = } \Pr ( X _ t > k \ , \vert \ , X _ 0 = 0 , \mathbf { p } _ { n + 1 } ) , k \in S , \end{align*}


\begin{align*} \int _ { ( T ^ { ( l ) } ) ^ i J _ l } | S ( l ) D \varphi ( x ) | d x = O ( e ^ { ( - \lambda _ 1 ( 1 - a ) + \tau ) k } ) ( c _ 0 + p _ a ) ( D \varphi ) . \end{align*}


\begin{align*} - M _ { x y } f ( x ) f ( y ) ( a ( x ) - a ( y ) ) ^ 2 = 0 . \end{align*}


\begin{align*} \begin{aligned} & \lambda _ { i j } v _ { i j } = 0 , \forall i , j \in \{ 1 , \ldots , n \} , \ i \neq j . \\ & \gamma _ i \left ( \sum _ { j \neq i } v _ { i j } - x _ 0 { _ i } \right ) = 0 , \forall i \in \{ 1 , \ldots , n \} . \end{aligned} \end{align*}


\begin{align*} \pi _ k ( X _ F , x ) ^ 0 = \varprojlim _ { F ' / F } \pi _ k ( X _ { F ' } , x _ { F ' } ) \ ; . \end{align*}


\begin{align*} g ( h _ i Z , Q X ) = \sum \nolimits _ { j } \eta ^ j ( X ) \ , \eta ^ j ( Z ) - \eta ^ i ( X ) \ , \bar \eta ( Z ) , \end{align*}


\begin{align*} \frac { d x } { d t } = \frac { \frac { \partial H } { \partial y } ( x , y ) } { V ( x , y ) } , \frac { d y } { d t } = - \frac { \frac { \partial H } { \partial x } ( x , y ) } { V ( x , y ) } \end{align*}


\begin{align*} C : = \sup _ { k \geq 1 } \| h _ k \| _ Y < \infty . \end{align*}


\begin{align*} \Phi ( t ) : = e ^ { - t ^ 2 / 2 } \end{align*}


\begin{align*} P ( E ) - P ( F ) = P ( E ) - P ( H ) + P ( H ) - P ( F ) \geq 2 \lambda \Bigl ( f \big ( | E \Delta H | \big ) - C ( n ) | E \Delta H | ^ { 3 } \Big ) . \end{align*}


\begin{align*} ( \delta - n _ { F ^ i } ) ( w ) = \sum _ { i = 1 } ^ p a _ i r - \langle n _ { F ^ i } , g \rangle \le r . \end{align*}


\begin{align*} s = \frac { t } { \lambda ^ { \frac { 1 } { 2 - \gamma } } } , \ \ y = \frac { x } { \lambda } . \end{align*}


\begin{align*} \ddot \theta = \sin ( \theta ) - \varepsilon \dot \theta - \cos ( \theta ) u , \end{align*}


\begin{align*} \zeta _ \delta = 0 \ , \partial \Omega , \zeta _ \delta w _ \delta = v \ \ \forall \ , \delta \end{align*}


\begin{align*} r ( i ) = \max \{ | S | : S \subseteq V _ e ^ { ( i ) } d ( v , w ) \neq 2 v , w \in S \} . \end{align*}


\begin{align*} A _ N ( i , j ) = \left \{ \begin{array} { l l } 1 , & , \\ 0 , & . \end{array} \right . \end{align*}


\begin{align*} \mathbf { K } _ { { \overline { \psi } _ 0 } } ^ 1 = \mathbf { K } _ { { \overline { \psi } _ 0 } } \cap \mathcal { G } ^ 1 ( \widehat { \mathbb { Z } } ) . \end{align*}


\begin{align*} \mbox { w e d e n o t e b y $ \Phi $ t h e p r i m i t i v e f u n c t i o n o f $ D $ , n a m e l y } \ , \Phi ( u ) = \int _ 0 ^ u D ( s ) \ , \dd s . \end{align*}


\begin{align*} A h = A ( \sum _ { S \subset [ n ] } \hat h ( S ) \chi _ S ) = \sum _ S \left ( - \frac { p } { 1 - p } \right ) ^ { | S | } \hat h ( S ) \chi _ S = \sum _ { d = 0 } ^ n \left ( - \frac { p } { 1 - p } \right ) ^ d h ^ { = d } . \end{align*}


\begin{align*} \overline { F } ( k - 1 , t , n ) - \overline { F } ( k , t , n ) \overset { ( 1 ) } { = } f ( k , t , n ) \overset { ( 2 ) } { = } \frac { n } { n - k } \Big ( \overline { F } ( k , t + 1 , n ) - \overline { F } ( k , t , n ) \Big ) , \end{align*}


\begin{align*} S _ { \omega , \mathbf { c } } ( \Psi ) = \mu _ { \omega , \mathbf { c } } \le S _ { \omega , \mathbf { c } } ( \Theta ) . \end{align*}


\begin{align*} a ( \theta ) = a ( - \theta ) \end{align*}


\begin{align*} | \nabla ^ 2 \psi ^ i _ m ( x ' ) | & \leq c \ , \sum _ { j = 1 } ^ N \bigg \{ \big | \nabla ^ 2 M _ m ( \phi ^ j ) ( H ^ { i j } _ m x ' ) \big | \ , \xi _ j \big ( x ' , \psi ^ i _ m ( x ' ) \big ) + 1 \bigg \} \\ & \leq c \ , \sum _ { j = 1 } ^ N \bigg \{ M _ m ( | \nabla ^ 2 \phi ^ j | ) ( H ^ { i j } _ m x ' ) \ , \xi _ j \big ( x ' , \psi ^ i _ m ( x ' ) \big ) + 1 \bigg \} \end{align*}


\begin{align*} \begin{aligned} \overline { F } ( k , t + 1 ; p , n ) & = p _ { k } ^ { } \overline { F } ( k - 1 , t ; p , n ) + ( 1 - p _ { k } ^ { } ) \overline { F } ( k , t ; p , n ) \\ & = \frac { ( n - k ) p } { n } \overline { F } ( k - 1 , t ; p , n ) + \frac { n - ( n - k ) p } { n } \overline { F } ( k , t ; p , n ) , \end{aligned} \end{align*}


\begin{align*} \tau ( \Sigma ^ s f ) = \mathrm { l c m } \left ( \tau ( f ) , \tau ( \Sigma ^ { s - 1 } [ 1 ] ) \right ) = 3 \cdot 2 ^ { 2 + k } \end{align*}


\begin{align*} \sum _ { p = 1 } ^ n u ^ { p \bar p } u _ { p \bar p j } = f _ j + f _ u u _ j , \end{align*}


\begin{align*} \begin{cases} \nabla \cdot ( \sigma \nabla u ) = 0 & \Omega , \\ u = f & \partial \Omega , \end{cases} \end{align*}


\begin{align*} \psi ( G ) \subseteq \ker \widehat { \phi ^ { ( p ) } } = \pi ( \ker \widehat { \phi } ) . \end{align*}


\begin{align*} \left . \frac { d } { d \alpha } \right | _ { \alpha = 0 } \Gamma _ T ( \alpha ) . \end{align*}


\begin{align*} \bar { S } _ { j , + } ^ k ( t ) \approx \nu \int _ 0 ^ t Y e ^ { \lambda s } p _ { j , + } ^ k ( t - s ) d s = : \hat { S } _ { j , + } ^ k ( t ) . \end{align*}


\begin{align*} J ^ 0 = \{ ( p , q ) \in X \times X \setminus \Delta _ X \ , | \ , X \cdot l _ { p , q } = 3 p + q \in Z _ 0 ( X ) \} \end{align*}


\begin{align*} T : = \left \{ ( s , t ) \in [ 0 , 1 ] ^ 2 \ , : \ , t \in [ s - \lambda ( s ) , s + \lambda ( s ) ] \right \} . \end{align*}


\begin{align*} [ X _ 1 , \dots , X _ j ] _ c = [ X _ 1 , \dots , X _ j ] + \sum _ { p \geq j + 1 } \sum _ { ( i _ 1 , \dots , i _ p ) \in \mathbb { I } _ j ^ p } \gamma _ { ( i _ 1 , \dots , i _ p ) } [ X _ { i _ 1 } , \dots , X _ { i _ p } ] . \end{align*}


\begin{align*} \overline { S } ( k , n ) = \begin{dcases} 2 \ , \overline { S } ( k , n - 1 ) & k = 0 ; \\ \overline { S } ( k , n - 1 ) \ , + \ , \overline { S } ( k - 1 , n - 1 ) & 1 < k < n ; \\ 1 \ , & k = n ; \end{dcases} \end{align*}


\begin{align*} \frac { 1 } { 4 } \triangle u = | \alpha | ^ 2 e ^ { 2 u } , \end{align*}


\begin{align*} L ^ { - 1 } \leq A ( x , y , t ) \leq L \quad A ( x , y , t ) = A ( y , x , t ) \quad ( x , y , t ) \in \mathbb { R } ^ { n } \times \mathbb { R } ^ { n } \times ( 0 , T ) \end{align*}


\begin{align*} \varphi = \pi ( f _ 0 ) \varphi + \pi ( f _ 1 ) P ( X ) \varphi . \end{align*}


\begin{align*} \left [ \sigma ^ { m } \left ( - i x \right ) e ^ { - 2 \pi i m x } \right ] = \cos \left ( 2 \pi m x + 2 m \arctan \left ( \frac { x } { p } \right ) \right ) . \end{align*}


\begin{align*} T _ p ( P \otimes \lbrace \alpha , \beta \rbrace ) = \sum _ { g \in R _ p } g ( P \otimes \lbrace \alpha , \beta \rbrace ) . \end{align*}


\begin{align*} ( K \cap \mathrm { E n d } _ { { \kappa } } ( \overline { A } _ g ) ) \otimes _ { \mathbb { Z } } \mathbb { Z } _ \ell = \mathcal { O } _ { K , \ell } . \end{align*}


\begin{align*} \frac { d } { d x } ( \tilde Q _ i ) = \frac { d } { d x } \left ( \frac { Q _ i } { a _ i } \right ) = \frac { Q _ i ' } { a _ i } \end{align*}


\begin{align*} \Tilde { \varepsilon } = \phi _ 1 - \Tilde { \phi _ { e } } = \frac { \phi _ 2 - \phi _ 1 } { r ^ { \Tilde { q } } - 1 } . \end{align*}


\begin{align*} \boldsymbol { d } _ { \mathcal { V } } ( ( s _ 1 , s _ 2 ) , ( s _ 1 ^ \prime , s _ 2 ^ \prime ) ) = \boldsymbol { d } _ { \mathcal { S } _ 1 } ( s _ 1 , s _ 1 ^ \prime ) + \boldsymbol { d } _ { \mathcal { S } _ 2 } ( s _ 2 , s _ 2 ^ \prime ) . \end{align*}


\begin{align*} h _ \psi ^ { i t } = [ D \psi : D \varphi ] _ t \ , \lambda ^ { \varphi } ( t ) , t \in \mathbb { R } , \end{align*}


\begin{align*} \alpha ( \mathcal { K } ^ \pm ( n , k ) ) \leq \frac { 2 \binom { n } { k } } { 1 - \frac { 2 \binom { n - k } { k } } { - 2 \binom { n - k - 1 } { k - 1 } } } = \frac { 2 \binom { n } { k } } { 1 + \frac { n - k } { k } } = 2 \binom { n - 1 } { k - 1 } = | \mathcal { S } | . \end{align*}


\begin{align*} ( R _ \gamma ) _ * X = X & & \gamma \in G , \end{align*}


\begin{align*} Z ( t ) & = N \Bigl ( y ( Z , t ) \Bigr ) . \end{align*}


\begin{align*} ( \varphi ( d ( q , \gamma ( s ) ) ) ) '' & \le \frac 1 r + \frac { 4 \epsilon \sigma } b = \frac 1 r + \frac { 4 \epsilon } { r } \le \frac 2 r . \end{align*}


\begin{gather*} \dim W ^ { \mathrm { u } } ( p ^ - ) = 2 n - k ^ - , \dim W ^ { \mathrm { s } } ( p ^ + ) = k ^ + + 1 . \end{gather*}


\begin{align*} g ( z ) = \sum _ { n \ge 1 } \rho _ { g } ( n ) n ^ { \frac { \l - 1 } { 2 } } e ( n z ) . \end{align*}


\begin{align*} h ^ { \sigma } = h \end{align*}


\begin{align*} [ v + L _ n ] = \sum _ { w \in L _ n / L _ { n m } } [ v + w + L _ { n m } ] . \end{align*}


\begin{align*} \kappa _ I ( \rho \cdot x ) = \varphi _ I ( \rho ) \cdot \kappa _ I ( x ) \end{align*}


\begin{align*} \mathcal { T } ( \ell ) = \bigsqcup _ { i = 1 } ^ { r ( \ell ) } \mathcal { T } ( \ell ) _ i \sigma , \sigma ' \in \mathcal { T } ( \ell ) _ i \Leftrightarrow \pi ( \sigma ) = \pi ( \sigma ' ) . \end{align*}


\begin{align*} \vect u = \vect u _ 0 + \sum _ { n = 1 } ^ \infty \vect u _ { n } , \end{align*}


\begin{align*} L ( \beta ) + \frac { 1 } { 2 } L ( \beta ^ 2 ) = \frac { 5 } { 7 } , \end{align*}


\begin{align*} \Sigma = \left [ \begin{array} { c c c } 2 & - 1 & - 2 \\ - 1 & 4 & - 2 \\ - 2 & - 2 & 4 \\ \end{array} \right ] . \end{align*}


\begin{align*} \left \{ \begin{aligned} K - G ' ( u ^ * ) u ^ * & \ \ = \ \ \mathcal { K } - G ( u ^ * ) ; \\ G ' ( u ^ * ) u ^ * \in K & \Longleftrightarrow G ( u ^ * ) \in \mathcal { K } \end{aligned} \right . \end{align*}


\begin{align*} P _ { \mu , x } ^ { \zeta } \left ( L _ { x } g , A \right ) = \sum _ { s \in G } { \bf 1 } _ { A } \left ( L _ { x } g s \right ) \frac { d g s . \bar { \lambda } } { d g . \bar { \lambda } } ( x , \zeta ) d \mu ( s ) . \end{align*}


\begin{align*} \sup _ { n \geq 1 } \| g _ n \| _ X = \sup _ { n \geq 1 } \| g _ n \| _ Y \leq \| f \| _ Y . \end{align*}


\begin{align*} F = F _ o + F _ e , \end{align*}


\begin{align*} \lambda = \frac { \langle g , M g \rangle } { \langle g , g \rangle } = \lambda + \frac { 1 } { \langle g , g \rangle } \sum _ { \{ x , y \} \in E } ( - M _ { x y } ) f ( x ) f ( y ) ( a ( x ) - a ( y ) ) ^ 2 . \end{align*}


\begin{align*} f _ j & = \sum _ { m \in \mathbb { Z } ^ 2 } \widetilde { \eta } _ m \eta _ m f _ j \\ & = \sum _ { m \in \mathbb { Z } ^ 2 } \widetilde { \eta } _ m f _ { j , m } + \sum _ { m \in \mathbb { Z } ^ 2 } \widetilde { \eta } _ m \left [ \eta _ m \left ( \psi ^ { ( j ) } \ast _ j f _ j \right ) - \psi ^ { ( j ) } \ast _ j \left ( \eta _ m f _ j \right ) \right ] \\ & = : f _ { j , \flat } + f _ { j , \sharp } + f _ { j , } \end{align*}


\begin{align*} \nu = \int _ { G / P } g . \lambda d \nu _ { 0 } ( g ) = \int _ { G / P _ { \mathsf { f } ( I ) } } g . \lambda d \bar { \nu } _ { 0 } ( g ) . \end{align*}


\begin{align*} \l ( a , b ) = \inf _ { ( u , v ) \in \mathcal { L } ( a , b ) } K ( u , v ) , \ \ \mathcal { L } ( a , b ) : = \{ ( u , v ) \in \mathcal { T } ( a , b ) : L ( u , v ) = 0 \} . \end{align*}


\begin{align*} \frac { d \Phi } { d U } ( U ; \lambda , \alpha ) = \frac { V _ { s } } { U _ { s } } = \alpha + \frac { H _ { \lambda } \left ( \phi _ { 2 } ( \lambda ) - U \right ) } { \Phi ( U ; \lambda , \alpha ) } . \end{align*}


\begin{align*} \| \varphi ( T ) \| _ { L ^ { p _ i , \infty } } \leq C _ i \| T \| _ { \mathcal { B } _ { p _ i } ( \mathcal { H } ) } \ \forall \ T \in \mathcal { B } _ { p _ i } ( \mathcal { H } ) , i = 0 , 1 \end{align*}


\begin{align*} h _ { \mu } ( Z _ { p } , \lambda _ { \ell , p } ) = h _ { \mu } ( M , \bar { \lambda } _ { \ell , p } ) . \end{align*}


\begin{align*} \lim _ { N \to \infty } N ^ { - 1 } E [ Z _ 0 ( r _ N ) | \Omega _ \infty ] = 1 . \end{align*}


\begin{align*} f _ \lambda = \frac { 1 } { \lambda } \bigl ( I - ( I - \lambda f ) ^ { - 1 } \bigr ) , \end{align*}


\begin{align*} G ( 0 , s ) = \sum _ j \frac { { a } _ j } { 1 - s { a } _ j } \nu ( j ) \end{align*}


\begin{align*} \lim _ { p \rightarrow \infty } \mathcal { K } _ { \nu , p } ( x ) = \frac { \Gamma \left ( \frac { 1 } { 2 } - \nu \right ) ( 2 x ) ^ { \nu } } { \sqrt { \pi } } \sum _ { n = 1 } ^ { \infty } n ^ { \nu } \ , K _ { \nu } ( x n ) , \end{align*}


\begin{align*} Q _ \eta = \frac { Q _ \eta } { Q _ { \eta - 1 } } Q _ { \eta - 1 } \leq \left ( 1 + \frac { \gamma } { 4 } \right ) Q _ n \left ( 1 + \frac { \gamma } { 2 } \right ) \leq Q _ n ( 1 + \gamma ) = \kappa Q _ n \end{align*}


\begin{align*} f = f _ 0 + \frac { f _ 1 } { a } \cdot ( a q ) + \ldots + \frac { f _ n } { a ^ n } \cdot ( a q ) ^ n \end{align*}


\begin{align*} \forall \ , p < r \leq q , \ ; \ ; \ ; \ ; & \sharp \{ r \leq s \leq q \ , | \ , D _ { \lambda } ( s ) = \times \} < \sharp \{ r \leq s \leq q \ , | \ , D _ { \lambda } ( s ) = \circ \} \\ & \sharp \{ p \leq s \leq q \ , | \ , D _ { \lambda } ( s ) = \times \} = \sharp \{ p \leq s \leq q \ , | \ , D _ { \lambda } ( s ) = \circ \} . \end{align*}


\begin{align*} \mbox { c o l i m } _ { \omega \subsetneq \tau } ( \underline { X } , \underline { A } ) ^ { \omega , c } = ( \underline { X } , \underline { A } ) ^ { \partial \tau , c } \overset { \imath ^ { \mu , c } _ { \partial \tau , c } } { \longrightarrow } ( \underline { X } , \underline { A } ) ^ { \mu , c } \end{align*}


\begin{align*} \alpha ^ H ( h , k ) & = \frac { ( 2 / \pi ) } { ( 2 \pi ) ^ 3 } \sum _ { r \in \Z ^ 3 } \chi ( r ) \chi ^ \perp ( r + k ) \delta _ { h \cdot k , r \cdot k } \ . \end{align*}


\begin{align*} d ( u , P ) = \psi \left ( \int _ { B ( P , R _ 0 ) } u ^ 2 \right ) , \end{align*}


\begin{align*} K _ 1 & = T ( \phi ) - T ( \phi _ 0 ) T ( \phi _ 1 ) \cdots T ( \phi _ R ) \\ K _ 2 & = T ( \tilde { \phi } ) - T ( \tilde { \phi } _ 0 ) T ( \tilde { \phi } _ 1 ) \cdots T ( \tilde { \phi } _ R ) \end{align*}


\begin{align*} \mathrm { H } ^ { - \infty } _ { \mathcal { H } _ { \hbar , V } } : = \mathcal { L } \left ( \mathrm { H } ^ { \infty } _ { \mathcal { H } _ { \hbar , V } } , \mathbb { C } \right ) , \end{align*}


\begin{align*} \mathrm { R e s } _ { t = 0 } \left ( \frac { \Gamma ( k + t ) ^ 2 } { ( 2 \pi ) ^ { 2 t } } \frac { X ^ t \zeta ( 2 t + 1 ) } { t } \right ) & = \lim _ { t \to 0 } \left ( \frac { d } { d t } \left ( \frac { t ^ 2 \Gamma ( k + t ) ^ 2 } { ( 2 \pi ) ^ { 2 t } } \frac { X ^ t \zeta ( 2 t + 1 ) } { t } \right ) \right ) \\ & = \frac { \left ( ( k - 1 ) ! \right ) ^ 2 } { 2 } \log X + c _ 0 ; \end{align*}


\begin{align*} \mathcal { F } \sqcup \mathcal { K } = K _ 1 \sqcup K _ 2 \sqcup \cdots \sqcup K _ N \end{align*}


\begin{align*} ( a , b , c , t ) \longmapsto \left \{ \begin{bmatrix} ( t - b ) / 2 & & a \\ c & & ( t + b ) / 2 \end{bmatrix} \right \} _ { \Gamma _ { p } } . \end{align*}


\begin{align*} \overline { \overline { C } } _ { X ^ { \ast } } ( g _ 1 , g _ 2 ) & = \overline { \overline { C } } _ { X ^ { \ast } } ( \omega , \omega ^ { - 1 } g _ 1 ) ^ { - 1 } \overline { \overline { C } } _ { X ^ { \ast } } ( \omega , \omega ^ { - 1 } g _ 1 g _ 2 ) \overline { \overline { C } } _ { X ^ { \ast } } ( \omega ^ { - 1 } g _ 1 , g _ 2 ) . \end{align*}


\begin{align*} \sigma = \hat { \varphi } _ { S } ( \pi ) = ( \hat { \varphi } _ { x _ { k } } \circ \cdots \circ \hat { \varphi } _ { x _ { 2 } } \circ \hat { \varphi } _ { x _ { 1 } } ) ( \pi ) . \end{align*}


\begin{align*} \tau _ N : = \inf \{ t \geq 0 : Z _ 0 ( t ) \geq N \} , \end{align*}


\begin{align*} \frac { 2 } { n } \sum _ { i = 1 } ^ n \left ( \boldsymbol { \hat { \tilde { H } } _ { k 2 } } ( t _ p ) - \boldsymbol { \hat { \tilde { H } } _ { k 2 } } ( t _ o ) \right ) ^ T \ ! \hat { \mathbf { \Sigma } } _ \mathbf { k } ^ { - 1 } \tilde { \mathbf { \Sigma } } _ \mathbf { k i } \hat { \mathbf { \Sigma } } _ \mathbf { k } ^ { - 1 } \left ( \boldsymbol { \hat { \tilde { H } } _ { k 2 } } ( t _ p ) - \boldsymbol { \hat { \tilde { H } } _ { k 2 } } ( t _ o ) \right ) , \end{align*}


\begin{align*} p _ 0 ( t ) & = \frac { p ( e ^ { \lambda t } - 1 ) } { e ^ { \lambda t } - p } , \\ p _ j ( t ) & = \frac { q ^ 2 e ^ { \lambda t } } { ( e ^ { \lambda t } - p ) ^ 2 } \cdot \left ( \frac { e ^ { \lambda t } - 1 } { e ^ { \lambda t } - p } \right ) ^ { j - 1 } , j \geq 1 , \end{align*}


\begin{align*} \varphi ( f ( \xi ) ) = \frac { 1 } { 2 \pi i } \int _ \Gamma \varphi ( z ) ( z - f ( \xi ) ) ^ { - 1 } d z . \end{align*}


\begin{align*} \begin{aligned} R _ N ^ 2 ( L , \alpha , \Delta ) & \leq \frac { N _ m } { N } R _ { N _ m } ^ 2 ( L N _ m / N , \alpha , \Delta ) \\ & \leq ( 1 + C / m ) R _ { N _ m } ^ 2 ( ( 1 + C / m ) L _ m , \alpha , \Delta ) . \end{aligned} \end{align*}


\begin{align*} \theta _ A \colon K / \Lambda \simeq A ( \mathbb { C } ) _ \mathrm { t o r } = A ( \overline { \mathbb { Q } } ) _ { \mathrm { t o r } } . \end{align*}


\begin{align*} \tilde { \omega } ^ + = n - | \mathcal { F } | - ( \tilde { k } + \tilde { r } - 1 ) , \ , \ , \ , \ , \ , \tilde { \omega } ^ - = \tilde { k } - 1 . \end{align*}


\begin{align*} \eta ^ { - 1 } S \eta = \gamma _ { p } ^ { m } \end{align*}


\begin{align*} D _ { V } [ \partial _ { t } \gamma _ { S } ] ( e ) + B _ { B } [ \partial _ { t } \gamma _ { B } ] ( e ) = G _ { V , 2 } ( e ) \end{align*}


\begin{align*} \overline { \overline { \Pi } } _ { \psi } ( g ) = \Pi _ { \psi } ( g ) \widetilde { s } ( g ) . \end{align*}


\begin{align*} F _ m < \varepsilon _ 5 ( m ) : = \frac { ( \pi ^ 2 B _ 1 a ) ^ { m / 3 } } { 3 1 6 \ , A ( \sigma ) } \Bigl ( \frac { m ! } { \Gamma ( m / 3 + 2 ) } \Bigr ) ^ { 1 / 2 } \ , \varepsilon _ 4 \end{align*}


\begin{align*} \sigma _ k ( \kappa ) = \sum \limits _ { m = 0 } ^ k \dfrac { ( - 1 ) ^ m \phi '^ { k - m } } { D ^ { k + 2 } \phi ^ m } { n - m \choose k - m } \rho ^ m \left [ \phi ^ 2 \sigma _ m ( D ^ 2 u ) + \dfrac { k + n - 2 m } { n - m } \rho ^ 2 u ^ i u _ j [ T _ m ] _ i ^ j ( D ^ 2 u ) \right ] . \end{align*}


\begin{align*} \deg ( [ x _ 1 ^ { \alpha _ 1 } \cdots x _ N ^ { \alpha _ N } ] ) : = \alpha _ 1 q _ 1 + \dots + \alpha _ N q _ N . \end{align*}


\begin{align*} \log Z _ { n } = S _ n + \log W _ { n } , \end{align*}


\begin{align*} \lim _ { t \to \infty } P ^ \omega ( C _ t ^ c \mid S ) = 0 . \end{align*}


\begin{align*} \int _ { a } ^ b x ^ { \lambda } \ , d x \simeq \begin{cases} ( b - a ) b ^ { \lambda } & \lambda > - 1 , \\ \log ( b / a ) & \lambda = - 1 , \\ ( b - a ) b ^ { - 1 } a ^ { \lambda + 1 } & \lambda < - 1 , \end{cases} 0 < a \le b < \infty . \end{align*}


\begin{align*} \partial _ t w = - i h ^ { - 1 } ( - i h X - i ( q _ 1 + W ) ) w , w | _ { t = 0 } = f , \end{align*}


\begin{align*} \vartheta _ L ( f ) _ P ( g ) = \int _ { \Gamma _ L \cap N \backslash N } \vartheta _ L ( f ) ( n g ) d n \end{align*}


\begin{align*} C _ r ^ 1 = \widehat \psi _ r ( 1 ) = \inf _ { x \in [ 0 , 1 ] } \int _ { [ 0 , 1 ] } \d y \ , \log \frac { 1 } { r ( x , y ) } . \end{align*}


\begin{align*} f ^ { + } ( z ) : = \sum _ { n \gg - \infty } C _ { f } ^ { + } ( n ) q ^ { n } , \end{align*}


\begin{align*} \Phi _ N ( t ) = U _ N ( t ) \Psi _ N ( t ) . \end{align*}


\begin{align*} \left | \frac { N ( y ) } { N ( 1 ) } \right | = | \alpha _ 0 | = \left | \frac { c _ 5 } { b _ 3 } \right | . \end{align*}


\begin{align*} \xi \cdot \nabla _ \xi H ( \xi ) = H ( \xi ) \end{align*}


\begin{align*} n - \delta ( G ) - 1 \leq n - \kappa ( G ) - 1 \leq n - ( n - k ) - 1 = k - 1 \end{align*}


\begin{align*} \pi _ { \psi } ( [ \omega , t ] ) f ( y ) = t \int _ { X ^ { \ast } } \psi ( \langle y , y ^ { \ast } \rangle ) f ( y ^ { \ast } \omega ^ { - 1 } ) d y ^ { \ast } , \end{align*}


\begin{align*} z ( s ) & = h ^ 2 \beta _ h ( s ) + \frac { h ^ 2 } { 2 } \overline { \beta _ h ( s ) } + \frac { \sigma ^ 2 } { 2 } \overline { \beta _ \sigma ( s ) } , \\ \zeta ( s ) & = i \beta _ h ( s ) ( K _ 1 + K _ 3 ) - i \overline { \beta _ h ( s ) } K _ 2 - i \overline { \beta _ \sigma ( s ) } K , \\ \gamma ( s ) & = - i s \beta _ h ( s ) ( | K _ 1 | ^ 2 + | K _ 3 | ^ 2 ) + i s \overline { \beta _ h ( s ) } | K _ 2 | ^ 2 + i s \overline { \beta _ \sigma ( s ) } | K | ^ 2 . \end{align*}


\begin{align*} L _ S T _ { \rho } f = T _ { \rho } L _ S f , \mbox { a n d } D _ { S , x } T _ \rho f = \rho ^ { | S | } T _ { \rho } D _ { S , x } f . \end{align*}


\begin{align*} & \lim _ { j \to \infty } \frac { 1 } { j } \log \big ( \| S ( j ) ( { \xi } _ { [ ( \sigma , k , l ) ] } ) \| _ { L ^ 1 ( I ^ { ( j ) } ) } / | I ^ { ( j ) } | \big ) = - \lambda _ 1 ( n _ { \sigma , k } - a _ { \sigma , k } ) = - \lambda _ 1 \mathfrak { o } ( \sigma , k ) \\ & \lim _ { j \to \infty } \frac { 1 } { j } \log \| S ( j ) ( { \xi } _ { [ ( \sigma , k , l ) ] } ) \| _ { \sup } = - \lambda _ 1 \mathfrak { o } ( \sigma , k ) \mathfrak { o } ( \sigma , k ) > 0 . \end{align*}


\begin{align*} \mathcal { E } _ { g } ^ { t } ( \chi ) = \zeta ( t ) , 0 \leq t \leq T . \end{align*}


\begin{align*} P _ l ( y ( \{ X _ { n i } \} ) ) = \begin{cases} 0 , & l \leq j - 1 , \\ Z _ j = Y ( \{ X _ { n i } \} ) , & l = j , \\ , & l \geq j + 1 . \end{cases} \end{align*}


\begin{align*} M _ 1 ( R , T ) & : = \int _ { T ^ { \beta } } ^ { T \log { T } } \left ( \int _ { - ( \log { t } ) ^ 2 } ^ { ( \log { t } ) ^ 2 } K ( u ) d u \right ) | R ( t ) | ^ { 2 } \Phi \left ( \frac { t } { T } \right ) d t , \\ M _ 2 ( R , T ) & : = \int _ { T ^ { \beta } } ^ { T \log { T } } \left ( \int _ { - ( \log { t } ) ^ 2 } ^ { ( \log { t } ) ^ 2 } e ^ { - i \theta } \log \zeta ( \sigma + i ( t + u ) ) K ( u ) d u \right ) | R ( t ) | ^ { 2 } \Phi \left ( \frac { t } { T } \right ) I ( \sigma , t ) d t , \end{align*}


\begin{align*} \tilde { S } = \oplus ^ { \operatorname { s } } _ i \begin{pmatrix} \tilde { S } _ { \alpha _ i \alpha _ i } & \tilde { S } _ { \alpha _ i \beta _ i } \\ - \tilde { S } _ { \alpha _ i \beta _ i } & \tilde { S } _ { \alpha _ i \alpha _ i } \end{pmatrix} + \mathcal { O } ( \| H \| ) . \end{align*}


\begin{align*} \varphi _ R ( z ) : = \int \limits _ { [ z , z + ( 1 , 1 ) ) } \varphi \big ( \tfrac { y } { \sqrt R } \big ) \ , \dd y \end{align*}


\begin{align*} w _ b = \sum _ { n } c _ n a _ n + T ( f ) _ b , \end{align*}


\begin{align*} \det ( M _ 2 ) = \frac { \partial x } { \partial r } \frac { \partial y } { \partial \omega } - \frac { \partial y } { \partial r } \frac { \partial x } { \partial \omega } . \end{align*}


\begin{align*} \widehat { \Delta } ( x ) = \int _ \mathbb { R } \Delta ( y ) e ( - x y ) \ , d y = \int _ { - 1 } ^ 1 \left ( 1 - | y | \right ) e ^ { - 2 \pi i x y } \ , d y = \begin{cases} 1 , & x = 0 \\ \frac { \sin ^ 2 { \pi x } } { \pi ^ 2 x ^ 2 } , & x \neq 0 \end{cases} \end{align*}


\begin{align*} d = \avg { w _ 0 \cdot { \bf e } _ k } _ { B } = \lim _ { h \to + \infty } \left ( \avg { w _ h \cdot { \bf e } _ k } _ { B } - \lambda _ h ^ { ( 1 / 2 ) } \right ) = 0 , \end{align*}


\begin{align*} \left \lbrace \begin{array} { l c r } L _ a ( u - v ) = H ( \cdot , u ) - H ( \cdot , v ) & \textnormal { i n } & \Omega \\ u - v = 0 & \textnormal { i n } & \partial \Omega \\ u - v \in C ^ { 2 } ( \Omega ) \cap C ^ 0 ( \bar \Omega ) . & & \\ \end{array} \right . \end{align*}


\begin{align*} \int _ { 0 } ^ { k _ \epsilon } | \tau | ^ { p - 2 } ( \tau - k ) _ - \ , d \tau \leq M \int _ { 0 } ^ { k _ \epsilon } | \tau | ^ { p - 2 } \ , d \tau = M | s | ^ { p - 2 } s \Big | _ { 0 } ^ { k _ \epsilon } \leq \boldsymbol \gamma ( p ) M ^ p \epsilon . \end{align*}


\begin{align*} = U ^ { j _ { 1 } - 1 } \alpha _ { p } U ^ { j _ { 2 } - 1 } \alpha _ { p } \dots U ^ { j _ { r } - 1 } \alpha _ { p } \iota \end{align*}


\begin{align*} \partial _ v ( a b ) : = \partial _ v ( a ) b + ( - 1 ) ^ { | v | \cdot | a | } a \partial _ v ( b ) . \end{align*}


\begin{align*} 2 ( - 1 ) ^ j \binom { n - k - j } { k - j } , \end{align*}


\begin{align*} [ D \widetilde { \psi } : D \widetilde { \varphi } ] _ t = s ( \psi ) [ D \widetilde { \chi } : D \widetilde { \varphi } ] _ t = s ( \psi ) [ D \chi : D \varphi ] _ t = [ D \psi : D \varphi ] _ t \end{align*}


\begin{align*} A = \left ( \frac { n } { n - m } \right ) \left ( \frac { n } { m } \right ) ^ { \frac { m } { n - m } } . \end{align*}


\begin{align*} f ( x ) & = \frac { 1 } { 2 K } \sum _ { j = L + 1 } ^ M j ^ { - \theta } = \frac { 1 } { 2 K } \sum _ { x - K x ^ \theta < j \leq x + K x ^ \theta } j ^ { - \theta } + O ( x ^ { - \theta } + x ^ { \theta - 1 } ) \\ & = \frac { 1 } { 2 K ( 1 - \theta ) } \big ( ( x + K x ^ \theta ) ^ { 1 - \theta } - ( x - K x ^ \theta ) ^ { 1 - \theta } \big ) + O ( x ^ { - \theta } + x ^ { \theta - 1 } ) \\ & = 1 + O ( x ^ { - \theta } + x ^ { \theta - 1 } ) , \end{align*}


\begin{align*} \nu _ i ( f ' ) - \nu _ i ( f ) = \alpha _ { i } . \end{align*}


\begin{align*} & E \left [ \left ( Y e ^ { \lambda s } - Z _ 0 ( s ) \right ) ^ 2 \right ] \\ & = E [ Z _ 0 ( s ) ^ 2 ] - 2 e ^ { \lambda s } E [ Y Z _ 0 ( s ) ] + e ^ { 2 \lambda s } E [ Y ^ 2 ] \\ & = e ^ { 2 \lambda s } E [ Y ^ 2 ] - E [ Z _ 0 ( s ) ^ 2 ] . \end{align*}


\begin{align*} \varphi _ n = ( - 1 ) ^ { n - 1 } \left . \frac { d } { d \alpha } \right | _ { \alpha = 0 } \sum _ { | T | = n } \chi _ T ( \alpha ) X _ T \end{align*}


\begin{align*} C _ { \mathrm { K P F } } = ( \Delta \log n ) ^ { \kappa } ( \log \log n ) ^ { 1 + \kappa } \cdot O \left ( \frac { e ^ { 1 3 \kappa } } { b ^ { 5 + 6 \kappa } } \right ) . \end{align*}


\begin{align*} \begin{aligned} T _ 1 \geq C s ^ 3 \lambda ^ 4 \iint _ Q \xi ^ 3 \left | A \nabla \eta \cdot \nabla \eta \right | ^ 2 | u | ^ 2 d x d t - C s ^ 3 \lambda ^ 3 \int _ 0 ^ T \int _ { \omega } \xi ^ 3 | u | ^ 2 d x d t . \end{aligned} \end{align*}


\begin{align*} \kappa ( ( \underbar { x } , \underbar { y } ) ) : = \sum _ { n = 1 } ^ { N } n \Big [ S _ n ( \mu ' ) ( x _ n ) - \rho ( x _ n ) - S _ n ( \mu ' ) ( y _ n ) - \rho ( y _ n ) \Big ] , ( \underbar { x } , \underbar { y } ) \in R , \end{align*}


\begin{align*} \mu ( \sigma ) = \frac { 1 } { Z _ H } e ^ { - H ( \sigma ) } , \end{align*}


\begin{align*} \overline g = E _ f ^ { - 1 } s ^ T . \end{align*}


\begin{align*} \lim _ { t \to \infty } \sup _ { x \in X ' } \varepsilon _ { x , n } ( t ) = 0 . \end{align*}


\begin{align*} A _ { I } x ^ * = b _ { I } , \mathcal { L } ( x ^ { * } , y ^ { * } ) \leq \mathcal { L } ( x , y ^ { * } ) \forall x \in X . \end{align*}


\begin{align*} \begin{array} { r c l } x & = & F _ { x } ( \varphi _ { [ 0 , R - 1 ] } ) \\ \varphi _ { c } & = & \varphi _ { c } \\ \varphi _ { [ A , R ] } & = & \varphi _ { [ A , R ] } \ , , \end{array} \end{align*}


\begin{align*} \sum _ { i = 1 } ^ D \alpha _ i \ , T _ i ( z ^ n \otimes \lbrace 0 , \infty \rbrace ) = 0 , \ , \ \forall \ , \ 0 \leq n \leq 2 k - 2 . \end{align*}


\begin{align*} \pi ^ + _ v \cdot \left ( \prod _ { i : v _ i \in V _ E } P _ i \right ) = \pi ^ + _ v , \end{align*}


\begin{align*} P _ { \mu , x } ^ { \zeta , n } \left ( L _ { x } , L _ { x } \omega _ { n } \right ) \le e ^ { - ( \delta - \epsilon ) n } = e ^ { - 2 n \epsilon } . \end{align*}


\begin{align*} \begin{cases} \nabla \cdot ( y ^ { 1 - 2 \gamma } \nabla U ) = 0 & \mathbb { R } ^ { n + 1 } _ + , \\ U ( x , 0 ) = u ( x ) & \mathbb { R } ^ n , \end{cases} \end{align*}


\begin{align*} \begin{cases} \displaystyle \dot h _ 1 ( t ) = - \omega \ , u _ 2 ( t ) , \\ \displaystyle \dot h _ 2 ( t ) = \omega \ , u _ 1 ( t ) . \end{cases} \end{align*}


\begin{align*} \log | f ( 0 ) | + \int _ 0 ^ r \frac { n ( r ) } { r } d r = \frac { 1 } { 2 \pi } \int _ 0 ^ { 2 \pi } \log | f ( r e ^ { i \theta } ) | d \theta , \end{align*}


\begin{align*} \beta \begin{bmatrix} \tfrac { \beta } { \alpha } M _ 1 ^ H + M _ 2 ^ H \\ \tfrac { \beta } { \alpha } M _ 1 ^ H - M _ 2 ^ H \end{bmatrix} = \begin{bmatrix} \tfrac { \beta } { \alpha } I _ n & I _ n \\ \tfrac { \beta } { \alpha } I _ n & - I _ n \end{bmatrix} \begin{bmatrix} M _ 1 ^ H \\ M _ 2 ^ H \end{bmatrix} . \end{align*}


\begin{align*} \left ( S ^ { - 1 } \tilde { S } \right ) _ { \gamma _ i \gamma _ i } = N _ { [ i ] } + \mathcal { O } ( \| H \| ^ 2 ) . \end{align*}


\begin{align*} w _ l : = \begin{cases} \phi ( w _ { l - 1 } ) & \mbox { o n } \cup _ { i \in \N } B ^ i _ l , \\ w _ { l - 1 } & \mbox { o t h e r w i s e } , \end{cases} \end{align*}


\begin{align*} \left ( \sum _ { j \in M ^ c } | a _ j | ^ p \right ) ^ \frac { 1 } { p } \leq \varepsilon \left ( \sum _ { j = 1 } ^ n | a _ j | ^ p \right ) ^ \frac { 1 } { p } . \end{align*}


\begin{align*} \lim _ { N \to \infty } N ^ { - 1 } M ( \tau _ N ) = \nu \int _ 0 ^ \infty e ^ { - \lambda s } ( 1 - p _ 0 ( s ) ) d s , \end{align*}


\begin{align*} \mho _ { H } ( v ) : = { \rm S h d } ( v ) \cap \partial \left ( H \backslash F \right ) . \end{align*}


\begin{align*} f ( x ) = \sum _ { n = 0 } ^ { \infty } C ( n ) x ^ { \alpha n + \beta - 1 } \end{align*}


\begin{align*} \sigma ( a ) = P _ H \pi ( a ) | _ H , a \in M . \end{align*}


\begin{align*} I ( R ) = \int _ { B _ R } \psi _ + ^ 2 w ^ 2 , \end{align*}


\begin{align*} r _ { J ^ - } [ I ( f ) ] ( g ) & = r _ { J ^ - } [ g ^ { - 1 } I ( f ) ] ( \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} ) \\ & = g ^ { - 1 } r _ { J ^ - } [ I ( f ) ] ( \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} ) \\ & = 2 \tfrac { \partial } { \partial \overline { z } } f ( g i ) . \end{align*}


\begin{align*} g l _ 1 ( K U ) ^ 1 ( X ) : = [ X , B G L _ 1 ( K U ) ] \ . \end{align*}


\begin{align*} \left ( \sigma \circ ( \sigma \circ f ) \ , , \ , M \ , , \ , ( ( \sigma \circ f ) ^ * h ) \circ ( f ^ * h \circ \phi ) \ , , \ , t \right ) & = \left ( f \ , , \ , M \ , , \ , ( f ^ * \sigma ^ * h ) \circ f ^ * h \circ \phi \ , , \ , t \right ) \\ & = \left ( f \ , , \ , M \ , , \ , f ^ * ( \sigma ^ * h \circ h ) \circ \phi \ , , \ , t \right ) \\ & = \left ( f \ , , \ , M \ , , \ , f ^ * ( \textnormal { I d } _ L ) \circ \phi \ , , \ , t \right ) , \ , \ , \ , \end{align*}


\begin{align*} t ^ \ast : = \min _ { r \in [ 0 , 1 ] } t ^ \ast ( r ) . \end{align*}


\begin{align*} \sum _ { x \in \Theta _ k ^ { n _ 1 } } | \phi _ E ( x ) | ^ 2 = 1 - \sum _ { x \in \Lambda _ { L _ { k - 1 } } \setminus \Theta _ k ^ { n _ 1 } } | \phi _ E ( x ) | ^ 2 \geq 1 - C L ^ { \beta d } e ^ { - m ' L ^ { \rho ^ { 2 n _ 1 - 1 } } } \geq 1 / 2 \end{align*}


\begin{align*} \widehat { P _ { \leq N } u } ( k ) : = \widehat { u _ { \leq N } } ( k ) : = \psi \left ( \frac { k } { N } \right ) \hat { u } ( k ) , \end{align*}


\begin{align*} T ( f \cdot g ) = T ( f ) \cdot g + f \cdot T ( g ) + 2 A ( f ) \cdot A ( g ) \end{align*}


\begin{align*} 1 2 \cdot \mu _ { D R } ^ { p \delta } = - [ ( \mu _ { D D , \delta } ) _ c ] \end{align*}


\begin{align*} \Omega _ 1 ( \mathcal { G } _ 1 ^ 1 ( \mathbb { Q } ) ) = \Omega _ 2 ( \mathcal { G } _ 2 ^ 1 ( \mathbb { Q } ) ) . \end{align*}


\begin{align*} ( - ) ^ \delta : M & \to M \\ m & \mapsto m ^ \delta : = \delta \ast m \end{align*}


\begin{align*} S ^ { - 1 } \left ( p / \sqrt { | u _ n | } - ( R _ 1 , R _ 2 ) \begin{bmatrix} \sqrt { | u _ n | } & 0 \\ 0 & \sqrt { | u _ n | } \end{bmatrix} \begin{bmatrix} u _ n + p / u _ n \\ u _ { \tau } \end{bmatrix} \right ) = G . \end{align*}


\begin{align*} 1 + q \rho d X + \binom { q } { 2 } \rho ^ 2 d ^ 2 X ^ 2 + e \frac { q ( q - 2 ) } { 6 } \rho ^ 3 | d X | ^ 2 , \end{align*}


\begin{align*} \xi _ F ( s ) = \omega \ , \overline { \xi _ F ( 1 - \bar { s } ) } , \end{align*}


\begin{align*} S _ { \mathrm { o f f } } = \sum _ { d _ 1 , \ , d _ 2 < D } \ , \ , \ , \ & \sum _ { I d _ 1 , \ , J d _ 2 < D } x _ { I d _ 1 } \bar { x } _ { J d _ 2 } \sqrt { I J } \sum _ { L , M \geq 1 } \frac { G _ k ( L M d _ 1 d _ 2 \slash p ) } { \sqrt { L M } } \\ & \times \left ( 2 \pi i ^ { 2 k } \sum _ { p | c } \frac { \mathcal { S } ( I L , J M , c ) } { c } \ , J _ { 2 k - 1 } \left ( \frac { 4 \pi \sqrt { I L J M } } { c } \right ) \right ) . \end{align*}


\begin{align*} p _ { ( M - \lambda _ k I ) } = p _ { ( N - \lambda _ k I ) } + | \mathcal { F } ^ c | , \end{align*}


\begin{align*} a _ { i j } = \frac 1 2 \epsilon _ { i m n } \epsilon _ { j k l } \partial _ { m } \eta _ k \partial _ { n } \eta _ l , \end{align*}


\begin{align*} \nu _ i \left ( ( f _ \gamma ) _ { \tilde { \bf Q } } \right ) = \nu \left ( ( f _ \gamma ) _ { \tilde { \bf Q } } \right ) . \end{align*}


\begin{align*} a _ 1 ^ 2 = O ( \varepsilon ) \| u \| _ { L ^ 2 ( \mathbb { S } ^ n ) } ^ 2 . \end{align*}


\begin{align*} \frac { \abs { w _ h - \lambda _ h ^ { ( 1 / 2 ) } } ^ 2 } { t _ h } + 2 \left ( w _ h - \lambda _ h \right ) \cdot \frac { \lambda _ h ^ { ( 1 / 2 ) } } { t _ h } = t _ h \left ( 1 - \frac { \lambda _ h ^ { ( 1 / 2 ) } } { t _ h } \right ) \left ( 1 + \frac { \lambda _ h ^ { ( 1 / 2 ) } } { t _ h } \right ) . \end{align*}


\begin{align*} \frac { 1 } { m ! } | P ^ { ( m ) } ( p ) - Q ^ { ( m ) } ( p ) | \le \sum _ { k = m } ^ { J - 1 } 2 ^ k \delta _ k < \frac 1 3 \frac { \varepsilon _ m } { m ! } \end{align*}


\begin{align*} \begin{array} { l l } g _ M ( x _ 1 , \dots , x _ k ) = & | R _ 1 | ^ { 1 - k } \left ( \sum _ { a _ 2 , \dots , a _ k \in R _ 1 } f ( x _ 1 + \sum _ { i = 2 } ^ k a _ i x _ i , x _ 2 , \dots , x _ k ) \right . \\ & \left . - \sum _ { a _ 2 , \dots , a _ k \in R _ 1 } f ( e x _ 1 + \sum _ { i = 2 } ^ k a _ i x _ i , x _ 2 , \dots , x _ k ) \right ) . \end{array} \end{align*}


\begin{align*} J ( u ) = \bar { \mathbb { E } } \bigg [ \int _ 0 ^ \infty e ^ { - \beta t } f \Big ( x _ t ^ u , y _ t ^ u , z _ t ^ u , \tilde { z } _ t ^ u , \int _ { \mathcal { E } } \gamma _ { ( t , e ) } ^ u \nu ( d e ) , u _ t \Big ) d t + \phi ( y _ 0 ^ u ) \bigg ] , \ \beta \ , \end{align*}


\begin{align*} X _ \varphi = \overline { [ \underline { \varphi } , \varphi ] } . \end{align*}


\begin{align*} \int _ { \Omega } | X u | ^ { p - 2 } X u \cdot X \varphi d x = \lambda \int _ { \Omega } | u | ^ { p - 2 } u \varphi d x \end{align*}


\begin{align*} | \mathcal { E } _ 1 | & = \int _ { \mathcal { E } _ 1 ' } \ ! \int _ { I } \ ! \mathbf { 1 } _ { \mathcal { E } _ 1 } ( \mathbf { z } - \mathbf { p } ( t ) , t ) \ , \mathrm { d } t \mathrm { d } \mathbf { z } + \int _ { \mathbb { R } ^ 2 \setminus \mathcal { E } _ 1 ' } \ ! \int _ { I } \ ! \mathbf { 1 } _ { \mathcal { E } _ 1 } ( \mathbf { z } - \mathbf { p } ( t ) , t ) \ , \mathrm { d } t \mathrm { d } \mathbf { z } \\ & \leq 2 | \mathcal { E } _ 1 ' | + c _ 1 C _ { P _ 1 , P _ 2 } | \mathcal { E } _ 1 | \end{align*}


\begin{align*} c _ { Y _ { \ell } } \left ( y _ { \ell } , y _ { \ell } ^ { \prime } \right ) = \inf _ { x _ { \ell } , x _ { \ell } ^ { \prime } \in \mathcal { X } } c _ { \ell } \left ( \left ( y _ { \ell } , x _ { \ell } \right ) , \left ( y _ { \ell } ^ { \prime } , x _ { \ell } ^ { \prime } \right ) \right ) . \end{align*}


\begin{align*} ( \bar x , y ) = Q _ { \psi _ 0 + ( y - y _ 0 ) \bar x } , y \in I . \end{align*}


\begin{align*} \frac { } { t } \frac { 1 } { 2 } \| u \| _ { H ^ { - 1 } ( \Omega ) } ^ 2 = \int _ { \Omega } \Delta w ( - \Delta _ N ) ^ { - 1 } u \ : x = - \int _ { \Omega } w u \ : x . \end{align*}


\begin{align*} w ( z ) = \Phi ( z ) + \widetilde { T } ( f ) ( z ) , \end{align*}


\begin{align*} \int _ { \mathbb { S } ^ n } u \mathrm { d } A = - \int _ { \mathbb { S } ^ n } \dfrac { n - j - 1 } { 2 } u ^ 2 \mathbb { d } A - \int _ { \mathbb { S } ^ n } \dfrac { j + 1 } { 2 n } | \nabla u | ^ 2 \mathrm { d } A + O ( \varepsilon ) \| u \| _ { L ^ 2 ( \mathbb { S } ^ n ) } ^ 2 + O ( \varepsilon ) \| \nabla u \| _ { L ^ 2 ( \mathbb { S } ^ n ) } ^ 2 . \end{align*}


\begin{align*} \| H ( f ) \| _ { L _ { | x | _ h } ^ p L _ { \theta } ^ { \bar { p } _ 1 } ( \mathbb H ^ n ) \rightarrow L _ { | x | _ h } ^ p L _ { \theta } ^ { \bar { p } _ 2 } ( \mathbb H ^ n ) } = \omega _ Q ^ { 1 / \bar { p } _ 2 - 1 / \bar { p } _ 1 } \frac { \omega _ Q Q } { ( Q - Q / p ) Q / p } . \end{align*}


\begin{align*} { T _ { \varepsilon _ i } ^ { \vec { v } } } ^ T = \begin{pmatrix} x _ i & - \alpha _ 1 y _ i - \alpha _ 0 z _ i & - \alpha _ 0 y _ i \\ y _ i & x _ i + \alpha _ 2 y _ i & - \alpha _ 0 z _ i \\ z _ i & y _ i + \alpha _ 2 z _ i & x _ i + \alpha _ 2 y _ i + \alpha _ 1 z _ i \end{pmatrix} . \end{align*}


\begin{align*} \Gamma ( t ) = \begin{cases} \{ 0 \} & \mbox { i f } t < 0 \\ \{ 1 \} & \mbox { i f } t > 0 \\ [ 0 , 1 ] & \mbox { i f } t = 0 \end{cases} . \end{align*}


\begin{align*} \xi : M \times \R ^ 2 \to T M , \xi ( x , y , z ; u _ 1 , u _ 2 ) = \bigg ( x , y , z ; u _ 1 , u _ 2 , \frac 1 2 ( u _ 2 x - u _ 1 y ) \bigg ) . \end{align*}


\begin{align*} \lambda \in \Lambda ^ { l } ( \sigma ) \iff \dim \ker S _ 1 = l . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ { M _ N } X _ { N , M _ N } ^ { ( i ) } \xrightarrow [ N \to \infty ] { d } \ , \mathcal { N } \big ( 0 \ , , \sigma _ { \varphi } ^ 2 \big ) \ , . \end{align*}


\begin{align*} \sum _ { h \in H } \chi ^ { [ n - 1 , 1 ] } ( x h ) = - \frac { | H | } { \binom { n - k } { k } } \binom { n - k - 1 } { k - 1 } . \end{align*}


\begin{align*} \vartheta _ L ( f ) _ P ( g ) = \vartheta _ L ( f ) _ { P _ { \le r } } ( g ) . \end{align*}


\begin{align*} G ( z ) : = H ( z ) e ^ { \alpha \vert z \vert ^ 2 } , \ , \ , z \in \Omega , \end{align*}


\begin{align*} \left \Vert \partial _ { t } ^ { s } \widetilde { u } \right \Vert _ { H ^ { 2 , 1 } \left ( Q _ { \varepsilon , T } \right ) } , \left \Vert \partial _ { t } ^ { s } \widetilde { m } \right \Vert _ { H ^ { 2 , 1 } \left ( Q _ { \varepsilon , T } \right ) } \leq C \delta ^ { 1 - \rho } , \forall \delta \in \left ( 0 , \delta _ { 0 } \right ) , s = 0 , 1 , 2 . \end{align*}


\begin{align*} \Re ( g _ { \lambda _ i } ( z ) ) = \Re ( \lambda _ i - 2 z + \lambda _ i z ^ 2 ) = \lambda _ i + \Re ( - 2 z + \lambda _ i z ^ 2 ) \geq \lambda _ i - | - 2 z + \lambda _ i z ^ 2 | \geq \frac { \lambda _ i } { 2 } , \end{align*}


\begin{align*} ( x _ 0 , t _ 0 ) + Q _ { \rho } ( \theta ) = K _ { \rho } ( x _ 0 ) \times ( t _ 0 - \theta \rho ^ { s p } , t _ 0 ] . \end{align*}


\begin{align*} \nu ( \{ 0 \} ) = 0 , \int _ { - \infty } ^ { \infty } \min ( 1 , \lambda ^ 2 ) \ , \nu ( d \lambda ) < \infty \end{align*}


\begin{align*} \begin{aligned} A ^ m = & \frac { \partial } { \partial \left ( | \nabla u | ^ 2 \right ) } \left . \left [ ( - 1 ) ^ m \rho ^ m { n - m \choose k - m } \frac { \phi '^ { k - m } } { D ^ { k + 1 } } \phi ^ { n - m + 1 } \right ] \right | _ { u = | \nabla u | ^ 2 = 0 } \\ = & ( - 1 ) ^ { m + 1 } \rho ^ { m + 2 } \frac { k + 1 } { 2 } { n - m \choose k - m } \phi '^ { k - m } ( \rho ) \phi ^ { n - m - k - 2 } ( \rho ) , \end{aligned} \end{align*}


\begin{align*} u _ { \alpha \bar \beta } ( 0 ) = - u _ { x _ n } ( 0 ) \rho _ { \alpha \bar \beta } ( 0 ) , ~ ~ \alpha , \beta \leq n - 1 . \end{align*}


\begin{align*} \zeta = \exp ( \sum _ j h _ j X _ j ) ( z ) , \end{align*}


\begin{align*} \L u : = \sum _ { i , j = 1 } ^ { m } a _ { i j } X _ i X _ j u - X _ 0 u , \end{align*}


\begin{align*} L _ N Y _ { s } ^ N ( \phi ) & = \frac { 1 } { \sqrt { N } } \sum _ { ( x , \sigma ) \in V } ( \eta _ s ( x , \sigma ) - \rho ) \cdot ( A \phi ) ( \tfrac { x } { N } , \sigma ) + R _ 1 ( \phi , N , s ) , \end{align*}


\begin{align*} \psi ( t ) = K _ { t \wedge \tau } ^ { - 1 } ( X ( t ) + ( t \wedge \tau ) \eta ) . \end{align*}


\begin{align*} \sum _ { k > L } \frac { R ^ k } { k } \min \{ q ^ { k / 2 } , d ( \chi ) \} \le d ( \chi ) \sum _ { k > L } \frac { R ^ k } { k } \le \frac { d ( \chi ) } { L + 1 } \sum _ { k > L } R ^ k = \frac { d ( \chi ) } { L + 1 } \frac { R ^ { L + 1 } } { 1 - R } , \end{align*}


\begin{align*} L p _ { \sigma ( j ) } ( y _ j ) = f ( y _ j ) , j = 1 , \ldots , M , \end{align*}


\begin{align*} \sigma _ { L ^ 2 _ { \rm p e r } ( 0 , N T ) } \left ( \mathcal { A } [ \phi ] \right ) \cap B ( 0 , r _ N ) = \{ 0 \} , \end{align*}


\begin{align*} \mathcal A [ \phi ] = - I + \mathcal { J } \mathcal { L } [ \phi ] , \end{align*}


\begin{align*} T _ { \varphi } ( x , i ) : = \begin{cases} ( x , i + 1 ) & 0 \leq i < h - 1 \\ ( \varphi ( x ) , 0 ) & i = h - 1 \end{cases} \end{align*}


\begin{align*} N _ 1 \coloneqq ( d + 1 ) \cdot \min \left \{ \prod _ { i = 1 } ^ n ( h _ i + 1 ) , \binom { n + h } { n } \right \} \end{align*}


\begin{align*} \frac { 1 } { n ! } \int _ { - R } ^ { R } \int _ { - R } ^ R \widehat { \gamma } _ n ( t , s , x - y ) d x d y = 4 \pi C _ H ^ n R \int _ { T _ n ( s ) } \big ( \ell _ R * \widehat { g } _ { \pmb { t } _ n } ^ { ( n ) } \big ) ( 0 ) d \pmb { t } _ n , \end{align*}


\begin{align*} F _ { K _ 1 K _ 2 K _ 3 K } ( s , x ) = e ^ { - z ( s ) | x | ^ 2 + \zeta ( s ) \cdot x + \gamma ( s ) } \end{align*}


\begin{align*} \widetilde { \rho } _ t ( t , \kappa ( t ) ) + 2 { \rm R e } ( \widetilde { \rho } _ w ( t , \kappa ( t ) ) \kappa ' ( t ) ) = 0 { \rm \ f o r \ } t > 0 . \end{align*}


\begin{align*} \overline { \Pi } _ { \psi } [ u ( b ) ] f ( [ \epsilon , x ] ) = e ^ { \pi i \epsilon x ^ 2 b } f ( [ \epsilon , x ] ) ; \end{align*}


\begin{align*} \left ( \sum _ { \theta = 1 } ^ { d } ( h _ { p \theta + p - 1 } t ^ { p \theta + p - 1 } - g _ { p \theta + p - 1 } ( a t + b ) ^ { p \theta + p - 1 } ) \right ) + f _ 0 ( t ) , \end{align*}


\begin{align*} \mu _ { j + 1 } ^ 2 ( A ) - \mu _ { j } ^ 2 ( A _ { \mathcal { S } } ) \le \mu _ 1 ^ 2 ( R ) = \| A h \| ^ 2 - | \langle h , A h \rangle | ^ 2 \end{align*}


\begin{align*} ( T f ) ( x ) = \Delta _ { h } f ( x ) = f ( x + h ) - f ( x ) \end{align*}


\begin{align*} F ^ { \prime } ( u ) = \mu G ^ { \prime } ( u ) \quad \enspace \mathcal { G } \end{align*}


\begin{align*} \xi ^ { \tau } ( x ) = \frac { 1 } { \tau ^ { 2 } } \xi \left ( \frac { x } { \tau } \right ) , \eta ^ { \tau } ( x ) = \frac { 1 } { \tau ^ { 2 } } \eta \left ( \frac { x } { \tau } \right ) , \quad \tau > 0 , \end{align*}


\begin{align*} \Upsilon _ { j } ( u ) = \frac { \int _ { B _ j } d ( u , P ) P d P } { \int _ { B _ j } d ( u , P ) d P } \in B _ j . \end{align*}


\begin{align*} \forall s , t \in [ 1 / 2 , 1 ) , z _ h ^ { ( s ) } = z _ h ^ { ( t ) } = v _ h \mbox { a . e . o n } B _ 1 \setminus \left ( \cup _ { \mathcal { F } ^ { ( s ) } _ h } B \bigcup \cup _ { \mathcal { F } ^ { ( t ) } _ h } B \right ) \end{align*}


\begin{align*} N ^ { \ , ( 1 ) } ( X , Y ) = [ { f } , { f } ] ( X , Y ) - 2 \sum \nolimits _ { i } d \eta ^ i ( X , Y ) \ , \xi _ i . \end{align*}


\begin{align*} \frac { \partial } { \partial h } \lambda ( y ^ u , y ^ h ) & = e ^ { \gamma } \log y \left ( \int _ { 1 } ^ { u / h } y ^ { h t - u } \omega ( t ) \ , d t - y ^ { h - u } \right ) - h ^ { - 1 } \lambda ( y ^ { u - h } , y ^ h ) \\ & = h ^ { - 1 } \lambda ( y ^ u , y ^ h ) - e ^ { \gamma } y ^ { h - u } \log y - h ^ { - 1 } \lambda ( y ^ { u - h } , y ^ h ) . \end{align*}


\begin{align*} \sigma ( C ) & = \left \{ \sqrt { n m } - 1 ^ { ( n + m ) } , 1 ^ { ( n + m ) } \right \} . \end{align*}


\begin{align*} \eta = \sum _ { l = 1 } ^ L \alpha _ { l , { n _ t } } \hat \beta _ l , \ \ \hat \eta = \sum _ { l = 1 } ^ L \alpha _ { l , { \hat { n } _ t } } \hat \beta _ l e ^ { - j ( \theta _ { l , n _ t } - \theta _ { l , { \hat { n } _ t } } ) } . \end{align*}


\begin{align*} D ^ { k - 1 } ( f ) = c F = c ^ { \sigma } F . \end{align*}


\begin{align*} u '' ( x ) = f ( x ) , x \in ( a , b ) , u ( a ) = u ( b ) = 0 , \end{align*}


\begin{align*} C s ^ 4 - C s ^ 2 + 1 = 0 \end{align*}


\begin{align*} \Upsilon _ { 0 , T } ^ { u , u ^ { \prime } ; p } = \mathbb { E } \left [ \int _ { 0 } ^ { T } \left \vert u _ { s } - u _ { s } ^ { \prime } \right \vert ^ { p } \mathrm { d } \theta \right ] . \end{align*}


\begin{align*} u = \prod b _ i ^ i , ( b _ i , b _ j ) = 1 i \neq j , b _ i . \end{align*}


\begin{align*} \widehat { P } ^ \omega \left ( ( \log t ) ^ { 2 / d } \left ( \frac { \log N _ t } { t } - \beta \right ) + c ( d , \nu ) < - \varepsilon \right ) = \widehat { P } ^ \omega \left ( N _ t < \exp \left [ t \left ( \beta - \frac { c ( d , \nu ) + \varepsilon } { ( \log t ) ^ { 2 / d } } \right ) \right ] \right ) . \end{align*}


\begin{align*} | \Psi ( H ) | = \frac { 1 } { 4 } \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} + \Psi ( | H | ) \end{align*}


\begin{align*} \lim _ { n \to \infty } \frac { \det T _ n ( \phi ) } { \prod _ { k = 0 } ^ R \det T _ n ( \phi _ k ) } = E \end{align*}


\begin{align*} \mu _ { \omega , \mathbf { c } } = \frac { 1 } { 6 } \inf \{ L _ { \omega , \mathbf { c } } ( \Psi ) | \ \Psi \in \mathcal { H } ^ 1 \backslash \{ ( \mathbf { 0 } , \mathbf { 0 } , \mathbf { 0 } ) \} , \ K _ { \omega , \mathbf { c } } ( \Psi ) = 0 \} . \end{align*}


\begin{align*} P ( x _ 1 , \dots , x _ n ) = 0 . \end{align*}


\begin{align*} L ( \mathcal { K } , \bar { u } ) = \{ t v \in \mathcal { U } : \ G ( \bar { u } ) + G ' ( \bar { u } ) v \in \mathcal { K } , \ \forall \ t > 0 \} . \end{align*}


\begin{align*} \left \{ \begin{aligned} d \xi _ t ^ u & = h ( x _ t ^ u , u _ t ) d t + d \tilde { W } _ t , \ t \in [ 0 , \infty ) , \\ \xi _ 0 & = 0 , \end{aligned} \right . \end{align*}


\begin{align*} S _ { x , \chi } = \sum _ { \substack { n \le x \\ p \mid n \implies p \le y } } \chi ( n ) + O ( x y ^ { - 1 / 2 } ( \log m ) ^ 4 ) . \end{align*}


\begin{align*} \phi = \Tilde { \phi } _ { e } + \sum _ { m = 1 } ^ { k } D _ { p _ m } \Delta x ^ { p _ m } \end{align*}


\begin{align*} h ^ { \sigma } = h \end{align*}


\begin{align*} ( \mathcal { L } _ h V ) ^ 2 = \mathrm { t r } \big ( ( \nabla V ) ^ 2 \big ) + e ^ h \mathrm { d i v } \Big ( e ^ { - h } \ , V \ , \mathcal { L } _ h V - e ^ { - h } \ , \nabla V \ , V \Big ) + \nabla ^ 2 h \ , V \cdot V \quad \end{align*}


\begin{align*} \bigl ( y ( \omega , t ) < s \bigr ) & = \bigl ( y ^ { * } ( \omega , \ , s ^ { - } ) > t \bigr ) \\ \bigl ( y ^ { * } ( \omega , \ , t ) < s \bigr ) & = \bigl ( y ( \omega , \ s ^ { - } ) > t \bigr ) . \end{align*}


\begin{align*} \sum _ { \substack { w \in V / L \\ w t = v } } \sigma ( w ) = \sigma ( v ) \mid t ^ { - 1 } . \end{align*}


\begin{align*} \alpha ' _ n = \frac { 1 + a } { 1 + a q ^ { 2 n } } q ^ n \alpha _ n ( a ^ 2 , q ^ 2 ) \quad \mbox { a n d } \quad \beta ' _ n = \sum _ { j \leq n } \frac { ( - a ) _ { 2 j } } { ( q ^ 2 ; q ^ 2 ) _ { n - j } } q ^ { j } \beta _ j ( a ^ 2 , q ^ 2 ) . \end{align*}


\begin{align*} V = V ^ { ( \nu _ 1 ) } \oplus W ^ { ( \nu _ 1 ) } \oplus \kappa ( W ^ { ( \nu _ 1 ) } ) . \end{align*}


\begin{align*} \Pi _ { \psi } ( g ) f ( [ \epsilon , w ] ) = ( ( \det g ) \epsilon , a ) _ { \R } f ( [ ( \det g ) \epsilon , w g ^ { s ( ( \det g ) \epsilon ) } ] ) . \end{align*}


\begin{align*} \sigma = B ^ { - 1 } \frac { \alpha ^ { p + 1 } } { \gamma ^ p } \end{align*}


\begin{align*} \tau _ { n - 1 } \le \tau _ n \le C _ r \ , \tau _ { n - 1 } , \mbox { f o r } \ ; 2 \le n \le N . \\ \end{align*}


\begin{align*} & b _ 2 + c _ 2 = u ( b _ 3 + c _ 3 ) \end{align*}


\begin{align*} \int _ M ( 1 + w ) | T ^ \varphi | ^ 2 = - \alpha \int _ M ( 1 + w ) | \tau ( \varphi ) | ^ 2 \ , . \end{align*}


\begin{align*} \left \{ \begin{alignedat} { 3 } u _ { t } + \mathcal { L } ^ { \Phi } _ { A } u & = ( - \Delta ) ^ { \frac { s } { 2 } } f + g & & \mbox { i n $ \Omega \times ( 0 , T ) $ } , \\ u & = h & & \mbox { i n $ \mathbb { R } ^ { n } \setminus \Omega \times [ 0 , T ] $ } , \\ u ( \cdot , 0 ) & = h ( \cdot , 0 ) & & \mbox { i n $ \Omega $ } \end{alignedat} \right . \end{align*}


\begin{align*} \underset { t \to \infty } { \lim } P ^ \omega ( A _ { m ( t ) } ^ c \mid S ) = 0 , \end{align*}


\begin{align*} ( \Delta + \mu - q ) v = f \ ; \mathrm { i n } \ ; M , \partial _ \nu v = \mp a w + \varphi . \end{align*}


\begin{align*} f ( q t ) = f ( t ) \sigma _ { q t } ( A ) \end{align*}


\begin{align*} \ell \sum _ { i = 1 } ^ N \sum _ { m \in \mathbb { Z } } \Delta \left ( \frac { m } { \ell } \right ) = L . \end{align*}


\begin{align*} ( \pounds _ { \xi _ i } \ , \Phi ) ( X , Y ) = ( \pounds _ { \xi _ i } \ , g ) ( X , { f } Y ) + g ( X , ( \pounds _ { \xi _ i } { f } ) Y ) . \end{align*}


\begin{align*} \eta ( t , x ) = A \ , \cos ( k x - \omega t ) \ , , \Phi ( t , x , y ) = A \ , \frac { \omega } { k } \ , \frac { \cosh k ( y + h _ 0 ) } { \sinh k h _ 0 } \ , \sin ( k x - \omega t ) \ , , \end{align*}


\begin{align*} m ( \lambda ) = \frac { 4 \pi } { 3 \vphantom { \lambda ^ 3 } } \left ( \frac { 1 } { 2 \lambda ^ 3 } - \frac { 2 } { ( \lambda + 2 ) ^ 3 } \right ) 1 _ { ( 0 , 2 ) } ( \lambda ) . \end{align*}


\begin{align*} \prod _ 0 ^ t \exp f ( s ) \ , d s = \exp _ G ( t - t _ j ) f ( t _ j ) \exp _ G ( t _ j - t _ { j - 1 } ) f ( t _ { j - 1 } ) \ldots \exp _ G t _ 1 f ( 0 ) . \end{align*}


\begin{align*} \Psi _ n \left ( \frac { A } { 1 - q } \right ) = \frac 1 { 1 - q t } \left . S _ n \left ( \frac { 1 - q t | } { | 1 - q } A \right ) \right | _ { t = \frac 1 q } . \end{align*}


\begin{align*} \sum _ { 1 \leq r \leq N ^ 2 } W _ N ( w _ r , \mathcal { A } ) ^ 2 = \# \{ ( i , j , k , l ) \in [ 1 , N ] ^ 4 : i \neq j , k \neq l , a _ i - a _ j = a _ k - a _ l \} \leq E _ N ( \mathcal { A } ) , \end{align*}


\begin{align*} \boldsymbol { d } _ { \mathcal { V } } ( ( s _ 1 , s _ 2 ) , ( s _ 1 ^ \prime , s _ 2 ^ \prime ) ) = \left [ \boldsymbol { d } _ { \mathcal { S } _ 1 } ( s _ 1 , s _ 1 ^ \prime ) ^ p + \boldsymbol { d } _ { \mathcal { S } _ 2 } ( s _ 2 , s _ 2 ^ \prime ) ^ p \right ] ^ { 1 / p } . \end{align*}


\begin{align*} L _ { \omega , \mathbf { c } } ( U _ n ) = K _ { \omega , \mathbf { c } } ( U _ n ) - 3 N ( U _ n ) \ \rightarrow \ 0 . \end{align*}


\begin{align*} \int _ 0 ^ { \rho } \int _ { \mathbb { S } ^ n } r ( 1 + u ( x ) ) ^ 2 \phi ^ n ( r ( 1 + u ( x ) ) ) x _ l \mathrm { d } A \mathrm { d } r = 0 , \ \ l = 1 , 2 , \cdots , n + 1 . \end{align*}


\begin{align*} d ( v , V ) + d ( V , v ) = 2 r , \deg ^ + ( v ) = n - d ( v , V ) \mbox { a n d } \deg ^ - ( v ) = n - d ( V , v ) \end{align*}