\begin{align*} \kappa ^ \pi _ n ( \gamma _ 1 \vee \cdots \vee \gamma _ n \otimes m ) = \sum _ { k = 0 } ^ \infty \frac { 1 } { k ! } \kappa _ { n + k } ( \pi \vee \cdots \vee \pi \vee \gamma _ 1 \vee \cdots \vee \gamma _ n \otimes m ) . \end{align*}


\begin{align*} u ( x ) = \bar { P } _ \rho \ast u ( x ) = \bar { P } _ \rho \ast u _ 1 ( x ) + \bar { P } _ \rho \ast u _ 2 ( x ) , | x - x _ 0 | < \rho / 2 . \end{align*}


\begin{align*} | B | \leq \dfrac { \left | A ^ { 1 / 2 } \dfrac { z } { | z | } \right | } { \left | A ^ { 1 / 2 } \dfrac { e } { | e | } \right | } = \sqrt { C } \leq \ ( \dfrac { \Lambda \ , \Phi } { \lambda \ , \Phi } \ ) ^ { 1 / 2 } = \sqrt { \dfrac { \Lambda } { \lambda } } . \end{align*}


\begin{align*} \rho ( A ) = \rho _ + ( A _ - ) + \rho _ - ( A _ + ) \end{align*}


\begin{align*} x _ { n + 1 } = \Phi ( \Delta t , x _ n ) \end{align*}


\begin{align*} D _ { \lambda , 1 } ( f ) = D _ { \sigma , 0 } \Big ( \frac { f ( z ) - f ^ { * } ( \lambda ) } { z - \lambda } \Big ) , \end{align*}


\begin{align*} \lim _ { N \to \infty } \frac { 1 } { N \theta } \log \left ( \int _ { \mathbb { R } ^ { d \times N } } \exp \left ( - N \theta f ( { \rm e m p } _ { N } ) \right ) d \gamma _ { N } \right ) = \inf _ { \mu \in \mathcal { P } ( \mathbb { R } ^ { d } ) } \{ f ( \mu ) + F ( \mu ) \} , \end{align*}


\begin{align*} 0 \neq q ^ { l + 1 } + \cdots + q ^ s = q ^ { l + 1 } ( 1 + q + \cdots + q ^ { s - l - 1 } ) \notin R ^ \times . \end{align*}


\begin{align*} \int _ { \Omega } { \rm i n t } [ \overline { \mathbf { P } } ^ { x } ] \ , d x = 1 . \end{align*}


\begin{align*} k m - a b - ( k - a ) ( m - b ) = 8 . \end{align*}


\begin{align*} d ( f ^ 2 ( x ) , x ^ * ) = d ( f ( f ( x ) ) , f ( x ^ * ) ) \leq c \cdot d ( f ( x ) , x ^ * ) \leq c ^ 2 \cdot d ( x , x ^ * ) . \end{align*}


\begin{align*} ( Y _ I ^ J ) ^ S = \{ w _ K \in W \mid K \subset \Delta , J \subset K \subset I \} . \end{align*}


\begin{align*} V _ g f ( x , \xi ) = \langle f , \pi ( x , \xi ) g \rangle . \end{align*}


\begin{align*} \tilde b _ { \epsilon } ( t , x ) : = \ , ( \tilde b _ { \epsilon , 1 } ( t , x ) , \dots , \tilde b _ { \epsilon , n } ( t , x ) ) \ , , \end{align*}


\begin{align*} T _ t f = \Psi ( i _ { \mathsf { b } } ( t ) ) \mbox { f o r a l l } t \in G \ , . \end{align*}


\begin{align*} S _ g = S _ { B } - \frac { 2 m } { f } \Delta _ { B } f + \frac { S _ { F } } { f ^ 2 } - m ( m - 1 ) \frac { | \nabla _ { B } f | ^ 2 } { f ^ 2 } . \end{align*}


\begin{align*} \lim _ { n \to \infty } \sqrt { n } \ , c _ n = \sqrt { 2 / \pi } . \end{align*}


\begin{align*} r = \frac { \sqrt { 4 p ^ 2 + 1 } + 2 p - 1 } { 2 p } . \end{align*}


\begin{align*} G _ k \subseteq \bigcap _ { j = 1 } ^ k \Big \{ \overline { U } _ j \ge - \frac { c _ 1 m } { ( \log m ) ^ 2 } \Big \} : = \bigcap _ { j = 1 } ^ k B _ j \ , , \end{align*}


\begin{align*} A : \{ i \in a : | i | = 0 \} \rightarrow K \ ; . \end{align*}


\begin{align*} 0 < \lim _ { t \to \infty } \rho ( t ) = \rho < \infty ; \\ \lim _ { t \to \infty } t \rho ' ( t ) \log t = 0 . \end{align*}


\begin{align*} F ( D ^ 2 P ) = 0 , \end{align*}


\begin{align*} { \downarrow } ^ { \mathbb { X } } p ^ { - 1 } [ Y \smallsetminus ( U \cap V ) ] = { \downarrow } ^ { \mathbb { X } } p ^ { - 1 } [ Y \smallsetminus U ] \cup { \downarrow } ^ { \mathbb { X } } p ^ { - 1 } [ Y \smallsetminus V ] , \end{align*}


\begin{align*} \phi \big ( h , \phi ( t , x , u ) , u ( t + \cdot ) \big ) = \phi ( t + h , x , u ) . \end{align*}


\begin{align*} \Lambda '' _ p : = \int _ { I } \int _ \Omega \exp \left ( \frac { \ell p ^ 2 } { p - n } \| D _ x b ( v , y ) \| \right ) \dd y \dd v . \end{align*}


\begin{align*} ( A \otimes B ) ( \vec { x } ) = A ( \vec { x } \rvert _ 0 ) \cdot B ( \vec { x } \rvert _ 1 ) \ ; . \end{align*}


\begin{align*} Y ^ { q ^ k } = f ( X ) \end{align*}


\begin{align*} \Gamma _ G ( f ) : = \frac { \sigma ^ 2 } { 2 } f '' ( x ) + ( \mu - G ( x ) ) f ' ( x ) - q f ( x ) = 0 , x > 0 , \end{align*}


\begin{align*} a \sqcup b = \{ ( 0 , i ) \ \ i \in a \} \cup \{ ( 1 , i ) \ \ i \in b \} \ ; . \end{align*}


\begin{align*} \log \| \langle s _ 0 , \dots , s _ d | Z \rangle \| _ v = \log \| \langle s _ 0 , \dots , s _ { d - 1 } | Z \cap \mathrm { d i v } ( s _ d ) \rangle \| _ v + \int _ { Z _ { \overline { K } _ v } ^ { \mathrm { a n } } } \log \| s _ d \| _ v \prod _ { i = 0 } ^ { d - 1 } c _ 1 ( \overline { L } _ { i , v } ) , \end{align*}


\begin{align*} \partial _ t \rho - \nabla \cdot ( \rho \nabla p ) = \rho G \ , p ( 1 - \rho ) = 0 , \rho \leq 1 , \end{align*}


\begin{align*} M _ { j } ^ { j - \pi ( \ell ) } ( \ell ) = \overline { M _ { j - \pi ( \ell ) } ^ { j } ( - \ell ) } \end{align*}


\begin{align*} m _ \delta ( s ) = \left \{ \begin{array} { l l } m ( \delta ) , & s \leq \delta \\ m ( s ) , & \delta \leq s \leq 1 - \delta \\ m ( 1 - \delta ) , & s \geq 1 - \delta . \end{array} \right . \end{align*}


\begin{align*} \vert \Delta _ { i } \vert = \vert \Delta _ { m / 2 } \vert = \frac { \vert G _ \alpha \vert } { \vert G _ { \alpha \beta } \vert } = \frac { 2 ( m ! ) ^ 2 } { 4 ( ( m - i ) ! i ! ) ^ 2 } = 2 ^ { - \lfloor \frac { 2 i } { m } \rfloor } \left ( \frac { m ! } { ( m - i ) ! i ! } \right ) ^ 2 . \end{align*}


\begin{align*} \left \| \hat { Z } _ i \hat { f } _ i \right \| _ { \infty } = \frac { A r _ i ^ { - 1 } } { r _ i ^ { - 2 } } \left \| \frac { \partial f _ i } { \partial z } \right \| _ \infty \le 2 r _ i A \rho ^ { i - 1 } = 2 \rho ^ { - 1 } . \end{align*}


\begin{align*} m \int _ 0 ^ 1 d \theta \log \left | 1 + e ^ { 2 \pi i \theta } + \frac { x + i y } { m } \right | = \max \{ x , 0 \} = m \max \left \{ \log | 1 + \frac { x + i y } { m } | , 0 \right \} . \end{align*}


\begin{gather*} \gamma _ { \mathfrak n } ( x , y , z ) = B \left ( [ x , y ] _ { \theta , \gamma } , z \right ) = B \left ( [ x , y ] + \theta ( x , y ) + \gamma ( x , y , \cdot ) , z \right ) = \gamma ( x , y , z ) . \end{gather*}


\begin{align*} Z ' ( \gamma ) = - i \ , \chi ( \rho ) ^ { 1 / 2 } \ , \zeta ' ( 1 - \rho ) . \end{align*}


\begin{align*} \lim _ { \delta \to 0 } \lim _ { N \to \infty } \left | \mathcal { E } ^ { \neq } ( { \rm e m p } _ { N } - \mu _ { \theta } ) - \mathcal { E } ( \rho - \mu _ { \theta } ) \right | = 0 . \end{align*}


\begin{align*} \frac { \partial r _ \epsilon } { \partial t } \ , - \ , \triangle r _ \epsilon \ , + \ , \frac { q } { \epsilon ^ { 1 - \kappa } } ( | D r _ \epsilon | ^ 2 \ , - \ , 1 ) \ ; = \ ; 0 . \end{align*}


\begin{align*} J _ \Sigma \ ; : = \ ; \bigl ( x _ \sigma ' \colon \sigma \in \Sigma \bigr ) \ ; = \ ; \bigl ( x _ \sigma ' \colon \sigma \in \Sigma [ \max ] \bigr ) \end{align*}


\begin{align*} X _ { c } ( w ) = \mathrm { R e } \int ^ w \left ( 1 - c G ^ 2 , - i ( 1 + c G ^ 2 ) , 2 G \right ) F d \zeta , w \in D , \end{align*}


\begin{align*} \# \left ( \{ x \in \mathbb { Z } _ q \mid s _ { h } x = 0 \} \bigcup \{ x _ i , \ - x _ i \mid 1 \leq i \leq h - 1 \} \right ) = d _ { h } + 2 h - 2 . \end{align*}


\begin{align*} \biggl ( 1 + \frac { 1 } { \sqrt { N } } \biggr ) ^ { N _ t } = \sum _ { k = 0 } ^ { N _ t } { N _ t \choose k } \frac { 1 } { N ^ { k / 2 } } , \end{align*}


\begin{align*} \delta ' _ { k , i } : = \begin{cases} \gamma _ { k , i } \Delta _ k \cap f ^ { - 1 } ( Y _ i ) \neq \emptyset \\ \infty \Delta _ k \cap f ^ { - 1 } ( Y _ i ) = \emptyset . \end{cases} \end{align*}


\begin{align*} \overline { \mathbf { F } } _ { R , N } ^ { i } ( C ) = \frac { 1 } { | K _ { i } | } \int _ { K _ { i } } \delta _ { \left ( x , \left ( \theta _ { N ^ { \frac { 1 } { d } } x } C \right ) \bigg | _ { \square _ { R } } \right ) } \ , d x , \end{align*}


\begin{align*} K _ { \ ! _ \mathcal W } ( z , w ) = \frac { \partial ^ { 2 n } } { \partial z ^ n \partial \overline { w } ^ n } \Big ( ( z - \lambda ) ( \overline { w } - \overline { \lambda } ) K _ { \sigma , n - 1 } ( z , w ) \Big ) , z , w \in \mathbb D . \end{align*}


\begin{align*} S ( v | \Psi ) ( x ) : = \bigg ( \iint _ { \widehat \Gamma ( x ) } | \nabla v | ^ 2 \Psi \frac { d s d y } { | s | ^ { n - d } } \bigg ) ^ { 1 / 2 } \end{align*}


\begin{align*} | M _ { n , \ell , { \bf t } } | = \binom { n + ( \ell - 1 ) - \sum _ { j = 1 } ^ { \ell - 1 } t _ j } { \ell } , \mbox { f o r $ 0 \le \ell \le d $ } . \end{align*}


\begin{align*} i \int _ 0 ^ t \Delta I _ h z ( s ) \d s = & I _ h z ( t ) - I _ h z _ 0 + \int _ 0 ^ t I _ h ( i \nu z ( r ) + \epsilon ( \gamma z ( r ) - \mu \overline { z } ( r ) ) ) \d r \\ & + \tfrac { 1 } { 2 } \int _ 0 ^ t I _ h ( z ( r ) F _ \Phi ) \d r - i \kappa \int _ 0 ^ t I _ h ( \theta _ R ( | z | _ { X ^ { \mathfrak { s } } _ r } ) | z ( r ) | ^ 2 z ( r ) ) \d r \\ & + i \int _ 0 ^ t I _ h ( z ( r ) \cdot ) \d W ( r ) \end{align*}


\begin{align*} v _ \kappa ( z ) \cdot z & = \rho _ \kappa ^ { - 1 } ( m _ \kappa \circ \ , \phi _ \kappa ^ { - 1 } ( z ) - z ) \cdot z = \rho _ \kappa ^ { - 1 } ( m _ \kappa \circ \ , \phi _ \kappa ^ { - 1 } ( z ) \cdot z - 1 ) = - \frac { 1 } { 2 \rho _ \kappa } | m _ \kappa \circ \ , \phi _ \kappa ^ { - 1 } ( z ) - z | ^ 2 . \end{align*}


\begin{align*} E [ G ( 0 , S _ \ell ) ] = \sum _ { i = \ell } ^ \infty P ^ 0 ( S _ i = 0 ) \le C _ 1 \big ( 1 + \sqrt { \ell } \big ) ^ { - 1 } \ , . \end{align*}


\begin{align*} c ( \lambda ; r ) : = \widehat r ( 0 ) - 1 + 2 \int _ 0 ^ 1 \alpha \ , \widehat r ( \alpha ) \ , d \alpha . \end{align*}


\begin{align*} K \coloneqq \left \{ x \in \mathbb { R } ^ { n } \left | \begin{array} { l } c _ { i } ( x ) = 0 ~ ( i \in \mathcal { E } ) , \\ c _ { j } ( x ) \geq 0 ~ ( j \in \mathcal { I } ) \end{array} \right \} , \right . \end{align*}


\begin{align*} \rho ( a \cdot X ) ( r ) & = \eta ( a ) \rho ( X ) ( r ) = a \cdot \rho ( X ) ( r ) , \\ \rho ( X ) ( a \cdot r ) & = \rho ( X ) \big ( \eta ( a ) r \big ) = \rho ( X ) \big ( \eta ( a ) \big ) r + \eta ( a ) \rho ( X ) ( r ) \\ & = \eta \big ( \omega ( X ) ( a ) \big ) r + a \cdot \rho ( X ) ( r ) = \omega ( X ) ( a ) \cdot r + a \cdot \rho ( X ) ( r ) \end{align*}


\begin{align*} \lambda ^ \star _ x ( n ) = \| D f ^ n ( x ) | _ { E ^ \star } \| \ : \ : \ : \ : \star = c , s , u , \end{align*}


\begin{align*} \int _ \Omega | g - h _ 1 | ^ p = \int _ { \Omega \setminus { E } } | h _ 2 - h _ 1 | ^ p = \int _ { \Omega \setminus { E } } | f _ E | ^ p = I \end{align*}


\begin{align*} \omega _ { N } = \int _ { \mathbb { R } ^ { d } } \exp \left ( - 2 N \beta \zeta ( x ) \right ) d x , \end{align*}


\begin{align*} a = b \land c a = b a = c . \end{align*}


\begin{align*} & \int _ \Sigma \partial _ { \boldsymbol { \nu } } u _ { \delta _ k } v _ { \delta _ k } \ , d \Gamma d t + { \delta _ k } \int _ \Sigma | \nabla _ \Gamma v _ { \delta _ k } | ^ 2 \ , d \Gamma d t \\ & { } + \int _ \Sigma \eta _ { \delta _ k } v _ { \delta _ k } \ , d \Gamma d t = \int _ \Sigma \bigl ( w _ { \delta _ k } - \pi _ \Gamma ( v _ { \delta _ k } ) + g \bigr ) v _ { \delta _ k } \ , d \Gamma d t . \end{align*}


\begin{align*} & d \mu _ t ^ \epsilon \ , = \ , \Bigl ( \frac { \epsilon ^ { 1 - \kappa } } 2 \vert D v _ \epsilon \vert ^ 2 \ , + \ , \frac { F ( v _ \epsilon ) } { 2 \epsilon ^ { 1 - \kappa } } \ , - \ , \frac { \epsilon ^ { 1 - \kappa } \dot \kappa } { K ( \epsilon ) } G ( v _ \epsilon ) \Bigr ) \ , d x , \\ & d \xi _ t ^ \epsilon \ , = \ , \Bigl ( \frac { F ( v _ \epsilon ) } { 2 \epsilon ^ { 1 - \kappa } } \ , - \ , \frac { \epsilon ^ { 1 - \kappa } } 2 \vert D v _ \epsilon \vert ^ 2 \Bigr ) \ , d x , \end{align*}


\begin{align*} f _ { \mathsf { b } } ( i _ { \mathsf { b } } ( t ) ) = f ( t ) \mbox { f o r a l l } t \in G \ , . \end{align*}


\begin{align*} F ( A ) _ M ( i , j ) = a A _ M ( i , j ) \ ; . \end{align*}


\begin{align*} \dot { f } _ t ( z ) = - p ( z , t ) f _ t ' ( z ) \end{align*}


\begin{align*} \phi : \frac { F _ { p ^ r } [ x ] } { \langle f ( x ) \rangle } \longrightarrow \frac { F _ { p ^ r } [ x ] } { \langle F ( x ) \rangle } \oplus \frac { F _ { p ^ r } [ x ] } { \langle F ^ { * } ( x ) \rangle } \cong \frac { F _ { p ^ r } [ x ] } { \langle { x ^ N } - 1 \rangle } P ( x ) \mapsto \phi ( P ( x ) ) = ( P ( x ) , 0 ) \end{align*}


\begin{align*} \xi '' ( x ) & + \xi ' ( x ) ( 2 - m ) \tanh x - \xi ( x ) ( m - 1 ) \big ( \tanh ^ 2 x - \frac { 1 } { \cosh ^ 2 x } \big ) \\ & + \frac { p - 2 } { \cosh x } \frac { d } { d x } \big ( \frac { \cosh x } { m } ( \xi ' ( x ) - ( m - 1 ) \tanh x \xi ( x ) ) \big ) \\ & + \frac { \hat { \lambda } } { \cosh ^ { 2 } x } \xi ( x ) = 0 , \end{align*}


\begin{align*} x = 1 2 N \frac { 1 } { v + u } , y = 3 6 N \frac { v - u } { v + u } \end{align*}


\begin{align*} \begin{aligned} \| \zeta \| _ { C ( \overline { S _ T } ) } & \leq c \Bigl ( \| \zeta ( \cdot , 0 ) \| _ { C ( \Gamma ( 0 ) ) } + \| f _ \zeta ^ \varepsilon \| _ { C ( Q _ { \varepsilon , T } ) } + \| \psi _ \zeta ^ \varepsilon \| _ { C ( \partial _ \ell Q _ { \varepsilon , T } ) } \Bigr ) \\ & + c \varepsilon \Bigl ( \| \zeta \| _ { C ( \overline { S _ T } ) } + \| \zeta _ 2 \| _ { C ( \overline { S _ T } ) } \Bigr ) . \end{aligned} \end{align*}


\begin{align*} \beta _ A \geq \max _ \ell \frac { 2 \dim A } { \dim G _ { A , \ell } ^ \circ } \geq \gamma _ A = \frac { 2 \dim A } { \dim G _ { A } } . \end{align*}


\begin{align*} { \bf E } _ v ^ { \otimes M } \ { \bf E } _ { y _ L } ^ { \otimes M } \sum _ { \gamma _ 1 , . . . , \gamma _ n = 1 } ^ M \prod _ { j = 1 } ^ n v _ { \gamma _ j } . \end{align*}


\begin{align*} & \frac { \partial w _ \epsilon } { \partial t } \ , - \ , \triangle w _ \epsilon \ , + \ , \frac 2 { \epsilon ^ { 2 ( 1 - \kappa ) } } f ( w _ \epsilon ) \ , - \ , \frac { \dot \kappa } { K ( \epsilon ) } g ( w _ \epsilon ) \\ & \ ; = \ ; \frac { \dot { q } } { \epsilon ^ { 1 - \kappa } } \left [ \frac { \partial r } { \partial t } \ , - \ , \triangle r \ , + \ , \frac { 2 q } { \epsilon ^ { 1 - \kappa } } ( | D r | ^ 2 \ , - \ , 1 ) \right ] \ ; \le \ ; 0 ( x \ne 0 ) . \end{align*}


\begin{align*} \begin{aligned} & I _ 1 \lesssim \bigl ( \varepsilon \norm { v } { H ^ 2 } \norm { \xi } { H ^ 2 } + \delta \bigr ) \norm { \nabla v _ { h h } } { 2 } ^ 2 + \bigl ( \varepsilon \norm { v } { H ^ 2 } \norm { \xi } { H ^ 2 } \\ & ~ ~ ~ ~ + \varepsilon ^ 4 \norm { v } { H ^ 1 } \norm { \xi } { H ^ 2 } ^ 4 + \varepsilon ^ 4 \norm { v } { H ^ 1 } ^ 4 \norm { \xi } { H ^ 2 } ^ 4 + \norm { v } { H ^ 2 } ^ 4 \\ & ~ ~ ~ ~ + 1 \bigr ) \norm { v } { H ^ 2 } ^ 2 . \end{aligned} \end{align*}


\begin{align*} P \bigg ( \limsup _ { n \to \infty } ( r ( s _ n ) ^ { - 1 } \sup _ { t \le s _ { n - 1 } } | B _ t | ) = 0 \bigg ) = 1 . \end{align*}


\begin{align*} \eta _ { n , k } = \frac { 1 } { k - 1 } { \displaystyle \sum \limits _ { j = 1 } ^ { k - 1 } } \left ( \eta _ { n - 1 , j } - \eta _ { n + 1 , j } \right ) \eta _ { n , k - j } , k \geq 2 . \end{align*}


\begin{align*} \underset { k = 0 } { \rm R e s } \breve { N } ^ { ( 2 ) } ( x , t , k ) = \frac { A } { 2 i } \breve { N } ^ { ( 1 ) } ( x , t , 0 ) , \end{align*}


\begin{align*} ( m + 1 ) \chi ( M ) \mp \chi ( N _ m ^ f M ) = - N ( a _ m ^ \pm ) . \end{align*}


\begin{align*} \| \Phi ^ { ( \mathfrak { s } ) } \| _ { \delta } : = \| \Phi ^ { ( \mathfrak { s } ) } \| _ { \mathcal { L } _ 2 } + \| \Phi ^ { ( \mathfrak { s } ) } \| _ { \gamma ( L ^ 2 ( \mathbb { R } ; \mathbb { R } ) ; W ^ { \mathfrak { s } , 2 + \delta } ( \mathbb { R } ; \mathbb { C } ) ) } . \end{align*}


\begin{align*} \sum _ { i \in F } | x _ i | & \ \le \ ( N + M ) - \min F + 1 \ = \ M - j _ 0 + 1 \\ & \ \stackrel { \eqref { r e e 3 1 } } { \le } \ M - g ( M , N ) + 1 \ \lesssim \ \sqrt { M + N } . \end{align*}


\begin{align*} V _ { \ell , k } ^ { \delta } : = A ^ { - T } ( \delta k / \sqrt { d } ) \left \langle \delta \cdot B _ 1 ( \ell / \sqrt { d } ) \right \rangle \times Q _ { k } ^ { \delta } , Q _ { k } ^ { \delta } : = Q _ { \Phi , k / \sqrt { d } } ^ { ( \delta , 1 ) } = \Phi ^ { - 1 } \bigl ( \delta \cdot B _ 1 ( k / \sqrt { d } ) \bigr ) , \end{align*}


\begin{align*} \kappa _ { \psi , y _ 0 } ^ { X | Y } ( Z ) = \kappa _ N ( \psi _ { y _ 0 } ( Z ) ) \cdot \kappa _ { \phi , y _ 0 } ^ Y ( Z _ Y ) . \end{align*}


\begin{align*} 2 \mathrm { R e } \langle u _ { \alpha } \otimes \Phi _ y ^ { ( 0 ) } \ , | \ , H _ y R _ y \rangle = 4 \alpha ^ { 1 / 2 } \mathrm { R e } ( \overline { \Phi } ^ { ( 0 ) } _ y \langle P u _ { \alpha } \otimes A ^ + _ y \Omega \ , | \ , R _ y \rangle ) = - 2 \mathrm { R e } ( \overline { \Phi } ^ { ( 0 ) } _ y \kappa ) \| \Phi _ { \# } ^ y \| ^ 2 _ { \# } , \end{align*}


\begin{align*} \left \{ \begin{aligned} & h _ 1 ( x , t ) = c _ 0 ( \xi ) + \frac { i \kappa } { k _ 0 } \left ( \mathfrak { B } _ { 1 2 } ^ r ( \xi , t ) + \overline { \mathfrak { B } _ { 1 2 } ^ r ( \xi , t ) } \right ) + R _ 3 ( \xi , t ) , \\ & h _ 2 ( x , t ) = i \kappa + \frac { c _ 0 ( \xi ) } { k _ 0 } \left ( \mathfrak { B } _ { 2 1 } ^ r ( \xi , t ) + \overline { \mathfrak { B } _ { 2 1 } ^ r ( \xi , t ) } \right ) + R _ 3 ( \xi , t ) , \end{aligned} \right . \end{align*}


\begin{align*} H ( v , Y ( a , z ) u ) = H ( Y ( A ( z ) a , z ^ { - 1 } ) v , u ) , u , v \in V . \end{align*}


\begin{align*} d _ Q \Sigma ( P , Q ) = \frac { F _ { d \Sigma } ( P , Q ) d x _ P \otimes d x _ Q } { ( x _ P - x _ Q ) ^ 2 h _ X ( x _ { P } , y _ { \bullet P } ) h _ X ( x _ { Q } , y _ { \bullet Q } ) } . \end{align*}


\begin{align*} \lim _ { j \to \infty } \frac { E [ \overline { U } _ j ^ 2 ] } { \rho _ { n _ j - n _ { j - 1 } } ^ 2 } = \sigma _ d ^ 2 \ , , \end{align*}


\begin{align*} \left | \frac { \Gamma ( p ) } { \Gamma ( p + q i ) } \right | ^ 2 : = \prod _ { k = 0 } ^ \infty \left ( 1 + \left ( \frac { q } { p + k } \right ) ^ 2 \right ) . \end{align*}


\begin{align*} \psi ( \xi ) = \int _ { \R ^ d \setminus \{ 0 \} } ( 1 - \cos ( \xi \cdot y ) \nu ( d y ) , \xi \in \R ^ d . \end{align*}


\begin{align*} \lim _ { \epsilon \to 0 } \lim _ { N \to \infty } \frac { 1 } { N } \log \left ( \rho _ { N } ^ { \otimes N } \left ( \overline { \mathbf { P } } _ { N } \in B ( \overline { \mathbf { P } } , \epsilon ) \right ) \right ) = - \overline { \rm E n t } [ \overline { \mathbf { P } } | \overline { \mathbf { \Pi } } ^ { 1 } ] - c _ { \omega , \Sigma } . \end{align*}


\begin{align*} \begin{array} { l } \mbox { f o r a l l } 1 \le j \not = k \le n \mbox { t h e r e e x i s t s } \beta _ { j k } \in C ^ 1 \\ \mbox { s u c h t h a t } b _ { j k } = \beta _ { j k } ( a _ k - a _ j ) . \end{array} \end{align*}


\begin{align*} w _ K = w _ 0 ^ { K _ 1 } \times w _ 0 ^ { K _ 2 } \times \cdots \times w _ 0 ^ { K _ m } \end{align*}


\begin{align*} \sum _ { x \in X _ n } \tau ( x ) \Lambda ^ { - 1 } _ { n } ( x ) \leq b _ { n } \int _ { - \infty } ^ \infty \Lambda ^ { - 1 } _ { n } ( x ) \mathrm { d } x = b _ { n } ( n + 1 ) . \end{align*}


\begin{align*} ( K s ) ( x _ 1 ) = \int _ X K ( x _ 1 , x _ 2 ) \cdot s ( x _ 2 ) d v _ X ( x _ 2 ) , s \in L ^ 2 ( X , L ^ p \otimes F ) . \end{align*}


\begin{align*} w ( \tau ) = \det ( \mathrm { D } \Phi ^ { - 1 } ( \tau ) ) = \prod _ { i \in \underline { d } } \mathrm { D } \Phi _ i ^ { - 1 } ( \tau _ i ) > 0 \forall \ , \tau \in \R ^ d , \end{align*}


\begin{align*} \frac { 2 } { p } + \frac { 3 } { q } \leq 1 , ~ ~ ~ \mathbf { X } ^ { p , q } : = \left \{ \begin{alignedat} { - 1 } L ^ p ( [ 0 , T ] ; L ^ q ( \Omega ) ) , ~ ~ p \neq \infty , \\ C ( [ 0 , T ] ; L ^ q ( \Omega ) ) , ~ ~ p = \infty . \end{alignedat} \right . \end{align*}


\begin{align*} I _ A \cap \{ l : \max a ( l ) \leq j - 2 \} & = \bigcup _ { l \in N _ { j - 2 } \cap J _ A } a ( l ) \setminus \{ l \} \end{align*}


\begin{align*} u ( x , t ) = A \delta ^ 2 ( 0 , \xi ) + o ( 1 ) , x > 0 , \ t < 0 \end{align*}


\begin{align*} X _ n = h ^ { ( n ) } ( \vec { Y } ^ { ( n ) } ) \end{align*}


\begin{align*} C _ i ^ j = \Gamma _ c ( G ^ { ( i ) } , s ^ * F ^ j ) , \end{align*}


\begin{align*} x ( d ) - x ( c ) = \int _ { c } ^ { d } D F ( x ( \tau ) , t ) , \textnormal { w h e n e v e r $ [ c , d ] \subseteq [ a , b ] $ } . \end{align*}


\begin{align*} \begin{array} { c c } ^ { \rho } D ^ { \beta ( 1 - \alpha ) } _ { 0 _ + } \Big ( f ( t , u ( t ) , ^ \rho D ^ { \alpha , \beta } u ( t ) ) - p ( t ) u ( t ) \Big ) \\ = \delta _ { \rho } \ ^ { \rho } I ^ { 1 - \beta ( 1 - \alpha ) } _ { 0 _ + } \Big ( f ( t , u ( t ) , ^ \rho D ^ { \alpha , \beta } u ( t ) ) - p ( t ) u ( t ) \Big ) \in C _ { 1 - \gamma , \rho } [ 0 , T ] . \end{array} \end{align*}


\begin{align*} \widetilde { \psi } \left ( x , \mathrm { i } \alpha \right ) = y \left ( \alpha , x \right ) , \ \ \ \alpha \in \operatorname * { S u p p } \sigma . \end{align*}


\begin{align*} | d _ 1 | , | d _ 2 | \leq \min \left \{ \eta \frac { 1 } { 4 } , \eta ^ 2 \frac { 1 } { 1 6 } \frac { 1 } { 2 s E } \right \} = : C ( s , \eta ) . \end{align*}


\begin{align*} d ( R , S ) & = \dim C o k e r ( p \vert _ { U } ) = \dim W ^ { \perp } - \dim I m ( p \vert _ { U } ) = \dim W ^ { \perp } - \dim U + \dim K e r ( p \vert _ { U } ) = \\ & = \dim W ^ { \perp } - \dim U + \dim U \cap W = \dim W ^ { \perp } - \dim U ^ { \perp } \cap W \\ & = \dim U ^ { \perp } \cap W ^ { \perp } = e ( S ^ { \perp } , R ^ { \perp } ) . \end{align*}


\begin{align*} [ A ] ( i ) = \sum _ { j \in B } A ( ( i , ( j , j ) ) ) \end{align*}


\begin{align*} \beta ( \mu ; P _ m ) = \sup _ { \eta } \beta ( \eta ; P _ m ) \ ; , \end{align*}


\begin{align*} x = x ' = ( \alpha _ 1 ' , . . . , \alpha _ m ' ) \ \ \ \mbox { a n d } \ \ \ A = \bigcap \limits _ { i = 1 } ^ m \Omega _ i ^ { * , 2 } , \end{align*}


\begin{align*} b _ n & = a _ n + b _ { n - 1 } - a _ { n - 1 } + f ( a _ n ) . \end{align*}


\begin{align*} H _ L ( u ) H _ L ( u ) ^ \top = \sum _ { k = 0 } ^ { N - L } u _ { [ k , k + L - 1 ] } u _ { [ k , k + L - 1 ] } ^ \top \succeq K . \end{align*}


\begin{align*} \Lambda _ { i } ( u ) = \varkappa _ { i } ^ { - } ( u ) \varkappa ^ { + } ( u + \frac { 1 } { 2 } h n ) ^ { - 1 } , i = 1 , \cdots , n . \end{align*}


\begin{align*} P = & \frac { u ^ 2 } { 2 F } \varphi , \\ Q ^ 0 = & u ^ 2 \left \{ \frac { 1 } { 2 \phi \Lambda } ( \phi - z \phi _ z ) ( \varphi _ z - 2 \phi _ { x ^ 0 } ) \Omega - ( r ^ 2 - s ^ 2 ) \left ( z \frac { \phi _ s } { \phi } U + \frac { \phi - z \phi _ z } { \phi } V \right ) \right \} , \\ Q ^ i = & - u ^ 2 \left \{ s U + ( r ^ 2 - s ^ 2 ) ( \frac { \phi _ s } { \phi } U - \frac { \phi _ z } { \phi } V ) + \frac { \phi _ z \Omega } { 2 \phi \Lambda } ( \varphi _ z - 2 \phi _ { x ^ 0 } ) \right \} \frac { y ^ i } { u } + { u ^ 2 } U x ^ i . \end{align*}


\begin{align*} \left \{ \begin{aligned} \delta \chi _ { \delta } ( y ) - \mathcal { L } ( \overline { x } , y , D \chi _ { \delta } , D ^ { 2 } \chi _ { \delta } ) + H ( \overline { t } , \overline { x } , y , \overline { p } , \overline { P } , 0 ) & = & 0 , & \ ; D _ { n } , \\ \chi _ { \delta } ( y ) & = & 0 , & \ ; \partial D _ { n } , \end{aligned} \right . \end{align*}


\begin{align*} u _ 1 = P _ N u + Q _ N v _ 1 , \ t \in I _ 0 . \end{align*}


\begin{align*} g ^ { T X } ( \cdot , \cdot ) : = \omega ( \cdot , J \cdot ) , \end{align*}


\begin{align*} 0 \leq p _ D ( t , x , y ) = p _ D ( t , y , x ) \leq p ( t , x , y ) \ , . \end{align*}


\begin{align*} e ^ + = \left ( \begin{array} { l l l l } 0 & 1 \\ 0 & 0 \end{array} \right ) \ , , \ \ \ e ^ - = \left ( \begin{array} { l l l l } 0 & 0 \\ 1 & 0 \end{array} \right ) \ , . \end{align*}


\begin{align*} \widehat { Q } ^ 1 _ 1 F = Q '^ 1 _ 1 \circ F - ( - 1 ) ^ { \abs { F } } F \circ Q \end{align*}


\begin{align*} \Lambda ( s , \xi _ { \ell } ) = \Lambda ( 1 - s , \overline { \xi _ { \ell } } ) . \end{align*}


\begin{align*} \Lambda = \binom { A } { B } = \binom { a _ 1 , a _ 2 , \ldots , a _ { m _ 1 } } { b _ 1 , b _ 2 , \ldots , b _ { m _ 2 } } \end{align*}


\begin{align*} a _ i : = \begin{cases} 0 & \\ 1 & . \end{cases} \end{align*}


\begin{align*} \beta ( \mu ; H ) = \sum _ { H ' \in \mathbf { C } ( H , n ) } \mu ( H ' ) \ ; , \end{align*}


\begin{align*} T ( x ) | I \rangle & = t ( x ) | I \rangle + 2 ( x + 1 ) \sum _ { j \in I } \frac { G _ j ^ I } { ( x - x _ j ) ( x + x _ j + 1 ) } B ( x ) \left | I \setminus \left \{ j \right \} \right \rangle \ , . \end{align*}


\begin{align*} b _ 0 : = \det ( f ^ { \partial } ) , \ b _ 1 : = ( F _ 1 ) ^ { \diamondsuit } , \ldots , b _ { \frac { g ( g + 1 ) } { 2 } } : = ( F _ { \frac { g ( g + 1 ) } { 2 } } ) ^ { \diamondsuit } , \end{align*}


\begin{align*} { \cal G } _ 0 ( \beta _ 1 , \beta _ 2 ) : = \Big \{ g \in { \cal G } _ 0 : \beta _ 1 \leq \| g \| _ 2 \leq \beta _ 1 \Big \} \ , . \end{align*}


\begin{align*} A \otimes B = A \oplus B \ ; . \end{align*}


\begin{align*} \Gamma ^ { E } = \nabla ^ E - \pi ^ * ( \nabla ^ E | _ { Y } ) , \end{align*}


\begin{align*} \lim _ { h \to \infty } \int _ I \int _ \Omega w ( x ) \exp ( c \| D _ x b _ h ( s , x ) \| ) \dd x \dd s = \int _ I \int _ \Omega w ( x ) \exp ( c \| D _ x b ( s , x ) \| ) \dd x \dd s . \end{align*}


\begin{align*} F = \lbrace x _ 0 ^ 7 + x _ 1 ^ 7 + x _ 2 ^ 7 = 0 \rbrace \subset \mathbb P ^ 2 \end{align*}


\begin{align*} ^ \rho I ^ { 1 - \gamma } u ( 0 ) = \displaystyle { \sum _ { i = 1 } ^ { m } \omega _ { i } u ( \xi _ i ) } . \xi _ i \in ( 0 , T ] \end{align*}


\begin{align*} u _ { x } ( a , t ) = u _ { x } ( b , t ) = 0 , \end{align*}


\begin{align*} \frac { d g _ * \hat \nu _ p ^ c } { d \hat \nu _ p ^ c } = c ( p , t ) . \end{align*}


\begin{align*} Y ^ n = f ( X ) : = ( X + r ) ^ m + ( X + 2 r ) ^ m + \dots + ( X + d r ) ^ m . \end{align*}


\begin{align*} V ^ { \omega } _ { 0 , b } ( x ) : = \mathrm { E } _ { x } \bigg [ \int _ 0 ^ { \infty } e ^ { - q t } \mathrm { d } D ^ { ( 0 , b ) } _ t - \phi \int _ 0 ^ { \infty } e ^ { - q t } \mathrm { d } R ^ { ( 0 , b ) } _ t + \lambda \int _ { 0 } ^ { \infty } e ^ { - q t } \omega ( U ^ { ( 0 , b ) } _ { t } ) \mathrm { d } t \bigg ] . \end{align*}


\begin{align*} G _ j ^ I = ( p + x _ j ) { \mathcal { A } } ^ { I } _ j - ( p - x _ j - 1 ) \tilde { \mathcal { D } } _ j ^ { I } \end{align*}


\begin{align*} m = 2 \cdot 2 ^ { v _ 1 } w _ 1 - 1 \end{align*}


\begin{align*} P _ k = J _ n + J _ m . \end{align*}


\begin{align*} \hat T = X ^ * U X = \begin{pmatrix} T & T _ { 1 2 } \cr T _ { 2 1 } & T _ { 2 2 } \cr \end{pmatrix} \in M _ { n + 2 } , \end{align*}


\begin{align*} \textrm { t h e l e a d i n g p r i n c i p a l m i n o r o f o r d e r } \ j \ \textrm { o f } \ g = 0 \ \ \ ( g \in X _ K ^ \circ ) . \end{align*}


\begin{align*} I _ F ( x ) = \frac { K } { q + K } \left ( x + \frac { \mu } { q } \right ) + \frac { R q } { q + K } . \end{align*}


\begin{align*} \mathcal { G } ^ { 0 } ( i , J , \emptyset ) = ( - p - x _ 1 + i ) { \mathcal { A } } _ { 1 } ^ { J } + ( p - x _ 1 - 1 - i ) \tilde { \mathcal { D } } _ { 1 } ^ { J } \ , , \end{align*}


\begin{align*} \pi ( x ) = \frac { \vartheta ( x ) } { \log x } + \int _ { 2 } ^ { x } { \frac { \vartheta ( t ) } { t \log ^ { 2 } t } \ t } , \end{align*}


\begin{align*} \theta _ t = x + \int _ 0 ^ t b ( s , \omega _ s + \theta _ s ) \ , d s . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ { t } r _ i \alpha _ i = \frac { ( K _ X + D ) \cdot \gamma } { P ^ 2 \cdot H ^ { n - 2 } } = \frac { ( P + N ) \cdot P \cdot H ^ { n - 2 } } { P ^ 2 \cdot H ^ { n - 2 } } = 1 . \end{align*}


\begin{align*} F _ { x ^ C y ^ 0 } y ^ C - F _ { x ^ 0 } & = u \left ( \varphi _ z - 2 \phi _ { x ^ 0 } \right ) , \\ F _ { x ^ C y ^ j } y ^ C - F _ { x ^ j } & = u \left ( \left [ - s ( \varphi _ s - \frac { 2 } { r } \phi _ r ) - z ( \varphi _ z - 2 \phi _ { x ^ 0 } ) \right ] u ^ j + \left [ \varphi _ s - \frac { 2 } { r } \phi _ r \right ] x ^ j \right ) . \end{align*}


\begin{align*} s = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \ , , \end{align*}


\begin{align*} \| A ( \upsilon + \tau ) \| = \| A ( \upsilon ) A ^ { - 1 } ( \upsilon ) A ( \upsilon + \tau ) \| \leq \| A ( \upsilon ) \| \cdot \| [ \phi _ { \upsilon } ( \tau ) ] ^ T \| \smash { \overset { \eqref { e q : P h i H i g h e r D e r i v a t i v e E s t i m a t e } } { \leq } } \| A ( \upsilon ) \| \cdot v _ 0 ( \tau ) , \end{align*}


\begin{align*} q _ { \sigma } \left ( x , t \right ) & = q \left ( x , t \right ) \\ & + 2 \left [ \int \psi _ { \sigma } \left ( x , t ; \mathrm { i } s \right ) \psi \left ( x , t ; \mathrm { i } s \right ) \mathrm { d } \sigma _ { t } \left ( s \right ) \right ] ^ { 2 } + 4 \int \psi _ { \sigma } \left ( x , t ; \mathrm { i } s \right ) \psi ^ { \prime } \left ( x , t ; \mathrm { i } s \right ) \mathrm { d } \sigma _ { t } \left ( s \right ) , \end{align*}


\begin{align*} \mathcal { H } ( \phi ) = - \nabla \mathcal { E } ( \phi ) = \partial _ x ^ 2 \phi - g ' ( \phi ) \ , , \end{align*}


\begin{align*} \partial _ i \partial _ j \bar { \eta } ( x ) = \sum _ { k = 1 } ^ n \partial _ i R _ { j k } ( x ) \overline { \underline { D } _ k \eta } ( x ) + \sum _ { k , l = 1 } ^ n R _ { j k } ( x ) R _ { i l } ( x ) \overline { \underline { D } _ l \underline { D } _ k \eta } ( x ) . \end{align*}


\begin{align*} & \int _ D | u ( x ) | ^ 2 \log _ { + } | u ( x ) | d x \\ \leq & \varepsilon \big \| u \big \| _ { V } ^ 2 + \left ( \frac { d } { 4 } \log \frac { 1 } { \varepsilon } \right ) \big \| u \big \| ^ 2 + \big \| u \big \| ^ 2 \log \big \| u \big \| + \frac { 1 } { 2 \mathrm { e } } m ( D ) , \end{align*}


\begin{align*} \lambda ( P ) : = \max \left \{ \lambda _ { \mathcal { E } } ( x _ 1 , \dots , x _ m ) \colon ( x _ 1 , \dots , x _ m ) \in \Delta _ { m - 1 } \right \} . \end{align*}


\begin{align*} | X ( n , a ) | = \frac 1 3 \left [ 1 + 2 \cdot \frac { ( \tau ( n _ q ) | 3 ) } { \tau ( n _ q ) } \right ] \cdot | Y ( n , a ) | , \end{align*}


\begin{align*} \begin{pmatrix} Z & - u _ 0 \mathbb { 1 } \\ - u _ 1 \mathbb { 1 } & 0 \end{pmatrix} ^ { - 1 } = \begin{pmatrix} 0 & - \frac { 1 } { u _ 1 } \mathbb { 1 } \\ - \frac { 1 } { u _ 0 } \mathbb { 1 } & - \frac { 1 } { u _ 1 u _ 0 } Z \end{pmatrix} \ ; . \end{align*}


\begin{align*} u \left ( x , 0 \right ) = q \left ( x \right ) , \end{align*}


\begin{align*} C _ k ( A ) : = A ^ { \hat { \otimes } { k + 1 } } = A \hat { \otimes } A \hat { \otimes } \ldots \hat { \otimes } A , k \in \N \end{align*}


\begin{align*} \abs { G } _ { M \otimes N } = \abs { G } _ M \cap \abs { G } _ N . \end{align*}


\begin{align*} Y ( x , t ) : = \frac { 1 6 c } { \beta ^ 4 } \sum _ { z \in \Z } \sum _ { s = 1 } ^ t p ( x - z , t - s ) K ( z , s ) ^ 4 . \end{align*}


\begin{align*} P ( u + v ) + Q ( u + \tilde \alpha v ) - P ( u - v ) - Q ( u - \tilde \alpha v ) = 0 , \ \ u , v \in Y . \end{align*}


\begin{align*} \alpha ( t , x ) \oplus \alpha ( s , x ) : = \alpha ( s + t , x ) \ , . \end{align*}


\begin{align*} \beta ( \theta ) : = \alpha ( \theta \mu _ 1 + ( 1 - \theta ) \mu _ 2 ) [ 0 , 1 ] . \end{align*}


\begin{align*} \rho _ n ( m ) = \sup _ { \mu } \rho ( \mu ; m ) \rho ( m ) = \sup _ { n \in \mathbb { N } } \rho _ n ( m ) \ ; , \end{align*}


\begin{align*} h '' ( x ) + & ( 2 - m ) \tanh x h ' ( x ) + \frac { m - 1 } { 2 } \sin 2 h ( x ) = 0 . \end{align*}


\begin{align*} M \sum \limits _ { | \alpha | = 0 } ^ { 3 } \left ( \| \sqrt { B _ 0 } \partial _ \tau ^ \alpha u _ 0 \| _ { L ^ 2 } + \| \sqrt { A _ 0 } \partial _ \tau ^ \alpha \tilde { h } _ 0 \| _ { L ^ 2 } \right ) + \sum \limits _ { | \alpha | = 0 } ^ { 2 } \left ( \| \sqrt { B _ 0 } \partial _ \tau ^ \alpha \partial _ y u _ 0 \| _ { L ^ 2 } + \| \sqrt { A _ 0 } \partial _ \tau ^ \alpha \partial _ y \tilde { h } _ 0 \| _ { L ^ 2 } \right ) \le \varepsilon , \end{align*}


\begin{align*} \prod _ { 0 \leq j \leq m - 1 } \left | 1 + e ^ { 2 \pi i \frac { j + \theta _ 1 / 2 } { m } } + \frac { z } { m } \right | = \exp \left \{ A _ m ( x + i y ) + B _ m ( \theta _ 1 , x + i y ) \right \} , \end{align*}


\begin{align*} \lim _ { t \rightarrow \infty } \norm { X ( \omega , t ) - x ^ { \star } } = \lim _ { k \rightarrow \infty } \tau _ { k } ( \omega ) , \quad \forall \omega \in \tilde { \Omega } , \end{align*}


\begin{align*} \mathbb \log \ ; { \mathbb P } \left \{ \sup _ { | z | = R _ k } \log | W _ \infty ( z ) | \ge \frac { a } 4 R _ k \log ^ { b } R _ k \right \} \lesssim - \log ^ { 2 ( b - 1 ) } R _ k . \end{align*}


\begin{align*} \langle a , b \rangle : = \mathrm { t r } ( a b ) , \ ; a , b \in \widehat { Z } _ d , \end{align*}


\begin{align*} \sum _ { \substack { 0 < \gamma , \gamma ' \le T \\ | \gamma - \gamma ' | \le \frac { 2 \pi \lambda } { \log T } } } \ ! \ ! \ ! \ ! \ ! \ ! 1 \ \ = \sum _ { 0 < \gamma _ d \le T } m ( \gamma _ d ) ^ 2 + \sum _ { \substack { 0 < \gamma , \gamma ' \le T \\ 0 < | \gamma - \gamma ' | \le \frac { 2 \pi \lambda } { \log T } } } \ ! \ ! \ ! \ ! \ ! \ ! 1 . \end{align*}


\begin{align*} \aleph _ { \alpha , L } : = \frac { 1 } { 6 \pi } \left \langle \alpha L \arctan \left ( \frac { 1 } { \alpha L ( h _ 1 - e _ 1 ) } \right ) \right \rangle _ { x u _ 1 } , \end{align*}


\begin{align*} D _ x X ( t , s , x ) = \begin{cases} 1 & x < 0 , \\ k ( t , s ) \left ( \frac { x } { e } \right ) ^ { { k ( t , s ) - 1 } } & 0 < x < e , \\ 1 & x > e . \end{cases} \end{align*}


\begin{align*} \partial _ t f = \mathcal { C } ( f ) , \end{align*}


\begin{align*} P ( \frac { 1 } { b } \sum _ { i = 1 } ^ { b } \bar T _ i < \eta ) \le e ^ { \lambda ^ 2 b \eta / 2 } E [ e ^ { - \lambda ^ 2 \bar { T } _ 1 / 2 } ] ^ b \le \Big ( c _ \delta e ^ { \lambda ^ 2 \eta / 2 - \lambda ( 1 - \delta ) } \Big ) ^ b \ , . \end{align*}


\begin{align*} f ( y ) = f ( \kappa y ) , \ \ y \in Y . \end{align*}


\begin{align*} \| u \| _ { 1 , \mathcal { H } _ i } = \| \nabla u \| _ { \mathcal { H } _ i } + \| u \| _ { \mathcal { H } _ i } , \end{align*}


\begin{align*} B _ n ( t , x ) \xi \cdot \eta = \int _ { \R ^ d } \rho _ n ( y ) B ( t , x - y ) \xi \cdot \eta \d y ( \xi , \eta \in \C ^ { d m } ) , \end{align*}


\begin{align*} \alpha ( P , t ) = \left \{ \sum _ { i = 1 } ^ { 4 } z _ i ^ { 2 } = t , \frac { z _ i } { \sqrt { t } } \in \mathbb { R } i \in \{ 1 , \dotsc , 4 \} \right \} . \end{align*}


\begin{align*} \mu ( F , x ) : = \max \left \{ 1 , \| F \| \| F ' ( x ) ^ { - 1 } \Delta _ F ( x ) \| \right \} \end{align*}


\begin{align*} \theta ' ( x ) = & - \frac { m } { 2 } + ( \frac { p } { 2 } - 1 ) \cos 2 \theta ( x ) + \frac { 1 } { 2 } ( m - p ) \big ( | \sin 2 \theta ( x ) | - \cos 2 \theta ( x ) \big ) + \delta _ p \\ = & - \frac { m } { 2 } + ( p - 1 - \frac { m } { 2 } ) \cos 2 \theta ( x ) + \frac { 1 } { 2 } ( m - p ) | \sin 2 \theta ( x ) | + \delta _ p , \end{align*}


\begin{align*} K ( x ; y ) = v ( x - y ) \varphi ( x ) \varphi ( y ) \end{align*}


\begin{align*} K \subset \bigcup ^ N _ { k = 1 } B ( z _ k , \delta _ 0 ) . \end{align*}


\begin{align*} h _ k ^ { ( Q , S ) } ( \mathcal { M } ) - h _ k ^ { \mu _ k ( Q , S ) } ( \mu _ k ( \mathcal { M } ) ) = g _ k ^ { ( Q , S ) } ( \mathcal { M } ) \end{align*}


\begin{align*} \sigma _ a ( b ) = ( \lambda _ a ( 1 ) ) ^ { - } \circ b = \begin{cases} ( 1 - 2 ^ { m - 1 } ) ( 1 - b ) , \quad { \rm i f } \ ; a \ ; { \rm i s \ ; e v e n , } \\ - 1 + ( 2 ^ { m - 1 } - 1 ) b , \quad { \rm i f } \ ; a \ ; { \rm i s \ ; o d d . } \end{cases} \end{align*}


\begin{align*} \lim _ { k \rightarrow + \infty } \mathbb { E } \left [ \| \partial _ x u _ { t _ k } \| ^ 2 _ { L ^ 2 } \right ] = 0 \ , . \end{align*}


\begin{align*} \Pi ( f \otimes \Psi ) : = f \otimes \Pi \Psi = f \otimes ( \Psi ^ { ( 0 ) } , \Psi ^ { ( 1 ) } ) , f \in L ^ 2 ( \mathbb { R } ^ 3 ; d x ) , \Psi \in \Gamma _ s ( \mathfrak { h } ) . \end{align*}


\begin{align*} u ( 0 , t ) = u ( 1 , t ) = 0 , 0 \leq t . \end{align*}


\begin{align*} \overline { \rm r k } \ , Z _ { \iota , \jmath } = \overline { \rm r k } \ , G _ { \iota , \jmath } ' , \end{align*}


\begin{align*} C ( \lambda _ 1 , \lambda _ 2 ) \int _ { 0 } ^ { t } \mathbb { E } \left [ \| w _ r \times \partial _ x w _ r \| ^ 2 _ { L ^ 2 } \right ] \dd r + & \lambda _ 2 \int _ { 0 } ^ { t } \mathbb { E } \left [ \| w _ r \times \partial ^ 2 _ x w _ r \| ^ 2 _ { L ^ 2 } \right ] \dd r = 0 \ , . \end{align*}


\begin{align*} g = \partial _ { \tau , \xi } w e ^ { - K \tau } - ( b \delta + C _ 1 ) ( 1 - \eta ) ^ { \alpha _ 0 } , \end{align*}


\begin{align*} \textbf { M } _ { x } ( v _ { x } , u _ { x } ) \rightarrow \textbf { M } _ { b x } ( \mathcal { A } _ { b } v _ { x } , \mathcal { A } _ { b } u _ { x } ) = \textbf { M } _ { x } ( v _ { x } , u _ { x } ) \end{align*}


\begin{align*} \partial _ x u + \partial _ y v = \frac { h ( h \partial _ x + g \partial _ y ) u + h \partial _ y ^ 2 \tilde { h } } { \gamma ( \frac 3 2 - \frac 1 2 h ^ 2 ) + h ^ 2 } . \end{align*}


\begin{align*} \left \{ \begin{alignedat} { 2 } - { \rm d i v } \ , A ( \nabla u ) & = f ( x , u , \nabla u ) + g ( x , u ) \ ; \ ; & & \mbox { i n } \ ; \ ; \Omega , \\ u & > 0 & & \mbox { i n } \ ; \ ; \Omega , \\ \frac { \partial u } { \partial \nu _ A } + \beta u ^ { p - 1 } & = 0 & & \mbox { o n } \ ; \ ; \partial \Omega , \end{alignedat} \right . \end{align*}


\begin{align*} W : = \{ ( z , w ) \in Y \times \C \colon | w - e ^ { i \theta ( z ) } | ^ 2 < 1 - \eta ( \theta ( z ) ) \} . \end{align*}


\begin{align*} ( f \otimes g ) ( x , y ) = ( f ( x ) , g ( y ) ) \end{align*}


\begin{align*} \delta _ a = ( - \Delta + y ^ 2 ) ( e ^ { i n x } u _ a ( y ) ) = y ^ 2 \left ( n ^ 2 - \frac { \partial ^ 2 } { \partial y ^ 2 } + { 1 } \right ) u _ a ( y ) = y ^ 2 \left ( - u _ a '' + ( n ^ 2 + { 1 } ) u _ a \right ) \end{align*}


\begin{align*} \min _ { \mu \in \mathcal { P } ( \mathbb { T } ^ { d } ) } \mathcal { E } _ { V } ( \mu ) = \lim _ { N \to \infty } \min _ { X _ { N } \in \mathbb { T } ^ { d \times N } } \left ( \frac { 1 } { N ^ { 2 } } \mathcal { H } _ { N } ( X _ { N } ) \right ) , \end{align*}


\begin{align*} \begin{gathered} \mathcal P _ \tau v _ \varepsilon = v _ \varepsilon - \mathcal P _ \sigma v _ \varepsilon \buildrel \ast \over \rightharpoonup 0 ~ ~ ~ L ^ \infty ( 0 , T ; H ^ 2 _ \tau ) , \\ \mathcal P _ \tau v _ \varepsilon = v _ \varepsilon - \mathcal P _ \sigma v _ \varepsilon \rightharpoonup 0 ~ ~ ~ L ^ 2 ( 0 , T ; H ^ 3 _ \tau ) . \end{gathered} \end{align*}


\begin{align*} g _ { H } ( N , \dot \gamma ) = g _ { H } ( N , X ) > 0 . \end{align*}


\begin{align*} \left | \langle v , w \rangle _ { L _ x ^ { p ^ \prime } \times L _ x ^ p } - \langle I _ h v , I _ h w \rangle _ { L _ x ^ { p ^ \prime } \times L _ x ^ p } \right | & = \left | \int _ { \R } v w - I _ h v I _ h w \d x \right | \leq \left | v w - I _ h v I _ h w \right | _ { L _ x ^ 1 } \to 0 \end{align*}


\begin{align*} 0 = \mathfrak { p } _ 0 \subset \mathfrak { p } _ 1 \subset \cdots \subset \mathfrak { p } _ n = \mathfrak { p } \end{align*}


\begin{align*} D _ k & = \{ \omega \in \Omega : \omega _ k \in A \} \end{align*}


\begin{align*} m ^ \mathrm { p a i r } ( x , y ) & = - \sum _ j { u _ j ( x ) \overline { p _ j ( y ) } } = \Big ( k \circ ( \delta - \overline { k } \circ k ) ^ { - 1 } \Big ) ( x , y ) \\ n ^ \mathrm { p a i r } ( x , y ) & = \sum _ k { p _ j ( x ) \overline { p _ j ( y ) } } = \Big ( ( k \circ \overline k ) \circ ( \delta - \overline { k } \circ k ) ^ { - 1 } \Big ) ( x , y ) . \end{align*}


\begin{align*} u \in \mathcal { W } \ : \ \langle \mathcal { A } ( u ) + \lambda \mathcal { B } ( u ) , v \rangle _ { \mathcal { W } } = \langle \mathcal { F } ( u ) , v \rangle _ { \mathcal { W } } + \langle \mathcal { G } ( u ) , v \rangle _ { \mathcal { W } } v \in \mathcal { W } , \end{align*}


\begin{align*} & \inf _ { \xi \in M } \inf _ { v \in C ( \xi ) } \| D _ \xi T _ t ^ { - n } v \| > \tilde c \tilde \nu ^ { - n } \| v \| \\ & \inf _ { \xi \in M } \inf _ { v \not \in C ( \xi ) } \| D _ \xi T _ t ^ n v \| > \tilde c \tilde \lambda ^ { n } \| v \| \end{align*}


\begin{align*} \Lambda ^ R _ \omega = \{ f \in \Lambda _ \omega : \mbox { \rm s u p p } f \subset B ( 0 , R ) \} , R > 0 . \end{align*}


\begin{align*} & u ( x , t ) = A + o ( 1 ) , & x \rightarrow + \infty , \\ & u ( x , t ) = o ( 1 ) , & x \rightarrow - \infty . \end{align*}


\begin{align*} F \left ( x \right ) = \int e ^ { - 2 s x } m \left ( x , \mathrm { i } s \right ) ^ { 2 } \mathrm { d } \sigma \left ( s \right ) = F _ { 0 } \left ( x \right ) \cdot \left ( 1 + o \left ( 1 \right ) \right ) , \ \ \ x \rightarrow \infty , \end{align*}


\begin{align*} F ( p , q ) : = \frac { 1 } { 2 } \sum _ { k \in \mathbb { Z } } \int _ { - 1 / 2 } ^ { 1 / 2 } \left \{ \log \left ( 1 + \left ( \frac { p + k } { q } \right ) ^ 2 \right ) - \log \left ( 1 + \left ( \frac { p + k + \phi } { q } \right ) ^ 2 \right ) \right \} \mathrm { d } \phi . \end{align*}


\begin{align*} S _ f ( d ; \alpha ) \ll \begin{cases} ( \log \log K ) ^ { \O ( 1 ) } \| f \| _ 2 ^ 2 , & \mbox { i f } \ , \alpha = 1 , \\ \exp ( C ( \alpha ) ( \log K ) ^ { 1 - \alpha } ) ( \log \log K ) ^ { - \alpha } ) \| f \| _ 2 ^ 2 , & \mbox { i f } \ , \frac { 1 } { 2 } < \alpha < 1 , \end{cases} \end{align*}


\begin{align*} C _ { n , A , L } [ z ; \psi _ 1 , \psi _ 2 ] & = \int _ { \R ^ d } d u _ 0 \prod _ { l \in I _ A } \left ( \int _ { \R ^ d } d v _ l \right ) \prod _ { j = 1 } ^ n E [ \widehat { B } _ \# ( - [ M _ A ( \cdot ) ] _ j ) ] ( v ) E [ \overline { \widehat { \psi } } _ { 1 } ] ( u _ 0 ) \\ & \times E [ \widehat { \psi } _ { 2 } ] ( u _ 0 ) E [ Q ] ( u _ 0 , v ) , \end{align*}


\begin{align*} \mathcal { K } _ { n , m } ^ { E R } [ A , C ] = \sum _ { \alpha , \alpha ' } Z _ N ^ { \alpha } \cdot Z ' _ N { } ^ { \alpha ' } \cdot \mathcal { K } _ { m , m } [ A ^ { \alpha } , C ^ { \alpha ' } ] . \end{align*}


\begin{align*} t = t ( i ) : = i / n ^ 2 . \end{align*}


\begin{align*} h _ { R , l } ( m , u ) \ \lesssim \ \sqrt { u } \mbox { a n d } h _ { R , l } ( m , u ) \ \gtrsim \ \begin{cases} m & \mbox { i f } u \ge m ^ 2 + m - 1 \\ \sqrt { u } & \mbox { i f } u \le m ^ 2 + m - 1 . \end{cases} \end{align*}


\begin{align*} W \left \{ \psi \left ( \mathrm { i } \alpha \right ) , \psi \left ( \mathrm { i } s \right ) \right \} & = - \left ( s ^ { 2 } - \alpha ^ { 2 } \right ) \int _ { x } ^ { \infty } \psi \left ( z , \mathrm { i } \alpha \right ) \psi \left ( z , \mathrm { i } s \right ) \mathrm { d } z \\ & = - \left ( s ^ { 2 } - \alpha ^ { 2 } \right ) K \left ( \alpha , s \right ) , \end{align*}


\begin{align*} h _ 2 \neq 0 , \partial _ x h _ 2 = 0 \ , . \end{align*}


\begin{align*} X _ { n + 1 } ^ { 0 , \Delta t } = \Phi ( \Delta \beta _ n , X _ n ^ { 0 , \Delta t } ) . \end{align*}


\begin{align*} D = \hat { Z } _ i D _ 0 = \sum _ { j + k \leq l } \hat { Z } _ i g _ { j , k } \cdot \hat { Z } _ i ^ k \hat { \partial } _ i ^ j + \sum _ { j + k \leq l } g _ { j , k } \hat { Z } _ i ^ { k + 1 } \hat { \partial } _ i ^ j . \end{align*}


\begin{align*} L _ { \infty , k + 1 } = \sum _ { \ell = 2 } ^ { k + 1 } Q _ { A , \ell } ^ 1 \circ P ^ \ell _ { k + 1 } - \sum _ { \ell = 1 } ^ { k } P _ \ell ^ 1 \circ Q ^ \ell _ { B , k + 1 } \end{align*}


\begin{align*} ( x + y ) \circ z = ( x \circ z ) - z + ( y \circ z ) . \end{align*}


\begin{align*} Q P = - P Q \ , . \end{align*}


\begin{align*} M = N \prod _ { \ell \in \mathcal { R } } \ell ^ { \alpha ( \ell ) } 0 \leq \alpha ( \ell ) \leq 2 , \end{align*}


\begin{align*} \zeta _ i = \frac { 1 } { g } \{ \nabla _ \Gamma g _ i \cdot \nabla _ \Gamma \eta - k _ d ^ { - 1 } ( \partial ^ \circ g _ i ) \eta + k _ d ^ { - 2 } g _ i V _ \Gamma ^ 2 \eta \} \end{align*}


\begin{align*} \begin{aligned} u ( h e _ n & + \tilde { h } e _ i , \hat { h } ) - u ( \tilde { h } e _ i , \hat { h } ) - \left ( u ( h e _ n , 0 ) - u ( 0 ) \right ) \\ = & h \left ( u _ n ( \xi _ 1 h e _ n + \tilde { h } e _ i , \hat { h } ) - u _ n ( \xi _ 2 h e _ n , 0 ) \right ) \\ = & h \left ( u _ n ( \tilde { h } e _ i , \hat { h } ) - u _ n ( 0 ) + O ( h ^ { \tilde \alpha } ) \right ) . \\ \end{aligned} \end{align*}


\begin{align*} m ( a ) = \{ 0 , \ldots , a - 1 \} \ ; . \end{align*}


\begin{align*} R \to R [ t ] \to U _ { R [ t ] / R } ^ { - 1 } ( R [ t ] ) = R ( t ) . \end{align*}


\begin{align*} \Theta _ { s ' } : = Q _ { s ' } B ^ { A _ 0 } ( \beta ) Q ^ H ( 2 s ' ) . \end{align*}


\begin{align*} \dim ( M ) = m = \dim _ { ( \textbf { d } ) } ( M ) . \end{align*}


\begin{align*} x \to y : = x * ( x \wedge y ) , \end{align*}


\begin{align*} ( \mathcal { T } _ { \mathfrak { s } } z ) ( t ) = & S ( t ) z _ 0 - \tfrac { 1 } { 2 } \int _ 0 ^ t S ( t - s ) ( z ( s ) F _ \Phi ) \d s + i \kappa \int _ 0 ^ t S ( t - s ) ( \theta _ R ( | z | _ { X ^ { \mathfrak { s } } _ s } ) | z ( s ) | ^ 2 z ( s ) ) \d s \\ & - i \int _ 0 ^ t S ( t - s ) ( z ( s ) \d W ( s ) ) , \end{align*}


\begin{align*} \prod _ { i = 1 } ^ 4 f _ i ( x _ 1 , x _ 2 ) = \big [ \omega - g ( x _ 3 ) \big ] \big [ \omega + g ( x _ 3 ) \big ] . \end{align*}


\begin{align*} \partial _ t \rho ^ \varepsilon ( x , t ) = - \varepsilon ^ { - 1 } V _ \Gamma \partial _ r \eta _ 0 ( r ) + \partial ^ \circ \eta _ 0 ( r ) - V _ \Gamma \partial _ r \eta _ 1 ( r ) + O ( \varepsilon ) \end{align*}


\begin{align*} \frac { d } { d s } \Big \lvert _ { s = 0 } g _ H ( T _ * N , N ^ \perp ) & = g _ H ( \nabla _ { N ^ \perp } T _ * N , N ^ \perp ) + g _ H ( T _ * N , \nabla _ { N ^ \perp } N ^ \perp ) \lvert _ { s = 0 } \\ & = g _ H ( T _ * N , N ) ^ 2 \kappa _ 1 \lvert _ { s = 0 } - g _ H ( T N , N ) \kappa _ \eta \lvert _ { s = 0 } \\ & = \kappa _ 1 - \kappa _ \eta < 0 . \end{align*}


\begin{align*} d _ { r , s } ( n , i ) = 0 , \mbox { i f } r \ne 0 , s \ne 0 , n \ne 0 \mbox { a n d } i \ne 0 . \end{align*}


\begin{align*} \tilde { X } ( t ) = x _ 0 - \int _ 0 ^ t \nabla f ( \hat { X } ( s ) ) d s + \int _ 0 ^ t \sigma ( s , \hat { X } ( s ) ) d B ( s ) . \end{align*}


\begin{align*} \varphi _ 1 = x _ { 1 } \land \dots \land x _ { p } \land \lnot \psi _ 1 \land \dots \land \lnot \psi _ q , \end{align*}


\begin{align*} \tau ( x ) = \Lambda _ n ( x ) , x \in X _ { n + 1 } , \end{align*}


\begin{align*} \sum _ { 0 < s \leq t } \langle \epsilon _ { J _ 1 } , \mathbb { X } _ { s ^ - } \rangle \Delta X _ s ^ { S ( J _ 2 ) } & = \int _ 0 ^ t \int _ { \mathbb R } \langle \epsilon _ { J _ 1 } , \mathbb { X } _ { s ^ - } \rangle x ^ { S ( J _ 2 ) } ( \mu - \nu ) ( d s , d x ) \\ & \qquad + \int _ 0 ^ t \langle \epsilon _ { J _ 1 } , \mathbb { X } _ { s ^ - } \rangle d s \int _ { \mathbb R } x ^ { S ( J _ 2 ) } F ( d x ) . \end{align*}


\begin{align*} \int _ \R \frac { 1 } { \lambda - z } f _ 0 ( \lambda ) \tanh ( \frac { \pi \lambda } { 2 } ) \ , d \lambda = \int _ \R \frac { 1 } { \lambda + z } f _ 0 ( \lambda ) \tanh ( \frac { \pi \lambda } { 2 } ) \ , d \lambda \ , . \end{align*}


\begin{align*} \chi ( A , B ) & : = \sum _ { y \in A } \sum _ { z \in B } P ^ y ( \tau _ { A \cup B } = \infty ) G ( y , z ) P ^ z ( \tau _ { B } = \infty ) , \end{align*}


\begin{align*} J _ m = \begin{bmatrix} b ( 0 ) & a ( 0 ) \\ a ( 0 ) & b ( 1 ) & \cdots \\ & \cdots & \cdots & \cdots \\ & & \cdots & b ( m - 2 ) & a ( m - 2 ) \\ & & & a ( m - 2 ) & b ( m - 1 ) \end{bmatrix} . \end{align*}


\begin{align*} S : = \{ Q = 0 \} \cap \{ R = 0 \} \subset \mathbb P ^ 3 \end{align*}


\begin{align*} C _ 1 ( \kappa ) = \frac { A \gamma _ 0 } { 2 i a _ 1 ' ( i \kappa ) \kappa ^ 2 } , C _ 2 ( \kappa ) = \frac { 2 i a _ 1 ' ( i \kappa ) } { \gamma _ 0 } , \kappa _ { \delta } \in ( 0 , \kappa ) , \gamma _ 0 ^ 2 = 1 . \end{align*}


\begin{align*} \partial _ t \pi ( x , t ) \cdot \nabla d ( \pi ( x , t ) , t ) & = - \partial _ t d ( x , t ) | \nabla d ( x , t ) | ^ 2 - \frac { 1 } { 2 } d ( x , t ) \partial _ t \bigl ( | \nabla d ( x , t ) | ^ 2 \bigr ) \\ & = - \partial _ t d ( x , t ) . \end{align*}


\begin{align*} x = \Phi + \frac { z ^ { 2 } } { 4 } \frac { 1 } { \Phi } . \end{align*}


\begin{align*} \sum _ { j , k = 0 } ^ n g _ { j k } ( x ) \frac { d x _ j } { d t } \frac { d x _ k } { d t } = 2 H ( y , \eta _ 0 , \eta ) = \sum _ { j , k = 0 } ^ n g ^ { j k } ( y ) \eta _ j \eta _ k . \end{align*}


\begin{align*} m ^ \epsilon ( t ) = e ^ { - \frac { t } { \epsilon ^ 2 } } m _ 0 ^ \epsilon + \frac { 1 } { \epsilon } \int _ { 0 } ^ { t } e ^ { - \frac { t - s } { \epsilon ^ 2 } } d \beta ( s ) , \end{align*}


\begin{align*} P _ 2 \left ( n \right ) = \sum _ { n = r - 1 } ^ { n - 1 } F ( n ) \end{align*}


\begin{align*} D _ { \lambda , n } ( \varphi f ) = D _ { \sigma , n - 1 } \Big ( \frac { \varphi f ( z ) - ( \varphi f ) ^ * ( \lambda ) } { z - \lambda } \Big ) . \end{align*}


\begin{align*} - \partial _ t u - \Delta D ^ 2 u + H ( x , t , D u ) = 0 \end{align*}


\begin{align*} \hat { U } = e ^ { i t \hat { H } } \ ; . \end{align*}


\begin{align*} \sigma ^ 1 = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} , \ \sigma ^ 2 = \begin{bmatrix} 0 & - i \\ i & 0 \end{bmatrix} , \ \sigma ^ 3 = \begin{bmatrix} 1 & 0 \\ 0 & - 1 \end{bmatrix} . \end{align*}


\begin{align*} \left \lbrace \begin{aligned} d X ^ \epsilon ( t ) & = \frac { \sigma ( X ^ \epsilon ( t ) ) m ^ \epsilon ( t ) } { \epsilon } d t \\ d m ^ \epsilon ( t ) & = - \frac { m ^ \epsilon ( t ) } { \epsilon ^ 2 } d t + \frac { 1 } { \epsilon } d \beta ( t ) , \end{aligned} \right . \end{align*}


\begin{align*} \omega _ \psi ( s ) = ( - 1 ) ^ { l ( T ^ F , s ) } \vartheta _ T ( s ) q ^ { \frac { 1 } { 2 } \dim V ^ s } , \end{align*}


\begin{align*} V = x \ , \partial _ x + 2 \ , y \ , \partial _ y + \partial _ z \end{align*}


\begin{align*} \langle \omega _ \alpha f _ s f _ { r - s } \omega _ \beta \rangle = \vec { W } ^ r ( \alpha , \beta ; s ) . \end{align*}


\begin{align*} \langle \cdot \ , | \ , \cdot \rangle _ { \# } : = \langle \cdot \ , | \ , ( h _ { \alpha } - e _ { \alpha } + H _ f ^ + ) \ , | \ , \cdot \rangle , \langle \cdot \ , | \ , \cdot \rangle _ { * } : = \langle \cdot \ , | \ , H _ f ^ + \ , | \ , \cdot \rangle . \end{align*}


\begin{align*} y ( t ) : = V ( \phi ( t , x , u ) ) , t \in [ 0 , t _ m ( x , u ) ) , \end{align*}


\begin{align*} f _ i ' = \left \{ \begin{matrix} - f _ k + \sum _ { j \in I , j \not = k } [ - \epsilon _ { k j } ] _ + f _ j & k = i \\ f _ i & k \not = i \end{matrix} \right . \end{align*}


\begin{align*} \phi _ x + i k \sigma _ 3 \phi = U _ { - } \phi , \phi _ t + 4 i k ^ 3 \sigma _ 3 \phi = V _ { - } ( k ) \phi , \end{align*}


\begin{align*} ( a ' , b ' ) \leq ( a , b ) a ' < a a ' = a b ' \leq b . \end{align*}


\begin{align*} \lambda u _ i h ( a _ i ) = v _ i ^ { \sqrt { q } + 1 } , \ 1 \leq i \leq n , \end{align*}


\begin{align*} \cos F ( \delta ) = \dfrac { \left \langle A ^ { 1 / 2 } \dfrac { x - x _ 0 } { | x - x _ 0 | } , A ^ { 1 / 2 } \dfrac { e } { | e | } \right \rangle } { \left | A ^ { 1 / 2 } \dfrac { x - x _ 0 } { | x - x _ 0 | } \right | \ , \left | A ^ { 1 / 2 } \dfrac { e } { | e | } \right | } , \end{align*}


\begin{align*} a ( n - 1 ) u _ E ( n - 1 ) + b ( n ) u _ E ( n ) + a ( n ) u _ E ( n + 1 ) = E u _ E ( n ) , \forall \ , 0 \le n \le N - 1 , \end{align*}


\begin{align*} I _ 2 \ge \frac 1 2 \frac d { d t } \sum \limits _ { | \alpha | = 0 } ^ { 3 } \left \| \sqrt { A } \partial _ \tau ^ \alpha \tilde { h } \right \| _ { L ^ 2 } ^ 2 - C D ( t ) ^ { \frac 1 4 } E ( t ) ^ { \frac 5 4 } . \end{align*}


\begin{align*} H _ { \ell , r - \ell } \left ( Q , \frac { r } { 2 } \right ) = \sum _ { \substack { 0 \neq u \in \Z ^ { \ell } \\ 0 \neq v \in \Z ^ { r - \ell } } } \frac { \exp ^ { ( 2 \pi i \cdot v Q u ^ \top ) } } { \sqrt { \det D } } \left ( \frac { v D ^ { - 1 } v ^ \top } { u A u ^ \top } \right ) ^ { \frac { \ell } { 4 } } \\ K _ { \frac { \ell } { 2 } } \left ( 2 \pi \left ( u A u ^ \top v D ^ { - 1 } v ^ { \top } \right ) ^ { 1 / 2 } \right ) , \end{align*}


\begin{align*} \delta _ * ( u ) = 1 _ { \{ 0 \} } ( u ) \end{align*}


\begin{align*} C ^ { \perp _ { \mathrm { M S } } } = \{ g \in R _ F \mid \mathrm { M S } ( g ) \star \mathrm { M S } ( c ) = 0 \ , \hbox { f o r a l l } \ , c \in C \} . \end{align*}


\begin{align*} \int _ { \mathbb { Z } ^ N } ( I _ \alpha \ast u ) ( x ) v ( x ) \ , d \mu : = \underset { x , y \in \mathbb { Z } ^ N , x \neq y } { \sum } \frac { u ( y ) v ( x ) } { | x - y | ^ { N - \alpha } } \leq C _ { r , s , \alpha , N } \| u \| _ r \| v \| _ s , \end{align*}


\begin{align*} \left ( \R \times Y , \omega = d \left ( e ^ s \lambda \right ) \right ) \end{align*}


\begin{align*} \begin{array} { l l } \mbox { t h e r e i s } T > 0 \mbox { a n d } k \in \N \mbox { s u c h t h a t , } \mbox { f o r a l l } w \in C _ h ( \Pi ) ^ n \mbox { a n d } x \in [ 0 , 1 ] , \\ \left [ Q ^ k w \right ] \left ( x , T \right ) \equiv 0 \mbox { w h e r e } Q = Q _ \varphi \mbox { f o r } \varphi ( x ) = w ( x , 0 ) . \end{array} \end{align*}


\begin{align*} w ( E _ { n } / \mathbb { Q } ) = \begin{cases} + 1 , & n \equiv 1 , 2 , 3 \mod 8 , \\ - 1 , & n \equiv 5 , 6 , 7 \mod 8 . \end{cases} \end{align*}


\begin{align*} 2 q m \cdot d ^ 1 _ { 0 , \frac q 2 } \left ( m , i - \frac { 3 q } 2 \right ) & = 0 , \\ 0 & = q m \cdot d ^ 1 _ { 0 , \frac q 2 } ( m , i ) . \end{align*}


\begin{align*} \tan \sigma = \frac { 2 \langle \alpha ^ g _ 3 ( E _ 1 , E _ 1 , E _ 1 ) , \alpha ^ g _ 3 ( E _ 1 , E _ 1 , E _ 2 ) \rangle } { \Vert \alpha ^ g _ 3 ( E _ 1 , E _ 1 , E _ 1 ) \Vert ^ 2 - \Vert \alpha ^ g _ 3 ( E _ 1 , E _ 1 , E _ 2 ) \Vert ^ 2 } , \end{align*}


\begin{align*} \mathbb E \left ( \log | W ( z ) | \right ) = & \mathbb E \left ( \sum _ { 1 } ^ { \infty } h _ k ( z ) \right ) = \sum _ { \lambda _ k < R } \log \frac R { \lambda _ k } = \int _ 1 ^ R \frac { { n ( t ) \rm d } t } { t } \\ = & \int _ 1 ^ R \frac { { ( t ^ 2 - { ( a + o ( 1 ) ) t \log ^ { b } t ^ 2 ) } \rm d } t } { t } = \frac { R ^ 2 } { 2 } - \int _ 1 ^ R { ( a + o ( 1 ) ) } \log ^ { b } t ^ 2 { \rm d } t \\ = & \frac { R ^ 2 } { 2 } - { ( a + o ( 1 ) ) R \log ^ { b } R ^ 2 . } \end{align*}


\begin{align*} u ^ \infty ( \hat x , d ) = \int _ { \partial { B _ R } } \left [ { \frac { \partial G ^ \infty ( \hat { x } , y ) } { \partial \nu ( y ) } } u ^ s ( y , d ) - \frac { \partial u ^ s ( y , d ) } { \partial \nu ( y ) } G ^ { \infty } ( \hat { x } , y ) \right ] d s ( y ) , \hat x \in \mathbb S ^ 1 _ + \cup \mathbb S ^ 1 _ - , \end{align*}


\begin{align*} \exp \bigl ( z D _ { ( r + r ' , d + d ' ) } \bigr ) = \exp \biggl ( \sum _ { l = 1 } ^ { \infty } \frac { z ^ l } { l ! } \ , ( r ' s _ { 1 , 0 , l } + d ' s _ { 1 , 2 , l + 1 } ) \biggr ) \exp \bigl ( z D _ { ( r , d ) } \bigr ) , \end{align*}


\begin{align*} \beta ( t ) = \left \{ \begin{array} { r r } 1 , & \left [ \frac { 1 } { k + 1 } , \frac { 1 } { k } - 2 ^ { - 4 k - 1 } \right ] \\ 2 ^ { k - 1 } , & t \in \left ( \frac { 1 } { k } - 2 ^ { - 4 k - 1 } , \frac { 1 } { k } \right ) \ , , \end{array} \right . \end{align*}


\begin{align*} & \left [ \Phi ( x + v ) , \Phi ( y + w ) \right ] _ { \theta ' } \\ & = d ' \left [ x - h ( x ) + v , y - h ( y ) + w \right ] _ { \theta ' } \\ & = [ x , y ] + \rho ( x ) w + \rho ( y ) v - \rho ( x ) h ( y ) - \rho ( y ) h ( x ) + \theta ' ( x , y ) \\ & = [ x , y ] + \rho ( x ) w + \rho ( y ) v - \rho ( x ) h ( y ) - \rho ( y ) h ( x ) + \theta ( x , y ) + d ^ 1 ( h ) ( x , y ) \\ & = [ x , y ] + \rho ( x ) w + \rho ( y ) v + \theta ( x , y ) - h \left ( [ x , y ] \right ) \\ & = \Phi \left ( [ x + v , y + w ] _ { \theta } \right ) . \end{align*}


\begin{align*} \lim _ { n \to \infty } \frac { \mathcal E _ { \kappa _ n } ( m _ { \kappa _ n } ) - 8 \pi } { \kappa _ n ^ 2 } = - \frac { 1 6 \pi ^ 2 } { T ( a _ 0 ) } . \end{align*}


\begin{align*} \begin{gathered} v _ \varepsilon \buildrel \ast \over \rightharpoonup v _ p ~ ~ ~ ~ L ^ \infty ( 0 , T ; H ^ 2 ( \Omega _ h \times 2 \mathbb T ) ) , \\ v _ \varepsilon \rightharpoonup v _ p ~ ~ ~ ~ L ^ 2 ( 0 , T ; H ^ 3 ( \Omega _ h \times 2 \mathbb T ) ) , \end{gathered} \end{align*}


\begin{align*} \varphi _ y : = u _ { \alpha } \otimes ( \Omega + 2 \alpha ^ { 3 / 2 } \tilde { \Phi } _ { * } ^ 1 ) + \Phi _ { \# } ^ y , \end{align*}


\begin{align*} \begin{cases} r K _ { 1 , d - 1 } + q K ' _ d , & d \\ r K _ { 1 , d - 1 } + q K _ d , & d , \end{cases} \end{align*}


\begin{align*} B : = \{ ( z , w ) \in \partial W \colon \theta ( z ) \in I , \ \partial _ z \theta ( z ) \neq 0 , \ w \neq 0 \} , \end{align*}


\begin{align*} \Tilde { H } : = \begin{cases} H & M \setminus \left ( D ^ 2 _ { \epsilon _ 0 } \times T ^ 2 \times \Sigma \right ) , \\ & \\ \phi ^ * ( H ) - d \Tilde { B } _ 0 & D ^ 2 _ 1 \times T ^ 2 \times \Sigma . \end{cases} \end{align*}


\begin{align*} d ' ( 1 7 3 ) = 1 0 . \end{align*}


\begin{align*} \textsf { p r i n } _ { \mathcal { I } } | _ { \tilde { A } } \ , : \ , \tilde { A } = A _ r \xrightarrow { \pi _ r } \ldots \rightarrow A _ 2 \xrightarrow { \pi _ 2 } A _ 1 \xrightarrow { \pi _ 1 } A _ 0 = A \end{align*}


\begin{align*} H [ 0 ] & : = \big ( - \Delta _ x + V _ \mathrm { t r a p } ( x ) \big ) \delta ( x - y ) + \overline { \phi _ { 0 } ( y ) } \upsilon _ N ( x - y ) \phi _ { 0 } ( x ) \\ & + ( \upsilon _ N * | \phi _ { 0 } | ^ 2 ) ( y ) \delta ( x - y ) - \mu _ 0 , \\ \Theta [ 0 ] & : = \upsilon _ N ( x , y ) \Big \{ \phi _ { 0 } ( x ) \phi _ { 0 } ( y ) \Big \} . \end{align*}


\begin{align*} \Phi = x \land y \leq z \ , \& \ , \lnot x \land y \leq z \Longrightarrow y \leq z \end{align*}


\begin{align*} \frac { 1 } { M } = \frac { q } { N } , \end{align*}


\begin{align*} A _ \kappa : = \{ x \in \Omega \ , : \ , \ u ( x ) > \kappa \} , \kappa \in \R \end{align*}


\begin{align*} \int _ { I } \operatorname { I m } \varphi \left ( \mathrm { i } s - 0 \right ) \mathrm { d } s & = - \sum _ { \kappa _ { n } \subset I } \pi c _ { n } ^ { 2 } \psi \left ( \mathrm { i } \kappa _ { n } \right ) \\ & = - \pi \int _ { I } \psi \left ( \mathrm { i } s \right ) \mathrm { d } \rho \left ( s \right ) , \end{align*}


\begin{align*} \hat { \mathbb { J } } ( ( \phi _ d ) _ { d | N } ) = \frac { 1 } { N } \sum _ { a , b , c \in \mathbb { Z } / N \mathbb { Z } } e ^ { - 2 \pi i a c / N } \sum _ { n \in \mathbb { Q } } \sum _ { r \in \mathbb { Z } / 2 t \mathbb { Z } } c _ { a , b } ( n , r ) q ^ { n - r ^ 2 / 4 t } \mathfrak { e } _ { ( c / N , r / 2 t , b / N ) } , \end{align*}


\begin{align*} ( u , v , \partial _ y h , g ) | _ { y = 0 } = { \bf { 0 } } , \qquad \theta | _ { y = 0 } = \theta ^ * ( t , x ) . \end{align*}


\begin{align*} \frac { \partial } { \partial M _ { x ^ 1 , y ^ 1 } } \ldots \frac { \partial } { \partial M _ { x ^ p , y ^ p } } \det \limits _ { i , j = 1 } ^ N M _ { i , j } = \det \limits _ { i , j = 1 } ^ N M _ { i , j } \det _ { i \in Y , j \in X } M ^ { - 1 } _ { i , j } , \end{align*}


\begin{align*} \overline { \mathbf { P } } | _ { K _ { i } \times { \rm C o n f i g } } = \overline { \mathbf { P } } _ { i } . \end{align*}


\begin{align*} \left \{ \begin{alignedat} { 2 } - \Delta _ p u = f ( x , u , v , \nabla u , \nabla v ) \ ; \ ; & \mbox { i n $ \R ^ N $ , } \ ; \ ; u > 0 \ ; \ ; & & \mbox { i n $ \R ^ N $ , } \\ - \Delta _ q v = g ( x , u , v , \nabla u , \nabla v ) \ ; \ ; & \mbox { i n $ \R ^ N $ , } \ ; \ ; v > 0 \ ; \ ; & & \mbox { i n $ \R ^ N $ . } \end{alignedat} \right . \end{align*}


\begin{align*} \mathrm { r t } _ A ( A _ + ) = \mathrm { r t } _ R ( A ) , \end{align*}


\begin{align*} \begin{cases} x ^ 2 + \widetilde \Delta ( t , w ) + y ( y + z ( b - a ) ) = 0 , \\ y w = ( t - a w ) z . \end{cases} \end{align*}


\begin{align*} \{ \omega - \phi _ { i j } = 0 , \ q _ i = 0 \} \qquad \{ \omega - \phi _ { i j } , \ q _ j = 0 \} \end{align*}


\begin{align*} A _ M : = \{ s \in [ 0 , T ] : \big \| \sigma ( Y ^ h _ s ) \big \| \leq M \} , A _ M ^ c = [ 0 , T ] \setminus { A _ M } . \end{align*}


\begin{align*} - \partial _ t v - \Delta v + h | D v | ^ \gamma = 0 , \end{align*}


\begin{align*} C _ { i , j } ^ k = \Gamma _ c ( G ^ { ( i ) } , s ^ * F _ j ^ k ) , \end{align*}


\begin{align*} E [ Y ^ p ] \le p ! \int _ { 0 \le s _ 1 \le \ldots \le s _ p \le 1 } \int _ { 0 \le t _ 1 \le \ldots \le t _ p \le 1 } E \Big [ \sum _ { \sigma } \prod _ { i = 1 } ^ p | \beta _ { s _ i } - \tilde { \beta } _ { t _ { \sigma ( i ) } } | ^ { - 2 } \Big ] d s _ 1 \cdots d s _ p d t _ 1 \cdots d t _ p \ , . \end{align*}


\begin{align*} \langle y _ { \ell , n } , y _ { \ell ' , n ' } \rangle _ { L ^ 2 _ \omega ( \mathbb S ^ 2 ) } = \int _ { \mathbb S ^ 2 } { y _ { \ell , n } ( \omega ) } \overline { y _ { \ell ' , n ' } ( \omega ) } \ , d \omega . \end{align*}


\begin{align*} A ( x , y ; \xi , \zeta ) = \int _ 0 ^ 1 \int _ 0 ^ 1 D ^ 2 h ( y - \zeta + s ( \zeta - \xi ) + t ( x - y ) ) d t \ , d s . \end{align*}


\begin{align*} \mathrm { m u l t } _ d ( 0 , 1 ) & = \frac { 2 } { d } \sum _ { t | d } \mu ( d / t ) = \begin{cases} 2 : & d = 1 ; \\ 0 : & d > 1 ; \end{cases} \end{align*}


\begin{align*} \int _ { Q _ T } \psi \partial _ t \rho = \lim _ { k \to \infty } \int _ { Q _ T } \psi \partial _ t \rho _ { \gamma _ k } = \lim _ { k \to \infty } \int _ { Q _ T } \psi \nabla \rho _ { \gamma _ k } \cdot \nabla p _ { \gamma _ k } \geq 0 . \end{align*}


\begin{align*} u _ N ( x , t ) = \sum _ { i = - 1 } ^ { N + 1 } \delta _ i B _ i ( x ) \end{align*}


\begin{align*} \rho _ 1 ( A ) = \mathrm { T r } ( A ^ 2 ) \rho _ 2 ( A ) = \mathrm { T r } ( A ^ 3 ) . \end{align*}


\begin{align*} \| u \| _ { \mathcal { H } _ i } = \inf \left \{ \tau > 0 \ , : \ , \rho _ { \mathcal { H } _ i } \left ( \frac { u } { \tau } \right ) \leq 1 \right \} , \end{align*}


\begin{align*} \mathsf { D } _ { d } Z _ { t } ( d ) = ( \log T ) ( Z _ { t } ^ { \ast } ( d ) - Z _ { t } ( d ) ) . \end{align*}


\begin{align*} \begin{cases} ( - \Delta ) ^ { \frac { n } { 2 } } h ( x , y ) = 0 , & y \in \Omega , \\ ( - \Delta ) ^ i h ( x , y ) = ( - \Delta ) ^ i \left ( C _ n \ln \left ( \frac { 1 } { | x - y | } \right ) \right ) , \ , \ , i = 0 , 1 , \cdots , \frac { n - 1 } { 2 } , & y \in \mathbb { R } ^ n \backslash \Omega . \end{cases} \end{align*}


\begin{align*} \eta _ a ( u _ w ) = - _ { s = w } a ^ { 1 - s } \cdot \frac { E _ s ( z ) } { \lambda _ s - \lambda _ w } = - a ^ { 1 - w } \cdot \frac { E _ w ( z ) } { w - 1 + w } . \end{align*}


\begin{align*} L = \sum _ { i , j = 1 } ^ n a _ { i j } \frac { \partial ^ 2 } { \partial x _ i \partial x _ j } + \sum _ { i = 1 } ^ n b _ i \frac { \partial } { \partial x _ i } + c . \end{align*}


\begin{align*} x * y = \max \{ u \colon ( u ] \cap ( x ] \cap [ y ) = \{ y \} \} . \end{align*}


\begin{align*} F ( r ) = 2 \pi i ^ k \int _ 0 ^ \infty f ( \rho ) J _ { - k } ( 2 \pi r \rho ) \rho \ , d \rho = 2 \pi ( - 1 ) ^ k i ^ k \int _ 0 ^ \infty f ( \rho ) J _ { k } ( 2 \pi r \rho ) \rho \ , d \rho \ , . \end{align*}


\begin{align*} d _ { 0 , s } ( m , i ) = 0 , \mbox { i f } m \ne 0 . \end{align*}


\begin{align*} \Lambda _ t : = \frac { L _ { z + t e _ j } - L _ z } { t } - L _ { z , j } , \ \ \ t \in \mathbb { C } \setminus \{ 0 \} . \end{align*}


\begin{align*} S _ \Lambda ( C ) = 0 . \end{align*}


\begin{align*} Q ^ { \widetilde { \pi } } = \exp ( [ g , \ , \cdot \ , ] ) \circ Q ^ \pi \circ \exp ( [ - g , \ , \cdot \ , ] ) , \end{align*}


\begin{gather*} \sum _ { \mu \in \N ^ d } e ^ { - t | \mu | } \Phi _ { \mu } ( x ) ^ 2 \lesssim t ^ { - \frac { d } { 2 } } , \ ; \forall \ ; x \in \R ^ d , \\ \int _ { \R ^ d } \Big ( \sum _ { \mu \in \N ^ d } e ^ { - t ( 2 | \mu | + d ) } \Phi _ \mu ( x ) ^ 2 \Big ) \dd x = C ( \sinh t ) ^ { - d } . \end{gather*}


\begin{align*} \alpha = \sigma _ { x _ 1 } ^ { \varepsilon _ 1 } \sigma _ { x _ 2 } ^ { \varepsilon _ 2 } \ldots \sigma _ { x _ { n - 1 } } ^ { \varepsilon _ { n - 1 } } \sigma _ { x _ n } ^ { \varepsilon _ n } \end{align*}


\begin{align*} d _ 1 ( \mu , \nu ) = \sup _ { \lvert \lvert \psi \rvert \rvert _ { L i p } \leq 1 } \left \{ \int \psi d ( \mu - \nu ) \right \} , \end{align*}


\begin{align*} s _ t ( x , y , z ) = ( t x , t y , t ^ 2 z ) . \end{align*}


\begin{align*} \Vert \nabla u _ { i } \Vert _ { p _ { i } ( x ) } \leq \tilde { L } , \ , \ , i = 1 , 2 . \end{align*}


\begin{align*} \langle R ^ \perp ( e _ 1 , e _ 2 ) e _ 3 , e _ 6 \rangle = 0 \langle R ^ \perp ( e _ 1 , e _ 2 ) e _ 4 , e _ 5 \rangle = 0 \end{align*}


\begin{align*} \| T \| ^ 2 & = \sup _ { \substack { f \in \mathbb { E } ^ { p } _ { q } \\ \| f \| _ { \mathbb { E } ^ { p } _ { q } } \leq 1 } } \left | \int _ { - \infty } ^ \infty f ( x ) W ( x ) \mathrm { d } x - \sum _ { x \in X _ n } \omega ( x ) f ( x ) \right | ^ 2 \\ & = \sum _ { x , y \in X _ n } \omega ( x ) \omega ( y ) \sum _ { k = 2 n } ^ \infty e ^ { q k ^ p } \ , h _ { k } ( x ) h _ { k } ( y ) \ , . \end{align*}


\begin{align*} \vec { y } = \sum \limits _ { k = 1 } ^ K \vec { H } _ 1 \mathbf { R } _ { \textrm { R I S } } ^ { \frac { 1 } { 2 } } \vec { \Phi } \vec { h } _ { 2 , k } x _ k + \vec { n } \end{align*}


\begin{align*} [ A ] ( i ) = \sum _ { j } A ( ( i , ( j , j ) ) ) \ ; . \end{align*}


\begin{align*} g ^ { ( 1 ) } ( t ) & = Q ^ 1 \left ( \exp ( \pi ( t ) ) \vee Q ^ 1 ( \lambda ( t ) \vee \exp ( \pi ( t ) ) ) \right ) \\ & = Q ^ 1 \left ( Q \left ( \lambda ( t ) \vee \exp ( \pi ( t ) ) \right ) + \lambda ( t ) \vee \exp ( \pi ( t ) ) \vee Q ^ 1 ( \exp ( \pi ( t ) ) ) \right ) \\ & = Q ^ 1 \left ( \lambda ( t ) \vee \exp ( \pi ( t ) ) \vee g ^ { ( 0 ) } ( t ) \right ) . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ s n _ i L _ { P _ i } \simeq 0 . \end{align*}


\begin{align*} \gamma ' _ x = f \gamma _ { f ^ { - 1 } ( x ) } f ^ { - 1 } . \end{align*}


\begin{align*} \mathrm { d i v } _ \Gamma u = \sum _ { i = 1 } ^ n \underline { D } _ i u _ i \quad \Gamma , \end{align*}


\begin{align*} c : = \frac { 1 } { 2 4 } \partial _ 1 ^ 4 \psi ( 0 , 0 ) + \frac { \beta ^ 3 } { 1 2 } . \end{align*}


\begin{align*} \alpha _ 0 ( t , s ) = \exp \left ( - 1 6 \ , e ^ 2 \int _ s ^ t \| \div _ { \gamma _ n } b ( v , \cdot ) \| _ { { \rm E x p } _ { \gamma _ n } \left ( \frac { L } { \log L } \right ) } \dd v \right ) , \end{align*}


\begin{align*} P ( G _ n ^ c \ , | \ , \hat { A } _ n ) = P ( | I _ n | - | \Lambda _ n | \ge \delta t _ n \ , | \ , \hat { A } _ n ) \le \frac { 1 } { \delta t _ n } \sum _ { i \in I _ n } P \ , \big ( \ , \hat { H } _ n ( i ) \ , \big | \ , \hat { A } _ n \big ) \le \frac { 2 \zeta _ { t _ n } } { \delta } \to 0 \ , . \end{align*}


\begin{align*} H _ 0 ( C _ \bullet ) & \cong D ^ s ( \Sigma _ { j , i } ) _ { , A } , & H _ k ( C _ \bullet ) & = 0 ( k > 0 ) . \end{align*}


\begin{align*} & x \mu ^ 3 + \lambda ^ 3 = x ^ 2 . \end{align*}


\begin{align*} \begin{cases} ( - \Delta ) ^ { \frac { n } { 2 } } u ( x ) = u ^ p ( x ) , & x \in \Omega , \\ u ( x ) = - \Delta u ( x ) = \cdots = ( - \Delta ) ^ { \frac { n - 1 } { 2 } } u ( x ) = 0 , & x \in \mathbb { R } ^ n \backslash \Omega , \ , \ , \mbox { i f } \ , \ , n \ , \ , \mbox { i s o d d } ; \\ u ( x ) = - \Delta u ( x ) = \cdots = ( - \Delta ) ^ { \frac { n } { 2 } - 1 } u ( x ) = 0 , & x \in \partial \Omega , \ , \ , \mbox { i f } \ , \ , n \ , \ , \mbox { i s e v e n } . \\ \end{cases} \end{align*}


\begin{align*} K _ { N , \beta } = \frac { Z _ { N , \beta } } { \exp \left ( - N ^ { 2 } \beta \mathcal { E } _ { V } ^ { \theta } ( \mu _ { \theta } ) \right ) } , \end{align*}


\begin{align*} ( \partial _ \nu ^ A \psi ) ^ 2 - ( A \nabla _ \Gamma \psi \cdot \nu ) ^ 2 = \left | A ^ { \frac { 1 } { 2 } } \nu \right | ^ 2 \left ( \left | A ^ { \frac { 1 } { 2 } } \nabla \psi \right | ^ 2 - \left | A ^ { \frac { 1 } { 2 } } \nabla _ \Gamma \psi \right | ^ 2 \right ) . \end{align*}


\begin{align*} \boldsymbol { \varphi } ^ k = ( \boldsymbol { \lnot \psi } ) ^ k \coloneqq \ ? \ ? \sim _ n \ ! \ ! ( \chi _ 1 , \dots , \chi _ n ) , \end{align*}


\begin{align*} \left \| \left ( p ( t ) * u _ 0 \right ) ( - x ) - \left ( p ( t ) * u _ 0 \right ) ( - z ) \right \| _ { L ^ k ( u _ 0 ) } & \le \mathbb { E } \left ( \left \| u _ 0 ( x + X ( t ) ) - u _ 0 ( z + X ( t ) ) \right \| _ { L ^ k ( u _ 0 ) } \right ) \\ & \le C ^ { 1 / k } \| x - z \| ^ \eta , \end{align*}


\begin{align*} x * y = \max \{ u \colon u \wedge _ y x = y \} \end{align*}


\begin{align*} f ( \overline { w } ) = R _ { \Gamma } \circ f ( w ) . \end{align*}


\begin{align*} \langle b _ x ^ \ast b _ y \rangle : = n ^ \mathrm { p a i r } ( x , y ) , \langle b _ x b _ y \rangle : = m ^ \mathrm { p a i r } ( x , y ) , \langle b _ x ^ \ast b _ x \rangle : = \rho ^ \mathrm { p a i r } ( x ) . \end{align*}


\begin{align*} \forall i = 1 , \ldots , d , \ ; \ ; \alpha _ i < 1 , \ ; \beta _ i < 1 , \ ; \ ; \rho ( \alpha _ i , \beta _ i ) : = \frac { | \alpha _ i \beta _ i | } { ( 1 - \alpha _ i ) ( 1 - \beta _ i ) } < 1 . \end{align*}


\begin{align*} \Omega _ { x ^ 0 } - \phi _ { s z } & = [ \varepsilon _ 1 ] _ { x ^ 0 } + [ \varepsilon _ 2 ] _ { x ^ 0 } - z [ \varepsilon _ 2 ] _ { x ^ 0 z } - [ \varepsilon _ 4 ] _ { s z } = 0 , \\ \Omega _ r - r \phi _ { s s } & = [ \varepsilon _ 1 ] _ r + [ \varepsilon _ 3 ] _ r - s [ \varepsilon _ 3 ] _ { r s } - r [ \varepsilon _ 3 + \varepsilon _ 4 ] _ { s s } = 0 . \end{align*}


\begin{align*} \bar { \eta } ( x , t , r ) = \eta ( \pi ( x , t ) , t , r ) , ( x , t ) \in \overline { N _ T } , \ , r \in [ \bar { g } _ 0 ( x , t ) , \bar { g } _ 1 ( x , t ) ] \end{align*}


\begin{align*} \mathcal { E } ^ { \neq } ( \mu ) & = \int _ { M \times M \setminus \Delta } g ( x - y ) d \mu _ { x } d \mu _ { y } \\ \mathcal { G } ( \mu , \nu ) & = \int _ { M \times M } g ( x - y ) d \mu _ { x } d \nu _ { y } \\ \mathcal { G } ^ { \neq } ( \mu , \nu ) & = \int _ { M \times M \setminus \Delta } g ( x - y ) d \mu _ { x } d \nu _ { y } , \end{align*}


\begin{align*} M u _ { [ k - n , k ] } = \begin{bmatrix} y _ { k - n } & \dots & y _ { k - 1 } & y _ k \end{bmatrix} d . \end{align*}


\begin{align*} D _ x X ( t , s , x ) = \exp \left ( - e \left ( \frac { 1 } { e } \log \frac { 1 } { x } \right ) ^ { k ( t , s ) } \right ) k ( t , s ) \exp \left ( - k ( t , s ) + 1 \right ) \left ( \log \frac { 1 } { x } \right ) ^ { k ( t , s ) - 1 } \frac { 1 } { x } . \end{align*}


\begin{align*} u ( t ) = U ( t , s ) u ( s ) , ( t , s ) \in \Omega _ { J } , \end{align*}


\begin{align*} \Big ( \sum _ { j , k = 0 } ^ n g _ 1 ^ { j k } ( y ) \eta _ j \eta _ k \Big ) ^ { \frac { 1 } { 2 } } T = \Big ( \sum _ { j , k = 0 } ^ n g _ 2 ^ { j k } ( y ) \eta _ j \eta _ k \Big ) ^ { \frac { 1 } { 2 } } T . \end{align*}


\begin{align*} \langle M _ { 1 , 2 } \psi , M _ { 2 , 1 } \psi \rangle _ { L ^ 2 ( \Omega _ T ) } & = - \frac { 1 } { 2 } s ^ 2 \lambda ^ 2 \int _ { \Omega _ T } \sigma \xi ^ 2 \partial _ t ( \psi ^ 2 ) \ , \d x \ , \d t = s ^ 2 \lambda ^ 2 \int _ { \Omega _ T } \sigma \partial _ t \xi \xi \psi ^ 2 \ , \d x \ , \d t \end{align*}


\begin{align*} a _ { i j } = h _ { x _ i x _ j } ( x - T x ) - \frac 1 2 \ ( \dfrac { \partial h _ { x _ i } ( x - T x ) } { \partial x _ j } + \dfrac { \partial h _ { x _ j } ( x - T x ) } { \partial x _ i } \ ) , \end{align*}


\begin{align*} \partial _ \eta g ( z _ { m i n } ) = - C _ 0 ( \sqrt { - \ln \mu } - \frac { 1 } { 2 \sqrt { - \ln \mu } } ) < 0 \end{align*}


\begin{align*} \zeta ^ \epsilon ( t ) = \frac { 1 } { \epsilon } \int _ { 0 } ^ { t } m ^ \epsilon ( s ) d s = \beta ( t ) + \epsilon ( m _ 0 ^ \epsilon - m ^ \epsilon ( t ) ) . \end{align*}


\begin{align*} N = \bigoplus _ { i = 1 } ^ t N _ i , \end{align*}


\begin{align*} \widetilde { \psi } \left ( k \right ) & = \psi \left ( k \right ) + \int \frac { W \left \{ \psi \left ( k \right ) , \psi \left ( \mathrm { i } s \right ) \right \} } { k ^ { 2 } + s ^ { 2 } } \mathrm { d } \mu \left ( s \right ) \\ & = \psi \left ( k \right ) - \int K \left ( \alpha , s \right ) \mathrm { d } \mu \left ( s \right ) \end{align*}


\begin{align*} \zeta _ { \theta } = - \frac { 1 } { \theta } \log ( \mu _ { \theta } ) . \end{align*}


\begin{align*} L \left [ P _ { n - 1 } \partial _ { x } P _ { n } \right ] = n h _ { n - 1 } \left ( z \right ) . \end{align*}


\begin{align*} \begin{gathered} ( A \otimes B ) _ M = A _ M \sqcup B _ M \\ ( A \otimes B ) _ H = A _ H \hat \sqcup B _ H \ ; . \end{gathered} \end{align*}


\begin{align*} T _ \sigma T _ s = \begin{cases} T _ { \sigma s } , & l ( \sigma s ) = l ( \sigma ) + 1 \\ ( u - u ^ { - 1 } ) T _ \sigma + T _ { \sigma s } , & l ( \sigma s ) = l ( \sigma ) - 1 , \end{cases} \end{align*}


\begin{align*} \langle \sqrt { \omega } \mathcal C ( f ) , \varphi \rangle = \sum _ { k = 1 } ^ 3 \langle \sqrt { \omega } \mathcal C ( f ) , \varphi _ k ^ \epsilon \rangle . \end{align*}


\begin{align*} L ^ { \mathcal { H } } ( \Omega ) = \left \{ u \in M ( \Omega ) \ , : \ , \rho _ { \mathcal { H } } ( u ) < + \infty \right \} , \end{align*}


\begin{align*} 2 \alpha ^ { 5 / 2 } \mathrm { R e } \langle u _ { \alpha } \tilde { \Phi } _ * ^ 1 \ , | \ , ( \| \lambda _ y \| ^ 2 + A ^ + _ y A ^ - _ y ) \Phi _ { \# } ^ y \rangle & = 0 , \\ 8 \alpha ^ 2 \mathrm { R e } \langle u _ { \alpha } \tilde { \Phi } _ * ^ 1 \ , | \ , P u _ { \alpha } A ^ + _ y \Omega \rangle & = 0 \end{align*}


\begin{align*} f ( r ) = \Bigl ( \frac { 1 + r ^ 2 } { 1 - r ^ 2 } \Bigr ) ^ { d - 1 } . \end{align*}


\begin{align*} \varphi \left ( x , k \right ) = \overline { \psi ( x , k ) } + R ( k ) \psi ( x , k ) , \operatorname { I m } k = 0 , \end{align*}


\begin{align*} \textrm { ( A ) } \quad \delta v _ { s , t } = \mathcal { D } ( v ) _ { s , t } + \int _ { s } ^ { t } v _ r \times h _ 1 \circ \dd B _ r \ , , \quad \textrm { ( B ) } \quad \delta \bar { v } _ { s , t } = \mathcal { D } ( \bar { v } ) _ { s , t } + \int _ { s } ^ { t } h _ 2 \bar { v } _ r \times \circ \dd \bar { B } _ r \ , , \end{align*}


\begin{align*} u _ { \varphi , f } ( t ) = U ( t , s ) \varphi + j ^ { - 1 } \int _ s ^ t U ^ { \odot \star } ( t , \tau ) f ( \tau ) d \tau , \forall t \in I , \end{align*}


\begin{align*} \mathcal { F } ^ { \alpha } _ { S } ( F ) = F ^ { \alpha } , ~ ~ \alpha \in ( - 1 , 1 ) . \end{align*}


\begin{align*} \begin{cases} \partial _ t \rho + ( u \partial _ x + v \partial _ y ) \rho + \rho ( \partial _ x u + \partial _ y v ) = 0 , \\ \rho \left ( \partial _ t u + ( u \partial _ x + v \partial _ y ) u \right ) + \partial _ x ( p + \frac { 1 } { 2 } h ^ 2 ) - ( h \partial _ x + g \partial _ y ) h - \mu \partial _ y ^ 2 u = 0 , \\ \partial _ y ( p + \frac { 1 } { 2 } h ^ 2 ) = 0 , \\ \partial _ t h + \partial _ y ( v h - u g ) - \kappa \partial _ y ^ 2 h = 0 , \\ \partial _ x h + \partial _ y g = 0 . \end{cases} \end{align*}


\begin{align*} \mathbb { E } \big [ \widetilde { v } _ { \textrm { N , T } , n } ( f ) ~ \widetilde { v } _ { \textrm { N , R } , m } ^ * ( f ) \big ] = 0 \quad \forall \ , n , m . \end{align*}


\begin{align*} \beta _ { i , i + j } ( I ) \ = \ \sum _ { u \in G ( I ) _ j } \binom { \max ( u ) - t ( j - 1 ) - 1 } { i } . \end{align*}


\begin{align*} & \int _ { \{ p _ V > n \} } p _ V \ , d \mu = n \mu ( x \colon p _ V ( x ) > n ) + \int _ { n } ^ { \infty } \mu ( x \colon p _ V ( x ) > t ) \ , d t \\ & \le n ( 2 ^ { - n } + \mu ( x \colon q _ n ( x ) > n \alpha _ n ) ) + \sum _ { k = n } ^ \infty \int _ { k } ^ { k + 1 } \mu ( x \colon p _ V ( x ) > t ) \ , d t \\ & \le n 2 ^ { - n } + \alpha _ n ^ { - 1 } \int _ { \{ x \colon q _ n ( x ) > n \alpha _ n \} } q _ n ( x ) \ , \mu ( d x ) + \sum _ { k = n } ^ \infty ( 2 ^ { - k } + \mu ( x \colon q _ k ( x ) > k \alpha _ k ) ) . \end{align*}


\begin{align*} ( 2 \partial _ z + w F ' ( z ) \partial _ w ) \tilde { r } = w F ' ( z ) e ^ { - F ( z ) } \neq 0 . \end{align*}


\begin{align*} P _ n g : = \sum _ { i = 1 } ^ { n } \ \langle g , e _ i \rangle e _ i . \end{align*}


\begin{align*} a = \frac 1 2 \ , \ ( D ^ 2 h ( x - T x ) \dfrac { \partial T } { \partial x } ( x ) + \ ( D ^ 2 h ( x - T x ) \dfrac { \partial T } { \partial x } ( x ) \ ) ^ t \ ) . \end{align*}


\begin{align*} [ x ^ n ] G ( x ) \prod _ { 1 \le i \le \ell } \sum \nolimits _ { j \ge 1 } j ^ { p _ i } C ( x ^ j ) \sim \frac { G ( q _ n ) \prod _ { 1 \le i \le \ell } \sum \nolimits _ { j \ge 1 } j ^ { p _ i } C ( q _ n ^ j ) } { \sqrt { 2 \pi \sum \nolimits _ { j \ge 1 } j q _ n ^ { 2 j } C '' ( q _ n ^ j ) } } \cdot q _ n ^ { - n } . \end{align*}


\begin{align*} \alpha _ { n , 2 } \left ( x \right ) = n \left ( n - 1 \right ) \frac { 8 n \left ( n - 1 \right ) x ^ { 2 } + \left ( 2 n + 1 \right ) \left ( 2 n - 1 \right ) \left ( n - 2 \right ) \left ( n - 3 \right ) } { 8 \left ( 2 n + 1 \right ) \left ( 2 n - 1 \right ) ^ { 2 } \left ( 2 n - 3 \right ) } x ^ { n - 4 } . \end{align*}


\begin{align*} \widetilde \Lambda ^ + ( [ \widetilde \omega , s ] ) = \widetilde B ^ E _ s \widetilde \Lambda ^ + ( \widetilde \omega ) , \widetilde \omega \in \widetilde \Omega , \ s \in [ 0 , 1 ] . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \frac { f ( z _ n ) } { g ( z _ n ) } \ = \ \infty . \end{align*}


\begin{align*} \mu _ { 0 } \left ( z \right ) = 2 { \displaystyle \int \limits _ { 0 } ^ { z } } e ^ { - x ^ { 2 } } d x = \sqrt { \pi } \operatorname { e r f } \left ( z \right ) , \end{align*}


\begin{align*} G _ 2 = \left ( \begin{array} { c c c c c c } 1 & 0 & 1 & 0 & 1 & 1 \\ 0 & 1 & 0 & 1 & 1 & 1 \\ \end{array} \right ) . \end{align*}


\begin{align*} X a \stackrel { \scriptscriptstyle \eqref { e q : c o m p L R a l g } } { = } a X + \omega ( X ) ( a ) . \end{align*}


\begin{align*} \alpha ( G _ A ) = \min _ { \emptyset \neq I \subseteq \{ 1 , \ldots . n \} } \frac { \dim G _ A - \dim ( G _ A ) _ { V _ I } } { \dim V _ I } , \end{align*}


\begin{align*} H = H _ 0 + P \ , \end{align*}


\begin{align*} \partial _ t u = \partial _ x ^ 2 u + u | \partial _ x u | ^ 2 \ , , \end{align*}


\begin{align*} p ( P ) = p ( r ' , \theta ' ) + \int _ { 0 } ^ { r ' } \frac { 1 } { \rho } [ ( v + b ) u _ { \theta ' } - u v _ { \theta ' } - \omega _ { \theta ' } ] d \rho . \end{align*}


\begin{align*} \partial _ t { \bf { v } } + { \bf { A _ 0 ( v ) } } \partial _ x { \bf { v } } + f _ 0 ( \partial _ y { \bf { v } } ) - { \bf { B _ 0 ( v } } ) \partial _ y ^ 2 { \bf { v } } = g _ 0 ( { \bf { v } } ) , \end{align*}


\begin{align*} y _ i : = a _ i e _ 1 - e _ i a _ i > 0 . \end{align*}


\begin{align*} \frac { w ( \Phi ( \omega ) ) } { w ( \Phi ( \eta ) ) } = \frac { w ( \Phi ( \eta ) ) + \int _ 0 ^ 1 \frac { d } { d t } \big | _ { t = s } \left [ w ( \Phi ( \eta ) + s \tau ) \right ] ~ d s } { w ( \Phi ( \eta ) ) } \leq 1 + \frac { \sup _ { \upsilon \in B _ { 2 \delta } ( \Phi ( \eta ) ) } \nabla _ { \tau } w ( \upsilon ) } { w ( \Phi ( \eta ) ) } , \end{align*}


\begin{align*} x _ t = x _ 0 - \omega _ t + \int _ 0 ^ t f ( s , x _ s ) \ , d s , \end{align*}


\begin{align*} i _ \omega = \sup \{ \alpha > 0 : \omega \in L R P ( \alpha ) \} . \end{align*}


\begin{align*} \varphi _ g ( \tau , z ) : = \eta _ g ( \tau ) \phi _ { - 2 , 1 } ( \tau , z ) , \end{align*}


\begin{align*} a \cdot _ L b \cdot _ R c : = a b \tau ( c ) , \end{align*}


\begin{align*} h [ n + 1 ] ( x , y ) & : = \big ( - \Delta _ x + V _ \mathrm { t r a p } ( x ) \big ) \delta ( x - y ) + ( \upsilon _ N \ast | \phi _ { n + 1 } | ^ 2 ) ( x ) \delta ( x - y ) \\ & + \frac { 1 } { N } ( \upsilon _ N \Big ( \frac { k _ { n + 1 } \circ \overline { k _ { n + 1 } } } { \delta - k _ { n + 1 } \circ \overline { k _ { n + 1 } } } \Big ) ) ( x , y ) + \frac { 1 } { N } ( \upsilon _ N \ast \rho ^ \mathrm { p a i r } [ n + 1 ] ) ( x ) \delta ( x - y ) . \end{align*}


\begin{align*} \int \limits _ { a } ^ c \alpha ^ { \ , \prime } ( t ) \ , d t = \alpha ( c ) - \alpha ( a ) \ , . \end{align*}


\begin{align*} d ^ 1 _ { r , s } ( m , k ) = 0 ( r , s ) ( m , k ) m \ne 0 . \end{align*}


\begin{align*} ( \tilde \Delta _ a - \lambda _ s ) u = - ( \Delta - \lambda _ s ) h _ s . \end{align*}


\begin{align*} \frac { d \hat { U } _ T ( T - t , 0 ) w } { d t } = - ( A ^ * - P ( T - t ) B B ^ * ) { U } _ T ( T - t , 0 ) w . \end{align*}


\begin{align*} d P ^ * ( \omega , x ) = d \mu _ \omega ( x ) d P ( \omega ) . \end{align*}


\begin{align*} \mathbb { P } \left \{ \bigcup _ { t \in \mathbb { Q } \cap [ 0 , 1 ] } \left \{ D X _ t = 0 \right \} \right \} = 0 . \end{align*}


\begin{align*} & S ( k ) = \begin{pmatrix} a _ 1 ( k ) & - \sigma \overline { b ( - k ) } \\ b ( k ) & a _ { 2 } ( k ) \end{pmatrix} , k \in \mathbb { R } \backslash \{ 0 \} , \end{align*}


\begin{align*} \begin{gathered} M ( A ) ( ( l , m ) ) = \begin{cases} A ( l ' ) ( m ) & \exists l ' : l = l ' \circ a _ \psi \\ 0 & \end{cases} \\ \forall l \in L ^ { a _ X } , m \in a _ M ( l ) \ ; . \end{gathered} \end{align*}


\begin{align*} \xi ' = \sum _ { \alpha \in \Gamma , i \in I } a _ { \alpha , i } \delta _ \alpha \otimes v _ i \eta ' = \sum _ { \beta \in \Gamma , j \in I } b _ { \beta , j } \delta _ \beta \otimes v _ j \end{align*}


\begin{align*} K _ 1 \cdot K _ 2 = \int _ \Lambda K _ 1 ( \bullet _ 1 , \lambda ) K _ 2 ( \lambda , \bullet _ 2 ) ~ d \mu ( \lambda ) . \end{align*}


\begin{align*} \zeta = \frac { s / f ^ a } { t / f ^ b } . \end{align*}


\begin{align*} \mu ( V _ C ) f ( t ) = e ^ { i \pi C t \cdot t } f ( t ) \end{align*}


\begin{align*} \lim _ { \epsilon \to 0 ^ + } P _ \Omega [ ( \phi _ { \epsilon } \ast u ) \mathbf { 1 } _ { V _ { \rho / 2 } } ] ( x ) = P _ \Omega [ u \mathbf { 1 } _ { V _ { \rho / 2 } } ] ( x ) . \end{align*}


\begin{align*} \widehat { L } _ { d , \lambda } ^ { \pm } : = \widehat { M } _ { d , \lambda } ^ { \pm } / \langle n _ { \pm } \rangle \end{align*}


\begin{align*} F ( \mu ) = \mathcal { E } ^ { * } ( \mu ) + \frac { 1 } { \theta } { \rm e n t } [ \mu | \pi ] . \end{align*}


\begin{gather*} \langle \partial _ t \bar { u } _ \delta , z \rangle _ { V ^ * , V } + \int _ \Omega \nabla \bar { \mu } _ \delta \cdot \nabla z \ , d x = 0 , \\ ( \bar { \mu } _ \delta , z ) _ H = ( \nabla \bar { u } _ \delta , \nabla z ) _ { H } - ( \partial _ { \boldsymbol { \nu } } \bar { u } _ \delta , z _ { | _ \Gamma } ) _ { H _ \Gamma } + ( \bar { \xi } _ \delta , z ) _ { H } + \bigl ( \pi ( u _ \delta ) - \pi ( u ) , z \bigr ) _ { \ ! H } \end{gather*}


\begin{align*} \Delta _ { l _ { 1 1 } } P ( u + v ) + \Delta _ { l _ { 1 2 } } Q ( u + \tilde \alpha v ) - \Delta _ { l _ { 1 3 } } P ( u - v ) = 0 , \ \ u , v \in Y , \end{align*}


\begin{align*} \begin{gathered} a \otimes b = a \times b \ ; , \\ | ( i , j ) | = | i | + | j | \ ; . \end{gathered} \end{align*}


\begin{align*} \sum _ { i , k = 1 } ^ n \nu _ i W _ { i k } \underline { D } _ k \eta = ( W \nu ) \cdot \nabla _ \Gamma \eta = 0 \quad \Gamma \end{align*}


\begin{align*} p = f _ 0 h _ 0 + \dots + f _ r h _ r . \end{align*}


\begin{align*} \exp \left ( \frac { \partial W } { \partial \ln X _ a } \right ) \ : = \ : 1 \end{align*}


\begin{align*} F ( \alpha ) = \big ( 1 + o ( 1 ) \big ) \ , T ^ { - 2 \alpha } \log T + \alpha + o ( 1 ) \end{align*}


\begin{align*} t \mapsto \psi ( t ) : = \ , \int _ { B ( o , 2 R ) } \Theta ( c \| D _ x b ( t , z ) \| ) \dd z \end{align*}


\begin{align*} | F ^ { ( i ) } ( x ) | \leq c _ { 1 , i } | x | ^ { p - i } + c _ { 2 , i } , \ ; i = 0 , 1 , 2 \textnormal { a n d } 4 \geq p \geq 2 , F ( x ) \geq - c _ 3 , F ^ { \prime \prime } ( x ) \geq - c _ 4 . \end{align*}


\begin{align*} \left ( \frac { \lambda ^ 2 } { \eta } \right ) ^ { n / 2 } \left ( 1 + \ln ( \eta ^ { - 1 } + 1 ) \right ) ^ n \eta ^ { - { 3 / 2 } } \leq C \lambda ^ { n - ( 2 - \epsilon ) \frac { n } { 2 } - n \frac { \epsilon } { 4 } - ( 2 - \epsilon ) \frac { 3 } { 2 } } = C \lambda ^ { \epsilon ( \frac { n } { 4 } + \frac { 3 } { 2 } ) - 3 } . \end{align*}


\begin{align*} u ( r ) : = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & r \\ 0 & 0 & 1 \end{bmatrix} . \end{align*}


\begin{align*} S ( n ) = \frac { 3 n + 1 } { 1 6 } \equiv - 1 \pmod { 4 } . \end{align*}


\begin{align*} H ( \gamma ( t ) ) & = \frac { 1 - \abs { \gamma ( t ) } ^ 2 } { 2 } H ^ ( \gamma ( t ) ) + ( n - 1 ) g _ { } ( \gamma ( t ) , \nu ) \\ & = \frac { 1 - \abs { \gamma ( t ) } ^ 2 } { 2 } \Bigl ( \kappa _ \gamma ^ + ( n - 2 ) \kappa ^ ( C _ t ) \Bigr ) + ( n - 1 ) g _ { } ( \gamma ( t ) , \nu ) \\ & = \kappa _ \gamma + ( n - 1 ) \kappa ( C _ t ) . \end{align*}


\begin{align*} \rho _ t = \ , X ( t , 0 , \cdot ) _ \# ( \bar \rho \ , \mathcal L ^ n ) t \in [ 0 , T ] \ , . \end{align*}


\begin{align*} \cos ( \tau \log a + \psi ( \tau ) ) \cdot J ( w ) = \sin ( \tau \log a + \psi ( \tau ) ) \cdot \frac { \theta E _ { 1 - w } \cdot \theta E _ w } { 2 \tau } . \end{align*}


\begin{align*} S _ { q , r } ( m ) = \frac { \left ( q - 1 \right ) m + r } { p ^ e } \end{align*}


\begin{align*} u \in \L _ t \big ( x _ 1 \big ( \textstyle \prod _ { s = \ell } ^ { d - 2 } x _ { n - s t - 2 } \big ) \big ( \prod _ { s = 0 } ^ { \ell - 1 } x _ { n - s t - 1 } \big ) , x _ 1 \big ( \prod _ { s = 0 } ^ { d - 2 } x _ { n - s t } \big ) \big ) . \end{align*}


\begin{align*} \max _ { \mathbf { W } , \boldsymbol { \theta } _ t , \boldsymbol { \theta } _ r } & \sum _ { k \in \mathcal { K } } R _ k \\ \mathrm { s . t . } & \mathrm { t r } ( \mathbf { W } \mathbf { W } ^ H ) \le P _ t , \\ & \beta _ { t , n } ^ 2 + \beta _ { r , n } ^ 2 = 1 , \forall n \in \mathcal { N } , \\ & \cos ( \phi _ { t , n } - \phi _ { r , n } ) = 0 , \forall n \in \mathcal { N } , \end{align*}


\begin{align*} \hat \mu _ 1 ( u + v ) \hat \mu _ 2 ( u + \tilde \alpha v ) = \hat \mu _ 1 ( u - v ) \hat \mu _ 2 ( u - \tilde \alpha v ) , \ \ u , v \in Y . \end{align*}


\begin{align*} g _ { \lambda } ( x ) = \int _ 0 ^ { \infty } e ^ { - \lambda t } p _ t ( x ) \d t , \lambda > 0 , \ x \in \R ^ d , \end{align*}


\begin{align*} \lambda _ i ( p ) = \frac { l } { d _ i ( p ) d ( p ) ^ \top \nu } | \nu _ i | , i = 1 , 2 , 3 , 4 . \end{align*}


\begin{align*} g ( { \bf { v } } ) = \left ( 0 , \frac { P - q } { a } r _ 2 + \theta r _ 3 , \theta r _ 2 + \frac { \theta ^ 2 } { 2 q } \frac { P + q } { P - q } r _ 3 \right ) ^ T . \end{align*}


\begin{align*} \displaystyle \lim _ { \lambda \rightarrow 1 / 2 } x _ * ( \lambda ) = + \infty , \ , \displaystyle \lim _ { \lambda \rightarrow 2 } x _ * ( \lambda ) = 0 . \end{align*}


\begin{align*} P \Big ( \sum _ { j = 1 } ^ k D _ j ^ { ( m ) } \ge \varepsilon k \log k \Big ) \le C ^ k e ^ { - \varepsilon c k ( \log k ) } \ , , \end{align*}


\begin{align*} \| | ( g _ 1 - g _ 0 ) \| | = \sup _ \tau \int \limits _ 0 ^ T | ( g _ 1 - g _ 0 ) ( \hat x _ \tau ) | d t + \sup _ \tau \int \limits _ 0 ^ T \Big | \Big ( \frac { \partial g _ 1 } { \partial x } - \frac { \partial g _ 0 } { \partial x } \Big ) ( \hat x _ \tau ) \Big | d t \end{align*}


\begin{align*} \tilde { \rho } = \rho - 1 , \tilde { h } = h - 1 . \end{align*}


\begin{align*} ( a \cdot X ) ( n ) = a \cdot X ( n ) X ( a \cdot n ) = X ( a ) \cdot n + a \cdot X ( n ) \end{align*}


\begin{align*} D _ H ( X , Y ) = \max ( \max _ { x _ j \in X } \mathrm { d i s t } ( x _ j , Y ) , \ , \max _ { y _ k \in Y } \mathrm { d i s t } ( y _ k , X ) ) , \end{align*}


\begin{align*} & M = F _ { - 1 } , \\ & C ^ \bullet \colon 0 = C ^ 2 \subseteq ( C ^ 1 _ F ) _ { - 1 } \subseteq F _ { - 1 } = C ^ 0 , \\ & D _ \bullet \colon 0 = D _ { - 1 } \subseteq ( D _ 0 ^ F ) _ { - 1 } \subseteq F _ { - 1 } = D _ 1 , \textup { a n d } \\ & \varphi _ 0 = \pi _ { - 1 } \varphi ^ F _ { 0 } , \ \varphi _ { 1 } = \pi _ { - 1 } \varphi ^ F _ { 1 } . \end{align*}


\begin{align*} \begin{aligned} \Omega _ F ( u _ 0 ) & = u _ 0 - i \int _ 0 ^ { \infty } e ^ { - i t \Delta } F ( u ( t ) ) \ , d t , \\ S _ F ( u _ - ) & = u _ - - i \int _ { - \infty } ^ \infty e ^ { - i t \Delta } F ( u ( t ) ) \ , d t . \end{aligned} \end{align*}


\begin{align*} I = 1 + \sum _ { d = 1 } ^ \infty \frac { z ^ d } { d ! } \sum _ { \alpha , \beta \vdash d } p _ \alpha ( A ) p _ \beta ( B ) \sum _ { r = 0 } ^ \infty ( - \hbar ) ^ r \sum _ { s = 0 } ^ r q ^ s \vec { W } ^ r ( \alpha , \beta ; s ) . \end{align*}


\begin{align*} F _ \mu ( r , y ) : = \mu ( \{ x \in \R ^ d : | x - y | \le r \} ) . \end{align*}


\begin{align*} \int _ { Q _ T } \Delta p | \nabla p | ^ 2 = \frac { 2 } { 3 } \int _ { Q _ T } p | D ^ 2 p | ^ 2 - p | \Delta p | ^ 2 . \end{align*}


\begin{align*} M ( A ) = A \ ; . \end{align*}


\begin{align*} \iota ( \L _ T h ) \varphi = ( \iota h ) ( \varphi \circ T ) . \end{align*}


\begin{align*} \frac { \partial ^ k } { \partial a ^ k } M _ f ( a ) = M _ f ( a ) \mathbb { E } [ ( \log X ) ^ k ] . \end{align*}


\begin{align*} & - \omega ^ { 1 / 2 } \mathcal C ( \mathfrak f _ { j _ M } ) = j _ M \delta _ \infty - j _ M \delta _ 0 \\ & - \omega ^ { 3 / 2 } \mathcal C ( \mathfrak f _ { j _ M } ) = 0 . \end{align*}


\begin{align*} \mathbb { M } ^ s & = \{ f \in L ^ 2 ( \mathbb { R } ) : \| f \| _ { \mathbb { M } ^ s } < \infty \} , s \geq 0 , \\ \mathbb { M } ^ s _ { \mathrm { e } } & = \{ f \in L ^ 2 ( \mathbb { R } ) : \| f \| _ { \mathbb { M } ^ s _ { \mathrm { e } } } < \infty \} , s \geq 0 , \\ \mathbb { M } ^ s _ { \mathrm { e } ^ 2 } & = \{ f \in L ^ 2 ( \mathbb { R } ) : \| f \| _ { \mathbb { M } ^ s _ { \mathrm { e } ^ 2 } } < \infty \} , \pi > s \geq 0 . \end{align*}


\begin{align*} \| g _ n \| _ { L ^ { q _ 0 } ( Q _ { R , \tau } ) } = \sigma _ n ^ { \gamma ' \frac { N + 1 } { N + 2 } } \| f _ n \| _ { L ^ { q _ 0 } ( \widetilde { Q } _ { n , R , \tau } ) } , \widetilde { Q } _ { n , R , \tau } = B _ { R r _ n } ( \bar x _ n ) \times ( \bar t _ n , \bar t _ n + r _ n ^ \gamma M _ n ^ { 1 - \gamma } \tau ) \end{align*}


\begin{align*} k + r = k ^ { \prime } + r ^ { \prime } . \end{align*}


\begin{align*} \sum _ { k = 1 } ^ 2 \lambda _ k \langle B _ k ( u _ k ) , ( u _ k - \overline { u } _ k ) _ + \rangle _ { \mathcal { H } _ k } \leq \sum _ { k = 1 } ^ 2 \langle A _ k ( \overline { u } _ k ) - A _ k ( u _ k ) , ( u _ k - \overline { u } _ k ) _ + \rangle _ { \mathcal { H } _ k } \leq 0 . \end{align*}


\begin{align*} ( - y ) ^ { - \gamma \cdot \mathrm c _ 1 ( S ) / 2 } \cdot \left \langle \gamma _ 1 , \dots , \gamma _ N \right \rangle ^ { \infty } _ { g , \gamma } = ( y ^ { 1 / 2 } - y ^ { - 1 / 2 { \tiny } } ) ^ { \gamma \cdot \mathrm c _ 1 ( S ) } \cdot \left \langle \gamma _ 1 , \dots , \gamma _ N \right \rangle ^ { 0 ^ + } _ { g , \gamma } . \end{align*}


\begin{align*} f ( a ) = \frac { 1 } { | \Omega | } \int _ \Omega { f } , \ \ \ \forall \ , f \in { A ^ p ( \Omega ) } . \end{align*}


\begin{align*} \frac { C _ 1 } { \omega + F _ n ( x _ 3 ) } = \frac { \omega - F _ n ( x _ 3 ) } { \prod _ { i = 2 } ^ { n / 2 } C _ i } \end{align*}


\begin{align*} \tilde { g } _ \theta \circ \sigma = \Psi _ \theta ( \sigma ) \circ \tilde { g } _ \theta \end{align*}


\begin{align*} \sigma _ x ( y ) : = ( \lambda _ x ( g ) ) ^ { - } \circ y \end{align*}


\begin{align*} \int _ { \R ^ d } ( D ^ j _ h f ) g \d x = - \int _ { \R ^ d } f ( D ^ j _ { - h } g ) \d x , \end{align*}


\begin{align*} f ( H _ b ^ \star ) = - \frac { 1 } { \pi } \int _ \mathcal { D } \Bar { \partial } f _ N ( z ) ( H _ b ^ \star - z ) ^ { - 1 } \d z _ 1 \d z _ 2 , \end{align*}


\begin{align*} \| f \| _ \omega = \sup _ { x \not = y } \frac { | f ( x ) - f ( y ) | } { \omega ( | x - y | ) } < + \infty . \end{align*}


\begin{align*} Y _ i ( r ) = f _ i ( r ) v _ i ( r ) , f _ i ( r ) = \begin{cases} r , & i = 0 , \\ \sinh ( r ) \cosh ( r ) , & i = 1 , \dots , d - 1 , \\ \sinh ( r ) , & i = d , \dots , n - 1 . \end{cases} \end{align*}


\begin{align*} \mathtt { m } : = \sum _ { j \in \Z ^ d } | \psi _ j | ^ 2 = \alpha ^ 2 + \sum _ { j \in \Z ^ d \setminus \{ 0 \} } | z _ j | ^ 2 \end{align*}


\begin{align*} d ^ t = \left \{ \begin{aligned} & \left ( n ^ { - 1 } \cos \theta _ d , - \sqrt { 1 - ( n ^ { - 1 } \cos \theta _ d ) ^ 2 } \right ) & & \textrm { i f } ~ n ^ { - 1 } \vert \cos \theta _ d \vert \leq 1 , \\ & \left ( n ^ { - 1 } \cos \theta _ d , - i \sqrt { ( n ^ { - 1 } \cos \theta _ d ) ^ 2 - 1 } \right ) & & \textrm { i f } ~ n ^ { - 1 } \vert \cos \theta _ d \vert > 1 . \end{aligned} \right . \end{align*}


\begin{align*} 0 ~ \leq ~ & y \sqrt { a b } - x ^ 2 \\ [ . 5 e m ] = ~ & x \sqrt { a b } - x ^ 2 + y \sqrt { a b } - x \sqrt { a b } \\ [ . 5 e m ] = ~ & x ( \sqrt { a b } - x ) + \sqrt { a b } ( y - x ) \\ [ . 5 e m ] \leq ~ & ( x + \sqrt { a b } ) ( y - x ) , \end{align*}


\begin{align*} S ^ i _ { \ell } \left ( 2 \cdot 2 ^ { \ell v } q + 1 \right ) = 2 \cdot \left ( 2 ^ { \ell } - 1 \right ) ^ i 2 ^ { \ell ( v - i ) } q + 1 \end{align*}


\begin{align*} \psi ( \kappa _ 2 ^ 2 - \mu _ 2 ^ 2 ) = c \kappa _ 2 ^ 2 \ , \ , \ , M \smallsetminus M _ 1 . \end{align*}


\begin{align*} \left \{ \begin{array} { l } ( - i \partial _ t + \theta | \nabla | ) u _ \theta = \theta | \nabla | ^ { - 2 } Q ( \overline u , u ) , \\ u _ { \theta } | _ { t = 0 } = u _ { 0 , \theta } . \end{array} \right . \end{align*}


\begin{align*} \frac { \int _ E | f _ E | ^ p } { \int _ \Omega | f _ E | ^ p } = s _ p ( E , \Omega ) = \frac { 1 } { 2 } . \end{align*}


\begin{align*} 2 - f ( a _ k ) \geq 1 \implies 1 \geq f ( a _ k ) \implies f ( a _ k ) = 1 \forall k < n \end{align*}


\begin{align*} \varphi _ n ( m ) = \left \{ \begin{array} { l l } \dfrac { n - | m | } { n } & | m | < n \\ 0 & \end{array} \right . \longrightarrow 1 \end{align*}


\begin{align*} E _ { \overline { k } } ^ { - 1 } I = E _ { \underline { k + 1 } } , \end{align*}


\begin{align*} \sum \limits _ { j = 1 } ^ m e ^ { i \theta _ j } \left ( \kappa _ 2 \langle T _ { \theta _ j } e _ 5 , v _ j \rangle - i \mu _ 2 \langle T _ { \theta _ j } e _ 6 , v _ j \rangle \right ) = 0 , \end{align*}


\begin{align*} \mathbb S _ n : = \bigoplus _ { w \in W _ + } \mathbb S _ n ( w ) . \end{align*}


\begin{align*} \mathcal { S } ^ { ( 0 ) } \stackrel { ? } { = } 4 m _ 3 ^ 2 ~ \mathcal { I } \end{align*}


\begin{align*} I _ { \lambda } ' ( \Psi _ { n } ^ { 1 } ) = I _ { \lambda } ' ( u _ n ) - I _ { \lambda } ' ( u ^ 0 ) + o ( 1 ) . \end{align*}


\begin{align*} \limsup _ { \varepsilon \to 0 ^ + } \varepsilon ^ { - a } I _ 1 ( \varepsilon ) = \infty . \end{align*}


\begin{align*} z _ { j 0 } & = \sum _ { i = 1 } ^ { n _ j - 1 } \sigma _ { j i } ^ + + \sum _ { i = 1 } ^ { n _ j - 1 } \sigma _ { j i } ^ - + \sigma _ j + \eta \\ \bar z _ { j } & = \bar s _ j - \bar y _ j = \sigma _ { j } ^ + - \sigma _ { j } ^ - - \sigma _ j \frac { \bar y _ { j } } { \| \bar y _ j \| } \end{align*}


\begin{align*} \left | \omega _ { \psi } ^ { ( \nu ) } ( s u ) \right | = q ^ { \frac { \nu } { 2 } \dim V ^ { s u } } \leq q ^ { \frac { \nu } { 2 } \dim V ^ { s } } = q ^ { \nu \cdot \overline { \rm r k } \ , G _ { \jmath , [ 1 ] } } . \end{align*}


\begin{align*} f ( x ) : = \limsup _ { r \to 0 } \dfrac { | S \cap B _ r ( x ) | _ * } { | B _ r ( x ) | } , \end{align*}


\begin{align*} | \hat { f } _ \# ( k ) - \hat { f } ( k ) | & = \left | \int _ { \R ^ d } ( 1 _ { \Lambda _ L } ( x ) - 1 ) e ^ { - i 2 \pi k \cdot x } f ( x ) d x \right | \leq \int _ { \R ^ d } | ( 1 - 1 _ { \Lambda _ L } ( x ) ) f ( x ) | d x . \end{align*}


\begin{align*} \langle \chi _ { \upsilon ( \jmath ) } \vartheta _ { Z _ \jmath } , \eta _ { \varsigma ( \jmath ) } \rangle _ { Z _ \jmath ^ F } = \prod _ j { | I _ j | \choose \nu _ j } { | I _ j ' | \choose \nu _ j ' } \nu _ j ! \nu _ j ' ! ( \kappa _ G \cdot j ) ^ { \nu _ j + \nu _ j ' } . \end{align*}


\begin{align*} q ^ \epsilon ( t ) - q ^ 0 ( t ) & = Q ^ \epsilon ( t ) - q ^ 0 ( t ) - P ^ \epsilon ( t ) \\ & = q _ 0 ^ \epsilon - q _ 0 ^ 0 + \int _ 0 ^ t \bigl ( f ( q ^ \epsilon ( s ) ) - f ( q ^ 0 ( s ) ) \bigr ) d s + \int _ 0 ^ t \bigl ( \sigma ( q ^ \epsilon ( s ) ) - \sigma ( q ^ 0 ( s ) ) \bigr ) d \beta ( s ) - \epsilon p ^ \epsilon ( t ) . \end{align*}


\begin{align*} \int _ { A _ { k } } \left \vert \nabla u \right \vert ^ { p ( x ) } d x = \int _ { A _ { k } } h \left ( x \right ) ( u - k ) ^ { + } d x \leq \left \Vert h \right \Vert _ { L ^ { N } ( \Omega ) } \left \Vert ( u - k ) ^ { + } \right \Vert _ { L ^ { N ^ { \prime } } ( A _ { k } ) } . \end{align*}


\begin{align*} L _ { \pi ( y ) } \pi L _ x = L _ { \pi ( x ) } \pi L _ y . \end{align*}


\begin{align*} \dim Z = \dim ( \textbf { d } ) \cdot M + 1 = \dim ( \textbf { d } ) - \dim ( M ) + 1 \leq \dim ( \textbf { d } ) . \end{align*}


\begin{align*} \mathring { R } _ { o s c } ^ B = \mathring { R } ^ B _ { o s c . 1 } + \mathring { R } ^ B _ { o s c . 2 } + \mathring { R } ^ B _ { o s c . 3 } , \end{align*}


\begin{align*} \biggl | \tfrac { 4 \pi } { 9 } \int _ \R & [ H ( k ) - \tilde H ( k ) ] w ( k + 2 \log A ) \ , d k \biggr | \\ & = \sigma ^ { - 4 } \biggl | \int _ 0 ^ \infty \int _ { \R ^ 2 } G ( | e ^ { i t \Delta } u _ 0 ^ \sigma | ^ 2 ) - \tilde G ( | e ^ { i t \Delta } u _ 0 ^ \sigma | ^ 2 ) \ , d x \ , d t \biggr | \lesssim _ A \sigma ^ 2 . \end{align*}


\begin{align*} \lim _ { k \rightarrow \infty } k ( E ( x , t , k ) - I ) = \mathfrak { B } ^ r ( \xi , t ) - \overline { \mathfrak { B } ^ r ( \xi , t ) } + R ( \xi , t ) . \end{align*}


\begin{align*} \chi ( k ) = \frac { 1 } { 2 \pi i } \int _ { - \infty } ^ { \infty } \frac { { \rm l o g } \frac { s ^ 2 } { 1 + s ^ 2 } \left ( 1 - b ^ 2 ( s ) \right ) } { s - k } d s . \end{align*}


\begin{align*} & \d z = ( i \Delta z - i \nu z - \epsilon ( \gamma z - \mu \overline { z } ) ) \d t - \tfrac { 1 } { 2 } z F _ \Phi \d t + i \kappa | z | ^ 2 z \d t - i z \Phi \d W _ H , \end{align*}


\begin{align*} \sum _ { y \in V _ x } u ^ \rho _ { x y } & = \sum _ { r = 1 } ^ k \sum _ { y \in V _ X } \delta _ { x \sigma ^ r ( y ) } p _ r + \sum _ { s = 1 } ^ k \sum _ { y \in V _ X } \delta _ { x \tau ^ s ( y ) } q _ s - 1 = \sum _ { r = 1 } ^ k p _ r + \sum _ { s = 1 } ^ k q _ s - 1 = 1 \end{align*}


\begin{align*} \alpha < \beta < \frac 1 { 2 H } - \frac d 2 , \frac 1 p < \eta = 1 - ( d + \beta ) H + \frac { d H } { 2 m } . \end{align*}


\begin{align*} \varphi ( t ) : = 2 \pi \sum _ { j \in \mathbb { Z } } f ( t + 2 \pi j ) . \end{align*}


\begin{align*} B _ { \mathfrak { a } } \left ( \rho ( x ) ( v ) , w \right ) & = B _ { \mathfrak { a } } \left ( v , \rho ( x ) ( w ) \right ) \end{align*}


\begin{align*} \mathcal { G } ( \Z _ 8 ) \cong D _ 8 = \langle \sigma _ 0 \sigma _ 1 \rangle \rtimes \langle \sigma _ 1 \rangle . \end{align*}


\begin{align*} \big \| u \big \| _ { V } ^ 2 = \int _ D | \nabla u ( x ) | ^ 2 d x . \end{align*}


\begin{align*} ( n , - 1 ) _ { p } = \left ( \frac { - 1 } { p } \right ) = \begin{cases} + 1 , & p \equiv 1 \mod 4 , \\ - 1 , & p \equiv 3 \mod 4 , \end{cases} \end{align*}


\begin{align*} \overline { R } _ n = k \varphi _ { n / k } - \varphi _ n + \sum _ { j = 1 } ^ k \overline { U } _ j - \sum _ { j = 1 } ^ { k - 1 } V _ { 0 , j m , ( j + 1 ) m } \ , . \end{align*}


\begin{align*} A = \sum _ { k \in \Lambda _ B } \gamma _ { ( k ) } ^ 2 ( A ) ( k _ 2 \otimes k _ 1 - k _ 1 \otimes k _ 2 ) . \end{align*}


\begin{align*} \left ( \prod _ { k = 0 } ^ { d - 1 } ( D _ \lambda - k ) + \lambda ^ d \prod _ { i = 1 } ^ n \prod _ { j = 0 } ^ { w _ i - 1 } \left ( D _ \lambda + \frac { 1 + \alpha _ i + d j } { w _ i } \right ) \right ) \omega = 0 . \end{align*}


\begin{align*} \left \{ \begin{alignedat} { 2 } - \Delta _ p u = f ( x , u , v , \nabla u , \nabla v ) \ ; \ ; & \mbox { i n $ \R ^ N $ , } \ ; \ ; u > 0 \ ; \ ; & & \mbox { i n $ \R ^ N $ , } \\ - \Delta _ q v = g ( x , u , v , \nabla u , \nabla v ) \ ; \ ; & \mbox { i n $ \R ^ N $ , } \ ; \ ; v > 0 \ ; \ ; & & \mbox { i n $ \R ^ N $ , } \end{alignedat} \right . \end{align*}


\begin{align*} \limsup _ { n \to \infty } \frac { M _ n } { \psi ( n ) } = 1 \ , , \liminf _ { n \to \infty } \frac { M _ n } { \hat { \psi } ( n ) } = 1 . \end{align*}


\begin{align*} \int _ { s } ^ { t } u _ r \times h \circ \dd W _ r = W _ { s , t } u _ s \times h + \mathbb { W } _ { s , t } u _ s \times h + ( u \times h ) ^ { \natural } _ { s , t } \ , . \end{align*}


\begin{align*} \begin{aligned} \Delta \rho ( x , t ) & = \varepsilon ^ { - 2 } \partial _ r ^ 2 \eta ( y , t , r ) - \varepsilon ^ { - 1 } H ( y , t ) \partial _ r \eta ( y , t , r ) \\ & - r | W ( y , t ) | ^ 2 \partial _ r \eta ( y , t , r ) + \Delta _ \Gamma \eta ( y , t , r ) + O ( \varepsilon ) . \end{aligned} \end{align*}