\begin{align*} \lambda _ { \widehat { \mathcal { E } } } ( x ) = r ! \left ( f ( x ) + g ( x ) \right ) \end{align*}


\begin{align*} \tilde { Q } ^ { \epsilon , \Delta t } ( t ) - \tilde { Q } ^ { \epsilon , \Delta t } ( t _ n ) = \int _ { s _ 1 } ^ { s _ 2 } f ( q _ n ^ { \epsilon , \Delta t } ) d s + \int _ { s _ 1 } ^ { s _ 2 } \sigma ( q _ n ^ { \epsilon , \Delta t } ) d \beta ( s ) , \end{align*}


\begin{align*} g ( z ) = z ^ 4 - z ^ 2 + \frac { 1 } { 8 } . \end{align*}


\begin{align*} X = { \downarrow } ^ { \mathbb { X } } \{ x \in X : { \uparrow } ^ { \mathbb { X } } x \subseteq \mathsf { d o m } ( p ) \} \end{align*}


\begin{align*} A \otimes B = A \oplus B \coloneqq \begin{pmatrix} A & 0 \\ 0 & B \end{pmatrix} \ ; . \end{align*}


\begin{align*} \frac { \mathbb { E } \varphi _ { t , a } \left ( Y \right ) } { \varphi _ { t , a } ( t ) } = \mathbb { P } \{ Y > t - a ^ { - 1 } \} \end{align*}


\begin{align*} \mathcal { E } [ k ; \ , H _ 0 , \Theta ] : = \mathrm { t r } \Big \{ ( \delta - \overline { k } \circ k ) ^ { - 1 } \circ \Big ( \overline { k } \circ H _ 0 [ \phi , k ] \circ k + \frac { 1 } { 2 } \overline { k } \circ \Theta [ \phi , k ] + \frac { 1 } { 2 } \overline { \Theta [ \phi , k ] } \circ k \Big ) \Big \} . \end{align*}


\begin{align*} c ' + m ' d ' = c + d - b + ( m - 1 ) d = c + m d - b \geq a - b = a ' , \end{align*}


\begin{align*} \xi - y _ 2 & = \ ( \ ( \xi - y _ 2 \ ) \cdot \dfrac { \zeta } { | \zeta | } \ ) \ , \dfrac { \zeta } { | \zeta | } + \ ( \ ( \xi - y _ 2 \ ) \cdot \dfrac { e } { | e | } \ ) \ , \dfrac { e } { | e | } \\ & = \sin \theta \ , \dfrac { \zeta } { | \zeta | } + \cos \theta \ , \dfrac { e } { | e | } ; \end{align*}


\begin{align*} - 2 \left ( \frac { r i } 3 + m j \right ) d ^ 1 _ { r , s } \left ( m - \frac r 3 , i + j \right ) = \left ( \frac { 2 r i } 3 - m ( j + s ) \right ) d ^ 1 _ { r , s } \left ( - \frac r 3 , j \right ) . \end{align*}


\begin{align*} L _ { P } ^ i : = \{ P \} \times ( 0 : 1 ) \times \cdots \times \mathbb P ^ 1 \times ( 1 : 0 ) \times \cdots \times ( 1 : 0 ) , \end{align*}


\begin{align*} \left \{ \begin{aligned} & u \in S ( \lambda , \Lambda , 0 ) & & ~ ~ \mbox { i n } ~ ~ Q _ 1 ^ + ; \\ & u = 0 & & ~ ~ \mbox { o n } ~ ~ S _ 1 . \end{aligned} \right . \end{align*}


\begin{align*} \log \mathbb P \left \{ \sup _ { | z | = R _ k } \log | W ( z ) | \ge \frac { R _ k ^ 2 } 2 - \frac { a } 8 R _ k \log ^ { b } R _ k \right \} \lesssim - \log ^ { 2 ( b - 1 ) } R _ k . \end{align*}


\begin{align*} F ( x ) : = \begin{cases} C _ { \max } , & | x | < 1 \\ C ' \cdot C _ { \max } \cdot ( 2 \pi | x | ) ^ { - k } , & \end{cases} \end{align*}


\begin{align*} d ' = c + d - b \geq a - b = q = b ' , \end{align*}


\begin{align*} \| \eta \| _ { C ^ { 2 , 1 } ( S _ T ) } = \| \eta \| _ { C ( S _ T ) } + \| \partial ^ \circ \eta \| _ { C ( S _ T ) } + \sum _ { i = 1 } ^ n \| \underline { D } _ i \eta \| _ { C ( S _ T ) } + \sum _ { i , j = 1 } ^ n \| \underline { D } _ i \underline { D } _ j \eta \| _ { C ( S _ T ) } \end{align*}


\begin{align*} I _ \alpha ( i ) = \sum _ { j \geq i } x _ { \alpha ^ { - 1 } ( j ) } \ , . \end{align*}


\begin{align*} G _ \delta ( 1 / 2 ) = 0 , G _ \delta ^ \prime ( 1 / 2 ) = 0 G _ \delta ^ { \prime \prime } ( s ) = \frac { 1 } { m _ \delta ( s ) } , s \in \mathbb { R } . \end{align*}


\begin{align*} c _ { n + 1 } = \begin{cases} 4 & \\ 2 & \end{cases} \end{align*}


\begin{align*} \prod _ { x \in \mathbb { T } _ { m _ 1 , m _ 2 } } K _ { A , \theta } ( x , \sigma ( x ) ) = ( - 1 ) ^ { \theta _ 1 h ( \sigma ) _ 1 + \theta _ 2 h ( \sigma ) _ 2 } \prod _ { x \in \mathbb { T } _ { m _ 1 , m _ 2 } } w _ A ( x , \sigma ( x ) ) . \end{align*}


\begin{align*} \mathcal { G } \left ( \delta _ { x } - \delta _ { x } ^ { \epsilon } , \mu _ { \theta } \right ) = \int _ { | y - x | \leq r } h ^ { \delta _ { x } - \delta _ { x } ^ { \epsilon } } \mu _ { \theta } ( y ) \ , d y + \int _ { | y - x | \geq r } h ^ { \delta _ { x } - \delta _ { x } ^ { \epsilon } } \mu _ { \theta } ( y ) \ , d y . \end{align*}


\begin{align*} V _ 0 ( x ) = I _ F ( x ) - I _ F ( 0 ) \varphi _ F ( x ) , x \geq 0 , \end{align*}


\begin{align*} \int _ { 0 } ^ { T } \big \| \sigma ( Y _ s ^ { h _ { \varepsilon } } ) - \sigma ( Y _ s ^ { h } ) \big \| ^ 2 d s = \int _ { 0 } ^ { T } \int _ { D } \big | \sigma ( Y _ s ^ { h _ { \varepsilon } } ( x ) ) - \sigma ( Y _ s ^ { h } ( x ) ) \big | ^ 2 d x d s \rightarrow 0 . \end{align*}


\begin{align*} \{ H x \mid x \in S \} = \{ H y \mid y \in H g _ 1 H \cup H g _ 2 H \} . \end{align*}


\begin{align*} T _ m f ( \rho , x ) & = \sum _ { k = 0 } ^ { \infty } \int _ { \R } e ^ { i \tau \rho } m ( \tau , k ) P _ k ( \mathcal { F } _ \rho f ) ( \tau , x ) \dd \tau , \ ; f \in C _ 0 ^ \infty ( \mathbb R ^ { d + 1 } ) . \end{align*}


\begin{align*} ( \omega _ 0 ( s ) g _ k ) ( r e ^ { i \theta } ) = ( - 1 ) ^ k r ^ { - 1 } e ^ { i k \theta } \ , , \end{align*}


\begin{align*} \pi : a = x _ 0 < x _ 1 < \hdots < x _ N = b \end{align*}


\begin{align*} \Psi _ \# ( \bar \rho \mu ) = \ , \bar \rho ( \Psi ^ { - 1 } ) \ , w \ , \mu \ , . \end{align*}


\begin{align*} W _ \mathcal { A } ( f , g ) ( x , \xi ) = e ^ { i \pi [ ( C _ { 1 1 } x + C _ { 1 2 } \xi ) \cdot x + ( C _ { 1 2 } ^ T x + C _ { 2 2 } \xi ) \cdot \xi ] } W _ { \mathcal { A } _ { F T 2 } \mathcal { D } _ L } ( f , g ) ( x , \xi ) , \end{align*}


\begin{align*} \sigma _ 0 ( A ) ( \vec { x } ) = A ( \vec { x } \circ \Phi _ \sqcup ^ \sigma ) \ ; , \end{align*}


\begin{align*} h ( z ) & = r \frac { ( z - \lambda ) g ( z ) - ( r \lambda - \lambda ) g ( r \lambda ) } { z - r \lambda } . \end{align*}


\begin{align*} \varphi ( t _ 1 + t _ 2 , \cdot ) = \varphi ( t _ 2 , \varphi ( t _ 1 , \cdot ) ) . \end{align*}


\begin{align*} \lim _ { x \to \infty } \dfrac { h ( x ) } { x } = \infty , \end{align*}


\begin{align*} \mathcal { E } _ { 2 } ( t ) + O ( \epsilon ) = \mathcal { E } _ { 2 } ^ { \ast } ( t ) . \end{align*}


\begin{align*} I _ c : = \{ t \in [ a _ 1 , a ) : \dot \gamma \} . \end{align*}


\begin{align*} S ^ j ( m ) = 2 ^ { v - 2 j } 3 ^ j w + 1 \end{align*}


\begin{align*} 2 \mathrm { R e } \langle u _ { \alpha } \otimes 2 \alpha ^ { 3 / 2 } \eta \Phi _ * ^ 1 \ , | \ , H _ y R _ y \rangle & = 2 \mathrm { R e } \langle u _ { \alpha } \otimes 2 \alpha ^ { 3 / 2 } \eta \tilde { \Phi } _ * ^ 1 \ , | \ , \alpha V _ y \Phi ^ y _ { \# } \rangle \\ & + 2 \mathrm { R e } \langle u _ { \alpha } \otimes 2 \alpha ^ { 3 / 2 } \eta \tilde { \Phi } _ * ^ 1 \ , | \ , \alpha V _ y R ^ { \# } _ y \rangle , \end{align*}


\begin{align*} \big [ \Omega \setminus \widehat { K \cup Q } \big ] \cup ( Q \setminus \widehat K ) = \Omega \setminus \widehat K \end{align*}


\begin{align*} \Delta _ \Gamma \eta = \sum _ { i = 1 } ^ n \underline { D } _ i \underline { D } _ i \eta , | \nabla _ \Gamma ^ 2 \eta | = \left ( \sum _ { i , j = 1 } ^ n | \underline { D } _ i \underline { D } _ j \eta | ^ 2 \right ) ^ { 1 / 2 } \quad \Gamma \end{align*}


\begin{align*} \begin{aligned} \lim _ { r \to \infty , \theta = \theta _ { 0 } } p ( r , \theta ) = 0 , \end{aligned} \end{align*}


\begin{align*} \nu _ M ( \bar f ) = \min _ \prec \{ \tilde \nu _ M ( f ) : f \in \bar f \} \end{align*}


\begin{align*} \alpha _ 0 ^ T \xi - \alpha _ 0 ^ t Y _ t & = \int _ t ^ T \alpha _ 0 ^ { s } Z _ s d B _ s \ , . \end{align*}


\begin{align*} e _ i ' : = \left \{ \begin{matrix} e _ i + [ \epsilon _ { i k } ] _ + e _ k & i \not = k \\ - e _ k & i = k \end{matrix} \right . \end{align*}


\begin{align*} A _ x = \{ ( h , g ) \mid h \in H ^ { ( n ) } , g \in G _ x , s ( h ) = r ( g ) \} . \end{align*}


\begin{align*} \mathrm { d } \rho \left ( k \right ) = { \displaystyle \sum \limits _ { n = 1 } ^ { N } } c _ { n } ^ { 2 } \delta _ { \kappa _ { n } } \left ( k \right ) \mathrm { d } k , \end{align*}


\begin{align*} _ Q ( I , \textbf { d } ) ^ { s s } _ { \theta } & = \{ M \in _ Q ( I , \textbf { d } ) \ , | \ , M \theta \} \\ _ Q ( I , \textbf { d } ) ^ { s } _ { \theta } & = \{ M \in _ Q ( I , \textbf { d } ) \ , | \ , M \theta \} \end{align*}


\begin{align*} K _ { \mathbb { H } ^ s } ( x , y ) = \sum _ { k \in \N } ( 1 + k ) ^ { - s } h _ k ( x ) h _ k ( y ) \ , . \end{align*}


\begin{align*} | \Lambda | & = | A | + | B | , \\ { \rm s i z e } ( \Lambda ) & = ( | A | , | B | ) , \\ { \rm r k } ( \Lambda ) & = \sum _ { i = 1 } ^ { m _ 1 } a _ i + \sum _ { j = 1 } ^ { m _ 2 } b _ j - \left \lfloor \biggl ( \frac { | A | + | B | - 1 } { 2 } \biggr ) ^ 2 \right \rfloor , \\ { \rm d e f } ( \Lambda ) & = | A | - | B | . \end{align*}


\begin{align*} \lim _ { k \rightarrow + \infty } \| u _ { t _ k } \times \partial _ x u _ { t _ k } \| ^ 2 _ { L ^ 2 } = \lim _ { k \rightarrow + \infty } \| \partial _ x u _ { t _ k } \| ^ 2 _ { L ^ 2 } = 0 \ , , \end{align*}


\begin{align*} A _ { x } ( \pmb { v } ) = \sum _ { j = 1 } ^ { N - 1 } \pmb { r } _ { j } \times ( \pmb { v } \times \pmb { r } _ { j } ) \end{align*}


\begin{align*} \Psi ( L ) = L ( s _ k ) + \widehat L = L _ { f r e e } + s _ k T ( a ) , \end{align*}


\begin{align*} \psi _ { \sigma } \left ( x , t ; k \right ) = \psi \left ( x , t ; k \right ) - \int K \left ( k / \mathrm { i } , s ; x , t \right ) y \left ( s , x , t \right ) \mathrm { d } \sigma _ { t } \left ( s \right ) , \end{align*}


\begin{align*} \psi _ { t t } + \Delta ^ 2 _ g \psi + m \psi = - \frac { \partial F } { \partial \psi } + \sum _ { l = 1 } ^ d \partial _ { x _ l } \frac { \partial F } { \partial ( \partial _ l \psi ) } \ , \end{align*}


\begin{align*} \rho _ { s _ { 1 : 2 m } } ^ { \epsilon , M _ 1 } = \rho _ { s _ { 1 : 2 m } } ^ { M _ 1 } \star \sigma _ { s _ { 1 : 2 m } } ^ \epsilon + \overline \rho _ { s _ { 1 : 2 m } } ^ { M _ 1 } \star \sigma _ { s _ { 1 : 2 m } } ^ \epsilon . \end{align*}


\begin{align*} L _ 0 = - \partial _ \tau - \eta \partial _ \xi + w ^ 2 \partial _ \eta ^ 2 . \end{align*}


\begin{align*} \left \{ \begin{aligned} & u \in S ( \lambda , \Lambda , 0 ) & & ~ ~ \mbox { i n } ~ ~ Q _ 1 ^ + ; \\ & u = 0 & & ~ ~ \mbox { o n } ~ ~ S _ 1 . \end{aligned} \right . \end{align*}


\begin{align*} \pi _ { \mathcal B } : S : = k [ x _ 1 , \dots , x _ N ] \to A , x _ i \mapsto b _ i . \end{align*}


\begin{align*} \partial _ t u _ 1 + \partial _ x u _ 1 = 0 , \partial _ t u _ 2 - \partial _ x u _ 2 = 0 \end{align*}


\begin{align*} \begin{multlined} Z ( \mathbf { o } ) = \sum _ { \mathbf { c } } e ^ { - \beta \sum _ p H _ p ( \mathbf { c } ) } \mathbf { O } ( \mathbf { o } | \mathbf { c } ) \\ = \sum _ { \mathbf { c } } \prod _ { p _ W \in P _ W } \prod _ { p _ O \in P _ O } W _ { p _ W } ( c _ { { p _ W } ^ { ( 0 ) } } , c _ { { p _ W } ^ { ( 1 ) } } , \ldots ) \\ \cdot O _ { p _ O } ( c _ { { p _ O } ^ { ( 0 ) } } , c _ { { p _ O } ^ { ( 1 ) } } , \ldots ) \ ; . \end{multlined} \end{align*}


\begin{align*} \mathrm { O P } _ { \hbar } ^ { T } ( \nu ) = \mathrm { O P } _ { \hbar } ^ { T } ( \nu ) ^ { \ast } \geq 0 \end{align*}


\begin{align*} \langle \dd P _ 1 , \ldots , \dd P _ r \rangle = \bigoplus _ { i = 1 } ^ { n } \ell ^ { e _ i } \Z _ \ell \ , \dd X _ i \end{align*}


\begin{align*} ( ( g f ) ( \pi ) ^ G _ i ) _ { x y } = \sum _ j f ( \pi ) _ { x _ i , y _ j } = \sum _ j ( \pi _ i ^ G ) _ { x y } \pi ^ I _ { i j } = ( \pi _ i ^ G ) _ { x y } \end{align*}


\begin{align*} c _ { \boldsymbol { r } } = & \displaystyle \int _ { - s / N ^ { 1 / d } } ^ { s / N ^ { 1 / d } } \cdots \int _ { - s / N ^ { 1 / d } } ^ { s / N ^ { 1 / d } } e { \Big ( - \sum _ { i \leq d } r _ i \alpha _ i \Big ) } d \alpha _ 1 \cdots d \alpha _ d \\ = & c _ { r _ 1 } \cdots c _ { r _ d } , \end{align*}


\begin{align*} \Delta _ i ^ h v ( x , t ) = \frac { v ( x + h e _ i , t ) - v ( x , t ) } { h } , \ \ \ h \neq 0 . \end{align*}


\begin{align*} \frac { d } { d t } u + \nu A u + B ( u , u ) = P f , \end{align*}


\begin{align*} \partial _ \pi \langle v \rangle = \begin{cases} 0 , \ \ \ \ \ \ \ \ \pi \ \ v , \\ \langle \overline { v / \pi } \rangle , \ . \end{cases} \end{align*}


\begin{align*} \left \{ \begin{aligned} & v _ t - \mathcal { M } ^ - ( D ^ 2 v ) \geq 0 & & ~ ~ \mbox { i n } ~ ~ \tilde Q ^ + _ { 1 } ; \\ & v \geq 0 & & ~ ~ \mbox { o n } ~ ~ \tilde S _ { 1 } ; \\ & v \geq 1 & & ~ ~ \mbox { i n } ~ ~ \partial \tilde Q ^ + _ { 1 } \backslash \tilde S _ { 1 } . \end{aligned} \right . \end{align*}


\begin{align*} Y _ { n + 1 } ^ { \epsilon , \Delta t } - \Phi ( \zeta ^ \epsilon ( t _ { n + 1 } ) - \zeta ^ \epsilon ( t _ n ) , Y _ { n } ^ { \epsilon , \Delta t } ) = R _ { n , 1 } ^ { \epsilon , \Delta t } + R _ { n , 2 } ^ { \epsilon , \Delta t } \end{align*}


\begin{align*} \Phi _ { n , a } ( A _ 2 , \theta _ 2 , \omega _ 2 , S ) = d ^ { - 1 + \frac { g \cdot \deg ( w ) } { 2 } } \cdot \Phi _ { n , a } ( A _ 1 , \theta _ 1 , \omega _ 1 , S ) . \end{align*}


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


\begin{align*} \hat { Z } _ i & = r _ i ^ { 2 } Z & \hat { \partial } _ i & = A r _ i \partial _ i & \hat { \nu } _ i & = A r _ i ^ { - 1 } \nu _ i & \hat { f } _ i & = A r _ i ^ { - 1 } f _ i . \end{align*}


\begin{align*} | I | = ( n - 1 ) \frac { g ( g + 1 ) } { 2 } + g \end{align*}


\begin{align*} \liminf _ { t \to 0 ^ + } f ( x , t ) > 0 \ ; \ ; \mbox { u n i f o r m l y w i t h r e s p e c t t o } \ ; \ ; x \in B _ \sigma ( x _ 0 ) , \\ f ( x , t ) \leq h ( x ) t ^ { - \gamma } \ ; \mbox { i n } \ ; \R ^ N \times \R ^ + , \ ; \mbox { w h e r e } \ ; h \in L ^ 1 ( \R ^ N ) \cap L ^ \eta ( \R ^ N ) , \end{align*}


\begin{align*} \left \{ \begin{array} { l l } \boldsymbol { w } ' ( x , y ) \cdot n = 0 , \\ 2 [ D ( \boldsymbol { w } ' ( x , y ) ) \cdot n ] _ { \tau } + \alpha ( x ) \boldsymbol { w } ' _ { \tau } = 0 , \end{array} \right . \end{align*}


\begin{align*} \partial _ t u - \Delta \mu = 0 & { \rm i n ~ } Q , \\ \mu = - \Delta u + \beta ( u ) + \pi ( u ) - f & { \rm i n ~ } Q , \\ \partial _ { \boldsymbol { \nu } } \mu = 0 & { \rm o n ~ } \Sigma , \\ u _ { | _ \Gamma } = v & { \rm o n ~ } \Sigma , \\ u ( 0 ) = u _ 0 & { \rm i n ~ } \Omega , \end{align*}


\begin{align*} h _ { n - 1 } \left ( z \right ) \gamma _ { n } \left ( z \right ) = - P _ { n } \left ( z ; z \right ) P _ { n - 1 } \left ( z ; z \right ) e ^ { - z ^ { 2 } } + \frac { 1 } { 2 } L \left [ \partial _ { x } P _ { n } P _ { n - 1 } \right ] . \end{align*}


\begin{align*} \left ( V ( h ) \psi - \rho \frac { V ( S _ F ) } { f ^ 2 } \right ) | \nabla _ B f | ^ 2 = 0 . \end{align*}


\begin{align*} R i c _ g + \nabla ^ 2 _ g \eta = ( \rho S _ g + \lambda ) g , \end{align*}


\begin{align*} \frac { \dd ^ N \coth t } { \dd t ^ N } & = ( - 1 ) ^ { N } 2 ^ { N + 1 } \operatorname { L i } _ { - N } ( e ^ { - 2 t } ) , \\ \frac { \dd ^ N \tanh t } { \dd t ^ N } & = - 2 ^ { N + 1 } \operatorname { L i } _ { - N } ( - e ^ { 2 t } ) , \end{align*}


\begin{align*} y _ k = \mathbf { h } _ k ^ H \mathbf { \Theta } _ i \mathbf { G } \sum _ { \ell \in \mathcal { K } } \mathbf { w } _ \ell s _ \ell + n _ k , \end{align*}


\begin{align*} E = \sum _ i D _ { i , 1 } [ g ] \dots D _ { i , k _ i } [ g ] \cdot C _ i \end{align*}


\begin{align*} \mathcal { G } ( \overline { \mathbf { P } } ) = \theta \overline { \mathbb { W } } ( \overline { \mathbf { P } } , \mu _ { V } ) + \overline { \rm E n t } [ \overline { \mathbf { P } } | \overline { \mathbf { \Pi } } ^ { 1 } ] . \end{align*}


\begin{align*} \int _ { x _ 1 } ^ x \frac { t } { \log ^ 7 t } = E ( x ) - E ( x _ 1 ) , \end{align*}


\begin{align*} f _ n ( t ) : = \frac { d } { d t } P _ N u _ n + \nu A P _ N u _ n + P _ N B ( u _ n , u _ n ) , \ t \in I _ n : = [ t _ { n - 1 } + \rho _ n , \infty ) , \ \ \rho _ n \gg 0 . \end{align*}


\begin{align*} \begin{cases} \displaystyle \frac { 1 - \nu ^ 2 } { E } \Delta ^ 2 w = 0 & A \ , , \\ [ 2 m m ] \displaystyle \nabla ^ 2 w \ , t = f & \textrm { o n } \partial A \ , , \end{cases} \end{align*}


\begin{gather*} \underbrace { [ \alpha ( x ) , [ y , z ] ] + [ \alpha ( y ) , [ x , z ] ] + [ \alpha ( z ) , [ y , z ] ] } _ { \in J } \\ + \underbrace { \theta \left ( \alpha ( x ) , [ y , z ] \right ) + \theta \left ( \alpha ( y ) , [ x , z ] \right ) + \theta \left ( \alpha ( z ) , [ x , y ] \right ) } _ { \in V } \\ + \underbrace { \rho ( \alpha ( x ) ) \theta ( y , z ) + \rho ( \alpha ( y ) ) \theta ( x , z ) + \rho ( \alpha ( z ) ) \theta ( x , y ) } _ { \in V } = 0 \end{gather*}


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


\begin{align*} d q ^ 0 ( t ) = f ( q ^ 0 ( t ) ) d t + \sigma ( q ^ 0 ( t ) ) d \beta ( t ) , \end{align*}


\begin{align*} ( ( \psi _ { y _ 0 } ^ { X | Y } ) ^ * d v _ X ) ( Z ) = \kappa _ { \psi , y _ 0 } ^ { X | Y } d Z _ 1 \wedge \cdots \wedge d Z _ { 2 n } . \end{align*}


\begin{align*} u ( t ) = U ( t , s ) u ( s ) + j ^ { - 1 } \int _ s ^ t U ^ { \odot \star } ( t , \tau ) R ( \tau , u ( \tau ) ) d \tau , - \infty < s \leq t < \infty . \end{align*}


\begin{align*} m \left ( 1 - e ^ { 2 \pi i \frac { j + \phi + I / 2 } { m } } + \frac { x + i y } { m } \right ) = x + ( - y + 2 \pi ( j + \theta _ 1 / 2 + \phi ) ) i + O ( m ^ { - 1 / 2 } ) . \end{align*}


\begin{align*} \beta _ A = \max _ { \substack { \ell \\ \emptyset \neq I \subseteq \{ 1 , \ldots , n \} } } \frac { 2 \dim A _ I } { \dim G _ { A _ I , \ell } } \end{align*}


\begin{align*} ( \{ F , G \} _ { \sigma , \tau } ) ^ { \diamondsuit } = p \cdot \det ( f ^ { \partial } ) ^ { 2 s } \cdot \{ F ^ { \diamondsuit } , G ^ { \diamondsuit } \} _ { \delta } \end{align*}


\begin{align*} g _ j ^ { ( Q , S ) } ( \mathcal { M } ) = b _ j - a _ j + v _ j . \end{align*}


\begin{align*} \sigma _ 0 ( A ) ( \vec x ) = A ( ( ( \vec x \circ \Phi _ { + 1 } ^ { a , b } ) \sqcup ( \vec x \circ \Phi _ { + 0 } ^ { a , b } ) ) \circ \bar \Phi _ { + \sqcup } ^ { b , a } ) \ ; . \end{align*}


\begin{align*} ( - \Delta ) ^ { \frac { 1 } { 2 } } \phi _ R ( x ) = c _ 0 , \forall \ , x \in B _ { R } ( 0 ) \end{align*}


\begin{align*} u v = \sum _ { \ell \geq 0 } \sum _ { \ell _ 1 + \ell _ 2 = \ell } u ^ { ( \ell _ 1 ) } v ^ { ( \ell _ 2 ) } \end{align*}


\begin{align*} \mathcal D u = ( D u , B u | _ { \partial X } ) \end{align*}


\begin{align*} X _ { k + 1 } = X _ k - \gamma \nabla f ( X _ k ) + \sqrt { 2 \gamma } \xi _ k , \end{align*}


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


\begin{align*} \check { f } ( x ) = \int _ { \R ^ d } f ( k ) e ^ { 2 \pi i k \cdot x } d k , x \in \R ^ d , \end{align*}


\begin{align*} \tilde b _ { \epsilon , i } ( t , x ) : = \ , \max \{ \{ \min \{ b _ i ( t , x ) , 1 / \epsilon \} , - 1 / \epsilon \} ( t , x ) \in I \times \R ^ n , \ , i = 1 , \dots , n \ , , \end{align*}


\begin{align*} ( - 1 ) ^ { k - 1 } q ( X _ a ) ^ k \prod _ { j = 1 } ^ { n } T _ j = \left ( \prod _ { b = 1 } ^ k X _ b \right ) \cdot \prod _ { i = 1 } ^ n ( T _ i - X _ a ) \ / . \end{align*}


\begin{align*} 6 K = \Delta \log { \kappa _ 1 ^ 2 } = \Delta \log ( 1 - K ) . \end{align*}


\begin{align*} f _ r ( x , y , u ) : = \frac { W ( \exp _ x ( r u ) , y ) + W ( \exp _ x ( - r u ) , y ) - 2 W ( x , y ) } { r ^ 2 } . \end{align*}


\begin{align*} \begin{multlined} | [ A ] ( \vec { x } ) | \leq \sum _ { \vec { y } \in \mathbb { N } ^ b } | A ( \vec { x } \sqcup ( \vec { y } \sqcup \vec { y } ) ) | \\ < C \sum _ { \vec { y } \in \mathbb { N } ^ b } \prod _ { i \in a } \phi ( \vec { x } ( i ) ) \prod _ { j \in b } \phi ( \vec { y } ( j ) ) ^ 2 \\ = C S _ \phi \prod _ { i \in a } \phi ( \vec { x } ( i ) ) \ ; . \end{multlined} \end{align*}


\begin{align*} & G ^ { ( 4 ) } _ { \mathcal R } ( x , y ) = \\ & - \frac { i } { 2 \pi } \left [ \int _ { \mathcal I _ { \theta _ c , \zeta _ o } } + \int _ { \mathcal D _ { \zeta _ o , \frac \pi 2 + \theta _ { \hat { x } } - i \infty } } \right ] \frac { 2 i \sin \zeta \widetilde { \mathcal S } ( \cos \zeta , n ) } { n ^ 2 - 1 } { e ^ { i k _ { + } \left ( - \vert y \vert \cos ( \zeta + \theta _ { \hat y } ) + \vert x \vert \cos ( \zeta - \theta _ { \hat x } ) \right ) } } d \zeta . \end{align*}


\begin{align*} \begin{cases} c ' _ { \ge \ell } ( z , \zeta ) & \textrm { i f } 1 \leq \ell \leq n - k \\ c ' _ { \ge \ell } ( z , \zeta ) + \bigl ( E \cdot C ^ \zeta _ { \ge n - k + 2 } \bigr ) _ \ell + ( - 1 ) ^ { n + k } q \binom { k - 1 } { n - \ell } c ^ \zeta & \textrm { i f } n - k + 1 \le \ell \le n \ / . \end{cases} \end{align*}


\begin{align*} S ( \lambda ) = \left [ \begin{matrix} \lambda I - A & B \cr - C & D \end{matrix} \right ] , \end{align*}


\begin{align*} u _ m ( t ) & = K _ { \varphi , f } u _ { m - 1 } ( t ) \\ & = T _ 0 ( t - s ) \varphi + j ^ { - 1 } \int _ s ^ t T _ 0 ^ { \odot \star } ( t - \tau ) [ B ( \tau ) u _ m ( \tau ) + f ( \tau ) + B ( \tau ) [ u _ { m - 1 } ( \tau ) - u _ m ( \tau ) ] ] d \tau . \end{align*}


\begin{align*} \kappa _ i ( x ) = u _ i , \kappa _ i ' ( x ) = u _ i ' . \end{align*}


\begin{align*} \partial _ i e ^ { \i R } = \i \int _ 0 ^ 1 \big ( e ^ { \i \alpha R } \otimes 1 \big ) \ \partial _ i R \ \big ( 1 \otimes e ^ { \i ( 1 - \alpha ) R } \big ) d \alpha . \end{align*}


\begin{align*} P ^ \epsilon ( s _ 2 ) - P ^ \epsilon ( s _ 1 ) = \bigl ( e ^ { - \frac { s _ 2 - s _ 1 } { \epsilon ^ 2 } } - 1 \bigr ) P ^ \epsilon ( s _ 1 ) + \int _ { s _ 1 } ^ { s _ 2 } e ^ { - \frac { s _ 2 - s } { \epsilon ^ 2 } } f ( q ^ \epsilon ( s ) ) d s + \int _ { s _ 1 } ^ { s _ 2 } e ^ { - \frac { s _ 2 - s } { \epsilon ^ 2 } } \sigma ( q ^ \epsilon ( s ) ) d \beta ( s ) . \end{align*}


\begin{align*} \left \{ \begin{aligned} & v _ s - \tilde { F } ( D ^ 2 v , y , s ) = \tilde { f } & & ~ ~ \mbox { i n } ~ ~ \tilde { \Omega } \cap Q _ 1 ; \\ & v = \tilde { g } & & ~ ~ \mbox { o n } ~ ~ \partial \tilde { \Omega } \cap Q _ 1 , \end{aligned} \right . \end{align*}


\begin{align*} u ^ { \leq } : = \Pi ^ { \leq } u \ , u ^ { \perp } : = \Pi ^ { \perp } u \ , \end{align*}


\begin{align*} \left \{ \begin{aligned} & u _ t - F ( D ^ 2 u ) = 0 & & ~ ~ \mbox { i n } ~ ~ Q _ 1 ^ + ; \\ & u = 0 & & ~ ~ \mbox { o n } ~ ~ S _ 1 . \end{aligned} \right . \end{align*}


\begin{align*} q _ { \rho } \left ( x , t \right ) & = 8 \left [ \int _ { 0 } ^ { 1 } s \sqrt { 1 - s ^ { 2 } } e ^ { - 2 s x } Y \left ( s ; x , t \right ) \mathrm { d } s \right ] ^ { 2 } \\ & - 8 \int _ { 0 } ^ { 1 } s ^ { 2 } \sqrt { 1 - s ^ { 2 } } e ^ { - 2 s x } Y \left ( s ; x , t \right ) \mathrm { d } s , \end{align*}


\begin{align*} \mathrm { m u l t } _ d ( 0 , 0 ) & = \frac { 1 } { d } \sum _ { t | d } \mu ( d / t ) \mathrm { t r } ( g ^ t ) - \frac { 4 } { d } \sum _ { t | d } \mu ( d / t ) = \begin{cases} b _ 1 - 4 : & d = 1 ; \\ b _ d : & d > 1 . \end{cases} \end{align*}


\begin{align*} x ^ { 2 } P _ { n } = P _ { n + 2 } + \left ( \gamma _ { n + 1 } + \gamma _ { n } \right ) P _ { n } + \gamma _ { n - 1 } \gamma _ { n } P _ { n - 2 } , \end{align*}


\begin{align*} \overline { N } _ G ( s , T ) ^ F : = G _ s ^ F \backslash N _ G ( s , T ) ^ F . \end{align*}


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


\begin{align*} \sum _ { A \in \mathcal { A } _ n } \tilde { \chi } ( z _ 1 , . . . , z _ n ) = 1 , \end{align*}


\begin{align*} \vartheta ( x y ) = \vartheta ( x ) + \vartheta ( y ) , \\ \vartheta ( x + y ) \geqslant _ { \boldsymbol { a } } \mathrm { m i n } \left \{ \vartheta ( x ) , \vartheta ( y ) \right \} . \end{align*}


\begin{align*} \| u \| _ { U ^ p } = \sup \left \{ \left | \int \langle u ' ( t ) , v ( t ) \rangle _ { L ^ 2 _ x } \ , d t \right | : v \in C ^ \infty _ 0 , \ \| v \| _ { V ^ { p ' } } = 1 \right \} . \end{align*}


\begin{align*} ( U \mathfrak { g } ) ^ G = \{ \alpha \in U \mathfrak { g } : g \alpha g ^ { - 1 } = \alpha , g \in G \} , \end{align*}


\begin{align*} \left \{ \begin{alignedat} { 2 } - \Delta u & = a ( x ) u ^ { - \gamma } \ ; \ ; & & \mbox { i n } \ ; \ ; \R ^ N , \\ u & > 0 & & \mbox { i n } \ ; \ ; \R ^ N . \\ \end{alignedat} \right . \end{align*}


\begin{align*} \hat { f } ( n ) : = \int _ { - \infty } ^ \infty f ( x ) e ^ { - i n x } \mathrm { d } x = \frac { 1 } { 2 \pi } \int _ 0 ^ { 2 \pi } \varphi ( t ) e ^ { - i n t } \mathrm { d } t . \end{align*}


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


\begin{align*} \begin{cases} x _ { k + 1 } = A _ 1 x _ k + A _ 2 \mathcal { E } x _ k + B u ^ F _ k \\ \ \ \ \ \ \ \ \ \ \ + ( C _ 1 x _ k + C _ 2 \mathcal { E } x _ k + D u ^ F _ k ) w _ k , \\ x _ 0 = \xi \in { \mathcal R } ^ n , \ k \in { \mathcal N } _ { T - 1 } , \end{cases} \end{align*}


\begin{align*} A ( \omega ^ { n } ) \omega ^ { n + 1 } = B ( \omega ^ { n } ) \omega ^ { n } + r \end{align*}


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


\begin{align*} \omega ( f ) \geq | A | \gamma _ { o i r 2 } ( H ) + | B | ( \gamma _ { o i r 2 } ( H ) + 1 ) = n \gamma _ { o i r 2 } ( H ) + | B | . \end{align*}


\begin{align*} \mathfrak { B } ( \xi , t ) = \begin{pmatrix} 0 & - i \eta \beta ( \xi ) e ^ { - t \varphi ( \xi , 0 ) } \tau ^ { - i \nu - \frac { 1 } { 2 } } \\ i \eta \gamma ( \xi ) e ^ { t \varphi ( \xi , 0 ) } \tau ^ { i \nu - \frac { 1 } { 2 } } & 0 \end{pmatrix} , \end{align*}


\begin{align*} - \Delta _ p u = \mu \ , u ^ { - \gamma } \ ; \ ; \mbox { i n } \ ; \ ; \Omega , \ ; \ ; u > 0 \ ; \ ; \mbox { i n } \ ; \ ; \Omega , \ ; \ ; u = 0 \ ; \ ; \mbox { o n } \ ; \ ; \partial \Omega , \end{align*}


\begin{align*} \mathrm { D } \Phi ^ { - 1 } ( \tau ) = \begin{pmatrix} e ^ { - \tau _ 2 } & - e ^ { - \tau _ 2 } \ , \tau _ 1 \\ 0 & 1 \\ \end{pmatrix} \end{align*}


\begin{align*} = \frac { p ( s + t , X ( t , s , x ) ) \partial _ t L _ X ( t , s , x ) } { 1 + p ( s + t , X ( t , s , x ) ) L _ X ( t , s , x ) } - \frac { L _ X ( t , s , x ) p ( s + t , X ( t , s , x ) ) u ( s + t , X ( t , s , x ) ) } { 1 + p ( s + t , X ( t , s , x ) ) L _ X ( t , s , x ) } \end{align*}


\begin{align*} K _ { p , \varphi _ p } ( z ) ^ { 1 / p } = f ( z ) \leq { | f ( w ) | + C _ S | z - w | } \leq { K _ { p , \varphi _ p } ( w ) ^ { 1 / p } } + C _ S | z - w | , \end{align*}


\begin{align*} \sup _ { \xi \in Q } \left \langle a + \frac { \xi } { L } \right \rangle ^ { - \alpha } = \left ( \inf _ { \xi \in Q } \left \langle a + \frac { \xi } { L } \right \rangle \right ) ^ { - \alpha } \leq \left ( c _ d \sup _ { \xi \in Q } \left \langle a + \frac { \xi } { L } \right \rangle \right ) ^ { - \alpha } = c _ d ^ { - \alpha } \inf _ { \xi \in Q } \left \langle a + \frac { \xi } { L } \right \rangle ^ { - \alpha } . \end{align*}


\begin{align*} \lim _ { i \to \infty } \int _ { X } \int _ { X } \frac { | f _ 0 ( x ) - f _ 0 ( y ) | } { | x - y | } \rho _ i ( x , y ) \ , d \mu ( x ) \ , d \mu ( y ) = \Vert D f _ 0 \Vert ( X ) , \end{align*}


\begin{align*} \begin{cases} y ' ( t ) & = - 2 \mu y ( t ) + \frac { \sigma _ { \infty } ^ 2 ( t ) } { 2 } , t > 0 \\ y ( 0 ) & = \frac { \norm { X _ 0 - x ^ { \star } } ^ 2 } { 2 } . \end{cases} \end{align*}


\begin{align*} ( S \otimes _ R M ) _ { 1 \otimes \mu } \subset S \otimes _ R M = \sum _ { \theta \in [ \l ] } S \otimes _ R M _ \theta \subset \sum _ { \theta \in [ \l ] } ( S \otimes _ R M ) _ { 1 \otimes \theta } . \end{align*}


\begin{align*} ( p _ a \pi _ * \underline { \Q } ) _ y = \Gamma ( \pi ^ { - 1 } ( y ) , \Q ) _ A = 0 \end{align*}


\begin{align*} \left \{ \begin{alignedat} { 2 } - \Delta _ p u = u ^ { \alpha _ 1 } \ , v ^ { \beta _ 1 } \ ; \ ; & \mbox { i n $ \Omega $ , } \ ; \ ; u > 0 \ ; \ ; & & \mbox { i n $ \Omega $ , } \ ; \ ; u \lfloor _ { \partial \Omega } = 0 , \\ - \Delta _ q v = v ^ { \alpha _ 2 } \ , u ^ { \beta _ 2 } \ ; \ ; & \mbox { i n $ \Omega $ , } \ ; \ ; v > 0 \ ; \ ; & & \mbox { i n $ \Omega $ , } \ ; \ ; v \lfloor _ { \partial \Omega } = 0 . \end{alignedat} \right . \end{align*}


\begin{align*} a = ( n - 2 ) s . \end{align*}


\begin{align*} m _ 0 ( \zeta ) = \left ( I + m ^ { p c } _ { - k _ 0 , 1 } \zeta ^ { - 1 } + \mathcal { O } ( \zeta ^ { - 2 } ) \right ) \zeta ^ { i \nu \sigma _ 3 } e ^ { - \frac { i } { 4 } \zeta ^ 2 \sigma _ 3 } , { \rm a s } \zeta \rightarrow \infty . \end{align*}


\begin{align*} u ( t ) ( x ) : = & u ( t , x ) , \quad \left ( u ( t ) \log | u ( t ) | \right ) ( x ) : = u ( t , x ) \log | u ( t , x ) | , \\ \sigma ( u ( t ) ) ( x ) : = & \sigma ( u ( t , x ) ) . \end{align*}


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


\begin{align*} 2 \lambda _ 1 \int _ { 0 } ^ { T } \int _ { D } \partial _ x ( u _ r \times ( A u _ r + b ) ) \cdot \partial _ x u _ r \dd x \dd r = 2 \lambda _ 1 \int _ { 0 } ^ { T } \int _ { D } u _ r \times A \partial _ x u _ r \cdot \partial _ x u _ r \dd x \dd r \\ \quad \leq 2 | \lambda _ 1 | \| A \| _ { L ^ \infty } \int _ { 0 } ^ { T } \| \partial _ x u _ r \| ^ 2 _ { L ^ 2 } \dd r \ , . \end{align*}


\begin{align*} H ( J ^ { \{ u _ \gamma \} } _ { 0 } J ^ { \{ u ^ { \gamma ' } \} } _ { 0 } v _ { \nu , \ell _ 0 } , v _ { \nu , \ell _ 0 } ) \ne 0 \Rightarrow \gamma = \gamma ' . \end{align*}


\begin{align*} X _ 1 \cdots X _ n a = \sum _ { \substack { t + s = n \\ \sigma \in W _ { t , s } } } \omega \left ( X _ { \sigma ( 1 ) } \right ) \cdots \omega \left ( X _ { \sigma ( t ) } \right ) ( a ) X _ { \sigma ( t + 1 ) } \cdots X _ { \sigma ( n ) } , \end{align*}


\begin{align*} \lim _ { \delta \to 0 } \lim _ { N \to \infty } \frac { 1 } { N } \log \left ( \mathfrak { Q } _ { N } ( B ( \overline { \mathbf { P } } , \delta ) ) \right ) = - \overline { { \rm E n t } } [ \overline { \mathbf { P } } | \overline { \mathbf { \Pi } } ^ { \phi } ] . \end{align*}


\begin{align*} \langle \mathbf { x } , \mathbf { y } \rangle _ e = \sum _ { i = 1 } ^ { n } x _ i y _ i ^ { p ^ e } , \ { \rm { w h e r e } } \ 0 \leq e \leq h - 1 . \end{align*}


\begin{align*} m ( \pi _ { \rm s s } , \bar \sigma _ { s , a } ) = m ( \pi _ { \rm s s } , I ( \tau , \bar \sigma _ { s , a } ) ) = \frac { 1 } { 2 } \left ( m ( R ^ G _ { T , s } , R ^ G _ { S , s ' _ a } ) + m ( R ^ G _ { T _ { \rm a } , s } , R ^ G _ { S , s ' _ a } ) \right ) . \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*} A ( x , y ; \xi , \zeta ) = \int _ 0 ^ 1 \int _ 0 ^ 1 | y - \zeta + s ( \zeta - \xi ) + t ( x - y ) | ^ { p - 2 } D ^ 2 h \left ( \frac { y - \zeta + s ( \zeta - \xi ) + t ( x - y ) } { | y - \zeta + s ( \zeta - \xi ) + t ( x - y ) | } \right ) d t \ , d s \end{align*}


\begin{align*} n ( i + q ) - m ( j + q ) = n ( i + q ) + n ( - i - 2 q + q ) = n ( i + q ) - n ( i + q ) = 0 , \end{align*}


\begin{align*} m \bigl ( \partial _ t u _ { \delta } ( t ) \bigr ) = 0 , m _ \Gamma \bigl ( \partial _ t v _ { \delta , \lambda } ( t ) \bigr ) = 0 \end{align*}


\begin{align*} | ( \tilde \Delta - \lambda _ s ) v | _ 2 ^ 2 = | ( \tilde \Delta - c ) v | _ 2 ^ 2 - i \cdot d \langle ( \tilde \Delta - c ) v , v \rangle _ 2 + i \cdot d \langle v , ( \tilde \Delta - c ) \rangle _ 2 + d ^ 2 | v | ^ 2 { \geq } d ^ 2 | v | ^ 2 . \end{align*}


\begin{align*} \left \{ \begin{array} { c c } \dot { x _ { k } } ( t ) = \xi _ { k } ( t ) , & x _ { k } ( 0 ) = x _ { k } ^ { i n } , \\ \dot { \xi _ { k } } ( t ) = - \frac { 1 } { \epsilon } \left ( \xi _ { k } ^ { \bot } + \frac { 1 } { N } \underset { j : j \neq k } { \sum } \nabla V ( x _ { k } ( t ) - x _ { j } ( t ) ) \right ) , & \xi _ { k } ( 0 ) = \xi _ { k } ^ { i n } , \end{array} \right . 1 \leq k \leq N \end{align*}


\begin{align*} I ( \tau , \bar \sigma _ { s , a } ) = R ^ G _ { S , s ' _ a } , \end{align*}


\begin{align*} x ^ { ' 1 } = \Re \frac { z ' - z _ 0 } { \partial _ { \gamma } \eta ' _ i ( l _ 0 ) } , x ^ 2 = \Im \frac { z ' - \eta ' _ i \circ \tilde { l } ^ { ' - 1 } ( x ^ 1 ) } { \partial _ { \gamma } \eta ' _ i ( l _ 0 ) } \end{align*}


\begin{align*} \delta _ p : = 2 ( n - m + k ) \delta _ Y - \epsilon p \alpha _ Y . \end{align*}


\begin{align*} P ( z ) = 2 \biggl [ S O ( z ) - \frac { z } { 2 } \biggr ] + \frac { z } { 2 } = 1 - \sqrt { 1 - z } , \end{align*}


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


\begin{align*} \begin{aligned} \omega _ j & = \sqrt { | j | _ g ^ 4 + \delta | j | _ g ^ 2 } = \sqrt { \beta ^ 4 | j | _ { \bar g } ^ 4 + \beta ^ 2 \delta | j | _ { \bar g } ^ 2 } = \beta ^ 2 \Omega _ j \ , , \\ \Omega _ j & : = | j | _ { \bar g } \sqrt { | j | _ { \bar g } ^ 2 + \frac { \delta } { \beta ^ 2 } } \ , , j \in \Z ^ d \setminus \{ 0 \} \ , . \end{aligned} \end{align*}


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


\begin{align*} ( \psi _ n ) _ * \hat { \nu } ^ c _ p = c _ n \hat { \nu } ^ c _ p , \end{align*}


\begin{align*} s _ \lambda ( B V ^ { - 1 } A C V D ) = s _ \lambda ( D B V ^ { - 1 } A C V ) , \end{align*}


\begin{align*} m _ { j \ , \pi _ 0 ( k ) } ^ { \omega } ( t ) \in E \ , ( j , \pi _ 0 ( k ) ) \in \mathcal { R } _ { E } , E = A , B , C , D . \end{align*}


\begin{align*} \left \{ \begin{aligned} & u _ t - F ( D ^ 2 u , x , t ) = 1 ~ ~ ~ ~ & & \mbox { i n } ~ ~ \Omega \cap Q _ 1 ; \\ & u = | D u | = 0 ~ ~ ~ ~ & & \mbox { o n } ~ ~ \partial \Omega \cap Q _ 1 . \\ \end{aligned} \right . \end{align*}


\begin{align*} H _ { \infty } = h _ { \alpha } + H _ f - 2 \alpha ^ { 1 / 2 } \mathrm { R e } ( P A _ { \infty } ( x ) ) + \alpha A ^ 2 _ { \infty } ( x ) , \end{align*}


\begin{align*} | u ( y ) | ^ { p - 1 } \leq \begin{cases} K _ 1 \ , \Delta ^ { ( p - 1 ) / ( n + p - 1 ) } & \\ K _ 2 \ , R ^ { - n } \ , \Delta & \end{cases} \end{align*}


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


\begin{align*} g \cdot \mu _ { s , t } = \lim _ { N \to \infty } ( g \cdot \mu _ { s , t } ) ^ { ( N ) } \end{align*}


\begin{align*} \begin{array} { r l } \phi = & g _ 1 ( z ) + x ^ 0 g _ 2 ( z ) + s g _ 3 ( z ) + z g _ 4 ( x ^ 0 ) + s g _ 5 ( r ) \\ & \\ & + \displaystyle \int ^ s _ 0 \left ( \int ^ \eta _ 0 g _ 6 ( r ^ 2 - \xi ^ 2 ) d \xi \right ) d \eta + \int ^ r _ 0 \xi g _ 6 ( \xi ^ 2 ) d \xi , \end{array} \end{align*}


\begin{align*} \Delta _ { \phi ( J ) | \Delta _ K ^ c \ne 0 } = \bigcup _ { a \in J } \{ t \in \Delta _ K ^ c : \ , \phi ( a ) ( t ) \ne 0 \} . \end{align*}


\begin{align*} = \hat \mu _ 1 ( s _ 1 - s _ 2 , h _ 1 - h _ 2 ) \hat \mu _ 2 ( s _ 1 - a s _ 2 , h _ 1 - \tilde \alpha _ { G } h _ 2 ) , \ \ s _ j \in \mathbb { R } , \ \ h _ j \in H . \end{align*}


\begin{align*} O ( X ) \subseteq \bigcup _ { j = 1 } ^ n U _ j \ , . \end{align*}


\begin{align*} \sigma _ a \sigma _ 0 ^ { - 1 } ( b ) = \begin{cases} b , \quad { \rm i f } \ ; a \ ; { \rm i s \ ; e v e n , } \\ ( 2 ^ { m - 1 } - 2 ) + b , \quad { \rm i f } \ ; a \ ; { \rm i s \ ; o d d } \end{cases} \end{align*}


\begin{align*} K = K _ 1 \sqcup K _ 2 \sqcup \cdots \sqcup K _ m . \end{align*}


\begin{align*} 0 = E ^ 2 _ { q + 2 , 0 } \rightarrow E ^ 2 _ { 0 , q + 1 } \rightarrow H _ { q + 1 } = 0 . \end{align*}


\begin{align*} \delta ( \langle u \rangle - & B ) ^ 2 _ { s , t } = - \int _ s ^ t \left [ \langle \lambda _ 2 u _ r \times [ u _ r \times \partial _ x ^ 2 u _ r ] - \lambda _ 1 u _ r \times \partial _ x ^ 2 u _ r \rangle \right ] \cdot ( \langle u _ r \rangle - B _ r ) \dd r \ , . \end{align*}


\begin{align*} R _ { 2 , 2 } ^ { ( d ) } ( s , \boldsymbol { \alpha } , N ) = \frac { 1 } { N } \displaystyle \sum _ { 1 \leq m \neq n \leq N } I _ { s , N } ( \boldsymbol { \alpha } ( \boldsymbol { a } _ m - \boldsymbol { a } _ n ) ) , \end{align*}


\begin{align*} \mathbf { R } _ { I W , b } = d i a g ( \mathbf { \hat { R } } _ { D , b } ) \end{align*}


\begin{align*} \widetilde { \eta } : = ( \eta \circ { F _ a ^ { - 1 } } ) | ( F _ a ^ { - 1 } ) ' | ^ 2 \in { A ^ 2 ( \mathbb { D } ) ^ \perp } . \end{align*}


\begin{align*} \begin{aligned} \int _ { 0 } ^ { \pi } [ r _ { n } ^ { 2 } \omega ( r _ { n } , \theta ) ^ { 2 } + 2 r _ { n } | \omega ( r _ { n } , \theta ) \omega _ { \theta } ( r _ { n } , \theta ) | ] d \theta \leq \frac { 1 } { \log 2 } \int _ { r > 2 ^ { n } , 0 < \theta < \pi } ( 2 \omega ^ { 2 } + | \nabla \omega | ^ { 2 } ) d x d y . \end{aligned} \end{align*}


\begin{align*} E _ { N \overline { k } } = \exp \sum _ { g = 0 } ^ k N ^ { 2 - 2 g } F _ { N g } . \end{align*}


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


\begin{align*} \alpha ^ { g _ \theta | _ U } ( X , Y ) = T _ \theta ^ U \circ \alpha ^ { g | _ U } ( J _ \theta X , Y ) , \ \ X , Y \in T M . \end{align*}


\begin{align*} \min _ { \mathbf { x } \in \mathcal { X } , \boldsymbol { \theta } _ t , \boldsymbol { \theta } _ r } & F ( \mathbf { x } , \boldsymbol { \theta } _ t , \boldsymbol { \theta } _ r ) \\ [ - 0 . 5 e m ] \mathrm { s . 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*} P ( \hat { A } _ n ^ c ) : = P ( Q _ n \ge ( \log n ) ^ 3 ) \le C _ 0 ( \log n ) ^ { - 2 } . \end{align*}


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


\begin{align*} T _ { \kappa _ N , p } ^ Y g : = B _ { k , p } ^ Y ( \kappa _ N \cdot B _ { k , p } ^ Y g ) . \end{align*}


\begin{align*} \mathcal { R } _ { N } ^ { ( n ) } : = B _ N ( Q _ N w _ n , Q _ N w _ n ) + D B _ N ( u ) Q _ N w _ n = g _ n . \end{align*}


\begin{align*} m ( \pi _ { \rm s s } , R ^ G _ { S , s ' } ) = 0 , \end{align*}


\begin{align*} p ( z , t ) \in U ( k ) = \left \{ w \in \mathbb H : \ ; \left | \frac { w - 1 } { w + 1 } \right | \leq k \right \} \end{align*}


\begin{align*} \begin{aligned} \sup _ { r \in [ 0 , t ] } \mathbb { E } \left [ \| \partial _ x u _ r \| ^ 2 _ { L ^ 2 } \right ] + \frac { 3 \lambda _ 2 } { 2 } \int _ { 0 } ^ { t } & \mathbb { E } \left [ \| u _ r \times \partial ^ 2 _ x u _ r \| ^ 2 _ { L ^ 2 } \right ] \dd r \\ & \quad \quad \quad \leq \mathbb { E } \left [ \| \partial _ x u ^ 0 \| _ { L ^ 2 } ^ 2 \right ] + t \left [ \| \partial _ x h \| ^ 2 _ { L ^ 2 } + \| g \| ^ 2 _ { L ^ \infty } C ( \lambda _ 1 , \lambda _ 2 ) \right ] \ , . \end{aligned} \end{align*}


\begin{align*} L \cdot = ( 2 w \partial _ { \eta } ^ 2 w ) \cdot - \partial _ \tau \cdot - \eta \partial _ \xi \cdot + w ^ 2 \partial _ { \eta } ^ 2 \cdot , \end{align*}


\begin{align*} \cap _ { n = 1 } ^ { \infty } A _ n ^ c & = \left \{ \omega \in \Omega : X _ n ( \omega ) \in B _ n ^ c \forall n \in \mathbb { N } \right \} \\ & = \left \{ \omega \in \Omega : \phi ^ { - 1 } \left ( ( X _ n ( \omega ) ) _ { n = 1 } ^ { \infty } \right ) \in \phi ^ { - 1 } \left ( \times _ { n = 1 } ^ { \infty } B _ n ^ c \right ) \right \} \\ & = X ^ { - 1 } ( B ) \end{align*}


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


\begin{align*} \mathcal H _ 1 = \mathcal H _ 1 ( \norm { \xi } { H ^ 2 } , \norm { v } { H ^ 2 } ) , ~ \mathcal H _ 2 = \mathcal H _ 2 ( \norm { \xi } { H ^ 2 } , \norm { v } { H ^ 2 } ) , \end{align*}


\begin{align*} ( x , y , z ) ( x ' , y ' , z ' ) = \left ( x + x ' , y + y ' , z + z ' + \frac { x y ' - x ' y } { 2 } \right ) \end{align*}


\begin{align*} \| \frac { 1 } { N } ( \upsilon _ N \ast \rho ^ \mathrm { p a i r } [ n + 1 ] ) \| _ { L ^ 2 } & \le \frac { g C } { 2 c } \frac { N ^ { 3 \beta } } { N } \cdot \Big ( \Big \| \Theta [ n - 1 ] \Big \| _ \mathrm { H S } + \mu [ n - 1 ] \Big ) ^ { 3 / 4 } \\ & \le \frac { g C ^ 2 } { c } \Big ( \frac { N ^ { 6 \beta } } { N } \Big ) . \end{align*}


\begin{align*} ( v | \Psi ) _ { W , a } ( z , r ) : = \bigg ( \frac { 1 } { | W _ a ( z , r ) | } \iint _ { W _ a ( z , r ) } | v | ^ 2 \Psi d y d s \bigg ) ^ { 1 / 2 } . \end{align*}


\begin{align*} ( \alpha ( u , \alpha ( s , x ) ) , \alpha ( 0 , x ) ) = ( \alpha ( u + s , x ) , x ) \in U \end{align*}


\begin{align*} y + \mathbb { K } y = \psi , \end{align*}


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


\begin{align*} \begin{aligned} & \| P _ { \lambda _ 0 } ( \varphi _ { \lambda _ 1 , N _ 1 } ^ \dagger \phi _ { \lambda _ 2 , N _ 2 } ) \| _ { L ^ 2 _ t L ^ 2 _ x } \\ & \lesssim \lambda _ 0 \left ( \frac { \min \{ \lambda _ 0 , \lambda _ 1 , \lambda _ 2 \} } { \max \{ \lambda _ 0 , \lambda _ 1 , \lambda _ 2 \} } \right ) ^ \delta ( \min \{ N _ 1 , N _ 2 \} ) ^ { 1 - \eta } \| \varphi _ { \lambda _ 1 , N _ 1 } \| _ { U ^ 2 _ { \theta _ 1 } } \| \phi _ { \lambda _ 2 , N _ 2 } \| _ { U ^ 2 _ { \theta _ 2 } } . \end{aligned} \end{align*}


\begin{align*} \lim _ { m \to \infty } B _ m ( \theta _ 1 , x + i y ) : = \log \left | 2 \sin \left ( \frac { \pi \theta _ 1 } { 2 } + \frac { y } { 2 } + \frac { x i } { 2 } \right ) \right | - \frac { 1 } { 2 } | x | . \end{align*}


\begin{align*} J ( w ) = \frac { h _ d ^ 2 } { - \lambda _ w \cdot \langle 1 , 1 \rangle } + \frac { 1 } { 4 \pi i } \int _ { ( 1 / 2 ) } \left | \frac { \zeta _ k ( s ) } { \zeta ( 2 s ) } \right | ^ 2 - \left | \frac { \zeta _ k ( w ) } { \zeta ( 2 w ) } \right | ^ 2 \frac { d s } { \lambda _ s - \lambda _ w } , \end{align*}


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


\begin{align*} h _ { \lambda } ( n ) = \begin{cases} \lambda n + 1 & \\ n / 2 & \end{cases} \end{align*}


\begin{align*} \frac { C _ 1 } { D _ 1 } = - \ , \frac { \prod _ { j > 1 } D _ j } { \prod _ { i > 1 } C _ i } \end{align*}


\begin{align*} \psi _ { j , E _ 0 } = \lim _ { x \rightarrow \infty } \lim _ { k \rightarrow \infty } \psi _ j = \lim _ { k \rightarrow \infty } \lim _ { x \rightarrow \infty } \psi _ j = \lim _ { k \rightarrow \infty } N _ { \pm } ( k ) = I , \end{align*}


\begin{align*} C _ c ( \Q _ q , \Z ) ^ { \Gamma _ k } \to C _ c ( \Q _ q , \Z ) ^ { \Gamma _ { k + 1 } } , f \mapsto \sum _ { i = 0 } ^ { N - 1 } \tau _ { \frac { i } { N ^ { k + 1 } } } f , \end{align*}


\begin{align*} \Phi ( t ) = \exp \left ( \frac { t } { \log ^ + t } \right ) - 1 , \end{align*}


\begin{align*} X ( t , s , x ) = \begin{cases} x & x \le 0 , \\ \exp \left ( - e \left ( \frac { 1 } { e } \log \frac { 1 } { x } \right ) ^ { k ( t , s ) } \right ) & 0 < x < e ^ { - e } , \\ x & x \ge e ^ { - e } , \end{cases} \end{align*}


\begin{align*} - \Delta _ { p _ { i } } \xi _ { i , \delta } ( x ) = \left \{ \begin{array} { l l } 1 & \Omega \backslash \overline { \Omega } _ { \delta } \\ - 1 & \Omega _ { \delta } \end{array} \right . , \xi _ { i , \delta } ( x ) = 0 \partial \Omega , \end{align*}


\begin{align*} W _ \mathcal { A } ( O p _ { w , 2 d } ( a ) f , g ) = O p _ { w , 4 d } ( b ) W _ \mathcal { A } ( f , g ) , \\ W _ \mathcal { A } ( f , O p _ { w , 2 d } ( a ) g ) = O p _ { w , 4 d } ( \tilde b ) W _ \mathcal { A } ( f , g ) , \\ W _ \mathcal { A } ( O p _ { w , 2 d } ( a ) f ) = O p _ { w , 4 d } ( c ) W _ \mathcal { A } ( f ) . \end{align*}


\begin{align*} \frac { d } { d t } \int f ( x ) d \mu _ t ( x ) = & \int \frac { d } { d t } f ( \exp _ x ( t v ( x ) ) ) d \mu _ 0 ( x ) \\ & = \int \langle \nabla f ( \exp _ x ( t v ( x ) ) ) , \Pi _ { t , v ( x ) } ( v ( x ) ) \rangle _ { \exp _ x ( t v ( x ) ) } d \mu _ 0 ( x ) , \end{align*}


\begin{align*} \| f \| _ { \alpha , \beta } = \sup _ { 0 < \omega < 1 } \omega ^ { \alpha } | f ( \omega ) | + \sup _ { \omega > 1 } \omega ^ { \beta } | f ( \omega ) | \end{align*}


\begin{align*} \langle u , v \rangle _ { x ^ \theta L ^ 2 ( \mathbb R _ + , d _ + x ) } = \langle x ^ { - \theta } u , x ^ { - \theta } v \rangle _ { L ^ 2 ( \mathbb R _ + , d _ + x ) } . \end{align*}


\begin{align*} \int _ { \R ^ d } B ( s , x ) \nabla \partial _ x ^ \alpha u ( s ) \cdot \overline { \nabla g } - B ( s , x ) \nabla u ( s ) \cdot \overline { \nabla \partial _ x ^ \alpha g } \d x = \int _ { \R ^ d } \Bigl [ B ( s , x ) , \partial _ x ^ \alpha \Bigr ] \nabla u ( s ) \cdot \overline { \nabla g } \d x . \end{align*}


\begin{align*} \omega '^ { a } = \theta ^ { a } + \sum _ { i = 1 } ^ { 3 N - 6 } \wedge '^ { a } _ { i } d q ^ { i } \end{align*}


\begin{align*} m ( \pi _ { \rm s s } , \sigma ) = 0 . \end{align*}


\begin{align*} K _ { q + 1 } \subseteq N _ { \delta _ { q + 2 } ^ \frac 1 2 } K _ { q } \subseteq \cdots \subseteq N _ { \sum \limits _ { j = 2 } ^ { q + 2 } \delta _ { j } ^ \frac 1 2 } K _ { 0 } . \end{align*}


\begin{align*} \bar r = 2 r - 1 \ , \end{align*}


\begin{align*} L _ z ( f ) = f ( z ) = \int _ \Omega { f } \overline { g } _ z , \ \ \ \forall \ , f \in { A ^ p ( \Omega ) } . \end{align*}


\begin{align*} x = ( x ^ 0 , \overline { x } ) , \ ; \overline { x } = ( x ^ 1 , \ldots , x ^ n ) , \ ; y = ( y ^ 0 , \overline { y } ) \ ; \mbox { a n d } \overline { y } = ( y ^ 1 , \ldots , y ^ n ) . \end{align*}


\begin{align*} P _ D ( \lambda ) = ( \lambda + 2 ) ^ { n - k } \left [ \prod _ { i = 1 } ^ k ( \lambda - n _ i + 2 ) - \sum _ { i = 1 } ^ k n _ i \prod _ { j = 1 , j \neq i } ^ k ( \lambda - n _ j + 2 ) \right ] . \end{align*}


\begin{align*} V ^ { \varepsilon } ( t , x , y ) : = \sup \limits _ { u \in \mathcal { U } } J ( t , x , y , u ) , \ ; \eqref { d y n a m i c s } \ , . \end{align*}


\begin{align*} \lim _ { x \rightarrow \infty } \frac { \# \{ \ell < x : a _ { \ell } ( f ) \equiv a \pmod { p } \} } { \pi ( x ) } = \frac { p ^ 2 - p - 1 } { ( p - 1 ) ^ 2 ( p + 1 ) } . \end{align*}


\begin{align*} 0 = E _ d \subset E _ { d - 1 } \subset \cdots \subset E _ 0 = E ' , \end{align*}


\begin{align*} \tilde b = \ , b I \times B ( o , R ) \ , . \end{align*}


\begin{align*} & c h \ , M ^ W ( \widehat \nu _ h ) = e ^ \nu q ^ { \ell ( h ) } F ^ { N S } ( q ) , \end{align*}


\begin{align*} \mathbb { E } ( \langle D Y , D Y \rangle _ \mathcal { H } ) & = \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 1 } g _ s g _ t d \mathbb { E } \langle D _ r X _ s , D _ r X _ t \rangle \\ & = n \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 1 } g _ s g _ t d \mathbb { E } ( X _ s X _ t ) . \end{align*}


\begin{align*} \gamma _ { A , B } ( ( a , b ) ) = ( b , a ) \ ; . \end{align*}


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


\begin{align*} k = f + u \ , . \end{align*}


\begin{align*} \ , X ( t , 0 , \cdot ) _ \# ( \mathcal L ^ n ) = \ , J _ { X ( 0 , t , \cdot ) } \mathcal L ^ n \ , . \end{align*}


\begin{align*} \frac { \partial \overline { H } _ 5 } { \partial \overline { z } } = 3 i \overline { H } _ 5 \omega _ { 1 2 } ( \overline { \partial } ) + \overline { H } _ 6 \omega _ { 5 6 } ( \overline { \partial } ) \end{align*}


\begin{align*} ( - i \partial _ t + \theta | \nabla | ) u _ \theta = \theta \frac { 1 } { 2 | \nabla | } F , \end{align*}


\begin{align*} \left | j ^ { - 1 } _ { G _ \jmath } ( { \rm c l } ( S , G ) ) \right | = 1 \quad { \rm a n d } \left | j ^ { - 1 } _ { G _ \jmath } ( { \rm c l } ( T , G ) ) \right | \leq 1 , \end{align*}


\begin{align*} \partial ^ \circ \eta ( \Phi _ \nu ( Y , t ) , t ) = \frac { \partial } { \partial t } \Bigl ( \eta ( \Phi _ \nu ( Y , t ) , t ) \Bigr ) , ( Y , t ) \in \Gamma _ 0 \times [ 0 , T ] , \end{align*}


\begin{align*} \frac { 1 } { 2 \pi i j + z } + \frac { 1 } { - 2 \pi i j + z } = \frac { 2 z } { ( 2 \pi j ) ^ 2 + z ^ 2 } \end{align*}


\begin{align*} e ^ { g ( x ) } = \frac { A } { \pi x } . \end{align*}


\begin{align*} S _ \ell ^ { ( k ) } = \sum _ { i = 0 } ^ \ell F _ { i } ^ { ( k ) } . \end{align*}


\begin{align*} \bar { g } ' _ j ( x ) = - f _ i ( X ^ x Z ^ { g _ j ( x ) } ) + \sigma x = \sigma x - f _ i ( W ^ x Z ^ { \bar { g } _ j ( x ) } ) . \end{align*}


\begin{gather*} E _ 2 ^ { 0 , q } = \begin{cases} \Q ( 0 ) & q = 2 j , \ , \ , 0 \leq j \leq 2 ( N - 1 ) \\ 0 & \end{cases} \end{gather*}


\begin{align*} 0 + B B ^ * w - B v = 0 . \end{align*}


\begin{align*} \lim _ { N \to \infty } \frac { \log \left ( K _ { N , \beta } \right ) } { N } = 0 . \end{align*}


\begin{align*} \sigma ( A ) ( ( x , y ) ) = A ( ( x , \bar { \Phi } _ \times ^ \sigma ( y ) ) ) \ ; , \end{align*}


\begin{align*} s = ( s _ { j 1 } , \ldots , s _ { j \lambda _ j } ; s ' _ { j 1 } , \ldots , s ' _ { j \lambda _ j ' } ) . \end{align*}


\begin{align*} D _ { 2 m } ( s , t ) : = \{ s _ 1 < \dots < s _ { 2 m } : s _ j \in [ s , t ] 1 \le j \le 2 m \} . \end{align*}


\begin{align*} f ( x , t ) : = f _ N ( x , t ) - \frac { t } { \beta } \log m ( N ^ { - 1 / 4 } \beta ) \end{align*}


\begin{align*} \langle j x , x ^ \odot \rangle : = \langle x ^ \odot , x \rangle , \forall x \in X , \ x ^ \odot \in X ^ \odot , \end{align*}


\begin{align*} & \psi _ { j , x } + i k [ \sigma _ 3 , \psi _ j ] = U \psi _ { j } \\ & \psi _ { j , t } + 4 i k ^ 3 [ \sigma _ 3 , \psi _ j ] = V \psi _ { j } , \end{align*}


\begin{align*} F ( ( x ^ 0 , { O } \overline { x } ) , ( y ^ 0 , { O } \overline { y } ) ) = F ( ( x ^ 0 , \overline { x } ) , ( y ^ 0 , \overline { y } ) ) \end{align*}


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


\begin{align*} \chi ( ( b , k ) ) = ( b + \sum _ { \ell = 1 } ^ { s } p _ \ell \cdot c _ { i _ \ell } + \overline { h } , k ) = ( b - a + \overline { h } , k ) \asymp ( b - a , k ) . \end{align*}


\begin{align*} b _ { i , j } ( A ) = b _ { c - i , r + n - j } ( A ) \end{align*}


\begin{align*} M _ { i j } v _ k = v _ k M _ { i j } \end{align*}


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


\begin{align*} x \to y : = \left \{ \begin{array} { l } x * y \mbox { i f } y \le x , \\ 1 _ y \mbox { o t h e r w i s e } . \end{array} \right . \end{align*}


\begin{align*} R [ I t , t ^ { - 1 } ] = \bigoplus _ { n = - \infty } ^ { \infty } I ^ n t ^ n \subset R [ t , t ^ { - 1 } ] , \end{align*}


\begin{align*} \begin{aligned} & g \{ k _ d ^ { - 1 } \partial ^ \circ \eta _ 0 + k _ d ^ { - 2 } V _ \Gamma ^ 2 \eta _ 0 - \Delta _ \Gamma \eta _ 0 - k _ d ^ { - 1 } V _ \Gamma H \eta _ 0 - k _ d ^ { - 1 } f \} \\ & = \nabla _ \Gamma g \cdot \nabla _ \Gamma \eta _ 0 - k _ d ^ { - 1 } ( \partial ^ \circ g ) \eta _ 0 + k _ d ^ { - 2 } g V _ \Gamma ^ 2 \eta _ 0 , \end{aligned} \end{align*}


\begin{align*} v - w = \sum _ { \mu \in \tilde \Lambda ^ { \tau , \pm } _ { \beta } } A _ \mu ( x , z , y ) , \end{align*}


\begin{align*} \hat { S } ( \boldsymbol { \ell } ) _ t ^ { - 1 } = t , \hat { S } ( \boldsymbol { \ell } ) _ t ^ 1 = S ( \boldsymbol { \ell } ) _ t . \end{align*}


\begin{align*} F _ w ( Z ) = \sum _ T z ^ T \end{align*}


\begin{align*} \prod _ { u \le p < z } \left ( 1 - \frac { 1 } { p } \right ) ^ { - 1 } = \exp \left ( - \sum _ { u \le p < z } \log \left ( 1 - \frac { 1 } { p } \right ) \right ) . \end{align*}


\begin{align*} l _ { n } \left ( z \right ) = \gamma _ { n } \left ( z \right ) + \gamma _ { n + 1 } \left ( z \right ) - n - \frac { 1 } { 2 } . \end{align*}


\begin{align*} \max \{ u \ge y \colon u \perp _ y x \} & = \max \{ u \ge u \colon ( u ] \cap ( x ] \cap [ y ) \subseteq \{ y \} \} \\ & = \max \{ u \ge y \colon ( u ] \cap ( ( x ] \cup ( y ] ) \cap [ y ) \subseteq \{ y \} \} \\ & = \max \{ u \colon ( u ] \cap ( ( x ] \cup ( y ] ) \cap [ y ) = \{ y \} \} . \end{align*}


\begin{align*} Y ^ 2 Z + a _ 1 X Y Z + a _ 3 Y Z ^ 2 = X ^ 3 \ , . \end{align*}


\begin{align*} F \left ( \{ x _ { i , r } \} \right ) = 0 \ \ \mathrm { o n c e } \ \ x _ { i , 2 } = q ^ 2 x _ { i , 1 } \ \mathrm { a n d } \ x _ { i + \epsilon , 1 } = q d ^ { - \epsilon } x _ { i , 1 } \ \ \mathrm { f o r } \ \ i \in [ n ] \ , , \ \epsilon = \pm 1 \ , . \end{align*}


\begin{align*} \hat { f } _ \theta ( k - a ) = e ^ { - d \theta / 2 } \left ( Q _ \lambda ( e ^ { - \theta } k ) e ^ { - \frac { \pi } { \lambda } \frac { \delta } { 4 } | k | ^ 2 } \right ) \left ( e ^ { - \frac { \pi } { \lambda } e ^ { - 2 \theta } | k | ^ 2 } e ^ { \frac { \pi } { \lambda } \frac { \delta } { 2 } | k | ^ 2 } \right ) \left ( e ^ { - 2 \pi i e ^ { - \theta } k \cdot x _ 0 } e ^ { - \frac { \pi } { \lambda } \frac { \delta } { 4 } | k | ^ 2 } \right ) . \end{align*}


\begin{align*} \Box A _ j = - \textrm { I m } \ , \mathbf P ( \phi \overline { \mathcal D _ j \phi } ) . \end{align*}


\begin{align*} \mathcal E _ 0 ( R _ 0 , r _ 0 , a _ 0 ) \geq \mathcal E _ 0 ( \mathrm { i d } , r _ 0 , a _ 0 ) = T ( a _ 0 ) \left ( r _ 0 - { 4 \pi \over T ( a _ 0 ) } \right ) ^ 2 - { 1 6 \pi ^ 2 \over T ( a _ 0 ) } . \end{align*}


\begin{align*} \begin{aligned} Z _ ( f ) ~ = ~ \frac { V _ 1 ( f ) } { I _ 1 ( f ) } ~ & = ~ \underbrace { \frac { 1 } { \textrm { j } 2 \pi f \frac { a } { c R } } } _ { Z _ C ( f ) } + \underbrace { \frac { 1 } { \frac { 1 } { \textrm { j } 2 \pi f \frac { a R } { c } } + \frac { 1 } { R } } } _ { Z _ { \rm L } ( f ) \ , \parallel \ , Z _ R ( f ) } \\ & = ~ \frac { c ^ 2 R + \textrm { j } 2 \pi f c a R - ( 2 \pi f a ) ^ 2 R } { \textrm { j } 2 \pi f c a - ( 2 \pi f a ) ^ 2 } \ , \ , [ \Omega ] . \end{aligned} \end{align*}


\begin{align*} E _ y = \langle \Psi _ y \ , | \ , H _ y \ , | \ , \Psi _ y \rangle . \end{align*}


\begin{align*} \lambda ( p ) & = \tau ^ 4 ( p ) + c ( p ) \nu , \\ \lambda ( q ) & = \tau ^ 4 ( q ) + c ( q ) \nu . \end{align*}


\begin{align*} \int _ c ^ \infty \frac { 1 } { L ( s ) } d s = + \infty \forall c > 0 . \end{align*}


\begin{align*} \int _ { a } ^ { b } D [ A ( t ) x ( \tau ) ] = \int _ { a } ^ { b } { \rm d } [ A ( s ) ] x ( s ) . \end{align*}


\begin{align*} \exp \left ( \frac { \partial W } { \partial \ln X _ i } \right ) = 1 \ / , 1 \le i \le k \ / . \end{align*}


\begin{align*} P \otimes P ' = ( P , P ' ) \ ; , \end{align*}


\begin{align*} \Lambda _ { M _ 1 } + \Lambda _ { M _ 2 } = \Lambda _ N \quad N = ( M _ 1 \cup M _ 2 ) \smallsetminus ( M _ 1 \cap M _ 2 ) . \end{align*}


\begin{align*} \left \{ \begin{aligned} & v \in S ( \lambda , \Lambda , \tilde { f } ) & & ~ ~ \mbox { i n } ~ ~ \tilde { \Omega } \cap Q _ 1 ; \\ & v = \tilde { g } & & ~ ~ \mbox { o n } ~ ~ \partial \tilde { \Omega } \cap Q _ 1 , \end{aligned} \right . \end{align*}


\begin{align*} \rho _ 0 = z _ 1 \ , \exp \left ( - \frac { p } { 4 } \xi ( \abs { z _ 1 } ) \frac { d z ^ { 1 \bar 1 } } { \abs { z _ 1 } ^ 2 } - \frac { b } { 2 } \ , d z ^ { 2 \bar 2 } + \frac { d z ^ 1 } { 2 z _ 1 } \left [ ( \frac { a } { 2 } - q ) d z ^ 2 - ( \frac { a } { 2 } + q ) d z ^ { \bar 2 } \right ] \right ) \wedge e ^ { i \phi ^ * ( \omega _ \Sigma ) } . \end{align*}


\begin{align*} C = - \sum _ { i = 1 } ^ s \lambda _ i T _ { L _ { P _ i } } , \end{align*}


\begin{align*} \overline { x } \in \overset { m } { \underset { i = 1 } { \bigcap } } X _ i . \end{align*}


\begin{align*} T = \sum _ { i = 1 } ^ s \lambda _ i T _ { L _ { P _ i } } \end{align*}


\begin{align*} \mathcal { E } ( \phi ) : = \frac { 1 } { 2 } \int _ { D } \left [ | \partial _ x \phi ( x ) | ^ 2 + g ( \phi ( x ) ) \cdot \phi ( x ) \right ] \dd x \ , , \end{align*}


\begin{align*} \lim \mathcal { G } ( { \rm e m p } _ { N } ( X _ { N } ) - \mu , \mu - \nu ) = 0 . \end{align*}


\begin{align*} \lim _ { \epsilon \to 0 } \int _ { K } \int _ I \ell _ { A ^ \epsilon } ^ { \alpha } ( v , y ) f ( v , y ) \dd v \dd y = \int _ { K } \int _ I \ell _ \Omega ^ { \alpha } ( v , y ) f ( v , y ) \dd v \dd y . \end{align*}


\begin{align*} \begin{aligned} & \sup _ { 0 \leq t \leq T _ \varepsilon } \norm { v ( t ) } { H ^ 2 } ^ 2 + \dfrac { c _ 1 \min \lbrace \mu , \lambda , 1 \rbrace } { 2 } \int _ 0 ^ { T _ \varepsilon } \norm { \nabla v } { H ^ 2 } ^ 2 \ , d t \leq 4 ( M _ 1 + M _ 1 / 2 ) \\ & ~ ~ ~ ~ = 6 M _ 1 \leq \min \biggl \lbrace \dfrac { 1 } { 3 } , \dfrac { c _ 1 \min \lbrace \mu , \lambda , 1 \rbrace } { 6 } \biggr \rbrace C _ 1 M _ 1 . \end{aligned} \end{align*}


\begin{align*} & T _ 0 [ E + i \eta ; \psi _ { 0 , q } , \psi _ { 0 , q } ] = \int d u _ 0 \overline { \widehat { \psi } } _ { 0 , 0 } ( u _ 0 + q ) \widehat { \psi } _ { 0 , 0 } ( u _ 0 + q ) \frac { 1 } { u _ 0 ^ 2 - E - i \eta } \end{align*}


\begin{align*} \Sigma = \left \{ E \subset \R ^ n : T ( E ) \right \} \end{align*}


\begin{align*} \sum _ { j \ge 1 } j ^ p \sum _ { k \ge 1 } c _ k r ^ { j k } \cos ( \eta j k ) \sim A _ { 0 , 1 + p } ( r ) - R _ 0 + R _ 1 , \quad R _ i : = \sum _ { j , k \ge 1 , j k \ge \chi ^ { - 1 - \delta / 2 } } j ^ p c _ k r ^ { j k } \cos ( \eta j k ) ^ i , \ , i = 0 , 1 . \end{align*}


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


\begin{align*} \rho _ s ( x ) = ( x / e ) ^ { \sigma } \left ( 1 + \sigma / x \right ) ^ { x + \sigma } \exp \left ( F _ { x + \sigma } ( t ) + i \psi ( t ) \right ) \left ( 1 + O \left ( x ^ { - 1 } \right ) \right ) \end{align*}


\begin{align*} \hat { \chi } ( \xi , k ) = - \frac { 1 } { 2 \pi i } \int _ { ( - \infty , - k _ 0 ) \cup ( k _ 0 , + \infty ) } \log \left ( k - s \right ) d _ { s } \log \left ( 1 + r _ 1 \left ( s \right ) r _ 2 \left ( s \right ) \right ) . \end{align*}


\begin{align*} \Psi _ \alpha ( t ) : = \frac { \overline { Q ^ { G _ \iota } _ { \upsilon , u } ( t ) } Q ^ { G _ \iota } _ { \varsigma , u } ( t ) } { | \overline { N } _ G ( \iota , S ) ^ F | \ , | P _ { \iota , u } ( t ) | } \end{align*}


\begin{align*} 2 \dim A _ I / \dim G _ { A _ I , \ell } ^ \circ \leq ( 2 \dim A _ I ) / 2 = \dim A _ I \leq \dim A , \end{align*}


\begin{align*} B _ n ( \phi , F ) _ x = \Gamma _ c ( ( x / \phi ) ^ { ( n ) } , s ^ * \pi _ x ^ * F ) . \end{align*}


\begin{align*} \gamma = 1 , \ , \alpha = \nu - 1 / 2 , \ , \beta ^ 2 / \gamma = \nu ^ 2 - 1 / 4 . \end{align*}


\begin{align*} \tilde { y } ^ i : = \tilde { x } ^ i + \mathrm { 1 } _ { i \in \mathcal { O } } \mathbf { e } _ 1 + \mathrm { 1 } _ { i \in \mathcal { B } } \mathbf { e } _ 2 \end{align*}


\begin{align*} \lim _ { N \to \infty } \frac { 1 } { N \theta } \log \left ( \int _ { \mathbb { R } ^ { d \times N } } \exp \left ( - \beta \mathcal { H } ^ { * } _ { N } \right ) d \pi ^ { \otimes N } \right ) = \inf _ { \mu \in \mathcal { P } ( \mathbb { R } ^ { d } ) } \{ \mathcal { E } ^ { * } ( \mu ) + \frac { 1 } { \theta } { \rm e n t } [ \mu | \pi ] \} . \end{align*}


\begin{align*} \begin{array} { l } R : = \underset { i = 1 , 2 } { \max } \{ 1 , k _ { p _ { i } } \} , \end{array} \end{align*}


\begin{align*} G _ { \frac { \hbar } { 2 } } ^ { 2 d } \star _ { x , \xi } g = \underset { 0 \leq n \leq \frac { m } { 2 } } { \sum } \frac { \hbar ^ { n } } { 4 ^ { n } n ! } \Delta _ { x , \xi } ^ { n } g \end{align*}


\begin{align*} \chi ^ { - 1 } J _ 2 = \int _ 1 ^ \infty g ( t ) d t \sim \chi ^ { - ( \gamma + 1 ) } \ln ( \chi ^ { - 1 } ) \end{align*}


\begin{align*} u ( x , 0 ) = \varphi ( x ) , 0 \le x \le 1 , \end{align*}


\begin{align*} | v ( h ) \cdot h | \le | v ( j ) \cdot h | = | v ( j ) \cdot ( j + h - j ) | \le 2 | j | ^ { \mu } + 2 | j | ^ \delta \max ( | j | , | h | ) ^ \delta < 4 | h | ^ { \mu } \end{align*}


\begin{align*} \beta _ \varepsilon = \frac { ( - 1 ) ^ { i + 1 } k _ d ^ { - 1 } } { \sqrt { 1 + \varepsilon ^ 2 | \bar { \tau } _ \varepsilon ^ i | ^ 2 } } \left \{ \varepsilon \ , \overline { \partial ^ \circ g _ i } + \varepsilon ^ 2 \bar { g } _ i \bar { \tau } _ \varepsilon ^ i \cdot \Bigl ( \nabla \overline { V _ \Gamma } + \overline { \nabla _ \Gamma V _ \Gamma } \Bigr ) \right \} \quad \Gamma _ \varepsilon ^ i ( t ) \end{align*}


\begin{align*} \max \big \{ \max ( u ) : u \in G ( I ) _ \ell \big \} \ \ge \ \max ( u _ 0 ) \ \ge \ k + \textstyle \sum _ { j = 1 } ^ { \ell - 1 } t _ j + 1 . \end{align*}


\begin{align*} f = F ^ { \diamondsuit } \cdot \det ( f ^ { \partial } ) ^ { w ' } \end{align*}


\begin{align*} \pm w ^ 2 = \pm \big ( y ^ 2 + g _ 2 ( x , z ) \big ) ^ 2 \pm x z ( x - z ) ( x - s z ) , \end{align*}


\begin{align*} \lambda \partial _ i Q ( \alpha ) + \mu \partial _ i R ( \alpha ) = 0 \end{align*}


\begin{align*} d _ i : = 2 ^ { - \lfloor \frac { 2 i } { m } \rfloor } \left ( \frac { m ! } { ( m - i ) ! i ! } \right ) ^ 2 i \in \{ 1 , 2 , \dots , \lfloor m / 2 \rfloor \} . \end{align*}


\begin{align*} \lim _ { t \to \pm \infty } \| u ( t ) - e ^ { i t \Delta } u _ \pm \| _ { H ^ 1 } = 0 . \end{align*}


\begin{align*} a \otimes ( b \otimes c ) = ( a \otimes b ) \otimes c \end{align*}


\begin{align*} q _ { \sigma } \left ( x , t \right ) = q \left ( x , t \right ) - 2 \partial _ { x } ^ { 2 } \log \det \left \{ I + \mathbf { K } \left ( x , t \right ) \right \} \end{align*}


\begin{align*} \lim _ { \substack { r \to \infty , \\ r e ^ { i t } \notin E } } \frac { \log | \widetilde W ( r e ^ { i t } ) | } { r ^ { \rho ( r ) } } = \frac a \rho < 1 . \end{align*}


\begin{align*} \sum _ { l \in N _ { j - 2 } \cap J _ A } \sum _ { s \in a ( l ) \setminus \{ l \} } - u _ s = \sum _ { \substack { l \in N _ { j - 2 } \cap I _ A \\ \max { a ( l ) } \leq j - 2 } } - u _ l . \end{align*}


\begin{align*} d ^ \star w ( \tau ) = U ^ { \odot \star } ( t , \tau ) d ^ { \star } ( j \circ u ) ( \tau ) - U ^ { \odot \star } ( t , \tau ) A ^ { \odot \star } ( \tau ) j u ( \tau ) , \forall \tau \in [ s , t ] . \end{align*}


\begin{align*} \int _ { G } | \nabla u _ n | ^ 2 \ , d \mu - \int _ { G } | \nabla ( u _ n - u ) | ^ 2 \ , d \mu = \int _ { G } | \nabla u | ^ 2 \ , d \mu + o ( 1 ) . \end{align*}


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


\begin{align*} G _ { \mathcal R , \zeta _ o , 1 } ( x , y ) : = \frac { i } { 2 \pi } \int _ { \mathcal I _ { \theta _ c , \zeta _ o } } \frac { 2 i \sin \zeta \mathcal S _ { - } ( \cos \zeta , n ) } { n ^ 2 - 1 } { e ^ { i k _ { + } \left ( - \vert y \vert \cos ( \zeta + \theta _ { \hat y } ) + \vert x \vert \cos ( \zeta - \theta _ { \hat x } ) \right ) } } d \zeta . \end{align*}


\begin{align*} \lim _ { s \to 1 } \left ( \zeta ( s ) - \frac { 1 } { s - 1 } \right ) = \gamma _ o , \end{align*}


\begin{align*} a _ { i j } ( \phi ) & = \frac 1 2 \ , \int _ { \R ^ n } \sum _ { k = 1 } ^ n \ ( h _ { x _ i x _ k } ( x - T x ) \frac { \partial T _ k x } { \partial x _ j } + h _ { x _ j x _ k } ( x - T x ) \frac { \partial T _ k x } { \partial x _ i } \ ) \ , \phi ( x ) \ , d x , \end{align*}


\begin{align*} \lim _ { q \to \infty } \dfrac { \ell _ { R } ( { R / I _ { q } } ) } { q ^ { d } } = \sum _ { i = 1 } ^ { n } \lim _ { q \to \infty } \dfrac { \ell _ { R _ { i } } ( R _ { i } / I _ { q } R _ { i } ) } { q ^ { d } } \end{align*}


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


\begin{align*} R ( x ) = \bigl ( R _ { i j } ( x ) \bigr ) _ { i , j } = \Bigl \{ I _ n - d ( x ) \overline { W } ( x ) \Bigr \} ^ { - 1 } , x \in \overline { N } . \end{align*}


\begin{align*} A _ X = - A _ X ^ T \ ; , \ A _ I = - A _ O ^ T \ ; . \end{align*}


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


\begin{align*} \frac { \gamma } { \Gamma ( 1 - \gamma ) } \int _ 0 ^ { \infty } ( 1 - e ^ { - \lambda u } ) u ^ { - 1 - \gamma } \d u = \lambda ^ { \gamma } , \lambda > 0 , \ \gamma \in ( 0 , 1 ) , \end{align*}


\begin{align*} \rho _ s ( x ) = \frac { \Gamma ( x + s + \tfrac 1 2 ) } { \Gamma ( x + \tfrac 1 2 ) } . \end{align*}


\begin{align*} \begin{aligned} \partial _ \mu F ^ { \mu \nu } & = - \textrm { I m } ( \phi \overline { \mathcal D ^ \nu \phi } ) , \\ \mathcal D _ \mu \mathcal D ^ \mu \phi & = 0 , \end{aligned} \end{align*}


\begin{align*} f ( A _ 2 , \theta _ 2 , \omega _ 2 , R ) = \det ( M ) ^ { \deg ( w ) } \cdot d ^ { - \frac { g \cdot \deg ( w ) } { 2 } } \cdot f ( A _ 1 , \theta _ 1 , \omega _ 1 , R ) . \end{align*}


\begin{align*} \frac { 2 \alpha } { \gamma } + \frac { 3 } { p } = 2 \alpha - 1 + s . \end{align*}


\begin{align*} \Lambda : = \{ \lambda \in Z _ { k ' } ( k ' ) \ , | \ , { \rm ( I ) } { \rm ( I I ) } \lambda \} . \end{align*}


\begin{align*} \bar { d } ( x , y ) = \bar { d } ( \alpha ( t , x ) , \alpha ( t , y ) ) \end{align*}


\begin{align*} \varphi ( [ x , y ] ) = \frac { 1 } { 2 } \left ( [ \varphi ( x ) , y ] + [ x , \varphi ( y ) ] \right ) . \end{align*}


\begin{align*} \sum _ { i j } \beta ( S v _ i , T w _ j ) \leq C \sum _ { i j } | S v _ i | ^ 2 \cdot | T w _ j | ^ 2 = C \cdot | S | ^ 2 _ { H S } \cdot | T | ^ 2 _ { H S } < \infty . \end{align*}


\begin{align*} d ^ { 3 } _ { r } B ( d ^ 1 _ { c } \tau \wedge \tau ) ( x , y , z , t ) = B ( ( d ^ 1 _ { c } \tau \circ \alpha ) \wedge d ^ { 1 } _ { c } \tau ) ( x , y , z , a ) . \end{align*}


\begin{align*} \begin{multlined} \sigma _ 0 ( A \otimes B ) ( ( i , j ) ) = ( - 1 ) ^ { | i | | j | } A ( i ) B ( j ) \\ = A ( i ) B ( j ) = ( A \otimes B ) ( ( i , j ) ) \ ; . \end{multlined} \end{align*}


\begin{align*} D _ a = \{ f \in C _ c ^ \infty ( \Gamma \backslash G / K ) : g \in Y _ a , c _ P f ( g ) = 0 , P \} . \end{align*}


\begin{align*} \theta ( u _ w ) \cdot \frac { a ^ w + c _ w a ^ { 1 - w } } { 1 - 2 w } \cdot a ^ { 1 - w } - \left ( \frac { \zeta _ k ( w ) a ^ { 1 - w } } { \zeta ( 2 w ) ( 1 - 2 w ) } \right ) ^ 2 = 0 . \end{align*}


\begin{align*} \sum _ { k = 0 } ^ { \infty } \frac { d _ 2 ( \mu _ k ^ { \tau } , \mu _ { k + 1 } ^ { \tau } ) ^ 2 } { \tau } \leq C . \end{align*}


\begin{align*} \delta \Phi ( t _ 1 , t _ 2 , x ) = \Phi ( t _ 1 + t _ 2 , x ) - \Phi ( t _ 1 , \Phi ( t _ 2 , x ) ) . \end{align*}


\begin{align*} g ( \phi ) \equiv g ( ( \phi ^ 1 , \phi ^ 2 , \phi ^ 3 ) ) = ( 0 , \beta _ 2 \phi ^ 2 , \beta _ 3 \phi ^ 3 ) \ , , \end{align*}


\begin{align*} \hat { f } _ { s , a , b } ( j ) = \int _ { - \infty } ^ \infty \frac { e ^ { 2 \pi i ( s - j ) x } } { a ^ 2 + ( x + b ) ^ 2 } \mathrm { d } x = \frac { \pi e ^ { - i ( s - j ) b } } { a } e ^ { - a | j - s | } . \end{align*}


\begin{align*} \Pi _ \theta ( \xi ) = \frac 1 2 \left ( I _ { 4 \times 4 } + \theta \frac { \xi _ j \gamma ^ 0 \gamma ^ j + m \gamma ^ 0 } { \langle \xi \rangle _ m } \right ) , \end{align*}


\begin{align*} \mathcal U _ \varepsilon ^ o - \mathcal L \bigl ( \dfrac t \varepsilon \bigr ) V ^ o = \mathcal L \bigl ( \dfrac t \varepsilon \bigr ) \bigl ( V _ \varepsilon ^ o - V ^ o \bigr ) \rightarrow 0 ~ ~ ~ L ^ \infty ( 0 , T ; H ^ 1 ( \mathbb T ^ 2 ) ) . \end{align*}


\begin{align*} \mu \phi ( x ) & = ( - \Delta _ x + V _ \mathrm { t r a p } ( x ) ) \phi ( x ) + ( \upsilon _ N \ast | \phi | ^ 2 ) ( x ) \phi ( x ) + \frac { 1 } { N } ( \upsilon _ N \ast \rho ^ \mathrm { p a i r } ) ( x ) \phi ( x ) \\ & + \frac { 1 } { N } \int { d y \{ \upsilon _ N ( x - y ) n ^ \mathrm { p a i r } ( x , y ) \phi ( y ) \} } + \frac { 1 } { N } \int { d y \{ \upsilon _ N ( x - y ) m ^ \mathrm { p a i r } ( x , y ) \overline { \phi ( y ) } \} } . \end{align*}


\begin{align*} ( b _ i \sigma ) ^ r = ( r \cdot \nu ) ( b ) ( \varpi _ { x _ i } ) \sigma ^ r \end{align*}


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


\begin{align*} b _ d = \frac { 1 } { d } \sum _ { k | d } \mu ( d / k ) \chi ( g ^ k ) . \end{align*}


\begin{align*} \Gamma _ s ( \mathfrak { h } ) = \bigoplus _ { n = 0 } ^ { \infty } \mathfrak { h } ^ { \otimes _ s n } , \end{align*}


\begin{align*} h ( x ) = \int \limits _ { \mathbb { S } ^ { n - 1 } } P ( x , \eta ) h _ b ( \eta ) d \sigma ( \eta ) \end{align*}


\begin{align*} \mathcal { H } ^ { * } _ { N } ( X _ { N } ) = \sum _ { i \neq j } g ( x _ { i } - x _ { j } ) , \end{align*}


\begin{align*} \left \{ \begin{aligned} & v _ t - \mathcal { M } ^ - ( D ^ 2 v ) \geq 0 & & ~ ~ \mbox { i n } ~ ~ Q ^ + _ { 1 } ; \\ & v \geq 0 & & ~ ~ \mbox { o n } ~ ~ S _ { 1 } ; \\ & v \geq 1 & & ~ ~ \mbox { i n } ~ ~ \partial Q ^ + _ { 1 } \backslash S _ { 1 } . \end{aligned} \right . \end{align*}


\begin{align*} \lim _ { x \to \infty } \frac { s _ 2 ( x ) } { \pi ( x ) } = \frac { 1 } { ( p - 1 ) ^ 2 } . \end{align*}


\begin{align*} \zeta ^ \epsilon ( t _ 2 ) - \zeta ^ \epsilon ( t _ 1 ) = \frac { 1 } { \epsilon } \int _ { t _ 1 } ^ { t _ 2 } m ^ \epsilon ( s ) d s \end{align*}


\begin{align*} Z _ r \left ( g g ^ \top , \frac { r s } { 2 } \right ) & = \zeta ( r s ) \sum _ { v \in \Lambda } { | v \cdot g | ^ { - r s } } = 2 \zeta ( r s ) \sum _ { v \in \{ \pm 1 \} \backslash \Lambda } { | v \cdot g | ^ { - r s } } \\ & = 2 \zeta ( r s ) \sum _ { \Psi ( v ) \in P \cap \Gamma \backslash \Gamma } \varphi _ s ^ P ( \Psi ( v ) g ) = 2 \zeta ( r s ) E ^ P _ { s , \varphi } ( g ) , \end{align*}


\begin{align*} v _ \epsilon \ , = \ , q \Bigl ( \frac { r _ \epsilon } { \epsilon ^ { 1 - \kappa } } \Bigr ) \end{align*}


\begin{align*} \sum _ { T < \gamma \le 2 T } \bigg | M \Big ( \frac { 1 } { 2 } + i \gamma + \frac { 2 \pi i \kappa } { \log T } , P \Big ) \bigg | ^ 4 & \ll T \log T \sum _ { n \le y ^ 2 } \frac { d ( n ) ^ 2 } { n } + T \sum _ { n \le y ^ 2 } \frac { ( \Lambda * d ) ( n ) \ , d ( n ) } { n } \\ & \ll T \log ^ 5 T \end{align*}


\begin{align*} & J ( x , t , k ) = \begin{pmatrix} 1 & r _ 2 ( k ) e ^ { - 2 i t \theta } \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ r _ 1 ( k ) e ^ { 2 i t \theta } & 1 \end{pmatrix} , \end{align*}


\begin{gather*} d ^ { 3 } _ { r } \gamma = B ( d ^ 2 f \wedge g ) + B \left ( f \wedge d ^ 1 g \right ) \end{gather*}


\begin{align*} \div \mathcal { R } ^ u ( v ) = v , \ \ \div \mathcal { R } ^ B ( f ) = f . \end{align*}


\begin{align*} \mathcal { C } ^ { \bot _ e } = \{ \mathbf { x } \in \mathbb { F } _ { q } ^ n : \ \langle \mathbf { x } , \mathbf { y } \rangle _ e = 0 \ { \rm { f o r \ a l l } } \ \mathbf { y } \in \mathcal { C } \} \end{align*}


\begin{align*} \lambda _ g = e ^ { \beta ( t ) g ( r ^ 2 ) } \left ( d t + \frac { 1 } { 2 } r ^ 2 \ , d \theta \right ) \end{align*}


\begin{align*} \mu ^ \sigma _ t ( \d { x } ) = e ^ { - | \sigma | t } \sum _ { k = 0 } ^ \infty \frac { t ^ k \sigma ^ { k * } ( \d { x } ) } { k ! } = e ^ { - | \sigma | t } \delta _ 0 ( \d { x } ) + e ^ { - | \sigma | t } \sum _ { k = 1 } ^ \infty \frac { t ^ k \sigma ^ { k * } ( x ) \d { x } } { k ! } . \end{align*}


\begin{align*} \mathcal { G } _ { 0 } : = \left \{ \left ( g _ { i j } \right ) _ { i \leq j } \in \R ^ { \frac { d ( d + 1 ) } { 2 } } \ ; : \ ; \inf _ { x \neq 0 } \frac { g ( x , x ) } { | x | ^ 2 } > K \right \} \end{align*}


\begin{align*} 0 = \liminf _ { t \rightarrow \infty } \norm { \nabla f ( X ( \omega , t ) ) } < \delta < \limsup _ { t \rightarrow \infty } \norm { \nabla f ( X ( \omega , t ) ) } , \end{align*}


\begin{align*} F _ m ( E ) = \# [ \sigma ( J _ m ) \cap ( E , \infty ) ] . \end{align*}


\begin{align*} l _ 3 & \coloneq l _ 3 ' + \underbrace { \left [ ( - 2 ) l _ 1 + g l _ 2 \right ] } _ { } \\ & = ( - 1 ) l _ 1 + \left ( \dfrac { g + 1 } { 2 } \right ) l _ 2 , \end{align*}


\begin{align*} \overline { f } _ i = \sum _ { j = 1 } ^ d \overline { l } _ { i j } \tau ' _ j + \overline { a } _ i \end{align*}


\begin{align*} 1 = \{ \bullet \} \ ; . \end{align*}


\begin{align*} L _ x ( y ) = y + c _ { i , j } \in A _ j { \rm a n d } \mathbf { R } _ y ( x ) = x - c _ { j , i } \in A _ i , \end{align*}


\begin{align*} \begin{gathered} B _ I G ^ B = A _ I \ ; , A _ I G ^ A = B _ I \ ; , \\ H ^ B B _ O = A _ O \ ; , H ^ A A _ O = B _ O \ ; . \end{gathered} \end{align*}


\begin{align*} & r ( 0 ) = 0 , r ( \pi ) = k \pi , k \in \Z , \end{align*}


\begin{align*} \delta a _ { s , t } = \mathcal { D } ( a ) _ { s , t } + A _ { s , t } a ^ 1 _ { s } + \mathbb { A } _ { s , t } a ^ { 2 } _ { s } + a ^ { \natural } _ { s , t } \ , , \end{align*}


\begin{align*} f ( u - n h ) = \frac { u ^ 2 - h ^ 2 } { u ^ 2 } f ( u ) . \end{align*}


\begin{align*} \partial _ r ^ 2 \eta _ 0 ( r ) = 0 , r \in ( g _ 0 , g _ 1 ) , \partial _ r \eta _ 0 ( g _ i ) = 0 , i = 0 , 1 . \end{align*}


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


\begin{align*} I _ N = 1 + \sum _ { d = 1 } ^ \infty \frac { z ^ d } { d ! } \sum _ { \alpha , \beta \vdash d } p _ \alpha ( a _ 1 , \dots , a _ M ) p _ \beta ( b _ 1 , \dots , b _ M ) H _ N ( \alpha , \beta ; q ) , \end{align*}


\begin{align*} P _ t | \varphi | ( x ) \leq \left ( \int _ { \R ^ d } p ( t , x , y ) \d y \right ) ^ { 1 / 2 } \left ( \int _ { \R ^ d } p ( t , x , y ) | \varphi ( y ) | ^ 2 \d y \right ) ^ { 1 / 2 } = \left ( \int _ { \R ^ d } p ( t , x , y ) | \varphi ( y ) | ^ 2 \d y \right ) ^ { 1 / 2 } . \end{align*}


\begin{align*} ( a \otimes b ) _ k = \sum _ { i , j } N _ { i j } ^ k a _ i b _ j \ ; . \end{align*}


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


\begin{align*} k \varphi _ { n / k } - \varphi _ n = ( 1 + o ( 1 ) ) h _ 4 ( n ) . \end{align*}


\begin{align*} B _ \eta = \{ f \in H ^ 1 : \| f \| _ { H ^ { s _ p } } < \eta \} . \end{align*}


\begin{align*} \max _ { \alpha = 1 , \dots , n - 1 } \max _ { y \in \Gamma } | \kappa _ \alpha ( y ) | , \max _ { i , j , k = 1 , \dots , n } \max _ { y \in \Gamma } | \underline { D } _ i W _ { j k } ( y ) | . \end{align*}


\begin{align*} 1 - K = | z | ^ { 2 k _ j } u _ 0 , \end{align*}


\begin{align*} \lim _ { m \to \infty } B _ m ( \theta _ 1 , x + i y ) = C ( \theta _ 1 , x + i y ) = F ( \theta _ 1 / 2 + y / 2 \pi , x / 2 \pi ) , \end{align*}


\begin{align*} & t ^ { 2 ^ h + 1 } = \alpha ^ { q - 1 } & & \mbox { i f } h \mbox { i s o d d , } \\ & t ^ { 2 ^ { n + h } + 1 } = \alpha ^ { q - 1 } & & \mbox { i f } h \mbox { i s e v e n } \end{align*}


\begin{align*} \lambda _ 2 : = \frac { v - 2 } { k - 2 } . \end{align*}


\begin{align*} d _ n = d ( ( \bar x _ n , \bar t _ n ) , \partial Q ) , M _ n = | u _ n ( x _ n , t _ n ) | , r _ n = | x _ n - \bar x _ n | + | t _ n - \bar t _ n | ^ { \frac 1 2 } . \end{align*}


\begin{align*} B C ^ 0 ( \R , B ) : = \{ f : \R \to B \ , : \ , f \mbox { b o u n d e d a n d c o n t i n u o u s } \} \end{align*}


\begin{align*} P [ Z ] & \leq P \left [ \omega \in \Omega : \omega _ k \in B \cup \tilde { R } \right ] \\ & = \mu ( B \cup \tilde { R } ) \\ & \leq \mu ( B ) + \mu ( \tilde { R } ) = 0 \end{align*}


\begin{align*} \begin{array} { c } h ^ 0 x | _ C = \sum \limits _ { i = 1 } ^ s h ( c _ i ) I _ { A _ i \cap C } = \sum \limits _ { i = 1 } ^ s \sum \limits _ { k = 1 } ^ \sigma h ( c _ i ) I _ { A _ i \cap B _ k \cap C } \\ = \sum \limits _ { i = 1 } ^ s \sum \limits _ { k = 1 } ^ \sigma h ( d _ k ) I _ { A _ i \cap B _ k \cap C } = \sum \limits _ { k = 1 } ^ \sigma h ( d _ k ) I _ { B _ k \cap C } \ \ P ^ * - \mbox { a . s . } \end{array} \end{align*}


\begin{align*} f ( [ n \alpha ] ) = f ( [ n \beta ] ) = ( [ n \beta ] - [ ( n - 1 ) \beta ] ) - ( [ n \alpha ] - [ ( n - 1 ) \alpha ] ) . \end{align*}


\begin{align*} \left \| g + \sum _ { j \in J } a _ j x _ j \right \| \ \le \ \mathbf { A _ p } \sup \left \{ \left \| g + \sum _ { j \in J } \varepsilon _ j x _ j \right \| \colon | \varepsilon _ j | = 1 \right \} . \end{align*}


\begin{align*} E _ j ( \overline { K } ) = \{ ( x _ 1 , \dots , x _ n ) \in H ( \overline { K } ) ~ | ~ ( x _ 1 , \dots , x _ { j - 1 } ) \times \mathbb { P } _ 1 ( \overline { K } ) \times ( x _ { j + 1 } , \dots , x _ n ) \subseteq H ( \overline { K } ) \} . \end{align*}


\begin{align*} \left | \partial _ t \rho _ \eta ^ \varepsilon - \overline { \partial ^ \circ \eta } - k _ d ^ { - 1 } \Bigl ( \overline { V _ \Gamma ^ 2 \eta } \Bigr ) \right | & \leq c \varepsilon \sum _ { \xi = \eta , \zeta _ 0 , \zeta _ 1 } \left ( | \bar { \xi } | + \Bigl | \overline { \partial ^ \circ \xi } \Bigr | + \Bigl | \overline { \nabla _ \Gamma \xi } \Bigr | \right ) \end{align*}


\begin{align*} q _ k ( x ) = \prod _ { i = 1 } ^ k ( x - x _ i ) ( x + x _ i + 1 ) \ , . \end{align*}


\begin{align*} b _ i m _ j = \begin{cases} [ 2 ] m _ i , & \ ; j = i ; \\ m _ i , & \ ; j = i \pm 1 ; \\ 0 , & , \end{cases} \end{align*}


\begin{align*} b _ { l 1 } = b _ { l 2 } = 0 , \ \ l \in L . \end{align*}


\begin{align*} 2 \mathrm { R e } \langle u _ { \alpha } \otimes \Phi _ y ^ { ( 0 ) } \ , | \ , H _ y ( u _ { \alpha } \otimes 2 \alpha ^ { 3 / 2 } \eta \tilde { \Phi } _ * ^ 1 ) \rangle = 0 , \end{align*}


\begin{align*} \int _ { \Omega _ T } \boldsymbol { \theta } \cdot \boldsymbol { \zeta } = 0 \boldsymbol { \zeta } \in L ^ { 5 / 2 } ( \Omega _ T ) ^ d . \end{align*}


\begin{align*} \sigma _ { \mathcal A } ( f ) = \{ h ( f ) : h \in \Sigma \} = \{ \tau ^ { - 1 } ( \lambda ) ( f ) : \lambda \in \overline { \mathbb D } \} = \{ f ( \lambda ) : \lambda \in \overline { \mathbb D } \} . \end{align*}


\begin{align*} \bar \theta ( \Lambda _ { N } ) = \begin{cases} \Lambda _ { \theta ( N ) } , & ; \\ \Lambda _ { \binom { a _ 1 } { - } \cup \theta ( N ) } , & . \end{cases} \end{align*}


\begin{align*} \widetilde P _ \tau \ , ( A - \lambda B + \tau \ , U \ , ( D _ A - \lambda D _ B ) \ , V ^ * ) \ , \widetilde Q _ \tau = \left [ \begin{array} { c c c } R ( \lambda ) & 0 & 0 \\ 0 & R _ { \rm p r e } ( \lambda ) & 0 \\ 0 & 0 & R _ { \rm r a n } ( \lambda ) \end{array} \right ] , \end{align*}


\begin{align*} d X _ t = b ( X _ t , Z _ t ) d t + d W _ t , \ ; t \geq 0 \ ; , X _ 0 = x , \ ; , Z _ 0 = z , \end{align*}


\begin{align*} 2 \alpha \mathrm { R e } \langle u _ { \alpha } \otimes R _ * \ , | \ , V _ y \Phi _ { \# } ^ y \rangle & \geq - C \alpha \varepsilon _ 8 \| R _ * \| ^ 2 - C \alpha \varepsilon _ 8 ^ { - 1 } | \kappa | ^ 2 \| V _ y \Phi _ { \# } ^ y \| ^ 2 \\ & = - C \alpha \varepsilon _ 8 \| R _ * \| ^ 2 + O ( \alpha ^ 4 ) . \end{align*}


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


\begin{align*} d q ^ 0 ( t ) = f ( q ^ 0 ( t ) ) d t + \sigma ( q ^ 0 ( t ) ) d \beta ( t ) \end{align*}


\begin{align*} \mathrm { r a d } ' ( \langle \cdot , \cdot \rangle ) = \left \{ m ' \in \widehat { M } _ { z ^ { - 1 } , \lambda ^ { - 1 } } \mid \langle m , m ' \rangle = 0 , \ ; \forall m \in \widehat { M } _ { z , \lambda } \right \} , \end{align*}


\begin{align*} 0 = \gamma . \left ( \sum _ { i = 1 } ^ s n _ i L _ { P _ i } \right ) = \sum _ { i = 1 } ^ s m _ i n _ i . \end{align*}


\begin{align*} \delta ^ n _ i | _ U \colon \Gamma ( g _ { n - 1 } ^ { - 1 } ( U ) , g _ { n - 1 } ^ * F ) = \Gamma ( U , F ^ { n - 1 } ) \to \Gamma ( g _ { n } ^ { - 1 } ( U ) , g _ n ^ * F ) = \Gamma ( U , F ^ n ) , \sigma \mapsto \sigma \circ d ^ n _ i , \end{align*}


\begin{align*} \ell _ k ( u ) \mapsto \sum _ { 1 \leq i _ { 1 } < \cdots < i _ { k } \leq n } \Lambda _ { i _ 1 } ( u ) \Lambda _ { i _ 2 } ( u + h ) \cdots \Lambda _ { i _ k } ( u + ( k - 1 ) h ) , k = 1 , \cdots , n , \end{align*}


\begin{align*} \nu ^ { c u } _ { n , x } ( A ) = \int _ { \pi ( A ) } \nu ^ { u } _ { n , s } ( \xi _ n ^ u ( y ) \cap A ) \ , d \tilde \nu ^ { c } _ { n , x } ( s ) , \end{align*}


\begin{align*} \Delta _ 1 = d _ 0 d _ 0 ^ * + d _ 1 ^ * d _ 1 . \end{align*}


\begin{align*} \sum _ i s _ i + \sum _ i s _ i ' = 0 \ ; , \end{align*}


\begin{align*} \mu _ { 2 n + 2 } \left ( z \right ) = \left ( n + \frac { 1 } { 2 } \right ) \mu _ { 2 n } \left ( z \right ) - z ^ { 2 n + 1 } e ^ { - z ^ { 2 } } . \end{align*}


\begin{align*} D _ { x } ( \dot { \pmb { \lambda } } ) : = \sum _ { j = 1 } ^ { N - 1 } \pmb { r } _ { j } ^ { 2 } \dot { \pmb { \lambda } } \end{align*}


\begin{align*} \boldsymbol { \beta } _ k = \frac { \left ( \boldsymbol { \Sigma } _ k + \sigma ^ 2 \vec { I } \right ) ^ { - 1 } \vec { g } _ k } { \left \vert \left \vert { \left ( \boldsymbol { \Sigma } _ k + \sigma ^ 2 \vec { I } \right ) ^ { - 1 } \vec { g } _ k } \right \vert \right \vert } . \end{align*}


\begin{align*} S _ \jmath = \{ s \in S : V ^ s = V ^ \jmath \} . \end{align*}


\begin{align*} \varphi ( t ) = ( C _ 0 + ( 1 - \beta - \delta - \alpha ) t ) ^ { \frac { 1 } { 1 - \beta - \delta - \alpha } } . \end{align*}


\begin{align*} M \rvert _ { \chi \cdot } ( \alpha , \beta ) = M ( ( \chi , \alpha ) , \beta ) \ ; , \end{align*}


\begin{align*} h _ c ( \theta ) : = \frac { { 2 } ^ { \frac { 9 } { 4 } } e ^ { { 5 \pi i } / { 8 } } } { k ^ { \frac { 3 } { 4 } } _ { + } \left [ \cos \left ( \frac { \theta _ c - \theta } 2 \right ) \right ] ^ { 3 / 2 } \left ( \tan \theta _ c \right ) ^ { 1 / 2 } } e ^ { - i k _ { + } \vert y \vert \cos ( \theta _ c + \theta _ { \hat y } ) } , \theta \in ( 0 , \pi / 2 ) . \end{align*}


\begin{align*} G \left ( f \left ( a \right ) , f \left ( b \right ) \right ) \left ( \underset { a } { \overset { b } { \int } } f \left ( u \right ) ^ { d u } \right ) ^ { \frac { 1 } { a - b } } = \left ( \underset { 0 } { \overset { 1 } { \int } } \left ( f ^ { \ast } \left ( \left ( 1 - t \right ) a + t b \right ) ^ { \left ( 2 t - 1 \right ) } \right ) ^ { d t } \right ) ^ { \frac { b - a } { 2 } } , \end{align*}


\begin{align*} Y _ t ^ z - U _ t ^ z = \int _ 0 ^ t \left ( F ( U _ s ^ z ) - l _ s ^ { Y } \right ) \mathrm { d } s . \end{align*}


\begin{align*} \rho \ , m _ j = \lambda z ^ { \delta _ { j , 0 } } m _ { j + 1 } , \end{align*}


\begin{align*} \tau ( s , t ) \tau ( s t , u ) = \tau ( s , t u ) \tau ( t , u ) s , t , u \in \mathcal { S } . \end{align*}


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


\begin{align*} \left ( 2 n + 3 \right ) \gamma _ { n + 1 } + 2 c _ { n + 1 } - \left ( n + 1 \right ) z ^ { 2 } = 2 \gamma _ { n + 1 } \left ( \gamma _ { n + 2 } + \gamma _ { n + 1 } + \gamma _ { n } - z ^ { 2 } \right ) , \end{align*}


\begin{align*} \overline { R } _ { n _ k } = \sum _ { j = 1 } ^ k \overline { U } _ j - \overline { \Delta } _ { n _ k , k } \ , , \end{align*}


\begin{align*} \widehat { B } _ { \Omega _ 1 , p } ( z ; X ) = \widehat { B } _ { \Omega _ 2 , p } ( F ( z ) ; F _ \ast X ) . \end{align*}


\begin{align*} \int _ { E } \phi d ( P ^ * _ t \nu ) = \int _ { E } P _ t \phi d \nu \ , . \end{align*}


\begin{align*} \hat { \nu } ^ c _ x ( J _ 0 ) \leq \liminf _ { n \to \infty } \hat { \nu } ^ c _ { x _ n } ( J _ 0 ) = 0 , \end{align*}


\begin{align*} k ( E ) = \int \ ! \chi _ { _ { ( - \infty , E ] } } \ , d \kappa . \end{align*}


\begin{align*} \mathcal { H } _ { N } ( X _ { N } ) = N ^ { 2 } \left ( \mathcal { E } _ { V } ^ { \theta } ( \mu _ { \theta } ) + { \rm F } _ { N } ( X _ { N } , \mu _ { \theta } ) + \int _ { M } \zeta _ { \theta } d { \rm e m p } _ { N } \right ) . \end{align*}


\begin{align*} \int _ { \mathbb { D } } { f \eta } = 0 , \ \ \ \forall \ , f \in { A ^ 2 ( \mathbb { D } ) } , \end{align*}


\begin{align*} \int _ { - \infty } ^ \infty K _ { t } ( x , y ) W ( x ) \mathrm { d } x & = \sum _ { k = 0 } ^ \infty t ^ { - k - 1 } \int _ { \infty } ^ \infty h _ k ( x ) \ , W ( x ) d x \ , h _ k ( y ) \\ & = \sum _ { k = 0 } ^ \infty t ^ { - k - 1 } 2 ^ { - 1 / 4 } \langle h _ k , h _ 0 \rangle h _ k ( y ) = t ^ { - 1 } W ( y ) \ , . \end{align*}


\begin{align*} Q _ { } ^ { \mathsf { c } } ( \varepsilon , n , q ) = \bigcup _ { \substack { x , y \in \Z _ { q } ; \\ \| x - y \| _ { q } > 2 n } } \{ \omega \in \Omega : _ { \omega } ^ * ( x , \varepsilon , n ) \cap _ { \omega } ^ * ( y , \varepsilon , n ) \neq \varnothing \} . \end{align*}


\begin{align*} H \cap H ^ g & = ( \langle h \rangle \times \langle h ^ g \rangle \times \cdots \times \langle h ^ { g ^ { s - 1 } } \rangle ) \cap ( \langle h ^ g \rangle \times \langle h ^ { g ^ 2 } \rangle \times \cdots \times \langle h ^ { g ^ s } \rangle ) \\ & = \langle h ^ g \rangle \times \langle h ^ { g ^ 2 } \rangle \times \cdots \times \langle h ^ { g ^ { s - 1 } } \rangle \\ & \cong C _ 2 ^ { s - 1 } , \end{align*}


\begin{align*} \begin{array} { l } \int _ { m } ^ { M } y d F _ { \epsilon } ( y ) = \left [ M F _ { \epsilon } ( M ) - m F _ { \epsilon } ( m ) \right ] - \int _ { m } ^ { M } F _ { \epsilon } ( y ) d y , \end{array} \end{align*}


\begin{align*} \alpha _ { e _ j } \leq e ^ { \nu _ { x _ j } + \nu _ { y _ j } } \cdot \beta _ { e _ j } \prod _ { e = ( u v ) \in E _ j } ( 1 + e ^ { w _ e - \nu _ u - \nu _ v } ) . \end{align*}


\begin{align*} \forall v , w , u \in V , \ ; \ ; v = w \Leftrightarrow v \dot { + } u = w \dot { + } u , \end{align*}


\begin{align*} \begin{cases} 2 [ \beta ] = [ \beta ] - 1 & \implies \beta < 0 \\ 2 [ \beta ] = [ \beta ] - 2 & \implies \beta < 0 \\ 2 ( [ \beta ] - 1 ) = [ \beta ] & \implies [ \beta ] = 2 \\ 2 ( [ \beta ] - 1 ) = [ \beta ] - 2 & \implies [ \beta ] = 0 \\ 2 ( [ \beta ] - 2 ) = [ \beta ] & \implies [ \beta ] = 4 \\ 2 ( [ \beta ] - 2 ) = [ \beta ] - 1 & \implies [ \beta ] = 3 . \end{cases} \end{align*}


\begin{align*} \lim _ { t \rightarrow \infty } X _ { \theta } ( t ) = x _ { \theta } ^ { \star } , a . s . \end{align*}


\begin{align*} f ( z ) = \int _ \Omega | m _ p ( \cdot , z ) | ^ { p - 2 } \ , \overline { K _ p ( \cdot , z ) } \ , f , \ \ \ \forall \ , f \in A ^ p ( \Omega ) . \end{align*}


\begin{align*} \mu _ n ( 3 ) = A _ n ( 3 ) n \in \mathbb N , \end{align*}


\begin{align*} [ E _ { i j } [ r ] , E _ { k l } [ s ] ] = \delta _ { k j } E _ { i l } [ r + s ] - \delta _ { i l } E _ { k j } [ r + s ] + r \delta _ { k j } \delta _ { i l } \delta _ { r , - s } K , \end{align*}


\begin{align*} \lim _ { { \rm R e } z \to 0 ^ + } \frac { 1 } { 2 } ( 2 { \rm R e } z ) ^ 2 \left | S h ( z ) \right | = 0 . \end{align*}


\begin{gather*} d ^ 2 \gamma ' ( x , y , z ) ( t ) = d _ { r } ^ { 3 } \gamma ( x , y , z , t ) + \frac { 1 } { 2 } B _ { \mathfrak a } \left ( \theta \wedge ( \theta \circ \alpha ) \right ) ( x , y , z , a ) . \end{gather*}


\begin{align*} { \rm E n t } [ \mathbf { P } | \mathbf { \Pi } ^ { \lambda } ] = \lim _ { R \to \infty } \frac { 1 } { R ^ { d } } { \rm E n t } [ \mathbf { P } | _ { \square _ { R } } | \mathbf { \Pi } ^ { \lambda } | _ { \square _ { R } } ] , \end{align*}


\begin{align*} \sum _ { i = 1 } ^ { \ell } \int _ { Q _ T } c _ i \omega ( p ) f ' ( u ) \Big ( \partial _ n G _ i ( p , n ) ( \nabla p \cdot \nabla n - \partial _ t n ) + \partial _ p G _ i ( p , n ) p u - G _ i ( p , n ) \Delta p \Big ) - c _ i G _ i ( p , n ) \nabla p \cdot \nabla ( \omega ( p ) f ' ( u ) ) \end{align*}


\begin{align*} \widetilde A - \lambda \widetilde B : = A - \lambda B + \tau \ , ( U D _ A V ^ * - \lambda \ , U D _ B V ^ * ) , \end{align*}


\begin{align*} \partial _ { x } ^ { \ast } L _ { H } + 2 x L _ { H } = 0 . \end{align*}


\begin{align*} & - C _ 1 ( 1 - \eta ) ^ { \alpha _ 0 } \leq \partial _ \xi w + \partial _ \tau w \leq \frac { ( \delta b ) ^ { \frac { 1 } { \alpha _ 0 } } } { K } ( 1 - \eta ) o n \xi = 0 a n d \tau = 0 , \\ & - C _ 1 ( 1 - \eta ) ^ { \alpha _ 0 } \leq \partial _ \tau w \leq b \delta ( 1 - \eta ) ^ { \alpha _ 0 } o n \xi = 0 a n d \tau = 0 , \end{align*}


\begin{align*} X ( t ) = P ( t ) e ^ { Q t } , , \end{align*}


\begin{align*} \| f \| _ { * , q } = \left ( \int _ { \Lambda _ L ^ * } | f ( k ) | ^ q d k \right ) ^ { 1 / q } \ ( 1 \leq q < \infty ) , \| f \| _ { * , \infty } = \sup _ { k \in \Lambda _ L ^ * } | f ( k ) | . \end{align*}


\begin{align*} G ( x , y ) & = \frac { e ^ { i k _ { - } \vert x \vert } } { \sqrt { \vert x \vert } } G ^ { \infty } ( \hat { x } , y ) + G _ { R e s } ( x , y ) , \\ \nabla _ y G ( x , y ) & = \frac { e ^ { i k _ { - } \vert x \vert } } { \sqrt { \vert x \vert } } H ^ { \infty } ( \hat x , y ) + H _ { R e s } ( x , y ) , \end{align*}


\begin{align*} ( ( \mathcal C ^ + - z ^ 2 ) ^ { - 1 } v ) ( u ) = \frac { 1 } { 2 \pi } \int _ \R \int _ { S ^ 1 } ( \lambda ^ 2 - z ^ 2 ) ^ { - 1 } v _ \lambda ( \sigma ) u _ { - \lambda } ( \sigma ) \ , d \sigma \ , d \lambda \ , . \end{align*}


\begin{align*} & \left ( \begin{array} [ c ] { c c c } \varphi \left ( \mathrm { i } s - 0 \right ) & , & \psi \left ( \mathrm { i } s - 0 \right ) \end{array} \right ) \\ & = \left ( \begin{array} [ c ] { c c c } \varphi \left ( \mathrm { i } s + 0 \right ) - 2 \mathrm { i } \pi \delta \left ( s , t \right ) \psi \left ( \mathrm { i } s + 0 \right ) & , & \psi \left ( \mathrm { i } s + 0 \right ) \end{array} \right ) , \end{align*}


\begin{align*} \int _ X \exp ( \kappa q ) \ , d \mu _ \alpha & = 1 + c \kappa \int _ { 0 } ^ { \infty } \exp ( c \kappa t ) \mu _ \alpha ( q \ge c t ) \ , d t \\ & \le \exp ( c \kappa ) + c \kappa \int _ { 1 } ^ { \infty } \exp ( c \kappa t ) \tau ^ t \ , d t . \end{align*}


\begin{align*} \ln ( f ( x ) ) = h ( d _ { H } ( o , x ) ) , \end{align*}


\begin{align*} x y C ' ( x ) + m c _ m \frac { x ^ m y } { 1 - x ^ m y } = n , y C ( x ) + c _ m \frac { x ^ m y } { 1 - x ^ m y } = N \quad x ^ m y < 1 . \end{align*}


\begin{align*} d ^ { 2 } ( f ) ( x , y , z ) & = f \left ( \alpha ( x ) , [ y , z ] \right ) + f \left ( \alpha ( y ) , [ x , z ] \right ) + f \left ( \alpha ( z ) , [ x , y ] \right ) \\ + & \rho ( \alpha ( x ) ) f ( y , z ) + \rho ( \alpha ( y ) ) f ( x , z ) + \rho ( \alpha ( z ) ) f ( x , y ) . \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*} f ( x ) = \int _ 0 ^ x g ( s ) \ , d s , x \in [ 0 , 1 ] . \end{align*}


\begin{align*} & P _ A \left ( \sigma ( x _ 1 ) = y _ 1 , \ldots , \sigma ( x _ p ) = y _ p \right ) \\ & = \frac { 1 } { 2 } \frac { \prod _ { i = 1 } ^ p w _ A ( x ^ i , y ^ i ) } { Y _ { m _ 1 , m _ 2 } ( A ) } \sum _ { \theta \in \{ 0 , 1 \} ^ 2 } ( - 1 ) ^ { ( \theta _ 1 + m _ 1 + 1 ) ( \theta _ 2 + m _ 2 + 1 ) } \frac { \partial } { \partial W _ { x ^ 1 , y ^ 1 } } \ldots \frac { \partial } { \partial W _ { x ^ p , y ^ p } } \det \limits _ { x , y \in \mathbb { T } _ { m _ 1 , m _ 2 } } K _ { A , \theta } ( x , y ) . \end{align*}


\begin{align*} G ( s ) : = \ , s \ , \log ^ + s s \ge \ , 0 \ , , \end{align*}


\begin{align*} \bar v ( x ) \coloneqq \begin{cases} \displaystyle \frac { E } { 1 - \nu ^ 2 } \frac { | x | ^ 2 } { 1 6 \pi } \log | x | ^ 2 & \\ [ 2 m m ] 0 & \end{cases} \end{align*}


\begin{align*} x \mapsto \lim \limits _ { n \to \infty } \sum _ { i = 0 } ^ n ( p x ) ^ i / i ! \end{align*}


\begin{align*} \left . \mathbb { K } _ { N } f _ { N } \right \vert _ { I _ { n } } & = { \displaystyle \sum \limits _ { m = 1 } ^ { N } } c _ { m } ^ { 2 } \left [ \int _ { S } g _ { n } \left ( s \right ) g _ { m } \left ( s \right ) \mathrm { d } s \right ] f _ { m } \\ & = \left ( \boldsymbol { K } _ { N } \boldsymbol { f } _ { N } \right ) _ { n } \end{align*}


\begin{align*} \phi = T _ { \theta } ^ { - 1 } A _ { \omega } T _ { \theta } + T _ { \theta } ^ { - 1 } d T _ { \theta } . \end{align*}


\begin{align*} \sum _ { x = 4 } ^ { a } \sum _ { y = b + 1 } ^ { \infty } u _ { x , y , a , b } ~ = ~ & \sum _ { x = 4 } ^ { a } \sum _ { y = b + 1 } ^ { \infty } \Bigl ( x \sqrt { ( x \wedge a ) ( x \wedge b ) } - y \sqrt { ( y \wedge a ) ( y \wedge b ) } \Bigr ) ^ 2 p ( x ) p ( y ) \\ [ . 5 e m ] = ~ & \sum _ { x = 4 } ^ { a } \sum _ { y = b + 1 } ^ { \infty } \bigl ( x ^ 2 - y \sqrt { a b } \bigr ) ^ 2 p ( x ) p ( y ) . \end{align*}


\begin{align*} - 1 \leq \sum _ { i = 1 } ^ { n } ( x _ i - \tfrac 1 2 ) \leq 0 . \end{align*}


\begin{align*} \pi _ \hbar = \exp ( \hbar [ X , \ , \cdot \ , ] _ S ) \widetilde { \pi } _ \hbar , \end{align*}


\begin{align*} \lim _ { w \to - i \infty } \rho _ { k + 1 , D } ( \tau , w ) = 0 , \lim _ { \tau \to i \infty } \rho _ { k + 1 , D } ( \tau , w ) = 0 . \end{align*}


\begin{align*} y \frac { \partial } { \partial y } { \rm L i } _ 2 ( y ) \ : = \ : - \ln ( 1 - y ) , \end{align*}


\begin{align*} I ( \epsilon ) = \int _ { \square _ { \epsilon } } g ( x ) \ , d x , \end{align*}


\begin{align*} \int \sqrt { \omega } \mathcal { C } ( f ) \ , \varphi \ , \dd \omega = \int J _ M ( f ) \ , \varphi ' \ , \dd \omega . \end{align*}


\begin{align*} \psi _ { j } ( x , t , k ) : = \phi _ { j } ( x , t , k ) e ^ { ( i k x + 4 i k ^ 3 t ) \sigma _ 3 } , j = 1 , 2 , \end{align*}


\begin{align*} J _ { 2 n } = \begin{pmatrix} & w _ n \\ - w _ n & \end{pmatrix} , \quad { \rm w i t h } w _ n = \begin{pmatrix} & & & 1 \\ & & 1 & \\ & \cdots & & \\ 1 & & & \end{pmatrix} _ { n \times n } . \end{align*}


\begin{align*} \nu _ \varepsilon ( x , t ) \cdot \nabla \sigma _ \varepsilon ( x , t ) = \varepsilon \bar { g } ( x , t ) \sqrt { 1 + \varepsilon ^ 2 | \bar { \tau } _ \varepsilon ^ i ( x , t ) | ^ 2 } \geq \varepsilon \bar { g } ( x , t ) \geq c \varepsilon \end{align*}


\begin{align*} \limsup _ { t \rightarrow \infty } \norm { \nabla f ( X ( \omega , t ) ) } = \liminf _ { t \rightarrow \infty } \norm { \nabla f ( X ( \omega , t ) ) } = \lim _ { t \rightarrow \infty } \norm { \nabla f ( X ( \omega , t ) ) } = 0 , \forall \omega \in \Omega _ { \mathrm { c o n v } } . \end{align*}


\begin{align*} \begin{aligned} \lim _ { n \to \infty } [ r _ { n } \max \limits _ { \theta \in [ 0 , \pi ] } \omega ( r _ { n } , \theta ) ^ { 2 } ] = 0 . \end{aligned} \end{align*}


\begin{align*} ( \gamma _ 1 * \gamma _ 2 ) ( t ) = \begin{cases} \gamma _ 1 ( t ) & \hbox { f o r } t \in [ 0 , \ell _ 1 ] \\ \gamma _ 2 ( t - \ell _ 1 ) & \hbox { f o r } t \in [ \ell _ 1 , \ell _ 1 + \ell _ 2 ] . \end{cases} \end{align*}


\begin{align*} r = \left ( \frac { q } { p } + \frac { n } { p - n } \right ) ^ { - 1 } \end{align*}


\begin{align*} A _ j = \mathfrak a ( v ( j ) , b ( j ) ) + \sum _ { k = 1 } ^ m r ^ { ( k ) } _ j \end{align*}


\begin{align*} \frac { 1 - \nu ^ 2 } E \Delta ^ 2 v ^ p = - \theta \end{align*}


\begin{align*} \tilde { D } ( x ) \left | I \right \rangle = { \tilde \delta } ( x ) \prod _ { k = 1 } ^ m \mathbf { h } ( x , x _ { k } ) | I \rangle + \frac { 2 + 2 x } { 1 + 2 x } \sum _ { j \in I } \left [ \frac { { \mathcal { A } } ^ { I } _ j } { 1 + x + x _ { j } } - \frac { \tilde { \mathcal { D } } ^ { I } _ j } { x - x _ { j } } \right ] B ( x ) \left | I \setminus \left \{ j \right \} \right \rangle \ , , \end{align*}


\begin{align*} A ( D , \mathtt { m } ) : = \omega ( D ) + \tfrac { \hbar } { 2 } ( - \Delta _ g ) + \tfrac { 1 } { 2 } \mathtt { m } p ' ( \mathtt { m } ) \ , , \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \frac { h _ { R , l } ( z _ n , v _ n ) } { h _ { R , r } ( z _ n , v _ n ) } \ = \ \infty \mbox { a n d } v _ n \ \le \ \psi ( z _ n ) . \end{align*}


\begin{align*} \mathcal { G } ( F ) = \left \{ ( u , F ( u ) ) : F ( u ) \in \mathcal { K } ( \mathbb { R } ) \right \} \subset [ 0 , 1 ] \times \mathcal { K } ( \mathbb { R } ) . \end{align*}


\begin{align*} C \left ( \mathbf { p } , \mathbf { q } \right ) ( \tau ) + C \left ( \mathbf { u } , \mathbf { v } \right ) ( \tau ) = 0 , \end{align*}


\begin{align*} H [ n ] \circ k _ { n + 1 } + k _ { n + 1 } \circ H [ n ] ^ T + \Theta [ n ] + k \circ \overline { \Theta [ n ] } \circ k = 0 , \end{align*}


\begin{align*} E < 0 H \varphi = E \varphi . \end{align*}


\begin{align*} R ^ T \bar { \nu } = R \bar { \nu } = \Bigl \{ I _ n - d \overline { W } \Bigr \} ^ { - 1 } \bar { \nu } = \bar { \nu } , \quad \sum _ { i = 1 } ^ n R _ { i j } \bar { \nu } _ i = \bar { \nu } _ j \quad \overline { N _ T } \end{align*}


\begin{align*} \mathbf { v } ( k ) = \left ( \begin{array} [ c ] { c c c } \varphi \left ( k \right ) & , & \psi \left ( k \right ) \end{array} \right ) , \end{align*}


\begin{align*} f ( \Phi _ x ( t , s ) ) = f \left ( \Phi ^ u _ { \Phi ^ c _ x ( s ) } ( \beta _ { x } ( s ) t ) \right ) = \Phi ^ u _ { f \circ \Phi ^ c _ x ( s ) } \left ( \lambda ^ u _ { \Phi ^ c _ x ( s ) } \beta _ { x } ( s ) t \right ) . \end{align*}


\begin{align*} y ^ { 0 0 } = & \phi \Omega ( ( \phi - z \phi _ z ) ^ 2 + z ^ 2 \phi \phi _ { z z } ) + ( r ^ 2 - s ^ 2 ) \phi \left ( \phi ^ 2 \phi _ { s s } + 2 z \phi ( \phi _ s \phi _ { s z } - \phi _ z \phi _ { s s } ) + z ^ 2 \delta _ 3 \right ) \\ y ^ { 0 i } = & \phi \left [ - ( \Omega + s \phi _ s ) ( \phi \Omega ) _ z + ( r ^ 2 - s ^ 2 ) ( \phi ( \phi _ s \phi _ { s z } - \phi _ z \phi _ { s s } ) + z \delta _ 3 ) \right ] u ^ i + \phi ^ 2 [ \phi _ s \Omega _ z - \phi _ { s z } \Omega ] x ^ i \end{align*}


\begin{align*} J _ { 0 , K } ^ E ( Z , Z ' _ Y ) = 0 . \end{align*}


\begin{align*} V = \bigoplus _ { \alpha \in J _ { 0 } } \bigoplus _ { i = 1 } ^ { k _ { \alpha } } \C _ { e _ { \alpha } q ^ { - i + 1 } } , W _ { 0 } = \bigoplus _ { \alpha \in J _ { 0 } } \C _ { e _ { \alpha } } , W _ { 1 } = \bigoplus _ { \alpha \in J _ { 1 } } \C _ { e _ { \alpha } } , \end{align*}


\begin{align*} G _ F = \{ ( u , w ) \in X \times Y : w \in F ( u ) \} . \end{align*}


\begin{align*} \delta \Phi ( t _ 1 , 0 , x ) = 0 \end{align*}


\begin{align*} \overline { X } = \left ( ( - \infty , 0 ] \times Y _ - \right ) \cup _ { Y _ - } X \cup _ { Y _ + } \left ( [ 0 , \infty ) \times Y _ + \right ) . \end{align*}


\begin{align*} x = \pi ( x ) + d ( x ) \nu ( \pi ( x ) ) , \nabla d ( x ) = \nabla d ( \pi ( x ) ) = \nu ( \pi ( x ) ) \end{align*}


\begin{align*} \int \omega ^ { 3 / 2 } \mathcal { C } ( f ) \ , \varphi \ , d \omega = \int J _ E ( f ) \ , \varphi ' \ , \dd \omega , \end{align*}


\begin{align*} \begin{aligned} \sum _ \rho \Big ( \frac { \sin \frac 1 2 ( \gamma - \tau ) \log x } { \frac 1 2 ( \gamma - \tau ) \log x } \Big ) ^ { \ ! 2 } & = \frac { \log \frac { \tau } { 2 \pi } } { \log x } - \frac 2 { \log x } \sum _ { n \le x } \frac { \Lambda ( n ) } { n ^ { 1 / 2 } } \Big ( 1 - \frac { \log n } { \log x } \Big ) \cos ( \tau \log n ) \\ & + O \Big ( \frac 1 { \tau \log x } \Big ) + O \Big ( \frac { x ^ { 1 / 2 } } { ( \tau \log x ) ^ 2 } \Big ) . \end{aligned} \end{align*}


\begin{align*} D f ( \Phi ^ c _ x ( s ) ) \cdot \beta _ x ( s ) v ^ u \left ( \Phi ^ c _ x ( s ) \right ) = \lambda ^ u _ x \beta _ { x _ 1 } ( \lambda ^ c _ x s ) v ^ u \left ( \Phi ^ c _ { x _ 1 } ( \lambda ^ c _ x s ) \right ) . \end{align*}


\begin{align*} \gamma = 1 , \ , \alpha = \nu + \frac 1 2 , \ , \beta ^ 2 / \gamma = \nu ^ 2 - 1 / 4 . \end{align*}


\begin{align*} \mu ^ f _ { s , t } = \lim _ { N \to \infty } \mu ^ { f , N } _ { s , t } \end{align*}


\begin{align*} \sum _ { j = 0 } ^ n d _ j a ^ \top y _ { k - j } = 0 . \end{align*}


\begin{align*} \iint _ \Omega | v | ^ 2 \Psi \ , \frac { d t } { | t | ^ { n - d - 2 } } \ , d x \approx \int _ { \R ^ d } \left ( \iint _ { ( y , t ) \in \widehat \Gamma ( x ) } | v ( y , t ) | ^ 2 \Psi ( y , t ) \ , \frac { d t } { | t | ^ { n - 2 } } \ , d y \right ) \ , d x \end{align*}


\begin{align*} \gamma = \frac { n } { 2 } - \beta . \end{align*}


\begin{align*} ( \Theta [ \Theta _ { i j } ] ) = \sum ^ { r \times n } _ { i = 1 } D _ { i i } = \sum ^ { r } _ { i = 1 } \sum ^ { n } _ { j = 1 } C _ j A _ { i i } D _ j ' = \sum ^ n _ { j = 1 } C _ j ( A ) D _ j ' . \end{align*}