\begin{align*} \delta _ { \pm } ( z ) = \exp [ \mathcal { P } ^ { \pm } \log \left ( 1 + \bar { r } _ 1 r _ 2 \right ) ] , z \in \mathbb { R } . \end{align*}


\begin{align*} K _ { j } \nu ( x ) = \int _ { X } k _ j ( x , y ) d \nu ( y ) , K ^ { * } \nu ( x ) = \sup _ { j } \vert K _ j \nu ( x ) \vert . \end{align*}


\begin{align*} \int _ { 0 } ^ { T } \int _ { \mathbb { R } ^ { 3 } } A _ t \cdot B 1 _ { ( 0 , \infty ) } ( \Phi ) \chi _ R \rho \dd x \dd t = - \int _ { 0 } ^ { T } \int _ { \mathbb { R } ^ { 3 } } \Phi _ + \frac { G } { r ^ { 2 } } \chi _ R \dot { \rho } \dd x \dd t - \int _ { 0 } ^ { T } \int _ { \mathbb { R } ^ { 3 } } ( \Phi _ + \nabla \theta \times A _ t ) \cdot \nabla \chi _ R \rho \dd x \dd t . \end{align*}


\begin{align*} \left \{ \begin{array} { c l } L ( 0 ) = & L _ 0 \\ \frac { d } { d t _ n } L = & \left [ ( L ^ n ) _ D , L \right ] \ ; . \end{array} \right . \end{align*}


\begin{align*} F ^ { \perp } : = \left \lbrace \varphi \in E ^ { \vee } \mid \varphi ( F ) = 0 \right \rbrace . \end{align*}


\begin{align*} \Big | \sum _ { x , y = 0 } ^ n ( \delta _ { x , y } - M _ { x , y } ( m ) ) f _ y ^ \star f _ x \Big | \ge { \frak C } \sum _ { x = 0 } ^ n | f _ x | ^ 2 , ( f _ 0 , \ldots , f _ n ) \in \mathbb C ^ { n + 1 } \end{align*}


\begin{align*} e ( - \rho , x ) = \overline { e ( \rho , x ) } , g ( - \rho , x ) = \overline { g ( \rho , x ) } . \end{align*}


\begin{align*} \lim _ { s \rightarrow 0 } s L _ { q } [ f ( t ) ] = \frac { f ( \infty ) } { 1 + ( 1 - q ) } . \end{align*}


\begin{align*} \nabla \times U = \mu 1 _ { ( 0 , \infty ) } ( \phi - \phi _ { \infty } ) U . \end{align*}


\begin{align*} h _ * ( \tau ) ( a ) = \int _ { X _ A } \hat a \circ h \ , d \mu _ \tau \quad a \in A . \end{align*}


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


\begin{align*} \frac { \partial \tilde y } { \partial x } = y \cdot \frac { 1 } { 4 } g ^ { - \frac { 3 } { 4 } } \cdot \biggl ( \frac { 1 } { 6 } \int \limits _ { 0 } ^ 1 f _ { x y ^ { 4 } } ^ { ( 5 ) } ( x , s y ) ( 1 - s ) ^ 3 \ , d s \biggr ) , \end{align*}


\begin{align*} \overline { X } ( t ) : = x + \int _ 0 ^ t \widetilde { \mathcal { A } } ( s ) d s + \int _ 0 ^ t \widetilde { \mathcal { B } } ( s ) d \widetilde { W } ( s ) . \end{align*}


\begin{align*} \tilde W \cap h _ r ( \tilde W ) = ( S _ 0 ^ - \cap E _ r ) \cap h _ r ( S _ 0 ^ - \cap E _ r ) = ( S _ 0 ^ - \cap h _ r ( S _ 0 ^ - ) ) \cap E _ r = \emptyset \cap E _ r = \emptyset , \end{align*}


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


\begin{align*} \sum \limits _ { x , y \in \mathbb { F } _ q , x \neq 1 } \overline { \chi _ 4 } ( x ) \chi _ 4 ( y ) \chi _ 4 ( 1 - y ) \overline { \chi _ 4 } ( x - y ) = 1 - \rho . \end{align*}


\begin{align*} L ( E , \epsilon ) = \max \{ \log | \lambda | + | \epsilon | , 0 \} , \forall \ E \in \Sigma _ { 2 \lambda \cos , \alpha } . \end{align*}


\begin{align*} S ( x ; N ) = \sum _ { n = 1 } ^ N x ^ n , S _ 1 ( x ; N ) = \sum _ { n = 1 } ^ N n x ^ n \textrm { a n d } S _ 2 ( x ; N ) = \sum _ { n = 1 } ^ N n ^ 2 x ^ n . \end{align*}


\begin{align*} \beta _ 1 \oplus \beta _ 2 = \begin{cases} \emptyset & i _ 1 = j _ 1 i _ 2 < j _ 1 - 1 j _ 2 \leq i _ 2 , \\ \{ ( \alpha _ { i _ 1 , j _ 2 } , \alpha _ { j _ 1 , i _ 2 } ) \} & j _ 1 - 1 \leq i _ 2 < j _ 2 . \end{cases} \end{align*}


\begin{align*} \mu _ n = \frac { 1 } { \left [ G : K \right ] } \sum _ { g \in G / K } g _ * \nu _ n \end{align*}


\begin{align*} r = \norm { A } + \norm { A _ d } , \end{align*}


\begin{align*} \lim _ { t \to + \infty } \mathrm { d i s t } _ { \mathbb { U } } ( \varphi ( t , \tau - t , \theta _ { - t } \omega ) \mathcal { D } ( \tau - t , \theta _ { - t } \omega ) , \mathcal { A } ( \tau , \omega ) ) = 0 . \end{align*}


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


\begin{align*} \lambda _ x | _ { E _ w } = \lambda _ y | _ { E _ w } , \lambda _ x ( w , w b ) = f \lambda _ x ( w b ) = s ( f ) \end{align*}


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


\begin{align*} \| \theta - \frac { 1 } { 2 } + \tilde { m } _ n \alpha \| = \inf _ { k \in [ - q _ n / 2 , q _ n / 2 ) } \| \theta - \frac { 1 } { 2 } + k \alpha \| . \end{align*}


\begin{align*} \Bigl ( \frac { 1 } { \lambda ( t ) } \mathcal { G } ( t ) \Bigr ) ^ { \prime } \leq - \frac { k _ 3 } { k _ 2 } \frac { \alpha ( t ) } { \lambda ( t ) } \Bigl [ \frac { 1 } { \lambda ( t ) } \mathcal { G } ( t ) \Bigr ] . \end{align*}


\begin{align*} \mathbf { v } ^ { 0 } = \sum \limits _ { k = 1 } ^ { N } \alpha _ { k } \mathbf { u _ { k } } , \end{align*}


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


\begin{align*} 0 = S ' ( h , v ) - S ' \big ( L _ X \bar g , X ( \bar u ) \big ) = S ' ( 0 , v - X ( \bar u ) ) = 0 \mbox { i n } M \setminus \Omega . \end{align*}


\begin{align*} X _ j ( X _ i ^ r b _ i ) = \left \{ \begin{array} { l l } b _ 1 & \\ 0 & \end{array} \right . \end{align*}


\begin{align*} \operatorname { d i v } ( \rho ^ { - Q + 2 } Z ) = 2 \rho ^ { - Q + 2 } \end{align*}


\begin{align*} \mu _ 0 = ( m _ 1 - m _ 2 ) \omega _ 1 + \dots + ( m _ { n - 1 } - m _ n ) \omega _ { n - 1 } . \end{align*}


\begin{align*} h _ c ( K _ { 1 , m } ) - h _ c ( K _ { 1 , m - b } ) & = f _ c ( m ) - f _ c ( m - b ) + b \Delta _ c \left ( 1 \right ) \\ & : = R H S ( m , b , c ) . \end{align*}


\begin{align*} \Pi _ { \tilde { V } } e ^ { J _ { z / | z | } ^ \perp t } x = e ^ { \lambda t } ( \Pi _ { \tilde { V } } x + t \Pi _ { V } x ) . \end{align*}


\begin{align*} \kappa ( t ) = h _ { \alpha } ( t ) \cdot \ , \kappa _ 1 ( t ) , \ \kappa _ 1 ( t ) = \sum _ { k = 0 } ^ { + \infty } \ , a _ k t ^ k , \ a _ 0 \not = 0 , \ 0 < \alpha < 1 , \end{align*}


\begin{align*} x _ 0 = 1 - \frac { x _ 0 \sum _ { l \in \mathbb Z _ 0 } { \lambda _ l z _ l } } { 1 + \sum _ { l \in \mathbb Z _ 0 } { \lambda _ l z _ l } } , \ \ \ x _ j = \frac { x _ 0 \lambda _ { j } z _ { j } } { 1 + \sum _ { l \in \mathbb Z _ 0 } { \lambda _ l z _ l } } , \ \ \ j \in \mathbb Z _ 0 . \end{align*}


\begin{align*} \mathcal { P } ' \left ( \partial _ Y \Phi ( X ) \right ) = - \sum _ { i = 1 , 2 } \sqrt { c _ i } \ \langle Y _ i , X \rangle \mathcal { P } ' ( N _ i ) + \Phi ( \mathcal { P } ( B ( Y , X _ T ) ) - \mathcal { P } ( B ^ * ( Y , X _ N ) ) ) ; \end{align*}


\begin{align*} \ln \frac { \sigma } { \tau } \int _ { \Omega } q ^ 2 ( K _ H ( x ) \nabla w ^ { \sigma , \tau } _ \varepsilon | \nabla w ^ { \sigma , \tau } _ \varepsilon ) d x = \frac { 1 } { \varepsilon ^ 2 } \int _ { \Omega } \left ( u _ \varepsilon - q \ln \frac { 1 } { \varepsilon } \right ) _ + ^ { p } q d x . \end{align*}


\begin{align*} L : = \sum _ { ( P , H ) \in \Omega _ 3 } ( | W | - q ^ { d - 1 } ) \le \delta q ^ { d + 1 } . \end{align*}


\begin{align*} \begin{array} { l l l } f ^ \sharp _ L ( L ' \circ L , R \circ R ' ) ( f _ L ( s ) ( t ) ) & = & f _ L ( L ' ( L ( s ) ) ) ( t ) \\ & = & f ^ \sharp _ L ( L ' , R ' ) ( f _ L ( L ( s ) ) ( t ) ) \\ & = & f ^ \sharp _ L ( L ' , R ' ) ( f ^ \sharp _ L ( L , R ) ( f _ L ( s ) ( t ) ) . \end{array} \end{align*}


\begin{align*} \phi _ 3 ' = - ( c \phi _ 3 + \phi _ 2 g ( \phi _ 1 ) ) , \end{align*}


\begin{align*} L ^ { i } = \{ p \in \Delta _ { m - 1 } \vert \ , A ^ { i } p = A ^ { i } d \} \end{align*}


\begin{align*} \ell ( x , \alpha ) : = \langle \mu , \alpha \rangle + \Phi ^ + ( \alpha ) - \Phi ^ + ( w \alpha ) . \end{align*}


\begin{align*} { { \bf { \bar X } } ^ { \star } } = { \left [ { { \bf { \tilde A } } } \right ] _ { : , 1 : { M _ t } } } , \end{align*}


\begin{align*} L _ { f _ 1 } ( z ) = ( e ^ { \lambda \Lambda _ 1 L _ { g _ 1 } ( z ) } - 1 ) / ( e ^ { \lambda \Lambda _ 1 } - 1 ) , \end{align*}


\begin{align*} A ( \tau _ y ( x ) ) + \sum _ { n = 1 } ^ N \left [ \hat { A } \circ \hat { \sigma } ^ { - ( n + 1 ) } ( y | x ) - \hat { A } \circ \hat { \sigma } ^ { - n } ( \sigma ( y ) | x ' ) - \hat { A } \circ \hat { \sigma } ^ { - n } ( y | x ) + \hat { A } \circ \hat { \sigma } ^ { - n } ( y | x ' ) \right ] & = \\ A ( \tau _ { y , N + 1 } ( x ) ) + \sum _ { n = 1 } ^ N \left [ \hat { A } \circ \hat { \sigma } ^ { - n } ( y | x ' ) - \hat { A } \circ \hat { \sigma } ^ { - n } ( \sigma ( y ) | x ' ) \right ] \ . \end{align*}


\begin{align*} \mathcal { I } - d ^ { - } ( p , H ) & = 1 - \mathcal { I } - d ^ { - } ( p , H ^ { c } ) = 1 - \mathcal { I } - \limsup y _ n = 1 + \mathcal { I } - \liminf ( - y _ n ) \\ & = \mathcal { I } - \liminf ( 1 - y _ n ) = \mathcal { I } - \liminf x _ n \\ & = \mathcal { I } - d _ { - } ( p , H ) \end{align*}


\begin{align*} W = \left \{ \begin{aligned} & 2 \lambda _ 1 W _ 1 \Im \left [ W _ 1 \overline { W _ 2 } \right ] & \ \lambda _ 6 = \lambda _ 1 , \\ & - 6 i \lambda _ 1 W _ 1 \Re \left [ W _ 1 \overline { W _ 2 } \right ] & \ \ \lambda _ 6 = 3 \lambda _ 1 . \end{aligned} \right . \end{align*}


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


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


\begin{align*} N ( x _ a ; x _ b , x _ c , x _ d ; \alpha , \beta ) - D ( x _ a ; x _ b , x _ c , x _ d ; \alpha , \beta ) = 0 , \end{align*}


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


\begin{align*} \tilde { \mathbf { U } } = \mathbf { U } _ { x _ w } - \omega \mathbf { U } _ { x _ o } . \end{align*}


\begin{align*} \begin{aligned} F ^ { i j } ( b _ { n - 1 } ( A _ 1 v - A _ 2 w ) \pm ( u - \varphi ) ) _ { i j } \geq \ , & 0 \mbox { i n } \omega \\ b _ { n - 1 } ( A _ 1 v - A _ 2 w ) \pm ( u - \varphi ) \leq \ , & 0 \mbox { o n } \partial \omega \end{aligned} \end{align*}


\begin{align*} \omega _ { i j } ^ k = \left \{ \begin{array} { l r } - 2 \Gamma _ { i j } ^ j & k = j \neq i , \\ 2 \Gamma _ { i j } ^ j & k = i \neq j , \\ 0 & \ , . \end{array} \right . \end{align*}


\begin{align*} \ell ^ { \frac \infty 2 } ( x ) : = \ell ( w ) + \langle \mu , 2 \rho \rangle . \end{align*}


\begin{align*} \int _ { - \infty } ^ \infty z ^ i \ , \phi _ s ( z ) \ , [ \Phi _ s ( z ) ] ^ { j + \alpha - 1 } \mathrm { d } z = & \sum _ { k = 0 } ^ \infty v _ k ( j + \alpha - 1 ) \ , \int _ 0 ^ \infty z ^ i \ , \phi _ s ( z ) \ , \Phi _ s ( z ) ^ k \mathrm { d } z \\ & + \sum _ { k = 0 } ^ \infty ( - 1 ) ^ { i + k } \ , \binom { j + \alpha - 1 } { k } \ , \int _ 0 ^ \infty z ^ i \ , \phi _ s ( z ) \ , \Phi _ s ( z ) ^ k \mathrm { d } z . \end{align*}


\begin{align*} \forall \tau \in [ - h , 0 ] , \varphi ( \tau ) = \underbrace { \ell _ n ^ \top ( \tau ) \Phi _ n } _ { \varphi _ n ( \tau ) } + \tilde { \varphi } _ { n } ( \tau ) , \end{align*}


\begin{align*} \rho _ E ( \lambda , z ) = \sum _ { j = 1 } ^ l m _ j n _ j z + \sum _ { i = 0 } ^ n k _ i \lambda _ i . \end{align*}


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


\begin{align*} B _ { \gamma x , \gamma y } ( \gamma ^ + ) = B _ { \gamma x , \gamma y } ( \gamma \gamma ^ + ) = B _ { x , y } ( \gamma ^ + ) . \end{align*}


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


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


\begin{align*} & \tilde p _ x ( \ell ) = \frac { 1 } { \theta _ n } \int _ 0 ^ { \theta _ n } e ^ { - 2 \pi i \ell t / \theta _ n } \bar p _ x ( t ) \dd t , \\ & \tilde q _ x ( \ell ) = \frac { 1 } { \theta _ n } \int _ 0 ^ { \theta _ n } e ^ { - 2 \pi i \ell t / \theta _ n } \bar q _ x ( t ) \dd t , \ell \in \mathbb Z . \end{align*}


\begin{align*} \Re \mathcal { P } [ \varrho ] = P ( a , b , X , Y ) + P ( b , a , Y , X ) , \end{align*}


\begin{align*} \varrho _ { Z Z } ( L , L ) = \overline { \varrho _ { Z Z } ( L , L ) } = \varrho _ { Z Z } ( N , N ) = \overline { \varrho _ { Z Z } ( N , N ) } = a ( 1 - \tau ^ 2 ) ( \gamma ^ 2 + \alpha ^ 2 ) , \end{align*}


\begin{align*} R \cdot R ( x , J x , J x , x ; u , J u ) = f ( p ) \ , \ , Q ^ c ( g , R ) R ( x , J x , J x , x ; u , J u ) , \end{align*}


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


\begin{align*} \tilde { Y } _ t = e ^ { | z | J ^ \perp t } \tilde { Y } _ 0 + \int _ 0 ^ t e ^ { | z | J ^ \perp ( t - s ) } P ^ { - 1 } \Pi _ { \mathrm { k e r } A ^ \perp } \sigma d W _ s . \end{align*}


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


\begin{align*} G ^ \prime = \begin{pmatrix} 1 & 1 & \cdots & 1 \\ g _ { 2 1 } ^ \prime & g _ { 2 2 } ^ \prime & \cdots & g _ { 2 n } ^ \prime \\ \vdots & \vdots & & \vdots \\ g _ { k 1 } ^ \prime & g _ { k 2 } ^ \prime & \cdots & g _ { k n } ^ \prime \end{pmatrix} = \begin{pmatrix} G _ 1 ^ \prime & G _ 2 ^ \prime & \cdots & G _ n ^ \prime \end{pmatrix} . \end{align*}


\begin{align*} X _ { p , q } = \{ f \in C ( X , M _ p \otimes M _ q ) \mid f ( x _ 0 ) \in M _ p \otimes 1 _ q , \ : f ( x _ 1 ) \in 1 _ p \otimes M _ q \} . \end{align*}


\begin{align*} & \| \alpha _ * u _ * + \lambda x _ v + ( 1 - \lambda ) x _ w \| ^ 2 = \\ & = \alpha _ * ^ 2 \| u _ * \| ^ 2 + \| \lambda x _ v + ( 1 - \lambda ) x _ w \| ^ 2 + 2 \alpha _ * \langle u _ * , \lambda x _ v + ( 1 - \lambda ) x _ w \rangle = \\ & = \lambda \| x _ v \| ^ 2 + ( 1 - \lambda ) \| x _ w \| ^ 2 \end{align*}


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


\begin{align*} & k ^ { - 1 } b ( k ) = \int _ { - \infty } ^ { \infty } \bar { u } _ y ( y ) \left ( \psi ^ - _ { 1 1 } - 1 \right ) e ^ { - 2 i z y } d y + \int _ { - \infty } ^ { \infty } \bar u _ y ( y ) e ^ { - 2 i z y } d y . \end{align*}


\begin{align*} D _ \mu & = \partial _ \mu - i g \frac { \lambda ^ a } { 2 } A ^ a _ \mu \\ F ^ a _ { \mu \nu } & = \partial _ \mu A ^ a _ \nu - \partial _ \nu A ^ a _ \mu + g f ^ { a b c } A ^ b _ \mu A ^ a _ \nu \end{align*}


\begin{align*} \varrho _ { Z Z } ( L , L ) = - a \varrho _ w ^ 2 - b \varrho _ z ^ 2 \end{align*}


\begin{align*} \partial _ t U ( t , s ) x & = A ( t ) U ( t , s ) x , \ t \ge s , \\ \partial _ s U ( t , s ) x & = - U ( t , s ) A ( s ) x , \ t \ge s , \end{align*}


\begin{align*} e _ { n + 1 } ^ i = e _ n ^ i + \int _ { t _ n } ^ { t _ { n + 1 } } ( b ^ i ( X _ t ) - b ^ i ( \tilde X _ n ) ) \d t . \end{align*}


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


\begin{align*} \P _ x ( \rho > n ) = \int _ { W ^ d } \det ( f _ { n } ( y _ j - x _ i ) ) _ { i , j = 1 } ^ d d y _ 1 \ldots d y _ d . \end{align*}


\begin{align*} H ( q , r ) = \frac { \hat W _ r ( 0 ) + \hat W _ r ( 2 q ) - 2 \hat W _ r ( q ) } { \hat W _ r ( q ) } = u ( q r ) , \end{align*}


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


\begin{align*} g _ s ( i ) : = \frac { c ' } { 2 \sqrt { m _ s } } 1 6 ^ { - i } m _ s n ^ { - \alpha } , \end{align*}


\begin{align*} \frac { e ^ { p ( \zeta - z ) } } { \zeta - z } = - \int _ { p - i e ^ { i \alpha } [ 0 , + \infty ) } e ^ { \omega ( \zeta - z ) } d \omega , \quad \zeta \in e ^ { - i \alpha } [ 0 , + \infty ) , \ ; z \in \Delta . \end{align*}


\begin{align*} \liminf _ { N \to \infty } \frac { 1 } { \# ( \Lambda _ N ^ { \vec { v } } ( 1 ) ) } \sum _ { ( m , n ) \in \Lambda _ N ^ { \vec { v } } ( 1 ) } \max _ { 1 \leq i < j \leq k } d ( T ^ { ( m , n ) } x _ i , T ^ { ( m , n ) } x _ j ) = 0 . \end{align*}


\begin{align*} x y ' - x ' y = n \ , . \end{align*}


\begin{align*} b ^ n _ l = - \frac { 1 } { n } a ^ n _ l \ , . \end{align*}


\begin{align*} \left ( \left ( I + \mathbf { H } ( \varphi ) \right ) \frac { \partial y } { \partial x } \right ) \left ( \rho \right ) = - \mathbf { H } ( \frac { \partial \varphi } { \partial x } ) \left ( 1 \right ) \left ( \rho \right ) - \mathbf { H } ( \frac { \partial \varphi } { \partial x } ) \left ( y \right ) \left ( \rho \right ) . \end{align*}


\begin{align*} P _ s ( x , t ) = \frac 1 { | x | ^ N } F _ s \left ( \frac { t } { | x | ^ { 2 s } } \right ) \end{align*}


\begin{align*} \partial ^ \square H = \Delta _ \Gamma V + V \big | \nabla _ \Gamma \nu \big | ^ 2 . \end{align*}


\begin{align*} I _ \varepsilon ( t ( u ) u ) = \max \{ I _ \varepsilon ( t u ) , t > 0 \} . \end{align*}


\begin{align*} \hat { k } _ d ( \Delta ) = \sum _ { \Upsilon \in \mathcal { T } _ d ( \Delta ) } x _ { \Upsilon } \ , { \mathbf { t } _ { d - 1 } ( \Upsilon ) ^ 2 } . \end{align*}


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


\begin{align*} F ( \mathbf { v } ) = \begin{bmatrix} D & 0 & \ell \sin \Psi & - \cos \Psi \\ 0 & D & - \ell \cos \Psi & - \sin \Psi \\ \frac { - \ell \sin \Psi } { R ^ 2 } & - \kappa \ell & D + \frac { \ell \cos \Psi } { R } & \frac { \sin \Psi } { R } - \kappa U \\ D ^ 0 _ { - 1 } & 0 & 0 & 0 \\ D ^ 0 _ { 1 } & 0 & 0 & 0 \\ 0 & 0 & D ^ 0 _ { - 1 } & 0 \\ 0 & 0 & D ^ 0 _ { 1 } & 0 \end{bmatrix} \mathbf { v } . \end{align*}


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


\begin{align*} \begin{aligned} & T ^ { \star } ( k _ 1 , \ldots , k _ n ; ( \alpha , \beta ) ) - T ^ { \star } ( k _ n , k _ 1 , \ldots , k _ { n - 1 } ; ( \alpha , \beta ) ) \\ = & ( k _ n - 1 ) Z ( n | k - n + 1 ; ( \alpha , \beta ) ) + Z ( n + 1 | k - n ; ( \alpha , \beta ) ) \\ & - \sum _ { j = 0 } ^ { k _ n - 2 } Z ^ { \star } _ { I } ( j + 1 , k _ 1 , \ldots , k _ { n - 1 } , k _ n - j ; ( \alpha , \beta ) ) \end{aligned} \end{align*}


\begin{align*} \varepsilon ^ { w s _ \alpha ( \mu - \alpha ^ \vee ) } = s _ { w \alpha } \varepsilon ^ { - w \alpha ^ \vee } x < x \end{align*}


\begin{align*} \lambda _ { + , 1 } = 2 \sqrt { \delta ( 1 - \delta ) } , E _ { + , 1 } = \left \{ \begin{pmatrix} - \bar { a } _ 1 \sqrt { \frac { \delta } { \delta - 1 } } \\ \bar { b } _ 1 \sqrt { \frac { \delta } { \delta - 1 } } \\ 0 \\ 1 \end{pmatrix} , \begin{pmatrix} \bar { b } _ 1 \sqrt { \frac { \delta } { \delta - 1 } } \\ \bar { a } _ 1 \sqrt { \frac { \delta } { \delta - 1 } } \\ 1 \\ 0 \end{pmatrix} \right \} , \end{align*}


\begin{align*} \sup _ { t < T _ { \rm m a x } } \int _ { \Omega } ( n \log n ( \cdot , t ) + e ^ { - 1 } ) \le \sum _ { i = 1 } ^ { N } \sup _ { t < T _ { \rm m a x } } \int _ { \Omega \cap B _ { \frac { \delta } { 2 } } ( q _ { i } ) } ( n \log n ( \cdot , t ) + e ^ { - 1 } ) \le C . \end{align*}


\begin{align*} B ( x , t ) = \sum _ { n = 0 } ^ { \infty } b _ { n } ( x ) L _ { n } ( x - t ) e ^ { - \frac { x - t } { 2 } } \end{align*}


\begin{align*} F ( t ) = \int _ { \mathcal { O } } f _ 1 ( t , x ) ^ { \frac { \alpha } { \alpha - 1 } } d x . \end{align*}


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


\begin{align*} & \sum _ { j = 0 } ^ { \infty } r _ { j } ^ { 2 - n } \int _ { B _ { r _ { j + 1 } } \setminus B _ { r _ { j + 2 } } } | \nabla u | ^ 2 \d x \\ & \leq C \int _ { B _ { 1 } } r ^ { 2 - n } u _ r ^ 2 \d x + C \varepsilon ^ 2 \int _ { B _ { 1 } } r ^ { 3 - n } | \nabla u | ^ 2 \d x \varepsilon \leq \varepsilon _ 0 . \end{align*}


\begin{align*} R _ { n + 1 } ( s ) = \beta R _ n ( s ) - { \frac { f '' \bigl ( \zeta _ n ( s ) \bigr ) } { 2 } } R _ n ^ 2 ( s ) \end{align*}


\begin{align*} x x ' + x y ' + y y ' = - n \ , . \end{align*}


\begin{align*} C _ m ( K ) = \int _ { K } ( \Delta \omega ^ * ( q , K , \Omega ) ) ^ m \wedge \beta _ n ^ { n - m } . \end{align*}


\begin{align*} u ( x , t ) = \lim _ { k \rightarrow 0 } ( k ^ { - 1 } \varphi ( x , t ; k ) ) _ { 1 2 } . \end{align*}


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


\begin{align*} \begin{aligned} & \mathbf { x } ( \alpha ) \triangleq \mathbf { x } + \alpha \Delta \mathbf { x } , \mathbf { y } ( \alpha ) \triangleq \mathbf { y } + \alpha \Delta \mathbf { y } , \mathbf { s } ( \alpha ) \triangleq \mathbf { s } + \alpha \Delta \mathbf { s } . \end{aligned} \end{align*}


\begin{align*} K _ M : = \bigg \{ f \in L ^ p ( [ 0 , T ] , H ) : \int _ 0 ^ T \Vert f ( t ) \Vert _ V ^ p d t \leq M \bigg \} . \end{align*}


\begin{align*} \sum _ { 1 \le i < j < k \le M } \left | \det \ ! \ ! \begin{pmatrix} x _ i & x _ j & x _ k \\ y _ i & y _ j & y _ k \\ z _ i & z _ j & z _ k \end{pmatrix} \right | \sum _ { i = 1 } ^ M | z _ i | \le \ ! \ ! \ ! \sum _ { 1 \le i < j \le M } \left | \det \ ! \ ! \begin{pmatrix} y _ i & y _ j \\ z _ i & z _ j \end{pmatrix} \right | \sum _ { 1 \le i < j \le M } \left | \det \ ! \ ! \begin{pmatrix} x _ i & x _ j \\ z _ i & z _ j \end{pmatrix} \right | \end{align*}


\begin{align*} \vert S _ k \vert \leq \sum _ { 2 ^ k / c _ k < p \leq 2 ^ k } \frac { 2 ^ k } { p } & = 2 ^ k ( \log \log 2 ^ k - \log \log ( 2 ^ k / c _ k ) + O ( 1 / \log ( 2 ^ k / c _ k ) ) ) \\ & = 2 ^ k \log \frac { k \log 2 } { k \log 2 - \log c _ k } + O ( 2 ^ k / k ) \\ & = 2 ^ k \log \left ( 1 + \frac { \log c _ k } { k \log 2 - \log c _ k } \right ) + O ( 2 ^ k / k ) \\ & = \frac { 2 ^ k \log c _ k } { k \log 2 } + o ( ( 2 ^ k \log c _ k ) / k ) \end{align*}


\begin{align*} \tilde { \varphi } ( t , x ) = \varphi ( t , x , \kappa _ t ( \cdot ) ) , ( t , x ) \in \mathbb [ \tau , \vartheta ] \times \mathbb R ^ n , \kappa ( \cdot ) \in \Lambda _ 0 ( \tau , z , w ( \cdot ) ) , \end{align*}


\begin{align*} \sigma ( x , T ( x ) ) = f ( x - T ( x ) ) + f ( - x - T ( x ) ) = 0 . \end{align*}


\begin{align*} h _ 2 ( t ) = \frac { | A \oplus _ 2 ( \sqrt { t } B ) | ^ 2 } { | P _ { u ^ \bot } ( A \oplus _ 2 ( \sqrt { t } B ) ) | ^ 2 _ { n - 1 } } \end{align*}


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


\begin{align*} \int _ { 1 } ^ { \infty } \frac { d s } { \left ( \int _ { 0 } ^ { s } F ( t ) d t \right ) ^ { \frac { p } { 2 p - q + 1 } } } < \infty \quad \mbox { a n d } \int _ { 1 } ^ { \infty } \frac { s \ d s } { \left ( \int _ { 0 } ^ { s } F ( t ) d t \right ) ^ { \frac { p } { 2 p - q + 1 } } } = \infty . \end{align*}


\begin{align*} \begin{aligned} \partial _ t u - \nabla \cdot ( A \nabla u ) = & \ F , & ( 0 , T ] \times \Omega , \\ u ( 0 , \cdot ) = & \ u _ 0 , \ , & \Omega , \\ u = & \ g _ D ^ { } , & ( 0 , T ] \times \Gamma _ D ^ { } , \\ \mathbf { n } \cdot A \nabla u = & \ g _ N ^ { } , & ( 0 , T ] \times \Gamma _ N ^ { } , \end{aligned} \end{align*}


\begin{align*} X ^ { \prime } \left ( 0 \right ) = Z \left ( x , 0 \right ) k ^ { \prime } \left ( 0 \right ) = Z \left ( 0 , x ^ { b } \right ) \end{align*}


\begin{align*} \mathcal { I } - d _ - ( x , A ) + \mathcal { I } - d _ - ( x , B ) \leq \mathcal { I } - \liminf \{ a _ n + b _ n \} & \leq \mathcal { I } - \liminf \{ 1 + p _ n \} \\ & = 1 + \mathcal { I } - \liminf p _ n \\ & = 1 + \mathcal { I } - d _ - ( x , A \cap B ) \end{align*}


\begin{align*} \mathbb E \left [ \langle \tilde Z _ n ^ N ( i ) , \tilde Z _ n ^ N ( j ) \rangle _ { \mathcal H } \right ] = & \mathbb E \left [ \mathbb E \left [ \langle \tilde Z _ n ^ N ( i ) , \tilde Z _ n ^ N ( j ) \rangle _ { \mathcal H } | \mathcal F _ { ( i - 1 ) \Delta _ n } \right ] \right ] \\ = & \mathbb E \left [ \langle \mathbb E \left [ \tilde Z _ n ^ N ( i ) | \mathcal F _ { ( i - 1 ) \Delta _ n } \right ] , \tilde Z _ n ^ N ( j ) \rangle _ { \mathcal H } \right ] = 0 . \end{align*}


\begin{align*} f ( x ) = \frac { 2 \theta } { \operatorname { B } ( a , b ) \ , x ^ 3 } \exp \left ( - \frac { a \theta } { x ^ 2 } \right ) \left [ 1 - \exp \left ( - \frac { \theta } { x ^ 2 } \right ) \right ] ^ { b - 1 } . \end{align*}


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


\begin{align*} \ln \left \lbrace \prod _ { l \neq j } \frac { | \sin \pi ( \theta - \theta _ l ) | } { | \sin \pi ( \theta _ j - \theta _ l ) | } \right \rbrace = \sum _ { l \neq j } \ln | \sin \pi ( \theta - \theta _ l ) | - \sum _ { l \neq j } \ln | \sin \pi ( \theta _ j - \theta _ l ) | < ( 2 s q _ { n - n _ 0 } - 1 ) \epsilon . \end{align*}


\begin{align*} \tilde z = \Psi _ 2 ^ { - 1 } ( \hat z ) = \tilde z _ 3 + \frac { ( \tilde z _ 3 - \tilde z _ 1 ) ( \tilde z _ 2 - \tilde z _ 3 ) ( \hat z _ 2 - \hat z _ 1 ) ( \hat z - \hat z _ 3 ) } { ( \tilde z _ 2 - \tilde z _ 1 ) ( \hat z _ 2 - \hat z _ 3 ) ( \hat z - \hat z _ 1 ) - ( \tilde z _ 2 - \tilde z _ 3 ) ( \hat z _ 2 - \hat z _ 1 ) ( \hat z - \hat z _ 3 ) } \ , . \end{align*}


\begin{align*} \begin{cases} H _ 0 \psi \ , \ , = - \triangle \psi , \\ H _ N \psi = - \triangle \psi + \displaystyle \sum _ { j = 1 } ^ N v _ j \langle \psi , \delta _ j \rangle \delta _ j , N \geq 1 , \end{cases} \end{align*}


\begin{align*} Q _ { j , + } ( x ; k ) = ( I + J ) Q _ { j , - } ( x ; k ) + D _ j , k \in \mathbb { R } \cup i \mathbb { R } , \end{align*}


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


\begin{align*} | \langle B ^ { \circ } S _ { T - t } ^ { \circ } x ^ { \circ } , u \rangle | = | \langle x ^ { \circ } , S _ { T - t } B u \rangle | \leq C p _ { \gamma _ U } ( u ) \end{align*}


\begin{align*} \lim _ { z \downarrow 0 } \int _ 0 ^ \infty w _ z ( \lambda ) \lambda ^ { 4 ( 1 - s + w ) } \frac { d \lambda } { \lambda } \int _ 0 ^ 1 d u \int _ 0 ^ \infty \frac { d t } { t ^ 3 } \phi ( n _ { u } ^ t a _ { \frac { 1 } { t } } ) \sum _ { m = 1 } ^ \infty \hat { F } _ { \pm } ( \lambda a _ { \frac { 1 } { t } } \cdot ( 0 , 0 , 0 , m ) ) \end{align*}


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


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


\begin{align*} \beta = \alpha - \alpha _ s a _ s = 1 \end{align*}


\begin{align*} p ^ { \varepsilon } ( x , t ) = \mathrm { e x p } \Big ( - \frac { 1 } { \varepsilon } y ( t ) h ( x ) \Big ) q ^ { \varepsilon } ( x , t ) , \end{align*}


\begin{align*} F ^ * x = ( x _ \epsilon : \epsilon \ne \vec 0 ) \end{align*}


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


\begin{align*} \delta _ x ( f ) : = \sup _ { \eta , \xi : \ \eta _ { x ^ c } = \xi _ { x ^ c } } \abs { f ( \eta ) - f ( \xi ) } \end{align*}


\begin{align*} \begin{aligned} \| \eta ( t ) - \eta _ a ( t ) \| _ 2 \le & \ , \| \eta ( t ) - z ( t ) \| _ 1 + \gamma \bar { k } _ 3 ( L _ r , \delta ) \\ = & \ , f ( t ) + \gamma \bar { k } _ 3 ( L _ r , \delta ) . \end{aligned} \end{align*}


\begin{align*} \sum _ { \gamma } f _ { \frac { 1 } { 2 } } ( t - \gamma ) \leq \frac { \widehat { h } ( 0 ) } { 2 \pi } \log t + \frac { 1 } { \pi } \sum _ { n = 2 } ^ \infty \frac { \Lambda ( n ) } { \sqrt { n } } \left | \widehat { h } \left ( \frac { \log n } { 2 \pi } \right ) \right | . \end{align*}


\begin{align*} \delta _ 1 ( X ) = 2 \mu F ( \mu ) - 2 J ( \mu ) \delta _ 2 ( X ) = \mu - 2 J ( m ) , \end{align*}


\begin{align*} Q ^ c ( g , R ) ( \pi ; \bar { \pi } ) = Q ( g , R ) ( \pi ; \bar { \pi } ) + Q ( g , R ) ( \pi ; J \bar { \pi } ) , \end{align*}


\begin{align*} \lim _ { n \to \infty } \int _ { t _ 1 } ^ { t _ 2 } \sum _ { i = m } ^ { n } \sum _ { j = n + 1 } ^ { \infty } g _ i V _ { i , j } \psi _ i ( s ) \psi _ j ( s ) d s = 0 \end{align*}


\begin{align*} a \mp _ 1 b & = \phantom { \pm _ 2 } x \\ b \pm _ 1 a & = \pm _ 2 1 \end{align*}


\begin{align*} x y ' - x ' y = 0 \ , . \end{align*}


\begin{align*} g ( x ) = \prod _ { j = 1 } ^ d \lambda _ j \int _ { x _ d } ^ \infty d b _ d \int _ { x _ { d - 1 } } ^ { b _ { d } } d b _ { d - 1 } \cdots \int _ { x _ 1 } ^ { b _ { 2 } } d b _ 1 g ( b _ 1 , \ldots , b _ d ) . \end{align*}


\begin{align*} \hat { k } _ d ( \Delta ) = \prod _ { q , i } x _ { q , i } ^ { e ( q , i ) } \prod _ { \rho \in \Gamma _ { d - 1 } } D _ { m ( \rho ) , k ( \rho ) } \end{align*}


\begin{align*} u = u _ { \infty } , B ( x , t ) = U ( x - u _ { \infty } t ) . \end{align*}


\begin{align*} \Phi _ s ( z ) = \begin{cases} \frac { \Gamma ( 1 / s , ( - z ) ^ s ) } { 2 \Gamma ( 1 / s ) } , & z \leq 0 , \\ 1 - \frac { \Gamma ( 1 / s , z ^ s ) } { 2 \Gamma ( 1 / s ) } , & z > 0 . \end{cases} \end{align*}


\begin{align*} d Y _ t = \kappa \partial _ { x x } Y _ t d t + Q ^ { \frac 1 2 } d W _ t . \end{align*}


\begin{align*} \Phi : \R \to \R , \qquad \Phi ( u ) : = \begin{cases} u - u \log ( u ) - 1 , & u > 0 , \\ \ - 1 , & u \leq 0 , \end{cases} \end{align*}


\begin{align*} \hat { A } ^ * ( y | x ) = A ( y _ 2 y _ 1 x ) + \left [ A ( y _ 1 x ' ) - A ( y _ 2 x ' ) \right ] . \end{align*}


\begin{align*} L _ X g = h . \end{align*}


\begin{align*} \mathbf { Y } = \mathbf { F X F } ^ \top , \end{align*}


\begin{align*} \left ( t _ i ^ 2 + c - c \ , ( U ^ \top U ) _ { i , i } \right ) U ^ \top _ { i , h } = \mu _ h U ^ \top _ { i , h } . \end{align*}


\begin{align*} \| v \| _ { - m , p ' } : = \sup _ { 0 \not = u \in W ^ { p , m } _ 0 } \frac { | \langle u , v \rangle | } { \| u \| _ { m , p } } . \end{align*}


\begin{align*} \Phi _ s ( z ) ^ \alpha = \sum _ { m = 0 } ^ \infty \sum _ { r = 0 } ^ m ( - 1 ) ^ { m + r } \ , \binom { \alpha } { m } \ , \binom { m } { r } \ , \Phi _ s ( z ) ^ r . \end{align*}


\begin{align*} S \cap [ q ] = ( U \cap [ q ] ) \cup ( T \cap [ q ] ) = ( T \cap F _ { \ell + 1 } ) \cup T = T . \end{align*}


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


\begin{align*} \mu = \frac { 1 } { \left [ G : K \right ] } \sum _ { g \in G / K } g _ * \nu = \frac { 1 } { \left [ G : H \right ] } \sum _ { g \in G / H } g _ * \nu . \end{align*}


\begin{align*} \lim _ { A \rightarrow \infty } \int _ { | x | > A } | g _ { b , \gamma } f ( x ) | ^ p w ( x ) d x = 0 , \end{align*}


\begin{align*} \pi _ j : = \lim _ { n \to \infty } { \mathcal { Q } } _ { i j } ( n ) = j q ^ { j - 1 } \nu _ { j } i , j \in { \mathcal { E } } , \end{align*}


\begin{align*} \pi ( B _ { d _ X } ( x , r ) ) = B _ { d _ Y } ( \pi ( x ) , r ) , \forall x \in X , \forall r > 0 . \end{align*}


\begin{align*} u _ N = z + v _ N \end{align*}


\begin{align*} p \ast q : = \sum _ { n \geq 1 } p _ n q _ n T ^ n . \end{align*}


\begin{align*} h ' ( x ) = \beta ^ { q + 1 } ( x + v ) ^ { q + 1 } h ( \mu _ { \alpha , \beta , u , v } ( x ) ) , \end{align*}


\begin{align*} \hat W _ r ( k ) = \frac { 2 } { \pi k } \sin ( 2 \pi k r ) = \frac { 1 } { k } f ( k r ) , \end{align*}


\begin{align*} e ( \rho , x ) = e ^ { i \rho x } + \int _ { x } ^ { \infty } A ( x , t ) e ^ { i \rho t } d t \end{align*}


\begin{align*} \mathcal O _ K ^ { \times , + } : = \{ a \in \mathcal O _ K ^ \times , \sigma _ i ( a ) > 0 , 1 \le i \le s \} , \end{align*}


\begin{align*} \rho ^ { + } ( A ) = \inf _ { G \supseteq A } \rho ( G ) , \end{align*}


\begin{align*} F ( x ) + F ( x + a ) = b \end{align*}


\begin{align*} \mu _ d ( n ) = \prod _ { i = 1 } ^ k ( - 1 ) ^ { m _ i } \binom { d } { m _ i } . \end{align*}


\begin{align*} \begin{pmatrix} w ( t ) \\ \overline { w ( t ) } \end{pmatrix} & = P Q ( t ) P ^ { - 1 } \begin{pmatrix} w ( 1 ) \\ \overline { w ( 1 ) } \end{pmatrix} \\ & + P Q ( t ) \int ^ t _ 1 Q ( \tau ) ^ { - 1 } P ^ { - 1 } \begin{pmatrix} S _ 2 + e ^ { 3 i \lambda _ 1 ( \Phi _ 1 - \theta _ 1 ) } R _ 2 \\ \overline { S _ 2 + e ^ { 3 i \lambda _ 1 ( \Phi _ 1 - \theta _ 1 ) } R _ 2 } \end{pmatrix} ( \tau ) \ d \tau \end{align*}


\begin{align*} { I } _ { l } ^ { k , \pm } = \int ( \psi ^ { k , \pm } _ l \psi _ j \mathfrak a ) ( s ) e ^ { i \lambda \mathcal P ( x , y , s ) } d s , \ \ { I } _ { \ast } = \int ( \psi _ \ast \psi _ j \mathfrak a ) ( s ) e ^ { i \lambda \mathcal P ( x , y , s ) } d s . \end{align*}


\begin{align*} | f ( \lambda ) | \leq \underbrace { \frac { 1 } { | a _ k | } } _ { = : C _ p } \sup _ { | z | = 1 } | p ( \lambda + z \cdot v ) f ( \lambda + z \cdot v ) | \leq C _ p \sup _ { \mu \in \C ^ 2 , | \mu | \leq 1 } | p ( \lambda + \mu ) f ( \lambda + \mu ) | , \end{align*}


\begin{align*} \operatorname { g r a d } \operatorname { d i v } \textbf { \textit { u } } = g ^ { i j } \Bigl ( \frac { \partial ^ 2 u ^ k } { \partial x _ j \partial x _ k } + \Gamma ^ l _ { k l } \frac { \partial u ^ k } { \partial x _ j } + \frac { \partial \Gamma ^ l _ { k l } } { \partial x _ j } u ^ k \Bigr ) \frac { \partial } { \partial x _ i } . \end{align*}


\begin{align*} r _ 1 ( z ) : = - \frac { b ( k ) } { 2 i k a ( k ) } , r _ 2 ( z ) : = \frac { 2 i k b ( k ) } { a ( k ) } , z \in \mathbb { R } , \end{align*}


\begin{align*} \begin{aligned} \min _ { \substack { { \alpha _ z , \ss _ { z } , { \boldsymbol { \theta } } } } } & \mathcal { L } , \end{aligned} \end{align*}


\begin{align*} \partial _ x U ( t , x , \nu ) = P _ t , \quad \partial _ { x x } U ( t , x , \nu ) = \partial _ { x \nu } U ( t , x , \nu ) = 0 , \quad \partial _ { \nu } U ( t , x , \nu ) = \partial _ { \nu } \Phi ( t , \nu ) , \quad \partial _ { \nu \nu } U ( t , x , \nu ) = \partial _ { \nu \nu } \Phi ( t , \nu ) \end{align*}


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


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


\begin{align*} \lambda = \lambda _ S ( G ) > 1 \end{align*}


\begin{align*} \begin{cases} d Y _ t = ( \Pi _ { V _ j ^ \perp } L _ { \Pi _ { V _ j } x _ 0 } \Pi _ { V _ j ^ \perp } ) Y _ t d t + \Pi _ { V _ j ^ \perp } \sigma d W _ t \\ Y _ 0 = \Pi _ { V _ j ^ \perp } x _ 0 . \end{cases} \end{align*}


\begin{align*} F _ k ( \theta , E ) = \left ( \begin{array} { c c } \tilde { P } _ k ( \theta , E ) & - \tilde { P } _ { k - 1 } ( \theta + \alpha , E ) \cos { \pi \theta } \\ \tilde { P } _ { k - 1 } ( \theta , E ) \cos { \pi ( \theta + ( k - 1 ) \alpha ) } & - \tilde { P } _ { k - 2 } ( \theta + \alpha , E ) \cos { \pi \theta } \cos { \pi ( \theta + ( k - 1 ) \alpha ) } \end{array} \right ) . \end{align*}


\begin{align*} \begin{aligned} \pi ( B ( x ' , r ) ) & = L _ { n } C _ { h ' } C _ { h ^ { - 1 } } L _ { n ^ { - 1 } } ( \pi ( B ( x , r ) ) ) , \end{aligned} \end{align*}


\begin{align*} w _ n ^ { ( i ) } ( s ) = \left [ { { \frac { f _ n ( q s ) } { q } } } \right ] ^ { i - 1 } w _ n ( s ) , \end{align*}


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


\begin{align*} \dfrac { \partial b _ { 1 , \delta } ( x ) } { \partial x } = \int _ { 0 } ^ { 1 } \left . \dfrac { \partial h ( s ) } { \partial s } \right | _ { x + \delta \sin ( 2 \pi t ) } \sin ( 2 \pi t ) d t \end{align*}


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


\begin{align*} p \Delta \left ( H - \frac { \mu } { n } \right ) ^ { p - 1 } + p \lvert A \rvert ^ 2 \left ( H - \frac { \mu } { n } \right ) ^ { p - 1 } - n ^ 2 H \left ( \left ( H - \frac { \mu } { n } \right ) ^ p + \varsigma \right ) = 0 , \end{align*}


\begin{align*} N ( u '' , V ' ) = N ( u , V ' ) . \end{align*}


\begin{align*} \phi _ { f ' } ( x ) = \alpha \left ( a ( t x + u ) ^ { q + 1 } + b ( t x + u ) ^ q ( v x + w ) + c ( t x + u ) ( v x + w ) ^ q + d ( v x + w ) ^ { q + 1 } \right ) , \end{align*}


\begin{align*} \lim _ { R \searrow 0 } F ( R ) & = \int _ { R _ 0 } ^ 0 \frac { 1 } { f ( z ) } \ , \mathrm { d } z = - \int _ 0 ^ { R _ 0 } \frac { 1 } { f ( z ) } \ , \mathrm { d } z > 0 , \\ \lim _ { R \nearrow \infty } F ( R ) & = \int _ { R _ 0 } ^ { \infty } \frac { 1 } { f ( z ) } \ , \mathrm { d } z < 0 \end{align*}


\begin{align*} [ b _ x ] = \max \{ [ y ] \mid y \leq x \} = \max \{ [ y ] \mid y \leq x y \} . \end{align*}


\begin{align*} L _ X g ( V , \cdot ) = h ( V , \cdot ) \mbox { i n t h e c o l l a r n e i g h b o r h o o d o f } \Sigma . \end{align*}


\begin{align*} | \sigma | = | \sigma ' | . \end{align*}


\begin{align*} \langle A ( t , u ) , v \rangle = - & \int _ { \mathcal { O } } \Big \{ \sum _ { i = 1 } ^ { d } a _ i \big ( t , x , u ( x ) , \nabla u ( x ) \big ) \partial _ i u ( x ) \\ & ~ ~ ~ ~ ~ ~ + a _ 0 \big ( t , x , , u ( x ) , \nabla u ( x ) \big ) v ( x ) \Big \} d x . \end{align*}


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


\begin{align*} a ^ { \varepsilon } _ { t } + ( B \times u ) ^ { \varepsilon } + \nabla Q ^ { \varepsilon } = - \mu \nabla \times b ^ { \varepsilon } \textrm { o n } \ L ^ { 2 } ( \mathbb { R } ^ { 3 } ) . \end{align*}


\begin{align*} - \nabla _ y \cdot \big ( D ( \nabla _ y \tilde { w } _ i + \nabla _ y ( S _ 0 ^ { - 1 } ) ^ T e _ i \big ) & = 0 & \mbox { i n } & Y ^ { \ast } ( t , x ) , \\ - D ( \nabla _ y \tilde { w } _ i + \nabla _ y ( S _ 0 ^ { - 1 } ) ^ T e _ i ) \cdot \nu & = 0 & \mbox { o n } & \Gamma ( t , x ) , \\ \int _ { Y ^ { \ast } ( t , x ) } J _ 0 ^ { - 1 } ( t , x , S _ 0 ^ { - 1 } ( t , x , y ) ) \tilde { w } _ i ( t , x , y ) d y & = 0 , & \tilde { w } _ i & \mbox { i s } Y \mbox { - p e r i o d i c } . \end{align*}


\begin{align*} H ( x , y ) : = ( x , y + h ( x ) ) \end{align*}


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


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


\begin{align*} \sum _ { k = 2 } ^ n \frac { \mu ( k ) } { k } ( I - S ) h _ k \to 1 - z \ H ^ p \end{align*}


\begin{align*} \zeta _ T = \zeta _ 0 + \sum _ { k = 0 } ^ { T - 1 } - \Theta _ k \Lambda _ k ' \log \gamma + ( 1 - \Theta _ k ) \Lambda _ k \log \gamma \ge 0 , \end{align*}


\begin{align*} x = \bigwedge \{ p \in P ( X ) \ : | \ : x \le p \} \end{align*}


\begin{align*} x \cdot x = x \cdot 1 = 1 , 1 \cdot x = x \end{align*}


\begin{align*} L \left ( | \varrho _ { Z Z } ( L , L ) | ^ 2 \right ) = - 2 \varrho _ { Z Z } ( L , L ) \overline { \varrho _ { Z Z } ( N , L ) } - \varrho _ { Z Z Z } ( L , L , L ) \overline { \varrho _ { Z Z } ( L , L ) } , \end{align*}


\begin{align*} A ^ { q - 1 } = \frac { r } { ( r + 1 ) ^ q } , \end{align*}


\begin{align*} \begin{aligned} \gamma _ 2 ^ \star : = & \ , \dfrac { d } { 6 M _ \mathcal { D } T e ^ { L _ \mathcal { D } \bar { t } } } \\ \gamma _ 3 ^ \star : = & \ , \dfrac { d } { 9 M _ \mathcal { D } T \left ( e ^ { L _ \mathcal { D } \bar { t } } - 1 \right ) } \\ \gamma _ 4 ^ \star : = & \ , \dfrac { d } { 6 M _ \mathcal { D } T } , \end{aligned} \end{align*}


\begin{align*} \mathbb E | X _ t ^ i | ^ 2 & = \mathbb E | X _ 0 ^ i | ^ 2 + 2 \int _ 0 ^ t X _ s ^ i \cdot b ^ i ( X _ s ) \d s + d \sigma ^ 2 t \\ & = \mathbb E | X _ 0 ^ i | ^ 2 + 2 \int _ 0 ^ t X _ s ^ i \cdot \big ( b ( X _ s ^ i ) + \gamma ^ i ( X _ s ) \big ) \d s + d \sigma ^ 2 t . \end{align*}


\begin{align*} \begin{aligned} \| f ( z ( \tau ) , \tau ) - f ( x _ a ( \tau ) , \tau ) \| \le & \ , L _ \mathcal { D } \| z ( \tau ) - x _ a ( \tau ) \| \\ \le & \ , \gamma 2 L _ \mathcal { D } M _ \mathcal { D } T \\ \| f ( x ( \tau ) , \tau ) - f ( z ( \tau ) , \tau ) \| \le & \ , L _ \mathcal { D } \| x ( \tau ) - z ( \tau ) \| . \end{aligned} \end{align*}


\begin{align*} J \varphi ( 1 ) = \frac { D } { 4 \gamma } \int _ 0 ^ 1 \varphi '' ( u ) e _ { \rm t h m } ( u ) d u , \end{align*}


\begin{align*} \frac { g _ u ^ { \ast 2 } \ast f _ r ( x ) } { g _ u ( x ) } = e ^ { \Lambda _ u } \frac { 2 } { \Lambda _ u ^ 2 } \frac { f ( x ) } { g _ u ( x ) } - \frac { 2 } { \Lambda _ u ^ 2 } \frac { f _ r ( x ) } { g _ u ( x ) } - \frac { 2 } { \Lambda _ u ^ 2 g _ u ( x ) } \sum _ { n \neq 2 } ^ \infty \frac { \Lambda _ u ^ n } { n ! } g _ u ^ { \ast n } \ast f _ r ( x ) \end{align*}


\begin{align*} g _ { \theta _ 1 } ( \omega ( s ) ) = g _ { \theta _ 2 } ( \omega ( s ) ) , s > > 1 . \end{align*}


\begin{align*} \nu ( b ) = \pi _ { J } ( \lambda _ G ( b ) ) \iff J _ 1 \subseteq J \subseteq J _ 2 . \end{align*}


\begin{align*} K _ p ( \theta ) = | | \kappa _ p ( \xi ) | | _ { \theta } \le v ( \theta , p , r ) \ | | \xi | | _ p ^ r , \ \end{align*}


\begin{align*} \lambda ^ 2 = \frac { a - 1 } { 1 - d } . \end{align*}


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


\begin{align*} \min \| F _ i ( x ) - \bar { F } _ i ( x ) \tilde { H } ( x ) \| i = 1 , \dots , n . \end{align*}


\begin{align*} N e w t o n ( F ) = C o n v ( \bigcup \limits _ { i = 0 } ^ m N e w t o n ( s _ { \lambda ^ { ( i ) } } ) ) . \end{align*}


\begin{align*} E = ( C _ i \setminus D ) \cup \{ c \} . \end{align*}


\begin{align*} \varphi ^ g ( n ) = \varphi ( n ^ g ) \forall g \in G , \ , n \in N . \end{align*}


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


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


\begin{align*} f ^ { \mathbf { s } } \cdot v _ \lambda = \sum _ { \mathbf { t } < \mathbf { s } } c _ { \mathbf { t } } f ^ { \mathbf { t } } \cdot v _ \lambda \end{align*}


\begin{align*} \mathcal { J } ' ( \overline { u } ; h ) & = \partial _ { y } J ( \overline { y } , \overline { u } ) S ' ( \overline { u } ; h ) + \partial _ { u } J ( \overline { y } , \overline { u } ) h \\ & = \langle \overline { y } - g , S ' ( \overline { u } ; h ) \rangle + \alpha \langle \overline { u } , h \rangle , \end{align*}


\begin{align*} x x ' + x ' y + y y ' = - 2 n \ , . \end{align*}


\begin{align*} U ( t , x , \nu ) : = P _ t x + \Phi ( t , \nu ) \end{align*}


\begin{align*} A _ { j , 2 k - j } ( g , h ) & = \sum _ { \ell + 2 p = j , \ \ell + 2 q = 2 k - j } c ^ { k , j } _ { \ell , p , q } ( \Lambda ^ \ell M ^ p g ) \ , \cdot \ , ( \Lambda ^ { \ell } M ^ q h ) , \end{align*}


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


\begin{align*} \Phi _ n ( x , y ) = \prod ^ n _ { \substack { j = 1 \\ ( j , n ) = 1 } } ( x - \zeta ^ j _ n y ) . \end{align*}


\begin{align*} [ T ] + \nu [ N ] = \frac { 2 } { \mu } h _ z J + \nu I , \end{align*}


\begin{align*} M = \left ( \begin{aligned} & - 1 2 . 4 6 \\ & 0 . 5 2 - 1 \end{aligned} \right ) , A = \left ( \begin{aligned} & 1 . 1 0 \\ & \ , \ , 0 5 0 \end{aligned} \right ) . \end{align*}


\begin{align*} \hat \Phi ^ \dagger ( 0 , 0 ) & = 0 , & \hat \Phi ^ \dagger _ a ( 0 , 0 ) & = 0 , \\ \hat \Phi ^ \dagger _ { a a } ( 0 , 0 ) & = 0 , & \hat \Phi ^ \dagger _ { a a a } ( 0 , 0 ) & = 6 \gamma _ 1 , \end{align*}


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


\begin{align*} { _ { 3 } } F _ { 2 } \left ( \begin{array} { c c c } \chi _ 4 , & \overline { \chi _ 4 } , & \overline { \chi _ 4 } \\ & \varphi , & \varepsilon \end{array} | 1 \right ) & = \frac { 1 } { q ^ 2 } J ( \chi _ 8 , \chi _ 8 ^ 2 ) \times 2 R e ( J ( \chi _ 8 , \chi _ 8 ) ) \times \chi _ 8 ( - 1 ) \\ & = \frac { 1 } { q ^ 2 } [ - 2 u ( - p ) ^ t ] , \end{align*}


\begin{align*} \sup _ { n \in \mathbb { N } } \mathbb { P } \left ( X _ n \notin \Upsilon \right ) \leq & \sup _ { n \in \mathbb { N } } \mathbb { P } \left ( X _ n \notin K _ M \right ) + \sum _ { k = 1 } ^ { \infty } \sup _ { n \in \mathbb { N } } \mathbb { P } \left ( X _ n \notin \Gamma _ k \right ) \\ \leq & \frac { \varepsilon } { 2 } + \sum _ { k = 1 } ^ { \infty } \frac { \varepsilon } { 2 ^ { k + 1 } } \leq \varepsilon . \end{align*}


\begin{align*} F _ { { \Phi } _ x } ( v ) = \sum _ { i = 1 } ^ m \big ( 4 \langle B ( v , e _ i ) , B ( v , e _ i ) \rangle - \langle B ( v , v ) , B ( e _ i , e _ i ) \rangle \big ) , \end{align*}


\begin{align*} \Psi ^ - _ 1 = e _ 1 + F \Psi ^ - _ 1 . \end{align*}


\begin{align*} | \varrho _ z | ^ 2 = \left ( \frac { a \sqrt { 5 } } { 2 } - 1 \right ) | w | ^ 2 . \end{align*}


\begin{align*} \Delta ^ g \left ( s ^ H - \| N \| ^ 2 \right ) = \Delta ^ g ( \| \theta \| ^ 2 ) - \delta ^ g D ^ g _ { \theta ^ \sharp } \theta - \delta ^ g D ^ g _ { J \theta ^ \sharp } J \theta , \end{align*}


\begin{align*} \frac { d | | R ^ { \nabla } | | _ { g _ s } ^ 2 } { d s } _ { \big { | } _ { s = 0 } } = - \sum _ { i , j } \langle i _ { e _ i } R ^ { \nabla } , i _ { e _ j } R ^ { \nabla } \rangle \delta g ( e _ i , e _ j ) \end{align*}


\begin{align*} m ( I _ k \cap ( H \setminus P ) ) & \leq \sum _ { n = n _ k } ^ { \infty } m ( I _ k \cap ( H _ n \setminus K ) \cap ( \bigcap _ { m = 1 } ^ { \infty } ( H _ n \setminus P _ m ) ) ) \\ & \leq \sum _ { n = n _ k } ^ { \infty } m ( H _ n \setminus P _ n ) \\ & < \sum _ { n = n _ k } ^ { \infty } \frac { 1 } { 2 ^ { n + 1 } } = \frac { 1 } { 2 ^ { n _ k } } \end{align*}


\begin{align*} f \square g ( s ) = \inf _ { a + b = s } f ( a ) + g ( b ) \mbox { f o r a l l } s \geq 0 . \end{align*}


\begin{align*} \begin{aligned} \left | ( h ( x _ a + \delta u ( \tau ) ) \pm h ( x _ a ) ) u ( \tau ) - \dfrac { b _ { 1 , \delta } ( x _ a ) } { 2 } \right | \le \\ \left | ( h ( x _ a + \delta u ( \tau ) ) - h ( x _ a ) ) u ( \tau ) \right | + \left | \dfrac { b _ { 1 , \delta } ( x _ a ) } { 2 } \right | \\ + | h ( x _ a ) | | u ( \tau ) | . \end{aligned} \end{align*}


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


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


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


\begin{align*} \int _ { F _ \nu ^ n } f _ \nu ( x _ \nu ) f _ \nu ( \delta _ \nu x _ \nu ) d \alpha _ \nu ^ n & = \alpha _ \nu ^ n ( A _ \nu \cap \delta _ \nu ^ { - 1 } A _ \nu ) \\ & \leq \min ( \alpha ^ n _ \nu ( A _ \nu ) , \| \delta _ \nu \| ^ { - n } _ \nu \alpha ^ n _ \nu ( A _ \nu ) ) . \end{align*}


\begin{align*} f ( ( a + 1 ) y + b ) - f ( a y + b ) & = y + y \int _ { u = a + \frac b y } ^ { a + 1 + \frac b y } \left ( \log ( u ) + \log ( y ) \right ) \d u \\ \end{align*}


\begin{align*} \lim _ { \delta \to 0 } \lim _ { m \to \infty } \limsup _ { \epsilon \to 0 } \lambda ^ 2 ( \epsilon ) \log \mathbb P \Big ( \sup _ { t \in [ 0 , 1 ] } | X ^ { \epsilon } ( t ) - \varphi ( t ) | < \delta , \tilde { \chi } < 1 \Big ) = - \infty . \end{align*}


\begin{align*} \rho _ { x , \sigma } ( u ) : = \rho _ x ( \prescript { \sigma ^ { - 1 } } { } ( u ) ) . \end{align*}


\begin{align*} \bigcup _ { \rho _ i > r _ i } K \ < \frac { x _ 1 } { \rho _ 1 } , \ldots , \frac { x _ 1 } { \rho _ n } \ > = \bigcup _ { \rho _ i > r _ i } T ( \rho _ 1 , \ldots , \rho _ n ) \end{align*}


\begin{align*} \gamma _ i + \bar { \gamma _ j } = 1 \ , \ , { \rm a n d } \ , \ , \gamma _ 3 + \bar { \gamma } _ 4 = 1 , \end{align*}


\begin{align*} k _ \theta \cdot f _ { - } & = \frac { 1 } { \sqrt { 2 } } ( s ( \theta ) , c ( \theta ) , s ( \theta ) , c ( \theta ) ) , \\ k _ \theta \cdot f _ { + } & = \frac { 1 } { ( 1 0 8 ) ^ { \frac { 1 } { 4 } } } ( s ( 3 \theta ) , 3 c ( 3 \theta ) , - 3 s ( 3 \theta ) , - c ( 3 \theta ) ) , \end{align*}


\begin{align*} \hat { k } _ d ( 1 \ast \Lambda ) = \prod _ { \sigma \in 1 \ast \Lambda _ { d - 1 } } \prod _ { i \in \sigma } x _ i = \prod _ { \rho \in \Lambda _ { d - 1 } } x _ 1 \prod _ { i \in \rho } x _ i = x _ 1 ^ { \abs { \Lambda _ { d - 1 } } } \prod _ i x _ i ^ { \deg _ \Lambda ( i ) } . \end{align*}


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


\begin{align*} \int _ { A _ \varepsilon } \left ( K _ H ( x ) \nabla \left ( u _ \varepsilon - q \ln \frac { 1 } { \varepsilon } \right ) | \nabla \left ( u _ \varepsilon - q \ln \frac { 1 } { \varepsilon } \right ) \right ) d x = \frac { 1 } { \varepsilon ^ 2 } \int _ { A _ \varepsilon } \left ( u _ \varepsilon - q \ln \frac { 1 } { \varepsilon } \right ) ^ { p + 1 } d x , \end{align*}


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


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


\begin{align*} \mathfrak { J } _ k : = \{ x \in E \ , : \ , d ( x ) = k \ , \ , \ , \ , i ( x ) < k \} \end{align*}


\begin{align*} \theta _ n ^ { u n i v } ( f ) ( \gamma _ 1 , \dots , \gamma _ n ) = f ( \rho ^ { u n i v } ( \gamma _ 1 ) , \dots , \rho ^ { u n i v } ( \gamma _ n ) ) . \end{align*}


\begin{align*} \prod _ { j = x _ 1 } ^ k | \cos ( \pi \theta _ j ) | \leq C ( \varepsilon ) e ^ { - ( \ln 2 - \varepsilon ) | x _ 1 - k | } , \prod _ { j = k } ^ { x _ 2 } | \cos ( \pi \theta _ j ) | \leq C ( \varepsilon ) e ^ { - ( \ln 2 - \varepsilon ) | x _ 2 - k | } c _ { n , \ell } . \end{align*}


\begin{align*} \lim _ { \Lambda \uparrow \Z ^ d } \frac { 1 } { \mu ( \eta _ { \Lambda } ) } \int \mathbf { 1 } _ { \eta _ { \Lambda } } ( \xi ) f ( \xi ) \mu ( d \xi ) = f ( \eta ) . \end{align*}


\begin{align*} f _ i : = \mathcal { F } ( ( \gamma ( N _ 3 ^ { ( 5 - 2 \alpha ) + \varepsilon } t - n ) \mathbf { 1 } _ { [ 0 , T ] } ( t ) \tilde { u _ i } ) ) , i = 1 , 2 , 3 . \end{align*}


\begin{align*} V & = g ( c ) H , \\ \partial ^ \square c & = \Delta _ \Gamma \big ( G ' ( c ) \big ) + c H V . \end{align*}


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


\begin{align*} g _ i ^ A = i \wedge A , \end{align*}


\begin{align*} | J _ { z _ { \infty } } ( s ) | \leq \sum _ { \{ i \geq 1 : s _ { \infty } ( i ) = s \} } | J _ { \infty } ( i ) | , \end{align*}


\begin{align*} \langle \cdot , \cdot \rangle : V ^ * \times V \rightarrow \mathbb { K } , a ^ * ( b ) = \langle a ^ * , b \rangle = \langle b , a ^ * \rangle , \ ; \ ; \forall a ^ * \in V ^ * , b \in V . \end{align*}


\begin{align*} \begin{aligned} \int _ { 0 } ^ { 1 / 2 } \left [ m ( x + \delta \sin ( 2 \pi t ) ) - m ( x ) \right ] \sin ( 2 \pi t ) \ , d t \\ \int _ { 0 } ^ { 1 / 2 } \left [ m ( x ) - m ( x - \delta \sin ( 2 \pi t ) ) \right ] \sin ( 2 \pi t ) \ , d t . \end{aligned} \end{align*}


\begin{align*} \Psi ^ q + v + \psi ( v , p ( s ) ) = \Psi ^ q + v + \frac 1 2 \psi _ { v v } ( 0 , p _ 0 ) [ v , v ] + O ( \norm { ( v , s ) } ^ 3 ) \end{align*}


\begin{align*} X _ { i ; j } V _ j & = \omega _ { i j } V _ j \\ \omega _ { i j ; \ell } V _ \ell & = - R _ { k \ell i j } X _ k V _ \ell + \tfrac { 1 } { 2 } \left ( h _ { i j , \ell } + h _ { i \ell , j } - h _ { j \ell , i } \right ) V _ \ell \end{align*}


\begin{align*} \nabla _ \Lambda f ( \eta ) : = \sum _ { \xi _ \Lambda } [ f ( \xi _ \Lambda \eta _ { \Lambda ^ c } ) - f ( \eta ) ] . \end{align*}


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


\begin{align*} \mu _ T ( e _ i , \mu _ T ( e _ j , e _ k ) ) = 0 \forall i , j , k \in \{ 1 , \ldots , n \} \end{align*}


\begin{gather*} P ( \abs { f } ) \le j + P \bigl \{ \abs { f } \ , 1 ( \abs { f } > j ) \bigr \} = j + \sup _ k P \bigl \{ \abs { f } \wedge k \ , 1 ( \abs { f } > j ) \bigr \} \\ = j + \sup _ k \int Q \bigl \{ \abs { f } \wedge k \ , 1 ( \abs { f } > j ) \bigr \} \ , P _ \mathcal { U } ( d Q ) \le j + \sup _ { Q \in \mathcal { U } } Q \bigl \{ \abs { f } \ , 1 ( \abs { f } > j ) \bigr \} . \end{gather*}


\begin{align*} g ( E ) = \frac { V ^ { N } ( 2 \pi m ) ^ { \frac { D N } { 2 } } } { h ^ { D N } N ! \left ( \frac { D N } { 2 } - 1 \right ) ! } E ^ { \frac { D N } { 2 } - 1 } , \end{align*}


\begin{align*} 2 \widetilde y _ n - \widetilde x _ 0 & = 2 e ^ { r y _ n } - e ^ { r x _ 0 } \\ & \leq e ^ { r ( y _ n + a ) } - e ^ { r x _ 0 } \\ & \leq e ^ { r x _ { n + 1 } } - e ^ { r x _ 0 } \\ & = \widetilde x _ { n + 1 } - e ^ { r x _ 0 } \\ & < \widetilde { x } _ { n + 1 } , \end{align*}


\begin{align*} \phi _ { \hat g } ( g H ) : = { \hat g } \phi ( { \hat g } ^ { - 1 } g H ) , \forall g H \in G / H . \end{align*}


\begin{align*} \Pi : \mathbb { R } ^ { 3 } \rtimes S ^ { 3 } \rightarrow \mathbb { R } ^ { 3 } \rtimes S O _ { 3 } \Pi \left ( x , q \right ) = \left ( x , I _ { q } \right ) \end{align*}


\begin{align*} a _ n ( u _ n , v _ n ) : = \sum _ { K \in { \mathcal T } _ n } \int _ K D ^ 2 u _ n : D ^ 2 v _ n d K , u _ n , v _ n \in X _ n . \end{align*}


\begin{align*} \psi = \sqrt { \frac { { 2 \Delta { t ^ 2 } \left ( { { p _ { E O S } } - \rho \hat c _ s ^ 2 } \right ) } } { { G \Delta { x ^ 2 } } } } . \end{align*}


\begin{align*} L = \frac { 0 . 9 \log n } { - \log \mu [ A , \infty ) } . \end{align*}


\begin{align*} \mu _ { ( \boldsymbol { x } , \boldsymbol { \phi } ( t , \boldsymbol { x } ) ) } ^ { N , i } : = k _ i ^ N ( t , \nu ^ { N , 1 } _ { \boldsymbol { x } } , \ldots , \nu ^ { N , N } _ { \boldsymbol { x } } , \lambda ^ { N , 1 } _ { \boldsymbol { \phi } ( t , \boldsymbol { x } ) } , \ldots , \lambda ^ { N , N } _ { \boldsymbol { \phi } ( t , \boldsymbol { x } ) } ) , \quad \mu ^ i : = k \big ( t , \nu ^ { N , i } _ { \boldsymbol { x } } , \Phi ( t , \nu ^ { N , i } _ { \boldsymbol { x } } ) \big ) , \end{align*}


\begin{align*} f ( [ V ] , x ) = a \circ \widehat { \varphi } ( [ I _ n , V ] , x ) = a ( D ( x ) V ) , \end{align*}


\begin{align*} \Psi ^ \pm _ 2 ( x ; z ) = \frac { 1 } { 2 i k } \begin{pmatrix} \psi ^ \pm _ { 1 2 } \\ - \bar { u } _ x \psi ^ \pm _ { 1 2 } + 2 i k \psi ^ \pm _ { 2 2 } \end{pmatrix} , \end{align*}


\begin{align*} \Phi _ { m , e } ^ { ( 1 ) } ( \lambda _ 1 , \lambda _ 2 ) \Phi _ { m , e } ^ { ( - 1 ) } ( \lambda _ 1 , \lambda _ 2 ) = \Phi _ { m e } ( \lambda _ 1 , \lambda _ 2 ) . \end{align*}


\begin{align*} c _ { k + 1 } - c _ { k } = ( c _ { k + 1 } - c _ { k } ) \left ( \sum _ { i \in \Omega _ t ^ o } a _ i - \sum _ { i \in \Omega _ t ^ e } a _ i \right ) . \end{align*}


\begin{align*} \mathbf { R } _ { X } = \frac { 1 } { L } \sum _ { l = 1 } ^ { L } \mathbf { x } \left [ l \right ] \mathbf { x } \left [ l \right ] ^ { H } = \mathbf { P } \mathbf { P } ^ { H } . \end{align*}


\begin{align*} \forall n \in \N : \ c ^ { B _ { n - 1 } ( x ( \Delta ) ) } _ { \Delta } ( \eta _ { \Lambda _ n } , \xi _ { \Delta } ) = 0 . \end{align*}


\begin{align*} \Theta _ { \epsilon } ( t ) = \frac { \epsilon } { 2 } - \epsilon ^ { 2 } \int ^ t _ 0 \theta _ { \epsilon } ( s - ) \widetilde { N } ^ { \epsilon ^ { - 2 } } ( d s ) - \frac { \epsilon ^ { 2 } } { 2 } \theta _ { \epsilon } ( t ) . \end{align*}


\begin{align*} \mathcal { I } ^ { \varphi } ( \varphi ) = \mathcal { I } ( \varphi ) . \end{align*}


\begin{align*} T _ { t } ^ { \alpha } f ( x ) = \sum _ { n = 0 } ^ { \infty } ( 2 n + d ) ^ { - \alpha } e ^ { ( 2 n + d ) i t } P _ { n } f ( x ) . \end{align*}


\begin{align*} A _ t \cdot B \dot { g } ( \Phi ) = \partial _ t \left ( g ( \Phi ) \frac { G } { r ^ { 2 } } \right ) + \nabla \cdot ( g ( \Phi ) \nabla \theta \times A _ t ) . \end{align*}


\begin{align*} 2 ( r - k ) \beta \ge s ' = \log t - s - 1 \ge \log t / 3 . \end{align*}


\begin{align*} [ J _ 0 ^ { \alpha } , J _ \pm ^ { \alpha } ] = \pm \omega ( \alpha ) J _ \pm ^ \alpha \end{align*}


\begin{align*} \lim _ { x \to 0 } \frac { h ( x ) } { \exp ( - a \theta / x ^ 2 ) / x ^ 3 } = & \frac { 2 \theta } { \operatorname { B } ( a , b ) } \lim _ { x \to 0 } \frac { \left [ 1 - \exp \left ( - \frac { \theta } { x ^ 2 } \right ) \right ] ^ { b - 1 } } { \operatorname { I } _ { 1 - \exp ( - \theta / x ^ 2 ) } ( b , a ) } \\ = & \frac { 2 \theta } { \operatorname { B } ( a , b ) } . \end{align*}


\begin{align*} ( \hat h , \hat v ) = \left \{ \begin{array} { l l } ( h , v ) & \mbox { i n } M \setminus \Omega \\ ( 0 , 0 ) & \mbox { i n } \overline U \end{array} \right . . \end{align*}


\begin{align*} F _ i ( x ) = f _ { i m } x ^ m + \dots + f _ { i 0 } x ^ 0 , i = 1 , \dots , n , f _ { 1 m } \neq 0 . \end{align*}


\begin{align*} \delta ( \eta , \gamma ^ + ) = \delta ( \eta \gamma ^ n , \gamma ^ + ) = \delta ( \eta \gamma ^ n , \gamma ^ + ) + \delta ( \eta \gamma ^ n , ( \eta \gamma ^ n ) ^ - ) . \end{align*}


\begin{align*} H ^ \ast ( - , - ) = \bigoplus _ i H ^ i ( - , - ) \end{align*}


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


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


\begin{align*} \begin{cases} - \delta ^ 2 ( K _ H ( x ) \nabla v ) = \left ( v - \left ( \frac { \alpha | x | ^ 2 } { 2 } + \beta \right ) \right ) ^ { p } _ + , \ \ & x \in B _ { R ^ * } ( 0 ) , \\ v = 0 , \ \ & x \in \partial B _ { R ^ * } ( 0 ) . \end{cases} \end{align*}


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


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


\begin{align*} \texttt { l e n g t h ( f i n d ( d i f f ( s i g n ( d i f f ( ( S 2 * v ( 2 * n + 1 : 3 * n ) ) ) ) ) ) ) > = 2 } \end{align*}


\begin{align*} \beta ( C _ S X , Y ) = \beta ( X , C _ S Y ) , \quad \forall S \in \Gamma , \quad \forall X , Y \in \Gamma ^ \perp . \end{align*}


\begin{align*} \widetilde G _ n ( x , z ) d z & : = \P _ x ( S _ n \in d z ; \rho > n ) \\ & = \prod _ { j = 1 } ^ d \lambda _ j e ^ { - \sum _ { j = 1 } ^ d \lambda _ j ( z _ j - x _ i ) } \det ( q _ n ( z _ j - x _ i ) ) _ { i , j = 1 } ^ d d z . \end{align*}


\begin{align*} T = F ( 0 ) = - \int _ 0 ^ { R _ 0 } \frac { 1 } { f ( z ) } \ , \mathrm { d } z = \frac { 1 } { d } \int _ 0 ^ { R _ 0 } \frac { z } { g \left ( \frac { m } { \alpha _ d } z ^ { - d } \right ) } \ , \mathrm { d } z \end{align*}


\begin{align*} y = \mathbf { X } \beta + \varepsilon , \varepsilon \sim \gamma _ { \sigma ^ 2 } , \end{align*}


\begin{align*} I ( A ) = \langle p ^ u - p ^ v \ , | \ , u , v \in \Z _ { \geq 0 } ^ m \ , \ , \ , \ , A u = A v \rangle \end{align*}


\begin{align*} h _ { M , K } : = h _ D + e ^ { 2 \varphi } \Tilde g _ K \end{align*}


\begin{align*} \frac { \mathrm { d } } { \mathrm { d } t } \Big ( x ( t ) - g ( t , x ( t - h ) ) \Big ) = f ( t , x ( t ) , x ( t - h ) , u ( t ) ) t \in [ \tau , \vartheta ] . \end{align*}


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


\begin{align*} f ( x ) = f _ { s ( x ) } ( x ) \end{align*}


\begin{align*} B _ { \infty , \infty } ^ { \Gamma _ 1 \times \Gamma _ 2 - l a } = \varinjlim _ { M \to \infty } B _ { \infty , \infty } ^ { p ^ n \Gamma _ 1 - a n , p ^ n \Gamma _ 2 } \langle z _ { k } - z _ { k , M } : \ ; 1 \leq k \leq d \rangle . \end{align*}


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


\begin{align*} \d Y _ t ^ i = b ^ i ( Y _ t ) \d t + \sigma \d W _ t ^ i , ~ ~ \tilde Y _ { n + 1 } ^ i = \tilde Y _ n ^ i + b ^ i ( \tilde Y _ n ) \tau + \sigma ( W _ { t _ { n + 1 } } ^ i - W _ { t _ n } ^ i ) , ~ ~ i = 1 , \cdots , N . \end{align*}


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


\begin{align*} \lim _ { \delta \rightarrow 0 } \sup _ { n \in \mathbb { N } } \mathbb { P } \Big ( | \langle Y _ n ( \zeta _ n + \delta ) - Y _ n ( \zeta _ n ) , e \rangle | > \varepsilon \Big ) = 0 , \end{align*}


\begin{align*} | \delta ( \zeta , \gamma ^ + ) | = | f ( \zeta \gamma ^ + , \gamma ^ - ) - f ( \gamma ^ + , \gamma ^ - ) | \leq 2 M . \end{align*}


\begin{align*} { h _ { \alpha } } = \log m _ \alpha , \forall \emptyset \neq \alpha \subseteq [ n ] \end{align*}


\begin{align*} \lim _ { \epsilon \rightarrow 0 } \frac { \lambda ( \epsilon ) } { \epsilon } = \kappa > 0 , \end{align*}


\begin{align*} 1 = \langle \varpi \otimes \varpi ( \cdot ) \xi ' _ i , \xi ' _ i \rangle i . \end{align*}


\begin{align*} I _ 2 = ( y - 1 ) \cdot f ( y ) + y \left ( f \left ( \left ( a + 1 + \frac { b } { y } \right ) ( y - 1 ) \right ) - f \left ( \left ( a + \frac { b } { y } \right ) ( y - 1 ) \right ) \right ) . \\ \end{align*}


\begin{align*} \dot { s } ( e - \varepsilon ( y ) ) \lambda _ i = \sum _ { j , \mu } g ^ i _ { j , \mu } ( e - \varepsilon ( y ^ { - 1 } x ^ i _ { j , \mu } ) ) \lambda _ j . \end{align*}


\begin{align*} u \ , = \ , F \big ( x _ 1 , x _ 2 , x _ 3 , x _ 4 , \dots , x _ n \big ) , \end{align*}


\begin{align*} f _ { a r } ( x ) = \frac { f _ { + a } ( x ) - e ^ { - \lambda _ s } f _ r ( x ) } { 1 - e ^ { - \lambda _ s } } = f _ r \ast f _ s ( x ) , \end{align*}


\begin{align*} \dot x _ a = - { \gamma \over 2 } b _ { 1 , \delta } ( x _ a ) \ , . \end{align*}


\begin{align*} 0 = V _ 0 \subset V _ 1 \subset V _ 2 \subset \cdots \subset V _ n \subseteq V , \end{align*}


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


\begin{align*} \mathbf { z } ( \alpha ) = ( 1 - \alpha ) ( \mathbf { w } _ { x s } - \mu \mathbf { e } ) + \alpha ( \mathbf { W } _ { x s } - \mathbf { R } _ { x s } ) \widehat { \Delta \mathbf { x } } + \alpha ^ 2 \widehat { \Delta \mathbf { x } } \circ \widehat { \Delta \mathbf { s } } , \end{align*}


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


\begin{gather*} \int f \ , d P = \sup _ k \int f \wedge k \ , d P . \end{gather*}


\begin{align*} \mu | _ { E _ { 2 , m } } = \lambda _ y = \lambda _ x | _ { E _ { 2 , m } } . \end{align*}


\begin{align*} \frac d { d s } \ , d v _ { g _ s } \ , _ { \big { | } _ { s = 0 } } = \frac 1 2 \langle g , \delta g \rangle d v _ g \ , . \end{align*}


\begin{align*} \lim _ { N \to \infty } \sup _ { n \in \mathbb N } \sum _ { k \geq N } \mathbb E \left [ \langle Y _ n , e _ k \rangle ^ 2 \right ] = 0 , \end{align*}


\begin{align*} \nabla ^ \perp _ T \eta _ i = 0 , \quad \forall T \in \Gamma . \end{align*}


\begin{align*} K _ m = B \ ! \left ( h _ m ( x ) , \frac 1 4 | h _ m ( x ) | \right ) \end{align*}


\begin{align*} G _ n ( x , z ) d z = \sum _ { \pi \in \mathcal S _ d } ( \pi ) \P _ x ( \widehat S _ i ( n + d - \pi ( i ) ) \in d z _ { \pi ( i ) } , i = 1 , \ldots , d ) . \end{align*}


\begin{align*} l . \psi \varphi = \psi \underbrace { c _ { g ' } ( l ) } _ { \in K } \varphi = \psi \varphi c _ { g } ( c _ { g ' } ( l ) ) = \psi \varphi c _ { g ' g } ( l ) \end{align*}


\begin{align*} u ( x ' , t ) : = P _ t ^ L * f ( x ' ) , ( x ' , t ) \in \R ^ n _ + , \end{align*}


\begin{align*} 0 = \varphi ' ( 0 ) = - \frac { \alpha \sqrt { M } } { 2 \sqrt { 2 H } } + \beta M ^ { 2 } \left ( z _ { 0 } ^ { 2 } - H ^ { 2 } \right ) . \end{align*}


\begin{align*} \int _ { \Gamma _ t } ( c + t w ) \circ \theta _ t ^ { - 1 } \ , \mathrm { d } \mathcal { H } ^ d _ { \phantom { x } | t = 0 } = \int _ \Sigma c \ , \mathrm { d } \mathcal { H } ^ d = m \end{align*}


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


\begin{align*} a x ^ q - b x = 0 \end{align*}


\begin{align*} \chi _ 1 = \psi _ 1 , \ldots , \chi _ { j - 1 } = \psi _ { j - 1 } , \chi _ j = \sigma < _ j \psi _ j . \end{align*}


\begin{align*} \mathbf { x } \left [ l \right ] = \mathbf { P } \mathbf { s } \left [ l \right ] = \mathbf { p } _ { \mathrm { c } } s _ { \mathrm { c } } \left [ l \right ] + \sum _ { k \in \mathcal { K } } \mathbf { p } _ { \mathrm { p } , k } s _ { \mathrm { p } , k } \left [ l \right ] , \end{align*}


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


\begin{align*} g = D ^ 2 u . \end{align*}


\begin{align*} \begin{aligned} | x _ a ( t ) - x ^ \star | & \leq \beta ( | x _ a ( 0 ) | _ { { \cal A } _ \delta } , 0 ) + \delta \leq \beta ( | x _ a ( 0 ) - x ^ \star | , 0 ) + \delta \\ & \leq \beta ( r _ 0 , 0 ) + \delta = r - d - \delta \end{aligned} \end{align*}


\begin{align*} \left ( \left ( I + \mathbf { H } ( \varphi ) \right ) y \right ) \left ( \rho \right ) = - \mathbf { H } ( \varphi ) \left ( 1 \right ) \left ( \rho \right ) \end{align*}


\begin{align*} ( \Theta ^ { r ( p _ 1 , \ldots , p _ { r - 1 } ) } ) ^ i { } _ { j _ 1 , \ldots , j _ { r - 1 } } & = - v ^ i { } _ \alpha u ^ \alpha { } _ { j _ 1 , \ldots , \underset { \stackrel { \frown } { p _ { r - 1 } } } { l } , \ldots , j _ { r - 1 } , \beta } v ^ \beta { } _ \gamma d u ^ \gamma \wedge v ^ l { } _ \delta d u ^ \delta \\ * & \hphantom { { } = { } } + ( \textit { t e r m s w h i c h d o n o t i n v o l v e $ u ^ i { } _ { j _ 1 , \ldots , j _ r } $ } ) . \end{align*}


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


\begin{align*} \varphi ( \tau ) = \left \{ \begin{array} { l l } \ell _ n ^ \top ( \tau ) \Phi _ n , & \forall \tau \in [ - h , 0 ) , \\ x , \quad & \tau = 0 . \end{array} \right . \end{align*}


\begin{align*} \frac { N ( x _ a ; x _ b , x _ c , x _ d ; \alpha , \beta ) } { D ( x _ a ; x _ b , x _ c , x _ d ; \alpha , \beta ) } = 1 , \end{align*}


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


\begin{align*} \Pi _ { \mathcal { K } , G } ( \theta ) = \big \{ \bar \theta \in \mathcal { K } \ \big | \ \norm { \bar \theta - \theta } _ { \mathcal { L } ^ 2 , G } = \min _ { \vartheta \in \mathcal { K } } \norm { \vartheta - \theta } _ { \mathcal { L } ^ 2 , G } \big \} . \end{align*}


\begin{align*} 2 \rho - s _ \alpha ( 2 \rho ) = & \sum _ { \beta \in I } \left ( \beta - s _ \alpha ( \beta ) \right ) + \sum _ { \beta \in \Phi ^ + \setminus I } \left ( \beta - s _ \alpha ( \beta ) \right ) \\ = & 2 \sum _ { \beta \in I } \beta . \end{align*}


\begin{align*} \dot { s } ( 0 ) = - \frac { \partial _ { \tau } j ( 0 , 0 ) } { \partial _ { s } j ( 0 , 0 ) } . \end{align*}


\begin{align*} \begin{aligned} y \left [ i \right ] & = \sum \limits _ { l = 1 } ^ L { { \bf { h } } _ l ^ H { \bf { d } } \left [ { i - { n _ l } } \right ] } + z \left [ i \right ] \\ & = \sum \limits _ { t = 0 } ^ { { n _ { { \rm { s p a n } } } } } { { { \bf { g } } ^ H } \left [ t \right ] { \bf { d } } \left [ { i - t - { n _ { \min } } } \right ] } + z \left [ i \right ] , \end{aligned} \end{align*}


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


\begin{align*} \nabla p = B \cdot \nabla B - \nu ( - \Delta ) ^ { \kappa } u . \end{align*}


\begin{align*} \nu _ { y _ 2 } = \mu * \delta _ { s ( y _ 2 ) } = \mu * \delta _ { g k _ 0 s ( y _ 1 ) } = g \left ( \mu * k _ 0 \delta _ { s ( y _ 1 ) } \right ) = g \nu _ { y _ 1 } . \end{align*}


\begin{align*} \displaystyle x ^ { p } = ( I _ { n } + p ^ { m } a ) ^ { p } = I _ { n } + \sum _ { k = 1 } ^ { p } { p \choose k } p ^ { m k } a ^ { m } \equiv I _ { n } \bmod p ^ { m + 1 } \ , \end{align*}


\begin{align*} b \left ( \ell \left ( \pi \right ) \alpha \left ( t \right ) \right ) \left ( z \right ) = R _ { \pi \alpha \left ( t \right ) } \left ( z \right ) = \alpha \left ( t \right ) z \overline { \alpha \left ( t \right ) } \end{align*}


\begin{align*} \Delta _ g \phi & = g ^ { p q } \phi _ { , p q } - \frac { 1 } { 2 } \xi ^ p \phi _ { , p } + \frac { 1 } { 2 } g ^ { p q } v _ p \phi _ { , q } \end{align*}


\begin{align*} a = - \frac { 1 } { \mu } \left ( \partial _ y \mu \ d x - \partial _ x \mu \ d y \right ) = ( \frac { 1 } { \mu } d \mu - \frac { 2 } { \mu } \mu _ z d z ) I \end{align*}


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


\begin{align*} v ^ k \gamma v ^ { - k } ( a , x ) = ( R _ v ^ k R _ \gamma R _ v ^ { - k } a + R _ v ^ k t _ \gamma , x ) . \end{align*}


\begin{align*} \log \int _ { \R ^ n } e ^ { f ( x ) } \ , d x - \int _ { \R ^ n } f \ , d Q + H ( Q ) = H ( Q \ , | \ , P ) \end{align*}


\begin{align*} g ( \omega ( Y ) , Z ) + g ( \omega ( Z ) , Y ) = h ( Y , Z ) \end{align*}


\begin{align*} c _ { 1 } : = \mathop { \inf } \limits _ { ( u , v ) \in E \setminus \{ ( 0 , 0 ) \} } \mathop { \max } \limits _ { t \geq 0 } I ( t u , t v ) . \end{align*}


\begin{align*} P _ j = \sum \limits _ { \forall v } \sum \limits _ { \forall q } p _ { v , j } ^ q + \sum \limits _ { \forall k , k \ne 0 } \sum \limits _ { \forall m } p _ { k , j } ^ m + \sum \limits _ { \forall n } p _ { 0 , j } ^ n , \end{align*}


\begin{align*} Z G \left ( X , Y \right ) = G \left ( \nabla _ { Z } X , Y \right ) + G \left ( X , \nabla _ { Z } ^ { \dag } Y \right ) , \end{align*}


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


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


\begin{align*} \Re \mathcal { P } [ \varrho ] = - \frac { 1 } { 4 } a \left ( 2 a - \left ( 5 a ^ 2 - 6 a + 4 \right ) \tau + ( 8 - 8 a ) \tau ^ 2 - \tau ^ 3 \right ) < 0 \end{align*}


\begin{align*} L ( \lambda \alpha ) = \lambda L ( \alpha ) = 0 \end{align*}


\begin{align*} f _ i ( x ) : = \begin{cases} & e ^ { - i / x } \\ & 0 \end{cases} \end{align*}


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


\begin{align*} \Delta u _ 1 \wedge \dots \wedge \Delta u _ k \wedge \beta _ n ^ { n - m } ( \omega ) = \int u _ k \Delta u _ 1 \wedge \dots \wedge \Delta u _ { k - 1 } \wedge \beta _ n ^ { n - m } \wedge \Delta \omega , k = 1 , \dots , m , \end{align*}


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


\begin{align*} \dot { { x } } _ a & = - \gamma \dfrac { b _ { 1 , \delta } ( x _ a ) } { 2 } \\ \dot { \bar { y } } _ a & = - \gamma \ , \bar { y } _ a + \gamma \ , \dfrac { a _ { 0 , \delta } ( { x } _ a ) } { 2 } \end{align*}


\begin{align*} \varrho ^ \star ( [ [ x , y ] , \alpha ( z ) ] ) ( \beta ^ { - 2 } ) ^ * & = \varrho ^ { \star } ( \alpha ^ { 2 } ( z ) ) \varrho ^ { \star } ( \alpha ( y ) ) \varrho ^ { \star } ( x ) - \varrho ^ { \star } ( \alpha ^ { 2 } ( y ) ) \varrho ^ { \star } ( \alpha ( x ) ) \varrho ^ { \star } ( z ) \\ & \quad - \varrho ^ { \star } ( \alpha ( [ z , x ] ) ) \varrho ^ { \star } ( \alpha ( y ) ) ( \beta ^ { - 1 } ) ^ * + \varrho ^ { \star } ( \alpha ^ { 2 } ( x ) ) \varrho ^ { \star } ( [ y , z ] ) ( \beta ^ { - 1 } ) ^ * . \end{align*}


\begin{align*} t = \sum _ { j = 1 } ^ \infty \frac { b _ j } { 2 ^ j } \end{align*}


\begin{align*} B _ { r } : = \left \{ g \in \mathbb { G } \ ; | \ ; \Gamma ( g , e ) > \frac { 1 } { r ^ { Q - 2 } } \right \} . \end{align*}


\begin{align*} { S } ^ { \rm P R \ , s t r u c t } ( { \cal A } ) = \kappa ( \rho ) W | _ { \mathcal S } \in \R ^ { N L \times N L } \ , \ , , \end{align*}


\begin{align*} \frac { \partial h } { \partial t } \leq \frac { 1 } { f _ { \min } } h [ h ^ { - \varepsilon } \beta _ { 0 } - f _ { \min } ] . \end{align*}


\begin{align*} \Lambda _ { g } = A \Bigl ( - \frac { \partial } { \partial x _ n } \Bigr ) - D , \end{align*}


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


\begin{align*} \lim _ { z \downarrow 0 } \hat { F } _ { \pm } ( 0 ) \int _ 0 ^ \infty w _ z ( \lambda ) \lambda ^ { 4 ( 1 - s + w ) } \frac { d \lambda } { \lambda } \int _ { G ^ 1 / \Gamma } \phi ( g ^ { - 1 } ) d g = 0 \end{align*}


\begin{align*} \left ( \begin{matrix} \phi ( k + 1 ) \\ \phi ( k ) \end{matrix} \right ) = A ( \theta + k \alpha , E ) \left ( \begin{matrix} \phi ( k ) \\ \phi ( k - 1 ) \end{matrix} \right ) . \end{align*}


\begin{align*} \lim _ { M \rightarrow \infty } \lim _ { n \rightarrow \infty } I I = 0 . \end{align*}


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


\begin{align*} - y ^ { \prime \prime } + q ( x ) y = \lambda y , \quad - \infty < x < \infty \end{align*}


\begin{align*} & \lim _ { n \rightarrow \infty } \int _ { \mathbb { T } ^ d } \langle a ( u _ n ( x ) ) \nabla u _ n ( x ) + b ( u _ n ( x ) ) , \nabla u _ n ( x ) - \nabla v ( x ) \rangle d x \\ = & \int _ { \mathbb { T } ^ d } \langle a ( u ( x ) ) \nabla u ( x ) + b ( u ( x ) ) , \nabla u ( x ) - \nabla v ( x ) \rangle d x . \end{align*}


\begin{align*} \langle a , b \rangle : = ( f | [ a , b ] ) . \end{align*}


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


\begin{align*} \frac { \mathrm { d } } { \mathrm { d } \sigma } \ell = & - \frac { 1 } { \sigma } - \frac { \alpha - 1 } { \sigma ^ 2 } ( x - \mu ) \frac { \phi _ s \left ( \frac { x - \mu } { \sigma } \right ) } { \Phi _ s \left ( \frac { x - \mu } { \sigma } \right ) } + \frac { \beta - 1 } { \sigma ^ 2 } ( x - \mu ) \frac { \phi _ s \left ( \frac { x - \mu } { \sigma } \right ) } { 1 - \Phi _ s \left ( \frac { x - \mu } { \sigma } \right ) } - \frac { s } { \sigma } \left | \frac { x - \mu } { \sigma } \right | ^ s , \end{align*}


\begin{align*} C ( G , F ) : = C ( G , K ) \widehat { \otimes } _ { K } F . \end{align*}


\begin{align*} s _ n ( \nu | \mu ) : = \sum _ { \eta \in \Omega _ { \Lambda _ n } } \sum _ { \Delta \subset \Tilde { \Lambda } _ n } \sum _ { \xi _ { \Delta } \neq \eta _ { \Delta } } f ( \nu , \Lambda _ n , \eta , \xi _ { \Delta } ) \hat { c } ^ { B _ { n - 1 } ( x ( \Delta ) ) } _ { \Delta } ( \xi _ { \Delta } \eta _ { \Delta ^ c } , \eta _ { \Delta } ) , \end{align*}


\begin{align*} \omega ( v _ H , - ) = d H \end{align*}


\begin{align*} \Box A _ \nu - \mu ^ 2 A _ \nu = 0 ~ , \nabla ^ \nu A _ \nu = 0 \ , , \end{align*}


\begin{align*} \epsilon ( \eta _ a , t ) : = \int _ 0 ^ t F ( \eta _ a , \tau ) - F _ a ( \eta _ a ) \ , d \tau \end{align*}


\begin{align*} \begin{aligned} & \sum _ { \begin{subarray} { l } k _ 1 + \cdots + k _ n = k \\ k _ i \in \mathbb { Z } _ { \ge 1 } ( i = 1 , \ldots , n - 1 ) , \\ k _ n \in \mathbb { Z } _ { \ge 2 } \end{subarray} } Z ^ { \star } _ { I } ( \{ k _ i \} _ { i = 1 } ^ { n } ; ( \alpha , \beta ) ) \\ = & \binom { k - 2 } { n - 1 } Z ( n - 1 | k - n + 1 ; ( \alpha , \beta ) ) + \binom { k - 2 } { n - 2 } Z ( n | k - n ; ( \alpha , \beta ) ) \end{aligned} \end{align*}


\begin{align*} \overline { I x I y I } = \overline { I ( x \ast y ) I } \end{align*}


\begin{align*} Z ( x , y ; N , K ) = 1 + S ( x , N ) + S ( y , K ) . \end{align*}


\begin{align*} \Phi ( x ; z ) = A ( x ; k ) \psi ( x ; k ) B ( x , k ) , \end{align*}


\begin{align*} M _ { N C } ( n , d ) \le \left ( 2 \sqrt { \frac { 2 } { d + 1 } } - \frac { 2 } { d + 1 } \right ) \cdot 2 ^ d \binom { n } { d } \enspace . \end{align*}


\begin{align*} \int _ { \R ^ n } f \ , d Q - H ( Q ) = - H ( Q \ , | \ , P ) + \log Z . \end{align*}


\begin{align*} { \tilde \Sigma } ^ t = \{ x \in M \mid { \rm d i s t } ( x , \tilde \Sigma ) \leq t \} . \end{align*}


\begin{align*} \P ( \tau > n ) & = \int _ { W ^ d } \det ( q _ { n + i - j } ( y _ j - x _ i ) ) _ { i , j = 1 } ^ d \prod _ { i = 1 } ^ d \lambda _ i ^ n e ^ { - \sum _ { i = 1 } ^ d \lambda _ i ( y _ i - x _ i ) } d y _ 1 \ldots d y _ d , \\ \P ( \rho > n ) & = \int _ { W ^ d } \det ( q _ { n } ( y _ j - x _ i ) ) _ { i , j = 1 } ^ d \prod _ { i = 1 } ^ d \lambda _ i ^ n e ^ { - \sum _ { i = 1 } ^ d \lambda _ i ( y _ i - x _ i ) } d y _ 1 \ldots d y _ d . \end{align*}


\begin{align*} g ( u ) = e \left ( \frac { m \sqrt { X } } { u ^ 2 + u v + v ^ 2 } - \frac { m u } { v ( u ^ 2 + u v + v ^ 2 ) } \right ) \ , , \end{align*}


\begin{align*} ( \mathbf { D } ^ { - T } \mathbf { d } _ { h x } ) ^ { T } ( \mathbf { D } \mathbf { d } _ { h s } ) = \mathbf { d } _ { h x } ^ { T } \mathbf { d } _ { h s } = 0 . \end{align*}


\begin{align*} \begin{cases} 2 a ^ 2 + 2 | v | ^ 2 + b \cdot v = 0 \\ 2 a b + A ^ t b + 2 A ^ t v = 0 \end{cases} \end{align*}


\begin{align*} g ( s ) : = G ( s , s ) = \frac { ( s - a ) ^ { \alpha - 1 } ( b - s ) ^ { \alpha - 1 - \beta } } { ( b - a ) ^ { \alpha - 1 - \beta } \Gamma ( \alpha ) } . \end{align*}


\begin{align*} \sup \{ \Re \lambda : \lambda \in \operatorname { S p } ( \mathcal { L } _ { \alpha } ) \} & = \sup \{ \Re \lambda : \lambda \in \operatorname { S p } ( \hat { \mathcal { L } } ) \} \\ & = \max ( \alpha ^ 2 - c \alpha , \epsilon \alpha ^ 2 - c \alpha - \kappa e ^ { - \kappa } ) \\ & = \alpha ^ 2 - c \alpha . \end{align*}


\begin{align*} \mathcal { H } ^ 2 ( t ; c , f , H ) : = \Bigg \{ \tilde { f } : \int _ { \{ \Psi < - t \} } | \tilde { f } | ^ 2 _ h c ( - \Psi ) < + \infty , \ \tilde { f } \in H ^ 0 ( \{ \Psi < - t \} , \mathcal { O } ( K _ M \otimes E ) ) \\ \& ( \tilde { f } - f ) _ { z _ 0 } \in \mathcal { O } ( K _ M ) _ { z _ 0 } \otimes ( J _ { z _ 0 } \cap H _ { z _ 0 } ) , \ z _ 0 \in Z _ 0 \Bigg \} , \end{align*}


\begin{align*} \frac { C } { \sqrt { R } } - \frac { C ^ 2 } { 4 R ^ 2 } = \frac { C } { \sqrt { R } } \left [ 1 - \frac { C } { 4 R \sqrt { R } } \right ] \geq 0 . \end{align*}


\begin{align*} V i r = \mathbb { C } [ \partial ] L , [ L _ \lambda L ] = ( \partial + 2 \lambda ) L . \end{align*}


\begin{align*} I V = - \Vert \Delta ( n - \widetilde { n } ) \Vert _ { L ^ 2 } ^ 2 = - \Vert n - \widetilde { n } \Vert _ { H ^ 2 } ^ 2 + \Vert n - \widetilde { n } \Vert _ { H ^ 1 } ^ 2 . \end{align*}


\begin{align*} w _ 1 ( \eta ) = \epsilon + \begin{cases} \frac { 3 } { 2 } - \frac { 3 \eta } { 2 } & ( 0 \leq \eta \leq \frac { 1 } { 2 } ) \\ 1 - \frac { \eta } { 2 } & ( \frac { 1 } { 2 } \leq \eta \leq 1 ) \\ \frac { 3 } { 2 } - \eta & ( 1 \leq \eta \leq \frac { 3 } { 2 } ) \\ 0 & ( \frac { 3 } { 2 } \leq \eta \leq 2 ) \end{cases} . \end{align*}


\begin{align*} a = \int _ { 0 } ^ 1 ( e ^ { \gamma x } - 1 ) x g ( x ) d x , b = 0 \nu ( d x ) = { \bf 1 } _ { \{ 0 < x \le c _ 1 \} } e ^ { \gamma x } g ( x ) d x . \end{align*}


\begin{align*} \eta \{ F \} \stackrel { d e f } { = } { \bf E } \eta / F . \end{align*}


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


\begin{align*} \tau = \frac { \partial _ v \Gamma _ { u v } ^ v - 2 \Gamma _ { u v } ^ u \Gamma _ { v u } ^ v } { \partial _ u \Gamma _ { v u } ^ u - 2 \Gamma _ { u v } ^ u \Gamma _ { v u } ^ v } \neq 1 \end{align*}


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


\begin{align*} d \mu _ { \beta , f , g } ( x ) = K _ { \beta , f } ( x ) g ( x ) \ , d x , \end{align*}


\begin{align*} h _ k = \frac { m ( H \cap I _ { k } ) } { m ( I _ { k } ) } = \frac { m ( ( A \cup ( H \setminus A ) ) \cap I _ k ) } { m ( I _ k ) } & = \frac { m ( A \cap I _ { k } ) } { m ( I _ { k } ) } + \frac { m ( ( H \setminus A ) \cap I _ { k } ) } { m ( I _ { k } ) } \\ & = \frac { m ( A \cap I _ { k } ) } { m ( I _ { k } ) } = a _ k \end{align*}


\begin{align*} & d _ { \mathrm { s t } } = \int d z \sum _ { \alpha \in \Delta _ + } e _ { \alpha } ( z ) \otimes \psi _ { \alpha _ i } ^ * ( z ) - \frac { 1 } { 2 } \sum _ { \alpha , \beta , \gamma \in \Delta _ + } c _ { \alpha , \beta } ^ \gamma : \psi _ \gamma ( z ) \psi _ \alpha ^ * ( z ) \psi _ \beta ^ * ( z ) : , \\ & d _ { \chi } = \int d z \sum _ { \alpha \in \Pi } \psi ^ * _ \alpha ( z ) = \sum _ { \alpha \in \Pi } \psi ^ * _ { \alpha , 1 } . \end{align*}


\begin{align*} ( i _ 1 \alpha _ 1 + i _ 2 \alpha _ 2 + \cdots + i _ n \alpha _ n ) ^ { p - 1 } = \sum _ { k _ 1 + \cdots + k _ n = p - 1 } \binom { p - 1 } { k _ 1 , \ldots , k _ n } i _ 1 ^ { k _ 1 } \cdots i _ n ^ { k _ n } \alpha _ 1 ^ { k _ 1 } \cdots \alpha _ n ^ { k _ n } \end{align*}


\begin{align*} \delta \in [ 0 , 1 ] \frac { 1 } { p } = \big ( \frac { 1 } { \alpha } - \frac { 1 } { d } \big ) \delta + \frac { 1 - \delta } { 2 } . \end{align*}


\begin{align*} \mu ( n ) = \frac { 1 } { f ( x _ 0 ) } \cdot \frac { x _ 0 ^ n } { n ! } \ 1 _ { [ k , l ] } ( n ) , \textrm { w h e r e } f ( x ) = \sum _ { i = k } ^ l \frac { x ^ i } { i ! } . \end{align*}


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


\begin{align*} \mathcal { G } = \left \lbrace u _ 0 ( x ) : \ u _ 0 ( x ) \in H ^ { 3 } ( \mathbb { R } ) \cap H ^ { 2 , 1 } ( \mathbb { R } ) , \ a ( k ) \neq 0 \mathbb { C } ^ - \right \rbrace , \end{align*}


\begin{align*} \lim _ { N \to \infty } \frac { 1 } { K _ { N } ^ { 2 } } \mathbb { E } \left ( | \hat { \Gamma } _ { N } ^ { D , M } ( 0 ) \cap A _ { N } | - e ^ { \pi \lambda b } | \hat { \Gamma } _ { N } ^ { D , M } ( b ) \cap A _ { N } | \right ) ^ { 2 } = 0 . \end{align*}


\begin{align*} \lim _ { x \to \infty } e ^ { \gamma x } ( f _ a ( x ) + f _ r ( x ) ) = 0 . \end{align*}


\begin{align*} R L ( b , c ) & = f _ c \left ( \binom { b } { 2 } + b \right ) - f _ c \left ( \binom { b } { 2 } \right ) + b \Delta _ c ( 1 ) \\ & = \sum _ { i = 1 } ^ b \left [ f _ c \left ( \binom { b } { 2 } + i \right ) - f _ c \left ( \binom { b } { 2 } + i - 1 \right ) \right ] + b \Delta _ c ( 1 ) \\ & = \sum _ { i = 1 } ^ b \Delta _ c \left ( \binom { b } { 2 } + i \right ) + b \Delta _ c ( 1 ) , \end{align*}


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


\begin{align*} \Gamma _ 0 ^ \varepsilon = \big \{ F ( z ) + \varepsilon \rho _ 0 \nu _ \Sigma ( z ) \ , \big | \ , z \in M \big \} \end{align*}


\begin{align*} \phi = 2 \pi \frac { d } { \lambda } { s i n } \theta _ 0 \end{align*}


\begin{align*} n ^ { 3 } + \left ( n + 1 \right ) ^ { 3 } - \left ( \beta n + \beta ^ { 2 } + \beta + 1 \right ) ^ { 3 } + \left ( \beta n + \beta ^ { 2 } + 1 \right ) ^ { 3 } = 0 \end{align*}


\begin{align*} k _ 3 ( \langle H \rangle ) = \frac { q - 1 } { 7 6 8 } & \left [ 2 ( q ^ 2 - 2 0 q + 8 1 ) + 2 u ( - p ) ^ t + 3 q ^ 2 \cdot { _ { 3 } } F _ { 2 } \left ( \begin{array} { c c c } \chi _ 4 , & \chi _ 4 , & \overline { \chi _ 4 } \\ & \varepsilon , & \varepsilon \end{array} | 1 \right ) \right ] . \end{align*}


\begin{align*} \begin{pmatrix} A & B \\ C & D \end{pmatrix} \begin{pmatrix} x \\ F ( x ) \end{pmatrix} + \begin{pmatrix} u \\ v \end{pmatrix} = \begin{pmatrix} \pi ( x ) \\ F ' ( \pi ( x ) ) \end{pmatrix} . \end{align*}


\begin{align*} K ( r , h ) = \sum \limits _ { \alpha \leq x \leq \beta \atop { ( x , r ) = 1 } } e \left ( \frac { h \overline { x } _ { | r | } } { r } \right ) \ , , \end{align*}


\begin{align*} \begin{aligned} & \dot y _ { b e n } = a y '' _ { b e n } + M y _ { b e n } , x \in ( 0 , R ) \\ & \dot y _ { n b } = b y '' _ { n b } + ( M - \mu \ 1 ) y _ { n b } , x \in ( R , R + r ) , \end{aligned} \end{align*}


\begin{align*} e ^ { i c _ + ( x ) } \partial _ { x } \left ( \bar { u } _ x ( x ) e ^ { i c _ + ( x ) } \right ) & = - \frac { 1 } { \pi } \int _ { \mathbb { R } } r _ 2 ( z ) e ^ { 2 i s x } \mathrm { d } z - \frac { 1 } { \pi } \int _ { \mathbb { R } } r _ 2 ( z ) e ^ { 2 i z x } \left [ M _ { + , 2 2 } - 1 \right ] d z \\ & : = I _ 3 ( x ) + I _ 4 ( x ) . \end{align*}


\begin{align*} \ T _ { \tau } ( t ) \stackrel { d e f } { = } { \bf P } ( | \tau | \ge t ) , \ t \ge 0 . \end{align*}


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


\begin{align*} G ( \Psi ) ( x ) = \int _ \S W _ r ( x - y ) \sin ( \Psi ( y ) - \Psi ( x ) ) \ \d y - \int _ \S W _ r ( y ) \sin ( \Psi ( y ) ) \ \d y . \end{align*}


\begin{align*} E _ 0 = \kappa \left ( \frac { B _ 3 } { 2 \sqrt { \kappa B _ 4 } } + L + 2 \right ) ^ 2 - \left ( \frac { Q } { 2 ( L + 1 ) } - \sqrt { \kappa B _ 4 } \right ) ^ 2 , \end{align*}


\begin{align*} \mathfrak X ( Z ) : = \overline { \mathfrak X ^ 0 ( Z ) } \subseteq \P ^ { n + 1 } \times \P ^ n . \end{align*}


\begin{align*} \left \{ \begin{array} { r l } \div ( A ( x ) \nabla \varphi _ k ) = 0 & B _ 1 \\ N \varphi _ k = g _ k & \partial B _ 1 \end{array} \right . \end{align*}


\begin{align*} \varphi ( t ) : = \exp \left ( - \int _ 0 ^ t \big [ f ( r ) + \rho ( X ( r ) ) + \eta ( X ^ { \prime } ( r ) ) \big ] d r \right ) . \end{align*}


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


\begin{align*} E ^ { \pm } ( \Q _ { p ^ g , \infty } ) = \bigcup _ { n \geq - 1 } E ^ { \pm } ( \Q _ { p ^ g , n } ) . \end{align*}


\begin{align*} S ( \theta ) = \frac { 1 } { \| \mathbf { a } ^ H ( \theta ) \mathbf { U } _ V \| ^ 2 } \end{align*}


\begin{align*} \left ( \int _ { B _ 1 } | v | ^ p \d x \right ) ^ { \frac { 1 } { p } } \leq C \left ( \int _ { B _ 1 } | \nabla v | ^ 2 \d x \right ) ^ { \frac { 1 } { 2 } } = C . \end{align*}


\begin{align*} \mathbb P \left [ \int _ 0 ^ T \| \alpha _ s \| ^ { \frac m 2 } + \| \sigma _ s \| _ { L _ { } ( U , H ) } ^ m d s < \infty \right ] = 1 . \end{align*}


\begin{align*} \Omega _ { \theta } : = \lbrace \omega \colon R e \left ( \omega e ^ { i \theta } \right ) < - I _ f ( \theta ) \rbrace , \end{align*}


\begin{align*} \lim _ { | z | \rightarrow \infty } z \mathcal { C } ( h ) ( z ) = - \frac { 1 } { 2 \pi i } \int _ { \mathbb { R } } h ( s ) d s . \end{align*}


\begin{align*} \left \langle l _ 1 , m _ 1 \right | \cos ( n \theta ) \left | l _ 2 , m _ 2 \right \rangle = \sum \limits _ { l = | l _ 1 - l _ 2 | } ^ { l _ 1 + l _ 2 } \sqrt { \left ( l _ 1 + \frac { 1 } { 2 } \right ) \left ( l _ 2 + \frac { 1 } { 2 } \right ) } ( - 1 ) ^ { m _ 1 } \end{align*}


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


\begin{align*} ( H _ { \lambda \exp , \alpha } \psi ) _ { n } = \psi _ { n + 1 } + \psi _ { n - 1 } + \lambda { \rm e } ^ { { \rm i } ( x + n \alpha ) } \psi _ { n } , \end{align*}


\begin{align*} & k _ 3 ( \langle H \rangle , a ) \\ & = \frac { 1 } { 2 \times 4 ^ 5 } \sum _ { x \neq 0 , a } \sum _ { y \neq 0 , a , x } [ ( 2 + h \chi _ 4 ( a - x ) + \overline { h } \overline { \chi _ 4 } ( a - x ) ) \\ & \times ( 2 + h \chi _ 4 ( a - y ) + \overline { h } \overline { \chi _ 4 } ( a - y ) ) ( 2 + h \chi _ 4 ( x - y ) + \overline { h } \overline { \chi _ 4 } ( x - y ) ) \\ & \times ( 2 + h \chi _ 4 ( x ) + \overline { h } \overline { \chi _ 4 } ( x ) ) ( 2 + h \chi _ 4 ( y ) + \overline { h } \overline { \chi _ 4 } ( y ) ) ] . \end{align*}


\begin{align*} \pi = \pi _ 1 \oplus \pi _ 2 \oplus \pi _ 3 \tau = t s _ 1 \tau _ 1 \oplus ( 1 - t ) s _ 2 \tau _ 2 \oplus ( t ( 1 - s _ 1 ) + ( 1 - t ) ( 1 - s _ 2 ) ) \tau _ 3 . \end{align*}


\begin{align*} & f ( t , y , z , u , \mu ) = \hat f \big ( t , y , z , u , \int \varphi d \mu \big ) , \ , \ , \ , \ , g ( t , y , z , u , \mu ) = \hat g \big ( t , y , z , u , \int \phi d \mu \big ) , \\ & h ( t , y , z , u , \mu ) = \hat h \big ( t , y , z , u , \int \psi d \mu \big ) , \ , \ , \ , \ , \Phi ( y , \mu _ y ) = \hat \Phi \big ( y , \int \gamma d \mu _ y \big ) , \end{align*}


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


\begin{align*} V _ { p , \alpha _ z } = \bigoplus _ { v \in S _ { i n } } V _ { v , \alpha _ { z , v } } \oplus \bigoplus _ { v \in S _ { s p } } V _ v ^ + . \end{align*}


\begin{align*} \norm { f } _ { L ^ 2 } ^ 2 & = \int _ \S \left ( \int _ \S W _ r ( x - y ) \prod _ { i = 1 } ^ n ( \eta _ i ( y ) - \eta _ i ( x ) ) \sin ^ { [ n ] } ( \tilde \Psi ( y ) - \tilde \Psi ( x ) ) \ \d y \right ) ^ 2 \d x \\ & \le \int _ \S \int _ \S \prod _ { i = 1 } ^ n ( \eta _ i ( y ) - \eta _ i ( x ) ) ^ 2 \ \d y \d x \\ & = \prod _ { i = 1 } ^ n \norm { \partial \eta _ i } _ { L ^ 2 } ^ 2 . \end{align*}


\begin{align*} \frac { x ^ { q + 1 } + x } { y ^ { q + 1 } + y } = d , \end{align*}


\begin{align*} v _ 1 \cdot v _ 2 = - \int _ X v _ 1 \wedge v _ 2 ^ \vee . \end{align*}


\begin{align*} k _ 3 ( \langle H \rangle , 1 ) = \frac { 1 } { 1 2 8 } \left [ 2 ( q ^ 2 - 2 0 q + 8 1 ) + 2 u ( - p ) ^ t + 3 q ^ 2 \cdot { _ { 3 } } F _ { 2 } \left ( \begin{array} { c c c } \chi _ 4 , & \chi _ 4 , & \overline { \chi _ 4 } \\ & \varepsilon , & \varepsilon \end{array} | 1 \right ) \right ] . \end{align*}


\begin{align*} \lim _ { N \to \infty } \sup _ { n \in \mathbb N } \sum _ { k \geq N } \langle \mathbb E [ X _ n ] e _ k , e _ k \rangle = 0 . \end{align*}


\begin{align*} 0 ^ p \cdot F _ 2 + 1 ^ p \cdot F _ 1 + 2 ^ p \cdot F _ 0 \ & = \ A _ p F _ 2 + B _ p F _ 3 - \sum _ { k = 0 } ^ p \binom { p } { k } B _ k 2 ^ { p - k } , \intertext { a n d i f w e s i m p l i f y t h i s a n d p u l l o u t t h e $ k = p $ t e r m f r o m t h e s u m , w e g e t } 1 \ & = \ A _ p + 2 B _ p - B _ p - \sum _ { k = 0 } ^ { p - 1 } \binom { p } { k } B _ k 2 ^ { p - k } . \end{align*}


\begin{align*} \Delta _ 1 ( m , n ) = \delta ( m , n ) + O ( ( m n ) ^ { 3 / 8 + \epsilon } k ^ { - 1 3 / 1 2 } ) . \end{align*}


\begin{align*} \forall \ t \in [ t _ 0 , 1 ] , \partial _ { \eta _ 1 } ^ { 2 k } \rho _ k = ( 2 k ) ! ( t - t _ 0 ) ^ k \leq ( 2 k ) ! \rho _ { 0 } . \end{align*}


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


\begin{align*} c _ { r } = r ^ { 1 - d } \int _ { 0 } ^ { r } \rho ^ { d - 1 } n c \ , d \rho < \frac { M _ { 0 } m _ { 0 } } { \sigma _ { d } } r \quad \mbox { f o r a l l } r \in ( 0 , R ] . \end{align*}


\begin{align*} S = \left ( \begin{array} { c c } 1 / 2 + \alpha & \beta \\ \beta & 1 / 2 - \alpha \end{array} \right ) \end{align*}


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


\begin{align*} \sigma _ { k - 1 } ( b ) u _ { n n } ( 0 ) - \sum _ { \alpha \leq n - 1 } u _ { n \alpha } ^ 2 ( 0 ) \sigma _ { k - 2 ; \alpha } ( b ) + \sigma _ k ( b ) = f ( 0 ) . \end{align*}


\begin{align*} D _ 2 ( { \frak y } ) = 4 \gamma \begin{bmatrix} T _ - & 0 & 0 & \dots & 0 \\ 0 & y _ 1 & 0 & \dots & 0 \\ 0 & 0 & y _ 2 & \dots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & \dots & y _ n \end{bmatrix} . \end{align*}


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


\begin{align*} - \Delta ( v \psi _ { m } ) ( x ) + F ' ( \overline { y } ; v \psi _ { m } ) ( x ) = 0 , x \in \mathcal { A } , \end{align*}


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


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


\begin{align*} U _ { i j } \subseteq U _ { i } \subseteq U U \backslash U _ { i } = B _ { i } \backslash U _ { i } U _ { j } \backslash U _ { i j } = B _ { i } \backslash U _ { i j } . \end{align*}


\begin{align*} 0 = \int _ \Sigma \Delta x _ { n + 1 } \ , d A = \int _ \Sigma \left ( \eta x _ { n + 1 } ^ m + \lambda \right ) \nu _ { n + 1 } \ , d A = \int _ \Sigma \eta x _ { n + 1 } ^ m \nu _ { n + 1 } \ , d A , \end{align*}


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


\begin{align*} ( r _ n \omega ) _ y = \begin{cases} \omega _ y , & y \in \Lambda _ n , \\ \ 1 , & . \end{cases} \end{align*}


\begin{align*} Z _ { V , [ \alpha ] } ( t ) = - \int _ { V } \alpha + \sqrt { - 1 } t \int _ { V } \omega . \end{align*}


\begin{align*} \Omega ( X , q , n ) = \frac { X } { q } + \mathcal { O } ( 1 ) \ , . \end{align*}


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


\begin{align*} \sigma _ b \left ( Q ( z ) \right ) = 0 \quad | b ( z ) | \geq C _ 1 \ . \end{align*}


\begin{align*} \lim _ { m _ i \in M _ 2 , m _ i \to \infty } \mu ( T ^ { m _ i } A \cap B ) = \mu ( A ) \mu ( B ) . \end{align*}


\begin{align*} \frac { \partial j } { \partial s } ( 0 , 0 ) = 2 \int _ { \mathbb { R } ^ { 3 } } ( \phi _ 0 1 _ { ( 0 , \infty ) } ( \phi - \phi _ { \infty } ) G + ( \phi - \phi _ { \infty } ) _ { + } G _ 0 ) \frac { 1 } { r ^ { 2 } } \dd x . \end{align*}


\begin{align*} I ( \lambda ) = \int _ a ^ b e ^ { i \lambda \phi ( x ) } \psi ( x ) d x \end{align*}


\begin{align*} \Lambda _ E = \left \{ ( \lambda , z ) \in { \Lambda } \mid \rho _ E ( \lambda , z ) = 0 \right \} \end{align*}


\begin{align*} \int _ { f ( [ 0 , 1 ] ) } \abs { \varphi \circ f ^ { - 1 } ( t ) } d t = \int _ { [ 0 , 1 ] } \abs { \varphi \circ f ^ { - 1 } ( f ( t ) ) } \ ; | f ' ( t ) | d t \leq \norm { \varphi } _ 1 \norm { f } _ L . \end{align*}


\begin{align*} \lim _ { k \rightarrow + \infty } ( \eta \gamma ^ n ) ^ k x = ( \eta \gamma ^ n ) ^ + . \end{align*}


\begin{align*} H ( \overline { Q } ) \le \frac 1 n \sum _ { i = 1 } ^ n H ( Q _ i ) . \end{align*}