\begin{align*} \left ( 2 \binom { n } { ( k - 1 ) n / p } \right ) ^ t \leq ( c p ) ^ { ( k - 1 ) n t / p } \end{align*}


\begin{align*} \Pi _ { \mathrm { s p a n } \{ P _ x ^ { - 1 } v \} } J _ x ^ \perp y = \lambda J _ x \Pi _ { \mathrm { s p a n } \{ P _ x ^ { - 1 } v \} } y \forall y \in \C ^ n . \end{align*}


\begin{align*} \theta _ 2 F _ { 1 } ( \theta _ 1 , \theta _ 2 , \theta _ 3 , \theta _ 4 ) = - \theta _ 1 \theta _ 2 + \theta _ 2 \theta _ 3 + \theta _ 2 \theta _ 4 . \end{align*}


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


\begin{align*} 2 \frac { F ^ { i i } h _ { 1 1 i } ^ 2 } { \kappa _ 1 ( \kappa _ 1 - \tilde { \kappa } _ i ) } - \frac { F ^ { i i } h _ { 1 1 i } ^ 2 } { \kappa _ 1 ^ 2 } = \frac { F ^ { i i } ( \kappa _ 1 + \tilde { \kappa } _ i ) } { \kappa _ 1 ^ 2 ( \kappa _ 1 - \tilde { \kappa } _ i ) } h _ { 1 1 i } ^ 2 \geq c ( n ) \frac { F ^ { i i } h _ { 1 1 i } ^ 2 } { \kappa _ 1 ^ 2 } , \end{align*}


\begin{align*} - \Delta w = w ^ p _ + , \ \ \ \mathbb { R } ^ 2 . \end{align*}


\begin{align*} \Phi ( x ) ^ { \delta } = \sum _ { r = 0 } ^ \infty \ , s _ r ( \delta ) \ , \Phi ( x ) ^ r , \end{align*}


\begin{align*} \begin{aligned} \vert J _ 1 \vert \lesssim ( 1 + \tau ) ( 1 + \tau ^ { - \frac { 2 } { 3 } } ) e ^ { - \zeta h \tau } \end{aligned} \end{align*}


\begin{align*} X = \sqrt { - \frac { \theta } { \log ( Y ) } } \end{align*}


\begin{align*} g _ 1 ' : = \frac { 1 } { 2 } ( 1 + i I ) v ' \end{align*}


\begin{align*} \tau ^ * = \bar { \tau } \Leftrightarrow \dfrac { | \lambda _ i | } { | \lambda _ { i + 1 } | } = \dfrac { | \lambda _ { i + 1 } | } { | \lambda _ { i + 2 } | } \quad \forall i \in [ 1 , p - 2 ] \quad \forall i \in [ p + 1 , n - 2 ] ) , \end{align*}


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


\begin{align*} ( B ^ { \circ } S _ { T - ( \cdot ) } ^ { \circ } x ^ { \circ } \odot \lambda ) ( \boldsymbol { \cdot } ) u = \int _ { ( \boldsymbol { \cdot } ) } \langle B ^ { \circ } S _ { T - t } ^ { \circ } x ^ { \circ } , u \rangle \d t \qquad ( u \in U ) \end{align*}


\begin{align*} P _ { \mu , f _ { 1 , \infty } } ( X , \psi ) : = \int P _ { \mu , f _ { 1 , \infty } } ( x , \psi ) \ d \mu ( x ) . \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*} - \Delta p _ { \epsilon } + F _ { \epsilon } ' ( y _ { \epsilon } ) p _ { \epsilon } & = y _ { \epsilon } - g , \\ \langle p _ { \epsilon } + \alpha u _ { \epsilon } , u - u _ { \epsilon } \rangle + \langle | u _ { \epsilon } - \overline { u } | ^ { p - 2 } ( u _ { \epsilon } - \overline { u } ) , u - u _ { \epsilon } \rangle & \ge 0 u \in \mathcal { C } _ { a d } . \end{align*}


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


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


\begin{align*} \frac { 1 } { 2 } B ( X ) : = \frac { 1 } { 2 } \sum _ { j = 1 } ^ p e _ j \cdot B ( X , e _ j ) \ \in C l ( T M \oplus E \oplus \mathcal { E } _ 2 ) \end{align*}


\begin{align*} v = 0 , \nu ( v ) = 0 \mbox { o n } \hat \Sigma . \end{align*}


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


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


\begin{align*} \mathcal { H } ^ 2 ( t ; c , f ) : = \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 } , z _ 0 \in Z _ 0 \Bigg \} , \end{align*}


\begin{align*} T _ { W _ \ell , \phi } ( \mu ) = \sum _ { i = 1 } ^ L c _ i \int _ { [ 0 , 1 ] ^ 2 } \bar a _ i ( u ) \bar b _ i ( v ) W _ \ell ( u , v ) d u d v , \end{align*}


\begin{align*} \bar { R } _ { c , k } & = \mathbb { E } \left [ R _ { c , k } | \mathbf { \hat { G } } \right ] & \bar { R } _ { k } = \mathbb { E } \left [ R _ { k } | \mathbf { \hat { G } } \right ] , \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*} F _ n ( \eta ) : = T _ n - \frac { 1 } { \eta } I _ n , \end{align*}


\begin{align*} d ^ a _ 1 ( x , y ) & = x - \eta ^ a y \\ d ^ a _ 0 ( x , y ) & = \prod _ { i \neq a } ( x - \eta ^ i y ) \end{align*}


\begin{align*} \lim _ { n \to \infty } T _ { W , \psi } ( \mu _ n ( Q ^ n ) ) = T _ { W , \psi } ( \mu ) , I ( \mu _ n ( Q ^ n ) ) \le I ( \mu ) , \ \forall n . \end{align*}


\begin{align*} \left \{ \begin{aligned} b = \mu x , \\ c x + ( d - a ) = \mu . \end{aligned} \right . \end{align*}


\begin{align*} t ( \Delta ) : = \min _ { n \geq 0 } \Big \{ n - ( \Delta _ n ) \ , \big | \ , \Delta = \sum _ { n \geq 0 } \hbar ^ n \Delta _ { n } \Big \} . \end{align*}


\begin{align*} W ( t ) = \sum _ { l = 1 } ^ { n _ 1 } \Psi _ { k l } \beta ^ c _ l ( t ) , k = 1 , \dots , n . \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*} \abs { c _ 1 ( S ^ 1 \setminus S _ i ) } \le \abs { c ( S \setminus S _ i ) } + \sum _ { h \in [ \chi ] \setminus \{ i \} } \abs { c _ 1 ( P _ i ' ) } \le \abs { S \setminus S _ i } + \sum _ { h \in [ \chi ] \setminus \{ i \} } q _ h = \abs { S } - \abs { S \cap S _ i } + q - q _ i . \end{align*}


\begin{align*} \partial _ i \varphi _ i = - 2 \sum _ { j \neq i } \Gamma _ { i j } ^ j \varphi _ j . \end{align*}


\begin{align*} \mathbf x ^ { t } \gets \underset { \mathbf z : \operatorname { s u p p } ( \mathbf z ) = \hat { S } ^ { t } } { \arg \min } f ( \mathbf z ) , \end{align*}


\begin{align*} \frac { d } { d s } \left ( \frac { \zeta ' ( s ) } { \zeta ( s ) } \right ) = \frac { 1 } { ( s - 1 ) ^ 2 } - \sum _ { n = 1 } ^ \infty \frac { 1 } { ( s + 2 n ) ^ 2 } - \sum _ \rho \frac { 1 } { ( s - \rho ) ^ 2 } . \end{align*}


\begin{align*} \langle A _ q ^ w , A _ q ^ z \rangle _ { L ^ 2 ( \mathbb { R } ) } = K _ q ( w , z ) . \end{align*}


\begin{align*} \varphi ( x , t ; k ) = I + k \begin{pmatrix} 0 & u \\ v & 0 \end{pmatrix} + \mathcal { O } \left ( k ^ 2 \right ) , k \rightarrow 0 , \end{align*}


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


\begin{align*} \exp \left ( a _ { \alpha + 1 } + d _ { \alpha + 1 } - d _ \alpha \right ) = 1 - \exp \left ( c _ { n + 1 } + b _ { n + 1 } - b _ n \right ) \ . \end{align*}


\begin{align*} g ( \nabla \Psi , \vec { \nu } ) = c o n s t \ \ \textnormal { o n e a c h l e a f } \Sigma _ { t } \end{align*}


\begin{align*} F ^ { q , \dagger } _ { v v } ( 0 , p ) [ u _ k , u _ k ] & = c _ 2 ( q , k , p ) u _ { 2 k } \end{align*}


\begin{align*} U A _ j U ^ * = \begin{bmatrix} A _ { j , 1 } & 0 & \ldots & 0 \\ 0 & A _ { j , 2 } & \ddots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & \ldots & 0 & A _ { j , \ell } \end{bmatrix} . \end{align*}


\begin{align*} F ^ q ( v , p ) : = F ( \Psi ^ q + v , p ) , F ^ q \colon X \times \mathcal P \to X . \end{align*}


\begin{align*} \Vert Y _ n ( t ) \Vert _ H ^ 2 = & \Vert P _ n x \Vert _ H ^ 2 + \int _ 0 ^ t \Big [ 2 \langle A ( s , Y _ n ( s ) ) , Y _ n ( s ) \rangle + \Vert P _ n B ( s , Y _ n ( s ) ) Q _ n \Vert _ { L _ 2 } ^ 2 \Big ] d s \\ & + 2 \int _ 0 ^ t \Big ( Y _ n ( s ) , B ( s , Y _ n ( s ) ) Q _ n d W ( s ) \Big ) . \end{align*}


\begin{align*} p ( 1 v , 0 w ) = p ( v 1 , w \bullet ) = \psi \left ( d _ { - } ^ { \ell + 1 } L ( v 1 , w \bullet ) \right ) . \end{align*}


\begin{align*} V ^ { - 1 } \rho \circ \varphi V = V ^ { - 1 } \varphi \circ \rho V = V ^ { - 1 } \rho V . \end{align*}


\begin{align*} \sum _ { n < x ^ 2 } \frac { \Lambda _ x ( n ) ^ 2 } { n ^ { 2 \sigma } } & < \sum _ { n = 1 } ^ \infty \frac { \Lambda ( n ) \log n } { n ^ { 2 \sigma } } = \frac { d } { d s } \left [ \frac { \zeta ' } { \zeta } ( s ) \right ] _ { s = 2 \sigma } . \end{align*}


\begin{align*} \chi ( E ) = \sum \limits _ { j = 1 } ^ n \chi ( E _ j ) - r ( n - 1 ) . \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*} \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*} s _ h : = \sum _ { j = 1 } ^ w x _ j ^ { h } = \frac { 1 } { 2 \pi \sqrt { - 1 } } \int _ { \mathcal C } \frac { p ' ( x ) } { p ( x ) } ~ x ^ h ~ d x . \end{align*}


\begin{align*} C _ m ( K ) = \sup \left \{ \int _ { K } \Delta u _ 1 \wedge \cdots \wedge \Delta u _ m \wedge \beta _ n ^ { n - m } ; u _ j \in Q S H _ m ( \Omega ) \cap C ( \Omega ) , - 1 \leq u _ j < 0 \right \} . \end{align*}


\begin{align*} \frac { \partial \textbf { \textit { U } } } { \partial x _ n } \bigg | _ { \partial \Omega } = - Q \textbf { \textit { U } } | _ { \partial \Omega } + \mathcal { R } \textbf { \textit { V } } . \end{align*}


\begin{align*} P ^ \infty & = \varprojlim P ^ r = \left \{ u = ( u ^ r ) _ { r \in \mathbb { N } } \in \prod _ { r \in \mathbb { N } } P ^ r \ ; \middle | \ ; \pi ^ r _ s u ^ r = u ^ s \ \right \} , \\ * \widetilde { P } ^ \infty & = \varprojlim \widetilde { P } ^ r , \\ * G ^ \infty & = P ^ \infty ( \mathbb { R } ^ n , o ) , \\ * \widetilde { G } ^ \infty & = \widetilde { P } ^ \infty ( \mathbb { R } ^ n , o ) , \end{align*}


\begin{align*} \theta ' _ i = \pi _ i ( X _ 2 , Y _ 2 , \dots , X _ { i - 1 } , Y _ { i - 1 } ) , i \ge 2 . \end{align*}


\begin{align*} G \frac { d } { d t } w ( t ) = L w ( t ) + f ( t ) \end{align*}


\begin{align*} \partial _ t U + \sum _ { j = 1 } ^ d A ^ j ( U ) \frac { \partial U } { \partial x _ j } + B ( x ) V = 0 . \end{align*}


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


\begin{align*} \begin{array} { l } \mu ^ { N , 1 } _ { \boldsymbol { \alpha } ^ * _ t } + h _ 1 ^ N ( \mu ^ { N , 1 } _ { \boldsymbol { \alpha } ^ * _ t } , \ldots , \mu ^ { N , N } _ { \boldsymbol { \alpha } ^ * _ t } ) = \Delta ^ { N , 1 } _ { * , t } , \\ \quad \vdots \qquad \qquad ~ ~ \vdots \qquad ~ ~ ~ ~ \vdots \\ \mu ^ { N , N } _ { \boldsymbol { \alpha } ^ * _ t } + h _ N ^ N ( \mu ^ { N , 1 } _ { \boldsymbol { \alpha } ^ * _ t } , \ldots , \mu ^ { N , N } _ { \boldsymbol { \alpha } ^ * _ t } ) = \Delta ^ { N , N } _ { * , t } , \end{array} \end{align*}


\begin{align*} \dot { { x } } _ a = \ , - \gamma c _ 1 \delta \left . \dfrac { \partial h } { \partial x } \right | _ { { x } _ a } - \gamma \sum _ { k = 2 } ^ { \infty } c _ k \ , { \delta ^ { 2 k - 1 } } \ , \left . \dfrac { \partial ^ { 2 k - 1 } h } { \partial x ^ { 2 k - 1 } } \right | _ { { x } _ a } \end{align*}


\begin{align*} X _ { t _ { n + 1 } } ^ i = X _ { t _ n } ^ i + \int _ { t _ n } ^ { t _ { n + 1 } } b ^ i ( X _ t ) \d t + \sigma W _ \tau ^ i , ~ ~ ~ ~ \tilde X _ { n + 1 } ^ i = \tilde X _ n + \int _ { t _ n } ^ { t _ { n + 1 } } b ^ i ( \tilde X _ n ) \d t + \sigma W _ \tau ^ i , \end{align*}


\begin{align*} \log \frac { 1 } { x } - \gamma + \sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } } { n \cdot n ! } x ^ { n } \ \ ( \gamma : \rm { E u l e r ' s \ c o n s t a n t } ) ; \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*} \begin{aligned} u ( \bar { x } ) & = \frac { 1 } { 2 } \sum _ { \beta = 1 } ^ { n - 1 } b _ \beta \bar { x } _ \beta ^ 2 + + \frac { 1 } { 6 } \sum _ { \xi , \beta , \gamma } \varphi _ { \xi \beta \gamma } ( 0 ) \bar { x } _ \xi \bar { x } _ \beta \bar { x } _ \gamma + O ( | \bar { x } ' | ^ 4 ) \\ & \leq u ( x _ 0 ) + \frac { 1 } { 2 } \sum _ { \beta = 1 } ^ { \alpha } b _ \beta \bar { x } _ \beta ^ 2 + C \delta ^ 3 b _ \alpha ^ 2 . \end{aligned} \end{align*}


\begin{align*} t ^ { \alpha } - s ^ { \alpha } = t ^ { \alpha - \varrho } t ^ { \varrho } - s ^ { \alpha - \varrho } s ^ { \varrho } \leq s ^ { \alpha - \varrho } t ^ { \varrho } - s ^ { \alpha - \varrho } s ^ { \varrho } \leq s ^ { \alpha - \varrho } ( t - s ) ^ { \varrho } , \end{align*}


\begin{align*} \int _ { \hat { B } _ { \delta } ( 0 ) \setminus \hat { B } _ { \varepsilon } ( 0 ) } q ^ 2 ( K _ { H } ) _ { 2 2 } ( x + \bar { x } ) \partial _ 2 \hat { U } \left ( \frac { x } { \varepsilon } \right ) \partial _ 2 \hat { U } \left ( \frac { x } { \varepsilon } \right ) d x = \frac { l _ 1 } { l _ 2 } \pi q ^ 2 ( K _ { H } ) _ { 2 2 } ( \bar { x } ) \ln \frac { \delta } { \varepsilon } + O ( 1 ) . \end{align*}


\begin{align*} \begin{aligned} & \left ( \Psi ^ - - e ^ { - i c _ - ( x ) \sigma _ 3 } \right ) - \left ( \widetilde { \Psi } ^ - - e ^ { - i c _ - ( x ) \sigma _ 3 } ( \tilde { u } ) \right ) \\ & = ( I - F ) ^ { - 1 } ( T - \widetilde { T } ) + ( I - F ) ^ { - 1 } ( F - \tilde { F } ) ( I - \tilde { F } ) ^ { - 1 } \widetilde { T } , \end{aligned} \end{align*}


\begin{align*} \Theta ^ { 2 ( 1 ) } & = d \theta ^ i { } _ j + \theta ^ i { } _ { \alpha j } \wedge \theta ^ \alpha + \theta ^ i { } _ \alpha \wedge \theta ^ \alpha _ j \\ * & = \Omega ^ i { } _ j + \theta ^ i { } _ { \alpha j } \wedge \theta ^ \alpha , \\ * \Theta ^ { 2 ( 2 ) } & = d \theta ^ i { } _ j + \theta ^ i { } _ { j \alpha } \wedge \theta ^ \alpha + \theta ^ i { } _ \alpha \wedge \theta ^ \alpha { } _ j \\ * & = \Omega ^ i { } _ j + \theta ^ i { } _ { j \alpha } \wedge \theta ^ \alpha . \end{align*}


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


\begin{align*} \widetilde \theta _ 1 & = 0 , \\ \widetilde \theta _ i & = \pi ^ { [ n - 1 ] } _ { i - 1 } ( \xi ^ { \widetilde \theta } _ 1 ( Y _ 1 ) , x + Y _ 1 , X _ 2 , Y _ 2 , \dots , X _ { i - 1 } , Y _ { i - 1 } ) , \end{align*}


\begin{align*} q _ X ( \alpha ) : = \frac { n } { 2 } \int _ X ( \sigma \overline \sigma ) ^ { n - 1 } \alpha ^ 2 + ( 1 - n ) \int _ X \sigma ^ n \overline \sigma ^ { n - 1 } \alpha \int _ X \sigma ^ { n - 1 } \overline \sigma ^ n \alpha \end{align*}


\begin{align*} \Psi ( r , \xi ) = \Phi ( r ) + \Gamma ( \xi ) \end{align*}


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


\begin{align*} \left ( \sqrt { q ' _ j } - \sqrt { p ' _ j } \right ) ^ 2 = q ' _ j \left ( 1 - \sqrt { 1 - \frac { q ' _ j - p ' _ i } { q ' _ j } } \right ) ^ 2 \leq \frac { ( q ' _ j - p ' _ i ) ^ 2 } { q ' _ j } . \end{align*}


\begin{align*} \widetilde { Q } = \frac { 1 } { 2 i } \begin{pmatrix} | u _ x | ^ 2 & u _ x \\ - 2 i \bar { u } _ { x x } - \bar { u } _ x | u _ x | ^ 2 & - | u _ x | ^ 2 \end{pmatrix} . \end{align*}


\begin{align*} W : = \max _ { \overline { \omega } _ \delta } ( A \Psi \pm T _ \alpha ( u - \varphi ) ) \end{align*}


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


\begin{align*} A ( X ) : = \left ( \begin{array} { c } P _ H [ \Delta u - ( u \cdot \nabla ) u - \nabla n \cdot \Delta n ] \\ \Delta n - ( u \cdot \nabla ) n - \varphi ( n ) \end{array} \right ) . \end{align*}


\begin{align*} \left \langle X , \Gamma _ { ( g , u ) } ( X ) \right \rangle = - \tfrac { 1 } { 2 } \Delta | X | ^ 2 - \tfrac { 1 } { 2 } u ^ { - 1 } { \nabla u } \cdot \nabla | X | _ g ^ 2 + | \nabla X | ^ 2 + u ^ { - 2 } ( X ( u ) ) ^ 2 . \end{align*}


\begin{align*} g _ k : = { \rm s g n } ( v ) \star \eta _ k , \end{align*}


\begin{align*} \lambda _ S = \langle \alpha , \beta \rangle \alpha - \langle \alpha , \alpha \rangle \beta . \end{align*}


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


\begin{align*} \frac { d } { d s } \left ( \frac { \zeta ' ( s ) } { \zeta ( s ) } \right ) = \frac { 1 } { ( s - 1 ) ^ 2 } - \sum _ { n = 1 } ^ \infty \frac { 1 } { ( s + 2 n ) ^ 2 } - \sum _ \rho \frac { 1 } { ( s - \rho ) ^ 2 } . \end{align*}


\begin{align*} D _ { n + 1 } ( x ) & = \sum _ { \lambda _ { n } \in \Lambda _ { n } } \sum _ { l \in L } e ^ { 2 \pi i \tau ( R ^ t \lambda _ { n } + l ) \cdot x } \\ & = \sum _ { l \in L } e ^ { 2 \pi i ( \tau l ) \cdot x } \sum _ { \lambda _ { n } \in \Lambda _ { n } } e ^ { 2 \pi i \tau ( R ^ t \lambda _ { n } ) \cdot x } \\ & = m _ \tau ( x ) D _ n ( R ^ t x ) . \end{align*}


\begin{align*} e _ { a _ j } \left [ t \right ] = a _ j \left [ t \right ] - a _ j ^ { \left ( o \right ) } , \end{align*}


\begin{align*} \sum _ { k = 2 } ^ n \frac { \mu ( k ) } { k } ( I - S ) h _ k ( z ) = \sum _ { k = 1 } ^ n \frac { \mu ( k ) } { k } \left [ \log ( 1 - z ^ k ) - \log ( 1 - z ) - \log k \right ] \end{align*}


\begin{align*} & \int _ { \mathbb { R } ^ { 3 } } \Phi _ { + } ^ { 2 } \chi _ R \dd x - \int _ { \mathbb { R } ^ { 3 } } \Phi _ { 0 , + } ^ { 2 } \chi _ R \dd x + 2 \mu \int _ { 0 } ^ { t } \int _ { \mathbb { R } ^ { 3 } } | \nabla \Phi _ { + } | ^ { 2 } \chi _ R \dd x \dd s \\ & = \mu \int _ { 0 } ^ { t } \int _ { \mathbb { R } ^ { 3 } } \Phi _ { + } ^ { 2 } \left ( \Delta + \frac { 2 } { r } \partial _ r \right ) \chi _ R \dd x \dd s + \int _ { 0 } ^ { t } \int _ { \mathbb { R } ^ { 3 } } u \cdot \nabla \chi _ R \Phi _ { + } ^ { 2 } \dd x \dd s . \end{align*}


\begin{align*} \lim \nolimits _ { k \to \infty } C ( l ' , r ' ; P _ { k , S } ^ { l , r } ) = \delta _ S ( l , r ; \ , l ' , r ' ) , \end{align*}


\begin{align*} S t _ { D N D C } \lim _ { n \rightarrow \infty } | | \mathcal { Y } _ { n } ( z ^ { 2 } , z ) - z ^ { 2 } | | _ { \infty } = 0 . \\ \end{align*}


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


\begin{align*} \lambda _ \mathrm { a } [ n ] = \left ( \frac { \Upsilon _ { \rm P D E } \zeta x _ { \rm t } [ n ] } { E _ { \rm p h } } + \vartheta _ { \rm D C R } + \vartheta _ { \rm B } \right ) \left ( 1 + \varphi _ { \rm A P } + \varphi _ { \rm C T } \right ) , \end{align*}


\begin{align*} \int _ { \Omega } - \delta _ N ^ 2 \left ( ( K ( x ) - K ( z _ { N , i } ) ) \nabla \frac { \partial V _ { \delta _ N , Z _ N , i } } { \partial z _ { i , h } } \right ) u _ N = O \left ( \frac { \varepsilon _ N ^ 2 } { | \ln \varepsilon _ N | ^ { p - 1 } } \right ) . \end{align*}


\begin{align*} B _ n \Big ( \frac { 2 \beta ^ j } { L _ j } \Big ) + B _ n \Big ( \frac { F _ j \sqrt 5 } { L _ j } \Big ) = 0 , \qquad \mbox { $ n $ o d d } . \end{align*}


\begin{align*} \abs { \mathcal { R } } \le C _ { V B } = C _ { V B } ( M , g ) . \end{align*}


\begin{align*} X _ { i ; j k } = - R _ { \ell k j i } X ^ \ell + \tfrac { 1 } { 2 } ( h _ { i j ; k } + h _ { i k ; j } - h _ { j k ; i } ) . \end{align*}


\begin{align*} \varphi _ i : = \frac { \langle \alpha ^ f ( X _ i , X _ i ) , \alpha ^ f ( X _ i , X _ i ) \rangle } { \langle \beta ( X _ i , X _ i ) , \beta ( X _ i , X _ i ) \rangle } \neq 0 , \end{align*}


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


\begin{align*} M _ { \pm } ( x ; z ) = I + \mathcal { P } ^ { \pm } \left ( M _ { - } ( x ; \cdot ) R ( x ; \cdot ) \right ) ( z ) , z \in \mathbb { R } , \end{align*}


\begin{align*} R _ { q _ 1 , \ldots , q _ 4 , \kappa } = \# \{ k _ { i j } : k _ { i j } \neq 0 , i < j \} \end{align*}


\begin{align*} | \phi ^ { N , i } ( t , \boldsymbol { x } _ t ) - \Phi ( t , \bar { \nu } _ t ^ i ) | = | \Phi ( t , \nu ^ { N , i } _ { \boldsymbol { x } _ t } ) - \Phi ( t , \bar { \nu } _ t ^ i ) | \leq C | \nu ^ { N , i } _ { \boldsymbol { x } _ t } - \bar { \nu } _ t ^ i | . \end{align*}


\begin{align*} \partial _ t \Phi _ { j } + u _ j \cdot \nabla \Phi _ { j } = \mu _ j \left ( \Delta - \frac { 2 } { r } \partial _ r \right ) \Phi _ j . \end{align*}


\begin{align*} I ( \widetilde { u } _ { k } , \widetilde { v } _ { k } ) = I ( \overline { u } _ { k } , \overline { v } _ { k } ) + o _ { k } ( 1 ) \geq \widetilde { E } _ { 1 } + \widetilde { E } _ { \omega } + o _ { k } ( 1 ) . \end{align*}


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


\begin{align*} \nu ^ { n } _ m ( t ) : = \sum _ { i = m } ^ { n } i \psi ^ { n } _ i ( t ) , \end{align*}


\begin{align*} L F = - \sum _ { q \geq 1 } q J _ q ( F ) . \end{align*}


\begin{align*} \begin{aligned} \Theta _ \gamma ( v ) = 1 \textrm { i f } 0 \leq \gamma \leq 1 \ , , \textrm { a n d } \Theta _ \gamma ( v ) = \nu ^ { - 1 } ( v ) \textrm { i f } - 3 < \gamma < 0 \ , . \end{aligned} \end{align*}


\begin{align*} \partial _ j \xi _ \tau ( 0 , \omega ) = 2 \pi i \ ; y _ j ( \xi ) \ ; \xi ( \omega ) . \end{align*}


\begin{align*} \begin{aligned} v _ t + ( w + u _ { \infty } ) \cdot \nabla v + \nabla p & = ( d + B _ { \infty } ) \cdot \nabla b + \nu \Delta v , \\ b _ t + ( w + u _ { \infty } ) \cdot \nabla b + \nabla q & = ( d + B _ { \infty } ) \cdot \nabla v + \mu \Delta b , \\ \nabla \cdot v = \nabla \cdot b & = 0 , \\ w = \chi _ { \varepsilon } * v , \ d & = \chi _ { \varepsilon } * b . \end{aligned} \end{align*}


\begin{align*} & \ \dot { s } \varepsilon ( z ) \dot { s } ( e - \varepsilon ( 1 ) ) \lambda _ i = \sum _ { j , \mu } g ^ i _ { j , \mu } \dot { s } ( \varepsilon ( z ) - \varepsilon ( x ^ i _ { j , \mu } + z ) ) \lambda _ j \\ & \ = \sum _ { j , \mu } g ^ i _ { j , \mu } \sum _ { k , \nu } g ^ j _ { k , \nu } ( \varepsilon ( z ^ { - 1 } x ^ j _ { k , \nu } ) - \varepsilon ( ( x ^ i _ { j , \mu } + z ) ^ { - 1 } x ^ j _ { k , \nu } ) ) \lambda _ k \end{align*}


\begin{align*} y \left [ n \right ] = \left ( { \sqrt { P \sum \nolimits _ { l = 1 } ^ L { { { \left \| { { { \bf { h } } _ l } } \right \| } ^ 2 } } } } \right ) s \left [ { n - { n _ { \max } } } \right ] + z \left [ n \right ] . \end{align*}


\begin{align*} a _ 1 x ^ q + a _ 1 x = c _ 1 y ^ q + b _ 1 y . \end{align*}


\begin{align*} x = \begin{pmatrix} - \sigma & - \tau & - 1 \\ - \tau & - 1 & - \sigma \\ 1 & \sigma & \tau \end{pmatrix} , \ y = \begin{pmatrix} 1 & \sigma & \tau \\ \sigma & \tau & 1 \\ \tau & 1 & \sigma \end{pmatrix} , \end{align*}


\begin{align*} f : = \varinjlim _ i f _ i \colon \varinjlim _ i \mathcal A _ i \to \varinjlim _ i \mathcal B _ i \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \inf _ { k \in { \mathbb N } } \left ( \frac { \mu ( f ^ { k + n } ( W ) ) } { \mu ( f ^ { k } ( W ) ) } \right ) = \infty \ \ \& \ \ \lim _ { n \rightarrow \infty } \inf _ { k \in - { \mathbb N } _ 0 } \left ( \frac { \mu ( f ^ { k - n } ( W ) ) } { \mu ( f ^ { k } ( W ) ) } \right ) = \infty \tag * { $ \mathcal { U E } 3 $ } \end{align*}


\begin{align*} - \left ( \partial _ { \rho } ^ { 2 } + \frac { 4 } { \rho } \partial _ { \rho } \right ) \tilde { \varphi } = \mu ^ { 2 } \tilde { \varphi } _ { + } , \rho > 0 . \end{align*}


\begin{align*} \int _ M \sum _ i \lambda _ i ^ + ( x ) \ , d x = h _ \mu ( f ) , \end{align*}


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


\begin{align*} f ( t ) = \frac { | A \oplus _ 2 \sqrt { t } B | ^ { \frac { 2 } { k } } } { | V ( ( A \oplus _ 2 \sqrt { t } B ) [ n - k ] , Z _ 1 , \dots , Z _ k ) | ^ { \frac { 2 } { k } } _ { n - k } } \end{align*}


\begin{align*} \rho ( F ) = \{ \eta \in \Omega : F ( \eta ) ^ { - 1 } \} \end{align*}


\begin{align*} \mathcal { I } ( x , t ) = \bigcup _ { i = 1 } ^ { n } [ t _ { i , e n } ( x , t ) , t ] . \end{align*}


\begin{align*} p ^ \mathfrak { t } ( s ) = ( r ^ \mathfrak { t } ( s ) , \lambda ^ \mathfrak { t } ( s ) , \mu ^ \mathfrak { t } ( s ) ) ^ \top = ( r _ 0 , 4 s - 2 \mathfrak { t } + H ( q , r _ 0 ) / 4 , 2 s + 4 \mathfrak { t } ) ^ \top , \end{align*}


\begin{align*} \tau ( [ L ] ) : = \log _ q [ \mathcal { O } _ F ^ d : \pi ^ i L ] , \end{align*}


\begin{align*} \log \frac { 1 } { x } - \gamma + \sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } } { n \cdot n ! } x ^ { n } \ \ ( \gamma : \rm { E u l e r ' s \ c o n s t a n t } ) ; \end{align*}


\begin{align*} \| x \| = \sup _ { p \in \Gamma _ { \tau } } p ( x ) , x \in X ; \end{align*}


\begin{align*} \tilde { h } _ d ( x ) = x \log d + \log \left ( \frac { \Gamma ( x ) } { \Gamma \left ( x + \frac { 1 } { 2 } \right ) \Gamma \left ( x + \frac { d - 1 } { 2 } \right ) } \right ) + \log \left ( \frac { 2 ^ { d - 2 } \Gamma \left ( \frac { d } { 2 } \right ) ^ 2 } { \sqrt { \pi } d ^ { \frac { d - 1 } { 2 } } } \right ) . \end{align*}


\begin{align*} \begin{aligned} \varphi _ t & = { \mathcal { L } } \varphi + \mathbb { P } \nabla \cdot F \textrm { o n } \ L ^ { 2 } _ { \sigma } ( \mathbb { R } ^ { 3 } ) . \\ \end{aligned} \end{align*}


\begin{align*} \mathbb { E } \left [ \mathbf { y ' } ^ H \mathbf { s } ^ { \left ( \right ) } \right . & \left . | \hat { \mathbf { H } } \right ] = a _ c \sum _ { i = 1 } ^ M \hat { \phi } ^ { \left ( i , c \right ) ^ * } + \sum _ { j = 1 } ^ { M } a _ j \hat { \phi } ^ { \left ( j , j \right ) ^ * } . \end{align*}


\begin{align*} f _ \xi ( z ) = \begin{cases} \frac { f ( z ) } { \xi - z } & , \\ 0 & \end{cases} \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*} x _ { 2 i , j } = \tfrac { 1 } { 2 } S - x _ { 2 i , n + 1 - j } . \end{align*}


\begin{align*} \boldsymbol { \lambda } _ k = \left ( \langle y _ k | A _ { 1 , k } y _ k \rangle , \dots , \langle y _ k | A _ { d , k } y _ k \rangle \right ) , \end{align*}


\begin{align*} \begin{cases} \Delta _ { \Psi } u = f ( u ) & \O \\ u = c _ { j } & ( \partial \O ) _ { j } \end{cases} \end{align*}


\begin{align*} \mathfrak { R } \int _ { \{ \Psi < - t \} } \langle F _ t , \bar { q } \rangle _ h c ( - \Psi ) = 0 , \end{align*}


\begin{align*} \rho \ , u _ { , t t } & = \ , \boldsymbol { \sigma } \ , , \\ \rho \eta \ , \tau _ { \rm c } ^ 2 \ , \mathbf { A } _ { , t t } & = - \ , \ , \boldsymbol { m } + \ , \boldsymbol { \sigma } \ , , \end{align*}


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


\begin{align*} \int _ M | | ( \eta + g ^ { - 1 } ) ^ { \frac { 1 } { 2 } } u _ 0 | | _ { \omega } + \int _ M | | \tau _ 0 | | _ { \omega } \le \int _ M \langle ( B + \lambda I ) ^ { - 1 } v , v \rangle _ { \omega , h } d V _ { \omega } . \end{align*}


\begin{align*} \partial _ { \theta _ i } \theta _ j = - \theta _ j \partial _ { \theta _ i } \hbox { i f } 1 \leq i \neq j \leq n \qquad \hbox { a n d } \partial _ { \theta _ i } \theta _ i = 1 \hbox { f o r } 1 \leq i \leq n ~ . \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*} \sigma _ W ^ 2 = 3 2 / \alpha ^ 2 \sigma _ Z ^ 2 = 3 2 T ^ 2 / \alpha ^ 2 \end{align*}


\begin{align*} [ a _ \lambda b ] = \sum _ { j \in \mathbb { Z _ { + } } } ( a _ { ( j ) } b ) \frac { \lambda ^ { j } } { j ! } . \end{align*}


\begin{align*} \overline { \mathcal F } ( X ) = \varinjlim _ { Y \in ( \mathcal C _ 0 ) _ { X / } } F ( Y ) , \end{align*}


\begin{align*} [ T ( f ) ] ( \xi ) : = \frac { f ( \xi ) } { \sqrt { ( 1 + \gamma ( \xi ) ^ 2 ) } } . \end{align*}


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


\begin{align*} \Phi _ p ( x ) = \begin{cases} \phi _ p ( x ) , & x \geq 1 , \\ 2 \phi _ p ( 1 ) - \phi _ p ( 2 - x ) , & 0 \leq x \leq 1 . \end{cases} \end{align*}


\begin{align*} \mathcal { C } \big ( \mathcal { U } _ { ( i , j ) } \big ) \cong \begin{cases} C ^ \infty \big ( \mathcal { U } _ { ( 0 , 0 ) } \big ) & \\ \mathcal { S } \big ( \mathcal { U } _ { ( i , j ) } \big ) & \\ \mathcal { A } \big ( \mathcal { U } _ { ( i , j ) } \big ) & \end{cases} \ , , \end{align*}


\begin{align*} \begin{cases} - \Delta \psi + \psi = \lambda ( \psi - W x _ 2 ) _ + , \quad & \mathrm { i n } \ \ \Pi , \\ \psi = 0 , & \mathrm { o n } \ \ \partial \Pi , \\ \psi \rightarrow 0 , & \mathrm { a s } \ \ | x | \rightarrow \infty . \\ \end{cases} \end{align*}


\begin{align*} \frac D 2 = \{ q _ 1 \mid q _ 1 \overline { q p } q \in D \} . \end{align*}


\begin{align*} \hat \Phi \left ( \begin{pmatrix} a \\ b \end{pmatrix} , s \right ) : = \begin{pmatrix} \langle z _ 1 ^ * , \Phi ( a u _ \ell + b w _ \ell , p ( s ) ) \rangle \\ \langle z _ 2 ^ * , \Phi ( a u _ \ell + b w _ \ell , p ( s ) ) \rangle \end{pmatrix} . \end{align*}


\begin{align*} A _ t & = a _ t , \\ B \times u & = b \times v + b \times u _ { \infty } + B _ { \infty } \times v , \\ \nabla \times B & = \nabla \times b . \end{align*}


\begin{align*} \mathbb { E } { \bf g } _ { \rm G E } ( G ) = + \infty . \end{align*}


\begin{align*} \left ( \psi ^ L _ { a ; S } \right ) _ { 0 j } \colon & f ( x , y , z ) \mapsto \eta ^ { - a \vert S \vert j } f ( x , \eta ^ { - a } x , z ) \\ \left ( \psi ^ R _ { S ; b } \right ) _ { i 0 } \colon & f ( x , y , z ) \mapsto f ( x , \eta ^ { b } z , z ) . \end{align*}


\begin{align*} \mathcal { U } = \mathrm { k e r } A \cap \S ^ { n - 1 } . \end{align*}


\begin{align*} { { \bf { F } } _ i } \left ( { { \bf { x } } , t } \right ) = f _ i ^ { \left ( { e q } \right ) } \left ( { { { \bf { u } } ^ { e q } } + \Delta { \bf { u } } } \right ) - f _ i ^ { \left ( { e q } \right ) } \left ( { { { \bf { u } } ^ { e q } } } \right ) , \end{align*}


\begin{align*} F _ { \delta , Z } = \{ u \in L ^ p ( \Omega ) \mid \int _ { \Omega } \frac { \partial V _ { \delta , Z , j } } { \partial z _ { j , h } } u = 0 , \ \ \forall j = 1 , \cdots , m , \ h = 1 , 2 \} , \end{align*}


\begin{align*} \begin{aligned} K f ( v ) = K _ 1 f ( v ) - K _ 2 f ( v ) \ , . \end{aligned} \end{align*}


\begin{align*} [ X \times _ k \mathbb A _ k ^ n \to S , \sigma ] = [ X \times _ k \mathbb A _ k ^ n \to S , \sigma ' ] \end{align*}


\begin{align*} - \Delta v - p w ^ { p - 1 } _ + v = 0 , \ \ v \in L ^ { \infty } ( \mathbb { R } ^ 2 ) . \end{align*}


\begin{align*} E ( G ) = \{ V _ i \cup V _ j : \ : \{ i , j \} \in F \} . \end{align*}


\begin{align*} a ( u _ n , v _ n ) = \lambda _ n ( u _ n , v _ n ) v _ n \in X _ n , \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*} \gamma _ { c , k } & = \frac { a _ c ^ 2 \lvert \mathbf { \hat { h } } _ k ^ { \textrm { T } } \mathbf { p } _ c \rvert ^ 2 } { \sum \limits _ { i = 1 } ^ K a _ i ^ 2 \lvert \mathbf { h } _ k ^ { \textrm { T } } \mathbf { p } _ i \rvert ^ 2 + \sigma _ w ^ 2 } . \end{align*}


\begin{align*} \frac { d ^ j } { d t ^ j } \tilde { X } _ t | _ { t = 0 } \end{align*}


\begin{align*} I _ { n + 1 } \frac { \partial ^ 2 } { \partial x _ n ^ 2 } + B \frac { \partial } { \partial x _ n } + C = \Bigl ( I _ { n + 1 } \frac { \partial } { \partial x _ n } + B - Q \Bigr ) \Bigl ( I _ { n + 1 } \frac { \partial } { \partial x _ n } + Q \Bigr ) . \end{align*}


\begin{align*} X = N \oplus X _ 0 \qquad Z = R \oplus Z _ 0 , \end{align*}


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


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


\begin{align*} D ^ k \varphi ( Z ) = \sum _ { q \geq k } \frac { b _ { q } } { ( q - k ) ! } H _ { q - k } ( Z ) D ^ k L \varphi ( Z ) = - \sum _ { q \geq k } q \ , \frac { b _ { q } } { ( q - k ) ! } H _ { q - k } ( Z ) , \end{align*}


\begin{align*} & ( i ) \ \ \ \nabla ( J + \sum \limits _ { k = 1 } ^ m \gamma _ k f _ k ) ( x ^ * ) = 0 \\ & ( i i ) \ \ \gamma _ k f _ k ( x ^ * ) = 0 \ \ k \in \{ 1 , . . . , m \} \\ & ( i i i ) \ A _ J + \sum \limits _ { k = 1 } ^ m \gamma _ k A _ k \succeq 0 \end{align*}


\begin{align*} \left ( \Delta K _ m \right ) ^ m \wedge \beta _ n ^ { n - m } = 0 \mathbb { H } ^ n \setminus \{ 0 \} . \end{align*}


\begin{align*} \mbox { $ b _ { m a x p r i m } ( G ) : = \max \{ b ( G ^ \Omega ) \mid G \Omega \} $ . } \end{align*}


\begin{align*} \dot { x } = - \nabla f ( x ) = - x ^ 3 , x ( 0 ) = 1 . \end{align*}


\begin{align*} \hat \Phi ^ \dagger _ a ( 0 , 0 ) & = \langle z _ 1 ^ * , \Phi _ v ( 0 , p _ 0 ) u _ \ell \rangle = 0 . \end{align*}


\begin{align*} v ^ { { \rm c o l } ( \mu ) } \prod _ { ( i , j ) \in \mu } X _ { a - j + 1 } = \prod _ { i = 1 } ^ { { \rm c o l } ( \mu ) } \bigg ( v \prod _ { j = 1 } ^ { \mu ' _ i } X _ { a - j + 1 } \bigg ) , \end{align*}


\begin{align*} d ( \lambda _ v ) = e = \iota _ { \star } = \lambda _ { d ^ 0 ( v ) } \end{align*}


\begin{align*} \begin{aligned} & \rho _ - \left \| Q _ { j , + } \right \| _ { L ^ 2 } ^ 2 \leq \operatorname { R e } \int _ { \mathbb { R } } Q _ { j , + } ( I + J ) ^ H Q _ { j , + } ^ { H } d z \\ & = \operatorname { R e } \int _ { \mathbb { R } } \mathcal { P } ^ + ( D _ j ) ( I + J ) ^ { H } Q _ { j , + } ^ { H } d z \leq \rho _ + \| D _ j \| _ { L ^ 2 } \left \| Q _ { j , + } \right \| _ { L ^ 2 } . \end{aligned} \end{align*}


\begin{align*} y _ n = \rho ( x ) - \rho ( x _ 0 ) \geq \sum _ { \beta = 1 } ^ { n - 1 } \rho _ \beta ( x ' _ 0 ) y _ \beta + \kappa | y ' | ^ 2 , \end{align*}


\begin{align*} \Psi = \rho ( x ' ) - x _ n - \theta | x ' | ^ 2 + K x _ n ^ 2 . \end{align*}


\begin{align*} \epsilon ( \pi ^ * D , y ) = \epsilon ( D , \pi ( y ) ) . \end{align*}


\begin{align*} H ( \widetilde { Q } ) \le \sum _ { i = 1 } ^ { n } \tilde { t } _ i H ( Q _ i ) , \end{align*}


\begin{align*} W = \frac { 1 } { 2 } \Big [ & ( \mu ^ * + \varkappa ) \ , { \rm e } ^ * _ { j i } { \rm e } ^ * _ { j i } + \mu ^ * \ , { \rm e } ^ * _ { j i } { \rm e } ^ * _ { i j } + \lambda \ , { \rm e } ^ * _ { i i } { \rm e } ^ * _ { j j } + \gamma \ , \mathfrak { K } _ { j i } \mathfrak { K } _ { j i } + \beta \ , \mathfrak { K } _ { j i } \mathfrak { K } _ { i j } + \alpha \ , \mathfrak { K } _ { i i } \mathfrak { K } _ { j j } \Big ] . \end{align*}


\begin{align*} \mathcal { G } \omega \left ( x \right ) = \int _ { \Pi } { G _ { \Pi } \left ( x , y \right ) \omega \left ( y \right ) \mathrm { d } y , x \in \Pi . } \end{align*}


\begin{align*} & \mathcal { P } ^ + \left ( f ( z ) e ^ { 2 i z x } \right ) = - \int _ { - \infty } ^ { 2 x } \widehat { f } ( \xi ) e ^ { - i z ( \xi - 2 x ) } \mathrm { d } \xi , \\ & \mathcal { P } ^ - \left ( f ( z ) e ^ { - 2 i z x } \right ) = \int _ { - \infty } ^ { 2 x } \widehat { f } ( \xi ) e ^ { i z ( \xi - 2 x ) } \mathrm { d } \xi . \end{align*}


\begin{align*} ( f _ 1 ( \mathbf { x } ) , f _ 2 ( \mathbf { x } ) , \cdots , f _ { n - 1 } ( \mathbf { x } ) ) = ( f _ 2 ( \mathbf { y } ) , f _ 3 ( \mathbf { y } ) , \cdots , f _ n ( \mathbf { y } ) ) = ( f _ 2 ( \mathbf { x } ) , f _ 3 ( \mathbf { x } ) , \cdots , f _ n ( \mathbf { x } ) ) . \end{align*}


\begin{align*} \partial ^ 2 _ { j k } v _ i - \Gamma _ { k j } ^ j \partial _ j v _ i - \Gamma _ { j k } ^ k \partial _ k v _ i = 0 . \end{align*}


\begin{align*} { { P } } _ k ^ { \mathrm { d i s c } } ( \mathcal { T } ) : = \left \{ q _ h \in L ^ 2 ( \Omega ) : q _ h | _ T \in P _ k ( T ) T \in \mathcal { T } \right \} . \end{align*}


\begin{align*} \phi ( t ) = f ( t ) + C ( \kappa \ , * \ , k _ 1 ) ( t ) , \ t > 0 . \end{align*}


\begin{align*} \begin{aligned} v _ 1 ' = v _ 1 - [ ( v _ 1 - v ) \cdot \omega ] \omega \ , , \ v ' = v + [ ( v _ 1 - v ) \cdot \omega ] \omega \ , , \end{aligned} \end{align*}


\begin{align*} { \rm V o l } ( { \tilde \Sigma } ^ { 3 r } ) \leq C ( n ) \sum _ { i = 1 } ^ s { \rm V o l } ( B _ { p _ i } ( r ) ) \leq C ( n ) \sum _ { i = 1 } ^ s { \rm V o l } ( A \cap B _ { p _ i } ( r ) ) , \end{align*}


\begin{align*} \int _ { B _ { \rho } } r ^ { 2 - n } u _ r ^ 2 \d x & \leq C \int _ { B _ { 2 \rho } \setminus B _ { \rho } } r ^ { 2 - n } | \nabla u | ^ 2 \d x + C \varepsilon \int _ { B _ { 4 \rho } } r ^ { 3 - n } | \nabla u | ^ 2 \d x \end{align*}


\begin{align*} j = a b , j ' = b c , \gcd ( a , c ) = 1 , a b c \le n . \end{align*}


\begin{align*} \lim _ { M \rightarrow \infty } \sup _ { n \in \mathbb { N } } \mathbb { P } ( \tau _ { n } ^ M < T ) = 0 . \end{align*}


\begin{align*} & N ( x ; k ) = I + \frac { 1 } { 2 \pi i } \int _ { \mathbb { R } \cup i \mathbb { R } } \frac { \varrho ( x ; s ) J } { s - k } d s = I + \mathcal { C } ( \varrho J ) ( z ) , \end{align*}


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


\begin{align*} X _ { r } = \xi _ r \quad { X } _ t = \xi _ 0 + \int _ { 0 } ^ { t } \mu ( s , X ) \ , d s + \int _ { 0 } ^ { t } \sigma ( s , X ) \ , d W _ s , \end{align*}


\begin{align*} \frac { \partial } { \partial t } \Theta ^ M ( t , x ) = \int _ \S W _ r ^ M ( x - y ) \sin ( \Theta ^ M ( t , y ) - \Theta ^ M ( t , x ) ) \ \d y , \end{align*}


\begin{align*} \beta ( z , w ) = \dfrac { 1 } { 2 } \log \dfrac { 1 + | \varphi _ z ( w ) | } { 1 - | \varphi _ z ( w ) | } , z , w \in \mathbb { B } _ n . \end{align*}


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


\begin{align*} g ( z ) = \sum _ { m = 0 } ^ { \infty } a _ m z ^ { q ( m + 1 ) } = \sum _ { n = 1 } ^ { \infty } a _ { n - 1 } z ^ { q n } = \sum _ { n = 0 } ^ { \infty } b _ n z ^ { q n } , b _ n = \begin{cases} a _ { n - 1 } & , n \geq 1 , \\ 0 & , . \end{cases} \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 ( E ) = \lim _ { \epsilon \rightarrow + \infty } L ( E , \epsilon ) = \log \Big | \frac { E } { 2 } + \frac { \sqrt { E ^ { 2 } - 4 } } { 2 } \Big | . \end{align*}


\begin{align*} f ( z ) = & \sum _ { n = 0 } ^ { \infty } \left ( x - \frac { \psi _ 0 } { \psi _ 1 } y \right ) ^ n a _ n = \sum _ { n = 0 } ^ { \infty } \left ( \frac { z + \bar z } { 2 } - \frac { \psi _ 0 } { \psi _ 1 } \frac { z - \bar z } { 2 } \right ) ^ n a _ n \\ = & \sum _ { n = 0 } ^ { \infty } \left ( ( 1 - \frac { \psi _ 0 } { \psi _ 1 } ) z + ( 1 + \frac { \psi _ 0 } { \psi _ 1 } ) \bar z \right ) ^ n \frac { a _ n } { 2 ^ n } , \end{align*}


\begin{align*} \dot { B } _ { p , r } ^ { s } = \Big \{ f \in \mathcal { S } ' _ h ( \mathbb { R } ^ { d } ) : \ ; \| f \| _ { \dot { B } _ { p , r } ^ { s } ( \mathbb { R } ^ { d } ) } < \infty \Big \} , \end{align*}


\begin{align*} C ( G , x ) : = 1 + \sum _ { k = 1 } ^ { \omega ( G ) } c _ { k } ( G ) , \end{align*}


\begin{align*} ( \C ^ n , \omega _ { \mathrm { s t } } ) , \omega _ { \mathrm { s t } } = \i \sum d z ^ i \wedge d \bar { z } ^ i \end{align*}


\begin{align*} q ^ \perp = 1 \oplus 0 \oplus 1 \oplus \begin{pmatrix} b ^ 2 & - a b \\ - a b & a ^ 2 \end{pmatrix} . \end{align*}


\begin{align*} \Phi : = { \sup } ^ * \left \{ \zeta : \begin{array} { l } \zeta \overline { \Omega } \times \mathbb R ^ m , \\ F \star \mathcal P \Omega \times \mathbb R ^ m \Phi | _ { \partial \Omega \times \mathbb R ^ m } \le \phi \end{array} \right \} . \end{align*}


\begin{align*} Z _ n = M _ n + r _ n N _ 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*} \begin{cases} \Delta _ { \Psi } u = f ( u ) & \O \\ u = \phi _ { j } & ( \partial \O ) _ { j } \end{cases} \end{align*}


\begin{align*} \gamma _ { t - 1 } \alpha ( \gamma ) ^ 2 = ( 1 - \alpha ( \gamma ) ) \alpha _ { t - 1 } ^ 2 \gamma + \mu \gamma \gamma _ { t - 1 } \alpha ( \gamma ) . \end{align*}


\begin{align*} \lim \nolimits _ { k \to \infty } C ( l ' , r ' ; P _ { k , S } ^ { l , r } ) = \delta _ S ( l , r ; \ , l ' , r ' ) , \end{align*}


\begin{align*} \mathcal { I } ( x , t ) = \bigcup _ { i = 1 } ^ p [ t _ { i , e n } ( x , t ) , t _ { i , e x } ( x , t ) ] \end{align*}


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


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


\begin{align*} T = \bigcup _ { i = 1 } ^ I A ( p _ i , s _ i , 2 s _ i ) \end{align*}


\begin{align*} M _ n & \ge \sup _ { Q \in \P ( \R ) } \bigg ( n \int _ { \R } V ( x ) \ , Q ( d x ) + \frac 1 2 \sum _ { i , j = 1 } ^ n J _ { i j } \int _ { \R } \int _ { \R } K ( x - y ) \ , Q ( d x ) Q ( d y ) - n H ( Q ) \bigg ) \\ & = n \sup _ { Q \in \P ( \R ) } \bigg ( \int _ { \R } V \ , d Q + \frac 1 2 \int _ { \R } \int _ { \R } K ( x - y ) Q ( d x ) Q ( d y ) - H ( Q ) \bigg ) . \end{align*}


\begin{align*} \varpi \partial _ i ( \partial _ j ( u _ k w _ \ell ) ) = \partial _ i ( \partial _ j ( \varpi u _ k w _ \ell ) ) - \partial _ i ( ( \partial _ j \varpi ) u _ k w _ \ell ) - \partial _ j ( ( \partial _ i \varpi ) u _ k w _ \ell ) - ( \partial _ i \partial _ j \varpi ) u _ k w _ \ell . \end{align*}


\begin{align*} ( d _ 1 : = d _ 1 ' , \ldots , d _ m ' : = d _ 2 ) , \end{align*}


\begin{align*} \int _ 0 ^ { T } \int _ { \mathbb { R } ^ { 3 } } \Phi _ j ( \partial _ t \varphi + ( v _ j + u _ { \infty } ) \cdot \nabla \varphi ) \dd x \dd s + \int _ { \mathbb { R } ^ { 3 } } \Phi _ 0 \varphi _ 0 \dd x = - \mu _ j \int _ 0 ^ { T } \int _ { \mathbb { R } ^ { 3 } } \left ( \Phi _ j \Delta \varphi - \frac { 2 } { r } \partial _ r \Phi _ j \varphi \right ) \dd x \dd s . \end{align*}


\begin{align*} R _ c ^ { s y m } = \pi / 0 . 8 6 3 \approx 3 . 6 4 . \end{align*}


\begin{align*} \cos ^ 2 \theta & = \frac { 1 + \cos ( 2 \theta ) } { 2 } = \frac { \sqrt { m } - \delta _ { m , l } } { 2 \sqrt { m } - \delta _ { m , k } - \delta _ { m , l } } , \\ \sin ^ 2 \theta & = \frac { 1 - \cos ( 2 \theta ) } { 2 } = \frac { \sqrt { m } - \delta _ { m , k } } { 2 \sqrt { m } - \delta _ { m , k } - \delta _ { m , l } } , \end{align*}


\begin{align*} \left . \frac { \partial s ( z ) } { \partial z } \right | _ { z = 0 } = - \frac { 1 } { 4 } + \frac { 1 - \alpha } { \pi } . \end{align*}


\begin{align*} \varphi ^ * x ^ { \mu } = y ^ { \mu } , \varphi ^ * \theta _ { i , a } = \eta _ { i , a } , \end{align*}


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


\begin{align*} R _ c = \pi \sqrt { 1 6 . 6 7 / 0 . 6 5 } \approx 1 5 . 9 \ , . \end{align*}


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


\begin{align*} \partial _ z ^ * = M _ z , \end{align*}


\begin{align*} \partial _ x T - \mathcal { P } ^ + \left ( \partial _ x T \right ) R _ + - \mathcal { P } ^ - \left ( \partial _ x T \right ) R _ - = \widetilde { F } , \end{align*}


\begin{align*} \sum _ { i = 1 } ^ m \mathcal { W } _ 2 ^ 2 ( P _ { S _ i } , Q _ { S _ i } ) \le \mathcal { W } _ 2 ^ 2 ( P _ { S _ 1 \cup \cdots \cup S _ m } , Q _ { S _ 1 \cup \cdots \cup S _ m } ) \le \mathcal { W } _ 2 ^ 2 ( P , Q ) . \end{align*}


\begin{align*} \partial _ t w + ( \mathbf { v } \cdot \nabla ) w = 0 . \end{align*}


\begin{gather*} \sum _ { i = 1 } ^ n \mu _ i ( f _ i ) = P \Bigl ( \oplus _ { i = 1 } ^ n f _ i \Bigr ) \ge \inf _ { Q \in \mathcal { U } } Q \Bigl ( \oplus _ { i = 1 } ^ n f _ i \Bigr ) \end{gather*}


\begin{align*} h ( x + \delta u ) = h ( x ) + R ( x , \delta u ) \end{align*}


\begin{align*} \int _ 0 ^ T \psi ( t ) d t = \int _ 0 ^ T \biggl ( 1 - \frac { t } { T } \biggr ) ^ m d t = C ( m ) T . \end{align*}


\begin{align*} C F ( F ) = \widehat { t e l } ( C F ( F ) , i d ) \rightarrow C ^ { L , k } = \varinjlim _ { p } \varprojlim _ { q } \varinjlim _ { n _ { i } } C F ^ { L , k } _ { [ p , q ] } ( H _ { n _ { i } , t } ) . \end{align*}


\begin{align*} e ^ { - t \Lambda _ g } = \frac { i } { 2 \pi } \int _ { \mathcal { C } } e ^ { - t \tau } ( \Lambda _ g - \tau I ) ^ { - 1 } \ , d \tau , \end{align*}


\begin{align*} & b _ k ^ T \Tilde { x } \le b _ k ^ T ( \lambda x _ v + ( 1 - \lambda ) x _ w ) , \\ & b _ k ^ T ( \Tilde { x } - ( \lambda x _ v + ( 1 - \lambda ) x _ w ) ) \le 0 . \end{align*}


\begin{align*} \frac { d } { d t } E \big [ [ M _ v ] ( t ) \big ] = \sum _ { k = 1 } ^ N e ^ { - 2 \lambda t } \lambda ^ 2 v _ { k } ^ 2 E | Z _ k ( t ) | \le 4 | v | ^ 2 e ^ { - 2 \lambda t } E \bigg [ \sum _ { k = 1 } ^ N | Z _ k ( t ) | \bigg ] . \end{align*}


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


\begin{align*} \begin{cases} \min \{ \frac { 1 } { 2 } + \frac { \alpha } { 2 \beta } ( r + \lambda - \kappa ) , 1 \} , & \mbox { i f } \alpha + \gamma = 1 , \\ \min \{ \frac { \alpha } { 2 \beta } \min \{ \kappa , r + \lambda \} + ( \gamma - \frac { 1 } { 2 } ) ^ { + } , 1 - \varepsilon \} , & \mbox { i f } \alpha + \gamma \neq 1 \end{cases} \end{align*}


\begin{align*} H = \Theta : L ^ 2 ( \R ^ { n - 1 } , \mathbb { C } ^ N ) \to L ^ 2 ( \R ^ n _ + , d x ' d t / t ) ^ N . \end{align*}


\begin{align*} \int _ { t _ 1 } ^ { t _ 2 } \sum _ { i = 1 } ^ { \infty } \sum _ { j = 1 } ^ { i } j ( g _ { i + 1 } - g _ i ) V _ { i , j } \psi _ j ( s ) \psi _ i ( s ) d s < \infty \end{align*}


\begin{align*} \left \{ \begin{array} { r l } \div ( A ( x ) \nabla v ) + \dd ( x ) \cdot \nabla u = 0 & B _ { \rho } \\ v = u & \partial B _ { \rho } . \end{array} \right . \end{align*}


\begin{align*} \operatorname { I } _ { y } ( a , b ) = \frac { 1 } { \operatorname { B } ( a , b ) } \int ^ { y } _ { 0 } \omega ^ { a - 1 } ( 1 - \omega ) ^ { b - 1 } \mathrm { d } \omega , \end{align*}


\begin{align*} & \ln \left \{ \Phi \left ( \frac { x - \mu } { \sigma } \right ) \right \} = \sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ n } { n } \ , \left [ 1 - \Phi \left ( \frac { x - \mu } { \sigma } \right ) \right ] ^ n \end{align*}


\begin{align*} X _ { t } = X _ 0 + \int _ 0 ^ t a _ s \ , d s + \int _ 0 ^ t b _ s \ , d W _ s , \end{align*}


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


\begin{align*} \Theta ^ M ( t , x ) = \phi _ i ( t ) x \in \left [ \frac { i - 1 } M , \frac i M \right ) . \end{align*}


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


\begin{align*} \Lambda _ C : = \{ ( x , y ) \in \Lambda : x \in C \} \end{align*}


\begin{align*} A ^ * ( y ) : = A ^ * ( y | x ) = A ( \tau _ y ( x ) ) + W \circ \hat { \sigma } ^ { - 1 } ( y | x ) - W ( y | x ) \ , \end{align*}


\begin{align*} H ( Q _ k \ , | \ , P ) = \frac { 1 } { Q ( B _ k ) } \int _ { B _ k } \log \frac { d Q } { d P } \ , d Q - \log Q ( B _ k ) \end{align*}


\begin{align*} K ^ L ( x ' , t ) : = P _ t ^ L ( x ' ) = t ^ { 1 - n } P ^ L ( x ' / t ) , \forall ( x ' , t ) \in \R ^ n _ + , \end{align*}


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


\begin{align*} 2 a = ( a + b ) + ( a - b ) \end{align*}


\begin{align*} \left \{ \begin{aligned} \Delta u & = v & & \quad \mbox { i n } \Omega , \\ \Delta v & = | \nabla u | ^ 2 & & \quad \mbox { i n } \Omega , \end{aligned} \right . \end{align*}


\begin{align*} F ( x ' , t ) : = \Phi _ t * f ( x ' ) \quad H ( x ' , t ) : = \widetilde { \Phi } _ t * h ( x ' ) , ( x ' , t ) \in \R ^ n _ + . \end{align*}


\begin{align*} E [ M ^ * _ v ( t ) ^ 2 ] \le e ^ { - 2 ( \lambda ^ * - \lambda ) t } E [ M _ v ( 0 ) ^ 2 ] + \left \{ \begin{aligned} & 4 | v | ^ 2 \| x \| t e ^ { - 2 ( \lambda ^ * - \lambda ) t } & & \lambda \ge 1 , \\ & 4 | v | ^ 2 \| x \| \frac { 1 } { 1 - \lambda } e ^ { - 2 ( \lambda ^ * - 1 ) t } & & \lambda < 1 . \end{aligned} \right . \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*} \begin{aligned} \mathbf { S } _ { \infty m } ( q , 1 ) = \big \{ ( q _ { i j } ) _ { \substack { i \geq 1 \\ 0 \leq j \leq m } } \big | \ , q < q _ { i j } < 1 \ , , q _ { i j } < q _ { i , j - 1 } ( \forall i \geq 1 , 0 \leq j \leq m ) \ , , q _ { i + 1 , 0 } < q _ { i m } ( \forall i \geq 1 ) \big \} \ , . \end{aligned} \end{align*}


\begin{align*} \mathcal { N } _ { \epsilon } ( x , y ) : = \int _ { \mathbb { R } } \rho _ { \epsilon } ( y - z ) \mathcal { N } ( x , z ) \ , d z , \end{align*}


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


\begin{align*} e m + \gamma = \widehat { \iota } ( D ) c \geq \widehat { \iota } ( D ) ( b _ 1 d + 1 ) . \end{align*}


\begin{align*} Y ^ n _ 0 = \xi _ 0 , \quad Y _ { k + 1 } ^ n = Y _ k ^ n + \mu ( \tfrac { k T } { n } , \mathcal { Y } ^ n ) \tfrac { T } { n } + \sigma ( \tfrac { k T } { n } , \mathcal { Y } ^ n ) ( W _ { \frac { ( k + 1 ) T } { n } } - W _ { \frac { k T } { n } } ) . \end{align*}


\begin{align*} \sigma ( \sqrt { a } ) = - \sqrt { a } , \ & \sigma ( \sqrt { d } ) = \sqrt { d } \\ \tau ( \sqrt { a } ) = \sqrt { a } , \ & \tau ( \sqrt { d } ) = - \sqrt { d } . \end{align*}


\begin{align*} X : = \{ \sum _ { n = 0 } ^ \infty a _ n z ^ n \mid ( a _ n n ^ { - \alpha } ) _ { n \in \mathbb { N ^ * } } \in \ell ^ 2 \} \end{align*}


\begin{align*} f ^ \star = \min f ( W _ 1 ) = \min f ( W _ 2 ) \ , . \end{align*}


\begin{align*} \nabla \Phi ^ g ( x , y ) = \begin{bmatrix} e ^ { y _ 1 } \nabla g _ 1 ( x ) & \dots & e ^ { y _ m } \nabla g _ m ( x ) \\ e ^ { y _ 1 } g _ 1 ( x ) & \dots & 0 \\ . & \dots & . \\ 0 & \dots & e ^ { y _ m } g _ m ( x ) \end{bmatrix} = \varphi ^ g ( x ) \times \begin{bmatrix} e ^ { y _ 1 } & \dots & 0 \\ . & \dots & . \\ 0 & \dots & e ^ { y _ m } \end{bmatrix} \ , . \end{align*}


\begin{align*} \prescript { } { l } { b } ( \lambda ) = \prescript { } { l } { \beta } ( \lambda ) \prescript { } { l } { \tilde { p } } ( \lambda ) \end{align*}


\begin{align*} \mathbb { E } _ { n , m } ' [ ( \alpha + \beta ) ( \gamma + \delta ) ] \\ = \mathbb { E } _ { n , m } [ \chi _ { n , m } ( w , x _ 1 ) \chi _ { n , m } ( w , x _ 2 ) ] \\ \\ = \mathcal { K } _ { n , m } ( x _ 1 , x _ 2 ) \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*} \begin{cases} | \mathcal { K } ( c ' , x ) | \leqslant ( c ' + 1 ) C , & | x | < 1 , \\ \mathcal { K } ( c ' , x ) \leqslant ( - \alpha + \frac { \beta } { | x | } + \frac { C } { | x | } + c ' C ) , & | x | \geqslant 1 , \end{cases} \end{align*}


\begin{align*} \mathcal U ( s , i ) = [ 0 , \bar L ] , ( s , i ) \in C . \end{align*}


\begin{align*} D ( x ) : = \det \left [ \begin{array} { c c c c c } \alpha _ 1 & \alpha _ 1 ^ q & \alpha _ 1 ^ { q ^ 2 } & \cdots & \alpha _ 1 ^ { q ^ { n } } \\ \alpha _ 2 & \alpha _ 2 ^ q & \alpha _ 2 ^ { q ^ 2 } & \cdots & \alpha _ 2 ^ { q ^ { n } } \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \alpha _ n & \alpha _ n ^ q & \alpha _ n ^ { q ^ 2 } & \cdots & \alpha _ n ^ { q ^ { n } } \\ x & x ^ q & x ^ { q ^ 2 } & \cdots & x ^ { q ^ { n } } \\ \end{array} \right ] \end{align*}


\begin{align*} - \Delta u = f & D , \\ [ 1 m m ] u = 0 & \partial D . \end{align*}


\begin{align*} \sup _ { M } v = \sup _ { \partial M } v . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ n { ( 1 \otimes a _ i \otimes a _ i + a _ i \otimes 1 \otimes a _ i + a _ i \otimes a _ i \otimes 1 ) } + 1 \otimes 1 \otimes y + 1 \otimes y \otimes 1 + y \otimes 1 \otimes 1 . \end{align*}


\begin{align*} d _ I ( \C ) = \min _ { \mathbf { a } , \mathbf { b } \in \C , \mathbf { a } \neq \mathbf { b } } \left \{ \ , \ , d _ I ( \mathbf { a } , \mathbf { b } ) \ , \ , \right \} . \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*} \left \{ \begin{aligned} a x + b = \lambda x , \\ d = \lambda . \end{aligned} \right . \end{align*}


\begin{align*} T _ u v = \sum _ j S _ { j - 1 } u \Delta _ j v , R ( u , v ) = \sum _ { | j - j ' | \leq 1 } \Delta _ j u \Delta _ j ' v . \end{align*}


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


\begin{align*} f _ { k } ( x , y ) : = \begin{cases} f ( x , - k ) , & y < - k , \\ f ( x , y ) , & | y | \le k , \\ f ( x , k ) , & y > k . \end{cases} \end{align*}


\begin{align*} x _ n - y _ n \geq ( 2 - \epsilon ) b _ { n - 1 } \mbox { o n } \partial \omega \cap \{ x _ n = 2 b _ { n - 1 } \} . \end{align*}


\begin{align*} \psi _ { \tau } ( p ) = | | \tau | | _ p \le C _ 1 ( \beta , \gamma , L ) \ ( \beta - p ) ^ { - ( \gamma + 1 ) / \beta } \ L ^ { 1 / \beta } ( 1 / ( \beta - p ) ) , \end{align*}


\begin{align*} j ^ 1 _ q ( F _ \varphi \circ f _ \varphi { } ^ { - 1 } ) & = ( q , F _ \varphi ( o ) , D F _ \varphi ( o ) ( D f _ \varphi ( o ) ) ^ { - 1 } ) \\ * & = ( q , F _ \varphi ( o ) , D F _ \varphi ( o ) ^ i { } _ { j \alpha } ( ( F _ \varphi ( o ) ) ^ { - 1 } ) ^ \alpha { } _ k ) \end{align*}


\begin{align*} F ( s , x ) = \left | \frac { \zeta ' } { \zeta } ( s ) + \sum _ { n < x ^ 2 } \frac { \Lambda _ x ( n ) } { n ^ s } \right | \leq | Z _ 1 | + | Z _ 2 | + | Z _ 3 | \end{align*}


\begin{align*} \overline { \mathrm { S } ( V ) } = \prod _ { n \in \mathbb { N } } \mathrm { S } ^ n ( V ) \end{align*}


\begin{align*} \begin{aligned} L ^ \Phi ( \theta ) [ \cdot ] : = & - \big ( { \rm d i v } _ { \ ! z } + 2 i \pi \theta \big ) \big ( A { ( \Phi ^ { - 1 } ( z , \omega ) , \omega ) } { \big ( \nabla _ { \ ! \ ! z } + 2 i \pi \theta \big ) } [ \cdot ] \big ) \\ & + V { \big ( \Phi ^ { - 1 } \left ( z , \omega \right ) , \omega \big ) } [ \cdot ] . \end{aligned} \end{align*}


\begin{align*} \mathrm { E } ( Z ^ s ) = \int _ { 0 } ^ { 1 } \ , \left ( \sum _ { i = 0 } ^ { \infty } f _ i \ , \ , u ^ { i / \alpha } \right ) ^ s \mathrm { d } u = \sum _ { i = 0 } ^ { \infty } \ , \frac { g _ { s , i } } { i / \alpha + 1 } , \end{align*}


\begin{align*} P _ { \lambda } ( Z ) : = \prod _ { k = 1 } ^ m \left ( k ! \ , \det \left ( ( z _ { i , j } ) _ { i , j = 1 } ^ { k } \right ) \right ) ^ { \lambda _ { k } - \lambda _ { k + 1 } } . \end{align*}


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


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


\begin{align*} \delta _ \square ( W _ 1 , W _ 2 ) : = \inf _ { \varphi } d _ \square ( W _ 1 , W _ 2 ^ \varphi ) , \end{align*}


\begin{align*} z : = x , ~ k = 2 ^ h , ~ { \rm a n d } ~ f _ k ( z ) : = p _ h ( x ) , \end{align*}


\begin{align*} \mathcal { P } ^ { \pm } ( h ) ( z ) : = \lim _ { \varepsilon \downarrow 0 } \frac { 1 } { 2 \pi i } \int _ { \mathbb { R } } \frac { h ( s ) } { s - ( z \pm i \varepsilon ) } d s z \in \mathbb { R } . \end{align*}


\begin{align*} \Delta _ { 1 } ( x , \xi _ 0 ) = E _ { \xi _ 0 } \left [ \varphi \left ( x + X _ { 1 } \right ) \right ] - E _ { \xi _ { 0 } } \left [ \varphi \left ( x + Y _ { 1 } \right ) \right ] \end{align*}


\begin{align*} \phi ( m , \ell ) = \sum _ { \widetilde K \ \ \pi ^ { - 1 } ( K ) } \phi ( m , \ell _ { \widetilde K } ) \equiv \sum _ { \widetilde K \ \ \pi ^ { - 1 } ( K ) } ( e _ { \widetilde K } - 1 ) d _ { \widetilde K } \mod 2 . \end{align*}


\begin{align*} & \chi ( \xi ) + \sum _ { j \geq 0 } \varphi ( 2 ^ { - j } \xi ) = 1 , ~ ~ \forall \xi \in \mathbb { R } ^ d , \\ & | j - j ' | \geq 2 \Rightarrow ~ \varphi ( 2 ^ { - j } \xi ) \cap ~ \varphi ( 2 ^ { - j ' } \xi ) = \emptyset . \end{align*}


\begin{align*} T ^ { - 1 } ( \hat \Phi ( T ( \rho , \chi ) , s ) ) = \begin{pmatrix} \hat h ( \rho , s ) \\ 0 \end{pmatrix} . \end{align*}


\begin{align*} \Delta _ { \Psi } v = X ( \Delta _ { \Psi } u ) = X ( f ( u ) ) = f ' ( u ) X ( u ) = f ' ( u ) v . \end{align*}


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


\begin{align*} C _ { \mathrm { o b s } } = \frac { C _ 1 } { T ^ { 1 / r } } \exp \left ( \frac { C _ 2 } { T ^ { \frac { \gamma _ 1 \gamma _ 3 } { \gamma _ 2 - \gamma _ 1 } } } + C _ 3 T \right ) , \end{align*}


\begin{align*} \hat A = A [ [ T ] ] / ( T - x ) \end{align*}


\begin{align*} h ( G ) \le ( m - y ) \log ( m - y ) + y \log ( y ) + m \left ( \log ( m ) - \log ( y ) \right ) - ( m - y ) \left ( \log ( m - y ) - \log ( y ) \right ) = m \log m . \end{align*}


\begin{align*} \begin{aligned} & E _ { P } \biggl [ \varphi \biggl ( x + \sum _ { i = 1 } ^ { n } Z _ i ^ \theta \biggr ) \biggr ] \\ = & \xi _ 0 E _ P \biggl [ \varphi \biggl ( x + \sum _ { i = 1 } ^ { n } Z _ i ^ \theta \biggr ) \vert H _ 1 \biggr ] + ( 1 - \xi _ 0 ) E _ P \biggl [ \varphi \biggl ( x + \sum _ { i = 1 } ^ { n } Z _ i ^ \theta \biggr ) \vert H _ 2 \biggr ] , \end{aligned} \end{align*}


\begin{align*} \alpha ^ \xi _ t : = - R ^ { - 1 } B ( P _ t x ^ \xi _ t + \varphi _ t ^ \xi ) - h ( \mu _ t ) \end{align*}


\begin{align*} E _ { \delta , Z } = \{ u \in W ^ { 2 , p } \cap H ^ 1 _ 0 ( \Omega ) \mid \int _ { \Omega } ( K ( x ) \nabla \frac { \partial V _ { \delta , Z , j } } { \partial z _ { j , h } } ) u = 0 , \ \ \forall j = 1 , \cdots , m , \ h = 1 , 2 \} . \end{align*}


\begin{align*} \min _ { r \in [ 0 , R ] } c ( r , t ) = c ( 0 , t ) . \end{align*}


\begin{align*} v _ { t _ 0 } ( t ) : = \lim _ { j \to + \infty } v _ { t _ 0 , B _ j } ( t ) = \left \{ \begin{aligned} & - t _ 0 & & x < - t _ 0 , \\ & \ t & & x \ge t _ 0 . \end{aligned} \right . \end{align*}


\begin{align*} E _ { \alpha , \beta } ( z ) = \sum _ { k = 0 } ^ { + \infty } \frac { z ^ k } { \Gamma ( \alpha \ , k + \beta ) } , \ \alpha > 0 , \ \beta , z \in \C . \end{align*}


\begin{align*} \Psi = \tau ^ { * } ( \alpha _ { \underline { S } \times \underline { X } } \circ \theta ) \circ \alpha _ { \mathfrak { X } } \circ \theta ^ { - 1 } \in \mathfrak { Z } ( S ) \end{align*}


\begin{align*} _ { \rm r e f } [ V ] ( y , z , q ) = _ { ^ F V } \left ( q ^ { L _ 0 - \frac { c } { 2 4 } } z ^ { h } y ^ { - 2 g } \right ) . \end{align*}


\begin{align*} \forall \theta > 0 \ \Rightarrow \sup _ { p \ge 1 } \left [ \ \frac { L ( p ^ { \theta } ) } { L ( p ) } \ \right ] = C ( \theta ) < \infty . \end{align*}


\begin{align*} ( \langle z _ 1 , \ldots , z _ n \rangle \in \mathfrak { C } ) \implies ( z _ 1 = z _ n ) z _ 1 \ldots , z _ n \in X . \end{align*}


\begin{align*} B _ n \Big ( \frac { \alpha ^ j } { L _ j } \Big ) & = \sum _ { k = 0 } ^ n { \binom n k \frac { { B _ k \alpha ^ { j ( n - k ) } } } { { L _ j ^ { n - k } } } } \\ & = \alpha \sum _ { k = 0 } ^ n \binom n k \frac { B _ k F _ { j ( n - k ) } } { L _ j ^ { n - k } } + \sum _ { k = 0 } ^ n \binom n k \frac { B _ k F _ { j ( n - k ) - 1 } } { L _ j ^ { n - k } } , \end{align*}


\begin{align*} \sigma _ \mathfrak { s } ( \boldsymbol { J } ; n , p ) = \sigma _ \mathfrak { s } ( \boldsymbol { J } ; q , p ) \quad \ | n - q | \leq 1 \quad \sigma _ \mathfrak { s } ( \boldsymbol { J } ; n , p ) = \sigma _ \mathfrak { s } ( \boldsymbol { J } ; n , m ) \quad \ | p - m | \leq 1 \ , . \end{align*}


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


\begin{align*} F ( \eta ) : = T - \frac { 1 } { \eta } I , \eta \in \Omega . \end{align*}


\begin{align*} ( h _ \alpha \ , * \ , h _ \beta ) ( t ) = h _ { \alpha + \beta } ( t ) , \ t > 0 , \ \alpha > 0 , \ \beta > 0 \end{align*}


\begin{align*} \mathbb { E } [ u ] = \int _ \Omega W ( x , u ( x ) , \nabla u ( x ) ) \ , d x . \end{align*}


\begin{align*} \gamma _ { u , v } ( t ) : = ( t ^ { \alpha } u ( t \cdot ) , t ^ { \alpha } v ( t \cdot ) ) , \ \ t \geq 0 , \end{align*}


\begin{align*} d = 4 + 4 n = 8 + 8 l \equiv 0 \mod 8 \end{align*}


\begin{align*} V ( K \cup L ) = V ( K ) + V ( L ) - V ( K \cap L ) \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*} ( \sigma _ F ) _ { i , j } = \ < D F _ i , D F _ j \ > _ { \mathcal { H } } , i , j = 1 , \ldots , n . \end{align*}


\begin{align*} \lim _ { x \to \infty } \frac { h _ { } ( x ) } { x } & = \lim _ { x \to \infty } \frac { \phi ( \frac { x - \mu } { \sigma } ) \frac { 1 } { x } } { \sigma [ 1 - \Phi ( \frac { x - \mu } { \sigma } ) ] } . \end{align*}


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


\begin{align*} a _ { k } ( x ) = \frac { i } { ( 2 \pi ) ^ { n } } \int _ { \mathbb { R } ^ { n - 1 } } \int _ { \mathcal { C } } e ^ { - t \tau } \operatorname { T r } \phi _ { - 1 - k } \ , d \tau \ , d \xi , 0 \leqslant k \leqslant n - 1 . \end{align*}


\begin{align*} \mathbb { E } \left [ \mathbf { y ' } ^ H \mathbf { s } ^ { \left ( \right ) } \right ] = a _ c \sum _ { j = 1 } ^ { M } \left ( \mathbf { h } _ { l , * } \mathbf { p } _ c \right ) ^ { * } + \sum _ { l = 1 } ^ M a _ l \left ( \mathbf { h } _ { l , * } \mathbf { p } _ l \right ) ^ { * } . \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*} \left \{ \begin{array} { l l } 0 = \Delta c - \tilde { n } c , & x \in \Omega , \\ \nabla c \cdot \nu = \gamma - c , & x \in \partial \Omega , \end{array} \right . \end{align*}


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


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


\begin{align*} \bigl ( T \xRightarrow { Q _ 1 ^ 0 } Q _ 1 \circ A \xRightarrow { \beta * 1 _ A } Q _ 2 \circ A \bigr ) = \bigl ( T \xRightarrow { Q _ 2 ^ 0 } Q _ 2 \circ A \bigr ) \end{align*}


\begin{align*} \int | f | ^ 4 \lesssim \sum _ { \substack { R ^ { \frac { 1 } { 2 } } \le W \le R \\ W \ , } } \sum _ { \ell ( \tau ) = \frac { W } { R } } \sum _ { U \| U _ { \tau , R } } | U | ^ { - 1 } \Big ( \int \sum _ { \theta \subset \tau } | f _ \theta | ^ 2 W _ U \Big ) ^ 2 . \end{align*}


\begin{align*} \widehat { \mathrm { S } ( V ) } _ I : = \varprojlim \mathrm { S } ( V ) / I ^ n \end{align*}


\begin{align*} \| x \| = \sup _ { p \in \Gamma _ { \tau } } p ( x ) , x \in X ; \end{align*}


\begin{align*} & \overline { \mu } ( \mathbb { J } ) = \sum \limits _ { ( \mathbb { J } ^ { ' } , t ^ { ' } ) \in \pi ^ { ' } } \overline { \mu } ( \mathbb { J } \cap \mathbb { J } ^ { ' } ) . \end{align*}


\begin{align*} c = \prod _ { i = 1 } ^ { n - 1 } \norm { \eta _ i } _ { H ^ 1 } ^ 2 \left ( 1 + ( n + 3 ) \left ( \frac { n + 6 } { 4 } + \frac 1 4 \norm { \tilde \Psi } _ { H ^ 1 } \right ) \right ) \end{align*}


\begin{align*} \mathbf { v } \cdot \overrightarrow { \zeta } = 0 , \end{align*}


\begin{align*} h _ 1 ( x ) = \prod _ { i = 1 } ^ n ( \eta _ i ( x + r ) - \eta _ i ( x ) ) \sin ^ { [ n ] } ( \tilde \Psi ( x + r ) - \tilde \Psi ( x ) ) \\ h _ 2 ( x ) = - \prod _ { i = 1 } ^ n ( \eta _ i ( x - r ) - \eta _ i ( x ) ) \sin ^ { [ n ] } ( \tilde \Psi ( x - r ) - \tilde \Psi ( x ) ) \end{align*}


\begin{align*} \mbox { \rm g r a p h } \left . \tau \right | _ { { \cal A } _ \delta } = \{ ( x _ a , y _ a ) \in { \cal A } _ \delta \times \mathbb { R } \ ; : \ ; y _ a = \tau ( x _ a ) \} \end{align*}


\begin{align*} [ T ] = \frac { 1 } { \mu } J ( \partial _ x h + I \partial _ y h ) = \frac { 2 } { \mu } h _ z J \end{align*}


\begin{align*} r _ { \ell - c } = s _ i \end{align*}


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


\begin{align*} 0 = \phi ^ \theta ( e ) = \tau ( u _ { l - 1 } , e ) \theta ( u _ { l - 1 } ) + \phi ( e ) + \tau ( u _ l , e ) \theta ( u _ l ) , \end{align*}


\begin{align*} \mathcal { J } ^ i ( \alpha ^ i , \boldsymbol { \alpha } ^ { - i } ) : = \frac { 1 } { 2 } \mathbb { E } \Big \{ \int _ 0 ^ T \Big [ Q \left ( x ^ i _ t + l ( \nu ^ { N , i } _ { \boldsymbol { x } _ t } ) \right ) ^ 2 + R \left ( \alpha ^ i _ t + h ( \mu ^ { N , i } _ { \boldsymbol { \alpha } _ t } ) \right ) ^ 2 \Big ] d t + G \left ( x ^ i _ T + g ( \nu ^ { N , i } _ { \boldsymbol { x } _ T } ) \right ) ^ 2 \Big \} , \end{align*}


\begin{align*} \mu _ N ( E ) = \dfrac { | A \cap \Phi _ N | } { | \Phi _ N | } \end{align*}


\begin{align*} \forall x \in E _ v \ , , \ \Vert x \Vert _ v = \begin{cases} \sqrt { h _ v ( x , x ) } v \in V _ { \infty } \\ \inf \left \lbrace \vert \alpha \vert _ v \ , , \ \alpha \in K _ v \ , , \ x \in \alpha L \right \rbrace v \in V _ f . \end{cases} \end{align*}


\begin{align*} s _ 1 ( z ) : = \mathcal { C } ( s J ) ( z ) s _ 2 ( z ) : = \mathcal { C } ( s J ) ^ { H } ( z ) , \end{align*}


\begin{align*} Q ( \mathcal { O } ^ { ( p ) } ) = d ( \mathcal { O } ^ { ( p - 1 ) } ) , \end{align*}


\begin{align*} W ( a u + b v , x ) = a W ( u , x ) + b W ( v , x ) . \end{align*}


\begin{align*} \frac { \partial } { \partial t } \Psi ( t , x ) & = - \int _ \S W _ r ( x - y ) \sin ( \Psi ( t , y ) - \Psi ( t , x ) ) \ \d y + \int _ \S W _ r ( y ) \sin ( \Psi ( y ) ) \ \d y . \end{align*}


\begin{align*} G \cdot R - \bar G \cdot \bar R - \bar R \cdot \nabla \bar G ( \nabla & \bar A ) ^ { - 1 } ( A - \bar A ) - \bar G \cdot ( P - \bar P ) = ( G - \bar G ) \cdot ( R - \bar R ) \\ & + \bar R \cdot G ( U | \bar U ; x , t ) + \bar G \cdot \left [ A _ t - \bar A _ t + f _ { \alpha , x _ \alpha - } \bar f _ { \alpha , x _ \alpha } \right ] \ , . \end{align*}


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


\begin{align*} { g _ i } \left ( { { { \bf { x } } _ b } , t } \right ) = g _ i ^ { \left ( { e q } \right ) } \left ( { { { \bf { x } } _ b } , t } \right ) + g _ i ^ { \left ( { n e q } \right ) } \left ( { { { \bf { x } } _ b } , t } \right ) , \end{align*}


\begin{align*} \omega _ { i j } V _ i V _ j = \tfrac { 1 } { 2 } h _ { i j } V _ i V _ j . \end{align*}


\begin{align*} M P A P P ( m , n ) = & \biggl \{ A \in P A P P ( m , n ) : A \mbox { i s t h e m i n i m a l e l e m e n t i n t h e s e t } \\ & \{ A , \sigma ( A ) \} \mbox { u n d e r t h e l e x i c o g r a p h i c o r d e r } \biggr \} \end{align*}


\begin{align*} \frac { 1 - e ^ { a _ { \alpha + n + 1 } + a _ { \alpha + n + 2 } } } { \left ( e ^ { - a _ { \alpha + 1 } } - 1 \right ) \left ( 1 - e ^ { a _ { n + 1 } } \right ) \exp \left [ \sum _ { j = 2 } ^ n ( a _ j - a _ { \alpha + j } ) \right ] } \end{align*}


\begin{align*} J \nabla _ g f = X _ v \end{align*}


\begin{align*} J ( x ) + \sum \limits _ { k = 1 } ^ m \gamma _ k f _ k ( x ) \ge J ( x ^ * ) + \sum \limits _ { k = 1 } ^ m \gamma _ k f _ k ( x ^ * ) = J ( x ^ * ) \ \ \forall \ , x \in \mathbb { R } ^ n . \end{align*}


\begin{align*} d ( \bar x , y ) & \le d ( \bar x , x ) + d ( x , y ) \le \bar K r _ { m ' } + ( 6 \lambda r _ i + 3 \lambda r _ m ) \le \left ( 6 \lambda + \frac 1 { \bar K } + \frac { 3 \lambda } { \bar K ^ 2 } \right ) r _ i \le 7 \lambda r _ i . \end{align*}


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


\begin{align*} \beta _ { \infty } ( n ) = \frac { 4 \pi ^ { 2 } n } { \sqrt { \det ( A ) } } . \end{align*}


\begin{align*} n = \frac { - 3 \left ( a ^ { 2 } + b ^ { 2 } - 1 \right ) \pm \sqrt { 3 \left ( \left ( a - b \right ) ^ { 4 } - \left ( a ^ { 4 } + b ^ { 4 } + \left ( a - 1 \right ) ^ { 4 } + \left ( b - 1 \right ) ^ { 4 } \right ) + 1 \right ) } } { 6 \left ( a + b - 1 \right ) } \end{align*}


\begin{align*} \psi _ { \tau } ( p ) = | | \tau | | _ p \le C _ 1 ( \beta , \gamma , L ) \ ( \beta - p ) ^ { - ( \gamma + 1 ) / \beta } \ L ^ { 1 / \beta } ( 1 / ( \beta - p ) ) , \end{align*}


\begin{align*} u ( x ) \geq \left ( \frac { 1 } { 2 } - C \delta \right ) \sum _ { \beta = \alpha + 1 } ^ { n - 1 } b _ \beta x _ \beta ^ 2 - C b _ \alpha | \hat { x } | ^ 2 \geq \frac { 1 } { 4 } \sum _ { \beta = \alpha + 1 } ^ { n - 1 } b _ \beta x _ \beta ^ 2 - C b _ \alpha | \hat { x } | ^ 2 \end{align*}


\begin{align*} f ( A ) = \begin{cases} 1 , & s < - \frac d 2 , \\ ( \log A ) ^ \frac 1 2 , & s = - \frac d 2 , \\ A ^ { \frac { d } { 2 } + s } , & s > - \frac d 2 . \end{cases} \end{align*}


\begin{align*} \C = \left \{ ( \lambda + \mu \theta ^ { i _ 1 } , \lambda + \mu \theta ^ { i _ 2 } , \cdots , \lambda + \mu \theta ^ { i _ n } ) \ , | \ , \lambda , \mu \in \mathbb { F } _ { p ^ e } \right \} \end{align*}


\begin{align*} G _ x ( \theta ) : = \sum _ { n \in \Z } g \left ( \pi _ N ( x ) ( \theta + n ) \right ) , \end{align*}


\begin{align*} \int _ { 1 } ^ \infty \frac { d s } { \Big ( \displaystyle \int _ 0 ^ s F ( t ) d t \Big ) ^ { p / ( 2 p - q + 1 ) } } = \infty . \end{align*}


\begin{align*} g ^ { q p } u _ { , p q i } = - v _ { i } + u _ { , i q } \xi ^ q \end{align*}


\begin{align*} \int _ 0 ^ \infty \frac { \exp [ - ( a + n ) y ] } { y } \mathrm { d } y = \operatorname { E } _ 1 ( 0 ) , \end{align*}


\begin{align*} f = \sum _ { \mathcal { J } } \sum _ { \beta } f _ { \mathcal { J } , \beta } ( x ^ { \mu } ) \prod _ { j \in \mathcal { J } } \theta _ j ^ { \beta ^ j } , \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*} P = P _ w = _ \gamma + \gamma I - B _ w : T L \rightarrow T L , \end{align*}


\begin{align*} \varphi _ { | N } = \frac { 1 } { n } ( \psi _ 1 + \cdots + \psi _ n ) \end{align*}


\begin{align*} 1 - e ^ { c _ 2 } = \frac { 1 - e ^ { a _ { \alpha + n + 1 } + a _ { \alpha + n + 2 } } } { \left ( e ^ { - a _ { \alpha + 1 } } - 1 \right ) \left ( 1 - e ^ { a _ { n + 1 } } \right ) \exp \left [ \sum _ { j = 2 } ^ n ( a _ j - a _ { \alpha + j } ) \right ] } \ , \ \forall n \in \N \ . \end{align*}


\begin{align*} \left | \sum _ { k = 1 } ^ m \mu ( k ) \left ( \left \lfloor \frac { n } { k } \right \rfloor - \left \lfloor \frac { n ' } { k } \right \rfloor \right ) \right | & = \left | \sum _ { t = 1 } ^ q \left ( M ( n / t ) - M ( n ' / t ) \right ) \right | \\ \leq \sum _ { t = 1 } ^ q \left ( \left \lfloor \frac { | d - d ' | } { t } \right \rfloor + 1 \right ) & \leq q + | d - d ' | \sum _ { t = 1 } ^ r \frac { 1 } { t } . \end{align*}


\begin{align*} \begin{aligned} u ( \bar { x } ) \leq \ , & u ( x ^ * ) + C \epsilon _ 0 \delta ^ 2 b _ { \alpha } ^ 2 + C b _ { \alpha } \bar { x } _ n + C \delta ^ 3 b _ \alpha ^ 2 \\ \leq \ , & u ( x ^ * ) + C \left ( \epsilon _ 0 + \delta \right ) \delta ^ 2 b _ \alpha ^ 2 \leq \left ( 1 + \frac { c _ 0 } { 3 } \right ) u ( x ^ * ) \end{aligned} \end{align*}


\begin{align*} \dot { s } \varepsilon ( z ) \dot { s } \varepsilon ( 1 ) \xi _ 2 = \varepsilon ( - z ^ { - 1 } ) \dot { s } h ( - z ) \varepsilon ( - z ^ { - 1 } ) \varepsilon ( 1 ) \xi _ 2 = \varepsilon ( - z ^ { - 1 } ) \dot { s } \varepsilon ( z ^ 2 - z ) \xi _ 2 . \end{align*}


\begin{align*} \dfrac { p } { p + 1 } \displaystyle \sum _ { n = 1 } ^ \infty \dfrac { 2 ^ { \omega ( \gcd ( n , p ) ) } \cdot 2 r _ { Q ^ \ast } ( n ) ^ 2 } { n } \displaystyle \sum _ { d = 1 } ^ \infty \psi \left ( d \sqrt { \dfrac { n } { p } } \right ) \leq \frac { 1 } { \min Q ^ { * } } + 3 2 1 6 . 6 5 2 4 \frac { M ( 2 5 . 0 9 p ^ { 3 5 / 6 } ) } { p ^ { 1 / 4 } } . \end{align*}


\begin{align*} g ( z ) = \sum _ { m = 0 } ^ { \infty } a _ m z ^ { q ( m + 1 ) } = \sum _ { n = 1 } ^ { \infty } a _ { n - 1 } z ^ { q n } = \sum _ { n = 0 } ^ { \infty } b _ n z ^ { q n } , b _ n = \begin{cases} a _ { n - 1 } & , n \geq 1 , \\ 0 & , . \end{cases} \end{align*}


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


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


\begin{align*} F ( x ' , t ) : = \Phi _ t * f ( x ' ) \quad H ( x ' , t ) : = \widetilde { \Phi } _ t * h ( x ' ) , ( x ' , t ) \in \R ^ n _ + . \end{align*}


\begin{align*} \mathop { \inf } \limits _ { \| ( u , v ) \| _ { E } = r } I ( u , v ) > 0 . \end{align*}


\begin{align*} I = ( I \cap X ) \vee ( I \cap Y ) = ( I \cap X ) \times ( I \cap Y ) . \end{align*}


\begin{align*} ( p _ n f , x _ n ) = ( f , x _ n ) x _ n \in X _ n . \end{align*}


\begin{align*} w _ n = \sum _ { i = 0 } ^ { r + \tilde { r } - 1 } c _ i w _ { n + i } \end{align*}


\begin{align*} \begin{aligned} \int _ { \Omega } & \delta n \xi ( \cdot , t ) - \int _ { 0 } ^ { t } \int _ { \Omega } \delta n \xi _ { t } + \int _ { 0 } ^ { t } \int _ { \Omega } \nabla \delta n \cdot \nabla \xi \\ & = \int _ { 0 } ^ { t } \int _ { \Omega } ( \delta n S ( x , \tilde { n } _ { 1 } , c _ { 1 } ) \cdot \nabla c _ { 1 } + n _ { 2 } Z \cdot \nabla c _ { 1 } + n _ { 2 } S ( x , \tilde { n } _ { 2 } , c _ { 2 } ) \cdot \nabla \delta c ) \cdot \nabla \xi , \end{aligned} \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*} F _ 1 = \frac { 1 } { \sqrt { c _ 1 } } \langle \langle \nu _ 1 \cdot \varphi , \varphi \rangle \rangle . \end{align*}


\begin{align*} \begin{cases} \frac { d } { d t } X _ t = B ( X _ t , X _ t ) \\ X _ 0 = x , \end{cases} \end{align*}


\begin{align*} \abs { c _ 1 ( S ^ 1 \setminus S _ i ) } \le \abs { c ( S \setminus S _ i ) } + \sum _ { h \in [ \chi ] \setminus \{ i \} } \abs { c _ 1 ( P _ i ' ) } \le \abs { S \setminus S _ i } + \sum _ { h \in [ \chi ] \setminus \{ i \} } q _ h = \abs { S } - \abs { S \cap S _ i } + q - q _ i . \end{align*}


\begin{align*} Y = \left ( \frac { \eta } { m + 1 } x _ { n + 1 } ^ { m + 1 } + \lambda x _ { n + 1 } \right ) e _ { n + 1 } . \end{align*}


\begin{align*} \langle e _ { i } , A e _ { j } \rangle = \langle A e _ { i } , A e _ { j } \rangle = \langle e _ { i } , e _ { j } \rangle = 0 \end{align*}


\begin{align*} F _ n - n ^ p \ \ = \ \ \mathcal { C } _ n ^ { ( p ) } \ - \ 2 \sum _ { j = 0 } ^ { p / 2 } \binom { p } { 2 j + 1 } \mathcal { C } _ n ^ { ( p - 2 j - 1 ) } \ + 2 \binom { p } { 2 \lfloor p / 2 \rfloor + 1 } F _ n \mbox { f o r $ p $ o d d } . \end{align*}


\begin{align*} \sum _ i F ^ { i i } h _ { i i 1 1 } = - \sum _ { p , q , r , s } F ^ { p q , r s } h _ { p q 1 } h _ { r s 1 } = - \sum _ { p \neq q } F ^ { p p , q q } h _ { p p 1 } h _ { q q 1 } + \sum _ { p \neq q } F ^ { p p , q q } h _ { p q 1 } ^ 2 . \end{align*}


\begin{align*} h ( \lambda ) = \frac { ( \lambda , \lambda + 2 \rho ) } { 2 ( \kappa + h ^ \vee ) } + \frac { ( \lambda , \lambda + 2 \rho ) } { 2 ( \kappa ^ * + h ^ \vee ) } - ( \lambda , \rho ^ \vee ) = \frac { ( \lambda , \lambda ) r ^ \vee n } { 2 } + ( \lambda , n r ^ \vee \rho - \rho ^ \vee ) . \end{align*}


\begin{align*} \begin{cases} \Delta _ \Psi ( - v ) \geq f ' ( u ) ( - v ) & A ( r _ { 1 } , r _ { 2 } ) \\ ( - v ) = 0 & \partial A ( r _ { 1 } , r _ { 2 } ) . \end{cases} \end{align*}


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


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


\begin{align*} \Phi \left ( \bigvee _ { l \geq j + 1 } p _ l \right ) ^ { \perp } \check { y } h _ n & = \Phi \left ( \bigvee _ { l \geq j + 1 } p _ l \right ) ^ { \perp } \check { y } ( h _ { n , 1 } + \cdots + h _ { n , j } ) \\ & = \Phi \left ( \bigvee _ { l \geq j + 1 } p _ l \right ) ^ { \perp } \check { y } ( p _ 1 + \cdots + p _ j ) h _ n . \end{align*}


\begin{align*} \kappa ( A ) : = \lVert A \rVert \ , \lVert A ^ { - 1 } \rVert = \frac { s _ { \max } ( A ) } { s _ { \min } ( A ) } . \end{align*}


\begin{align*} \left \{ \begin{array} { l l } \varphi '' + \Big ( ( m - 1 ) \frac { \sigma ' } { \sigma } - \Phi ' \Big ) \varphi ' + B = 0 & \textnormal { i n } \ I \\ \varphi ( r _ 1 ) = 1 \\ \varphi ' ( r _ 1 ) = b < 0 \end{array} \right . \end{align*}


\begin{align*} K f = \int _ { - \infty } ^ x u _ y ( y ) e ^ { 2 i z y } d y \int _ { - \infty } ^ { y } \bar { u } _ s ( s ) e ^ { - 2 i z s } f d s , \end{align*}


\begin{align*} \Delta _ { \ 1 } ^ + = \{ e _ i \pm f _ j | 1 \leq i \leq m , 1 \leq j \leq n \} \cup \{ e _ i | 1 \leq i \leq m \} . \end{align*}


\begin{align*} \lambda = a + b x \in \sigma _ l ( A ) , \ , v = ( 1 , x ) ^ T \in V _ l ( \lambda ) , \forall x \in S _ Q ( b , a - d , - c ) . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ m U _ i \geq \sum _ { i = 1 } ^ m V _ i . \end{align*}


\begin{align*} \sum _ { k \leq j < m } \delta _ { 0 , j } + \sum _ { 0 < i < k } \delta _ { i , k } - \sum _ { k < j \leq m } \delta _ { k , j } - \sum _ { 0 < i < k } \delta _ { i , m } = 1 + \sum _ { 0 \leq i < m } ( \delta _ { 0 , i } + \delta _ { i , k } - 1 ) , \end{align*}


\begin{align*} \tilde \Sigma = \{ x \in M \mid { \rm V o l } ( A \cap B _ x ( r ) ) = { \rm V o l } ( B \cap B _ x ( r ) ) \} , \end{align*}


\begin{align*} \Delta _ { \Psi } X ( u ) = X ( \Delta _ { \Psi } u ) . \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*} \begin{cases} \mathcal { C } _ { \rm s } = \Upsilon _ { \rm P D E } \zeta \left ( 1 + \varphi _ { \rm A P } + \varphi _ { \rm C T } \right ) / E _ { \rm p h } , \\ [ 2 p t ] \mathcal { C } _ { \rm n } = ( \vartheta _ { \rm D C R } + \vartheta _ { \rm B } ) \left ( 1 + \varphi _ { \rm A P } + \varphi _ { \rm C T } \right ) , \end{cases} \end{align*}


\begin{align*} & \mathbb { E } \widetilde { g } _ { \delta , L , \mu } ( x ) = g _ { \delta , L , \mu } ( x ) , \\ & \mathbb { E } \| \widetilde { g } _ { \delta , L , \mu } ( x ) - g _ { \delta , L , \mu } ( x ) \| ^ 2 \leq \frac { \sum _ { i = 1 } ^ n \sigma _ { f , i } ^ 2 } { n ^ 2 r } = \frac { \sigma _ f ^ 2 } { n r } . \end{align*}


\begin{align*} d u ( t ) = \big [ - \Delta ^ 2 u + \Delta \varphi ( u ) \big ] d t + B ( t , u ) d W ( t ) , \end{align*}


\begin{align*} \frac { d } { d x } f ( x ) = & \frac { \theta ^ { 2 } } { B ( a , b ) \ , x ^ 6 } \exp \left ( - \frac { a \theta } { x ^ 2 } \right ) \left [ 1 - \exp \left ( - \frac { \theta } { x ^ 2 } \right ) \right ] ^ { b - 1 } \\ & \times \left [ 4 a - \frac { 6 x ^ 2 } { \theta } + \frac { 4 ( b - 1 ) } { 1 - \exp \left ( \frac { \theta } { x ^ 2 } \right ) } \right ] , x > 0 . \end{align*}


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


\begin{align*} { \mathcal { I } } _ h \leq \int _ { \Omega } | B | ^ { 2 } \dd x \leq \liminf _ { n \to \infty } \int _ { \Omega } | B _ n | ^ { 2 } \dd x = { \mathcal { I } } _ h , \end{align*}


\begin{align*} c _ 1 ( q ) = \begin{cases} c _ { 1 , 2 } ( q ) , & 0 < q \leq q _ 1 ^ * , \\ c _ { 1 , \infty } ( q ) , & q _ 1 ^ * \leq q < 2 , \end{cases} \end{align*}


\begin{align*} \sum _ { i = 1 } ^ 3 \arctan \lambda _ i = \hat \theta _ 0 \in \left ( \pi , \frac { 3 \pi } { 2 } \right ) . \end{align*}


\begin{align*} Z _ { i } ^ { \theta } : = \theta _ { i } X _ { i } + ( 1 - \theta _ { i } ) Y _ { i } , ~ 1 \leq i \leq n . \end{align*}


\begin{align*} \prod _ { p \leq X } \left ( 1 - \frac { 1 } { p } \right ) \left ( 1 + \frac { \Delta _ K ( p ) \iota _ { K , f } ( p ) } { p } \right ) \asymp \exp \left ( \sum _ { p \leq X } \frac { \Delta _ K ( p ) \iota _ { K , f } ( p ) - 1 } { p } \right ) & = \exp \left ( \sum _ { p \leq X } \frac { \Delta _ K ( p ) - 1 } { p } + O _ { K , f } ( 1 ) \right ) \\ & \asymp _ { K , f } \prod _ { p \leq X } \left ( 1 + \frac { \Delta _ K ( p ) - 1 } { p } \right ) , \end{align*}


\begin{align*} \xi ( t ) = \sigma _ { \xi ( u ) \xi ( v ) } ( ( t - u ) / ( v - u ) ) \in \Sigma _ { \Gamma ( u ) \Gamma ( v ) } ( ( t - u ) / ( v - u ) ) . \end{align*}


\begin{align*} W ( \xi _ 0 , n , x , \mathsf { L } ^ n ) = W ( 1 - \xi _ 0 , n , x , \mathsf { R } ^ n ) , ~ \forall \xi _ 0 \in [ 0 , 1 ] , \forall x \in \mathbb { R } . \end{align*}


\begin{align*} I = ( I \cap X ) \vee ( I \cap Y ) = ( I \cap X ) \times ( I \cap Y ) . \end{align*}


\begin{align*} \sum _ { \gamma _ { \chi } } \frac { \frac { 1 } { 2 } } { \frac { 1 } { 4 } + \gamma _ { \chi } ^ 2 } & \leq \left ( \dfrac { 1 } { 2 } + \dfrac { 2 } { \log q - 2 } \right ) \log { \dfrac { q } { \pi } } + 2 \left ( 1 - \dfrac { 2 } { \log q } \right ) ^ { - 2 } \log \log q \leq \dfrac { 1 } { 2 } \log { \dfrac { q } { \pi } } + 2 . 6 \log \log q . \end{align*}


\begin{align*} \phi _ { t } ^ { \ast } g = g . \end{align*}


\begin{align*} \sum _ { \tau \in D ( n ) } \prod _ { j = 1 } ^ { n } \dfrac { 1 } { 1 - \zeta ^ { j - \tau ( j ) } } = \dfrac { \left ( \left ( n - 1 \right ) ! ! \right ) ^ { 2 } } { 2 ^ n } = \dfrac { n ! } { 4 ^ n } \binom { n } { n / 2 } , \end{align*}


\begin{align*} G _ { 0 \alpha } & = G _ \alpha ( \theta _ 2 , \theta _ 3 , \ldots , \theta _ n ) \hbox { a n d } \\ G _ { 1 \alpha } & = \theta _ 1 G _ { 0 \alpha } + P _ { \alpha } ( \theta _ 2 , \theta _ 3 , \ldots , \theta _ n ) \end{align*}


\begin{align*} \frac { u ^ k - u ^ { k - 1 } } { \tau _ k } + \mathcal { L } u ^ k = F ^ k , \end{align*}


\begin{align*} \prod _ { i = 1 } ^ j a _ f ( p _ i ^ { 2 m _ i } ) \prod _ { i = j + 1 } ^ r a _ f ( p _ i ^ { l _ i } ) \prod _ { i = 1 } ^ j a _ f ( q _ i ^ { m _ i ' } ) \prod _ { i = j + 1 } ^ r a _ f ( q _ i ^ { l ' _ i } ) = \prod _ { u = 1 } ^ t \left ( \prod _ { i \in \mathcal I ( s _ u ) } a _ f ( s _ u ^ { 2 i } ) \right ) , \end{align*}


\begin{align*} \frac { 1 } { m } \sum _ { i = 1 } ^ m V _ i > \left ( \frac { \pi } { 4 } + \frac { 1 9 9 \pi } { 3 2 0 0 k } \right ) \| \mathbf { x } \| . \end{align*}


\begin{gather*} P ( c ) = \inf _ { Q \in \Gamma } Q ( c ) ; \end{gather*}


\begin{align*} D ( a b ) = D ( a ) b + ( - 1 ) ^ { p ( D ) p ( a ) } a D ( b ) . \end{align*}


\begin{align*} \pi ( [ X ] ) \wedge \pi ( [ Y ] ) = [ f _ \alpha '' X \cap ( ( \lambda \setminus A _ \alpha ) \cup f _ \alpha '' ( Y \cap A _ \alpha ) ] = [ f _ \alpha '' X \cap f '' _ \alpha ( Y \cap A _ \alpha ) ] = [ f _ \alpha '' ( X \cap Y ) ) ] . \end{align*}


\begin{align*} \sigma _ { k } ( \lambda ) = \sum _ { i _ { 1 } < \ldots < i _ { k } } \lambda _ { i _ { 1 } } \ldots \lambda _ { i _ { k } } , \ \ k = 1 , \ldots , n , \end{align*}


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


\begin{align*} g _ i ^ { ( e q ) } = { { w } _ i } T , \end{align*}


\begin{align*} \nabla _ V ( \nabla X ) = - R ( X , V ) + T _ h ( V ) \end{align*}


\begin{align*} \mu ^ { \pm } ( x , t ; k ) = I + \int _ { \pm \infty } ^ { x } \mathrm { e } ^ { - \mathrm { i } k ^ 2 \left ( x - y \right ) \widehat { \sigma } _ 3 } \left ( P _ 1 \mu ^ { \pm } \right ) \left ( y , t ; k \right ) d y , \end{align*}


\begin{align*} [ F , G ] ( x ) = \sum _ { { \bf k , n } \in \Gamma _ { r } } [ \widehat { F } _ { \bf k } , \widehat { G } _ { \bf n } ] { \rm e } ^ { { \rm i } \langle { \bf k + n } , x \rangle } . \end{align*}


\begin{align*} \tau ( x ) \omega : = \varphi ( x ) \ , \omega , \end{align*}


\begin{align*} \phi ^ { N , i } ( t , \boldsymbol { x } ) = \Phi ( t , \nu ^ { N , i } _ { \boldsymbol { x } } ) ~ ~ ~ ~ \nu ^ { N , i } _ { \boldsymbol { x } } = \frac { 1 } { N - 1 } \underset { j \neq i } { \sum } x ^ j , \end{align*}


\begin{align*} S \textbf { \textit { u } } : = \nabla \textbf { \textit { u } } + \nabla \textbf { \textit { u } } ^ t , \textbf { \textit { u } } \in ( C ^ 1 ( \Omega ) ) ^ 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*} J ( t , j ) = \int _ { - \infty } ^ { \infty } \ , x ^ { j } \ , \exp \left ( - t x - \frac { x ^ 2 } { 2 } \right ) \mathrm { d } x = ( - 1 ) ^ j \ , \sqrt { 2 \pi } \ , \ , \ , \frac { \partial ^ j } { \partial t ^ j } \left \{ \exp \left ( \frac { t ^ 2 } { 2 } \right ) \right \} \end{align*}


\begin{align*} u ( x ) \geq \left ( \frac { 1 } { 2 } - C \delta \right ) \sum _ { \beta = \alpha + 1 } ^ { n - 1 } b _ \beta x _ \beta ^ 2 - C b _ \alpha | \hat { x } | ^ 2 \geq \frac { 1 } { 4 } \sum _ { \beta = \alpha + 1 } ^ { n - 1 } b _ \beta x _ \beta ^ 2 - C b _ \alpha | \hat { x } | ^ 2 \end{align*}


\begin{align*} ( a - d ) x = c . \end{align*}


\begin{align*} X _ n = ( \mathfrak X , \mathcal O _ { \mathfrak X } \otimes _ R R _ n ) , \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*} \begin{aligned} [ D _ * ^ q , M _ { z ^ q } ] & = D ^ q _ * M _ { z ^ q } ( f ) ( z ) - M _ { z ^ q } D _ * ^ q ( f ) ( z ) \\ & = D ^ q _ * ( z ^ q f ) ( z ) - z ^ q D _ * ^ q ( f ) ( z ) . \end{aligned} \end{align*}


\begin{align*} \mathfrak { R } \int _ { \{ \Psi < - t \} } \langle F _ t , \bar { q } \rangle _ h c ( - \Psi ) = 0 , \end{align*}


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


\begin{align*} D ( A ^ { \circ } ) & = \{ x ^ { \circ } \in X ^ { \circ } \ | \ \exists y ^ { \circ } \in X ^ { \circ } \ \forall x \in D ( A ) : \langle - A x , x ^ { \circ } \rangle = \langle x , y ^ { \circ } \rangle \} , \\ - A ^ { \circ } x ^ { \circ } & = y ^ { \circ } \quad ( x ^ { \circ } \in D ( A ^ { \circ } ) ) , \end{align*}


\begin{align*} q ' _ j & = \sum _ { i \in A _ j } q _ i = \sum _ { i \in [ k ] \setminus [ m ] : \tilde { \delta } _ { i - m } \in ( \nu _ { j + 1 } , \nu _ { j } ] } q _ i = \sum _ { i \in [ k ] \setminus [ m ] : \tilde { \delta } _ { i - m } \in ( \nu _ { j + 1 } , \nu _ { j } ] } 0 . 5 \tilde { q } _ { i - m } = 0 . 5 \P \{ \tilde { X } \in ( \nu _ { j + 1 } , \nu _ { j } ] \} . \end{align*}


\begin{align*} w _ \varepsilon ( x _ 1 , x _ 2 , 0 , t ) = \frac { 1 } { \varepsilon ^ 2 } \left ( u _ \varepsilon ( \bar { R } _ { - \alpha | \ln \varepsilon | t } ( x _ 1 , x _ 2 ) ) - \left ( \frac { \alpha | ( x _ 1 , x _ 2 ) | ^ 2 } { 2 } + \beta \right ) \ln \frac { 1 } { \varepsilon } \right ) ^ p _ + . \end{align*}


\begin{align*} B _ n \Big ( \frac { \beta ^ j } { L _ j } \Big ) = ( - 1 ) ^ n B _ n \Big ( \frac { \alpha ^ j } { L _ j } \Big ) . \end{align*}


\begin{align*} \dot { s } \varepsilon ( z ) \dot { s } \varepsilon ( 1 ) \xi _ 2 = \dot { s } ( \varepsilon ( z - 1 ) - \varepsilon ( z ) ) \xi _ 2 + \dot { s } \varepsilon ( z ) \varpi . \end{align*}


\begin{align*} \widetilde { E } ( x , t ) = \log \lambda _ { m a x } ( \{ w ^ { i j } \} ) - d \log h + \frac { l } { 2 } \rho ^ { 2 } , \end{align*}


\begin{align*} _ i H _ j \coloneqq \langle \alpha _ { i j } ^ { ( k ) } \mid k = 1 , \dots , g _ { i j } \rangle \subset H . \end{align*}


\begin{align*} \abs { a ^ \textrm { a p p } ( s ( \tau ) ) - a ( \tau ) } = \abs { \sqrt { \frac { - \gamma _ 2 s ( \tau ) } { \gamma _ 1 } } - \tau } = \mathcal O ( \tau ^ 3 ) = \mathcal O ( s ^ \frac 3 2 ) . \end{align*}


\begin{align*} ( x _ 1 , x _ 2 , \cdots , x _ k ) \bar { G } = { \bf 0 } , \end{align*}


\begin{align*} V _ { \mathrm { o r i g } } = \sup _ { Q \in \P ( \R ^ n ) } \left ( \int _ { \R ^ n } g \ , d Q - \frac { 1 } { n } H ( Q \ , | \ , \gamma _ T ) \right ) = \frac { 1 } { n } \log \int _ { \R ^ n } e ^ { n g } \ , d \gamma _ T , \end{align*}


\begin{align*} M _ I : = M _ { \varrho ( I ) } \textrm { a n d } R _ I : = R _ { \varrho ( I ) } . \end{align*}


\begin{align*} d _ x T ( x , y ) = - \frac { d _ x \Phi ( x , y , T ( x , y ) ) } { \partial _ z \Phi ( x , y , T ( x , y ) ) } . \end{align*}


\begin{align*} A _ { \mu } f ( x ) = a ( x ) L _ { \mu } ^ { \rm d i f } f ( x ) , \end{align*}


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


\begin{align*} \phi ^ q _ i = 2 \pi q i / M + \alpha , i = 1 , \dots , M , \end{align*}


\begin{align*} F = \bigcup _ { n = 0 } ^ \infty K _ n , \end{align*}


\begin{align*} \sum _ { n = 0 } ^ \infty \frac { a _ { i , n } } { \prod _ { k = 0 } ^ n \beta _ { i + k } } = \frac { 1 } { q _ i } , \end{align*}


\begin{align*} \lim _ { p \to \infty } \frac { \psi ( p ) } { p } = 0 . \end{align*}


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


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


\begin{align*} & \lim _ { m ' \to + \infty } \int _ { X _ j } v '' _ { \epsilon } ( \Psi ) | f F | ^ 2 _ { h _ { m ' } } e ^ { - u ( - v _ { \epsilon } ( \Psi ) ) } \\ = & \int _ { X _ j } v '' _ { \epsilon } ( \Psi ) | f F | ^ 2 _ h e ^ { - u ( - v _ { \epsilon } ( \Psi ) ) } < + \infty . \end{align*}


\begin{align*} J _ 1 = \int _ { \Lambda _ { h } } u _ { 0 } \partial _ { \nu } ( w - v ) \mathrm d \sigma , J _ 2 = - \int _ { \Lambda _ { h } } ( w - v ) \partial _ { \nu } u _ { 0 } \mathrm d \sigma . \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*} \sum _ s \phi _ { i s } ( \partial _ r ) \varphi _ s & = \phi _ { i i } ( \partial _ r ) \varphi _ i + \phi _ { i r } ( \partial _ r ) \varphi _ r = \frac { 1 } { 2 } \partial _ r ( \varphi ^ { - 1 } _ i ) \varphi _ i + \Gamma _ { r i } ^ i = 0 , \end{align*}


\begin{align*} \partial _ t u ( t , x ) = \nabla \cdot a \big ( t , x , u ( t , x ) , \nabla u ( t , x ) \big ) - a _ 0 \big ( t , x , , u ( t , x ) , \nabla u ( t , x ) \big ) , \end{align*}


\begin{align*} I _ n : = \left < F _ { 1 ^ r } ( \theta _ 1 , \theta _ 2 , \ldots , \theta _ n ) : 1 \leq r \leq n \right > . \end{align*}


\begin{align*} D \sigma _ T ( x ) = \frac { 1 } { \pi } \Big ( \log \Big ( \frac { \sin ( \pi x _ 1 ) } { \sin ( \pi ( x _ 1 + x _ 2 ) ) } , \log \Big ( \frac { \sin ( \pi x _ 2 ) } { \sin ( \pi ( x _ 1 + x _ 2 ) ) } \Big ) \Big ) . \end{align*}


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


\begin{align*} d G = \sqrt { \tau _ 0 } / 4 \left \{ \left ( \left ( 1 + 4 Q _ 0 / \tau _ 0 \right ) d z + \left ( 1 + 4 \overline { Q _ 0 } / \tau _ 0 \right ) d \overline { z } \right ) u _ 1 \right . \\ \left . - i \left ( \left ( 1 - 4 Q _ 0 / \tau _ 0 \right ) d z - \left ( 1 - 4 \overline { Q _ 0 } / \tau _ 0 \right ) d \overline { z } \right ) u _ 2 \right \} \end{align*}


\begin{align*} \sum _ { i = 1 } ^ s { \rm V o l } ( \Sigma \cap B _ { p _ i } ( 3 r ) ) \leq C _ o ( n ) { \rm V o l } ( \Sigma ) . \end{align*}


\begin{align*} \varphi = \begin{bmatrix} v \\ b \\ \end{bmatrix} , F = \begin{bmatrix} ( d + B _ { \infty } ) \otimes b - ( w + u _ { \infty } ) \otimes v \\ ( d + B _ { \infty } ) \otimes v - ( w + u _ { \infty } ) \otimes b \\ \end{bmatrix} , { \mathcal { L } } = \begin{bmatrix} \nu & 0 \\ 0 & \mu \\ \end{bmatrix} \Delta . \end{align*}


\begin{align*} \aligned e _ 1 & \ , : = \ , ( 1 - y ) \ , \partial _ x - z \ , \partial _ y + x \ , \partial _ u , \\ e _ 2 & \ , : = \ , ( 1 - y ) \ , \partial _ y - 2 z \ , \partial _ z + u \ , \partial _ u , \\ e _ 3 & \ , : = \ , u \ , \partial _ x - \tfrac { 4 } { 3 } \ , x \ , \partial _ y + ( 1 - y ) \ , \partial _ z , \\ e _ 4 & \ , : = \ , x \ , \partial _ x - z \ , \partial _ z + 2 \ , u \ , \partial _ u . \endaligned \end{align*}


\begin{align*} y _ i = a _ c s _ c \left ( \hat { \mathbf { h } } _ { i , * } + \tilde { \mathbf { h } } _ { i , * } \right ) \mathbf { p } _ c + \sum _ { j = 1 } ^ { M } a _ j s _ j \left ( \hat { \mathbf { h } } _ { i , * } + \tilde { \mathbf { h } } _ { i , * } \right ) \mathbf { p } _ j + n _ i . \end{align*}


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


\begin{align*} \lim \limits _ { j \to + \infty } \frac { G ( t _ 0 ) - G ( t _ 0 + B _ j ) } { \int _ { t _ 0 } ^ { t _ 0 + B _ j } c ( t ) e ^ { - t } d t } = \liminf \limits _ { B \to 0 + 0 } \frac { G ( t _ 0 ) - G ( t _ 0 + B ) } { \int _ { t _ 0 } ^ { t _ 0 + B } c ( t ) e ^ { - t } d t } \end{align*}


\begin{align*} \int _ { D ( 0 , 2 ) \backslash D ( 0 , 1 ) } | \phi _ n | ^ { 2 } \frac { 1 } { r } \dd z \dd r = 1 , \int _ { D ( 0 , 2 ) \backslash D ( 0 , 1 ) } | \nabla \phi _ n | ^ { 2 } \frac { 1 } { r } \dd z \dd r \to 0 . \end{align*}


\begin{align*} \sigma = ( y _ 1 + \i \theta _ 1 ) \wedge \dots \wedge ( y _ n + \i \theta _ n ) . \end{align*}


\begin{align*} \Delta _ { \Psi } u = \Delta u - g ( \nabla \Psi , \nabla u ) \end{align*}


\begin{align*} \int _ { A _ { n , j } \cap A _ n } & \int _ { \eta _ j ( K ) } ^ { \tau _ { n + 1 } ( \omega ) - \tau _ n ( \omega ) } D ^ p ( x _ { \tau _ n + t } ( \omega ) ) d t d \P \\ & \ge c \delta ^ p K ^ { 2 r p } \int _ { A _ { n , j } \cap A _ n \cap \{ \bar { \tau } \ge T \} } ( \tau _ { n + 1 } ( \omega ) - \tau _ n ( \omega ) ) d \P - c \eta _ j ( K ) \delta ^ p K ^ { 2 r p } \P ( A _ { n , j } \cap A _ n ) . \end{align*}


\begin{align*} \lim \limits _ { | x | \nearrow R } u ( x ) = \lim \limits _ { | x | \nearrow R } v ( x ) = \infty \end{align*}