\begin{align*} A ^ 0 ( U ) U _ t + A ^ 1 ( U ) U _ x + B ( U ) U = F ( U ) , \end{align*}


\begin{align*} \phi _ { x , \beta } ^ { t } \left ( \mho ( v ) \right ) : = \beta \left ( \left \{ \xi \in \partial F : L _ { x } \xi _ { t } \in { \rm S h d } ( v ) \right \} \right ) . \end{align*}


\begin{align*} \gamma _ { \tau } = \frac { 2 n + 4 \tau } { n + 2 s + 2 \tau } . \end{align*}


\begin{align*} v _ { \gamma , 1 } ( x ) = 2 ^ { \alpha } x ^ { 1 + \alpha } ( \log 2 + \log x ) \forall x \in ( 0 , \tfrac 1 2 ] \end{align*}


\begin{align*} \sigma _ k ( A ) = \dfrac { 1 } { k ! } \delta _ { i _ 1 i _ 2 \cdots i _ k } ^ { j _ 1 j _ 2 \cdots j _ k } A ^ { i _ 1 } _ { j _ 1 } A ^ { i _ 2 } _ { j _ 2 } \cdots A ^ { i _ k } _ { j _ k } . \end{align*}


\begin{align*} \Lambda ( x ) = \begin{cases} \dfrac { S ^ \varphi u ( x ) [ 1 - u ^ { \frac { 4 } { m - 2 } } ( x ) ] } { c _ m [ 1 - u ( x ) ] } & \ , u ( x ) \neq 1 \\ [ 0 . 3 c m ] \dfrac { S ^ \varphi } { m - 1 } & \ , u ( x ) = 1 . \end{cases} \end{align*}


\begin{align*} g = & \ , ( d \psi + \beta x ( d \phi + \bar \sigma ) ) ^ 2 + \hat { g } , \\ X = & \ , \alpha \frac { \partial } { \partial \psi } , \\ \beta = & \ , \sqrt { \frac { 2 \alpha ^ 2 } { m n } + \frac { 2 \lambda } { n } } . \end{align*}


\begin{align*} \underset { t \rightarrow \infty } { \lim } P ^ t = \left [ O _ { ( n + 1 ) \times n } \ , \Big \vert \ , \mathbf { 1 } _ { n + 1 } \right ] . \end{align*}


\begin{align*} \lim _ { n \to \infty } \frac { 1 } { n } \log \frac { d \omega _ { n } \lambda } { d \lambda } \left ( \psi \left ( x , { \rm b n d } ( \omega ) \right ) \right ) = h _ { \mu } ( Z , \lambda ) . \end{align*}


\begin{align*} \frac { U A U ^ * + V A V ^ * } { 2 } = \begin{pmatrix} X & 0 \\ 0 & T \end{pmatrix} . \end{align*}


\begin{align*} \operatorname { v o l } ( B _ { c c } ( R ) ) = \operatorname { v o l } ( B _ { c c } ( 1 ) ) R ^ Q \end{align*}


\begin{align*} V _ e ^ { ( i ) } : = \{ v \in V _ e \mid m ( v ) = i \} . \end{align*}


\begin{align*} S _ { k \ell } [ a ] : = \left ( \begin{array} { c c c c } a _ 1 & a _ 2 & \ldots & a _ { \ell } \\ \ldots & \ldots & \ldots & \ldots \\ a _ 1 & a _ 2 & \ldots & a _ { \ell } \\ \end{array} \right ) . \end{align*}


\begin{align*} \Delta ^ * ( f , g ) & = [ f , g ] , \\ \varphi _ L ^ * : L ^ * \rightarrow L ^ * \ ; \ & \eta \mapsto \varphi _ L ^ * ( \eta ) = \eta \circ \varphi _ L , \\ \alpha \circ \eta & = \eta \circ \alpha . \end{align*}


\begin{align*} & ( a ) \ - \tau ^ { - 1 } \langle c , A \psi - \phi \rangle - \tau ^ { - 1 } \sum _ { i = 1 } ^ m \langle c , B _ i \psi \rangle [ C \chi ] _ i \\ & + \sum _ { j = 1 } ^ J a _ j ( x ) \sigma _ j ( x ) + \sum _ { \ell = 1 } ^ L b _ \ell ( x ) \rho _ \ell ( x ) \ \mathrm { i s \ S O S } , \\ & ( b ) \ \sigma _ j ( x ) \ \mathrm { i s \ S O S } , \end{align*}


\begin{align*} \textup { M M D } _ { n , m } ^ { 2 } = \Delta _ { 0 } + \sum ^ { 2 ( l - 1 ) - 1 } _ { s = 1 } \Delta _ { s } + \widetilde { \Delta } _ { 2 ( l - 1 ) } . \end{align*}


\begin{align*} \chi ( X _ { - \alpha } ) = 1 \alpha \in \Pi _ 0 \chi ( X _ \beta ) = 0 \beta \in \Phi _ 0 \backslash \Pi _ 0 \end{align*}


\begin{align*} \tilde { S } = S Q + \mathcal { O } ( \| H \| ) . \end{align*}


\begin{align*} h _ i \sum \limits _ { j = 1 } ^ { s _ { { i _ { } i } } } b _ { i _ { } i j } { \textbf { X } } ^ { \lambda _ j } \in \mathcal I , \end{align*}


\begin{align*} S _ { \alpha , \beta } f ( x ) = \sup _ { t > 3 x } t ^ { - \beta } \int _ { \frac { t - x } 2 } ^ { t - x } ( t + x - z ) ^ { - \alpha - 1 / 2 } ( t - x - z ) ^ { \alpha + \beta - 1 / 2 } \abs { f ( z ) } \ , d z , x > 0 . \end{align*}


\begin{align*} { \delta _ { \underline { f } } = f _ 1 * \dots * f _ p : \mathcal { A } [ \underline { n } ] \longrightarrow \mathcal { A } [ \underline { n } ] } . \end{align*}


\begin{align*} u _ 0 ( x ) : = \begin{cases} \alpha ( \log \frac { 1 } { | x | } ) ^ { \frac 1 p } , & ~ ~ ~ | x | < 1 \\ 0 , & ~ ~ ~ ~ | x | \geq 1 . \end{cases} \end{align*}


\begin{align*} P _ + = - R , P _ - = I - P _ + , \end{align*}


\begin{align*} \int _ { s } ^ t ( \delta h ) _ { s , \tau } \otimes \mathrm { d } X _ { \tau } : = \lim _ { | \pi | \rightarrow 0 } \int _ { \pi } h _ { s , \tau } \circ \mathrm { d } X _ { \tau } = \lim _ { | \pi | \rightarrow 0 } \sum _ { 0 \leq i < m } h _ { s , \kappa _ i } \otimes ( \delta X ) _ { \kappa _ i , \kappa _ { i + 1 } } , \end{align*}


\begin{align*} \Theta _ { S , u } = a _ 0 + \sum _ { T \in \mathcal { Q } _ m , T \neq S } \left ( \prod _ { i \in T } \alpha _ i \right ) f _ T ( u _ T ) \in \C . \end{align*}


\begin{align*} D \left ( \alpha \parallel \beta \right ) = \sup _ { n } H _ { \alpha \parallel \beta } \left ( \mathcal { P } _ { n } \right ) . \end{align*}


\begin{align*} \mathcal { H } ^ 1 \left ( \Omega _ \delta \cap \left ( \overline { J _ u } \setminus J _ u \right ) \right ) = 0 \end{align*}


\begin{align*} ( f , g ) & = S _ { \chi \oplus \xi } S _ { \chi \oplus \xi } ^ { - 1 } ( f , g ) \\ & = \int _ { \Theta } v ^ { 2 } ( w ) \pi _ { F ( w ) \oplus G ( w ) } ( \chi _ { w } \oplus \xi _ { w } ) ^ { \ast } ( \chi _ { w } \oplus \xi _ { w } ) \pi _ { F ( w ) \oplus G ( w ) } S _ { \chi \oplus \xi } ^ { - 1 } ( f , g ) d \mu ( w ) \end{align*}


\begin{align*} \langle f , g \rangle & : = \langle - \ln { \mathfrak R } \sum \limits _ { j = - \infty } ^ \infty \sum \limits _ { Q \in Q ^ j } w ( Q ) \psi _ Q ( \cdot , x _ { Q } ) q _ { Q } h ( x _ { Q } ) , g \rangle \\ & = - \ln { \mathfrak R } \sum \limits _ { j = - \infty } ^ \infty \sum \limits _ { Q \in Q ^ j } w ( Q ) \psi _ Q g ( x _ { Q } ) q _ { Q } h ( x _ { Q } ) , \end{align*}


\begin{align*} L ( x ) + L ( y ) = L ( x y ) + L \left ( \frac { x ( 1 - y ) } { 1 - x y } \right ) + L \left ( \frac { y ( 1 - x ) } { 1 - x y } \right ) . \end{align*}


\begin{align*} \begin{bmatrix} W - \epsilon \Lambda ^ { - 1 } T ^ T \tilde F \\ \epsilon T ^ T \tilde F \end{bmatrix} ^ T \begin{bmatrix} \Lambda & 0 \\ 0 & - \Lambda ^ { - 1 } \end{bmatrix} \begin{bmatrix} W - \epsilon \Lambda ^ { - 1 } T ^ T \tilde F \\ \epsilon T ^ T \tilde F \end{bmatrix} . \end{align*}


\begin{align*} H _ 1 ( z ) : = A _ 1 z ^ { n ' + m ' } + B _ 1 \overline { z } ^ { m ' } + C _ 1 \textrm { f o r a l l } z \in \mathbb { C } \end{align*}


\begin{align*} \mathcal { J } _ { \lambda } ^ t \coloneqq \{ X \in \mathcal { J } _ { \lambda } \mid X ( t ) = 0 \} \end{align*}


\begin{align*} y < 1 \iff 0 = f ( y ) < f ( 1 ) = 1 + \alpha _ 2 + \alpha _ 1 + \alpha _ 0 . \end{align*}


\begin{align*} \hat { I } _ { \varepsilon , j } ^ 1 + \Delta \big ( \varphi _ j c \big ) = \psi _ j \mathcal { L } ^ { \mathbb { R } ^ n _ { \gamma _ j } } _ \varepsilon \big ( \varphi _ j c \big ) + \Delta \big ( \varphi _ j c \big ) = \psi _ j \Big ( \mathcal { L } ^ { \mathbb { R } ^ n _ { \gamma _ j } } _ \varepsilon \big ( \varphi _ j c \big ) + \Delta \big ( \varphi _ j c \big ) \Big ) . \end{align*}


\begin{align*} b _ { i , j } ' = b _ { g ( i ) , b ( j ) } . \end{align*}


\begin{align*} L _ { u ( t ) } \ , S ^ k \varrho ( t ) = ( \nu _ { u _ 0 } + k ) S ^ k \varrho ( t ) \ , , \end{align*}


\begin{align*} \displaystyle \left ( \Pi _ { i = 1 } ^ t \sigma _ { ( 1 , 1 ) } \right ) \star \sigma _ { \lambda } = q \sigma _ { ( \lambda _ 2 + t ) } - \left ( \sum _ { | \mu | + 2 n = | \lambda | } a _ { \mu } q \sigma _ { ( \mu _ 1 + t , \mu _ 2 + t ) } \right ) . \end{align*}


\begin{align*} \frac { \partial f } { \partial \theta } = f \frac { \partial g } { \partial \theta } , \end{align*}


\begin{align*} K \left ( y \left ( t _ { j } \right ) , t _ { j } \right ) & = K \left ( y _ { j } , x _ { j } \right ) \\ K \left ( y ^ { \prime } \left ( t _ { j } \right ) , t _ { j } \right ) & = K \left ( y _ { j } ^ { \prime } , x _ { j } \right ) . \end{align*}


\begin{align*} ( \mathrm { d i v } \ , V _ k ) \ , V _ k \cdot \nu - \nabla V _ k \ , V _ k \cdot \nu = ( \mathrm { d i v } _ T V _ k ) \ , V _ k \cdot \nu - \nabla _ T V _ k \ , ( V _ k ) _ T \cdot \nu \quad \partial \Omega \ , . \end{align*}


\begin{align*} ( \Lambda _ 0 , \xi _ 0 , \psi _ 0 ) \leftrightarrow x _ 0 = ( A _ 0 , \lambda _ 0 , \psi _ 0 ) \leftrightarrow [ z _ 0 , g _ 0 ] _ { \mathbf { K } } \in X _ { \mathbf { K } } ( \overline { \mathbb { Q } } ) . \end{align*}


\begin{align*} I = \int _ { 0 } ^ { \infty } f ( x ) \ , d x \end{align*}


\begin{align*} | X | ^ 2 \equiv c ^ 2 = \frac { m \lambda } { 2 } + \frac { 3 | m \lambda | } { 2 } . \end{align*}


\begin{align*} \mathrm { s p a n } \{ \mathrm { d } \varphi _ { [ 0 , R ] } \} = \mathrm { s p a n } \{ \mathrm { d } \varphi _ { [ 0 , A - 1 ] } , \mathrm { d } \varphi _ { [ A , R ] } \} = \mathrm { s p a n } \{ \mathrm { d } x , \mathrm { d } \varphi _ { c } , \mathrm { d } \varphi _ { [ A , R ] } \} \ , . \end{align*}


\begin{align*} \sum \limits _ { \substack { m = 1 \\ ( m . n ) = 1 } } ^ n ( m - 1 , n ) = \phi ( n ) \tau ( n ) . \end{align*}


\begin{align*} \lambda _ i \lambda _ j = \lambda _ i \lambda _ { j ^ * } = \lambda _ j \lambda _ { i ^ * } = \lambda _ { j ^ * } \lambda _ { i ^ * } = \lambda _ i \lambda _ { i ^ * } = \lambda _ j \lambda _ { j ^ * } , \end{align*}


\begin{align*} L H = u ^ { j \bar k } H _ { j \bar k } \leq - \alpha \ , u ^ { j \bar k } ( \delta _ { j k } - \alpha \bar z _ j z _ k ) H . \end{align*}


\begin{align*} \limsup _ { h \to + \infty } F _ h ( v _ h , \gamma _ h , B _ \rho ) \leq \limsup _ { h \to + \infty } F _ h \left ( w _ h ^ { ( \rho ) } , \gamma _ h , B _ \rho \right ) \end{align*}


\begin{align*} T = \begin{pmatrix} W & X \\ Y & Z \end{pmatrix} , \end{align*}


\begin{align*} \varrho ^ t = e ^ { - t \left ( \frac { \nu ^ 2 A } { 2 } + L \right ) } \varrho ^ 0 - \int _ 0 ^ t e ^ { - ( t - \tau ) \left ( \frac { \nu ^ 2 A } { 2 } + L \right ) } B ^ \tau ( \varrho ^ \tau , \varrho ^ \tau ) d \tau . \end{align*}


\begin{align*} \kappa = \frac { S ^ \varphi } { m ( m - 1 ) } \ , . \end{align*}


\begin{align*} & \gamma ' ( t ) G ( t , \sigma ( t ) ) + \gamma ( t ) \partial _ t G ( t , \sigma ( t ) ) + \gamma ( t ) \sigma ' ( t ) \partial _ s G ( t , \sigma ( t ) ) \\ & = G ( t , \sigma ( t ) ) \left ( \gamma ' ( t ) + \varphi ( t ) \gamma ( t ) \right ) + \gamma ( t ) \partial _ s G ( t , \sigma ( t ) ) \left ( \sigma ' ( t ) + q + \varphi ( t ) \sigma ( t ) \right ) \\ & = 0 . \end{align*}


\begin{align*} g = \frac { m _ g } { f ^ { k _ g } } \quad h = \frac { m _ h } { f ^ { k _ h } } \rlap { . } \end{align*}


\begin{align*} y ^ 3 + \alpha _ 2 y ^ 2 + \alpha _ 1 y + \alpha _ 0 = 0 \end{align*}


\begin{align*} \frac { 1 } { n ^ { d } } \sum \limits _ { r \le i _ 1 \leqslant i _ 2 \leqslant . . . \leqslant i _ { d } \le k } i _ 1 \cdot i _ 2 \cdots i _ d = \frac { 1 } { n ^ { d } } h _ { d } ( r , r + 1 , . . . , k ) \end{align*}


\begin{align*} \Theta ^ { \{ \partial W ^ 1 , \partial W ^ 2 , \dots , \partial W ^ m \} } ( x ) = \sum _ { i = m + 1 } ^ n \Theta ( x , \partial W ^ i ) . \end{align*}


\begin{align*} \mathcal { Q } _ { \ell , \vec { v } } = \mathcal { Q } _ { \ell , \alpha \vec { v } } \end{align*}


\begin{align*} f _ t = f _ 0 + \lambda ^ 2 t \ Q _ t [ f _ 0 ] + \lambda ^ 2 t \ B _ t [ f _ 0 ] \end{align*}


\begin{align*} c = - N \left ( 1 + \frac { 2 \alpha } { \beta } \right ) \ , . \end{align*}


\begin{align*} G ' ( 0 ) = - 2 I m \int _ { \mathbb { R } ^ N } \left ( \dfrac { a _ 2 } { m _ 1 } x \phi _ 0 \nabla \overline { \phi _ 0 } + \dfrac { a _ 1 } { m _ 2 } x \psi _ 0 \nabla \overline { \psi _ 0 } \right ) d x . \end{align*}


\begin{align*} \mathbb { G } _ { t , T } ^ { t , \mu ; u } \left [ \Phi ^ { 0 } \left ( \rho _ { T } ^ { t , \mu ; u } \right ) \right ] = \mathbb { G } _ { t , t + \delta } ^ { t , \mu ; u } \left [ Y _ { t + \delta } ^ { t , \mu ; u } \right ] . \end{align*}


\begin{align*} \pi _ 0 ^ { - 1 } \left ( f ^ { - 1 } ( S _ 0 ) \cap \Gamma _ 0 \right ) \cap \Gamma = \left ( f \circ \pi _ 0 \right ) ^ { - 1 } ( S _ 0 ) \cap \Gamma , \end{align*}


\begin{align*} \vec { v } ^ T T _ \lambda ^ { \vec { v } } \vec { e _ i } = \lambda v _ i \end{align*}


\begin{align*} \Sigma _ { \mathrm { D } } ( \delta ) = \left \{ \gamma \in \mathcal { P } ( \mathcal { Y } _ 1 \times \mathcal { Y } _ 2 ) : \boldsymbol { K } _ { Y _ 1 } ( \mu _ { Y _ 1 } , \gamma _ { 1 } ) \le \delta _ 1 , \boldsymbol { K } _ { Y _ 1 } ( \mu _ { Y _ 2 } , \gamma _ { 2 } ) \le \delta _ 2 \right \} \end{align*}


\begin{align*} L _ { \hat { n } } = L _ 0 + 2 J \sum _ { k = 1 } ^ { \hat { n } } l _ k \leq ( 1 + 2 J \hat { n } ) L _ 0 . \end{align*}


\begin{align*} x _ i = \sum _ { \substack { 1 \le j \le n \\ j \neq i } } ( m _ i \ss _ { i j } ^ { 3 } ( x _ i - x _ j ) ) . \end{align*}


\begin{align*} \hat { f } ( a ^ s \Lambda _ u ) = \sum _ { ( \bar { p } , \bar { q } ) \in \Z ^ { m + n } } f \left ( \frac { N p _ 1 + v _ 1 + \langle \bar { u } _ 1 , N \bar { q } + v '' \rangle } { N } , \ldots , \frac { N p _ m + v _ m + \langle \bar { u } _ m , N \bar { q } + v '' \rangle } { N } , \frac { N \bar { q } + v '' } { N } \right ) . \end{align*}


\begin{align*} P U _ t + \frac { 1 } { 2 } \left [ ( A U ) _ x + A U _ x + ( B U ) _ y + B U _ y \right ] = \epsilon ( ( P U _ x ) _ x + ( P U _ y ) _ y ) \end{align*}


\begin{align*} \log \nu _ t ^ { ( n ) } ( x ) & = e ^ { - \frac { \sigma ^ 2 } { 2 } t } \log \nu _ 0 ( x ) \\ & - \int _ 0 ^ t \frac { \sigma ^ 2 } { 2 } e ^ { - \frac { \sigma ^ 2 } { 2 } ( t - s ) } \left ( \frac { 2 } { \sigma ^ 2 } \frac { \delta F } { \delta \nu } ( \nu _ s ^ { ( n - 1 ) } , \mu _ s ^ { ( n - 1 ) } , x ) - \log \pi ( x ) - \operatorname { D _ { K L } } ( \nu _ s ^ { ( n - 1 ) } | \pi ) \right ) \mathrm { d } s , \\ \end{align*}


\begin{align*} \lim _ { k \to \infty } \frac { \nu _ k } { \sigma ^ 2 _ k } = 0 . \end{align*}


\begin{align*} \varphi \circ f = f \circ \varphi ^ { \otimes n } \quad \psi \circ f = f \circ \psi ^ { \otimes n } . \end{align*}


\begin{align*} \int \chi ( x ) e ^ { - i x \cdot \xi / h } \Tilde { R } _ h ( z ) ( \psi ( x ) e ^ { i x \cdot \eta / h } ) d x = \mathcal { O } ( h ^ \infty ) \langle \xi , \eta \rangle ^ { - \infty } , \quad \langle \xi , \eta \rangle \to \infty . \end{align*}


\begin{align*} \left ( u _ { i } , m _ { i } , k _ { i } \right ) \in S _ { 1 } ^ { 2 } \left ( N _ { 2 } \right ) \times S _ { 2 } \left ( N _ { 3 } \right ) , i = 1 , 2 \end{align*}


\begin{align*} h = \Delta + O ( \Delta \alpha ^ 2 ) , \quad \ ; \ ; \alpha \leq \overline C _ 1 \max _ { q \in S ^ 2 : \ , q \cdot v = 0 } \ ; \max _ { 0 \le s \le t } \ ; ( q \cdot W _ s ) . \end{align*}


\begin{align*} \| X - \tau I \| _ { \infty } - \left \| \frac { | X - \tau I | + | X ^ * - \overline { \tau } I | } { 2 } \right \| _ { \infty } & = \left \| \begin{pmatrix} | x | & 0 \\ 0 & | y | \end{pmatrix} \right \| _ { \infty } - \frac { 1 } { 2 } \left \| \begin{pmatrix} | x | + | y | & 0 \\ 0 & | x | + | y | \end{pmatrix} \right \| _ { \infty } \\ & = \left | | x | - | y | \right | \\ & = \varepsilon ( X ) \end{align*}


\begin{align*} \Phi _ { k , i } ( \alpha _ 1 , \ldots , \alpha _ k ) = 0 , \forall i = 1 , \ldots , n , \end{align*}


\begin{align*} B ( u , v ) = \langle u , v \rangle = \sum _ { i = 1 } ^ { N } u _ { i } v _ { i } \left ( u , v \in \mathbb { R } ^ { N } \right ) , \end{align*}


\begin{align*} U ( \vec { a } , \Lambda ) & \phi ^ { \alpha , n } ( f ) U ( \vec { a } , \Lambda ) ^ { - 1 } \left ( S , g _ \beta \otimes \rho _ n ( E ^ \beta ) \right ) \\ & = \phi ^ { \alpha , n } [ f ( \Lambda ^ { - 1 } ( \cdot - \vec { a } ) ) ] \left ( S , g _ \beta \otimes \rho _ n ( E ^ \beta ) \right ) . \end{align*}


\begin{align*} \lambda = \lambda ( a ) = a ^ { 2 } ( 1 - a ) ^ { - 2 } \ , . \end{align*}


\begin{align*} I : = \{ i : \Vert B _ { i } \Vert _ { 2 } > C \log ^ { - \frac { 5 } { 2 } } n \} . \end{align*}


\begin{align*} \limsup _ { N \to \infty } \ , \frac { \theta _ N ^ 2 } { N ^ 4 } \ , \Xi ( 2 ) \le \frac { C } { M ^ 2 } \ , , \end{align*}


\begin{align*} S _ { \varepsilon } A _ { \varepsilon } - A _ { \varepsilon } ^ { * } S _ { \varepsilon } = 0 . \end{align*}


\begin{align*} \left ( \left ( Q _ { 1 , \vec { v } } \right ) _ i \right ) _ { j , k } = \begin{cases} 1 & i \leq j , k = j - i + 1 \\ \alpha _ { n - i + 1 + j - k } & 2 \leq i \leq j , k \in \{ j - i + 2 , \dots , j \} \\ - \alpha _ { n - i + 1 + j - k } & j < i , j + 1 \leq k \leq n + j - i + 1 \\ 0 & \end{cases} . \end{align*}


\begin{align*} \lim _ { k \to \infty } \nu _ k = 0 , \lim _ { k \to \infty } \sigma _ k = 0 , \lim _ { k \to 0 } \frac { \nu _ k } { \sigma ^ 2 _ k } = 0 . \end{align*}


\begin{align*} \{ \underline { a } \colon \underline { a } / \varpi ^ k \in L ( \underline { c } ) \} = \{ a \underline { c } ^ \bot \colon ( a , \varpi ) = 1 , | a | < | \varpi | ^ k \} . \end{align*}


\begin{align*} ( f g ) x = \lambda _ 1 ( f g ) x & + \lambda _ 2 ( f x ) g + \lambda _ 3 ( g f ) x + \lambda _ 4 ( x f ) g \\ & + \lambda _ 5 x ( f g ) + \lambda _ 6 y ( f x ) + \lambda _ 7 x ( g f ) + \lambda _ 8 g ( x f ) \\ \intertext { a n d } x ( f g ) = \mu _ 1 ( f g ) x & + \mu _ 2 ( f x ) g + \mu _ 3 ( g f ) x + \mu _ 4 ( x f ) g \\ & + \mu _ 5 x ( f g ) + \mu _ 6 g ( f x ) + \mu _ 7 x ( g f ) + \mu _ 8 g ( x f ) . \end{align*}


\begin{align*} \sum _ { p \in \Z } \mathrm { I } _ { \mathrm { p } } ( 2 t ) = e ^ { 2 t } . \end{align*}


\begin{align*} \sup _ { k \leq n } | S _ k - T _ k | = o ( n ^ { 1 / p } ( \log n ) ^ { \eta } ) \end{align*}


\begin{align*} x ^ 3 \theta ( [ x ] ) \leq 1 + 8 \sum _ { k = 1 } ^ { [ x ] } k ^ 2 \theta ( k ) \leq 1 + 8 \sum _ { k \geq 1 } k ( k \wedge x ) \theta ( k ) \ , . \end{align*}


\begin{align*} \Phi _ b = \sum _ { n } c _ n a _ n . \end{align*}


\begin{align*} \begin{cases} \dot { \xi } _ x ( t ) = \mathbf { v } ( \xi _ x ( t ) ) , & \\ \xi _ x ( 0 ) = x . \ \ & \end{cases} \end{align*}


\begin{align*} \nabla = \partial _ z + u ( z ) \check { \alpha } + L ( z ) e \end{align*}


\begin{align*} r _ { X / G } ( h G ) = r _ X ( h ) G , \iota _ { X / G } ( h G ) = \iota _ X ( h ) G . \end{align*}


\begin{align*} \frac { 2 \pi \delta _ j } { 3 } \le \frac { t \delta _ j } { 3 j } = \frac { 2 \pi K \delta _ j } { 3 j } \le \beta ^ { - 1 } \frac { 2 \pi \delta _ j } { 3 } = \Big ( 1 + \frac { \alpha } { 4 \pi } \Big ) \frac { 2 \pi \delta _ j } { 3 } \le \frac { 2 \pi \delta _ j } { 3 } + \frac { \alpha } { 2 } \end{align*}


\begin{align*} \mathcal { D } _ x f ( x ) = \frac { d } { d x } f ( x ) . \end{align*}


\begin{align*} \mathcal { H } \oplus \mathcal { H } = \operatorname { G r } ( T ) \oplus W \operatorname { G r } ( T ^ * ) \end{align*}


\begin{align*} \uparrow ^ i X ^ \mathbf { n } = X ^ { \mathbf { n } + e _ i } \end{align*}


\begin{align*} J _ { F , \Q } = \left ( \frac { q _ 0 } { \Delta } , \dots , \frac { q _ { n - 2 } } { \Delta } \right ) \ . \end{align*}


\begin{gather*} \forall h \in \N , \mathcal { L } ^ 2 \left ( \left \{ f _ h \neq g _ h \right \} \cap B _ 1 \right ) \lesssim \left ( \mathcal { H } ^ 1 \left ( \left \{ f _ h \neq g _ h \right \} \cap B _ 1 \right ) \right ) ^ 2 , \\ \left ( \mathcal { H } ^ 1 \left ( \left \{ f _ h \neq g _ h \right \} \cap B _ 1 \right ) \right ) ^ 2 = { \rm o } ( \gamma _ h ) \mbox { a s } h \to + \infty . \end{gather*}


\begin{align*} \mathrm { d e s c d } ( v ) & = \# \{ j \in [ i + 1 , n ] : \sigma ( i ) > \sigma ( j ) \} , \\ \mathrm { d e p t h } ( v ) & = \# \{ j \in [ i - 1 ] : \sigma ( j ) > \sigma ( i ) \} . \end{align*}


\begin{align*} \P ( \omega _ i ^ w = \omega _ i \mid E _ 1 , \dots , E _ { i - 1 } ) \le \ell / ( n - i \ell ) . \end{align*}


\begin{align*} \| T _ { \eta } f \| _ { L ^ p ( x ^ \gamma d x ) } ^ p & \lesssim \int _ 0 ^ \infty x ^ { - \tau - 1 + \gamma } \int _ x ^ \infty \abs { f ( z ) } ^ p z ^ { \tau } \ , d z \ , d x = \int _ 0 ^ \infty \abs { f ( z ) } ^ p z ^ { \tau } \int _ 0 ^ z x ^ { - \tau - 1 + \gamma } \ , d x \ , d z \\ & \simeq \| f \| _ { L ^ p ( x ^ \gamma d x ) } ^ p , \end{align*}


\begin{align*} \Vert Z \Vert _ { \mathcal { D } _ { \mathbf { X } , \tilde { \alpha } } ^ { \gamma } ( I ) } : = \Vert Z \Vert _ { C ( I ; \mathcal { B } _ { \tilde { \alpha } } ) } + \Vert Z ^ { \prime } \Vert _ { \mathcal { E } ^ { 0 , \gamma } _ { \tilde { \alpha } - \gamma ; I } } + \Vert Z ^ { \# } \Vert _ { \mathcal { E } ^ { \gamma , 2 \gamma } _ { \tilde { \alpha } ; I } } < \infty \end{align*}


\begin{align*} \psi ( x , t ) = \phi ( x ) + \tilde { v } ( x , t ) \end{align*}


\begin{align*} \rho [ G _ \alpha ( u ) ] _ t = \rho G _ \alpha ' ( u ) u _ t = - G _ \alpha ' ( u ) [ - \Delta u + ( - \Delta ) ^ s u ] \textrm { i n } \ ; \ , S _ T \ , . \end{align*}


\begin{align*} C _ n ^ \mu ( - z ) = ( - 1 ) ^ n C _ n ^ \mu ( z ) . \end{align*}


\begin{align*} \partial _ t E ( t ) = - 2 E ( t ) + E _ 1 ( t ) + E _ 2 ( t ) + E _ 3 ( t ) , \end{align*}


\begin{align*} \prod _ { \substack { m , n \geq 1 \\ m \leq n ; \ , ( m , n ) = 1 } } \left ( \frac { 1 } { 1 - z ^ n } \right ) ^ { \frac { m ^ 2 } { n ^ 3 } } = \exp \left \{ \sum _ { n = 1 } ^ { \infty } \left ( \frac { n } { 6 } + \frac { n ^ 2 } { 2 } + \frac { n ^ 3 } { 3 } \right ) \frac { z ^ n } { n ^ 3 } \right \} \end{align*}


\begin{align*} \int _ { \Omega ' } [ a ^ { i j , k l } ( D ^ { 2 } u ) u _ { i j } ] ^ { h _ { m } } \eta _ { k l } d x = 0 . \end{align*}


\begin{align*} \mu _ N \coloneqq \frac { 1 } { N + 1 } \sum _ { i = 1 } ^ { N + 1 } \delta _ { x _ i } \end{align*}


\begin{align*} f ( z _ 1 , \ldots , z _ { m } ) = a _ 0 + \left ( \sum _ { S \in \mathcal { Q } _ m } \left ( \prod _ { i \in S } z _ i \right ) f _ S ( z _ S ) \right ) + f ^ * ( z _ 1 , \ldots , z _ { m } ) , \end{align*}


\begin{align*} ( 1 - \varepsilon - \delta ) \| x \| \leq \| V \theta _ g x \| , \forall x \in \mathcal { X } \implies ( 1 - \varepsilon - \delta ) \| y \| = ( 1 - \varepsilon - \delta ) \| \theta _ g ^ { - 1 } y \| \leq \| V y \| , \forall y \in \ell ^ p ( [ n ] ) , \end{align*}


\begin{align*} \nu _ { _ { \Omega _ h } } ( 0 ) : = \frac { T _ h \nu _ { \Omega } ( 0 ) } { | T _ h \nu _ { \Omega } ( 0 ) | } . \end{align*}


\begin{align*} \deg \left ( \frac { \textbf { Q } ^ { \lambda _ j } } { Q _ i ^ { \lambda _ j ( Q _ i ) } } \right ) = \deg \left ( \frac { \tilde { \textbf { Q } } ^ { \lambda _ j } } { \tilde Q _ i ^ { \lambda _ j ( Q _ i ) } } \right ) < \deg ( Q _ i ) \mbox { f o r e v e r y } j , 1 \leq j \leq r . \end{align*}


\begin{align*} \zeta _ { p } ( \sigma + i t ) = \begin{cases} O \left ( | t | ^ { 1 - \sigma } \log | t | \right ) & \frac { 1 } { 2 } \leq \sigma \leq 1 , \\ O \left ( | t | ^ { \frac { 1 } { 2 } } \log | t | \right ) & 0 < \sigma \leq \frac { 1 } { 2 } , \\ O \left ( | t | ^ { \frac { 1 } { 2 } - \sigma } \log | t | \right ) & \sigma \leq 0 . \end{cases} \ , \ , \ , \ , \ , | t | \rightarrow \infty . \end{align*}


\begin{align*} ( 1 + w ) ( R ^ \varphi _ { k j , i } - R ^ \varphi _ { k i , j } ) = w _ t R _ { t k j i } + \frac { S ^ \varphi } { m - 1 } ( w _ i \delta _ { j k } - w _ j \delta _ { i k } ) - ( w _ i R ^ \varphi _ { j k } - w _ j R ^ \varphi _ { i k } ) \ , . \end{align*}


\begin{align*} ( \prod _ { i = 1 } ^ m a ( i ) \ast f ( t ( i ) ) ) \ast a ( m + 1 ) \in B \cap \sigma ( W ) . \end{align*}


\begin{align*} W : = \left \{ \left ( x , L _ { x } \omega \right ) : x \in X , \omega \in G \right \} = \bigsqcup _ { x \in X } \{ x \} \times L _ { x } \backslash G . \end{align*}


\begin{align*} \sigma _ K = \{ ( q , p ) \mid p ^ i = 0 , \forall i \in K , \ q ^ j = 0 , \forall j \notin K \} . \end{align*}


\begin{align*} \frac { \Sigma _ N ^ 2 ( L , \alpha ) - L } { L } = X _ N ( L , \alpha ) - \frac { L } { N } , \end{align*}


\begin{align*} p \widetilde { \mathbb { S } } _ { \lambda ^ i } p \widetilde { \mathbb { S } } _ { \lambda ^ j } = \sum _ { \lambda ^ \ell \succeq \lambda ^ i , \lambda ^ j } \sum _ { P : \delta P = \Delta _ { \sigma _ r \lambda ^ i ~ \sigma _ r \lambda ^ j } ^ { \sigma _ r \lambda ^ \ell } } \sigma _ r w t ( P ) p \widetilde { \mathbb { S } } _ { \lambda ^ \ell } , \end{align*}


\begin{align*} P _ { n } ( \epsilon , q ) \leq \sum \limits _ { c = r } ^ { m } P \left \{ \omega : \sup _ { k \geq n } { k \choose r } ^ { - \frac { 1 } { ( q + 1 ) } } \left | \left | C _ { m , k , c } ( \omega ) \right | \right | _ { { \cal { B } } \otimes { \cal { B } } } \geq \epsilon _ { c } \right \} , \end{align*}


\begin{align*} \sum _ { k = 2 } ^ d { k \choose 2 } \abs { A _ k } & = \sum _ { k = 2 } ^ d \sum _ { i = 1 } ^ { k - 1 } i \abs { A _ k } = \sum _ { i = 1 } ^ { d - 1 } i \sum _ { k = i + 1 } ^ { d } \abs { A _ k } = \abs { A _ 2 } + \biggl \lvert \bigcup _ { k \geq 3 } A _ k \biggr \rvert + \sum _ { i = 2 } ^ { d - 1 } i \cdot \biggl \lvert \bigcup _ { k \geq i + 1 } A _ k \biggr \rvert . \end{align*}


\begin{align*} \theta _ { s } : = \frac { \tau } { \tau - s } M _ { \tau } ( \mathrm { P } _ 0 ) ^ { \frac { s } { \tau } } \end{align*}


\begin{align*} Z ( G ) = M ( G ) = n - 2 T + s _ 1 + 2 s _ 0 , \end{align*}


\begin{align*} i _ a = \inf _ { t > 0 } \frac { t \ , a ' ( t ) } { a ( t ) } \quad s _ a = \sup _ { t > 0 } \frac { t \ , a ' ( t ) } { a ( t ) } , \end{align*}


\begin{align*} P \psi ^ h _ { \mu , \xi } ( x ) = \psi ^ h _ { \mu , \xi } ( x ) - \alpha _ n \mu ^ { \frac { n } { 2 } } \partial _ { \xi _ h } H ( x , \xi ) + O ( \mu ^ { \frac { n + 2 } { 2 } } ) , \end{align*}


\begin{align*} \mu _ { a \cup b } ( \Pi ) = \Pi - P _ a - P _ b + P _ a \cup P _ b . \end{align*}


\begin{align*} g _ { \varepsilon } ( t , k ) : = \left ( \tilde { a } _ { \varepsilon } - a _ { \varepsilon } \right ) ( t ) \mathcal { H } _ { \hbar , V } \tilde { u } _ { \varepsilon } ( t , k ) + \left ( \tilde { q } _ { \varepsilon } - q _ { \varepsilon } \right ) ( t ) \tilde { u } _ { \varepsilon } ( t , k ) + ( f _ { \varepsilon } - \tilde { f } _ { \varepsilon } ) ( t , k ) . \end{align*}


\begin{align*} \pi _ { \varphi _ 1 \bar { \otimes } \varphi _ 2 } ( a \otimes b ) & = \pi _ { \varphi _ 1 } ( a ) \otimes \pi _ { \varphi _ 2 } ( b ) , a \in M _ 1 , b \in M _ 2 , \\ \lambda ^ { \varphi _ 1 \bar { \otimes } \varphi _ 2 } ( t ) & = \lambda ^ { \varphi _ 1 } ( t ) \otimes \lambda ^ { \varphi _ 2 } ( t ) , t \in \mathbb { R } . \end{align*}


\begin{align*} \overset \cdot b _ i = b _ i ( \sum _ { j = 1 } ^ p b _ { i + j } - \sum _ { j = 1 } ^ p b _ { i - j } ) , i \in \mathbb { Z } . \end{align*}


\begin{align*} \left \Vert u \right \Vert _ { \alpha , p } = \left ( \sum \limits _ { i = 1 } ^ { m } \alpha _ { i } \left \Vert u _ { i } \right \Vert ^ { p } \right ) ^ { 1 / p } . \end{align*}


\begin{align*} \Lambda _ \chi ^ { \rm s t i f f } \vect g : = \vect h \in H ^ { - 1 / 2 } ( \Gamma ; \C ^ 3 ) , \mbox { w h e r e : } \left \{ \begin{array} { c c } \vect u = \Pi _ \chi ^ { \rm s t i f f } \vect g & \\ \vect h = \frac { \partial \vect u } { \partial \nu } | _ \Gamma . & \end{array} \right . \end{align*}


\begin{align*} \left \{ \begin{aligned} & \ h ' ( r , x ) > h ( r , x ) 0 < r \leq \delta . \\ & \ h ' ( r , x ) = \big ( \frac r \delta \big ) ^ 2 h ' ( \delta , x ) r \geq \delta . \end{aligned} \right . \end{align*}


\begin{align*} \gamma = \begin{bmatrix} a & b \\ c & d \end{bmatrix} . \end{align*}


\begin{align*} g = \widetilde g _ 0 \cdot g _ 1 \cdots g _ p , \end{align*}


\begin{align*} D = \sqrt { \phi ^ 2 + \rho ^ 2 | \nabla u | ^ 2 } , \end{align*}


\begin{align*} 0 \leq u _ { n \bar n } ( 0 ) = u _ { x _ n x _ n } + u _ { y _ n y _ n } ( 0 ) \leq C . \end{align*}


\begin{align*} \frac { 1 } { \left ( \lambda _ { n } ^ { 2 } + x ^ { 2 } \right ) ^ { s } } = \sum _ { m = 0 } ^ { \infty } \left ( \begin{array} { c } - s \\ m \end{array} \right ) \ , \frac { x ^ { 2 m } } { \lambda _ { n } ^ { 2 m + 2 s } } . \end{align*}


\begin{align*} \overline { \mathrm { d i m } } _ B Y + \overline { \mathrm { u m d i m } } _ Z \left ( G , d , \nu ^ \mathbb { N } \right ) = \overline { \mathrm { u m d i m } } _ { Y ^ { \mathbb { N } } \times Z } \left ( F , D \right ) . \end{align*}


\begin{align*} \left \{ \begin{array} { l c } u _ { t t } ( x , t ) = \Delta u ( x , t ) + f ( x , t ) , & ( x , t ) \in \Omega \times ( 0 , T ] , \\ u = 0 , & ( x , t ) \in \partial \Omega \times ( 0 , T ] , \\ u ( x , 0 ) = \psi _ 0 , ~ u _ t ( x , 0 ) = \psi _ 1 , & x \in \Omega . \end{array} \right . \ , \end{align*}


\begin{align*} \sum _ { t \in ( a , b ) } \dim ( \sigma _ 0 ( t ) \cap \sigma ) - \sum _ { \lambda \in ( \lambda _ 0 , 0 ) } \dim ( \sigma _ \lambda ( b ) \cap \sigma ) = 0 . \end{align*}


\begin{align*} \limsup _ { n \rightarrow \infty } G ( U _ n ) = \limsup _ { n \rightarrow \infty } \left \{ ( 4 - 2 d ) \omega Q ( U _ n ) + ( 3 - d ) \mathbf { c } \cdot \mathbf { P } ( U _ n ) \right \} \ge \eta . \end{align*}


\begin{align*} \begin{cases} \dot \gamma ( t ) = \displaystyle u _ 1 ( t ) X _ 1 ( \gamma ( t ) ) + u _ 2 ( t ) X _ 2 ( \gamma ( t ) ) , \\ \displaystyle u ( t ) \in B ^ { \norm { \cdot } } _ 1 ( 0 ) , \quad \gamma ( 0 ) = q _ 0 , \quad \quad \gamma ( T ) = q _ 1 , \\ T \to \min . \end{cases} \end{align*}


\begin{align*} \nabla _ { { f } X } \ , \xi _ i - { f } \ , \nabla _ { X } \ , \xi _ i = 0 , X \in \mathfrak { X } _ M . \end{align*}


\begin{align*} \prod _ { \substack { ( a , b , c ) = 1 \\ a , b , c \geq 1 } } \left ( \frac { 1 } { 1 - x ^ a y ^ b z ^ c } \right ) ^ { \frac { c ^ 3 } { a ^ 2 b ^ 2 } } = \exp \left \{ L i _ 2 ( x ) L i _ 2 ( y ) L i _ { - 3 } ( z ) \right \} \end{align*}


\begin{align*} v ( T ) : = \sup \{ | x ^ * ( T x ) | \colon ( x , x ^ * ) \in \Pi ( X ) \} . \end{align*}


\begin{align*} \int _ 0 ^ \infty \mathrm { e } ^ { \beta t } t ^ \nu K _ \nu ( \alpha t ) \ , \mathrm { d } t = \frac { \sqrt { \pi } ( 2 \alpha ) ^ \nu } { ( \alpha - \beta ) ^ { 2 \nu + 1 } } \frac { \Gamma ( 2 \nu + 1 ) } { \Gamma ( \nu + 3 / 2 ) } { } _ 2 F _ 1 \bigg ( 2 \nu + 1 , \nu + \frac { 1 } { 2 } ; \nu + \frac { 3 } { 2 } ; - \frac { \alpha + \beta } { \alpha - \beta } \bigg ) , \end{align*}


\begin{align*} \Delta ( 2 , 2 n ) = \Big \{ e _ 1 + e _ j \ : : \ : \{ i , j \} \in \binom { [ 2 n ] } { 2 } \Big \} \subset H \end{align*}


\begin{align*} G ( t ) = G ( 0 ) + G ' ( 0 ) t + \int _ 0 ^ t ( t - s ) G '' ( s ) d x , 0 \leq t < + \infty . \end{align*}


\begin{align*} I _ H [ u ] = \int _ \Omega B ( H ( \nabla u ) ) \ , d x - \int _ \Omega f u \ , d x \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \overline { S } _ n / \mathbb { E } ( \overline { S } _ n ) = 1 \end{align*}


\begin{align*} \overline { \Pi } _ { \psi } ( p ) B ( [ \epsilon , x ] ) = a ^ { \tfrac { 3 } { 2 } } x e ^ { - ( \det p ) \epsilon \pi x ^ 2 a ^ 2 } \cdot e ^ { \pi i \epsilon ( \det p ) x ^ 2 a b } = a ^ { \tfrac { 3 } { 2 } } x e ^ { i \epsilon \pi x ^ 2 z _ p } . \end{align*}


\begin{align*} \mathcal { R } \Phi ( A \circ B ) & = \Phi ( \mathcal { R } ( A \circ B ) ) & & \\ & \geq \Phi ( \mathcal { R } A \circ \mathcal { R } B ) & & \\ & = \Phi ( \mathcal { R } A ) \circ \Phi ( \mathcal { R } B ) & & \\ & = \mathcal { R } ( \Phi ( A ) ) \circ \mathcal { R } ( \Phi ( B ) ) . & & \end{align*}


\begin{align*} = \int \limits _ { R _ * } ^ { R _ 0 } \int \limits _ { S ( z _ 1 , t ) } Q ( y ) \cdot \eta ^ { \ , q } ( | y - z _ 1 | ) \ , d \mathcal { H } ^ { n - 1 } \ , d t = \frac { \omega _ { n - 1 } } { I ^ { q - 1 } } < \infty \ , . \end{align*}


\begin{align*} [ ( \varphi + \psi ) ^ { \otimes p } ] = \sum ^ p _ { i = 0 } { p \choose i } [ \varphi ^ { \otimes i } \otimes \psi ^ { \otimes p - i } ] = [ \varphi ^ { \otimes p } ] + [ \psi ^ { \otimes p } ] \ ; . \end{align*}


\begin{align*} O ( R \log ^ \ast n + R ( 2 \Delta ) ^ { 4 R } + R q ) \ , = \ , O ( R \log ^ \ast n + R ( 2 \Delta ) ^ { 4 R } + R ( 2 \Delta ) ^ { 2 R } ) . \end{align*}


\begin{align*} \Pi _ { \psi } ( s ( - 1 ) ) f ( [ \epsilon , y ] ) = f ( [ - \epsilon , y ] ) , \end{align*}


\begin{align*} \mathcal { A } _ \chi : = \left ( \left ( \widehat { \Pi } _ \chi ^ { \rm s t i f f } \right ) ^ * \right ) ^ { - 1 } \widehat { \Lambda } _ \chi ^ { \rm s t i f f } \left ( \widehat { \Pi } _ \chi ^ { \rm s t i f f } \right ) ^ { - 1 } , \mathcal { B } _ \chi ( z ) : = z I + \left ( \left ( \widehat { \Pi } _ \chi ^ { \rm s t i f f } \right ) ^ * \right ) ^ { - 1 } \widehat { M } _ \chi ^ { \rm s o f t } ( z ) \left ( \widehat { \Pi } _ \chi ^ { \rm s t i f f } \right ) ^ { - 1 } , \end{align*}


\begin{align*} \Lambda - \Delta = \{ \lambda - \delta \mid \lambda \in \Lambda \mbox { a n d } \delta \in \Delta \} . \end{align*}


\begin{align*} \lim _ { \alpha \to 0 } s _ \alpha [ Y ] ( x ) = \sigma [ Y ] ( x ) \quad \lim _ { \alpha \to 0 } t _ \alpha [ Y ] ( x ) = \sigma [ Y ] ( x ) \end{align*}


\begin{align*} \| u _ \varepsilon - \mathfrak { U } _ { M } ^ { ( \varepsilon ) } \| _ { C ( \overline { \Omega } _ \varepsilon \times [ 0 , T ] ) } : = \max _ { \overline { \Omega _ \varepsilon } \times [ 0 , T ] } | u _ \varepsilon - \mathfrak { U } _ { M } ^ { ( \varepsilon ) } | \le C _ M \ , \varepsilon ^ { \gamma ( M + \lfloor \alpha \rfloor - 1 ) } . \end{align*}


\begin{align*} Z ^ 0 _ n ( L ) = \{ ( x , y ) \in X \times X \ , | \ , \exists \ , C \in | L | \mbox { s m o o t h s . t . } x , y \in C , \ , n [ x - y ] = 0 \in J C \} , \end{align*}


\begin{align*} Q ( r _ 1 , \theta ) \approx & \frac { \pi ^ 2 \lambda R _ S ( R _ { m a x } - r _ 1 ) } { N R _ E } \sum _ { n = 1 } ^ N \sqrt { 1 - \phi _ n ^ 2 } c _ n \\ & \times \left ( 1 - \prod _ { m = 1 } ^ { M } { \left ( 1 + \frac { m \eta \theta r _ 1 ^ { \alpha } } { M c _ n ^ { \alpha } } \right ) ^ { - M b _ m } } \right ) \end{align*}


\begin{align*} \mathcal J = ( d f _ j \mid j \in J ) . \end{align*}


\begin{align*} f _ { \tilde { \textbf { Q } } } = g ( x ) ^ r h ( x ) \mbox { f o r s o m e } h \in K [ x ] , r \in \N \mbox { a n d } ( g , h ) = 1 . \end{align*}


\begin{align*} \bar { \mathcal I } = ( \mathcal I _ 1 + ( g ( X _ 0 ) ) ) K [ \textbf { X } ] . \end{align*}


\begin{align*} \Gamma _ { \varepsilon , K } ( u ) = \frac { 1 } { 2 } \int _ { \R ^ N } ( | \nabla u | ^ 2 + \widetilde V ( \varepsilon x ) u ^ 2 ) - \int _ { \mathbb R ^ N } F _ K ( u ) + \Phi _ { \varepsilon } ( u ) + \Psi _ \varepsilon ( u ) , u \in Z ( \frac { \rho _ 1 } { 3 2 } , 3 \delta _ 0 ) . \end{align*}


\begin{align*} \Delta \chi _ T ( t ) = \chi _ { T _ 1 } ( t ) \cdots \chi _ { T _ k } ( t ) \end{align*}


\begin{align*} \mathbb { P } _ { \lambda } ^ { \vee } = \mathbb { I } _ { \lambda } , \Delta _ { \lambda } ^ { \vee } = \nabla _ { \lambda } , ( \overline { \Delta } _ { \lambda } ) ^ { \vee } = \overline { \nabla } _ { \lambda } . \end{align*}


\begin{align*} \overline { v } _ \lambda ( t ) = v _ 0 ( t ) + \int _ 0 ^ t S ( t - s ) F _ \lambda ( s ) \overline { v } _ \lambda ( s ) \ , d s + \int _ 0 ^ t S ( t - s ) \sigma ' ( u ( s ) ) \overline { v } _ \lambda ( s ) B \ , d W ( s ) , \end{align*}


\begin{align*} \mu _ k = \left ( \underbrace { \frac { k } { n } , \dots , \frac { k } { n } } _ { n - k } , \underbrace { \frac { k - n } { n } , \dots , \frac { k - n } { n } } _ { k } \right ) \end{align*}


\begin{align*} p _ 1 ( X _ 1 , \dots , X _ n ) & = 0 \rlap { , } \\ \vdots \quad \quad \ ; \ ; \\ p _ m ( X _ 1 , \dots , X _ n ) & = 0 \rlap { , } \end{align*}


\begin{align*} L _ { \Sigma } \phi = \lambda \phi \mbox { i n } \partial \Sigma \mbox { a n d } \nabla _ n \phi = h ( \nu , \nu ) \phi \mbox { o n } \partial \Sigma . \end{align*}


\begin{align*} \sup _ { \gamma \in \mathcal { P } ( \mathcal { S } ) } \left \{ \int _ { \mathcal { S } } g \ , d \gamma - \lambda _ 1 \boldsymbol { K } _ 1 ( \mu _ { 1 3 } , \gamma _ { 1 3 } ) - \lambda _ 2 \boldsymbol { K } _ 2 ( \mu _ { 2 3 } , \gamma _ { 2 3 } ) : \boldsymbol { K } _ \ell ( \mu _ { \ell 3 } , \gamma _ { \ell 3 } ) < \infty \ell = 1 , 2 \right \} , \end{align*}


\begin{align*} \cos \alpha _ { 1 0 2 } = \cos \alpha _ { 3 0 4 } , \end{align*}


\begin{align*} \int \nu _ V ^ { \widetilde { W } ^ { ( u ) } , \widetilde \theta ^ { ( u ) } , \widetilde \eta ^ { ( u ) } } ( d \widetilde { \beta } ^ { ( u ) } ) = 1 . \end{align*}


\begin{align*} \langle f , \zeta \rangle = F ( [ \zeta ] ) \forall \ \zeta \in \mathcal { X } . \end{align*}


\begin{align*} \lambda _ 1 : = \inf \{ \gamma _ 1 ( a ) : ~ ~ a \in \mathcal { A } ( \Omega ) \} . \end{align*}


\begin{align*} \eta ( B ^ { \prime } ) = \eta ( B ) + 1 \ge T + \max _ { 1 \le i \le s - 1 } a _ i . \end{align*}


\begin{align*} \begin{cases} \tilde { P } ( ( - \Delta ) ^ { \tilde { s } } ) u = 0 & \Omega , \\ \mathcal { L } u = 0 & \Omega ^ c , \\ u = 0 & W , \end{cases} \end{align*}


\begin{align*} \alpha _ r ( u ( s , t ) ) u ( r , s t ) = u ( r , s ) u ( r s , t ) , r , s , t \in G . \end{align*}


\begin{align*} \sigma \left ( b \right ) : = \left \{ g _ { i } ( x ) \leq b _ { i } , \ ; i = 1 , \ldots , m \right \} , \end{align*}


\begin{align*} \mathrm { d i v } \Big ( V \ , \mathrm { d i v } V \Big ) = \big ( \mathrm { d i v } V \big ) ^ 2 + V \cdot \nabla ( \mathrm { d i v } V ) = \big ( \mathrm { d i v } V \big ) ^ 2 + V ^ i \ , \partial _ i \partial _ j V ^ j . \end{align*}


\begin{align*} \Big \langle V + \phi - i ^ * [ f _ \epsilon ( V + \phi ) ] , P \psi ^ h _ { \mu _ { j _ \epsilon } , \xi _ { j _ \epsilon } } \Big \rangle = 0 , \quad \mbox { f o r } \ \ h = 0 , \cdots , n , \end{align*}


\begin{align*} \langle S _ N ( \ell ) \rangle = \int _ 0 ^ 1 S _ N ( \ell ) \ , d x _ 0 = \sum _ { j = 1 } ^ N \sum _ { n \in \mathbb { Z } } \int _ 0 ^ 1 \chi \left ( \frac { x _ j - x _ 0 + n } { \ell } \right ) \ , d x _ 0 = L . \end{align*}


\begin{align*} \gamma _ 1 ^ \star = \left ( \frac { \eta } { \eta _ 0 } \right ) \gamma _ 1 ^ { \eta _ 0 , \delta } + \left ( \frac { \eta _ 0 - \eta } { \eta _ 0 } \right ) \mu _ 1 . \end{align*}


\begin{align*} \mathcal { B } ^ H = \nabla _ T \big ( \nabla _ \xi H ( \nu ) \big ) = \nabla _ \xi ^ 2 H ( \nu ) \ , \nabla _ T \nu . \end{align*}


\begin{align*} \mathbb { K } : = \mathbb { E } - \mathbb { I } , \end{align*}


\begin{align*} P _ { n } ( \lambda ) = & \left ( \lambda - \frac { 1 } { 2 } \right ) ^ n \left [ \left ( ( \lambda - 3 ) ^ 2 - 4 \right ) \left ( \lambda - \frac { n + 4 } { 2 } \right ) \right ] - 2 \left ( \lambda - \frac { 1 } { 2 } \right ) ^ n ( \lambda - 5 ) \\ & - \frac { n } { 4 } \left ( \lambda - \frac { 1 } { 2 } \right ) ^ { n - 1 } \left [ ( \lambda - 3 ) ^ 2 - 4 \right ] \end{align*}


\begin{align*} \begin{bmatrix} x _ { 1 , k + 1 } \\ x _ { 2 , k + 1 } \end{bmatrix} = \hat A \begin{bmatrix} x _ { 1 , k } \\ x _ { 2 , k } \end{bmatrix} , \| \hat A z \| _ { \hat { \mathcal { X } } } \leq \| z \| _ { \hat { \mathcal { X } } } , \quad \| z \| _ { \hat { \mathcal { X } } } : = \left \| \begin{bmatrix} \mathcal { X } _ 1 & 0 \\ 0 & \mathcal { X } _ 2 \end{bmatrix} z \right \| . \end{align*}


\begin{align*} \phi = \phi _ { e } + \sum _ { m = 1 } ^ { \infty } D _ { q _ m } \Delta x ^ { q _ m } , \end{align*}


\begin{align*} \omega ( \xi ) = \lim _ \lambda \omega ( e _ \lambda ) = 1 \end{align*}


\begin{align*} \gamma = \begin{bmatrix} a & b \\ 0 & d \end{bmatrix} . \end{align*}


\begin{align*} \varphi _ 1 ( p ) = \begin{cases} \dfrac 1 2 , & p = 0 , \\ - \dfrac { p + q \log ( q ) } { p \log ( q ) } , & 0 < p < 1 , \end{cases} \end{align*}


\begin{align*} B _ { c _ 0 } ( y ) \cap \bigcup _ { Q ' \in \mathcal { F } } Q ' = \emptyset \ , . \end{align*}


\begin{align*} \nu ( \{ n \} ) : = \phi ^ 2 ( n ) . \end{align*}


\begin{align*} J ^ F _ \lambda : = \frac { 1 } { 1 - \lambda F } , \end{align*}


\begin{align*} \lim _ { c _ 1 , c _ 2 \to \infty } F _ { \lambda , k , r } ( c _ 1 , c _ 2 ; b _ 1 , \dots , b _ \lambda ; a ; q ) = \lim _ { c _ 1 , c _ 2 \to \infty } G _ { \lambda , k , r } ( c _ 1 , c _ 2 ; b _ 1 , \dots , b _ \lambda ; a ; q ) . \end{align*}


\begin{align*} z _ t & : = M _ t ( \hat { x } , \bar { y } ) ( \hat { x } - \bar { y } ) , \\ M _ t ( \hat { x } , \bar { y } ) & : = A _ \gamma ( \hat { x } , \hat { x } ) + t \big ( A _ \gamma ( \hat { x } , \bar { y } ) - A _ \gamma ( \hat { x } , \hat { x } ) \big ) \end{align*}


\begin{align*} A ( X _ i / \pi ) _ { 2 1 } = \left | \left \{ ( x H , y H ) : \ y H \in \mathcal { S } \right \} \right | . \end{align*}


\begin{align*} \gamma : = \min \{ \alpha _ l \mid Q _ l \mbox { a p p e a r s i n t h e e x p a n s i o n o f s o m e } f _ j \} , \end{align*}


\begin{align*} r \geq \tilde { r } = e _ 0 - 2 z + c + | \mathcal { F } | , \end{align*}


\begin{align*} \Pi _ { \psi } ( \omega ) f ( [ \epsilon , y ] ) = \nu ( \epsilon , g ) \int _ { X ^ { \ast } } \psi ( \langle \epsilon y , y ^ { \ast } \rangle ) f ( [ \epsilon , y ^ { \ast } \omega ^ { - 1 } ] ) d y ^ { \ast } . \end{align*}


\begin{align*} \Phi ( t ) = \big ( \log ( Z _ i ( t ) ) , ( T _ i ( u ) + t ) \wedge T _ i ( v ) \big ) _ { i \in V } \in \R ^ V \times \R ^ V \end{align*}


\begin{align*} E \left [ \left ( S ^ k _ { j , + } ( t ) - \bar { S } ^ k _ { j , + , \Delta } ( t ) \right ) ^ 2 \right ] & = \nu \Delta \sum _ { \ell = 0 } ^ { \lfloor t / \Delta \rfloor } p ^ k _ { j , + } ( t - \ell \Delta ) E [ Z _ 0 ( \ell \Delta ) ] \\ & + 2 \sum _ { \ell _ 1 < \ell _ 2 } \left ( I _ 1 ( \ell _ 1 , \ell _ 2 ) - I _ 2 ( \ell _ 1 , \ell _ 2 ) \right ) + C _ 3 \Delta e ^ { 3 \lambda t } \\ & \leq C _ 1 \theta ^ k t e ^ { \lambda t } + C _ 2 \theta ^ k t ^ 2 e ^ { \lambda t } + C _ 3 \Delta e ^ { 3 \lambda t } . \end{align*}


\begin{align*} \Rightarrow \ A \geq \sqrt { q \cdot A } \geq \frac { 1 } { C ' \ , r } \frac { 1 } { \frac { p } { 1 - p } } = \frac { 1 } { C ' \ , r } \frac { 1 - p } { p } > \frac { 1 } { 2 \ , C ' \ , r } \frac { 1 } { p } \end{align*}


\begin{align*} ( \alpha \otimes \Delta ) \Delta ( l ) & = ( \Delta \otimes \alpha ) \Delta ( l ) \\ ( \alpha \otimes \Delta ) ( l _ 1 \otimes l _ 2 ) & = ( \Delta \otimes \alpha ) ( l _ 1 \otimes l _ 2 ) \\ \alpha ( l _ 1 ) \otimes l _ { 2 1 } \otimes l _ { 2 2 } & = l _ { 1 1 } \otimes l _ { 1 2 } \otimes \alpha ( l _ 2 ) \end{align*}


\begin{align*} D ^ { k - 1 } ( f ) = c F + D ^ { k - 1 } ( h ^ { \sigma } ) + d ^ { \sigma } g \end{align*}


\begin{align*} \rho _ { i } = 2 ^ { j _ { 0 } } \rho _ { z _ { i } } \quad i . \end{align*}


\begin{align*} D ( U \otimes v _ 1 ) & = ( - \sqrt { n } + 1 ) U \otimes v _ 1 , \\ D ( U \otimes v _ 2 ) & = ( - \sqrt { n } + 1 ) U \otimes v _ 2 , \\ D ( U \otimes u _ i ) & = ( - \sqrt { n } + 1 ) ( U \otimes u _ i ) , \mbox { f o r a n y } 2 \leq i \leq n \\ D ( V \otimes v _ 1 ) & = - V \otimes v _ 1 , \\ D ( V \otimes v _ 2 ) & = - V \otimes v _ 2 , \\ D ( V \otimes u _ i ) & = - ( V \otimes u _ i ) , \mbox { f o r a n y } 2 \leq i \leq n . \end{align*}


\begin{align*} x _ i = \left ( A \prod _ { j = 1 } ^ n u _ j ^ { \alpha ' _ j } \right ) ^ { z _ i } \left ( B \prod _ { j = 1 } ^ n u _ j ^ { \gamma ' _ j } \right ) ^ { t _ i } w ^ { - z _ i - t _ i } \cdot u _ i , i = 1 , \dots , n , \end{align*}


\begin{align*} \left [ \mathcal { B } ( z ) \right ] _ { i j } : = z \delta _ { i j } + z ^ 2 \sum _ { k = 1 } ^ { \infty } \frac { [ \overline { \varphi _ k } ] _ i [ \overline { \varphi _ k } ] _ j } { \eta _ k - z } , i , j \in { 1 , 2 , 3 } , \end{align*}


\begin{align*} & 0 = \min _ { 0 \le s \le t } \ , ( R ^ { \overline { X } } _ s - \widetilde { \Lambda } _ { s } ) = \overline { X } + \min _ { 0 \le s \le t } \bigg ( \int _ 0 ^ s \frac { 1 } { R ^ { \overline { X } } _ r } \ , \mathrm { d } r + B _ s - \widetilde { \Lambda } _ { s } \bigg ) \ ; \Longrightarrow \ ; \overline { X } \le \max _ { 0 \le s \le t } \ , ( - B _ s + \widetilde { \Lambda } _ { s } ) , \\ & 0 = \min _ { 0 \le s \le t } \ , ( R ^ { \underline { X } } _ s - \Lambda _ { s } ) . \end{align*}


\begin{align*} f ( k , t , n ) = \frac { ( n ) _ k } { n ^ t } \frac { 1 } { k ! } \Delta ^ { k } \left [ x ^ t \right ] _ { x = 0 } ^ { } . \end{align*}


\begin{align*} | \eta _ \epsilon | ^ 2 _ 2 = O ( \epsilon ^ 2 | \ln \epsilon | ) \ \ N = 4 , | \eta _ \epsilon | ^ 2 _ 2 = O ( \epsilon ) \ \ N = 3 , \end{align*}


\begin{align*} \mathcal { I } ^ \star ( \lambda ) = \sup _ { \gamma \in \Pi \left ( \mu _ 1 , \mu _ 2 \right ) } \int _ { \mathcal { V } } f _ \lambda ( v ) \ , d \gamma ( v ) , \end{align*}


\begin{align*} \sum _ { i = 1 } ^ n \int _ 0 ^ t \frac { \tilde { H } _ { k 1 } ^ * ( u , t ) } { \sqrt { n } \ ! \cdot \ ! S ^ { ( 0 ) , * } ( \boldsymbol { \hat { \beta } _ k } , u ) } M _ { k i } ^ * ( d u ) \end{align*}


\begin{align*} \varphi \colon B \rightarrow [ X ] _ 2 , \varphi ( b ) = b \ast - \end{align*}


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


\begin{align*} Y ^ N _ { t } = \frac { 1 } { \sqrt { N } } \sum _ { x \in \mathbb { Z } } \big ( \eta _ t ( x , \sigma ) - \rho \big ) \delta _ { ( \tfrac { x } { N } , \sigma ) } . \end{align*}


\begin{align*} \vec { u } ( A _ { 0 } , A _ { i } ) \cdot \vec { u } ( A _ { 0 } , A _ { j } ) = \cos \alpha _ { i 0 j } , \end{align*}


\begin{align*} [ \widehat { P } ] _ { i , j } ^ { } = \hat { p } _ { i , j } ^ { } = \begin{dcases} \frac { n - i } { n } & \mbox { i f } \ , j = i \neq n ; \\ \frac { n } { n } & \mbox { i f } \ , j = i = n ; \\ \frac { i } { n } & \mbox { i f } \ , j = i + 1 ; \\ 0 & \mbox { o t h e r w i s e } ; \end{dcases} \end{align*}


\begin{align*} \min \{ p , q \} > \frac { 6 - \mu } { 2 } , N = 3 , \end{align*}


\begin{align*} \| u _ n \| _ { C ^ { 2 , \alpha } ( \overline { \mathcal { O } } } & \leq C \big ( \| u _ n \| _ { L ^ \infty ( \R ^ N ) } + \| g \| _ { C ^ \gamma ( \overline { O } _ { \frac { 7 \rho } { 8 } } \times \R } \big ) \big ( 1 + \| g \| _ { C ^ \gamma ( \overline { O } _ { \frac { 7 \rho } { 8 } } \times \R } \big ) \\ & \leq C ( | \eta | + \| c \| _ { C ^ \gamma ( \R ^ N ) } ) ( 1 + \| c \| _ { C ^ \gamma ( \R ^ N ) } ) = : \vartheta \end{align*}


\begin{align*} \left \{ Q _ { 2 ^ { j _ { 0 } } \rho _ { z _ { i } } } ( z _ { i } ) \right \} _ { i \in \mathbb { N } } D _ { \kappa \lambda } \subset \bigcup \limits _ { i = 1 } ^ { \infty } Q _ { 5 \times 2 ^ { j _ { 0 } } \rho _ { z _ { i } } } ( z _ { i } ) . \end{align*}


\begin{align*} \begin{cases} ( \Phi ( \xi ) - \Phi ( \xi ' ) ) ( \xi - \xi ' ) \geq L ^ { - 1 } | \xi - \xi ' | ^ { 2 } , \\ | \Phi ( \xi ) - \Phi ( \xi ' ) | \leq L | \xi - \xi ' | \quad \xi , \xi ' \in \mathbb { R } . \end{cases} \end{align*}


\begin{align*} W ^ - = \begin{bmatrix} U _ 1 ^ 2 + U _ n ^ 2 \\ U _ n U _ { \tau } \end{bmatrix} , \Lambda ^ - = \begin{bmatrix} \frac { 1 } { 2 U _ n \sqrt { U _ 1 } } & 0 \\ 0 & \frac { 1 } { 2 U _ n \sqrt { U _ 1 } } \end{bmatrix} , W ^ + = U _ 1 ^ 2 , \Lambda ^ + = - \frac { 1 } { 2 U _ n \sqrt { U _ 1 } } . \end{align*}


\begin{align*} \lim _ { t \to \infty } e ^ { - \lambda t } \hat { S } _ { j , + } ( t ) = \nu Y \int _ 0 ^ \infty e ^ { - \lambda s } \left ( \sum _ { k = 1 } ^ \infty p _ { j , + } ^ k ( s ) \right ) d s . \end{align*}


\begin{align*} \bigcup _ { q _ { 1 } , \cdots , q _ { m } } \prod _ { j = 1 } ^ { m } A _ { p _ { j } , q _ { j } } & \subsetneq A _ { \vec { P } , q } , \end{align*}


\begin{align*} \mathcal { F } = \{ A \subset [ n ] : \ | A | = k \mbox { a n d } S \subset A \} . \end{align*}


\begin{align*} s ( X ) = \frac { 1 2 } { p - 1 } \sum _ { j \in J } w _ j ^ { - 1 } . \end{align*}


\begin{align*} \langle f , \mu \rangle = \int _ X f ( x ) \ , d \mu ( x ) \end{align*}


\begin{align*} \rho _ { \theta } ( \theta , f ( \theta ) ) + 2 { \rm R e } ( \rho _ w ( \theta , f ( \theta ) ) f ( \theta ) \frac { \partial g } { \partial \theta } ( \theta ) ) = 0 \end{align*}


\begin{align*} S _ { n \left ( i \right ) } = \left ( S _ { n \left ( i - 1 \right ) } \setminus k _ { e x \left ( i - 1 \right ) } \right ) \cup k _ { n e w \left ( i - 1 \right ) } \end{align*}


\begin{align*} Y : = \xi \left ( Q / P \times _ { \alpha } X _ { 0 } \right ) , \end{align*}


\begin{align*} r _ 2 ^ { q ^ m } = x f _ 2 . \end{align*}


\begin{align*} v _ 1 ( x ) = v _ 0 ( x _ W ) , x \in L _ \R . \end{align*}


\begin{align*} \int _ { \R ^ N } { \phi } { H _ j } = \int _ { \R ^ N } { g } { Z _ j } . \end{align*}


\begin{align*} \begin{cases} - \Delta u = f ( u ) , \ \ u > 0 , & \ \ D , \\ u = 0 , \ \ | \nabla u | = c , & \ \ \partial D , \end{cases} \end{align*}


\begin{align*} S _ { \mathrm { m a i n } } = \sum _ { L = 1 } ^ { \infty } L \sum _ { U , \ , V < \frac { D } { L } } \tau ( U ) \tau ( V ) \ , x _ { U L } \bar { x } _ { V L } \sum _ { T | L } \frac { \mu ( T ) } { T } \sum _ { A = 1 } ^ { \infty } \frac { G _ k ( A ^ 2 T ^ 2 U V \slash p ) } { A } , \end{align*}


\begin{align*} \mathcal { I } _ { \mathrm { D } } ( \delta ) = \sup _ { \gamma \in \Sigma _ { \mathrm { D } } ( \delta ) } \int _ { \mathcal { V } } g d \gamma , \end{align*}


\begin{align*} \mathcal { S } = \left \{ \left ( A , \underline { A } \right ) ^ \pm : \ A \in \binom { [ n ] } { k } \mbox { a n d } n \in A \right \} . \end{align*}


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


\begin{align*} S ^ { - 1 } ( \sqrt { | \Lambda ^ - | } W ^ - - R \sqrt { \Lambda ^ + } W ^ + ) = G \sqrt { | \Lambda ^ - | } W ^ - = R \sqrt { \Lambda ^ + } W ^ + + S G \end{align*}


\begin{align*} & [ \alpha ( a ) , \alpha ( b ) , [ x , y , z ] ] = [ [ a , b , x ] , \alpha ( y ) , \alpha ( z ) ] + [ \alpha ( x ) , [ a , b , y ] , \alpha ( z ) ] + [ \alpha ( x ) , \alpha ( y ) , [ a , b , z ] ] , \end{align*}


\begin{align*} g \left ( h ( X , Y ) , H \right ) = g ( X , Y ) g ( H , H ) \end{align*}


\begin{align*} \rho = \frac { - 1 } { p } \alpha _ 1 \alpha _ 2 = \frac { 1 } { p } \widehat { \phi _ 1 } \widehat { \psi _ 1 } \widehat { \pi } \pi \phi _ 1 ^ { ( p ) } \widehat { \phi _ 2 ^ { ( p ) } } \psi _ 2 \phi _ 2 = \widehat { \phi _ 1 } \widehat { \psi _ 1 } \phi _ 1 ^ { ( p ) } \widehat { \phi _ 2 ^ { ( p ) } } \psi _ 2 \phi _ 2 . \end{align*}


\begin{align*} \textup { P L } _ { m , k } = \left \{ \begin{matrix} - \textup { D } - 3 5 \log _ { 1 0 } \left ( d _ { m , k } \right ) , \textup { i f } d _ { m , k } > d _ { 1 } & \\ - \textup { D } - 1 0 \log _ { 1 0 } \left ( d _ { 1 } ^ { 1 . 5 } d _ { m , k } ^ 2 \right ) , \textup { i f } d _ { 0 } < d _ { m , k } \leq d _ { 1 } & \\ - \textup { D } - 1 0 \log _ { 1 0 } \left ( d _ { 1 } ^ { 1 . 5 } d _ { 0 } ^ 2 \right ) , \textup { i f } d _ { m , k } \leq d _ { 0 } & \end{matrix} \right . \end{align*}


\begin{align*} \sum _ { n = 1 } ^ \infty \mathbb { P } \left ( A _ { n , k } ^ c \right ) \leq c ( n _ 0 ) + \sum _ { n = n _ 0 } ^ \infty \left \lceil e ^ { a ( n + 1 ) ^ { 3 / 2 } } \right \rceil e ^ { - n ^ { 1 4 / 9 } } < \infty , \end{align*}


\begin{align*} & \max \{ k \geq 0 : \exists _ { \sigma \in \mathrm { S d } ( \psi _ \R ) \cap M ' } \mathfrak { o } ( \sigma , k ) < r \} + 1 = \lceil m r + ( m - 2 ) \rceil \leq k _ r \\ & \max \{ k \geq 0 : \exists _ { \sigma \in \mathrm { S d } ( \psi _ \R ) \cap M ' } \widehat { \mathfrak { o } } ( \sigma , k ) < r \} + 1 = \lceil r + ( m - 2 ) \rceil \leq k _ r . \end{align*}


\begin{align*} X = G / Q \times _ { \beta } \left ( \Omega _ { 0 } \times _ { \sigma } X _ { 0 } \right ) \mbox { e q u i p p e d w i t h m e a s u r e } \eta _ { \rho } ^ { \mu } = \nu _ { Q } \times m _ { \Omega _ { 0 } } \times \rho , \end{align*}


\begin{align*} \mathcal { P } _ { \textrm { o u t } } & = 1 - \mathcal { P } _ 1 - \mathcal { P } _ 2 - \mathcal { P } _ 3 , \end{align*}


\begin{align*} B A = B \overline { A } + B \tilde { A } , \end{align*}


\begin{align*} \mathcal { N } _ { L , x _ 0 } = \bigcap _ { n = 1 } ^ { n _ 1 } \mathcal { N } _ { \Lambda _ { L _ { n - 1 } , L _ n ( x _ 0 ) } } \end{align*}


\begin{align*} \left \{ \sum _ { i = 1 } ^ n t _ i v _ i : t _ i \in D ( X ) \right \} \subseteq V \end{align*}


\begin{align*} \bigl [ S ( t ) \phi \bigr ] ( x ) = \int _ G K _ t ( x , y ) \phi ( y ) \ , d y \end{align*}


\begin{align*} | I m o ( 1 ) | = & \left | - I m \left [ \omega \left ( n \left ( r _ n , \frac { 1 } { f } \right ) + \frac { k - 1 } { 2 k } n \left ( r _ n , \frac { 1 } { A } \right ) \right ) \right ] \right | \\ & \geq \frac { k - 1 } { 2 k } | I m ( \omega ) | \\ & \geq \frac { k - 1 } { 2 k } \Delta , \end{align*}


\begin{align*} f ( \Lambda _ { t _ 0 } - x ) = 1 - \psi ( x ) , x \in ( 0 , \Lambda _ { t _ 0 } - \Lambda _ T \wedge \widetilde { \Lambda } _ T + \varepsilon _ 2 ] \end{align*}


\begin{align*} \mathring { \phi } ( x , t ) = \phi \left ( x + \gamma ( x , t ) + \frac { 1 } { N } \sigma ( t ) \right ) , \end{align*}


\begin{align*} u ^ { \ast } \left ( s \right ) = \hat { u } \left ( t , \mathbb { P } _ { \bar { X } _ { s } } ^ { W } \right ) , t \leq s < T \end{align*}


\begin{align*} \rho _ \ell ( y _ \ell , y _ \ell ^ \prime ) = \left ( y _ { \ell } - y _ { \ell } ^ { \prime } \right ) ^ { \top } V _ { \ell , Y Y } ^ { - 1 } \left ( y _ { \ell } - y _ { \ell } ^ { \prime } \right ) . \end{align*}


\begin{align*} \alpha _ { i , j } = \sum _ { v = 0 } ^ { N + q } D _ v \alpha _ { i - v - 1 , j } , D _ v \in \mathbb { C } , i \ge N + q + 1 , \end{align*}


\begin{align*} \norm { A _ n \phi - A \phi } ^ 2 = \norm { \Phi _ n \phi - \Phi \phi } ^ 2 = \int _ \Omega \abs { \phi } ^ 2 \abs { \Phi _ n - \Phi } ^ 2 \ , d x \leq \norm { \phi } _ \infty ^ 2 \int _ { \operatorname { s u p p } \phi } \abs { \Phi _ n - \Phi } ^ 2 \ , d x \end{align*}


\begin{align*} \sum _ { d = t + 1 } ^ \infty \lambda ^ d = \frac { \lambda ^ { t + 1 } } { 1 - \lambda } 0 \le \lambda < 1 . \end{align*}


\begin{align*} \oint \limits _ { \partial \Omega } U ^ T ( n _ i A _ i ) \\ \ U \\ \ d s = \oint \limits _ { \partial \Omega } \frac { 1 } { 2 } U ^ T ( ( n _ i A _ i ) + ( n _ i A _ i ) ^ T ) U \\ \ d s \geq 0 . \end{align*}


\begin{align*} G ( X , Y ) \ge \begin{cases} c | X - Y | ^ { 2 s - n } & { \mbox { i f } } n > 2 s , \\ c | \log | X - Y | | & { \mbox { i f } } n = 2 s . \end{cases} \end{align*}


\begin{align*} ( p ( u ) - p ( v ) ) ^ \top S ( p ( u ) - p ( v ) ) = 0 u v \in E ( G ) \ell ^ \top S \ell = 0 . \end{align*}


\begin{align*} & \overline X : = \sup \big \{ y \in [ r _ 0 - \varepsilon _ 2 , \Lambda _ 0 + \varepsilon _ 2 ] : \ , \max _ { [ 0 , t - s ] } \ , ( x + B _ { \cdot \wedge \tau ^ B _ { r _ 0 } } - \Lambda _ { s + \cdot } ) < 0 \big \} \vee ( r _ 0 - \varepsilon _ 2 ) , \\ & \underline X : = \inf \big \{ y \in [ r _ 0 - \varepsilon _ 2 , \Lambda _ 0 + \varepsilon _ 2 ] : \ , \max _ { [ 0 , t - s ] } \ , ( x + B _ { \cdot \wedge \tau ^ B _ { r _ 0 } } - \widetilde \Lambda _ { s + \cdot } ) \ge 0 \big \} . \end{align*}


\begin{align*} \partial _ t m = D \Delta m - \chi \nabla \cdot \left ( \frac { m } { 1 + \delta m } \nabla c \right ) + m \left ( 1 - m \right ) . \end{align*}


\begin{align*} \lim _ { k \rightarrow \infty } \frac { \log \norm { S ( k ) h } _ { \sup } } { k } = \max \big ( & \{ \lambda _ i - l \lambda _ 1 : 0 \leq l \leq n , 1 \leq | i | \leq g , d ( h , h _ { i , l } ) \neq 0 \} \\ & \cup \{ - l \lambda _ 1 : 0 \leq l \leq n , 1 \leq s < \gamma , d ( h , c _ { s , l } ) \neq 0 \} \big ) . \end{align*}


\begin{align*} z _ 1 & = \frac { e ^ s \cos \theta + i ( z \cos \theta - p _ \theta \sin \theta ) } { \sqrt { e ^ { 2 s } + z ^ 2 + p _ \theta ^ 2 } } , \\ z _ 2 & = \frac { e ^ s \sin \theta + i ( z \sin \theta + p _ \theta \cos \theta ) } { \sqrt { e ^ { 2 s } + z ^ 2 + p _ \theta ^ 2 } } , \\ e ^ u & = e ^ { 2 s } + z ^ 2 + p _ \theta ^ 2 . \end{align*}


\begin{align*} \Vert f \Vert _ { H ^ { s } } : = \left \| \langle \nabla \rangle ^ s f \right \| _ { L ^ 2 } = \left \Vert \langle \xi \rangle ^ { s } \hat { f } \right \Vert _ { L ^ { 2 } } , \end{align*}


\begin{align*} { \rm W F } ' ( B ) \cap \Delta ( T ^ * \mathcal { M } ) = \varnothing , \end{align*}


\begin{align*} \begin{cases} w _ { j i } = ( 1 + w ) T ^ \varphi _ { j i } - w \frac { S ^ \varphi } { m ( m - 1 ) } \delta _ { j i } \\ \varphi ^ a _ s w _ s = - ( 1 + w ) \varphi ^ a _ { t t } \ , . \end{cases} \end{align*}


\begin{align*} \begin{gathered} \left | y _ t ( x ) - y _ s ( x ' ) \right | \le C ( | x - x ' | + | t - s | ) , \\ \left | y _ t ^ { - 1 } ( x ) - y _ s ^ { - 1 } ( x ' ) \right | \le C ( | x - x ' | + | t - s | ) , \end{gathered} \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ N } f ( x ) d \omega ( x ) = 0 . \end{align*}


\begin{align*} A _ q ( u , v ) : = \left \langle \alpha _ 1 , ( ( - \Delta ) ^ { a _ 1 } u ) v \right \rangle + \left \langle \alpha _ 2 , ( ( - \Delta ) ^ { a _ 2 } u ) v \right \rangle + \cdots + \left \langle \alpha _ p , ( ( - \Delta ) ^ { a _ p } u ) v \right \rangle + ( q u , v ) _ \Omega . \end{align*}


\begin{align*} b ( \xi ) = | \xi - 1 | ^ { \delta _ 1 } | \xi + 1 | ^ { \delta _ { - 1 } } \widetilde { b } ( \xi ) \end{align*}


\begin{align*} \| u \| _ { \exp L ^ p } = \inf \left \{ \lambda > 0 : \int _ { \R ^ d } \left ( e ^ { \frac { | u ( x ) | ^ p } { \lambda ^ p } } - 1 \right ) d x \leq 1 \right \} . \end{align*}


\begin{align*} r \colon V _ e \to V _ e , r ( v ) = w v ) . \end{align*}


\begin{align*} \lim _ { t \uparrow \tau _ { \max } } \left \| \left ( \gamma , \gamma _ t , \sigma , \sigma _ t \right ) \right \| _ { H ^ 4 _ N \times H ^ 2 _ N \times \R \times \R } = \infty . \end{align*}


\begin{align*} \varphi _ { \lambda , d } ( x _ 1 , x _ 2 ) = \min _ { u \in \mathcal { X } : d ( u ) = d } \sum _ { 1 \leq \ell \leq 2 } \lambda _ \ell \| x _ \ell - u \| _ 2 , \end{align*}


\begin{align*} \mathcal { T } = \bigsqcup _ { i = 1 } ^ r \mathcal { T } _ n \end{align*}


\begin{align*} F _ { I } = \left \{ x \in \mathbb { R } ^ { m n } \ | \ A _ { I } x = b _ { I } \right \} \ \mbox { a n d } \ D _ { I } = \left \{ x \in X \ | \ A _ { I } x = b _ { I } \right \} = X \bigcap F _ { I } , \end{align*}


\begin{align*} x ^ 2 ( x ^ q y - x y ^ q ) + y ^ 2 ( y ^ q z - y z ^ q ) + ( z ^ 2 + k x ^ 2 ) ( z ^ q x - z x ^ q ) = 0 . \end{align*}


\begin{align*} ( T _ \varepsilon ^ { \vec { v } } ) _ { n , j - 1 } = - ( T _ \varepsilon ^ { \vec { v } } ) _ { 1 , j } \alpha _ 0 , \end{align*}


\begin{align*} y ^ { ( k ) } + A _ 1 y = 0 \end{align*}


\begin{align*} a \boxplus b = \max \{ a , b \} , a \odot b = a + b . \end{align*}


\begin{align*} \mathbf { P } \mathbf { G } ^ 1 ( \mathbb { Q } _ p ) = \mathbf { G } ^ 1 ( \mathbb { Q } _ p ) / Z ( \mathbf { G } ^ 1 ( \mathbb { Q } _ p ) ) , \end{align*}


\begin{align*} K _ { \omega , c } ( \lambda \Phi _ { \omega } ) & = 2 \lambda ^ 2 L ( \Phi _ { \omega } ) + 3 \lambda ^ 3 N ( \Phi _ { \omega } ) + 2 \lambda ^ 2 \omega Q ( \Phi _ { \omega } ) + 2 \lambda ^ 2 c P ( \Phi _ { \omega } ) \\ & = \lambda ^ 2 K _ { \omega , 0 } ( \Phi _ { \omega } ) + \lambda ^ 2 \{ 3 ( \lambda - 1 ) N ( \Phi _ { \omega } ) + 2 c P ( \Phi _ { \omega } ) \} \\ & = 2 \lambda ^ 2 \{ - 3 ( \lambda - 1 ) \mu _ { \omega , 0 } + c P ( \Phi _ { \omega } ) \} . \end{align*}


\begin{align*} \lambda _ k = \min \limits _ { P \in P _ k } \max \limits _ { 0 \neq g \in P } \frac { \langle g , M g \rangle } { \langle g , g \rangle } = \max \limits _ { P \in P _ { k - 1 } ^ { \perp } } \min \limits _ { 0 \neq g \in P \in P _ k } \frac { \langle g , M g \rangle } { \langle g , g \rangle } , \end{align*}


\begin{align*} \mathcal { R } _ \varepsilon c ( x ) : = \int _ { \mathbb { R } ^ n _ - } J _ \varepsilon ( | x - y | ) ( c ( x ) - \tilde { c } ( y ) ) \ : y \end{align*}


\begin{align*} \mathcal { P } ( \widetilde { X } _ { \eta } ) = \mathcal { P } ( \widetilde { X } _ { \eta ' } ) . \end{align*}


\begin{align*} \limsup _ { n \rightarrow \infty } n ^ { - 1 / 2 } \big \Vert \sum _ { i = 1 } ^ n d _ { i , N } \big \Vert _ 2 \leq \limsup _ { n \rightarrow \infty } n ^ { - 1 / 2 } \big \Vert \sum _ { i = 1 } ^ n Y _ { i , N } \big \Vert _ 2 \ , . \end{align*}


\begin{align*} M _ b ( \theta ) = \mathbb { E } _ { \Phi } \{ P _ s ^ b ( \theta ) \} \end{align*}


\begin{align*} { { \boldsymbol { X } } _ { i ; k | k - 1 } } = { \boldsymbol { f } } \left ( { { { \boldsymbol { \xi } } _ { i ; k - 1 | k - 1 } } } \right ) , \left ( { i = 1 , 2 , \cdots 2 n } \right ) . \end{align*}


\begin{align*} & = \frac { | h _ 1 | ^ 2 } { \sum _ { x _ i \in \Phi \backslash \{ x _ 1 \} \cap \mathcal { A } } | h _ i | ^ 2 } \overset { \Delta } { = } \frac { | h _ 1 | ^ 2 } { I } , \end{align*}


\begin{align*} \mathrm { V o l } ( \Omega ) = \mathrm { V o l } ( \overline { B } _ { \Omega } ) = \mathrm { A r e a ( \mathbb { S } ^ n ) } \int _ { 0 } ^ R \phi ^ n ( r ) \mathrm { d } r . \end{align*}


\begin{align*} \frac { v _ j } { \lambda } \sum _ { k \in [ n ] , k \neq j } ^ n ( t _ k t _ k ^ * ) = \lambda v _ j . \end{align*}


\begin{align*} d = \sum _ { E } \delta ( \mathbf { u } ( E ) ) , \end{align*}


\begin{align*} J _ H ( u ) = \int _ \Omega B ( H ( \nabla u ) ) \ , d x - \int _ \Omega f u \ , d x \end{align*}


\begin{align*} \mathcal { P } _ { \textrm { o u t } } = \begin{cases} 1 - \sum \limits _ { m = 1 } ^ M \mathcal { V } ^ { ( M , m ) } \xi ^ m J _ { + } ^ { ( m ) } & \textrm { i f ~ } \mu _ { \textrm { D } } ^ 2 \Gamma _ 0 \leq \rho - 1 \\ 1 - \sum \limits _ { m = 1 } ^ M \mathcal { V } ^ { ( M , m ) } \xi ^ m \big ( I _ + ^ { ( m ) } + I _ - ^ { ( m ) } \big ) & \textrm { i f ~ } \mu _ { \textrm { D } } ^ 2 \Gamma _ 0 > \rho - 1 \end{cases} \end{align*}


\begin{align*} g _ { i } ( x ) = \left \{ \begin{array} { l l } f _ k ( x ) , & \hbox { i f $ x \in \Omega _ i $ ; } \\ 0 , & \hbox { o t h e r w i s e . } \end{array} \right . \end{align*}


\begin{align*} \widetilde { B } ( \xi ) = \left \{ \begin{array} { r l } e ^ { - i \frac { \pi } { 4 } } B ( \xi ) \ , \overline { \xi } ^ { \frac { 1 } { 2 } } = e ^ { i \frac { 2 \theta - \pi } { 4 } } , & { \rm i f \ } 0 \le \theta \le \pi \\ e ^ { i \frac { 3 \pi } { 4 } } \ , \overline { \xi } ^ { \frac { 1 } { 2 } } = e ^ { i \frac { 3 \pi - 2 \theta } { 4 } } , & { \rm i f \ } \pi < \theta < 2 \pi . \end{array} \right . \end{align*}


\begin{align*} \mu ^ 2 = \pi \psi \pi \psi = \pi \pi \psi ^ { ( p ) } \psi = \pi _ E \epsilon \widehat { \psi } \psi = \epsilon d \pi _ E . \end{align*}


\begin{align*} \lim _ \lambda \| e ^ * _ \lambda ( b - \omega ( b ) I ) e _ \lambda \| = 0 , b \in B . \end{align*}


\begin{align*} \sum _ { n = 1 } ^ { \infty } \frac { p ^ { 2 } + \lambda _ { n } ^ { 2 } } { p \left ( p + \frac { 1 } { \pi } \right ) + \lambda _ { n } ^ { 2 } } \ , \left \{ \frac { 1 } { \left ( \lambda _ { n } ^ { 2 } + x ^ { 2 } \right ) ^ { s } } - \sum _ { m = 0 } ^ { N - 1 } \left ( \begin{array} { c } - s \\ m \end{array} \right ) \ , \frac { x ^ { 2 m } } { \lambda _ { n } ^ { 2 s + 2 m } } \right \} + \sum _ { m = 0 } ^ { N - 1 } \left ( \begin{array} { c } - s \\ m \end{array} \right ) x ^ { 2 m } \zeta _ { p } \left ( 2 s + 2 m \right ) . \end{align*}


\begin{align*} \sum _ { \delta = 2 } ^ n \varepsilon ^ \delta + \sum _ { \delta = 1 } ^ n s \delta \varepsilon ^ \delta & \leq \frac { \varepsilon ^ 2 } { 1 - \varepsilon } + \frac { s \varepsilon } { ( 1 - \varepsilon ) ^ 2 } \end{align*}


\begin{align*} \rho _ t ( 0 , y ) = t ^ { - ( d + 2 \beta ) / \alpha } \rho _ 1 ( 0 , t ^ { - 1 / \alpha } y ) , t > 0 , \ , y \in \Gamma , \end{align*}


\begin{align*} \widetilde { \varphi } : = \widehat { \varphi } \circ T _ M , \end{align*}


\begin{align*} \xi _ { 2 - k } : = 2 i y ^ { 2 - k } \overline { ( \dfrac { \partial } { \partial \overline { z } } ) } : H _ { 2 - k } ( \Gamma _ { 0 } ( N ) ) \rightarrow S _ { k } ( \Gamma _ { 0 } ( N ) ) . \end{align*}


\begin{align*} \mathcal { U } ^ { ( i ) } _ { p \alpha + k - 1 } ( x , t ; \varepsilon ) : = w _ { p \alpha + k - 1 } ^ { ( i ) } ( x _ i ) + u _ { p \alpha + k - 1 } ^ { ( i ) } \big ( x _ i , \tfrac { \overline { x } _ i } { \varepsilon } , t \big ) , \mathcal { P } ^ { ( i ) } _ { p \alpha + k - 1 } ( x _ i , t ; \varepsilon ) : = \chi _ \delta ^ { ( i ) } ( x _ i ) \ , \Pi _ { p \alpha + k } ^ { ( i ) } \big ( \tfrac { \ell _ i - x _ i } { \varepsilon } , t \big ) , \end{align*}


\begin{align*} e ^ h \mathrm { d i v } \Big ( e ^ { - h } ( V \cdot \nabla ) V \Big ) & = e ^ h \partial _ j \Big ( e ^ { - h } V ^ i \ , \partial _ i V ^ j \Big ) = \partial _ j V ^ i \ , \partial _ i V ^ j + V ^ i \ , \partial _ j \partial _ i V ^ j - \partial _ j h \ , V ^ i \ , \partial _ i V ^ j \\ & = \mathrm { t r } ( ( \nabla V ) ^ 2 ) + V ^ i \ , \partial _ j \partial _ i V ^ j - \partial _ j h \ , V ^ i \ , \partial _ i V ^ j . \end{align*}


\begin{align*} \overline j _ f = \left ( \frac { 1 } { 4 } , \frac { 1 } { 2 } , \frac { 1 } { 1 2 } , \frac { 1 } { 6 } \right ) . \end{align*}


\begin{align*} \underset { t \rightarrow \infty } { \lim } ( \log t ) ^ { 2 / d } \left ( \frac { \log N _ t } { t } - \beta \right ) = - c ( d , \nu ) \ : \ : \widehat { P } ^ \omega . \end{align*}


\begin{align*} g _ w ( y ) = \prod _ { i = 1 } ^ n h _ { b _ i } ( n y + i ) . \end{align*}


\begin{align*} \binom { t - k ( s ( t ) , t ) - \left \lfloor \frac { t - k ( s ( t ) , t ) } { s ( t ) } \right \rfloor s ( t ) } { 2 } & = \left ( 1 - \left ( \frac { 1 } { 2 } - \frac { 1 } { 4 i } + o ( 1 ) \right ) - ( i + o ( 1 ) ) \left ( \frac { 1 } { 2 i } + o ( 1 ) \right ) \right ) ^ 2 \frac { t ^ 2 } { 2 } \\ & = \left ( \frac { 1 } { 1 6 i ^ 2 } + o ( 1 ) \right ) \binom { t } { 2 } . \end{align*}


\begin{align*} w ( E ) : = \int _ E \ ! w \ , \mathrm { d } \mu , \end{align*}


\begin{align*} \sum _ { i = 1 } ^ n \alpha _ i t _ i = \sum _ { i = 1 } ^ n \beta _ i t _ i = \sum _ { i = 1 } ^ n \gamma _ i t _ i + 1 , t _ i \geq 0 , i = 1 , \dots , n , \end{align*}


\begin{align*} S _ { i , 0 } & = \sum _ { \substack { n \leq x _ i \\ P ( n ) \leq y _ 0 } } \frac { f ( n ) } { \sqrt { n } } , \\ S _ { i , j } & = \sum _ { \substack { y _ { j - 1 } < m \leq x _ i \\ p | m \Rightarrow p \in ( y _ { j - 1 } , y _ j ] } } \frac { f ( m ) } { \sqrt { m } } \sum _ { \substack { n \leq x _ i / m \\ P ( n ) \leq y _ { j - 1 } } } \frac { f ( n ) } { \sqrt { n } } . \end{align*}


\begin{align*} \mathcal { H } _ { \hbar , V } u _ { \xi } = \lambda _ { \xi } u _ { \xi } , \xi \in \mathcal { I } _ { \hbar } . \end{align*}


\begin{align*} \Theta _ { \mathrm { D } } ( \delta ) = \left [ \min _ { \gamma \in \Sigma _ { \mathrm { D } } ( \delta ) } \int _ { \mathcal { S } _ 1 \times \mathcal { S } _ 2 } g \ , d \gamma , \max _ { \gamma \in \Sigma _ { \mathrm { D } } ( \delta ) } \int _ { \mathcal { S } _ 1 \times \mathcal { S } _ 2 } g \ , d \gamma \right ] , \end{align*}


\begin{align*} S ^ { - 1 } r S = r ^ { - 1 } . \end{align*}


\begin{align*} M ( t ) : = \sum _ { j = 1 } ^ \infty S _ j ( t ) . \end{align*}


\begin{align*} \Lambda \vect g = \Gamma _ 1 \Pi \vect g \forall \vect g \in \mathcal { D } ( \Lambda ) , \Pi ^ * \vect f = \Gamma _ 1 \mathcal { A } _ 0 ^ { - 1 } \vect f \forall \vect f \in \mathcal { H } . \end{align*}


\begin{align*} \widehat { \sigma x } _ { \sigma z } = \begin{cases} \hat { x } _ z , & z ( 0 ) = 0 , \\ \sigma \hat { x } _ z , & z ( 0 ) = 1 , \end{cases} \end{align*}


\begin{align*} u _ 1 u _ 2 = e ^ { - i \eta \cdot x } + \varrho , \varrho = e ^ { - i \eta \cdot x } ( w _ 1 + w _ 2 + w _ 1 w _ 2 ) , \end{align*}


\begin{align*} L i _ 2 ( 0 ) = 0 , \end{align*}


\begin{align*} \{ \mathcal { D } _ { \eta ^ { k } } ( \hat { \eta } , \eta ^ { k } ; \lambda ^ { k } ) \} _ { k = 1 } ^ { + \infty } \ \ \mbox { i s \ a \ C a u c h y \ s e q u e n c e } , \end{align*}


\begin{align*} \left \lbrace \begin{array} { l c r } ( d d ^ c u ) ^ n = ( 1 - \lambda u ) ^ n f ^ n \omega ^ n & \textnormal { i n } & \Omega , \\ u \equiv 0 & \textnormal { o n } & \partial \Omega , \end{array} \right . \end{align*}


\begin{align*} F ( a , b , c , - r ^ 2 ) = \frac 1 { ( 1 + r ^ 2 ) ^ b } F \left ( c - a , b , c , \frac { r ^ 2 } { 1 + r ^ 2 } \right ) \textrm { f o r a l l } \ , \ ; r > 1 \ , . \end{align*}


\begin{align*} \omega ( b ) = \langle \pi ( b ) \xi , \xi \rangle , b \in B . \end{align*}


\begin{align*} \boldsymbol { K } _ 1 ( \gamma _ 1 , \gamma _ 1 ^ { \eta _ 0 , \delta } ) = \mathbb { E } \left [ c _ 1 ( S _ 1 , \widetilde { S } _ 1 ) \right ] \leq \eta _ 0 - \eta . \end{align*}


\begin{align*} | \varphi ( x ) | & \leq \frac { \norm { \varphi } _ { L ^ 1 } } { | I | } + p _ a ( \varphi ) \Big ( \frac { 1 } { a \min \{ x - l _ \alpha , r _ \alpha - x \} ^ { a } } + \frac { 2 ^ { a + 2 } } { a ( 1 - a ) | I _ \alpha | ^ { a } } \Big ) 0 < a < 1 , \\ | \varphi ( x ) | & \leq \frac { \norm { \varphi } _ { L ^ 1 } } { | I | } + p _ a ( \varphi ) \Big ( \log \frac { | I _ \alpha | } { 2 \min \{ x - l _ \alpha , r _ \alpha - x \} } + 2 \Big ) a = 0 . \end{align*}


\begin{align*} & \frac 1 { \omega ( B ) \ell ( B ) ^ \beta } \int _ B | g _ { 2 1 } ( y ) - ( g _ { 2 1 } ) _ B | d \omega ( y ) + \frac 1 { \omega ( B ) \ell ( B ) ^ \beta } \int _ B | g _ { \sigma _ j 1 } ( y ) - ( g _ { \sigma _ j 1 } ) _ B | d \omega ( y ) \\ & \le C \| b \| _ { \Lambda ^ \beta } \big ( M _ t ( R _ { { \rm D } , \ell } f _ 1 ) ( x ) + M _ t ( R _ { { \rm D } , \ell } f _ { \sigma _ j } ) ( x ) \big ) \qquad \mbox { f o r } j = 1 , 2 , \cdots , | G | - 1 . \end{align*}


\begin{align*} \sum _ { t \in ( a , b ) } \dim ( \sigma _ 0 ( t ) \cap \sigma ) = \sum _ { \lambda < 0 } \dim ( \sigma _ { \lambda } ( b ) \cap \sigma ) . \end{align*}


\begin{align*} \frac { u } { d ^ s } ( x _ 0 ) = 0 . \end{align*}


\begin{align*} \alpha _ { \tau , i } + \alpha _ { \tau , i + 2 } = 2 \alpha _ { \tau , i + 1 } \end{align*}


\begin{align*} \lim _ { n \to \infty } ~ \frac { \delta _ { A , C } ( n ) } { \delta _ { A , D } ( n ) } = \frac { 1 } { 2 } \left ( \alpha + \sqrt { \alpha ^ 2 - 4 } \right ) \end{align*}


\begin{align*} \int _ M d d ^ c \omega _ 0 ^ { n - k - 1 } \wedge \omega ^ k = \int _ M \omega _ 0 ^ { n - k - 1 } \wedge d d ^ c \omega ^ k = 0 , \end{align*}


\begin{align*} c ( z ) = \frac { c _ \alpha } { | z | ^ { 1 + \alpha } } , \textrm { ~ w h e r e ~ } c _ \alpha = \pi ^ { - 1 } \Gamma ( \alpha + 1 ) \sin \left ( \frac { \alpha \pi } { 2 } \right ) , \end{align*}


\begin{align*} \int _ { \mathcal { X } } \Psi ( \nu , \mu ) \left ( \mathrm { d } x \right ) = \int _ { \mathcal { X } } \Psi ( \nu , \mu ) \left ( x \right ) \mathrm { d } x = \frac { 1 } { Z ( \nu , \mu ) } \int _ { \mathcal { X } } \exp \left ( - \frac { 2 } { \sigma ^ 2 } \frac { \delta F } { \delta \nu } ( \nu , \mu , x ) - U ^ { \pi } \left ( x \right ) \right ) \mathrm { d } x = 1 , \end{align*}


\begin{align*} \frac { 1 } { t - s } ( f ( t ) - f ( s ) ) & = \frac { 1 } { t - s } \psi \left ( e ^ { s a * } e ^ { ( t - s ) a * } e ^ { ( t - s ) a } e ^ { s a } - e ^ { s a * } e ^ { s a } \right ) \\ & = \psi \left ( e ^ { s a * } \frac { 1 } { t - s } \left ( e ^ { ( t - s ) a * } e ^ { ( t - s ) a } - I \right ) e ^ { s a } \right ) . \end{align*}


\begin{align*} L w ( x ) = L \psi ( x ) - L u ^ - ( x ) \le - \frac { \alpha _ r } { r ^ { 2 s } } + C _ { \star } . \end{align*}


\begin{align*} g ' ( s ) = \rho \left ( 2 t s \cos ( \gamma + \beta ) + \frac { 1 - 2 s ^ 2 } { \sqrt { 1 - s ^ 2 } } \right ) \end{align*}


\begin{align*} x _ \tau = | { \rm A u t } ( \tau ) | \sum _ { \bar T = \tau } X _ T \end{align*}


\begin{align*} \mathcal P ^ \tau : = \{ ( u , s ) : u \notin \Lambda , s \in \Omega ^ \tau _ u \} \end{align*}


\begin{align*} \prod _ { \substack { ( a , b ) = 1 \\ a , b \geq 1 } } \left ( \frac { 1 } { 1 - y ^ a z ^ b } \right ) ^ { \frac { b ^ 2 } { a ^ 3 } } = \exp \left \{ \frac { z ( 1 + z ) L i _ 3 ( y ) } { ( 1 - z ) ^ 3 } \right \} . \end{align*}


\begin{align*} \Phi : = w - \widetilde { T } ( f ) \end{align*}


\begin{align*} \Sigma _ t = \psi ^ { - 1 } ( B \cap \{ x _ 3 = t \} ) , \end{align*}


\begin{align*} { \mathcal { C } } _ \beta : = \left \{ x \in \Omega { \mbox { s . t . } } \frac { x - x _ 0 } { | x - x _ 0 | } \cdot \bar \nu > c _ \beta \right \} , \qquad { \mbox { w h e r e } } \ , c _ \beta : = \cos \left ( \frac \pi 2 - \beta \right ) > 0 . \end{align*}


\begin{align*} \frac { d } { d t } P _ k ( U ) & = - \sum _ { j = 1 } ^ 3 { \rm R e } ( i \partial _ t u _ j , \partial _ k u _ j ) _ { L ^ 2 ( \R ^ d ) } \\ & = { \rm R e } ( \nabla \cdot u _ 3 , \partial _ k u _ 1 \cdot u _ 2 ) _ { L ^ 2 ( \R ^ d ) } + { \rm R e } ( \nabla \cdot \overline { u _ 3 } , \overline { u _ 1 } \cdot \partial _ k u _ 2 ) _ { L ^ 2 ( \R ^ d ) } - { \rm R e } ( \nabla ( u _ 1 \cdot \overline { u _ 2 } ) , \partial _ k u _ 3 ) _ { L ^ 2 ( \R ^ d ) } \\ & = 0 \end{align*}


\begin{align*} | \underline { \alpha } | \coloneqq \max _ { i = 1 , \dots , n } | \alpha _ i | \dd \underline { \alpha } \coloneqq \dd \alpha _ 1 \dots \dd \alpha _ n \end{align*}


\begin{align*} K _ r = \sigma _ \infty \widehat { P } ^ { - 5 } , \end{align*}


\begin{align*} \Sigma _ { \Pi } ( \delta _ 0 ) : = \left \{ \gamma \in \Pi ( \mu _ 1 , \mu _ 2 ) : \boldsymbol { K } _ c ( \gamma , \mu ) \le \delta _ 0 \right \} , \end{align*}


\begin{align*} \mathcal { H } _ { V } u ( x ) : = \left ( - \mathcal { L } + V \right ) u ( x ) , x \in \mathbb { R } ^ { n } , \end{align*}


\begin{align*} \left \{ A ^ \pm : \ A \in \binom { [ n ] } { k } \mbox { a n d } x \in A \right \} , \end{align*}


\begin{align*} \lim _ { n \to \infty } \frac { \det T _ n ( \phi ) } { G [ \phi _ 0 ] ^ n \prod _ { k = 1 } ^ R \det T _ n ( \phi _ k ) } = E \end{align*}


\begin{align*} S = \begin{pmatrix} 0 & - 1 \\ 1 & 0 \end{pmatrix} , T = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} . \end{align*}


\begin{align*} I = \int _ M J \theta \wedge d \Omega ^ { n - 1 } = ( n - 1 ) \int _ M J \theta \wedge \theta \wedge \Omega ^ { n - 1 } = - 2 ( n - 1 ) \int _ M \sqrt { - 1 } \theta ^ { ( 1 , 0 ) } \wedge \overline { \theta ^ { ( 1 , 0 ) } } \wedge \Omega ^ { n - 1 } . \end{align*}


\begin{align*} \int _ { \rho } ^ { 2 \rho } \frac { \exp \left ( - \frac { u ^ 2 } { 4 t ^ 2 } \right ) } { \sqrt { \cosh u - \cosh \rho } } \d u \leq \exp \left ( - \frac { \rho ^ 2 } { 4 t ^ 2 } \right ) \int _ { \rho } ^ { 2 \rho } \frac { \d u } { \sqrt { ( u - \rho ) \rho } } = 2 \exp \left ( - \frac { \rho ^ 2 } { 4 t ^ 2 } \right ) . \end{align*}


\begin{align*} F = F ( Q _ { 2 \rho } ) : = \begin{cases*} \mathbf { P } ^ f _ { p ' _ * } ( Q _ r ( x _ 0 , t _ 0 ) ) \mbox { i f } p < 2 , \\ \mathbf { M } ^ f _ p ( Q _ r ( x _ 0 , t _ 0 ) ) \mbox { i f } p \geq 2 . \end{cases*} \end{align*}


\begin{align*} a _ { i , k } = 0 k = i - q + 1 , \dots , i + r - 1 \end{align*}


\begin{align*} S ( I ) = \bigsqcup _ { J \ge I } W ( J ) . \end{align*}


\begin{align*} \begin{aligned} j _ 1 & \colon X \hookrightarrow X \times X : x \mapsto ( x , x _ 0 ) \\ j _ 2 & \colon X \hookrightarrow X \times X : x \mapsto ( x _ 0 , x ) \end{aligned} \end{align*}


\begin{align*} \mathrm { t r } \big ( ( X Y ) ^ 2 \big ) = \mathrm { t r } ( X Y X Y ) \geq \l _ { { \rm m i n } } \ , \mathrm { t r } ( Y X Y ) = \l _ { { \rm m i n } } \ , \mathrm { t r } ( X Y ^ 2 ) \geq \l _ { { \rm m i n } } ^ 2 \ , \mathrm { t r } ( Y ^ 2 ) = \l _ { { \rm m i n } } ^ 2 \ , | Y | ^ 2 , \end{align*}


\begin{align*} { \rm E n t S p } ( S , \mu ) = \bigcup _ { I \subseteq \{ 1 , \ldots , d - 1 \} } \left [ h _ { \mu } \left ( G / P _ { I } , \nu _ { I } \right ) , h _ { \mu } \left ( G / P _ { \mathsf { f } ( I ) } , \nu _ { \mathsf { f } ( I ) } \right ) \right ] , \end{align*}


\begin{align*} Z ( I \otimes T ^ * T ) Z ^ * = T ^ * Y Y ^ * T = T ^ * T . \end{align*}


\begin{align*} & 2 \underline { \widehat { r } } < | x | + \underline { r } = r _ * + \widehat { r } _ * . \end{align*}


\begin{align*} \mathcal { N } _ { t } = \sigma \{ N ( s ) , \ s \le t \} . \end{align*}


\begin{align*} \eta ( t , x ) = A ( t ) x , \ \ A \in ^ + ( 3 , \mathbb R ) , \end{align*}


\begin{align*} \int _ { y } ^ { z } \log \left ( 1 - t ^ { - 1 } \right ) \frac { d t } { \log t } = - \log \frac { \log z } { \log y } + \int _ { y } ^ { z } \left ( t ^ { - 1 } + \log \left ( 1 - t ^ { - 1 } \right ) \right ) \frac { d t } { \log t } . \end{align*}


\begin{align*} d = \left [ \frac { 3 } { 2 } + \frac { \left ( 1 + \beta \right ) \left ( T / 2 - \varepsilon \right ) ^ { 2 } } { \left ( \varepsilon \left ( T - \varepsilon \right ) \right ) } \right ] b ^ { 2 } . \end{align*}


\begin{align*} ( f * \psi ) ( x ) = \int _ { G _ { \tau ( x ) } } f ( \gamma ^ { - 1 } ) g ( \gamma \cdot x ) \ d \lambda _ { \tau ( x ) } . \end{align*}


\begin{align*} i \hbar \frac { \partial \psi } { \partial t } = - \frac { \hbar ^ 2 } { 2 m } \Delta \psi + V ( x ) \psi - g ( | \psi | ) \psi = 0 . \end{align*}


\begin{align*} 0 < a _ 1 < C _ \epsilon h ^ { \frac { 2 } { 3 } - \epsilon } , \ \ C _ \epsilon h ^ { \frac { 1 } { 2 } + \epsilon } < a _ i < C _ \epsilon h ^ { \frac { 1 } { 3 } - \epsilon } \ \ i = 2 , \cdots , n . \end{align*}


\begin{align*} \Omega : = \left \{ \begin{array} { c } \forall k \ge \log n \Gamma \subseteq [ - n ^ 2 , n ^ 2 ] ^ d \\ k | \{ e \in \Gamma : t _ e \in B _ { \delta _ 0 } \} | \ge k / 2 \end{array} \right \} . \end{align*}


\begin{align*} h _ b = \left ( \frac { 1 } { 2 \pi } \langle h _ b , P _ r ( \theta - \cdot ) \rangle \right ) _ b \end{align*}


\begin{align*} \| U ( a ) \| _ 2 ^ 2 = \sum _ { i = 1 } ^ n \langle \omega _ i , a \rangle ^ 4 . \end{align*}


\begin{align*} \mathcal { P } _ { \alpha } ^ { - { 1 } / { 2 } } \mathcal { Y } \mathcal { T } \mathcal { P } _ { \alpha } ^ { - { 1 } / { 2 } } = \mathcal { Q } _ { \alpha } - \mathcal { E } _ \alpha , \end{align*}


\begin{align*} \begin{pmatrix} 0 & 0 & \dots & 0 & 0 & t + \varepsilon \\ 1 & 0 & \dots & 0 & 0 & 0 \\ 0 & 1 & \dots & 0 & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\ 0 & 0 & \dots & 1 & 0 & 0 \\ 0 & 0 & \dots & 0 & 1 & 0 \end{pmatrix} . \end{align*}


\begin{align*} ( f , g ) & = S _ { \chi \oplus \xi } ^ { - \frac { 1 } { 2 } } S _ { \chi \oplus \xi } S _ { \chi \oplus \xi } ^ { - \frac { 1 } { 2 } } ( f , g ) \\ & = \int _ { \Theta } v ^ { 2 } ( w ) S _ { \chi \oplus \xi } ^ { - \frac { 1 } { 2 } } \pi _ { F ( w ) \oplus G ( w ) } ( \chi _ { w } \oplus \xi _ { w } ) ^ { \ast } ( \chi _ { w } \oplus \xi _ { w } ) \pi _ { F ( w ) \oplus G ( w ) } S _ { \chi \oplus \xi } ^ { - \frac { 1 } { 2 } } ( f , g ) d \mu ( w ) . \end{align*}


\begin{align*} \tau _ 1 ( \theta ) ( \rho q _ t + \rho u \cdot q _ x ) + q + \kappa ( \theta ) \theta _ x = 0 , \end{align*}


\begin{align*} ( \underline { X } , \underline { A } ) ^ { \emptyset } = ( \underline { X } , \underline { A } ) ^ { \emptyset , c } = \prod ^ m _ { j = 1 } A _ j \qquad \mbox { a n d } ( \underline { Y } , \underline { B } ) ^ { \emptyset } = ( \underline { Y } , \underline { B } ) ^ { \emptyset , c } = \prod ^ m _ { j = 1 } B _ j , \end{align*}


\begin{align*} \Delta ^ r Y _ F = \sum _ { u \in C ( F ) \cap [ r ] ^ n } Y _ { F _ { ( 1 ) } } \otimes Y _ { F _ { ( 2 ) } } \cdots \otimes Y _ { F _ { ( r ) } } \end{align*}


\begin{align*} S _ { \mathrm { m a i n } } = \sum _ { L = 1 } ^ { \infty } L \sum _ { U , \ , V < \frac { D } { L } } \tau ( U ) \tau ( V ) \ , x _ { U L } \bar { x } _ { V L } \sum _ { T | L } \frac { \mu ( T ) } { T } \sum _ { A = 1 } ^ { \infty } \frac { G _ k ( A ^ 2 T ^ 2 U V \slash p ) } { A } , \end{align*}


\begin{align*} & \forall \ , x : n \forall \ , y : n \forall \ , z : m ( ( ( x + _ n y ) \ast _ { n m } z ) = ( x \ast _ { m n } z + _ { n + m } y \ast _ { n m } z ) ) \\ & \forall \ , x : n \forall \ , y : m \forall \ , z : m ( ( x \ast _ { m n } ( y + _ m z ) ) = ( x \ast _ { n m } y + _ { n + m } x \ast _ { n m } z ) ) \\ & \forall \ , x : n \forall \ , y : m ( h _ { n + m } ( x \ast _ { n m } y ) = h _ n ( x ) \ast _ { n m } h _ m ( y ) ) \end{align*}


\begin{align*} \begin{aligned} \phi _ \lambda ( v , s ^ { \prime } ) & : = f ( s ^ { \prime } ) - \lambda _ 1 c _ 1 ( s _ 1 , s _ 1 ^ { \prime } ) - \lambda _ 2 c _ 2 ( s _ 2 , s _ 2 ^ { \prime } ) , \end{aligned} \end{align*}


\begin{align*} \mathcal { I } _ \Pi ( \delta _ 0 ) : = \sup _ { \gamma \in \Sigma _ { \Pi } ( \delta ) } \int _ { \mathcal { V } } g \ , d \gamma , \end{align*}


\begin{align*} E = \{ \left ( ( k , \ell ) , ( k ' , \ell ' ) \right ) \colon k ' - k \in Q , \ell ' - \ell \in P \} \subset \Lambda \times \Lambda . \end{align*}


\begin{align*} \rho _ 0 = \sum _ { i = 1 } ^ m \frac { m - n + 1 - 2 i } { 2 } \epsilon _ i + \sum _ { j = 1 } ^ n \frac { m + n + 1 - 2 j } { 2 } \delta _ { j } , \end{align*}


\begin{align*} \Theta ^ { \{ \partial W ^ 1 , \partial W ^ 2 , \dots , \partial W ^ m \} } ( x ) = \Theta ^ { \{ \partial W ^ 1 , \partial W ^ 2 , \dots , \partial W ^ { m + 1 } \} } ( x ) + \Theta ( x , \partial W ^ { m + 1 } ) . \end{align*}


\begin{align*} \| g _ n \| = 1 \ ; \ ; \max _ { \partial _ A \setminus V _ n } | g _ n | \leq 1 / n . \end{align*}


\begin{align*} \lim _ { T \rightarrow \infty } \mathbb { E } \big [ | x _ T | ^ 2 e ^ { - \beta T } \big ] = 0 . \end{align*}


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


\begin{align*} \begin{pmatrix} f & 0 \\ 0 & p e \end{pmatrix} \ , \ , ( p \in \mathcal H ) \end{align*}


\begin{align*} \frac { \partial \phi } { \partial t } + a \frac { \partial \phi } { \partial x } = \nu \frac { \partial \phi } { \partial x ^ 2 } + 4 \pi ^ 2 \nu ^ 2 \cos [ 2 \pi ( x - a t ) ] \end{align*}


\begin{align*} \mathcal { D } = \{ x \in \mathbb { R } ^ d | \ a _ j ( x ) \geq 0 , b _ \ell ( x ) = 0 , \forall j , \ell \} . \end{align*}


\begin{align*} \mathcal M _ { L , x _ 0 } ^ { ( E ) } : = \bigcup _ { x \in \mathcal N _ l \cap \Lambda _ { 2 L _ + , L _ - } } \left \{ \omega : \Lambda _ l ( x ) ( \omega , E _ 0 , m _ 0 , \eta _ 0 ) - \right \} . \end{align*}


\begin{align*} Q ( x , y , z ) = p d d _ 2 ^ { 2 } x ^ 2 + p d d _ 1 ^ { 2 } y ^ 2 + z ^ 2 - t p x y , \end{align*}


\begin{align*} \Pi _ { p \alpha + k } ^ { ( i ) } ( \eta _ 1 , t ) = \mathcal { O } \big ( e ^ { - \theta _ i \ , \eta _ i } \big ) \eta _ i \to + \infty \end{align*}


\begin{align*} \delta = \frac { v _ + - v _ - } { 4 ( \beta + 1 ) } . \end{align*}


\begin{align*} \mu = \begin{cases} 1 / 2 & \widehat { P } \widehat { \Theta } _ 1 \ll \widehat { \Theta } _ 2 \\ 0 & \end{cases} \end{align*}


\begin{align*} \begin{cases} \dot { \sigma } _ y ( t ) = \nabla u ( \sigma _ y ( t ) ) , & \\ \sigma _ y ( 0 ) = y . \ \ & \end{cases} \end{align*}


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


\begin{align*} \div ( T ^ \varphi ( \nabla w , \ , \cdot \ , ) ^ \sharp ) & = ( T ^ \varphi _ { i k } w _ k ) _ i \\ & = T ^ \varphi _ { i k , i } w _ k + T ^ \varphi _ { i k } w _ { k i } \\ & = \left ( R ^ \varphi _ { i k , i } - \frac { S ^ \varphi _ i } { m } \delta _ { i k } \right ) w _ k + T ^ \varphi _ { i k } w _ { k i } \ , . \end{align*}


\begin{align*} \| \hat \lambda \| _ * = \| N ^ { - 1 } ( \hat \lambda ) \| , \end{align*}


\begin{align*} X _ i = \left \{ 1 , 2 , \ldots , k - i , x _ 1 , x _ 2 , \ldots , x _ i \right \} , \ \mbox { f o r } 0 \leq i \leq k . \end{align*}


\begin{align*} M = \begin{pmatrix} a _ { 0 0 } & \dots & 1 _ { 0 \ , r } & 0 & 0 & \dots & \dots & \dots \\ a _ { 1 0 } & a _ { 1 1 } & \dots & 1 & 0 & \dots & \dots & \dots \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \dots \\ a _ { q \ , 0 } & a _ { q \ , 1 } & a _ { q \ , 2 } & \dots & \dots & 1 _ { q \ , q + r } & 0 & \dots \\ 0 & a _ { q + 1 \ , 1 } & a _ { q + 1 \ , 2 } & a _ { q + 1 \ , 3 } & \dots & \dots & 1 & \dots \\ 0 & 0 & \ddots & \vdots & \vdots & \vdots & \vdots & \ddots \end{pmatrix} ; \end{align*}


\begin{align*} ( e ^ { i t \Delta } g _ { K , \epsilon } ) ( x ) = \frac { \beta _ \epsilon ( t ) } { ( 2 \pi ) ^ 2 } e ^ { - \frac { \epsilon ^ 2 \beta _ \epsilon ( t ) } { 2 } | x | ^ 2 + i \beta _ { \epsilon } ( t ) x \cdot K - i t \beta _ { \epsilon } ( t ) | K | ^ 2 } \end{align*}


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


\begin{align*} \times \exp \left \{ \sum _ { n = 1 } ^ { \infty } \left ( + \frac { y ^ 4 + 1 1 y ^ 3 + 1 1 y ^ 2 + z } { ( 1 - y ) ^ 5 } \right ) \frac { z ^ n } { n ^ 5 } \right \} \end{align*}


\begin{align*} E _ s : = \Big \{ n \in \mathbb N : 0 \leq n < p ^ { k + \ell } , \ & n = \lfloor a _ { k + \ell - 1 } \dots a _ 0 \rfloor _ p \\ a _ { k - m - 1 } = p - 1 & a _ { k - m - 2 } \neq 0 \\ a _ { k - m - i } = 0 \forall \ , 3 \leq i \leq \ell & a _ { k - m - \ell - 1 } < b _ { k - m - \ell - 1 } \\ a _ j \ge b _ j \forall \ , j \in \{ 0 , \dots , & \ , k - m - \ell - 2 \} \cup \{ k - m , \dots , k \} \Big \} . \end{align*}


\begin{align*} N _ { p \alpha + k } ( \xi , t ) = w ^ { ( i ) } _ { p \alpha + k } ( 0 , t ) + \Psi ^ { ( i ) } _ { p \alpha + k } ( \xi _ i , t ) + \mathcal { O } ( \exp ( - \beta _ 0 \ , \xi _ i ) ) \mbox { a s } \ \ \xi _ i \to + \infty , \ \ \xi \in \Xi ^ { ( i ) } , \end{align*}


\begin{align*} k _ i : = | \left \{ y H \in G / H : \ ( H , y H ) \in O _ i \right \} | . \end{align*}


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


\begin{align*} \left \{ \begin{array} { l } \partial _ { t } ^ { 2 } u ( t , k ) + a ( t ) \mathcal { H } _ { \hbar , V } u ( t , k ) + q ( t ) u ( t , k ) = f ( t , k ) , ( t , k ) \in ( 0 , T ] \times \hbar \mathbb { Z } ^ { n } , \\ u ( 0 , k ) = u _ { 0 } ( k ) , k \in \hbar \mathbb { Z } ^ { n } , \\ \partial _ { t } u ( 0 , k ) = u _ { 1 } ( k ) , k \in \hbar \mathbb { Z } ^ { n } , \end{array} \right . \end{align*}


\begin{align*} \mu _ { \omega , c } & \le S _ { \omega , c } ( \lambda _ 0 \Phi _ { \omega } ) \\ & = \lambda _ 0 ^ 2 L ( \Phi _ { \omega } ) + \lambda _ 0 ^ 3 N ( \Phi _ { \omega } ) + \lambda _ 0 ^ 2 \omega Q ( \Phi _ { \omega } ) + \lambda _ 0 ^ 2 c P ( \Phi _ { \omega } ) \\ & = \lambda _ 0 ^ 2 S _ { \omega , 0 } ( \Phi _ { \omega } ) + \lambda _ 0 ^ 2 \{ ( \lambda _ 0 - 1 ) N ( \Phi _ { \omega } ) + c P ( \Phi _ { \omega } ) \} \\ & = \lambda _ 0 ^ 3 \mu _ { \omega , 0 } < 8 \mu _ { \omega , 0 } . \end{align*}


\begin{align*} \sigma _ 0 : = \min \left \{ \frac { 2 } { \alpha } , \frac { 1 } { \beta } , \frac { 1 } { \gamma } \right \} . \end{align*}


\begin{align*} \delta _ t \Bigg ( \sum _ { j = 1 } ^ k X _ j \Bigg ) = \sum _ { j = 1 } ^ k t ^ j X _ j . \end{align*}


\begin{align*} \frac { \sqrt { \pi } \ , \Gamma ( s - \frac { 1 } { 2 } ) } { \Gamma ( s ) } = \pi - 2 \pi \log ( 2 ) \ , ( s - 1 ) + O \left ( s - 1 \right ) ^ { 2 } \end{align*}


\begin{align*} J _ 6 = - \frac { s } { 2 } \iint _ Q \sigma _ { t t } | u | ^ 2 d x d y d t \geq - C s \iint _ Q \xi ^ { 3 / 2 } | u | ^ 2 d x d y d t . \end{align*}


\begin{align*} z _ 0 = \frac { e _ 1 + e _ b } { 2 } \ \hbox { a n d } \ z _ 0 ^ \ast = e _ 1 - e _ b , \end{align*}


\begin{align*} \left ( \prod _ { x : X } M _ x \right ) _ f = \prod _ { x : X } \left ( M _ x \right ) _ { f ( x ) } = \prod _ { x : X } \left ( M _ x \right ) ^ { D ( f ) } = \left ( \prod _ { x : X } M _ x \right ) ^ { D ( f ) } \rlap { . } \end{align*}


\begin{align*} f ( x ) = \prod x _ i = \prod \left ( \sqrt { p ( 1 - p ) } \chi _ i + p \right ) \end{align*}


\begin{align*} \rho _ { s } ^ { t , \mu ; u } = \rho _ { s } ^ { \tau , \rho _ { \tau } ^ { t , \mu ; u } ; u ^ { \tau } } , \end{align*}


\begin{align*} r ( p ) \ = \ f q ^ \beta , \end{align*}


\begin{align*} \int _ { B _ r } ( b ^ { i j , k l } ) _ r v _ { i j } v _ { k l } = \int _ { B _ r } [ ( b ^ { i j , k l } ) _ r - b ^ { i j , k l } ( x ) ] f _ { i j } v _ { k l } . \end{align*}


\begin{gather*} \lim _ { k \to \infty } \frac { 1 } { k } \log \| S ( k ) ( D ^ l \mathfrak { r } _ { a , n } ( \varphi ) ) \| _ { \sup } = - \lambda _ 1 ( n - l - a ) 0 \leq l < n \\ \lim _ { k \to \infty } \frac { 1 } { k } \log \big ( \| S ( k ) ( D ^ l \mathfrak { r } _ { a , n } ( \varphi ) ) \| _ { L ^ 1 ( I ^ { ( k ) } ) } / | I ^ { ( k ) } | \big ) = - \lambda _ 1 ( n - l - a ) 0 \leq l \leq n . \end{gather*}


\begin{align*} L i _ 2 \left ( \frac { x } { 1 - x } . \frac { y } { 1 - y } \right ) = L i _ 2 \left ( \frac { x } { 1 - y } \right ) + L i _ 2 \left ( \frac { y } { 1 - x } \right ) + L i _ 2 \left ( \frac { - x } { 1 - x } \right ) \end{align*}


\begin{align*} \alpha = \alpha ( \{ \beta _ j \mid j \in I ' \} ) = \alpha ( \{ \tilde { \beta } _ j \mid j \in I ' \} ) . \end{align*}


\begin{align*} v = \begin{bmatrix} 1 - \frac { | \mathcal { S } | } { | \Omega | } \\ - \frac { | \mathcal { S } | } { | \Omega | } \end{bmatrix} . \end{align*}


\begin{align*} z ^ { * } \bigl ( Y , \ , t \bigr ) = y ( N , \ , t ) , \ \forall t , \ \pi \end{align*}


\begin{align*} ( - \Delta ) ^ s [ f ( x ) g ( x ) ] = f ( x ) ( - \Delta ) ^ { s } g ( x ) + g ( x ) ( - \Delta ) ^ s f ( x ) - \mathcal B ( f , g ) ( x ) , \end{align*}


\begin{align*} \| E \| = \sup _ { \substack { u \in \mathcal F _ { r , R } \\ r \in ( 0 , R ) } } \frac { \| u \| _ { p ^ * , q , \mu } } { \| \nabla u \| _ { p , q , \mu } } . \end{align*}


\begin{align*} \left ( S _ { [ 1 ] } \diamond \cdots \diamond S _ { [ j ] } \right ) ^ T J _ { 2 | \alpha _ i | } S _ { [ j + 1 ] } & = \left ( S _ { [ 1 ] } \diamond \cdots \diamond S _ { [ j ] } \right ) ^ T J _ { 2 | \alpha _ i | } W _ { [ j + 1 ] } R _ { [ j + 1 ] } ^ { - 1 } . \end{align*}


\begin{align*} \sup _ { \partial \Omega } \vert \nabla u \vert \leq C _ 1 : = \max \{ \sup _ { \partial \Omega } \vert \nabla \underline u \vert , \Vert \nabla w \Vert _ { L ^ { \infty } ( \partial \Omega ) } \} \cdot \end{align*}


\begin{align*} M _ H ( h , x ) & = ( \Phi \circ \Psi ) ( h , x ) , \\ ( \widetilde { R } \circ \Psi ) ( h , x ) & = ( \Psi \circ ( R \times \sigma ) ) ( h , x ) , \\ ( \sigma \circ \Phi ) ( h , x ) & = ( \Phi \circ \widetilde { R } ) ( h , x ) . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ { 3 } \int _ { \Xi ^ { ( i ) } } \big ( F _ { p \alpha + k } - \partial _ t { { N } } _ { p \alpha + k - 1 } \big ) \ , d \xi = \int _ { \Xi ^ { ( 0 ) } } \partial _ t { { N } } _ { p \alpha + k - 1 } \ , d \xi + \delta _ { p \alpha + k , \alpha } \int _ { \Gamma _ 0 } \varphi ^ { ( 0 ) } \ , d \sigma _ \xi . \end{align*}


\begin{align*} x _ 1 \oplus x _ 2 \oplus \cdots \oplus x _ n = x _ 1 ( 1 + \eta _ n ) + x _ 2 ( 1 + \eta _ { n - 1 } ) + \cdots + x _ n ( 1 + \eta _ 1 ) . \end{align*}


\begin{align*} z _ { 2 } & = x _ 2 + \iota x _ 2 - \langle w _ 1 , x _ 2 + \iota y _ 2 \rangle w _ 1 \\ & = y _ 2 + \iota y _ 2 - \langle x _ 1 + \iota y _ 1 , x _ 2 + \iota y _ 2 \rangle w _ 1 + \mathcal { O } ( \| H \| ) \\ & = x _ 2 + \iota y _ 2 + \mathcal { O } ( \| H \| ) . \end{align*}


\begin{align*} \| B _ { \tilde { h } } W u \| _ { H ^ N _ { h \tilde { h } } } = \mathcal { O } ( h ^ \infty \tilde { h } ^ \infty ) \| u \| _ { H ^ { - N } _ { h \tilde { h } } } . \end{align*}


\begin{align*} \left [ U ^ { - 1 } \right ] _ { { i , j } } = \begin{cases} ( - 1 ) ^ { j - i } \binom { n - i } { n - j } & \textnormal { i f } \ , i \le j ; \\ 0 & \textnormal { o t h e r w i s e } . \end{cases} \end{align*}


\begin{align*} \texttt { L H S } \eqref { i d : 9 } \simeq ( b - a ) ^ { \gamma } \int _ { b - a } ^ { b - a + a / 4 } x ^ { - \gamma } \ , d x \simeq ( b - a ) ^ { \gamma } \begin{cases} b ^ { - \gamma + 1 } & \gamma < 1 , \\ \log \big ( 1 + a / ( b - a ) \big ) & \gamma = 1 . \end{cases} \end{align*}


\begin{align*} \delta _ k = \frac { \varepsilon _ 4 a ^ k } { 4 } \quad \frac { T _ k } { a ^ k } \cdot 1 . 0 6 \cdot ( 2 k + 3 ) \cdot 2 ^ { - d _ k } < \frac { \varepsilon _ 4 } { 2 } . \end{align*}


\begin{align*} \| \vec U \| _ { P \otimes P _ { \Omega } } ^ 2 \leq \| \vec F \| _ { P \otimes P _ { \Omega } } ^ 2 + 2 \int _ 0 ^ t \sum _ { j = 1 , N } \lbrack \vec G ^ T \vec G ) \rbrack _ j d s _ j \ d t . \end{align*}


\begin{align*} \Delta w + \frac { S } { m - 1 } w = 0 , \end{align*}


\begin{align*} \| u _ \theta \| _ { W ^ { 1 , \ , \infty } ( \mathbb S ^ { n - 1 } ) } \le \theta ^ 2 , \| \Delta _ { \mathbb S ^ { n - 1 } } u _ \theta \| _ { L ^ \infty ( \mathbb S ^ { n - 1 } ) } \leq \theta ^ 2 , \| D ^ 2 _ { \tau } u _ \theta \| _ { L ^ \infty ( \mathbb S ^ { n - 1 } ) } = \theta ^ { - 1 } . \end{align*}


\begin{align*} t _ q ( \Gamma ) \ : = \ \sum ^ n _ { i = 0 } ( - 1 ) ^ i \mathrm { g r a n k } ( \tau ^ { - i } ( \Gamma ) ) \in \Z [ q , q ^ { - 1 } ] , \end{align*}


\begin{align*} ( f \sharp g ) _ { m + m ' } ( \xi , \mu ) = f _ m ( \xi , \mu ) g _ { m ' } ( \xi , \mu ) . \end{align*}


\begin{align*} \theta ( \bar { u } ) + I _ K ( \bar { \zeta } ) = \lim \limits _ { k \rightarrow + \infty } [ \theta ( u ^ k ) + I _ K ( \zeta ^ k ) ] . \end{align*}


\begin{align*} \left . \begin{array} { l l } \displaystyle R e \int _ { \mathbb { R } ^ N } \psi \overline { \phi } ^ 2 d x < 0 A \mu ^ 2 = \dfrac { N } { 2 } B \mu ^ \frac { N } { 2 } . \end{array} \right . \end{align*}


\begin{align*} \rho ( A ) = \lim _ { n \to \infty } \left \| \sum _ { | \alpha | = n } A ^ { \alpha } A ^ { \alpha * } \right \| ^ { \dfrac { 1 } { 2 n } } . \end{align*}


\begin{align*} f ( z ) = \frac { 1 } { \pi } \langle f _ b , P ( x - \cdot , y ) \rangle , \end{align*}


\begin{align*} f ( t ) = f ( 0 ) + f ' ( u ) t \end{align*}


\begin{align*} w ^ h _ 1 : = \begin{cases} \phi ^ i _ 1 ( u ) & \mbox { o n } B _ h ^ i , \ , \ , i \leq h , \\ u & \mbox { o t h e r w i s e } . \end{cases} \end{align*}


\begin{align*} N ( I , \delta _ 1 , \chi , \mathbf { X } _ \omega ) \coloneqq \inf \lbrace n > 0 : \ \tau _ { n , \omega } ^ { I } ( \chi ) = b \rbrace . \end{align*}


\begin{align*} f ^ { B , \alpha } ( \xi , \mu ) : = \partial _ x ^ \alpha | _ { x = 0 } \alpha _ { - x } \big ( f ( \xi + B x , \mu ) \big ) = \sum _ { j = 0 } ^ { | \alpha | } \sum _ { \substack { \beta + \gamma = \alpha \\ | \gamma | = j } } \binom \alpha \beta ( i \delta ) ^ \beta \partial _ { B , \xi } ^ \gamma f ( \xi , \mu ) . \end{align*}


\begin{align*} \mathcal { M } _ { q , \alpha ^ 1 } f = \mathcal { M } _ { q , \alpha ^ 2 } f \end{align*}


\begin{align*} \mu _ { \omega , \mathbf { c } } = S _ { \omega , \mathbf { c } } ( \Psi _ { \omega } ) = \omega ^ { 2 - \frac { d } { 2 } } S _ { 1 , \frac { \mathbf { c } } { \sqrt { \omega } } } ( \Psi ) = \omega ^ { 2 - \frac { d } { 2 } } \mu _ { 1 , \frac { \mathbf { c } } { \sqrt { \omega } } } . \end{align*}


\begin{align*} P _ 1 ( r ) \coloneqq \{ \varpi ^ k \parallel r \colon ( \varpi , \Delta _ { F _ 2 } c _ 2 ) = 1 , \varpi ^ k \nmid d \} \quad P _ 2 ( r ) = \{ \varpi ^ k \parallel r \colon ( \varpi , \Delta _ { F _ 2 } ) = 1 , \varpi ^ k \mid d \} . \end{align*}


\begin{align*} \hat { w _ 0 } ( \xi ) = \int _ 0 ^ \infty \frac { e ^ { - 3 \tau } } { 2 } \cdot e ^ { - i \xi \tau } \ , d \tau = \frac { 1 } { 6 + 2 i \xi } \ , . \end{align*}


\begin{align*} \left \| \ , \left | \sum _ { i = 1 } ^ n B ^ * _ i A _ i \right | ^ { \alpha } \ , \right \| ^ 2 \le \left \| \ , \left | \sum _ { i = 1 } ^ n B ^ * _ i B _ i \right | ^ { \alpha } \ , \right \| \left \| \ , \left | \sum _ { i = 1 } ^ n A ^ * _ i A _ i \right | ^ { \alpha } \ , \right \| \end{align*}


\begin{align*} \alpha \big ( ( \ell _ E ) _ { E \in \mathbf { E } } , ( n _ E ) _ { E \in \mathbf { E } } \big ) = { } & \big ( ( n _ E ) _ { E \in \mathbf { E } } , ( 0 ) _ { 1 \le i \le r } , ( ( - 1 ) ^ { \delta _ { v , v ' _ E } } \ell _ E ) _ { v \in E \in \mathbf { E } } \big ) , \\ \beta \big ( ( n _ E ) _ { E \in \mathbf { E } } \big ) = { } & \big ( ( n _ E ) _ { v \in E \in \mathbf { E } } \big ) . \end{align*}


\begin{align*} E _ 1 ^ { p , q } = \begin{cases} \bigoplus _ { \lvert I \rvert = p + 1 } R _ F ( G _ I ) \otimes \Q & q , \\ 0 & q \ . \end{cases} \end{align*}


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


\begin{align*} e _ x v _ x + e _ y v _ x ^ 2 v _ y - e _ y v _ y u _ z ^ 2 v _ z ^ 2 = 0 . \end{align*}


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


\begin{align*} [ [ x _ \mu [ { y _ 1 } _ \eta y _ 2 ] ] _ { \mu + \gamma } \varphi _ { \lambda } ( w , v ) ] = [ ( \varphi _ \lambda ( x , [ { y _ 1 } _ \eta y _ 2 ] ) ) _ { \lambda + \mu } [ w _ \gamma v ] ] . \end{align*}


\begin{align*} B ( \phi _ 0 + \phi _ + + U \phi _ + ) = A \phi _ 0 + i \phi _ + - i U \phi _ + \end{align*}


\begin{align*} \sigma _ * ( \varphi ) \circ \sigma _ { E _ \bullet } = \sigma _ { F _ \bullet } \circ \varphi . \end{align*}


\begin{align*} x _ p = \frac { m - l } { ( k , m - l ) } v _ p , y _ p = \frac { k } { ( k , m - l ) } v _ p , \end{align*}


\begin{align*} { \rm I m } ( e ^ { i \pi ( 1 - \beta _ 1 ) } \kappa ' ( t ) ) = 0 { \rm \ f o r \ } t < 0 \end{align*}


\begin{align*} \sup _ { x \in [ 0 , 1 ] } \P ( N ^ { - 1 } d _ { i _ x } \geq \beta ) \leq \P \left ( N ^ { - 1 } \max _ { i \in [ N ] } d _ i \geq \beta \right ) \leq \sum _ { i \in [ N ] } \P ( N ^ { - 1 } d _ i \geq \beta ) \leq N \sup _ { x \in [ 0 , 1 ] } \P ( N ^ { - 1 } d _ { i _ x } \geq \beta ) . \end{align*}


\begin{align*} \bar { \rm P } _ { j , j + i } = & \sum ^ { M - j } _ { m = i + 1 } { M - j \choose m } \mathbb { P } _ { \rm T X } ^ m \left ( 1 - \mathbb { P } _ { \rm T X } \right ) ^ { M - j - m } \\ & \times m \cdots ( m - i + 1 ) \mathbb { P } _ K ^ { m } \gamma _ i \gamma _ { m , i } \\ & + { M - j \choose i } \mathbb { P } _ { \rm T X } ^ i \left ( 1 - \mathbb { P } _ { \rm T X } \right ) ^ { M - j - i } i ! \mathbb { P } _ K ^ { i } \gamma _ i \end{align*}


\begin{align*} \sum _ { l = k } ^ { \infty } \left ( \partial ^ 2 u _ l ( 0 ) - \partial ^ 2 u _ { l + 1 } ( 0 ) \right ) = \partial ^ 2 u _ k ( 0 ) - \partial ^ 2 u ( 0 ) . \end{align*}


\begin{align*} \mathcal { V } \left ( t , \mu \right ) = \inf _ { u \in \mathcal { U } _ { t } ^ { 2 } } J \left ( t , \mu ; u \right ) . \end{align*}


\begin{align*} \Delta ^ \ell = \left \{ ( t _ 1 , \dots , t _ \ell ) \in \R ^ { \ell } \ \mid \ \sum _ { i = 1 } ^ \ell t _ i \leq 1 t _ j \geq 0 \ \forall j \in \{ 1 , \dots , \ell \} \right \} \end{align*}


\begin{align*} \frac { d } { d x } ( f _ { \tilde { \textbf { Q } } } ) = \frac { g ' } { b } \cdot ( b h ) + g \cdot \frac { d } { d x } ( b h ) = \frac { g ' } { b } \cdot ( b h ) \end{align*}


\begin{align*} \aligned \left \{ \begin{array} { l l l } - \Delta u + \lambda _ 1 u = ( I _ \mu \ast | u | ^ { r _ 1 } ) | u | ^ { r _ 1 - 2 } u + \nu p ( I _ \mu \ast | v | ^ q ) | u | ^ { p - 2 } u \ & \mathbb { R } ^ N , \\ - \Delta v + \lambda _ 2 v = ( I _ \mu \ast | v | ^ { r _ 2 } ) | v | ^ { r _ 2 - 2 } v + \nu q ( I _ \mu \ast | u | ^ p ) | v | ^ { q - 2 } v \ & \mathbb { R } ^ N , \\ \int _ { \mathbb { R } ^ N } u ^ 2 = a ^ 2 , \ \int _ { \mathbb { R } ^ N } v ^ 2 = b ^ 2 . \end{array} \right . \endaligned \end{align*}


\begin{align*} f ^ 3 - f Q = 0 , Q \ , \xi = \xi ( \xi \in \ker f ) . \end{align*}


\begin{align*} w _ 1 - w _ { j } = b _ { r _ j } - b _ { s _ j } ; 2 \leq j \leq n . \end{align*}