\begin{align*} f = \sum _ { i = 1 } ^ s a _ i \textbf { Q } ^ { \lambda _ i } \end{align*}


\begin{align*} - \Delta w _ t = \lambda w _ t + f ( u _ t ) - f ( u ) . \end{align*}


\begin{align*} \Omega _ + = \big \{ x \in M \ : \ w ( x ) > 0 \big \} , \end{align*}


\begin{align*} \frac { \partial } { \partial x _ i } \phi _ { i , k } = \begin{cases} 0 & \mbox { i f } d _ { i , k } - x _ i \leq 0 \\ \frac { x _ i - d _ i ( \omega _ k ) } { \bar { y } } & \mbox { i f } 0 \leq d _ { i , k } - x _ i \leq \bar { y } \\ - 1 & \mbox { i f } d _ { i , k } - x _ i \geq \bar { y } \end{cases} \end{align*}


\begin{align*} \frac { 1 } { \ell _ i } \sum _ { n \leq \ell _ i } \mathbf { 1 } _ { \underline { A } \times A } ( \underline { \varphi } ( R ^ n ( \Delta ( 0 ) ) ) , \varphi ( R ^ n ( \Delta ( 0 ) ) ) ) = \frac { 1 } { \ell _ i } \sum _ { n \leq \ell _ i } \mathbf { 1 } _ { \underline { C } } ( R ^ n ( \Delta ( 0 ) ) ) \mathbf { 1 } _ { C } ( R ^ n ( \Delta ( 0 ) ) ) . \end{align*}


\begin{align*} \tfrac { d } { d t } E x ( t ) = ( J - R ) Q x ( t ) , J = - J ^ H , R = R ^ H \geq 0 , Q ^ H E = E ^ H Q \end{align*}


\begin{align*} - \varepsilon ^ 2 \Delta w + V ( x ) w = w \log w ^ 2 , w \in H ^ 1 ( \R ^ N ) . \end{align*}


\begin{align*} { \rm P } _ { j , j } = & 1 - \sum ^ { K } _ { i = 1 } \bar { \rm P } _ { j , j + i } \\ = & 1 - \sum ^ { K } _ { i = 1 } \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 \choose i } \mathbb { P } ( E _ { i , 1 } ) \mathbb { P } ( E _ { i , 2 } ) . \end{align*}


\begin{align*} 2 \lambda \left ( \delta \right ) = \ln \left ( \delta ^ { - 2 \rho / d } \right ) . \end{align*}


\begin{align*} h ( x , x ^ { \ast } ) + h ( y , y ^ { \ast } ) = \left \langle x , x ^ { \ast } \right \rangle + \left \langle y , y ^ { \ast } \right \rangle \Rightarrow h ( x , x ^ { \ast } ) + h ( y , y ^ { \ast } ) = \left \langle x , y ^ { \ast } \right \rangle + \left \langle y , x ^ { \ast } \right \rangle . \end{align*}


\begin{align*} L ( \tau ) = \sum _ { i = 1 } ^ { R } V _ { i } \end{align*}


\begin{align*} ( \partial _ j \partial _ i V ^ i ) \ , V ^ j = ( \partial _ j \partial _ i V ^ j ) \ , V ^ i . \end{align*}


\begin{align*} h _ i d \left ( \sum \limits _ { j = 1 } ^ { s _ \ell } b _ { i _ { } i j } { \textbf { X } } ^ { \lambda _ j } \right ) \end{align*}


\begin{align*} f ^ { p ^ k - \gamma } = [ e _ { 0 } ] ^ { p ^ k - \gamma } + [ e _ { 1 } ] ^ { 1 + p ^ k - \gamma } + \dots + [ e _ { \gamma } ] ^ { \gamma + p ^ k - \gamma } + \cdots + [ e _ { \eta - 1 } ] ^ { \eta - 1 + p ^ k - \gamma } . \end{align*}


\begin{align*} f _ { k } ( z , \lambda ) : = \frac { 1 } { 2 \pi i } \intop _ { \sigma - i \infty } ^ { \sigma + i \infty } \Gamma ( s ) \ , ( s , \lambda ) _ { k } \ , z ^ { - s } d s = \left ( \frac { \lambda - z } { \lambda + z } \right ) ^ { k } \ , e ^ { - z } , \ , \ , \ , \ , \sigma > 0 , \ , \ , ( z ) > 0 . \end{align*}


\begin{align*} \lambda : = \inf \left \{ Q _ { \Sigma } [ \phi ] \ , : \ , \int _ { \Sigma } \phi ^ 2 = 1 \mbox { a n d } \nabla _ n \phi = h ( \nu , \nu ) \phi \mbox { o n } \partial \Sigma \right \} . \end{align*}


\begin{align*} \chi e ^ { - ( t _ 0 - t ) X } ( Q _ \infty + q _ 1 ) e ^ { - i t h ^ { - 1 } \tilde { P } _ h } \tilde { R } _ h ( z ) \chi = 0 . \end{align*}


\begin{align*} P ^ \omega ( C _ t ^ c \mid A _ { m ( t ) } ) \leq ( 1 - p ( t ) ) ^ { I ( t ) } = e ^ { - p ( t ) I ( t ) } , \end{align*}


\begin{align*} B ( x _ 1 t , \ldots , x _ m t ) = 1 + t B _ 1 ( x _ 1 , \ldots , x _ m ) + \cdots + t ^ d B _ d ( x _ 1 , \ldots , x _ m ) . \end{align*}


\begin{align*} \phi = \Tilde { \phi } _ { e } + D _ { 2 } \Delta x ^ { 2 } + D _ { 3 } \Delta x ^ { 3 } . \end{align*}


\begin{align*} \sum _ { x _ i \in X _ j ^ \mathrm { n d f } } w _ { j i } \hat u _ i = f ( y _ j ) , j = 1 , \ldots , M , \end{align*}


\begin{align*} \{ y = s \} \cap { \rm G r a p h } ( f ) = \{ \left ( \alpha ( s ) , s \right ) ; \left ( \beta ( s ) , s \right ) \} . \end{align*}


\begin{align*} \left \{ \begin{aligned} \Delta _ p u & = f _ 1 ( | x | ) \cdot g _ 1 ( v ) \cdot | \nabla u | ^ { \alpha } & \quad \mbox { i n } \Omega , \\ \Delta _ p v & = f _ 2 ( | x | ) \cdot g _ 2 ( v ) \cdot g _ 3 ( | \nabla u | ) & \quad \mbox { i n } \Omega , \end{aligned} \right . \end{align*}


\begin{align*} \hat { y } _ { i ; t } = \prod _ { j \in [ m ] } x _ { j ; t } ^ { b ^ { ( t ) } _ { i , j } } \end{align*}


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


\begin{align*} \phi _ 1 & = \Tilde { \phi } _ { e } + C _ { \Tilde { q } } h ^ { \Tilde { q } } , \\ \phi _ 2 & = \Tilde { \phi } _ { e } + C _ { \Tilde { q } } ( r h ) ^ { \Tilde { q } } , \\ \phi _ 3 & = \Tilde { \phi } _ { e } + C _ { \Tilde { q } } ( r ^ 2 h ) ^ { \Tilde { q } } . \end{align*}


\begin{align*} ( 2 ^ * - 1 ) \alpha _ n \sum _ { i = 1 } ^ k \mu _ i ^ { \frac { n - 2 } { 2 } } \int _ \Omega U _ { \mu _ i , \xi _ i } ^ { 2 ^ * - 2 } ( x ) H ( x , \xi _ i ) \psi ^ h _ { \mu _ j , \xi _ j } d x = o \Big ( \frac { \epsilon } { | \ln \epsilon | ^ 2 } \Big ) . \end{align*}


\begin{align*} \Big \| \sum _ { j = 1 } ^ \infty \alpha _ j \tilde { u } _ j \Big \| _ { p ^ * , q , \mu } \geq \frac { \lambda } { 1 + \varepsilon _ 1 } \Big ( \sum _ { j = 1 } ^ \infty | \alpha _ j | ^ q \Big ) ^ { \frac 1 { q } } . \end{align*}


\begin{align*} \overline { S } ( k , n ) = \sum _ { j = k + 1 } ^ n \binom { n } { j } = \frac { 1 } { 2 } \sum _ { j = k } ^ { n - 1 } { 2 ^ { n - j } } \binom { j } { k } . \end{align*}


\begin{align*} w _ b = \sum _ { n } c _ n a _ n , \end{align*}


\begin{align*} g ( m , u , x + 1 ) - g ( m , u , x ) = \frac { \partial } { \partial u } g ( m - 1 , u , x ) . \end{align*}


\begin{align*} \imath ^ { \sigma , c } _ { \tau , c } \circ \imath ^ { \tau , c } _ { \mu , c } = ( \imath ^ { \sigma , c } _ { \partial \sigma , c } \circ \imath ^ { \partial \sigma , c } _ { \tau , c } ) \circ \imath ^ { \tau , c } _ { \mu , c } = \imath ^ { \sigma , c } _ { \partial \sigma , c } \circ ( \imath ^ { \partial \sigma , c } _ { \tau , c } \circ \imath ^ { \tau , c } _ { \mu , c } ) = \imath ^ { \sigma , c } _ { \partial \sigma , c } \circ \imath ^ { \partial \sigma , c } _ { \mu , c } = \imath ^ { \sigma , c } _ { \mu , c } . \end{align*}


\begin{align*} G _ { \d , \nu , \l } ( H ) = 0 . \end{align*}


\begin{align*} 1 0 ^ m - 6 \ \le \ v ( f ) \ = \ v ( 2 ^ \delta ) + v ( 3 ^ \gamma ) \ \le \ 2 + \delta + 3 + \gamma . \end{align*}


\begin{align*} \nu _ { y ' } ^ n ( x ) & = \langle y ' \sigma _ n ( x ) \Omega _ \rho , \Omega _ \rho \rangle _ { \rho } \\ & \leq \norm { y ' } \langle \sigma _ n ( x ) \Omega _ \rho , \Omega _ \rho \rangle _ { \rho } \\ & = \norm { y ' } \rho ( \sigma _ n ( x ) ) \\ & = \norm { y ' } \rho ( x ) \\ & = \norm { y ' } \langle x \Omega _ \rho , \Omega _ \rho \rangle _ { \rho } . \end{align*}


\begin{align*} R ( 0 ) ^ 2 & = \sum _ { m , n \in \mathcal { M } ' } r ( m ) r ( n ) \le | \mathcal { M } ' | \sum _ { m \in \mathcal { M } ' } r ( m ) ^ 2 \end{align*}


\begin{align*} { \rm e } ^ { { \rm i } \chi y } \bigl ( \mathcal { A } _ { 0 , \chi } ^ { \rm s o f t } - z I \bigr ) ^ { - 1 } { \rm e } ^ { - { \rm i } \chi y } = \bigl ( \mathcal { A } _ { 0 , 0 } ^ { \rm s o f t } - z I \bigr ) ^ { - 1 } , \end{align*}


\begin{align*} a x _ i + P ( x _ 1 , \dots , x _ { i - 1 } , x _ { i + 1 } , \dots , x _ n ) = 0 , a \neq 0 , \end{align*}


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


\begin{align*} \xi _ \phi ( X _ i ) : = \frac { k _ i } { | H | } \sum _ { h \in H } \phi ( x _ \ell h ) , \end{align*}


\begin{align*} \int _ \Omega \big ( \mathrm { d i v } \ , V \big ) ^ 2 \ , d x = \int _ \Omega \mathrm { t r } \big ( ( \nabla \ , V ) ^ 2 \big ) d x + \int _ { \partial \Omega } \Big ( ( \mathrm { d i v } _ T V ) \ , V \cdot \nu - \nabla _ T V \ , V _ T \cdot \nu \Big ) d \mathcal { H } ^ { n - 1 } . \end{align*}


\begin{align*} u & = \left ( ( s _ m , f _ m ) , \dots , ( s _ 1 , f _ 1 ) \right ) \\ v & = \left ( ( s ' _ n , f ' _ n ) , \dots , ( s ' _ 1 , f ' _ 1 ) \right ) , \end{align*}


\begin{align*} \rho _ { a d _ { \alpha } } = ( \alpha \otimes \Delta ) - \xi \circ ( \Delta \otimes \alpha ) : L \otimes L \rightarrow L \otimes L \otimes L \otimes L . \end{align*}


\begin{align*} Y ( x , y ) = 1 \Rightarrow y \in \tilde { \mathcal { A } } \Rightarrow x \in \tilde { \mathcal { A } } ^ \uparrow = \mathcal { B } \Rightarrow X ( x , y ) = 1 . \end{align*}


\begin{align*} M _ p ( k _ 1 k _ 2 & k _ 3 k _ 4 , k _ 3 k _ 4 k _ 1 k _ 2 ) \\ & = 2 \cos \big [ ( t - s ) \Delta E ( \vec k ) \big ] | \Phi ( \vec k ) | ^ 2 \big ( \delta ( p - k _ 1 ) + \delta ( p - k _ 2 ) - \delta ( p - k _ 3 ) - \delta ( p - k _ 4 ) \big ) \ . \end{align*}


\begin{align*} \lim _ { l \to \infty } \sup _ { x \in M _ + , x \neq 0 } \abs { ( B _ l ( \nu ) - \bar { \nu } ) ( x ) } / \rho ( x ) = 0 . \end{align*}


\begin{align*} \alpha ( Y _ T ) = \begin{cases} - a & \\ 0 & \end{cases} \end{align*}


\begin{align*} W ^ + = \begin{bmatrix} u _ n + p / u _ n \\ u _ { \tau } \end{bmatrix} , \Lambda ^ + = \begin{bmatrix} u _ n & 0 \\ 0 & u _ n \end{bmatrix} , W ^ - = p , \Lambda ^ - = - 1 / u _ n . \end{align*}


\begin{align*} \begin{array} { l l } & c ^ { \prime } x \\ & a _ { t } ^ { \prime } x \leq b _ { t } , \ , \ , \ , t \in T : = \left \{ 1 , 2 , . . . , m \right \} , \end{array} \end{align*}


\begin{align*} u : = \begin{cases} { \bf e } _ 1 & \mbox { i n } \Omega , \\ 0 & \mbox { i n } B _ 1 \setminus \overline { \Omega } . \end{cases} \end{align*}


\begin{align*} g ( z ) = z \exp \left ( \int _ { 0 } ^ { z } \frac { x ^ 2 + \sqrt { 1 + x ^ 4 } - 1 } { x } d x \right ) = z + \frac { z ^ 3 } { 2 } + \frac { z ^ 5 } { 4 } + \cdots , \end{align*}


\begin{align*} \min \limits _ { \ell \in I _ 0 ( g , i ) } \alpha _ \ell = \alpha _ i , \end{align*}


\begin{align*} u ( r ) = C _ \lambda r ^ \lambda , v ( r ) = C _ \mu r ^ \mu . \end{align*}


\begin{align*} N _ { 0 } : = | I | \le C ^ { - 2 } n \log ^ { 5 } n \le n ^ { 1 + \varepsilon / 1 0 } , \end{align*}


\begin{align*} f _ \psi ( s ) = C ' _ \circ ( \psi ) \cdot s ^ 2 + o ( s ^ 2 ) , s \to 0 . \end{align*}


\begin{align*} e ^ { ( \gamma _ p + \gamma _ q - 2 ) t ( u , v ) } = \frac { | \nabla u | ^ 2 _ 2 + | \nabla v | ^ 2 _ 2 } { ( \gamma _ p + \gamma _ q ) \int _ { \mathbb { R } ^ N } ( I _ \mu * | u | ^ { p } ) | v | ^ { q } } . \end{align*}


\begin{align*} \abs { \frac { 1 } { \sqrt { n } } \frac { \partial \Re f _ r } { \partial x } } & \le \abs { \frac { 1 } { \sqrt { n } } \frac { \partial f _ r } { \partial x } } = \abs { \frac { \partial ( f - f _ 0 ) } { \partial x } } = \abs { \frac { \partial ( \log h - \log h _ 0 ) } { \partial x } } \\ & \le \abs { \frac { 1 } { h _ 0 } \cdot \frac { \partial h _ 0 } { \partial x } - \frac { 1 } { h } \cdot \frac { \partial h } { \partial x } } \le \frac { C } { \sqrt { n } } , \end{align*}


\begin{align*} ( \nabla u _ \mu ^ \bigstar ) _ \mu ^ * ( t ) = \phi ^ * ( t ) . \end{align*}


\begin{align*} c _ { \Phi } ( \xi , \tau ) + C _ 1 \lambda \frac { \left | p _ { \Phi } ( x _ 0 , \xi , \tau ) \right | ^ 2 } { | \xi | ^ 2 + \tau ^ 2 } & = \lambda \left ( c _ { \Psi } ( \xi , \lambda \tau ) + 2 \lambda \left | \{ p _ { \Psi } , \Psi \} ( x _ 0 , \xi , \lambda \tau ) \right | ^ 2 + C _ 1 \frac { \left | p _ { \Psi } ( x _ 0 , \xi , \lambda \tau ) \right | ^ 2 } { | \xi | ^ 2 + \tau ^ 2 } \right ) . \end{align*}


\begin{align*} O _ { B } ^ { H } ( M ) & = \min \left \{ \sum _ { i = 1 } ^ k h ^ i x _ i \mid \sum _ { i = 1 } ^ k ( h ^ i - h + 1 ) x _ i = M , \ x _ i \in \mathbb { N } , 1 \leq i \leq k \right \} \\ & = \min \left \{ M + ( h - 1 ) \cdot \sum _ { i = 1 } ^ k x _ i \mid \sum _ { i = 1 } ^ k ( h ^ i - h + 1 ) x _ i = M , \ x _ i \in \mathbb { N } , 1 \leq i \leq k \right \} . \end{align*}


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


\begin{align*} H = \cap _ { \varrho } \ker \sigma \end{align*}


\begin{align*} \sum \limits _ { k = n } ^ { m } \left ( \mathcal { D } _ { \eta ^ { k } } ( \eta ^ { k + 1 } , \eta ^ { k } ; \lambda ^ { k } ) + \beta \| r ^ { k + 1 } \| ^ 2 \right ) = \mathcal { D } _ { \eta ^ { n } } ( \hat { \eta } , \eta ^ { n } ; \lambda ^ { n } ) - \mathcal { D } _ { \eta ^ { m } } ( \hat { \eta } , \eta ^ { m } ; \lambda ^ { m } ) . \end{align*}


\begin{align*} \Psi _ \Omega ( r ) = \begin{cases} \sup \limits _ { x \in \partial \Omega } \| \mathcal B \| _ { L ^ { n - 1 , \infty } ( \partial \Omega \cap B _ r ( x ) ) } \quad n \geq 3 , \\ \\ \sup \limits _ { x \in \partial \Omega } \| \mathcal B \| _ { L ^ { 1 , \infty } \log L ( \partial \Omega \cap B _ r ( x ) ) } \quad n = 2 \end{cases} \end{align*}


\begin{align*} \sum _ { n \in \mathbb { Z } } \widehat { \Delta } ( a n ) = \frac { 1 } { a } \sum _ { m \in \mathbb { Z } } \Delta \left ( \frac { m } { a } \right ) . \end{align*}


\begin{align*} f ( x ) = \sum \limits _ { j = - \infty } ^ \infty \sum \limits _ { Q \in Q ^ j } \omega ( Q ) \psi _ { { \mathfrak R } ^ j } ( x , x _ { Q } ) q _ { { \mathfrak R } ^ j } h ( x _ { Q } ) , \end{align*}


\begin{align*} \norm { \cdot } _ { ( H ^ \alpha ( \Omega ) ) ^ * } : = \sum _ { i = 1 } ^ p \norm { F } _ { ( H ^ { \alpha _ i } ( \Omega ) ) ^ * } . \end{align*}


\begin{align*} \nabla ( e ^ { - \tau \Phi } ) & = - \tau \nabla \Phi e ^ { - \tau \Phi } \\ \Delta ( e ^ { - \tau \Phi } ) & = \div ( \nabla ( e ^ { - \tau \Phi } ) ) = - \tau \div ( \nabla \Phi e ^ { - \tau \Phi } ) = - \tau ( \Delta \Phi ) e ^ { - \tau \Phi } - \tau \nabla \Phi \cdot \nabla ( e ^ { - \tau \Phi } ) \\ & = - \tau ( \Delta \Phi ) e ^ { - \tau \Phi } + \tau ^ 2 | \nabla \Phi | ^ 2 e ^ { - \tau \Phi } . \end{align*}


\begin{align*} ( \pi ( f ) \psi ) ( m ) = \int _ \R f ( - t ) \psi ( e ^ { t X } m ) \ d t . \end{align*}


\begin{align*} \nu ( x ) = \langle y _ 1 ' x \Omega _ { \rho } , \Omega _ { \rho } \rangle _ { \rho } x \in M . \end{align*}


\begin{align*} S \bar { u } = \bar { \zeta } . \end{align*}


\begin{align*} \Re \left \langle \mathcal { A } x , x \right \rangle = \left \langle \Re \mathcal { A } x , x \right \rangle , \Im \left \langle \mathcal { A } x , x \right \rangle = \left \langle \Im \mathcal { A } x , x \right \rangle . \end{align*}


\begin{align*} r _ n ( G ) \left ( \overline \bigotimes ^ { s - 1 } _ { i = 0 } [ \langle 1 , - g _ { 1 , i } \rangle ] \overline \otimes . . . \overline \otimes [ \langle 1 , - g _ { n , i } \rangle ] \overline \otimes I ^ { n + 1 } ( G ) \right ) : = \\ \sum ^ { s - 1 } _ { i = 0 } l ( - 1 ) ^ { 2 ^ { n - 1 } - n } l ( g _ { 1 , i } ) \otimes . . . \otimes l ( g _ { n , i } ) + \mathcal Q _ { 2 n - 1 } ( G ) \end{align*}


\begin{align*} \mathcal { L } = \textrm { s u p p } ( f ) \setminus ( M \cup L ) \end{align*}


\begin{align*} A _ { n } = \left \{ x \in H _ { 0 } \backslash F : - \frac { 1 } { n } \log P _ { \mu , H _ { 0 } } ^ { n } ( o , x ) \ge h - \epsilon \right \} , \end{align*}


\begin{align*} P _ s ( \theta ) & \overset { \Delta } { = } \left ( > \theta , \Phi ( \mathcal { A } ) > 0 | \Phi \right ) \\ & = \left ( > \theta | \Phi , \Phi ( \mathcal { A } ) > 0 \right ) \mathbf { 1 } ( \Phi ( \mathcal { A } ) > 0 | \Phi ) , \end{align*}


\begin{align*} \tilde z ( Y ^ { t } ( \omega ) , t ) = y ^ { * } ( \omega , t ) , \forall t \in \Q , \forall \omega \in A . \end{align*}


\begin{align*} | v ( y ) - v ( x ) | \leq \sum _ { q = 1 } ^ { q ( k ) } \mathrm { o s c } ( v , J ^ { ( k ) } ( q ) ) + \sum _ { l > k } \sum _ { \epsilon = \pm } \sum _ { q = 1 } ^ { q _ \epsilon ( l ) } \mathrm { o s c } ( v , J _ \epsilon ^ { ( l ) } ( q ) ) \end{align*}


\begin{align*} \begin{pmatrix} W _ { \mathcal { I } \mathcal { I } } & X _ { \mathcal { I } \mathcal { I } } \\ Y _ { \mathcal { I } \mathcal { I } } & Z _ { \mathcal { I } \mathcal { I } } \end{pmatrix} . \end{align*}


\begin{align*} f ( w , z ) = e ^ { w + z } \quad \mbox { a n d } g ( w , z ) = e ^ { w z } , \end{align*}


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


\begin{align*} \mathcal { L } _ { D } \left ( b , d \right ) : = \left \{ x \in \mathbb { R } ^ { n } : a _ { t } ^ { \prime } x \leq b _ { t } , \ , t \in T ; \ ; - a _ { t } ^ { \prime } x \leq d _ { t } , \ , t \in D \right \} . \end{align*}


\begin{align*} \tau \partial _ t c = \alpha \Delta c - c + \lambda d + \beta m . \end{align*}


\begin{align*} \lim \limits _ { k \rightarrow \infty } \| p ^ { k } - y ^ { k } \| = \lim \limits _ { k \rightarrow \infty } \| p ^ { k } - y ^ { k - 1 } \| = \lim \limits _ { k \rightarrow \infty } \| x ^ { k } - x ^ { k - 1 } \| = 0 , \end{align*}


\begin{align*} \frac { 1 } { t ^ s } \rho ( t , t ^ s \kappa ( s ) ) = A t ^ { 1 - s } + { \rm I m } ( e ^ { i \pi ( 1 - \beta _ 1 ) } \kappa ( t ) ) + B t ^ { 2 - s } + C t ^ s | \kappa ( t ) | ^ 2 + \end{align*}


\begin{align*} \sum _ { j = 2 } ^ n b _ j ( a ' _ 1 a _ j - a _ 1 a ' _ j ) \equiv \lambda ( a _ 1 ' - a _ 1 ) \pmod q \ , , \end{align*}


\begin{align*} \left | \int _ s ^ t L _ N \pi _ { r } ^ N ( \phi _ j ) d r \right | ^ 2 & = \left ( \int _ s ^ t \left ( \pi _ { r } ^ N ( A \phi _ j ) + R _ 3 ( \phi _ j , N , r ) \right ) d r \right ) ^ 2 \\ & \leq 2 \left ( \int _ s ^ t \pi _ { r } ^ N ( A \phi _ j ) d r \right ) ^ 2 + 2 \left ( \int _ s ^ t R _ 3 ( \phi _ j , N , r , \sigma ) d r \right ) ^ 2 . \end{align*}


\begin{align*} x _ { n + 1 } = \frac { 1 } { \| x _ n + \alpha _ n u \| } ( x _ n + \alpha _ n u ) \in U . \end{align*}


\begin{align*} \hat S _ k ^ l ( t ) = \frac { S _ k ^ l ( t ) } { S _ { l - 1 } ^ l ( t ) } , l = 1 , \dots , p ; \end{align*}


\begin{align*} t \log \Big ( 1 + \frac { \delta _ j } { 3 j } \Big ) = \frac { t \delta _ j } { 3 j } + O \Big ( \frac { t } { j ^ 2 } \Big ) = \frac { t \delta _ j } { 3 j } + O \Big ( \frac { 1 } { K } \Big ) , j \ge \beta K . \end{align*}


\begin{align*} \left ( \bigcup _ { \ell < m } K _ { \ell } \right ) \cap K _ { m } = \bigcup _ { \ell < m } \left ( K _ { \ell } \cap K _ { m } \right ) = K _ { 1 } \cap K _ { m } \in 2 ^ { K _ { 1 } } \subset \bigcup _ { \ell < m } 2 ^ { K _ { \ell } } . \end{align*}


\begin{align*} S = \begin{bmatrix} 0 & - 1 \\ 1 & 0 \end{bmatrix} ; T = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} . \end{align*}


\begin{align*} x _ 0 y _ 0 + x _ 1 y _ 1 + x _ 2 y _ 2 = 0 \end{align*}


\begin{align*} L _ { \omega , \mathbf { c } } ( V ) & = L _ { \omega , \mathbf { c } } ( V _ n ) - L _ { \omega , \mathbf { c } } ( V _ n - V ) - ( L _ { \omega , \mathbf { c } } ( V _ n ) - L _ { \omega , \mathbf { c } } ( V _ n - V ) - L _ { \omega , \mathbf { c } } ( V ) ) \\ & < L _ { \omega , \mathbf { c } } ( V _ n ) - 6 \mu _ { \omega , \mathbf { c } } - ( L _ { \omega , \mathbf { c } } ( V _ n ) - L _ { \omega , \mathbf { c } } ( V _ n - V ) - L _ { \omega , \mathbf { c } } ( V ) ) \\ & \rightarrow 0 \end{align*}


\begin{align*} \vect u = ( \mathcal { A } _ 0 - z I ) ^ { - 1 } \vect f + \left ( I - z \mathcal { A } _ 0 ^ { - 1 } \right ) ^ { - 1 } \Pi ( \overline { \beta _ 0 + \beta _ 1 M ( z ) } ) ^ { - 1 } \left ( \vect g - \beta _ 1 \Pi ^ * \left ( I - z \mathcal { A } _ 0 ^ { - 1 } \right ) ^ { - 1 } \vect f \right ) . \end{align*}


\begin{align*} \delta ( n ) = \begin{cases} 1 , & n = 0 \\ 0 , & n \neq 0 . \end{cases} \end{align*}


\begin{align*} \widetilde { D } _ { \alpha , z } ( \psi _ 1 \bar { \otimes } \psi _ 2 | | \varphi _ 1 \bar { \otimes } \varphi _ 2 ) = \widetilde { D } _ { \alpha , z } ( \psi _ 1 | | \varphi _ 1 ) + \widetilde { D } _ { \alpha , z } ( \psi _ 2 | | \varphi _ 2 ) , \end{align*}


\begin{align*} h _ n ( \eta ) : = h _ n ( \rho ( a _ 1 ) \otimes . . . \otimes \rho ( a _ n ) ) = l _ R ( f ( a _ 1 ) ) \ast . . . \ast l _ R ( f ( a _ n ) ) = 0 \in R _ n . \end{align*}


\begin{align*} \gamma _ { n } ( \Omega _ { i } ) = \gamma _ { n } ( \Omega _ { i } ' ) , \qquad \forall \ , 1 \leq i \leq 3 . \end{align*}


\begin{align*} \sum _ { s = 1 } ^ { m } \ , \mu _ s \ , ^ { c } _ { 0 } { D } ^ { \alpha _ s } _ { t } u ( x , t ) - a ( l ( u ) ) \ : \Delta u ( x , t ) & = f ( x , t ) & \mbox { i n } & \Omega \times ( 0 , T ] , \\ u ( x , t ) & = 0 & \mbox { o n } & \partial \Omega \times ( 0 , T ] , \\ u ( x , 0 ) & = u _ 0 ( x ) & \mbox { i n } & \Omega , \end{align*}


\begin{align*} L ( f , k + t ) ^ 2 = \sum _ { l , \ , m \geq 1 } \frac { \lambda _ f ( l ) \lambda _ f ( m ) } { \sqrt { l m } } ( l m ) ^ { - t } . \end{align*}


\begin{align*} x ^ a y ^ b + y ^ a z ^ b + z ^ a x ^ b = 0 , \end{align*}


\begin{align*} | V ( T ) | = ( n - 1 ) m + 2 . \end{align*}


\begin{align*} & \sum _ { \substack { m , n \in \mathcal { M } ' \\ m \not = n } } r ( m ) r ( n ) \Phi \left ( T \log { \frac { m } { n } } \right ) \\ & \le \sum _ { \substack { j , l \in \mathcal { J } \\ j \not = l } } r ( m _ j ) r ( n _ l ) \Phi \left ( T ( | j - l | - 1 ) \log { ( 1 + T ^ { - 1 } ) } \right ) \\ & \ll \sum _ { \substack { j , l \in \mathcal { J } \\ j \not = l } } r ( m _ j ) r ( n _ l ) \Phi \left ( | j - l | - 1 \right ) \end{align*}


\begin{align*} \theta _ { 3 / 2 } ( z , \epsilon ) & = \sum _ { n \in \Z } n e ^ { i \epsilon \pi n ^ 2 z } , \end{align*}


\begin{align*} \widetilde { c } _ { X ^ { \ast } } ( g _ 1 , g _ 2 ) = \gamma ( \psi ( q ( g _ 1 , g _ 2 ) / 2 ) ) . \end{align*}


\begin{align*} \left \{ \begin{aligned} & \left [ r ^ \delta u ' ( r ) ^ { p - 1 - \alpha } \right ] ' = \frac { \delta } { n - 1 } \ , r ^ \delta f _ 1 ( r ) g _ 1 ( v ( r ) ) & & \\ & \left [ r ^ { n - 1 } v ' ( r ) ^ { p - 1 } \right ] ' = r ^ { n - 1 } f _ 2 ( r ) g _ 2 ( v ( r ) ) \cdot g _ 3 ( u ' ( r ) ) & & \end{aligned} \right . \end{align*}


\begin{align*} u ^ { ( i , k , n ; L ) } ( x , t ) = 0 . \end{align*}


\begin{align*} \Delta _ r w ( r ) + \pi w ( r ) ^ 3 = - \frac 3 4 \delta . \end{align*}


\begin{align*} \displaystyle \mathbf { M } = 2 \boldsymbol { \Pi } \mathbf { W } \mathbf { R } ^ T - 2 \boldsymbol { \mathit { S } } , \end{align*}


\begin{align*} \varphi _ L \circ R = R \circ \varphi _ L \end{align*}


\begin{align*} d \overline { \Pi } _ { \psi } ( h ) f ( [ \epsilon , x ] ) & = \frac { d } { d t } \Big \{ \overline { \Pi } _ { \psi } ( [ e ^ { t h } , 1 ] ) f ( [ \epsilon , x ] ) \Big \} \Big | _ { t = 0 } \\ & = \frac { d } { d t } \Big \{ e ^ { \tfrac { t } { 2 } } f ( [ \epsilon , x e ^ t ] ) \Big \} \Big | _ { t = 0 } \\ & = \{ \tfrac { 1 } { 2 } + x \tfrac { d } { d x } \} f ( [ \epsilon , x ] ) . \end{align*}


\begin{align*} I _ K ( \zeta ) = \left \{ \begin{aligned} & 0 , & & \mbox { i f } \ \zeta \in K , \\ & + \infty , & & \mbox { i f } \ \zeta \not \in K . \end{aligned} \right . \end{align*}


\begin{align*} \sum _ { I \in \mathbb { N } _ 0 ^ n } c _ I \eta ^ I , \eta ^ I : = \eta ^ { i _ 1 } _ 1 \dots \eta ^ { i _ n } _ n , \end{align*}


\begin{align*} A \phi ( x , \sigma ) = \left ( \tfrac { \kappa } { 2 } \partial _ { x x } + \sigma \lambda \partial _ x \right ) \phi ( x , \sigma ) + \sum _ { \sigma ' \in S } c ( \sigma , \sigma ' ) \big ( \phi ( x , \sigma ' ) - \phi ( x , \sigma ) \big ) . \end{align*}


\begin{align*} \mathcal { F } _ n : = \left \{ \mathcal { M } \subset \mathcal { F } : \mathcal { M } \right . \left . \gamma ( \mathcal { M } ) \geq n \right \} \end{align*}


\begin{align*} V ( X / G ) = V ( X ) / G = \{ v G : v \in V ( X ) \} , H ( X / G ) = H ( X ) / G = \{ h G : h \in H ( X ) \} , \end{align*}


\begin{align*} G ( d , n ) = \bigsqcup _ { M \in M ^ d _ { S _ n } ( K ) } G _ M , \end{align*}


\begin{align*} \tilde f ( x ) = f ( x ) + 2 0 \delta \cdot \sum _ { i = 1 } ^ m \left ( d ( x , p _ i ) - d ( p , p _ i ) \right ) . \end{align*}


\begin{align*} \left \{ v _ S : S \in \binom { [ n + m ] } { n } \right \} \end{align*}


\begin{align*} \left ( \mathord { \operatorname { p r o j } } _ { Q } \circ { \operatorname { p r o j } _ { K _ i } } ^ { - 1 } \right ) \# \mu _ i = \left ( \mathord { \operatorname { p r o j } } _ { Q } \circ { \operatorname { p r o j } _ { K _ j } } ^ { - 1 } \right ) \# \mu _ j . \end{align*}


\begin{align*} F ( s ) = \sum _ { n = 1 } ^ { \infty } \frac { a _ F ( n ) } { n ^ s } \end{align*}


\begin{align*} F ( u ) : = \int _ { \Omega } | u | ^ p d x G ( u ) : = \int _ { \Omega } | X u | ^ p d x . \end{align*}


\begin{align*} D ^ t = \pm \left ( \begin{array} { c c } 2 \pi a & s \\ 0 & 2 \pi a \end{array} \right ) , \end{align*}


\begin{align*} { \mathcal M } \big ( e ^ { \psi _ m } \big ) ( z ) & = \dfrac 1 { \Omega _ { 2 n } \delta ^ { 2 n } } \int _ { B ( z , \delta ) } ( 1 + | \zeta | ^ 2 ) ^ { a _ m } e ^ { 2 m V ( \zeta ) } \ , d \lambda ( \zeta ) \\ & \leq ( 1 + ( | z | + \delta ) ^ 2 ) e ^ { 2 m ( V ( z ) + \varepsilon ) } , \end{align*}


\begin{align*} - \Delta u = u ^ { 2 ^ * - 1 } , \ \ u > 0 \quad \mbox { i n } \ \ \mathbb { R } ^ n . \end{align*}


\begin{align*} \bar \rho : = \rho \big ( E _ t ^ { - 1 } \big ( L _ { \gamma ( \bar s ) ^ { - 1 } } ( \gamma ( t + \bar s ) ) \big ) \big ) > 0 . \end{align*}


\begin{align*} Z _ k X _ { - \beta _ k } ^ { \pi ( \beta _ k ) } \otimes v _ \lambda = 1 \otimes v _ \lambda . \end{align*}


\begin{align*} \widetilde { q } _ { y _ 0 } ( r ) = \frac { 1 } { \omega _ { n - 1 } r ^ { n - 1 } } \int \limits _ { S ( y _ 0 , r ) } \widetilde { Q } ( y ) \ , d \mathcal { H } ^ { n - 1 } ( y ) \end{align*}


\begin{align*} \mathcal { H } _ { \hbar , V } u ( k ) : = \left ( - \hbar ^ { - 2 } \mathcal { L } _ { \hbar } + V \right ) u ( k ) , k \in \hbar \mathbb { Z } ^ { n } , \end{align*}


\begin{align*} \varphi _ j = \frac { W _ j } { 3 } \arctan [ \sinh ( x ) ] , j = 1 , 2 \ , \end{align*}


\begin{align*} \R ^ 3 _ + : = \left \{ ( \alpha , \beta , \gamma ) \in \R ^ 3 \ | \ \alpha > 0 , \ \beta > 0 , \ \gamma > 0 \right \} , \ \ \R ^ 3 _ - : = \left \{ ( \alpha , \beta , \gamma ) \in \R ^ 3 \ | \ \alpha < 0 , \ \beta < 0 , \ \gamma < 0 \right \} . \end{align*}


\begin{align*} ( f \sharp g ) _ { m + m ' - j } ( \xi , \mu ) = \sum _ { k + l + | \alpha | = j } \frac { ( - i ) ^ { | \alpha | } } { \alpha ! } \partial _ \xi ^ \alpha f _ { m - k } ( \xi , \mu ) g _ { m ' - l } ^ { B , \alpha } ( \xi , \mu ) , j \geq 0 . \end{align*}


\begin{align*} y \gets c _ 2 A r + c _ 1 r = c _ 2 ( A - I ) r + ( c _ 1 - c _ 2 ) r = - c _ 2 E r + ( c _ 1 - c _ 2 ) r \end{align*}


\begin{align*} & q _ { R } ( x , \xi , \tau ) = \frac { p _ { \Phi } + \overline { p _ { \Phi } } } { 2 } ( x , \xi , \tau ) = \Re p _ { \Phi } ( x , \xi , \tau ) = p _ 2 ( x , \xi ) - \tau ^ 2 p _ 2 ( x , d \Phi ( x ) ) , \\ & q _ { I } ( x , \xi , \tau ) = \frac { p _ { \Phi } - \overline { p _ { \Phi } } } { 2 i } ( x , \xi , \tau ) = \Im p _ { \Phi } ( x , \xi , \tau ) = 2 \tau \widetilde { p _ 2 } ( x , \xi , d \Phi ( x ) ) . \end{align*}


\begin{align*} J _ r ( x , \beta ) = \sup _ { \theta \in \R } [ \theta \beta - \Lambda _ r ( x , \theta ) ] . \end{align*}


\begin{align*} \varphi \circ \xi \left ( g P , x _ { 0 } \right ) = \varphi \circ \xi \left ( g . ( P , \alpha ( g , P ) ^ { - 1 } . x _ { 0 } ) \right ) = g . \left ( \varphi \circ \xi \left ( P , \alpha ( g , P ) ^ { - 1 } . x _ { 0 } \right ) \right ) = g Q . \end{align*}


\begin{align*} \lim _ { r \downarrow 0 } \frac { g ( r ) } { h ( r ) } = 0 \ , . \end{align*}


\begin{align*} \overset \cdot c _ i = c _ i ^ 2 \left ( \prod _ { j = 1 } ^ { r } c _ { i + j } - \prod _ { j = 1 } ^ { r } c _ { i - j } \right ) . \end{align*}


\begin{align*} H = ( \alpha _ { i , j } ) _ { i , j = 0 } ^ \infty . \end{align*}


\begin{align*} \Big \Vert \sup _ { k \leq n } | S _ k - T _ k | \Big \Vert _ 2 = O ( n ^ { 1 / p } ( \log n ) ^ { 1 / 2 } ) \ , . \end{align*}


\begin{align*} & \frac { ( r + x + \varepsilon _ 2 ) ^ { 1 / 2 } ( r + x - \varepsilon _ 2 ) ^ { 1 / 2 } ( r + \varepsilon _ 2 - x ) ^ { 1 / 2 } ( x + \varepsilon _ 2 - r ) ^ { 1 / 2 } } { 2 r } . \end{align*}


\begin{align*} \begin{pmatrix} \tau _ { k , Y _ { 2 , 1 } } \\ \tau _ { k , Y _ { 2 , 2 } } \end{pmatrix} = \begin{pmatrix} 1 / 2 & 1 / 2 \\ 1 / 2 & - 1 / 2 \end{pmatrix} \begin{pmatrix} T _ 1 \tau _ { k , Y _ { 1 , 1 } } \\ T _ 2 \tau _ { k } \end{pmatrix} , \end{align*}


\begin{align*} & E [ \hat \beta _ l ( \alpha _ { l , { { n } _ t } } - \alpha _ { l , { \hat { n } _ t } } e ^ { - j \omega } ) ] = \frac { \sqrt { \pi } } { 2 } E ( \hat \beta _ l ) , \\ & V a r [ \hat \beta _ l ( \alpha _ { l , { { n } _ t } } - \alpha _ { l , { \hat { n } _ t } } e ^ { - j \omega } ) ] = 2 - \frac { \pi } { 4 } E ^ 2 ( \hat \beta _ l ) . \end{align*}


\begin{align*} A _ \gamma ( \hat { x } , \hat { x } ) = F _ \gamma ( \hat { x } ) = U ( \hat { x } ) \left ( \begin{array} { c c c | c } & & & 0 \\ & A ^ \prime ( \hat { x } ) & & \vdots \\ & & & 0 \\ \hline 0 & \ldots & 0 & 1 \end{array} \right ) , \end{align*}


\begin{align*} ( \partial _ n ) _ \sigma ^ { \sigma ' } : = \begin{cases} ( - 1 ) ^ i & \\ 0 & . \end{cases} \end{align*}


\begin{align*} d _ q < d _ { q + 1 } < \cdots < d _ l < x \qquad \qquad \tilde { \delta } = ( x , d _ l , d _ { l - 1 } \dots , d _ 2 ) ; \end{align*}


\begin{align*} \left \lbrace \begin{array} { l c r } ( d d ^ c u ) ^ n = \psi ( \cdot , u ) \ , \omega ^ n & \textnormal { i n } & \Omega , \\ u = \varphi & \textnormal { i n } & \partial \Omega , \end{array} \right . \end{align*}


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


\begin{align*} \tilde { \mathcal { L } } ( \varphi ) = \frac { \tilde { \mathcal { K } } ( \varphi ) - \varphi } { \tau } . \end{align*}


\begin{align*} N ^ { \ , ( 2 ) } _ i ( X , Y ) = \eta ^ i ( [ \widetilde { Q } X , \ , { f } Y ] ) ; \end{align*}


\begin{align*} \xi \eta ' + \eta \xi ' = \xi ' \eta + \eta ' \xi = - ( \xi , \eta ) _ { \mathbb { F } } , \qquad \mathrm { w h e r e } \qquad ( \xi , \eta ) _ { \mathbb { F } } : = ( u , v ) - 2 a b \end{align*}


\begin{align*} D ^ t = \left ( \begin{array} { c c } \sqrt { \lambda } + b & b \\ - b & - ( \sqrt { \lambda } + b ) \end{array} \right ) \end{align*}


\begin{align*} P _ { N } u : = u _ { N } : = u _ { \leq 2 N } - u _ { \leq N } \\ P _ { A \leq \cdot \leq B } : = u _ { A \leq \cdot \leq B } : = u _ { \leq B } - u _ { \leq A } . \end{align*}


\begin{align*} & \frac { 1 } { ( q ; q ) _ \infty ( x ^ \alpha ; q ) _ \infty ( q x ^ { - \alpha } ; q ) _ \infty } \\ & = \frac { 1 } { \big ( ( 1 - q ) ( 1 - q ^ 2 ) \dotsm \big ) \big ( ( 1 - x ^ \alpha ) ( 1 - q x ^ { \alpha } ) \dotsm \big ) \big ( ( 1 - q x ^ { - \alpha } ) ( 1 - q ^ 2 x ^ { - \alpha } ) \dotsm \big ) } \\ & = ( 1 + x ^ \alpha + x ^ { 2 \alpha } + x ^ { 3 \alpha } + \dotsm ) + q ( x ^ { - \alpha } + 2 + 3 x ^ \alpha + 3 x ^ { 2 \alpha } + \dotsm ) + q ^ 2 \dotsm . \end{align*}


\begin{align*} h ( 0 ) & = \mu ( \omega , \mathbf { c } ) = S _ { \omega , \mathbf { c } } ( \Phi ) , \\ h ' ( 0 ) & = - \frac { 1 } { \sqrt { \omega } } ( 2 \omega Q ( \Phi ) + \mathbf { c } \cdot \mathbf { P } ( \Phi ) ) , \\ h '' ( 0 ) & = \frac { 3 - d } { \omega } ( 2 \omega Q ( \Phi ) + \mathbf { c } \cdot \mathbf { P } ( \Phi ) ) . \end{align*}


\begin{align*} \widetilde { { S } } _ { \lambda ^ 3 } \widetilde { { S } } _ { \lambda ^ 3 } & = ( y _ { 4 } - y _ { 2 } ) ( y _ { 4 } - y _ { 3 } ) \widetilde { { S } } _ { \lambda ^ 3 } + ( y _ { 4 } - y _ { 3 } ) \widetilde { { S } } _ { \lambda ^ 1 } + \widetilde { { S } } _ { \lambda ^ 0 } \\ & = ( Y _ { \lambda ^ 5 } - Y _ { \lambda ^ 3 } ) ( Y _ { \lambda ^ 2 } - Y _ { \lambda ^ 1 } ) \widetilde { { S } } _ { \lambda ^ 3 } + ( Y _ { \lambda ^ 2 } - Y _ { \lambda ^ 1 } ) \widetilde { { S } } _ { \lambda ^ 1 } + \widetilde { { S } } _ { \lambda ^ 0 } . \end{align*}


\begin{align*} Z = \sum _ { j = 1 } ^ k Z _ j = \prod _ { j = 1 } ^ k y \big ( \bigl \{ X _ { n i } ^ { ( j ) } \bigr \} \big ) . \end{align*}


\begin{align*} { ( J _ \varepsilon ( u ^ { \varepsilon , \alpha } ) ) ^ 2 - ( J _ \varepsilon ( u ^ { \varepsilon } ) ) ^ 2 \over J _ \varepsilon ( u ^ { \varepsilon , \alpha } ) + J _ \varepsilon ( u ^ { \varepsilon } ) } = J _ \varepsilon ( u ^ { \varepsilon , \alpha } ) - J _ \varepsilon ( u ^ { \varepsilon } ) \ge - \sqrt { \varepsilon } d ( u ^ { \varepsilon , \alpha } , u ^ { \varepsilon } ) \ge - C \alpha \sqrt { \varepsilon } , \end{align*}


\begin{align*} \widehat { \psi } _ r ( \beta ) = \inf _ { x \in [ 0 , 1 ] } J _ r ( x , \beta ) . \end{align*}


\begin{align*} \delta ^ { 2 } e ^ { 2 \lambda \left ( \delta \right ) d } = \delta ^ { 2 - 2 \rho } . \end{align*}


\begin{align*} H ( \lambda ) = \max _ { u \in \R ^ k } \left ( - \frac { \norm { u } ^ 2 } { 2 } \right ) = 0 . \end{align*}


\begin{align*} \Sigma = W _ 1 \cup W _ 2 \cup _ Y W _ 3 \cup W _ 4 . \end{align*}


\begin{align*} u ^ \nu _ z : = u ( z + t \nu ) , z : = \left ( { \rm I d } - \nu \otimes \nu \right ) x , \nu : = \frac { x - y } { \abs { x - y } } . \end{align*}


\begin{align*} f ^ * _ n ( z _ 1 , \ldots , z _ m ) : = f ^ * _ { n , L } ( z _ 1 , \ldots , z _ m ) \end{align*}


\begin{align*} | I | = i _ 1 + \dots + i _ n , d _ A ( I ) = v _ 1 i _ 1 + \dots + v _ n i _ n . \end{align*}


\begin{align*} \textbf { Q } ^ \lambda : = \prod _ { q \in \textbf { Q } } q ^ { \lambda ( q ) } \in K [ x ] . \end{align*}


\begin{align*} f _ { r _ 1 | \Phi ( \mathcal { A } ) > 0 } ( r ) = \upsilon ( \lambda , R _ S ) r e ^ { - \lambda \pi \frac { R _ S } { R _ E } r ^ 2 } , R _ { m i n } \leq r \leq R _ { m a x } , \end{align*}


\begin{align*} \Big \| \widehat { \mathcal { L } ^ { \mathbb { R } ^ n } _ \varepsilon c } + \widehat { \Delta c } \Big \| _ { L ^ 2 ( \mathbb { R } ^ n ) } ^ 2 = \frac { 1 } { ( 2 \pi ) ^ n } \int _ { \mathbb { R } ^ n } \big | \big ( - \mathcal { F } ( J _ \varepsilon ) ( \xi ) + \mathcal { F } ( J _ \varepsilon ) ( 0 ) - | \xi | ^ 2 \big ) \mathcal { F } ( c ) ( \xi ) \big | ^ 2 \xi , \end{align*}


\begin{align*} x \cdot p = 1 \cdot a & \mbox { a n d } l _ R ( x ) l _ R ( p ) = l _ R ( 1 ) l _ R ( a ) = 0 \\ y \cdot q = 1 \cdot a & \mbox { a n d } l _ R ( y ) l _ R ( q ) = l _ R ( 1 ) l _ R ( a ) = 0 . \end{align*}


\begin{align*} C _ { g } ( p ^ { 2 m + 1 } ) = 0 \end{align*}


\begin{align*} \beta _ j < \tilde { \beta } _ j + \epsilon < \alpha _ { i _ - } + \epsilon = \alpha _ i . \end{align*}


\begin{align*} [ { f } , { f } ] ( X , \xi _ i ) = { f } ( \nabla _ { \xi _ i } { f } ) X + \nabla _ { { f } X } \ , \xi _ i - { f } \ , \nabla _ { X } \ , \xi _ i , X \in \mathfrak { X } _ M . \end{align*}


\begin{align*} u _ \lambda = \arg \underset { u \in H } { \min } \big ( \lambda \lVert u \rVert _ H ^ 2 + \frac { 1 } { n } \sum _ { i = 1 } ^ n L ( y _ i , u ( x _ i ) \big ) , \end{align*}


\begin{align*} \frac { L } { N } \sum _ { n _ 1 , n _ 2 \neq 0 } \widehat { \Delta } \left ( \frac { L n _ 1 } { N } \right ) \widehat { \Delta } \left ( \frac { L n _ 2 } { N } \right ) \delta ( n _ 1 w _ r - n _ 2 w _ s ) = \frac { L } { N } \sum _ { n _ 0 \neq 0 } \widehat { \Delta } \left ( \frac { L w _ r n _ 0 } { N d } \right ) \widehat { \Delta } \left ( \frac { L w _ s n _ 0 } { N d } \right ) . \end{align*}


\begin{align*} \mu \big ( [ [ N , T + T ^ * ] , S ] \big ) = 2 \mathrm { R e } \ , \mu \big ( [ [ N , T ] , S ] \big ) \ . \end{align*}


\begin{align*} a _ 0 ^ 2 = O ( \varepsilon ) \| u \| _ { L ^ 2 ( \mathbb { S } ^ n ) } ^ 2 . \end{align*}


\begin{align*} U = \begin{pmatrix} \frac { 1 } { \sqrt { 2 } } & 0 & \frac { 1 } { \sqrt { 2 } } \\ 0 & 1 & 0 \\ \frac { 1 } { \sqrt { 2 } } & 0 & - \frac { 1 } { \sqrt { 2 } } \end{pmatrix} . \end{align*}


\begin{align*} \overline { F } ( k , t + 1 , n ) = \begin{dcases} \frac { k } { n } \overline { F } ( k , t , n ) + \frac { n - k } { n } \overline { F } ( k - 1 , t , n ) & \textnormal { i f } \ ; k \le t ; \\ 0 & \textnormal { i f } \ ; k > t ; \end{dcases} \end{align*}


\begin{align*} \begin{aligned} & \left \| \partial _ x ^ l \partial _ t ^ j s _ { p , N } ( t ) \partial ^ k _ x v \right \| _ { L _ N ^ 2 } \leq C _ { j , l , k } ( 1 + t ) ^ { - \frac { 1 } { 4 } - \frac { l + j } { 2 } } \| v \| _ { L _ N ^ 1 } , & & v \in H ^ k _ N , \\ & \left \| \partial _ x ^ l \partial _ t ^ j s _ { p , N } ( t ) \partial ^ k _ x v \right \| _ { L _ N ^ 2 } \leq C _ { j , l , k } ( 1 + t ) ^ { - \frac { l + j } { 2 } } \| v \| _ { L _ N ^ 2 } , & & v \in H ^ k _ N , \end{aligned} \end{align*}


\begin{align*} \Lambda ( \theta ) \Phi : = ( e ^ { 2 i \theta } \varphi _ 1 , e ^ { i \theta } \varphi _ 2 , e ^ { i \theta } \varphi _ 3 ) . \end{align*}


\begin{align*} a x ^ n + b y ^ m = c z ^ k , \end{align*}


\begin{align*} Q + Q ^ * & \simeq 2 P \oplus \begin{pmatrix} 2 I & R \\ R & 0 \end{pmatrix} \\ & \simeq 2 P \oplus \left \{ \begin{pmatrix} I & I \\ I & I \end{pmatrix} + \begin{pmatrix} R & 0 \\ 0 & - R \end{pmatrix} \right \} . \end{align*}


\begin{align*} \Sigma ( \psi ) ( \omega _ r ) = \{ m \in \bar \omega \ , | \ , \langle \delta - n _ { F ^ i } , m \rangle \le r , 1 \le i \le p \} . \end{align*}


\begin{align*} P ^ \omega ( D _ t ^ c \mid S ) & = \frac { 1 } { P ^ \omega ( S ) } \left [ P ^ \omega ( D _ t ^ c \cap S \cap C _ t ) + P ^ \omega ( D _ t ^ c \cap S \cap C _ t ^ c ) \right ] \\ & \leq c ( \omega ) \left [ P ^ \omega ( D _ t ^ c \cap S _ t \cap C _ t ) + P ^ \omega ( D _ t ^ c \cap S \cap C _ t ^ c ) \right ] \\ & \leq c ( \omega ) \left [ P ^ \omega ( D _ t ^ c \mid C _ t ) + P ^ \omega ( C _ t ^ c \mid S ) \right ] , \end{align*}


\begin{align*} ( E , \sigma ( H _ { \Lambda _ 2 } ) ) = \Vert ( H _ { \Lambda _ { L _ 2 } } - E ) ^ { - 1 } \Vert ^ { - 1 } = \inf _ { \psi \in \ell ^ 2 } \frac { \Vert ( H _ { \Lambda _ { L _ 2 } } - E ) \psi \Vert } { \Vert \psi \Vert } . \end{align*}


\begin{align*} ( \lambda , \ell ) + \rho = ( \lambda _ 1 + n - d - 1 , \lambda _ 2 + n - d - 2 , \ldots , \lambda _ { n - d - 1 } + 1 , \ell ) . \end{align*}


\begin{align*} \Delta u = \frac { 1 } { \sqrt { | g | } } \sum _ { j , k = 1 } ^ n \frac { \partial } { \partial x _ j } \left ( \sqrt { | g | } g ^ { j k } \frac { \partial } { \partial x _ k } u \right ) . \end{align*}


\begin{align*} p _ { Y _ k , 1 } ( X _ f ) = p _ { Z , k } ( X _ f ) . \end{align*}


\begin{align*} \Gamma ( S _ i ) \le \left ( \frac { 5 e ^ 2 \Delta ( S _ i ) } { b ^ 2 } \right ) ^ { \kappa } b _ 1 ( 3 \log _ 2 | S _ i | ) ^ { 1 + \kappa } = b _ 2 \Delta ( S _ i ) ^ \kappa ( \log _ 2 | S _ i | ) ^ { 1 + \kappa } \le b _ 2 | S _ i | ^ \kappa ( \log _ 2 | S _ i | ) ^ { 1 + \kappa } , \end{align*}


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


\begin{gather*} J \begin{bmatrix} { { \alpha } ^ { n } } \\ d \end{bmatrix} = \begin{bmatrix} A & b \\ c & - 1 \end{bmatrix} \begin{bmatrix} { { \alpha } ^ { n } } \\ d \end{bmatrix} = \begin{bmatrix} \bar { F } \\ F _ { M + 1 } \end{bmatrix} , \end{gather*}


\begin{align*} \| [ \zeta ] \| _ q = \inf \{ \| \zeta - \zeta ' \| : \ \zeta ' \in \mathrm { K e r } ( \lambda ^ * ) \cap \overline { R ( S ) } \} . \end{align*}


\begin{align*} \overline { b } + \widetilde { \mu } _ { i } \left ( \widetilde { b } - \overline { b } \right ) = \overline { b } + \mu _ { i } \left ( b - \overline { b } \right ) , i = 1 , . . . , N - 1 . \end{align*}


\begin{align*} \sup _ { \omega \in P } \omega ( a ^ * a ) = \| a \| ^ 2 , a \in A . \end{align*}


\begin{align*} \sum _ { \exp ( \log ^ \beta ( n - n ' ) ) \le p \le \exp ( \log ^ \beta n ) } \log ( 1 - p ^ { - 1 } ) \ll & \sum _ { \exp ( \log ^ \beta ( n - n ' ) ) \le p \le \exp ( \log ^ \beta n ) } p ^ { - 1 } \\ = & \beta \log \log n - \beta \log \log ( n - n ' ) + O ( \log ^ { - \beta } n ) \\ \ll & \log ^ { - \beta } n . \end{align*}


\begin{align*} f _ 1 ( z ) = i c _ 0 - I _ 0 + \frac { 1 } { 2 \pi } \langle ( h _ 0 ) _ b , P _ r ( \theta - \cdot ) \rangle \end{align*}


\begin{align*} \rho _ { s } ^ { t , \mu ; u } = \mathbb { P } _ { X _ { s } ^ { t , \xi ; u } } ^ { W } . \end{align*}


\begin{align*} \widetilde { \mathcal { T } } ( V , q ) : = \{ \alpha \in \mathcal { C } | \alpha ( \mathbb { F } \oplus V ) \alpha ^ { * } \subseteq \mathbb { F } \oplus V , \ N ( \alpha ) \in \mathbb { F } \} . \end{align*}


\begin{align*} D u _ \lambda ( t , x ) & = v _ { 0 , \lambda } ( t ) + \int _ 0 ^ t \ ! \ ! \int _ G K _ { t - s } ( x , y ) f ' _ \lambda ( u _ \lambda ( s , y ) ) D u _ \lambda ( s , y ) \ , d y \\ & + \int _ 0 ^ t \ ! \ ! \int _ G K _ { t - s } ( x , y ) \sigma ' _ \lambda ( u _ \lambda ( s , y ) ) D u _ \lambda ( s , y ) \ , \bar { W } ( d y , d s ) , \end{align*}


\begin{align*} \eta : = \frac { 1 } { t } \sqrt { \log \left ( \sqrt { \frac e 2 } \ , t ( b - a ) \right ) } \end{align*}


\begin{align*} \sigma _ r : = \begin{pmatrix} 1 & 2 & 3 & \cdots & n - 1 & n \\ n & n - 1 & n - 2 & \cdots & 2 & 1 \end{pmatrix} . \end{align*}


\begin{align*} \| f ^ { = d } \| _ { 2 } ^ { 2 } \leq \left ( \frac { C r } { d } \right ) ^ { 2 d } \alpha ^ { 2 } \log ^ { d } \left ( 1 / \alpha \right ) . \end{align*}


\begin{align*} \begin{aligned} T _ 3 \ge & - C s ^ 2 \lambda ^ 2 \iint _ { Q } \xi ^ 3 \left | A \nabla \eta \cdot \nabla \eta \right | ^ 2 u ^ 2 d x d t - C s ^ 2 \lambda ^ 2 \int _ { 0 } ^ { T } \int _ { \omega } \xi ^ 3 u ^ 2 d x d t , \end{aligned} \end{align*}


\begin{align*} Y = Y ^ { ( u ) } : t \mapsto \big ( X _ i ( T _ i ( u ) + t ) \big ) _ { i \in V } \end{align*}


\begin{align*} F ( D ^ { 2 } u , D u , u ) & = \int _ \Omega \tilde { g } \left ( \det ( u _ { i j } ) \right ) d x + \int _ \Omega \Tilde { G } ( x ) \det ( u _ { i j } ) d x \\ & + \int _ \Omega \int _ \Omega W ( x , y ) \det ( u _ { i j } ( y ) ) \det ( u _ { i j } ( x ) ) d y d x , \end{align*}


\begin{align*} \begin{aligned} \boldsymbol { K } _ 1 ( \widehat { \gamma } _ { 1 , 3 } , \mu _ 1 ) & \leq \left ( \frac { \eta } { \eta _ 0 } \right ) \boldsymbol { K } _ 1 ( \gamma ^ { \eta _ 1 , \delta } _ { 1 , 3 } , \mu _ 1 ) + \left ( 1 - \frac { \eta } { \eta _ 0 } \right ) \boldsymbol { K } _ 1 ( \widetilde { \gamma } _ { 1 , 3 } , \mu _ 1 ) \leq \eta . \end{aligned} \end{align*}


\begin{align*} Z _ 0 ( G ) = M _ 0 ( G ) = T . \end{align*}


\begin{align*} \overline { F } ( k , t + 1 ; p , n ) = \overline { F } ( k , t ; p , n ) + p _ { k } ^ { } f ( k , t ; p , n ) . \end{align*}


\begin{align*} M _ { [ 1 ] } \diamond \cdots \diamond M _ { [ j + 1 ] } & = S _ { [ 1 ] } \diamond \cdots \diamond S _ { [ j + 1 ] } + \mathcal { O } ( \| H \| ^ 2 ) . \end{align*}


\begin{align*} H _ { q , n } ( F _ { f } / 2 ) = \begin{vmatrix} \gamma _ { n } & \gamma _ { n + 1 } & \cdots & \gamma _ { n + q - 1 } \\ \gamma _ { n + 1 } & \gamma _ { n + 2 } & \cdots & \gamma _ { n + q } \\ \vdots & \vdots & \vdots & \vdots \\ \gamma _ { n + q - 1 } & \gamma _ { n + q } & \cdots & \gamma _ { n + 2 ( q - 1 ) } \end{vmatrix} . \end{align*}


\begin{align*} p ( x ) = M \mathrm { e } ^ { \beta ( x - \mu ) } | x - \mu | ^ { \nu } K _ { \nu } ( \alpha | x - \mu | ) , x \in \mathbb { R } , \end{align*}


\begin{align*} F _ 0 > \frac { 4 c _ 2 } { c _ 3 } = \frac { 3 2 \sigma \max \rho _ 0 M ^ 2 } { 3 - \gamma } . \end{align*}


\begin{align*} \alpha = \frac { n q } { n q - n + 2 - \theta } \quad \mbox { a n d } \beta = \frac { n q } { n - 2 + \theta } , \end{align*}


\begin{align*} N _ { \Q ( \beta ) | \Q } ( \gamma ) = \det ( T _ \gamma ) \in \Q . \end{align*}


\begin{align*} ( \Delta + \lambda - q _ 1 ) w = \psi ( q _ 2 - q _ 1 ) u _ 2 + [ \Delta , \psi ] u . \end{align*}


\begin{align*} { n + m \brack m } _ { q } & = \sum _ { k = \ell } ^ n { n - \ell \brack k - \ell } _ { q } { k + m - ( k - \ell ) \brack m - ( k - \ell ) } _ { q } q ^ { k ( k - \ell ) } \\ & = \sum _ { k = 0 } ^ { n - \ell } { n - \ell \brack k } _ { q } { m + \ell \brack m - k } _ { q } q ^ { k ( k + \ell ) } , \end{align*}


\begin{align*} \psi _ \infty ( w ) + \psi _ \infty ( v ) = \int _ { \R ^ 2 \setminus ( Q _ 1 \cup Q _ 2 \cup Q _ 3 ) } g ( b - w ) + g ( b - v ) \ , d \mu \geq - 6 \eta \ , . \end{align*}


\begin{align*} M = \begin{pmatrix} a & \frac { a \overline { a } - 1 } { c } \\ c & \overline { a } \end{pmatrix} , \end{align*}


\begin{align*} U _ n ^ { - 1 } \Sigma = \left [ \begin{array} { c | c } U _ { n - 1 } ^ { - 1 } & \mathbf { 0 } \\ \hline \mathbf { 0 } ^ \top & 1 \end{array} \right ] . \end{align*}


\begin{align*} \| v _ t \| _ { L ^ 2 ( \mathrm { M } _ 0 ) } ^ 2 = ( u | ( \Delta + \lambda - q ) w _ t ) _ { L ^ 2 ( \mathrm { M } _ 0 ) } . \end{align*}


\begin{align*} \tilde \gamma _ { S } ( e ) = \partial _ { e } \Big [ \phi _ { F } ( z _ { S } ( e ) ) - \phi _ { A } ( z _ { S } ( e ' ) ) \Big ] = \partial _ { e } \mu _ { S } ( e ) \tilde \gamma _ { B } ( e ) = - \partial _ { e } \phi _ { F } ( z _ { B } ( e ) ) = \partial _ { e } \mu _ { B } ( e ) \ , , \end{align*}


\begin{align*} \begin{aligned} \frac { r } { 2 } \| \lambda ^ k \| _ { ( \tilde { X } , \| \cdot \| _ * ) ' } & \leq \langle \lambda ^ k , r x ^ k \rangle _ * = - \langle \lambda ^ k , S v ^ k - t _ k ( \zeta ^ k - \zeta ^ * ) \rangle _ * < + \infty . \end{aligned} \end{align*}


\begin{align*} \left \{ \begin{array} { r l l l } w _ t + \mathcal { L } ^ { \Phi } _ A w & = 0 \quad \mbox { i n } Q _ 3 , \\ w & = u \quad \mbox { i n } ( \mathbb { R } ^ n \setminus B _ 3 ) \times \Lambda _ 3 \cup B _ 3 \times \{ - 3 ^ { 2 s } \} . \end{array} \right . \end{align*}


\begin{align*} \mathbb P \Big ( \big ( g ^ + _ { \tau _ 1 } ( X _ 1 ) , \dots , g ^ + _ { \tau _ n } ( X _ n ) \big ) \in A \Big ) \ge \exp \Big ( - \frac { p \| \tau \| ^ 2 } { 2 ( p - 1 ) } \Big ) \cdot \mathbb P \big ( ( X _ 1 , \dots , X _ n ) \in A \big ) ^ p , \end{align*}


\begin{align*} X = A \cdot \{ b _ 1 , b _ 2 \} = ( A \cdot b _ 1 ) \cup ( A \cdot b _ 2 ) . \end{align*}


\begin{align*} L u ( x ) = f ( x ) , \forall x \in G , \end{align*}


\begin{align*} P ^ \omega ( E _ t \mid S _ t ) = \frac { P ^ \omega ( E _ t \cap S _ t ) } { P ^ \omega ( S _ t ) } \leq c _ 1 P ^ \omega ( E _ t \cap S _ t ) . \end{align*}


\begin{align*} & \limsup _ { j \to \infty } \frac { 1 } { j } \log \| S ( j ) \mathfrak { r } _ r ( f ) \| _ { \sup } \leq - \lambda _ 1 r r > 0 \\ & \limsup _ { j \to \infty } \frac { 1 } { j } \log \| S ( j ) \mathfrak { r } _ r ( f ) \| _ { L ^ 1 } \leq - \lambda _ 1 r r \leq 0 . \end{align*}


\begin{align*} \overline { h } _ 2 ( z ) : = \overline { A _ 2 } z ^ { n + m } + \overline { B _ 2 } \overline { z } ^ m + \overline { C _ 2 } \textrm { f o r a l l } z \in \mathbb { C } . \end{align*}


\begin{align*} \left [ C \right ] _ { i , j } ^ { } = \begin{cases} \dfrac { n } { n } & \textnormal { i f } j = i = 1 ; \\ \dfrac { j } { n } & \textnormal { i f } j = i \neq 0 ; \\ \dfrac { n - j } { n } & \textnormal { i f } j = i + 1 ; \\ 0 & \textnormal { o t h e r w i s e } ; \end{cases} \end{align*}


\begin{align*} H _ { 2 , 1 } ( F _ { f } / 2 ) : = \begin{vmatrix} \gamma _ 1 & \gamma _ 2 \\ \gamma _ 2 & \gamma _ 3 \end{vmatrix} , \end{align*}


\begin{align*} \lim _ { n \to \infty } \overline { \mathbb { P } } _ { \mu } \left ( \exists m \ge n : \ L _ { x } \omega _ { m } \in A _ { n } ( x , \zeta ) | ( x , \zeta ) \in U \right ) = 0 . \end{align*}


\begin{align*} B ^ { k + \ell } = B ^ { k + \ell - 1 } \cup \{ u \} \setminus \{ v \} = B ^ { k + \ell - 2 } \cup \{ e , u \} \setminus \{ f , v \} . \end{align*}


\begin{align*} \sup _ { [ z _ 0 - \ell ( t ) , z _ 0 + \ell ( t ) ] ^ d } V ( \ : \cdot \ : , \omega ) = o ( \log \ell ( t ) ) , t \to \infty . \end{align*}


\begin{align*} P \mathbf { u } _ 0 = \lambda _ 0 \mathbf { u } _ 0 = \mathbf { 0 } . \end{align*}


\begin{align*} \tilde { \Lambda } _ i ( z ) = p _ + ^ i ( z ) = q _ + ^ i ( z ) ~ { \rm f o r } ~ i \in I _ w \cup I _ b . \end{align*}


\begin{align*} \liminf _ { m \rightarrow \infty } \| u _ m \| = C > 0 . \end{align*}


\begin{align*} P _ V a | _ V = \phi ( a ) I , a \in M \end{align*}


\begin{align*} \| ( f - p _ m ) e ^ { - m V } \| _ K = \| u _ m e ^ { - m V } \| _ K \leq \dfrac { C _ { \varepsilon , \gamma } e ^ { m \varepsilon } \| f \| _ { X _ { R - \gamma / 2 } } } { d _ m ^ { 1 / 2 } ( R - \gamma ) ^ m } . \end{align*}


\begin{align*} M ( z ) = \Lambda + z \left ( \Pi ^ * \Pi \right ) + z ^ 2 \left ( \Pi ^ * ( \mathcal { A } _ 0 - z I ) ^ { - 1 } \Pi \right ) . \end{align*}


\begin{align*} f ( t ) = \left \{ \begin{array} { l l } \frac { t } { | \log t | } & \ 0 < t < \frac 1 e \\ \frac 1 e & \ t \ge \frac 1 e \end{array} \right . . \end{align*}


\begin{align*} \mathbf { y } _ t = \beta _ t \mathbf { x } _ t + \mathbf { w } _ t , \end{align*}


\begin{align*} \nu _ i ( g ' ) = \nu ( g ' ) . \end{align*}


\begin{align*} ( \partial _ t - \partial _ x ^ 2 ) u = u \xi , \end{align*}


\begin{align*} a = a ( z ) = \sum _ { n \ge 0 } a _ n z ^ { n - 1 } . \end{align*}


\begin{align*} \mu _ { \omega , \mathbf { c } } : = \inf \{ S _ { \omega , \mathbf { c } } ( \Psi ) \ | \ \Psi \in \mathcal { H } ^ 1 \backslash \{ ( \mathbf { 0 } , \mathbf { 0 } , \mathbf { 0 } ) \} , \ K _ { \omega , \mathbf { c } } ( \Psi ) = 0 \} \end{align*}


\begin{align*} \begin{cases} \rho \ , u _ t - \Delta u + ( - \Delta ) ^ s u = 0 & \ , \ , \ , S _ T : = \R ^ N \times ( 0 , T ] \\ u = 0 & \ , \ , \ , \R ^ N \times \{ 0 \} . \end{cases} \end{align*}


\begin{align*} \mathcal { L } _ \varepsilon c ( x ) : = - \int _ { \Omega } J _ \varepsilon ( | x - y | ) c ( y ) \ : y + \int _ { \Omega } J _ \varepsilon ( | x - y | ) c ( x ) \ : y \ ; \ ; \ ; x \in \Omega . \end{align*}


\begin{align*} \frac { P ( f + T ) } { A ( f + T ) } = \frac { g ( T ) } { T } + \frac { g ( T ) Q ' ( T ) } { Q ( T ) } \ , \cdot \end{align*}


\begin{align*} C _ { \Tilde { q } } h ^ { \Tilde { q } } = \sum _ { m = 1 } ^ { \infty } C _ { q _ m } h ^ { q _ m } , \end{align*}


\begin{align*} h ( x + i y ) = \frac { 1 } { \pi } \langle h _ b , P ( x - \cdot , y ) \rangle . \end{align*}


\begin{align*} Q = \log _ { 2 } C \end{align*}


\begin{align*} \omega _ { n , a + 1 } = \omega _ { n , a } , \omega _ { n , a + \tau } = \sum _ { k = 0 } ^ n { ( 2 \pi \mathrm { i } ) ^ { n - k } \over ( n - k ) ! } \omega _ { k , a } . \end{align*}


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


\begin{align*} \int ( \omega + | z _ { S , e } | ^ { - 1 } \gamma _ { S } \delta _ { \Gamma _ { S } } + | z _ { B , e } | ^ { - 1 } \gamma _ { B } \delta _ { \Gamma _ { B } } ) = 0 \ , , \end{align*}


\begin{align*} C ^ \circ = \{ v \in \mathcal { U } : \ \langle v , w \rangle _ { \mathcal { U } } \leq 0 \forall \ w \in C \} . \end{align*}


\begin{align*} q _ e : = \max \left \{ p _ { e ' } \cdot \exp \bigg ( - \frac { \| e ' - e \| } { \log n } \bigg ) \colon e ' \in E ( \mathbb Z ^ d ) \right \} \end{align*}


\begin{align*} f = \sum _ { j = 1 } ^ r b _ j \textbf { Q } ^ { \lambda _ j } \mbox { w i t h $ b _ j \in K $ a n d } \lambda _ j ( Q _ k ) = 0 \mbox { i f } k \geq i \end{align*}


\begin{align*} \mu _ { K S } ^ \delta ( f ) ( \tau ) = \prod _ { D \mid N } \mu _ { K S } ( f ) ( D \tau ) ^ { n _ D } = \prod _ { D \mid N } \zeta ^ { \pi _ c ( f ) } \cdot u _ c ( f ) ( D \tau ) . \end{align*}


\begin{align*} y _ { } & = \sqrt { P } \sum _ { n = 1 } ^ N h _ { } ^ { ( n ) } \exp ( j \theta _ n ) h _ { } ^ { ( n ) } x + \epsilon _ { } , \end{align*}


\begin{align*} \psi ( b ) ( t ) = b ( t ) \xi ( t ) , b \in F ( \alpha , u ) , t \in G , \end{align*}


\begin{align*} \lim _ { n \to \infty } \| u _ n \| _ { p ^ * , q , \mu } = \| u _ \mu ^ \bigstar \| _ { p ^ * , q , \mu } . \end{align*}


\begin{align*} \widetilde { D } _ \alpha ( \psi | | \varphi ) : = \frac { 1 } { \alpha - 1 } \log \frac { \widetilde { Q } _ \alpha ( \psi | | \varphi ) } { \psi ( 1 ) } , \end{align*}


\begin{align*} q _ u ( t ) \geq \exp \left [ - \frac { ( 2 \ell ( t ) ) ^ 2 } { 2 \ell ( t ) } ( 1 + o ( 1 ) ) - \ell ( t ) \log \ell ( t ) \right ] \geq \exp \left [ - 2 \ell ( t ) \log \ell ( t ) \right ] = : p ( t ) , \end{align*}


\begin{align*} q ^ { k ^ 2 } [ 2 ] _ q ^ { k } S _ B [ n , k ] = \sum _ { \pi \in B _ { \subseteq } ( [ n ] , k ) } q ^ { m ( \pi ) } \end{align*}


\begin{align*} \bigg \langle \nabla V ( x ) , f ( x ) + \sum _ { i = 1 } ^ m g _ i ( x ) u _ i \bigg \rangle < 0 , \forall x \in \mathcal { D } \setminus \{ x _ * \} . \end{align*}


\begin{align*} \int _ \Omega f \ , d x = 0 . \end{align*}


\begin{align*} \partial \left ( \sum _ { i = 1 } ^ n c p _ i \max \{ - x _ i - y _ i , - d _ i \} \right ) & = \sum _ { i = 1 } ^ n c p _ i \partial \left ( \max \{ 0 , - x _ i - y _ i + d _ i \} \right ) \\ & = \sum _ { i = 1 } ^ n c p _ i \Gamma ( - x _ i - y _ i + d _ i ) \nabla _ v ( - x _ i ) \\ & = \sum _ { i = 1 } ^ n c p _ i \Gamma ( - x _ i - y _ i + d _ i ) ( e _ { i \bullet } - e _ { \bullet i } ) \end{align*}


\begin{align*} \left . \begin{array} { l l } \displaystyle ( 1 + 2 \Lambda ) \dfrac { a _ 1 } { 4 m _ 2 } \int _ { \mathbb { R } ^ N } | \nabla \eta | ^ 2 d x + \dfrac { a _ 1 } { 2 } \omega _ 2 \int _ { \mathbb { R } ^ N } \eta ^ 2 d x - \dfrac { 1 } { 2 } \left ( 1 + \dfrac { N } { 2 } \Lambda \right ) a _ 1 a _ 2 \int _ { \mathbb { R } ^ N } \xi ^ 2 \eta d x = 0 . \end{array} \right . \end{align*}


\begin{align*} \begin{cases} - \Delta u _ j = f ( u _ j ) + \lambda u _ j \ \ \ \ \mathbb R ^ N , \\ u _ j ( x ) > 0 , \ \lim _ { | x | \to \infty } u _ j ( x ) = 0 , i = 1 , 2 , \cdots , \ell \\ \sum _ { i = 1 } ^ { \ell } | u _ j | _ 2 ^ 2 = \alpha . \end{cases} \end{align*}


\begin{align*} f ( z ) = z + \sum _ { n = 2 } ^ { \infty } a _ n z ^ n , \ ; \mbox { f o r } \ ; z \in \mathbb { D } . \end{align*}


\begin{align*} \lim _ { N \to \infty } e ^ { - \lambda ( \tau _ N - t _ N ) } \cdot N ^ { - 1 } S _ j ( \tau _ N ) & = Y ^ { - 1 } \lim _ { N \to \infty } e ^ { - \lambda \tau _ N } S _ j ( \tau _ N ) \\ & = \nu \int _ 0 ^ \infty e ^ { - \lambda s } p _ j ( s ) d s , \end{align*}


\begin{align*} 0 = - | A | ^ 2 + 4 c ^ 2 \left ( \lambda + \frac { c ^ 2 } { m } \right ) , \end{align*}


\begin{align*} \big \langle z ^ m \lbrace 0 , \infty \rbrace , f \big \rangle = \frac { m ! \ , i ^ { m + 1 } } { ( 2 \pi ) ^ { m + 1 } } \ , L ( f , m + 1 ) \ , \ \ , \ 0 \leq m \leq k - 2 . \end{align*}


\begin{align*} \| u \mid L ^ 1 _ p ( D \| = \| \nabla u \mid L _ p ( D ) \| . \end{align*}


\begin{align*} P U _ t + \frac { 1 } { 2 } \left [ ( A U ) _ x + A U _ x + ( B U ) _ y + B U _ y \right ] = 0 . \end{align*}


\begin{align*} G ( z ) : = \log \beta ( z ) + u ^ 2 \slash 2 + B \vert z \vert ^ 2 \end{align*}


\begin{align*} K _ { \omega , \mathbf { c } } ( V _ n - V ) & = - K _ { \omega , \mathbf { c } } ( V ) + K _ { \omega , \mathbf { c } } ( V _ n ) - ( K _ { \omega , \mathbf { c } } ( V _ n ) - K _ { \omega , \mathbf { c } } ( V _ n - V ) - K _ { \omega , \mathbf { c } } ( V ) ) \\ & \rightarrow - K _ { \omega , \mathbf { c } } ( V ) < 0 . \end{align*}


\begin{align*} s _ n ( G ) \left ( \sum ^ { s - 1 } _ { i = 0 } l ( g _ { 1 , i } ) \otimes . . . \otimes l ( g _ { n , i } ) + \mathcal Q _ n ( G ) \right ) : = \overline \bigotimes ^ { s - 1 } _ { i = 0 } [ \langle 1 , - g _ { 1 , i } \rangle ] \overline \otimes . . . \overline \otimes [ \langle 1 , - g _ { n , i } \rangle ] \overline \otimes I ^ { n + 1 } ( G ) . \end{align*}


\begin{align*} \mathbb { E } [ H ( G ) ] = \frac { 1 } { 2 n } \sum _ { \pi \in \L } p ^ { f ( \pi ) } \ ; . \end{align*}


\begin{align*} ( - \Delta ) ^ s u ( \varphi ) & = \frac { C _ { N , s } } { 2 } \iint _ { \R ^ { 2 N } } \frac { ( u ( x ) - u ( y ) ) ( \varphi ( x ) - \varphi ( y ) ) } { | x - y | ^ { N + 2 s } } \ , d x \ , d y . \end{align*}


\begin{align*} E _ 1 & = \det T ( \phi ) T ( \phi _ R ) ^ { - 1 } \cdots T ( \phi _ 0 ) ^ { - 1 } , \\ E _ 2 & = \det T ( \tilde { \phi } ) T ( \tilde { \phi } _ R ) ^ { - 1 } \cdots T ( \tilde { \phi } _ 0 ) ^ { - 1 } , \end{align*}


\begin{align*} a _ { i + q , i } = \frac { \Delta _ { i + q } \Delta _ { i - 1 } } { \Delta _ { i + q - 1 } \Delta _ { i } } , i \ge 0 ; \end{align*}


\begin{align*} d = h ' h _ 1 h _ 2 ^ 2 \cdots h _ k ^ k h _ { k + 1 } , ( h ' , h _ i ) = ( h _ i , h _ j ) = 1 , h ' \mid ( N \Delta _ { F _ 2 } ) ^ \infty i \neq j , \end{align*}


\begin{align*} \displaystyle \limsup _ { n \rightarrow \infty } G ( U _ n ( t _ n ) ) = \displaystyle \limsup _ { n \rightarrow \infty } G ( \Phi _ { \omega , \mathbf { c } , n } ) \ge \eta . \end{align*}


\begin{align*} X Y Z = i \left ( \begin{array} { c c } A & 0 \\ 0 & - A \end{array} \right ) . \end{align*}


\begin{align*} | x _ 1 \bar { \otimes } x _ 2 | ^ p = | x _ 1 | ^ p \bar { \otimes } | x _ 2 | ^ p = h _ { \psi _ 1 } \bar { \otimes } h _ { \psi _ 2 } = h _ { \psi _ 1 \bar { \otimes } \psi _ 2 } . \end{align*}


\begin{align*} F \xi = B \xi + C \xi \end{align*}


\begin{align*} b _ { j } = & \left ( \sinh ( 2 \alpha x _ j ) - 2 \alpha h _ j - 2 \tanh ( \alpha x _ { j - 1 } ) ( \cosh ^ 2 ( \alpha x _ j ) - \alpha ^ 2 h _ { j } ^ 2 ) \right ) / F _ { j - 1 } \\ & - \left ( \sinh ( 2 \alpha x _ j ) + 2 \alpha h _ { j + 1 } - 2 \tanh ( \alpha x _ { j + 1 } ) ( \cosh ^ 2 ( \alpha x _ j ) - \alpha ^ 2 h _ { j + 1 } ^ 2 ) \right ) / F _ { j } \\ = & \dfrac { h _ { j } + h _ { j + 1 } } { 3 } + O \left ( ( \alpha \overline { h } ) ^ 2 \right ) ; \alpha \to 0 . \end{align*}


\begin{align*} B _ j = B _ { \rho _ j } ( X ) , \rho _ j = \left ( \frac { \sigma } { 2 } \right ) ^ { j - 1 } \rho . \end{align*}


\begin{align*} \hat { M } _ + ( t ) : = \nu \int _ 0 ^ t Y e ^ { \lambda s } d s \end{align*}


\begin{align*} w \rho _ { w } ( j , w ) = | w | ^ 2 - 2 i \frac { R _ j } { | w | } { \rm I m } ( w ( R _ j ) _ { z } ) \end{align*}


\begin{align*} \langle \lambda _ 0 , w \rangle = 0 , \forall \ , w \in T _ { \gamma ( 0 ) } \Sigma . \end{align*}


\begin{align*} L ( V ) U = G ( \vec x , t ) , t \geq 0 , \vec x = ( x _ 1 , x _ 2 , . . , x _ k ) \in \partial \Omega . \end{align*}


\begin{align*} d : = \left \{ \begin{array} { l l } \frac { p } { 2 } & \ p \geq 2 , \\ \frac { 2 p } { p ( n + 2 ) - 2 n } & \ p \leq 2 , \\ \end{array} \right . \end{align*}


\begin{align*} \alpha _ n = - \frac { v ( a ) } { p ^ n } \mbox { a n d } \beta _ n = - \frac { v ( a ) } { p ^ { n - 1 } } . \end{align*}


\begin{align*} \sum _ { k = 1 } ^ \infty s ^ { k - 1 } R _ k ( t ) = \sum _ j \nu ( j ) \sum _ { k = 1 } ^ \infty s ^ { k - 1 } M _ j ( \boldsymbol { a } , t ) ^ k = G ( t , s ) . \end{align*}


\begin{align*} { \phi } ( X E _ { i i } X ^ * \otimes B ) = ( Z _ X E _ { i i } Z _ X ^ * ) \otimes \varphi _ { i , X } ( B ) \hbox { f o r a l l \quad } i = 1 , \ldots , m \hbox { a n d } B \in M _ n . \end{align*}


\begin{align*} \begin{cases} \Delta _ g u = - \lambda _ u , & \Omega , \\ u = 0 , & \partial \Omega , \end{cases} \end{align*}


\begin{align*} t \cdot [ x _ { i , j } ] = [ t _ i t _ j x _ { i , j } ] , \ ; t _ i = \begin{cases} \mu _ i & i \in [ n ] , \\ \mu _ { i ^ * } & i \in [ n ^ * ] . \end{cases} \end{align*}


\begin{align*} + t ^ s { \rm R e } ( D \kappa ( t ) ^ 2 ) + t { \rm R e } ( E \kappa ( t ) ) + \sqrt { t ^ { 2 - 2 s } + | \kappa ( t ) | ^ 2 } \ , g ( t , t ^ s \kappa ( t ) ) = 0 . \end{align*}


\begin{align*} P ( z _ 1 , \ldots , z _ m , f ( z _ 1 , \ldots , z _ m ) ) = 0 , \ \forall ( z _ 1 , \ldots , z _ m ) \in \Omega , \end{align*}


\begin{align*} w _ m ( p ) \equiv \begin{cases} \ < p \ > ^ m \ , & p \in \S \\ 1 \ , & p \in \Lambda ^ * \backslash \S \end{cases} \ . \end{align*}


\begin{align*} \frac { d \bar { \mathbb { P } } } { d \mathbb { P } } \bigg | _ { \mathcal { F } _ t } = \mathcal { Z } _ t , \end{align*}


\begin{align*} { \rm t r } ^ \flat ( P ( h ) ) = \mathcal { O } ( h ^ { - 2 n - m } ) . \end{align*}


\begin{align*} r \gtrsim t ^ { 1 / s } \frac { 1 } { p } \approx ( n p ) ^ { p / \log n } \frac { 1 } { p } = e ^ { p ( 1 - \frac { \log ( 1 / p ) } { \log ( n ) } ) } \frac { 1 } { p } \approx \frac { 1 } { p } . \end{align*}


\begin{align*} \Psi _ { p \alpha + k } ^ { ( i ) } ( \xi _ i , t ) = \sum \limits _ { j = 1 } ^ { k + 1 } \frac { \xi _ i ^ j } { j ! } \ , \dfrac { \partial ^ j w _ { p \alpha + k - j } ^ { ( i ) } } { \partial x _ i ^ j } ( 0 , t ) i \in \{ 1 , 2 , 3 \} . \end{align*}


\begin{align*} ( q ) _ \lambda = \prod _ { i = 0 } ^ n \prod _ { j = 1 } ^ { \lambda _ i - \delta _ { i , \sigma } } \left ( 1 - q ^ j \right ) \end{align*}


\begin{align*} \Psi _ 1 \left ( \boldsymbol { d } _ { \mathcal { S } _ 1 } ( s _ 1 , s _ 1 ^ \prime ) ^ p , \rho _ 2 ( y _ 1 , y _ 1 ^ \prime ) ^ { p } \right ) = \Psi _ 1 \circ \psi _ 1 \left ( \boldsymbol { d } _ { \mathcal { Y } _ 1 } ( y _ 1 , y _ 1 ^ \prime ) ^ { p } , \boldsymbol { d } _ { \mathcal { Y } _ 2 } ( y _ 2 , y _ 2 ^ \prime ) ^ p , \boldsymbol { d } _ { \mathcal { X } } ( x , x ^ \prime ) ^ p \right ) , \end{align*}


\begin{align*} \lim _ { N \to \infty } \ , \sum _ { a N < n \le b N } \ , \frac { q _ { 2 n } ^ { \varphi _ N , \varphi _ N } } { N ^ 2 } \ , = \ ! \ ! \int \limits _ { \R ^ 2 \times \R ^ 2 } \varphi ( x ) \ , \varphi ( x ' ) \ , \bigg ( \int _ { a } ^ { b } \frac { 1 } { u } \ , g \bigg ( \frac { x - x ' } { \sqrt { u } } \bigg ) \ , \mathrm { d } u \bigg ) \ , \mathrm { d } x \ , \mathrm { d } x ' \ , , \end{align*}


\begin{align*} j _ m : = \max \{ j \in \{ 0 , \dotsc , n \} | \ | w _ { j } | < \ell \} . \end{align*}


\begin{align*} \frac { x } { y } = t = \frac { 1 } { 2 } \left ( \alpha - \sqrt { \alpha ^ 2 - 4 } \right ) \end{align*}


\begin{align*} \chi _ { I , \pm } ( H _ \omega ) : = i _ - \chi _ I ( H _ \omega ) i _ + \in \mathcal B ( \mathcal H _ + , \mathcal H _ - ) , \end{align*}


\begin{align*} ( \vec U , { \bf D _ { x _ i } } \vec V ) = \vec U ^ T ( I _ n \otimes P _ { \Omega } ) ( { \bf D _ { x _ i } } \vec V ) = - ( { \bf D _ { x _ i } } \vec U , \vec V ) + ( E \vec U ) ^ T ( I _ n \otimes P _ { \partial \Omega } ) N _ { i } ( E \vec V ) . \end{align*}


\begin{align*} \widehat { \xi } _ { [ ( \sigma , k , l ) ] } ( s ) & = \frac { ( s - l _ \alpha ) ^ { \frac { k - ( m _ \sigma - 2 ) } { m _ \sigma } } } { m _ \sigma ^ 2 k ! } k \neq m _ \sigma - 2 \ \operatorname { m o d } m _ \sigma \\ \widehat { \xi } _ { [ ( \sigma , k , l ) ] } ( s ) & = - \frac { ( s - l _ \alpha ) ^ { \frac { k - ( m _ \sigma - 2 ) } { m _ \sigma } } \log ( s - l _ \alpha ) } { m _ \sigma ^ 2 k ! } k = m _ \sigma - 2 \ \operatorname { m o d } m _ \sigma ; \end{align*}


\begin{align*} \omega \left [ 2 \pi \iota n \left ( r _ n , \frac { 1 } { f } \right ) + \frac { k - 1 } { 2 k } 2 \pi \iota n \left ( r _ n , \frac { 1 } { A } \right ) \right ] + o ( 1 ) = 2 \pi \iota n \left ( r _ n , \frac { 1 } { y } \right ) + \frac { k - 1 } { 2 k } 2 \pi \iota n \left ( r _ n , \frac { 1 } { A } \right ) . \end{align*}


\begin{align*} Q ( k , l ) = Z ( l ) \cdot Z ( l - 1 ) \cdot \dotsc \cdot Z ( k + 2 ) \cdot Z ( k + 1 ) = A ^ { ( n _ { l } - n _ { k } ) } ( \mathcal { R } ^ { n _ k } ( T ) ) ^ t . \end{align*}


\begin{align*} f ( a , b ) : = ( p - 1 ) \left ( 2 a b \kappa ( n - 1 ) - a ^ { \frac { p } { p - 1 } } - a b ^ 2 p \right ) \ge C . \end{align*}


\begin{align*} \left ( \mathcal { F } _ { \mathcal { H } _ { \hbar , V } } f \right ) ( \xi ) = \widehat { f } ( \xi ) : = \sum \limits _ { k \in \hbar \mathbb { Z } ^ { n } } f ( k ) \overline { u _ { \xi } ( k ) } , \xi \in \mathcal { I } _ { \hbar } , \end{align*}


\begin{align*} \int x ^ \mu K _ \nu ( x ) \ , \mathrm { d } x = - 2 ^ { \mu - 1 } \Gamma \bigg ( \frac { \mu - \nu + 1 } { 2 } \bigg ) \Gamma \bigg ( \frac { \mu + \nu + 1 } { 2 } \bigg ) G _ { \mu , \nu } ( x ) , \end{align*}


\begin{align*} F _ { N } ( t , x ) \coloneqq - \frac { 1 } { 2 } + \frac { 1 } { N } \stackrel [ k = 1 ] { N } { \sum } m _ { k } ( t ) H ( x - x _ { k } ( t ) ) , \end{align*}


\begin{align*} X ( w ) ' = X ' ( w ^ { - 1 } ) . \end{align*}


\begin{align*} \langle f , g \rangle = - f \circ g . \end{align*}


\begin{align*} \left ( ( \mathcal G : ) \times ( u : \mathcal G ^ { L } ) \times ( i , j : L ) \to u _ i = u _ j \right ) \to \rlap { . } \end{align*}


\begin{align*} \Gamma ( x ) : = \int _ { 0 } ^ { \infty } e ^ { - t } t ^ { x - 1 } d t , \Psi ( x ) : = \frac { \Gamma ' ( x ) } { \Gamma ( x ) } . \end{align*}


\begin{align*} \gamma \ast m ^ { \delta } = \sum _ { g } n _ g \cdot ( \gamma g ) \ast m = \sum _ g n _ g \cdot ( g \gamma _ g ) \ast m = \sum _ g n _ g \cdot g \ast m = m ^ \delta . \end{align*}


\begin{align*} \mathsf { F l } _ { 1 2 } = \{ L _ 1 \subset L _ 2 \subset \C ^ 4 | \dim L _ i = i \} , \ \ \mathsf { F l } _ { 1 2 3 } = \{ L _ 1 \subset L _ 2 \subset L _ 3 \subset \C ^ 4 | \dim L _ i = i \} \end{align*}


\begin{align*} \Phi \left ( x , x ^ { 1 / 3 } \right ) \ge \frac { x } { \log x ^ { 1 / 3 } } \left ( \omega \left ( \frac { \log x } { \log x ^ { 1 / 3 } } \right ) + \frac { \Delta _ 3 ^ - } { \log x ^ { 1 / 3 } } \right ) = \frac { 3 x } { \log y } \left ( \frac { \omega ( 3 ) } { u } + \frac { 3 \Delta _ 3 ^ - } { u ^ 2 \log y } \right ) . \end{align*}


\begin{align*} \left ( T _ \lambda ^ { \vec { v } } \right ) _ { i , j } = \displaystyle \beta _ { - i + j } + \sum _ { k = j } ^ { n - 1 } \beta _ k \sum _ { r = 0 } ^ { \min \{ k - j , n - i \} } \alpha _ { n - j - r } \sum _ { \mathclap { \substack { p _ 1 \dots p _ m \\ m \leq k - j - r \\ p _ s \geq 1 \\ \sum _ { s = 1 } ^ m p _ s = k - j - r } } } ( - 1 ) ^ { m + 1 } \alpha _ { n - p _ 1 } \dots \alpha _ { n - p _ m } . \end{align*}


\begin{align*} \mathcal { B } _ T \left ( \mathcal { A } [ \phi ] v \right ) ( \xi , x ) = \mathcal { A } _ \xi [ \phi ] \mathcal { B } _ T ( v ) ( \xi , x ) , v \in L ^ 2 _ N , \end{align*}


\begin{align*} D _ { I } = X \bigcap F _ { I } , \ X = X _ { 1 } \times \dots \times X _ { m } = \prod \limits _ { i = 1 } ^ { m } X _ { i } , \ X _ { i } \subseteq \mathbb { R } ^ { n } , \ i = 1 , \dots , m ; \end{align*}


\begin{align*} \langle \xi , \eta \rangle : = \sum _ { t \in G } \xi ( t ) ^ * \eta ( t ) . \end{align*}


\begin{align*} \int _ { U ^ - U ^ 0 U } \varphi ( g ) \ , d m _ G ( g ) = \int _ { U ^ - \times U ^ 0 \times U } \varphi ( u ^ - u ^ 0 u ) \Delta ( u ^ 0 ) \ , d \nu ^ - ( u ^ - ) d \nu ^ 0 ( u ^ 0 ) d \nu ( u ) \end{align*}


\begin{align*} \Bigl ( y ( Z , t ) \le s \Bigr ) & = \Bigl ( \alpha _ { m } ( Z ^ { t _ { m } } ) \le \frac { s - y ( t _ { m } ) } { t - t _ { m } } \Bigr ) \\ & = \Bigl ( \alpha _ { m } ( Z ^ { t _ { m } } ) \le \frac { s - y ( t _ { m } ) } { t - t _ { m } } \Bigr ) \cap ( y ( Z , t _ { m } ) \le s ) \end{align*}


\begin{align*} H = ( 1 , 2 , 3 , 2 , 1 ) , P _ H = \begin{array} { c c c } \bullet & \bullet & \bullet \\ \bullet & \bullet & \\ \bullet & \bullet & \\ \bullet & & \\ \bullet & & \end{array} H ^ \vee = ( P _ H ) ^ \vee = ( 5 , 3 , 1 ) : \begin{array} { c c c c c } \bullet & \bullet & \bullet & \bullet & \bullet \\ \bullet & \bullet & \bullet & & \\ \bullet & & & & \end{array} \end{align*}


\begin{align*} \Lambda _ { u } = \left \{ \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 ) : ( \bar { p } , \bar { q } ) \in \Z ^ m \times \Z ^ n \right \} \end{align*}


\begin{align*} \sum _ { \substack { K = K _ 1 - K _ 2 + K _ 3 \\ \Delta \omega _ { K K _ 1 K _ 2 K _ 3 } \neq 0 } } & \eta _ { K _ 1 } \overline { \eta _ { K _ 2 } } \eta _ { K _ 3 } \frac { 1 - e ^ { - i t \Delta \omega _ { K K _ 1 K _ 2 K _ 3 } } } { i \Delta \omega _ { K K _ 1 K _ 2 K _ 3 } } \\ & = L ^ 4 \int _ { K = k _ 1 - k _ 2 + k _ 3 } \eta ( k _ 1 ) \overline { \eta ( k _ 2 ) } \eta ( k _ 3 ) \frac { 1 - e ^ { - i t \Delta \omega _ { K k _ 1 k _ 2 k _ 3 } } } { i \Delta \omega _ { K k _ 1 k _ 2 k _ 3 } } \dd k _ 1 \dd k _ 2 \dd k _ 3 + \mathcal { O } ( L ^ 3 ) \end{align*}


\begin{align*} \Omega _ \infty : = \{ Z _ 0 ( t ) > 0 \ ; \} . \end{align*}


\begin{align*} \sum _ { | d | \leq \widehat { T } ^ { - 1 } \widehat { Y } ^ { 1 / 2 } } \widehat { C } _ 1 \widehat { C } _ 2 | d | = \widehat { Y } \sum _ { | d | \leq \widehat { T } ^ { - 1 } \widehat { Y } ^ { 1 / 2 } } | d | ^ { - 1 } \ll \widehat { Y } ^ { 1 + \varepsilon } . \end{align*}


\begin{align*} d = \texttt { w p i n i t i a l } + 1 + \texttt { m a g } \Bigl \{ 1 2 \frac { t } { 2 \pi } \log \frac { t } { 2 \pi } \Bigr \} \end{align*}


\begin{align*} A _ { t w } = \frac { \rho _ { F } - \rho _ { A } } { \rho _ { F } + \rho _ { A } } \end{align*}


\begin{align*} \mathbb { E } ( X ) \leq 0 + \frac { 2 t ^ 2 } { \ell + 2 } + \sum _ { \substack { h < i < j \leq k \\ j - i \leq t } } \mathbb { P } ( s _ i = s _ j ) . \end{align*}


\begin{align*} \varphi ( x ) = \inf _ { \lambda \in \mathbb { R } ^ n _ + } \left \{ \langle \lambda , x \rangle + \varphi ^ \star ( \lambda ) \right \} . \end{align*}


\begin{align*} & \psi _ r ( \beta ) = I _ r ( \bar { h } ) \geq \int _ { S _ r ( \beta ) } \d x \ , J _ r ( x , \beta ) . \end{align*}


\begin{align*} ( \rho , u , \mathbb { F } ) ( x , t ) \mid _ { t = 0 } = ( \rho _ 0 , u _ 0 , \mathbb { F } _ 0 ) ( x ) , x \in \Omega \end{align*}


\begin{align*} \chi _ { t t } + \frac { G ( r ) } { \chi ^ 2 } = 0 , \end{align*}


\begin{align*} A _ { n } ( t ) = \sum _ { k = 0 } ^ { \left \lfloor ( n - 1 ) / 2 \right \rfloor } \gamma _ { k } t ^ { k } ( 1 + t ) ^ { n - 1 - 2 k } . \end{align*}


\begin{align*} ( D ^ 2 - \alpha ^ 2 ) ^ k s = 0 \quad [ a , b ] \setminus X , \end{align*}


\begin{align*} \gamma _ { \mathrm { n l } } ( x , t ) = \gamma ( x , t ) + \frac 1 N \sigma ( t ) , \end{align*}


\begin{align*} x ^ 2 + y ^ 3 = z ^ 5 . \end{align*}


\begin{align*} \sigma _ Y ( g \cdot y ) = g ^ { - 1 } \sigma _ Y ( y ) \ , \ , \forall \ , y \in Y \ , . \end{align*}


\begin{align*} x _ i = \sum _ { k = 1 } ^ \infty x _ { i , k } 2 ^ { - k } , \end{align*}


\begin{align*} x ^ r \cdot x ^ q + r y ^ { r - 1 } \cdot ( y ^ q z - y z ^ q ) + y ^ r \cdot ( - z ^ q ) = 0 \end{align*}


\begin{align*} - d y _ t = g ( x _ t , y _ t , z _ t , \tilde { z } _ t , \gamma _ { ( t , e ) } ) d t - z _ t d W _ t - \tilde { z } _ t d \xi _ t - \int _ { \mathcal { E } } \gamma _ { ( t , e ) } \tilde { N } ( d e , d t ) , \ t \in [ 0 , \infty ) , \end{align*}


\begin{align*} \lim _ { d \to \frac { 1 } { 2 } } \bigg \{ \sec ^ 2 ( d \pi ) - \frac { 2 } { \pi } \frac { \tan ( d \pi ) } { 1 - 2 d } \bigg \} = \frac { 2 } { 3 } . \end{align*}


\begin{align*} f _ t ( p ) = | \Lambda | ^ { - 1 } { \nu _ t ( a _ p ^ * a _ p ) } \ . \end{align*}


\begin{align*} \widetilde { d } ( \theta , f ( \theta ) ) = { \rm R e } ( q ( \theta , f ( \theta ) ) ) + \widetilde { e } ( \theta , f ( \theta ) ) \end{align*}


\begin{align*} \lim _ { j \to + \infty } \int _ \Omega ( - u _ j ) ^ { n + 1 } d V _ g = \int _ \Omega ( - u ) ^ { n + 1 } d V _ g . \end{align*}


\begin{align*} \lim \limits _ { k \rightarrow \infty } \| y ^ { k } - y ^ { k - 1 } \| = 0 . \end{align*}


\begin{align*} \ker \begin{bmatrix} D _ 1 ^ { 1 1 } & - I \\ - I & D _ 2 ^ { 1 1 } \end{bmatrix} = \{ 0 \} . \end{align*}


\begin{align*} \theta \mathrm { d } \eta = \mathrm { d } e + p \mathrm { d } v - \frac { Z ( \theta ) } { \theta } q \mathrm { d } q , \end{align*}


\begin{align*} U _ n = [ X _ { n 1 } , \dots , X _ { n j } ] + \sum _ { m \geq j + 1 } \sum _ { ( i _ 1 , \dots , i _ m ) \in \mathbb { I } _ j ^ m } \gamma _ { ( i _ 1 , \dots , i _ m ) } [ X _ { n i _ 1 } , \dots , X _ { n i _ m } ] . \end{align*}


\begin{gather*} g _ { r , s } ( x ) = r ^ { 2 - m } h ( x / r ) - s ^ { 2 - m } h ( x / s ) . \end{gather*}


\begin{align*} \int _ { \mathbb { R } ^ n } J _ \varepsilon ( | x | ) x _ m ^ 2 \ : x = \int _ { \mathbb { R } ^ n } J _ \varepsilon ( | x | ) x _ 1 ^ 2 \ : x \end{align*}


\begin{align*} C f - ( 1 - \lambda \underline u ) f & = \lambda C f ( \Vert u _ 0 \Vert _ { C ^ 0 ( \bar \Omega ) } + u _ 0 ) \\ & \geq 0 . \end{align*}


\begin{align*} \ddot A ( t ) & = \delta \det ( A ( t ) ) ^ { 1 - \gamma } A ( t ) ^ { - \top } , \\ A ( 0 ) & = A _ 0 , \ \ \dot A ( 0 ) = A _ 1 , \\ \nabla _ x w & = - \delta \frac { \gamma - 1 } { \gamma } x . \end{align*}


\begin{align*} \tilde { a } = \frac { \frac { p + \gamma - 1 } { 2 } - \frac { ( p + \gamma - 1 ) ( q - 1 ) } { 2 q ( p - \gamma + 1 ) } } { \frac { p + \gamma - 1 } { 2 } + \frac { 1 } { 2 } } = \frac { ( q ( p - \gamma ) + 1 ) ( p + \gamma - 1 ) } { q ( p + \gamma ) ( p - \gamma + 1 ) } . \end{align*}


\begin{align*} & \lim _ { t \to \infty } \frac { S _ j ( t ) } { M ( t ) } = \lim _ { N \to \infty } \frac { S _ j ( \tau _ N ) } { M ( \tau _ N ) } = \frac { \int _ 0 ^ \infty e ^ { - \lambda s } p _ j ( s ) d s } { \int _ 0 ^ \infty e ^ { - \lambda s } ( 1 - p _ 0 ( s ) ) d s } , j \geq 1 , \end{align*}


\begin{align*} Z _ { N , \beta } ^ \omega ( \varphi ) : = \int _ { \R ^ 2 } Z _ { N , \beta } ^ { \omega } ( \lfloor \sqrt { N } x \rfloor ) \ , \varphi ( x ) \ , \dd x = \frac { 1 } { N } \sum _ { z \in \Z ^ 2 } Z _ { N , \beta } ^ { \omega } ( z ) \ , \varphi _ N ( z ) \ , , \end{align*}


\begin{align*} \widetilde { s } ( g ) = \widetilde { s } ( k _ g ) \textrm { a n d } \overline { s } ( g ) = \overline { s } ( k _ g ) . \end{align*}


\begin{align*} \int _ 0 ^ { \infty } e ^ { - \lambda s } ( 1 - p _ 0 ( s ) ) d s = \dfrac 1 { \lambda } \int _ 0 ^ { 1 } \frac { q } { 1 - p x } d x = \begin{cases} \dfrac 1 \lambda , & p = 0 , \\ - \dfrac { q \log ( q ) } { \lambda p } , & 0 < p < 1 . \end{cases} \end{align*}


\begin{align*} 1 \geq M _ k = \max _ { 1 \leq l \leq k } u _ l m _ k = \max _ { 1 \leq l \leq k - 1 } u _ l \geq 0 . \end{align*}


\begin{align*} \mathcal C = \bigg \{ z \in \R ^ N : \ , 4 R \leq | z | \leq 5 R , \ , \frac { z \cdot z ^ + } { | z | \cdot | z ^ + | } \geq \cos ( \pi / 1 5 ) \bigg \} \ , , \end{align*}


\begin{align*} \| T _ { s , v } ( x ) \| & = \| P _ s ( \lambda ( x ) ( v \delta _ e ) ) \| \leq \| \lambda ( x ) ( v \delta _ e ) \| _ 2 \leq \| \lambda ( x ) \| \| v \delta _ e \| _ 2 \\ & = \| x \| _ \lambda \| v \| \leq \| x \| _ { \lambda } . \end{align*}


\begin{align*} \gamma '' : = \left ( \frac { 1 0 0 r } { \sqrt { d - 1 } } \right ) ^ { d - 1 } r \gamma _ 1 \log ^ { ( d - 1 ) / 2 } \left ( \frac { \gamma _ 2 } { \gamma _ 1 } \right ) \end{align*}


\begin{align*} E ( a , b , c ) : = \left \{ \frac { x ^ 2 } { a ^ 2 } + \frac { y ^ 2 } { b ^ 2 } + \frac { z ^ 2 } { c ^ 2 } \leq 1 \right \} \subset \mathbb { R } ^ 3 . \end{align*}


\begin{align*} R _ \ell : = R _ 0 ( \log \ell ) ^ { 1 / d } - ( \log \log \ell ) ^ 2 , \ : \ : \ell > 1 . \end{align*}


\begin{align*} \lim _ { t \to \tau _ i } U _ i ( t ) = + \infty , \end{align*}


\begin{align*} k Y + \frac { R _ { k - 2 } ( Y , M ) } { M Y ^ { k - 2 } } = \frac { h } { M Y ^ { k - 2 } } . \end{align*}


\begin{align*} A ( X _ i / \pi ) = \begin{bmatrix} a _ i & k _ i - a _ i \\ \frac { ( k _ i - a _ i ) | \mathcal { S } | } { | \Omega | - | \mathcal { S } | } & \frac { d | \Omega | + a _ i | \mathcal { S } | - 2 k _ i | \mathcal { S } | } { | \Omega | - | \mathcal { S } | } \end{bmatrix} . \end{align*}


\begin{align*} X ( K ) \cap \Gamma = \left \{ p ^ n Q \colon n \ge 0 \right \} . \end{align*}


\begin{align*} I _ n = \{ i \in I \mid Q _ i \in \textbf { Q } _ n \} . \end{align*}


\begin{align*} T ' = \begin{pmatrix} W ' & X ' \\ Y ' & Z ' \end{pmatrix} , \end{align*}


\begin{align*} u _ { x _ n } ( 0 ) \leq w _ { x _ n } ( 0 ) = : - \delta _ 0 < 0 . \end{align*}


\begin{align*} D _ j S _ { \omega , \mathbf { c } } ( \Psi ) = 0 , \ \ j = 1 , 2 , 3 . \end{align*}


\begin{align*} \rho _ { + } = e ^ { - \phi } , \end{align*}


\begin{align*} f ^ n ( x , t ) = \begin{cases} f ^ { \mathcal { K } _ n } ( x , t ) & t \in [ 0 , 5 0 ] , \\ f ^ { \mathcal { K } _ n } ( x , 1 0 0 - t ) & t \in [ 5 0 , 1 0 0 ] . \end{cases} \end{align*}


\begin{align*} D u ^ { k \bar q } = - u ^ { q \bar q } u ^ { k \bar k } D u _ { q \bar k } . \end{align*}


\begin{align*} \rho ( a ^ * a ) = \rho ( a ) ^ * \rho ( a ) , a \in X . \end{align*}


\begin{align*} \Pi _ { \psi } ( g ) f ( [ \epsilon , w ] ) & = \widetilde { C } _ { X ^ { \ast } } ( g , h _ { \epsilon } ) \widetilde { \beta } ( g ^ { s ( \epsilon ) } ) ^ { - 1 } f ( [ \epsilon , w g ^ { s ( \epsilon ) } ] ) . \end{align*}


\begin{align*} \mathrm { t r } \ , \mathcal { B } ^ H = \mathrm { d i v } _ T \big ( \nabla _ \xi H ( \nu ) \big ) \ , . \end{align*}


\begin{align*} \mathrm { c l } ( X ) = ( \bigcup X ) ^ \supseteq . \end{align*}


\begin{align*} F _ { 2 i } ( V ) = K _ X ^ { n - 1 } \oplus \cdots \oplus K _ X ^ { n + 1 - 2 i } \ , \ , \ , ( i = 1 , \ldots , n ) . \end{align*}


\begin{align*} I _ r ( h ) = \int _ { [ 0 , 1 ] ^ 2 } \d x \ , \d y \ , \ , \mathcal { R } \big ( h ( x , y ) \mid r ( x , y ) \big ) , h \in \mathcal { W } , \end{align*}


\begin{align*} [ b , R _ { { \rm D } , \ell } ] f ( x ) & = b ( x ) R _ { { \rm D } , \ell } f ( x ) - R _ { { \rm D } , \ell } ( b f ) ( x ) \\ & = \int _ { \mathbb R ^ N } ( b ( x ) - b ( y ) ) R _ { { \rm D } , \ell } f ( y ) d \omega ( y ) , \end{align*}


\begin{align*} Y _ A = \{ ( w , z ) \in A \times S ^ 1 \setminus \{ 1 \} \ | \ z \neq w _ i i \in \{ 1 , . . . , n \} \} \ , \end{align*}


\begin{align*} Y ^ { y ( a ) } ( t ) & = Y \left ( y ( a ) \wedge t \right ) \\ & = N \left ( y ^ { * } ( N , \ , y ( a ) \wedge t ) \right ) \\ & = N \left ( a \wedge y ^ { * } ( N , t ) \right ) . \end{align*}


\begin{align*} = \frac { P ' } { P } + U ' . \end{align*}


\begin{align*} a x ^ n y ^ q + b x ^ k y ^ l + c x ^ r y ^ m = 0 , \end{align*}


\begin{align*} F = \frac { P ' } { P } + P _ 0 + Y - \frac { Q ' } { Q } = Y + M , \end{align*}


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


\begin{align*} & \sigma ^ { ( \mathcal { I } , r e d ) } ( H _ { \omega , \Lambda _ L ( x _ 0 ) } ) : = \\ & \left \{ E \in \sigma ^ { ( \mathcal { I } ) } ( H _ { \omega , \Lambda _ L ( x _ 0 ) } ) : \left ( E , \sigma ^ { ( \mathcal { I } ) } ( H _ { \omega , \Lambda _ { L _ n } ( x _ 0 ) } \right ) \leq 2 e ^ { - \frac { \hat { m } } { K } L _ n } , n = 1 , \cdots , n _ 1 \right \} \end{align*}


\begin{align*} \int _ S \ d \acute { \rho } : = & \sum _ { 0 \leq a < b \leq 3 } \int _ { I ^ 2 } \acute { \rho } _ \sigma ^ { a b } ( s , t ) [ \det \acute { J } _ { a b } ^ \sigma ( \hat { s } ) ] \ d \hat { s } , \\ \int _ S d | \acute { \rho } | : = & \int _ { I ^ 2 } \left | \sum _ { 0 \leq a < b \leq 3 } \acute { \rho } _ \sigma ^ { a b } ( \hat { s } ) [ \det \acute { J } _ { a b } ^ \sigma ] ( \hat { s } ) \right | \ d \hat { s } . \end{align*}


\begin{align*} \partial _ t w = - i h ^ { - 1 } \tilde { P } _ h ( 0 ) w = - i h ^ { - 1 } ( - i h X - i ( q _ 1 + W + Q _ \infty ) ) w , w | _ { t = 0 } = f . \end{align*}


\begin{align*} \mathcal { O } ^ - _ i : = \left \{ \left ( ( A ^ + , \underline { A } ^ + ) , ( B ^ + , \underline { B } ^ - ) \right ) , \left ( ( A ^ + , \underline { A } ^ + ) , ( B ^ - , \underline { B } ^ + ) \right ) : A , B \in \binom { [ n ] } { k } \mbox { a n d } | A \cap B | = k - i \right \} . \end{align*}


\begin{align*} P = \big ( p _ 1 ^ { n _ 1 } , \ldots , p _ c ^ { n _ c } , ( d - n ) ^ { d - n + k - 1 } \big ) , \end{align*}


\begin{align*} h ' ( \tau ) & = - ( 4 - d ) ( \sqrt { \omega } - \tau ) ^ { 3 - d } \mu \left ( 1 , \frac { \mathbf { c } } { \sqrt { \omega } } \right ) , \\ h '' ( \tau ) & = ( 3 - d ) ( 4 - d ) ( \sqrt { \omega } - \tau ) ^ { 2 - d } \mu \left ( 1 , \frac { \mathbf { c } } { \sqrt { \omega } } \right ) , \end{align*}


\begin{align*} h ( \alpha _ 1 , \ldots , \alpha _ n ) = \log H ( \alpha _ 1 , \ldots , \alpha _ n ) . \end{align*}


\begin{align*} \nu _ j ( g ' ) = \nu ( g ' ) \mbox { f o r e v e r y } j \in I ' , j \geq j _ 0 . \end{align*}


\begin{align*} ( \Delta + \lambda - q _ 1 ) \tilde { v } _ 2 = 0 \ ; \mathrm { i n } \ ; \mathrm { M } ( \partial _ \nu - i a ) \tilde { v } _ 2 = ( \partial _ \nu - i a ) v _ 2 . \end{align*}


\begin{align*} \Sigma = W _ 1 \cup _ { K _ 1 } \cdots \cup _ { K _ { 2 m - 1 } } W _ { 2 m } \end{align*}


\begin{align*} \frac { d } { d t } \left [ \begin{array} { c c } u \\ u _ { t } \end{array} \right ] = \left [ \begin{array} { c c } u _ { t } \\ - \alpha f ( u _ { t } ) - H _ { \lambda } ( u ) \end{array} \right ] . \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ N } \left ( \left | \nabla v _ n \right | ^ 2 + ( V ( \epsilon x ) + 1 ) \left | v _ n \right | ^ 2 \right ) d x + \int _ { \mathbb { R } ^ N } F ' _ 1 ( v _ n ) v _ n d x - \lambda _ \epsilon \int _ { \mathbb { R } ^ N } \left | v _ n \right | ^ 2 d x = \int _ { \mathbb { R } ^ N } F ' _ 2 ( v _ n ) v _ n d x + o _ n ( 1 ) . \end{align*}


\begin{align*} E _ f ^ { - 1 } = \left ( \overline \rho _ 1 | \dots | \overline \rho _ N \right ) . \end{align*}


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


\begin{align*} \sigma \curvearrowright ^ { \mathbf { n } } X ^ { \mathbf { n } ' } = \sigma \delta _ \mathbf { n } ^ { \mathbf { n } ' } \end{align*}


\begin{align*} \| S u ^ k - \zeta ^ k \| = \| r ^ k \| \ \rightarrow 0 \ \ \mbox { a s } \ \ k \rightarrow + \infty . \end{align*}


\begin{align*} \tfrac { 1 } { 2 } \nabla _ \xi H ^ 2 ( \nabla v ) \cdot ( \partial _ \nu v ) \nu = H ^ 2 ( \nabla v ) \ , . \end{align*}


\begin{align*} P ^ \omega \left ( A _ t ^ c \cap S _ { \alpha t } \right ) & = P ^ \omega \left ( A _ t ^ c \cap S _ { \alpha t } \cap E _ t ^ c \right ) + P ^ \omega \left ( A _ t ^ c \cap S _ { \alpha t } \cap E _ t \right ) \\ & \leq P ^ \omega \left ( E _ t ^ c \mid S _ { \alpha t } \right ) + P ^ \omega \left ( A _ t ^ c \mid E _ t \right ) . \end{align*}


\begin{align*} \begin{cases} \tilde { P } ( ( - \Delta ) ^ { \tilde { s } } ) u = 0 & \mathbb { R } ^ n , \\ u = 0 & W , \end{cases} \end{align*}


\begin{align*} & ( a ) \ - \bigg \langle \nabla V ( x ) , f ( x ) + \sum _ { i = 1 } ^ m g _ i ( x ) u _ i ( x ) \bigg \rangle \\ & + \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*} \left ( \mathord { \operatorname { p r o j } } _ { K _ i \cap K _ j } \circ { \operatorname { p r o j } _ { K _ i } } ^ { - 1 } \right ) \# \mu _ i = \left ( \mathord { \operatorname { p r o j } } _ { K _ i \cap K _ j } \circ { \operatorname { p r o j } _ { K _ j } } ^ { - 1 } \right ) \# \mu _ j . \end{align*}


\begin{align*} \mathcal { L } V = \bigg \langle \nabla V ( x ) , f ( x ) + \sum _ { i = 1 } ^ m g _ i ( x ) u _ i \bigg \rangle , \end{align*}


\begin{align*} \begin{pmatrix} y _ 1 \\ y _ 2 \end{pmatrix} a = \sigma ( a ) \begin{pmatrix} y _ 1 \\ y _ 2 \end{pmatrix} + \delta ( a ) \end{align*}


\begin{align*} \varphi ^ a _ s w _ s = - ( 1 + w ) \varphi ^ a _ { t t } = 0 \end{align*}


\begin{align*} \left \langle S , \begin{pmatrix} \hat { P } & \ell \end{pmatrix} F _ e \begin{pmatrix} \hat { P } & \ell \end{pmatrix} ^ \top \right \rangle = 0 & e \in \hat { E } ^ \circ \end{align*}


\begin{align*} Z ( N _ { U _ r } ) = \{ [ \mathrm { i d } _ { U _ r } , \mathrm { i d } _ { L _ { r + 1 } } , 0 , \psi ] \} \end{align*}


\begin{align*} h _ { \beta } ( x , y ) = \begin{cases} r ( x , y ) , & x , y \not \in S _ r ( \beta ) \\ \hat { r } _ \beta ( x , y ) , & x \in S _ r ( \beta ) y \in S _ r ( \beta ) ( x , y ) \not \in S _ r ( \beta ) ^ 2 , \\ \min \{ \hat { r } _ \beta ( x , y ) , \hat { r } _ \beta ( y , x ) \} , & x , y \in S _ r ( \beta ) . \end{cases} \end{align*}


\begin{align*} { \rm d i v } \ , \overline v = 0 , { \rm c u r l } \ , \overline v = \omega _ \theta e _ \theta \end{align*}


\begin{align*} \varphi _ V \circ \rho _ A ( x ) - \rho _ A ( \alpha ( x ) ) \circ \varphi _ V = \rho _ A ( \varphi _ L ( x ) ) \circ A \end{align*}


\begin{align*} \mu _ { X _ 1 } ( x _ 1 , y _ 1 ) > 0 \wedge \mu _ { X _ 1 } ( y _ 1 , x _ 1 ) > 0 \Rightarrow x _ 1 = y _ 1 \end{align*}


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


\begin{align*} \mathfrak { h } _ { i _ a } ( \varphi ) = \sum _ { 1 \leq i < i _ a } d ^ + _ { i , 0 } ( \varphi ) h _ i . \end{align*}


\begin{align*} ( K _ { \leq n } ) _ p = \begin{cases} K _ p & \hbox { i f } p \leq n \\ \varnothing & \hbox { i f } p > n . \end{cases} \end{align*}


\begin{align*} \pi _ { \nu } ( f ) = f ( \pi _ { \nu } ( S ) ) = ( I \otimes c ^ * ) \left ( I - \sum _ { j = 1 } ^ d \pi _ { \nu } ( S _ j ) \otimes T _ { 0 , j } ^ * \right ) ^ { - 1 } \left ( \sum _ { j = 1 } ^ d \pi _ { \nu } ( S _ j ) \otimes T _ j ^ * x \right ) . \end{align*}


\begin{align*} Y ( F ) = \bigcup _ { M > 1 } \bigcup _ { k > 1 } \bigcup _ { l > 1 } Y ( F ; M , k , l ) , \end{align*}


\begin{align*} \mathcal { F } : = \{ u \in S _ { 0 } ^ { 1 , p } ( \Omega ) : \enspace F ( u ) = 1 \} . \end{align*}


\begin{align*} \mu ^ 2 = \frac { 1 } { ( \deg \tau ) ^ 2 } \tau \pi \psi \widehat { \tau } \tau \pi \psi \widehat { \tau } = \frac { 1 } { \deg \tau } \tau \pi \psi \pi \psi \widehat { \tau } = \frac { - d p } { \deg \tau } \tau \widehat { \tau } = - d p , \end{align*}


\begin{align*} \partial _ z q ^ i ( z ) + \langle Z , \alpha _ i \rangle q ^ i ( z ) = L _ i ( z ) \Big [ { p ^ i ( z ) } \Big ] ^ { 2 } \prod _ { j \neq i , j \in I _ w \cup I _ g } p ^ j ( z ) ^ { a _ { j i } } \prod _ { j \neq i , j \in I _ b } p ^ j ( z ) ^ { a _ { j i } / 2 } . \end{align*}


\begin{align*} c ^ 2 t ^ { k - 1 } t _ 2 t _ 3 t _ 4 t _ 5 = p q . \end{align*}


\begin{align*} h ( \tau ) : = \mu \bigg ( ( \sqrt { \omega } - \tau ) ^ 2 , \frac { \mathbf { c } } { \sqrt { \omega } } ( \sqrt { \omega } - \tau ) \bigg ) = ( \sqrt { \omega } - \tau ) ^ { 4 - d } \mu \left ( 1 , \frac { \mathbf { c } } { \sqrt { \omega } } \right ) , \end{align*}


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


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


\begin{align*} \Delta _ { y _ { 1 } , \ldots , y _ { m } } A ^ { \ast } ( x ) = \left \{ \begin{array} { r c l } 0 & & m > n \\ n ! A ( y _ { 1 } , \ldots , y _ { m } ) & & m = n . \end{array} \right . \end{align*}


\begin{align*} J _ { a b } ^ \sigma ( s , t ) = \left ( \begin{array} { c c } \sigma _ a ' ( s , t ) & \dot { \sigma } _ a ( s , t ) \\ \sigma _ b ' ( s , t ) & \dot { \sigma } _ b ( s , t ) \\ \end{array} \right ) , \ a \neq b , \end{align*}


\begin{align*} R ^ \varphi _ { i j , i } = \frac { 1 } { 2 } S ^ \varphi _ j - \alpha \varphi ^ a _ { s s } \varphi ^ a _ j \end{align*}


\begin{align*} \tilde { \mathcal { K } } ( \varphi ) : = \langle c , A \psi \rangle + \sum _ { i = 1 } ^ m \langle c , B _ i \psi \rangle u _ i , \end{align*}


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


\begin{align*} \| \phi ( A \otimes B ) \| _ { ( p , k ) } = \| A \otimes B \| _ { ( p , k ) } \hbox { f o r a l l \quad } A \in M _ m \hbox { a n d } B \in M _ n . \end{align*}


\begin{align*} 0 = K _ { \omega , \mathbf { c } } ( \Psi ) = \partial _ { \lambda } S _ { \omega , \mathbf { c } } ( \lambda \Psi ) \big | _ { \lambda = 1 } = \sum _ { j = 1 } ^ 3 \langle D _ j S _ { \omega , \mathbf { c } } ( \Psi ) , \psi _ j \rangle = \eta \sum _ { j = 1 } ^ 3 \langle D _ j K _ { \omega , \mathbf { c } } ( \Psi ) , \psi _ j \rangle = \eta \partial _ { \lambda } K _ { \omega , \mathbf { c } } ( \lambda \Psi ) \big | _ { \lambda = 1 } . \end{align*}


\begin{align*} U _ { \lambda } : = \left \{ ( x , y , t ) \in \mathcal { Q } _ { r _ { 1 } } \ , : \ , | D ^ { \tau } d _ { s } u ( x , y , t ) | \geq \lambda \right \} \subset \left ( \bigcup \limits _ { i \in \mathbb { N } } \mathcal { Q } _ { 5 \rho _ { { i } } } \left ( z _ { i } \right ) \right ) \bigcup \left ( \bigcup \limits _ { \mathcal { K } \times I \in \mathcal { A } } \mathcal { K } \times I \right ) \end{align*}


\begin{align*} V ( p , t , \rho ) = \inf _ { ( r , \sigma , \pi ) } J ( p , t , ( r , \sigma , \pi ) , \rho ) \end{align*}


\begin{align*} f _ { N , j , n } ( b _ 1 , \ldots , b _ N ) = \sum _ { M \in \Z } \left ( ( - 1 ) ^ M \sum _ { u \in \mathbb { Z } } a ^ { j - u } q ^ { ( M - u ) ( n - u ) + ( j - u ) ( n - N ) } \begin{bmatrix} M \\ u \end{bmatrix} \begin{bmatrix} N - M \\ j - u \end{bmatrix} \right ) \sum _ { \substack { I \subseteq \{ 1 , \dots , N \} \\ s . t . | I | = M } } \prod _ { i \in I } b _ { i } . \end{align*}


\begin{align*} g = \prod _ { k \in I _ g ^ c } \rho _ k ^ { s _ k } , \end{align*}


\begin{align*} \left . \begin{array} { l l } \delta _ { \xi } \left [ \mathcal { S } ( \xi , - \eta ) + \Lambda \mathcal { Q } ( \xi , - \eta ) \right ] = 0 , \quad \delta _ { - \eta } \left [ \mathcal { S } ( \xi , - \eta ) + \Lambda \mathcal { Q } ( \xi , - \eta ) \right ] = 0 , \end{array} \right . \end{align*}


\begin{align*} \Phi _ s ( n ) = n \prod \limits _ { \substack { p ^ s \mid n \\ p } } \left ( 1 - \frac { 1 } { p ^ s } \right ) . \end{align*}


\begin{align*} \theta _ { 1 / 2 } ( \tfrac { a z + b } { c z + d } , \epsilon ) = \lambda ^ { \pm } ( \gamma , \epsilon ) \sqrt { \det \gamma ( c z + d ) } \sum _ { n \in \Z } e ^ { i ( \det \gamma ) \epsilon \pi n ^ 2 z } , \end{align*}


\begin{align*} E _ { \varepsilon } ( t , \xi ) : = ( S _ { \varepsilon } ( t ) W _ { \varepsilon } ( t , \xi ) , W _ { \varepsilon } ( t , \xi ) ) , \end{align*}


\begin{align*} \Big \| \sum _ { n = 1 } ^ N f _ n \Big \| _ X \leq \sum _ { n = 1 } ^ N K _ X ^ n \| f _ n \| _ X . \end{align*}


\begin{align*} \aligned \left \{ \begin{array} { l l l } - \Delta u + \mu _ 1 u = ( I _ \mu \ast H _ { 1 1 } u ) K _ { 1 1 } + ( I _ \mu \ast H _ { 1 2 } v ) K _ { 1 2 } \ & \mathbb { R } ^ N , \\ - \Delta v + \mu _ 2 v = ( I _ \mu \ast H _ { 2 1 } v ) K _ { 2 1 } + ( I _ \mu \ast H _ { 2 2 } u ) K _ { 2 2 } \ & \mathbb { R } ^ N , \end{array} \right . \endaligned \end{align*}


\begin{align*} \mathcal { G } = \{ u \in X : G ( u ) = 1 \} . \end{align*}


\begin{align*} g _ \beta ( x , t _ 1 , t _ 2 ) = \sum _ { n = 0 } ^ \infty \sum _ { m = 0 } ^ \infty \frac { t _ 1 ^ n t _ 2 ^ m } { n ! m ! } I _ { 2 n + 3 m + \beta } ( 2 \sqrt { x } ) . \end{align*}


\begin{align*} \lambda _ \chi ^ { \rm s t i f f ( s o f t ) } ( \vect g , \vect h ) : = a _ { \chi } ^ { \rm s t i f f ( s o f t ) } \bigl ( \Pi _ \chi ^ { \rm s t i f f ( s o f t ) } \vect g , \Pi _ \chi ^ { \rm s t i f f ( s o f t ) } \vect h \bigr ) , \vect g , \vect h \in H ^ { 1 / 2 } ( \Gamma ; \C ^ 3 ) . \end{align*}


\begin{align*} L _ S [ f ] = \sum _ { T \subseteq S } ( - 1 ) ^ { | T | } E _ T [ f ] . \end{align*}


\begin{align*} x _ j + \left ( \sum _ { k = 1 } ^ n a _ { j k } x _ k \right ) ^ d , j = 1 , \ldots , n . \end{align*}


\begin{align*} \Delta u ( x ) = f ( x ) , x \in \Omega = ( 0 , 1 ) ^ 2 , u | _ { \partial \Omega } = 0 , \end{align*}


\begin{align*} \left ( \begin{array} { c c c } x = 0 : & 0 & e _ 1 \\ x = 1 : & 0 & e _ 2 \\ x = \infty : & e _ 3 & e _ 4 \end{array} \right ) \to \left ( \begin{array} { c c c } 0 & - e _ 1 \\ 0 & - e _ 2 \\ 1 - e _ 3 & 1 - e _ 4 \end{array} \right ) . \end{align*}


\begin{align*} \mathbf { T } _ { K ^ \circ } ( R ) & = \{ x \in K \otimes _ { \mathbb { Q } } R \mid c ( x ) x \in R ^ \times \} , \\ \mathbf { T } _ { K ^ \circ } ^ 1 ( R ) & = \{ x \in K \otimes _ { \mathbb { Q } } R \mid c ( x ) x = 1 \} . \end{align*}


\begin{align*} \boldsymbol { W } _ { p } ( \mu , \nu ) & = \left [ \inf _ { \pi \in \Pi ( \mu , \nu ) } \int _ { \mathcal { X } \times \mathcal { X } } \boldsymbol { d } ^ p \ , d \pi \right ] ^ { 1 / p } . \end{align*}


\begin{align*} ( \partial _ t - \Delta ) \Phi = - \lambda \Phi ^ 3 + \xi ; \end{align*}


\begin{align*} w ( x ) : = \frac { \phi ( x ) } { \sqrt { x + 4 } } , \check { h } ( x ) : = 4 \cosh \left ( \tfrac { x } { 2 } \right ) \int _ { 0 } ^ { \infty } w \left ( 4 \sinh ^ 2 \left ( \tfrac { x } { 2 } \right ) + y ^ 2 \right ) \d y , \end{align*}


\begin{align*} \left \{ \begin{aligned} & \langle D _ u \theta ( u ^ * ) , v \rangle _ \mathcal { U } + \langle \tilde { \lambda } , S v \rangle _ { * } = 0 \forall \ v \in \mathcal { U } ; \\ & - \langle \tilde { \lambda } , \zeta - S u ^ * \rangle _ * \geq 0 \forall \ \zeta \in K , \\ \end{aligned} \right . \end{align*}


\begin{align*} \mathbb { \overline { P } } _ { \mu , x , g ^ { - 1 } } ( A ) = \overline { \mathbb { P } } _ { \mu , g . x , i d } \left ( g . A \right ) . \end{align*}


\begin{align*} \begin{aligned} \Pr ( T _ k > t ) = \Pr ( X _ t \le k - 1 ) , \Pr ( T _ k \le t ) = \Pr ( X _ t > k - 1 ) . \end{aligned} \end{align*}


\begin{align*} \ddot A ( t ) x + \frac \gamma { \gamma - 1 } A ( t ) ^ { - \top } \det ( A ( t ) ) ^ { 1 - \gamma } \nabla _ x w & = 0 , \ \ \gamma > 1 , \end{align*}


\begin{align*} y ' ( t ) = f ( t , y ( t ) ) , y ( t _ 0 ) = y _ 0 , \end{align*}


\begin{align*} \int _ { \R ^ d } e ^ { - z | x | ^ 2 + \xi \cdot x } \dd x = \frac { \pi } { z } e ^ { \frac { \xi \cdot \xi } { 4 z } } . \end{align*}


\begin{align*} \Pi _ { \psi } ( g ) f ( [ \epsilon , w ] ) = \Upsilon ( g ) f ( [ ( \det g ) \epsilon , w g ^ { s ( ( \det g ) \epsilon ) } ] ) , \end{align*}


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


\begin{align*} \sup _ { \gamma \in \mathcal { P } _ { \mathrm { D } } } I _ { \mathrm { D } , \lambda } [ \gamma ] = \sup _ { \gamma \in \mathcal { P } _ { \mathrm { D } } } \sup _ { \pi \in \Pi ( \mu _ 1 , \mu _ 2 , \gamma ) } \int _ { \mathcal { V } \times \mathcal { V } } \varphi _ \lambda \ , d \pi . \end{align*}


\begin{align*} \begin{aligned} J _ 1 \geq C s ^ 3 \lambda ^ 4 \iint _ Q \xi ^ 3 \left | A \nabla \eta \cdot \nabla \eta \right | ^ 2 | u | ^ 2 d x d y d t - C s ^ 3 \lambda ^ 3 \int _ 0 ^ T \int _ { \omega _ 0 } \xi ^ 3 | u | ^ 2 d x d y d t , \end{aligned} \end{align*}


\begin{align*} f ( x ) = \frac { 1 } { 2 } c x ^ 2 + o ( x ^ 2 ) f ' ( x ) = c x + o ( x ) . \end{align*}


\begin{align*} B _ \vee \Biggl ( \sum _ { | \tau | = \ell } \omega ( \tau ) \cdot \tau \Biggr ) = \sum _ { | \tau | = \ell + 1 } \omega ( \tau ) \cdot \tau , \end{align*}


\begin{align*} \int _ { \tilde { X } } g ^ { - 1 } . f d \tilde { \lambda } = \int _ { G / \Gamma } \int _ { X } f _ { z } d \boldsymbol { \beta } ( \tau ( z ) ^ { - 1 } g P ) d m _ { G / \Gamma } ( z ) . \end{align*}


\begin{align*} Y = D Y _ 2 + ( 1 - D ) Y _ 1 . \end{align*}


\begin{align*} C : 9 x ( x ^ 2 + 2 0 r ^ 2 ) = y ^ 3 . \end{align*}


\begin{align*} L ( u , \chi ) = \exp \left ( \sum _ { k \ge 1 } \frac { u ^ k } { k } \sum _ { f \in \mathcal { M } _ { k , q } } \chi ( f ) \Lambda ( f ) \right ) . \end{align*}


\begin{align*} A ( u , v ) = \max _ { ( \sigma , \tau ) \in S _ { u , v } } d _ { T V } ( \mu _ { v } ( \cdot \mid \sigma ) , \mu _ v ( \cdot \mid \tau ) ) , \end{align*}


\begin{align*} & U _ 1 = \{ z \in \mathbb R ^ N : \| z - x \| > 5 { \mathfrak R } ^ k , \| z - \sigma _ 1 ( x ) \| \le 5 { \mathfrak R } ^ k \} , \\ & U _ { j + 1 } = \{ z \in \mathbb R ^ N : \| z - x \| > 5 { \mathfrak R } ^ k , \| z - \sigma _ { j + 1 } ( x ) \| \le 5 { \mathfrak R } ^ k \} \setminus \Big ( \bigcup _ { i = 1 } ^ j U _ i \Big ) \quad \mbox { f o r } 1 \leq j \leq | G | - 1 . \end{align*}


\begin{align*} \Delta _ x u - u _ { t t } = 0 , u ( x , 0 ) = f ( x ) , u _ t ( x , 0 ) = g ( x ) , \end{align*}


\begin{align*} \tilde D _ \epsilon : = \{ ( \bar q , \lambda ) : \bar q \in V _ q , \ , 0 < \sqrt { 2 H ( \lambda ) } < \epsilon \} , \end{align*}


\begin{align*} \operatorname { s y s } ( M , g ) = \inf \{ l ( c ) \mid c \colon \} , \end{align*}


\begin{align*} S ( \rho _ A ) = \lim _ { \epsilon \to 0 } \frac 1 { 4 \pi i } \int _ { \Gamma ( \epsilon ) } e ( 1 + \epsilon , \lambda ) \frac { d \log D _ L ( \lambda ) } { d \lambda } d \lambda , \end{align*}


\begin{align*} d _ \lambda h = \sigma ( \cdot , \vec h ( \lambda ) ) , \qquad \forall \ , \lambda \in T ^ * M , \end{align*}


\begin{align*} ( \sharp _ 1 ) \left \{ \begin{array} { @ { \ , } l l l } p = 0 \\ q = - a \\ a \ne 0 \\ b = 0 \\ c = a \\ d = 0 \end{array} \right . \ , \ , { \rm { o r } } \ , \ , \ , \ , ( \sharp _ 2 ) \left \{ \begin{array} { @ { \ , } l l l } p \ne 0 \\ q = - a \\ a \ne 0 \\ b = 0 \\ c = a \\ d = 0 . \end{array} \right . \end{align*}


\begin{gather*} Q _ { l j } ( 0 , \dots , 0 ) = 0 \bigl \| B _ l ^ { ( j ) } \bigr \| _ l \leq Q _ { l j } \big ( \nu _ 1 , \sqrt { \nu _ 2 } , \dots , \sqrt [ k ] { \nu _ k } \big ) . \end{gather*}


\begin{align*} \| u \| _ { H ^ s _ h } = \| \langle h D \rangle ^ s u \| _ { L ^ 2 } . \end{align*}


\begin{align*} V = N \oplus N ^ { - 1 } K _ X , \gamma = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} , \beta = \begin{pmatrix} \nu & q _ 2 \\ q _ 2 & \mu \end{pmatrix} . \end{align*}


\begin{align*} \left . \begin{array} { l l } \phi ( 0 , x ) = \phi _ 0 ( x ) , \quad \psi ( 0 , x ) = \psi _ 0 ( x ) , x \in \mathbb { R } ^ N , \end{array} \right . \end{align*}


\begin{align*} \widehat { P } _ \chi - \widehat { P } _ 0 = \widehat { P } _ \chi ( \widehat { P } _ \chi - \widehat { P } _ 0 ) + ( \widehat { P } _ \chi - \widehat { P } _ 0 ) \widehat { P } _ 0 . \end{align*}


\begin{align*} \lim _ { k \rightarrow \omega } \| \phi _ k ( a b ) - \phi _ k ( a ) \phi _ k ( b ) \| = 0 \lim _ { k \rightarrow \omega } \mathrm { t r } _ \mathcal Q ( \phi _ k ( a ) ) = \mathrm { t r } _ G ( a ) . \end{align*}


\begin{align*} \sum \limits _ { \substack { 1 \leq m _ 1 , m _ 2 , \ldots , m _ k \leq n ^ s \\ ( m _ 1 , n ^ s ) _ s = 1 \\ ( m _ 2 , n ^ s ) _ s = 1 \\ \cdots \\ ( m _ k , n ^ s ) _ s = 1 \\ 1 \leq b _ 1 , b _ 2 , \ldots , b _ r \leq n ^ s } } ( m _ 1 - a _ 1 , m _ 2 - a _ 2 , & \ldots , m _ k - a _ k , b _ 1 , b _ 2 , \cdots , b _ r , n ^ s ) _ s \\ & = \Phi _ s ( n ^ s ) ^ k \sum \limits _ { \substack { d ^ s | n ^ s } } \frac { ( d ^ s ) ^ r } { \Phi _ s ( \frac { n ^ s } { d ^ s } ) ^ { k - 1 } } . \end{align*}


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


\begin{align*} \sup \left \{ n : H _ { k } \cap K _ { n } = \bar { H } \cap K _ { n } \right \} \underset { k \to \infty } { \longrightarrow } \infty . \end{align*}


\begin{align*} h ( t ) = t ^ { \ell + y } f ( t ) - a t ^ { 2 k + \ell + x } f ( t ^ { - 1 } ) . \end{align*}