harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\delta^2(P; Z) = -2h \langle \mathbf{v}_1, \mathbf{z}\rangle=(\langle \mathbf{z}, \mathbf{v}_2 \rangle +\langle \mathbf{z}, \mathbf{v}_1 \rangle)\langle \mathbf{v}_1,\mathbf{z}\rangle)\ge\langle \mathbf{v}_1, \mathbf{z}\rangle\langle \mathbf{v}_2, \mathbf{z}\rangle.
0 =\frac{d}{ds}[(\Delta-\lambda_s)(s-1)E_s]= (\Delta-\lambda_s)\frac{d}{ds}[(s-1)E_s]- 2(2s-1)(s-1)E_s &= (\Delta-\lambda_s)\frac{d}{ds}[(s-1)E_s]-2(2s-1)[3/\pi+(s-1)E_1^*+O(s-1)].
\leq \left\langle \prod_{b = 1}^{N}2 \cosh \left( 2 \beta^{( b )}\int_{\Lambda}d^{2}x t ( x ) : \cos \left( 2 \sqrt{\pi}\varphi^{( b )}( x ) \right) :_{Q^{W}}\right) \right\rangle_{Q^{\mu}}
\mathbf{\tilde{z}}=-\operatorname*{i}k\int_{\Gamma}g_{-k}\left( \left\Vert\mathbf{\cdot}-\mathbf{y}\right\Vert \right) T_{-k}\left( \left( L_{\Omega}\mathbf{v}\right)^{\operatorname*{curl}}\right) \left( \mathbf{y}\right)d\Gamma_{\mathbf{y}}.
|(h^{*(H-1)}*h')(p^k)| = \sum_{j=0}^{\min(k,H-1)}|h'(p^{k-j}) h^{*(H-1)}(p^j)|\leq \sum_{j=0}^{\min(k,H-1)}(2H)^{k-j}H^{j}\leq 2(2H)^k
\|T\|- \|Tg\|_{q}\geq \|T\|-\|T^\ast G\|_{p'}\left(1-\frac{p-1}{4}\left\|g-D_{p'}\overline{T^\ast G}\right\|_{p}^2\right) & = \|T\|-\|T^\ast G\|_{p'}+\frac{p-1}{4}\|T^\ast G\|_{p'}\left\|g-D_{p'}\overline{T^\ast G}\right\|_{p}^2 & \gtrsim \inf_{G_\ast}\|G-G_\ast\|_{q'}^{2}+\|T^\ast G\|_{p'}\left\|g-D_{p'}\overline{T^\ast G}\right\|_{p}^2,
\left\|{A}\xi- \hat{A}\xi \right\|_2^2 =\frac{1}{2\pi}\int_{-\pi}^{\pi}\left|\sum\limits_{j=0}^{\infty}(\bold{B}^{-1}\bold{a})_je^{ij\theta}\right|^2 f^{-1}(\theta)d\theta=<\bold{B}^{-1}\vec{\bold{a}}, \vec{\bold{a}}>,
\int_I S_1\left( u_{\varepsilon,n}-u_{\varepsilon,m}\right)(t) dx \leq \int_I | u_{0,n}-u_{0,m}| dx + \int^t_{0}\int_I | g_{\varepsilon}(u_{\varepsilon,n}) -g_{\varepsilon}(u_{\varepsilon,m}) | dxds \overset{(\ref{plap36})}{=}o(n,m),
\widehat{c}_1(\overline{\mathcal{E}}) =\widehat{c}_1(\overline{\mathcal{O}_X(a+b)}) \widehat{c}_2(\overline{\mathcal{E}}) =\widehat{c}_1(\overline{\mathcal{O}_X(a)})\widehat{c}_1(\overline{\mathcal{O}_X(b)})+log(\#H^0(Z,\mathcal{O}_Z)).
\int_{0}^{2 \pi}d \sigma \left( - \frac{1}{2 \pi}p^{+}\partial_{\sigma}X^{-}+{\cal P}^{I}\partial_{\sigma}X^{I}+ \frac{i}{2}\psi_{+}^{A}\partial_{\sigma}\psi_{+}^{A}+ \frac{i}{2}\psi_{-}^{A}\partial_{\sigma}\psi_{-}^{A}\right) = 0 ~ .
V_{eff}(\sigma)={\sigma^2\over 2\lambda}-\left[ \frac{\Lambda \sigma^2}{2\pi^{3/2}}+\frac{\sqrt{2}}{\pi}(eB)^{3/2}\zeta \left( -\frac{1}{2}, 1+ \frac{\sigma^2}{2eB}\right)+\frac{1}{2\pi}eB \sigma \right]
-\int_{\mathbb R^3}\partial_t\nabla_\beta u^i\nabla_\beta\Delta u^i\phi^6dx=\frac{1}{2}\frac{d}{dt}\int_{\mathbb R^3}|\nabla^2 u|^2\phi^6dx+6\int_{\mathbb R^3}\partial_t\nabla_\beta u^i\nabla_{\beta\alpha}^2u^i\phi^5\nabla_\alpha \phi dx.
\Gamma (2-\alpha)h^\alpha y^{(\alpha)}_n \approx \sum_{k=1}^n \rho_k^{(\alpha)}(y_k-y_{k-1})=\sum_{k=1}^n y_k\rho_k^{(\alpha)}-\sum_{k=1}^n y_{k-1}\rho_{k}^{(\alpha)}& =\rho_n^{(\alpha)}y_n+\sum_{k=1}^{n-1}y_k\left(\rho_k^{(\alpha)}-\rho_{k-1}^{(\alpha)}\right)-\rho_1^{(\alpha)}y_0.
\mathcal{O}_{1}= (1,3),(2,4),(1,4),(2,3),(1,2),(3,4), \mathcal{O}_{2}= (1,4),(2,3),(1,3),(2,4),(1,2),(3,4), \mathcal{O}_{1}' = (1,2),(3,4),(1,3),(2,4),(1,4),(2,3), \mathcal{O}_{2}' = (1,2),(3,4),(1,4),(2,3),(1,3),(2,4), \mathcal{O}_{1}'' = (1,4),(2,3),(1,2),(3,4),(1,3),(2,4), \mathcal{O}_{2}'' = (1,3),(2,4),(1,2),(3,4),(1,4),(2,3).
C(\bar r,t|{{\bar r}_{tx}},{t_0}) = \frac{1}{{{\sqrt{(4\pi D(t -{t_0}))}}}}{e^{\frac{{-{{(z -{z_{\rm tx}}- v(t -{t_0}))}^2}}}{{4D(t -{t_0})}}-{k_d}(t -{t_0})}}\times \sum\limits_{m = 1}^\infty{\frac{{{J_0}({\lambda_{0m}}{\rho_{tx}}){J_0}({\lambda_{0m}}\rho )}}{{{N_{0m}}}}}{e^{- \gamma_{0m}^2(t -{t_0})}}u(t-t_0).
\mathcal{X}= \left\{X \in{\rm St}(m,n) \mid X ={\rm BlkDiag}(P_1,\ldots,P_{n_A}) \cdot E(h) \cdot{\rm BlkDiag}(Q_1,\ldots,Q_{n_B}) \right. &\quad\left. \mbox{for some}\, P_i \in \mathcal{O}^{s_i-s_{i-1}}\, (i=1,\ldots,n_A), \, Q_j \in \mathcal{O}^{t_j-t_{j-1}}\, (j=1,\ldots,n_B), \mbox{and}h\in\mathcal{H}\right\}.
{\Vert \phi' \Vert}_{-r}^2 = \sum_{k=1}^{\infty}\langle \phi', \phi_{r,k}\rangle^2 = \sum_{k=1}^{\infty}(1 + k^2)^{-r}\langle \phi', \phi_{k}\rangle^2, \phi' \in H_{-r}([0, \pi]).
\underset{\lambda=1}{Res}\Bigg\{-\frac{1}{4}\frac{1}{(1-t\lambda)^{2}}\left(\frac{1+\lambda}{1-\lambda}\right)\frac{1}{\lambda^{2l+1}}\frac{(1-t\lambda^{2})^{2}}{(1+t\lambda^{2})^{4}}d\lambda\Bigg\}.
S = \int d^{4}x \sqrt{\hat{g}_{4}}\Bigl \{\hat{R}_{4}+ \frac{2 \hat{\nabla}_{\mu}{\cal A}\hat{\nabla}^{\mu}{\cal A}^{\dagger}}{({\cal A}-{\cal A}^{\dagger})^{2}}+ \frac{1}{8}T r \bigl ( l \hat{\nabla}m \bigr )^{2}- \frac{Q^{2}}{2}\frac{e^{\Phi_{4}}}{b^{6}}\Bigr \}
\lim_{\rho\rightarrow0^{+}}\frac{1}{\rho^{n}}\int_{Q_{\rho}^{\nu}}|u(x+x_{0})-w^{\nu}(x+x_{0})|~dx=0~, & g(x_{0})=\lim_{\rho\rightarrow0^{+}}\frac{\mu(x_{0}+\overline{Q_{\rho}^{\nu}})}{\rho^{n-1}}<+\infty~.
\begin{array}{rl}\displaystyle R(3,\mathbb{Z}_n,2) \displaystyle \leq 36 \cdot 3{(n+6)/24 \choose 2}+ 36{(n-18)/24 \choose 2}+ \frac{n}{3}+n & \displaystyle \leq 36 \cdot 3{(n-6)/24 \choose 2}+ 36{(n+18)/24 \choose 2}+ \frac{n}{3}+n & \displaystyle = \frac{n^2}{8}-\frac{5n}{3}+\frac{27}{2}.\end{array}
( m_{1}; m_{2}, m_{3}, m_{4}; n^{2}, n^{3}, n^{4}; n^{1}) = ( m ; m^{3 4}, m^{4 2}, m^{2 3}; m^{1 2}, m^{1 3}, m^{1 4}; m^{1 2 3 4}) \ .
\chi_{k,n}^{(t_{1},...,t_{k+n})}(x_{1},...,x_{n+k}|\rho)=\sum_{j\geq0}\rho^{j}\prod_{m=1}^{k}T_{j+t_{m}}(x_{m})\prod_{m=k+1}^{n+k}U_{j+t_{m}}(x_{m}),
\mu(\{h>1\}) = \int_S \int_0^\infty 1_{\{h(ru)>1\}}\alpha r^{-\alpha-1}dr \sigma(du) & = \int_S \int_0^\infty 1_{\{h(u)> 1/r\}}\alpha r^{-\alpha-1}dr \sigma(du) & = \int_S \int_0^\infty 1_{\{h^\alpha(u) > x \}}dx \sigma(du)=\int_S h^\alpha(u) \sigma(du),
\begin{array}{c c c}{{x_{\pm}^{\prime \mu}= - \displaystyle{\frac{x_{\mp}^{\mu}}{x_{\mp}^{2}}}~}}&{{~ \theta^{\prime}{}^{a t}= \displaystyle{- i \frac{1}{x_{-}^{2}}\, \epsilon^{- 1}x_{-}{\cdot \sigma}\bar{\theta}_{b}\, \zeta^{b a}}~}}&{{~ \bar{\theta}_{a}^{\prime}= i \displaystyle{\frac{1}{x_{+}^{2}}\, \bar{\epsilon}^{- 1}x_{+}{\cdot \sigma^{t}}\theta^{b t}\, \bar{\zeta}_{b a}}}}\\ \end{array}
m^{2}L^{2}={\frac{( p - 1 )^{2}}{( q - 1 )^{2}}}( k + 1 ) ( k + q - 2 ) + ( p - 1 ) \left( 1 \pm \sqrt{1 + 2{\frac{p - 1}{( q - 1 )^{3}}}( p + q - 2 ) ( k + 1 ) ( k + q - 2 )}\right) \, .
\begin{array}{r c l}{{{\tilde{S}}[ X^{\mu}, b_{i}]}}&{{=}}&{{- T_{M 2}l \int d^{2}\xi \sqrt{| D_{i}X^{\mu}D_{j}X^{\nu}g_{\mu \nu}|}}}\\{{}}&{{}}&{{}}\\{{}}&{{}}&{{-{\textstyle \frac{T_{M 2}l}{2}}\int d^{2}\xi \epsilon^{i j}D_{i}X^{\mu}D_{j}X^{\nu}B_{\mu \nu}\, .}}\\ \end{array}
E \sim \int^{\infty}d s \left( c_{1}( h^{\prime})^{2}+ c_{2}e^{- 2 s}( \sin h )^{2}+ c_{3}( \sin h )^{2}( h^{\prime})^{2}+ c_{4}e^{- 2 s}( \sin h )^{4}\right) .
\frac{\left(qz^{2};q\right)_{\infty}}{(q;q)_{\infty}}z^{\nu}=\sum_{m=0}^{\infty}\frac{q^{m^{2}}}{\left(q;q\right)_{m}}J_{\nu+m}^{(3)}(2zq^{m/2};q)\left(\frac{q^{\nu}}{z}\right)^{m}.
\ln R(n+1,E)- \ln R(n,E)= -\vert \varphi(n,E)\vert^2 \frac{C}{n-v}+O(1)\vert \varphi(n,E)\vert^2 \frac{\cos 4 \theta(n,E)}{n-v}+\frac{O(1)}{(n-v)^2}.
g_{2}((\nabla\pi_*)(U_{2},V_{2}),\pi_*Z) =-g_{1}(\nabla^{^{M_1}}_{U_{2}}\phi V_{2},\varphi Z)-g_{1}(\nabla^{^{M_1}}_{U_{2}}\omega V_{2},\mathcal{B}Z) &-g_{1}(\nabla^{^{M_1}}_{U_{2}}\omega V_{2},\mathcal{C}Z)-g_{1}(V_{2},\phi U_{2})\eta(Z).
T^{-1}=\begin{pmatrix}A -a^{-1}\beta d^{-1}-a^{-1}\beta d^{-1}\gamma a^{-1}\beta d^{-1}\cr-d^{-1}\gamma a^{-1}-d^{-1}\gamma a^{-1}\beta d^{-1}\gamma a^{-1}D \end{pmatrix}
\gamma_{n+1}(K_l)=\frac{1}{(\sqrt{2\pi})^{n+1}}\Pi_{i=1}^{n+1}\left\{\int_{-h_{i, l}}^{h_{i, l}}e^{\frac{-x_i^2}{2}}dx_i\right\}\rightarrow 0, \, \, \, \, l\rightarrow \infty
\rho\frac{Du}{Dt}=-\nabla\overline{p}+\mu\nabla^{2}u+\frac{1}{3}\mu\nabla(\nabla\cdot u)+\rho g-\rho[2\Omega\times u+\Omega\times(\Omega\times x)+\frac{dU}{dt}+\frac{d\Omega}{dt}\times x].
G_2(x;d) =\frac{x^{d+2}}{(1-x)^{d+1}}+\frac{x^{d+3}}{(1-x)^{d+2}}+\sum_{s=2}^{d}\frac{x^{d+3}}{(1-x)^{s+1}}&+\frac{x^{d+2}}{(1-x)^{d+1}}(F_T(x)-1)+\sum_{t=1}^{d}\left(\frac{x^{d+3}}{(1-x)^{t+2}}\big(F_T(x)-1\big)+\frac{x^{d+5}}{(1-x)^{t+3}(1-2x)}\right),
H_{2}=H\otimes H=\frac{1}{\sqrt{2}}[\begin{matrix}1&1\\ 1&-1\end{matrix}]\otimes\frac{1}{\sqrt{2}}[\begin{matrix}1&1\\ 1&-1\end{matrix}]=\frac{1}{2}[\begin{matrix}1&1&1&1\\ 1&-1&1&-1\\ 1&1&-1&-1\\ 1&-1&-1&1\end{matrix}]
\int_0^1 \left( \frac{1}{n}\sum_{i=k+1}^{n+k}\log f_i^{(k)}(x) - \lambda_k\right)^2 d \mu =\frac{1}{n^2}\sum_{i,j= k+1}^n \left( I(i,j)-\lambda_k^2\right) & = O\left( \frac{1}{n^2}\lambda_k^2\sum_{\substack{i,j=k+1 j-i>k}}^n q^{\sqrt{j-i-k}}\right) +O\left( \frac{k}{n}\right) & = O\left( \frac{k}{n}\right).
\langle \delta(\mathcal{T}),\mathcal{T}'\otimes \mathcal{T}''\rangle =\frac{1}{|\min(\mathcal{T})|}\sum_{\substack{Y\circledcirc \mathcal{T}\mathcal{T}_{|Y}\approx \mathcal{T}', \mathcal{T}_{|X \setminus Y}\approx \mathcal{T}''}}\sigma(\mathcal{T}')\sigma(\mathcal{T}'')=\frac{1}{|\min(\mathcal{T})|}|A|
\int_0^\infty G(y)(y^2 -1)\varphi(y)dy &\geq G(y_0) \int_0^1 (y^2-1)\varphi(y)dy + G(1)\int_1^\infty (y^2-1)\varphi(y)dy & = -G(y_0) \int_1^\infty (y^2-1)\varphi(y)dy +G(1)\int_1^\infty (y^2-1)\varphi(y)dy & =(G(1)-G(y_0)) \int_1^\infty (y^2-1)\varphi(y)dy>0\;.
\left[ \, \bar{\psi}_{n}^{( K )}\, , \, \Phi_{\chi_{k}\otimes \chi_{\bar{k}}}^{\beta}\, \right] \ = \, P h i_{\chi_{k}\otimes \bar{k}\, \left( \delta_{1}\, \left( \bar{\psi}^{( K )}\right)_{n}\right) \, \chi_{\bar{k}}}^{\beta}\ ,
\begin{aligned}|f(x)-f(x')| \leqslant C \Big(\frac{\|x-x'\|}{r}\Big)^\beta \Big\{\frac{1}{V(x,x_0,r+d(x,x_0))}\Big(\frac{r}{r+\boldsymbol{d(x,x_0)}}\Big)^\gamma \qquad+ \frac{1}{V(x',x_0,r+d(x',x_0))}\Big(\frac{r}{r+\boldsymbol{d(x',x_0)}}\Big)^\gamma\Big\}; \end{aligned}
U_{g}^{a\prime\prime}=V^{\prime}(y^a+L^a)U_{g}^{\prime}V^{\prime\dagger}(y^a)=\left(\begin{array}{cccc}{\rm e}^{-i\theta_{a, 1}}& \ddots & \ddots &{\rm e}^{-i\theta_{a, N}}\end{array}\right)\equiv U_{g}^{a:inv}.
\mathbb{P}_{n}^{(N)}(f)-\mathbb{P}_{n}^{(N-1)}(f) =\sum_{j=1}^{m_{N}}\mathbb{P}_{n}^{(N-1)}(f1_{A_{j}^{(N)}})\frac{P(A_{j}^{(N)})}{\mathbb{P}_{n}^{(N-1)}(A_{j}^{(N)})}-\sum_{j=1}^{m_{N}}\mathbb{P}_{n}^{(N-1)}(f1_{A_{j}^{(N)}}) & =\sum_{j=1}^{m_{N}}\mathbb{E}_{n}^{(N-1)}(f\;|\;A_{j}^{(N)})\left(P(A_{j}^{(N)})-\mathbb{P}_{n}^{(N-1)}(A_{j}^{(N)})\right) .
\lim_{\Omega \to 0}\Phi^{\Delta , J}( t , \bar{t}, \psi ; \tau , \bar{\tau}, \phi ) = e^{i ( \mu \hat{\psi}+ \lambda \hat{\phi})}\, \sum_{m , n = 0}^{\infty}c_{m n}t^{m}\tau^{n}e^{- \frac{1}{2}| t |^{2}- \frac{1}{2}| \tau |^{2}}\, .
C^{\diamond}=\left(\begin{array}{ccc}C_{1}^{\diamond}\cdots C_{n}^{\diamond}\end{array}\right)\colon Y\ni\left(\begin{array}{c}y_{1}\vdots y_{n}\end{array}\right)\mapsto\sum_{k=1}^{n}C_{k}^{\diamond}y_{k}\in X'_{1}
\frac{\Gamma ( - \alpha^{\prime}s ) \Gamma ( - \alpha^{\prime}t )}{\Gamma ( 1 - \alpha^{\prime}s - \alpha^{\prime}t )}= \frac{1}{{\alpha^{\prime}}^{2}s t}- \frac{\pi^{2}}{6}- \zeta ( 3 ) ( s + t ) \alpha^{\prime}+{\cal O}({\alpha^{\prime}}^{2}) \, .
\begin{array}{l}{{( m , - m , 1 , p ) \longleftrightarrow \tilde{\beta}^{\frac{2 k^{\prime}}{\delta}m - \frac{k}{\delta}( p - k^{\prime})},}}\\{{( m , - m , 0 , p ) \longleftrightarrow \tilde{\beta}^{\frac{2 k^{\prime}}{\delta}m + \frac{k}{\delta}p}\tilde{\gamma}.}}\\ \end{array}
D_{\mu_{1},m_{1}}^{j_{1}}D_{\mu_{2},m_{2}}^{j_{2}}={\displaystyle\sum\limits_{j}}C_{\mu_{1},\mu_{2},\mu_{1}+\mu_{2}}^{j_{1},j_{2},j}C_{m_{1},m_{2},m_{1}+m_{2}}^{j_{1},j_{2},j}D_{\mu_{1}+\mu_{2},m_{1}+m_{2}}^{j}
\frac{F_{k+l}}{(k+l)^{\frac{4}{n}}}=\Big(1+\frac{2}{n}\Big)\left(\frac{\frac{1}{k+l}\sum_{i=1}^{k+l}\upsilon_i}{(k+l)^{\frac{2}{n}}}\right)^2-\frac{\frac{1}{k+l}\sum_{i=1}^{k+l}\upsilon_i^2}{(k+l)^{\frac{4}{n}}}.
\left[\frac{2\mathbf K(\sqrt{\lambda})}{\pi}\right]^6\frac{16(1-\lambda)^2}{\lambda}={}\frac{}{}\frac{\eta^{36}\left(i\frac{\mathbf K(\sqrt{1-\lambda})}{\mathbf K(\sqrt{\lambda})}\right)}{\eta^{24}\left(2i\frac{\mathbf K(\sqrt{1-\lambda})}{\mathbf K(\sqrt{\lambda})}\right)}<e^{\frac{\pi\mathbf K(\sqrt{1-\lambda})}{\mathbf K(\sqrt{\lambda})}}
\tilde{\sigma}_i(a,b) = \sum_{k=0}^{b-a-1}t_i^{a+k}t_{i+1}^{b-k}= t_i^{a}t_{i+1}^{b}+ t_i^{a+1}t_{i+1}^{b-1}+ \cdots + t_i^{b-1}t_{i+1}^{a+1}.
\hat{G}_{x x}^{( 1 0 )}\to \hat{G}_{x x}^{( 1 0 ) \, \prime}={\frac{1}{\hat{G}_{x x}^{( 1 0 )}}}, \hat{G}_{\mu \nu}^{( 1 0 )}\to \hat{G}_{\mu \nu}^{( 1 0 )}\ \ ( \mu , \nu \neq x ) , e^{2 \hat{\Phi}}\to e^{2 \hat{\Phi}^{\prime}}={\frac{e^{2 \hat{\Phi}}}{\hat{G}_{x x}^{( 1 0 ) \, \prime}}}.
\sum_{n=0}^{\infty}\frac{x^{n}}{n!}z^{n}=\sum_{m=0}^{\infty}\frac{z^{m}}{m!}A_{m}^{\lambda,\beta}(x)\cdot\sum_{k=0}^{\infty}\frac{z^{k}}{k!}M_{\lambda,\beta}(n) =\sum_{n=0}^{\infty}\frac{z^{n}}{n!}\left(\sum_{k=0}^{n}\binom{n}{k}A_{k}^{\lambda,\beta}(x)M_{\lambda,\beta}(n-k)\right).
Tr~(T_x) =\frac{1}{4}\int_{\Theta}d\theta \int_{K_1}d \mu_l (k) ~{\mid k_{22}\mid}^{-2\sigma -1}~x(\underline{\theta}^{-1}~\underline{k}~\underline{\theta}) ~+~ \frac{1}{4}\int_{\Theta}d \theta \int_{K_2}d \mu_l (k) ~{\mid k_{22}\mid}^{-2\sigma -1}~x(\underline{\theta}^{-1}~\underline{k}~\underline{\theta}).
\mathbf{E}^\nu\Big[\sup_{t_k \le s \le t \wedge \tau^{\nu}_N}\big|z^{u, N}(s) - \tilde{z}^u(s)\big|\Big] \le\Big(\big|z_N - z\big| + C_{h,N}\Big)\,e^{L_F (t-t_k)}\,, t\in [t_k, t_{k+1}]\,, \mathbf{P}(\tau^{\nu}_N < t_{k+1}) \le 3 \gamma^{-1}e^{L_Fh}C_{h,N}\,,
\begin{array}{rcl}{\sf \Omega}^1_N(U) := \left\{\beta\,:\, \beta\in{\sf \Omega}^{1}(U)\,:\,i^*_{\partial U}\left(\star\beta\right)=0\,\right\}, \mathfrak{H}_N^1(U) := \mathfrak{H}^1(U)\cap{\sf \Omega}^k_N(U). \end{array}
\varphi^{s,t,u}(\bar{x}; \bar{y}_1, \ldots, \bar{y}_n; z_1, \ldots, z_n) := \bigwedge_{(i,I,q) \in [n] \times K \times \mathbb{Q}_n^{[0,1]}, s(i,I,q)=1}\varphi_{I,q}\left(\bar{x},{\bar{y}_i}\right) \land \bigwedge_{(i,I,q) \in [n] \times K \times \mathbb{Q}_n^{[0,1]}, s(i,I,q)=-1}\neg \varphi_{I,q}\left(\bar{x},\bar{y}_i \right) \land
K-\mathcal{B}(t)=p^{E}(t,\mathcal{B}(t)) + \int_{0}^{t}[rKe^{-r(t-s)}\aleph(-d_{2}(\mathcal{B}(t),t-s,\mathcal{B}(s))) - \delta \mathcal{B}(t) e^{-\delta (t-s)}\aleph(-d_{1}(\mathcal{B}(t),t-s,\mathcal{B}(s)))]\mathrm{d}s,
\omega_1 =c a_1^*\wedge \cdots \wedge a_n^*, \omega_2 = b_1^*\wedge \cdots \wedge b_n^*, \omega = c a_1^*\wedge \cdots \wedge a_n^* -b_1^*\wedge \cdots \wedge b_n^*,\Theta = a_1^* \wedge b_1^* + \cdots + a_n^* \wedge b_n^*.
\begin{array}{lll}T_{\alpha{\dot{\alpha}}}= -2 D_\alpha G{\bar D}_{\dot{\alpha}}G{\mathcal F}''+ G [D_\alpha,{\bar D}_{\dot{\alpha}}]{\mathcal F}'\ , begin{align*}3mm]K_{\alpha{\dot{\alpha}}}= 2G\partial_{\alpha{\dot{\alpha}}}{\mathcal F}'=-2\partial_{\alpha{\dot{\alpha}}}({\mathcal F}-G{\mathcal F}')\ , begin{align*}3mm]J = 0\ .\end{array}
\limsup_{n\to\infty}\frac{1}{a_c^{(n)}}\log P(S_n(\kappa_n(x))\leq\lfloor\kappa_n(x)\rfloor-a_n) \leq-(1-r^{-1})^{-1}h(x)\sup_{\varepsilon>x}\inf_{y\in \left(-\infty,\frac{\varepsilon}{h(x)}\right]}H(y) &=-(1-r^{-1})^{-1}h(x)\inf_{y\in \left(-\infty,\frac{x}{h(x)}\right]}H(y) &=-J(x),
\lim_{x\rightarrow y}p_{j}(|x-y|,t)=p_{j}(0,t) =p_{j,\mathbb{R}}(0,t)+2\sum_{n\in \mathbb{N}\setminus \{0\}}p_{j,\mathbb{R}}(|2na_j|,t) -\sum_{n\in \mathbb{Z}}p_{j,\mathbb{R}}(|2(y-na_j)|,t)
\left( \begin{array}{c}{{\dot{\Phi}}}\\{{\Phi^{\prime}}}\\ \end{array}\right) \; \; \rightarrow \; \; \left( \begin{array}{c c}{{c o s h \, \theta}}&{{s i n h \, \theta}}\\{{s i n h \, \theta}}&{{c o s h \, \theta}}\\ \end{array}\right) \; \; \left( \begin{array}{c}{{\tilde{\dot{\Phi}}}}\\{{\tilde{\Phi}^{\prime}}}\\ \end{array}\right) \; \; .
\int_0^T\|u^{\Delta t}_{h}-v^{\Delta t}_{h}\|_{2}^2\, dt =\sum_{m=1}^N\|u_h^{m}-u_{h}^{m-1}\|_{2}^2 \int_{t_{m-1}}^{t_m}\left(\frac{1}{2}-\frac{t-t_{m-1}}{\Delta t}\right)^2 dt \leq\frac{\Delta t}{12}\sum_{m=1}^N\|u_{h}^m-u_{h}^{m-1}\|_{2}^2.
\mathfrak{L}(u)=\frac{1}{p}\int\limits_{\mathbb{G}}|\mathcal{R}_{1}^{\frac{a_{1}}{\nu_{1}}}u(x)|^{p}dx+\frac{1}{p}\int\limits_{\mathbb{G}}|\mathcal{R}_{2}^{\frac{a_{2}}{\nu_{2}}}u(x)|^{p}dx-\frac{1}{q}\int\limits_{\mathbb{G}}|u(x)|^{q}dx
\mathcal L_{\overline{I}_{n,\ell}}(s)=\exp\left(-\frac{2\pi}{\alpha}\Big(1-\sum_{j=\ell}^L T_{n,j}\Big)\lambda_hs^{\frac{2}{\alpha}}B\left(\frac{2}{\alpha},1-\frac{2}{\alpha}\right)\right).
{\Phi_{5}}({\rho}) = 4 \pi^{2}g_{s}\left({\frac{m^{I}d_{I}}{| \Gamma |}}+{\frac{g_{s}}{2 \pi}}{\widehat C_{I J}m^{I}m^{J}}\ln{\frac{\rho}{\rho_{0}}}\right) = 4 \pi^{2}g_{s}{Q}({\rho}) ,
S_{0}= - N \int d \tilde{t}S T r \left( - \frac{1}{2}\partial_{\tilde{t}}\tilde{\Phi}\partial_{\tilde{t}}\tilde{\Phi}+ \frac{1}{4}[ \tilde{\Phi}, \tilde{\Phi}]^{2}+{\frac{1}{2}}\tilde{\Theta}\dot{\tilde{\Theta}}-{\frac{1}{2}}\tilde{\Theta}\gamma^{i}[ \tilde{\Phi}_{i}, \tilde{\Theta}] \right)
\sum_{r \in\mathbb{Z}}c^F_r(\widetilde{W_v^l},\widetilde{U_v^l})q_F^{-r/2}X^{-r}= \omega_{\sigma^{(l)}}(-1)^{n-2}\gamma(X,\pi^{(l)},\sigma^{(l)},\psi^l)\sum_{r \in \mathbb{Z}}c^F_r(W_v^l,U_v^l)q_F^{r/2}X^r.
\epsilon_{e a c d}\, t_{L}^{c d}\, h_{b}^{\, \, \bar{a}}\, F^{+ + a b}= 0 \, , \qquad \epsilon_{\bar{e}\bar{b}\bar{c}\bar{d}}\, t_{R}^{\bar{c}\bar{d}}\, h_{\, \, \bar{a}}^{a}\, F^{+ + \bar{a}\bar{b}}= 0
\mathcal{T}_{\neq}=- \int_1^T \int \langle D \rangle^{N-1}t^2 g \langle D \rangle^{N-1}t \Big[ \underbrace{(U^1_{\neq}\partial_X U^1_{\neq})_0}_0 + (U^2_{\neq}\partial_Y^t U_{\neq}^1)_0 + (U^3_{\neq}\partial_Z^t U_{\neq}^1)_0 \Big]\,dV\,dt.
U_{N, 2k}^l \leq e^{2k}\, (2k)! \,\Big(\sum_{\substack{a_1+\cdots + a_l=2k, a_1 \geq 2,\cdots a_l \geq 2}}1 \Big)= e^{2k}\,(2k)!\, \binom{2k-l-1}{l-1}.
\begin{array}{r c l}{{q ( t )}}&{{=}}&{{\frac{1}{\sqrt{2 m \omega}}\left[ \alpha \, e^{- i \omega ( t - t_{0})}+ \alpha^{*}\, e^{i \omega ( t - t_{0})}\right] \ ,}}\\{{}}&{{}}&{{}}\\{{p ( t )}}&{{=}}&{{- \frac{i m \omega}{\sqrt{2 m \omega}}\left[ \alpha \, e^{- i \omega ( t - t_{0})}- \alpha^{*}\, e^{i \omega ( t - t_{0})}\right] \ .}}\\ \end{array}
\begin{cases}D^{\alpha_n}_ay(t) = f(t,y(t),D^{\alpha_1}_ay(t),D^{\alpha_2}_ay(t),\cdots,D^{\alpha_{n-1}}_ay(t)), y^{(j)}(a) = y^{(j)}_{a},~j = 0,1,\cdots,\lfloor \alpha_n \rfloor,\end{cases}
[\xi,x^z_{-\alpha}] =p_{\alpha}(x)p_{-\alpha}(y)p_{\alpha}(-x)p_{-\alpha}(-y) &=p_{\alpha}(x)p_{\alpha}(\frac{-x}{1-axy})\check{\alpha}(1+axy)p_{-\alpha}(\frac{y}{1-axy})p_{-\alpha}(-y) &=\check{\alpha}(1+axy)p_{-\alpha}(axy^2).
e_{0}\lrcorner (\delta^{\ast}\alpha) =\frac{1}{2}\Big( d\alpha_{0i}(e_{0})e^{i}+ d\alpha_{00}+ \alpha_{0i}\omega_{ij}(e_{0})e^{j}+ \alpha_{0i}\omega_{i0}\Big) &=\frac{1}{2}\Big( (d\alpha_{0i}(e_{0}))e^{i}+ d\alpha_{00}+ \alpha_{0i}\omega_{i0}\Big),
{{{\boldsymbol{\hat x}}}_{k;t + 1}}{}{\boldsymbol{f}}\left({{{{\boldsymbol{\hat x}}}_{k;t}}}\right) ={\left({{\boldsymbol{W}}_k^T{{\boldsymbol{\Omega}}_{k;t}}{{\boldsymbol{W}}_k}}\right)^{- 1}}{\boldsymbol{W}}_k^{}{{\boldsymbol{\Omega}}_{k;t}}{{\boldsymbol{d}}_k},
\begin{cases}\displaystyle\frac{dy(t)}{dt}= Ay(t) + Fy_t + f(t),\,\,\,\,t\in (0, T], y(0)=\phi_0,\,\, y(t)=\phi_1(t),\,\,t\in [-r, 0],\,\,\,\Phi=(\phi_0, \phi_1)\in{\cal X}=W\times L^2([-r, 0]; Z),\end{cases}
g(z_1, z_2, z_3, z_4) = [z_1, z_2][z_3, z_4] + [z_1, z_3][z_2, z_4] = [z_2, z_1] [z_4, z_3] & + [z_2, z_4] [z_1, z_3] - \bigl[ [z_2, z_4], [z_1, z_3] \bigr] = g(z_2, z_1,z_4, z_3) - \bigl[ [z_2, z_4], [z_1, z_3] \bigr]
\gamma^0 = \left( \begin{array}{rr}0 -i i 0 \end{array}\right)\; \; \; \; \;\gamma^1 = \left( \begin{array}{rr}0 i i 0 \end{array}\right)\; \; \; \; \;\gamma^2 = \left( \begin{array}{rr}i 0 0 -i \end{array}\right)
\rho(i,j,t-s) =\frac{1}{(d\delta)^{2}}\mathbb{E}[N^{i}(dt)N^{j}(ds)] &=\frac{1}{d\delta}\mathbb{E}[N^{i}(dt)\lambda^{j}_{s}] &=\frac{1}{d\delta}\mathbb{E}\left[N^{i}(dt)\int_{-\infty}^{s}\gamma_{j}(s-u)N^{j-1}(du)\right] &=\int_{-\infty}^{s}\gamma_{j}(s-u)\rho(i,j-1,t-u)du.
= \|F(u_k^{\delta})-v^{\delta}\|^{p}\bigg[-\frac{\mu}{2}+ \frac{\mu}{2}LC_F^2\bigg(\frac{p}{C_p}\bigg)^{2/p}\|F(u_k^{\delta})-v^{\delta}\|^{\epsilon}\bigg] \leq \frac{\mu}{2}LC_F^2\bigg(\frac{p}{C_p}\bigg)^{2/p}\delta^{\epsilon}\| F(u_k^{\delta})- v^{\delta}\|^{p}.
V_{i}= \left[ v_{L i}^{0}( v_{L i}^{1}v_{L i}^{2}v_{L i}^{3}) ( v_{L i}^{4}v_{L i}^{5}v_{L i}^{6}) ( v_{L i}^{7}v_{L i}^{8}v_{L i}^{9}) | | v_{R i}^{0}( v_{R i}^{1}v_{R i}^{2}v_{R i}^{3}) ( v_{R i}^{4}v_{R i}^{5}v_{R i}^{6}) ( v_{R i}^{7}v_{R i}^{8}v_{R i}^{9}) \right] ~ .
\varepsilon\int_{\Gamma^{\varepsilon}}\partial_{t}v_{i}^{0}\psi_{i}dS_{\varepsilon}=\varepsilon\int_{\Gamma^{\varepsilon}}\left(A_{i}u_{i}^{0}-B_{i}v_{i}^{0}\right)\psi_{i}dS_{\varepsilon}.
e^{-it\mathcal{L}_{\mathbf{A},a}}u_0(x)=\frac{e^{\frac{i|x|^{2}}{4t}}}{i(2t)^{{N}/{2}}}\int_{{\mathbb{R}}^{N}}K\bigg(\frac{x}{\sqrt{2t}},\frac{y}{\sqrt{2t}}\bigg)e^{i\frac{|y|^{2}}{4t}}u_{0}(y)\,dy,
a_x^{\dagger}=\frac{1}{\sqrt{2}}(m x + D_{x}^{\mu_x}), \ \a_y^{\dagger}=\frac{1}{\sqrt{2}}(n y + D_{y}^{\mu_y}), a_x =\frac{1}{\sqrt{2}}(m x - D_{x}^{\mu_x})\ \a_y=\frac{1}{\sqrt{2}}(n y - D_{y}^{\mu_y})
K =\{({\rm sec}(\theta),\theta): \theta\in (0,\frac{\pi}{4})\cup(\frac{7\pi}{4}, 2\pi)\}\bigcup \{(\sqrt{2},\theta): \theta\in(\frac{\pi}{4}, \frac{3 \pi}{4}) \cup (\frac{5\pi}{4}, \frac{7\pi}{4})\}&\bigcup \{(-{\rm sec}(\theta), \theta): \theta \in(\frac{3\pi}{4}, \frac{5 \pi}{4})\}.
(\ell_1')^3= k_0\,\ell_1^2(m_1\ell_1+m_0)^2\quad, \ell_2=\frac{1}{m_1\ell_1+m_0}b_{11}^3= \frac{k_1}{\ell_1(m_1\ell_1+m_0)}\quad, b_{22}=k_2 b_{11}(m_1\ell_1+m_0) b_{12}= k_3 b_{11}(m_1\ell_1+m_0)\quad, b_{13}=k_4 b_{11}\ell_1\quad, b_{23}=k_5 b_{11}\ell_1\ .
2(2x-1)^{k}>(x+1)^k+(x+2)^k+\cdots +(2x-1)^k=y^n-(2x)^k=y^n-(2x)^{Bn}&=(y-(2x)^B)(y^{n-1}+\cdots + (2x)^{B(n-1)}) \ge (2x)^{B(n-1)}.
\psi_x \left[\begin{matrix}\rho u \rho u^2+p \rho uv \rho uw \end{matrix}\right]-\begin{bmatrix}\rho v \rho uv \rho v^2+p \rho vw\end{bmatrix}+\psi_z\begin{bmatrix}\rho w \rho uw \rho vw \rho w^2+p\end{bmatrix}=0,
\|u\|_{(\kappa_1+1)p^*}=\|u^{\kappa_1+1}\|_{p^*}^{\frac{1}{\kappa_1+1}}\leq M_7(\kappa_1, u) \left [\|u\|_{p^*}^{p^*}+1\right]^{\frac{1}{(\kappa_1+1)p}}<\infty.
\Delta \left( X_{\pm}^{-}\right) \left( V_{\alpha}( z_{1}) V_{\beta}( z_{2}) \right) = X_{\pm}^{-}\left( V_{\alpha}( z_{1}) \right) V_{\beta}( z_{2}) + k_{\pm}^{- 1}\left( V_{\alpha}( z_{1}) \right) X_{\pm}^{-}\left( V_{\beta}( z_{2}) \right){}~ .
\begin{matrix}(x_{i},x_{j})(x_{k}):=\begin{cases}x_{j}\mbox{if $k=i$,}x_{i}\mbox{if $k=j$,}x_{k}\mbox{if $k\neq i,$ $j$.}\end{cases}e_{i}(x_{k}):=\begin{cases}x^{-1}_{i}\mbox{if $k=i$,}x_{k}\mbox{if $k\neq i$}. \end{cases}\end{matrix}
S^{\vee}(g)(\psi,\Psi)(z,\zeta^{-1}) =([\gamma_{\psi(z)}\circ \rho_L(g^{-1})], \gamma_{\psi(z)}\circ \rho_L(g^{-1})\zeta^{-1}) &=([\gamma_{\psi(T_gz)}], \gamma_{\psi(T_gz)}\phi_g(z)^{-1}\zeta^{-1}),
\sum_{\substack{0<\gamma,\gamma'\le T |\gamma-\gamma'|\le \frac{2\pi\lambda}{\log T}}}\!\!\!\!\!\!1 \ \&\ge \N(T)\int_{-1}^1\widehat r(\alpha)F(\alpha)\,d\alpha &= N(T) \bigg(\widehat r(0) + 2\int_0^1 \alpha \, \widehat r(\alpha)\,d\alpha +o(1) \bigg) &= N(T) \bigg( 1 + c(\lambda;r) +o(1) \bigg).
i \sqrt{2}\partial_{+}^{x}G ( x ; y ) + g^{2}\int d w^{-}G ( x^{+}, x^{-}; x^{+}, w^{-}) | x^{-}- w^{-}| G ( x^{+}, w^{-}; y^{+}, y^{-}) = i \delta ( x^{-}- y^{-}) .
\xi =(K\ast |u|^{p_{r;s}^{\uparrow *}}) |u|^{p_{r;s}^{\uparrow *}}+\sum_{i\in \mathcal{I}}\xi_i\delta_{x_i}; \mu \geq \int_{\mathbb{R}^N}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}dy +\sum_{i\in \mathcal{I}}\mu_i \delta_{x_i}; \nu =|u|^{p_s^*}+\sum_{i\in \mathcal{I}}\nu_i\delta_{x_i};
\Sigma(\delta):=\Sigma(\mu_{13}, \mu_{23}, \delta) = \left\{\gamma \in \mathcal{P}(\mathcal{S}): \ \boldsymbol{K}_1( \mu_{13}, \gamma_{13}) \leq \delta_1, \ \boldsymbol{K}_2( \mu_{23}, \gamma_{23}) \leq \delta_2 \right\}
\left\vert \int(\psi\circ\mathsf{R}_{k})\,\left( u-\frac{1}{M}{\textstyle\sum_{j=0}^{M-1}}\widehat{T}^{j}u\right) \,d\mu\right\vert \leq2\sup\left\vert\psi\right\vert \,\int_{\{\varphi_{E_{k}}\leq M\}}u\,d\mu & +\mathrm{Lip}(\psi)\,a_{T}(M)\,\mu(E_{k})
L^{q,\infty}(\Omega) \coloneqq\{f\mid , \sup_{t>0}t\lambda_f (t)^{1/q}<\infty\}&= \{f\mid , \|f\|_{L^{q,\infty}(\Omega)}\coloneqq\sup_{s>0}s^{1/q}f^*(s)<\infty\},
\Sigma(Q,N;\mathcal{S}):=\sum\limits_{\substack{q\in \mathcal{S}Q/2<\mathcal{N}(q)\le Q}}\sum\limits_{\substack{r \bmod{q}(r,q)=1}}\left| \sum\limits_{\substack{n\in \mathbb{Z}[i] \mathcal{N}(n)\le N}}a_{n}\cdot e\left(\mbox{Tr}\left(\frac{nr}{2q}\right)\right)\right|^2,
E_{n}= \hbar \omega \left[ \left( n + \frac{1}{2}\right) \sqrt{1 + \frac{\beta^{2}\mu^{2}\hbar^{2}\omega^{2}}{4}}+ \left( n^{2}+ n + \frac{1}{2}\right) \frac{\beta \mu \hbar \omega}{2}\right] \; .