harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
MRIG_n=\int_{C_W}e^{-\d\,( a^*M_xa+b^*M_x^{-1}b)}\frac{dx}{\sqrt{\det M_x}}=\left(\frac{\pi}{2}\right)^{n/2}\frac{e^{-(a_1b_1+\cdots+a_nb_n)}}{a_1\ldots a_n}
F_{\mu \nu}\equiv \partial_{\mu}A_{\nu}- \partial_{\nu}A_{\mu}\; , \; D_{\mu}\varphi_1 \equiv \partial_{\mu}\varphi_1 + eA_{\mu}\varphi_2\; , \; D_{\mu}\varphi_2 \equiv \partial_{\mu}\varphi_2 - eA_{\mu}(\varphi_1 + \frac me)
\beta_i = \binom{d}{i+2}- d \binom{d-1}{i+1}+ \binom{d}{2}\binom{d-2}{i}+ \sum_{j=3}^{i+2}(-1)^j \left [\sum_{\ell \in L}\binom{\# \ell}{j}\right ] \binom{d-j}{i+2-j}= \binom{d}{i+2}- d \binom{d-1}{i+1}+ \binom{d}{2}\binom{d-2}{i}+ \sum_{\ell \in L}\sum_{j = 3}^{i+2}(-1)^{j}\binom{\# \ell}{j}\binom{d -j}{i+2-j}.
\begin{array}{l}\displaystyle b_{\Lambda}:=\Pi\left (\frac{-\lambda}{2}\right )b_{0}, \displaystyle c_{\Lambda,\tau}=2\pi^{\frac{3}{2}}\Pi\left (\frac{-\lambda}{2}\right )\tau^{-\frac{3-\lambda}{2}}b_{0},\end{array}
w_i^{(N)}:= \begin{cases}({M}_i^{(N)},Y_i^{(N)})_{i=1}^{N-1}, i<N, 0,{i\geq N},\end{cases}w_i^{(\infty)}:= ({M}_i^{(\infty)},Y_i^{(\infty)})_{i\in\mathbb{N}},
g(s)= \frac{\int_{-s}^{s}f_{nn}(x_0+\sigma e_n,t_0)d\sigma}{\psi(s,t_0)}= \frac{2f_{nn}(x_0,t_0)+\frac{1}{3}f_{nnnn}(x_0,t_0)s^2+\cdots}{\psi_s(0,t_0)+\frac{1}{6}\psi_{sss}(0,t_0)s^2+\cdots}= m+\frac{1}{3\psi_s(0,t_0)}(f_{nnnn}(x_0,t_0)-m\psi_{sss}(0,t_0)/2)t^2+\cdots
(Y^{M(\mu,t)^\vee}(b,z)f_m)(m')=H_\mu(m',Y^\mu(\prod_{n=1}^{\infty}e^{\frac{2(-t+\sqrt{-1}\Im(\mu))}{n}(-z)^{-n}a_n}b,z)m) = f_{Y^\mu(\prod_{n=1}^{\infty}e^{\frac{2(-t+\sqrt{-1}\Im(\mu))}{n}(-z)^{-n}a_n}b,z)m}(m').
\begin{aligned}I_s ( (X_s - (T-t)V_s,V_s)) \geq I_s ((X_s - (T-t)V_s^*,V_s^*)) & \textnormal{whenever}t \in [0,T]\textnormal{and}Z_s =(X_s,V_s) \in \partial \mathcal{D}_s\textnormal{is \emph{pre}-collisional}\end{aligned}
RO^{\kappa_H}_{\gamma}(\Phi_1 \times \Phi_2)=\mathcal{O}^{\kappa_H}_{y_{\gamma}}(\Phi_1 * \Phi_2^{-1}) TRO^{\kappa}_{\delta}(f_1 \times f_2)=\mathcal{O}^{\kappa}_{x_{\delta}\rtimes \tau}(f_1 *f_2^{-\tau})
f_{\xi,\xi}(z,w)V_\xi(z)V_{\xi}(w)=f_{\xi,\xi}(w,z)V_{\xi}(w)V_\xi(z)\,, &f_{\xi,\eta}(z,w)V_\xi(z)V_{\eta}(w)-f_{\eta,\xi}(w,z)V_{\eta}(w)V_\xi(z) &=\sum_{k=0}^ns_3^{-1}a_{n,k}\delta\bigl(a^{2n-2ks_1z/w}\bigr)V_{\chi_{k,1}}(z)+\sum_{k=0}^ns_3a_{n,k}\delta\bigl(a^{2n-2ks_1w/z}\bigr)V_{\chi_{\bar k,1}}(w)\,.
\sum_{A\in\mathcal{F}}2^{-|A|/(2q-1)}\leqslant\binom{n}{\lceil n/2\rceil}2^{-\binom{\lceil n/2\rceil}{k}/(2q)}\leqslant\left(\frac{en}{\lceil n/2\rceil}\right)^{\lceil n/2\rceil}2^{-((1+1/\ln 2)n+1)}<\frac{1}{2}\left(\frac{e}{2^{1/\ln 2}}\right)^n=\frac{1}{2},
e^{zL_{-1}+xJ^-_0}(J^+_0+D^+_{\jmath_0}) e^{-zL_{-1}-xJ^-_0}=J^+_0-2xJ^0_0+2\jmath_0 x -x^2\partial_x=(J^+_0-D^+_{\jmath_1})-2x(J^0_0-D^0_{\jmath_1}-\jmath_0)
\int_{\Omega_t}(\partial_t\mathcal B_0+\partial_1\mathcal B_1-2\partial_2\mathcal B_2-\mathcal B_3){\mathbf V}_{x_2}\cdot{\mathbf V}_{x_2}d{\bf x}ds\le C_1\Vert{\mathbf V}_{x_2}\Vert_{L^2(\Omega_t)}^2=C_1\int_0^t I_2(s)ds\,,
A_8 \lesssim \sum_{p> Q_1+2}\|\nabla u_{\leq Q_1}\|_\infty\|b_p\|_2\sum_{|p-q|\leq 2}\lambda_q^{2s}\|b_q\|_2 \lesssim Q_1f(t)\sum_{p> Q_1+2}\|b_p\|_2\sum_{|p-q|\leq 2}\lambda_q^{2s}\|b_q\|_2 \lesssim Q_1f(t)\sum_{q> Q_1}\lambda_q^{2s}\|b_q\|_2^2;
\begin{array}{l l}{g_{\mu \nu}= \eta_{\mu \nu}+ h_{\mu \nu},}\\{\Lambda ( \phi ) = \Lambda ( \phi_{0}) + \Lambda^{\prime}( \phi_{0}) \phi_{( 1 )},}\\{T_{\mu \nu}= T_{( 0 ) \mu \nu}+ T_{( 1 ) \mu \nu},}\\{\phi = \phi_{0}+ \phi_{( 1 )},}\\ \end{array}
| v |^{2}\left({\frac{n}{2}}\right) + \bar{v}v_{-}^{n / 2 - 1}\sqrt{{\frac{n}{2}}}\left({\frac{n}{2}}- c \right) + c . c . + | v_{-}^{n / 2 - 1}|^{2}\left({\frac{n}{2}}- c \right)^{2}.
\left[ \frac{a ( v_{j}) a^{\iota_{2}}( \bar{v}_{j})}{b ( v_{j}) b^{\iota_{2}}( \bar{v}_{j})}\right]^{N}= - \prod_{i \neq j}^{n}\frac{a ( v_{j}- v_{i})}{a ( v_{i}- v_{j})}, \qquad j = 1 , . . . n .
\int_{B_R}(x\cdotp\nabla K)e^{2m\bar{w}}dx + 2m\int_{B_R}Ke^{2m\bar{w}}dx - 2mt\alpha\int_{B_1}(x\cdotp\nabla \bar{w})(-\Delta)^mu_0dx =R\int_{\partial B_R}Ke^{2m\bar{w}}d\sigma -mR\int_{\partial B_R}|\Delta^{\frac{m}{2}}\bar{w}|^2d\sigma - 2m\int_{\partial B_R}fd\sigma,
\frac{s_k (V_{1:k}V_{1:k}^\dagger \mathbb{E}[(A^{(1)})^{\dagger}] W)}{\|(I-V_{1:k}V_{1:k}^\dagger) \mathbb{E}[(A^{(1)})^{\dagger}] W\|_2+\|((A^{(1)})^{\dagger}- \mathbb{E}[(A^{(1)})^{\dagger}]) W\|_2}= \omega(1).
F \Big ( m_{N}, H - \sum_{\ell = 1}^{N - 1}{U}_{\ell}\Big ){\cal A}_{k}^{\dagger}{\cal A}_{\ell}= F \Big ( m_{N}, H - \sum_{\ell = 1}^{N - 1}{U}_{\ell}+ 1 \Big ){\cal A}_{\ell}{\cal A}_{k}^{\dagger}.
\begin{aligned}t_{k}=\tau_{k}^{0}<\tau_{k}^{1}\cdots <\tau_{k}^{J_{k}-1}<\tau_{k}^{J_{k}}=t_{k+1}, \tau_{k}^{j+1}-\tau_{k}^{j}=\frac{\delta_{1}}{4}\lambda(t_{k+1})^{2}, \forall j\leq J_{k}-2 \tau_{k}^{J_{k}}-\tau_{k}^{J_{k}-1}\leq \frac{\delta_{1}}{4}\lambda(t_{k+1})^{2}, \forall j\leq J_{k}-2.\end{aligned}
|\eta(t,s)-\xi| =|p(s,t;y(s,t),\xi)-\xi| &=\left|\int_s^t \nabla_x V_\rho(\tau,q(\tau,t;y(s,t),\xi)) d\tau\right| &\leq C\rho^{\varepsilon_0}\langle s\rangle^{-\varepsilon_1}.
\frac{1}{n_{1}}\log \lVert M^{z}_{n_{1}}(x_{0}+j \omega)\rVert + \frac{1}{n_{1}}\log \lVert M^{z}_{n_{1}}(x_{0}+(-j-n_{1}) \omega)\rVert > 2L_{n_{1}}(z) + O \Big ( \frac{1}{n_{1}}\Big),
\mathbb{E}\Big[\sum_{q=1}^{N_1}\mathbb{P}\big(|\Delta_{i}^n X^c| \geq v_n \sqrt{\Delta_n}\big)\Big] \leq \mathbb{E}\left[N_1\right]2e^{-v_n^2/2{\bar{\sigma}}^2}= 2 \lambda_n e^{-v_n^2/2{\bar{\sigma}}^2}\rightarrow 0.
\int_{\{2\|\mathbf{x}\|>{\eta}_{n}\}}\bigl|{g}_{\infty}(\mathbf{x})-g_{{\lambda}_{n}}(\mathbf{x}) \bigr|^2 \,\mathrm{d}\mathbf{x}= \mathrm{o}(1)\mbox{as}n\to\infty.
\sup_{y\in U_\epsilon}|H(y)-H(e)| &\leq D_\epsilon \sum_{\stackrel{1\leq |\alpha|\leq n}{|\delta|=|\alpha|}}\int_{[-\epsilon,\epsilon]^{|\delta|}}|\langle\lambda,\rho(\tau_\delta(\mathbf{t})) \rho(X_n)^{\delta_n}\cdots\rho(X_1)^{\delta_1}u)| (dt_1)^{\delta_1}\dots (dt_n)^{\delta_n},
\lim_{\varepsilon \to 0}\int_{-\frac{1}{2}}^{\frac{1}{2}}v(\sigma) \int_{{\mathbb{R}}^n}u(s-\varepsilon(1-2\sigma),x)\overline{\phi}(s-\varepsilon(1-2\sigma),x)\,{\rm d}x\,{\rm d}\sigma = \int_{{\mathbb{R}}^n}u(s,x)\overline{\phi}(s,x)\,{\rm d}x,
\tilde{\Gamma}_{s}= \left\{z = z^{( s )}+ \vartheta^{( s )}\varepsilon_{s}( z^{( s )}) , \quad \vartheta = \vartheta^{( s )}\left( 1 + \frac{\varepsilon_{s}\varepsilon_{s}^{\prime}}2 \right) + \varepsilon_{s}( z^{( s )}) \right\},
\left( up\right) \left( x\right) =\sum\nolimits_{n=0}^{m}\left(\sum\nolimits_{\nu =n}^{m}a_{\nu}\left( u\right)_{\nu -n}\right)x^{n}=\left\langle u,\frac{xP_{n+1}\left( x\right) -\xi P_{n+1}\left( \xi \right)}{x-\xi}\right\rangle,
\mathcal T^u=\cos(\hat ft)u-\sin(\hat ft)v-\hat f(\sin(\hat ft)x+\cos(\hat ft)y), \mathcal T^v=\sin(\hat ft)u+\cos(\hat ft)v+\hat f(\cos(\hat ft)x-\sin(\hat ft)y), 2\mathcal T^\phi=2\phi+\hat f^2(x^2+y^2)+T\mathcal T^\phi_T,T\mathcal T^T_T=\mathcal T^T,
J_{2,2}=\int_{\theta_0}^{\frac{3\pi}{2}}\int_{0}^{\frac{R}{\gamma}}\frac{d\rho d\theta}{|\rho \sin\theta+1|^{\tau}\rho^{q-1}}=\int_{\theta_{0}}^{\frac{3\pi}{2}}\int_{1}^{\frac{R}{\gamma}\sin\varphi+1}\frac{1}{\sin\varphi}\frac{dt d\varphi}{|t|^{\tau}\Big(\frac{t-1}{\sin\varphi}\Big)^{q-1}}
m - \frac{( - \det h )}{2}\, \frac{v}{\mu}- \frac{\Lambda_{1}^{3}}{4}\left( \frac{\mu}{v}\right)^{2}= 0 , \qquad - ( - \det h ) + \Lambda_{1}^{3}\left( \frac{\mu}{v}\right)^{3}= 0 \; .
Z_f(s, \chi, A_3) = q^{-\omega}\int_{B_3}\chi(ac (\pi^{n+l}x^p+\alpha y^nz^l+\beta z^{n+l}))|\pi^{n+l}x^p+\alpha y^nz^l+\beta z^{n+l}|^s|dxdydz|,
\boldsymbol{\varsigma}_{k}\!\triangleq\! \textbf{I}_S\left(\hat{\boldsymbol{\psi}}_{k-1}, \textbf{W}_k\right)^\frac{\partial \, p_S \left(\textbf{y}_k |\boldsymbol{\psi},\textbf{W}_k \right)}{\partial \boldsymbol{\psi}}\Bigg|_{{\boldsymbol{\psi}}\!=\! \hat{\boldsymbol{\psi}}_{k\!-\!1}}.
\small{\begin{aligned}\dot{\hat{\tau}}= \dot{\hat{\tau}}^*\left(K_0(\hat{\tau}^*) - K_{\bar{\tau}}(\hat{\tau}^*) \right) + \hat{\tau}^*(\dot{K}_0(\hat{\tau}^*) - \dot{K}_{\bar{\tau}}(\hat{\tau}^*)) + \bar\tau \dot{K}_{\bar\tau}(\hat\tau^*),\end{aligned}}\normalsize
+\frac{1}{\sqrt{2\pi}}\sum_{n=1}^{\infty}\left(\sum\limits_{k=0}^{N-1}\frac{(\varphi_{1n}-q_{k,n}\varphi_{0n})e^{-q_{k,n}t}}{l^{(1)}_n(-q_{k,n})}\right)\psi_n(x),
\times D_{Q}I_{C D}D_{J + L + T + U - P - Q - R - S}{\frac{\partial^{2}f}{\partial \phi_{A}^{( U )}\partial \phi_{C}^{( T )}}}\biggl [ D_{R}E_{B}^{J}( h ) D_{S}E_{D}^{L}( g ) - ( H \leftrightarrow G ) \biggr ] .
P_{Y^{i-1},{\bf S}}(y^{i-1},{\bf s}) = P_{Y^{i-1},{\bf S}}(\tilde{y}^{i-1},{\bf s}), &\sum_{i=t}^L H(Y_i |{\bf S}, y^{t-1}, Y_t^{i-1}) = \sum_{i=t}^L H(Y_i |{\bf S}, \tilde{y}^{t-1}, Y_t^{i-1})
\ \ \frac{d}{dt}\left( \int_{\mathbb{T}_{\alpha}}(|\omega_{s2}^{\nu}|^{2}-|\partial_{y}\psi_{s2}^{\nu}|^{2})dxdy+\left\Vert \omega_{n}^{\nu}\right\Vert_{X}^{2}\right) & =-2\nu\left( \int_{\mathbb{T}_{\alpha}}(|\partial_{y}\omega_{s2}^{\nu}|^{2}-|\omega_{s2}^{\nu}|^{2})dxdy+\left\Vert \omega_{n}^{\nu}\right\Vert_{X^{1}}^{2}\right) .
f_1(n,a,b,c,d) = (a+b-c-d-1)(a+b-c-d-2) \quad\cdot [c^3(a+b-c-d)+(a-c)(3c^2+3c+1)(n-c-d)] + (a-c)(a-c-1)(n-c-d)(n-c-d-1) \quad\cdot [3(a+b-c-d-2)(c+1)+(a-c-2)(n-c-d-2)]
\omega_{12}=\frac{B_{12,1}}{b_1-b_2}\omega_1,~\omega_{13}=\frac{B_{13,1}}{b_1-b_3}\omega_1,~\omega_{23}=\frac{B_{23,2}}{b_2-b_3}\omega_2+\frac{B_{23,3}}{b_2-b_3}\omega_3.
\Lambda^{(m)}_2(\lambda)(f,g) = -m!\sum_{k = 1}^\infty \partial_\nu(\Delta\phi_k)\Big\vert_{\partial B_R}\frac{1}{(\lambda_k - \lambda)^{m + 1}}\Big( \int_{\partial B_R}\partial_\nu\phi_k f{\rm d}s(y) &\quad+ \int_{\partial B_R}\partial_\nu(\Delta\phi_k )g{\rm d}s(y)\Big),
N(k) =\frac{q^2-1}{q^k-q}\frac{1}{q^{k-2}}q^{2(k-2)}q^2(q-1)\times &\times\frac{(q^k-1)(q^{k-1}-1)}{(q^2-1)(q-1)}N(k-2)= &=q^{k-1}(q^k-1)N(k-2).
\left(\frac{H_{k+1}^{(s,t)}H_{k-1}^{(s,t+1)}}{H_{k}^{(s,t)}H_{k}^{(s,t+1)}}\right){\cal H}_{k-1}^{(s,t+1)}(z)={\cal H}_{k}^{(s,t)}(z)-{\cal H}_{k}^{(s,t+1)}(z),
H ( x_{1}, x_{2}) = \left( \begin{array}{c c}{\bar{q}_{D}( x_{1}) q_{C}( x_{2})}&{D_{A B}( x_{1}) q_{C}( x_{2})}\\{\bar{q}_{D}( x_{1}) \bar{D}_{E F}( x_{2})}&{D_{A B}( x_{1}) \bar{D}_{E F}( x_{2})}\\ \end{array}\right)
p_{\omega}(n)=\begin{cases}1, n = 0, 2n, 1\leq n\leq N-1, n + 1 + \sum_{j=-1}^{k-1}(p_{j}+ q_{j})(N - 1) , t_k< n \leq s_{k+1}, 2n + 1 + \sum_{j=-1}^{k-1}(p_{j}+ q_{j}) (N - 1) - s_k, s_k < n \leq t_k\end{cases}
\tilde{U}(b_s,b_tec,h) =b_sb_tec\otimes h=b_sb_t\otimes \pi(ec)h =b_sb_t\otimes \pi(e)\pi(c)h =b_sb_te\otimes \pi(c)h =\tilde{U}(b_s,b_t, \pi(c)h).
E_{\pi|_{V_1}}((T_1, \ldots, T_n)|_{V_1}) \cdots E_{\pi|_{V_p}}((T_1, \ldots, T_n)|_{V_p}) E_{\pi|_{Y}}((T_1, \ldots, T_n)|_{Y}) = E_{\pi|_{V}}((T_1,\ldots, T_n)|_{V}),
v\in E^s_x \Leftrightarrow \|Df^k_x(v)\|\leq C\lambda_+^k\|v\| \textrm{for all}k\geq 0, v\in E^u_x \Leftrightarrow \|Df^k_x(v)\|\leq C\mu_-^{-k}\|v\| \textrm{for all}k\leq 0, v\in T_x\Lambda \Leftrightarrow \|Df^k_x(v)\|\leq C\eta_+^k \|v\|, \|Df^{-k}_x(v)\| \leq C\eta_-^{-k}\|v\|,\textrm{for all}k \geq 0.
\frac{\rho_{0}u_{0}^{2}}{r_{0}}\frac{\partial\rho^{*}u^{*}}{\partial t^{*}}+\frac{\nabla^{*}}{r_{0}}\cdot(\rho_{0}u_{0}^{2}\rho^{*}u^{*}\otimes u^{*}+p_{0}p^{*})=-\frac{\tau_{0}}{r_{0}}\nabla^{*}\cdot\tau^{*}+f_{0}f^{*}
\begin{cases}\partial_t\widehat \rho = \;\; \Delta \widehat\rho \,+\, \widehat F(\widehat \rho) \;\;\; \hbox{in}\Lambda \times (0,T) , \widehat \rho (0,\cdot) = \;\; \widehat \gamma (\cdot) \;\;\; \hbox{in}\Lambda , \widehat \rho(t,\cdot) |_\Gamma = \;\; \widehat b(\cdot) \;\;\; \hbox{for}\; 0\leq t\leq T \, ,\end{cases}
\begin{array}{ll}\frac{(\partial_{t,x}^{k}w)_t}{\beta-\eta|_{y=0}}e^{-2\lambda t}\partial_{t,x}^{k}w begin{align*}7pt]- \partial_{t,x}^{k}\Big(w_y + 2\eta|_{y=0}w + \tilde{f}+ \zeta|_{y=0}\int\limits_0^{+\infty}w(t,x,\tilde{y}) \,\mathrm{d}\tilde{y}\Big)\, e^{-2\lambda t}\partial_{t,x}^{k}w\end{array}
\sum_{k=1}^{r_1+r_2}\frac{1}{2\pi i}\int_{(c)}\frac{\Gamma'_k}{\Gamma_k}(s)\hat{g}(s)ds+ \sum_{k=1}^{r_1+r_2}\frac{1}{2\pi i}\int_{(c')}\frac{\Gamma'_k}{\Gamma_k}(1-s)\hat{g}(s)ds,
\lambda_{2}(a) = \lambda_{2}^{(0)}(a) + k^{-2}\lambda_{2}^{(2)}(a) g_a^{\mu \nu}\partial_{\mu}\partial_{\nu}+ k^{-4}\lambda_{2}^{(4)}(a) g_a^{\mu \nu}g_a^{\rho \sigma}\partial_\mu \partial_\nu \partial_\rho\partial_\sigma + \ldots ,
S \to{\frac{a S + b}{c S + d}}, \left( \begin{matrix}{{{\cal F}_{\mu \nu}^{i}}}\\{{\tilde{\cal F}_{\mu \nu}^{i}}}\\ \end{matrix}\right) \to \omega_{S}^{- 1}\left( \begin{matrix}{{{\cal F}_{\mu \nu}^{i}}}\\{{\tilde{\cal F}_{\mu \nu}^{i}}}\\ \end{matrix}\right) ; \ \omega_{S}= \left( \begin{matrix}{{a}}&{{b}}\\{{c}}&{{d}}\\ \end{matrix}\right) \in S L ( 2 ,{\bf R})_{S},
I\lesssim\big{\|}e^{-t\partial_x^3}\big{(}\tilde{u}(0)\big{)}\big{\|}_{L_{[0,t]}^{\infty}\dot{H}^{\sigma_0}}=\frac{1}{\lambda(0)^{\frac{2}{p-1}-\frac{1}{p_0}}}\|\varepsilon(0)\|_{\dot{H}^{\sigma_0}}\leq \frac{b_c^{10}}{\lambda(t)^{\frac{2}{p-1}-\frac{1}{p_0}}}.
r_n(x)-r(x)= ~(g_n-rf_n)(x)-\mathrm{E}((g_n-rf_n)(x)) &- (r(x)-\mathrm{E}g_n(x)) + (r(x)-\mathrm{E}g_n(x))(f_n(x)-\mathrm{E}f_n(x)) &-(g_n(x)-\mathrm{E}g_n(x))(f_n(x)-\mathrm{E}f_n(x))+r_n(x)(f_n(x)-\mathrm{E}f_n(x))^2.
\lim_{\mu_{f}, \mu_{g}\to 0}\int_{0}^{\infty}d |{\bf p}| \: \frac{\partial}{\partial |{\bf p}|}{\tilde{g}}_{{\mu}_{g}}( |{\bf p}| ) \cdot{\tilde{f}}_{{\mu}_{f}}( |{\bf p}| ) = - \frac{1}{2},
d s^{2}= - \frac{f^{2}}{l^{2}}\left( r_{+}d \phi - r_{-}\frac{d t}{l}\right)^{2}- \frac{f{' 2}d r^{2}}{( \frac{f^{2}}{l^{2}}+ 1 )}+ \left( \frac{f^{2}}{l^{2}}- 1 \right) \left( r_{-}d \phi - r_{+}\frac{d t}{l}\right)^{2}
\hat{B}=\sum_{j_{1}=1}^{n_{1}}\sum_{j_{2}=1}^{n_{2}}\cdot\cdot\cdot\sum_{j_{d}=1}^{n_{d}}b_{j_{1},j_{2},...,j_{d}}e_{j_{1}}^{1}\otimes e_{j_{2}}^{2}\otimes\cdot\cdot\cdot\otimes e_{j_{d}}^{d},
(\mathbf{u}_{tt},\mathbf{v})+\mu a(\mathbf{u}_t,\mathbf{v})+a_1(\mathbf{u}_t,\mathbf{u},\mathbf{v})+a_1(\mathbf{u},\mathbf{u}_t,\mathbf{v})-d(\mathbf{v},p_t) &\qquad+d(\mathbf{u}_t,q)+J(t;\mathbf{u}_t,\mathbf{v})=-\rho t^{-\beta}e^{-\delta t}a(\mathbf{u}(0),\mathbf{v})+(\mathbf{f}_t(t),\mathbf{v}).
{}_0\mathbf{F}^2_q(b;X,Y) = \int_{\mathcal{S}(r)}{}_0\mathbf{F}_{q}\left(b;xq_r'Y q_r\right) ({\rm d}q_r) = \frac{1}{(2\pi i)^q}\int_{-\infty}^{(0+)}\cdots \int_{-\infty}^{(0+)}e^{\sum_{j=1}^q w_j}\int_{\mathcal{S}(r)}e^{\frac{x}{w}q_r'Y q_r}({\rm d}q_r) \frac{{\rm d}w}{w^b}.
\gamma\alpha =\gamma\alpha\varepsilon_Pr=\varepsilon_ML(h)\varepsilon_N^{-1}\alpha\varepsilon_Pr=\varepsilon_ML(h)\varepsilon_N^{-1}\varepsilon_NLR(\alpha)r =\varepsilon_ML(hR(\alpha))r=\varepsilon_MLR(\beta)r=\beta\varepsilon_Pr=\beta.
a(x,\xi) = a(x, d\varphi(x)) + \int_0^1 \partial_\xi a(x, t\xi + (1-t)d\varphi(x)) dt \, (\xi -d\varphi(x)) & = a(x, d\varphi(x)) + b(x,\xi) (x+p'(\xi)) \, ,
ds_{\rm BTZ}^2 = -{{(\rho^2 - \rho_+^2)(\rho^2 -\rho_-^2)}\over{\rho^2 R^2}}dt^2 + \rho^2 ( d \varphi -{J \over 2 \rho^2}dt )^2 +{\rho^2 R^2 \over{(\rho^2 - \rho_+^2)(\rho^2 -\rho_-^2)}}d \rho^2.
\overline{PL}_{\rm Prob}(d) = P_{\mathrm{LOS}}(d) \overline{PL}_{\rm LOS}(d)+P_{\mathrm{NLOS}}(d) \overline{PL}_{\rm NLOS}(d) X_{\sigma}(d) = P_{\mathrm{LOS}}(d) X_{\sigma,\rm LOS}+ P_{\mathrm{NLOS}}(d) X_{\sigma,\rm NLOS}
\int_{\tau_\ell}^{\tau_{\ell+1}}G_\ell(v) \: dv = s\ell(\tau_{\ell+1}- \tau_{\ell}) - s \int_{\tau_{\ell}}^{\tau_{\ell+1}}M(v) \: dv - \mu(\tau_{\ell+1}- \tau_{\ell}).
q=\frac{c_1}{\sqrt{c_1^2-2c_2}}\ln{\left(\frac{\gamma+c_1-\sqrt{c_1^2-2c_2}}{\gamma+c_1+\sqrt{c_1^2-2c_2}}\right)}-\ln{\left(\frac{1}{2}\gamma^2+c_1\gamma+c_2\right)}
T_Q = \begin{bmatrix}Q_0 Q_{-1}Q_{-2}\cdots \cdots Q_{1}Q_0\otimes I_d Q_{-1}\otimes I_d \cdots \cdots Q_{2}Q_{1}\otimes I_d Q_0\otimes I_d\otimes I_d \cdots \cdots \vdots \vdots \vdots \ddots \end{bmatrix},
\tilde{m}'_{-,-}(\xi-\eta, \eta):= \tilde{m}_{-,-}(\xi-\eta, \eta), \widetilde{Q}_{\mu, \nu}(\Phi_{\mu}, \Phi_{\nu}):= \tilde{Q}_{\mu, \nu}(\Phi_{\mu}, \Phi_{\nu}), (\mu, \nu)\in\{(+,+), (-,-)\}.
\frac{H}{p}w w^{\prime}\Bigg |_{r_{1}}^{r_{2}}= \int_{r_{1}}^{r_{2}}d r \, \Bigg \{\frac{H}{p}( w^{\prime})^{2}- \frac{p}{H}u^{2}w^{2}- \frac{1}{p r^{2}}w^{2}( 1 - w^{2}) \Bigg \}~ ~ .
\phi_{n}^{\pm}(x)=\frac{e^{i \left[\left(\varphi \pm 2n\pi \right) x/a\pm \theta \right]}}{\sqrt{a}}; E_{n}^{\pm}=\frac{\hslash^{2}}{2m}\left(\frac{2n\pi \pm \varphi}{a}\right)^{2}+\frac{\beta \hslash^{4}}{3m}\left(\frac{2n\pi \pm \varphi}{a}\right)^{4},
\mathbb E \bigl[(f(s) - \widetilde f(s))g(s)\bigr] = \mathbb E\Bigl[ \mathbb E\bigl((f(s) - \widetilde f(s))g(s)\big|\mathcal F_{\frac{n}{2^m}}\bigr) \Bigr] =\mathbb E\Bigl[ \mathbb E\bigl((f(s) - \widetilde f(s))\big|\mathcal F_{\frac{n}{2^m}}\bigr)g(s) \Bigr] = 0.
\tilde{\cal L}_{\kappa=0}= -8(D-2)^2\int dr \left({\partial W\over\partial\phi}\right)^2\left(1+{1\over 2}\partial_{\mu}\tilde r\partial^{\mu}\tilde r\ -{1\over 2}\bar{\tilde\eta}_{-i}\tilde\Gamma^{\mu}\partial_{\mu}\tilde\eta^i_-\right)\ .
\begin{cases}\partial_t h = |\partial_x|\omega - \partial_x T_V h + M_2(\omega,h) + G_{\geq 3}, & \partial_t \omega = \partial_x^2 h + \partial_x T_\sigma h_x - T_V \omega_x + T_{|\partial_x|^3 h}P_{\geq 1}h + Q_2(\omega,\omega) + \Omega_{\geq 3},\end{cases}
\exp \left[ - i g \tilde{g}\epsilon_{\mu \nu \rho \sigma}( \bar{\tilde{\psi}}\gamma^{\rho}\tilde{X}^{\alpha}\tilde{\psi}) \dot{\xi}^{\sigma}\tilde{X}_{\alpha}\delta \xi^{\mu}\delta \xi^{\nu}\right] = \exp 2 \pi i / N .
s_1(k_1, k_2, k_3) := (-k_3, -k_1, -k_2), s_2(k_1, k_2, k_3) := (k_1 + k_2, k_2, k_3), t(k_1, k_2, k_3) := (k_1 + 1/2, k_2, k_3)
\left[ - \nabla_{\widehat{\mbox{\boldmath $\scriptstyle{\xi}$}}, D}^2 - \widehat{{\mathcal W}}^{({D})}( \widehat{\mbox{\boldmath $ \xi$}})\right]\, \widehat{\Phi}( \widehat{\mbox{\boldmath $ \xi$}})= \widehat{\eta}\, \widehat{\Phi}( \widehat{\mbox{\boldmath $ \xi$}})\; .
B_{s=i^+}=\, \begin{cases}\dim(\overline{W}_{j_-(s)}/\overline{W}_{j^+(s)}) \,, s_- + \dim(\overline{W}_{j_-(s)}/\overline{W}_{j^+(s)})< i\,, t - s_- \,, s_- +\dim(\overline{W}_{j_-(s)}/\overline{W}_{j^+(s)}) \ge i \end{cases}=\, \min\left\{\dim(\overline{W}_{j_-(s)}/\overline{W}_{j^+(s)}), i-i_-\right\}\,.
(zxe_{1}+x^{2}e_{2})[k(1-\alpha)] =\Big(zxp_{1}+\frac{x^{2}}{2}(p_{1}^{2}-p_{2})\Big)[k(1-\alpha)] =kzx(1-\alpha)+\frac{x^{2}}{2}(k^{2}(1-\alpha)^{2}-k(1-\alpha^{2})) =kzx-kzx\alpha+{k \choose 2}x^{2}+{k+1 \choose 2}x^{2}\alpha^{2}-k^{2}x^{2}\alpha.
\hat{R}_{i j , k l}^{\epsilon}= \epsilon_{j s}^{t}\hat{R}_{i s , k t}( \epsilon^{t})_{t l}^{- 1}\quad ( \hat{R}^{\epsilon}= ({\bf 1}\otimes \epsilon^{t}) \hat{R}({\bf 1}\otimes ( \epsilon^{- 1})^{t}) ) \; ,
III = \ \sum_{2^\ell = \alpha^{-2}t^2}^\infty \min\{2^{(k+\frac{1}2)\ell}|\alpha|^{-1}(\|(w,r)\|_{H^{-1}}+ \|(g,k)\|_{H^{-1}}),2^{-(k+\frac32) \ell}\|(w,r)\|_{\dot H^{2k+2}}\}\lesssim \ |\alpha|^\frac12 |t|^{-1}(\|(w,r)\|_{H^{-1}}+ \|(g,k)\|_{H^{-1}})^\frac12 \|(w,r)\|_{\dot H^{2k+2}}^\frac12
\ddot{\phi}+ 3 H \dot{\phi}+{\frac{\dot{\phi}^{2}}{\phi}}={\frac{1}{( 1 + 6 \epsilon ) \phi}}[ 4 V - \phi V^{\prime}] -{\frac{1}{( 1 + 6 \epsilon ) H \phi}}[ \dot{K}+ 4 H K ] .
{\frac{\partial}{\partial x^{\alpha}}}\; \Gamma_{r}^{\alpha \beta \gamma}\ ( x , y , z ) = \{\frac{i}{4 \pi^{2}}}\; \varepsilon^{\beta i \gamma j}{\frac{\partial}{\partial y^{i}}}{\frac{\partial}{\partial z^{j}}}\delta ( x - y ) \delta ( x - z ) .
\Lambda(\alpha, 0)=\int\frac{d^D k}{(2\pi)^D}\frac{1}{\left[ k^2-m_0^2\right]^\alpha}=\frac{i}{(4\pi)^{D/2}}\left(-m_0^2\right)^{2-\alpha}m_0^{2\epsilon}\frac{\Gamma(\alpha-2-\epsilon)}{\Gamma(\alpha)}
g^{\mu\nu}\langle i_{\partial_\mu}G,i_{\partial_\nu}G\rangle =g^{\mu\nu}\frac{1}{(k-1)!}\sum_{\mu_2,\ldots,\mu_k}G_{\mu\mu_2\ldots\mu_k}G_{\nu}^{\,\,\,\mu_2\ldots\mu_k}&=\frac{1}{(k-1)!}\sum_{\mu,\mu_2,\ldots,\mu_k}G_{\mu\mu_2\ldots\mu_k}G^{\mu\mu_2\ldots\mu_k}&=\frac{k!}{(k-1)!}|G|^2 &=k|G|^2.
\max_{j\in B}\left\{\Pr_{v\sim\mu_t}\big[|vH|=j \big]\cdot \frac{2^N}{\binom{N}{j}}\right\}\leq \sqrt{\frac{e^4 N}{8\pi}}\cdot 2^{(1-h(\beta))\eta N}\leq \sqrt{\frac{e^4 N}{8\pi}}\cdot 2^{4\tilde{\epsilon}(1-\tilde{\epsilon})\eta N}.
d \tilde{s}^{2}= \eta_{A B}\, d X^{A}\, d X^{B}= - ( d X^{0})^{2}+ ( d X^{1})^{2}+ \dots + ( d X^{d + 1})^{2}\equiv - ( d X^{0})^{2}+ d \vec{X}\cdot d \vec{X}\, ,
J_{\nu - 1}^{2}( i x ) - e^{2 \theta}J_{\nu}^{2}( i x ) ={J_{\nu}^{\prime}}^{2}( i x ) - \left({\frac{\nu^{2}}{x^{2}}}+ e^{2 \theta}\right) J_{\nu}^{2}( i x ) +{\frac{2 \nu}{i x}}J_{\nu}( i x ) J_{\nu}^{\prime}( i x ) .
\frac{1}{p}\|(u_k,v_k)\|^p-\frac{1}{q}\int_\Omega(\lambda|u_k|^q+\mu|v_k|^q)dx-\frac{2}{p_s^\ast}\int_\Omega|u_k|^\alpha|v_k|^\beta dx = c+o_k(1), \|(u_k,v_k)\|^p- \int_\Omega(\lambda|u_k|^q+\mu|v_k|^q)dx-2\int_\Omega|u_k|^\alpha|v_k|^\beta dx = o_k(1).
\Phi(W) = \Phi\left(V_1 L_m^{g_2}R_k^{g_1}V_2\right) + \Phi(V_1 P_0 V_2) = \Phi\left(V_1 L_m^{g_2}R_k^{g_1}V_2\right) + \Phi(V_1)\Phi( V_2)
\begin{array}{l l}{{\}}&{{e^{- 2 \Phi}={\frac{M}{\lambda}}- \lambda^{2}x^{+}x^{-}={\frac{M}{\lambda}}+ e^{2 \lambda \sigma}}}\\{{\Rightarrow}}&{{e^{- 2 \left( \Phi + \lambda \sigma \right)}= 1 +{\frac{M}{\lambda}}e^{- 2 \lambda \sigma}}}\\{{\Rightarrow}}&{{\Phi \sim - \lambda \sigma -{\frac{M}{2 \lambda}}e^{- 2 \lambda \sigma}\; .}}\\ \end{array}
+ \frac{1}{2 e^{2}\mid \varphi \mid^{2}}\{( \partial_{\nu}\Lambda^{\nu \mu}) + \frac{e}{2}( \varphi^{\dagger}\partial_{\mu}\varphi - \partial_{\mu}\varphi^{\dagger}\varphi ) \}^{2}+ \frac{1}{2}\mid \partial_{\mu}\varphi \mid^{2}.
z^{3^{\nu-1}}\left(1+z^{3^{\nu-1}}\right)^{2}=\left(1+z^{3^\nu}\right)-\left(1+z^{3^{\nu-1}}\right)^{2}3 z^{3^{\nu-1}}\left(1+z^{3^{\nu-1}}\right)^3 =z^{3^{\nu-1}}\left(1+z^{3^\nu}\right)3
\begin{array}{r c c c l c r c c c l}{\left( \frac{G}{H}\right)_{7}}&{=}&{\frac{S O ( 8 )}{S O ( 7 )}}&{\equiv}&{S^{7}}&{;}&{\left( \frac{G}{H}\right)_{4}}&{=}&{\frac{S O ( 5 )}{S O ( 4 )}}&{\equiv}&{S^{4}}\\ \end{array}
d\omega =\frac{i}{2\pi}\frac{(z_{1}dz_{0}-z_{0}dz_{1})\wedge (\bar{z}_{1}d\bar{z}_{0}-\bar{z}_{0}d\bar{z}_{1})}{\left( \left\vert z_{0}\right\vert^{2}+\left\vert z_{1}\right\vert^{2}\right)^{2}}
\left. \frac{s h{\omega}^{'}x_{0}}{{\omega}^{'}}\Theta \left({\mu}^{2}-{\vec{p}}^{\, 2}\right) \right] = \frac{- 1}{\left( 2 \pi \right)^{\nu / 2}}\int d^{\nu - 1}p e^{i \vec{p}\cdot \vec{r}}\frac{s i n \Omega x_{0}}{\Omega}
T_j \mathbf{e}^{i{\xi}}=_j(s_j\boldsymbol{\xi}) \mathbf{e}^{i\boldsymbol{\xi}}+_j(s_j\boldsymbol{\xi}) \mathbf{e}^{i s_j \boldsymbol{\xi}}=_j(-\boldsymbol{\xi}) \mathbf{e}^{i\boldsymbol{\xi}}+_j(-\boldsymbol{\xi}) \mathbf{e}^{i s_j \boldsymbol{\xi}},
\psi_{2}\rightarrow \exp \left( ( i \gamma_{i}\partial^{i}+ e \gamma_{i}A^{i}) / ( 2 m_{2}) \right) \psi_{2}, \ \overline{{\psi}}_{2}\rightarrow \overline{{\psi}}_{2}\exp \left( ( - i \gamma_{i}\stackrel{\leftarrow}{\partial^{i}}+ e \gamma_{i}A^{i}) / ( 2 m_{2}) \right) .
\hat h = \frac{\hbar^2}{2}\left(-\frac{1}{\mu(q)}\Delta +\frac{\alpha}{\mu^2c^2}C_1\frac{{\bf q}}{q^3}\frac{\partial}{\partial{\bf q}}+ \frac{4\pi\alpha\lambda}{\mu^2c^2}C_1\delta({\bf q})\right)+ v(q)\,,
\mathfrak{X}_{[m]\times P}(L_d^{n_0}, L_{i_1}^{n_1}, \dots, L_{i_s}^{n_s})=(L_{i_1+1}^{n_0+1}, L_{i_2+1}^{n_1}, \dots, L_{i_s+1}^{n_{s-1}}, L_0^{n_s-1}),
\tilde\phi_2(X,Y,t) :=u(X_{j_l},Y_{j_l},t_{j_l})-w(Y_{j_l},t_{j_l})-\varphi_{j_l}( X_{j_l}, Y_{j_l},t_{j_l},\tilde X_{j_l},\tilde Y_{j_l},\tilde t_{j_l}) &+\varphi_{j_l}(X,Y,t,\tilde X_{j_l},\tilde Y_{j_l},\tilde t_{j_l})+w(Y,t),
-(M_0)^{-1}M_1 =\begin{pmatrix}0 0 \cdots 0 (-1)^{(r-1)r/2+1}c_0 1 0 \cdots 0 (-1)^{(r-1)r/2+1}c_1 0 1 \cdots 0 (-1)^{(r-1)r/2+1}c_2 0 0 \ddots 0 \vdots 0 0 \cdots 1 (-1)^{(r-1)r/2+1}c_{r-1}\end{pmatrix}.
= \sum_{n=2^{2^N}+1}^{2^{2^{N+1}}}\left(\psi(n)-\psi(n+1) \right) \sum_{j=2^{2^N}+1}^n \frac{\varphi(j)}{j|{j}|_{\mathcal{D}}}+ \psi(2^{2^{N+1}}+1) \sum_{j=2^{2^N}+1}^{2^{2^{N+1}}}\frac{\varphi(j)}{j|{j}|_{\mathcal{D}}}.
h^{i}\equiv f_{+}^{- i}+ f_{-}^{+ i}= 2 f_{+}^{+ i}={\frac{1}{2}}u^{\underline{{{m}}}i}u^{\underline{{{n}}}-}u^{\underline{{{k}}}+}H_{\underline{{{k}}}~ \underline{{{n}}}~ \underline{{{m}}}}( X ( \xi ) )
\left| (w_{m})_{B_{r_j}}- (w_{m})_{B_{r_0}}\right| \leq C \sum_{i=1}^{j}\left\| w_{m}- (w_{m})_{B_{r_i}}\right\|_{\underline{L}^2 \left( B_{r_i}\right)}\leq C r_j r_{m}^{-\beta}.
\left\Vert v_{n}\right\Vert_{1,2}^{2}=\int_{\Omega}g(x,\mathrm{T}_{1}(z_{1,n}),\mathrm{T}_{2}(z_{2,n}))v_{n}\,\mathrm{d}x\leq M\int_{\Omega}v_{n}\,\mathrm{d}x.
(\bar{A}\Gamma_{\hat{a}}B)(\bar{C}\Gamma^{\hat{a}}D)=-\frac{1}{2}(\bar{A}\Gamma_{\hat{a}}e_{l}D)(\bar{B}\Gamma^{\hat{a}}e_{l}C)+\frac{1}{2}(\bar{A}\Gamma_{\hat{a}}e_{l}C)(\bar{B}\Gamma^{\hat{a}}e_{l}D),
\zeta + a_{0}\frac{1}{\zeta}x_{1}^{4 - n}z^{6 - n}\to \zeta + a_{0}\frac{1}{\zeta}\prod_{i = 1}^{N_{s}}( x_{1}- m_{i}^{4}) \prod_{j = 1}^{N_{f}}( z - m_{j}^{2}) ,