harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
T^{\mathcal N}_{\rho_{\varepsilon}}:=(\pi^{\sharp,\mathcal N}_{\rho_{\varepsilon}})^{-1}\circ (\mathcal P^{\mathcal N}_{\rho_{\varepsilon}})^{-1}\circ \mathcal J^{\mathcal N}_{\rho_{\varepsilon}}\circ i \circ \pi^{\sharp,\mathcal N}_{\rho_{\varepsilon}}.
\phi_1 * \phi_2 (g) = \int_{0}^{\infty}\int_{0}^{\infty}\int_{G}^{}1_{L_1 (t_1)}(g') 1_{L_2 (t_2)}(g'^{-1}g) dg' dt_1 dt_2 = \int_{0}^{\infty}\int_{0}^{\infty}1_{L_1 (t_1)}* 1_{L_2 (t_2)}(g) dt_1 dt_2.
B_{R}^{2}[u] =\mu\frac{-2N(q-2)}{q}\int_{\mathbb{R}^{N}}|u|^{q}-\mu\frac{2(q-2)}{q}\int_{|x|\geq R}(\Delta \varphi_{R}-N)|u|^{q}&=\mu\frac{-2N(q-2)}{q}\int_{\mathbb{R}^{N}}|u|^{q}+O(R^{-\frac{(q-2)(N-1)}{2}}\|\Delta u\|_{2}^{\frac{q-2}{4}}),
-1-\frac{v}{v_{1}}+\frac{f(x,y_{1},v_{1})}{f(x,y,v)}+\frac{v}{v_{1}}\frac{f(x,y,v)}{f(x,y_{1},v_{1})}&=\left(1-\frac{f(x,y,v)}{f(x,y_{1},v_{1})}\right)\left(\frac{f(x,y_{1},v_{1})}{f(x,y,v)}-\frac{v}{v_{1}}\right)\leq 0.
1 - \frac{1}{8}x^{2}y^{2}+ \frac{1}{1 9 2}x^{4}y^{4}- \left( \frac{1}{9 2 1 6}x^{6}+ \frac{1}{2 4 x^{2}}\right) y^{6}+ o ( y^{8}) - \frac{4 a}{\pi y^{2}x^{2}}+ . . .
q_{g,n,\ell}(\mathbf p)\triangleq \Pr\left[C_{g,n,\ell}\geq R_{g,\ell}\right]= \Pr\left[\frac{1}{K_{g,n,\ell}}\log_2(1+{\rm SIR}_{g,n,\ell})\geq \theta_g\right],g\in\{{\rm SVC},{\rm DASH}\},
\Gamma^{P}( g^{\mu \nu}) = \frac{1}{9 6 \pi}\int d^{2}x \partial_{\mu}\sigma \gamma^{\mu \nu}\partial_{\nu}\sigma + \frac{1}{4 8 \pi}\int d^{2}x \partial_{\mu}\sigma{\cal R}^{\mu}+ \Gamma ( \gamma^{\mu \nu})
\wp(p|\tau)=\frac{-4(C\eta_{1}-\eta_{2})^{3}-g_{2}(C\eta_{1}-\eta_{2})(C-\tau)^{2}+2g_{3}(C-\tau)^{3}}{(C-\tau)[12(C\eta_{1}-\eta_{2})^{2}-g_{2}(C-\tau)^{2}]},
a_k\leq Cb^{-\frac{s}{Q}}\bigg(\sum_j\int_{2B_0}g_j^p\,d\mu\bigg)^{\frac{s}{Q}}\sum_{n=k_0}^{k-1}2^{n(1-\frac{sp}{Q})}+C'r_0^{s}2^{k_0}.
(\underline{s}_i \circ \widehat{\pi}_i)(x) := \min \{\max \{(\underline{r}_i \circ \widehat{\pi}_i)(x),-R\},R\}(\overline{s}_i \circ \widehat{\pi}_i)(x) := \min \{\max \{(\overline{r}_i \circ \widehat{\pi}_i)(x),-R\},R\}.
S(\omega_{{\sf f}_1 \cdots{\sf f}_m}\|\omega) = i\left.\frac{d}{dt}\right|_{t = 0}{\rm Pf}({\bf A}(t))= \frac{1}{2}\,{\rm Tr}\left({\bf A}^{-1}\left(\begin{array}{cc}0 -{\bf a}{\bf a}^T 0 \end{array}\right) \right)
R_{2n}^{\lambda}(x) =\frac{a_n^\lambda}{(1+x^2)^{\frac{\lambda+1}{2}}}\,{}_{2}F_{1}\Big(\!-n,n+\lambda;\lambda+\frac{1}{2};\frac{1}{1+x^2}\Big) begin{align*}2pt] =a_n^\lambda\, \sum_{k=0}^{n}\frac{\left(-n\right)_{k}\left(n+\lambda\right)_{k}}{\left(\lambda+\frac{1}{2}\right)_{k}k!}\frac{1}{\left(1+x^2\right)^{k+\frac{\lambda+1}{2}}},
\sum_{\pi(1,2,...n)}\mbox{Tr}\Big[\tau^{(l_0)}\tau^{(l_{\pi(1)})}.....\tau^{(l_{\pi(n)})}\Big]\frac{1}{\tilde{x}_0 - \tilde{x}_{\pi(1)}}\frac{1}{\tilde{x}_{\pi(1)}- \tilde{x}_{\pi(2)}}.....\frac{1}{\tilde{x}_{\pi(n)}- \tilde{x}_0}\; ,
u_1(x,x_0) = e^{i\tau\Phi}(1 - \frac{1}{4}e^{-i\tau (\bar\Phi+\Phi)}\partial^{-1}(e^{i\tau (\bar\Phi+\Phi)}(\bar\partial^{-1}q_1-\bar\partial^{-1}q_1(x_0)))+ r_1(x,x_0)),
\int_{Q_t}\partial_t \mu_{\varepsilon}\partial_t \varphi_{\varepsilon}= \nu \int_{Q_t}|\nabla \partial_t \varphi_{\varepsilon}|^2 +\int_{Q_t}\beta_{\varepsilon}'(\varphi_{\varepsilon}) |\partial_t \varphi_{\varepsilon}|^2
\begin{array}{c c c c c}{{\alpha + \beta + \gamma}}&{{=}}&{{\bar{\alpha}+ \bar{\beta}+ \bar{\gamma}}}&{{=}}&{{k n - m - n - 1 ,}}\\{{\alpha^{\prime}+ \beta^{\prime}+ \gamma^{\prime}}}&{{=}}&{{\bar{\alpha}^{\prime}+ \bar{\beta}^{\prime}+ \bar{\gamma}^{\prime}}}&{{=}}&{{k m - m - n - 1 ,}}\\ \end{array}
\lim_{t \to 0}\left[\int_{\partial F^{-1}(D_t) \setminus \Gamma}A \nabla u_0 \cdot \nu \, \bar u_0 + \int_{\partial D_t \setminus \Gamma}A \nabla u_0 \cdot \nu \, \bar u_0 \right]= 0.
\int_{-1}^{0}\frac{ln(1-t)}{t}dt=\int_{1}^{2}\frac{ln(u)}{1-u}du=\int_{1}^{2}ln(u)-\frac{ln(u)}{u}du=[uln(u)-u-\frac{(ln(u))^{2}}{2}]_{1}^{2}
U'(x(T,y))\mathbb{E}[\delta_Y(y)|\mathcal{F}_T] =p(0,y) \exp \big( \int_0^T \ln[1-\frac{b_0(s,y)}{\lambda\gamma_0(s,y)}]d\tilde{N}(s) &+ \lambda \int_0^T (\ln[1-\frac{b_0(s,y)}{\lambda\gamma_0(s,y)}]+\frac{b_0(s,y)}{\lambda\gamma_0(s,y)})ds \big).
\frac{- 1}{4}\left( \overrightarrow{\partial}_{x}^{2}- \overrightarrow{\partial}_{X}^{2}\right) K_{i \left( P - p \right)}( e^{x + X}) \, K_{i \left( P + p \right)}( e^{x - X}) = - e^{2 x}\, K_{i \left( P - p \right)}^{\prime}( e^{x + X}) \, K_{i \left( P + p \right)}^{\prime}( e^{x - X}) \; ,
S_{\|}\left[ g , h \right] ~ = ~ -{\frac{1}{G}}\int \sqrt g \left( \sqrt h ~ R_{h}+{\frac{1}{4}}\sqrt h h^{i j}\partial_{i}g_{\alpha \beta}\partial_{j}g_{\gamma \delta}{\epsilon}^{\alpha \gamma}{\epsilon}^{\beta \delta}\right) ,
I_1(N) = \int_0^{U(N; \alpha)}e^{-x}\left(1 - \frac{\ln x}{\ln N}\right)^r dx \,+ \, o\left(\frac{1}{N^{\theta}}\right) \int_0^{U(N; \alpha)}e^{-x}\left(1 - \frac{\ln x}{\ln N}\right)^r dx.
\mathcal{B}^+_{IV}= \left\{\begin{aligned}\left(\tau,v_{s+k+1},\omega_{k+1}\right)\in\mathcal{A}^+ \backslash \mathcal{B}^+_{II}\; \textnormal{such that}& \inf_{i\in\left\{1,\dots,s,s+1,\dots,s+k\right\}\backslash \left\{i_{k+1}\right\}}\left|v_{i_{k+1}}^{\prime *}- v_i^\prime \right| \leq \eta\end{aligned}\right\}
\|\psi_N(A)f - f \|_{\dot{B}^{\theta,A}_{p,q}}^q \cong \int_0^\infty \left( t^{-\theta}\| t^2 A^2 e^{-tA}(\psi_N(A) - 1) f\|_p \right)^q \frac{dt}{t}& = \int_0^\epsilon \ldots \frac{dt}{t}+ \int_\epsilon^{\frac{1}{\epsilon}}\ldots \frac{dt}{t}+ \int_{\frac{1}{\epsilon}}^\infty \ldots \frac{dt}{t},
\int_{-\delta}^{\delta}f(v| x-e^{\xi}y|)\,d\xi -\int_{-\delta}^{\delta}f(v|x-y|)\,d\xi = \int_{-\delta}^{\delta}\int_1^{e^{\xi}}\frac{d}{dt}f(v|x-t y|)\,dt\,d\xi & \ll |vy| \int_{-\delta}^{\delta}|\xi|\,d\xi \ll |vy|\delta^2.
\tan(\Delta\psi) =\frac{\sin(\psi_+)\cos(\psi_-)-\cos(\psi_+)\sin(\psi_-)}{\cos(\psi_+)\cos(\psi_-)+\sin(\psi_+)\sin(\psi_-)}&=\frac{(r-1)S_+L_{as}}{L_a^2+rS_+^2+S_+L_{ac}(1+r)}.
\widetilde w_{r_1}\tilde{u_1}\widetilde{w}_{r_1}^{-1}=\widetilde w_{\gamma_1}\tilde{v}\widetilde{w}_{\gamma_1}^{-1}\qquad\widetilde w_{s_1}\tilde{u_2}\widetilde{w}_{s_1}^{-1}=\widetilde w_{\gamma_1}\tilde{v}\widetilde{w}_{\gamma_1}^{-1}
(\psi_\varepsilon)_t(\hat{x},\hat{t}) \le J[u^*,p_\varepsilon](\hat{x},\hat{t})+K_{(0,\hat{x}-\delta)}[\psi_\varepsilon,p_\varepsilon](\hat{x},\hat{t}) &\quad+K_{(\hat{x}-\delta,\hat{x})}[u^*,p_\varepsilon](\hat{x},\hat{t})+f(\hat{x},\hat{t})
\mathbb{L}(t)=C(1+\|\phi_1\|_{H^2(\bar{\Omega},d\sigma)}^2+\|\phi_2\|_{H^2(\bar{\Omega},d\sigma)}^2+\|\nabla u_2\|_{L^2(\Omega)}^2+\|\nabla\mu_2\|_{L^2(\Omega)}^2).
t_{n}= \l_{n-1,1}(L_1+\dots+L_k)+l_{n-1,2}+\dots+l_{n-1,k}= \t_{n-1}+l_{n-1,1}(L_1+\dots+L_k-1), l_{n,1}= \l_{n-1,2}+L_1l_{n-1,1}, &\vdots l_{n,k-1}= \l_{n-1,k}+L_{k-1}l_{n-1,1}, l_{n,k}= \L_{k}l_{n-1,1}
\frac 3 4 \ln \left( 1 + \frac{L^{\prime 2}}{m^{2}}\right) + \frac{L^{\prime}}{2 m}\arctan \frac m{L^{\prime}}- \frac 1 8 \frac{L^{\prime 2}/ m^{2}}{1 + L^{\prime 2}/ m^{2}}= 1 , 1 1 5 .
\omega<0; (\rho_{0},\rho_{1},v_{0}) =(14.59210391\cdot\omega^{2},-1.302577877\cdot\omega^{-1},-0.07988549281\cdot\omega^{-3}) \omega>0; (\rho_{0},\rho_{1},v_{0}) =(2.763075661\cdot\omega^{2},1.227863217\cdot\omega^{-1},0.5505487785\cdot\omega^{-3})
M_1(X,Y,Z) = \sum_{i=0}^{q-3}\sum_{j=0}^{q-3}\sum_{k=0}^{q-3}b_{ijk}X^iY^jZ^k, M_2(X,Y) = \sum_{i=1}^{q-2}\sum_{j=1}^{q-2}c_{ij}X^i Y^j.
\mathbb{E}[\Vert\Theta(k+1)\Vert] \leq h^k\mathbb{E}[\Vert\Theta(0)\Vert]+\vartheta\bar{d_\eta}\sum_{t=0}^{k-1}h^{k-1-t}\bar{q}^t &\leq h^k\mathbb{E}[\Vert\Theta(0)\Vert]+\vartheta\bar{d_\eta}h^{k-1}\sum_{t=0}^{k-1}\left(\frac{\bar{q}}{h}\right)^t &\leq (\mathbb{E}[\Vert\Theta(0)\Vert]+\frac{\vartheta\bar{d_\eta}}{h-\bar{q}})h^k.
(n+1) \Delta_{P, n}^{( \alpha, \beta; \,\ell+1)}(x) + \ell \Delta_{P, n+1}^{(\alpha, \beta; \,\ell)}(x) + \ell (1-x) \Delta_{P, n+1}^{( \alpha+1, \beta; \ell-1 )}(x) + (n + \frac{\alpha + \beta}{2}+ 1) (1-x) \Delta_{P, n}^{( \alpha + 1, \beta; \ell)}(x)= \alpha \Delta_{P, n}^{(\alpha, \beta; \,\ell)}(x).
\Pi_{i j}^{\psi}\, = \, \frac{\partial{\cal L}}{\partial ( \partial_{0}\psi_{i j})}\, = \, \frac{i}{4 \lambda^{2}}\bar{\psi}_{j i}\gamma_{0}\, = \, \frac{i}{4 \lambda^{2}}{{\psi}^{\dagger}}_{j i}\; ,
\begin{cases}\partial_{t}u_i-d_i\partial_{xx}u_i=f_i(x,t,u), x\in \Omega,~ t>0, \partial_xu_i(0,t)=\partial_xu_i(L,t)=0, t>0, u_i(x,0)=u_{i,0}(x), x\in\Omega, \end{cases}
(\partial^{\lambda\dot{\mu}})=\left[\begin{array}{cc}\partial^{1\dot{1}}\partial^{1\dot{2}}\partial^{2\dot{1}}\partial^{2\dot{2}}\end{array}\right]=\left[\begin{array}{cc}\partial_{0}+\partial_{3}\partial_{1}+i\partial_{2}\partial_{1}-i\partial_{2}\partial_{0}-\partial_{3}\end{array}\right],
\mathcal{I}(\beta,\mu) = \lim_{\Lambda}\frac{1}{V}\sum_{k \in \Lambda^{*}}\omega_{\beta,\mu,\Lambda}^{0}(N_k)= \frac{1}{(2\pi)^3}\int_{\mathbb{R}^3}d^3 k \ \frac{1}{e^{\beta \left(\varepsilon_{k}-\mu \right)}-1}\ ,
p ( x , \xi ; h ) = 1 - a_{0}^{2}G - 2 a_{0}a_{1}G h - \frac{h}{2 i}a_{0}\{G , a_{0}\}- \frac{h}{2 i}\{a_{0}, a_{0}G \}+ S^{0}( h^{2}) &= 1 - a_{0}^{2}G - 2 a_{0}a_{1}G h + S^{0}( h^{2}) .
g(A^{*}_{E}X,Y)=B(X,Y),\;\;\;\;\; \overline{g}(A^{*}_{E}X,N)=0, g(A_{W}X,Y) = \varepsilon D(X,Y)+ \phi(X)\lambda(Y), g(A_{N}X,PY) = C(X,PY ),\;\;\overline{g}(A_{N}X,N) = 0, \overline{g}(A_{W}X,N) = \varepsilon \rho(X),\;\;\mbox{where}\;\; \varepsilon=\overline{g}(W,W),
v(p,w)=\prod_{i=1}^{n}(\frac{w\alpha_{i}}{p_{i}})^{\alpha_{i}}=\frac{\Pi_{i=1}^{n}w^{\alpha_{i}}\cdot\Pi_{i=1}^{n}\alpha_{i}^{\alpha_{i}}}{\Pi_{i=1}^{n}p_{i}^{\alpha_{i}}}=K(\frac{w}{\Pi_{i=1}^{n}p_{i}^{\alpha_{i}}})
\left[ \bar{P}_{a},\bar{P}_{b}\right] =\frac{1}{2}\left[ P_{a}+Z_{a},P_{b}+Z_{b}\right] , & =\frac{1}{2}\left( \left[ P_{a},P_{b}\right] +\left[ P_{a},Z_{b}\right] +\left[ Z_{a},P_{b}\right] +\left[ Z_{a},Z_{b}\right]\right) , & =\frac{1}{2}\left( J_{ab}+Z_{ab}-Z_{ba}-J_{ab}\right) , & =Z_{ab},
\frac{\partial D_{1}^{\prime}}{\partial t}(t,t^{\prime};w,w^{\prime})=i\lambda \int{\cal K}(w,w^{\prime \prime})e^{i[\omega (w)-\omega(w^{\prime \prime}))]t}D_{1}^{\prime}(t,t^{\prime};w^{\prime \prime},w^{\prime})dw^{\prime \prime}
\times \exp \left( - 2 e \sum_{b = 1}^{N}\sum_{j = 1}^{n_{b}}\Big ( \frac{\varepsilon_{\mu \nu}\partial_{\mu}}{\triangle}a_{\nu}^{( b )}, \delta_{n}( x_{j}^{( b )}) - \delta_{n}( y_{j}^{( b )}) \Big ) \right)
f_{(t,s)}'(z) =t^{-b_v}\sum_{\alpha\in \Delta_v\cap A}\xi_{\alpha}t^{\nu (\alpha ) -<\alpha ,a_v>}z^{\alpha}\,\, +\,\, \sum_{\alpha\in A\setminus \Delta_v}\xi_{\alpha}t^{\nu (\alpha ) -<\alpha ,a_v>-b_v}z^{\alpha}.
{}^{(4)}\!R(g+\langle\delta_2g\rangle)={}^{(4)}\!R(g)-2\kappa_5^2\langle T^{\rm GW}{}^5_5\rangle + \frac{\dot B^2}{a^2}\left(\frac{\langle\delta_2 V\rangle}{V}-\frac{\langle\dot \delta_2 V\rangle}{\dot V}\right),
\int_{\{\Psi_1<-t_1\}\cap D_4}|f|^2_he^{-\Psi_1}= \int_{\{\Psi_1<-t_1\}\cap D_4}|fF^2|^2_he^{-\varphi_1-\Psi_1}\le \int_{\{\Psi_1<-t_1\}\cap D_4}|\tilde{F}|^2_he^{-\varphi_1-\Psi_1}+\int_{\{\Psi_1<-t_1\}\cap D_4}|\tilde{F}-fF^2|^2_he^{-\varphi_1-\Psi_1}< +\infty.
U \approx \lambda \left[ \left( \frac{2}{q^{- 2}+ 1}\right)^{3 / 2}\frac{2}{5}\mu_{0}^{5 / 2}+ \frac{0 . 6 4 \sqrt{2 \mu_{0}}}{( q^{- 2}+ 1 )^{3 / 2}\beta^{2}}\right] ,
J_{\mu}^{L}= \frac{F_{\pi}^{2}}{4}U \partial_{\mu}U^{\dag}= - \frac{F_{\pi}^{2}}{4}( \partial_{\mu}U ) U^{\dag}= i \mathrm{\boldmath \tau}\cdot ( \mathrm{\boldmath V}_{\mu}- \mathrm{\boldmath A}_{\mu})
\begin{aligned}W\Bigl(\frac{\xi_0f_1(u)}{\xi_0f_1(u)+(1-\xi_0)f_2(u)},k,x+u,\mathsf{M}^{k}\Bigr) = W\Bigl(\frac{(1-\xi_0)f_2(u)}{\xi_0f_1(u)+(1-\xi_0)f_2(u)},k,x+u,\mathsf{M}^{k}\Bigr).\end{aligned}
\delta_{\bf v}{\cal L}_{0}=\mu\dot{\bf x}\cdot\delta_{\bf v}\dot{\bf x}+\frac{\mu}{R^{2}}\,{\bf x}\cdot\delta_{\bf v}{\bf x}=\mu{\bf v}\cdot\frac{d}{dt}\left[{\bf x}\,\cosh\left(\frac{t}{R}\right)\right],
\mathrm{SIR}_{\mathrm{U}_{0}}=\frac{P_{\mathrm{BS}}g_{n}\left(d_{0}\right)H_{\mathrm{U}_{0},\mathrm{BS}_{0}}}{\underset{\mathrm{BS}_{i}\in\Pi_{\mathrm{BS}}^{\dagger}}{\sum}P_{\mathrm{BS}}g_{n}\left(d_{i}\right)H_{\mathrm{U}_{0},\mathrm{BS}_{i}}},\: n\in\left\{1,2\right\}
\Big\|e^{t\Delta}u_0\Big\|_{\dot{H}^{\frac{d}{p}- 1}_{\mathcal{L}^{\tilde p,1}}}=\big\|e^{-t\left| \xi \right|^2}|\xi|^{\frac{d}{p}- 1}\hat u_0(\xi)\big\|_{L^{\tilde p', 1}_\xi}.
-\frac{1}{\pi d}\int_{- \alpha_{N}}^{\alpha_{N}}\frac{x\, w'(x)}{w(x)}\d x= \frac{2 \beta^{\rho}\rho}{\pi d (\rho + 1)}\alpha_{N}^{\rho + 1}, & - \frac{4(\pi \alpha_{N}+ 2d)}{\pi^{3}}\frac{w'(\alpha_{N})}{w(\alpha_{N})}= \frac{4\beta^{\rho}\rho\, \alpha_{N}^{\rho - 1}(\pi \alpha_{N}+ 2d)}{\pi^{3}}.
y_q(u)= \sum_{\mathcal{R}:|\mathcal{R}|=r+1,\mathcal{R}\ni q}\sum_{\mathcal{T}:|\mathcal{T}|=t}\sum_{i=1}^t\left[\sum_{p\in \mathcal{T}}h_{qp}(u)\left(\mathbf{v}_{{\mathcal{R}},{\mathcal{T}},p}^i(u)\right)^T\right]\mathbf{x}_{{\mathcal{R}},{\mathcal{T}}}^i+\sum_{m(u)\in \mathcal{M}_q[N+1](u)}m(u)x_{m(u)},
s^{-k}\int_{B_s(x)}D_{\mu'}(y,t)\,d\mu'(y) \leq Cs^{-k}\int_{B_s(x)}t^{-k}\int_{B_t(y)}W_t(z)\,d\mu'(z)\,d\mu'(y) &= Cs^{-k}t^{-k}\int_{B_s(x)}\mu'(B_t(y)) W_t(y)\,d\mu'(y) &\leq Cs^{-k}\int_{B_s(x)}W_t(y)\,d\mu'(y)\, ,
L_{\textrm{S R}}\equiv \dot{x}^{\mu}\dot{x}_{\mu}\longrightarrow L_{\textrm{E S R}}\equiv \dot{X}^{\mu}\dot{X}_{\mu}= ( \dot{x}^{\mu}\dot{x}_{\mu}+ \dot{u}^{\mu}\dot{u}_{\mu}) + 2 I \dot{x}^{\mu}\dot{u}_{\mu},
\begin{aligned}B_\mathbf{u}(c^{n-1};(\mathbf{u}^{n},p^{n},p^{n});(0,0,\mu_h))=\langle\mathbf{u}^{n}\cdot\mathbf{n},\mu_h\rangle_{\partial\mathcal{T}_h}=0,\end{aligned}
{(*_{\varepsilon})}\,\,\,_M\left(\dfrac{\nabla u_{\varepsilon}}{\sqrt{|\nabla u_{\varepsilon}|^2+\varepsilon^2}}\right)-\left(g^{ij}-\dfrac{\nabla^iu_{\varepsilon}\nabla^ju_{\varepsilon}}{|\nabla u_{\varepsilon}|^2+\varepsilon^2}\right)K_{ij}=-\frac{1}{\sqrt{|\nabla u_{\varepsilon}|^2+\varepsilon^2}}.
\sum_{t\in [N]^r}\prod_{\substack{\{l,l'\}\in E \{l,l'\}\ne \{a, b\}}}x_{q_l(t)q_{l'}(t)}\prod_{\substack{\{l,l'\}\in E \{l,l'\}\ne \{k-1,k\}}}y_{s_l(t)s_{l'}(t)}&= \sum_{t\in [N]^r}\prod_{\substack{\{l,l'\}\in E \{l,l'\}\ne \{a, b\}}}x_{t_lt_{l'}}\prod_{\substack{\{l,l'\}\in E \{l,l'\}\ne \{k-1,k\}}}y_{t_{\phi(l)}t_{\phi(l')}}&= R(x,y, A, B).
< \rho_{-}> = \frac{k_{-}}{\sqrt{K}\alpha^{\prime}}, < P_{-}> = \frac{1}{2 \sqrt{K}\alpha^{\prime}}[ \frac{k_{-}^{2}- 2}{k_{-}}+ \frac{2}{k_{-}}\frac{E ( k_{-})}{K ( k_{-})}] ,
a_{i}^{[ 2 ] e f f}( x ) = e_{i j}\partial_{j}\int d^{2}x^{\prime}{\frac{\ln ( \vec{x}- \vec{x}^{\prime})^{2}}{4 \pi}}\{-{\frac{e}{\mu c}}\sqrt{g ( x^{\prime})}J_{0}( x^{\prime}) + b^{( 2 )}( x^{\prime}) \}.
u_{2j+1}(x_1,x_2)|_{|x|=R_0}&=(-1)^j\sin(2j+1)\theta-2\sum_{k=1}^\infty\sum_{l=0}^\infty \frac{\alpha^k{{2l+2j+1}\choose{2j}}(-1)^l\sin(2l+1)\theta}{(kR_0)^{2j+2l+2}};
\prod_{\ell=0}^j \big(e^{i\eta n_{k_\ell}}-1\big)-\prod_{\ell=0}^j\big(e^{i\xi n_{k_\ell}}-1\big)=\big( e^{i\eta n_{k_0}}-e^{i\xi t_{k_0}}\big)\prod_{\ell=1}^j\big( e^{i\eta n_{k_\ell}}-1 \big) &+\big(e^{i\xi n_{k_0}}-1\big)\Bigg(\prod_{\ell=1}^j\big(e^{i\eta n_{k_\ell}}-1\big)-\prod_{\ell=1}^j\big(e^{i\xi n_{k_\ell}}-1\big) \Bigg),
\lim_{t_1,t_2\rightarrow0^+}\mathcal C_t^{p1}g=\mathcal C_\Gamma^{p1}g \lim_{t_1,t_2\rightarrow0^+}\mathcal C_t^{p2}g=\mathcal C_\Gamma^{p2}g L^p(\Gamma).
\sum_{j=1}^n\sum_{k=1}^n\|\langle \tau_j, \tau_k\rangle \|^{2m}\geq \sum_{j=1}^n\sum_{k=1}^n\langle \tau_j, \tau_k\rangle^{m}\langle \tau_k, \tau_j\rangle^{m}\geq \frac{n^2}{{d+m-1\choose m}}, \forall m \in \mathbb{N}.
\big( q^3\alpha_1 x -1\big) \big(q^2 \alpha_2 x -1\big) \big(q^3 \alpha_3 x -1\big) \bar{g}_1\big(q^4 x\big) +c_3(x) \bar{g}_1\big(q^3 x\big) +c_2(x) \bar{g}_1\big(q^2 x\big) &+c_1(x) \bar{g}_1(q x) + q^{\mu +4}\big( \beta_1 x - q^{\lambda}\big) \big(\beta_2 x-q^{\lambda}\big) \big(\beta_3 x-q^{\lambda}\big) \bar{g}_1( x) =0,
\phi(-\frac{q}{2}x_{n-1}) \cdots \phi(-\frac{q}{2}x_1)=\sum_{k_1,\cdots,k_{n-1}\geq 0}\frac{q^{k_1^2/2+ \cdots + k_{n-1}^2/2-k_1 k_2 -\cdots-k_{n-2}k_{n-1}}x_1^{k_1}\cdots x_{n-1}^{k_{n-1}}}{(q)_{k_1}\cdots (q)_{k_{n-1}}}.
(H-\epsilon)u_{k+1}=\sum_{l=1}^{k-1}C_{k-l}\frac{\partial_{\epsilon}^{l-1}u}{(l-1)!}+\sum_{l=1}^{k-1}D_{k-l}\frac{\partial_{\epsilon}^{l-1}v}{(l-1)!}+\frac{\partial_\epsilon^{k-1}u}{(k-1)!},
g_{a b}= \left( \begin{array}{c c c c c}{0}&{- \omega_{1}}&{}&{}&{}\\{\omega_{1}}&{0}&{}&{}&{}\\{}&{}&{\ddots}&{}&{}\\{}&{}&{}&{0}&{- \omega_{\frac{p + 1}{2}}}\\{}&{}&{}&{\omega_{\frac{p + 1}{2}}}&{0}\\ \end{array}\right) .
W_{\nu}^{M}( z , z^{\prime}) = e^{( v + v^{\prime}) ( \frac{d}{2}+ \nu )}\widetilde{W}_{\nu}\left( \eta \left( v , z \right) , \eta^{\prime}\left( v^{\prime}, z^{\prime}\right) \right) = \frac{1}{2 \pi^{\frac{d}2}}\frac{\Gamma ( \nu + \frac{d}2 )}{\Gamma ( \nu + 1 )}\frac 1{{[ - ( z - z^{\prime})^{2}]}^{\frac{d}2 + \nu}}.
\sum_{m\in M_j(Q_0^{-1})}Q_0^{-1}m m^t Q_0^{-1}= Q_0^{-1}\Big( \sum_{m\in M_j(Q_0^{-1})}m m^t \Big) Q_0^{-1}= Q_0^{-1}d_j Q_0 Q_0^{-1}= d_j Q_0^{-1}.
|H(x', t)| = \bigg|\int_{Q}(\theta(x', t; y') - \theta(x', t; x'_Q) ) h(y') \, dy'\bigg| = \sup_{y' \in Q}|\nabla_{y'}\theta(x', t; y')| \, \|h\|_{L^1(Q)}&\lesssim \sup_{y' \in Q}\frac{t}{|(x'-y', t)|^{n+1}}\, \|h\|_{L^1(Q)}& \lesssim \frac{t \, \ell(Q)}{(2^k \ell(Q))^{n+1}}\|h\|_{L^1(Q)},
E_{2}= \frac{v^{2}}{2}\int \frac{d^{3}q}{( 2 \pi )^{2}}\left\{G_{h}^{- 1}( - \vec{q}\,^{2}) h ( \vec{q}) h ( - \vec{q}) + G_{p}^{- 1}( - \vec{q}\,^{2}) \vec{p}( \vec{q}) \cdot \vec{p}( - \vec{q}) \right\}
D(A_{Y,U}) = \{u\in D(A_0^*);\; u(0-) = - u(0+),\, u''(0+)\alpha_+ + u''(0-)\alpha_- = \frac{\beta_+ - \beta_-}{2}u(0-),\, u'(0+) = U u'(0-)\}, A_{Y,U}u = -A_0^*u.
Z \left( \delta \ddot{\phi}_{k}+ 3 H \delta \dot{\phi}_{k}\right) + Z^{\prime}\left( \ddot{\phi}+ 3 H \dot{\phi}\right) \delta \phi_{k}+ \left( Z \frac{k^{2}}{a^{2}}+ V^{\prime \prime}\right) \delta \phi_{k}= 0
\prod_{i=1}^t a_ib_i -\prod_{i=1}^t a_i'b_i'=\sum_{i=1}^t \left(\prod_{j=1}^t \left(a_ib_j-a_i'b_j'\right)\right)\left(\prod_{1 \leq j \leq t,\j\neq i}\left(\frac{a_i}{a_j}-\frac{a_i'}{a_j'}\right)^{-1}\right).
\Pi_{B \cdot P}^{\mathrm{o n e - l o o p}}( p^{2}) ={\frac{N g^{2}m}{8 \pi}}\left[ \left( - \frac{2 7}{1 6}\frac{p^{2}}{m^{2}}+ \frac{9}{4}\right) \frac{2 m}{p}\tan^{- 1}\frac{p}{2 m}- \frac{3}{4}\right]
\int_{\partial B_n}f(x)^{-\frac{2}{n-1}}\,d\mu_{\partial B_n}(x) \geq \mu_{\partial B_n}(\partial B_n)^{\frac{n+1}{n-1}}=\int_{\partial B_n}f_{\rm unif}(x)^{-\frac{2}{n-1}}\,d\mu_{\partial B_n}(x).
|\nabla_{g_0}^j(\Phi^{*}g-g_{0})|_{g_{0}}=O(r^{\lambda_1-j}), |\nabla_{g_0}^j(\Phi^{*}J-J_{0})|_{g_{0}}=O(r^{\lambda_2-j}), |\nabla_{g_0}^j(\Phi^{*}\Omega-\Omega_{0})|_{g_{0}}= O(r^{\lambda_2-j}).
\sum_i([{f}_{11}(a_i'),{g}_{11}(a_i'')]-(-1)^{|a_i'||a_i''|}[{f}_{11}(1),{g}_{11}(a_i''a_i')])=e_{11}\left(\sum_i([a_i',a_i'']-\overline{[a_i',a_i'']})\right)=2e_{11}(a),
\nu \int_{0}^{\infty}f_{1}^{\prime \prime}( x ) f_{2}( x ) \, d x = - \nu f_{2}^{\prime}( 0 ) + \nu f_{1}^{\prime}( 0 ) + \nu \int_{0}^{\infty}f_{2}( y ) f_{1}^{\prime \prime}( y ) \, d y \, .
I_2^{(5)}= r^{2-n}\Big[\frac{(n-6)(n-4)}{12(n-1)}|W(p)|^2r^2-\frac{32}{9(n-2)}\sum_{k,l,s}\big((W_{ikls}(p)+W_{ilks}(p))x^i\big)^2 & +\frac{4}{3(n-2)(n-1)}|W(p)|^2r^2-\frac{16(7n-8)}{n-2}\sigma_1(A)_{,ij}(p)x^ix^j\Big]+O(r^{5-n}).
\int \biggl[\int \Xi_k f \,dM^{N}_k-t\sum_{\ell\neq k}\iint \partial_x D_{k\ell}F^a_0 (\mu^*,\tau_B^N)[\delta_y,\delta_x]f(x)\,d\mu^{*}_{k,t}(x) \,dM^{N}_\ell(y)\biggr]\,dQ^{N,aV}_{t}=\sum_{\gamma=1}^4 R^N_\gamma (f)
\frac{\partial\psi}{\partial t}=i(\omega_{1}(\begin{matrix}0&e^{i\omega_{r}t}(cos\omega_{r}t-isin\omega_{r}t)\\e^{-i\omega_{r}t}(cos\omega_{r}t+isin\omega_{r}t)&0\end{matrix})+(w_{0}+\frac{\omega_{r}}{2})\sigma_{z})\psi,
\xi_{i}(t):= G(q(t),\Psi_{i})\ \textup{for}\i\in \mathbb{Z}\setminus\{0\}, &z_j(t):= G(q(t),\Upsilon_j),\ \bar{z}_j(t):=G(q(t),\overline{\Upsilon}_j) \ \textup{for}\ 1\leq j\leq J,
S^{A}_{R^4}= \int d^{10}x\sqrt{-g}~2\zeta(3) e^{-2\phi}(t_8t_8 + \frac{1}{8}\epsilon_{10}\epsilon_{10}) R^4+\frac{2\pi^2}{3}(t_8t_8 - \frac{1}{8}\epsilon_{10}\epsilon_{10}) R^4
X_{(0)}^{\mu}(x,y)=x^{\mu}+\frac{1}{2}\alpha_{0}^{\mu}\left((x+y)-(x-y)\right) +i\sum \frac{1}{m}\alpha_{m}^{\mu}e^{im(x-y)}+i\sum\frac{1}{m}\alpha_{m}^{\mu}e^{-im(x+y)}
([\nabla_t,D]\xi)(X,X_1,\ldots,X_k) =R^{SM}(\partial_t,X)\cdot\xi+R^P(\nabla_t\phi,D_X\phi)(\xi(X_1,\ldots,X_k)) &-\frac{1}{2}D_{\dot{g}(X)}\xi(X_1,\ldots,X_k)+\frac{1}{2}\sum_{i=1}^k\xi(X_1,\ldots,D_X\dot{g}(X_i),\ldots,X_k).
f(\psi)= \sum_{j=1}^k\left\{\big[-(-\Delta)^{\frac{s}{2}}\eta_{j, R},-(-\Delta)^{\frac{s}{2}}\tilde{\phi}_j\big] + \tilde{\phi}_j\big(-(-\Delta)^{s}-\partial_t\big)\eta_{j, R}\right\}+ \tilde{N}_{\mu, \xi}(\tilde{\phi}) + S_{out}.
S_{F}^{a}( x , y ) = - S_{F}( x , - y ){\cal M}, \hspace{0 . 4 c m}S_{F}^{b}( x , y ) = -{\cal M}S_{F}( - x , y ) , \hspace{0 . 4 c m}S_{F}^{c}( x , y ) ={\cal M}S_{F}( - x , - y ){\cal M}.
{\frac{d n^{( \lambda )}( \omega )}{d \omega}}= \int_{\Sigma_{r}}d \Sigma^{\mu}\sum_{k}\tilde{j}_{\mu}( \phi_{\omega , k}^{( \lambda )}) = \int_{\Sigma_{r}}\sqrt{h}d^{D - 1}x \sum_{k}2 \omega | \phi_{\omega , k}^{( \lambda )}|^{2}.
\zeta_{\rho}( s ) = - \frac{2 \, R^{2 s}}{\Gamma ( s + 1 ) \, \Gamma ( - s )}\sum_{l = 1}^{\infty}\nu^{1 - 2 s}\int_{0}^{\infty}d k \, k^{- 2 s}\frac{d}{d k}\ln \Delta_{\rho , l}( \nu k ) , \quad \nu = l + \frac{1}{2}, \; \; \rho = \pm 1
x = c_1v_1 + \cdots + c_{a-1}v_{a-1}+ c_a y_1 + \cdots + c_{a+b-2}y_{b-1}+ c_{a+b-1}v_a &+ c_{a+b}u_1 + \cdots + c_{a+2b-2}u_{b-1}+ c_{a+2b-1}z + c_{a+2b}w_{1,1}+ c_{a+2b+1}w_{2,1}+ \cdots + c_n w_{l_k,k}.
\begin{array}{llllllllll}A^1_1 = \langle h_1, x_1, y_1 \rangle, A^3_1 = \langle h_1+h_2, x_1+x_2, y_1+ y_2 \rangle, A^2_1 = \langle h_2, x_2, y_2 \rangle. \end{array}
\begin{array}{l}\partial^{(a_{j+1}}K^{a_1 a_2 ...a_j)}= -2M{\dot K}^{a_1 a_2...a_{j+1}},j=0,1,...n-1, \partial^{(a_{n+1}}K^{a_1 a_2 ...a_n)}=0,{\dot K}=0,j=0.\end{array}
\left( T_{\Sigma}(s) \star_{H, \ell}T_{\Sigma}(s') \right)[\psi] = \frac{1}{4}\psi{(s)}^2 \psi{(s')}^2 + \frac{\hbar}{4 \pi}\frac{\psi(s) \psi(s')}{{(s - s')}^2}+ \frac{\hbar^2}{32 \pi^2}\frac{1}{{(s - s')}^4}.
lim_{\theta\rightarrow0^{-}}\frac{sin\theta}{\theta}=lim_{\theta\rightarrow0^{+}}\frac{sin(-\theta)}{-\theta}=lim_{\theta\rightarrow0^{+}}\frac{-sin\theta}{-\theta}=lim_{\theta\rightarrow0^{+}}\frac{sin\theta}{\theta}=1.
\mu_{\mathrm{I}}( \eta ) = \frac{1}{\sqrt{2 n}}\biggl \vert \frac{\eta}{\eta_{\mathrm{i}}}\biggr \vert^{1 / 2}\cos \biggl ( \frac{2 \pi}{\epsilon}\ln \biggl \vert \frac{\eta}{\eta_{\mathrm{i}}}\biggr \vert \biggr ) .
X= \Gamma\left( x_0 + \int_0^{\cdot}T^{X - \tilde{W}^H + \tilde{W}^H}f(dr, x_0) + \tilde{W}^H \right) = \Gamma\left( x_0 + \int_0^{\cdot}T^{\tilde{W}^H}f(dr, x_0 + \theta_r ) + \tilde{W}^H \right),
I(a,b,g_1,g_2,Q_1,Q_2) &\qquad=\sum_{T<\gamma \leq 2T}Q_1\Big(-\frac{d}{da}\Big) \zeta\Big(\rho+\frac{a}{\log T}\Big) \, Q_2\Big(-\frac{d}{db}\Big)\zeta\Big(1-\rho+\frac{b}{\log T}\Big)\,M(\rho,g_1) \, M(1-\rho,g_2).
\,\varphi_{\underline{d}''',\underline{d}}(V_{\underline{d}'''})= r+1-\,V_{\underline{d}'''}^{X_{3}^{c},0}= r+1-(\,V_{\underline{d}''}^{X_1,0}-\,V_{\underline{d}''}^{X_{2}^{c},0}).
V_{1}^{[r,1]}(x) - g_{1,[r,1]}(x) = M^{x}_{1}(D_{[0,\ell_{r}]},D_{[r,1]}) - g_{1,[r,1]}(x) & = \begin{cases}f_{1}(x) - g_{1,[r,1]}(x), \forall x \le \ell_{r}f_{1}(\ell_{r})\frac{r-x}{r-\ell_{r}}+ g_{1}(r)\frac{x-\ell_{r}}{r-\ell_{r}}- g_{1,[r,1]}(x), \forall \ell_{r}< x \le r\end{cases}
K_{1}(\varepsilon)\overset{\triangle}{=}H_{3}P_{10}^{*}+\mathcal{\varepsilon}H_{1}\left(P_{20}^{*}\right)^{T},\ \ \K_{2}(\varepsilon)\overset{\triangle}{=}H_{3}P_{20}^{*}+H_{1}P_{30}^{*};
\mathcal{L}_{p,F'}= \dfrac{l_\lambda e'Tr'_1\left(\left(Tr'_2(\mathcal{E'}_c^{\chi\chi_{-1}}*\Theta'_\chi\mid[\mathfrak{l'}^2])\right)\mid\Psi'_2\right)}{\Delta'(X',Y')\mathcal{E'}_2(X',Y')H'},
=\frac{1}{\beta-i\alpha t}\left(g(a_{j+1})e^{(\beta-i\alpha t)a_{j+1}}-g(a_j)e^{(\beta- i \alpha t)a_j}\right) -\frac{1}{\beta-i \alpha t}\int_{a_j}^{a_{j+1}}e^{(\beta-i\alpha t)x}g^{\prime}(x)dx
sinx_{deg}=siny_{rad}=\frac{\pi}{180}x-(\frac{\pi}{180})^{3}\frac{x^{3}}{3!}+(\frac{\pi}{180})^{5}\frac{x^{5}}{5!}-(\frac{\pi}{180})^{7}\frac{x^{7}}{7!}+\cdot\cdot\cdot.
\langle \nabla_i\nabla_j \Psi_t-\nabla_l\Psi_t\Gamma_{ij}^l, \nabla_i\nabla_j \Psi_t-\nabla_l\Psi_t\Gamma_{ij}^l\rangle =\frac{1}{2t}\Big(1+O(t)\Big)+\sum_l\Gamma_{ij}^l\Gamma_{ij}^l.
(\iota_{\ll \alpha,\beta\gg_\ast}\widetilde{\mathcal{M}}_\Pi^\nu)\nu=\big(((a\circ\Pi)(\alpha))\iota_\beta\widetilde{\mathcal{M}}_\Pi^\nu\big)\nu-\big(((a\circ\Pi)(\beta))\iota_\alpha\widetilde{\mathcal{M}}_\Pi^\nu\big)\nu.
L_\mathbf{g}(\psi)^{\mu a}=\psi^{\mu a}\ \ \ \Rightarrow \ \ \ \psi^{\mu a}(\mathbf{x}, \mathbf{z}\mathbf{g})= \psi^{\mu a}(\mathbf{x}, \mathbf{e})\equiv \psi^{\mu a}(\mathbf{x}) \;.