harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
P^{h,h}((\textbf{y}\textbf{x})_h) =P^{h,\mathcal{P}(\textbf{x}, h)}(\textbf{y}'_h)*P^{h,h}(\textbf{x}_h) =P^{\mathcal{P}(\textbf{x}, h),\mathcal{P}(\textbf{x}, h)}(\textbf{y}'_{\mathcal{P}(\textbf{x}, h)})*P^{h,h}(\textbf{x}_h),
\begin{array}{rcccc}\hbox{d}\theta(e_a,\partial_y)/\theta_4 = \Theta_a = 0 \hbox{(a)}\Theta_{ab}+ \pi_{ab}= 0 \hbox{(b)}\hbox{d}\pi = \frac{1}{2}\|\pi\|^2e^{(4)}\hbox{(c)}\end{array}
(\begin{matrix}M+QN^{-1}Q^{T}&0\\ 0&N\end{matrix})=(\begin{matrix}I&-QN^{-1}\\ 0&I\end{matrix})(\begin{matrix}M&Q\\ -Q^{T}&N\end{matrix})(\begin{matrix}I&0\\N^{-1}Q^{T}&I\end{matrix}).
\Delta_2d(x,y) = d(2x+y)+d(2x-y) & -\frac{2d(x)d(y)}{\left(4d(y)^{\frac{1}{5}}-d(x)^{\frac{1}{5}}\right)^{10}}\Big[1024d(y)+11520d(x)^{\frac{1}{5}}d(y)^{\frac{4}{5}}+13440d(x)^{\frac{2}{5}}d(y)^{\frac{3}{5}}&\qquad\qquad\qquad\qquad+3360d(x)^{\frac{3}{5}}d(y)^{\frac{2}{5}}+180d(x)^{\frac{4}{5}}d(y)^{\frac{1}{5}}+d(x)\Big]
|V| = \left| \bigcup_{xH\in \Omega\setminus \mathcal{S}}\{(xH,yH):\y H \in \mathcal{S}\}\right| = A(X_i/\pi)_{12}|\Omega\setminus \mathcal{S}| = A(X_i/\pi)_{21}\left(|\Omega|-|\mathcal{S}|\right).
\begin{aligned}\sum_\gamma\Big(\frac{\sin\frac12(\gamma-\tau)\log x}{\frac12(\gamma-\tau)\log x}\Big)^{\!2}= \frac{\log\frac{\tau}{2\pi}}{\log x}- \frac2{\log x}\sum_{n\le x}\frac{\Lambda(n)}{n^{1/2}}\Big(1-\frac{\log n}{\log x}\Big)\cos (\tau\log n) &+O\Big(\frac1{\tau\log x}\Big) +O\Big(\frac{x^{1/2}}{(\tau\log x)^2}\Big).\end{aligned}
\omega_1 =a_2a_1 +a_4a_3+a_6a_5+a_8a_7; \omega_2 =a_3a_1+a_2a_4+a_7a_5+a_6a_8; \omega_3 =a_4a_1+a_3a_2+a_8a_5+a_7a_6.
\mathfrak{Z}_{q,p}^{\#}\left( S\right) :=\sum_{\sigma\in\mathfrak{P}\left(S\right)}\Psi_{\#}\left( \sigma\right) =\sum_{\sigma\in\mathfrak{P}\left(S\right)}\prod_{\mathsf{s}\in\sigma}\left( \frac{1-p}{p}\right)^{\left\vert \mathsf{s}\right\vert}q^{\left\Vert \mathsf{s}\right\Vert_{\#}}\;,\quad\#=
- \epsilon^{q}_{bk}\hat{R}_{ib,am}\epsilon^{q}_{cm}K_{ac}=(\epsilon^{q})_{kb}^{t}\hat{R}_{ib,am}(\epsilon^{q})_{mc}^{t-1}K_{ac}= \hat{R}^{\epsilon}\,_{ik,ac}K_{ac}= K_{ik}^{\epsilon}
{C^{\textrm{proposed}}_{\textrm{secrecy}}}=\mathop{\max}\limits_{i \in{\cal{S}}}{\log_2}\left({\dfrac{{1 + \frac{{|{h_{is}}{|^2}P_i}}{{{N_0}}}}}{{1 + \frac{{|{h_{ie}}{|^2}P_i}}{{{N_0}}}}}}\right).
\small \ddot{r}( t ) + [ ( 2 n - 2 p - 1 ) \cot t - ( 2 p + 1 ) \tan t ] \, \dot{r}( t ) begin{align*}1ex] + \left[ \frac{p}{\cos^2 t}- \frac{( n - p - 1 )}{\sin^2 t}\right] \, \sin 2 r ( t ) - \frac{\sin 4 r ( t )}{\sin^2 2 t}= 0
\int_{\mathcal S^n}Y_l(y)\,f(x\cdot y)\,d\sigma(y) &=Y_l(x)\cdot\frac{(4\pi)^\lambda\Gamma(l+1)\Gamma(\lambda)}{\Gamma(2\lambda+l)}\int_{-1}^1 f(t)\,C_l^\lambda(t)\left(1-t^2\right)^{\lambda-1/2}dt.
\left( \frac{8 \pi^{2}}{g_{c}^{2}( M^{\prime})}+ C_{A}\ln g_{c}^{2}( M^{\prime}) \right) = \left( \frac{8 \pi^{2}}{g_{c}^{2}( M )}+ C_{A}\ln g_{c}^{2}( M ) \right) + b_{0}\ln \frac{M}{M^{\prime}}
\begin{aligned}\log (t^3+1) < n \log \left(\frac{\theta''-t}{t-\theta}\right ) -m \log \left(\frac{\theta''}{\lvert \theta \rvert}\right ) < \log (t^3+2) t^3+1 < \frac{\theta''-t}{t-\theta}<t^3+2 ,\ \t^9+6t^6< \frac{\theta''}{\lvert \theta \rvert}< t^9+7t^6+6 \end{aligned}
\left\| \left( A(x^{1}_{i},x^{2}_{j})\right)_{i,j=0}^{m}\right\|_{Rad^{(2)}(F)}\le C\prod_{k=1}^{2}\Vert(x_{j}^{k})_{j=0}^{m}\Vert_{\ell_{2}^{w}(E_{k})}
\mathcal{B}_I = \left\{\begin{aligned}(\tau,v_{s+k+1},\omega_{k+1}) \in [0,\infty) \times \mathbb{R}^d \times \mathbb{S}^{d-1}\textnormal{such that}\tau = 0 \textnormal{or}& \exists t^\prime, t^{\prime \prime}\geq 0 \; : \;d_X \left( Z_{s+k}^\prime (\tau; t^\prime),Z_{s+k}^\prime (\tau;t^{\prime \prime}) \right) \leq y\end{aligned}\right\}
\|\boldsymbol{\vartheta}-\boldsymbol{\vartheta}_0\|_\infty\left\|\sum_{m_1=0}^{2^{N_1}-1}\cdots\sum_{m_d=0}^{2^{N_d}-1}|D^{\boldsymbol{r}}\varphi_{\boldsymbol{N},\boldsymbol{m}}|\right\|_\infty\lesssim\|\boldsymbol{\vartheta}-\boldsymbol{\vartheta}_0\|\lesssim\epsilon_{n,\boldsymbol{r}},
(\Gamma+1)\,\Upsilon_{s,\pm}^{(N)}=(N+\mu_1+\mu_2+\mu_3+1)\,\Upsilon_{s,\pm}^{(N)}, K_1\,\Upsilon_{s,\pm}^{(N)}=(-1)^{s}(s+\mu_2+\mu_3+1/2)\,\Upsilon_{s,\pm}^{(N)}, \sigma_3 R_3\,\Upsilon_{s,\pm}^{(N)}=\pm (-1)^{N-s}\,\Upsilon_{s,\pm}^{(N)}.
\Big\| \sum_{j = 1}^\infty \alpha_j u_j \Big\|_{p^*,q,\mu}\geq \Bigg( \frac{\lambda}{1 + \varepsilon_1}- \varepsilon_2 \Bigg)\Big( \sum_{j = 1}^\infty |\alpha_j|^q \Big)^{\frac1{q}}.
\tilde{R}_{1 2}( \lambda_{1 2}) \tilde{R}_{1 3}( \lambda_{1 3}) \tilde{R}_{2 3}( \lambda_{2 3}) = \tilde{R}_{2 3}( \lambda_{2 3}) \tilde{R}_{1 3}( \lambda_{1 3}) \tilde{R}_{1 2}( \lambda_{1 2}) \ .
\sigma((u^i,u^i{}_j).a) =\sigma(u^i,u^i{}_\alpha a^\alpha{}_j) * =(u^i,u^i{}_\alpha a^\alpha{}_j,-\Gamma^i{}_{\alpha\beta}u^\alpha{}_\gamma u^\beta{}_\delta a^\gamma{}_ja^\delta{}_k) * =\sigma(u^i,u^i{}_j).a
\Gamma_2^{\mathrm{hori}}(f,f) :=\frac{1}{2}(L \Gamma^{\mathrm{hori}}(f,f) -2 \Gamma^{\mathrm{hori}}(f, Lf)), \Gamma_2^{\mathrm{vert}}(f,f) := \frac{1}{2}(L \Gamma^{\mathrm{vert}}(f,f) -2 \Gamma^{\mathrm{vert}}(f, Lf)), \Gamma_2^{\mathrm{mix}}(f,f) := \frac{1}{2}(L \Gamma^{\mathrm{elli}}(f,f) -2 \Gamma^{\mathrm{elli}}(f, Lf)).
P_{\mu, \kappa}(\lambda; \tau, \eta) := e^{-\pi i \frac{4\eta + \tau}{2\kappa}(\mu + 2)^2 + \pi i 3\tau/4}\frac{\Delta_{\mu + 2, \kappa}(\lambda; \tau, \eta)}{\theta(\lambda - 2\eta; \tau)\theta(\lambda; \tau)\theta(\lambda + 2\eta; \tau)}.
\begin{aligned}G(v)-G(u^n)+\langle \lambda^n, B(v-u^n)\rangle+ r\langle Bu^n-\eta^{n},B(v-u^n)\rangle \geq 0, \forall v\in V, u^n\in V, F(q)-F(\eta^n)-\langle\lambda^n, q-\eta^n\rangle+ r\langle\eta^n-Bu^n,q-\eta^n\rangle \geq 0, \forall \, q\in H, \eta^n \in H,\end{aligned}
{\cal H}= \zeta^{\, 0}+ \frac{1}{2}\left\{( \dot{\phi})^{2}+ ( \partial_{\, z}\phi + F )^{2}+ i \psi_{2}( \partial_{\, z}- F^{\prime}) \psi_{1}+ i \psi_{1}( \partial_{\, z}+ F^{\prime}) \psi_{2}\right\}\, ,
\widetilde{\alpha}_{m,\,\varepsilon}^{n-1}\wedge\beta_{\varepsilon}= \frac{1}{n}\,(\Lambda_{\widetilde{\alpha}_{m,\,\varepsilon}}\beta_{\varepsilon})\,\widetilde{\alpha}_{m,\,\varepsilon}^n = \frac{1}{n}\,(\Lambda_{\widetilde{\alpha}_{m,\,\varepsilon}}\beta_{\varepsilon})\,\lambda_1\dots\lambda_n\,\gamma_{m,\,\varepsilon}^n.
\sum_{k=0}^{n-1}\beta_{\zeta}(k)=\sum_{j=0}^{n/t-1}\sum_{k=0}^{t-1}\beta_{\zeta}(jt+k)=\sum_{j=0}^{n/t-1}\beta_{\zeta}(jt)\sum_{k=0}^{t-1}\beta_{\zeta}(k)=0,
A_{3 , \zeta}= Q \rho \tau_{, \zeta}, A_{3 , \bar{\zeta}}= - Q \rho \tau_{, \bar{\zeta}}, \kappa^{2}= I^{3}={\frac{h^{3}e^{\tau_{0}\tau}}{( 1 -{\frac{2 m}{r}}) g_{2 2}^{\beta}}}
\frac{\partial}{\partial t}|\nabla w|^2= \big(R+\frac{\alpha'}{4}R^2-r\big)|\nabla w|^2+2\langle\nabla\frac{\partial w}{\partial t},\nabla w\rangle = \big(R+\frac{\alpha'}{4}R^2-r\big)|\nabla w|^2+2\big(1+\frac{\alpha'}{4}R+a\frac{\alpha'}{4}\big)\langle\nabla\Delta w,\nabla w\rangle &+\frac{\alpha'}{2}\nabla R\Delta w\nabla w+2a\langle\nabla f,\nabla w\rangle.
r(G_{\lambda}) \geq r(G_{\lambda'}) \geq \frac{1}{2}( d_1/2 -1 + d_k/2 - 1 + \sum_{j \neq 1,k}d_j -1) = \frac{1}{2}(\dim(V_{\lambda}) - k - d_1/2 - d_k/2).
A^h(x)^{-1}= \bar A(x')^{-1}+ \bar A(x')^{-1}\Big(-hA_1\big(x', \frac{x_3}{h}\big) + h^2 A_1\big(x', \frac{x_3}{h}\big) \bar A(x')^{-1}A_1\big(x', \frac{x_3}{h}\big) -\frac{h^2}{2}A_2\big(x',\frac{x_3}{h}\big)\Big) \bar A(x')^{-1}+ o(h^2),
\Sigma^{0}( p_{f}) ={\frac{e^{2}}{\beta}}\int{\frac{d^{2}k}{( 2 \pi )^{2}}}{\frac{{\cal M}({\bf k}, p_{0 f})}{{\bf k}^{2}+ p_{0 f}^{2}+{\cal M}^{2}({\bf k}, p_{0 f})}}[ D_{L}^{0}({\bf q}) + D_{T}^{0}({\bf q}) ]
{\mit \Xi}_{[ 1 / 2 ]}= \left( \begin{array}{c c}{e^{i \phi}}&{0}\\{0}&{e^{- i \phi}}\\ \end{array}\right) \quad , \qquad{\mit \Xi}_{[ 1 ]}= \left( \begin{array}{c c c}{e^{i 2 \phi}}&{0}&{0}\\{0}&{1}&{0}\\{0}&{0}&{e^{- i 2 \phi}}\\ \end{array}\right) \quad .
\int_{\{\Psi<-t\}}|F_{t_0,\tilde{c}}|^2_hc(-\Psi)= \int_{\{\Psi<-t\}}|\tilde{F}|^2_hc(-\Psi) + \int_{\{\Psi<-t\}}|F_{t_0,\tilde{c}}-\tilde{F}|^2_hc(-\Psi)
d \tilde{\mathbb{P}}_{\mathrm{Gibbs}}^{f}(\varphi):=\frac{1}{\tilde{z}_{\mathrm{Gibbs}}^{f}}\,e^{-H^{\mathrm{Wick}}(\varphi)}\,f\bigl(:\|\varphi\|_{\mathfrak{h}}^2:\bigr)\,d\varphi\,.
\langle V_{\mu}(x) V_{\nu}(y)\rangle_{0}=e^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\left(g_{\mu\nu}+\frac{k_{\mu}k_{\nu}}{m_{V}^{2}}\right) \frac{1}{k^{2}+m_{V}^{2}}e^{ik\cdot(x-y)},
\frac{d}{dt}(\frac{\partial\bar{L}}{\partial\dot{\rho}})=\frac{\partial\bar{L}}{\partial\rho}\frac{d}{dt}(\frac{\partial\bar{L}}{\partial\dot{\varphi}})=\frac{\partial\bar{L}}{\partial\varphi},\text{\}\frac{d}{dt}(\frac{\partial\bar{L}}{\partial\dot{\theta}})=\frac{\partial\bar{L}}{\partial\theta}
\frac{{\partial}f_k}{{\partial}x_k}\left(x,u_\mathrm{ss}\right) =-\sum_{\substack{j=1 j\neq{k}}}^{{n}}\frac{2}{\left|x_k-x_j\right|^3}-\sum_{j=0}^{c}\frac{2u_\mathrm{ss}^j}{\left|x_k-q_j\right|^3}\,, \frac{{\partial}f_k}{{\partial}x_i}\left(x,u_\mathrm{ss}\right) =\frac{2}{\left|x_k-x_i\right|^3}\,,\quad{i\neq{k}}.
g(W_{i},\hat{\gamma}_{\ell},\theta_{0})+\phi(W_{i},\hat{\gamma}_{\ell},\hat{\alpha}_{\ell},\tilde{\theta}_{\ell})-\psi(W_{i},\gamma_{0},\alpha_{0},\theta_{0})=\hat{R}_{1\ell i}+\hat{R}_{2\ell i}+\hat{R}_{3\ell i}+\hat{\Delta}_{\ell}(W_{i}).
\begin{aligned}f_{y}\, \omega = f_{y}\left( \psi_i \, dx + \varphi_i \, dy \right) = ( \psi_i f_{y}- \varphi_i f_{x}) \, dx f_{x}\, \omega = f_{x}\left( \psi_i \, dx + \varphi_i \, dy \right) = ( \varphi_i f_{x}- \psi_i f_{y}) \, dy\end{aligned}
\begin{cases}r_{\alpha_1,\lambda_1,\alpha_2,\lambda_2}(x, 0) = \delta(x) \lim_{|x|\rightarrow \infty}r_{\alpha_1,\lambda_1,\alpha_2,\lambda_2}(x, t)=0, \lim_{|x|\rightarrow \infty}\frac{\partial}{\partial x}r_{\alpha_1,\lambda_1,\alpha_2,\lambda_2}(x, t)=0. \end{cases}
F(u)=\frac{uz^2(1+z)H(0)+2 u z^{3}-u^{2}z +u^{2}+z^{2}-u}{u^{2}z^{2}+2 u z^{3}+u^{2}+z^{2}-u}, G(u)=-\frac{z \left(H(0)(u z^{2}+u-1)+2 u z^{2}+z \right)}{u^{2}z^{2}+2 u z^{3}+u^{2}+z^{2}-u}, H(u)=\frac{z \left(zH(0)(u z-u+1) -u^{2}z +u^{2}-u \right)}{u^{2}z^{2}+2 u z^{3}+u^{2}+z^{2}-u}.
LU^{\pm}= \pm (L J_0(w_1) - LJ_0(w_2)) - | w |_{0,\mathcal{M}}L e^{- \epsilon t}= \pm (f - f) - | w |_{0,\mathcal{M}}Le^{- \epsilon t}= - | w |_{0,\mathcal{M}}L e^{- \epsilon t}\geq 0 \textnormal{in}\mathcal{M}\times (0,T).
e^{-i2t\log\frac{t}{2\pi e\sqrt{n_1n_2}}}=\left(\frac{e^{-i2t\log\frac{t}{2\pi e\sqrt{n_1n_2}}}}{2i\log\frac{2\pi\sqrt{n_1n_2}}{t}}\right)'-\frac{e^{-i2t\log\frac{t}{2\pi e\sqrt{n_1n_2}}}}{2it(\log\frac{2\pi\sqrt{n_1n_2}}{t})^2}
\sum_{n=1}^N \frac{1}{n^{2\alpha-1}}=(1-2\alpha)\sum_{n=1}^N\int_1^n\frac{dt}{t^{2\alpha}}+N &=(1-2\alpha)\int_1^N\bigg(\sum_{t<n\le N}1\bigg)\frac{dt}{t^{2\alpha}}+N &=(1-2\alpha)\int_1^N (N-\lfloor t\rfloor)\frac{dt}{t^{2\alpha}}+N.
\left[e^{-tL}\varphi^{k,i}\right](x) =\left[e^{-tL_k}\phi^{k,i}\right](x)\,Q_{k,i}\left(\frac{x}{|x|}\right), \left[e^{-tL}\varphi\right](x) =\sum_{k=0}^\infty\sum_{i=1}^{\ell_k}\left[e^{-tL_k}\phi^{k,i}\right](x)\,Q_{k,i}\left(\frac{x}{|x|}\right) \qquad\mbox{in $L^2({\bf R}^N)\cap L^\infty({\bf R}^N)$ for any $t>0$}.
|g(s_2)| \le \Theta(c_1'+\sigma_2+1,h) X^{c_1'}\int_{-\infty}^{\infty}\min\Big( \frac{1}{|s_1|}, \frac{\epsilon^{-1}}{|s_1(s_1+1)|}\Big) |\zeta(s_1+1)|^{k}dt_1 s_1=c_1'+it_1.
\sum_{j=i}^{i+m-1}H(\mu,\mathcal{D}_{i+1}|\mathcal{D}_{i}) =\int H(\mu^{x,i},\mathcal{D}_m)d\mu(x) &=\int_{x : \, H(\mu^{x,i},\mathcal{D}_m)\leq \varepsilon}H(\mu^{x,i},\mathcal{D}_m)d\mu(x) +\int_{x: \, H(\mu^{x,i},\mathcal{D}_m)>\varepsilon}H(\mu^{x,i},\mathcal{D}_m)d\mu(x).
(\begin{matrix}C_{l}^{m}\\S_{l}^{m}\end{matrix})\equiv\sqrt{2}r^{l}\Theta_{l}^{m}(\begin{matrix}cosm\varphi\\sinm\varphi\end{matrix})=\frac{1}{\sqrt{2}}(\begin{matrix}(-1)^{m}&1\\ -(-1)^{m}i&i\end{matrix})(\begin{matrix}R_{l}^{m}\\R_{l}^{-m}\end{matrix}),m>0.
b_3:= |B-A|_2^2\|(-B)^{-1}\|^2 \leq O\left(\frac{(\ln n)^{3}}{n^2}\right)O\left(n^2(\ln n)^{-1}e^{-(\ln n)^{\frac{1}{2}}}\right) = O\left({(\ln n)^{2}}e^{-(\ln n)^{\frac{1}{2}}}\right)=O\left({(\ln n)^{-2}}\right).
Z_{N+1}\ =\ \int d \Phi' \ \ \exp\left[-(N\ +\ 1)^2 \left(g{\makebox{tr}\Phi'^4 \over 4 (N\ +\ 1)}\ +\ \psi\left({\makebox{tr}{\Phi'}^2 \over 2 (N\ +\ 1)}\right)\right)\right]
\frac{a}{\sqrt{n}}W_j \eta_j + \frac{b}{n}W_j \eta_j^2 = a \frac{\eta_j^2}{\sqrt{n}}+ \bigg( a \frac{g^{\prime \prime}(0 )}{2 g^\prime (0 )}+ b \bigg) \frac{\eta_j^3}{n}+ O (n^{-3/2}) .
\zeta_i=\frac{\mu P_t\left(\sum_{j\in\mathcal{M}_i}\beta_j^{(i)}|\alpha_{ij}|^2\right)^2}{N \sum_{j\in\mathcal{M}_i}\left(\beta_j^{(i)}\right)^2\left[|\alpha_{ij}|^2\sigma_{Z_{s,j}^{(i)}}^2+\frac{N}{\mu P_t}\sigma_{Z_{0,j}^{(i)}}^2\right]},
\begin{array}{c c}{\alpha^{\prime}M^{2}= 4 \left( N_{R}- 1 \right) + \alpha^{\prime}Q_{+}^{2}= 4 \left( N_{L}- 1 \right) + \alpha^{\prime}Q_{-}^{2},}&{N_{R}- N_{L}= m_{0}w_{0},}\\ \end{array}
{\bar{{\cal G}}}_{({\rm I})}^{\left( \rho \right) a}\Gamma^{\left( \rho\right)}=\Delta_{{\rm cl}({\rm I})}^{\left( \rho \right) a}+O\left( \rho^2\right) \,\,,\;\;\;{\bar{{\cal G}}}_{({\rm II})}^{\left( \rho \right)a}\Gamma^{\left( \rho \right)}=\Delta_{{\rm cl}({\rm II})}^{\left( \rho\right) a}+O\left( \rho^2\right) \,\,,
{\cal A}_{\Lambda}^{a}\left( \phi , \phi^{*}, \overline{{{\phi}}}, \pi \right) = \left[ \overline{{{\Delta}}}^{a}W_{\Lambda}+ \frac i{2 \hbar}\left( W_{\Lambda}, W_{\Lambda}\right)^{a}\right] \left( \phi , \phi^{*}, \overline{{{\phi}}}, \pi \right) =
\prod_{\substack{m,n \geq 1 m \leq n ; \, (m,n)=1}}\left( \frac{1}{1-z^n}\right)^{\frac{m^2}{n^3}}= \exp\left\{\sum_{n=1}^{\infty}\left( \frac{n}{6}+ \frac{n^2}{2}+ \frac{n^3}{3}\right) \frac{z^n}{n^3}\right\}
[\begin{matrix}\hat{r}\\ \hat{\theta}\\ \hat{\phi}\end{matrix}]=[\begin{matrix}sin\theta cos\phi&sin\theta sin\phi&cos\theta\\cos\theta cos\phi&cos\theta sin\phi&-sin\theta\\ -sin\phi&cos\phi&0\end{matrix}][\begin{matrix}\hat{x}\\ \hat{y}\\ \hat{z}\end{matrix}]
Z_{q}(\beta) = \frac{1}{h^{DN}}\displaystyle \int_{0}^{\infty}\; d\bar{x}\;d\bar{p}\; \exp_{q}\left( - \beta \displaystyle \sum_{i=1}^{DN}\left(\frac{p_{i}^{2}}{2m}+ \frac{1}{2}m \omega^{2}x_{i}^{2}\right) \right),
P\{\max_{0\le s \le t}\mathcal{T}(t)<\beta\ |\V(0) = c_1,\N(t) = 3\}= \frac{\beta^3}{(c_1t)^3}+\frac{3\beta(c_1t-\beta)(c_2t+\beta)}{(c_1+c_2)\,c_1^2\,t^3}
N ( h ) = \left[ ( h^{0})^{3}- \frac{3}{2}h^{0}\delta_{i j}h^{i}h^{j}+ \frac{1}{\sqrt{2}}( h^{1})^{3}+ \frac{3}{2 \sqrt{2}}h^{1}[ ( h^{2})^{2}+ \ldots + ( h^{n})^{2}] \right] .
\phi(s)\,=\,1\,+\,\frac{2}{\Gamma(\beta)}\,\sum_{\ell=1}^\infty\, \bigg|\,\Gamma\bigg(\beta-\,\i\,\frac{2\pi\ell}{h}\bigg)\bigg|\,\sin\bigg(\frac{2\pi\ell}{h}\,(s-s_\ell)\bigg),
\frac{d}{dz}\left[ e^{-\frac{\lambda}{\xi}z}z^{\frac{\mu}{\xi}- 1}P_{1}(z) \right] = e^{-\frac{\lambda}{\xi}z}z^{\frac{\mu}{\xi}- 1}\left[ -\frac{\gamma}{\xi(1-z)}P_{0}(z) + p_{0,0}\left( \dfrac{(\mu - \xi)\gamma}{\xi \lambda z}+ \frac{1}{A(1)(1-z)}\right) \right],
b_j =q\beta_{j-1}+2=pqb_{j-2}+q+2=(q+2)\sum\limits_{k=0}^{(j-3)/2}(pq)^k+(pq)^{\frac{j-1}{2}}b_1 &=\left(\frac{q+2}{pq-1}+b_1\right)(pq)^{\frac{j-1}{2}}-\frac{q+2}{pq-1}=\left(\frac{q+2}{pq-1}+1\right)(pq)^{\frac{j-1}{2}}-\frac{q+2}{pq-1}.
0\leq\sum_{j=0}^{n}\sum_{k=0}^{n}\frac{\Gamma(j+k+a)}{\Gamma(j+k+b+c)}\,\xi_{j}\xi_{k}\leq\left\Vert B(a,b,c)\right\Vert \left(\sum_{k=0}^{n}\frac{\Gamma(k+a)\, k!}{\Gamma(k+b)\Gamma(k+c)}\,\xi_{k}{}^{2}\right)\!,
\phi^{-1}_{h'\circ g',f'}\bullet (F(\Pi)\lhd F(f')) \bullet \phi_{h\circ g'', f'}= F(\Pi \lhd f') \phi^{-1}_{h,g''\circ f'}\bullet (F(h)\rhd F(\Theta)) \bullet \phi_{h, g\circ f}= F(h \rhd \Theta)
{\cal A}\simeq \frac{G_{o}^{4}}{\alpha_{e}^{\prime}}\frac{\mathrm{T r}( \xi \cdot \xi^{\prime}) ( \zeta \cdot \zeta^{\prime})}{\alpha_{e}^{\prime}( \Delta n^{2}/ R_{2}^{2}+ q^{2})}\frac{R_{1}^{2}m^{2}w^{2}}{R_{2}^{2}}( 1 + v_{\perp}^{2}+ v_{2}^{2}+ 2 v_{2}) ( 1 + v_{\perp}^{2}+ v_{2}^{2}- 2 v_{2}) +{\cal O}( p^{\prime}+ p ) ~ .
S = \int d \mu [ e^{- \phi_{B}}\partial_{z}\phi_{M}\partial_{\bar{z}}\phi_{M}+ \frac{e^{\phi_{0}+ \phi_{M}}}{2}+ 2 ( \alpha - 1 ) e^{- \phi_{B}}\phi_{M}\partial_{z}\partial_{\bar{z}}\phi_{B}] =
\mathbb{E}[U_t]\leq e^{-2\kappa\tilde{\gamma}t}\left[U_0+\frac{(\mu-\kappa)}{a\lambda_2}\overline{V}_0\right] +\frac{m\tau^2\tilde{\gamma}}{4\kappa N}\left[1+\frac{(\mu-\kappa)(N-1)}{a\lambda_2}\right](1-e^{-2\kappa\tilde{\gamma}t}).
\left|{{\lambda_0}}+\sum_{i=1}^m\sum_{s=0}^{r-1}{{\lambda_{i,s}}}F_{s}(x,\alpha_i/\beta)\right|_{v_0}>C(\boldsymbol{\eta},\boldsymbol{\zeta},\boldsymbol{\alpha},\beta,\varepsilon){\mathrm{H}}_{v_0}({{\boldsymbol{\lambda}}}){\mathrm{H}}({{\boldsymbol{\lambda}}})^{-\mu(\boldsymbol{\eta},\boldsymbol{\zeta},\boldsymbol{\alpha},\beta,\varepsilon)}\enspace,
S_{r}(x_{1},x_{2},\cdots)=\frac{1}{r!}\det\left(\begin{array}[c]{ccccc}x_{1}x_{2}\cdots \ x_{r}-r+1 x_{1}x_{2}\cdots x_{r-1}0 -r+2 x_{1}\cdots x_{r-2}\vdots \ddots \ddots \ddots \vdots 0 \cdots 0 -1 x_{1}\end{array}\right)
\langle f,g\rangle^{\mathrm{H}}=\sum_{I\in\mathcal{I}}\frac{f(z_{i_1},...,z_{i_d};z_1,...,z_n;h)g(z_{i_1},...,z_{i_k};z_1,...,z_n;h)}{R^{\mathrm{H}}_IQ^{\mathrm{H}}_I}, \langle f,g\rangle^{\mathrm{K}}=\sum_{I\in\mathcal{I}}\frac{f(z_{i_1},...,z_{i_d};z_1,...,z_n;h)g(z_{i_1},...,z_{i_k};z_1,...,z_n;h)}{R^{\mathrm{K}}_IQ^{\mathrm{K}}_I}.
\frac{\partial}{\partial \eta^{\prime \prime}{}^{\mu}}\frac{\partial}{\partial \overline{{{\psi}}}^{\prime \prime}}\Omega ( \zeta^{\prime \prime}) = \frac{\partial}{\partial \overline{{{\psi}}}^{\prime \prime}}\frac{\partial}{\partial \eta^{\prime \prime}{}^{\mu}}F ( \zeta^{\prime \prime}) \, ,
\frac{\partial G(\vec z^*)}{\partial \vec t}\ge 0, \,\,\, \forall\, \vec t\in T \Leftrightarrow \begin{cases}\displaystyle \frac{\partial G(\vec z^*)}{\partial z_i}\le 0,\,\,\,\, i = 1, 2, \ldots,m_0, \displaystyle \frac{\partial G(\vec z^*)}{\partial z_i}= 0, \,\,\,\, i = m_0+1,\ldots, n.\end{cases}
\frac{\exp(4 \pi i (\tilde{\Delta}-\eta_{S_l})_1(\eta_k)_2)}{\exp(4 \pi i (\tilde{\Delta}-\eta_{S_m})_1(\eta_k)_2)}= \exp(4 \pi i (\eta_{S_m}-\eta_{S_l})_1(\eta_k)_2) = \exp(4 \pi i (\eta_{m}-\eta_{l})_1(\eta_k)_2).
a_1=\int_{\mathbb{S}^n}uY_1 \mathrm{d}A = \dfrac{\sqrt{n+1}}{\sqrt{\mathrm{Area}(\mathbb{S}^n)}}\sum\limits_{l=1}^{n+1}v_l \int_{\mathbb{S}^n}ux_l \mathrm{d}A = O(\varepsilon) \|u\|_{L^2(\mathbb{S}^n)}.
d_{r}\omega_{h_{k}....h_{1},j}^{i}=\widetilde{d}_{r}\omega_{h_{k}....h_{1},j}^{i}-T_{ar}^{i}\omega_{h_{k}....h_{1},j}^{a}+T_{rj}^{a}\omega_{h_{k}....h_{1},a}^{i}
aba^{-1}b^{-1}= nh n^{-1}bn h^{-1}n^{-1}b^{-1}& = nh n^{-1}h^{-1}h b n h^{-1}b^{-1}bn^{-1}b^{-1}= n (h n^{-1}h^{-1}) (b h n h^{-1}b^{-1}) (b n^{-1}b^{-1}).
I = \int_{\Omega}\frac{1}{2}\left( \epsilon_{\mu \nu \lambda}B_{\mu}^{*}( \partial_{\nu}- i g a_{\nu}) B_{\lambda}+ M B_{\mu}^{*}B^{\mu}+ \epsilon_{\mu \nu \lambda}a_{\mu}\partial_{\nu}a_{\lambda}\right) \; ,
\mathrm{R e}\, \, I_{n}( \beta ) = - \frac{M}{2{\pi^{2}}}\left( p^{2 n}\beta + \frac{\beta^{3}}{3}p^{2 n - 2}+ \ldots + \frac{\beta^{2 n + 1}}{2 n + 1}\right) +{p^{2 n}}\frac{M p}{4{\pi^{2}}}\log \left( \frac{\beta + p}{\beta - p}\right) ,
\frac{1}{nd_n}\sum\limits_{i=1}^{\lfloor n\lambda\rfloor}\sum\limits_{j=\lfloor n\lambda\rfloor+1}^n\left(1_{\left\{G(\xi_i)\leq G(\xi_j)\right\}}-\frac{1}{2}\right)\overset{\mathcal{D}}{\longrightarrow}\left(Z(\lambda)-\lambda Z(1)\right)\int_{\mathbb{R}}J(x)dF(x)
\lim_{\epsilon \rightarrow 0}\int_{-\infty}^{\infty}dp \frac{f(p)}{p-k}\lim_{z \rightarrow 1}(1-z)^{2 \epsilon}\left\{(1-z)^{-i(\sqrt{k^2-\mu^2}-\sqrt{p^2-\mu^2})}-(1-z)^{i(\sqrt{k^2-\mu^2}-\sqrt{p^2-\mu^2})}\right\}
\| f \|_{C^{k,\alpha}_{\phi, \varphi}(B_{a\phi(x)}(x))}= \sum_{j=0}^k \varphi(x) \phi^j(x)\| \nabla^j f \|_{C^0(B_{a\phi(x)}(x))}+ \varphi(x) \phi^{k+\alpha}(x) [ \nabla^{k}f]_{0, \alpha; B_{a\phi(x)}(x)}.
Q_{m_{1},m_{2}}^{(j)}=Q_{m_{1}}\cap\left[x_d=\frac{2}{j-1}x_{d-1}-\frac{2}{j-1}\left(x_{d-1}^{0}+\frac{j+1}{2}x_d^{0}+m_{2}\right)\right],|m_{2}|<M.
f(u, |x|) = k_1(|x|)|u|^{q_1-1}, f(u, |x|) = k_1(|x|)|u|^{q_1-1}+ k_2(|x|)|u|^{q_2-1}, f(u, |x|) = k_1(|x|) \min \{u^{q_1-1}, u^{q_2-1}\},
\mathbb{E}\bigg[\tanh\frac{u^{(l)}}{2}\bigg] =\mathbb{E}\bigg[ \prod^{d_{c}-1}_{j=1}\tanh\frac{\gamma^{(l)}_{j}}{2}\bigg] &=\bigg(\mathbb{E}\bigg[\tanh\frac{\gamma^{(l)}}{2}\bigg]\bigg)^{d_{c}-1},
\gamma_{\xi_1\times\xi_2}\big(\max\{\varphi_1\circ \pi_1,\varphi_2\circ \pi_2\}, \psi_1\circ\pi_1+\psi_2\circ \pi_2\big)=\gamma_{\xi_1}(\varphi_1,\psi_1)+\gamma_{\xi_2}(\varphi_2,\psi_2),
P_{WU_1U_2V_1V_2X_1X_2YY_{12}Y_1 Y_2}=P_{\tilde W \tilde U_1\tilde U_2\tilde V_1\tilde V_2\tilde X_1\tilde X_2\tilde Y\tilde Y_{12}\tilde Y_1 \tilde Y_2},
\tilde{\lambda}_{\varepsilon_{n_k}}= \tilde{\lambda}_0 + \Phi \, e^{-\beta_{\varepsilon_{n_k}}\tilde{\Psi}_{\varepsilon_{n_k}}(\mathbf{x}_{\varepsilon_{n_k}})}+ o\biggl(e^{-\beta_{\varepsilon_{n_k}}\tilde{\Psi}_{\varepsilon_{n_k}}(\mathbf{x}_{\varepsilon_{n_k}})}\biggr) k \to +\infty \; ,
(\sigma_{\widetilde i})^r=(L_{\widetilde i}\,\pi)^r \stackrel{\eqref{piLxpi}}{=}\prod_{k=0}^{r-1}(L_{\widetilde k}^{-1}L_{\widetilde{i+k}}) \pi^r= \prod_{k=0}^{r-1}L_{\widetilde k}^{-1}\prod_{k=0}^{r-1}L_{\widetilde{i+k}}= \mathrm{id}
\sum_{i=1}^t \left(\frac{a_i'}{a_{t+1}'}\frac{-1}{a_i/a_{t+1}-a_i'/a_{t+1}'}\prod_{j=1}^t d(i,j) \right)=\prod_{j=1}^t d(t+1,j) - \prod_{i=1}^t a_i'b_i'.
\lVert f \rVert_{\mathrm{L}^p(\partial\Omega)}^p := \int_{\partial\Omega}\lvert f(x)\rvert^p ~\mathrm{d}\sigma_x \qquad \lVert f \rVert_{\dot{\mathrm{B}}^{s}_{p,p}(\partial\Omega)}^p:= \int_{\partial\Omega}\int_{\partial\Omega}\frac{\lvert f(x)-f(y) \rvert^p}{\lvert x-y\rvert^{ps+n-1}}~\mathrm{d}\sigma_x\mathrm{d}\sigma_y
\begin{array}{l l}\Phi(L_1)=(\frac{n!}{2},\frac{n!}{2},\frac{(n-3)\cdot n!}{8}), \Phi(L_2)=(0,n!,\frac{(n-1)\cdot n!}{4}), begin{align*}15pt]\Phi(L_3)=(n!,0,\frac{(n-1)\cdot n!}{4}), \Phi(L_4)=(0,0,\frac{(n+1)!}{2}).\end{array}
\int pdf_{MJ}(p)dp_{1}\cdot\cdot\cdot dp_{d}\propto\int e^{-\frac{E(p)}{k_{B}T}}(p(\gamma)^{d-1}\frac{dp(\gamma)}{d\gamma})d\Omega_{d}d\gamma\equiv\int_{\Omega_{d}}d\Omega_{d}\int P_{MJ}(\gamma)d\gamma
\frac{n^{4d}}{m^4}\mathrm{Var}_0\left[\sum_{l=3}^m\sum_{1\le p<q<l}\mathbb{E}_0\left[ \bar{U}^{(pl)}_h\bar{U}^{(ql)}_h \middle| \mathcal{F}_{n, l-1}\right] \right]\;\longrightarrow\; 0.
\tilde{\gamma}_{i_{1}\ldots i_{p - 2 k - 2}}^{( 1 )}\equiv \pi_{i_{1}\ldots i_{p - 2 k - 2}}- \pi_{i_{1}\ldots i_{p - 2 k - 2}}^{( 1 )}\approx 0 , k = 0 , \ldots , c ,
\lambda_{(j-1)k+i}-\lambda_{(j-1)k+(i+1)}=\phi(a_{i,j})-\phi(a_{i+1,j}) &=\sum\limits_{s=i}^ka_{s,j}+\sum\limits_{s=1}^{i-1}a_{s,j+1}-\left(\sum\limits_{s=i+1}^ka_{s,j}+\sum\limits_{s=1}^{i}a_{s,j+1}\right) &=a_{i,j}-a_{i,j+1}.
\tilde{{\cal M}}_0(i) = \bigg\{m : \frac{|{\cal T}_{\bar{X}}^n|}{|\tilde{{\cal M}}_0|}2^{(i-1)}< |\tilde{\varphi}_0^{-1}(m) \cap{\cal T}_{\bar{X}}^n | \le \frac{|{\cal T}_{\bar{X}}^n|}{|\tilde{{\cal M}}_0|}2^i \bigg\}
\frac{1}{(1 - \overline{\zeta}^t z)^{k+p}}= \sum_{n=0}^\infty \binom{n + k + p - 1}{k + p - 1}\overline{\zeta}^{tn}z^n &= \sum_{n=0}^\infty \frac{(n + k + p - 1) \cdots (n + 1)}{(k + p - 1)!}\overline{\zeta}^{tn}z^n = \frac{1}{(k + p - 1)!}\sum_{n=0}^\infty \big(n^{k+p-1}+ v(n)\big) \overline{\zeta}^{tn}z^n,
\beta^{\Gamma^{(j)}_i}_{i+1}=\frac{p^{(j)}_{i+1}D^{(j)}_{i+1}}{p}\Big(\sum_{i'=1}^{i}\big(k_{i'}^{(j)}\cdot a^{(j)}_{i'}p^{(j)}_{i'}\dots p^{(j)}_{i}-x^{(j)}_{v_{i'}}\cdot a^{(j)}_{i'}p^{(j)}_{i'+1}\dots p^{(j)}_{i}\big) - x^{(j)}_{v_{0}}\cdot p^{(j)}_{1}\dots p^{(j)}_{i}\Big).
B(\{s_n\}, \{t_n\},1) = \bigcap_{\ell=1}^\infty \bigcup_{\substack{a_1,\ldots,a_\ell a_i \in [s_i-t_i, s_i+t_i]}}I_\ell(a_1,\ldots,a_\ell) = \bigcap_{\ell=1}^\infty \bigcup_{\substack{a_1,\ldots,a_\ell a_i \in [s_i-t_i, s_i+t_i]}}D_\ell(a_1,\ldots,a_\ell).
\|g\varphi_\delta\|_\infty=\|\theta g\varphi_\delta\|_\infty\leq\|\theta g\varphi_\delta- \omega g \psi_\delta\|_\infty + \|\omega g \psi_\delta\|_\infty \leq \|g\|_\infty CC_1+\|\omega\|_\infty C_1 & \delta>0.
G_{\tau}(\chi) =\sum_{(x_1,\ldots, x_r) \in X_{\tau}}\prod_{i=1}^r\chi(1-x_ix) &=(\sum_{a\in{{\bf{F}}_q}}\chi(1+ax))^{c_1}(\sum_{a\in{{\bf{F}}_q}}\chi^2(1+ax))^{c_2}\cdots(\sum_{a\in{{\bf{F}}_q}}\chi^{r}(1+ax))^{c_{r}}&=\prod_{i=1}^{r}(\sum_{a\in{{\bf{F}}_q}}\chi^i(1+ax))^{c_i}.
S_{E}= - \frac{1}{1 6 \pi G_{d - 1}}\int_{\Sigma}d^{d - 1}x \sqrt{g}[ R - \frac{4}{d - 3}( \partial \phi )^{2}- m e^{\frac{4 \phi}{\sqrt{d - 2}( d - 3 )}}- \frac{1}{4}e^{\frac{- 4 \sqrt{d - 2}}{d - 3}\phi}F^{2}] .
u(X,Y,t) =\frac 1 2 \biggl\{\sup_{\tilde X\in{{B_\epsilon(X)}}}u(\tilde X,Y+{\epsilon^2}\tilde X/2,t-{\epsilon^2}/2)+\inf_{\tilde X\in{{B_\epsilon(X)}}}u(\tilde X,Y+{\epsilon^2}\tilde X/2,t-{\epsilon^2}/2)\biggr \}&+{o}(\epsilon^2),\mbox{as}\epsilon\to 0.