harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
p_m\,p_{m'}=k_m\,k_{m'}=\delta(m-m')\:p_m p_m\,k_{m'}=k_m\,p_{m'}=\delta(m-m')\:k_m k_m \,b_m^>\, b_{m'}^<\, k_{m'}= \delta(m-m') \Big( p_m+ \pi^2\,k_m\, b_m\, p_m\, b_m\, k_m \Big) \:.
-d\mathcal{X}_r^{\alpha,x}=f(\alpha_r,r,X_r^{\alpha,x},\mathcal{X}_r^{\alpha,x},Z_r^{\alpha,x},K_r^{\alpha,x}(\cdot))dr-Z_r^{\alpha,x}dW_r-\int_{{\bf E}}K_r^{\alpha,x}(e)\Tilde{N}(dr,de); \mathcal{X}_S^{\alpha,x}=\eta,
\lim_{n\rightarrow\infty}\frac{1}{n}\log|f_{n}(E,x+\mathrm{i}y)|=\lim_{n\rightarrow\infty}\frac{1}{(2\pi)^{d}}\int_{\mathbb{T}^{d}}\frac{1}{n}\log|f_{n}(E,x+\mathrm{i}y)|\mathrm{d}x=L(E,\mathrm{i}y).
J_m =\prod_{i=0}^{r-1}\frac{\Gamma_p(\langle\frac{-2mp^i}{q-1}\rangle)\Gamma_p(\langle\frac{2mp^i}{q-1}\rangle)}{\Gamma_p(\langle(1-\frac{m}{q-1})p^i\rangle)\Gamma_p(\langle\frac{mp^i}{q-1}\rangle)}.
(W_N,v_N)\in\mathfrak{A}_{\mathsf{m}+1}^{\sigma,\tau}\,\wedge\,\mathfrak{O}_{0,N}=\bigsqcup\limits_{n=1}^{N-1}\Big[\,(W_N,v_N)\in\mathfrak{A}_{\mathsf{m}+1}^{\sigma,\tau}\,\wedge\,\mathfrak{O}_{0,N}\,\wedge\,n=n_{\textnormal{min}}\,\Big]\,.
\frac{\partial G^{( \alpha )}}{\partial \delta}= \frac{1}{i}\sum_{n = 1}^{\infty}( 2 n + 1 )^{\alpha + 1}\int_{C}d s \, [ 2 \pi ( 2 n + 1 ) \delta ]^{- s}\, \Gamma ( s ) \zeta ( s ) .
\mathcal{L}y =\mathcal{L}y^{[0]}+\mathcal{L}y^{[1]}&=-(1-x)\triangle_{[0]}y^{[0]}+\Big(\frac{N}{2}+L_1\Big)Dy^{[0]}+L_0y^{[0]}+ &+x\triangle_{[1]}y^{[1]}+\Big(-\frac{5}{2}(1-x)+L_1\Big)Dy^{[1]}+L_0y^{[1]}.
{1\over{4\pi G_4}}\int_{\partial\Sigma}dS_{\mu\nu}e\hat{E}^{\mu\nu}= \bar{\varepsilon}_{\infty}\left[ P^{ADM}_{\mu}\gamma^{\mu}+{1\over{2\sqrt{2}G_4}}e^{-\varphi_{\infty}/2}\{V_{R\,\infty}L(\vec{Q}-i\gamma^5\vec{P})\}^a\Gamma^a\right]\varepsilon_{\infty},
Tr[\,Q_{2}\lambda^{2}_{SU(2)}\,] = - \frac{2}{3}\sum_{k=1}^{3}Tr[\,\gamma_{k, 7_4, (0)}\lambda_{U(1)_{2, (0)}}\,]\, Tr[\,\gamma^{-1}_{k, 7_4, (0)}\lambda^{2}_{SU(2)}\,]
(\tilde{y}(1)-\hat t)\hat{t}+(\tilde{y}(3)-\sqrt{1-\hat{t}^2})\sqrt{1-\hat t^2}- \frac{\tilde{y}(3)-\sqrt{1-\hat{t}^2}}{\sqrt{1-\hat{t}^2}}=\tilde{y}(1)\hat{t}-\frac{\hat{t}^2}{\sqrt{1-\hat{t}^2}}\tilde{y}(3).
S^{\nu}\left(\psi ;z_1 ,\dots ,z_n ,\left\{v,x\right\},\tau \right) =\sum_{\sigma \in I(n)}\prod_{j<\sigma (j)}\langle v_j ,v_{\sigma (j)}\rangle P_2 \left( z_{\sigma (j),j},\tau \right) \sum_{U\subseteq f(\sigma)}S^{\nu}\left( \prod_{r\in U}o(v_r); \left\{\left(\prod_{s\in f(\sigma)\setminus U}\phi (v_s)\right) v,x\right\},\tau \right).
\frac{dz}{d\lambda}=\mathcal{B}\frac{\omega(\zeta,-1)^2}{\left[\omega(\zeta,\zeta_\infty)\omega(\zeta,1/\bar\zeta_\infty)\right]^2\prod_{j=1}^{M}\omega(\zeta,\gamma_{1}^{(j)}) \omega(\zeta,\gamma_{2}^{(j)})}\prod_{j=0}^{M}\prod_{k=1}^{n_j}\left[\omega(\zeta,a_{k}^{(j)})\right]^{\beta_k^{(j)}}.
\lambda((p-2)\nabla_X^2\phi(\tilde X_j^k, \tilde Y_j^k, \tilde t_j^k) ) + \Delta_X \phi(\tilde X_j^k, \tilde Y_j^k, \tilde t_j^k) \leq (p-1)\Lambda(E^X_k) + \sum^{m-1}_{i=1}\lambda_i(E^X_k).
\mathcal{A}_n(y) = \left\{|u| \leq n : V(u_j) \geq f_n(j) - y, j \leq |u| \right\}, \bar{\mathcal{A}}_n(y,h) = \left\{|u| = n : u \in \mathcal{A}_n(y), V(u) - f_n(n) + y \in [h-1,h] \right\}\mathcal{B}_n(y,z) = \left\{|u| \leq n : \xi(u_j) \leq z + (V(u_j) - f_n(j) + y)/2 , j \leq |u| \right\}.
T = \left(\begin{array}{cccc}0 T^{(1)}0 0 0 0 T^{(2)}0 0 0 0 T^{(3)}0 0 0 0\end{array}\right),T^* = \left(\begin{array}{cccc}0 0 0 0 T^{(1)*}0 0 0 0 T^{(2)*}0 0 0 0 T^{(3)*}0\end{array}\right).
v(x,t)-\int_\tau^t\int_{{\bf R}^N}G(x-y,t-s)v(y,s)^p\,dy\,ds =\int_{{\bf R}^N}G(x-y,t-\tau)u(y+z,\tau+(2\rho)^\theta)\,dy \ge c_*\int_{B(z,\rho)}u(y,\tau)\,dy\int_{{\bf R}^N}G(x-y,t-\tau)G(y,\rho^\theta)\,dy =c_*\int_{B(z,\rho)}u(y,\tau)\,dy\,G(x,t-\tau+\rho^\theta)
\left(\widehat{i\ 1}\right)=V_{\mathbb{Z}\beta+\frac{i}{4k}\beta}\otimes(V_{\mathbb{Z}\beta}^{T_{2}})^{-}+V_{\mathbb{Z}\beta+\frac{2k+i}{4k}\beta}\otimes(V_{\mathbb{Z}\beta}^{T_{2}})^{+}.
\mu\Big(\bigcup_{i \in I}A_i(z_\infty)\Big) = \lim_{n \to \infty}\mu\Big(\bigcup_{i \in I}\bigcup_{m \ge n}A_i(z_m) \cup A_i(z_\infty)\Big) \ge \lim_{n \to \infty}\lim_{\ell \to \infty}\mu_\ell\Big(\bigcup_{i \in I}\bigcup_{m \ge n}A_i(z_m) \cup A_i(z_\infty)\Big) \ge \sum_i q_i,
R_{n,p}={\pi^{\frac{n}{2 p}+n}{\Gamma \left(\frac{n+2}{2}\right)^{-2}}\left(\frac{2 \Gamma \left(\frac{1}{2}(n+p+2)\right)}{\Gamma \left(\frac{n+2}{2}\right) \Gamma \left(\frac{p+1}{2}\right)}\right)^{\frac{n}{p}}}
d_{i}^{n}p^{6}d_{j}^{n}\rightarrow d_{i}^{n}p^{5}d_{j}^{n+1}\rightarrow d_{i}^{n-1}p^{6}d_{j}^{n+1}\rightarrow d_{i}^{n}p^{5}d_{j}^{n+1}\rightarrow d_{i}^{n}p^{6}d_{j}^{n}
\frac{\partial L}{\partial t}=(1+\frac{\alpha'}{2}R)\Delta L+(1+\alpha'R)|\nabla\log R|^2+\frac{\alpha'^2}{4}|\nabla R|^2+\big(1+\frac{\alpha'}{2}R\big)\big(R+\frac{\alpha'}{4}R^2-r\big)
a(t,x, y_1,\cdots )=\Phi_0(t, x, y_1,\cdots)\mbox{for}0<x\ll 1, &a(t,x,y_1,\cdots)=\Phi_1(t,1-x, (1-x)^{N/2}, y_1,\cdots)\mbox{for}0<1-x\ll 1.
\left ( \begin{array}{c}\tilde{f_{1}}(x) \vdots \tilde{f_{N}}(x) \end{array}\right ) = \sum_{k = k_{2}}^{k_{3}}\tilde{A_{k}}\left ( \begin{array}{c}\tilde{f_{1}}(2x - k) \vdots \tilde{f_{N}}(2x - k) \end{array}\right ) .
\zeta'(x)=-\frac{2}{\tilde{\gamma}_1}x^{-1/\tilde{\gamma}_1-1/{\gamma}_0-1}- \frac{\gamma_1}{\tilde{\gamma}_1 \gamma_0}\left(x^{-1/\tilde{\gamma}_1-1/{\gamma}_0-1}-x^{-1/\gamma_0-1}\right)<0.
d s_{\mathrm{c a n}}^{2}= \left( 1 +{\frac{4 M}{r}}-{\frac{4 g^{2}q^{2}}{r^{2}}}\right)^{-{\frac{1}{2}}}d t^{2}- \left( 1 +{\frac{4 M}{r}}-{\frac{4 g^{2}q^{2}}{r^{2}}}\right)^{\frac{1}{2}}d \vec{x}^{2}\ .
\hat{h}= \frac{\hbar^{2}}{2}\left( - \frac{1}{\mu ( q )}\Delta + \frac{\alpha}{\mu^{2}c^{2}}C_{1}\frac{{\bf q}}{q^{3}}\frac{\partial}{\partial{\bf q}}+ \frac{4 \pi \alpha \lambda}{\mu^{2}c^{2}}C_{1}\delta ({\bf q}) \right) + v ( q ) \, ,
\lim_{r\to 0}\begin{vmatrix}f^+(r) \overline{\widetilde{f}^+(r)}f^-(r) \overline{\widetilde{f}^-(r)}\end{vmatrix}= \begin{vmatrix}E\begin{pmatrix}A^+ A^-\end{pmatrix}\overline{E\begin{pmatrix}\widetilde A^+ \widetilde A^-\end{pmatrix}}\end{vmatrix},
|E_{i_{\ell_t}}|~ q^{-\ell_t+t}=|E_{i_{\ell_t}}|^{1-\frac{(\ell_t-t)}{d}}|E_{i_{\ell_t}}|^{\frac{(\ell_t-t)}{d}}q^{-\ell_t+t}&\le |E_{i_{\ell_t}}|^{\frac{d-\ell_t+t}{d}}\le 2^{\frac{(d+1)(d-\ell_t+t)}{d(k+1)}i_{\ell_t}}.
\int_{0}^{\infty}|\theta_{t}f(x)|^{2}\, \frac{dt}{t}= \sum_{R \in \mathcal{D}(w)}1_R(x) \int_{\ell(R)/2}^{\ell(R)}|\theta_{t}f(x)|^{2}\, \frac{dt}{t}= \lim_{s \to \infty}\mathop{\sum_{R \in \mathcal{D}(w)}}_{\ell(R) \le 2^s}1_R(x) \int_{\ell(R)/2}^{\ell(R)}|\theta_{t}f(x)|^{2}\, \frac{dt}{t}.
\int_0^\infty\frac{(e^{-\lambda_{2}s}-e^{-\lambda_{1}s})_+}{\sqrt{s}}\Big[\int_0^\infty \tau e^{-\tau^2}d\tau\Big]ds =\frac{\sqrt{\pi}(\sqrt{\lambda_1}-\sqrt{\lambda_2})_{+}}{2\sqrt{\lambda_2\lambda_1}}
{\sum_{i , j , k , l = 1}^{h}\omega_{l}^{2}\partial_{z}\left({\omega_{i}/ \omega_{l}}\right) A_{i j}A_{l k}\overline{{\omega}}_{k}^{2}\partial_{\bar{z}}\left({\overline{{\omega}}_{j}/ \overline{{\omega}}_{k}}\right)}= \left( \sum_{i , j = 1}^{h}\omega_{i}A_{i j}\overline{{\omega}}_{j}\right)^{3}.
r_{n+1}=m^{2}r_{n},\ \ \nu_{n+1}=\nu_n-b\tilde M_n^4\epsilon_n^{\frac{1}{m}}, \delta_{n+1}={b}^{-1}R_n^{-(2m-1)}\nu_{n}^{m}\delta_{n}, \ \ \tilde{M}_{n+1}= \tilde{M}_{n}+20m\tilde M_n^2\epsilon_n, c_{n+1}=({b}^{-1}R_{n}^{-1}r_{n}^{-1}\delta_{n}c_{n})^{\kappa^{r_{n}}}-({b}R_n \tilde{M}_{n}\delta_{n}^{-1}r_{n})^{\kappa r_{n}}\epsilon_{n},
H'(t) \leq -a_1 \int_U |u_t(x,t)|^2 \, dx - a_1 \int_U |u_t(x,t)|^m \, dx + (\frac{3}{2}+ (a_2^2 B))\epsilon \int_U (u_t(x,t))^2 \, dx - \frac{1}{4}\epsilon \int_U (u_{xx}(x,t))^2 \,dx - a_2 \epsilon \int_U |u_t|^{m-1}|u|(x,t) \, dx -\epsilon E(t).
N_p(x) = \frac x{p \log x}\left(1 + O\left(\frac1{\sqrt{\log_2 x}}\right)\right) \sum_{k \le K \log_2 x}\frac{e^{-\gamma \xi}}{\Gamma(1+\xi)}\frac{(\log u)^{k-1}}{(k-1)!}\sum_{\substack{P^+(A) \le p \Omega(A) = k}}\frac 1A,
V ( 0 ) ={\frac{1}{4 \alpha}}[ n_{1}n_{1}^{\prime}\mathrm{T r}( Y^{2}) \mu_{0}+ n_{3}n_{3}^{\prime}\mathrm{T r}( \lambda_{i}\lambda_{j}) \mu_{3}+ n_{8}n_{8}^{\prime}\mathrm{T r}( T_{a}T_{b}) \mu_{8}] \ < \ 0 \ .
\pi_{*}(\sum_{i_{1}<\cdot\cdot\cdot<i_{k-1}}\alpha_{i}dt\wedge dx^{i}+\sum_{j_{1}<\cdot\cdot\cdot<j_{k}}\beta_{j}dx^{j})=(\int_{0}^{1}\alpha_{i}(\cdot,t)dt)dx^{i}
E\left[\max_{1\leq k\leq d_n}\left|\sum_{j=1}^N\gamma_k(i,j)W^{(i)}_j\right|^3\right]\leq \left(5a^2\log(2d_n-1+\sqrt{e})\max_{1\leq k\leq d_n}\sum_{j=1}^{N_n}\gamma_{n,k}(i,j)^2\right)^{3/2}
\sqrt{r^{2}\sin \theta \sqrt{B ( r )}}\: \: \Psi^{T}= ( 0 , \chi ) , \hspace{1 c m}\chi_{\alpha \ell}( r , t ) = \delta_{\alpha \ell}\chi_{1}( r , t ) - i ( \tau^{1})_{\alpha \ell}\chi_{2}( r , t ) .
\overline{\overline{C}}_{X^{\ast}}(g,k)=\overline{\overline{C}}_{X^{\ast}}(p_gk_g,k) =\overline{\overline{C}}_{X^{\ast}}(p_g, k_g)^{-1}\overline{\overline{C}}_{X^{\ast}}(p_g,k_gk)\overline{\overline{C}}_{X^{\ast}}(k_g,k)=1.
L_{\mathrm{S}}= \mathrm{}- \frac{1}{2}\partial_{\mu}A_{\nu}\partial^{\mu}A^{\nu}- \frac{m^{2}}{2}A_{\mu}A^{\mu}- \frac{1}{2}\partial_{\mu}B \partial^{\mu}B - \frac{m^{2}}{2}B^{2},
\left|\log\frac{\mu[\pmb{a}_1^n]}{\mu_\ell[\pmb{a}_1^n]}\right|\leq\sum_{j=1}^{n-\ell-1}\left\{\left|\log\frac{\mu[\pmb{a}_j^n]}{\mu[\pmb{a}_{j+1}^n]}- \log\frac{\mu[\pmb{a}_j^{j+\ell}]}{\mu[\pmb{a}_{j+1}^{j+\ell}]}\right| \right\}.
\tilde T_4(z) = \frac{i}{2\pi}\int_{\xi_1+\xi_2=\eta_1+\eta_2}\frac{1}{(2z+\xi_1)(2z+\eta_1)(2z+\eta_2)}\real \{\overline{\hat u(\xi_1) \hat u(\xi_2)}\hat u(\eta_1) \hat u(\eta_2) \}
(\overline{\rho}(aw_2,w_2a,w_1,w_2) w_1 \overline{\lambda}(aw_2,w_2a,w_1,w_2))^{\omega+1}\quad=\overline{\rho}(aw_2,w_2a,w_1,w_2) w_1 \overline{\lambda}(aw_2,w_2a,w_1,w_2) \qquad\qquad\qquad\Rightarrow\overline{\lambda}(aw_2,w_2a,w_1,w_2) =\overline{\rho}(aw_2,w_2a,w_1,w_2).
h_j(\beta,p,q) = \Big ( -{2 \over \beta}\Big )^{j+1}h_j\Big ({4 \over \beta},-{\beta p \over 2},-{2q \over \beta}\Big ), \tilde{h}_j(\beta,p,q) = \Big ( -{2 \over \beta}\Big )^{j+1}\tilde{h}_j\Big ({4 \over \beta},-{\beta p \over 2},-{2q \over \beta}\Big ).
P_{\mu}^5(k)=i\int \frac{d^2p}{(2\pi )^2}\frac{{\rm tr}\{\gamma_{\mu}\left[ (p\!\!\!\slash +k\!\!\!\slash) + m\right]\gamma_5(p\!\!\!\slash +m)\}}{\left[ (p+k)^2 -m^2\right] (p^2 - m^2 )}.
F_{F}(\beta)=\int_{S}\frac{d^{2}\tau}{\tau_{2}^{2}}\tau_{2}^{-\frac{d-2}{2}}\frac{\bar{\theta}_{2}^{4}}{\bar{\eta}^{12}}(z_{s}+z_{c})\theta^{'}_{4}\left(0\left|\frac{i\beta^{2}}{2\pi^{2}\tau_{2}}\right.\right)
{\mathrm{\boldmath ~ e ~}}( u = 0 , t ) = \lim_{\epsilon \rightarrow + 0}{\mathrm{\boldmath ~ e ~}}( \epsilon , t ) ={\mathrm{\boldmath ~ e ~}}( t ) \equiv \frac{\dot{\mathrm{\boldmath ~ x ~}}( t )}{| \dot{\mathrm{\boldmath ~ x ~}}|}\; ;
F^{\ast}(\chi_{j}) = F^{\ast}(\alpha_{u_j}+ \cdots + \alpha_{u_m}) &= \sum_{l=1}^m \langle\alpha_{u_j}+ \cdots + \alpha_{u_m}, \alpha_{u_l}^{\vee}\rangle \varpi_l &= -\varpi_{j-1}+ \varpi_{j}+ \varpi_m,
H=\int_0^\tau H_\tau d \tau' + H(0) = \int_0^\tau a^{-\frac{3 \gamma-3}{2}}a^{\frac{3 \gamma -3}{2}}H_\tau d \tau' + H(0) \lesssim \sup_{0 \leq \tau' \leq \tau}a^{\frac{3 \gamma -3}{2}}H_\tau + H(0),
\sum_{i=0}^k i\binom{i+n-2}{n-2}=\frac{1}{n}k(k-1)\binom{k+n-2}{n-2}+k\binom{k+n-2}{n-2}&= \frac{1}{n}k(k+n-1)\binom{k+n-2}{n-2}&= \frac{1}{n}k(k+1)\binom{k+n-1}{n-2}.
\mathbb{P}^{\otimes n}\biggl( \biggl\vert\frac{\tilde{\sigma}_{t}^2(h)}{\sigma_{t}^2(h)}-1\biggr\vert\geq\eta\biggr) \leq2 \exp\Biggl( -\frac{3 \eta^2}{2(3+2\eta) \Vert K \Vert_\mathrm{sup}^2}\sigma_t^2(h) \cdot n^2\prod_{i=1}^dh_i^2 \Biggr),
{\rm E}|G_T^{d,p}| =\sum_{x\in T^d(r)}P\big(\zeta_T^{d,p}(x)=1\big) &=\sum_{x\in T^d(r)}P\big(\zeta_0^{d,p}(x)=1, T_x>T\big) &=\sum_{x\in T^d(r)}pe^{-T}=r^dpe^{-T}.
\mathrm{div}^{\nabla'}(T_{r}) =(-1)^{r}\mathrm{div}^{\nabla'}(S_{r}\mathbb{I})+\mathrm{div}^{\nabla'}(\mathcal{A}_{\widehat{W}}\circ T_{r-1}) =(-1)^{r}\nabla'S_{r}+\mathcal{A}_{\widehat{W}}\mathrm{div}^{\nabla'}(T_{r-1})+\sum_{i=1}^{n}\epsilon_{i}(\nabla'_{Z_{i}}\mathcal{A}_{\widehat{W}})T_{r-1}Z_{i}.
\Delta_q(u\cdot\nabla v)= \sum_{q-1\leq p \leq q+2}\Delta_q(u_{\leq{p-2}}\cdot\nabla v_p)+\sum_{q-1\leq p \leq q+2}\Delta_q(u_{p}\cdot\nabla v_{\leq{p-2}}) &+\sum_{p\geq q-2}\sum_{\substack{|p-p'|\leq 1 p' \geq q-2}}\Delta_q(u_p\cdot\nabla v_{p'}).
w_G(s_1^{-1}t_1u_1,\ldots,s_d^{-1}t_du_d) =w_G(\vec{s}^{-1}\vec{t}\vec{u})=\varphi(\vec{s})^{-1}\varphi(\vec{t})\varphi(\vec{u})=w_G(\vec{s})^{-1}w_G(\vec{t})w_G(\vec{u}) &=w_G(s_1,\ldots,s_d)^{-1}w_G(t_1,\ldots,t_d)w_G(u_1,\ldots,u_d).
\omega_{( - )}\partial_{-}\omega_{( - )}^{- 1}= \left( g_{( 0 )}^{- 1}\, y_{( 0 ) +}\right)^{- 1}\, L_{( - 1 )}\, g_{( 0 )}^{- 1}y_{( 0 ) +}= y_{( 0 ) +}^{- 1}\, N_{( - 1 )}^{- 1}\, \partial_{-}N_{( - 1 )}\, y_{( 0 ) +}.
F=\left[ \left\{\sum_{i=1}^{2k}\left(\frac{dT_i}{dx^i}\right)^2-m^2T^2_i+\lambda T_i^4+\frac{m^2}{4\lambda}\right\}+\left\{\partial_{\mu}\mathcal{T}\partial^{\mu}\mathcal{T}-m^2\mathcal{T}^2+\lambda\mathcal{T}^4 +f(\mathcal{T})\overline{\theta}_R\theta_L\right\}\right]
s^1=\begin{pmatrix}0 1 1 0\end{pmatrix},s^2=\begin{pmatrix}0 -i i 0\end{pmatrix},s^3=\begin{pmatrix}1 0 0 -1\end{pmatrix},s^4=\begin{pmatrix}1 0 0 1\end{pmatrix}
\tfrac{L_X^{(\alpha)}(s+1,t,z)}{L_X^{(\alpha)}(s,t,z)}+\tfrac{L_Y^{(\beta)}(t+1,s,z)}{L_Y^{(\beta)}(t,s,z)}=\tfrac{L_U^{(\alpha)}(t+1,s,z)}{L_U^{(\alpha)}(t,s,z)}+\tfrac{L_V^{(\beta)}(s+1,t,z)}{L_Y^{(\beta)}(s,t,z)}.
lim_{n\rightarrow\infty}\frac{\int_{0}^{1}x^{n}(1-x)^{n}\sum_{k=0}^{\infty}\frac{(x-1/2)^{k}}{k!}f^{(k)}(1/2)dx}{\int_{0}^{1}x^{n}(1-x)^{n}dx}
\frac{d\rho}{dt}=-\frac{i}{\hbar}[\hat{H}_{RWA},\rho]+\frac{\gamma}{2}(2\hat{a}\rho\hat{a}^{\dagger}-\hat{a}^{\dagger}\hat{a}\rho-\rho\hat{a}^{\dagger}\hat{a})=L(\rho),
t_0^{(2)}= \frac{2}{\sqrt{\tau}}\left( 1+\rho_0 \cos (\beta \pi) -i \rho_0 \sin (\beta \pi) \right)^{\frac{1}{2}}-1 &= \frac{2}{\sqrt{\tau}}\left( i \sqrt{A^+_{\beta}}+ \sqrt{A^-_{\beta}}\right)-1,
a_{i_k}^*\cdots a_{i_{p-1}}^*+\sum_{\substack{\tau\cr (\tau(1),\dots,\tau(k-1))\ne(k-1,\dots,1)}}a_{i_k}\cdots a_{i_{p-1}}a_{i_{p-1}+\tau(p-1)}^*\cdots a_{i_1+\tau(1)}^*=0.
\left[ - \nabla_{\widehat{\mathrm{\boldmath ~ \scriptstyle{\xi}~}}, D}^{2}- \widehat{{\mathcal W}}^{({D})}( \widehat{\mathrm{\boldmath ~ \xi ~}}) \right] \, \widehat{\Phi}( \widehat{\mathrm{\boldmath ~ \xi ~}}) = \widehat{\eta}\, \widehat{\Phi}( \widehat{\mathrm{\boldmath ~ \xi ~}}) \; .
\mathcal{X}=L^2(J;V)\cap H^1(J;V'), \|u\|_{\mathcal{X}}^2=\|u\|_{L^2(J;V)}^2+\|u'\|_{L^2(J;V')}^2, &\mathcal{Y}_1=L^2(J;V), \|v_1\|_{\mathcal{Y}_1}^2=\|v_1\|_{L^2(J;V)}&\mathcal{Y}=\mathcal{Y}_1 \times H, \|v\|_{\mathcal{Y}}^2 =\|v_1\|_{L^2(J;V)}^2+\|v_2\|_H^2,
A_{j}(\textbf{u}) = \left( \begin{array}{ccccc}v_{j}\delta_{1j}\rho \delta_{2j}\rho \cdots \delta_{dj}\rho \delta_{j1}f(\textbf{u}) v_{j}0 \cdots 0 \delta_{j2}f(\textbf{u}) 0 v_{j}\cdots 0 \cdots \cdots \cdots v_{j}\cdots \delta_{jd}f(\textbf{u}) 0 0 \cdots v_{j}\end{array}\right).
E(a^3b^4h^2m,a^3b^2hn;l^2)=dE(a^3b^4h^2m_1\overline{l_1}^2,a^3b^2hn_1\overline{l_1}^2;d)E(a^3b^4h^2m_1\overline{d},a^3b^2hn_1\overline{d};l_1^2).
g_1 ={}3\alpha_1 + \alpha_2 +2\alpha_3 + r, g_2 ={}3\alpha_1 + \alpha_2 -2\alpha_3 + r , g_3 ={}-\tfrac{\alpha_3^2}{2\alpha_2}+ 3\alpha_1 -\alpha_2 + r, g_4:={}-\tfrac{\alpha_3^2}{\alpha_1+ \alpha_2}+ 2\alpha_1 + r.
\frac{\delta K_{\lambda+1}(\delta\gamma)}{\gamma K_{\lambda}(\delta\gamma)}+\frac{\beta^{2}\delta^{2}}{\gamma^{2}}(\frac{K_{\lambda+2}(\delta\gamma)}{K_{\lambda}(\delta\gamma)}-\frac{K_{\lambda+1}^{2}(\delta\gamma)}{K_{\lambda}^{2}(\delta\gamma)})
\mathcal{L}(\vec{x},\vec{y})= -\sum_{i=1}^M\alpha_i\log_2\left( 1+ \frac{h_{ss}^iE_s^i}{\alpha_i(\sigma_s^2+h_{ps}^iP_p)}\right) &+\sum_{i=1}^M \lambda_i \left[\sum_{j=1}^iE_s^j-\sum_{j=1}^i(1-\alpha_j)\eta P_p \right] &+\sum_{i=1}^M\gamma_i[h_{sp}^iE_s^i-P_{int}\alpha_i]+\sum_{i=1}^M\mu_i(\alpha_i-1)
\left( \sum_{i_{1},\dots ,i_{k}=1}^{n}\left\vert T\left(e_{i_{1}}^{n_{1}},\dots ,e_{i_{k}}^{n_{k}}\right) \right\vert^{r}\right)^{\frac{1}{r}}\leq D_{m,r,p,k}^{\mathbb{K}}\cdot n^{s}\left\Vert T\right\Vert .
\psi|_{S_1}= \pi_1^*[p], \lambda|_{S_1}= 0, \delta_0|_{S_1}= -2\pi_1^*[q_1] -2\pi_2^*[q_2], \delta_{1,1}|_{S_1}= \pi_1^*[q_1], \delta_{2,1}|_{S_1}= \pi_2^*[q_2].
P_{{\bf b},d}:=(-\theta)^{\frac{-\sum_{j=1}^s(b_j-1)q^{d-1}}{q-1}}\prod_{l=1}^{\infty}\prod_{\substack{j=1 b_j\geq 2}}^s\Biggl\{\biggl(1-\frac{t}{\theta^{q^l}}\biggr)^{q^{b_j-2}}\biggl(1-\frac{t}{\theta^{q^{l+1}}}\biggr)^{q^{b_j-3}}\cdots\biggl(1-\frac{t}{\theta^{q^{l+b_j-2}}}\biggr) \Biggr\}^{q^{d-b_j}}
\Vert u_M \Vert_{L^{\theta' \beta_{n+1}}(\mathbb{R}^N)}\leq C^{\displaystyle\sum_{j=1}^{n+1}\frac{1}{\beta_j}}\left( \displaystyle\prod_{j=1}^{n+1}\beta_j^{\frac{1}{\beta_j}}\right)^{\beta_0}\Vert u_M \Vert_{L^{p_{s}^*}(\mathbb{R}^N)}^{\displaystyle\prod_{j=0}^{n}\sigma_j}
\hat \lambda_{{\cal A},j}^{(i)}= \frac{{1 - \alpha}}{{1 -{\alpha^i}}}\left({\alpha \sum\limits_{l = 1}^{i - 1}{{\alpha^{K - l - 1}}u_j^{(l)}}+ u_j^{(i)}}\right) = \frac{{\alpha -{\alpha^i}}}{{1 -{\alpha^i}}}\hat \lambda_{{\cal A},j}^{(i - 1)}+ \frac{{1 - \alpha}}{{1 -{\alpha^i}}}u_j^{(i)},
\mathfrak{D}(f,id) =\{(x_1,x_2,1-x_2,1-x_1)\mid 0\le x_1\le x_2<1/2\}, D_1 =\{(x_1,x_2,1-x_2,1-x_1)\mid 0\le x_1\le x_2<1/2,x_1\le A\}, D_2 =\{(x_1,x_2,1-x_2,1-x_1)\mid 0\le x_1\le x_2<\min(1/2, A)\}, D_3 =\{(x_1,x_2,1-x_2,1-x_1)\mid 0\le x_1\le x_2<1/2,1-x_2\le A\}, D_4 =\{(x_1,x_2,1-x_2,1-x_1)\mid 0\le x_1\le x_2<1/2,1-x_1\le A\},
\|u\|^p_{L^{p,2}_{i,j}}=\int_{\mathbb{R}^3_{d_{i,j}}}\left(\int_{(\mathbb{R}^3)^{N-1}}|\mathcal{R}_{i,j}u|^2\,dD_{i,j}dx_1\cdots\widehat{dx_i}\cdots\widehat{dx_j}\cdots dx_N\right)^{p/2}d r_{i,j}.
\mathbb{E}\left[ \left\| M_{n+1}\right\|^{2p}\right] \leq \mathbb{E}\left[ \left\| M_{n}\right\|^{2p}\right] + A_{p}n^{p-1}& \leq \left\| \xi_{2}\right\|^{2p}+ A_{p}\sum_{k=2}^{n}k^{p-1}& \leq M^{2p}+ A_{p}n^{p},
\lVert \widetilde{\Phi}_n(a u_h)\widetilde{\Phi}_n(a' u_{h'}) - \widetilde{\Phi}_n(au_h\,a'u_{h'}) \rVert =\, \lVert \Phi_n(a)\,\gamma_h(\Phi_n(a')) v_{hh'}- \Phi_n(a\,\alpha_{h}(a')) v_{hh'}\rVert =\, \lVert \Phi_n(a)\,\Phi_n(\alpha_h(a')) - \Phi_n(a\,\alpha_h(a')) \rVert
\sum_{n=0}^{\infty}\frac{q^{\binom{n+1}{2}}}{\left(q;q\right)_{n}}\left(-x\right)^{n}A_{q}\left(-q^{n}\right)=\frac{\left(q^{1/2};q\right)_{\infty}}{\left(q^{1/2}x;q\right)_{\infty}}\sum_{n=0}^{\infty}\frac{q^{\left(n^{2}+n\right)/4}\prod_{k=0}^{n-1}\left(x-q^{k}\right)}{\left(q^{1/2},q^{1/4},-q^{1/4};q^{1/2}\right)_{n}}.
S_{\mathrm{D 6}}= - T_{\mathrm{K K 1 1}}\int d^{7}\hat{\xi}e^{- \hat{\phi}}\sqrt{| \mathrm{d e t}\ ( \hat{g}_{\hat{\imath}\hat{\jmath}}+{\hat{\cal F}}_{\hat{\imath}\hat{\jmath}}) |}+ \mathrm{W Z},
{}_{p+1}F_{q+1}\left[\begin{matrix}\alpha_{0},\ \vec{\alpha}\beta_{0},\ \vec{\beta}\end{matrix};c\right] =\frac{\Gamma ( \beta_{0})}{\Gamma( \alpha_{0})\Gamma( \beta_{0}-\alpha_{0})}\int_{0}^{1}t^{\alpha_{0}-1}( 1-t)^{\beta_{0}-\alpha_{0}-1}\,{}_{p}F_{q}\left[\begin{matrix}\vec{\alpha}\vec{\beta}\end{matrix};ct\right] dt.
\det \begin{bmatrix}\int_{y_N}^{\infty}\phi_{\omega}(x_{N+1})dx_{N+1}\cdots -\phi^{(N-1)}_{\omega}(y_N) &\vdots \ddots \vdots &\int_{y_1}^{y_2}\phi_{\omega}(x_2)dx_2 \cdots \phi^{(N-1)}_{\omega}(y_2)-\phi^{(N-1)}_{\omega}(y_1) &\int_{-\infty}^{y_1}\phi_{\omega}(x_1)dx_1 \cdots \phi^{(N-1)}_{\omega}(y_1)\end{bmatrix}_{(N+1) \times (N+1)}.
\overline{{{\mathrm{\Large ~ v ~}}}}_{4}( x ) ={\frac{4 \pi^{2}}{g^{2}\Lambda^{2}}}x - \left({\frac{x^{2}}{2}}-{\frac{{\cal E}^{2}}{3}}\right) E_{i}( - x ) +{\frac{1}{2}}( 1 - x ){\mathrm e}^{- x}-{\frac{1}{2}}\ ,
\sqrt{\frac{K_{s}\overline{\gamma}_{s}}{(K_{s}+1)M_{T}M_{R}}}\sum_{m=1}^{M_{R}}\underbrace{\left(\sum_{l=1}^{M_{T}}e^{j(\theta_{T,l}(k)+\phi_{s}^{l,m})}\right)}_{a_{m}}e^{j\theta_{R,m}(k)},
E[e^{uY^x_t}]= \big(1-\frac{\sigma^2}{2a}u(1-e^{-at})\big)^{-\frac{2a\theta}{\sigma^2}}\cdot \exp \Big(\frac{xue^{-at}}{1-\frac{\sigma^2}{2a}u(1-e^{-at})}\Big),
\begin{aligned}\left|T_2\right| \lesssim\begin{cases}\frac{1}{D}\int h^2 \left<\xi\right>^{\gamma+1}\varrho\, m_0, \gamma\in[-1,1], \alpha\delta \int h^2 \left<\xi\right>^{2(1+\gamma)}\varrho\, m_0, \gamma \in [-2,-1)\end{cases}& \lesssim \int h^2 \left<\xi\right>^{\gamma+1}\varrho\, m_0.\end{aligned}
\bigwedge_{i=0 \atop i \neq u_1}^{n}dw_{iu_1}\otimes P \longmapsto \bigwedge_{i=0 \atop i \neq u_2}^{n}dw_{iu_2}\cdot (w_{u_1 u_2})^{-n-1}\otimes \iota_{u_1 u_2}(P)w_{u_1 u_2}^{n+1}
\int d^{4}y \frac{\delta}{\delta{\cal O}( y )}\int d^{4}x \, \underbrace{{\cal O}( x ) \star{\cal O}( x ) \star \ldots \star{\cal O}( x )}_{n \mathrm{\; f a c t o r s}}= n \int d^{4}x \, \underbrace{{\cal O}( x ) \star{\cal O}( x ) \star \ldots \star{\cal O}( x )}_{n - 1 \mathrm{\; f a c t o r s}}.
\pi_{e,p}(r,t)=\frac{P_{0}e^{i(kz-\omega t)}}{4\pi\epsilon_{0}}\int_{0}^{\infty}dz^{\prime}e^{ik(z^{\prime}-z)}\int_{-\infty}^{\infty}dx^{\prime}\int_{-\infty}^{\infty}dy^{\prime}\frac{e^{ik_{0}|r-r^{\prime}|}}{|r-r^{\prime}|}.
\left|\sum_{\substack{D \in P D\leq Z}}(-1)^{\omega(D)}\sum_{\substack{n=1 g'(n) \equiv 0 D}}^X e^{2\pi \i g(n)\alpha}\right| \ll_k (g)^2\sum_{D \in P}k^{\omega(D)}\max_{0\leq b \leq D}\left| \sum_{n=0}^{X/D}e^{2\pi \i g(Dn+b)\alpha}\right|.
E(a^{(i)}) \geq \frac12\sum_{j=1}^k (a^{(i)}_j)^2- 2\log(x^{(i)}_{j_1}) -2 \log(x^{(i)}_{j_3}+ x^{(i)}_{j_3+1})-\ldots & -2\log(x^{(i)}_{j_m}+ \ldots+ x^{(i)}_{j_m+m-1}) .
\begin{array}{lcr}\{\{L_{m_1n_1},L_{m_2n_2}\},L_{m_3n_3}\}=(m_2n_1-m_1n_2)[m_3(n_1+n_2)-n_3(m_1+m_2)]L_{\sum\limits_{i=1,2,3}m_i,\sum\limits_{i=1,2,3}n_i},\end{array}
t_c \cdot (t_c!)^{r-1}\cdot (t_f!)^{\ell - r}\cdot \exp\left(-2i\right)\cdot (t_c-1)!\exp\left(\pm O\left(\frac{k^{2}}{\ell}\right)\right) = (t_c!)^{r}\cdot(t_f!)^{\ell - r}\cdot \exp(-2i)\cdot\exp\left(\pm O \left(\frac{k^{2}}{\ell}\right)\right).
\left[ \begin{array}{c c}{\Psi_{1}( t )}\\{\Psi_{2}( t )}\\ \end{array}\right] = \left[ \begin{array}{c c}{\cos{\theta_{v}}}&{- \sin{\theta_{v}}}\\{\sin{\theta_{v}}}&{\cos{\theta_{v}}}\\ \end{array}\right] \left[ \begin{array}{c c}{\Psi_{e}( t )}\\{\Psi_{\mu}( t )}\\ \end{array}\right] ,
f(t,m,x) \ = \ \int_0^\infty (z + x) \frac{2}{\sqrt{T-t}}\varphi\Big(\frac{z}{\sqrt{T-t}}\Big) dz &= \ 2\sqrt{T-t}\int_0^\infty z\varphi(z)dz + 2x\int_0^\infty \varphi(z)dz \ = \ \sqrt{\frac{2(T-t)}{\pi}}+ x.
\|[f'(s|D|),\delta(a)]\|_1 = 2s\|[|D|e^{-s^2D^2},\delta(a)]\|_1 \leq 2s\|\delta^2(a)e^{-s^2D^2}\|_1+2s\||D|[e^{-s^2D^2},\delta(a)]\|_1.
r^{2}\frac{\partial^{2}\psi}{\partial r^{2}}=r^{2}cos^{2}\phi\frac{\partial^{2}\psi}{\partial x^{2}}+2r^{2}sin\phi cos\phi\frac{\partial^{2}\psi}{\partial x\partial y}+r^{2}sin^{2}\phi\frac{\partial^{2}\psi}{\partial y^{2}}
\nu(\hat{A}_y) \leq \nu(\hat{A}_y|_{B(0, 4r_{t_k})}) = \lim_\gamma \nu(\hat{A}_{y_{j_\gamma}}|_{B(0, 4r_{t_k})}) &\leq \lim_\gamma \nu(\hat{A}_{x_{j_\gamma}}) + \frac{1}{2^{k-1}}= \lim_{j \to \infty}\nu(\hat{A}_{x_j}) + \frac{1}{2^{k-1}}.
u^{s}\left( x \right) = \frac{^{ik_pr}}{\sqrt{r}}u^{\infty}_p \left( \hat{x}\right) \hat{x}+ \frac{^{ik_sr}}{\sqrt{r}}u^{\infty}_s \left( \hat{x}\right) \hat{x}^{\perp}+ \mathcal{O}\left( \frac{1}{r^{\frac{3}{2}}}\right) r \to \infty.
S = \frac{k}{1 6 \pi}\int d z d \bar{z}d \theta d \bar{\theta}\left[ - D g^{s}\bar{D}g^{s \dagger}+ \int d t g^{s}\frac{\partial g^{s \dagger}}{\partial t}( D g^{s}\bar{D}g^{s \dagger}+ \bar{D}g^{s}D g^{s \dagger}\right]
\alpha_{\mu}=|E(uw)|^{2(n+k)}E(|w|^2) -(1+n)\mu^n|E(uw)|^{2(k-1)}\chi_{S_0}(E(|w|^2))^2E(|u|^2) &+n\mu^{n+1}|E(uw)|^{2(k-1)}\chi_{S_0}E(|w|^2).
\begin{cases}\lim_{t\to 0}\mathbf{F}(t,t)=1, \lim_{t\to 0}\mathbf{F}(s,t)=0, \end{cases}\mathbf{F}(\cdot,t)\ \text{is decreasing, for every fixed $t>0$;}\ \mathbf{F}(s,t)\ \text{is uniformly continuous on $s\geq t\geq \tau$.}
\delta_{\xi}\left< T_{z \overline{{z}}}\left( z , \overline{{z}}\right) \right> ={\frac{1}{9 6}}\left( 2 \, \partial_{z}^{\, 2}\partial^{z}\xi^{z}\left( z , \overline{{z}}\right) + 2 \, \partial_{\overline{{z}}}^{\, 2}\partial^{\overline{{z}}}\xi^{\overline{{z}}}\left( z , \overline{{z}}\right) \right) \; .
\sum_{l\in\mathcal{S}^{\left(i\right)}}q_{ml}\mathbf{w}_{l,j}=\psi^{-\left\langle j\right\rangle}\left(\left[\sum_{l\in\mathcal{S}^{\left\langle j\right\rangle}}a_{ml}\mathbf{t}_{l,j}\right]\bmod\Lambda_{A,j+1}\right).
\sup_{t\in[B^2_{m(n)},B^2_n]}\left|\frac{U_g(t)}{U_g(B_n^2)}-1\right|=\max_{m(n)\leq k\leq n}\left|\frac{\mathbf{E}Z^*_{k}}{\mathbf{E}Z^*_{n}}-1\right|\le\frac{\alpha_{m,n}^*}{\mathbf{E}Z^*_{n}}\le 40(\rho_n^{2/3}+\lambda_n^{3/4}) \to 0.
f(\rho) := J_\mu(\bar{a}^{1/2}B_{\rho},\bar{a}) = \lambda_1(B_\rho,\textup{id}) + \mu \ \textup{det}(\bar{a})^{1/2}|B_1| \rho^d = \rho^{-2}\lambda_1(B_1,\textup{id})+ \mu \ \textup{det}(\bar{a})^{1/2}|B_1| \rho^d