harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
S_{i}= - \frac{k}{2 \pi}\int_{\cal M}d^{2}x \left[ \frac{1}{2 y_{( i )}^{2}}\partial_{\mu}y_{( i )}\partial^{\mu}y_{( i )}+ \frac{1}{4 y_{( i )}^{2}}f_{( i ) \mu \nu}f_{( i )}^{\mu \nu}\right] + ( S U ( 2 ) \mathrm{~ W Z W ~ w i t h ~ l e v e l ~}k )_{i}
M^{\otimes_{w^*h}\eta}= [L^2(M)_r\otimes_{h M'}(\overline{(.\#S^{(1)})_*[L^2(M,\eta_1)_{L^2(M)}]}^{\ell^\infty(I_1)\overline{\otimes}B(L^2(M))}\cdots \otimes_{h M'}L^2(M)_c ]^{* M' normal}&\cap [L^2(M)_r\otimes_{h M'}(\overline{(.\#S^{(1)})_*[{}_{L^2(M)}L^2(M,\eta_1)]}^{\ell^\infty(I_1)\overline{\otimes}B(L^2(M))}\cdots \otimes_{h M'}L^2(M)_c ]^{* M' normal}
\int_{Q_0}u\cdot\varphi=\int_{Q_0}u\cdot{\rm curl}\,q+\int_{Q_0}({\rm curl}\,p+\nabla\psi)\cdot\nabla\rho=\int_{Q_0}{\rm curl}\,u\cdot q\ \ \ \forall\varphi\in[H^1(Q_0)]^3.
\delta =\begin{cases}1-2\alpha+\alpha\theta+\alpha\theta{\hbox{if}}\alpha\leq\tfrac{1}{2-\theta}-(1-2\alpha+\alpha\theta)+\alpha\theta{\hbox{if}}\alpha\geq\tfrac{1}{2-\theta}\end{cases}=\begin{cases}1-2(1-\theta)\alpha{\hbox{if}}\alpha\leq\tfrac{1}{2-\theta}2\alpha-1{\hbox{if}}\alpha\geq\tfrac{1}{2-\theta}\end{cases}
g_2(t) = \int_0^t e^{is\Lambda^{1/2}}L(w)\, ds, g_3(t)= \int_0^t e^{is\Lambda^{1/2}}N_1(u,w)\, ds, g_4(t) = \int_0^t e^{is\Lambda^{1/2}}N_2(w,w)\, ds, g_5(t)= \int_0^t e^{is\Lambda^{1/2}}N_3(u,w)\, ds,
G ( z , \bar{z}) = \lim_{R \rightarrow \infty}\mid{R}\mid^{4 \tilde{\bigtriangleup}_{4}}\langle \Phi_{H . W .}^{( \lambda_{1}, \mu_{1})}( 0 , 0 ) \Phi_{H . W .}^{( \lambda_{2}, \mu_{2})}( z , \bar{z}) \Phi_{T_{z}^{3}, Y^{3}}^{( \lambda_{3}, \mu_{3})}( 1 , 1 ) \tilde{\Phi}_{L . W .}^{( \lambda_{4}, \mu_{4})}( R , R ) \rangle =
\begin{array}{c c c}{{\pi}}&{{:}}&{{\Omega^{*}({\cal A}) \longrightarrow{\cal L}({\cal H})}}\\{{\pi_{p}( ( D F_{1}) . . . ( D F_{p}) F )}}&{{=}}&{{\prod_{i = 1}^{p}[ D , \pi_{0}( F_{i}) ] \pi_{0}( F ) ,}}\\ \end{array}
\;\partial_t (1-\alpha^2\Delta)u + u\cdot \nabla (1-\alpha^2\Delta)u - \alpha^2\nabla u^T\cdot \Delta u +\nabla p = -\nu (1-\alpha^2\Delta)A^s u, &\; \mathrm{div}\,u = 0,u|_{t = 0}= u_0.
\begin{bmatrix}y_1 y_2 \vdots y_M\end{bmatrix}=\begin{bmatrix}h_{11}h_{12}h_{21}h_{22}\vdots \vdots h_{M1}h_{M2}\end{bmatrix}\begin{bmatrix}x_1 x_2\end{bmatrix}+\begin{bmatrix}n_1 n_2 \vdots n_M\end{bmatrix}
\widetilde{\varphi}_b^{\sigma}\left(\frac{N}{b^j}\right)= \widetilde{\varphi}_b^{\sigma}\left(\left\{\frac{N}{b^j}\right\}\right)=\widetilde{\varphi}_b^{\sigma}\left(\left\{N_{n-1}b^{n-j-1}+\dots+N_j+N_{j-1}b^{-1}+\dots+N_0b^{-j}\right\}\right) = \widetilde{\varphi}_b^{\sigma}\left(N_{j-1}b^{-1}+\dots+N_0b^{-j}\right)
b^{(n)}_\ell(x_\ell)\quad=\begin{cases}0 |x_\ell|<k^{(n)}_\ell\,, \frac{x_\ell-k^{(n)}_\ell}{2k^{(n)}_\ell-x_\ell}k^{(n)}_\ell\leq|x_\ell|\leq 2k^{(n)}_\ell\,, \infty 2k^{(n)}_\ell<|x_\ell|\,,\end{cases}
\mathbf{U}_t(\vartheta)=\mathbf{Y}_0^t(\vartheta)+\sum_{z\in\vartheta^t}\mathbf{Y}^t_z(\vartheta) \boldsymbol{\upsilon}^\star_\nu \mathbf{L}(z) \boldsymbol{\upsilon}_\nu \mathbf{U}_z(\vartheta),
B=\frac{1}{e}Q=-\frac{i}{e}\int_{-\infty}^{+\infty}dx~J^{EM}_2(x,0)=\frac{1}{\sqrt{\pi}}\int_{-\infty}^{+\infty}dx\sum_i\chi'_i=\frac{1}{\sqrt{\pi}}\sum_i[\chi_i(\infty,0)-\chi_i(-\infty,0)].
\begin{array}{rcl}\left[M_{ab},M_{cd}\right] = -\eta_{ac}M_{bd}-\eta_{bd}M_{ac}+\eta_{ad}M_{bc}+\eta_{bc}M_{ad}\, , & \left[P_{c},M_{ab}\right] = -2P_{[a}\eta_{b]c}\, ,\hspace{1.5cm}\left[P_{a},P_{b}\right] = -g^{2}M_{ab}\, . \end{array}
Z = \int D A ~ D B D a ~ e^{- S^{\prime}[ A , B ]}d e t ( i \raise . 1 5 e x \mathrm{~ / ~}\kern - . 5 7 e m \mathrm{~ \partial ~}+ \not \! \! A ) d e t ( i \gamma_{0}\partial_{t}+ a \gamma_{0}+ \not \! \! B ) .
\sum_{\sigma \in S_n}\prod_{i=1}^n x_{\sigma(i)}^{a_i}= \sum_{\sigma \in S_n}\prod_{i=1}^n x_{\sigma(i)}^{b_i+c_i}= \sum_{\sigma \in S_n}\prod_{i=1}^n x_{\sigma(i)}^{b_i}\prod_{i=1}^n x_{\sigma(i)}^{c_i}.
\epsilon(x_{0})=\hat{Z}(x_{0})-Z(x_{0})=[\begin{matrix}W^{T}&-1\end{matrix}]\cdot[\begin{matrix}Z(x_{1})&\cdot\cdot\cdot&Z(x_{N})&Z(x_{0})\end{matrix}]^{T}=\sum_{i=1}^{N}w_{i}(x_{0})\times Z(x_{i})-Z(x_{0}).
1 = \frac{\partial q_j}{\partial q_j}=- \frac{\partial^2 F}{\partial p_j \, \partial Q_j}\frac{\partial Q_j}{\partial q_j}- \frac{1}{2}\frac{\partial^3 F}{\partial{p_j}^2 \,\partial Q_j}\frac{\partial Q_j}{\partial q_j}\Delta p_j \; .
\bigcap_{\substack{N_l\leq j_l\leq J_{n,l}-1 0\leq k_l\leq2^{j_l}-1 l=1,\dotsc,d}}\left\{\frac{|\beta_n(\widetilde{\Theta})-\widetilde{\beta}_n(\widetilde{\Theta})|}{\sqrt{\sigma_0^2(\boldsymbol{\Psi}_{\boldsymbol{j}}^T\boldsymbol{\Psi}_{\boldsymbol{j}})_{\boldsymbol{k},\boldsymbol{k}}}}\leq\left(2\log{\prod_{l=1}^d2^{j_l}}+c\log{n}\right)^{1/2}\right\}.
dX_t = \left( X_t - \left| \left( \begin{array}{c}X_t Y_t \end{array}\right) \right|^{2}X_t \right) dt + \sigma \, dW_t \textrm{on}\; \mathbb{R}^n dY_t = \left( Y_t \; - \left| \left( \begin{array}{c}X_t Y_t \end{array}\right) \right|^{2}Y_t \; \right) dt \textrm{on}\; \mathbb{R}^{d-n}
A \leq \frac{1}{N^2}\sum_{k =1}^{\infty}\frac{2}{\pi^2 k^2}\left| \sum_{n=1}^{N}{e^{2 \pi i k x_n}}\right|^4 +1 = \frac{1}{N^2}\sum_{k=1}^{\infty}\frac{2}{\pi^2 k^2}\left| \sum_{m, n=1}^{N}{e^{2 \pi i k (x_n-x_m)}}\right|^2 +1.
R\left(f(z,1),f_t(z,1),f\left(z,\frac{1}{1-z}\right), z, t_1(z)\right) = 0, R\left(f(z,1),f_t(z,1),f\left(z,\frac{1}{1-z}\right), z, t_2(z)\right) = 0, R\left(f(z,1),f_t(z,1),f\left(z,\frac{1}{1-z}\right), z, t_3(z)\right) = 0.
\varpi_k = 1 + \gamma_k, \upsilon_k = \frac{\boldsymbol{\theta}_i^T \mathrm{diag}(\mathbf{h}_k^H) \mathbf{G}\mathbf{w}_k}{\sum_{\ell \in \mathcal{K}}|\boldsymbol{\theta}_i^T \mathrm{diag}(\mathbf{h}_k^H) \mathbf{G}\mathbf{w}_\ell|^2 + \sigma_k^2}.
\Gamma_3=\frac{\lambda^3}{8\left(2\pi\right)^5}\int \frac{{\rm d}^5 Q}{|{\bf Q}|}N\left(|{\bf Q}|\right)\left\{\frac{k_1^2\,k_2\cdot k_3}{\left(Q\cdot k_1\right)^2 Q\cdot k_2 Q\cdot k_3}+permutations\right\},
\varphi_{2}\left( n\right) :=2\,B_{1}C_{3}\,c\,P_{n}^{\beta-\kappa\left(\beta-\mu\right)}+C_{1}\left( P_{n}^{\sigma-\delta/\kappa-\kappa\left(\beta-\mu\right)}g^{\delta/\kappa}+P_{n}^{(\sigma-\delta)_{+}-(\delta-\sigma)_{+}/\kappa-\kappa\left( \beta-\mu\right)}\right)
\Bigg(\bigoplus_{i\in I}\varphi_i\Bigg)^{\sim}\big(\eta,(y_i)_{i\in I}\big)=\eta\Bigg(\bigoplus_{i\in I}\varphi_i\bigg)(y_i/\eta)_{i\in I}=\sum_{i\in I}\eta\varphi_i(y_i/\eta)=\Bigg(\bigoplus_{i\in I}\widetilde{\varphi_i}\Bigg)\Big((\eta,y_i)\Big)_{i\in I}.
( F_{\mu \nu}) = \left( \begin{array}{c c c}{{0}}&{{B}}&{{E_{1}}}\\{{- B}}&{{0}}&{{E_{2}}}\\{{- E_{1}}}&{{- E_{2}}}&{{0}}\\ \end{array}\right) = i ( E_{2}T_{1}- E_{1}T_{2}+ B T_{3}) \ ,
\sigma^{2}\sqrt{\gamma^{n - 2}}( \sqrt{\gamma})^{\prime}\big |_{0}^{\epsilon}= - \frac{1}{( n + 2 )}\frac{1}{M_{D}^{n + 2}}\bigg ( 4 \mu_{0}+ \mu_{\rho}- 3 \mu_{\theta}\bigg ) ~ ,
\sqrt{2\pi (1-\rho^2)}\int_{0}^{\infty}\sigma^{-(m+h+1)}\exp \left( -\left(\frac{m \hat{\sigma}^2+(\theta-\hat{\theta})^2}{2}\right){\sigma}^{-2}\right) d\sigma^2 =\sqrt{2\pi (1-\rho^2)}\, \Gamma\left(\frac{m+h-1}{2}\right) \left(\frac{m \hat{\sigma}^2+(\theta-\hat{\theta})^2}{2}\right)^{-(m+h-1)/2}
\displaystyle\frac{1}{\mu -\hat{x}}=\displaystyle\frac{1}{\nu}=\displaystyle\frac{(-1)^{\delta}e^{iZ_{N}(\hat{x}+\nu )}iZ_{N}'(\hat{x}+\nu )}{1+(-1)^{\delta}e^{iZ_{N}(\hat{x}+\nu )}}+...
\begin{pmatrix}Y_1 Y_2 \end{pmatrix}= Y = L|_{\hat t\times\hat t}X = \begin{pmatrix}L_{11}L_{21}L_{22}\end{pmatrix}\begin{pmatrix}X_1 X_2 \end{pmatrix}= \begin{pmatrix}L_{11}X_1 L_{21}X_1 + L_{22}X_2 \end{pmatrix}.
\{L_{1}( u ) , L_{2}( v ) \}= [ r ( u - v ) , L_{1}( u ) + L_{2}( v ) ] , \quad r ( u ) ={\frac{1}{u}}\left( \begin{array}{l l l l}{{1}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{1}}&{{0}}\\{{0}}&{{1}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{1}}\\ \end{array}\right) ,
\frac12 \int\limits_{\mathbb{R}^d}v_\phi^2 (x,s) \, v_0^\varepsilon(x) \, dx \ - \ \int\limits_0^s \int\limits_{\mathbb{R}^d}\phi (x,t) \, v_\phi(x,t) \, v_0^\varepsilon(x) \, dx \, dt \ = \ \int\limits_0^s (L^\varepsilon v_\phi, v_\phi)_{v_0}\, dt \ \le \ 0.
|M_\xi^2 (0)| - |{\cal M}_\xi^2 (X_0)| = \lambda e_\xi \vec{\varphi}^{\, 2}(X_0) < < |M_\xi^2 (0)| \;\;\;\;\; (e_\pi = 1 / 6, \, e_\sigma = 1 / 2) ,
\prod_{\substack{m,n \geq 1 m \leq n ; \, (m,n)=1}}\left( \frac{1}{1- y^m z^n}\right)^{\frac{m^1}{n^2}}= \left(1-yz\right)^{\frac{y}{1-y}}\exp\left\{\frac{y}{(1-y)^2}\left(Li_2(z)- Li_2(yz)\right) \right\},
\sigma_{\hspace{. 3 t r u e c m}\alpha \beta}^{\mu}= \frac{1}{\sqrt{- g}}( e_{\alpha}^{a}e_{\beta}^{b}- e_{\beta}^{a}e_{\alpha}^{b}) \frac{\delta}{\delta \Omega_{\hspace{. 2 t r u e c m}\mu}^{a b}}( \sqrt{- g}I_{m}) .
d x^{i}d x^{j}\wedge{}^{*}( d x^{0}d x^{l}d x^{m}) = x_{0}^{- ( d - 5 )}\delta_{l m}^{i j}d x^{1}. . . \, d x^{d}, \delta_{l m}^{i j}= \delta_{l}^{i}\delta_{m}^{j}- \delta_{m}^{i}\delta_{l}^{j},
g(x) :=c_4\left(\frac{\alpha^{-x(d+1)}}{1-\alpha^{-(d+1)}}+\frac{\exp\bigl(-\alpha^{x-2}\bigr)}{1-\exp\bigl(-\alpha^{x-2}\bigr)}+\exp\bigl(-\alpha^{x-2}\bigr)\right) &\,+2\left(\exp\bigl(-b\alpha^x\bigr)+\frac{\exp\bigl(-b\alpha^x\bigr)}{1-\exp\bigl(-b\alpha^x\bigr)}\right)
\frac{1}{C_1}\sum_{j=0}^{n-1}\mu(\Delta_{0,j}\setminus\Delta_{0,j+1})\log\frac{1}{|\Delta_{0,j}|}\leq\int_{I_0}\!\psi_n\,d\mu\leq C_1\sum_{j=0}^{n-1}\mu(\Delta_{0,j}\setminus\Delta_{0,j+1})\log\frac{1}{|\Delta_{0,j+1}|}.
f\left(\boldsymbol{P}^\prime\right) =\ln\left(1+\frac{P_m^\prime}{\sigma^2+\sum_{j\neq m, n}^KP_j^\prime+P_n^\prime}\right)+\ln\left(1+\frac{P_n^\prime}{\sigma^2+\sum_{j\neq m, n}^KP_j^\prime+P_m^\prime}\right) &+\sum_{i=1}^K\ln\left(1+\frac{P_i^\prime}{\sigma^2+\sum_{j\neq i, m, n}^K P_j^\prime+P_m^\prime+P_n^\prime}\right).
\begin{aligned}r\int_{\partial B_r}\left(\frac{\left|\nabla u\right|^2}{2}-(\nu\cdot\nabla u)^2\right)\mbox{d}\sigma &=\frac{r}{a}\int_{\partial B_r}e^{au}\mbox{d}\sigma +r^{2N+1}\int_{\partial B_r}e^u\mbox{d}\sigma-\frac{2}{a}\int_{B_r}e^{au}-2(N+1)\int_{B_r}\left|x\right|^{2N}e^{u}\end{aligned}
\mathbf{l}_k (\mathbf{x}) = \mathbf{l}^0_k + b(\mathbf{D}) \mathbf{w}_k(\mathbf{x}), \mathbf{l}^0_k \in \mathbb{C}^m, \mathbf{w}_k \in \widetilde{H}^1 (\Omega; \mathbb{C}^n), k = 1, \ldots,m,
\int g^2_z (|p|) d^2p \leq (2\pi)\delta \int_0^{r_\varepsilon}\frac{tdt}{(t^2+z^2)|\ln (t^2 + z^2)|}= \pi \delta \left[ \bigl|\ln|\ln z^2|\bigr| - \bigl|\ln|\ln (z^2 + r_\varepsilon^2)|\bigr| \right]
q_{\pm, j}(\theta_j^0) = Q_{\theta_0}(Q_j(-\theta_j^0)\geq e^{\pm 2\alpha \theta_j^0}) = \Phi\big(-D_j^{1/2}\mp \alpha \theta_j^0 D_j^{-1/2}\big)= \Phi(-\alpha \mp \theta_j^0)+ O\Big(\frac{\alpha^2}{nF_j}\Big),
\frac{d N(t)}{dt}= \int_0^{x_{}}\int_u^{x_{}}\frac{u}{x}\beta(u,x-u)n(t,u)n(t,x-u) dx du - \int_0^{x_{}}\int_0^{x_{}}\beta(x,v)n(t,x)n(t,v) dv dx &+ \int_{0}^{x_{}}\int_{0}^{v}b(x,v) S(v) n(t,v) dx dv- \int_{0}^{x_{}}\int_0^x \frac{u}{x}b(u,x) S(x) n(t,x) du dx.
\begin{aligned}u(t) = \zeta E_{\alpha}\big[-\gamma t^{\alpha}\big]u(0) &+ \frac{\alpha c_{0}\zeta}{B(\alpha)}\int_{0}^{t}\bigg( E_{\alpha,\alpha}\big[-\gamma(t-s)^{\alpha}\big] + \frac{(1-\alpha)}{\alpha}\gamma^{-2\alpha}E_{\alpha,\alpha}\big[-\gamma^{-2\alpha}(t-s)^{\alpha}\big]\bigg)(t-s)^{\alpha-1}h(s)ds .\end{aligned}
\rho^{\square}(\gamma)=\begin{pmatrix}1+t_{\gamma}+x_{11}x_{12}x_{21}1+t_{\gamma}-x_{11}\end{pmatrix}, \rho^{\square}(\delta)=\begin{pmatrix}1+t_{\delta}+y_{11}y_{12}y_{21}1+t_{\delta}-y_{11}\end{pmatrix}.
={\displaystyle\sum\limits_{i=1}^{2s}}C_{i}\Delta_{h,x_{N}}^{i}\left[ \left( x_{N}^{0}+h_{\theta}\right)^{n-j}\right] \Delta_{h,x_{N}}^{2s-i}D_{x}^{\alpha}v(x_{0}^{\prime},x_{N}^{0}+h_{\theta},t_{0})+
U_{5}=(\begin{matrix}0&1&0&0&0\\ 0&0&1&0&0\\ 0&0&0&1&0\\ 0&0&0&0&1\\ 0&0&0&0&0\end{matrix})L_{5}=(\begin{matrix}0&0&0&0&0\\ 1&0&0&0&0\\ 0&1&0&0&0\\ 0&0&1&0&0\\ 0&0&0&1&0\end{matrix}).
\lambda_{1,2}(\sigma) = \frac{2 \int_0^1 | \phi_\sigma' |^2 (1-t)^{n-\frac{1}{2}}\,dt}{\int_0^1 | \phi_\sigma |^2 (1-t)^{n-\frac{3}{2}}\,dt}= \frac{1}{2}\cdot j_{n-\frac{3}{2},1}^2
G_t :=cP_t e^{|\rho|_t}\Big(F_t^\theta -\frac{ce^{|\rho|_t}}{2}|Z_t^\theta|^2\Big), H_t := \int_0^t cP_s e^{|\rho|_s}\Big(g_s d\langle N \rangle_s -\theta g^\prime_s d\langle N^\prime \rangle_s - \frac{ce^{|\rho|_s}}{2}d\langle N^\theta \rangle_s \Big).
|Q_k^t|\lesssim t^n,1\!{\rm l}_{\Omega}\le \sum_k1\!{\rm l}_{Q_k^t}\le N\,1\!{\rm l}_{\Omega}, &\sup_j\sum_ke^{-\varepsilon\,{\rm dist}\,(Q_k^t,Q_j^t)/t}=\sup_k\sum_je^{-\varepsilon\,{\rm dist}\,(Q_k^t,Q_j^t)/t}\le C_\varepsilon
\begin{array}{l}\epsilon(k_i)=1~,~~~~ \epsilon(X^-_i)=0=\epsilon(X^+_i)~, S(k_i)=k_i^{-1}~,~~~S(X^-_i)=-X^-_ik_i^{-1}~,~~~S(X^+_i) = -k_iX^+_i~.\end{array}
\begin{array}{rl}p(x_a,x_b,x_c) = \displaystyle \frac{(x_a+1)(x_b+1)(x_c+1)-(x_a-1)(x_b-1)(x_c-1)}{8}&= \displaystyle \frac{x_ax_b+x_ax_c+x_bx_c+1}{4},\end{array}
D_{\mu \nu}= \frac{1}{p^{2}+ M^{2}}( \epsilon_{\mu \nu \lambda}p_{\lambda}+ \frac{p_{\mu}p_{\nu}}{M}+ M \delta_{\mu \nu}) \; ,{}~ D_{\mu \nu}^{- 1}= - \epsilon_{\mu \nu \lambda}p_{\lambda}+ M \delta_{\mu \nu}\; .
Q_k(x) = (\delta_k x + \theta_k) Q_{k-1}(x) + \gamma_k \beta_{k-1}Q_{k-2}(x) + \gamma_k \alpha_{k-1}\beta_{k-2}Q_{k-3}(x) + \gamma_k \alpha_{k-1}\alpha_{k-2}\beta_{k-3}Q_{k-4}(x) + \ldots + \gamma_k \alpha_{k-1}\cdots \alpha_2 \beta_1 Q_0(x).
[\Omega_\lambda]=\pi_*([Y_{\lambda}])=\pi_{1*}\cdots \pi_{d*}(\alpha_1\cdots\alpha_d)=\phi_1\left( t_1^{\lambda_1}\cdots t_{d}^{\lambda_{d}}{\prod_{1\leq i<j\leq d}( 1-\bar t_i/\bar t_{j})}\right).
\frac{\prod\limits_{r=1}^m{\prod\limits_{j=\theta(N - m+1)}^{\theta(N-r+1)-1}{\big(z_r - q^{j-\theta}\big)}}}{\prod\limits_{\substack{1\leq i < j\leq m 0\leq k < \theta}}{(z_i - q^k z_j)}}.
\sum_{|\lambda/\mu|=n}f_{\lambda/\mu}\frac{L_{1}(\lambda)}{H_\lambda}=\frac{L_{1}(\mu)}{H(\mu)}+n D\Bigl(\frac{L_{1}(\mu)}{H(\mu)}\Bigr)+\binom{n}{2}D^2\Bigl(\frac{L_{1}(\mu)}{H(\mu)}\Bigr) &=\frac{L_{1}(\mu)}{H(\mu)}+n \frac{|\mu|}{H(\mu)}+\binom{n}{2}\frac{1}{H(\mu)},
P_{2n}=\sum_{k=1}^{2n}\sum_{j=\lceil\frac k2\rceil}^{n}{2j-1\choose k-1}{n+j-1\choose n-j}, P_{2n-1}=\sum_{k=1}^{2n-1}\sum_{j=\lceil\frac{k+1}{2}\rceil}^{n}{2j-2\choose k-1}{n+j-2\choose n-j}.
u(x,t)= ~\int \limits_{\mathbb{H}^{n}}\mathcal{K}(x,y,t) \, u_{0}(y)~{\rm d}v_{\mathbb{H}^{n}}(y) & +\int \limits_{0}^{t}\int \limits_{\mathbb{H}^{n}}\mathcal{K}(x,y,t-s)e^{\mu s}u(y,s)\big( e^{\beta u^{p}(y,s)}-1\big)~{\rm d}v_{\mathbb{H}^{n}}(y) \,{\rm d}s.
\Omega(p)=\frac{4\Lambda^2c^2}{\pi}\left[2\lambda_0+8\pi\frac{\epsilon}{\Lambda^2}\partial_1^2I_1(0)\right]e^{-8\pi I_1(0)}+\frac{4c^2\epsilon}{\pi}e^{-8\pi I_1(0)}p_1^2
{\cal O}^{[m,0,0]}_{1/2}= [W^{1}(\theta_{2,3,4}, \bar\theta^{1})]^m\quad \mbox{and}\quad{\cal O}^{[0,0,m]}_{1/2}= [\bar W_4(\theta_{4}, \bar\theta^{1,2,3})]^m\,.
\biggl(\frac{1+4\mu}{.96d}\biggr)^{h}\leq \exp\Biggl[ \biggl(-\log\Bigl(\frac{d}{1+4\mu}\Bigr) + \log\Bigl(\frac{25}{24}\Bigr)\biggr) \frac{C\bigl(j + \log(1+\mu)\bigr)}{\log \bigl(\frac{d}{1+4\mu}\bigr)}\Biggr] \leq \exp\biggl[ -.9C\bigl(j + \log(1+\mu)\bigr)\biggr]=e^{-.9Cj}(1+\mu)^{-.9C}.
\log g_m(r)-\log g_m(\sqrt{m-1}) =\int_r^{\sqrt{m-1}}\int_{t}^{\sqrt{m-1}}\left( \frac{d^2}{ds^2}\log g_m(s) \right) ds dt &\leq \int_r^{\sqrt{m-1}}\int_{t}^{\sqrt{m-1}}(-1) ds dt =-\frac12 (\sqrt{m-1}-r)^2,
\lambda(\mathcal{A},\mathcal{B}) := \sum_{i=0}^{\dim_{\mathbb{R}}X(\mathcal{A},\mathcal{B})}(-1)^i Tr ((\psi_{\mathcal{A},\mathcal{B}})_i (\phi_{\mathcal{A},\mathcal{B}})^i ).
\mu_0(\alpha_t,\gamma_t)\longmapsto\mu_0(\alpha,\gamma)=\frac{H(\gamma)-H(\alpha)}{\gamma-\alpha}\equiv{\mu}_0,~~~\mu_1(\beta_t,\delta_t)\longmapsto\mu_1(\beta,\delta)=\frac{H(\beta)-H(\delta)}{\beta-\delta}\equiv\mu_1,~\forall{t}.
\begin{aligned}\Vert \chi_k \langle t \rangle^{{1 \over 2}+ \delta}\bar \partial (r \partial^\gamma \phi) \Vert_{L_u^2 L_v^\infty L_\omega^\infty}\le C \langle \tau_k \rangle^{10 \delta}\epsilon^{{3 \over 4}}. \end{aligned}
\dim_{\mathbb R}\Delta(Z_{\mathbf i}, \eta_{{\bf i}, \mathcal I}^{\ast}\mathcal L,v) = \dim_{\mathbb R}\Delta(Z_{\mathcal I}, \mathcal L, v) = \dim_{\mathbb C}Z_{\mathcal I}= \dim_{\mathbb C}Z_{\mathbf i}
\langle{\cal C}^{11}(x_{1}){\cal C}^\dagger_{11}(x_{3}){\cal K}_{{\bf 1}}(x_{2}) \rangle_{_{(0)}}={N^{2}-1 \over 3 (2\pi)^{6}(x_{13})^{2}(x_{12})^{2}(x_{23})^{2}}\; .
{}_{\mu ,\sigma \ast}D_{v,q;z}^{\alpha ,\eta ,p}(a^{(z-\xi )^{r}}):=e^{\frac{\ln a(-\xi )^{r}B(\eta ,\alpha -1)z^{\eta +\alpha}}{\Gamma (\alpha )}F_{v,q;p}^{(\mu ,\sigma )}(-r,\eta ;\eta +\alpha -1;\frac{z}{\xi}).}.
\delta x^{\mu}= \dot{x}^{\mu}\, \xi \, , \; \; \delta e = \frac{d}{d \tau}( e \xi ) \, , \; \; \delta \psi^{n}= \dot{\psi}^{n}\xi \, , \; \; \delta \chi = \frac{d}{d \tau}( \chi \xi ) \, ,
\mathcal{B}_\eta\left(\hat{D}_A(\mathbf{x})\,\hat{D}_B(\mathbf{y})\,\hat{\rho}_{AB}\,{\hat{D}_A(\mathbf{x})}^\dag\,{\hat{D}_B(\mathbf{y})}^\dag\right) &= \hat{D}_C\left(\sqrt{\eta}\,\mathbf{x}+ \sqrt{1-\eta}\,\mathbf{y}\right)\,\mathcal{B}_\eta(\hat{\rho}_{AB})\,{\hat{D}_C\left(\sqrt{\eta}\,\mathbf{x}+ \sqrt{1-\eta}\,\mathbf{y}\right)}^\dag
\int_{0}^{\pi}e^{ig(t)}{\rm d}t=\int_{0}^{\varphi_+-\delta}e^{ig(t)}{\rm d}t+\int_{\varphi_+-\delta}^{\varphi_++\delta}e^{ig(t)}{\rm d}t+\int_{\varphi_++\delta}^{\pi}e^{ig(t)}{\rm d}t
u^{i}=\sum_{k}\langle u^{i},\ \Phi_{N_{0},k}\rangle\Phi_{N_{0},k}+\sum_{N\geq N_{0},}\sum_{k}\langle u^{i},\ \Psi_{N,k}\rangle\Psi_{N,k}\left\Vert u\right\Vert_{L^{2}}^{2}=\sum_{i=1}^{5}\left( \sum_{k}\langle u^{i},\ \Phi_{N_{0},k}\rangle^{2}+\sum_{n\geq N_{0},}\sum_{k}\langle u^{i},\ \Psi_{n,k}\rangle^{2}\right)
\left( -{\mbox{exp}}( - \frac{\epsilon P_0}{2}) \gamma_{i}P_{i}+\gamma_{4}\frac{1}{\epsilon}\sinh (\epsilon P_{0}) - \frac{i\epsilon}{2}\gamma_{4}P_{i}P_{i}\right) \psi = m (1 + \frac{\epsilon^{2}m^{2}}{4})^{\frac{1}{2}}\psi,
\lim_{n \to \infty}\lim_{t \to t_0+}\frac{\eta_n(1-\sigma-it)}{\eta_n(\sigma+it)}=\lim_{n \to \infty}\frac{R_n(1-\sigma-it)}{R_n(\sigma+it)}&=\lim_{n \to \infty}\left [\frac{\dfrac{(-1)^n}{2(n+0.5)^{1-\sigma-it}}}{\dfrac{(-1)^n}{2(n+0.5)^{\sigma+it}}}\right] &=\lim_{n \to \infty}(n+0.5)^{2\sigma-1+2it}
X\in L^\infty([0,t];C^s),~\omega\in L^\infty([0,t]; H^s),~\partial_X\omega\in L^\infty([0,t];L^p). \theta\in L^\infty([0,t];H^{1+s}) , ~\partial_1\theta\in L^2([0,t];H^{1+s}).
F_{1}=x^{2}+wx+\frac{1}{2}w^{2}+\frac{1}{2}c-\frac{1}{2}\cdot\frac{c^{2}w}{d}-\frac{1}{2}\cdot\frac{w^{5}}{d}-\frac{cw^{3}}{d}+2\frac{ew}{d}
P = \begin{bmatrix}p_{0, 0}p_{0, 1}\dots p_{0, r - 1}p_{1, 0}p_{1, 1}\dots p_{1, r -1}\vdots \vdots \vdots \vdots p_{r - s - 1, 0}p_{r - s - 1, 1}\dots p_{r - s - 1, r - 1}\end{bmatrix}.
\mathcal{R}_{p,\alpha}(g_k,g_k)=\frac{|\langle g_k,g_k\rangle_\alpha|}{\|g_k\|_{p, \alpha}\|g_k\|_{p'\!, \alpha}}=\frac{\|g_k\|^2_{2,\alpha}}{\|g_k\|_{p, \alpha}\|g_k\|_{p'\!, \alpha}}&=\left(\frac{pp'}{4}\right)^{k/2}\frac{\Gamma(k+1)}{\Gamma(kp/2+1)^{1/p}\Gamma(kp'/2+1)^{1/p'}}.
( - )^{{\bar{\alpha}}+{\bar{\beta}}+ \bar{\alpha}\bar{\beta}}\; \frac{{\bar{\vartheta}}[_{\bar{\beta}}^{\bar{\alpha}}]}{{\bar{\eta}}}\rightarrow \frac{{\bar{\vartheta}[_{\bar{\beta}}^{\bar{\alpha}}]}^{1 3}}{{\bar{\eta}}^{1 3}}.
\begin{gathered}\sum_{j=0}^{(k-1)/2}(-1)^{j}U_{1,0}(2ja+t)=\sum_{j=1}^{(k-1)/2}(-1)^{j}U_{1,0}(2ja-t)\;(0,a), \sum_{j=0}^{(k-1)/2}(-1)^{j}V_{1,0}(2ja+t)+\sum_{j=1}^{(k-1)/2}(-1)^{j}V_{1,0}(2ja-t)=0\;(0,a).\end{gathered}
\tau_{\mathbf{x}}(pg_{1})(-\mathbf{y}) =\tau_{\mathbf{x}}\left(\sum_{\ell=0}^{d}\sum_{\|\alpha\| \leq \ell}c_{\ell,\alpha}T^{\alpha}(g_{1+\ell})(-\mathbf{y})\right)=\sum_{\ell=0}^{d}\sum_{\|\alpha\| \leq \ell}c_{\ell,\alpha}T^{\alpha}\tau_{\mathbf{x}}(g_{1+\ell})(-\mathbf{y}),
\frac{( - 1 )^{p}}{\prod_{i = 0}^{q}( w_{i}- v )^{\beta - 1}\prod_{i = 1}^{q - 1}( \xi_{i}- v )}\frac{d^{p}}{d v^{p}}\left[ \prod_{i = 0}^{q}( w_{i}- v )^{\beta - 1}\prod_{i = 1}^{q - 1}( \xi_{i}- v ) \right] .
{\cal L}_{\rm GF}=-\frac{1}{2\alpha}\displaystyle\sum_{a=1}^2 (\partial^\mu A_\mu^a+\alpha M_W\chi^a)^2 -\frac{1}{2\alpha}(\partial^\mu Z_\mu+\alpha M_Z\chi^3)^2 -\frac{1}{2\alpha}(\partial^\mu A_\mu)^2.
\mathbf{y}^{(g)}\triangleq \underbrace{\left(\mathbf{I}_T \otimes \left[\mathbf{S}_D^{(g)}\right]^H\right)}_{\triangleq \left(\boldsymbol{\Upsilon}_S^{(g)}\right)^H}\mathbf{y},\;g=1,\ldots,G,
\frac{1}{\tau}\big\|(e_n-e_{n-1})_{n=k}^N\big\|_{L^p(X)}+ \big\|(e_n )_{n=k}^N\big\|_{L^p(D)}\le C\tau^k , & \big\|(e_n )_{n=k}^N\big\|_{L^\infty(W)}\le C\tau^k ,
K\big(\eta^{(1)}\big) -\sigma K\big( \eta^{(2)}\big) \sigma=\alpha_{0}( x(u_{1},0) )^{\ast}( A_{1}-A_{2}) \alpha_{0}( x(u_{1},0)) +O\big(\vert z_{1}\vert^{1-2\vert \kappa \vert}\big) .
\int\limits_{\{s_1<\cdots< s_n\}}\bar{g}_{\varepsilon/2}(s) \,ds - \int\limits_{\{s_1<\cdots< s_n\}}\bar{g}_{\varepsilon}(s) \,ds = O\Bigl(\varepsilon^{n-1 - \mu \phi(-in)}\Bigr).
\Pi(c,d) = \begin{bmatrix}(I + \Sigma^T\Sigma)^{-1}(I + \Sigma^T\Sigma)^{-1}\Sigma^T (I + \Sigma\Sigma^T)^{-1}\Sigma (I + \Sigma\Sigma^T)^{-1}\Sigma\Sigma^T\end{bmatrix}\begin{bmatrix}c d\end{bmatrix},
\delta(\jmath, r_0+\frac{1}{2})=\delta(\jmath, r_0-\frac{1}{2})=\delta_{\imath_0\jmath}\ \ \jmath\in I_{\mathrm{A}}, &\delta(\imath_0-1, r_0)=1,\ \delta(\imath_0+1, r_0)=0\ (\delta(2n, r_0):=0).
X_{m-i,n-m-j}^\xi= \frac{A_{m-i,n-m-j+1}^\xi A_{m-i+1,n-m-j}^\xi A_{m-i-1,n-m-j-1}^\xi}{A_{m-i+1,n-m-j+1}^\xi A_{m-i-1,n-m-j}^\xi A_{m-i,n-m-j-1}^\xi}= \pm \frac{A_{i,j-1}^vA_{i-1,j}^vA_{i+1,j+1}^v}{A_{i-1,j-1}^vA_{i+1,j}^vA_{i,j+1}^v};
{\rm Re}\left\{1+z\frac{T_g''(z)}{T_g'(z)}\right\}={\rm Re}\left\{z\frac{f'(z)}{f(z)}+z\frac{g''(z)}{g'(z)}+1\right\}>\frac{(1-\alpha)r^2-kr+1+\alpha}{1-r^2}=\frac{\lambda(r)}{1-r^2}
\mathbb{E}|X_1^p\xi_{\theta}^q|=\mathbb\mathbb{E}\bigg|X_1^p\bigg(S^{(1)}+\theta\frac{X_1}{\sqrt{n}}\bigg)^q\bigg|\leq 2^{q-1}\bigg[\mathbb{E}|X|^p\mathbb{E}|S^{(1)}|^q+\frac{\mathbb{E}|X|^{p+q}}{n^{q/2}}\bigg],
\Vert BA^{\prime}\Vert^{2}=\Vert (BA^{\prime})^{*}\Vert^{2}\le 4\max_{x\in \mathcal{N}}\Vert(BA^{\prime})^{*}x\Vert_{2}^{2}=4\max_{x\in \mathcal{N}}\sum_{j\le n}\langle W_{j}^{\prime}, x\rangle^{2}.
S = \int_0^{-1}\frac{(1+u)^{-x}-1}{x(1-u)}\,du = \frac{1}{2}\int_0^1 \frac{1-u^{-x}}{x(1-u/2)}\,du = \sum_{k\ge 1}2^{-k}\bigg(\frac{1}{k}-\frac{1}{k-x}\bigg),
\left[ \partial^{\tau}\partial_{\tau}+\kappa\left( u^{\tau}\partial_{\tau}\right)^{2}\right] A_{\sigma}-\partial_{\sigma}\left(\partial^{\tau}A_{\tau}+\kappa u^{\nu}u^{\tau}\partial_{\nu}A_{\tau}\right) & \qquad=\frac{4\pi\mu}{c}\left( g_{\sigma\lambda}-\frac{\kappa}{1+\kappa}u_{\sigma}u_{\lambda}\right) j^{\lambda}.
\mathcal{D}(w^{\theta}) =k\langle x,y,z\rangle/(\partial_{x}w,\partial_{y}w,\partial_{z}w) &=k\langle x,y,z\rangle/(\alpha^{2}\beta yz-\alpha^{2}\gamma zy,\beta^{2}\gamma zx-\alpha\beta^{2}xz,\alpha\gamma^{2}xy-\beta\gamma^{2}yx) &=k\langle x,y,z\rangle/(\beta yz-\gamma zy,\gamma zx-\alpha xz,\alpha xy-\beta yx).
\frac{d}{dx}h(x,y)=H'_{0}(x)\frac{1}{2\sqrt{H_{0}(x)-U(y)}}\frac{1}{F(y)}-\frac{v'(x)\sqrt{m(x)}}{\sqrt{2}F(x)}+\frac{v(x)F'(x)\sqrt{m(x)}}{\sqrt{2}F^{2}(x)}-\frac{v(x)m'(x)}{2\sqrt{2}F(x)\sqrt{m(x)}}
\sum_{i=0}^{p-1}\frac{2p-1-2i}{2p-1}\left( A_{i}. e_j - A_{2p-1-i}. e_j \right) = \sum_{i=0}^{p-1}\frac{2p-1-2i}{2p-1}e_{l_i(j)}+ \frac{2p-1-2(2p-1-i)}{2p-1}e_{l_{2p-1-i}(j)}, &= \sum_{i=0}^{2p-1}\frac{2p-1-2d_{j,l_i(j)}}{2p-1}e_{l_{i}(j)}, =\sum_{k=0}^{2p-1}\Delta_{jk}e_{k},
b_j (x,\xi' + \zeta', i |\xi'|_x+ \tilde{\delta}) = b_j (x,\xi' + \zeta', i |\xi'|_x+ \delta) + ( \tilde{\delta}- \delta) \partial_{\xi_d}b_j (x,\xi' + \zeta', i |\xi'|_x+ \delta) + ( \tilde{\delta}- \delta)^2 \int_0^1 (1- s) \partial_{\xi_d}^2 b_j (x,\xi' + \zeta', i |\xi'|_x+ \delta_s ) \, ds,
s^{(\pi)}_\lambda(X) = [s_\lambda(Z)]\M(XZ)\L_\pi(Z) \,; \cr s^{*(\pi)}_\lambda(X) = \begin{cases}[s_\lambda(Z)]\L(XZ)\L_{\pi'}(Z) \mbox{if $|\pi|$ is even};\cr [s_\lambda(Z)]\L(XZ)\M_{\pi'}(Z)) \mbox{if $|\pi|$ is odd}.\cr\end{cases}
M_{f}(t)= f(x^{N}(t))-f(x^{N}(0)) -\left[\frac{1}{2}\int_{0}^{t}\Delta f(x^{N}(s))ds+\sum_{i\neq j}\int_{0}^{t}\mathfrak{U}_{ij}^{\lambda}f(x^{N}(s))dA_{ij}^{N}(s)\right]
\sum_{i}z_{i}= \frac{1}{4 i \sqrt{\pi}}\int_{- \infty}^{( 0 + )}\frac{d t}{t^{3 / 2}}\mathrm{T r}\exp ( - t [ A_{\mu}, A_{\nu}]^{2}) = \mathrm{T r}\sqrt{- [ A_{\mu}, A_{\nu}]^{2}},
r^2I''(r,f') = t^2\int_0^{2\pi}|f'|^t\left|(z\frac{f''}{f'}-\frac{2r^2}{1-r^2})+\frac{2r^2}{1-r^2}\right|^2\mathrm{d}\theta-rI'(r,f') = t^2\int_0^{2\pi}|f'|^t\left|(z\frac{f''}{f'}-\frac{2r^2}{1-r^2})+\frac{2r^2}{1-r^2}\right|^2\mathrm{d}\theta-r\Re(z\frac{f''}{f'})I(r,f')
\displaystyle\left\vert\frac{p-x}{\vert p-x\vert}-\frac{z'-x}{\vert z'-x\vert}\right\vert\ge\left\vert\frac{p-x}{\vert p-x\vert}-\frac{p'-x}{\vert p-x\vert}\right\vert \displaystyle=\frac{(p-x)\cdot\nu_x}{\vert p-x\vert}.