harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
W[\Phi_{0},a,M]=-{1 \over 64\pi^2}\tau ({4 \over 3}\pi a^3)\Phi_{0}^4{1 \over 35}Log{M^2 \over \Phi_{0}^2}(1+{70 \over 63}{1 \over\Phi_{0}^2 a^2}+{35 \over 2}{m^2 \over \Phi_{0}^2}) + ...
J_{{\mu_{0}}{\mu_{1}}}^{a b}\left( x \right) = \int d \tau d \sigma \sum_{i}z_{i}^{a b}\, \delta^{d}\left( x_{\mu}- X_{\mu}^{i}\left( \tau , \sigma \right) \right) \, \, \partial_{\tau}X_{[{\mu_{0}}}^{i}\left( \tau , \sigma \right) \, \partial_{\sigma}X_{{\mu_{1}}]}^{i}\left( \tau , \sigma \right) \, \, .
e_\lambda(x, y)~=~\frac{2\sqrt 2}{(2\pi)^{(d+2)/2}}\int_r^\infty \frac{\sinh s}{\sqrt{\cosh s -\cosh r}}\cdot \left( \frac{-1}{\sinh s}\partial_s \right)^{\frac{d}{2}}\frac{\sin (s \cdot \sqrt{\lambda - b_d})}{s}\,ds\,.
\frac{d}{dt}\int_{V}\Re(\alpha_{t}+\sqrt{-1}\omega)^{p}-\cot\Theta_{0}\cdot\Im(\alpha_{t}+\sqrt{-1}\omega)^{p}={}p\int_{V}\left(\Re(\alpha_{t}+\sqrt{-1}\omega)^{p-1}-\cot\Theta_{0}\cdot\Im(\alpha_{t}+\sqrt{-1}\omega)^{p-1}\right)\wedge \omega> 0.
\left( \begin{array}{c}-A_{KW}U+\nabla_{U}^{s}KW+D^{l}(U,KW) -A_{LW}U+\nabla_{U}^{l}LW+D^{s}(U,LW)\end{array}\right) =\left( \begin{array}{c}K\nabla_{U}W+L\nabla_{U}W+\breve{J}h^{l}(U,W) +Bh^{s}(U,W)+Ch^{s}(U,W)\end{array}\right) ,
T_{m}(x) = \sum_{i=0}^{\lfloor m/2 \rfloor}(-1)^{i}\frac{m}{m-i}{m-i\choose i}2^{m-2i-1}x^{m-2i}= \sum_{i=0}^{2^{s-1}q-1}(-1)^{i+1}\frac{2^{s}q-1}{2^{s-1}q+i}{2^{s-1}q+i\choose 2i+1}2^{2i}x^{2i+1}.
\mbox{LHS}{>}\frac{1}{2}\sum_{j=1}^m \Bigg(\sqrt{\left(\gamma_j{+}\frac{1{-}\alpha}{m/2}\right)^2{+}\frac{(1{-}\alpha)^2}{m^2}\left(\frac{\gamma_j}{\min_i \gamma_i}{-}1\right)}{-}\gamma_j\Bigg) \geq \frac{1}{2}\sum_{j=1}^m \left(\sqrt{\left(\gamma_j + \frac{1-\alpha}{m/2}\right)^2}-\gamma_j\right) = 1-\alpha,
\widehat A_j=\frac{s_{j+1}-s_{j}}{2\pi i}G_{j}(y_{j})\sigma_{+}G_{j}(y_{j})^{-1}=\frac{s_{j+1}-s_j}{2\pi i}\begin{pmatrix}-G_{j,11}(y_j)G_{j,21}(y_j) G_{j,11}^2(y_j) -G_{j,21}^2(y_j) G_{j,11}(y_j)G_{j,21}(y_j)\end{pmatrix},
\|CT(t)x\| =\|C(-A)^{\alpha}T(t/2) T(t/2) (-A)^{-\alpha}x\| &\leq \frac{M_1}{\sqrt{t/2}}\,\| T(t/2) (-A)^{-\alpha}x\| &\leq \frac{2^{(1+\beta)/2}M_1M_2}{\sqrt{t^{1+\beta}}}\|x\|
{\bf J}'\frac{\partial}{\partial x_1}=\frac{\partial}{\partial p_m},\qquad{\bf J}'\frac{\partial}{\partial x_m}=\frac{\partial}{\partial p_1},\qquad{\bf J}'\frac{\partial}{\partial p_1}=-\frac{\partial}{\partial x_m},\qquad{\bf J}'\frac{\partial}{\partial p_m}=-\frac{\partial}{\partial x_1},\qquad
O^{T}T O = \left( \begin{array}{c c c c}{{0}}&{{1}}&{{0}}&{{0}}\\{{- 1}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{- 1}}\\{{0}}&{{0}}&{{1}}&{{0}}\\ \end{array}\right) , ~ O^{T}Q O = \left( \begin{array}{c c c c}{{0}}&{{- 1}}&{{0}}&{{0}}\\{{1}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{- 1}}\\{{0}}&{{0}}&{{1}}&{{0}}\\ \end{array}\right) .
w_1(\eta) = \epsilon+ \begin{cases}\frac{3}{2}-\frac{3\eta}{2}(0\leq\eta\leq\frac{1}{2}) 1-\frac{\eta}{2}(\frac{1}{2}\leq\eta\leq 1) \frac{3}{2}-\eta (1\leq\eta\leq\frac{3}{2}) 0 (\frac{3}{2}\leq\eta\leq 2) \end{cases}.
80352000 n(5 n-1)(5 n-2)(5 n-4) c_{n}+ 25\left(2592000 n^{4}-16588800 n^{3}+39118320 n^{2}-39189168 n+14092603\right) c_{n-1}+ 20\left(4500 n^{2}-18900 n+19739\right) c_{n-2}+ c_{n-3}= 0,
= \frac{i \beta}{2 \pi}4 i m \varepsilon^{\mu \nu \alpha}\frac{k_{\alpha}}{\sqrt{k^{2}}}\operatorname{l o g}\left( \frac{1 - \sqrt{\frac{4 m^{2}}{k^{2}}}}{1 + \sqrt{\frac{4 m^{2}}{k^{2}}}}\right) \, \, ,
\phi_{n+1}\big(\psi_{n+1}(rd\bar{r}\wedge d\bar{r}')\big)=\phi_{n+1}\big(\{e^{r\Pi'},e^{\bar{r}},\bar{r}'\}\big)= \log\big(e^{r\Pi'}\big)d\log(e^{\bar{r}})\wedge \frac{d\bar{r}'}{\Pi'}=rd\bar{r}\wedge d\bar{r}'.
B_{3,1}(9) =\{(9),\, (6,2,1),\, (6,1^3),\, (5,3,1),\, (4,3,2),\, (4,3,1^2),\, & &\quad\ (3^2,2,1),\, (3^2,1^3),\, (3,2^3),\, (3,2^2,1^2),\, (3,2,1^4),\, (3,1^6)\}.
h(G) =h(G')+f(y)+\sum_{j=1}^y f(x_j) -\sum_{j=1}^y f(x_j-1) \le h(K_{1,m-y}) +f(y) + y\left( f\left( \frac{m}{y}\right) - f\left( \frac{m}{y}-1 \right) \right).
W^0_{l+2}(p,p_1,\dots,p_{l+1})\ast W^0_{m+2}(p',p^{'}_{1}\dots,p^{'}_{m+1})= \underset{t_1\in Y^p,t_2\in Y^q, p+q+1=l}{\sum}\underset{t^{'}_1\in Y^{p'},t^{'}_2\in Y^{q'}, p'+q'+1=m}{\sum}t_1\vee (t_2\ast t')+(t\ast t^{'}_1)\vee t^{'}_2.
\widehat{h}= \sum_{j = -1}^{J_n}\sum_{k \in I_j}\widehat{d}_{j,k}\psi_{j,k}= \sum_{j = -1}^{J_n}\sum_{k \in I_j}Y_{j,k}\mathbf{1}(|Y_{j,k}| > \tau \sqrt{\log n/n}) \psi_{j,k}
\int_{U_0}|x-y|^{2-n}\partial_\nu\hat \zeta_\delta(y)\, dy \leq C\delta^{n-1}\int_{U_0}|x-y|^{2-n}\, dy=C\int_{\delta U_0}|x-\delta^{-1}z|^{2-n}\, dz &\leq C\delta^{n-2}\sup_{w\in B}\int_{\partial B}|w-z|^{2-n}\, dz=O(\delta^{n-2}).
S^{obs}=i\hbar\int d^{3}k(\hat{a}_{k,2}^{\dagger}\hat{a}_{k,1}-\hat{a}_{k,1}^{\dagger}\hat{a}_{k,2})\frac{k}{|k|}=\epsilon_{0}\int d^{3}xE_{\perp}\times A_{\perp},
{\mathrm d}M^{\varphi}_{t}\, :=\, \varphi^{\prime}( \widetilde{X}_{t}){\mathrm d}B_{t, 2}\, ,{\mathrm d}A_{t}^{\varphi}\, :=\, \varphi^{\prime}( \widetilde{X}_{t}) b(t, \widetilde{X}_{t}, F_{t, 2}){\mathrm d}t + \frac{1}{\, 2\,}\varphi^{\prime\prime}( \widetilde{X}_{t}){\mathrm d}t \, ,
\frac 2{1-2\delta_{x,i}}\left( \delta_{x,i}+ \frac{r_{\tilde x,k}}{r_{x,i}}\delta_{\tilde x,k}\right) \leq 4 (\delta_{x,i}+ 2 C_x \delta_{\tilde x,k}) \leq 4 \left( \frac \varepsilon{24C_1}+ 2 C_x \frac \varepsilon{48C_1C_x}\right) = \frac \varepsilon{3C_1}< \frac 1{\sqrt 2}.
S_{D1}= -\int d^2\sigma \left(\frac rR\right)^2\left(\sqrt{-\det( g + F )}- 1\right) + 2 R^2 \sin^2\theta_1 \sin\theta_2\theta_3 \varepsilon^{\mu\nu}\partial_\mu\theta_1 \partial_\nu \theta_2\,,
U^{\nu}\cdot\nabla\omega^{\nu}=\left( U_{s1}^{\nu}+U_{n1}^{\nu}\right)\cdot\nabla\left( \omega_{n2}^{\nu}-\psi_{n2}^{\nu}\right) +U_{n1}^{\nu}\cdot\nabla\left( \omega_{s2}^{\nu}-\psi_{s2}^{\nu}\right) & \ \ \ \ \ \ \ +U_{s2}^{\nu}\cdot\nabla\omega_{n2}^{\nu}+U_{n2}^{\nu}\cdot\nabla\omega_{s2}^{\nu}+U_{n2}^{\nu}\cdot\nabla\omega_{n2}^{\nu},
\overline{{{\Gamma}}}_{5}= - \varepsilon^{\widetilde{0}, 0}- \varepsilon^{0 , \widetilde{0}}- \varepsilon^{\widetilde{\mu}, \mu}- \varepsilon^{{\mu}, \widetilde{\mu}}- \frac 1 2 e_{\mu \nu \rho \omega}\varepsilon^{[ \mu \nu ] , [ \rho \omega ]}.
\exp\left( X\right) \exp\left( Y\right) =\exp\left( X+Y+\frac{1}{2}\left[ X,Y\right] \right) =\exp\left[\begin{array}[c]{ccc}0 x_{1}+y_{1}x_{3}+y_{3}+\frac{x_{1}y_{2}-x_{2}y_{1}}{2}0 0 x_{2}+y_{2}0 0 0\end{array}\right] .
\frac{\alpha_{1,\ell}(\phi)}{\ell (\ell+1)}=h_0(0) \sin \psi_{0,\ell}\frac{1}{\ell^{1+1/2}\sin^{1+1/2}\phi}+ \phi^{-2-1/2}O(\ell^{-2-1/2})+ \phi^{-1}O(\ell^{-2}).
r_{\ell}\leq e^{50\varepsilon q_n}\frac{e^{-q_nL}}{\max(|\ell|, 1)}\max(r_{\ell-1}, r_{\ell+1})\times\begin{cases}\max(|\ell|, e^{\delta_n q_n}), \beta_n\geq \delta_n+200\varepsilon, e^{\beta_n q_n}, \beta_n< \delta_n+200\varepsilon\end{cases}
c(K_{\lfloor n/2\rfloor, \lceil n/2\rceil})=\sum_{k=2}^{\lfloor n/2\rfloor}\frac{\lfloor n/2\rfloor ! \lceil n/2\rceil !}{2k(\lfloor n/2 \rfloor -k)! (\lceil n/2 \rceil -k)!}.
x^{(l+1)}= x^{(l)}+ \alpha^{(l+1)}\Delta x^{(l+1)}, s^{(l+1)}= s^{(l)}+ \alpha^{(l+1)}\Delta s^{(l+1)}, \lambda^{(l+1)}= \lambda^{(l)}+ \alpha^{(l+1)}\Delta \lambda^{(l+1)}, v^{(l+1)}= v^{(l)}+ \alpha^{(l+1)}\Delta v^{(l+1)}, l = l + 1.
{\cal L}= - \frac{\epsilon_{0}}{4}F_{\mu \nu}F^{\mu \nu}- \frac{\epsilon_{0}}{4}G_{\mu \nu}G^{\mu \nu}- \frac{1}{c}{j_{e}}_{\mu}A^{\mu}- \frac{1}{c}{j_{m}}_{\mu}W^{\mu}+ ( g a u g e f i x i n g t e r m s ) \, .
\begin{array}{rcll}G_iG_j = G_jG_i \mbox{for all $i,j=1,\ldots,n-1$ with $\vert i-j\vert > 1$,}G_iG_{i+1}G_i = G_{i+1}G_iG_{i+1}\mbox{for all $i=1,\ldots,n-2$,}\end{array}
\Sigma_{2,3}(x) \leq \sum_{\substack{1\leq i\leq \kappa d_i=3}}\#\{(n,p,m): H_i(n)=p^2m, ~n\leq x, ~p>x^{1+\delta}, ~m\ll x^{1-2\delta}\}&\leq \sum_{\substack{1\leq i\leq \kappa d_i=3}}\sum_{m\ll x^{1-2\delta}}\#\{n\leq x: H_i(n)\equiv 0 \bmod{m}, ~H_i(n)/m=\square\}.
\begin{cases}a_{\alpha + n +1}- a_{n+1}= d_{\alpha + n}- d_{\alpha + n +1}b_n = a_{\alpha + 1}+ (d_{\alpha + 1}- d_\alpha) = b d_{\alpha + n}= d_\alpha + (b_1 - a_{\alpha+1}) + \sum_{j=2}^n (a_j - a_{\alpha + j}) d = d_\alpha + b - a_{\alpha + 1}+ \sum_{j=2}^\infty (a_j - a_{\alpha+j})\end{cases}
|f_\eta^\varepsilon| \leq c\varepsilon\sum_{\xi=\eta,\zeta_0,\zeta_1}\left(|\bar{\xi}|+\Bigl|\overline{\partial^\circ\xi}\Bigr|+\Bigl|\overline{\nabla_\Gamma\xi}\Bigr|+\Bigl|\overline{\nabla_\Gamma^2\xi}\Bigr|\right)
a\left(4\frac{C^1_{2,1}}{C^1_1}-3\frac{C^1_1}{\mu}\right)\omega_1\wedge\omega_2-c\omega_1\wedge\theta_{56}=0, c\left(4\frac{C^1_{2,1}}{C^1_1}-3\frac{C^1_1}{\mu}\right)\omega_1\wedge\omega_2+a\omega_1\wedge\theta_{56}=0.
\delta{\cal E}=\delta{\cal E}_0 -\frac{3\hbar^2}{4m}\int d^3\vec{r}\,\varphi^{\dagger}[(\vec{\sigma}\cdot\vec{\epsilon})(\vec{\sigma}\cdot\vec{\nabla})+(\vec{\sigma}\cdot\vec{\nabla})(\vec{\sigma}\cdot\vec{\epsilon})]\varphi\,{,}
\frac{3}{10}\tilde{u}_{j-\frac{1}{2}}+\frac{6}{10}\tilde{u}_{j+\frac{1}{2}}+\frac{1}{10}\tilde{u}_{j+\frac{3}{2}}=\frac{1}{30}\bar{u}_{j-1}+\frac{19}{30}\bar{u}_{j}+\frac{10}{30}\bar{u}_{j+1}.
E^*_{i+k-1}A_1E^*_{i+k-2}A_1E^*_{i+k-3}\cdots E^*_{i+1}A_1E^*_{i}=\big((\frac{k-2}{2})!\big)^2\frac{k}{2}M^{\frac{k-2}{2},0}_{\frac{2m-i-k+1}{2},\frac{i-1}{2}}&M^{0,0}_{\frac{i+k-1}{2},\frac{2m-i-k+1}{2}}M^{\frac{k-2}{2},0}_{\frac{2m-i-k+1}{2},\frac{i-1}{2}}=\frac{k}{2}M^{\frac{2m-k}{2},\frac{i-1}{2}}_{\frac{i+k-1}{2},\frac{i-1}{2}}
\delta_{\xi}A_{i}^{a}= - \xi^{j}F_{i j}^{a}, \hspace{7 m m}\delta_{\xi}A_{0}^{a}= - \xi^{j}F_{0 j}^{a}, \hspace{7 m m}\delta_{\xi}N^{i}= \dot{\xi}^{i}+ [ \xi , N ]^{i},
\overline{\alpha}_{f}(x)=\limsup_{n \to \infty}\max\{1, h_{H}(f^{n}(x))\}^{1/n}, & \underline{\alpha}_{f}(x)=\liminf_{n \to \infty}\max\{1, h_{H}(f^{n}(x))\}^{1/n}.
\zeta_5=\frac{\mu^{2}}{2(4\pi)^{2}(s-1)}\int d^{4}k\ \tilde S(k)\tilde S(-k)\left(\frac{k^2}{8\mu^2}\right)^{-s+1}F(\frac{s}{2},\frac{s-1}{2};1;1)=
\sum_{i}\frac{i}{2}\nabla_{a}({\bf V}\cdot{\bf K}_{a}) \psi^{a}\psi^{b}= \frac{2 i \pi^{2}}{\sqrt{3}e^{2}}\frac{( \hat{\bf v}^{i}\cdot \hat{\bf v}^{i})^{\frac{1}{2}}}{( 1 + 2 \mu r )^{2}}\left( \cos \theta \, \psi^{r}\psi^{\phi}+ \psi^{r}\psi^{\psi}- r \sin \theta ( 1 + 2 \mu r ) \psi^{\theta}\psi^{\phi}\right)
A ( \omega ) = \left( \begin{array}{c c c}{{\alpha ( \omega )}}&{{\beta ( \omega )}}&{{\lambda ( \omega )}}\\{{j^{2}\lambda ( \omega )}}&{{\alpha ( \omega )}}&{{j \beta ( \omega )}}\\{{j^{2}\beta ( \omega )}}&{{j \lambda ( \omega )}}&{{\alpha ( \omega )}}\\ \end{array}\right)
\mathbf{Pr}\left(\hat{x}_{i,min,\gamma}> x_i + \epsilon\right) \leq \left[1-\gamma\frac{2}{\pi}\tan^{-1}\left(\frac{\sqrt{\epsilon}}{\gamma\sum_{t\neq i}^N x_t^{1/2}}\right) \right]^M \leq \left[1-\gamma\frac{2}{\pi}\tan^{-1}\left(\frac{\sqrt{\epsilon}}{\gamma\sum_{t=1}^N x_t^{1/2}}\right) \right]^M
[\mathrm{x}_2,\mathrm{x}_3]_{-}= 2 \, \mathrm{x}_0 \mathrm{x}_1 \, , [\mathrm{x}_1,\mathrm{x}_3]_{-}= 2 \, \mathrm{x}_0 \mathrm{x}_2 \, , [\mathrm{x}_1,\mathrm{x}_2]_{-}= 2 \, \mathrm{x}_0 \mathrm{x}_3 \, .
(k-0.1)nc < \sum_{u\sim z}\sum_{\substack{w\sim u w\in N_2(z)}}\mathbf{\mathrm{v}}_w = \sum_{u\sim z}\sum_{\substack{w\sim u w\in M_2}}\mathbf{\mathrm{v}}_w + \sum_{u\sim z}\sum_{\substack{w\sim u w\in S_2\setminus M_2}}\mathbf{\mathrm{v}}_w < 0.9nc + e(N_1, S_2\setminus M_2)\frac{\alpha}{3},
\Sigma^{\pm}(s-w, \phi, F_{\pm})= \lim_{z \downarrow 0}\sum_{\varepsilon = \pm}\int_0^\infty w_z(\lambda)\lambda^{4(1-s+w)-1}\int_0^\infty \times \sum_{\ell, m=1}^\infty \rho_\phi(3\ell m)K_\nu(6\pi \ell m t^2)\hat{F}_4\left(0,0, \frac{3m\lambda}{t}, \frac{\varepsilon 3\ell m t^3}{\lambda}\right)\frac{d\lambda}{\lambda}tdt.
\theta\bigl((\operatorname{id}\otimes\omega_{\Lambda_{\varphi}(q),\Lambda_{\varphi}(p)})(W)\bigr)=\bigl\langle\Lambda_{\tilde{\varphi}}(\pi(q)),\Lambda_{\tilde{\varphi}}(\pi[(\bar{\theta}\otimes\operatorname{id})(\Delta p)])\bigr\rangle,
\psi^{(t)}(y_{-k_1}\cdots y_{-k_n})(z) = \sum_{m_1,\ldots,m_n\geq 0}\frac{B_{m_1}}{m_1!}\cdots \frac{B_{m_n}}{m_n!}C^{k_1,\ldots,k_n}_{m_1,\ldots,m_n}(t) z^{m_1+\cdots+m_n-(k_1+\cdots+k_n)-n}
\int [ d \varphi^{I}] \, \epsilon^{a}\, \lbrace \, \,{\frac{\partial}{\partial \chi^{a}}}\Big ( S_{1}\, + \, \hbar M_{1}\Big ) \, + \, J^{r}{\overline{{R}}}_{a}^{r}\, \rbrace \, e x p{\frac{i}{\hbar}}\Big ( S_{_\Sigma}+ \hbar M_{1}\, + \, J^{A}\, \phi_{A}\Big ) \, = \, 0
\begin{gathered}x_ix_j=x_jx_i, x_i\tau_j=\tau_jx_i~(|i-j|>1),\tau_i\tau_j=\tau_j\tau_i~(|i-j|>1), \tau_i^2=0,x_i\tau_i-\tau_ix_{i+1}=1=\tau_ix_i-x_{i+1}\tau_i,\tau_{i}\tau_{i+1}\tau_1=\tau_{i+1}\tau_i\tau_{i+1}.\end{gathered}
JZB^{2p^{d-2}+p-1+\lambda}\subseteq 1_BJZFG^{2p^{d-2}+p-1+\lambda}\subseteq I_{\le D}(G)\cdot JZFG^{2p^{d-2}+p-1+\lambda}&\subseteq I_{<D}(G)\cdot JZFG^{p^{d-2}+p-1+\lambda}=\sum_{Q<D}{I_{\le Q}(G)\cdot JZFG^{p^{d-2}+p-1+\lambda}}&\subseteq \sum_{Q<D}{I_{<Q}(G)\cdot JZFG^{\lambda}}=0.
\sum_{k=1}^\infty k\tau_kY^k =Y\tau'=\tau\sum_{k=1}^n\frac{X_kY}{1-X_kY}=\tau\sum_{k=1}^n\sum_{i=1}^\infty(X_kY)^i &=\tau\sum_{i=1}^\infty\rho_iY^i=\sum_{k=1}^\infty\Bigl(\sum_{i=1}^k\rho_i\tau_{k-i}\Bigr)Y^k.
\frac{1}{2}|C_k|e^{\frac{r_i^2}{4}}r_i^{-m}\le |C_k||1+g_k(\frac{r_i^2}{4})|e^{\frac{r_i^2}{4}}r_i^{-m}&=|\int_{S^{m-1}}u(r_i,\theta)\varphi_k| &\le \big(\int_{S^{m-1}}( u(r_i,\theta))^2\big)^{\frac{1}{2}}(\int_{S^{m-1}}\varphi_k^2)^{\frac{1}{2}}&\le C\sqrt{\omega_n}e^{\frac{r_i^2}{4}}r_i^{-(m+\epsilon)},
\left[ W_{\mu}, P_{\nu}\right] = 0 \, \, \, \, \, \, \, \, \, \, \left[ W_{\mu}, M_{\nu \kappa}\right] = i ( g_{\nu \mu}W_{\kappa}- g_{\kappa \mu}W_{\nu}) \, \, \, \, \, \, \, \, \, \left[ W_{\mu}, W_{\nu}\right] = i \epsilon_{\mu \nu \kappa \lambda}W^{\kappa}P^{\lambda}
{\bf H}_{21}{\bf V}_{r_{13}t_{1}}={\bf H}_{22}{\bf V}_{r_{13}t_{2}}={\bf H}_{23}{\bf V}_{r_{13}t_{3}}\triangleq{\bf V}_{2},{\bf H}_{31}{\bf V}_{r_{12}t_{1}}={\bf H}_{32}{\bf V}_{r_{12}t_{2}}={\bf H}_{33}{\bf V}_{r_{12}t_{3}}\triangleq{\bf V}_{3}.
\delta M_{intermediate}\lesssim \delta M + \delta M_{\partial}+ \|\delta \nu_{\partial}\|_{2\alpha -1}+ \| U_1 - U_0\|_{\alpha}+ \| u_1 - u_0\|_{\alpha}+ \max_{i = 0,1}(\| U_i \|_{\alpha}+ \| u_i \|_{\alpha}) \|a_1 - a_0 \|_{\alpha}+ \|v_1 - v_0\|_{\alpha}
2 M^\dag J(k) M= -{ik}M^\dag(U+I_n) M+iM^\dag(U-I_n) M &+M^\dag Q(0)M M^\dag(U+I_n) M+o(1) = -ik{\rm diag}\{I_{n-n_D},0_{n_D}\}T^{-1}+i{\rm diag}\{(X_1-I_{n-n_D}),-2I_{n_D}\}&+Q'{\rm diag}\{I_{n-n_D},0_{n_D}\}T^{-1}+o(1),
(G_{ij}^E)_{ij=1, \cdots, M+1}=\left(\begin{array}{cc}\displaystyle g_{rs}=(\rho\cdot\cos\theta)^{\frac{2}{M}}{\bar g}_{rs}\displaystyle 0 & \vdots 0 \cdots 0{\dot{\vec x}}^2=\sin^2\theta \end{array}\right) \qquad .
\check{F}_t:=h(F_{\tilde{a}^{-1}(2t)}),\ \check{g}_t:=\tilde{g}_{\tilde{a}^{-1}(2t)}, check{\xi}(t):=h_{\tilde{a}^{-1}(2t)}(\xi(\tilde{a}^{-1}(2t))).
\sum\limits_{n=K+1}^{\infty}\mathbb{P}(S_{(n)}>\mathit{xy})\mathbb{P}(\eta=n) \leqslant c_3\overline{F}_{\xi_1}(\mathit{xy})\sum\limits_{n=K+1}^{\infty}n^{p+1}\mathbb{P}(\eta=n),
f(t,q^{\prime}) = \displaystyle \sum_{n=0}^{\infty}\left(- \frac{q^{\prime}}{1-q^{\prime}}\right)_{2n}(1-q^{\prime})^{2n}\frac{(\alpha t)^{2n+1}}{(2n+1)!}= \sinh_{q^{\prime}}\left(\alpha t\right).
S_l\leq S_{{p_1}^{h_1}}\cdots S_{{p_{n-1}}^{h_{n-1}}}\lneq S_{{p_1}^{h_1}}\cdots S_{{p_{n-1}}^{h_{n-1}}}S_{p_n}= S_{{{p_1}^{h_1}}\cdots p_{n-1}^{h_{n-1}}p_n}
P_{x_0}^{\omega}(X_0 = x_0) = 1, & P_{x_0}^{\omega}(X_{n+1}= X_n + 1|X_0, \ldots, X_n) = \omega(X_n, I_n), & P_{x_0}^{\omega}(X_{n+1}= X_n - 1|X_0, \ldots, X_n) = 1 - \omega(X_n, I_n),
\lim_{\rho\searrow0}\int_{B_r}\big( |\nabla v_\rho|^2-|\nabla u_\rho|^2\big)\le \int_{B_r}\nabla(v_0+u_0)\cdot\nabla(v_0-u_0)=\int_{B_r}\big( |\nabla v_0|^2-|\nabla u_0|^2\big).
\left( [ \delta_{\varepsilon_1}\,,\,[ \delta_{\varepsilon_2}\,,\,\delta_{\varepsilon_3}] ] + [ \delta_{\varepsilon_3}\,,\,[\delta_{\varepsilon_1}\,,\,\delta_{\varepsilon_2}] ]+ [ \delta_{\varepsilon_2}\,,\,[ \delta_{\varepsilon_3}\,,\,\delta_{\varepsilon_1}] ] \right) q^r = 0 ,
w_{ab}= A\alpha(\alpha-1)d^{\alpha-2}d_{a}d_{b}+A\alpha d^{\alpha-1}d_{ab}+2B\delta_{ab}, w_{an}= A\alpha(\alpha-1)d^{\alpha-2}d_{n}d_{a}+A\alpha d^{\alpha-1}d_{na}, w_{nn}= A\alpha(\alpha-1)d^{\alpha-2}d_{n}^{2}+A\alpha d^{\alpha-1}d_{nn}.
\alpha = k \beta = 2[k(k-2) - 2c(k-1)^2(k+1)] \gamma = 2 c^2 k( k-1)^2 ( 3 k^2-4) - c (6 k^4 - 8 k^3 - 4 k^2 + 18k -12) + k (k^2-6k+6) \delta = -2 ( k -1) [2 c^3 k^2 (k+1)(k-1)^3 -c^2 k (3k^4+k^3-8k^2-6k+10) +c (k^4-k^3-k^2+6k-6)+k(k-2)] \varepsilon =( k- 1)^2 (c k^2-1)^2 [c^2 k (k-1)^2 - 2 c (k^2+k-2)+k].
\displaystyle{\frac{\phi^{( 1 )}( a , b + 1 , c | u , v , w + 1 / 4 )}{\phi^{( 1 )}( a , b - 1 , c | u , v , w + 1 / 4 )}}= \displaystyle{\frac{\phi^{( 0 )}( a , b + 1 , c | u - \eta , v - \eta , w + 1 / 4 )}{\phi^{( 0 )}( a , b - 1 , c | u - \eta , v - \eta , w + 1 / 4 )}}.
\begin{multlined}I_{3,\varepsilon,t,\xi}=-\int_M F_\varepsilon(u_0-W_{\delta_\varepsilon (t),\xi})-F_\varepsilon(u_0)-F_\varepsilon(W_{\delta_\varepsilon (t),\xi}) +f_\varepsilon(u_0)W_{\delta_\varepsilon (t),\xi}+f_\varepsilon(W_{\delta_\varepsilon (t),\xi})u_0dV.\end{multlined}
g=\sum_{w\in\tilde{W}}(-1)^{\ell(w)}q^{-\ell(w)}T_w=\sum_{w\in\tilde{W}}(-1)^{\ell(w)}q^{-\ell(w)}\left(\sum_{y\leq w}q^{\frac{\ell(y)}{2}}(-1)^{\ell(w)-\ell(y)}C'_y\right),
\begin{cases}\tilde h_1 (x,u,\varphi) = h_1(x,xu,\varphi) \tilde h_2 (x,u,\varphi) = \frac{h_2 (x,xu,\varphi)}{h_1 (x,xu,\varphi)}\tilde h_3 (x,u,\varphi) = h_3 (x,xu,\varphi)\end{cases}
\langle \phi_{1}( x_{1}) \phi_{2}( x_{2}) \phi_{3}( x_{3}) \rangle = \frac{C_{1 2 3}}{x_{1 2}^{\Delta_{1}+ \Delta_{2}- \Delta_{3}}x_{2 3}^{\Delta_{2}+ \Delta_{3}- \Delta_{1}}x_{1 3}^{\Delta_{3}+ \Delta_{1}- \Delta_{2}}},
(\mu \otimes id)\mu (\gamma) = \sum_{i,j=1}^{n}\mu(\gamma_i) \otimes \gamma_j \gamma(p_jp_i) = \sum_{i,j=1}^{n}\sum_{r,t=1}^{n}\gamma_r \otimes \gamma_t \gamma_i(p_tp_r) \otimes \gamma_j \gamma(p_jp_i) = \sum_{i,j,r,t=1}^{n}\gamma_r \otimes \gamma_t \otimes \gamma_i(p_tp_r) \gamma_j \gamma(p_jp_i);
\begin{aligned}\phi\left( \frac{z}{c \tau + d}, \frac{a \tau + b}{c \tau + d}\right) = (c \tau + d)^k e\left( \frac{c z^t L z}{c \tau + d}\right) \phi(z,\tau) \phi\left( z + \lambda \tau + \mu, \tau \right) = e\left( - \lambda^t L \lambda \tau - 2 \lambda^t L z \right) \phi(z,\tau)\end{aligned}
\hat{\mathrm{P}}_{\pm}= \pm \hat{\cal H}_{\mathrm{F}}^{\pm}( 0 ) + \sum_{n > 0}\hat{\mathrm{G}}_{\pm}^{a}( - n ){A}_{1 , \pm}^{a}( n ) + \sum_{n \leq 0}{A}_{1 , \pm}^{a}( n ) \hat{\mathrm{G}}_{\pm}^{a}( - n ) ,
\begin{gathered}\Gamma (\tau \circ X^{-1}) (t) = \int_0^t \Delta g_{\nu (t-s)}* \left ( \left ( \tau \circ X^{-1}\right ) (s) - \left (\tau \circ X^{-1}\right ) (t) \right ) ds + \int_0^t \Delta g_{\nu (t-s)}* \left (\tau \circ X^{-1}\right ) (t) ds.\end{gathered}
\frac{i^2\frac{1}{2}d_1(p_1^2+p_2^1 + (p_1+p_2)^2) + i^2c_1}{(p_1+p_2)^2-m^2}= -\frac{2a_1(2m^2 + (p_1+p_2)^2) - m^26a_1}{(p_1+p_2)^2-m^2}= -\frac{2a_1((p_1+p_2)^2 - m^2)}{(p_1+p_2)^2-m^2}= -2a_1
\overline{r_1}<\underset{t}{\min}\Big(\frac{b_1(t)}{b_2(t)}\Big)\overline{r_2},\ \ \overline{r_2}<\underset{t}{\min}\Big(\frac{a_2(t)}{a_1(t)}\Big)\overline{r_1}, \overline{r_1}+\overline{r_2}>\underset{t}{\max}\Big(\frac{a_2(t)}{a_1(t)}\Big)\overline{r_1},\ \ \overline{r_1}+\overline{r_2}>\underset{t}{\max}\Big(\frac{b_1(t)}{b_2(t)}\Big).
V ( x_{-}) \equiv - \frac{1}{2 f}\frac{d^{2}f}{d x_{-}^{2}}+ \frac{3}{4}\left( \frac{1}{f}\frac{d f}{d x_{-}}\right)^{2}- \frac{1}{4{\hbar}^{2}c^{2}}f^{2}+ \frac{4 m^{2}c^{4}P_{c m}^{2}}{{\hbar}^{2}( f^{2}- 4 c^{2}P_{c m}^{2})}
P=\left[ \begin{array}{rr}0 0 0 2 1 1 2 0 -3 -3 \end{array}\right] \mbox{and}\Omega^0 =\left[ \begin{array}{rr}0 1 -2 1 0 \end{array}\right]\left[ \begin{array}{rrrrr}0 1 -2 1 0 \end{array}\right].
\nu_1 = -\frac{1}{2d}\sum_{ij}A_{ii}A_{jj}\pi_i \pi_j, \nu_2 = -\frac{1}{4d^2}\sum_{ijk\ell}A_{ik}^2 A_{j\ell}^2 \pi_i \pi_j \pi_k \pi_\ell, \mbox{and}\xi_n = O(n^{-1/2}) \sum_{ij}|X_{ij}| + O(n^{-1}) \left(\sum_{ij}|X_{ij}|\right)^2 + O(n^{-1}).
X_\tau^{0,x}= e^{\tau A}x+e^{\tau A}v(0,x)-v(\tau,X_\tau^{x})+\int_0^\tau e^{(\tau-s)A}\nabla^G v(s,X_s^x)\;dW_s +\int_0^\tau e^{(\tau-s)A}GdW_s,
\left( k_{3}-k_{2}\right)_{\lambda}\Gamma_{\lambda \mu \nu}^{AVV}=2\varepsilon_{\mu \nu \alpha \beta}\left[ (k_{1}-k_{3})_{\beta}(k_{1}+k_{3})_{\xi}+(k_{2}-k_{1})_{\beta}(k_{1}+k_{2})_{\xi}\right] triangle_{\xi \alpha},
\big{(}\partial_t f_t(x)\big{)}^{\perp_{f_t}}= -\frac{1}{2}\frac{1}{|A^0_{f_t}(x)|^4}\,\Big{(}\triangle_{f_t}^{\perp}\vec H_{f_t}(x) + Q(A^{0}_{f_t})(\vec H_{f_t})(x) \Big{)}
\begin{aligned}({''E}^{+}_{0})^{\cdot, q}:0\longrightarrow C^{q}(W_{\overline{Z_{1}}}^{u}, \overline{F})^{+}\overset{\psi^{+}_{1}}\longrightarrow C^{q}(W^{u}_{Z_{1}}, F) \longrightarrow 0.\end{aligned}
\mathbb{E}[fg] =\sum_{x,y}p(x,y)f(x)g(y) &=\sum_{x,y}\frac{p(x,y)}{\sqrt{p(x)p(y)}}\cdot\sqrt{p(x)}f(x)\cdot \sqrt{p(y)}g(y) &=v_f^T \tilde P v_g
\sum_{k}(\mathsf{N}_{m}^{n})_{k}^{i}(\mathsf{N}_{o}^{p})_{j}^{k}=u_{m}^{i}\left(\sum_{k}u_{n}^{k*}u_{o}^{k}\right)u_{p}^{j*}=\delta_{o}^{n}u_{m}^{i}u_{p}^{j*}=\delta_{o}^{n}(\mathsf{N}_{m}^{p})_{j}^{i}.
\begin{array}{rl}\hat{\Gamma}(k_1,k_2,k_3,k_4) = \displaystyle -g_{\pi}^2(2\pi)^4\delta^4(\sum_ik_i)\sum_{\{\lambda_j\}}\sum_{\lambda}\prod_{j=1}^4\left[\eta_{\lambda_j}(k_j)\right]\left(A_s+A_t+A_u\right), \end{array}
\pi_{1}\left( L_{i}\right) \overset{p_{i}^{-1}}{\longrightarrow}\underleftarrow{\lim}\left\{\pi_{i}(L_{i}),(g_{i})_{\ast}\right\}\rightarrow\underleftarrow{\lim}\left\{\pi_{i}(K_{i}),(f_{i})_{\ast}\right\}=\check{\pi}_{1}\left( \left\{K_{i},f_{i}\right\}\right)
[\begin{matrix}1&-1&0&0&0&0&0\\ 0&1&-1&0&0&0&0\\ 0&0&1&-1&0&0&0\\ 0&0&0&1&-1&0&0\\ 0&0&0&0&1&1&0\\ -\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&\frac{\sqrt{2}}{2}\\ 0&0&0&0&1&-1&0\\ \end{matrix}].
\omega_0 \equiv -{{\textstyle\frac{1}{2}}}\left({\displaystyle\sum_{d>1}\omega_d}\right) \ + \F_1/2 \ ,\quad\omega_1 \equiv -{{\textstyle\frac{1}{2}}}\left({\displaystyle\sum_{d>1}\omega_d}\right) \ - \F_1/2 \ .
\langle T_{z z}\rangle^{\mathrm{i n f}}= - \int \frac{d \kappa \, \kappa^{3}}{4 \pi^{2}}, \quad \langle T_{0 0}\rangle^{\mathrm{i n f}}= - \int \frac{d \kappa \, \kappa^{3}}{1 2 \pi^{2}}.
\mathsf{R}_{k}-\mathsf{R}_{k}\circ T^{j}=\mu(E_{k})\left( a_{T}(\varphi_{E_{k}}\circ T^{j}+j)-a_{T}(\varphi_{E_{k}}\circ T^{j})\right) & \leq\mu(E_{k})\,a_{T}(j)\{\varphi_{E_{k}}>j\}
\bigg(1 + \sum_{k=1}^\infty \frac{n_k}{y^kx^{n_k+1}}\bigg) \bigg(1 + \sum_{\ell=1}^\infty \frac{\tilde{n}_\ell}{x^\ell y^{\tilde{n}_\ell+1}}\bigg) - \sum_{k=1}^\infty \frac{k}{y^{k+1}x^{n_k}}\, \sum_{\ell=1}^\infty \frac{\ell}{x^{\ell+1}y^{\tilde{n}_\ell}}> 0.
S = \frac{1}{2}\int \F_{t a 1}D_{a 1}+ F_{t a 2}D_{a 2}- \frac{1}{2}( E_{a 1}E_{a 1}+ D_{a 2}D_{a 2}+ B_{a 1}B_{a 1}+ B_{a 2}B_{a 2}) \ .
\int_K \int_H \frac{\Delta_G (h)}{\Delta_H(h) \Delta_K(k^{-1})}f(k^{-1}x h) d h d k = \int_H \int_K \frac{\Delta_G (h)}{\Delta_H(h) \Delta_K(k^{-1})}f(k^{-1}x h) d k d h & = \int_{K \times H}\frac{\Delta_G (h)}{\Delta_H(h) \Delta_K(k^{-1})}f(k^{-1}x h) d (k \times h).
R_L^H(\chi)(g) = \frac{1}{|L||C_{\mathbf{H}}^{\circ}(s)^F|}\sum_{\{h \in H \, \mid \, s \in h \mathbf{L}h^{-1}\}}|C_{h \mathbf{L}h^{-1}}^{\circ}(s)^F| \sum_{v \in C_{h \mathbf{L}h^{-1}}^{\circ}(s)^F_{\mathrm{u}}}Q_{C_{h \mathbf{L}h^{-1}}^{\circ}(s)}^{C_{\mathbf{H}}^{\circ}(s)}(u, v^{-1}) \chi^h(sv),
\int_s^{2R}\frac{g_{R,1}(t)}t dt = \Big(\Big(p'-\frac32\Big) R^{1-1/p}- \frac{s_0^{1-1/p}}{p-1}\Big) \Big(\frac1s-\frac1{2R}\Big) & +2R^{-1/p}\ln \frac{2R}s -\frac{R^{-1-1/p}(2R-s)}2.
|K_\alpha(x,y)-K_\alpha(x,y')| \leqslant c \int_0^\infty |h_t(x,y)-h_t(x,y')|\frac{dt}{t^{1-\alpha/2}}&\leqslant c\bigg(\int_0^{d(x,y)^2}+\int_{d(x,y)^2}^\infty\bigg)|h_t(x,y)-h_t(x,y')|\frac{dt}{t^{1-\alpha/2}}&=: I\!I_1 +I\!I_2.
{\cal Z}= \frac{1}{2 \pi i}\int_{{\cal C}_{0}}\frac{d z}{z + e^{- \frac{\pi}{g_{n c}^{2}\theta}}}\exp \left[ z \Phi ( - z ; \frac{3}{2}; 1 ) \frac{A_{n c}}{\sqrt{2 \pi g_{n c}^{2}\theta^{3}}}\right] ,
b =F(x+a)+cF(x)=(A_1 \circ F'')(x+a)+c(A_1\circ F'')(x)=A_1(F''(x+a))+cA_1(F''(x)) &=u_1 F''(x+a)+v_1 +c(u_1 F''(x)+v)=u_1(F''(x+a)+cF''(x))+(c+1)v_1
L(\zeta)=\left[\begin{pmatrix}1+\frac{n^2+\frac{n}{2}}{\zeta}\frac{-2\pi\i\n}{\Gamma(n+\frac12)\, n!\ \zeta}\frac{-n!\ \Gamma(n+\frac32)}{2\pi\i\ \zeta}1-\frac{n^2+\frac{n}{2}}{\zeta}\end{pmatrix}+\mathrm{O}(\zeta^{-2})\right]\zeta^{-n\sigma_3},\quad\zeta\to\infty,
\Delta_{M}= \frac{1}{\sqrt{a}}\frac{\partial}{\partial r}\left( \sqrt{a}\frac{\partial}{\partial r}\right)+ \Delta_{\partial B(o,r)}= \frac{\partial^2}{\partial r^2}+ m(r, \theta) \frac{\partial}{\partial r}+ \Delta_{\partial B(o,r)},