harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sum_{A,B}\xi_{A}\xi_{B}\Delta_{AB}^{(\ell+1)}= \sum_{Q}\Delta_{Q}^{(\ell+1)}\sum_{Q^{(1)}+\cdots + Q^{(n_{\ell+1})}=Q}\prod_{i=1}^{n_{\ell+1}}\binom{q_i}{Q^{(i)}}\xi_{i;Q^{(i)}},
\{\varphi ( x ) , \varphi ( y ) \}_{x^{+}= y^{+}}^{\mathrm{D B}}= \frac{\mu^{2}}{N}\frac{1}{2}\left\{- \frac{1}{2}\epsilon ( x^{-}- y^{-}) + \frac{( x^{-}- y^{-})}{2 L}\right\},
A_{ij}=\frac{\partial\beta_{j}}{\partial\theta_{i}}=\frac{\partial\tau}{\partial\theta_{i}\partial\theta_{j}}=\delta_{ij}-\frac{\partial\psi}{\partial\theta_{i}\partial\theta_{j}}=[\begin{matrix}1-\kappa-\gamma_{1}&\gamma_{2}\\ \gamma_{2}&1-\kappa+\gamma_{1}\end{matrix}]
\|u\|_{L^{p}_{\sigma}(\mathbb{R}^{3})}:=\|u\|_{L^{p}(\mathbb{R}^{3})},\quad\|u\|_{\dot{H}^{1}_{\sigma}(\mathbb{R}^{3})}=\|\nabla u\|_{L^{2}(\mathbb{R}^{3})}.
\int_{\partial D_{\epsilon}(z_{0})}\frac{\tilde{w}(\zeta)}{\zeta - z_{0}}d \zeta =\int_{0}^{2\pi}\frac{\tilde{w}(z_{0}+\epsilon e^{i\theta})}{\epsilon e^{i\theta}}i\epsilon e^{i\theta}d\theta \to 2\pi i \tilde{w}(z_{0}), \ \ \mbox{as}\ \ \epsilon \to 0.
\begin{aligned}M =B^{-1}M_cB, S_1 =B^{-1}S_{c,1}B, H =B^{-1}H_{c}B, (S_2)^\top =B^{-1}(S_{c,2})^\top B,\end{aligned}\begin{aligned}P^{[1]}_n =B^{-1}P^{[1]}_{c,n}B, (P^{[2]}_n)^\top =B^{-1}(P^{[2]}_{c,n})^\top B.\end{aligned}
I-\cal{F}_P(X)^*\cal{F}_P(X)= \,P^*_{21}(I-X^*P^*_{22})^{-1}(I-X^*X)(I-P_{22}X)^{-1}P_{21}&\quad+\begin{bmatrix}I P^*_{21}(I-X^*P^*_{22})^{-1}X^* \end{bmatrix}(I-P^*P)\begin{bmatrix}I X(I-P_{22}X)^{-1}P_{21}\end{bmatrix}.
\Gamma_{\mu}= \left( \begin{array}{c c}{0}&{\gamma_{\mu}}\\{\gamma_{\mu}}&{0}\\ \end{array}\right) , \Gamma_{5}= \left( \begin{array}{c c}{0}&{i I}\\{- i I}&{0}\\ \end{array}\right) , \Gamma_{6}= \left( \begin{array}{c c}{0}&{\gamma_{5}}\\{\gamma_{5}}&{0}\\ \end{array}\right)
\mathcal{F}^{-1}[i\mathrm{sign}(\xi)\hat{f}(\xi)] =i\mathcal{F}^{-1}[\mathrm{sign}(\xi)]\ast\mathcal{F}^{-1}[\hat{f}(\xi)], &=i\Big( \frac{-i}{\pi x_1}\Big) \ast f(x_1), &=\frac{1}{\pi}\int^\infty_{-\infty}\frac{f(\eta)}{x_1-\eta}\mathrm{d}\eta.
\hat{H}_{coul}=-\sum_{i=1}^{D}\frac{\partial^2}{\partial x_i^2}-\frac{\eta}{r}+\frac{g_1(\Omega_1)}{x_1^2+\cdots+x_{n_1}^2}+\frac{g_2(\Omega_2)}{x_{n_1+1}^2+\cdots+x_{n_2}^2}+\cdots+\frac{g_{N-1}(\Omega_{N-1})}{x_{n_{N-2}+1}^2+\cdots+x_{n_{N-1}}^2},
\sup_{z\in\mathbb{B}_n}\beta(\varphi_{kj}(z),\rho(z)) \leq\sup_{z\in V^{-1}\mathbb{B}_n}\omega(\Big{|}((V^{-1}\varphi_{kj}V)^1,...,(V^{-1}\varphi_{kj}V)^s)(z)\Big{|},0) &= \dfrac{1}{2}\sup_{z\in V^{-1}\mathbb{B}_n}\tanh^{-1}(\Big{|}((V^{-1}\varphi_{kj}V)^1,...,(V^{-1}\varphi_{kj}V)^s)(z)\Big{|})\rightarrow 0,
\sum_{m=0}^{\infty}\sum_{l=1}^{h(m,n)}\left[\|\mathbf{B_m^*\Psi_m^*\Psi_m a_m^l}\|^2\right] \rightarrow inf, F_m(\lambda)=d_m(\lambda) (d_m(\lambda))^* -G_m(\lambda) \in D_F.
Z_{4}= \frac{( m L )^{4}}{4 !}I_{1}^{4}- ( m L )^{3}I_{1}^{2}I_{2}+ \frac{( M L )^{2}}{2}\left[ \frac{2}{3}I_{1}I_{3}+ \left( \frac{I_{2}}{2}\right)^{2}\right] - \frac{( m L )}{4}I_{4}\, \, \, .
\bigg\|\sum_{j=1}^n L_j^2 f \bigg\|_{2}^2 = \bigg\|\frac{1}{|x|^2}\Delta_{\mathbb{S}^{n-1}}f \bigg\|_{2}^2 & = \int_0^\infty \| (\Delta_{\mathbb{S}^{n-1}}f)(r(\cdot)) \|_{L^2(\mathbb{S}^{n-1})}^2 r^{n-5}\, \mathrm{d}r & = \sum_{k=1}^\infty \mu_k^2\int_0^\infty \| (P_k f)(r(\cdot)) \|_{L^2(\mathbb{S}^{n-1})}^2 r^{n-5}\, \mathrm{d}r
\lim_{\alpha\rightarrow 0^+}\left(\lim_{n\rightarrow\infty}\sqrt[n]{\binom{n+ \lceil \alpha n \rceil}{\lceil\alpha n\rceil}}\right) = \lim_{\alpha\rightarrow 0^+}\left(\lim_{n\rightarrow\infty}\sqrt[n]{\binom{n}{\lceil\alpha n\rceil}}\right)=1.
V_{1}( t ) = \exp \{i{\frac{e}{\hbar c}}\int \int d^{2}x d^{2}x^{\prime}\sqrt{g ( x )}J_{0}( x ) G ( \vec{x}, \vec{x}^{\prime}) \partial_{i}^{\prime}a_{i}^{( 1 )}( x^{\prime}) \}.
\int_\Omega \phi\, (f_1-f_0)\, dx\le W_\frac{p}{p-q}(\rho_0,\rho_1)\,\|\nabla \phi\|_{L^{p}(\Omega)}\,\left(\frac{\|f_0\|^{q'}_{L^{q'}(\Omega)}+\|f_1\|^{q'}_{L^{q'}(\Omega)}}{2}\right)^\frac{q-1}{p},
\;\|A^{1+s/2}u_*(t_{K+2})\|_{L^2}^2 \leq \;C(\delta^{-1}+1)(\|u_0\|^2_{D(A)}+1)^{K+1}\|u_0\|^2_{D(A)}\leq \;C(\varepsilon^{-1}+1)(\|u_0\|^2_{D(A)}+1)^{K+1}\|u_0\|^2_{D(A)},
g_n = \lambda g_{n-1}- g_{n-2}begin{align*}.2em] = \lambda \left(\phi(P_{l-1})g_{n-l}- \phi(P_{l-2}) g_{n-l-1}\right) - \left(\phi(P_{l-2})g_{n-l}- \phi(P_{l-3})g_{n-l-1}\right) begin{align*}.2em] = \left(\lambda\phi(P_{l-1}) - \phi(P_{l-2}) \right) g_{n-l}-\left(\lambda\phi(P_{l-2}) - \phi(P_{l-3}) \right) g_{n-l-1}begin{align*}.2em] = \phi(P_l)g_{n-l}- \phi(P_{l-1})g_{n-l-1},
\ln|\chi(\exp(C_E)\exp(C_D))| = \ln|\chi(\exp(C_E))|+\ln|\chi(\exp(C_D))| & = \ln|e^{\chi(C_E)}|+\ln|e^{\chi(C_D)}| \intertext{where we still denote by $\chi$ the Lie algebra character $\mathfrak{h}\rightarrow \mathbb{C}$ induced by $\chi$,}= \mathrm{Re}(\chi(C_E)+\chi(C_D)) & = \mathrm{Re}(\chi(C_E+C_D)).
\begin{aligned}\Vert e^{itH}P_a^{{\varepsilon}_a}(t)J_ae^{-itH_a}P^a_Mf-e^{isH}P_a^{{\varepsilon}_a}(s)J_ae^{-isH_a}P^a_Mf\Vert &\le\hskip-2pt M_2\hskip-2pt\left(\int_s^t\left\Vert\frac{1}{\sqrt{\tau}}Q_a^{{\varepsilon}_a}(\tau)J_ae^{-i\tau H_a}P^a_Mf\right\Vert^2d\tau\hskip-2pt\right)^{1/2}\hskip-14pt+Cs^{-{\delta}}\Vert f\Vert\to 0.\end{aligned}
\left( \begin{array}{l}{\tilde{\mathcal{F}}^{+}}{\tilde{G}_{+}}\end{array}\right) = \mathcal{S}\left( \begin{array}{l}{\mathcal{F}^{+}}{G_{+}}\end{array}\right) = \left( \begin{array}{l l}{A}{B}{C}{D}\end{array}\right) \left( \begin{array}{l}{\mathcal{F}^{+}}{G_{+}}\end{array}\right) \, .
N_{r,\varphi}^{+}= \left[\Lambda^{+}\right]^{-1}\left\{\tilde{\Omega}_{r - 1}^{+}[M_{r - 1,\varphi}] - C_{r - 1}\right\},\;N_{r,\varphi}^{-}= - \left\{\tilde{\Omega}_{r - 1}^{-}[M_{r - 1,\varphi}] - C_{r - 1}\right\},
\overline{c}_{X^{\ast}}(g_1, g_2)= (x(g_1), x(g_2))_F(-x(g_1)x(g_2), x(g_1g_2))_F ((-1)^t, \det(2q))_F (-1, -1)_F^{\tfrac{t(t-1)}{2}}\epsilon(2q),
\sum_{k=n}^\infty (1+k)^{-s}|h_{k}(x)|^2 \lesssim (1+n)^{-s+1-\frac{1}{\alpha}}, \sum_{k=n}^\infty \mathrm{e}^{- q k^p}|h_{k}(x)|^2 \lesssim \mathrm{e}^{-q n^p}(1+n)^{\frac{1}{3}-\frac{1}{\alpha}+\max(1-p,0)}.
q(q^4;q^4)^3_\infty B(q)= \sum_{n=0}^\infty \left(\frac{q^{4n+1}}{(1-q^{4n+1})^2}+\frac{q^{4n+3}}{(1-q^{4n+3})^2}\right)-\sum_{\substack{n=-\infty n\neq0}}^\infty\frac{(-1)^nnq^{2n(n+1)}}{1-q^{4n}}
(1+X)^n =\sum_{I\subseteq N}X^{|I|}=\sum_{k=0}^n\binom{n}{k}X^k, (1-X)^{-n}=\sum_{k_1,\ldots,k_n\ge 0}X^{k_1+\ldots+k_n}=\sum_{k=0}^\infty\binom{n+k-1}{k}X^k,
\frac{\partial Y}{\partial x}=\left(\frac{A_0}{x}+\frac{A_{t_1}}{x-t_1}+\frac{A_{t_2}}{x-t_2}+N \right)Y , \frac{\partial Y}{\partial t_1}=\left( N_1-\frac{A_{t_1}}{x-t_1}\right)Y ,\frac{\partial Y}{\partial t_2}=\left( N_2-\frac{A_{t_2}}{x-t_2}\right)Y .
\mu^p(p(l)=k, \xi_1=x_1, \ldots, \xi_k=x_k, \tau_1^{*}\in dt^1, \ldots , \tau_k^{*}\in dt^k) = Q_{x_2}^{x_1}\cdots Q_{x_1}^{x_k}\frac{t^1}{t^1+\cdots+t^k}(-L^{x_1}_{x_1})e^{L_{x_1}^{x_1}t^1}\cdots (-L^{x_k}_{x_k}) e^{L_{x_k}^{x_k}t^k}\,dt^1\cdots dt^k.
\tau (r_i)=\frac{1}{2\kappa^2}\frac{H'}{H}=\frac{1}{2\kappa^2}\frac{(p-7) +\frac{Q^2}{(3-p)\rho r_i^{3-p}}}{1-\frac{Q^2}{\rho (3-p)(10-2p)r_i^{3-p}}}\frac{1}{r_i}sim frac{1}{2\kappa^2}\Bigg(\frac{p-7}{r_i}+\frac{Q^2}{(10-2p)\rho}\frac{1}{r_i^{4-p}}\Bigg)
\langle r(\tilde{x}_k), \epsilon_{out}^k \rangle \leq \langle s_{\delta,L}(\lambda_k), \lambda_{\delta k}- \lambda_k \rangle & \leq \Phi_\rho(\lambda_{\delta k}) - \Phi_{\delta,L}(\lambda_k) + \frac{L}{2}B_{out}^2 + \delta.
S_{E}= - \frac{1}{16\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].
\sum_{\substack{j=1 j\neq k}}^{n-1}\frac{2}{t_k-t_j}-\sum_{j=1}^{n-1}\left(\frac{1}{t_k-\zeta_j}+\frac{1}{t_k-\overline{\zeta}_j}\right)=0,k=1,2,\dots, n-1.
S(\xi,\eta):=\frac{1}{(\sqrt{n+1}+\sqrt{m+1})^{2}t^\delta}\partial_{\xi}\phi(\xi,\eta) \theta_{s^\delta}(|\eta|)\theta_{s^\delta}(|\xi-\eta|)\psi^{s}(\xi,\eta)\frac{1}{\phi(\xi,\eta)}.
M= -\int^t_{t_0}|\partial_{v_1}+(r-t_0) \partial_{x_1}|^2 dr=- (t-t_0)\partial_{v_1}^2-(t-t_0)^2 \partial_{x_1}\partial_{v_1}-\frac{(t-t_0)^3}{3}\partial_{x_1}^2,
A_i = \frac{1}{\mu_i^+ - \mu_i^-}\left(u_i'(1) -\mu_i^- u_i(1) + \int_{\delta_i}^0 s^{1-\mu_i^+}f_i(s)\,ds\right), \bar{u}_i(r) = \frac{1}{\mu_i^+ - \mu_i^-}\left( r^{\mu_i^+}\int_{1-\delta_i}^r s^{1-\mu_i^+}{f}_i(s) \, ds - r^{\mu_i^-}\int_0^r s^{1-\mu_i^-}{f}_i(s)\, ds\right).
\begin{gathered}\textbf{i}^{2}=\textbf{j}^{2}=\textbf{k}^{2}=\textbf{i}\textbf{j}\textbf{k}=-1, \textbf{i}\textbf{j}=-\textbf{j}\textbf{i}=\textbf{k},\ \textbf{j}\textbf{k}=-\textbf{k}\textbf{j}=\textbf{i}, \textbf{k}\textbf{i}=-\textbf{i}\textbf{k}=\textbf{j}. \end{gathered}
M^{0,0}_{\frac{i+k-1}{2},\frac{2m-i-k+1}{2}}M^{\frac{2m-k+1}{2},\frac{2m-i-k+1}{2}}_{\frac{2m-i-k+1}{2},\frac{2m-i}{2}}=\frac{k+1}{2}M^{\frac{k-1}{2},\frac{k-1}{2}}_{\frac{i+k-1}{2},\frac{2m-i}{2}}
\mathbf{P}(\Omega_0)= \delta^{N+1}+O\left(\frac{N(\log{\Delta})^{N+1}}{\Delta}\right), \mathbf{E}_{\Omega_0}\left[\|\zeta(s,\mathbb{X}_\alpha)-\zeta_N(s,\mathbb{X}_\alpha)\|^2\right] \ll \mathbf{P}(\Omega_0) N^{1-2\sigma_0}+ L^{1-2\sigma_0}+ \frac{NL (\log{\Delta})^{N+1}}{\Delta}
0 =\gamma_2, 0 =\dfrac{u_1}{\delta'}+ \gamma_0 u_2^2 0 =u_1^2 + \dfrac{u_2}{\delta'}+ \gamma_1 u_2^2 + \dfrac{\alpha_2 \gamma_0 u_1 + \alpha_1 \gamma_0 u_2}{\delta'^2}= u_1^2 + \gamma_1 u_2^2 +\dfrac{\alpha_2 \gamma_0 u_1}{\delta'^2}+\left(\frac{1}{\delta'}+ \dfrac{\alpha_1\gamma_0}{\delta'^2}\right)u_2.
\tilde{m}_b={\frac{\beta^2}{8\pi}}(1-{\frac{\tilde{m}_0^2}{\tilde{\mu}^2(\varphi_0)}}) [{\frac{1}{\sqrt{1-\tilde{m}_b^2}}}tan^{-1}\sqrt{{\frac{1+\tilde{m}_b^2}{1-\tilde{m}_b^2}}}-{\frac{\pi}{4}}]
\Theta^n_s = \sigma(X_s) \int_0^s \psi_{n,1}(u,s) dW_u + X^n_{\eta_n(s)}- X^n_s = \sigma(X_s) \int_0^s \psi_{n,1}(u,s) dW_u + \int^s_{\eta_n(s)}\left(s-\eta_n(u)\right)^{\alpha}\sigma(X^n_{\eta_n(u)})\, dW_u + \int^{\eta_n(s)}_0 \psi_{n,2}(u,s)\, \sigma(X^n_{\eta_n(u)})\, dW_u.
|\hat{M}(\tau,i)-\hat{M}((\tau,i)-1)|\leq 1+\frac{\hat{\pi}(\tau-1)}{1-\hat{\pi}(\tau-1)}=\frac{1}{1-\hat{\pi}(\tau-1)}\stackrel{\tau\leq t}{\leq}\frac{1}{1-\hat{\pi}(t)}.
\frac{\partial \varrho (x,t)}{\partial t}= p(t) \left[ \delta_0 - \frac{\partial \varrho (x,t)}{\partial x}\right] - \varrho (x,t) f(x) - (u(t) - x) \frac{\partial \varrho ( x, t)}{\partial x}+ \varrho (x,t ).
q^{(n-1)(k-1)}\mathfrak{S}_{t_{k,n-k}}\big(1,q^{-1},q^{-2},q^{-3},\ldots\big) \qquad{}= \sum_{j=1}^{k}(-1)^{j-1}q^{{j \choose 2}}{n-1 \brack k-j}_{q}~{n -2+j\brack k+j-1}_{q}=\sum_{j=1}^{n-2}q^{j}\left({j+k-2 \brack k-1}_{q}\right)^{2}.
\epsilon ( g = 0 ) = H^{- 1 / 8}e^{-{\frac{1}{2}}\gamma_{\overline{{{\theta r}}}}}e^{-{\frac{1}{2}}\gamma_{\overline{{{\phi \theta}}}}}( 1 - \eta \gamma_{\overline{{{0}}}}\epsilon^{i j}) \epsilon_{0},
I_{p_1,q_1}\otimes\delta_1 u_{h_1}\oplus\cdots\oplus I_{p_m,q_m}\otimes\delta_m u_{h_m}\oplus\begin{bmatrix}0 I_{k_{m+1}}I_{k_{m+1}}0\end{bmatrix}\otimes 1\oplus\cdots\oplus\begin{bmatrix}0 I_{k_l}I_{k_l}0\end{bmatrix}\otimes 1
M'_{k, \ell}= \sum_{\substack{0 \leq s_1, \dots, s_q \leq m+1 0 \leq s_{q+1}\leq r}}\binom{r}{s_{q+1}}\left(\prod_{i=1}^q \binom{m+1}{s_i}\right) \left(\prod_{i=1}^{q+1}(-\lambda_i)^{s_i}\right)\binom{k- \sum_{i=1}^{q+1}s_i + f}{r_\ell}\lambda_{q_\ell+1}^{k-\sum_{i=1}^{q+1}s_i+f-r_{\ell}}.
Z \, = \, \int \prod_{x}\, D \varphi ( x ) \, \exp \left\{\int d^{D}x \, T r \left( - \frac 1 2 \, ( \partial_{\mu}\varphi )^{2}- \frac 1 2 M^{2}\varphi^{2}- V ( \varphi ) \right) \right\}
- i \left( \begin{matrix}{a^{\alpha}}&{b}\\ \end{matrix}\right) \left( \begin{matrix}{\langle U_{\alpha}, \bar{U}_{\bar{\beta}}\rangle}&{\langle U_{\alpha}, V \rangle}\\{\langle \bar{V}, \bar{U}_{\bar{\beta}}\rangle}&{\langle \bar{V}, V \rangle}\\ \end{matrix}\right) \left( \begin{matrix}{\bar{a}^{\bar{\beta}}}\\{\bar{b}}\\ \end{matrix}\right) = 0 \ .
\|x\|_k =\left\|\left(I-T\right)^k \sum\limits_{n=1}^{\infty}c_n e_n\right\| &=\left\|\sum\limits_{n=1}^{\infty}c_n \left(e_{n}-C_{k}^{1}e_{n+1}+\dots + (-1)^{k-1}C_{k}^{k-1}e_{n+k-1}+(-1)^{k}e_{n+k}\right)\right\| &=\left\|\sum\limits_{n=1}^{\infty}\left(c_{n}-C_{k}^{1}c_{n-1}+\dots+ (-1)^{k+1}C_{k}^{k-1}c_{n-k+1}+(-1)^{k}c_{n-k}\right) e_n\right\|,
\int_0^x \frac{(x-z)^{-\alpha}}{\Gamma(1-\alpha)}z^{\mu-1}(1-z)^\nu dz =\sum_{n=0}^\infty (-1)^n \begin{pmatrix}\nu \cr n \end{pmatrix}\frac{\Gamma(\mu+n)}{\Gamma(\mu-\alpha +n+1)}x^{\mu-\alpha+n}.
\sum_{n=0}^{\infty}p_{-7}(5n+3)q^{n}=140\dfrac{E_{5}^{5}}{E_{1}^{12}}+49\times5^{4}q\dfrac{E_{5}^{11}}{E_{1}^{18}}+21\times5^{7}q^{2}\dfrac{E_{5}^{17}}{E_{1}^{24}}\quad+91\times5^{8}q^{3}\dfrac{E_{5}^{23}}{E_{1}^{30}}+7\times5^{11}q^{4}\dfrac{E_{5}^{29}}{E_{1}^{36}}+5^{13}q^{5}\dfrac{E_{5}^{35}}{E_{1}^{42}}
\lim_{y \to \frac{1}{\beta}^{-}, \,\gamma \to \beta^{-}}T_{\gamma}(y) = 1,\lim_{y \to \frac{1}{\beta}^{+}, \, \gamma \to \beta^{+}}T_{\gamma}(y) = T_{\beta}(\frac{1}{\beta}) = 0;
\Sigma_q =\left(\begin{matrix}1-\frac{r}{r_2}1 1 1 \cdots 1 1 1 1-\frac{r}{r_4}1 \vdots \vdots 2-\frac{r}{r_2}2-r 2 2 \cdots 2 2 2-\frac{r}{r_4}2-r \vdots 2 2 2 2 \cdots 2-r\end{matrix}\right)
A_{j, i_j}:= \sum_{h=0}^L \left| \binom{L + i_j}{h}a_j^{L-h}\right|< \frac{1}{| a_j |^{i_j}}\sum_{h=0}^{L+i_j}\binom{L + i_j}{h}\left| a_j \right|^{L+i_j-h}&= \frac{1}{| a_j |^{i_j}}\left( | a_j | + 1 \right)^{L+i_j}= \left( | a_j | + 1 \right)^L \left( 1 + \frac{1}{| a_j |}\right)^{i_j},
u\big(\sigma^{-1}\big)(z)=\frac1{L\big(\frac1{z^*}\big)^*},l\big(\sigma^{-1}\big)(z)=\frac1{U\big(\frac1{z^*}\big)^*}, m\big(\sigma^{-1}\big)=m(\sigma)^*,a\big(\sigma^{-1}\big)=a(\sigma).
+ q^{x}\cdot \Phi_{\alpha , s}\Phi_{\bar{\alpha}, r s}) + ( q^{- x}\cdot \Phi_{\beta , r}\Phi_{\bar{\beta}, s s}+ q^{x}\cdot \Phi_{\bar{\beta}, s}\Phi_{\beta , r s}) - ( q^{- x}\cdot \Phi_{\beta , s}\Phi_{\bar{\beta}, r s}+ q^{x}\cdot \Phi_{\bar{\beta}, r}\Phi_{\beta , s s}) ]
q_N^{-1}(\tilde{\nabla}^2 u_{1,i})(x,t)=[M_{1,i}+o(1)]t^{-\frac{N+2A_1}{2}}\tilde{\nabla}^2[U_1(|x|)\theta_i]+O(t^{-\frac{N+2A_1}{2}-1})
u(x)= \begin{pmatrix}1 u_1 * \cdots * 1 u_2 \cdots * \cdots \cdots \cdots 1 u_{n-1}1 \end{pmatrix}, u'(x)= \begin{pmatrix}1 u_1' * \cdots * 1 u_2' \cdots * \cdots \cdots \cdots 1 u_{n-1}' 1 \end{pmatrix}.
w_l^{(n)}(\rho(t))=a_l^{(n)}\left(1+\sum_{k=1}^{(n-|l|)/2}\left[\prod_{s=1}^k\frac{n+2-(2s+|l|)}{l^2-(2s+|l|)^2}\right]\rho^{2k}(t)\right)\rho^{|l|}(t)\;,
\sum_{j=k}^N \delta_j^2 j^q = \sum_{j=k}^N b_j (a_j - a_{j+1}) = b_k a_k - b_{N}a_{N+1}+ \sum_{j=k}^{N-1}a_{j+1}( b_{j+1}- b_j ) &\leq k^q \, \sum_{j=k}^{\infty}\delta_j^2 + C\, \sum_{j=k}^{\infty}j^{- \rho}\, j^{q-1}\, .
d_M(z_0,z_{m_{N}})\leq \sum_{n=0}^{N}d_M(z_{m_{n+1}},z_{m_n})\leq \sum_{n=0}^{N}\left(\frac{3}{4}\right)^n d_M(z_{m_0},z_{m_1})=\frac{4}{5}\eta \ .
\left\Vert \varphi_k\right\Vert_{L^{2}}^{2}=\frac\pi4 \sum_{\substack{j=1 j\neq k}}^\infty \frac{\left(a_{c,\left\vert k-j\right\vert}-a_{c,k+j}\right)^{2}}{(k-j)^{2}(k+j)^{2}}\leq \frac{C}{k^{2}}\left\Vert a\right\Vert_{L^{2}}^{2}
(n+2)^{2}b_{n+2}=\frac{m_{0}^{2}}{8b^{2}}b_{n+1}-\sum_{k=0}^{n}(k+1)(-1)^{n-k}(\frac{1}{2^{n-k+1}}-1)b_{k+1}+\frac{m_{0}^{2}}{8}\sum_{k=0}^{n}(-1)^{n-k}b_{k}(\frac{1}{2^{n-k+1}}-2).
I^{( 4 )}= -{\frac{1}{1 6 \pi}}\int_{M^{4}}^{}R^{( 4 )}\sqrt{g}d^{4}x -{\frac{1}{8 \pi}}\int_{\partial M^{4}}^{}( K^{( 4 )}- K_{0}^{( 4 )}) \sqrt{h}d^{3}x ,
\left (\int_{G_{n,m}}|P_F(D)|^{-n}\,d\nu_{n,m}(F)\right )^{-\frac{1}{mn}}=|D|^{\frac{1}{n}}\left (\int_{G_{n,m}}|P_F(\tilde{D})|^{-n}\,d\nu_{n,m}(F)\right )^{-\frac{1}{mn}}.
\mathbb K^{(\alpha, \frac{1}{2})}(x,y)= \frac{1}{2\pi i (x-y)}\begin{pmatrix}-1 1 0\end{pmatrix}\Phi_{\alpha,+}^{-1}(y) \Phi_{\alpha,+}(x) \begin{pmatrix}1 1 0\end{pmatrix},
f_{1}\left( e^{- \frac{\pi}{s}}\right) = \sqrt{s}f_{1}\left( e^{- \pi s}\right) , \; \; f_{3}\left( e^{- \frac{\pi}{s}}\right) = f_{3}\left( e^{- \pi s}\right) , \; \; f_{2}\left( e^{- \frac{\pi}{s}}\right) = f_{4}\left( e^{- \pi s}\right) .
V ( \overline{{\rho}}) = - \frac{1}{1 2 \pi}[ | M_{+}( \overline{{\rho}}^{2}) |^{3}+ | M_{-}( \overline{{\rho}}^{2}) |^{3}] + \frac{\lambda}{4 !}\overline{{\rho}}^{4}+ \frac{\eta}{6 !}\overline{{\rho}}^{6}.
I_2(r)=C_{\nu}e^{-i(\frac{\nu\pi}2+\frac{\pi}4)}\int_0^\infty(r\rho)^{-\frac{n-1}2}e^{i(r\rho+t\rho^2)}\chi_{N}(\rho)\rho^{n-1}\mathrm{d}\rho,
\frac{1}{A^{r}B^{s}}= \frac{\Gamma ( r{+}s )}{\Gamma ( r ) \Gamma ( s )}\int_{0}^{1}\frac{x^{r - 1}( 1{-}x )^{s - 1}\, d x}{( A x + B ( 1{-}x ) )^{r + s}}
\pi_{\mu \nu}= - \frac{1}{4}\tau_{\mu \nu}\tau_{\nu}^{\alpha}+ \frac{1}{1 2}\tau \tau_{\mu \nu}+ \frac{1}{8}q_{\mu \nu}\tau_{\alpha \beta}\tau^{\alpha \beta}- \frac{1}{2 4}q_{\mu \nu}\tau^{2}
f'(x) = \frac{b' \left( a^{-1}(x) \right)}{a' \left( a^{-1}(x) \right)}= - \frac{a \left( a^{-1}(x) \right) b(a^{-1}(x))^2}{a \left( a^{-1}(x) \right)^3/2 + a \left( a^{-1}(x) \right) b \left( a^{-1}(x) \right)^2/2}= \frac{2xf(x)^2}{x^3 + xf(x)^2}= \frac{2}{(x/f(x))^2 + 1}\;.
s^C-s=R_{\alpha\enskip\gamma}^{\enskip\alpha\enskip\gamma}+ R^{\alpha\quad\gamma}_{\enskip\gamma\alpha}= (\nabla_{\bar\alpha}T)_\gamma^{\enskip\bar\alpha\gamma}- T^{\alpha\qquad\;\gamma}_{\enskip T(\gamma,\alpha)}= (\nabla_\alpha T)_{\bar\gamma}^{\enskip\alpha\bar\gamma}- T^{\bar\alpha\qquad\;\bar\gamma}_{\enskip T(\bar\gamma,\bar\alpha)}.
\big(\big(n,\big(\frac{r_1}{r_2},w\big)\big),b\big) = \big(\big(bn,\big(\frac{\tilde{b}r_1}{r_2},w\big)\big),1\big) = \big(\big(bn,\big(\frac{\tilde{b}r_1}{r_2},0\big)\big),1\big) &= \big((bn,(r',0)),1\big)
\lim_{s\rightarrow\infty}\left( \max_{j,m\in\left\{-k_{s},k_{s}\right\}}\int\nolimits_{[0,h]}\left\vert \left( \alpha_{k_{s}}(t)\right)^{-1}u_{j}(t)u_{m}^{\ast}(t)\right\vert dt\right) =\infty,
\begin{array}[c]{rl}i) \ \;A\,\;(H_{1})\medskip ii) u_{0}\in\overline{D\left( A\right)},\medskip iii) f\in L^{1}\left( 0,T;H\right) ,\medskip iv) M\in C\left( \left[ 0,T\right] ;X\right) ,\;\;M\left(0\right) =0.\end{array}
J = \frac{1}{i ( 2 \pi )^{2}}\left[ \int D v D p \delta^{4}\left( \int v d \tau \right) \exp \left\{i \int d \tau \left( - \frac{v^{2}}{2}+ \frac{p^{2}}{2}\right) \right\}\right]^{- 1}
\prod_{pprime}\frac{1}{1-p^{-s}}=\frac{1}{1-2^{-s}}\cdot\frac{1}{1-3^{-s}}\cdot\frac{1}{1-5^{-s}}\cdot\frac{1}{1-7^{-s}}\cdot\cdot\cdot\frac{1}{1-p^{-s}}\cdot\cdot\cdot
H^f_{111}= \frac{\partial H^f_{11}}{\partial x_1}-2\Gamma^1_{11}H^f_{11}-2\Gamma^2_{11}H^f_{12}, & H^f_{121}= \frac{\partial H^f_{12}}{\partial x_1}-\Gamma^1_{11}H^f_{12}-\Gamma^2_{11}H^f_{22}-\Gamma^1_{12}H^f_{11}-\Gamma^2_{12}H^f_{12}, & H^f_{221}= \frac{\partial H^f_{22}}{\partial x_1}-2\Gamma^1_{12}H^f_{12}-2\Gamma^2_{12}H^f_{22}.
\sum_{M\geq0}g_1(M)\,z^M =\sum_{k\geq1}\frac{z^{2k}}{(1-z)^{k+1}}+\sum_{k\geq0}\frac{z^{2k+2}k}{(1-z)^{k+2}}&=\sum_{k\geq0}\left(\frac{z^{2k+2}}{(1-z)^{k+2}}+\frac{z^{2k+2}k}{(1-z)^{k+2}}\right) &=\sum_{k\geq0}\sum_{j\geq0}z^{2k+2+j}\binom{k+1+j}j(k+1).
\frac{172}{441}f \left(\frac{1}{2}\right)-\frac{1625}{1611}f \left(\frac{1}{5}\right) + \frac{42592}{26313}f \left(-\frac{27}{44}\right)= -\frac{1}{4 \pi \sqrt{-1}}\int_{S^1}f(z) e^{-\frac{2}{z}}dz
\bar{G}_h(s_1,\xi)= \frac{\alpha_n}{2}a_2\partial_{\xi_h}\varphi(\xi)s_1^{n-2},\quad\mbox{for}h=1,\cdots,n, \bar{G}_0(\bar{s},\bar{\sigma},\xi) = \alpha_n a_1 s_1^{n-2}\varphi(\xi) + a_3 \sum\limits_{i=2}^{k}s_i^{\frac{n-2}{2}}g(\sigma_i) - a_4\sum\limits_{i=1}^k \frac{2}{2i-1}|\ln s_i|.
F^{ij}\{D_{ij}(\bar u_\epsilon -u)-[A_{ij}(x,\bar u_\epsilon,D\bar u_\epsilon)-A_{ij}(x,u,Du)]\}\geq F[\bar u_\epsilon]-F[u]= F[\bar u_\epsilon]- \tilde B(x,u,Du)\geq - C_1,
\langle X(v_{1}, \dotsc, v_{k}), v_{i}\rangle = 0 \, , 1 \leq i \leq k \, . &\langle X(v_{1}, \dotsc, v_{k}), X(v_{1}, \dotsc, v_{k}) \rangle = \det (\langle v_{i}, v_{j}\rangle) \, .
\sum\limits_{\substack{(A,B)\in \mathbb{Z}^2 H(E_{A,B})\le X}}\mu^2\left(\Delta_{A,B}\right)=4X^{10}\prod\limits_{p}\left(1-\frac{\sigma\left(p^2\right)}{p^4}\right) + O\left(X^{7+\varepsilon}\right).
d{\cal L}_{4}= d{\cal L}_{4}^{0}+ d{\cal L}_{4}^{W Z}= i ( \hat{E}^{( - )}\gamma^{( 3 )}\hat{E}^{( - )}) - 2 ( \hat{E}^{( - )}\tilde{\gamma}^{( 4 )}\hat{\Lambda}) ,
x_{1}[\begin{matrix}a_{11}\\a_{21}\\ \vdots\\a_{m1}\end{matrix}]+x_{2}[\begin{matrix}a_{12}\\a_{22}\\ \vdots\\a_{m2}\end{matrix}]+...+x_{n}[\begin{matrix}a_{1n}\\a_{2n}\\ \vdots\\a_{mn}\end{matrix}]+[\begin{matrix}b_{1}\\b_{2}\\ \vdots\\b_{m}\end{matrix}]=0
\mathbb{P}[X^n=\mathbf{x}_1, Y^n=\mathbf{y}_1]=\mathbb{P}[X^n=\mathbf{x}_2, Y^n=\mathbf{y}_2]=p_{11}^{n/2}p_{22}^{n/2}&\mathbb{P}[X^n=\mathbf{x}_2, Y^n=\mathbf{y}_1]=\mathbb{P}[X^n=\mathbf{x}_1, Y^n=\mathbf{y}_2]=p_{12}^{n/2}p_{21}^{n/2}.
\sum_{Q-Q_1\in \Delta^-}\Big(\big\lfloor w(pQ)\big\rfloor-\big\lfloor w(Q)\big\rfloor- w(pQ)+w(Q)\Big)\geq 0 \textrm{\for every\}Q_1=\sum\limits_{i=1}^n m_i\mathbf V_i\in \Lambda_{\Delta}.
\limsup_{n \rightarrow \infty}\|\sum_{\nu=0}^{n+|\textup{\textbf{m}}|-m_\alpha}b_{\nu,n}^{(\alpha)}\Phi_\nu\|_K^{1/n}\leq \frac{\|\Phi\|_K}{\rho_{|\textup{\textbf{m}}|}(\textup{\textbf{F}})}.
-[R]^{2}=|R|^{2}[E_{3}]-[RR^{T}]=[\begin{matrix}x^{2}+y^{2}+z^{2}&0&0\\ 0&x^{2}+y^{2}+z^{2}&0\\ 0&0&x^{2}+y^{2}+z^{2}\end{matrix}]-[\begin{matrix}x^{2}&xy&xz\\yx&y^{2}&yz\\zx&zy&z^{2}\end{matrix}],
\Pi(q^2) = -{1\over 48\pi^2}\left[{1\over{\bar\epsilon}}+{1\over 2}\ln\left({\mu^2\over 4kT}\right)\right]+{1\over 16\pi^2}\int_0^1 dx x \sqrt{1-x^2}\ln N_{++}\left({x\sqrt{q^2}\over 2}\right),
F\!\left(\sum_{j=1}^n q_j( \xi_I - \rho\cdot \xi_I)t_j\right) \cdot (\rho \ast \kappa_I(x)) = \ \psi(c_I(\rho))\cdot (\rho \ast \kappa_I(x)) = \varphi_I(\rho) \cdot \kappa_I(x)\ ,
z^{\lambda}G_{p q}^{m n}\left( z \left| \begin{array}{l}{{a_{r}}}\\{{b_{s}}}\\ \end{array}\right. \right) = G_{p q}^{m n}\left( z \left| \begin{array}{l}{{a_{r}+ \lambda}}\\{{b_{s}+ \lambda}}\\ \end{array}\right. \right) .
\frac{F_d(x + c_d{\rm d}t) - F_d(x)}{{\rm d}t}= c_d \frac{F_d(x + c_d{\rm d}t) - F_d(x)}{c_d{\rm d}t}\underset{{\rm d}t \downarrow 0}{\longrightarrow}c_d F_d'(x),
\varlimsup_{n\to\infty}2^{n}\mathbb{P}\Big(\bigcup_{x\le z^{*}}\bigcup_{z}B^{1}(x,z,y,2^{-n})\,\Big|\,\sigma_{0}\Big)\le\varlimsup_{n\to\infty}2^{n}\mathbb{P}(B^{1}(z^{*},y,2^{-n})\,|\,\sigma_{0})=0.
G=G_{0}+R_{3}+4R_{2}=(R_{11}:2R_{3})+R_{3}+4R_{2}=(\zeta_{6}+\zeta_{6}^{5})(\zeta_{11}+\zeta_{11}^{10})+\sum_{i\neq1,10}\zeta_{11}^{i}+R_{3}+4R_{2}.
q_1(\alpha,\beta,z)=\frac{(1-|z|^2)^{\beta}}{(1-|\varphi(z)|^2)^{\alpha}}|\psi(z)||\varphi'(z)|; q_2(\beta,z)=(1-|z|^2)^{\beta}\log \frac{1}{1-|\varphi(z)|^2}|\psi'(z)|; q_3(\alpha,\beta,z)=\frac{(1-|z|^2)^{\beta}}{(1-|\varphi(z)|^2)^{\alpha -1}}|\psi'(z)|,
d(\pi(g),\pi(h)) = d(\pi(hs_1 \dots s_k s_{k+1}),\pi(h)) & \leq d(\pi(hs_1 \dots s_k s_{k+1}), \pi(hs_1 \dots s_k)) + d(\pi(hs_1 \dots s_k), \pi(h)) & \leq \ell(\lambda)+B + (\ell(\lambda)+B)k & = (\ell(\lambda)+B)(k+1).
v_p(f^{n-1}_{\alpha, \beta}(f_{\alpha, \beta}(0))) = \infty, v_p(f^{n-1}_{\alpha, \beta}(f_{\alpha, \beta}(1))) = 0, v_p(f^{n-1}_{\alpha, \beta}(\beta)) = \infty, \quad v_p(f^{n-1}_{\alpha, \beta}(\alpha B_{d,k}(1) + \beta)) = 0.
\delta \Gamma_{\sf{a n o m .}}^{\sigma}= \frac{- 3 6 g^{2}}{( 4 \pi )^{2}M^{2}}\int_{0}^{1}d x d y \; \delta ( x + y - 1 ) \Biggl \{\epsilon m Q^{\sigma}\left[ \frac{2}{\epsilon}- \gamma + \ln \left( \frac{4 \pi \mu^{2}}{\Delta}\right) \right] \Biggr \}.
(\begin{pmatrix}A 0 a+r(t)^{\top}0\end{pmatrix},\begin{pmatrix}B-tId 0 b 0\end{pmatrix},\begin{pmatrix}I X+Y^{\top}(t)\cdot\Omega^{-1}\end{pmatrix},\begin{pmatrix}1\!\!1_{n-1}0 0 0\end{pmatrix}).
\beta_2(\partial_1(a) \cdot b) - \beta_1(\partial_2(a) \cdot b) + \partial_1(a) \cdot \partial_2(b) - \partial_2(a) \cdot \partial_1(b) \qquad{}\overset{\eqref{eq:qadm1}}{=}\beta_2(\partial_1(a)) \cdot b - \beta_1(\partial_2(a)) \cdot b, \forall a,b \in A.
={\alpha \over \pi}\sum_{k=0}^{\infty}\sum_{n=0}^{\infty}{\Gamma(z+k)\Gamma(2z+2k+2n-1) \over 2^{2k}k!~\Gamma(z)\Gamma(2z+2k)}~\zeta_R(2z+2k+2n-1,1/2)~{B_{2n}\over (2n)!}\left({2\pi \over \alpha}\right)^{2n}
E\vert_{S}=\left \{\left(q^i,\frac{\partial W}{\partial q^i};\frac{\partial F}{\partial p_i},-\frac{\partial F}{\partial q^i}\right )\in TT^*Q:\frac{\partial F}{\partial \lambda^a}=0,\frac{\partial W}{\partial \mu^\beta}(q,\mu) =0 \right\},
\begin{cases}c'_{i1}=\sum_{0\le u\le \xi-2-i}\binom{i+u}{i}t^uc_{i+u,1}+\binom{\xi-1}{i}t^{\xi-1-i}\sigma &+t^{\xi-i}\left(\binom{\xi}{i}+O(t)\right)+O(t^{\xi-i},\overline c),i=0,...,\xi-2, c'_{i1}=\sum_{u\ge0}\binom{i+u}{i}t^uc_{i+u,0},i=0,...,\xi+\eta-1, \sigma' =\sigma+t(\xi+O(t,\overline c)).\end{cases}
\begin{array}{c}{e_{1}e_{2}= e_{3}\ , \e_{2}e_{3}= e_{1}\ , \e_{3}e_{1}= e_{2}}\\{e_{3}e_{2}= - e_{1}\ , \e_{2}e_{1}= - e_{3}\ , \e_{1}e_{3}= - e_{2}\ .}\\ \end{array}