harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left( \Sigma^{\mu \nu}\right)^{\rho \tau \alpha \beta}= \frac 1 2 \left[ \eta^{\mu \rho}\eta^{\nu \alpha}\eta^{\tau \beta}+ \eta^{\mu \tau}\eta^{\nu \beta}\eta^{\rho \alpha}+ \left( \alpha \leftrightarrow \beta \right) - \left( \mu \leftrightarrow \nu \right) \right] ,
\sum_{j=1}^h \Big((k_2(0)+1)k_1(j)-(k_1(0)+1)k_2(j)\Big)d^{\widehat{Q}_j}&-(k_2(0)+1)(1+\sum_{i=0}^hk_1(i))d^{\widehat{Q}^{+r}}+(k_1(0)+1)(1+\sum_{i=0}^hk_2(i))d^{\widehat{Q}^{+s}}.
|\varphi_i (z)-U_1(z)| \le C \begin{cases}\max\{\Theta_im_i^{-2}m_i^{\frac{2}{n-4}}(1+|z|)^{-1}, m_i^{-2}\}, \mbox{if}n=8, \max\{\Theta_i m_i^{-2}m_i^{\frac{2(n-8)}{n-4}}(1+|z|)^{8-n}, m_i^{-2}\}, \quad\mbox{if}n\ge 9.\end{cases}
\int_{B_{\rho}}r^{2-n}|\nabla u|^2 \d x \leq \theta^k \int_{B_{8^k\rho}}r^{2-n}|\nabla u|^2 \d x \leq \frac{1}{\theta}\rho^{2\alpha}\int_{B_1}r^{2-n}|\nabla u|^2 \d x,
x_{0}=0 y_2 = y_1 \cdot y_1 z_2 =y_k \cdot z_1 x_{1}=1 y_3 = y_2 \cdot y_1 z_3 = z_1 \cdot y_k & \;\;\vdots & y_{m-1}= y_{m-2}\cdot y_1 & x_1 = y_{m-1}\cdot y_1
H_{p}= \frac{1}{2}( \pi_{1}^{2}( 1 + g^{2}x_{2}^{2}) + \pi_{2}^{2}( 1 + g^{2}x_{1}^{2}) - 2 g^{2}x_{1}x_{2}\pi_{1}\pi_{2}) + V ( \rho^{2})
\sum_{1\leq i_1<\ldots <i_r \leq n}\ddot{f}_{i_1}\ldots \ddot{f}_{i_r}\Big(1+\sum_{{1\leq k\leq n}\atop{k\notin \{i_1,\ldots i_r\}}}\dot{f}_{k}^{2}\Big)=0.
\|p_{n,\xi}\|_{X^2(\xi)}=\sum_{k=1}^{q+M}\delta_{n,k}(\xi)\|u_{n,k}(\xi)\|_{H^1_\xi}^2 \leq (q+M) \sup_{k\geq 1}\|\delta_{n,k}(\xi)u_{n,k}(\xi)\|_{H^1_\xi}^2.
A_{i}^{\prime}\,\left( A+\delta_{\lambda}A\right) -A_{i}^{\prime}\,\left( A\right) -\partial_{i}\lambda^{\prime}-i\left[ \lambda^{\prime},A_{i}\right] -i\left[ \lambda ,A_{i}^{\prime}\right] =-\frac{1}{2}\theta^{kl}\left( \partial_{k}\lambda \partial_{l}A_{i}+\partial_{l}A_{i}\partial_{k}\lambda \right) +o \left( \theta^{2}\right) ,
\lambda_j(u_j,u^\psi)_{L^2(M,e^\phi d\mu_g)}= - \int_M \Delta_g^Fu_j(x) u^\psi(x) e^{\phi(x)}d\mu_g(x) = \int_M u_j(x) \Delta_g^Fu^\psi(x) e^{\phi(x)}d\mu_g(x) + \langle u_je^\phi\arrowvert_{\partial M}, \psi\rangle.
A(-1/2)={}_4F_3\biggl(\begin{matrix}\frac12, \, \frac12, \, \frac12, \, \frac12 1, \, 1, \, 1 \end{matrix}\biggm|1\biggr)=\sum_{k=0}^\infty\frac{(\frac12)_k^4}{k!^4}
\delta s^{2}= \Omega^{2}( t , w , y ) [ d x^{2}+ g_{2}\left( t , w \right) d t^{2}+ g_{3}\left( t , w \right) d w^{2}+ h_{4}( t , w , y ) \delta y^{2}+ h_{5}\left( t , w \right) d z^{2}] ,
\frac{4\pi v^{1-s}}{2s-1}\bigg(\delta_{p,q}y_{p}^{s}+\alpha_{p,q}(s)y_{p}^{1-s}+\sum_{n\not = 0}\alpha_{p,q}(n,s)W_{s}(n\sigma_{p}^{-1}z)\bigg)+\frac{4\pi\delta_{p,q}}{2s-1}\big(v^{s}y_{p}^{1-s}-v^{1-s}y_{p}^{s}\big).
\Phi = i \sum_{i = 1}^{N}\left({\frac{\partial H}{\partial q_{i}}}{\frac{\partial}{\partial p_{i}}}-{\frac{\partial H}{\partial p_{i}}}{\frac{\partial}{\partial q_{i}}}\right) + \sum_{i , j = 1}^{N}{\frac{\partial H}{\partial p_{i}}}G_{i j}{\frac{\partial}{\partial p_{j}}}
\tilde M_2(p,q,\tau,s,u_0)= C_6 \Big( \int_\Omega u^{-p}(\tau,x)dx+M_1(p)+\tilde M_1(p,\tau,s,u_0)\Big)^{\frac{2(q+1)(q+1-\varepsilon_0/2)}{\varepsilon_0 p}}&- C_6 \Big( 2 M_1(p)\Big)^{\frac{2(q+1)(q+1-\varepsilon_0/2)}{\varepsilon_0 p}}.
b_s^{}= (-1)^{2n+1-m-s}q (j^* \sigma_{2n+1-m,2n-m,\dots,2n+3-2m,2n+2-2m-p}^+)^\vee &= (-1)^{2n+1-m-s}q \sigma_{2n-m,2n-1-m,\dots,2n+2-2m,2n+1-2m-p}^\vee,
\Sigma_{\mathrm{RDJ}}(\delta):= \left\{\gamma \in \mathcal{P}(\mathcal{S}): \ \boldsymbol{K}_1( \widehat{\mu}_{13}, \gamma_{13}) \leq \delta_1, \ \boldsymbol{K}_2(\widehat{\mu}_{23}, \gamma_{23}) \leq \delta_2 \right\},
\mu\bigl((\theta\otimes\operatorname{id})(E)\bigr) =\mu\bigl((\nu\otimes\operatorname{id})(E(b\otimes1))\bigr)=\mu\bigl(\gamma_B(b)\bigr) &=\mu\bigl((R\circ\sigma^{\nu}_{i/2})(b)\bigr)=\nu(b)=\theta(1).
X \equiv <\!0|\chi^2|0\!> = \int d^3 x d^3 y{F(\mathbf{x}) F(\mathbf{y}) + \mathbf{\nabla}G(\mathbf{x})\!\cdot\!\mathbf{\nabla}G(\mathbf{y}) \over 4\pi^2 |\mathbf{x}-\mathbf{y}|^2}
\Phi\wedge F_{A}=-\left[3Q^{2}a^{i}(r)\theta^{234567}+\left(Q^{2}\frac{da^{i}}{dr}+2PQ\alpha(r)\right)dr\wedge\left(\theta^{13467}+\theta^{12457}+\theta^{12356}\right)\right]\otimes e_{i},
\frac{p_{n-1}^{\prime}(x)}{p_n(x)}=\sum_{k=1}^{n}\frac{p_{n-1}^{\prime}(x_k)}{p_n^{\prime}(x_k)}\cdot\frac{1}{x-x_k}=\sum_{k=1}^{n}\frac{x_k}{x-x_k}&=\frac{1}{2}\sum_{k=1}^{n}\Big(\frac{x_k}{x-x_k}-\frac{x_k}{x+x_k}\Big)=\sum_{k=1}^{n}\frac{x_k^2}{x^2-x_k^2}\,.
\int \frac{x(qx;q)_\infty}{(q r_n x;q)_\infty}p_n(x|q) d_qx &= \frac{q^{n}x(x;q)_\infty}{[n]_q [n+1]_q(r_nx;q)_\infty}\left(\frac{1}{q}(1-x)\,_2\phi_1(q^{-n+1}, q^{n+2};q^2;q,x)- p_n(\frac{x}{q}|q)\right).
\langle \widetilde Y_{\tau}^{t,x}, h \rangle = \langle \widetilde Y_{0}^{t,x}, h \rangle - \int_0^\tau \langle e^{-s{A}}G B(s,\Xi^{t,x}_s), h \rangle \,ds - \int_0^\tau \langle e^{-s{A}}Z_s^{t,x}\, B(s,\Xi^{t,x}_s), h \rangle \,ds + \int_0^\tau \langle d W_s , (Z^{t,x}_{s})^* e^{-s A^*}h \rangle_U
a_{n+1}=\sqrt{\frac{\|Q_{2^s}\left(\cdot;\mu_{K(\gamma)}\right)\|_{L^{2}\left(\mu_{K(\gamma)}\right)}^2-a_{2^{s+1}k}^2\cdots a_{2^{s+1}k-2^s+1}^2}{a_{2^s(2k+1)-1}^2\cdots a_{2^{s+1}k+1}^2}},
{\bf \Phi}_{1}^{( n_{\rho}, m )}={\bf \Phi}_{1}^{( n_{\rho}, m )}( \rho , k ) = \left( \begin{array}{c}{{\phi_{1}^{( n_{\rho}, m )}( \rho , k )}}\\{{\phi_{2}^{( n_{\rho}, m )}( \rho , k )}}\\ \end{array}\right)
I_6\leq c\int_0^\tau\frac{1}{|\Omega_R|}\int_{\Omega_R}p(\rho)\varepsilon^{-2}\int_{\Omega_R}\left(P(\rho)-P(r)-P'(r)(\rho-r)\right) \leq c\int_0^\tau\frac{1}{|\Omega_R|}\int_{\Omega_R}p(\rho) \mathcal E(\rho, u|r,U)(t)
R_{u,vs_j}^{J,q}(q)= (-1)^{\ell(vs_j)-\ell(u)}(1-q)^2\prod_{k\in D(u,vs_j)}\left(1-q^{a_k(u,vs_j)}\right) begin{align*}5pt]= (-1)^{\ell(v)-\ell(u)-1}(1-q)^2\prod_{k\in D(u,v)}\left(1-q^{a_k(u,v)}\right).
\frac{1}{\epsilon}(a \,\sharp^{\epsilon}\, b\,-\,a \, \sharp^{0}\, b) (X) &=\,-\frac{4i}{(2\pi)^{4}\epsilon}\int_{\Xi\times\Xi}e^{-2i\sigma(Y,Z)}\left(\int_0^1e^{-4itF_{\epsilon}(x,y,z)}dt\right)F_\epsilon(x,y,z) \, a(X-Y)\,b(X-Z)\,dYdZ\,.
c_{n_1}(x_2, y_2, \ldots, x_k, y_k)=\frac{\partial^{n_1}f(0, y_1, x_2, y_2, \ldots, x_k, y_k)}{(b_1; q)_{n_1}n_1! \partial{y_1}^{n_1}}\Big|_{y_1=0}.
\int_{-L}^L \max(|t-s|,2)^p \, ds = \int_{-L}^{t-2}\! \left( t-s \right)^{p}{ds}+4\cdot{2}^{p}+\int_{t+2}^{L}\! \left( s-t \right)^{p}{ds}={\frac{\left( t+L \right)^{p+1}-{2}^{p+1}}{p+1}}+4\cdot{2}^{p}+{\frac{\left( L-t \right)^{p+1}-{2}^{p+1}}{p+1}},
\|\Box_k^{\alpha_2}f\|_{M_2}=\|\Box_k^{\alpha_2}f\|_{M_{2,2}^{0,\alpha_2}}=\|\Box_k^{\alpha_2}f\|_{M_{2,2}^{0,\alpha_1}}=\left(\sum_{l\in \Gamma_k^{\alpha_1,\alpha_2}}\|\Box_l^{\alpha_1}\Box_k^{\alpha_2}f\|^2_{L^2}\right)^{1/2}.
0=F_u^{\prime\prime}+\frac{2}{r}F_u^\prime \left[1+rL_-^\prime(kr)\right]-\frac{s^\prime}{s+\frac{\omega}{m}}\left[F_u^\prime+F_u L_-^\prime(kr)\right] +W(r,-m) F_u -\left[K,F_u\right]
\mathcal E_{L,R}(u) \leq\frac{L^{-2/3}}{a_2}\left(g\big((1+\eta)a_2L^{2/3}\big)+\frac{C\sqrt{1+\eta}\,L^{1/3}}{\ell}\right)(\sqrt{a_2}\ell)^2+C\eta^{-1}\ell^6 &=L^{-2/3}\left(g\big((1+\eta)a_2L^{2/3}\big)+\frac{C\sqrt{1+\eta}\,L^{1/3}}{\ell}\right)\ell^2+C\eta^{-1}\ell^6\,,
{{\mathcal B}_i^{DF}}\left({\pmb \rho}^\prime \right) = \left[{\left({{X_i}{W_i^\prime}+{X_i}+{Y_i}{Z_i}+{Z_i}}\right) -}\right. &\left.{\sqrt{{{\left({{X_i}{W_i^\prime}+{X_i}-{Y_i}{Z_i}+{Z_i}}\right)}^2}+ 4{Y_i}Z_i^2}}\right]/\left({2{Y_i}{Z_i}}\right),
\frac{\pi_E}{\tau(\pi_E)}\mathrm{Fr}^{d_0}(\tilde{f}_0) - \tilde{f}_0 = \left(\frac{\pi_E}{\tau(\pi_E)}- 1\right)\mathrm{Fr}^{d_0}(\tilde{f}_0) + \mathrm{Fr}^{d_0}(\tilde{f}_0 - [\tilde{\mu}]) - (\tilde{f}_0 - [\tilde{\mu}]).
P_{\texttt{a};0}(\boldsymbol{\xi};q) =P_{\texttt{b}; 0}(\boldsymbol{\xi};q,q_0) = \prod_{1\leq j<k\leq n}\frac{1-q^{1+k-j}}{1-q^{k-j}}= \prod_{1\leq j\leq n}\frac{1-q^j}{1-q}.
I_1 = \int_{-1/2}^{1/2}\int_{\sqrt{1-x^2}}^\infty |F_{k,m}(x+iy)|^2 y^{k-2}dy dx \leq \int_{\sqrt{3}/2}^\infty y^{k-2}\int_{-1/2}^{1/2}|F_{k,m}(x+iy)|^2 dx dy = \int_{\sqrt{3}/2}^\infty y^{k-2}\left(e^{-4\pi m y}+ \sum_{n=\ell}^\infty |A_k^{(1)}(m,n)|^2 e^{-4\pi n y}\right)dy .
\left(\eta_{\mu\nu}\partial_\mu\partial_\nu + m^2 +\xi R ++{1\over 3}R^\nu_\alpha y^\alpha \partial_\nu-{1\over 3}R^{\mu\ \nu}_{\ \alpha\ \beta}y^\alpha y^\beta\partial_\mu\partial_\nu+ O(R^2)\right)\phi =0
(\begin{matrix}1&1&1&1&1&0&0&0&0&0\\ 1&1&0&0&0&1&1&1&0&0\\ 1&0&1&0&0&1&0&0&1&1\\ 0&1&0&1&0&0&1&0&1&1\\ 0&0&1&0&1&0&1&1&1&0\\ 0&0&0&1&1&1&0&1&0&1\\ \end{matrix})
\begin{array}{l}{C l ( \eta_{1}) > 0 \Longrightarrow C l ( \eta_{1}\oplus \eta_{2}) = C l ( \eta_{1}) \otimes C l ( \eta_{2})}\\{C l ( \eta_{1}) < 0 \Longrightarrow C l ( \eta_{1}\oplus \eta_{2}) = C l ( \eta_{1}) \otimes C l ( - \eta_{2}) .}\\ \end{array}
I_{m}^{(2)}\left(2q^{-\ell/2};q\right)=I_{|m|}^{(2)}\left(2q^{-\ell/2};q\right)=\frac{q^{\ell|m|/2}S_{\ell}\left(-q^{-|m|-\ell};q\right)}{\left(q^{\ell+1};q\right)_{\infty}}.
\mathbb{P}\Big( \Big| \Big| \frac{1}{n}\sum_{i=b+1}^n \mathbf{y}^M_i(\mathbf{y}_i^M)^*-\Sigma \Big| \Big| \geq t \Big) \leq CM \sup_i \mathbb{P}\Big( \Big|\Big| \frac{1}{n}\sum_{k \in \mathcal{I}_i}\mathbf{y}_k^M (\mathbf{y}_k^M)^*-\Sigma \Big|\Big| \geq t/M \Big) & \leq CMbc \exp\Big(\frac{-t^2/(2M^2)}{\sigma_M^2+R_Mt/3M}\Big),
\mu:=1+\sum_{j\ge1}\prod_{i=0}^{j-1}\alpha_i\,\,\,\,\,\,\,\textrm{and}\,\,\,\,\,\,\sigma^2:= \sum_{k\ge1}k^2(1-\alpha_{k-1})\prod_{i=0}^{k-2}\alpha_i-\mu^2.
V^{n,r}= V^{n,0}+ \sum\limits_{k=1}^r \frac{1}{m^n_0 + p^n_0 + k + 2}\left( \left( \bar v^{n,m^n_0 + p^n_0 + k+1}- \beta^0_{(1)}\right) - V^{n,k-1}\right).
\frac{1}{\kappa^{n}(s) r_{s}(x)}\mathbb{E}\Big[ r_{s}(X_{n}^{x}) |G_nx|^{s}\varphi \big( X_{1}^{x}, \log |G_1 x|,\dots, X_{n}^{x}, \log |G_n x| \big) \Big] &=\mathbb{E}_{\mathbb{Q}_{s}^{x}}\Big[ \varphi \big( X_{1}^{x}, \log |G_1 x|,\dots, X_{n}^{x}, \log |G_n x| \big) \Big].
\left| \sigma_n^{(p+1)}\right| \le 2R^{n+p+3}\frac{(n+2+p)!}{(n+2)!}\: + & \quad\quad\sum_{j=0}^{n-1}\sum_{k=0}^p \binom{p}{k}R^{n+p+3}\frac{(j+1+k)!}{(j+1)!}\frac{(n-j+p-k)!}{(n-j)!}.
\sum_{n = e v e n}^{\infty}H ( n ) = \frac{1}{4 i}\oint d z \cot ( \frac{\pi z}{2}) H ( z ) \hspace{. 1 i n}\sum_{n = o d d}^{\infty}H ( n ) = - \frac{1}{4 i}\oint d z \tan ( \frac{\pi z}{2}) H ( z ) ,
S_N[\mathbf{g}] = \frac{1}{\alpha(\alpha-1)}\sum_{i=0}^{N-1}\int_{\omega_{i}}^{\omega_{i+1}}\big(g_i\phi_i + g_{i+1}\phi_{i+1}+ g_{N+i}\phi_{N+i}\big)^{1-\alpha}\, d\omega.
m_{0}^{2}={\frac{4 \kappa}{3}}\nu_{2}^{2}\varphi_{( - )}^{2}e^{- 2 d / \ell}\left[{\frac{\sqrt{1 + ( M^{2}\ell^{2}/ 4 )}}{1 - e^{- 4 \sqrt{1 + ( M^{2}\ell^{2}/ 4 )}d / \ell}}}\right] .
{\frac{\partial \, \overline{{{\mathrm{\Large ~ v ~}}}}_{D}( x )}{\partial ( m / \Lambda )}}={\frac{2 m}{\Lambda}}\left[{\frac{( 4 \pi )^{D / 2}}{{\mathrm{t r}}{\bf 1}\, g^{2}\Lambda^{D - 2}}}+ f_{D}( x ) \right] = 0 \ ,
\big\|\mathcal{M}_{\mathcal{R},\alpha}^{(k)}(\vec{f})\big\|_{L^q(\nu)}\lesssim \frac{1}{|B_k|}\int_{B_k}\prod_{i=1}^m \big\| \tau_t f_i \big\|_{L^{p_i}(\tau_t \omega_i)}dt &=\frac{1}{|B_k|}\int_{B_k}\prod_{i=1}^m \big\| f_i \big\|_{L^{p_i}(\omega_i)}dt=\prod_{i=1}^m \big\| f_i \big\|_{L^{p_i}(\omega_i)}.
\mathcal{A}_{\theta}= \left\{a \in A_{\theta}\ | \a = \sum_{r,s}a_{rs}u^r, v^s \ \bigwedge \ \mathrm{sup}_{r, s \in \mathbb{Z}}\left(1 + r^2 + s^2\right)^k|a_{rs}| < \infty, \ \forall k \in \mathbb{N}\right\}.
={\displaystyle \sum_{i , j}}\eta \left({\phi}_{i},{\phi}_{j}, \xi_{1}, \ldots ,{\xi}_{q - 2}\right) \left({\phi}_{1}, \ldots , \widehat{{\phi}_{i}}, \ldots , \widehat{{\phi}_{j}}, \ldots ,{\phi}_{p + 2}\right) = 0
A_{2}= g^{\mu \nu}g^{\alpha \beta}\nabla_{\mu}\nabla_{\nu}\nabla_{\alpha}\nabla_{\beta}+ b^{\mu \nu \alpha}\nabla_{\mu}\nabla_{\nu}\nabla_{\alpha}+ C^{\mu \nu}\nabla_{\mu}\nabla_{\nu}+ d^{\mu}\nabla_{m u}+ X ,
V(k):=\sup_{\sigma\in\Sigma}\int_{T_k}^{T_{k+1}}Y_\sigma(\tau)^\frac{1}{p}\,d\tau,\k\in \mathbb{N}\cup\{0\}, &W(k):=\left(\frac{V(k)}{T_{k+1}-T_k}\right)^\frac{1}{p},\k\in \mathbb{N}\cup\{0\},
S_{latency}=\frac{1}{\frac{p1}{s1}+\frac{p2}{s2}+\frac{p3}{s3}+\frac{p4}{s4}}=\frac{1}{\frac{0.11}{1}+\frac{0.18}{5}+\frac{0.23}{20}+\frac{0.48}{1.6}}=2.19.
q^{\frac{p}{12}-\frac{a}{2}+\frac{a^2}{2p}}\left[a,p;q\right]_{\infty}=A(a,p;q)=\frac{\sqrt{m_1^{*}(pz)}\tau^{*}(a,p;z)}{\textrm{dn}\left((p-2a)zK[pz],q^p\right)}.
f^{j',h'}_{j,h}= \begin{cases}\operatorname{Id}_{D_j}~~h'=h+1,\, j'=j q_jx_{j-1}\ldots x_{h'+1}x_{h'}\bar{x}_{h'}\bar{x}_{h'+1}\ldots \bar{x}_{j'-1}p_{j'}~~h=j, 0,~.\end{cases}
\sum_{n=0}^{\infty}\bigg(\sum_{k=0}^{\infty}x^{k}(k)_{n,\lambda}\bigg)\frac{t^{n}}{n!}=\sum_{k=0}^{\infty}x^{k}e_{\lambda}^{k}(t)=\frac{1}{1-xe_{\lambda}(t)}&=\frac{1}{1-x-x(e_{\lambda}(t)-1)}=\frac{1}{1-x}\frac{1}{1-\frac{x}{1-x}(e_{\lambda}(t)-1)}=\frac{1}{1-x}\sum_{n=0}^{\infty}F_{n,\lambda}\bigg(\frac{x}{1-x}\bigg)\frac{t^{n}}{n!}.
\widehat T^{\mu \nu}( q ) = - \,{\frac{e^{2}}{q^{2}}}\, \left( \eta^{\mu \nu}q^{2}- q^{\mu}q^{\nu}\right) \left[ \left. \hat{h}\right|_{q^{2}= 0}+ \left( \hat{h}( q^{2}) - \left. \hat{h}\right|_{q^{2}= 0}\right) \right] .
S(x) y = x\times y = - y\times x = - S(y) x, S(x)^2 = -(x^T x) I_{3\times 3}+ x x^T, S(x)^3 = -(x^T x) S(x), x\cdot S(y)z = y\cdot S(z)x = z\cdot S(x)y, S(x) S(y) z=(x\cdot z) y - (x\cdot y)z=(yx^T-x^TyI_{3\times 3})z, S(x\times y) = S(x) S(y) -S(y)S(x) = yx^T-xy^T,
(\delta c)(p_0, \ldots, p_n) = \phi(p_0) c(p_1, \ldots, p_n) + \sum_{i = 1}^{n-1}(-1)^i c(p_0, \ldots, p_{i-1}, p_{i+1}, \ldots, p_n) &+ (-1)^n c(p_0, \ldots, p_{n-1}) p_n.
\ddot{q}_{i}=\sum\limits_{j=1,\textrm{}j\neq i}^{n}\frac{m_{j}(q_{j}-\sigma(q_{i}\odot_{k}q_{j})q_{i})}{(\sigma-\sigma(q_{i}\odot_{k}q_{j})^{2})^{\frac{3}{2}}}-\sigma(\dot{q}_{i}\odot_{k}\dot{q}_{i})q_{i},\textrm{}i\in\{1,...,n\}.
\cosh(\pi t/2) K_{it}(u) \ll \begin{cases}t^{-1/4}(t -u)^{-1/4}, 0 < u < t - C t^{1/3}t^{-1/3}, |u-t| \leq C t^{1/3}, u^{-1/4}(u -t)^{-1/4}\exp\Big(- c (\frac{u}{t})^{3/2}\big(\frac{u-t}{t^{1/3}}\big)^{3/2}\Big), u > t + C t^{1/3}. \end{cases}
\{q_{T}:V\otimes\mathcal{O}_{X_{T}}(-m)\twoheadrightarrow\tilde{\mathcal{E}},\alpha_{T}:\tilde{\mathcal{E}}\rightarrow\mathcal{D}_{T}\mid\tilde{\mathcal{E}}\,\,T-\mbox{flat},\,P_{\tilde{\mathcal{E}}}=P\},
\left\{\sum_{n = 0}^{\infty}\frac{( - 1 ) \ \rho_{n}^{( 2 )}( s )}{( t - n - \frac{1}{2})^{2}}- \sum_{n = 0}^{\infty}\frac{( - 1 )^{2}\ 2 ! \r_{n}^{( 1 )}( s )}{( s - n - \frac{1}{2})^{3}}\right\}_{s = 0 , \t = 0}= 0 .
\Xi_{1}\triangleq \sum^{m-1}_{\rho_{1}=0}\sum^{\rho_{1}}_{\rho_{2}=0}\cdots\sum^{\rho_{n-2}}_{\rho_{n-1}=0}\frac{m!}{\rho_{n-1}!p^{n+\phi}(n-1)!}&\times \prod^{n-1}_{t=1}\left[\frac{1}{(\rho_{t-1}-\rho_{t})!(t!)^{\rho_{t}-\rho_{t+1}}}\right],\ \i=1,
{\rm Pr}(\textbf{y}_R=y| b_0,{b}_1,\cdots,{b}_M)=\frac{e^{-\mathbb{E}(\textbf{y}_R| b_0,{b}_1,\cdots,{b}_M)}(\mathbb{E}(\textbf{y}_R| b_0,{b}_1,\cdots,{b}_M))^y}{y!},
\check{R}( u ) = P_{j = k}+ \frac{y^{2}- q^{2 k}}{1 - y^{2}q^{2 k}}P_{j = k - 1}+ \ldots + \frac{y^{2}- q^{2 k}}{1 - y^{2}q^{2 k}}\frac{y^{2}- q^{2 ( k - 1 )}}{1 - y^{2}q^{2 ( k - 1 )}}\ldots \frac{y^{2}- q^{2}}{1 - y^{2}q^{2}}P_{j = 0}\; ,
Hz\Phi =(H_0+vUv^*)\Bigl[\|v^*\mathbf n\|^{2\dagger}\langle M_0v^*\mathbf n,\Phi\rangle \mathbf n -G_{0,0}v\Phi\Bigr] &=-v\Phi &\quad+\|v^*\mathbf n\|^{2\dagger}\langle M_0v^*\mathbf n,\Phi\rangle vUv^*\mathbf n -vU(M_0-U)\Phi &=-vUS M_0\Phi.
\prod_{j=1}^k e^{-\alpha_j\eta(t_{j-1},t_j]}= \prod_{j=1}^k \sum_{\ell_j =0}^\infty \frac{(1-\alpha_j)^{\ell_j}}{\ell_j !}\eta(t_{j-1},t_j]^{\ell_j}e^{-\eta(t_{j-1},t_j]}a.s.,
G_{\mu \nu}= \left( \frac{\partial G_{\mu \nu}}{\partial F_{\rho \sigma}}( - F_{\rho \sigma}+ B_{\rho \sigma}^{N}) + 2 \chi \frac{\partial G_{\mu \nu}}{\partial \chi}+ 2 \chi \frac{\partial G_{\mu \nu}}{\partial \phi}\right) + * B_{\mu \nu}^{R}.
J^{( N )}( n ; 1 , \ldots , 1 ) = \frac{1}{\Lambda^{( N )}}\; \left( \prod m_{i}^{2}\right) \; \sum_{i = 1}^{N}\frac{1}{m_{i}^{2}}\; F_{i}^{( N )}\; J_{i}^{( N )}( n ; 1 , \ldots , 1 ) .
\frac{\partial(\rho\omega)}{\partial t}+\frac{\partial(\rho u_{j}\omega)}{\partial x_{j}}=\frac{\gamma}{\nu_{t}}P-\beta\rho\omega^{2}+\frac{\partial}{\partial x_{j}}[(\mu+\sigma_{\omega}\mu_{t})\frac{\partial\omega}{\partial x_{j}}]+2(1-F_{1})\frac{\rho\sigma_{\omega2}}{\omega}\frac{\partial k}{\partial x_{j}}\frac{\partial\omega}{\partial x_{j}}
\langle \phi^{2}( x ) \rangle_{\beta , R e n .}= \langle \phi^{2}( x ) \rangle_{\infty , R e n .}+ \frac{1}{4 \pi^{2}r^{2}}\sum_{n = 1}^{\infty}\sum_{l = 0}^{\infty}( 2 l + 1 ) Q_{\nu_{l}- 1 / 2}\left( 1 + \alpha^{2}\frac{n^{2}\beta^{2}}{2 r^{2}}\right) .
\begin{aligned}\tilde{Z}_{s,s+k}\left[ Z_s,t+\tau;\left\{t_j + \tau, v_{s+j},\omega_j,i_j \right\}_{j=1}^k \right] = & = \tilde{\psi}_{s+k}^{-\tau}\tilde{Z}_{s,s+k}\left[ Z_s,t;\left\{t_j,v_{s+k},\omega_j,i_j \right\}_{j=1}^k \right]\end{aligned}
\nabla_{\bar E_{\alpha}}^{\perp\psi}H^{\psi}=\frac{q}{p+q}\bar\nabla^{\perp}_{\bar E_{\alpha}}H_2+\frac{p}{p+q}B^j(\bar E_{\alpha},H_1),A^{\psi}_{H^{\psi}}\bar E_{\alpha}=\frac{q}{p+q}\bar A_{H_2}\bar E_{\alpha}.
\frac{\partial^2 G}{\partial h_1'\partial h_1'}\le0,\begin{vmatrix}\frac{\partial^2 G}{\partial h_1'\partial h_1'}\frac{\partial^2 G}{\partial h_1'\partial h_2'}\frac{\partial^2 G}{\partial h_2'\partial h_1'}\frac{\partial^2 G}{\partial h_2'\partial h_2'}\end{vmatrix}\le0.
\iint_{\Omega_T}\partial_tv_h(x,t)\, \varphi(x,t) \,dxdt+\frac{1}{2}\int_0^T\iint_{\mathbb{R}^n \times \mathbb{R}^n}\dfrac{\overline{U}_h(x,y,t)}{|x-y|^{n+sp}}(\varphi(x,t)-\varphi(y,t)) \,dxdydt =0
H_{2}=\sum_i D_i^2+\beta\sum_{i,j,a,b}E^{ab}_iE^{ba}_jD_j-\beta s\sum_i D_i+ +\frac{\beta^2}{3}\sum_{a,b,c}\mathbb{E}^{ab}\mathbb{E}^{bc}\mathbb{E}^{ca}-\frac{2s\beta^2}{3}\sum_{a,b}\mathbb{E}^{ab}\mathbb{E}^{ba}+\beta^2\frac{2s^2+p_0-1}{6}p_0,
\Phi(T, x) := \sup_{\mathcal{M}\in \mathfrak{M}(x)}\mathfrak{u}_{\mathcal{M}}(T, x){\mathrm{and}}\widehat{\Phi}(T, x) := \sup_{\mathcal{M}\in \widehat{\mathfrak{M}}(x)}\mathfrak{u}_{\mathcal{M}}(T, x)
Z (x_T,z) =\sum_{i=1}^{n_1}{b^{(0)}\over (x_T^2+|z-z_i^{(1)}|^2)^2}+ \sum_{i=1}^{n_2}{1-b^{(0)}\over (x_T^2+|z-z_i^{(2)}|^2)^2}+{\rm const}\int d^2 w{|\partial\gamma (w)|^2\over (x_T^2+|z-w|^2)^2}.
\begin{array}{c}{{\langle e_{a_{k}. . . a_{1}}\mid = ( \mid e_{a_{1}. . . a_{k}}\rangle )^{+}}}\\{{\langle e_{a_{k}. . . a_{1}}\mid e_{b_{1}. . . b_{k}}\rangle = \delta_{a_{1}b_{1}}. . . \delta_{a_{k}b_{k}}}}\\ \end{array}
w(X) \ge w(f) &= b(n-1)w(V(G))-(3b(n-1)-1)w(v) &> b(n-1)w(V(G))-\frac{3(3b(n-1)-1)\varepsilon}{n}w(V(G)) &= \left[b(n-1)-\frac{9\varepsilon^2}{n(n-1)}\right]w(V(G)) &= \left[b(n)+\frac{\varepsilon}{n(n-1)}-\frac{9\varepsilon^2}{n(n-1)}\right]w(V(G)) &= b(n)w(V(G))=w(c_{b(n)}).
{\rm Re}\left<\mathbb{A}\mathbf{u},\mathbf{u}\right>_X=\left<\left[\begin{smallmatrix}0 -u Au \end{smallmatrix}\right],\left[\begin{smallmatrix}u 0 -u \end{smallmatrix}\right]\right>_X=-{\rm Re}\langle Au,u\rangle_\mathcal{H}<0.
\left.{\frac{\partial}{\partial s}}\right|_{s=0}(I_{X}-J_{X})(\phi_{t\,s})=-{\frac{1}{1-t}}\int_MY(h_t)\omega_{\phi_t}^n-{\frac{1}{1-t}}\int_M\theta_X(\omega_{\phi_t})\theta_Y(\omega_{\phi_t})\omega_{\phi_t}^n=0.
Q_1\colon (\mu-\mu\overline{\mu}+\overline{\mu})y_1^2-2y_1y_2+y_2^2+(1-\overline{\mu}+\mu\overline{\mu}-\mu)y_3^2+y_4^2=0, Q_2\colon \mu\overline{\mu}y_1^2-2\mu\overline{\mu}y_1y_2+(\mu-1+\overline{\mu})y_2^2+\mu\overline{\mu}y_4^2+(1-\overline{\mu}+\mu\overline{\mu}-\mu)y_5^2=0,
{\frac{1}{T}}={\frac{d S}{d E}}={\frac{d ( S_{L}+ S_{R})}{d E}}= 2 \left({\frac{d S_{L}}{d E_{L}}}+{\frac{d S_{R}}{d E_{R}}}\right) = 2 \left({\frac{1}{T_{L}}}+{\frac{1}{T_{R}}}\right) ,
P_{\textrm{sout}}^{\textrm{CSANC}}= \textrm{Pr}^{\textrm{NOMA}}\left\lbrace \textrm{NHF=0,FDF=0,NDF=1,NDN=1}\right\rbrace +\textrm{Pr}^{\textrm{NOMA}}\left\lbrace \textrm{NDN=0}\right\rbrace.
S(x) := \lbrace d\in\mathcal{D}_D(x) : h(d) \geq 2 e^\gamma \frac{\sqrt{d}}{\log d}\cdot \tau \rbrace= \lbrace d\in\mathcal{D}_D(x) : L(1,\chi_d) \geq e^\gamma \tau_0(d) \rbrace
g|_k = a^{\mathbf{n}_{0}\cdot \mathbf{e}_{k}}a^{\mathbf{n}_{1}\cdot \mathbf{e}_{{k-1}}}\dots a^{\mathbf{n}_{k-1}\cdot \mathbf{e}_{{1}}}b^{\mathbf{n}_{k}}a^{\mathbf{n}_{k+1}\cdot \mathbf{e}_{{p-1}}}\dots a^{\mathbf{n}_{p-1}\cdot \mathbf{e}_{{k-1}}}&= a^{s_k}b^{\mathbf{n}_{k}}\tilde y_k
(-q;q^2)_\infty\sum_{k=0}^{\infty}\frac{q^{2kr+\ell}}{1+q^{2kr+\ell}}= \sum_{k=0}^{\infty}(-q;q^2)_\infty(1-q^{2kr+\ell}) \frac{q^{2kr+\ell}}{1-q^{2(2kr+\ell)}}.
h(g)= \sum_{j=1}^M\sum_{i=1}^Q c_{ij}e^{ik_{ij}^1\phi_1}\sin^{m_{ij}^1}(\psi_1)\cos^{n_{ij}^1}(\psi_1)\cdots e^{ik_{ij}^{N-1}\phi_{N-1}}\sin^{m_{ij}^{N-1}}(\psi_{N-1})\cos^{n_{ij}^{N-1}}(\psi_{N-1}) &\cdot (h_{SU(N-1)})_{ij}(g_{SU(N-1)}) e^{il^N_{ij}\omega_{N-1}},
\dfrac{\partial c}{\partial t}-\sum_{i=1}^n\sum_{j=1}^n a_{ij}(x,t)D_{ij}c+\sum_{i=1}^n b_i(x,t)D_ic+d(x,t)c=g(x,t),~~(x,t)\in Q_T=\Omega\times (0,T),
s^{(j)}(x)=\left ( \frac{p(x|t_1)}{p(x|t_j)},\ldots, \frac{p(x|t_{j-1})}{p(x|t_j)},\frac{p(x|t_{j+1})}{p(x|t_j)}, \ldots, \frac{p(x|t_k)}{p(x|t_j)}\right ), j\neq 1,
\big(\log(k^{2^i})+1\big)^{n/k^{2^{i-1}}}\exp\Big(\frac n{k^{2^{i-1}}}(\log(k^{2^{i-1}})+1)\Big) = \exp\Big(\frac n{k^{2^{i-1}}}\big(\log(k^{2^{i-1}})+\log(\log(k^{2^i})+1)+1\big)\Big).
\sigma=1-\frac{M^{2}}{L_{P}L_{S}}=1-k^{2}\approx\frac{L_{sc}}{L_{oc}}\approx\frac{L_{sc}^{sec}}{L_{P}}\approx\frac{L_{sc}^{pri}}{L_{S}}\approx\frac{i_{oc}}{i_{sc}}
\begin{array}{l l c c l}{J_{L}}&{:}&{T_{L}^{*}({\cal M}_{n})}&{\rightarrow}&{T_{L}({\cal M}_{n})}\\{}&{}&{U}&{\mapsto}&{\ ( L U )_{+}L - L ( U L )_{+}+ L \Phi_{U}( L ) + \bar{\Phi}_{U}( L ) L}\\ \end{array}
F_{s}(\alpha_i/z)=\sum_{k=0}^{\infty}\dfrac{1}{(k+x+1)^{r-s}}\cdot\dfrac{\alpha^{k+1}_i}{z^{k+1}}=\Phi_{r-s}(x,\alpha_i/z) \ \ (1\le i \le m, 0\le s \le r-1)\enspace,
a ( t ) = b_{0}t - \frac{b_{0}g}{9}t^{3}+ \frac{1}{2 7 0}\frac{b_{0}g^{2}( 7 b_{0}^{2}+ k )}{( b_{0}^{2}+ k )}t^{5}+ O ( t^{7}) , \quad t \to 0 ,
\sum_{k=0}^{b-m}\binom{a}{k}\binom{b}{m+k}\binom{m+k}{d}H_k H_{a-k}={}\binom{a+b-d}{m+a-d}\binom{b}{d}[H_a^{(2)}-H_{a+b-d}^{(2)}+ (H_a-H_{a+b-d}+H_{m+a-d})(H_a-H_{a+b-d}+H_{b-m})]
\left[ \frac{d^{2}}{d \varrho^{2}}+ \widetilde{E}- \mathrm{s g n}\left( \widetilde{\beta}\right) \widetilde{\lambda}\varrho^{\widetilde{\beta}}- \frac{\left( \widetilde{l}+ \widetilde{\nu}\right)^{2}- 1 / 4}{\varrho^{2}}\right] \widetilde{u}_{l}( \varrho ) = 0 ,
| v_{\pm}( \lambda ) \rangle \, = \, \sqrt{\frac{\omega ( \lambda ) \mp M}{2 \, \omega ( \lambda )}}\, \left( \begin{array}{c}{{| v_{1}( \lambda ) \rangle}}\\{{\displaystyle{\frac{C}{\omega ( \lambda ) \mp M}}\, | v_{1}( \lambda ) \rangle}}\\ \end{array}\right) \; ,
z_1 = \frac{(q-1)(q^d-1)(x_1^2 - q^d)}{x_1^2 - q}, z_i = \frac{q^{i-1}(q^i-1)(q^{d-i+1}-1)(x_1^2 - q^{i-2})(x_1^2 - q^{d+i-1})}{(x_1^2 - q^{2i-3})(x_1^2 - q^{2i-1})}(2 \leq i \leq d-1), z_d = \frac{q^{d-1}(q-1)(q^d-1)(x_1^2 - q^{d-2})}{x_1^2 - q^{2d-3}}.
\begin{array}{l}E_{\rm vib}=\displaystyle hc\omega_e(v+{1\over 2})-hc\omega_ex_e(v+{1\over2})^{2}+hc\omega_ey_e(v+{1\over 2})^3+\cdots~, begin{align*}2mm]E_{\rm rot}=BhcJ(J+1)-DhcJ^2(J+1)^2+HhcJ^3(J+1)^3+\cdots~.\end{array}
\frac{\Delta\hat{g}}{\hat{g}}\approx\frac{1}{\hat{g}}\frac{\partial\hat{g}}{\partial L}\Delta L+\frac{1}{\hat{g}}\frac{\partial\hat{g}}{\partial T}\Delta T+\frac{1}{\hat{g}}\frac{\partial\hat{g}}{\partial\theta}\Delta\theta
t^{\rho}_{H^{n+1}}(a\wedge b) = - \theta^{n+1}([X_{a}^{n+1}, X_{b}^{n+1}]_{H^{n+1}}) & = - \theta^{n+1}([(\xi^{a})^{P^{n+1}}, X_{b}^{n+1}]_{H^{n+1}}) - \theta^{n+1}( [ Y, X_{b}^{n+1}]_{H^{n+1}}).
d s^{2}= g_{y y}d y^{2}+ g_{\, i j}^{( d )}d x^{i}d x^{j}={\frac{l^{2}d y^{2}}{4 y^{2}}}+{\frac{1}{y}}\tilde{g}_{i j}( x , y ) d x^{i}d x^{j}\,
S_{\mathrm{c t}}= - \frac{1}{8 \pi G}\int_{\cal \partial M}d^{n}x \sqrt{- h}\left[ \frac{n - 1}{l}+ \frac{l}{2 ( n - 2 )}{\cal R}+ \frac{l^{3}}{2 ( n - 4 ) ( n - 2 )^{2}}\left({\cal R}_{a b}{\cal R}^{a b}- \frac{n}{4 ( n - 1 )}{\cal R}^{2}\right) \right] ,
S(A'B')_{\hat{\gamma}^{(n)}_{CA'B'}}= g\left(\frac{n^2}{a}-\frac{1}{2}\right) + g\left(\frac{n^2}{b}-\frac{1}{2}\right) = \ln\frac{n^4}{a\,b}+ 2 + \mathcal{O}\left(\frac{1}{n^4}\right)\;.
\Tilde{c_1}(T_1)\Tilde{\gamma_1}(T_1)\dfrac{\partial T_1}{\partial t}=\dfrac{1}{r^{\nu}}\dfrac{\partial}{\partial r}\bigg(\Tilde{\lambda_1}(T_1)r^{\nu}\dfrac{\partial T_1}{\partial r}\bigg),\;\;\; \alpha(t)<r<\beta(t),\;\;t>0,
u^{(t)}_{nm}= \binom{2t}{t-n}^{\frac{1}{2}}_{q^{-2}}\binom{2t}{t-m}^{-\frac{1}{2}}_{q^{-2}}\sum^{(t-n) \land (t+m)}_{i=0 \lor (m-n)}q^{(t-n-i)(n-m+2i)}q^{-i(n-m+i)}&\;\;\;\; \times \binom{t-n}{i}_{q^{-2}}\binom{t+n}{t+m-i}_{q^{-2}}(-qc^*_q)^ic^{n-m+i}_qa^{t-n-i}_q(a^*_q)^{t+m-i},
{\left.{I_{{0^+}}^\alpha \,I_{{b^-}}^\alpha \left(\lambda + q \left(t\right) \right)f\left( t \right)}\right|_{t ={t_i}}}\approx \sum\limits_{j = 0}^i{{w_{i,j}}\sum\limits_{k = j}^n{{\left(\lambda + q_k\right) f_k}\,{v_{j,k}}}}.