harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
A^{-1}= \begin{pmatrix}(U_{1,1})^{-1}(L_{1,1})^{-1}+ (U_{1,1})^{-1}U_{1,2}S^{-1}_{2,2}L_{2,1}(L_{1,1})^{-1}- (U_{1,1})^{-1}U_{1,2}S^{-1}_{2,2}-S^{-1}_{2,2}L_{2,1}(L_{1,1})^{-1}S_{2,2}^{-1}\end{pmatrix}.
t^N\langle 1 \cdots M|\Bigg(\frac{\mathcal{B}(t^{-1}z_1)}{t^{M+1}}\Bigg) \cdots\Bigg(\frac{\mathcal{B}(t^{-1}z_N)}{t^{M+1}}\Bigg)| \overline{x_1}\cdots \overline{x_N}\rangle = \prod_{j=1}^N z_j^{j-1-N}(1+t^{-1}z_j^2)\prod_{1 \le j<k \le N}(1+t^{-1}z_j z_k)(t^{-1}+z_j z_k^{-1})sp_{\overline{\lambda}}( \{z \}_N ).
s=b[F(gdv|-\frac{a^{2}}{b^{2}})-E(gdv|-\frac{a^{2}}{b^{2}})+\sqrt{1+\frac{a^{2}}{b^{2}}tanh^{2}v}sinhv]_{arcosh\frac{x_{1}}{a}}^{arcosh\frac{x_{2}}{a}}
\sum_{i = 0}^{s}\frac{(q^{-s})_{i}}{(q)_{i}}q^{i}(-1)^{\delta(i \leq m)}\frac{b^{1 - s}}{(q^{1 - s})_{s - 1}}\sum_{i_{1}= 0}^{n_{1}- i}\dotsm \sum_{i_{r}= 0}^{n_{r}- i}\sum_{j = \max\left\{0,\, I'\right\}}^{\infty}H (i, i_{1}, \dotsc, i_{r}, j).
\begin{aligned}\|u_{0,\lambda}\|_{\dot{B}^{d/q-1/(h-1)}_{q,p}}\eqsim \lambda^{1/[2(h-1)]}(\lambda^{1/2})^{d/q-1/{(h-1)}-d/q}\|u_0\|_{\dot{B}^{d/q-1/(h-1)}_{q,p}}&=\|u_0\|_{{\dot{B}^{d/q-1/(h-1)}_{q,p}}};\end{aligned}
\mathbb E_{P_\sigma}\left[\left(H_n\left(\frac{1}{\|f_1\|_\sigma}W^{(\sigma)}(f_1)\right)\right)\left(H_k\left(\frac{1}{\|f_2\|_\sigma}W^{(\sigma)}(f_2)\right)\right)\right]=n!\delta_{k,n}\left(\frac{\langle f_1,f_2\rangle_\sigma}{\|f_1\|_\sigma\|f_2\|_\sigma}\right)^n.
\begin{aligned}W_k(F):= \big\{v\in H^1(F): \;\; \Delta_F v \in \mathbb{P}_{k}(F), v|_{\partial F}\in B_k(\partial F), & (v,q)_{F}=(\Pi_F^k v,q)_{F}, \; \forall q\in \widetilde{\mathbb{P}}_k(F)\cup \widetilde{\mathbb{P}}_{k-1}(F)\big\}.\end{aligned}
|\lambda|^{1/2}\|\partial_z(\beta v)\|_{L^\infty_H L^p_z(\Omega)}\le C_{\theta,p,\lambda_0}\left(\eta^{2/p-1}+\eta^{3/p}|\lambda|^{-1/2p}\big( 1 + |\log(\eta|\lambda|^{-1/2})| \big) \right)\|f\|_{L^\infty_H L^p_z(\Omega)}&+ C_{\theta,p}\eta^{2/p-1}|\lambda|^{1/2}\|\partial_zv\|_{L^\infty_H L^p_z(\Omega)}.
M_n(k) = \frac{1}{n}\sum_{\mathbf{p}_k\in \mathcal{P}(d,k)}\sum_{i\neq j}^n\left\{\prod_{\ell=1}^k I^{(p_\ell)}_{i,j}-\left( \frac{-1}{6n}\right)^{k}\right\}.
2 \, \Re \left( X \left( t \right) \overline{Y \left( t \right)}\right) =\frac{4 \, d \, g}{\gamma}\, \Re \left( X \left( t \right) \overline{\frac{\gamma}{2 \, d \, g}Y \left( t \right)}\right) \leq\frac{2 \, d \, g}{\gamma}\left\vert X \left( t \right) \right\vert^2 + \frac{\gamma}{2 \, d \, g}\left\vert Y \left( t \right) \right\vert^2,
\rho\,d q =-\tfrac{2}{1-2 m\rho}\bigl(\tfrac{d\rho_0}{\rho_0}+\tfrac{d\rho_I}{\rho_I}\bigr) + \rho_I\tfrac{d\rho_0}{\rho_0}, \ \ \rho\,d s = \rho_I\tfrac{d\rho_0}{\rho_0}.
|\nabla y(x) - R|^4 \le \Big|\frac{1}{|B_r(x)|}\int_{B_r(x)}(\nabla y(t) - R) \, dt \Big|^4 & \le C\Big((sm)^{-2}\int_{\hat{F}}|\nabla y -R|^2 \Big)^2 \le C l^{8}m^{-4}s^{-4}\gamma(\hat{F})^2
U(x,t) = \frac{1}{\sqrt{8\,\pi^3\,K_{1/2}\,t^{1/2}}}\,\sum\limits_{j=0}^{\infty}\frac{(-2\,k_\ast\,t^{1/2})^j}{j!}\,G^{3,0}_{0,3}\left[ \left(\frac{x^2}{16\,K_{1/2}\,t^{1/2}}\right)^2 \left| \; \left( 0,\frac{1}{4}+ \frac{j}{2}, \frac{1}{2}\right)\right.\right].
dX_t=\frac{-(1+t)^{\frac{1}{2}}|X_t|^{2\gamma-2}X_t-2K_1X_t}{2(1+t)}dt+\sqrt{\frac{|X_t|^{2\gamma}}{(1+t)^\frac{1}{2}}+\frac{C}{(1+t)^{K_1}}}dB_t,
(x_i^+\otimes 1)w_\mu=0,\ \ \ (h_i\otimes t^r)w_\mu=\mu(h_i)\delta_{r,0}w_\mu,\ \ \ \ (x_i^-\otimes 1)^{\mu(h_i)+1}w_\mu=0, (x^-_{i,j}\otimes t^{s_{i,j}}) w_\mu=0,\ \ 1\le i\le j\le n, (x^-_{i,j}\otimes t^{s_{i,j}-1})^{m_{i,j}+1}w_\mu=0,
A^{*(p+q)}A^{p}A^{*p}A^{p+q}=A^{*q}(A^{*p}A^{p})^2A^q=A^{*q}(A^{*}A)^{2p}A^q &=A^{*q}(A^{*2p}A^{2p})A^q=A^{*2p+q}A^{2p+q}=(A^*A)^{2p+q}.
X^i_a = \left(\begin{array}{cc}X^i_{a,11}X^i_{a,12}X^i_{a,21}X^i_{a,22}\end{array}\right) =\left(\begin{array}{cc}C^i_a +{{\cal O}(t)}{{\cal O}(t^\gamma)}{{\cal O}(t^\gamma)}{T^i_a \over{s-\mu_a}}\end{array}\right) .
(\Phi \circ \beta)(T_i)(x_i) = q x_i - E_i F_i (x_i) &= qx_i + q^{-1}x_{i+1}(1 - q^{2}x_i x_{i+1}^{-1}) &= q^{-1}x_{i+1}= \chi(T_i)(x_i).
I = \int_{\cal{M}}d^{4}\! x \sqrt{\tilde{g}}e^{\Phi}\left( \tilde{R}( \tilde{g}) + \left( \nabla \Phi \right)^{2}-{\frac{1}{1 2}}H_{\mu \nu \rho}H^{\mu \nu \rho}- \left( \nabla T \right)^{2}+ V ( T ) \right) \ ,
f_i \cdot f_{i+1}\cdots f_{j-1}=\frac{u_{i,i}}{v_{i,i}}\frac{v_{i+1,i}}{u_{i+1,i}}\cdot\frac{u_{i+1,i+1}}{v_{i+1,i+1}}\frac{v_{i+2,i+1}}{u_{i+2,i+1}}\cdots\frac{u_{j-1,j-1}}{v_{j-1,j-1}}\frac{v_{j,j-1}}{u_{j,j-1}}=1.
{}_{3}\langle \tilde{I}| V_{3}\rangle_{1 2 3}={}_{3}\langle 0 | c_{- 1}^{3}c_{0}^{3}\exp \left[ c^{1}C b^{2}+ c^{2}C b^{1}+ ( c^{1}- c^{2}) \Omega ( b_{0}^{1}- b_{0}^{2}) + ( c^{1}- c^{2}) \Gamma b_{0}^{3}\right] ( c_{0}c_{1}| 0 \rangle )_{1 2 3}\nonumber
f_{ra}-r^{-1}f_a+(f_{rr}+r^{-2}\tilde{\Delta}f+2r^{-1}f_r) f_r f_a &= A_a(p-1) r^{p-1}+\left((p(p-1) A+\tilde{\Delta}A) r^{p-2}+2 p Ar^{p-2}\right)(pA r^{p-1})(A_a r^p) &=A_a(p-1) r^{p-1}+( p(p+1) A+\tilde{\Delta}A)p A A_a r^{3p-3}.
\int dt' G(0,t;x',t')= i\int dt'\int\int\frac{d\omega'}{2\pi}\frac{dk'}{2\pi}\left(\frac{e^{-i\omega'(t-t')}}{\omega'^{2}-k'^{2}-4+i\rho}\right)\left(\frac{2ik'}{ik'-2\sigma_{1}}\right)e^{-ik'x'}.
\begin{aligned}(I-F)\left[z\left(\Psi^{-}_1-e^{-ic_-}e_1\right)-\left(\widehat{\Psi}^-_{11}e_1+\widehat{\Psi}^-_{21}e_{2}\right)\right] = zde_{2}-(I-F)\left(\widehat{\Psi}^-_{11}e_1+\widehat{\Psi}^-_{21}e_2\right),\end{aligned}
(D_x)^{\ell}\alpha_x(a) = (D_x)^{\ell}\alpha_x\left(\sum_ma_m\prod_{j=1}^nU_j^{m_j}\right) &= (D_x)^{\ell}\sum_ma_me^{ix\cdot m}\prod_{j=1}^nU_j^{m_j}&= \sum_m a_m m^{\ell}e^{ix\cdot m}\prod_{j=1}^nU_j^{m_j}&= \delta^{\ell}\alpha_x(a).
~ [T_\pi\mathbf{v}]_{i}= \bar{r}_{i}^{\pi_i}+\gamma{\mathbf{p}_{i}^{\pi_i}}^\top \mathbf{v}= r_{i}^{\pi_i}+\gamma \sum_{j\in\mathcal{S}}p_{ij}^{\pi_i}v_j.
u_{\underline{{m}}}^{i}= G_{{\cal R}}^{i k}H_{L}^{k j}( - l_{\underline{{m}}}^{j}+ e^{2 W}r^{( + + ) \underline{{n}}}l_{\underline{{n}}}^{j}l_{\underline{{m}}}^{( -- )}) , \qquad
J_{12}\mapsto\widetilde{J}_{12}= e^{-B}S^{-1}\left(-i\frac{\partial}{\partial \theta_1}+\frac{1}{2}\Sigma_{12}\right)S e^{B}=-i\frac{\partial}{\partial \theta_1}+\frac{1}{2}\Sigma_{12}+(-i)\left(\frac{-i}{2}\Sigma_{12}\right)=-i\frac{\partial}{\partial \theta_1}
f^{(n-k)}(z) =\sum_{j=0}^{n-k}\frac{(-1)^{j}j!}{(l_{\pi_{\lambda,\beta}}(z))^{j+1}}B_{n-k,j}\left(\left(\sum_{p=i}^{\infty}\frac{p!}{(p-i)!}\frac{\lambda^{p}z^{p-i}}{\Gamma(p\beta+1)}\right)_{i=1}^{n-k-j+1}\right)
\left\| -\Delta\phi_n - \lambda\;\!\phi_n \right\| = \left\| -\Delta\varphi_n + 2 i k \cdot \nabla \varphi_n \right\| \leq \|\Delta\varphi_n\| + 2 \, |k| \;\! \|\nabla\varphi_n\| \xrightarrow[n \to \infty]{}0 \,.
H( v, u_{(1)},\ldots , u_{(m)}):=\left[\begin{matrix}H_1( u_{(1)}, v) H_1( u_{(2)}, v) \ldots H_1( u_{(m)}, v) H_2( u_{(1)}, v) H_2( u_{(2)}, v) \ldots H_2( u_{(m)}, v) \vdots \vdots \ddots \vdots H_m( u_{(1)}, v) H_m( u_{(2)}, v) \ldots H_m( u_{(m)}, v) \end{matrix}\right].
{\breve{F}}^{- 1}\simeq 1 +{\frac{m_{0}}{\breve{R}^{2}}}\int_{- \infty}^{\infty}d v{\frac{1}{( 1 + v^{2})^{{\frac{3}{2}}}}}= 1 +{\frac{2 m_{0}}{\breve{R}^{2}}},
D(p)\cdot \nabla_p H =\frac{mc}{\sqrt{m^2c^2+|p|^2}}\left(I+\frac{p\otimes p}{m^2c^2}\right)\cdot \frac{c\,p}{\sqrt{m^2c^2+|p|^2}}&=\frac{1}{m(m^2c^2+|p|^2)}(m^2c^2 I+p\otimes p)\cdot p &=\frac{1}{m(m^2c^2+|p|^2)}(m^2c^2 +|p|^2)\,p=\frac{p}{m}.
\frac1T \sum \limits_{k=0}^{T-1}\langle \hat{g}_{k+1}, x_k -x^* \rangle \leq \frac12 \frac{R_{0}^{2}}{\nu T}+ \frac{\nu}{2}\frac{1}{T}\sum \limits_{k=0}^{T-1}\|\hat{g}_{k+1}\|^{2}_q.
g_{\alpha \beta}^j(f_{jk}^1(z_k,t),...,f_{jk}^n(z_k,t))=\sum_{r,s=1}^n g_{rs}^k(z_k,t)\frac{\partial f_{jk}^{\alpha}}{\partial z_k^r}\frac{\partial f_{jk}^{\beta}}{\partial z_k^s}
\delta\left( \alpha\right) \left[ \psi\left( \mathrm{i}\alpha\right)-\int K\left( \alpha,s\right) \mathrm{d}\mu\left( s\right) \right]+\mu^{\prime}\left( \alpha\right) =\widetilde{\delta}\left( \alpha\right)\widetilde{\psi}\left( \mathrm{i}\alpha\right) .
J^{\pm}=\left[\begin{array}{cccccc}1 0 0 -\hat{H}^{\pm}_1 -\hat{H}^{\pm}_{\tau}0 0 1 \partial_2\hat{\Psi}^{\pm}0 0 0 0 0 1 0 0 0 0 0 0 1 \partial_2\hat{\Psi}^{\pm}0 0 0 0 0 1 0 0 0 0 0 0 1 \end{array}\right],
\int_{1}^{\infty}cos(ax)\frac{lnx}{x}dx=-\frac{\pi^{2}}{24}+\gamma(\frac{\gamma}{2}+lna)+\frac{ln^{2}a}{2}+\sum_{n\ge1}\frac{(-a^{2})^{n}}{(2n)!(2n)^{2}},
\frac{\partial}{\partial \pi_n}\Big\{\sum_{x_n,y_{n}}\log\Big(\frac{r_{n}(x_n|y^{n-1}_{n-M},y_n)}{\pi_{n}(x_n|y^{n-1}_{n-J})}\Big)q_n(y_n|y^{n-1}_{n-M},x_n)\pi_n(x_n|y^{n-1}_{n-J}) &-s\sum_{x_n}\gamma_n(x_n,y^{n-1}_{n-N})\pi_n(x_n|y^{n-1}_{n-J})+\lambda_n(y^{n-1}_{n-J})\Big{(}\sum_{x_n}\pi_{n}(x_n|y^{n-1}_{n-J})-1\Big{)}\Big\}=0,~\forall{x_n}\in{\cal X}_n
{\cal V}[ \phi_{0}] = V [ \phi_{0}, \lambda ( \mu ) ] +{\frac{\hbar}{6 4 \pi^{2}}}\left\{{\frac{\lambda ( \mu )^{2}\phi_{0}^{4}( \mu )}{4}}\right\}\ln{\frac{\phi_{0}^{2}}{\mu^{2}}}+ \hbar \alpha_{3}( \lambda ( \mu ) ) \; \phi_{0}^{4}+ O ( \hbar^{2}) .
\int \frac{\alpha}{\beta}= 2 \frac{\alpha}{\beta}\beta\int R^2 = \frac{\beta}{\pi R_*^2 L_*}\int R = 2 \frac{R^\frac{3}{2}}{L}\int R^2/\sqrt{...}dR = 2\frac{R^{3/2}}{L}\left( \frac{4\alpha + 147}{8 \beta^{5/2}}(-\frac\pi 2 + \arctan(7/(2\sqrt{\alpha}))) + \frac{21 \sqrt{\alpha/\beta}}{4\beta^2}\right)
d X^{N, \kappa}_t = \big[ P_N A X^{N, \kappa}_t + P_N F_{\kappa}( X^{N, \kappa}_t ) \big] \, dt + P_N B_{\kappa}( X^{N, \kappa}_t ) \, dW_t , \;\; t \in [0,T] , \;\; X^{N, \kappa}_0 = P_N( \xi ) .
\int Ku^{\frac{n+2}{n-2}}\phi_{k,i}= \int K (u_{\alpha, \beta}+\alpha^{j}\varphi_{j})^{\frac{n+2}{n-2}}\phi_{k,i}& +\frac{n+2}{n-2}\int K(u_{\alpha, \beta}+\alpha^{j}\varphi_{j})^{\frac{4}{n-2}}\phi_{k,i}v +O(\Vert v \Vert^{2})
H=\left( \begin{array}{ccccc}1 \alpha \alpha^{2}... \alpha^{n-1}1 \alpha^{8}\alpha^{16}... \alpha^{8(n-1)}... ... ... ... ... 1 \alpha^{7k+1}\alpha^{2\left( 7k+1\right)}... \alpha^{(7k+1)\left( n-1\right)}\end{array}\right) ,
\tilde x\frac{\lambda_1}{\tilde\lambda_1-\lambda_1}\sum_{l>1}\sqrt{\lambda_l}\langle \tilde u_1,u_l\rangle=\Big(1-\tilde x\frac{\lambda_1}{\tilde\lambda_1-\lambda_1}\Big)\sqrt{\lambda_1}\langle\tilde u_1,u_1\rangle,
Z_{k}( \lambda_{1}^{2}, \lambda_{2}^{2}, q^{2}; \lambda^{2}) = \int_{0}^{1}d z z^{k}l n \left( \frac{q^{2}z ( 1 - z ) + \left( \lambda_{1}^{2}- \lambda_{2}^{2}\right) z - \lambda_{1}^{2}}{- \lambda^{2}}\right) ,
|\alpha\rangle=e^{-\frac{|\alpha|^{2}}{2}}\sum_{n=0}^{\infty}\frac{\alpha^{n}}{\sqrt{n!}}|n\rangle=e^{-\frac{|\alpha|^{2}}{2}}e^{\alpha\hat{a}^{\dagger}}e^{-\alpha^{*}\hat{a}}|0\rangle,
\| \phi_1 * \phi_2 \|_{q (p_1,p_2)}\geq \| \phi_1^* * \phi_2^* \|_{q (p_1,p_2)}\geq C (p_1,p_2) \| \phi_1^* \|_{p_1}\| \phi_2^* \|_{p_2}= C (p_1,p_2) \| \phi_1 \|_{p_1}\| \phi_2 \|_{p_2}
S^{\mp , a , p}( x , x^{\prime}) = \int_{- \infty}^{+ \infty}\, d z \, \int_{- \infty}^{+ \infty}\, \frac{d p_{3}}{2 \pi}\, e^{- i p_{3}y^{3}}S_{Q}^{\mp , a , p}\; , \quad y_{\mu}= x_{\mu}- x_{\mu}^{\prime}\; ,
\check{G}_{k-1,x}=-\displaystyle\frac{1}{u}\sum_{l=0}^{k-1}\check{G}_{l-1,x}\check{F}_{k-1-l}-\sum_{l=0}^{k+1}\left(\check{G}_{l-1}\check{G}_{k-l}+\frac{v}{u}\check{F}_{l-1}\check{F}_{k-l}\right)
K_{\xi_1\times\xi_2,\max\{\varphi_1\circ \pi_1,\varphi_2\circ \pi_2\}}^{\psi_1\circ\pi_1+\psi_2\circ \pi_2}(t)=K_{\xi_1,\varphi_1}^{\psi_1}(t)\cdot K_{\xi_2,\varphi_2}^{\psi_2}(t)
\begin{aligned}\mathbb{E}\big[ |A_0(n)|^2\big] = \sum_{\substack{|\lambda|=n \lambda_1 \leqslant y_0}}\mathbb{E}\big[|a(\lambda)|^2\big] = \sum_{\substack{|\lambda|=n \lambda_1 \leqslant y_0}}\prod_k \frac{1}{k^{m_k}m_k!}.\end{aligned}
\langle A + | A - \rangle \;=\; \det_{\lambda ,\lambda^{\prime}}\, \langle v_+ (\lambda) | v_- (\lambda^{\prime}) \rangle\;\;,\;\; \langle \pm | A \pm \rangle \;=\; \det_{\lambda ,\lambda^{\prime}}\, \langle v^0_\pm (\lambda) | v_\pm (\lambda^{\prime})\rangle\;,
(I+\xi E_i)(I+\xi E_{i+1})(I+\xi E_i)=(I+\xi E_{i+1})(I+\xi E_i)(I+\xi E_{i+1}) = (I+2 \xi E_i+\xi E_{i+1}+\xi^2 E_i^2 + \xi^2 E_i E_{i+1}+\xi^2 E_{i+1}E_i + \xi^3 E_i E_{i+1}E_i ) = (I+ \xi E_i+ 2\xi E_{i+1}+\xi^2 E_{i+1}^2 + \xi^2 E_i E_{i+1}+\xi^2 E_{i+1}E_i + \xi^3E_{i+1}E_i E_{i+1})
m_i v'+m_j v'_* = m_i v+m_j v_*,\frac 12 m_i\,|v'|^2 + \frac 12 m_j\,|v_*'|^2 = \frac 12 m_i\,|v|^2 + \frac 12m_j\,|v_*|^2.
\frac{\partial V_{eff}}{\partial \chi}= \frac{1}{8 \pi \chi}\left[\chi -m^2 - \lambda_r\left(\phi^2+\frac{1}{4 \pi}{\ln}[\frac{m^2}{\chi}] \right)- \eta \left( \phi^2+\frac{1}{4 \pi}{\ln}[\frac{m^2}{\chi}] \right)^2 \right]
\begin{array}{l c r}{( | W_{1}|^{2}+ | Z_{2}|^{2}) - ( | Z_{1}|^{2}+ | W_{2}|^{2}) = R_{2}}{Z_{1}W_{1}+ Z_{2}W_{2}= \overline{{Z}}_{1}\overline{{W}}_{1}+ \overline{{Z}}_{2}\overline{{W}}_{2}= 0 ,}\end{array}
\sum_{k=1}^{n}\frac{y^k}{m^{n-k}}\prod_{h=0}^{k-1}\left(1-\frac{h}{m}\right)a_{n,k}=\sum_{k=1}^{n-1}\frac{y^{k+1}}{m^{n-k-1}}\prod_{h=0}^{k-1}\left(1-\frac{h}{m}\right)\left(1-\frac{k}{m}\right)a_{n-1,k}+\sum_{k=1}^{n-1}\frac{y^{k}}{m^{n-k}}k\prod_{h=0}^{k-1}\left(1-\frac{h}{m}\right)a_{n-1,k}
\sup_{\substack{i \in \{1, \dots, N^d\}x \in I_{\lambda_i}}}\left| f(x) - p_{i,\alpha}(x) \right| \leq C B \left(\frac{d}{N}\right)^\beta p_{i,\alpha}(x) := \sum_{|\alpha| \leq n}\frac{\partial^\alpha f (x_i)}{\alpha!}(x-x_i)^\alpha \, .
\vec{B}= \frac{\Phi_{0}}{2 \pi \lambda^{2}}\, \vec{b}\ \ , \ \ \frac{\vec{E}}{c}= \frac{\Phi_{0}}{2 \pi \lambda^{2}}\, \vec{e}\ \ , \ \F_{\mu \nu}= \frac{\Phi_{0}}{2 \pi \lambda^{2}}\, f_{\mu \nu}\ \ , \ \A^{0}= \frac{\Phi_{0}}{2 \pi \lambda}\, \varphi \ \ , \ \ \vec{A}= \frac{\Phi_{0}}{2 \pi \lambda}\, \vec{a},
\sum\limits_{j=-\lambda}^\lambda \binom{2\lambda}{\lambda+j}(-1)^j \frac{1}{\mu_1+\lambda+j}\frac{1}{\mu_2+\lambda-j}= \frac{(-1)^\lambda}{1+2\lambda}\frac{1}{\mu_1+\mu_2+2\lambda}\Biggl[\frac{1}{\binom{\mu_1+2\lambda}{1+2\lambda}}+ \frac{1}{\binom{\mu_2+2\lambda}{1+2\lambda}}\Biggr].
\Phi(y)=h(c(y))\geq h(c(x))+\langle w,c(y)-c(x)\rangle \geq \Phi(x)+\langle w,\nabla c(x)(y-x)\rangle -\frac{\beta\|w\|}{2}\|y-x\|^2 &\geq \Phi(x)+\langle v,y-x\rangle-\frac{\mu}{2}\|y-x\|^2.
{\bf P}_+\left(\bar b^2_m\bar j\check{w}_m/w_m\right)=b_{N-s_\ell}{\bf P}_+\left(\bar\beta^N\bar b^2_m\bar j\check{u_\ell}/u_\ell\right)+{\bf P}_+\left(b_{N-s_\ell}{\bf P}_-\left(\bar\beta^N\bar b^2_m\bar j\check{u}_\ell/u_\ell\right)\right).
\langle \bar{\psi}\psi \rangle_{T 0}~ \sim ~ \vert e H \vert^{\frac{3}{2}}\left( 1 - \frac{T}{T_{c}}\right)^{\frac{1}{2}}\operatorname{l n}\left( 1 - \frac{T}{T_{c}}\right)^{\frac{1}{2}}
\frac{|D_\ell(a_1,\ldots,a_\ell)|}{|I_\ell(a_1,\ldots,a_\ell)|}\ \stackrel{{e}}{\sim}\ \sum_{i=s_{\ell+1}-t_{\ell+1}}^{s_{\ell+1}+t_{\ell+1}}i^{-d}\ \stackrel{{e}}{\sim}\t_{\ell+1}s_{\ell+1}^{-d}.
e^{izH_0}V_{n+1}(A;z',z'') e^{-izH_0}\Psi = -i \int_{z''}^{z'}d\zeta e^{izH_0}A(\zeta ) V_n (A;\zeta , z'') e^{-izH_0}\Psi & = -i \int_{z''}^{z'}d\zeta\, A(\zeta + z) V_n (A;\zeta + z , z'' +z) \Psi & = -i \int_{z''+z}^{z'+z}d\zeta \,A(\zeta ) V_n (A;\zeta , z'' +z) \Psi & = V_{n+1}(A;z' +z , z'' +z) \Psi ,
\begin{aligned}|D^2 u_j|^2-2 \theta a_{0,j}a_{1,j}\geq \ \left[ (1-4 \varepsilon) (a_{0,j}^2+a_{1,j}^2) - \frac{\sigma^2 z_j}{\varepsilon|x|^2}- \frac{(p-2)^2}{4\varepsilon}\frac{|\nabla z_j|^2}{z_j}\right] \theta.\end{aligned}
B_{i,1}=\left\{Q\colon Q\in\bigcup_{j\geq1}\Pi_{j+N},\vert Q\cap\Omega_{i,1}\vert>\frac{1}{2}\vert Q\vert,\vert Q\cap\Omega_{i+1,1}\vert\leq\frac{1}{2}\vert Q\vert\right\}.
b_{p,q}(z)=\begin{cases}\sqrt{\frac{q!}{p!}}(-1)^{q}\,z^{p-q}L_q^{(p-q)}(|z|^2), \p\ge q; begin{align*}1ex]\sqrt{\frac{p!}{q!}}(-1)^{p}\,\overline{z}^{q-p}L_p^{(q-p)}(|z|^2), \p\le q.\end{cases}
\mathfrak{R}_1 = W_{\alpha\beta\gamma\delta, i_1\cdots i_{k}t}- W_{\alpha\beta\gamma\delta, i_1\cdots ti_{k}}&= \underbrace{W_{p\beta\gamma\delta, i_1\cdots i_{k-1}}R_{p\alpha i_{k}t}+W_{\alpha p \gamma\delta, i_1\cdots i_{k-1}}R_{p\beta i_{k}t}+\ldots + W_{\alpha\beta\gamma\delta, i_1\cdots i_{k-2}p}R_{p i_{k-1}i_{k}t}}_{k+3},
\mathrm{SNDR}= \frac{\alpha^2\sigma_x^2}{\varepsilon_d + \sigma_v^2}&= \frac{(E[xh(x)])^2/\sigma_x^2}{E[h^2(x)]-(E[xh(x)])^2/\sigma_x^2-E^2[h(x)] + \sigma_v^2}.
\beta_h^{W_t}(\psi) = \mathrm{i}\int_0^t \left[L_1^{\prime}\left(h(\psi)(\tau) + q, \frac{d h(\psi)}{d\tau}(\tau) \right)h(\psi)(\tau) \right. & + \left. L_2^{\prime}\left(h(\psi)(\tau) + q, \frac{d h(\psi)}{d\tau}(\tau)\right) \frac{d h(\psi)}{d\tau}(\tau) \right]d\tau + \mathrm{tr}\,h^{\prime}(\psi).
\mathbb{E}\left[Z_{t}^{\kappa}\right]=\int_{\mathbb{R}_{\geqslant0}}z^{\kappa}\mu_{Z_{t}}(\mathrm{d}z)\leqslant\int_{\mathbb{R}_{\geqslant0}}(y+z)^{\kappa}\mu_{Z_{t}}(\mathrm{d}z)<\infty.
P_{n}\left(g \cdot v, \ell_{f,v}(g,y) \right) = \int_{- \infty}^{\ell_{f,v}(g,y)}H_n(g \cdot v, t) dt + o \left( \frac{1}{\sqrt{n}}\right) & = \int_{- \infty}^{y}H_n(g \cdot v, t) dt - \int_{\ell_{f,v}(g,y)}^{y}H_n'(g \cdot v, t) dt + o \left( \frac{1}{\sqrt{n}}\right).
\alpha\le \alpha_{j_0},\ \mathcal{E}^0_{}\big(Q_\infty\big)=2m(2j_0+1)+\frac{\alpha^2m}{g'_1(0)^2}E_{\mathbf{t}X^{\ell_0}}^{nr}+\mathcal{O}\big(\alpha^3 K(j_0)\big).
\begin{array}{lll}r_1 := uv - qvu, r_2 := u'v' - qv'u' , r_3 := uu'-q^{-\frac{1}{2}}u'u, r_4 := vu'-q^{\frac{1}{2}}u'v - (q^{-\frac{1}{2}}- q^{\frac{3}{2}}) v'u, r_5 := uv' - q^{\frac{1}{2}}v'u, r_6 := vv' - q^{-\frac{1}{2}}v'v .\end{array}
- \beta{\cal F}( \beta ) \simeq - \frac{2^{- 1 5}\pi^{- 2 6}}{( 2 \alpha^{\prime})^{1 3}}\int_{0}^{\varepsilon}d \tau_{2}\tau_{2}^{- 1 4}\left[ \theta_{3}\left( 0 , \frac{i \beta^{2}}{4 \pi^{2}\tau_{2}\alpha^{\prime}}\right) - 1 \right] e^{\frac{4 \pi}{\tau_{2}}}\tau_{2}^{5 1 / 2}
{\bf P}_{c_L}= \sum\limits_{q=1}^{R}({\cal Q}\sum\limits_{k_1=1}^L{\cal W}_{\nu({k_1})}{\bf p}^{(1)}_{q}) \otimes ({\cal Q}\sum\limits_{k_2=1}^L{\cal W}_{\nu({k_2})}{\bf p}^{(2)}_{q}) \otimes ({\cal Q}\sum\limits_{k_3=1}^L{\cal W}_{\nu({k_3})}{\bf p}^{(3)}_{q}),
2{\frac{\mu}{R^{2}}}u = \arctan{\frac{\mu r^{2}/ R^{2}- E^{2}R^{2}/ 2 \mu}{\sqrt{( E^{2}-{\frac{\mu^{2}}{R^{2}}}) r^{2}-{\frac{\mu^{2}}{R^{4}}}r^{4}+{\frac{\mu^{2}M}{R^{2}}}}}}.
\mathcal{I}_{1,\theta}(f_1,f_2)=\int_\Sigma \left<P_1\nabla f_1,\nabla f_2\right>-2H_1\left(H_2+c\right)f_1f_2\,d\mu_\Sigma+\int_{\partial\Sigma}\left\vert{P_1\nu}\right\vert\alpha_\theta f_1f_2\,d\mu_{\partial\Sigma},
r_{m,\mu}(x) =-(-i)^mx^{m/2}h_{m-2,\mu+2}((2\sqrt{x}i)^{-1}) &=(-1)^m(\mu+2)_{m-2}{\;}_2 F_3 \left({-(m-2)/2,-(m-3)/2 \atop \mu+2,-m+2,1-m-\mu}\Big{|}4x \right)x &=(-1)^m\sum_{j=0}^{\lfloor (m-2)/2 \rfloor}\binom{m-j-2}{j}(\mu+j+2)_{m-2j-2}x^{j+1},
f^{-}( z ) ={\frac{1}{P^{+}}}\int_{0}^{z}\left({\frac{\alpha}{2}}{f^{\prime}}_{\mathrm{t r}}^{2}+{\frac{\beta}{2}}{h^{\prime}}^{2}- 2 \beta h^{\prime \prime}\right) \, d z^{\prime}\; + \; c^{-}.
\sum_{n\geq 1}\frac{(-1)^n q^{2n}}{1-q^{2n}}\frac{(-q/z;q^2)_{n-1}}{(q^2,q^2)_{n-1}}(zq)^{n-1}= \frac{(q^2;q^2)_\infty}{(-qz;q^2)_\infty}- \frac{\tiny 1}{1+zq}
[\mathcal{A},\Omega] =\mathcal{B}\mathcal{A}-\mathcal{A}\mathcal{C}+(\lambda_2-\lambda_3)(\lambda_4-\lambda_1), [\mathcal{B},\Omega] =\mathcal{C}\mathcal{B}-\mathcal{B}\mathcal{A}+(\lambda_3-\lambda_1)(\lambda_4-\lambda_2), [\mathcal{C},\Omega] =\mathcal{A}\mathcal{C}-\mathcal{C}\mathcal{B}+(\lambda_1-\lambda_2)(\lambda_4-\lambda_3),
\sigma^N_\alpha(\bar{a}_{\bar{u}^\smallfrown \bar{v}_2}) = \sigma^N_\beta (\bar{a}_{\bar{u}^\smallfrown \bar{w}}) \mbox{and}\sigma^N_\alpha(\bar{a}_{\bar{u}^\smallfrown \bar{v}}) = \sigma^N_\beta (\bar{a}_{\bar{u}^\smallfrown \bar{w}_2})
\mbox{range map:}r(\xi,[\eta]) :=[\xi\eta], \mbox{domain map:}d(\xi,[\eta]) :=[\eta], \mbox{composition:}(\xi_{1},[\eta]) \circ(\xi_{2},[\xi_{2}^{-1}\eta]):=(\xi_{1}\xi_{2},[\xi_{2}^{-1}\eta]), \mbox{and inversion:}\quad(\xi,[\eta])^{-1}:=(\xi^{-1},[\xi\eta]).
\phi_a(x,t):=\sum_i\mathbf{1}_{I_i}(x)\sum_j\mathbf{1}_{[(j-1)\Delta t,j\Delta t)}(t)\left[\frac{a_{i,j}-a_{i,j-1}}{\Delta t}t+j\cdot a_{i,j-1}-(j-1)\cdot a_{i,j}\right],
{}[e_1,e_2]\,=\,0,\ \ \ \ \ \ \ \ \ \ \ [e_1,e_3]\,=\,\mp\,\tfrac{4}{15}\,e_4,\ \ \ \ \ \ \ \ \ \ \ [e_1,e_4] \,=\,\pm\,\tfrac{5}{4}\,e_1,{}[e_2,e_3]\,=\,0,\ \ \ \ \ \ \ \ \ \ \ [e_2,e_4] \,=\,0,{}[e_3,e_4] \,=\,\mp\,\tfrac{5}{4}\,e_3.
T := \left\{t \in \mathbb Z_{(p)}\,\middle| \,\begin{tabular}{c}\textup{there is a $\lambda_t\in \mathbb Z_{>0}$ and a $\mathbb Z_{(p)}$-Weil divisor $E_t\ge 0$ on $U$}\textup{such that $B|_U +\lambda_tD|_U -t h^*K_V \sim_{\mathbb Z_{(p)}}E_t$}\end{tabular}\right\}.
(D^t)^N(\psi) = \sum_{\substack{\alpha = (\alpha_0, ..., \alpha_k) |\alpha| = N}}C_{N, \alpha}\frac{\partial^{\alpha_0}\psi \partial^{\alpha_1+1}\phi \partial^{\alpha_2+1}\phi ... \partial^{\alpha_k+1}\phi}{\lambda^N(\phi')^{N+k}}.
d(\rho_j \alpha_j)=\eta_j\wedge \iota_{\xi_j}(d\rho_j \wedge \alpha_j+\rho_j d\alpha_j)=\eta_j\wedge (\pm d\rho_j \wedge \iota_{\xi_j}\alpha_j+\rho_j \iota_{\xi_j}d\alpha_j)=0.
S_M = N\frac{\coth\frac{3}{4}\beta}{1+2{\rm e}^{-\frac{3}{2}\beta}}\frac{1}{2}{\rm Tr}\left[M^2-\frac{2}{1+{\rm e}^{\frac{3}{2}\beta}}{\cal L}M{\cal L}^2M\right]+N{\rm Tr}\ln (1-gM).
M_{a,b}(r)\; =\;\frac{\,e^r\,r^{a-b}\,}{\Gamma(a)}\,(1+O(r^{-1}))\qquad\textrm{as}\,r\to+\infty M_{a,b}(r)\; =\;1+O(r)\qquad\qquad\;\;\textrm{as}\,r\downarrow 0\quad\textrm{and}-\!b\notin\mathbb{N}
\mbox{\textbf{P1-1:}}\;\; \max_{\{x_{k_j}\}}\sum_{k=1}^K{\eta_{k}}\mbox{s.t.}\;\; \sum_k x_{k_j}\leq L_j \leq M_j,~j=1,2,\cdots,J \;\; \sum_j x_{k_j}\leq 1,~k=1,2,\cdots,K \;\; \mbox{Constraints~(\ref{eqn:mmhn95}),~(\ref{eqn:mmhn98nn}), (\ref{eqn:mmhn98}),~(\ref{eqn:mmhn87})}.
\begin{cases}\sinh(2\lambda +2\tilde{\Lambda})= \sinh(2\lambda)+2\cosh(2\lambda)\tilde{\Lambda}+ 4 \sinh(2\lambda +2\xi_1)\tilde{\Lambda}^2, \cosh(2\lambda +2\tilde{\Lambda})= \cosh(2\lambda)+2\sinh(2\lambda)\tilde{\Lambda}+4 \cosh(2\lambda +2\xi_2) \tilde{\Lambda}^2,\end{cases}
\begin{aligned}-(m+1)\pi_{m+1,P}(X)+\sum_{p\leq X}\pi_{m,P}\left(\frac{X}{p}\right) \geq -(m+1)\pi_{m+1,A}(X)+ \sum_{\substack{p\leq X p\in A}}\pi_{m,A}\left(\frac{X}{p}\right) &\geq 0.\end{aligned}
J_{12}= K_2u_{22}+\sum_{3\leq i\leq n}K_iu_{ii}= \frac{K_2}{d_2}\big[-\sum_{3\leq i\leq n}d_iu_{ii}+f v^3+\frac{h'}{2(1-G_u)}\frac{Au_1}{w_1}-\frac{D}{1-G_u}\big]+\sum_{3\leq i\leq n}K_iu_{ii}= \sum_{3\leq i\leq n}[K_i-\frac{K_2d_i}{d_2}]u_{ii}+\frac{K_2}{d_2}[f v^3+\frac{h'}{2(1-G_u)}\frac{Au_1}{w_1}-\frac{D}{1-G_u}].
\int_{\Delta_{\theta,t}^n}f_1(s_1) \dots f_k(s_k) \int_{\Delta_{\theta, s_k}^p}g_1(r_1) \dots g_p(r_p) dr_p \dots dr_1 f_{k+1}(s_{k+1}) \dots f_n(s_n) ds_n \dots ds_1 = \sum_{\sigma \in A_{n,p}}\int_{\Delta_{\theta,t}^{n+p}}h^{\sigma}_1(w_1) \dots h^{\sigma}_{n+p}(w_{n+p}) dw_{n+p}\dots dw_1,
P\left( a_{\hbar},a_{\hbar}^{\dag}\right) =\sum_{k=0}^{d}\sum_{\mathbf{b}=\left( b_{1},\dots,b_{k}\right) \in\Lambda_{1}^{k}}c_{k}\left(\mathbf{b}\right) \Xi_{\hbar}\left( b_{1}\right) \dots\Xi_{\hbar}\left(b_{k}\right)
\hspace{. 4 i n}= c \times \frac{\delta_{m^{\prime}m}( q^{2}; q^{2 n})_{\infty}( q^{2}; x )_{\infty}^{n - 1}\varphi ( z ; x )}{( q^{2 n}; q^{2 n})_{\infty}t r_{V ( \Lambda_{i})}( x^{- d}y^{2 \overline{{{\rho}}}})}\left( \prod_{k = 1}^{\infty}\frac{( q^{2}x^{k}; q^{2 n})_{\infty}}{( q^{2 n}x^{k}; q^{2 n})_{\infty}}\right)^{2}
\biggl< \sum_{j=1}^m \pi(\tau_j)\eta_j \biggm| \sum_{i=1}^n \pi(\phi*\psi_i)\xi_i \biggr> = \biggl< \sum_{j=1}^m \pi(\phi^**\tau_i)\eta_i \biggm| \sum_{i=1}^n \pi(\psi_i)\xi_i \biggr>
k_{i+1}^{\pm}(u+\frac{1}{2}hi)^{-1}E_{i+1}(v)k_{i+1}^{\pm}(u+\frac{1}{2}hi)=\frac{u_{\pm}-v+\frac{1}{2}h}{u_{\pm}-v-\frac{1}{2}h}E_{i+1}(v), k_{i}^{\pm}(u+\frac{1}{2}hi)E_{i+1}(v)k_{i}^{\pm}(u+\frac{1}{2}hi)^{-1}=E_{i+1}(v),
\frac{|X_{\kappa}(n_1,\hdots,n_N)|}{|X(n_1,\hdots,n_N)|}=t^{c_{\kappa}}\cdot (1-O(t))c_{\kappa}=\sum_{i=1}^N\sum_{j=1}^m(j-1)\kappa^{(i)}_j.
|I_{2,\rho}(x)| \leq\int_{|xy^{-1}|\geq R}\frac{\rho(|xy^{-1}|)}{(1+|xy^{-1}|)^{\gamma}}|f(y)|dy\leq\int_{|xy^{-1}|\geq R}\frac{\rho(|xy^{-1}|)}{|xy^{-1}|^{\gamma}}|f(y)|dy = \sum_{k=0}^{\infty}\int_{2^{k}R\leq|xy^{-1}|<2^{k+1}R}\frac{\rho(|xy^{-1}|)}{|xy^{-1}|^{\gamma}}|f(y)|dy.
\frac{\|Q^{\frac{1}{2}}s_{\overline{j}}/t_{\overline{j}}+Q^{-\frac{1}{2}}b_{i}\|}{\sqrt{u_{i}-d_{i}+\|Q^{-\frac{1}{2}}b_{i}\|^2}}\le \sqrt{2},~i=1,\ldots,p.
R_1+R_2+R_3 \leq \max_{\alpha\in[0,1]}\left\{\mathcal{C}\left(\tfrac{\bar{\alpha}P_1}{1+\alpha P_1}\right) + \mathcal{C}(a\alpha P_1+bP_2) \right\}+ 0.5, &=\max_{\alpha\in[0,1]}\frac{1}{2}\log\left( a(1+P_1) -\tfrac{a-1-bP_2}{1+\alpha P_1}(1+P_1) \right) + 0.5, &=\mathcal{C}(aP_1+bP_2)+0.5.
\mathbb{R}^{n}\setminus \Omega^{*}\subset \mathbb{R}^{n}\setminus \big(Q_{1,k_{1}}^{*}\bigcup Q_{2,k_{2}}^{*}\big) \subset \bigcup_{h=1}^{\infty}\bigcup_{i=1}^{\infty}\big(\mathcal{Q}_{1,k_{1}}^{i}\bigcap \mathcal{Q}_{2,k_{2}}^{h}\big).