harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
P^+_{\gamma_{*},\xi}-P^+_{\gamma_n,\xi}=\frac{1}{2\pi}\int_{-\infty}^{+\infty}(D_{\gamma_{*},\xi}-iz)^{-1}V_{\gamma_n-\gamma_{*},\xi}(D_{\gamma_n,\xi}-iz)^{-1}dz,
h^*(p_1,p_2) =n_1p_1+n_2p_2+ne^{-p_1A_{21}-p_2A_{12}}&=\frac{n_1}{A_{21}}(p_1A_{21}+p_2A_{12})+ne^{-p_1A_{21}-p_2A_{12}}&=\frac{n_1}{A_{21}}\ln \frac{nA_{21}}{n_1}+ne^{-\ln \frac{nA_{21}}{n_1}}=\frac{n_1}{A_{21}}(1+\ln \frac{nA_{21}}{n_1})
g_{4}= d \hat{r}^{2}+ \frac{\hat{r}^{2}}{4}\left[ d \theta^{2}+ ( \sin \theta )^{2}d \varphi^{2}+ \left( \frac{2 d z}{l_{s}g N}+ ( 1 - \cos \theta ) d \varphi \right)^{2}\right] ~ .
\sum_{\lambda\subseteq [n+1,\infty)}sp_\lambda(\mathbf{x}^{\pm};z)s_\lambda(y)=\prod^{n}_{i}\prod^{n+1}_{j}\left[(1-x_iy_j)(1-x^{-1}_iy_j)\right]^{-1}\prod^{n+1}_{j}(1-zy_j)^{-1}\prod_{1\leq k< l\leq n+1}(1-y_ky_l),
f^{B,\alpha}(\xi,\mu) := \partial_x^\alpha|_{x=0}\alpha_{-x}\big( f(\xi+Bx,\mu) \big) = \sum_{j=0}^{|\alpha|}\sum_{\substack{\beta+\gamma = \alpha |\gamma| = j}}\binom \alpha \beta (i\delta)^\beta \partial_{B,\xi}^\gamma f(\xi,\mu) .
e^{-\pi (-Q_0^{-1}H Q_0^{-1}+ 1/2 Q_0^{-1}HQ_0^{-1}H Q_0^{-1}+ o(\lVert H\rVert))[m]t}= 1 + \pi Q_0^{-1}H Q_0^{-1}[m] + o(\lVert H\rVert) ,
\|\varphi_k(x)-\varphi_n(\psi_n(\varphi_k(x)))\| =\|(\Psi^{-1}\circ\rho_k)(x)-(\Psi^{-1}\circ\rho_n)(\psi_n(\varphi_k(x)))\| =\|\rho_k(x)-\rho_n(\psi_n(\varphi_k(x)))\| < \|\rho_k(x)-\rho_n(\rho_{n,k}(x)))\|+ \varepsilon =\varepsilon.
\lim_{\delta\rightarrow 0}\frac{1}{\delta^{\frac{2}{n+1}}}\cdot \frac{\|x\|_2-\|x_{\delta}\|_2}{\|x\|_2}=\frac{(n+1)^{\frac{2}{n+1}}}{2}\left(\frac{|K|_n}{|B_2^{n-1}|_{n-1}}\right)^{\frac{2}{n+1}}\frac{\kappa(x)^{\frac{1}{n+1}}}{\langle x, u(x)\rangle}=G(x),
|e^{i\frac{t}{2\pi R}\Delta}f_{dk}(x_1)| =\frac{R^{-1/2}}{(2\pi)^{1/2}}\Big|\int \widehat{\phi}\big(R^{-1/2}(\xi_1-\pi R)\big)e^{ix_1\xi_1 -i\frac{t}{2\pi R}\xi_1^{2}}d \xi_1\Big| &=\frac{1}{(2\pi)^{1/2}}\Big|\int_{-\rho}^\rho e^{iR^{1/2}(x_1-t)y-i\frac{t}{2\pi}y^{2}}d y\Big|\simeq 1 \, ,
\dim V(r \varpi_n) = F(r) \dfrac{F(n+r)}{F(n+2r)}\dfrac{F(2n+2r)}{F(2n+r)}\dfrac{F(n)}{F(n+r)}\times \dfrac{F(2n+r)}{F(2n+1)}\dfrac{\Phi(2n+1)}{\Phi(2n+2r-1)}\dfrac{\Phi(2r-1)}{F(r)}&= \dfrac{\Phi(2r-1)\Phi(2n+2r)F(n)}{\Phi(2n)F(n+2r)}.
0 < \delta \leq \frac{d-\alpha}{2}\kappa = \frac{2^\alpha \Gamma\left(\frac{\alpha+\delta}{2}\right) \Gamma\left(\frac{d-\delta}{2}\right)}{\Gamma\left(\frac{\delta}{2}\right)\Gamma\left(\frac{d-\alpha-\delta}{2}\right)}.
\big(1+(b_i-z+\omega_B) m_{\mu_A\boxplus \mu_B}+O_\prec(N^{-\frac{\gamma}{4}})\big)T_{i}= O_\prec(N^{-\frac{\gamma}{4}})+O_\prec(\Psi)=O_\prec(N^{-\frac{\gamma}{4}})\,,
\delta_{0}: f \rightarrow \left( \begin{array}{c}{{- z f}}\\{{2 \left( \frac{\partial}{\partial \bar{s}}+ \frac{1}{2}\chi \right) f}}\\ \end{array}\right) , \ \ \delta_{1}: \left( \begin{array}{c}{{g_{0}}}\\{{g_{1}}}\\ \end{array}\right) \rightarrow \left( - 2 \left( \frac{\partial}{\partial \bar{s}}+ \frac{1}{2}\chi \right) g_{0}- z g_{1}\right) .
A\left(1,(1,0)\right) ={1 \choose 1,0}\frac{1}{4}\int_{0}^1(1-s)sds=\frac{1}{4!}, A\left(1,(0,1)\right) ={1 \choose 0,1}\frac{1}{4\cdot 2}\int_{0}^1s^2ds=\frac{1}{4!}.
R_{j;\,m+1}(\tau, v-\tau) - e^{2\pi i (m+1) (\tau-2v)}R_{j;\, m+1}(\tau, v) begin{align*}1ex] = 2 e^{-\frac{\pi i \tau}{2(m+1)}(2m+2-j)^2 + 2\pi i (2m+2-j)\tau}e^{-2\pi i (2m+2-j)v}.
\begin{aligned}w^{n+1}_i=w_i^n+\frac{\Delta t}{2}P^n_{ij}(w_j^n-w^n_i)+\Delta t u^n, &w^{n+1}_j=w_j^n+\frac{\Delta t}{2}P^n_{ji}(w_i^n-w^n_j)+\Delta t u^n, \end{aligned}
\begin{array}{lll}x_{\varepsilon}(t) = \alpha \int_0^t\frac{ds}{x_{\varepsilon}(s)}+ \alpha^m \beta \int_0^t\psi_{\varepsilon}(x_{\varepsilon}(s), z_{\varepsilon}(s))ds z_{\varepsilon}(t) = \alpha y + \alpha(\alpha \delta-\beta \gamma) \int_0^t\psi_{\varepsilon}(x_{\varepsilon}(s), z_{\varepsilon}(s))ds . \end{array}
\psi_1^{T}( -\log(T-t) + \log T,\tfrac{x}{T-t}) :=(T-t)^{\frac{2}{p-1}}u(t,x), \psi_2^{T}( -\log(T-t) + \log T,\tfrac{x}{T-t}) :=(T-t)^{\frac{2}{p-1}+1}\partial_t u(t,x),
\Delta_{a}(x) = \frac{x^{\frac14 +\frac{a}{2}}}{\pi\sqrt{2}}\sum_{n\leq N}\frac{\sigma_{a}(n)}{n^{\frac{3}{4}+\frac{a}{2}}}\cos\left(4\pi\sqrt{nx}-\frac{\pi}{4}\right) + O\left(x^{\frac12 + \varepsilon}N^{-\frac12}\right)
\mathcal{K}[f](\mathrm{y},\tau) := \frac{1}{\alpha}\displaystyle \int_{0}^{\tau}\int_{\partial\Omega}\partial_{\nu_\mathrm{y}}\Phi(\mathrm{y},\tau; \mathrm{v},\mathrm{s}) f(\mathrm{v},\mathrm{s}) d\sigma_{\mathrm{v}}d\mathrm{s}.
\sup_{J_j}|\psi_\lambda| \leq C_7 \left ( 2^{-4/3 \ell}\right )^{\frac{c_0}{2}m \sqrt{\lambda}}e^{\sqrt{\lambda}}= C_7 e^{(1 - \frac{2}{3}(\log 2){c_0}m \ell) \sqrt{\lambda}}\leq C_7 e^{(1 -{c_0}m \ell) \sqrt{\lambda}}.
I_2 \leq C\int_{(t-s)^{\frac{1}{\alpha}}<|z| \leq \frac{|x-y|}{2}}\left[\rho^0(t-s,y-x-z)+ \rho^0(t-s,y-x+z) + \rho^0(t-s,y-x) \right] \, \frac{dz}{|z|^{d + \alpha}}+ C\int_{|z| > \frac{|x-y|}{2}}\left[\rho^0(t-s,y-x-z) + \rho^0(t-s,y-x+z) + \rho^0(t-s,y-x) \right] \, \frac{dz}{|z|^{d + \alpha}}=: I_{2,1}+ I_{2,2}.
\int_{\mathbb{R}^{n}}f(x)g(x)dx=\int_{\mathbb{R}^{n}}\int_{0}^{\infty}\chi_{f(x)>r}dr\int_{0}^{\infty}\chi_{g(x)>s}dsdx=\int_{\mathbb{R}^{n}}\int_{0}^{\infty}\int_{0}^{\infty}\chi_{f(x)>r}\chi_{g(x)>s}drdsdx
p_j(i,l) = \begin{cases}c, \textrm{if}l \cdot \sum_{t=2}^{j}e_t=0, l+e_j = i, 1, \textrm{if}l \cdot \sum_{t=2}^{j}e_t\neq 0, l+e_j = i, 0, \textrm{o.w.}\end{cases}
\widetilde{\psi}_{k}^{(n^{2})}\big(x;\omega ,N,- \Vert \psi_{n}\Vert^{2}\big) = [ E_{n}(\omega )-E_{k}(\omega ) ] \psi_{k}(x;\omega,N) -\frac{W(\psi_{n},\psi_{k}\,|\, x)}{\Psi_{n}^{(n)}(x;\omega ,N,- \Vert \psi_{n}\Vert^{2})},\!\!\! k\neq n,
V_1 (x,t) \sim-\frac{R}{4}\frac{y_\omega^{2}}{\sin(y_\omega)^{2}}+\frac{R}{2}\frac{y_\omega^{2}}{\sin(y_\omega)^{2}}=\frac{R}{4}\frac{y_\omega^{2}}{\sin(y_\omega)^{2}}\asymp d(x,t)^2
\sum_{I \subset [1,n] \atop |I|=k}\mathfrak{D}_{I}\prod_{i \notin I, j \in J \atop i < j}\big(1-t^2 q_{ij}\big)= \sum_{I \subset [1,n] \atop I=\{1 \le i_1,\ldots,i_k \le n \}}\prod_{a=1}^{k}\bigl(z_{i_{a}}+1 +t h_{i_{a}}(i_{a}-a) \bigr).
\phi_{\varepsilon}\left(x\right)+\psi_{\varepsilon}\left(Lx\right) = a \lVert x \rVert^2_X +\langle u_\varepsilon,x\rangle_X +b \lVert Lx \rVert^2_Y +\langle v_\varepsilon,Lx\rangle_Y \geq b \lVert Lx \rVert^2_Y-\varepsilon,
\sup_{x\geq 20}\left\|(u,h) x^{\frac{1}{4}}\right\|_{L_y^{\infty}}\leq \sup_{x\geq 20}\left\|(u_x,h_x) x^{\frac{5}{4}}\right\|_{L_y^{\infty}}\lesssim \|(u ,v ,h ,g)\|_{X_{1}\cap Y_{1}\cap Y_{2}}
\begin{aligned}S_{\{b_{i,i+1}\}, \{c_{i,i+1}\}}(\psi_p,\psi_p';c,w_{G_n}) < p^{-(n-1)\ell}(1-p^{-1})^{-(n-1)}\cdot & \cdot \sum_{(\{b_{i,j}\}, \{c_{i,j}\},j-i \geq 2)\in\mathcal{S}}\#(X_{\{b_{i,j}\}}^{\{c_{i,j}\}}( w_{G_n}c)) S_{w_{G_n}}(\theta_{\{b_{i,j}\}}^{\{c_{i,j}\}};\ell),\end{aligned}
\left|r\cos\theta\int_{[0,+\infty]}\frac{1-t^{2}}{|re^{i\theta}-t|^{2}}\,d\sigma(t)\right| \le r\int_{[0,+\infty]}\frac{1+t^{2}}{|re^{i\theta}-t|^{2}}\,d\sigma(t) =\frac{\theta}{\sin\theta}I_{r}(\theta)\le\frac{\pi}{2}I_{r}(f(r))<2
\log \left( \frac{d\mathbb{Q}_{n}}{d\mathbb{P}_{n}}\right)- \sum_{k=2}^{m_{n}}\frac{2\mu_{k}\left( C_{n,k}- (n-1)\mathbb{I}_{k=2}\right)-\mu_{k}^2}{4k}\left| \mathbb{P}_{n}\right. \stackrel{p}{\to}0.
\mathrm{P}_{m,k}^1 = 1- \int_{\delta\mathcal{R}_c}^{\infty}\int^{z+ \mathcal{R}_c}_{z- \mathcal{R}_c}e^{- \frac{{\epsilon}_0r^{\alpha}}{\rho\zeta_0}}g(z,r) d r \bar{f}_{r_m}(z)dz,
F (v) := \begin{pmatrix}f_{e_1}(0) \vdots f_{e_l}(0) f_{e_{l + 1}}(L (e_{l + 1})) \vdots f_{e_{m}}(L (e_{m})) \end{pmatrix}F' (v) := \begin{pmatrix}f'_{e_1}(0) \vdots f'_{e_l}(0) - f'_{e_{l + 1}}(L (e_{l + 1})) \vdots - f'_{e_{m}}(L (e_{m})) \end{pmatrix}.
A_{j}A_{k}\mathcal{L}^{n-2}=[A_{j}]_{A_{k}}\mathcal{L}_{A_{k}}^{n-2}=\sum_{h_{1}+...+h_{r}=n-2}\frac{(n-2)!}{h_{1}!...h_{r}!}a_{1k}^{h_{1}}...a_{jk}^{h_{j}+1}...a_{kk}^{h_{k}}...a_{rk}^{h_{r}}l_{k}.
\Phi(n_{-}(c)) = \frac{\psi( c^{-1}s_\lambda)}{\mathrm{Vol}(U^{d(k,\lambda)}_{F,\mathfrak{p}})}\cdot \int_{U^{d(k,\lambda)}_{F,\mathfrak{p}}}\psi(- c^{-1}s_\lambda y)dy,
\begin{aligned}\int_{0}^{t}\Gamma_{11}(t+\tau,t-\tau)d\tau=\int_{0}^{t}E(t+\tau,t-\tau)d\tau -\int_{B}\int_{0}^{t}\biggl[\rho\ddot{u}_{i}(t+\tau)u_{i}(t-\tau)+\frac{1}{\beta}\gamma\dot{\theta}(t+\tau)A\theta(t-\tau) \ -k_{ji}\Theta(t+\tau)A\theta_{,i}(t-\tau)\biggr]d\tau dv,\end{aligned}
\sum_{n=0}^\infty\frac{BC_{2,n}}{\Pi(n)}x^n =\dfrac{\dfrac{x^9}{D_2}}{\sum_{i=2}^\infty\dfrac{x^{3^i}}{D_i}}&=1-\frac{D_2}{D_3}x^{18}+\frac{D_2^2}{D_3^2}x^{36}-\frac{D_2^3}{D_3^3}x^{54}&\quad+\left(\frac{D_2^4}{D_3^4}-\frac{D_2}{D_4}\right)x^{72}-\left(\frac{D_2^5}{D_3^5}-\frac{2 D_2^2}{D_3 D_4}\right)x^{72}+\cdots\,.
\Pi ( t , \Theta ) = \frac{\partial}{\partial \frac{d \phi}{d t}}L = \frac{i}{2}\frac{\partial}{\partial \frac{d \phi}{d t}}\int d \Theta ( D_{\Theta}\phi ) \dot{\phi}= \frac{1}{2}\int d \Theta \{\Theta \dot{\phi}+ i D_{\Theta}\phi \}, \qquad \dot{\phi}\equiv \frac{d \phi}{d t},
\begin{cases}\partial_t\psi =-(-\Delta)^{s}\psi + V_{\mu, \xi}\psi + f(x, t)\quad \quad\Omega\times (t_0, \infty), \psi = g\quad (\mathbb{R}^n\setminus\Omega)\times (t_0,\infty), \psi(\cdot,t_0) = h\quad \quad\mathbb{R}^n,\end{cases}
n(\rho_{-},\rho_{+},t)=\frac{(\rho_{+}-\rho_{-})K(\rho_{-},\rho_{+},t)}{\sqrt{2\pi t (H'(\rho_{+})-H'(\rho_{-}))^3}}\frac{\rho_{+}+\int_{\rho_{+}}^{\infty}\frac{H''(\rho)-K(\rho_{+},\rho,t)}{\sqrt{2\pi t (H'(\rho)-H'(\rho_{+}))^3}}d\rho}{\rho_{-}+\int_{\rho_{-}}^{\infty}\frac{H''(\rho)-K(\rho_{-},\rho,t)}{\sqrt{2\pi t (H'(\rho)-H'(\rho_{-}))^3}}d\rho}
M=\{(Z^0,Z^1)=Z^0(1,z^1)\in \mathbb{C}^2 \; | \; Z^0\in \mathbb{C}^{\times}, \;\; z^1\in \overline{M}^{}, \;\; \frac{t^3}{3}> \frac{\chi\zeta(3)}{2(2\pi)^3}\}\,.
-\frac{1}{\kappa}R_{\mu\nu}(\Gamma)(\Phi +b_{g}\sqrt{-g})+\frac{1}{2}\phi_{,\mu}\phi_{,\nu}(\Phi +b_{k}\sqrt{-g})-\frac{1}{2}\sqrt{-g}g_{\mu\nu}\left[-\frac{b_{g}}{\kappa}R(\Gamma ,g)+\frac{b_{k}}{2}g^{\alpha\beta}\phi_{,\alpha}\phi_{,\beta}\right]=0
\underset{\pmb{U}_{K+1}}{}\mathrm{tr}\left\{\pmb{U}_{K+1}^H\left(\sum\limits_{\ell=1}^{K}\pmb{H}_{K+1,\ell}\pmb{F}_\ell\pmb{F}_\ell^H\pmb{H}_{K+1,\ell}^H\right)\pmb{U}_{K+1}\right\}\pmb{U}_{K+1}^H\pmb{U}_{K+1}= \pmb{I}_d.
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}\hline$p\setminus$ q 0 1 2 3 4 5 6 7 8 \hline0 + + + + + + + + + \hline1 + + + + + + \hline2 + + + + + + + \hline3 + + + + + \hline4 + + + + + + + + + \hline5 + + + + + + \hline6 + + + + + + + \hline7 + + + + + \hline8 + + + + + + + + + \hline \end{tabular}
\tilde{p}_0(z;q)= \frac{1}{\tilde{\beta}_0}z, \tilde{p}_{n+1}(z;q)= \frac{1}{\tilde{\pi}_{n(n+1)}}\Big[z^{2}\tilde{p}_{n}(z;q)-\sum_{k=0}^{n}\tilde{\pi}_{nk}\, \tilde{p}_k(z;q)\Big],\; \left(n\in \mathbb{N}\right).
\begin{array}{c c c c c c c}{{\nabla_{i}\tilde{\Phi}}}&{{=}}&{{( \partial_{i}+ p \partial_{i}{\cal K}) \tilde{\Phi}}}&{{;}}&{{\nabla_{i^{*}}\tilde{\Phi}}}&{{=}}&{{\partial_{i^{*}}\tilde{\Phi}}}\\ \end{array}
\displaystyle\tfrac{1}{2}\,\tfrac{d}{d t}\left(\left\| z_k(t)\right\|_2^2+2\alpha\left\|Dz_k(t)\right\|_2^2\right) +2\nu \left\|D z_k(t)\right\|_2^2=\left( w(t),z_k(t)\right)-\left( \boldsymbol{curl}\,\sigma(\boldsymbol z_k(t))\times \boldsymbol y_1(t),\boldsymbol z_k(t)\right).
\sum_{j=1}^J k_n^{(j)}B^{(j)}(n) \leq \frac{n}{2}\log \left( \sum_{j=1}^J k_n^{(j)}\right) - \sum_{j=1}^J \beta^{(j)}\ell_n H_2 \left( \alpha_n^{(j)}\right).
A_t^{W,W}(l_n) = \int_0^t l_n(r, W_r) \mathrm dr &= \int_0^t \left( \partial_t \phi_n(r, W_r) +\frac12 \Delta \phi_n (r, W_r)\right) \mathrm dr &= \phi_n(t, W_t)- \phi_n (0, W_0) - \int_0^t \nabla \phi_n^*(r, W_r) \mathrm d W_r,
U_H=(v_H,y_H)=\left(\frac{-4\alpha^2 v_L^2+2\alpha \alpha_2 v_L(\sqrt{y_L^2-4v_L}-y_L)}{\alpha^2(\sqrt{y_L^2-4v_L}-y_L)^2-2\alpha_1 v_L(\sqrt{y_L^2-4v_L}-y_L)}, -\frac{\alpha_1v_H}{\alpha^2}-\frac{\alpha_2}{\alpha}\right);
|\xi_{m-1}(x')-f_0^m(w_0)| \leq |(f_0^{m-1})'(f_0(w_0))|W(x',y_1,m-1) &\leq |(f_0^{m-1})'(f_0(w_0))| \sum_{i=1}^{m-1}\frac{4|\lambda^{k(i-1)}(\lambda z_0)^k|}{|(f_0^i)'(f_0(w_0))|}&= Q|(f_0^{m-1})'(f_0(w_0))| |z_0|^k,
\sum_{\rho\in R_\epsilon}(\rho) =\frac{1}{2\pi}\int_{0}^{T}\left[\log \left| \frac{\frac{1}{2}-\epsilon+it}{2}\right|-\log \left|\frac{\frac{1}{2}+\epsilon+it}{2}\right|\right]\dt+ \frac{1}{2\pi}\int_{\frac{1}{2}-\epsilon}^{\frac{1}{2}+\epsilon}\left[\arg \left(\frac{\sigma+iT}{2}\right)-\arg \frac{\sigma}{2}\right]\d\sigma
\mathcal{F}(R,M) : D_{\delta, \mu}\times \mathbb{R}^{n \times n}\rightarrow \mathbb{R}, \mathcal{F}(R,M) =\sum_{i,j,k,\ell}\frac{\partial^2 F}{\partial R_{ij}\partial R_{k\ell}}M_{ij}M_{k\ell}& = - \sum_{i,j,k,\ell}R^{\ell i}R^{jk}M_{ij}M_{k\ell}.
\frac{1}{t^{\eta}}\int_0^{t}z^{\eta -1}|f(z)|\, dz \le \frac{1}{x^{\eta}}\int_0^{x}z^{\eta-1}|f(z)|\, dz + \frac{1}{t^{\eta}}\int_x^t z^{\eta-1}|f(z)|\, dz & \le \frac{1}{x^{\eta}}\int_0^{x}z^{\eta-1}|f(z)|\, dz + \int_x^{\infty}\frac{|f(z)|}{z}\, dz= H_{\eta}|f|(x) + H_0^{\infty}|f|(x).
\max_{y\in \Omega_\varepsilon}|y-x_\varepsilon| |\nabla u_\varepsilon(y)| u_\varepsilon(y)=|y_\varepsilon-x_\varepsilon| |\nabla u_\varepsilon(y_\varepsilon)| u_\varepsilon(y_\varepsilon):=C_\varepsilon\to +\infty
\frac{1}{N}\sum_{n=0}^{N-1}\prod_{k=1}^{j-1}f_k(T_{g(k)}^{p_k(a_n)}x) (f_j-\mathbb{E}(f_j|\mathcal{A}_{g_{j_0}}))(T_{g(j)}^{p_j(a_n)}x)\prod_{l=j+1}^{m}\mathbb{E}(f_{l}|\mathcal{A})(T_{g(l)}^{p_{l}(a_n)}x)\rightarrow 0
A{\pmb{X^1}}D\tilde{A}^{-1}= A{\pmb{X^1}}A^{-1}(AD\tilde{A}^{-1}) = LD, AD{\pmb{X^1}}\tilde{A}^{-1}= (AD\tilde{A}^{-1}) \tilde{A}{\pmb{X^1}}\tilde{A}^{-1}= DM.
H(x; n) = u(x; n) \cdot n^{\gamma(n)}\cdot \Phi(x)^n \cdot \{1+ e(x; n) \}, begin{align*}1mm]|e( x; n)| \le C_0/n,{}^{\forall}n \ge N_0, \,\, r_0 \le{}^{\forall}x \le r_1.
1 \asymp \int_{\prod_{k=1}^rx_{k}\leq (nb)^{1/m},x_{k}\geq1}\left(b\prod_{k=1}^rx_{k}^m - n^{-1}\prod_{k=1}^rx_{k}^{2m}\right) \left(1+\sum_{k=1}^rx_k^2\right)^{-1}dx_{1}\cdots dx_{r}& \asymp \int_{\prod_{k=1}^rx_{k}\leq (nb)^{1/m},x_{k}\geq1}b\prod_{k=1}^rx_{k}^m \left(1+\sum_{k=1}^rx_k^2\right)^{-1}dx_{1}\cdots dx_{r}
\frac{1}{4}(Z_1+Z_2)^3+\frac{3}{4}(Z_1+Z_2)(Z_1-Z_2)^2 =-9Z_3^3 \frac{1}{4}+\frac{3}{4}V^2 = 9U^3 V^2 =12U^3-\frac{1}{3}
L_{f_0(g)}= \left(\frac{p^{j/2 -1}}{\alpha_p(f)}\right)^{\mu-1}\frac{\left\langle f_0^\tau \bigg |_j \left( \begin{array}{cc}0 -1 N p^\mu 0 \end{array}\right), g \right\rangle_{Np^\mu}}{\left\langle f_0^\tau \bigg |_j \left( \begin{array}{cc}0 -1 N p 0 \end{array}\right), f_0 \right\rangle_{Np}}.
E=\langle d_1^{16}\rangle \times\prod_{l=1}^7\langle d_{2l+1}^{-1}d_{32-(2l+1)}\rangle\times &\times\langle d_3^{-2}d_{13}^2\rangle\times\langle d_5^{-2}d_{11}^2\rangle\times\langle d_7^{-2}d_9^2\rangle\times &\times\langle d_1^{-4}d_7^4\rangle\times\langle d_3^{-4}d_5^4\rangle\times\langle d_1^{-8}d_3^8\rangle.
\ll X^{\varepsilon}\sum_{\substack{p \leq z^{1/2}p \nmid q}}S(p^{e_0}, 1, 1, 1) \ll X^{1+\varepsilon}\Big(\sum_{p \leq z^{1/4}}\frac{1}{z^{1/2}}+ \sum_{z^{1/4}\leq p \leq z^{1/2}}\frac{1}{p^2}\Big) + z^{1/2}X^{\varepsilon}Y & \ll \frac{X^{1+\varepsilon}}{z^{1/4}}+ z^{1/2}X^{\varepsilon}Y.
L_k := \frac{1}{\sqrt{N}}L\left( \left[ l(h^k_{i,j})\right] \right), L^*_k := \frac{1}{\sqrt{N}}L\left( \left[ l^* (h^k_{j,i})\right] \right) R_k := \frac{1}{\sqrt{N}}R\left( \left[r(h^k_{i,j})\right] \right), R^*_k := \frac{1}{\sqrt{N}}R\left( \left[ r^*(h^k_{j,i})\right] \right).
\frac{d}{dt}[(\tilde{\rho}(t,q(t,x))+1)q_x(t,x)] &= (\tilde{\rho_t}(t,q(t,x))+\tilde{\rho_x}(t,q(t,x))q_t(t,x))q_x(t,x)+(\tilde{\rho}(t,q(t,x))+1)q_{xt}(t,x) &= \Bigl(\tilde{\rho_t}(t,q(t,x))+\tilde{\rho_x}(t,q(t,x))u(t,q(t,x))+\tilde{\rho}(t,q(t,x))u_x(t,q(t,x))+u_x(t,q(t,x))\Bigr)q_x(t,x) &= 0.
\begin{aligned}\zeta_{n,j}^*(r) =\zeta_{n,j}^*(\Lambda_{n,j,R}^{-})+o(1)+o\Big(\frac{1}{\lambda_{n,j}^{(1)}}\Big)R+O({R^{-2}}) \ \ \textrm{for all}\ \r\in (\Lambda_{n,j,R}^{-},{\delta}).\end{aligned}
(-q;q^2)_\infty\left(\sum_{k=1}^{\infty}\frac{q^{2kr}}{1-q^{4kr}}-\sum_{k=0}^{\infty}\frac{q^{2(2kr+1)}}{1-q^{2(2kr+1)}}\right) &+(-q;q^2)_\infty\left( \sum_{k=0}^{\infty}\frac{q^{2kr+1}}{1-q^{2(2kr+1)}}-\sum_{k=1}^{\infty}\frac{q^{4kr}}{1-q^{4kr}}\right).
- \int_\mathbb{R}W^T \alpha L(u^a) E_2dx = - a^2 \int_\mathbb{R}W^T \alpha L(u^a) r_p(u^p)\rho_{xt}\leqslant O(1) \int_\mathbb{R}|W||\rho_{xt}| dx &\leqslant O(1) \|W\| \|\rho_{xt}\|\leqslant O(1)\varepsilon_0 \delta (1+t)^{-{5}/{4}}.
t^{\dim{F'}-\dim{F}}g(B;t^{-1})=\sum_{F''\in[F,F']}(t-1)^{\dim{F'}-\dim{F''}}g([F,F''];t). ({\rm resp.}~t^{\dim{F'}-\dim{F}}g(B;t^{-1})=\sum_{F''\in[F,F']^{*}}(t-1)^{\dim{F''}-\dim{F}}g([F'',F']^{*};t).)
\sum_{x\in \eta_{A_J^c}}g_J(x)=\sum_{x\in \eta_{A_J^c}}\sum_{j\in J}\tilde{l}_\alpha(|y_{c}(B_j)-x|)=\sum_{j\in J}\sum_{x\in \eta_{A_J^c}}\tilde{l}_\alpha(|y_{c}(B_j)-x|)=\sum_{j\in J}\tilde{\Psi}_{\eta_{A_J^c}}(y_{c}(B_j)),
{\mathcal Z}_{( \alpha \beta )}= \int d^{3}x \ \sigma_{\alpha \beta}^{0 i}\partial_{i}\left( - \frac{4 i}{3}{\kern . 3 5 e m \bar{\kern - . 3 5 e m{\widetilde J}}}+ \frac{i}{3}D^{2}K \right) \displaystyle{\biggl \vert}= - 3 \int d^{3}x \ \sigma_{\alpha \beta}^{0 i}\partial_{i}\{\bar{\mathcal W}}\displaystyle{\biggl \vert}\ .
\Lambda_{q}^{n}:=(\alpha-\lambda_{q})\bigg(u-\sum_{k=1}^{n}\sigma_{k}e^{-\sigma_{k}\alpha}\bigg)+\sum_{k\geq n+1}e^{-\sigma_{k}\alpha}-\sum_{k=n+1}^{n+q}e^{-\sigma_{k}\lambda_{q}};
\begin{bmatrix}1 * 1 1\end{bmatrix}\begin{bmatrix}0 0 0\end{bmatrix},\begin{bmatrix}1 1 * 1\end{bmatrix}\begin{bmatrix}0 0 *\end{bmatrix},\begin{bmatrix}1 1 * 1\end{bmatrix}\begin{bmatrix}0 0 0\end{bmatrix}.
M_{ADM}^2 = (8G_N)^{-1}\left({}{\cal M}_{\infty\, ab}({\vec v}^{a\, T}\mu_+{\vec v}^b) +\left[{2}{\cal L}_{ac}{\cal L}_{bd}({\vec v}^{a\, T}\mu_+{\vec v}^b) ({\vec v}^{c\, T}\mu_+{\vec v}^d)\right]^{1\over 2}\right),
\Big [ \begin{array}{c}{{v}}\\{{\emptyset}}\\ \end{array}\Big ] , \qquad \Big [ \begin{array}{c}{{\emptyset}}\\{{v}}\\ \end{array}\Big ] , \qquad \Big [ \begin{array}{c}{{v}}\\{{v + F}}\\ \end{array}\Big ] \sim \Big [ \begin{array}{c}{{v}}\\{{v}}\\ \end{array}\Big ] ,
0 = \ \int_M Z (Z f) e^{-f}dV_g = \ \int_M \left\{Z \left( e^{-f}Zf \right) + e^{-f}(Z f)^2 \right\}dV_g = \ \int_M L_Z \left( e^{-f}Zf dV_g \right) + \int_M (Zf)^2 e^{-f}dV_g = \ \int_M (Z f)^2 e^{-f}dV_g.
\lVert Px \rVert^2 + \lVert Qx \rVert^2 = \langle (P+Q)x, x \rangle = 2 \langle x , x \rangle = \lVert Px \rVert^2 + \lVert x - Px \rVert^2 + \lVert Qx \rVert^2 + \lVert x - Qx \rVert^2 ,
A_{m,0}:=\Bigl\{n_{j_{m}}+kd_{j_{m}}+k'\textrm{per}(x_{l})\,;\,0\le k'\le \dfrac{\alpha d_{j_{m}}}{\textrm{per}(x_{l})},\ 0\le k\le \dfrac{\alpha d_{j_{m}+1}}{d_{j_{m}}}-2\Bigr\},
\underset{\epsilon,\delta,\kappa \rightarrow \infty}{lim}\mathbb{E}\left|\langle:X\otimes X:,\Psi_{\epsilon,\delta,\psi_\kappa}^f \rangle - \int_{\mathbb{R}^2}^{''}\widehat{\psi}(x+y)|y|^{-\gamma}Z_G(dx)Z_G(dy)\right|^2=0,
\begin{matrix}1&2&3&4\\ 2&1&4&3\\ 3&4&1&2\\ 4&3&2&1\end{matrix}\begin{matrix}1&2&3&4\\ 4&3&2&1\\ 2&1&4&3\\ 3&4&1&2\end{matrix}\begin{matrix}1&2&3&4\\ 3&4&1&2\\ 4&3&2&1\\ 2&1&4&3\end{matrix}.
{}_{1}F_{1}(a,2a,x)=e^{x/2}{}_{0}F_{1}(;a+\frac{1}{2};\frac{x^{2}}{16})=e^{x/2}(\frac{x}{4})^{1/2-a}\Gamma(a+\frac{1}{2})I_{a-1/2}(\frac{x}{2}).
\mathbf{h}^0 - (\mathbf{I}- \beta \, \mathbf{P}^0) \, \mathbf{v}^S = \left[\begin{array}{c}\mathbf{c}_S^S \mathbf{0}_{N \setminus S}\end{array}\right] (\mathbf{I}- \beta \, \mathbf{P}^1) \, \mathbf{v}^S - \mathbf{h}^1 = \left[\begin{array}{c}\mathbf{0}_{S}\mathbf{c}_{N \setminus S}^S\end{array}\right].
f_{-L}(t)=\sum_{l=1}^{l}\frac{(-1)^{l-1}}{l}\sum_{\cup_{s=1}^l\mathbf{I}_s=\mathbf{L}}\prod_{s=1}^lh_{-I_s}^{\mathcal{L}_{\mathbf{I}_s}}(t)=0.
\lim\limits_{M\rightarrow+\infty}\frac{\lambda_1}{a}= \lim\limits_{M\rightarrow+\infty}\left [1-\frac{\left \lfloor 4(MN-1)N\sin^2 \frac{\pi}{2(2N-1)}\right \rfloor+1}{4N(MN-1)\sin^2 \frac{\pi}{2(2N-1)}}\right ] \rightarrow ~ 0^{-},
D_{1}'(m \otimes [a_{1}| \dots | a_{n}]) = (-1)^{\deg m}m a_{1}\otimes [a_{2}| \dots | a_{n}] - (-1)^{\tilde{\epsilon}_{n}(\deg a_{n}+ 1)}a_{n}m \otimes [a_{1}| \dots | a_{n-1}],
A_{\varepsilon}(\theta, y, z):=\varepsilon\,v_{\varepsilon}(\theta, y)+\varepsilon^b z, v_{\varepsilon}(\theta, y):=\sum_{j\in S}\sqrt{\lvert j \rvert}\,\sqrt{\xi_j+\varepsilon^{2(b-1)}y_j}\,e^{i \theta_j}e^{i j x}.
\sum_{k=1}^{n}\frac{z(1-z^{k})}{1-z}=\sum_{k=1}^{n}\frac{z}{1-z}-\sum_{k=1}^{n}\frac{z^{k+1}}{1-z}=\frac{nz}{1-z}-\frac{1}{1-z}\sum_{k=1}^{n}z^{k+1}=\cdot\cdot\cdot
\theta_{1,1}(\tau,z) =\sum\limits_{n\in\mathbb{Z}}\mathrm{e}^{\pi i(n+1/2)}q_{N}^{N/2(n+1/2)^{2}}\zeta^{n+1/2)}begin{align*}1mm] =\sum\limits_{n\in\mathbb{Z}}\mathrm{e}^{\pi(n+1/2)}u_{\nu}^{N/2(n+1/2)^{2}+ (\nu+1)(n+1/2)}v_{\nu}^{N/2(n+1/2)^{2}+\nu(n+1/2)}.
m^2(u)|\dot{\sigma}_u(\tau)(\tau_n-\tau)| = m^2(u)|(\dot{\sigma}_u(\tau)-\dot{\sigma}_u(\tau(u)))(\tau_n-\tau)| &= m^2(u)|\tau_n-\tau||\ddot{\sigma}(v_1)(\tau(u)-\tau)| &\leq\frac{2B}{A}m^2(u)q(u)\tau^*(u) &=\theta \frac{2B}{A}\frac{m(u)\Delta(u)}{u}\log m(u),
h(R_1,R_2,R_3)=\frac{L(1)-L(0)}{L(0)}=\frac{\alpha_1 X_1(0)}{L(0)}R_1 + \frac{\alpha_2 X_2(0)}{L(0)}R_2 + \frac{\alpha_3 X_3(0)}{L(0)}R_3 .
(C)_{[(k-1)/2]}\begin{cases}(C)_1 \mbox{holds}\,, \displaystyle{\mbox{For}\alpha \in \mathbb{N}^d \mbox{such that}2 \le |\alpha| \le 2 [(k-1)/2] -1}\,, \displaystyle{\partial^{\alpha}c=0 \mbox{on}\Gamma}\,.\end{cases}
t_{b}^{-4n}t_{t_{b}^{-6n}(a)}^{4n}t_{b}^{-4n}t_{t_{b}^{-2n}(c)}^{4n}= [t_{b}^{-4n}t_{t_{b}^{-6n}(a)}^{4n},f_3], t_{v}t_{d}^{-1}t_{t_{b}(v)}t_{d}^{-1}= [t_{v}t_{d}^{-1},f_4].
\lvert F(\psi(s))(y)\rvert \lesssim \frac{1}{y^{2}}\begin{cases}Q(e^{\omega_{l}s}y) 0\le y < Ke^{-\omega_{l}s}e^{-3\lambda_{l}}(y^{-3\gamma}+y^{6\lambda_{l}}) Ke^{-\omega_{l}s}\le y < e^{\sigma s}1 e^{\sigma s}<y \end{cases}
\Psi_2(g,yH) = \left(\theta(g^p,y^pH)\right)^a\left(\theta(g^q,y^qcH) \right)^{b}= g^{pa}\left(s(g^py^pH)^{-1}s(y^pH)\right)^a g^{qb}\left(s(g^qy^qcH)^{-1}s(y^qcH)\right)^{b}= g\left(s(g^py^pH)^{-1}s(y^pH)\right)^a \left(s(g^qy^qcH)^{-1}s(y^qcH)\right)^{b}.
b_{n}(\omega)=\frac{\mu_{1}[\rho j_{n}(\rho)]^{\prime}j_{n}(\rho_{1})-\mu[\rho_{1}j_{n}(\rho_{1})]^{\prime}j_{n}(\rho)}{\mu_{1}[\rho h_{n}(\rho)]^{\prime}j_{n}(\rho_{1})-\mu[\rho_{1}j_{n}(\rho_{1})]^{\prime}h_{n}(\rho)},
\mathsf{E}(M_{i_1i_2}M_{i_2i_3}\ldots M_{i_{k-1}i_k}M_{i_ki_1})=\prod_{\ell=1}^{p}\frac{1}{K_{m_{\ell}}^{s_{\ell}/2}}\mathsf{E}\left(g_{m_\ell}^{s_{\ell}}\right),
U_{\hbar}\left( \alpha\right)^{\ast}a_{\hbar}U_{\hbar}\left(\alpha\right) =a_{\hbar}+\alpha, U_{\hbar}\left( \alpha\right)^{\ast}a_{\hbar}^{\dag}U_{\hbar}\left(\alpha\right) =a_{\hbar}^{\dag}+\bar{\alpha},
\|v\|_{p,\Omega}\leq \|v\|_{n/(n-1),\Omega}^\lambda \|v\|_{1,\Omega}^{1-\lambda}(\lambda=n(p-1)/p<1) \leq S_{1}^\lambda |\Omega|^{\lambda(1-1/p)}\|\nabla v\|_{p,\Omega}^\lambda\|v\|_{1,\Omega}^{1-\lambda}+S_1^\lambda\| v\|_{1,\Omega}.
M_{2 \gamma \to \gamma}= - \frac{4}{1 5 \sqrt{\pi}}\biggl ( \frac{e^{4}}{3 m^{4}}\sin \theta \left[ \frac{7}{4}( \phi_{1}^{2}+ \phi_{2}^{2}) - \phi_{1 2}^{2}\right] + \frac{Q g ( Q^{2}- g^{2})}{8 M^{4}}\cos \theta \, \phi_{1 2}^{2}\biggr ) \omega \omega_{1}\omega_{2}H \cos 2 \beta
j^{+}= \left( \begin{array}{c c c}{{0}}&{{- 1}}&{{0}}\\{{0}}&{{0}}&{{0}}\\{{- 1}}&{{0}}&{{0}}\\ \end{array}\right) j^{-}= \left( \begin{array}{c c c}{{0}}&{{0}}&{{- 1}}\\{{1}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}\\ \end{array}\right)
\sup_{B_1(0)}u \leq \lim_{j \to +\infty}\Phi_\rho(\mu_{j+1}, h^j_1)\leq C_1 \Phi_\rho(p_1,2), C_1=C^\frac{\kappa}{p_1(\kappa-1)}(2^{\beta}\kappa^{\nu})^{\frac{1}{p_1}\sum_j \frac{j}{\kappa^j}}
\begin{array}{rclcl}\dot{\varphi}^{0}= [ \varphi^{0}, H ] = \varphi^{0}_{1}, \dot{\varphi}^{m}= [ \varphi^{m}, H ] = 2\varphi^{m}_{1}, \dot{\varphi}^{0}_{1}= [ \varphi^{0}_{1}, H ] = - \partial_{m}\varphi^{m}_{1}, \dot{\varphi}^{m}_{1}= [ \varphi^{m}_{1}, H ] = 0 .\end{array}
{(\begin{matrix}n\\m\end{matrix})}^{-1}[z^{n}]\frac{1}{m}\frac{z^{m}}{(1-z)^{m+1}}={(\begin{matrix}n\\m\end{matrix})}^{-1}\frac{1}{m}[z^{n-m}]\frac{1}{(1-z)^{m+1}}
\begin{array}{cc}\Big(\hat{C}^{-1}(\hat{C}C\hat{C}^{-1}-I)\underline{v}+ \hat{C}^{-1}\hat{C}(Dp - U(J^{(r)}X) - \mathbf{1}\frac{\beta}{2}) \Big)_i =\Big( (C\hat{C}^{-1}-\hat{C}^{-1})\underline{v}+ Dp - U(J^{(r)}X) - \mathbf{1}\frac{\beta}{2}\Big)_i\leq 0. \end{array}
m(0) \cdot s^-=\lambda s_1^-<0 \implies m(x^-)\cdot s^-=m_1(x^-) s_1^- +m_2(x^-)s_2^- \leq 0 \implies m_2(x^-)\leq -\frac{m_1(x^-) s^-_1}{s^-_2}\leq -\tilde{C}\alpha_1^-.
\tilde{\mathcal{E}}_{i,1}(t):= G( \tilde{\mathcal{E}}(t),\psi_{i,1}),\ \tilde{\mathcal{E}}_{i,2}(t):= G(\tilde{\mathcal{E}}(t),\psi_{i,2})\ \textup{for}\i\in I_1, &\tilde{\mathcal{E}}_{i}(t):= G(\tilde{\mathcal{E}}(t),\Psi_{i})\ \textup{for}\i\in \mathbb{Z}\setminus\{0\}.