harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
W(R) =W(R')f(u,v)\Bigg(\prod_{z\in Y}f(u,z)f(z,v)\Bigg)\Bigg(\prod_{z\notin Y}f(u,z)f(z,v)\Bigg) &\leq W(R')f(u,v) (m(r)^2-1)^{s}m(r)^{2(t-2-s)}\leq W(R')r (m(r)^2-1)^{s}m(r)^{2(t-2-s)}.
({\cal E}_{\tau}^{\underline{{{D - 1}}}})^{2}+ \sum_{i = 1}^{i = D - 2}({\cal E}_{\tau}^{i})^{2}= ({\cal E}_{\tau}^{\underline{{0}}})^{2}\equiv ( \partial_{\tau}X^{\underline{{0}}}- i \partial_{\tau}\bar{\Theta}\Gamma^{\underline{{0}}}\Theta )^{2},
\Omega^{\scriptscriptstyle 1}(M)=ker(\mathrm{1\mkern-5muI\!}\hskip2pt + \ast\hskip 2pt d_{\scriptscriptstyle 1}) \oplus ker(\mathrm{1\mkern-5muI\!}\hskip2pt + \ast\hskip 2pt d_{\scriptscriptstyle 1})^{\perp}
[x^n]T(x)^m = [x^{n-m}]\left( \frac{T(x)}{x}\right)^m = [x^{n-m}] (z+1)^m &= \sum_{k=0}^{m}{m \choose k}[x^{n-m}] z^k = \sum_{k=0}^m{m \choose k}\frac{k}{n-m}[x^{n-m-k}](x+1)^{3(n-m)},
s_{i,j}^x \;\! s_{k,\ell}^y = \begin{cases}s_{k,\ell}^y \;\! s_{i,j}^x \; i\neq \ell j \neq k, s_{k,\ell}^y \;\! s_{i,j}^x \;s_{i,\ell}^{xy}\; j = k, s_{k,\ell}^y \;\! s_{i,j}^x \;s_{k,j}^{-xy}\; i = \ell,\end{cases}
\zeta(s) = \begin{cases}\zeta_E(s) = \sum_{n=1}^\infty \frac{1}{n^s}\Re(s)>1, \zeta_R(s) = 2^s\pi^{s-1}\sin\left(\frac{\pi s}{2}\right)\Gamma(1-s)\zeta(1-s) s\in\mathbb{C}\setminus\lbrace 0,1\rbrace, \end{cases}
f(x,y) = y^6-4(x^2+1)y^5+\left(12x^2+6x^4+\frac{41}{4}\right)y^4-\left(4x^6+\frac{25}{2}+12x^4+\frac{99}{4}x^2\right)y^3 +\left(x^8+4x^6+\frac{75}{4}x^4+\frac{59}{4}x^2\right)y^2+\left(-\frac{17}{4}x^6+\frac{75}{4}x^2+4x^4+\frac{25}{4}\right)y -\frac{25}{2}x^2-\frac{25}{4}x^6-\frac{71}{4}x^4.
\tilde{A}(\varepsilon)^\top \tilde{Q}+ \tilde{Q}\tilde{A}(\varepsilon) + \tilde{c}\tilde{c}^\top + \tilde{H}^{(2)}(\tilde{P}\otimes \tilde{Q}) (\tilde{H}^{(2)})^\top + \displaystyle \sum_{j=1}^{n_{\rm in}}\tilde{N}_j^\top \tilde{Q}\tilde{N}_j = 0 .
G= \left[\bigsqcup_{C\in \frak A}\bigsqcup_{C^*\in A_{t_2^C}}\dot{g}_{t_2^C}E_{t_2^C}(C^*,C)\right]\bigsqcup\left[\bigsqcup_{C\in \frak B}\bigsqcup_{C^*\in A_{t^C}}\dot{g}_{t^C}E_{t^C}(C^*,C)\right].
[\xi, [\xi,\nabla G(\xi)]] =\left[\xi^p{\bf e}_p,\left[\xi^s{\bf e}_s,k^{\beta j}\frac{\partial G}{\partial \xi^j}{\bf e}_{\beta}\right]\right] &=\left[\xi^p{\bf e}_p,C_{s\beta}^{\tau}k^{\beta j}\xi^s\frac{\partial G}{\partial \xi^j}{\bf e}_{\tau}\right] &=C_{p\tau}^i C_{s\beta}^{\tau}k^{\beta j}\xi^p\xi^s\frac{\partial G}{\partial \xi^j}{\bf e}_i.
\rho_{p \rightarrow 1 + 2 + 3}= \frac{( 3 2 \pi )^{2 - 2 \ell}}{( \Gamma ( \ell - 1 / 2 ) )^{2}( p^{2})^{\ell - 1}}\int_{( m_{1}+ m_{2})^{2}}^{( \sqrt{p^{2}}- m_{3})^{2}}s^{1 - \ell}{\cal D}^{\ell - 3 / 2}d s ,
\overline L^i_{\underline{jk}}= L^i_{\underline{jk}}+\delta^i_k\psi_j+\delta^i_j\psi_k- \frac12\overline F^i_k\overline\sigma_j- \frac12\overline F^i_j\overline\sigma_k+ \frac12F^i_k\sigma_j+\frac12F^i_j\sigma_k.
\|x\|_p=\|(x_i)_{i=1}^n\|_p=\begin{cases}\displaystyle \Bigl(\sum_{i=1}^n|x_i|^p\Bigr)^{1/p}\,, 0< p < \infty\,; \displaystyle\max_{1\leq i \leq n}|x_i|\,, p=\infty.\end{cases}
-2\bar{R}_{\mu\nu}F^{\rho\mu}F_\rho~^\nu-\frac 12 (F_{\mu\nu}F^{\mu\nu})^2+\frac 12 F^{\lambda\nu}F^{\mu\rho}F_{\lambda\mu}F_{\nu\rho}-2(F_{\rho}~^\nu F^{\rho\mu})_{;\nu\mu}+(F_{\nu\rho}F^{\nu\rho})^{;\mu}~_{\mu}~~~.
\rho -\rho_{c}(\beta)= \lim_{\eta \rightarrow +0}\lim_{\Lambda}\frac{1}{V}\sum_{\left\{k\in \Lambda^{\ast}, \left\| k\right\|\leq \eta \right\}}\left\{e^{\beta(\varepsilon_{k}-{\mu_{\Lambda}(\beta,\rho)})}- 1\right\}^{-1}, \{\rm{for}}\ \ \rho > \rho_{c}(\beta) \ .
\begin{bmatrix}q_{m+1}^{[1]}q_{m+1}^{[2]}\end{bmatrix}= \begin{bmatrix}q_{m-1}^{[1]}q_{m-1}^{[2]}\end{bmatrix}+ 2h \begin{bmatrix}q_{m}^{[2]}-sin(q_{m}^{[1]})\end{bmatrix},
f(0) \prod_{j=1}^{\alpha_1}p^{(a_j-b_j)b_j}\le \prod_{j=1}^{\alpha_1}p^{(a_j-b_j)b_j}(1+\theta_jp^{-1}) \le f(6) \prod_{j=1}^{\alpha_1}p^{(a_j-b_j)b_j}.
A_p := \frac{u_{p}(y_{j,p})}{p}\int_{B_{\frac{r}{\varepsilon_{j,p}}}(0)}H(y_{j,p},y_{j,p}+\varepsilon_{j,p}z) \left(1+\frac{w_{j,p}(z)}{p}\right)^{p}dz=o_p(1).
\eta^2(\mathcal{T},\mathcal{M}):= \big(\eta(\mathcal{T},\mathcal{M})\big)^2:=\sum_{T\in\mathcal{M}}\eta^2(\mathcal{T},T) \quad\eta^2(\mathcal{T}):=\eta(\mathcal{T},\mathcal{T}).
\int\int \hat{f}_{g,\delta}(-\omega)\hat{\nu}_{g}(\omega)\psi(2^{-j}\omega)d\omega dg=\int\int |\hat{\mu}(\omega)|^2|\hat{\mu}(g\omega)|^2 \hat{\phi^D_\delta}(\omega)\psi(2^{-j}\omega)d\omega dg.
I(\Lambda) = \frac{1}{2}\left(3d-2-\sqrt{7d^2-6d+4F}\right)\left(3d-2+\sqrt{7d^2-6d+4F}\right) + 2F + 6d-2 = 2-3d+d^2-2F + 2F + 6d-2 = d^2 + 3d = I(e_{1,1}^d)
\Big(\mathbb{E}\big[\sup_{1\leq m\leq M}\|Y_m^{M,N}\|^{2p}\big]\Big)^{\frac1p}+\Big\|k\sum_{j=1}^M\|AY_j^{M,N}\|^2\Big\|_{L^p(\Omega;\mathbb{R})}+\Big\|k\sum_{j=1}^M\|\nabla( Y_j^{M,N})^2\|^2\Big\|_{L^p(\Omega;\mathbb{R})}<\infty.
E(x)=\frac{1}{{\partial}_-^{\;\;2}}{\Delta}(x;m^2)-\frac{x^-}{m^2-n_-{\partial}_-^{\;\;2}}{\delta}(x^+)+\frac{2n_+{\partial}_+}{(m^2-n_-{\partial}_-^{\;\;2})^2}{\delta}(x^+).
\left( \begin{array}{c c}{A}&{B}\\{C}&{D}\\ \end{array}\right) = \left( \begin{array}{c c}{A}&{0}\\{C}&{1}\\ \end{array}\right) \left( \begin{array}{c c}{1}&{A^{- 1}B}\\{0}&{D - C A^{- 1}B}\\ \end{array}\right) .
| M_{[ \alpha , \psi_{\alpha}]}^{i}| \leq A_{[ \alpha , \psi_{\alpha}] [ \alpha , \psi_{\alpha}]}^{i}\mathrm{a n d}M_{[ \alpha , \psi_{\alpha}]}^{i}= A_{[ \alpha , \psi_{\alpha}] [ \alpha , \psi_{\alpha}]}^{i}\mathrm{m o d}2 .
\sum_{j=1}^r\frac{a_j t e^{a_j t}}{e^{a_j t}-1}-r =\frac{\sum_{j=1}^r\prod_{i\ne j}(e^{a_i t}-1)(a_j t e^{a_j t}-e^{a_j t}+1)}{\prod_{j=1}^r(e^{a_j t}-1)}&=\frac{\frac{1}{2}\bigl(\sum_{j=1}^r a_1\cdots a_{j-1}a_j^2 a_{j+1}\cdots a_r\bigr)t^{r+1}+\cdots}{(a_1\cdots a_r)t^r+\cdots}&=\frac{1}{2}\left(\sum_{j=1}^r a_j\right)t+\cdots
(1-\alpha)(x_i-x_{i-1}) =[\lambda_1- \alpha (m+2)](x_{i+1}-x_i)-(1-\alpha)(x_{i+2}-x_{i+1})=0, begin{align*}0.1cm](1-\alpha)(x_{i-1}-x_{i-2}) =[\lambda_1- \alpha (m+2)](x_i-x_{i-1})-(1-\alpha)(x_{i+1}-x_i)=0, begin{align*}0.1cm]\vdots (1-\alpha)(x_2-x_1) =[\lambda_1- \alpha (m+2)](x_3-x_2)-(1-\alpha)(x_4-x_3)=0,
((r_{i,1})^{n-2}(1-a+\varepsilon ) +a|\nabla \tilde u|^{n-2}) \frac{\partial \tilde u}{\partial \tilde t}= \mbox{div}\left ( |\nabla\tilde u|^{n-2}{\nabla}\tilde u\right )+|\nabla\tilde u|^{n-2}A(\tilde u)(\nabla\tilde u,\nabla \tilde u)
\ln W \equiv \left( - \int_{x}\frac{1}{\tilde{W}^{2}}\frac{\delta \tilde{W}}{\delta f ( x )}\frac 1 i \frac{\delta \tilde{W}}{\delta j_{3}( x )}+ \int_{x}\frac{1}{\tilde{W}}\frac{\delta^{2}\tilde{W}}{\delta j_{3}( x ) \delta f ( x )}\right)_{f = f^{( 0 )}}
- \partial^{K}\partial_{K}C_{R}^{M N}={\delta^{M}}_{\mu}{\delta^{N}}_{\nu}\left[ 2 \epsilon^{\mu \nu}{}_{\sigma \rho}F_{-}^{\sigma \rho}+ L^{2}\partial^{\sigma}\partial_{\sigma}C_{R}^{\mu \nu}\right] \delta^{( 2 )}( x^{i}) .
\hat{x}_t = \sum_{i=1}^n \sigma^i(\theta_t^i,\lambda_t)=\frac{\gamma}{\eta}\left[\left(\frac{\eta}{\lambda_t}\right)^{\frac{1}{\gamma}}\sum_{i=1}^n(\theta_t^i)^{\frac{1}{\gamma}}-\hat{\phi}\right],
\phi_{(a)}=\sqrt{{N\, V\over g_0^2}\, \mathbf{F^{(0)}}{}_{cd\, \mu\nu}\mathbf{F}^{(0)}{}^{cd\,\mu\nu}}\equiv \sqrt{\,{N\,V\over g_0^2}\,\mathrm{Tr}\,\mathbf{F^{(0)}}{}_{\mu\nu}\mathbf{F}^{(0)}{}^{\mu\nu}}
Q_{+}^{I}= \frac{\partial}{\partial \theta_{I}^{+}}-i\eta^{IJ}\theta_{J}^{+}\frac{\partial}{\partial \sigma^{+}}, \ \ \ \Q_{-}^{I'}=\frac{\partial}{\partial \theta_{I'}^{-}}- i\eta^{I' J'}\theta^{-}_{J'}\frac{\partial}{\partial \sigma^{+}},
\mathcal L_v\psi(w,[u, e_1],e_2,e_3) =\psi(w,{\nabla_{[u, e_1]}v},e_2,e_3) +\psi(\nabla_wv,{[u, e_1]},e_2,e_3)+\psi(w,{[u, e_1]},\nabla_{e_2}v,e_3) +\psi(w,{[u, e_1]},e_2,\nabla_{e_3}v).
\phi_{1}(z)=\dfrac{2a}{\pi z}\dfrac{\vartheta_{1}\left( \left.\frac{\pi z}{2a}\right| i\right)}{\vartheta'_{1}\left( 0 \vert i\right)}\ \ \ \ , \ \ \ \ \phi_{2}(z)=\dfrac{\vartheta_{3}\left( \left.\frac{\pi z}{2a}\right| i\right)}{\vartheta_{3}\left( 0 \vert i\right)}.
a_{d,k}^{\chi_{r}^{2}}(\mathcal{D})=\left[ \int_{\mathcal{D}}\int_{\mathcal{D}}\frac{1}{\Vert \mathbf{x}-\mathbf{y}\Vert^{2k\alpha}}d\mathbf{x}d\mathbf{y}\right]^{1/2},k\geq 1.
k_m\,|_t\, k_m = k_m = p_m \,|_t\, p_m k_m\,|_t\, p_m = p_m = p_m\,|_t\, k_m \mp k_m\,|_{\pm \infty}\, s_m = i \pi\, k_m = \pm s_m\,|_{\pm \infty}\, k_m \mp p_m\,|_{\pm \infty}\, s_m = i \pi\, p_m = \pm s_m\,|_{\pm \infty}\, p_m s_m\,|_{\pm \infty}\, s_m = \pi^2 \,k_m \:.
\nu_p ((b^{(\ge r, \ell, \theta)})_{\lambda,\lambda}) = \nu_p \left( \prod_{i\ge r}[\ell]_{p^i}^{m_{p^i (\lambda)}}\right) =r \ell(\lambda^{\ge r}) = \nu_p ((\widetilde{y}^{(r)})_{\lambda,\lambda}).
T_u(\mathcal{S}) = \overline{W}_u + D_u(\mathcal{S}_p) + \sum_{t \in \mathcal{S}_s}D_u(t,\kappa) &= \overline{W}_u + D_u(\mathcal{S}_p) + D_u(\mathcal{S}_s) &= \overline{W}_u + D_u(\mathcal{S}).
G^{T}:=(\begin{matrix}1&1&0&1\\ 1&0&1&1\\ 1&0&0&0\\ 0&1&1&1\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\\ \end{matrix}),H:=(\begin{matrix}1&0&1&0&1&0&1\\ 0&1&1&0&0&1&1\\ 0&0&0&1&1&1&1\\ \end{matrix}).
z^3 \frac{d^3}{dz^3}\Psi_{N,\sigma}-L^2 \frac{3 N(N+1) z^L}{(z^L-1)^2}z \frac{d}{dz}\Psi_{N,\sigma}+L^3 \frac{N(N^2-1)(z^L+1) z^L}{(1-z^L)^3}\Psi_{N,\sigma}+3L^2 \frac{N(N+1) z^L}{(1-z^L)^2}\Psi_{N,\sigma}=0,
(1+w)xwx^4w x^4 = \left(xwx^4w + w\left(s-\frac{1}{6}x^3\right)wx^4w\right) x^4 & = \left(xwx^4w- swcx^4w- cwsx^4w-\frac{1}{6}wx^3wx^4w\right) x^4 & = - \frac{1}{6}(1+w)x^3wx^4w x^4
\int_{c_i}^{2^{-i+1}}|x-x_i|^{-b_i}|x-x_{i-1}|^{-b_{i-1}}\, dx \leq \left( 2^{-i-1}\right)^{-b_{i}}\cdot \int_{0}^{2^{-i-1}}t^{-b_{i-1}}\, dx & = \left( 2^{-i-1}\right)^{-b_{i}}\cdot \frac{(2^{-i-1})^{1-b_{i-1}}}{1-b_{i-1}}.
t_{i}=\alpha^{k_{1}}s_{i}-s_{i+1}=\alpha^{k_{1}}\sum_{j=0}^{n-1}e_{j}\alpha^{ij}-\sum_{j=0}^{n-1}e_{j}\alpha^{j}\alpha^{ij}=\sum_{j=0}^{n-1}e_{j}(\alpha^{k_{1}}-\alpha^{j})\alpha^{ij}.
\begin{cases}\frac{\partial}{\partial t}p^{r_0}(t;x,y) = \Delta_{\mu}p^{r_0}(t;x,y) t>0,\x \in (0,r_0),\y \in (0,r_0), p^{r_0}(t;x,r_0) = 0 t>0,\x \in (0,r_0), \lim_{t \to 0}p^{r_0}(t;x,y) = \delta_x(y) x \in (0,r_0),\y \in (0,r_0). \end{cases}
|(T_{n+1}, v_{n+1})_{\nu_i(n)\downarrow}| = n - |(T_n, \nu_i(n))_{u_1\downarrow}| &\geq n - \psi_{T_n}(\nu_i(n)) &\geq n - \psi_{T_n}(\nu_K(n) &\geq n-Mn &= n(1-M).
-\frac{\tilde{g}_y}{\tilde{g}_z}(x,y,z) = \frac{1}{g^*_u(0,y,h-z)}\frac{g_y}{g_z}(x,y,g^*(0,y,h-z)) + \frac{g^*_y}{g^*_u}(0,y,h-z),
w'(e)^{p'-r}\prod_{v\in e}B'(v,e)=w(e)^{p'-p}\alpha \leq (\alpha\Delta^r)^{(p'-p)/(p-r)}\alpha & =\alpha^{(p'-r)/(p-r)}\Delta^{r(p'-p)/(p-r)}& =r^{p'-r}\bigg(\frac{\Delta^{r(p'-p)}}{(\lambda^{(p)}(G))^{p(p'-r)}}\bigg)^{1/(p-r)}.
\dot{J}_{ab}=-\frac{\theta^{2}}{9}h_{ab}-\frac{2\theta}{3}(\sigma_{ab}+\omega_{ab})-(\sigma_{am}{\sigma^{m}}_{b}+\omega_{am}{\omega^{m}}_{b})-(\sigma_{am}{\omega^{m}}_{b}+\omega_{am}{\sigma^{m}}_{b})-{E[\vec{X}]}_{ab}
\allowdisplaybreaks{\langle a \rangle}\mathcal{E}_{1}(\tau) = \frac{1}{8}\sum_{\chi(-1) = -1}\chi(a)E_{\chi,1}(\tau), \mathcal{E}_{k}(\tau)= \pm{\langle 3 \rangle}^{k-1}\mathcal{E}_{1}(\tau)
\sum_{j=1}^n \bigg\|\frac{L_j f}{|x|}\bigg\|_{2}^2 = \sum_{k=0}^\infty \mu_k \int_0^\infty \|P_k f(r(\cdot))\|_{L^2(\mathbb{S}^{n-1})}^2 r^{n-5}\, \mathrm{d}r = \sum_{k=1}^\infty \mu_k \bigg\|\frac{P_k f}{|x|^2}\bigg\|_{2}^2
h_2 =\sum_{i=1}^{d/2}\frac{1}{(2i-1)(2(d+1-i)-1)}+\underset{i+j\neq d+1}{\sum_{1\leq i<j\leq d}}\frac{1}{(2i-1)(2j-1)}&=\frac{1}{2d}h_1+\sum_{1\leq i<j\leq d/2}\frac{4d^2}{(2i-1)(2j-1)(2d-2i+1)(2d-2j+1)}.
\beta = \frac{2}{3}, R_h^j = \frac{4}{3}X_h^{j-1}- \frac{1}{3}X_h^{j-2}+ g(X_h^{j-1}) \Delta_h W^j - \frac{1}{3}g(X_h^{j-2}) \Delta_h W^{j-1},
z =\sqrt{\frac{a}{a+b}}\left(\sqrt{\frac{1-b}{1+a}}\cos (t) + i \sqrt{\frac{1+b}{1-a}}\sin(t)\right), w = \pm\sqrt{\frac{b}{a+b}}\left(\sqrt{\frac{1-a}{1+b}}\sin (t) - i \sqrt{\frac{1+a}{1-b}}\cos(t)\right).
\mathcal{L}_{\Pi\alpha}\nu^\prime = \big(\iota_\alpha \widetilde{\mathcal{M}}_\Pi^{\nu^\prime}+ (-1)^{n-1}\iota_{d_A\alpha}\Pi\big)\nu^\prime &= f\big(i_\alpha \widetilde{\mathcal{M}}_\Pi^{\nu^\prime}+ (-1)^{n-1}i_{d_A\alpha}\Pi\big)\nu.
(\Phi\otimes\mathbb{I}_M)\left(\hat{D}_A(\mathbf{x})\,\hat{\rho}_{AM}\,{\hat{D}_A(\mathbf{x})}^\dag\right) = \hat{D}_B(K\mathbf{x})\,(\Phi\otimes\mathbb{I}_M)(\hat{\rho}_{AM})\,{\hat{D}_B(K\mathbf{x})}^\dag\;.
({\rm I}\,+z D_{{}^\Vert})^{-1}:\begin{cases}{\rm{\sf R}}^r(d,\Omega){\rm{\sf R}}^r(\underline{\delta},\Omega)\end{cases}\longrightarrow L^r(\Omega,\Lambda)\qquad\forall z\in{\mathbb{C}}\setminus S_\theta,
A_{n}=\left\{r_{n}^{(N)}\leqslant(d_{0}+\varepsilon)v_{n}\right\},B_{n}=\left\{\left\Vert \mathbb{G}_{n}^{(N)}\right\Vert_{\mathcal{F}}\leqslant a_{n}\right\}, C_{n,k}=\left\{\theta^{k-1}a_{n}<\left\Vert\mathbb{G}_{n}^{(N)}\right\Vert_{\mathcal{F}}\leqslant\theta^{k}a_{n}\right\}.
\begin{array}{c}{{{\bf P}_{A}= \bar{p}_{A}\ , \quad{\bf M}^{A B}= x^{A}\bar{p}^{B}- x^{B}\bar{p}^{A}+ S^{A B}\ ,}}\\{{S^{A B}= 2 i \{Z_{i}^{A}\partial^{i B}- \overline{{{Z}}}_{i}^{A}\overline{{{\partial}}}^{i B}- ( \Sigma^{A B})_{b}^{\a}( \psi_{a}\partial^{b}+ \overline{{{\psi}}}_{a}\overline{{{\partial}}}^{b}) \}\Phi}}\\ \end{array}
\xi^{h}_{L+\theta,t}(n)=\begin{cases}\frac{1}{t^{h}_{\theta,s^*}(n)-\tau(t^{h}_{\theta,s^*}(n))}, \tau(t^{h}_{\theta,s^*}(n)) \leq t < t^{h}_{\theta,s^*}(n) t^{h}_{\theta,s^*}(n)=\min\limits_{\begin{subarray}~k''\in\mathbb{N}_+\end{subarray}}\{t^{h}_{\theta,k''}(n)|t^{h}_{\theta,k''}(n)> t\}, 0, .\end{cases}
\|u_1\|_{L^p(\Omega)}=\|(\lambda+A_p)^{-1}P_{av}s\|_{L^p(\Omega)}&=\|(\lambda+A_p|_{L^p_{av}(\Omega)})^{-1}P_{av}s\|_{L^p(\Omega)}&\le \frac{C}{|\lambda|+1}\|P_{av}s\|_{L^p(\Omega)}
\mathcal{V}_2\left( \frac{m}{p_2(p_1q)^2}\right) = 2\pi i^k \int_0^\infty V_2\left( \frac{y}{N}\right) y^{-i(t+\nu)}e\left( -\frac{uy}{p_1qQ}\right)\, J_{k-1}\left( \frac{4\pi}{p_1q}\sqrt{\frac{my}{p_2}}\right) \{d}y.
\Phi^{\pm}(r)\;:=\;e^{-r}r^{-B}\big( \textstyle{\frac{\pm(1+\nu)+B}{1+\nu}}\,U_{-B,1-2B}(2r)-\textstyle{\frac{2rB}{1+\nu}}\,U_{1-B,2-2B}(2r)\big)\,,
\sum_{k = 1}^\infty{\frac{{\zeta (2k)}}{{4^{k}(2k + 1)}}}= \frac{1}{2}- \frac{\ln 2}{2}, \sum_{k = 1}^\infty{\frac{{\zeta (2k)}}{{16^{k}(2k + 1)}}}= \frac{1}{2}- \frac{\ln 2}{4}- \frac{G}{\pi}, \sum_{k = 1}^\infty \Bigl(\frac{9}{16}\Bigr)^{k}\frac{\zeta (2k)}{2k + 1}= \frac{1}{2}- \frac{\ln 2}{4}+ \frac{G}{3\pi}.
\rho(a\cdot T) =(_{\phi}\rho^{\alpha}(P_{\alpha}\circ (a\cdot T)))_{\alpha\in \Gamma}=(_{\phi}\rho^{\alpha}(a\cdot(P_{\alpha}\circ T)))_{\alpha\in \Gamma}&=(a\cdot_{\phi}\rho^{\alpha}(P_{\alpha}\circ T))_{\alpha\in \Gamma}=a\cdot(_{\phi}\rho^{\alpha}(P_{\alpha}\circ T))_{\alpha\in \Gamma}&=a\cdot\rho(T).
[\xi,\eta]^{r}=-{\mathrm{ad}^{\ast}}^{(r)}_{\underline{r}^{t}(\eta)}\xi, &[[r,r]](\xi,\eta,\zeta)=\langle \xi,[\underline{r}^{t}\eta,\underline{r}^{t}\zeta]\rangle+\langle\eta,[\underline{r}\xi,\underline{r}^{t}\zeta]\rangle,\quad\forall\xi,\eta,\zeta\in\mathfrak{g}^{\ast}.
\begin{array}{r c l}{{d{\tilde{s}}^{2}}}&{{=}}&{{( H^{2}+ \theta^{2}/ ( 2 \pi )^{2})^{1 / 2}[ H^{- 1 / 2}d s_{1 , 7}^{2}- H^{1 / 2}d \omega d{\bar{\omega}}] \, ,}}\\{{}}&{{}}&{{}}\\{{{\tilde{\lambda}}}}&{{=}}&{{- 2 \pi i / \log{\omega}\, .}}\\ \end{array}
\left\{\varphi ( x ) , \pi ( x^{\prime}) \right\}= ( 1 -{\cal P}) ( x , x^{\prime}) \, , \qquad \left\{P , \varphi \right\}= \left\{P , \pi \right\}= 0 \, , \qquad \left\{\varphi , \varphi \right\}= \left\{\pi , \pi \right\}= 0 \, .
\zeta = \frac{a \tilde{\zeta}+ b}{- \overline{{{b}}}\tilde{\zeta}+ \overline{{{a}}}}, \qquad \eta = \frac{\tilde{\eta}}{( - \overline{{{b}}}\tilde{\zeta}+ \overline{{{a}}})^{2}}, \qquad | a |^{2}+ | b |^{2}= 1 ,
\psi^{c}| 0 > = \int d^{3}p ~ e x p \{\frac{i}{2}\mathrm{\boldmath ~ \g a m m a ~}\cdot \mathrm{\boldmath ~ \l a m b d a ~}({\bf p}) \}|{\bf p}> ~ e^{i \mathrm{\boldmath ~ p ~}\cdot \mathrm{\boldmath ~ x ~}}
{\cal V}\equiv \left( \begin{array}{c}{{U_{\alpha}^{T}}}\\{{\bar{U}^{\alpha \, T}}}\\ \end{array}\right) \equiv \left( \begin{array}{c c}{{e_{\alpha}^{A}}}&{{h_{A \alpha}}}\\{{g^{\alpha \bar{\beta}}\bar{e}_{\bar{\beta}}^{A}}}&{{g^{\alpha \bar{\beta}}\bar{h}_{A \bar{\beta}}}}\\ \end{array}\right) \ ,
\sum_{k,|k|\geq n}\rho_k \geq \rho_i + \sum_{k,|k|> n}\rho_k = \chi_i(1-\sum_{k,|k|> n}\rho_k) + \sum_{k,|k|> n}\rho_k = \chi_i+(1-\chi_i)\sum_{k,|k|> n}\rho_k =1+0=1.
\frac{\delta_{i}^{(j)}}{2^{R^{th}}-1}- \sum_{m=1}^{i-1}\delta_{m}^{(j)}> 0 ~~ \Leftrightarrow ~~ \frac{\delta_{i}^{(j)}}{\sum_{m=1}^{i-1}\delta_{m}^{(j)}}> 2^{R^{th}}-1, i = 2, \cdots, \ell
\widehat\Theta\le\frac{1}{\theta^{n+1}}\max_{1\le r\le n}\ \min\limits_{\substack{r_1+\dots+r_m=r begin{align*}0.5ex] 0\le r_j\le \ell_j\ (1\le j\le m)}}~\prod_{j=1}^m\prod_{i=0}^{r_j-1}\theta_{j,i}\,.
f(w,x,y,z)+f(z,w,x,y) = (wx,y,z)-x(w,y,z)-(x,y,z)w+(zw,x,y)-w(z,x,y)-(w,x,y)z = (wx,y,z)-x(w,y,z)-(x,y,z)w+(zw,x,y)-(wx,y,z)+(w,xy,z)-(w,x,yz) = -x(w,y,z)-(x,y,z)w+(zw,x,y)+(w,xy,z)-(w,x,yz) = -x(w,y,z)-(x,y,z)w+(xy,z,w)-(x,yz,w)+(x,y,zw)=0.
\tilde{sec}(U, V) = sec(U, V) + \frac{\lambda^2}{2}\left( -g(T_U U, \nabla_{\mathcal{H}}\frac{1}{\lambda^2}) - g(T_V V, \nabla_{\mathcal{H}}\frac{1}{\lambda^2})\right) - \frac{\lambda^4}{4}\mid \nabla_{\mathcal{H}}\frac{1}{\lambda^2}\mid^2
\delta_H\Theta^2(x,y)=-\sum_{i+j=N+1,~i,j>0}~~\sum_{m+n=i,~m,n>0}[f_m^{\lambda_j(x,y)},\alpha_n]-[\alpha_i,f_0^{\lambda_j(x,y)}]-[\alpha_0,[f_i^y,f_j^x]]
v_n(\cdot) = \frac{v_n^{n_1,n_2}+h^{n_1,n_2}(v_n^{n_1,n_2},\omega_n)}{\|v_n^{n_1,n_2}+h^{n_1,n_2}(v_n^{n_1,n_2},\omega_n)\|}
|\nabla \psi|^2 + 2g \rho (y+d) = Q, \textrm{on}\{y = \eta(x)\}, \psi = 0, \textrm{on}\{y = \eta(x)\}, \psi = -p_i, \textrm{on}\{y = \eta_i(x)\}, \psi = -p_0, \textrm{on}\{y = -d\};
\hat{\beta}_j^{k,\lambda}(\xi) = \frac{1}{2i}k^{-\frac{1}{2}}\lambda^{-3+\frac{2}{r_j}}\sum_{\varepsilon_1, \varepsilon_2 = \pm 1}\sum_{m=1}^2 (-1)^{j+1}\varepsilon_1 \varepsilon_2 \hat{\rho}\big( \lambda^{-1}\xi_m^k \big) e^{-2\pi i \langle x_\varepsilon^\ast, \xi_m^k \rangle}
{\cal S}_{C S}= \frac{1}{4 ( 2 \pi )^{3}}\int \mathrm{t r}_{q}\left( 2 \pi{\cal F}^{\prime}\wedge A^{( 2 )}+ \frac{1}{2}2 \pi{\cal F}^{\prime}\wedge 2 \pi{\cal F}^{\prime}\wedge A^{( 0 )}\right) .
\int_{B_{2\,R}(x_0)}\frac{|u(y)|^p}{(1+|y|)^{N+s\,p}}\,dy\le \int_{B_{2\,R}(x_0)}|u(y)|^p\,dy \le \int_{\Omega}\frac{|u|^p}{d_\Omega^{s\,p}}\,dy\,\|d_\Omega\|^{s\,p}_{L^\infty(B_{2\,R}(x_0))}&\le C_\Omega\,\|u\|^p_{\mathcal{X}^{s,p}(\Omega;d_\Omega)},
E_C^{(2)\;ren}(R)=\frac{1}{2\pi}\int\limits_{0}^{\infty}d\omega\ln\left(1-e^{-2R\sqrt{\omega^2+\omega_0^2}}\right)=-\frac{\omega_0}{2\pi}\sum\limits_{n=1}^{\infty}n^{-1}K_1(2n\omega_0 R),
(-1)^{k+1}\sum_{0 \le s \le (k-1)d/n}\binom{d}{s}(-1)^s \sum_{j=0}^{n-1}j \omega_n^{ikj}= \omega_n^{i\binom{k}{2}}\Big( 1_{n \mid dk}\cdot \binom{k}{2}\binom{d}{dk/n}+ \omega_n^i{n \brack k}^{'}_{\omega_n^i}\Big).
\Phi(\varphi(G^{\mathrm{low}}(b)).v_{-\lambda_1}) =\omega(\varphi(G^{\mathrm{low}}(b))).\Phi(v_{-\lambda_1}) &=\ast(G^{\mathrm{low}}(b)).v_{\lambda_1}=G^{\mathrm{low}}(\ast b).v_{\lambda_1}=G_{\lambda_1}^{\mathrm{low}}(\pi_{\lambda_1}(\ast b)).
\Phi(r)=\frac{q}{4\pi\epsilon r^{\prime}}\sum_{l=0}^{\infty}(\frac{r}{r^{\prime}})^{l}(\frac{4\pi}{2l+1})\sum_{m=-l}^{l}Y_{lm}(\theta,\phi)Y_{lm}^{*}(\theta^{\prime},\phi^{\prime})
\|\phi\|_{L^{\frac{2np}{2n-p}}(\mathbb C^n)}\le \frac{C_n}{p-1}\sum_{j=1}^n \|\partial \phi/\partial\bar{z}_j\|_{L^p(\mathbb C^n)},\ \ \ \forall\,\phi\in C^1_0(\mathbb C^n),
\|\left(\Phi(X^*AY)\Phi(Y^*AY)^{-1}\Phi(Y^*AX)\right)^{\frac{p}{2}}\Phi(X^*AX)^{-\frac{p}{2}}\| &\le \frac{1}{4}\left(\left(\frac{M-m}{M+m}\right)^2M+\frac{1}{m}\right)^p, p\ge 2.
\vec{\psi}_{i+1}(n)=\begin{pmatrix}\lambda^n 0\end{pmatrix}+\sum_{s=n_0+1}^n\Phi_1(n-s)R(s-1)\vec{\psi}_{i}(s-1)-\sum_{s=n+1}^\infty\Phi_2(n-s)R(s-1)\vec{\psi}_{i}(s-1).
\frac 1x \sum_{n\leq x}f(n) = \frac{x^{i\phi}}{1+i\phi}\cdot \frac 1x \sum_{n\leq x}f_\phi(n) +O\Big( \frac{\log \log x}{\log x}\exp\Big( \sum_{p\le x}\frac{|1-f(p)|}{p}\Big)\Big).
\|\varTheta(t_\gamma)-\varTheta(\tau)\| ={\sup}_{x\in X}|\vartheta(t_\gamma x)-\vartheta(\tau x)|={\sup}_{x\in X}|\vartheta(t_\gamma(a_jx))-\vartheta(\tau(a_jx))| &=\|\varTheta(t_\gamma a_j)-\varTheta(\tau a_j)\| &<2\varepsilon
\inf_{z\in\mathbb{X}}\!\sup_{u\in\mathcal{A}(F(\overline{x}),F'(\overline{x})w)}\!\big\{\!\langle u,D^2F(\overline{x})(w,w)\rangle\!+\!\langle\nabla\!F(\overline{x})u-\overline{v},z\rangle\big\}\!=\!\sup_{\xi\in\Lambda_{\overline{x},\overline{v}}}\langle\xi, D^2F(\overline{x})(w,w)\rangle.
E\psi=-\frac{\hbar^{2}}{2L_{1}}\frac{d^{2}\psi}{dQ_{1}^{2}}-\frac{\hbar^{2}}{2L_{2}}\frac{d^{2}\psi}{dQ_{2}^{2}}-\frac{\hbar^{2}}{m}\frac{d^{2}\psi}{dQ_{1}dQ_{2}}+\frac{1}{2}L_{1}\omega^{2}Q_{1}^{2}\psi+\frac{1}{2}L_{2}\omega^{2}Q_{2}^{2}\psi
F(\alpha, p)= \sum_{\substack{n\geq 1 P^{+}(n)\leq y}}\frac{\psi_p(n)}{n}- \sum_{\substack{n\geq 1 P^{+}(n)\leq y}}\frac{\psi_p(n)\cos(2\pi n\alpha)}{n}+O\left(\frac{1}{\log y}\right) \leq -\frac{C_2}{\log y},
\sum_{x\in B}\nabla^{e_1}\varphi(x) \nabla^{e_1}\xi_o (x) = -\sum_{x\in l_3}\xi_o ( x) \nabla^{-e_1}\varphi (x)- \sum_{x\in l_1}\xi_o (x+ e_1) \nabla^{e_1}\varphi(x)+ \sum_{x\in B}\xi_o (x) \nabla^{e_1}\nabla^{-e_1}\varphi(x).
H_{(-4,0,-1)}(4,0,1) = (4,3)!(5,4)!(1,0)! = 4\cdot 3\cdot 2\cdot 1\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1\cdot 1 = 2880 H_{(-2,0,-2)}(4,0,1) = (4,1)!(5,3)!(1,1)! = 4 \cdot 3 \cdot 5\cdot 4\cdot 3\cdot 2\cdot 1\cdot 0 = 0 H_{(-3,-2,0)}(4,0,1) = (0,1)!(4,2)!(5,2)!(1,1)! = 0\cdot -1\cdot 4\cdot 3\cdot 2\cdot 5\cdot 4\cdot 3\cdot 1\cdot 0 = 0
\frac{\varphi_{s}(n)\zeta(s+1)}{n^{s}}=\frac{\mu(1)c_{1}(n)}{\varphi_{s+1}(1)}+\frac{\mu(2)c_{2}(n)}{\varphi_{s+1}(2)}+\frac{\mu(3)c_{3}(n)}{\varphi_{s+1}(3)}+\cdot\cdot\cdot.
\mathrm{DF}_{\zeta}(\mathcal{X},\mathcal{L}) = \mathrm{Fut}_{X_0',\zeta}(\xi) & = \mathrm{Fut}_{X_0,\zeta}(\xi) \intertext{by Corollary~\ref{cor_const_fut}, then by Theorem~\ref{thm_fut_horo}, this is}= C \left<2\rho_Q-\mathrm{bar}_{DH,\tilde{\zeta}}(\Delta^+), \tilde{\xi}\right>
h_{\mu \nu}^{\mathrm{e v e n}}= \left( \begin{array}{c c c c}{{H_{0}\left( 1 -{\frac{2 M}{r}}\right)}}&{{H_{1}}}&{{0}}&{{0}}\\{{H_{1}}}&{{H_{2}\left( 1 -{\frac{2 M}{r}}\right)^{- 1}}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{r^{2}K ( r )}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{r^{2}\sin^{2}\theta K ( r )}}\\ \end{array}\right) e^{- i \omega t}P_{l}( \theta ) .
\Omega^{\scriptscriptstyle 1}( M ) = k e r ( \mathrm{1 \m k e r n - 5 m u I \!}+ \ast d_{\scriptscriptstyle 1}) \oplus k e r ( \mathrm{1 \m k e r n - 5 m u I \!}+ \ast d_{\scriptscriptstyle 1})^{\perp}
\begin{array}{lcl}[ X_{\underline{a}\underline{b}ij\cdots}Y_{\underline{c}\underline{d}kl\cdots}]_s \equiv ~~ X_{abij\cdots}Y_{cdkl\cdots}+ X_{cdij\cdots}Y_{abkl\cdots}& + X_{acij\cdots}Y_{bdkl\cdots}+ X_{bdij\cdots}Y_{ackl\cdots}& + X_{adij\cdots}Y_{bckl\cdots}+ X_{bcij\cdots}Y_{adkl\cdots}\ .\end{array}
J^{w_1}(\textbf{X}_{RSS}^{(n)}) =- \frac{Q_n}{2}\prod_{i=1}^{n}E\left(\Lambda_X^{w_1}(B_{2i-1:2n-1})\right) &\le - \frac{Q_n}{2}\prod_{i=1}^{n}E\left(\Lambda_Y^{w_2}(B_{2i-1:2n-1})\right) &=J^{w_2}(\textbf{Y}_{RSS}^{(n)}),
\widetilde{\beta_k}(t,\xi)=\widetilde{A_k}(t,\xi)^{1/2}f'(t,k,\xi),(\widetilde{\beta_h},\widetilde{\beta_\theta})(t,\xi)=\widetilde{A_*}(t,\xi)^{1/2}(h',\theta')(t,\xi).
J_{g}(u) =\frac{1}{2}\int_{M}\left[ a|\nabla_{g}u|_{g}^{2}+bu^{2}\right]dV_{g}-\frac{1}{2^{\ast}}\int_{M}c|u|^{2^{\ast}}dV_{g}& =\frac{1}{2}\Vert u\Vert_{g,a,b}^{2}-\frac{1}{2^{\ast}}|u|_{g,c,2^{\ast}}^{2^{\ast}}
\Lambda(\alpha) := \sup_{z \in \mathbb{C}_+}\frac{\left\|h \cdot \left(k^{\mathcal{B}^2_\alpha(\mathbb{C}_+)}_z \circ \varphi\right) \right\|_{A^p_\nu}}{\left\|k^{\mathcal{B}^2_\alpha(\mathbb{C}_+)}_z \right\|_{A^p_\nu}}< \infty.
\Psi^0(\widehat{\alpha}_{})-\Psi^0(L-\widehat{\alpha}_{})-\log(1-\widehat{\alpha}_{})+{}\frac{\widehat{\alpha}_{}}{1-\widehat{\alpha}_{}}+\frac{1}{n}\sum_{i=1}^n{\log(1-\widehat{\alpha}_{}+Lz_i)}-{}\frac{\widehat{\alpha}_{}-L}{n}\sum_{i=1}^n \frac1{1-\widehat{\alpha}_{}+Lz_i}= 0 ,
(\partial^{*}F_{\ell})_{\bar{z}}=_{\mathcal{H}^{n-1}}\overline{B^{n-1}\left( 0 , r_{\ell}^{\vee}(\bar{z})\right)}\backslash B^{n-1}\left( 0 , r_{\ell}^{\wedge}(\bar{z})\right) &=_{\mathcal{H}^{n-1}}B^{n-1}\left( 0, r_{\ell}^{\vee}(\bar{z})\right) \backslash B^{n-1}\left( 0 , r_{\ell}^{\wedge}(\bar{z})\right),
d\Psi=-[(b_1-b_2)^2R_{1212}+(b_1-b_3)^2R_{1313}+(b_2-b_3)^2R_{2323}]dv_g &+[\frac{18b_1b_2C_3^2}{(b_1-b_3)^2(b_2-b_3)^2}+\frac{18b_2b_3C_1^2}{(b_1-b_2)^2(b_1-b_3)^2}+\frac{18b_1b_3C_2^2}{(b_1-b_2)^2(b_2-b_3)^2}]dv_g,
{P ( x , T , T^{*}) ={\frac{{\mid H ( T ) \mid}^{2}}{{T_{r}}^{3}{\mid \eta ( T ) \mid}^{6}}}2 x A^{2}e^{- 2 \alpha x}{\left({\frac{\alpha - \beta}{1 + 2 x \beta}}\right)}^{2}\left( 1 + \alpha + \beta + 2 x \alpha \beta -{1 /{4 x^{2}}}\right)}