harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
S =\frac{2l_n-n}{b^2}\sum_{a=0}^{b-1}\sigma(a)a+\frac{n-l_n}{2b}(b-1)^2+\frac{1}{4}\left(\frac{1}{N}+N-2+n\left(2-\frac{1}{b}-b\right)\right) =(n-2l_n)\left(\frac{(b-1)^2}{4b}-\frac{1}{b^2}\sum_{a=0}^{b-1}\sigma(a)a\right)+\frac{1}{4N}+\frac{N}{4}-\frac{1}{2}.
\rho(t,x)=\rho^{(d)}(t,x):={\Phi\left(\left(\frac{1}{\Phi^{-1}(t^{-1})}+|x|\right)^{-1}\right)}\left(\frac{1}{\Phi^{-1}(t^{-1})}+|x|\right)^{-d}\, .
\lambda^2 |A|^2 \le CH_2 + Cm^{-2}G \le CC_m\delta_2(Q^k_i(q)\cap \hat{W}) + Cm^{-2}G =: \hat{H}_2, \lambda^2 |A|^4 \le CH_4 + C\lambda^{-2}m^{-4}G^2 \le CH_4 +C\lambda^{-1}m^{-5}C_m \epsilon G & \le C C_m\delta_4(Q^k_i(q)\cap \hat{W}) +C\vartheta G =: \hat{H}_4.
\sum_{\delta}\eta_{\delta}Q_{\delta}S ( q_{1}, q_{2}) \langle \psi ( q_{1}) \cdot \partial \psi ( q_{1}) \prod_{i = 1}^{3}k_{i}\cdot \psi ( z_{i}) \epsilon_{i}\cdot \psi ( z_{i}) \rangle_{\delta}= 0 .
|A_R(x)| =|C_{\alpha,\beta}|R^{-2(\alpha+\beta)}\Big|\int^R_0(R^2-t^2)^{\beta-1}t^{2\alpha+1}(B^\alpha_t(x)-B^{\alpha+1}_t(x)) dt\Big| &\leq|C_{\alpha,\beta}|R^{-2(\alpha+\beta)}\Big(\int^R_0|(R^2-t^2)^{\beta-1}t^{2\alpha+1}|^2dt\Big)^{\frac12}&\quad\times\Big(\int^R_0|B^\alpha_t(x)-B^{\alpha+1}_t(x)|^2dt\Big)^{\frac12}\lesssim G^\alpha(x),
c(t,x)= \frac{\partial b}{\partial x}(t,x)+ \frac{\partial b_1}{\partial x}(t,x) \in X_0((0,T_1] \times D_{R_1}), \gamma(t,x)= \frac{\partial \lambda}{\partial x}(t,x)+ \frac{\partial \lambda_1}{\partial x}(t,x) \in X_0((0,T_1] \times D_{R_1}).
\|u_{\epsilon}\|_{4^*}^{4^*}=R^{4^*}\omega\epsilon^{N}\int_{0}^{2}\frac{(\varphi^{4^*}(r)-1)r^{N-1}}{(\epsilon^{2}+r^{2})^{N}}dr+R^{4^*}\omega\epsilon^{N}\int_{0}^{2}\frac{r^{N-1}}{(\epsilon^{2}+r^{2})^{N}}dr &=I_{1}(\epsilon)+I_{2}(\epsilon).
\begin{cases}\partial_t\boldsymbol u + \boldsymbol u_2\cdot \nabla \boldsymbol u +\boldsymbol u\cdot\nabla\boldsymbol u_1+ \nabla\pi = \theta \boldsymbol e_2,\boldsymbol e_2=(0,1), \partial_t \theta - \Delta\theta + \boldsymbol u_2\cdot\nabla\theta + \boldsymbol u\cdot\nabla\theta_1 +\boldsymbol u\cdot\nabla S = 0, \end{cases}
\delta_{l}\sigma(g_1, \cdots, g_{l+1}) ={\rho}(g_1)({\sigma}(g_2, \cdots, g_{l+1}))+ &+ \sum_{i}(-1)^i{\sigma}( g_1, \cdots, g_i g_{i+1}, \cdots, g_{l+1})+ (-1)^{l+1}{\sigma}(g_1, \cdots, g_l).
\begin{array}{cc}f_{n+1,k}(z_1,...,z_n,\eta)=(-1)^{i-1}z_1^i...z_n^i\eta^iq^{-\frac{i(i-1)}{2}}f_{n,k}(z_1,...,z_n)- -z_1^{ki}f_{n,k}(\eta z_1,z_2,...,z_n)-...-z_n^{ki}f_{n,k}(z_1,...z_{n-1},\eta z_n) \end{array}
U_{\omega}(x) = \sum_{n=0}^{+\infty}\langle U_{\omega},\phi_n\rangle_{L^2}\phi_n(x) = \sum_{n=0}^{+\infty}\bigg(\int_{\mathcal{D}}U_{\omega}(y)\phi_n(y)dy\bigg)\phi_n(x) \ \L^2(\mathcal{D})
S^{(1)}\left(g_1,g_2;\mathfrak{p}^{j}\right) =\#\{1\leq n_1,n_2\leq p^{j}, \left(n_1,n_2,p\right)=1, \mathfrak{p}^{j}|(n_1\omega_1+n_2\omega_2)\}&=(p-1)p^{j -1}.
\begin{aligned}\vert J_1 \vert \leq \|u_{0}\|_{H^{\frac{1}{2}}(\Lambda_{h})}\|\partial_{\nu}(w-v)\|_{H^{-\frac{1}{2}}(\Lambda_{h})}\leq C\|u_{0}\|_{H^{1}(\Lambda_{h})}\|\partial_{\nu}(w-v)\|_{H^{1}(\Lambda_{h})}\lesssim \|u_{0}\|_{H^{1}(\Lambda_{h})}\end{aligned}
\frac{x^{\frac{k-2}{2}-m}y^{\frac{k-2}{2}+m}}{2^{2-k}}=(z+\bar z)^{\frac{k-2}{2}-m}(z-\bar z)^{\frac{k-2}{2}+m}i^{\frac{2-k}{2}-m}=i^{\frac{2-k}{2}-m}\sum_{\frac{2-k}{2}\leq n\leq\frac{k-2}{2}}C(n)\cdot z^{n+\frac{k-2}{2}}\bar z^{\frac{k-2}{2}-n}
S_{2,\lambda}(k,p)= \sum_{m=p}^{k}\bigg\{\binom{k}{m}(-1)^{k-m}\langle 1\rangle_{k-m,\lambda}S_{2,\lambda}(m+1,p+1) &+\lambda k\binom{k-1}{m}(-1)^{k-m-1}\langle 1\rangle_{k-m-1,\lambda}S_{2,\lambda}(m+1,p+1)\bigg\}.
\phi_{j}^{(q)}(z) :=\langle(zI-\mathcal{H}_{q})^{-1}e_{j},e_{0}\rangle=\sum_{n=0}^{\infty}\frac{\langle\mathcal{H}_{q}^{n}\, e_{j},e_{0}\rangle}{z^{n+1}},0\leq j\leq p,q\geq 0, \beta_{j}^{(q)}(z) :=\langle(zI-\mathcal{B}_{q})^{-1}e_{j},e_{0}\rangle =\sum_{n=0}^{\infty}\frac{\langle\mathcal{B}_{q}^{n}\,e_{j},e_{0}\rangle}{z^{n+1}},0\leq j\leq p,q\geq 0,
\sum_{[\xi]\in \widehat{G}:|\xi|\leq \lambda}d_\xi\smallint\limits_G\textnormal{Tr}({\tilde\sigma}_{\textnormal{glob}}(x,\xi))dx=(2\pi)^{-n}\left( \smallint\limits_{T^*\mathbb{S}(G)}{\tilde\sigma}_{\textnormal{loc}}(x,\xi')d\mu_L(x,\xi')\right)\lambda^n+O(\lambda^{n-1}),
\mathbb{E}_{\lambda}^d\xi_t(O)=e^{-t}\sum_{n=0}^{\infty}\frac{t^n\lambda^n}{n!}\Big(\sum_{\overrightarrow{x}\in V_n}E\prod_{j=0}^{n-1}\rho(x_j,\omega)\rho(x_{j+1},\omega)\Big).
D_{-}^{\alpha}\equiv \partial_{z}+ \frac{i}{2}( A_{z}^{1}+ \frac{1}{\sqrt{3}}A_{z}^{2}+ \cdots + \frac{1}{\sqrt{( \alpha - 1 ) ( \alpha - 2 ) / 2}}A_{z}^{\alpha - 2}+ \frac{\alpha}{\sqrt{\alpha ( \alpha - 1 ) / 2}}A_{z}^{\alpha - 1})
= \int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\psi(x)\psi(y)\left( \left( \frac{\tilde{u}(s,x)}{\psi(x)}\right)^{\frac{1-q}{2}}- \left( \frac{\tilde{u}(s,y)}{\psi(y)}\right)^{\frac{1-q}{2}}\right)^{2}k(x,y)dxdy,
^{\rho}I^{1-\gamma}_{0_+}u(0)= \displaystyle\frac{\Gamma(\gamma)}{\Gamma(\alpha)}\Omega\ \sum_{i=1}^{m}\omega_{i}\int^{\xi_{i}}_{0}\left(\frac{\xi_{i}^{\rho}-s^{\rho}}{\rho}\right)^{\alpha-1}s^{\rho-1}\left(f(s,u(s),^\rho D^{\alpha,\beta}u(s))-p(s)u(s)\right) ds
{\varphi ''_n}={{\left({\varphi_{n,B}^{{\rm{CFSK}}}+{\varphi_B}- \varphi_{n,A}^{{\rm{CFSK}}}-{\varphi_A}}\right)}\mathord{\left/{\vphantom{{\left({\varphi_{n,B}^{{\rm{CFSK}}}+{\varphi_B}- \varphi_{n,A}^{{\rm{CFSK}}}-{\varphi_A}}\right)}2}}\right. \kern-\nulldelimiterspace}2}
\left(\frac{\partial\varphi}{\partial t}\right)_{\overline{k}}=\tilde{g}^{i\overline{i}}\left(\overline{e}_{k}e_{i}\overline{e}_{i}(\varphi)-\overline{e}_{k}[e_{i},\overline{e}_{i}]^{(0,1)}(\varphi)\right)-F_{\overline{k}}.
\Big|\frac{d}{dt}|z_j(t)-z_k(t)|^{-1}\Big|= \Big|\frac{(z_j(t)-z_k(t))\cdot (\dot{z}_j(t)-\dot{z}_k(t))}{|z_j(t)-z_k(t)|^3}\Big|\leq |\dot{z}_j(t)-\dot{z}_k(t)|d_p(t)^{-2},
L_{e f f}^{( 1 )}= 2 \pi \kappa \frac{\sigma_{1}\sigma_{2}}{q_{1}q_{2}}\sum_{p_{1}= 1}^{n_{1}}\sum_{p_{2}= 1}^{n_{2}}\frac{d}{d t}A r g ( \vec{R}_{p_{1}}^{( 1 )}- \vec{R}_{p_{2}}^{( 2 )}) .
N(\tau) = \frac{\operatorname{vol}(\mathbb{S}^{n-2})\operatorname{vol}(\partial \Omega)}{n-1}\biggl[ \biggl( \frac{\lambda + 3\mu}{2\mu (\lambda + \mu)}\biggr)^{n-1}+ \frac{1}{(2\mu)^{n-1}}+ \frac{n-2}{\mu^{n-1}}+ \frac{1}{\alpha^{n-1}}\biggr] \tau^{n-1}+ o(\tau^{n-1}) \ \tau \to +\infty.
f_{\phi}\:=\: \left(ac-bd\right)+\left(a-\left(b+c\right)\right)+\left(c-\left(a+d\right)\right) & \:=\: ab^{\prime}c+acd^{\prime}+ab^{\prime}c^{\prime}+a^{\prime}cd^{\prime}& \:=\: \left(a-b\right)+\left(c-d\right)
I_3= -\frac{2A\gamma}{\delta-1}\int \phi^{\frac{2\gamma-\delta-1}{\delta-1}}\nabla \phi\cdot u \leq C|\phi|^{\frac{2\gamma-\delta-1}{\delta-1}}_\infty|\nabla \phi|_2| u|_2 \leq C|\phi|^{\frac{4\gamma-2\delta-2}{\delta-1}}_\infty|\nabla \phi|^2_2+C| u|^2_2 \leq Cc^{\frac{4\gamma-4}{\delta-1}}_0+C| u|^2_2.
A=\begin{bmatrix}0 1 0 0 0 0 1.000002542612694 0 0 2 0 0 0 0 0 1 0 0 0 -0.000294117647059 0 0 0 0 0 0 0 0 0 1 0 0 0 0 -0.00000127311747 0\end{bmatrix},
\{F , G \}_{\mathrm{n e w}}= \{F , G \}_{\mathrm{o l d}}+ \oint_{\partial \Sigma}d^{n - 2}x \frac{\delta^{\vee}F}{\delta \phi_{A}}\delta_{G}\phi_{A}- \oint_{\partial \Sigma}d^{n - 2}x \frac{\delta^{\vee}G}{\delta \phi_{A}}\delta_{F}\phi_{A}+ ? ! ,
d s^{2}= - \left( r^{2}/ l^{2}- \frac{\mu}{r}- 1 \right) d t^{2}+ \frac{d r^{2}}{\left( r^{2}/ l^{2}- \frac{\mu}{r}- 1 \right)}+ \frac{r^{2}}{z^{2}}\left( d z^{2}+ d \varphi^{2}\right) ,
{\mathfrak{n}}_1(\lambda_1) = \frac{1}{\sqrt{1+ 9 \xi_1^4 + 9 \eta_1^4}}\left(-1, \ 3 \xi_1^2, \ 3 \eta_1^2 \right), &{\mathfrak{n}}_2(\lambda_2) = \frac{1}{\sqrt{1+ 9 \xi_2^4 + 9 \eta_2^4}}\left(-1, \ 3 \xi_2^2, \ 3 \eta_2^2 \right), &{\mathfrak{n}}_3(\lambda_3) = \frac{1}{\sqrt{1+ 9 \xi^4 + 9 \eta^4}}\left(-1, \ 3 \xi^2, \ 3 \eta^2 \right).
\langle \psi \alpha_1,\alpha_2,\dots,\alpha_k\rangle_{1,2}^0=\mathrm{deg}\big(\mathrm{ev}_{[k]}^*(p_1^*\alpha_1\cup\cdots\cup p_k^* \alpha_k)\cap \Phi_{k*}(\psi\cap [\overline{\mathcal{M}}^0_{1,k}(X,2)]^{\mathrm{vir}})\big)=0.
(U_1^{q+1}\cup\cdots\cup U_n^{q+1})^c\cap U^{q+1}=\{(x_0,\ldots,x_q)\in U^{q+1}:\forall i\in\{1,\ldots,n\}\exists j\in\{0,\ldots,q\}x_j\not\in U_i\}&=\bigcup_{f:\{1,\ldots,n\}\to \{0,\ldots,q\}}A_f.
\overline{\mu}_1^* =\overline{\mu}_1+N(e^{F_1}-e^{F_1-F_2}-F_2)\mu_H &\leq \overline{\mu}_1+N\left(e^{F_1}(1-e^{-F_2})\right)\mu_H\leq \overline{\mu}_1+e^{\|F_1\|_{\infty}}N(F_2)\mu_H\in S_{D_0}^1({\bf Y}).
-2 \sqrt{h}D_{j}\Bigl(\frac{1}{\sqrt{h}}\frac{\delta S}{\delta h^{i}_{\hspace{0.5em}j}}\Bigr)+ \partial_{i}\phi^{I}\frac{\delta S}{\delta \phi^I}+D_{i}\tilde{A}^{a}_{j}\frac{\delta S}{\delta \tilde{A}^{a j}}- \sqrt{h}D_{j}\Bigl( \frac{1}{\sqrt{h}}\frac{\delta S}{\delta \tilde{A}^{a}_{j}}\tilde{A}^{a}_{i}\Bigr) = 0.
S_{w o r m}= - 6 \pi^{5}\int d \rho \big [{\frac{R^{7}}{N}}( \partial_{\rho}R )^{2}+ N R^{7}-{\frac{1}{1 4 4}}{\frac{R^{9}}{N}}( \partial_{\rho}\phi )^{2}-{\frac{Q^{2}}{\pi^{1 0}}}{\frac{N}{R^{9}}}e^{- 2 \phi}\big ] .
(at+b)^{p\theta+p-1}= (at+b)^{p\theta}(at+b)^{p-1}& = \begin{cases}a^{p\theta}t^{p\theta+p-1}b=0, \sum_{m=0}^{p-1}\binom{p-1}{m}\left( a^{p\theta-m}b^m t^{p\theta+p-1-m}+ a^{-m}b^{p\theta+m}t^{p-1-m}\right). \end{cases}& = a^{p\theta}t^{p\theta+p-1}+ r_{\theta}(t),
\operatorname{Trace}(T) =\sum_{j=1}^mg_j(\omega_j)=\sum_{j=1}^mg_j\left(\sum_{k=1}^nf_k(\omega_j)\tau_k\right)=\sum_{k=1}^n\sum_{j=1}^mf_k(\omega_j)g_j(\tau_k) &=\sum_{k=1}^nf_k\left(\sum_{j=1}^mg_j(\tau_k)\omega_j\right)=\sum_{k=1}^nf_k(T\tau_k).
p_{\beta,\mu}(\{n_i\}) = \frac{1}{\mathcal{Z}(\beta,\mu)}\prod_j \exp\{- \beta n_j (\varepsilon_j - \mu)\}&= \exp\left\{- \ln \mathcal{Z}(\beta,\mu) - \beta \sum_j n_j\varepsilon_j + \beta \mu \sum_j n_j\right\},
\theta_{p+k,q+k}(\theta_{n}(\pi)) =\pi_{1}(\lambda_{d},1,\Psi,\mu,\nu,(\varepsilon|(1,1,\dots,1)), \qquad(\kappa|(n-\frac{p+q}{2},n-1-\frac{p+q}{2},\dots,n-k+1-\frac{p+q}{2})))
\psi_j^{(K)}= \mathop{\max}\limits_{1 \le{k_j}\le K}\sum\limits_{i ={k_j}}^K{\phi_{ji}^{\left( 4 \right)}}= \max \bigg\{{\mathop{\max}\limits_{1 \le{k_j}\le K - 1}\sum\limits_{i ={k_j}}^K{\phi_{ji}^{\left( 4 \right)}},\phi_{jK}^{\left( 4 \right)}}\bigg\}= \max \left\{{\psi_j^{(K - 1)},0}\right\}+ \phi_{jK}^{\left( 4 \right)}
E_N(x) = -{R \over 2\pi}\, \ln \left|{g_N^x(iR) \over g_N^{x=1}(iR)}\right| +{1 \over 2\pi}\int_0^R \ln \left|{g_N^x(i\xi) \over g_N^{x=1}(i\xi)}\right| \,d\xi \;.
\overline{p(z)}= \overline{\sum_{k = n_{1}}^{n_{2}}a_{k}z^{k}}= \sum_{k = n_{1}}^{n_{2}}\overline{a_{k}}\overline{z^{k}}= \sum_{k = n_{1}}^{n_{2}}\overline{a_{k}}( \overline{z})^{k}= \sum_{k = n_{1}}^{n_{2}}\overline{a_{k}}z^{-k}.
r_i=\kappa_i(-\lambda_i,1,\rho(\lambda_i u-v), \frac{\rho (\lambda_i u-v)}{c^2},0)^\top, i=1,5; &r_2=(u,v,0,0,0)^\top, r_3=(0,0,0,\rho,0)^\top, r_4=(0,0,0,0,1)^\top,
\tilde{\phi}_{m}:=\delta_{m}^{\frac{n-2}{2}}\phi_{\delta_{m},q_{m}}(\psi_{q_{m}}^{\partial}(\delta_{m}y))\chi(\delta_{m}y)y=(z,t)\in\mathbb{R}_{+}^{n},z\in\mathbb{R}^{n-1}t\ge0
\Omega_{2}=\hat{R}_{MNPQ}\hat{R}^{MNPQ}-4\hat{R}_{MN}\hat{R}^{MN}+\hat{R}^2 \stackrel{\mathrm{4D}}{=}\frac{1}{4}\hat{R}_{MN}^{\ \ \ \AB}\hat{R}_{PQ}^{\ \ \ \CD}\eta^{MNPQ}\eta_{ABCD}.
\mu_1 =\max_{f}\frac{(\delta f,\delta f)_H}{(f,f)_V}&=\max_{f}\frac{\sum_{h\in H}\delta f(h)^2}{\sum_{v\in V}\deg v\cdot f(v)^2}&=\max_{f:\sum_{v\in V}\deg v\cdot f(v)^2=1}\sum_{h\in H}\delta f(h)^2 &=\max_{f:\sum_{v\in V}\deg v\cdot f(v)^2=1}\sum_{h\in H}\biggl(\sum_{v_i h}f(v_i)-\sum_{v^j h}f(v^j)\biggr)^2
4\big(\frac{\partial^2 F}{\partial s_1^2}(0)\big)^2+2\frac{\partial F}{\partial s_2}(0)\frac{\partial^2 F}{\partial s_1^2}(0)-8\frac{\partial^2 F}{\partial s_1^2}(0)-2 \frac{\partial F}{\partial s_2}(0)+4=0.
x\times y=(x^{2}y^{3}-x^{3}y^{2})\frac{h_{2}h_{3}}{h_{1}}e_{1}+(x^{3}y^{1}-x^{1}y^{3})\frac{h_{1}h_{3}}{h_{2}}e_{2}+(x^{1}y^{2}-x^{2}y^{1})\frac{h_{1}h_{2}}{h_{3}}e_{3}
\Theta_j-M\frac{\langle \nabla z_j, \nabla w_j \rangle}{z_j \phi^{2\alpha_j}}= \Psi_j-M \frac{|\nabla z_j|^2}{z_j}-2 \alpha_j (M+2) \frac{\langle \nabla z_j, \nabla \phi\rangle}{\phi}\omega.
\sum_{s\in D^{n}}e^{\lambda_{1}s}h(\widetilde{\xi}_{s})=\sum_{s\in D^{n}}e^{\lambda_{1}s}h(\widetilde{\xi}_{s})1_{\{h(\widetilde{\xi}_{s})>e^{\varepsilon s}\}}+\sum_{s\in D^{n}}e^{\lambda_{1}s}h(\widetilde{\xi}_{s})1_{\{h(\widetilde{\xi}_{s})\le e^{\varepsilon s}\}}=:\mathrm{I}+\mathrm{II}.
\int_{I_{j^1}}\int_{I_{j^2}}\Pi w dx^1 dx^2 = \; \int_{I_{j^1}}\int_{I_{j^2}}w dx^1 dx^2, \widetilde{\Pi_h}(\Pi w,v_h)_{j^1,j^2}= \; \widetilde{\Pi_h}(w,v_h)_{j^1,j^2}, \forall v_h\in P^k(I_{j^1}\times I_{j^2}),\forall j^1,j^2,
F_\lambda\big(z_1^{\pm 1},\dots,z_n^{\pm 1};b;q,t;p\big)=\prod_{1\le i\le n,1\le j\le \lambda_1}\frac{\theta_p\big((p/b) t^{-\lambda'_j}q^j z_i^{\pm 1}\big)}{\theta_p\big((p/b) q^j z_i^{\pm 1}\big)}\hphantom{F_\lambda\big(z_1^{\pm 1},\dots,z_n^{\pm 1};b;q,t;p\big)}{}=R^{*(n)}_\lambda\big(z_1,\dots,z_n;pq/t^nb,b;q,t;p\big).
M = \frac{i}{\sqrt{{\mathrm{\boldmath ~ \b e t a ~}}^{2}}}\left( \begin{matrix}{{E_{\mathrm{\boldmath ~ \b e t a ~}}}}&{{- E_{-{\mathrm{\boldmath ~ \m u ~}}}}}\\{{E_{\mathrm{\boldmath ~ \m u ~}}}}&{{\, \, \, E_{-{\mathrm{\boldmath ~ \b e t a ~}}}}}\\ \end{matrix}\right) .
\forall k \in \mathbb{N}^\star, \, T_{a^k}= \frac{1}{k}\left( \right) T_{a'^k}\underset{k \rightarrow \infty}{\rightarrow}P'_{A}T_{a''^k}\underset{k \rightarrow \infty}{\rightarrow}P''_{A}\gamma'(k) \underset{k \rightarrow \infty}{\sim}\frac{k}{t}\gamma''(k) \underset{k \rightarrow \infty}{\sim}\frac{k}{1-t}.
g_{k,r}(Z) = \begin{cases}\log Z \log \log Z (k,r) \ne (2,6) (3,8) (\log Z)^{\frac{(d-2)(0.7043)}{d}}(k,r) = (2,6) \left(\dfrac{\log \log Z}{\log \log \log Z}\right)^{1 - \frac{2}{d}}(k,r) = (3,8). \end{cases}
\psi_u(t_*)= \frac{1}{(2n-1-q) t_*^{q+1}}\bigg[ B(t_*u)- (2n-1) \int_{\Omega}(|x|^{-\mu}*F(t_*u)) f(t_*u)t_*u ~dx + nb \|t_* u\|^n \bigg].
{\cal L}_{p}^{\hbar}\left[ H \right] = - \sum_{i = 0}^{p}\, \frac{2}{\hbar}\, H \left( \phi_{i}\right) \operatorname{s i n}\left[ \frac{\hbar}{2}\, \stackrel{\leftarrow}{\partial}_{a}^{( i )}\omega^{a b}\, \stackrel{\rightarrow}{\partial}_{b}^{( i )}\right]
\mathbb{P}\left( \prod_{l=1}^m \Tilde{Y_l}\geq 1 \right) = \mathbb{P}\left( \prod_{l=1}^m \Tilde{Y_l}^{\theta}\geq 1 \right) \leq \mathbb{E}\left[ \Tilde{Y}^{\theta}\right]^{m}\leq \left(\frac{\mu}{C}\right)^{m}.
B_{i}=\frac{k}{\kappa_{s}}\int_{\Omega_{1}}N_{i}p(x,y)d\Omega+\frac{k}{\kappa_{s}}\int_{v}N_{i}gdV-kT_{o}\sum_{j=0}^{N_{D}}\int_{v}\nabla N_{j}^{D}.\nabla N_{i}dV
\sum_{i=0}^{k}\binom{rp+\sigma -i}{k-i}A_{a,b,r}^{\alpha_{s},\beta_{s},r_{s},p_{s}}\left( p,i\right) =\sum_{j=0}^{k}\left( -1\right)^{j}\binom{k}{j}\binom{\left( k-j\right) a+b}{r}^{p}\prod_{s=2}^{L}\binom{\left(k-j\right) \alpha_{s}+\beta_{s}}{r_{s}}^{p_{s}}.
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}^{}\}, \mathfrak{F}^{}=-\frac{1}{6}\frac{(Z^1)^3}{Z^0}\,.
\begin{array}{l}{{d{\hat{s}}^{2}= 2 d u \, \left( d v - \frac{1}{2}K \, d u \right) - F^{- 1}\, d x_{I}d x_{I}\qquad , \qquad e^{- 2 \hat{\phi}}= F}}\\{{\hat{H}_{I J K}= \frac{1}{6}\epsilon_{I J K L}\partial^{L}F^{- 1}\, , \qquad \hat{H}_{I J u}= \hat{H}_{I J v}= \hat{H}_{I u v}= 0 \, .}}\\ \end{array}
X_1 \triangleq \beta_{2,2}^{-1}\Bigl(a,b_0=x_1,\,b_1=b_2=(x_2-x_1)/2\Bigl), X_2 \triangleq \beta_{2,2}^{-1}\Bigl(a,b_0=(x_1+x_2)/2,\,b_1=a_1/2,\,b_2=a_2/2\Bigr), X_3 \triangleq \beta_{2,2}^{-1}\Bigl(a, b_0=a_1+a_2,\,b_1=b_2=(x_1+x_2-a_1-a_2)/2\Bigl).
d_{i\,j}(s) = q_{ij}\sum_{0 \le t <s}d_{i\,j+1}(t)d_{j\,j+2t+2}(s-t-1), b_{i\,j}(s) = \sum_{0 \le t <s}b_{i+1\,j}(t)d_{i\,i+2t+2}(s-t-1).
\boldsymbol{w}_{k+1}= \boldsymbol{w}_k + \mu e_k \boldsymbol{x}_k- \rho_{\rm p}\frac{\left( \| \boldsymbol{w}_k\|_{p}\right)^{1-p}(\boldsymbol{w}_k )}{\epsilon_{\rm p}+| \boldsymbol{w}_k |^{(1-p)}}
\prod_{v\nmid\infty}\left(\beta_v(\Phi_{1,v},\Phi_{2,v})\cdot\alpha_{v}(W_{\mathfrak{f},v},W^-_{\mathfrak{f},v})\right)=\frac{{\rm vol}(U_0(N))}{{\rm vol}(U)}|c_1^2 D|^{-\frac{1}{2}}2^{\#S_D}L_{c_1}(1,\eta_{T})^2\prod_{v\in S}L(1/2,\Pi_v,\chi_v)^{-1},
\|\hat{x}-x\|_2 = \| F_{sob}(q- y)\|_2 = \|F_{sob}\|_2 \|q-y\|_2 \leq \sqrt m \delta \|F_{sob}\|_2 \leq \frac{\sqrt m \delta}{\sigma_{\min}(D^{-r}F)}\leq \frac{\sqrt m \delta}{\sigma_l(D^{-r})\sigma_{min}(V_l^T F)}\leq c \delta \left(\frac{m}{l}\right)^{-r}\frac{m}{\sigma_{\min}(V_l^TF)},
G_{E}^{+}\left( x,x^{\prime}\right) =\left\langle E\right| \phi \left(x\right) \phi \left( x^{\prime}\right) \left| E\right\rangle =\frac{\Gamma\left( h_{+}\right) \Gamma \left( h_{-}\right)}{16\pi^{2}}F\left(h_{+},h_{-};2;\frac{1+P}{2}\right) ,
\lim_{r\to 0}\,\sup_{x\in U}\int_{|x-y|<r}\frac{|b(y)|^2}{|x-y|^{n-2}}dy=0=\lim_{r\to 0}\,\sup_{x\in U}\int_{|x-y|<r}\frac{|b(y)|}{|x-y|^{n-1}}dy,
A_{k_{\bullet}}:=\mathrm{Sym}\left(\frac{z_1^{k_1}\cdots z_d^{k_d}}{(1-q^{-1}z_1^{-1}z_2)\cdots (1-q^{-1}z_{d-1}^{-1}z_d)}\cdot \prod_{j>i}w(z_i z_j^{-1}) \right)
\int_{t_l}^{t_{l+1}}|\tilde G(\mu^{N,M}(t),z^{N,M}(t))dt - \tilde G( \mu^{N,M}(t_l), z^{N,M}(t))|dt\leq \Delta(\epsilon)\nu(\Delta(\epsilon)) \ \ \{\rm if}\ \ \ \t_l\notin\cup_{i=1}^k ( \mathcal{T}_i - \delta(\epsilon) , \mathcal{T}_i+ \delta (\epsilon)).
\begin{cases}w_{tt}-(1+t)^{2\ell}\Delta w = g(t,x), x\in \mathbb{R}^n, t>0, w(0,x)= w_0(x), x\in \mathbb{R}^n, w_t(0,x)= w_1(x), x\in \mathbb{R}^n, \end{cases}
\lim_{\lambda \to \infty}\frac{p_{\lambda p(p+1)+\rho+1,k}(z)}{p_{\lambda p(p+1)+\rho,k}(z)}=\kappa_k^{(\rho)}\widetilde{F}_{k}^{(\rho)}(z),z\in\mathbb{C}\setminus[a_{k},b_{k}], k=0,\ldots,p-1,
\Bigg|{\; \atop{\mbox{det}\atop{\scriptstyle (i,j)}}}\Bigg(\frac{1}{w_i - z_j}\Bigg) \Bigg| \; = \;\Bigg|\frac{\prod_{1 \leq i < j \leq n}(w_i - w_j )( z_i - z_j )}{\prod_{i,j = 1}^n ( w_i - z_j )}\Bigg| \; ,
q_k=q_{0\left( k\right)}-\frac \nu{2W\left( q_{0\left( k\right)}\right)},\;\psi_{+\left( k\right)}=\lambda_{+}W^k\left( q_{0\left( k\right)}\right) ,\;\psi_{-\left( k\right)}=\lambda_{-}W^{-k}\left( q_{0\left(k\right)}\right) ,
{y^{[j]}}({t_3}) ={h^{[j1]}}({t_3})u_3^{[1]}+{h^{[j2]}}({t_3})\frac{{{h^{[32]}}({t_1})}}{{{h^{[32]}}({t_3})}}u_1^{[2]}+{h^{[j3]}}({t_3})\frac{{{h^{[33]}}({t_1})}}{{{h^{[33]}}({t_3})}}u_1^{[3]}.
S = \int_{E}d^{2}x \, \Bigr \{\mathrm{~ \frac{1}{4}~}F^{\mu \nu}( A ) F_{\mu \nu}( A ) + \mathrm{~ \frac{1}{2}~}\partial^{\mu}A_{\mu}\partial^{\nu}A_{\nu}+ \partial^{\mu}\bar{C}\partial_{\mu}C \Bigr \}\, ,
D_{11}= \frac{(at_3)^{1-s_1}}{s_1+s_3-1}\int_1^{at_3}\frac{1}{v^{s_2}(at_3+v)^{s_3}}dv &+ \frac{s_3}{s_1+s_3-1}\int_1^{at_3}\frac{1}{v^{s_2-1}}\int_{at_3}^\infty \frac{1}{u^{s_1}(u+v)^{s_3+1}}dudv &= D_{111}+D_{112},
J(u):=\left(\int_{\mathbb{G}}|\mathcal{R}_{1}^{\frac{a_{1}}{\nu_{1}}}u(x)|^{p}dx\right)^{\frac{Q(q-p)-a_{2}pq}{(a_{1}-a_{2})p^{2}}}\left(\int_{\mathbb{G}}|\mathcal{R}_{2}^{\frac{a_{2}}{\nu_{2}}}u(x)|^{p}dx\right)^{\frac{a_{1}pq-Q(q-p)}{(a_{1}-a_{2})p^{2}}}\left(\int_{\mathbb{G}}|u(x)|^{q}dx\right)^{-1}.
{\cal W}\, ( \phi , \widetilde \phi ) = \frac{m^{2}}{4 \lambda}\, \phi - \frac{\lambda}{3}\, \phi^{3}+ \lambda \, \phi \, \widetilde \phi^{\, 2}+ \frac{p \, m}{2}\, \widetilde \phi^{\, 2}+ \frac{q \lambda}{3}\, \widetilde \phi^{\, 3}\, ,
\zeta_{p,p^{\prime}}\left(\frac{1}{2},c\right)=\frac{2C_{p}^{(1)}+\log\left(\frac{c}{4}\right)-4e^{2\pi p^{\prime}}Q_{2\pi p^{\prime}}(0)}{1+\frac{1}{\pi p^{\prime}}}+2C_{p^{\prime}}^{(2)}-2\log(2\pi)-2\gamma+\frac{2^{5/2}\theta e^{-\pi\sqrt{c}}}{c^{1/4}}.
\Phi_{v^o_n}(z^o_n, x^o_n) = \frac{k_n}{2n}+o\left(\frac{k_n}{n}\right) + 1-\frac{k_n}{n}+o\left(\frac{1}{n}\right) = 1 - \frac{k_n}{2n}+o\left(\frac{k_n}{n}\right).
\sum_{m \geq j}2p^m m \frac{\Gamma(m+\rho)}{\Gamma(m+1)}= \sum_{m=j}^{j'}2p^m m \frac{\Gamma(m+\rho)}{\Gamma(m+1)}+ \sum_{m = j'+1}^{\infty}2p^m m \frac{\Gamma(m+\rho)}{\Gamma(m+1)}.
W_{k,k-1}[f](\lambda)=\int_{r_{0}}^{\infty}\frac{J_{k}(\lambda s)Y_{k-1}(\lambda r_{0})-Y_{k}(\lambda s)J_{k-1}(\lambda r_{0})}{\sqrt{J_{k-1}^{2}(\lambda r_{0})+Y_{k-1}^{2}(\lambda r_{0})}}f(s)sds,
Z_{c}= \lim_{L \rightarrow \infty}\int \exp \{i \sum_{\alpha}[ \frac{( \chi^{\alpha})^{2}L}{m_{\alpha}}+ \frac{\chi^{\alpha}}{m_{\alpha}}\sin ( 2 m_{\alpha}L ) ( A_{\alpha}+ A_{\alpha}^{+}) ] \}d \chi
R_{m n}{}^{p q}\rightarrow W_{m n}{}^{p q}- \frac{1 6}{d - 2}\delta_{[ m}{}^{[ p}( \bar{\psi}_{| r |}\Gamma^{| r |}\psi_{n ]}{}^{q ]}- \bar{\psi}^{| r |}\Gamma^{q ]}\psi_{n ] r}) .
||T^nTh||=||T^{n+1}h||=||h||=||Th|| \ \ (n=0,1,2, \dotsc), ||T^{*n}Th|| =||T^{*n-1}T^*Th||=||T^{*n-1}h||=||h||=||Th|| \ \ (n=1,2, \dotsc),
{_{3}}F_{2}\left(\begin{array}{ccc}\chi_4, \chi_4, \chi_4 \varepsilon, \varepsilon\end{array}\mid 1\right) =\binom{\chi_8}{\chi_8^2}\binom{\chi_8}{\chi_8^3}+\binom{\chi_8^5}{\chi_8^2}\binom{\chi_8^5}{\overline{\chi_8}}&=\frac{\chi_8(-1)}{q^2}[J(\chi_8,\chi_8^6)J(\chi_8,\chi_8^5)+J(\chi_8^5,\chi_8^6)J(\chi_8^5,\chi_8)].
\overline{x}_1^{k+1}= x_1^k - \frac{\lambda}{q_1M}\left(\frac{\gamma_1}{N(N+1) p_{i_k1}}\nabla f_{i_k}\left(\hat{x}_1^k\right) - \frac{1}{(N+1) p_{i_k1}}y_{i_k, 1}^k + \frac{1}{N+1}\sum_{i=1}^{N+1}y_{i, 1}^k \right); \left( \forall j > 1\right) \qquad x_j^{k+1}= x_j^k.
2 \pi nE(1)\left( \begin{matrix}0 1 0 0\end{matrix}\right) E^{-1}(1) \left( \frac{3}{16 \pi in^2 \log^2 n}+ O\left(\frac{1}{n^2\log^3n}\right)\right) \qquad{}=\frac{3}{8 i n\log^2n}E(1)\left( \begin{matrix}0 1 0 0\end{matrix}\right) E^{-1}(1) + O\left(\frac{1}{n\log^3n}\right).
M_{4g}= \frac{q^{2g+1}}{\zeta_q(2)}\Big[ \frac{1}{2 \pi i}\oint_{|u|=\frac{1}{q^{2+\epsilon}}}\frac{1}{(1-q^2u)(q^2u)^y}P_u(2g) \, \frac{du}{u}+ \frac{1}{(2 \pi i)^2}\oint_{|w|=\frac{1}{q^{\frac{1}{2}+\epsilon}}}\oint_{|u|=\frac{1}{q^{2+\epsilon}}}\frac{\mathcal{H}(w,u)}{(1-qw)^{11}(qw)^{2g}(1-q^2u)(q^2u)^y}\, \frac{du}{u}\, \frac{dw}{w}\Big],
\begin{array}{l}\left.X_1^+\right.^*=X_1^+~,~~~~~\left.X_1^-\right.^*=X_1^-~,~~~~~H_1^*=-H_1~, \left.X_2^+\right.^*=-X_2^+~,~~~~~\left.X_2^-\right.^*=-X_2^-~,~~~~~H_2^*=-H_2~.\end{array}
f(x)= \displaystyle\sum_{i=0}^{\frac{s}{2}}\left[\binom{s}{s-2i}\,k+\,\binom{s}{s-(2i+1)}\,(2-k)\right]\,x^\frac{p^l+s-2i-1}{2}\cr +\displaystyle\sum_{i=0}^{\frac{s}{2}}\left[\binom{s}{s-2i+1}\,k+\,\binom{s}{s-2i}\,(2-k)\right]\,x^\frac{s-2i}{2}
\begin{aligned}\binom{d(u) - 1}{a}\binom{d(v) - 1}{b}+ \binom{d(u) - 1}{b}\binom{d(v)- 1}{a}< \binom{\lfloor n/2 \rfloor - 1}{a}\binom{\lceil n/2 \rceil - 1}{b}+ \binom{\lfloor n/2 \rfloor - 1}{b}\binom{\lceil n/2 \rceil - 1}{a}\end{aligned}
\big|M^*_v(t)-M^*_v(\delta n)\big| \le e^{-(\lambda^*-\lambda)t}\big|M_v(t)-M_v(\delta n)\big|+\big(e^{-(\lambda^*-\lambda)\delta n}-e^{-(\lambda^*-\lambda)t}\big)|M_v(\delta n)| &\le\delta e^{-(\lambda^*-\lambda)(t-\delta n)}+\delta\big(1-e^{-(\lambda^*-\lambda)(t-\delta n)}\big)=\delta,
\textsf{P}\{\inf_{z\in_{J}(K_{1}, K_{2})}\Vert \tilde{W}z\Vert<t\sqrt{d}\}&\le (C_{1}t)^{d}+\sum_{s=2^{k}, k\le k_{1}}((C_{2}t)^{d}e^{-cs}+e^{-c_{1}\sqrt{N}})+\textsf{P}\{\Vert \tilde{W}\Vert>C_{0}\sqrt{N}\}&\le (C_{3}t)^{d}+k_{1}e^{-c_{1}\sqrt{N}}+\textsf{P}\{\Vert A\Vert>C_{0}\sqrt{N}\}.
\mathcal{F}_d^{\mathbf{t}}(\mathbf{x},\mathbf{y})= G_d+\frac{q}{2}\mathbf{t}\cdot \mathbf{y};~\mathcal{G}_d^{\mathbf{t}}(\mathbf{x}, \mathbf{y})=G_{d}+\frac{q}{2}x_{m-2}+\frac{q}{2}\mathbf{t}\cdot \mathbf{y},
\int_{0}^{1}(1-r)^{\alpha\cdot(\mathbf a+\mathbf 1)}r^{c}dV=\frac{\Gamma\big(1+\alpha\cdot(\mathbf a+\mathbf 1)\big)\Gamma(c+1)}{\Gamma\big(2+c+\alpha\cdot(\mathbf a+\mathbf 1)\big)},
t^{1/2}T_i f_{\ldots,\lambda_i,\lambda_{i+1},\ldots}= tT_i^2 f_{\ldots,\lambda_{i+1},\lambda_{i},\ldots}= t \left(1+(t^{1/2}-t^{-1/2})T_i \right) f_{\ldots,\lambda_{i+1},\lambda_{i},\ldots}&= t f_{\ldots,\lambda_{i+1},\lambda_{i},\ldots}+(t-1) f_{\ldots,\lambda_{i},\lambda_{i+1},\ldots}.
\phi\big((K^*X^{rs}K)^{\frac{1}{s}}\big) = \ \phi\big((M^*C^{-\frac{rs}{2}}X^{rs}C^{-\frac{rs}{2}}M)^\frac{1}{s}\big) = \ \phi\big((|M|Q^*C^{-\frac{rs}{2}}X^\frac{rs}{2}X^\frac{rs}{2}C^{-\frac{rs}{2}}Q|M|)^\frac{1}{s}\big) = \ \phi\big(|G_X(s)|^\frac{2}{s}\big).
\langle\langle B, \bar{V}\rangle\rangle_{M+1}=\langle b_q, v_{2M+1}\rangle= b^+_q p_{(2M+1)}+b^-_q \bar{p}_{-(2M+1)}= 0 \langle\langle B^*, \bar{V}\rangle\rangle_{M+1}=\langle \hat{b}_1, v_{2M+1}\rangle= \hat{b}^-_1 p_{(2M+1)}+\bar{b}^+_1 \bar{p}_{-(2M+1)}= 0,
E^p[H_0](\tau_2)+\int_{\tau_1}^{\tau_2}E^{p-1,\deg}[H_0](\tau) d\tau \leq C \left( E^p[H_0](\tau_1)+\int_{\tau_1}^{\tau_2}\int_{\Sigma_\tau}r^{p-5}|P|^2 dVol_\tau d\tau +\int_{\tau_1}^{\tau_2}\int_{\Sigma_{\tau}\cap \{2M\leq r \leq 4M\}}\sqrt{e [H_0]}|P| dVol_\tau d\tau \right).
L_2^{-1}=1+T^2, L_3^{-1}=(1+T) (3T^2 - T + 1), L_7^{-1}=(1-T) (7T^2 + 3T + 1), L_{101}^{-1}=(1+T) (101T^2 + 3T + 1), L_{163}^{-1}=(1-T) (163T^2 + 11T + 1).