harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
g_n(x) := \mathbb E_x\left[\frac{1}{n}\sum_{i=0}^{n-1}h\circ X_i \mid \tau >n\right]= \frac{\lambda^n}{\mathcal P^n(x,M)}\frac{1}{n}\sum_{i=0}^{n-1}\frac{1}{\lambda^i}\mathcal P^i\left(h(\cdot) \frac{1}{\lambda^{n-i}}\mathcal P^{n-i}(\cdot,M)\right)(x).
\frac{\varphi_x(w_1,\dots,w_k)}{\phi(w_1)\cdots\phi(w_k)}\quad=\frac{\varphi_x(w_1,\dots,w_{k-1})}{\phi(w_1)\cdots\phi(w_{k-1})}\,\frac{\varphi_x(w_k|w_1,\dots, w_{k-1})}{\phi(w_k)}.
[\begin{pmatrix}A 0 0 \alpha\end{pmatrix},\begin{pmatrix}B tb^{\top}b \beta+t\chi\end{pmatrix}]-\begin{pmatrix}I X\end{pmatrix}\Omega\begin{pmatrix}I^{\top}X^{\top}\end{pmatrix}\begin{pmatrix}1\!\!1_{k}0 0 t1\!\!1_{n-k}\end{pmatrix}=
{\bf E}\Bigl[\Bigl(\int\limits_0^1 f(s)\,M_\mu(ds)\Bigr)^n\Bigr] = n! \int\limits_{\{s_1<\cdots< s_n\}}\prod_{r=1}^n f(s_r) \, \lim\limits_{\varepsilon\rightarrow 0}\exp\Bigl(\mu\sum_{p=1}^n\sum_{k=1}^{p-1}d(p-k) \,\rho_{\varepsilon}(s_p-s_k)\Bigr)ds,
\lim_{T\rightarrow\infty}\frac{1}{T}\int_0^T \mu\left(X_t^{Z^{b}}\right)X_t^{Z^{b}}\,dt = \int_0^{b}\mu(x)x \frac{m'(x)}{m((0,b))}\,dx = \frac{1}{S'(b)m((0,b))}.
\left[\frac{\mathcal{C}_1 (-,?)}{I_1}\otimes \frac{J_1}{J_2}\right]^2=\left[\frac{\mathcal{C}_1 (-,?)}{I_1}\right]^2\otimes \left[ \frac{J_1}{J_2}\right]^2=\frac{\mathcal{C}_1 (-,?)}{I_1}\otimes \frac{J_1}{J_2}
\tilde{g}_1= (I-\mathfrak{H})\partial_t g_c+(I-\mathfrak{H})\partial_t g_d+(I-\mathfrak{H})ia_t(I-\mathfrak{H})(z-\bar{z})_{\alpha}:= \tilde{g}_{11}+\tilde{g}_{12}+\tilde{g}_{13}.
\sum_{k=0}^{\infty}Q(k)x^{k}=\frac{1}{(x;x^{2})_{\infty}}=\theta_{3}(x)^{1/6}\theta_{4}(x)^{-1/3}[\frac{\theta_{3}(x)^{4}-\theta_{4}(x)^{4}}{16x}]^{1/24}
B = \frac{\pi}{2}+ \gamma + \log{8\pi}- \log \operatorname{rd}(K) - \int_0^\infty \frac{1-F(x)}{2}\left( \frac{1}{\sinh{\frac{x}{2}}}+\frac{1}{\cosh{\frac{x}{2}}}\right) \, dx + 2 \sum_{p \in S}\sum_{m=1}^\infty \frac{\log p}{p^{m/2}}F(m \log p).
H^{0}(\pi^*{\cal S},{\cal O}_{\pi^{*}{\cal S}}(A^{\prime}\sigma|_{\pi^*{\cal S}}- B^{\prime}F))=H^{0}({\cal S}, \pi_{*}{\cal O}_{\pi^{*}{\cal S}}(A^{\prime}\sigma|_{\pi^*{\cal S}}- B^{\prime}F)).
A= \begin{pmatrix}a_0 a_1 a_2 \cdots a_{n-1}a_{n-1}a_0 a_1 \cdots a_{n-2}\vdots \ddots \ddots \ddots \vdots a_2 a_3 \cdots a_0 a_1 a_1 a_2 \cdots a_{n-1}a_0\end{pmatrix}
<{\cal J}_{\mu_{1}\cdots \mu_{s}}^{( s )}( x ){\cal J}_{\nu_{1}\cdots \nu_{s}}^{( s )}( 0 ) > ={\prod}_{\mu_{1}\cdots \mu_{s}, \nu_{1}\cdots \nu_{s}}^{( s )}\left({\frac{c_{s}( t )}{x^{4}}}\right) + \sum_{s^{\prime}= 0}^{s - 2}
-\lambda_{i_1 ,\dots ,i_{n-1}}=(-1)^{n-1+i_1+\dots +i_{n-1}}\frac{(\alpha \beta +(-1)^{i_1+\dots +i_{n-1}}\prod_{k=1}^{n-1}\gamma_k^{i_k})\prod_{k=1}^{n-1}\gamma_k^{1-i_k}}{\alpha \beta}.
\begin{cases}2\lambda^{(i)}+1 := [2\lambda_1,2\lambda_2,\ldots , 2\lambda_i+1,\ldots,2\lambda_t], \ \mbox{for}i\in \mathcal{I}(\lambda) 2\lambda^{(t+1)}+1 := [2\lambda_1,2\lambda_2,\ldots,2\lambda_t,1].\end{cases}
\|T_g^{\varphi}\|_{\mathcal{X}\to H_v^{\infty}}\asymp \|D \circ T_g^{\varphi}\|_{\mathcal{X}\to H_w^{\infty}}= \| (g \circ \varphi)^{\prime}C_{\varphi}\|_{\mathcal{X}\to H_w^{\infty}}&= \sup_{z \in \mathbb{D}}(1-|z|)v(z)|(g \circ \varphi)^{\prime}(z)|\|\delta_{\varphi(z)}\|_{\mathcal{X}\to \mathbb{C}}
G_{0}^{( 2 )}( \vec{\rho}, v ) = - \frac{g}{L}\frac{1}{( 2 \pi )^{d - 1}}\sum_{n = 1}^{\infty}\cos ( \frac{n \pi v}{L}) \int d^{d - 1}p \frac{e^{i \vec{p}. \vec{\rho}}}{( \vec{p}^{\, 2}+ ( \frac{n \pi}{L})^{2}+ m^{2})}.
\vert[\tilde{\bf T}^2u](z)-[\tilde{\bf T}^2v](z)\vert \le{2\over \vert 2\Lambda-1\vert}\int_{z_0}^z\vert g(t)\vert\vert [\tilde{\bf T}u](t)-[\tilde{\bf T}v](t)\vert\vert dt\vert &\le \left\vert{[2(z-z_0)]^2\over 2(2\Lambda-1)^2}\right\vert\,\vert\vert g\vert\vert_\infty^2\,\vert\vert u-v\vert\vert_\infty
p\left({{d_{p2}}\leq \delta}\right) = \frac{{M{\delta^{2nk -{k^2}}}}}{{k!}}\int_{{{\mathbf{\Omega}}_2}}{\prod\limits_{i = 1}^k{{x_i}^{n - 2k}}\prod\limits_{1 \leq i < j \leq k}{{{\left({{x_i}-{x_j}}\right)}^2}}d{x_1}\cdots d{x_k}}
\rho_{\varepsilon}(x)=\sum_{\vec{k}\in\mathbb{Z}^{3}}\widetilde{\rho}_{\varepsilon}(x+\vec{k})=\sum_{\vec{k}\in\mathbb{Z}^{3}}\frac{1}{\varepsilon^{3}}\widetilde{\rho}\left(\frac{x+\vec{k}}{\varepsilon}\right).
\left\lbrace\begin{array}{c}\lambda_{+}\phi (\pm a/2)=-\lambda_{-}\phi (\pm a/2) \lambda_{+}\partial_{x}\phi (\pm a/2)=-\lambda_{-}\partial_{x}\phi (\pm a/2)\end{array}\right. \Rightarrow \left\lbrace\begin{array}{c}\phi (a/2)=\phi (-a/2)=0 \partial_{x}\phi (a/2)=\partial_{x}\phi (-a/2)=0\end{array}\right. ,
{\mathcal L}_i(f):=\frac{d^{m_i}z^{q+m_i+k}f_{i}}{dz^{m_i}}(z) - CG\left(z,\frac{d^lz^{m_j+k}f_{j}}{dz^l}(z),\;1\leq j\leq r,\; 0\leq l\leq m_j\right).
\max_{j\in\{1,\dots,N_h\}}\Big( \big\| \varrho^j_h(X|_{\tau_h}) \big\|^2 + \frac{1}{h}\big\|P_h^{j-1}\varrho^j_h(X|_{\tau_h})\big\|^2 \Big) \leq C h^2.
\Big|\sum_{k=p}^q a_k{\rm e}^{it\lambda_k}- \sum_{k=v_{p'}}^{v_{q'}}b_k{\rm e}^{it \log_2 u_k}\Big| \le \sum_{k\, :\, u_k=u_p}|a_k| + \sum_{k\, :\, u_k=u_q}|a_k| + \sum_{\ell=p'}^{q'}\sum_{k\, :\, u_k= v_\ell}|a_k|\, |{\rm e}^{it \lambda_k}-{\rm e}^{it \log_2 u_k}|
\int_{B^{g_{\theta}}(x_{\theta}',a_{\theta}^{-\frac{2}{n-2}}r)}\tilde{u}_{\theta}^p\,dv^{\tilde{g}_{\theta}}=\int_{B^{g_{\theta}}(x_{\theta}',a_{\theta}^{-\frac{2}{n-2}}r)}u_{\theta}^p\,dv^{g_{\theta}}&\leq\int_{N}u_{\theta}^p\,dv^{g_{\theta}}&\leq C_1
m_{I}^{2}= \Bigl [{\frac{2 \pi^{3}}{3}}-{\frac{\pi}{3}}\Big ( \mathrm{l n}\Big ({\frac{2 \pi^{2}}{3}}\Big ) \Big )^{2}\Big ] T^{2}{\hat{g}}^{4}+{\frac{4 \pi}{3}}T^{2}\mathrm{l n}\Big ({\frac{1}{\hat{g}}}\Big ) -{\frac{4 \pi}{3}}T^{2}\Big ( \mathrm{l n}\Big ({\frac{1}{\hat{g}}}\Big ) \Big )^{2}+ 0 ({\hat{g}}^{5})
\sharp\{\underline{g}= \underline h_k \dots \underline h_1 \in G_{kp}:\underline{g}\mbox{not $\gamma$-acceptable}\}\leq\! \displaystyle\sum_{l\geq [\gamma k]}^k\sharp\{\underline{g}\in G_{kp}:\sharp\{0\leq j\leq k:\underline{h}_j\in G_p\backslash\tilde G_p\}=l\}.
\frac{\mathrm d}{\mathrm dx}\!\left(\frac{2\sqrt{n\left(x+T\right)}}k-\frac{2\sqrt{mx}}k\right)=\frac{\sqrt n}{k\,\sqrt{x+T}}-\frac{\sqrt m}{k\,\sqrt x}&=\frac{nx-mx-mT}{k\,\sqrt{x\left(x+T\right)}\left(\sqrt{nx}+\sqrt{m\left(x+T\right)}\right)}\gg\frac{\left(n-m\right)M}{k\,M\left(\sqrt n+\sqrt m\right)\sqrt M}\gg\frac{n-m}{k\,\sqrt{nM}}.
G\left(|a|^\frac{p-2}{2}\,a\right)-G\left(|b|^\frac{p-2}{2}\,b\right) \le \max\left\{G'\left(|a|^\frac{p-2}{2}\,a\right),\, G'\left(|b|^\frac{p-2}{2}\,b\right)\right\}\,\Big(V_p(a)-V_p(b)\Big).
\dim \delta H_3^g = \; 4 + 6(k+2) + 4\,(k+1)(k+2)/2+k(k+1)(k+2)/6 \;-(k+4)(k+5)(k+6)/6 = \; 0 \dim \delta E_3^g = \; I_M(V_3^g\times W_3^g) = \; 0,
\overline\Lambda_n \,\le\, \sum_{i=0}^{n-1}\binom{n}{i}(n-i)\,\frac{i!}{\alpha^i}\,=\, \frac{n!}{\alpha^n}\,\alpha\,\sum_{i=0}^{n-1}\frac{\alpha^{n-i-1}}{(n-i-1)!}\,=\, \frac{n!}{\alpha^n}\,\alpha\,\sum_{k=0}^{n-1}\frac{\alpha^k}{k!}\,\le\, \frac{n!}{\alpha^n}\,\alpha\, e^\alpha \,\le\, \frac{n!}{\alpha^n},
\Phi =\left(A+O_4(\rho^{-1}) \right)\left(\rho + \alpha +O_4(\rho^{-1}) \right)=A\rho + O_4(\rho^{0}) \nabla_i\Phi =A\frac{z^i}{\rho}+ O_3(\rho^{-1}), \Delta_{\tilde{g}}\Phi =\frac{2A}{\rho}+ O_2(\rho^{-2}), \nabla_i\Delta_{\tilde{g}}\Phi =-2A\frac{z^i}{\rho^{3}}+ O_1(\rho^{-3}),
\| (\mathcal{F}^{-1}(|\psi|))^{k_0}\mathcal{F}^{-1}(|h_0|)\|_{L^2}\leq \|\mathcal{F}^{-1}(|\psi|)\|_{L^\infty}^{k_0}\| \mathcal{F}^{-1}(|h_0|)\|_{L^2}\|= \|\mathcal{F}^{-1}(|\psi|)\|_{L^\infty}^{k_0}\| h_0\|_{L^2}\|.
\int_{[-1,1]}\xi^{2k}m_{\omega}(\xi) \ \mathrm{d}\xi = - \frac{(-1)^k \cdot 2 \pi^3}{\alpha^3}(\pi^2 / \alpha^2 - \omega)^{k-1/2}+ \int_{\gamma}\xi^{2k}m_\omega (\xi) \ \mathrm{d}\xi .
M_t[u-v](t,x) &=\frac{(u-v)(t,x)-(u-v)(0,x)}{t^\alpha\Gamma(1-\alpha)}+\frac{\alpha}{\Gamma(1-\alpha)}\int_0^t\frac{(u-v)(t,x)-(u-v)(s,x)}{(t-s)^\alpha}ds.
(x_1,\dots,x_N)= (u_1^{(i)}u_i^{(i)}+h_1, \dots, u_{i-1}^{(i)}u_i^{(i)}+h_{i-1},u_i^{(i)}+h_i, &u_{i+1}^{(i)}u_i^{(i)}+h_{i+1}\dots, u_k^{(i)}u_i^{(i)}+h_k,x_{k+1},\dots,x_N).
T_jm_{x_2,y,h_2}= T_{j}T_{x_1}T_{x_2}T_y T_{d_i}T_{h_1}T_{h_2}= T_{x_1}T_{x_2}T_{j}T_y T_{d_i}T_{h_1}T_{h_2}& = \begin{cases}T_{x_1}T_{x_2}T_{s_j y}T_{d_i}T_{h_1}T_{h_2}\ell(s_j y) > \ell(y), (q-1)T_{x_1}T_{x_2}T_{y}T_{d_i}T_{h_1}T_{h_2}+ qT_{x_1}T_{x_2}T_{s_jy}T_{d_i}T_{h_1}T_{h_2}\ell(s_j y) < \ell(y),\end{cases}
\sum_{j=0}^J d_j^{-1+N/2}\|\tilde g - g_h\|_{L^2(\Omega_h \cap A_j)}\le C(hd_0^{-1}) \Big( \sum_{j=0}^J d_j^{N/2}\|\tilde g - g_h\|_{H^1(\Omega_h\cap A_j)}+ \|\tilde g - g_h\|_{W^{1,1}(\Omega_h)}\Big) &+ Ch^{5/2-m}d_0^{-1}+ Ch^2 d_0^{-1/2-m}+ Ch^{2-m}|\log h|^{1-m}|\log d_0|,
2(2n)!\left(\frac{2}{\pi}\right)^{2n+1}\left(1+\frac{1}{3^{2n+1}}\right)\left(1-\frac{1}{5^{2n+1}}\right)\left(1+\frac{1}{7^{2n+1}}\right)\ldots= E_n^* = E_{2n},
\prod_{n=1}^{\infty}(1-x^{n})=\sum_{k=-\infty}^{\infty}(-1)^{k}x^{k(3k-1)/2}=1+\sum_{k=1}^{\infty}(-1)^{k}(x^{k(3k+1)/2}+x^{k(3k-1)/2}).
\begin{aligned}|\mathcal{R}_{2,\epsilon}(t)| \leq\vartheta\lambda^{2}(\epsilon)\int^{t}_{0}1+|\varphi(s)|+|\varphi'_m(s)|ds &+\lambda(\epsilon)\epsilon^{2}\Big|\int^{t}_{0}\big(\sigma(\varphi_{m}(s))-\sigma(\hat{X}^{\epsilon}(s-))\big)\theta_{\epsilon}(s-)\tilde{N}^{\epsilon^{-2}}(ds)\Big|,\end{aligned}
I_{pk}\left( M_{p}\right) =\int_{M_{p}}^{\infty}\, d\omega \, \omega \left( \omega +M_{p}\right)^{-1/2}\left( \omega -M_{p}\right)^{-1/2}\left( \omega +\mu \right)^{-k}\left( \omega -\mu \right)^{k}e^{-2k\omega}.
ic^{++-}(\xi,0,-\xi) = 2m_{++}(\xi,0) q_{+-}(0,-\xi)+ m_{+-}(\xi,-\xi) q_{++}(2\xi, \xi) - m_{+-}(\xi,0) q_{+-}(0, \xi) & - 2m_{--}(\xi,-\xi) q_{--}(2\xi, \xi)+ m_{++-}(\xi,0,-\xi) & = m_{+-}(\xi,-\xi) q_{++}(2\xi, \xi) - 2m_{--}(\xi,-\xi) q_{--}(2\xi, \xi) + i d_1|\xi|^{3/2}.
c^{-(i_1+1)}=c^{-2}\cdot c^{-(i_1-1)}\geq c^{-2}\cdot\frac{a}{\sqrt{p+1}}> c^{-2}\cdot\frac{c^{-i_0}}{\sqrt{p+1}}=\frac{c^{-3}}{\sqrt{p+1}}\cdot c^{-i_0+1}\geq \delta c^{-i_0+1}>c^{-i_2}.
h \phi_{n}( \varphi ) \; = \; \lambda_{n}^{2}\phi_{n}( \varphi ) \; \Rightarrow \;{\tilde{h}}\; [ \frac{1}{\lambda_{n}}a \phi_{n}( \varphi ) ] \; = \; \lambda_{n}^{2}\; [ \frac{1}{\lambda_{n}}a \phi_{n}( \varphi ) ]
\omega(z)=\frac{i}{2\pi}\partial \overline{\partial}\log\bigl(\frac 1{1-|z|^2}\bigr)=\frac{1}{(1-|z|^2)^2}\frac{i}{2\pi}\sum^{n}_{j,k=1}[(1-|z|^2)\delta_{j,k}+z_k\overline{z_j}]dz_j\wedge d\overline{z_k}\ .
\lambda_{(n+1,k)}= \,(1-q^{k+n}r)(k+n+1)_q\lambda_{(n,k)}, \beta_{(i,m,k)}= \,(q^{m+2k-i}r-r^{-1})\beta_{(i-1,m,k)}, \beta_{(i,m+1,k)}= \,(q^{m+2k}r-r^{-1})\beta_{(i-1,m,k)}.
\partial_k g_{i\bar{j}}=\Gamma^{(\alpha)}_{ki,\bar{j}}+\Gamma^{(-\alpha)}_{k\bar{j},i}\partial_{\bar{k}}g_{i\bar{j}}=\Gamma^{(\alpha)}_{\bar{k}i,\bar{j}}+\Gamma^{(-\alpha)}_{\bar{k}\bar{j},i}.
2^{jn\alpha_1(1/p_1-1/p_2)}=2^{jA_1(\mathbf{p},q,\alpha_1,\alpha_2)}=2^{jA_2(\mathbf{p},q,\alpha_1,\alpha_2)}=2^{jA_3(\mathbf{p},q,\alpha_1,\alpha_2)}.
\mathfrak{P}_{1}= \mathfrak{B}_{1}; \mathfrak{P}_{2}= \mathfrak{B}_{2}; \mathfrak{P}_{3}= \mathfrak{B}_{3}; \mathfrak{P}_{4}= \mathfrak{P}_{5}= (x^{2}, xy, y^{2}, xz, yz, z^{2}).
(i)\ \ \phi^2+\mathcal{B}\omega=-id,\,\,\,\,\,\,\, (ii)\ \ \mathcal{C}^2+\omega\mathcal{B}=-id, &(iii)\ \ \omega\phi+\mathcal{C}\omega=0,\,\,\,\,\,\,\, (iv)\ \ \mathcal{B}\mathcal{C}+\phi\mathcal{B}=0,
\begin{array}{rcl}\left\{\overline{\Psi}_{1}(R_0) > e^x\right\}= \left\{\overline{\Psi}_{1}(R_0)>e^x,\: M_1 > x-\log(2)\right\}= \left\{M_1> x-\log(2) \right\}\setminus \left\{\overline{\Psi}_{1}(R_0)\leq e^x, M_1 > x-\log(2)\right\}. \end{array}
(a,b,c) = \left(\frac{l^2-n^2}{2ln},\frac{m^2-n^2}{2mn},\frac{lm-n^2}{(l+m)n}\right),\ \left(\frac{l^2-n^2}{2ln},\frac{m^2-n^2}{2mn},-\frac{(l+m)n}{lm-n^2}\right)
\mathbb{P}(Z' = 0) = \sum_{s =0}^\ell \mathbb{P}(Z' = 0, Z=s) = \sum_{s =0}^\ell \mathbb{P}(Z' = 0\ \big| \Z=s) \mathbb{P}(Z=s) \le \sum_{s =0}^\ell (1 - \beta_k)^s \mathbb{P}(Z=s) = \sum_{s < c\ell}(1 - \beta_k)^s \mathbb{P}(Z=s) + \sum_{s \ge c\ell}(1 - \beta_k)^s \mathbb{P}(Z=s) &\le \mathbb{P}(Z < c\ell) + (1 - \beta_k)^{c\ell}\mathbb{P}(Z \ge c\ell).
\begin{aligned}p=0 : w = (h(j,j+1) \; a_{n+1})^k \; w_r , p>0 :w = h(i_1, r_1) \; a_{n+1}\dots h(i_p, r_p) \; a_{n+1}(h(j,j+1) \; a_{n+1})^k \; w_r , \end{aligned}
H =\prod^{2d-1}_{j=0}(\xi^{j}x+\xi^{-j}y+(-1)^{j}z) = \prod^{d-1}_{j=0}(\xi^{j}x+\xi^{-j}y+(-1)^{j}z)(\xi^{j+d}x+\xi^{-j+d}y+(-1)^{j+d}z) = \prod^{d-1}_{j=0}(\xi^{j}x+\xi^{-j}y+(-1)^{j}z)(\xi^{j}x+\xi^{-j}y+(-1)^{j+d}z) = \prod^{d-1}_{j=0}(\xi^{j}x+\xi^{-j}y-z)(\xi^{j}x+\xi^{-j}y+z)=F.
\Phi ( r ) = m ( r ) \left( \begin{array}{c c c}{{1}}&{{0}}&{{0}}\\{{0}}&{{- \frac{1}{2}}}&{{0}}\\{{0}}&{{0}}&{{- \frac{1}{2}}}\\ \end{array}\right) + q ( r ) \left( \begin{array}{c c c}{{0}}&{{0}}&{{0}}\\{{0}}&{{\frac{3}{2}}}&{{0}}\\{{0}}&{{0}}&{{- \frac{3}{2}}}\\ \end{array}\right)
\begin{cases}g <_{\sf base}g' g, g' \in G g<_{G}g' h <_{\sf base}h' h, h' \in H h<_{H}h' h <_{\sf base}g h \in H-A, g \in G-A a(h) \leq_{A}a(g) g <_{\sf base}h h \in H-A, g \in G-A a(g) <_{A}a(h) \end{cases}
{I_R}= \int\limits_{\left[{1, - k}\right]\left({\sqrt{\textbf{R}}{\textbf{t}}+{\textbf{p}}}\right) \ge 0}{\frac{{\exp \left({- \frac{{{{\bf{t}}^T}{\textbf{t}}}}{2}}\right)\det \left({\sqrt{\textbf{R}}}\right)}}{{2\pi \sqrt{\det \left({\textbf{R}}\right)}}}d}{\textbf{t}}
\sup_{1\leq l\leq \beta}\left|\frac{\partial^lk\cdot\tilde\omega}{\partial m^l}(m)\right| =\sup_{1\leq l\leq \beta}|k\cdot{\bf M}_l(m)| &\geq \xi=\tilde c |k|_2c_\star^{\beta(\beta-1)},
\mathrm{d}\varepsilon = \frac{1}{60\pi^2}|J| \, \left( \mathrm{d}^{(1/30,1/15)}+ \mathrm{d}^{(1/30,4/15)}+ \mathrm{d}^{(1/15,11/30)}+ \mathrm{d}^{(2/15,7/30)}\right) + \frac{1}{8\pi^2}|J| \, \mathrm{d}_6 \times \mathrm{d}_6,
\left(\partial_{s}-\partial_{yy}^{2}+\frac{1}{y}\left(2\left(n-1\right)-\frac{y^{2}}{2}\right)\partial_{y}-\frac{1}{y^{2}}\left(2\left(n-1\right)+\frac{y^{2}}{2}\right)\right)v\left(y,\,s\right)=\frac{h\left(y,\,s\right)}{y^{2}}
\| u \|^2_{H^k_{\rho_g}(\Omega, g)}= \sum_{j=0}^k \| \nabla^j_g u \|^2_{L^2_{\rho_g}(\Omega, g)}= \sum_{j=0}^k \int_{\Omega}| \nabla_g^{j}u |^2 \rho_g \, d\mu_g.
\langle\langle v, v\cdot g \rangle\rangle =\langle (\bar f^g f^g)\cdot (v\cdot g), v\cdot g \rangle[g]=\langle f^g\cdot (v\cdot g), f^g\cdot (v\cdot g)\rangle[g]= &=\langle v\cdot g, v \cdot g\rangle[g]=\langle v, v \rangle[g]=[g].
\rightarrow[\begin{matrix}\frac{2}{\sqrt{6}}&-\frac{1}{\sqrt{6}}&-\frac{1}{\sqrt{6}}\\ 0&\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{3}}&\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{3}}\end{matrix}]
\frac{n2^{Q(n,n)}}{\sum_{i=1}^{n}\sum_{p = 0}^{\lfloor (n-i)/2 \rfloor}\binom{n}{i}\binom{n-i}{2}^p (2^i - 1)^n 2^{Q(n-i+1,n-i+1-2p)}}= \frac{1}{(\sum_{i=1}^{n}\sum_{p = 0}^{\lfloor (n-i)/2 \rfloor}\binom{n}{i}\binom{n-i}{2}^p (2^i - 1)^n 2^{Q(n-i+1,n-i+1-2p)-Q(n,n)})/n}
T^{\mu\nu}=\frac{\hbar^{2}}{m}(g^{\mu\alpha}g^{\nu\beta}+g^{\mu\beta}g^{\nu\alpha}-g^{\mu\nu}g^{\alpha\beta})\partial_{\alpha}\overline{\phi}\partial_{\beta}\phi-g^{\mu\nu}mc^{2}\overline{\phi}\phi,
I_{A B}( \hat{{\cal X}}_{1 2}) = \textstyle{\frac{1}{4}}\mathrm{t r}( \gamma_{A}\hat{{\cal X}}_{1 2}\gamma_{B}\hat{{\cal X}}_{2 1}) = \textstyle{\frac{1}{4}}\mathrm{t r}( \tilde{\gamma}_{A}\hat{{\cal X}}_{2 1}^{- 1}\tilde{\gamma}_{B}\hat{{\cal X}}_{1 2}^{- 1})
\|B_{N}f\|^{2}_{L^{2}(G/\Gamma)}= \int_{-\infty}^{\infty}\left\lvert \frac{1}{N}\sum_{n=0}^{N-1}e^{2\pi i p(n) t}-\frac{1}{N}\int_{0}^{N}e^{2\pi i p(n) t}dn \right\rvert^{2}\lvert \hat{f}\rvert^{2}\lvert t \rvert^{1-2s}dt.
\overrightarrow{\xi}= \frac{1}{\textsf{P}_{0}}\biggl ( ( c h ( l m ) P_{0}+ s h ( l m ) m ) \overrightarrow{\bigtriangledown}+ m^{- 1}s h ( l m ) \overrightarrow{P}( \overrightarrow{P}. \overrightarrow{\bigtriangledown}) \biggr )
\mathcal{L}_{1}^{\mu}[\beta b]\nabla_X\psi = - \frac{1}{\beta}\sinh{(\beta b(X) \sqrt{\mu}|\mathrm{D}|)}\mathrm{sech}(\sqrt{\mu}|\mathrm{D}|)\dfrac{1}{\sqrt{\mu}|\mathrm{D}|}\nabla_X\psi,
X_{i,\ell}=\frac{\displaystyle \sum_{k=1}^{d_i}\beta_k^{\ell}(Z_k^++Z_k^-)}{2d_i}, Y_{i,\ell}=\frac{\displaystyle \sum_{k=1}^{d_i}\beta_k^{\ell}(Z_k^+-Z_k^-)}{2\sqrt{a}d_i}0 \leqslant \ell \leqslant d_i-1.
\gamma_{o}(T') =\gamma_{o}(G)- \gamma_{o}(G^{\prime\prime}) & =\frac{n-l(G)+s(G)}{3}-\frac{n^{\prime\prime}-l^{\prime\prime}+s^{\prime\prime}}{3}& =\frac{(n-n^{\prime\prime})-(l(G)-l^{\prime\prime})+(s(G)-s^{\prime\prime})}{3}=\frac{n'-(l'-1)+s'}{3}=\frac{n'-l'+s'+1}{3}.
\mathfrak{p}_{4,1}:= \left\{(i,j,\pi,1) \in \mathfrak{p}_4 \; ; \; \pi(1)=1 \right\}\mathfrak{p}_{4,2}:= \left\{(i,j,\pi,1) \in \mathfrak{p}_4 \; ; \; \pi(1)\neq 1 \right\}.
\,\, \Phi^{(R_1,R_2),\beta,D_1\cup_{p}D_2}_{p_1,\ldots,p_n}(\{\tau_i\}_{i=1}^{|Q|},Q)(-) =\sum_{\beta_1+\beta_2=\beta}\left(\Phi^{R_1,\beta_1,D_1}_{p_1,\ldots,p_s,p}(\{\tau_i\}_{x_i\in Q\cap D_1},Q\cap D_1)\otimes \Phi^{R_2,\beta_2,D_2}_{p_{s+1},\ldots,p_n,p}(\{\tau_i\}_{x_i\in Q\cap D_2},Q\cap D_2) \right)(-\boxtimes\eta),
F_{\beta}(t) = \exp\left(-\beta\frac{|t|^3}{24}+ \frac{\sqrt{2}}{3}(\beta/2 - 1 )|t|^{3/2}\right. \left.\hphantom{F_{\beta}(t) =}{}+ \frac{1}{8}(\beta/2 + 2/\beta - 3 )\log|t|+ c_0 + O\left(\frac{1}{|t|^{3/2}}\right)\right), t \to -\infty.
\bigwedge_{i=1}^k\bigwedge_{j=1}^{n-k-1}\neg a_{i,j}\lor a_{i,j+1}\qquad\bigwedge_{i=0}^k\bigwedge_{j=1}^{n-k}\neg a_{i,j}\lor a_{i+1,j}\lor\neg b_{i+j}\bigwedge_{i=1}^{k-1}\bigwedge_{j=1}^{n-k}a_{i,j}\lor\neg a_{i+1,j}\qquad\bigwedge_{i=1}^k\bigwedge_{j=0}^{n-k}a_{i,j}\lor\neg a_{i,j+1}\lor b_{i+j}
{\frac{m^{2}( S )}{m^{2}( V )}}={\frac{1}{6 ( 4 \pi )^{2}}}\left[ 3 6 e^{2}+{\frac{\lambda^{2}\xi^{2}}{e^{2}}}\left( 1 5 +{\frac{3}{4 \omega^{2}}}-{\frac{9 \xi}{\omega^{2}}}+{\frac{2 7 \xi^{2}}{\omega^{2}}}\right) \right] .
A^{\Omega}_t(x,y) :=\inf_{\xi\in\Gamma^{0,t}_{x,y}(\overline{\Omega})}\int^t_0L(\xi(s),\dot{\xi}(s))\ds, \Phi^{\Omega}_L(x,y) :=\inf_{t>0}A_t^{\Omega}(x,y),c_{\Omega}(L):=-\inf_{t>0,x\in\overline{\Omega}}\frac 1tA^{\Omega}_t(x,x).
P_{W{\underline X}^n{\underline S}^{\ell}Y^n}( w,{\underline x}^n,{\underline s}^{\ell}, y^n) =P_W(w) &\times \prod_{i=1}^{\ell}\left( P_{\underline S}({\underline s}_i ) \prod_{j=1}^L P_{Y|{\underline X \underline S}}( y_{i,j}|{\underline x}_{i,j}{\underline s}_i ) P({\underline x}_{i,j}| w,{y}_{i}^{j-1}) \right)
|j *_{\{0, p \}}A| \ge |A + ((j - 1) *_{\{0, p \}}A)| + |J| \ge |A| + |(j - 1) *_{\{0, p \}}A| + |J| - |H| \ge |A| + |(j - 1) *_{\{0, p \}}A| \ge |A| + (j-1) |A| = j |A|,
\| P_j f\|_p = \| P_j D^{-2}\partial J^{1-s}\cdot J^{s-1}\partial f \|_p & \lesssim_{d}2^{-j}\| P_{<10}J^{1-s}\cdot J^{s-1}\partial f \|_p & \lesssim_{s,d}2^{-j}\| J^{s-1}\partial f \|_p.
i \int_{\mathbb R^n}u_0(x) dx + \int_0^T \int_{\mathbb R^n}|u(t,x)|^{(n+1)/n}\phi_r(t,x) dx dt &= \int_0^T \int_{\mathbb R^n}u(t,x) \bigg( - i \partial_t \phi_r(t,x) + (-\Delta)^{1/2}\phi_r(t,x) \bigg) dx \thinspace dt.
g_{x_k}(\tilde{u},\underline{\tilde{r}}^\prime,\tilde{r}_k):=|\tilde{u}-u|+\sum_{x^\prime\in S}|r^\prime(x^\prime)-\tilde{r}^\prime(x^\prime)|+\mathbb{I}_{x_k}(\tilde{u},\underline{\tilde{r}}^\prime,\tilde{r}_k)
2^d \prod_{i=1}^d\frac{\ell_i}{r/\alpha_1(T_{x\wedge y})}\leq (2c_2)^d \alpha_1(T_{x\wedge y})^d r^{-d}\prod_{i=1}^d\min\left\{1,\; \frac{r}{\alpha_i(T_{x\wedge y})}\right\}& = (2c_2)^d \alpha_1(T_{x\wedge y})^d r^{-d}Z_{x\wedge y}(r) \mbox{(by \eqref{ef-4.2})}
{\cal V}^{( 0 )}( \Phi ) ={\frac{1}{2}}I_{0}( \mu^{2}) -{\frac{1}{2}}I_{0}({\cal M}^{2}) +{\frac{{\cal M}^{2}}{4}}I_{1}({\cal M}^{2}) +{\frac{m^{2}}{\beta^{2}}}\; .
{\cal S}^{D 0}= \frac{1}{2 g_{S}}\int d t \mathrm{T r}\left( G_{i j}\dot{X}^{i}\dot{X}^{j}+ \frac{1}{2}\frac{1}{( 2 \pi )^{2}}G_{i j}G_{k l}[ X^{i}, X^{k}] [ X^{j}, X^{l}] + \right.
\frac{d^{2}\xi}{d r^{2}}+ \left( \frac{\omega^{2}}{\lambda^{4}}- \frac{\frac{l ( l + 1 )}{r^{2}}+ \frac{2 M}{r^{3}}- \frac{2 Q^{2}}{r^{4}}+ m^{2}}{\lambda^{2}}- \frac{1}{\lambda}\frac{d^{2}\lambda}{d r^{2}}\right) \xi = 0 .
\sum_{k=0}^{M}(-1)^k[4k-1]\frac{(q^{-1};q^2)_k^4(q^{-3};q^2)_k}{(q^2;q^2)_k^4(q^{4};q^2)_k}q^{k^2+7k}\equiv\frac{\Omega_q(n)}{[2]^2[4][6]}\sum_{k=0}^{(n-3)/2}\frac{(q^3;q^2)_k^3}{(q^2,q^6,q^8;q^2)_k}q^{2k}.
C_{n,i}= \sum \limits_{\substack{\lambda_1 \vdash n \alpha_1, \xi_1}}\sum\limits_{(\Lambda^{\prime}, A^{\prime}, \Xi^{\prime})}\left (\prod_{j=1}^q m(\lambda_j) \sum \limits_{(d^j_k)}\prod \limits_{k=1}^{\ell(\lambda_j)}s(b_k^j , d_k^j) \right ).
|\nabla \psi -\nabla \hat{\psi}| =O(|\hat{\psi}| \, |\nabla \hat{\psi}|) + O(|\hat{\psi}|^2)= O(|\psi| \, |\nabla \psi|) + O(|\psi|^2), |\nabla^2 \psi -\nabla^2 \hat{\psi}| =O(|\hat{\psi}| |\nabla^2 \hat{\psi}|) + O(|\nabla \hat{\psi}|^2) + O(|\hat{\psi}|^2)= O(|\psi| |\nabla^2 \psi|) + O(|\nabla \psi|^2) +O(|\psi|^2)
C_E = O + \frac{-e^{2\alpha_2(a)}}{2}\bar{z}_2z_1[\theta(e_{\alpha_2}),e_{\alpha_1}] & - e^{2(\alpha_1-\alpha_2)(a)}\theta(\frac{-e^{2\alpha_1(a)}}{2}\bar{z}_1z_2[\theta(e_{\alpha_1}),e_{\alpha_2}]).
\begin{cases}y_-\mkern-2mu= -\lceil y_{1+}\rceil + 1 1-[-y_{1+}] \ge \frac{q^*}{p}y_-\mkern-2mu\notin{\mathbb{Z}}1-[-y_{1+}] < \frac{q^*}{p},\end{cases}\;\;\;\;\;\;\;\begin{cases}y_+\mkern-2mu= -\lfloor y_{1-}\rfloor [y_{1-}] \le \frac{q^*}{p}y_+\mkern-2mu\notin{\mathbb{Z}}[y_{1-}] > \frac{q^*}{p}.\end{cases}
\Sigma={\displaystyle\sum_{k=0}^{\frac{l-m-1}{2}}}{\displaystyle\sum_{i=0}^{l_{1}-m\,_{1}}}{\displaystyle\sum_{j=0}^{l_{2}-m_{2}}}(-1)^{i+j+k}\binom{l-m}{k}\binom{l_{1}-m_{1}}{i}\binom{l_{2}-m_{2}}{j}\cdot\{...\}
qL_{X\wedge Y}i_Z\omega =(-1)^{l(m-1)}(-1)^{kl}\left(\nabla_X\omega(Z\wedge Y)+\omega(\nabla_X(Z\wedge Y))\right) &+(-1)^{k(m-1)}\left(\nabla_Y\omega(Z\wedge X)+\omega(\nabla_Y(Z\wedge X))\right) &-q(-1)^l\omega(Z\wedge [X,Y]).
\Omega_{H}= - \left.{\frac{1}{{\omega^{0}}_{\phi}}}\right|_{H}={\frac{a^{2}}{2 J}}\left[ r_{+}- M +{\frac{\left( M^{2}+ \Sigma^{2}- P^{2}- Q^{2}\right) \left( M + \Sigma / \sqrt{3}\right)}{\left( M + \Sigma / \sqrt{3}\right)^{2}- Q^{2}}}\right]^{- 1}.
E_1 = f\big(u^a(x,t)\big) - \Bigg[\sum_{i = 1}^n f\big(u^i(x-x_i,t)\big) - \sum_{i=2}^n f(u_i)\Bigg] \ \{\rm and}\ \E_2 = a^2 u^p_{xt}(x-x_i,t).
C[\Delta,\omega]={1\over 16\pi^2}\int_{S^4}\left\{\alpha_0\omega+\alpha_1\omega_{zzzz}+\alpha_2\omega_{zzz}\omega_{z}+\alpha_3\omega_{zz}^2+\alpha_4\omega_{z}^2\omega_{zz}+\alpha_5\omega_{z}^4+\alpha_6\omega_{z}^2+\alpha_7\omega_{zz}\right\}.
|p(t)| =\left| \int_{-1}^t \dot{p}(\tau) \; d\tau \right| =\left| \sum_{i=1}^N \dot{p}(\tau_i) \int_{-1}^t l_i(\tau) \; d\tau \right|\le \sum_{i=1}^N \left| \int_{-1}^t l_i (\tau) \; d\tau \right| .
u(s)-u(t) =[e^{-(t-s)A(s)}u(t)-u(t)]+\int_{s}^{t}e^{-(r-s)A(s)}(\mathcal{A}(s)- A(r))u(r) dr &+\int_{s}^{t}e^{-(r-s)A(s)}((B(r)+I) A(r)u(r)) dr &-\int_{s}^{t}e^{-(r-s)A(s)}[f(r)-P(r)u(r)] dr &=: I_{1}(s,t)+I_{2}(s,t)+I_{3}(s,t)+I_{4}(s,t).
\uppercase\expandafter{\romannumeral1}\leq \exp({-\frac{1}{8}\lambda\delta kN_1})\exp\left({-l\lambda(\int_{\Omega}\log X dP + \frac{\delta}{2})}\right) = \exp({-\frac{1}{8}\lambda\delta n}) \exp\left(-l\lambda(\int_{\Omega}\log X dP + \frac{3}{8}\delta)\right).
U_{\sigma}\chi_{\Lambda_{L/2}(\sigma)}\{h(g(H)_{\Lambda_L})-f_0 \}U_{\sigma}= \chi_{[-L,0]^d}U_{\sigma}\{h(g(H)_{\Lambda_L})-f_0 \}U_{\sigma}&= \chi_{[-L,0]^d}\{h(g(H^{R_{\sigma}})_{\Lambda_L})-f^{R_{\sigma}}_0 \}.
f ( x^{0}) = \frac{1}{1 + ( \lambda_{1}+ \lambda_{2}) \frac{\sin ( \pi \sqrt{\lambda_{1}^{2}- \lambda_{2}^{2}})}{\sqrt{\lambda_{1}^{2}- \lambda_{2}^{2}}}e^{x^{0}}}+ \frac{1}{1 + ( \lambda_{1}- \lambda_{2}) \frac{\sin ( \pi \sqrt{\lambda_{1}^{2}- \lambda_{2}^{2}})}{\sqrt{\lambda_{1}^{2}- \lambda_{2}^{2}}}e^{- x^{0}}}- 1 .
\frac{\partial}{\partial s_1}F(s_1,s_2) = \frac 12 f'(s_1) (s_2-s_1) + \frac 12 \big( f(s_1) - f (s_2)\big) \frac{\partial}{\partial s_2}F(s_1,s_2) = \frac 12 f'(s_2) (s_2-s_1) + \frac 12 \big( f(s_1) - f (s_2)\big).
\hat{y}_{i}\in[b_{0}+b_{1}x_{i}\pm t_{\alpha/2,df=n-2}\sqrt{(\frac{1}{n-2}\sum_{j=1}^{n}e_{j}^{2})\cdot(\frac{1}{n}+\frac{(x_{i}-\overline{x})^{2}}{\sum_{j=1}^{n}(x_{j}-\overline{x})^{2}})}],
\mathcal{B}\mathcal{T}_{U}V+\phi\hat{\nabla}_{U}V =\hat{\nabla}_{U}\phi V+\mathcal{T}_{U}\omega V, g_1(U,V)\xi+\mathcal{C}\mathcal{T}_{U}V+\omega\hat{\nabla}_{U}V =\mathcal{T}_{U}\phi V+\mathcal{H}\nabla^{^{M_1}}_{U}\omega V,
\zeta_{{\mbox{Robin}},u=1}'(0)=-{41\over 864}-{7\over 45}\ln(2)-{1\over 2}\ln(\pi)-{31\over 90}\ln(a)+{1\over 3}\zeta_{R}'(-3)+{1\over 2}\zeta_{R}'(-2)-{11\over 6}\zeta_{R}'(-1) .
u(x,t)e^{2i(c_-(x)+c)}=\lim_{k\rightarrow 0}(k^{-1}\psi^+(x,t;k))_{12}, &u(x,t)e^{-i(2c_-(x)+c)}=\lim_{k\rightarrow 0}(k^{-1}\psi^-(x,t;k))_{12}.
\int_{D}|\nabla u_+|^2\le \liminf_{p\to\infty}\int_{D}|\nabla v_p^+|^2 = J_\infty\left(\liminf_{p\to\infty}\int_{D}|\nabla v_p^+|^2\ ;\,\liminf_{p\to\infty}\int_{D}|\nabla v_p^-|^2\right) &= J_\infty\left(\int_{D}|\nabla u_+|^2\ ;\,\int_{D}|\nabla u_-|^2\right)=\int_{D}|\nabla u_+|^2,