harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathcal{D}= \frac{1}{2}\sum_{(m, p) = 1}\frac{A(p, m)}{m p^{\frac{1}{2}}}H_{m, p}+ \frac{1}{2}\sum_{(m, p) > 1}\frac{A(p, m)}{m p^{\frac{1}{2}}}H_{m, p}= \frac{1}{2}\sum_{(m, p) = 1}\frac{A(p, m)}{m p^{\frac{1}{2}}}H_{m, p}+ \frac{1}{2}\sum_{m \geq 1}\frac{A(p, m p)}{(m p) p^{\frac{1}{2}}}H_{m p, p}.
\chi(u) = \tilde{\varepsilon}_p(u)\chi(g) + \sum_{i = 1}^k \varepsilon_{q_i}(u)\chi(q_i) \psi(u) = \tilde{\varepsilon}_p(u)\psi(g) + \sum_{i = 1}^k \varepsilon_{q_i}(u)(\chi(q_i) + 1).
\frac{\partial^2}{\partial x_2^2}V(x_2)\;=\; \phi_0(x_2)\phi^{\dagger}_0(x_2) \int \frac{d^2k}{2\pi}\frac{-i\;k_0+ie\;V_{0,0}}{(-i\;k_0+ie \gamma_0 V_{0,0})^2}\;.
\pi ( d \alpha ) = \left( \begin{matrix}{{- g^{\mu \nu}\partial_{\mu}a_{i 1}\partial_{\nu}b_{i 1}}}&{{- 2 \gamma^{\mu \nu}\gamma^{5}V_{\nu}a_{i 1}\partial_{\mu}b_{i 2}}}\\{{2 \gamma^{\mu \nu}\gamma^{5}V_{\nu}a_{i 2}\partial_{\mu}b_{i 1}}}&{{- g^{\mu \nu}\partial_{\mu}a_{i 2}\partial_{\nu}b_{i 2}}}\\ \end{matrix}\right)
\tau =\frac{1}{L}\Bigl(h-\frac{\mu}{\lVert \vec r\rVert}\vec u\cdot\vec r\Bigr), \vec \Xi= (\vec r\times \vec v)\times\vec u +\frac{1}{L}\Bigl(h-\frac{\mu}{\lVert \vec r\rVert}\vec u\cdot\vec r\Bigr)\vec v, f=h.
p(h_{l}^{(2t)}=1|h^{(1t)},h^{(3)})=g(\sum_{j=1}^{F_{1}^{t}}W_{jl}^{(2t)}h_{j}^{(1t)}+\sum_{p=1}^{F_{3}}W_{lp}^{(3t)}h_{p}^{(3)}+b_{l}^{(2t)})
\sum_{i=0}^{\infty}\binom{m}{i}(a_{l+i}b)_{m+n-i}& = \sum_{i=0}^{\infty}\binom{l}{i}(-1)^i\big(a_{l+m-i}b_{n+i}+(-1)^{l+1}b_{l+n-i}a_{m+i}\big).
S^x_{k}=S^x_{k-1}+R_k\cos\Phi_k \sin\Theta_k, S^y_{k}=S^y_{k-1}+R_k(-S^z_{k-1}\sin\Phi_k\sin\Theta_k-S^y_{k-1}\cos\Theta_k), S^z_{k}=S^z_{k-1}+R_k(S^y_{k-1}\sin\Phi_k\sin\Theta_k-S^z_{k-1}\cos\Theta_k).
\textbf{g}\left(\Phi_1(q), \Phi_2(q)\right) = g\left(\Phi_1(q^{-1}), \Phi_2(q)\right) \textbf{G}_\tau\left(\Phi_1(q), \Phi_2(q)\right) = G_\tau \left(\Phi_1(q^{-1}), \Phi_2(q)\right)
i \, \lim_{r \rightarrow 0}\left( M r \right)^{\nu + 1}g_{n}( r ) \sin \left( \frac{\pi}{4}+ \frac{\Theta}{2}\right) = \lim_{r \rightarrow 0}\left( M r \right)^{- \nu}f_{n}( r ) \cos \left( \frac{\pi}{4}+ \frac{\Theta}{2}\right)
\Sigma_{H}=\{\lambda \in \Sigma \mid \langle \lambda, H \rangle \in \pi\mathbb{Z}\}, W_{H}=\{\alpha \in W \mid \langle \alpha , H \rangle \in (\pi/2) +\pi \mathbb{Z}\}, \tilde{\Sigma}_{H}=\Sigma_{H}\cup W_{H}, \Sigma_{H}^{+}= \Sigma^{+}\cap \Sigma_{H}, W_{H}^{+}=W^{+}\cap W_{H}, \tilde{\Sigma}_{H}^{+}=\Sigma_{H}^{+}\cup W_{H}^{+}.
\mathcal{D}_{G}( a ) = N_{G}( a ) \exp \left( \int_{0}^{\infty}\frac{d t}{t}\left[ \frac{a^{2}}{2}e^{- 2 t}- \sum_{\alpha > 0}\Psi_{\alpha}( t ) \mathcal{F}_{\alpha}( a , t ) \right] \right)
\{\varepsilon^{\mu \nu}\}\rightarrow \{\varepsilon'^{\mu \nu}\}= \{\varepsilon^{\mu \nu}\}+ \omega \left( \begin{array}{cccc}0 f^1 f^2 f^0 - f^3 -f^1 0 0 -f^1 -f^2 0 0 - f^2 f^3 - f^0 f^1 f^2 0\end{array}\right)
\lambda^{\alpha}( x_{+}, \theta ) = \xi^{\alpha}+ \textstyle{\frac{1}{2}}( \lambda + i \tau ) \theta^{\alpha}+ \textstyle{\frac{1}{4}}w^{\mu \nu}( \theta \sigma_{\mu}\tilde{\sigma}_{\nu})^{\alpha}+ 2 \theta^{2}\zeta^{\alpha}-
u(x,t) = \frac{1}{2 \pi t^2}\int_{B(x,t)}\frac{t u_0(y) + t^2 u_1 (y)+ t \nabla u_0 (y) \cdot (y-x)}{[t^2 -|y-x|^2]^{1/2}}dy + \frac{1}{2 \pi}\int_0^t \int_{B(x,t-s)}\frac{F(y,s)}{\left[(t-s)^2 -|y-x|^2\right]^{1/2}}dy ds.
(a)\ -\tau^{-1}\langle c, A\psi - \phi\rangle - \tau^{-1}\sum_{i = 1}^m\langle c, B_i\psi\rangle [C\chi]_i + \sum_{j = 1}^Ja_j(x)\sigma_j(x) + \sum_{\ell = 1}^L b_\ell(x)\rho_\ell(x)\ \mathrm{is\SOS}, (b)\ \sigma_j(x)\ \mathrm{is\SOS},
A_{[n,j]}=\sum_{\gamma\in\mathcal{D}_{[n,j]}}w(\gamma)=\sum_{k=0}^{n-j}\sum_{\gamma\in\mathcal{D}_{[n,j,k]}}w(\gamma)=\sum_{k=0}^{n-j}A_{[k,0]}\,A_{[n-k-1,j-1]}^{(1)},n\geq j.
[[\begin{matrix}0&a&b\\ 0&0&c\\ 0&0&0\end{matrix}],[\begin{matrix}0&a^{\prime}&b^{\prime}\\ 0&0&c^{\prime}\\ 0&0&0\end{matrix}]]=[\begin{matrix}0&0&a^{\prime\prime}\\ 0&0&0\\ 0&0&0\end{matrix}]
\left\{\begin{array}{ll}-\Delta_pu(z)+|u(z)|^{p-2}u(z)=\hat{f}(z,u(z),Du(z))+\epsilon e(z) \mbox{in}\ \Omega, \frac{\partial u}{\partial n_p}+\beta(z)|u|^{p-2}u=0 \mbox{on}\ \partial\Omega.\end{array}\right\}
K = \frac{\sqrt{G^{2}M^{2}- G q^{2}}}{G^{2}M^{2}( 1 + \sqrt{1 - \frac{q^{2}}{G M^{2}}})^{2}}\approx \frac{1}{4 G M}\Bigl ( 1 - \frac{q^{2}}{2 G M^{2}}\Bigr ) \Bigl ( 1 + \frac{q^{2}}{2 G M^{2}}\Bigr ) \approx \frac{1}{4 G M}
e^{iyp_x}x^k=\sum_{l=0}^{\infty}\frac{(iy)^l}{l!}(-i)^l\frac{d^l}{dx^l}x^k=\sum_{l=0}^{k}(y)^l\left(\begin{array}{c}k l\end{array}\right) x^{k-l}=(x+y)^k,
\frac{|G|}{|\Phi|}\left(\sum_{i=1}^{k_1}\frac{1}{\lambda_{i}}+ \sum_{i=1}^{k_2}\frac{\nu_i}{\mu_i}+\sum_{i=1}^{k_3}\frac{1}{\omega_i}- 1 \right) = \sum_{i=1}^{k_3}\frac{|G|}{|\Phi|}\frac{1}{q_i}- 1 .
\begin{array}{ll}g((\nabla_U A)_{X_j}X_j, V) = g(\nabla_U g(X_j, X_j) H' - g(\mathcal{H}\nabla_U X_j, X_j) H' &- g(X_j, \mathcal{H}\nabla_U X_j) H', V).\end{array}
A_2(\beta)= \frac{1}{1-m}\left\{1-\frac{\beta}{2\rho_1}\left(\sqrt{A(\beta)^2-8(n-2-nm)(1-m)}+A(\beta)\right)\right\}\le \frac{1}{1-m}\left\{1-\frac{\beta A(\beta)}{2\rho_1}\right\}\le \frac{1}{1-m}\left\{1-\frac{\beta^2}{2\rho_1^2}(n-2-nm)\right\}< 0\qquad\qquad\qquad\qquad\qquad\qquad\forall \beta>\beta_2.
g = b^{2}\left[ A^{2}( d \psi + \cos ( \theta_{1}) d \phi_{1}+ \cos ( \theta_{2}) d \phi_{2})^{2}+ \sum_{i = 1}^{2}[ d \theta_{i}^{2}+ \sin^{2}( \theta_{i}) d \phi_{i}^{2}] \right]
y_q(u) =\sum_{\mathcal{R}:|\mathcal{R}|=r+1}\sum_{i=1}^\rho\left[\sum_{p\in [N_T]}h_{qp}(u)v_{{\mathcal{R}},{[N_T]},p}^i(u)\right]x_{{\mathcal{R}},{[N_T]}}^i
\nabla_x^k\Phi_\tau(x, y) = \frac{{e}^{-{\tau}l(x, y)}}{8\pi\gamma_+\gamma_-\sqrt{{\rm det}H(x, y)}}\Big(\frac{-\tau}{\sqrt{\gamma_-}}\Big)^k\Big(\sum_{j = 0}^{N}{\tau^{-j}}\Phi_{j}^{(k)}(x, y) + Q_{N, \tau}^{(k)}(x, y)\Big),
z(\delta) = z(0) + z'(0) \delta + \tfrac{1}{2}z''(c) \delta^2 \left( \gamma_\circ+\delta \right)^{1-q}=\gamma_\circ^{1-q}+(1-q) \gamma_\circ^{-q}\delta -\tfrac{1}{2}q(1-q)(\gamma_\circ+c)^{-q-1}\delta^2 ,
l_{2}'(\iota(s^{-1}f),\iota(s^{-1}g)) = l_{2}'(s^{-1}\widetilde{f},s^{-1}\widetilde{g}) &= (-1)^{|f|}s^{-1}[\widetilde{f},\widetilde{g}]_{\Omega}&= (-1)^{|f|}s^{-1}\left(0,\dots,0, [f,g]_{\Omega}; [f,g]_{\Omega}, \dots, [f,g]_{\Omega},0\right) &= \iota \left((-1)^{|f|}s^{-1}[f,g]_{\Omega}\right),
( \Delta \gamma )_{1 2}= - i \hbar ( \Gamma_{i}^{( 2 )}- \Gamma_{i}^{( 1 )}) d x^{i}+ 1 \otimes \frac{1}{2}\omega_{i j}\left( ( \Gamma_{s}^{( 2 )})_{k l}^{i}- ( \Gamma_{s}^{( 1 )})_{k l}^{i}\right) y^{j}y^{k}d x^{l}
e^{\frac{1}{2}imt}c'+e^{\frac{1}{2}imt}\frac{1}{2}imc =e^{\frac{1}{2}imt}e^{\frac{1}{2}im\left(\frac{\lambda-(n+3)}{1-\lambda}\right)t}\left(e^{\frac{1}{2}imt}c\right)' =e^{\frac{1}{2}imt}e^{\frac{1}{2}im\left(\frac{\lambda-(n+3)}{1-\lambda}\right)t}.
\frac{var_{P_{\alpha}}\left[ \chi_{P_{\alpha},a,eff}^{1}(\mathbf{V};\mathcal{G})\right]}{var_{P_{\alpha}}\left[ \psi_{P_{\alpha},a}(\mathbf{O};\mathcal{G})\right]}=1-\frac{var_{P_{\alpha}}\left[ \Delta_{P_{\alpha}}\left( \mathbf{O}\right) \right]}{var_{P_{\alpha}}\left[ \psi_{P_{\alpha},a}(\mathbf{O};\mathcal{G})\right]}.
Q_1={\rm Diag}\left(Q^{1}_1,\dots,Q^{r_1}_{1}\right)\in{\mathbb P}^{m_1}{\rm and}Q_2={\rm Diag}\left(Q_{2}^{r_{1}+1},\dots,Q_{2}^{r_{1}+r_{2}},Q^{r_1+r_2+1}_{2}\right)\in\pm{\mathbb P}^{m_2}\,.
\begin{aligned}Z\left(z_1(k_1)\left\{\prod_{i=2}^{p-1}z_{0}(k_i)\right\}z_{0}(k_p+r);\alpha\right) = \sum_{\begin{subarray}{c}\sum_{i=1}^{q-1}c^{'}_{i}r_i+r_{q}=r c^{'}_{i}r_i, r_q\ge0\end{subarray}}Z\left(\left\{\prod_{i=1}^{q-1}z_{c^{'}_{i-1}}(1+c^{'}_ir_i)\right\}z_{c^{'}_{q-1}}(2+r_q);\alpha\right)\end{aligned}
\phi ( r , \theta ) \stackrel{r \gg 1}{\longrightarrow}\mathrm{e}^{- i p r \operatorname{c o s}\theta}+ \mathrm{e}^{i \pi / 4}{\cal A}_{\mathrm{A B}}( |{\bf p}| , \theta ) \frac{\mathrm{e}^{i p r}}{\sqrt{r}}
|\varphi(S)-\varphi(\alpha_u S\alpha_u^*)|\leq 2\|S\|\varphi(\sum_{t \not\in F}p_t)^{1/2}+\|S\|\varphi(1-\sum_{t \in F}\alpha_{t^{-1}}(p_t))^{1/2}= 3 \| S \| \varphi( \sum_{t \not\in F}p_t )^{1/2}.
F(z,w) =\left(z_{1}+\mbox{O}(2),z_{2}+\mbox{O}(2),\dots,z_{N}+\mbox{O}(2),\mbox{O}(2),\dots,\mbox{O}(2)\right), & G(z,w) =\left( w_{1}+\mbox{O}(2),w_{2}+\mbox{O}(2),\dots,w_{N}+\mbox{O}(2),\mbox{O}(2),\dots,\mbox{O}(2)\right).
\zeta_{\mathfrak f_0}(0,\sigma|_{K_{\mathrm{CM}}}) =[K:K_{\mathrm{CM}}]\zeta_{\mathfrak f_0}(0,\sigma) (\sigma \in G), \sum_{\sigma \in \mathrm{Gal}(K_{\mathrm{CM}}/F)}\zeta_{\mathfrak f_0}(0,\sigma)\sigma =[K:K_{\mathrm{CM}}]\,\mathrm{Inf}^{-1}(\sum_{\sigma \in G}\zeta_{\mathfrak f_0}(0,\sigma)\sigma).
{\cal K}^{(4)}_{{\hat \imath}_1 \dots{\hat \imath_4}}={1 \over 2! \sqrt{{\rm det \Pi^{(A)}}}\sqrt{1+e^{2 \phi}k^{-2}(i_k C^{(1)})^2}}\Pi^{(A)}_{{\hat \imath}_1{\hat \jmath}_1}\dots \Pi^{(A)}_{{\hat \imath}_4{\hat \jmath}_4}\epsilon^{{\hat \jmath}_1\dots{\hat \jmath}_6}{\cal K}^{(2)}_{{\hat \jmath}_5{\hat \jmath}_6}\, ,
\Sigma_{d i v}^{0}={\frac{C e^{2}}{\beta}}\int{\frac{|{\bf k}| d |{\bf k}|}{( 2 \pi )}}{\frac{{\cal M}({\bf k}, p_{0 f})}{{\bf k}^{2}+ p_{0 f}^{2}+{\cal M}^{2}({\bf k}, p_{0 f})}}{\frac{1}{|{\bf k}^{2}-{\bf p}^{2}|}}
\frac 1{2\pi i}\int\limits_{c-i\infty}^{c+i\infty}ds\frac{(-s)}{\mu}\left(\frac{t}{\mu}\right)^{-s-1}\frac{2^{s-1}}{\sqrt{\pi}}\Gamma \left( \frac{s+1}2\right) \left[ \mu^{-s}\Gamma \left(\frac s2\right) \zeta \left( \frac s2, \frac{D_B}{\mu^2}\right)\right]\,,
(-1)^{j+r-1-l}\frac{1}{2}S(n,r)\frac{\binom{2(j+r-1-l)}{j+r-1-l}\binom{2(n-j+l+1)}{n-j+l+1}\binom{2n-j-r+l+1}{n}}{2\binom{2n-j+l+1}{n}}
\mu_0(\varepsilon,B \setminus B_J)\geq \dim S_3\sim \sum\limits_{j=1}^J 2^{n(l_j-j)}\sim J\sum\limits_{j=1}^J \frac{1}{j}\sim J\log J \sim \varepsilon^{-\frac{n}{m}}|\log \varepsilon|.
\begin{array}{l}{\mathbf{F}= ( q \cos \theta + g \sin \theta ) [ ( \mathbf{E}\cos \theta + \mathbf{B}\sin \theta ) + \mathbf{v}\times ( - \mathbf{E}\sin \theta + \mathbf{B}\cos \theta ) ]}\\{= q \mathbf{E}+ g \mathbf{B}+ \mathbf{v}\times ( q \mathbf{B}- g \mathbf{E}) .}\\ \end{array}
\frac{\partial N_{ia}}{\partial \lambda_{jb}}= s_aL\left\{\rho(\lambda_{ia})\delta_{ia,jb}+ \frac{1}{2}\int_{-\infty}^{\infty}d\lambda\, s_a \mathrm{sgn}(\lambda_{ia}-\lambda)\frac{d \rho}{d \lambda_{jb}}(\lambda) \right\}.
G_\ell(n) =4\zeta_2(n)-6\zeta_1(n)+4\zeta_0(n)-n(n+1)(n+2) &=4\zeta_2(n)-6\zeta_1(n)+4\zeta_0(n)-3\zeta_2(n)+3\zeta_1(n)-2\zeta_0(n) &=\zeta_2(n)-3\zeta_1(n)+2\zeta_0(n)\,.
d_{<}\left( \vec{x}^{(1)},\dots,\vec{x}^{(r)}\right) =\left\{x \in \vec{x}^{(i)}\Big| x+1 \in \vec{x}^{(j)}, \i<j \right\}, d_{>}\left( \vec{x}^{(1)},\dots,\vec{x}^{(r)}\right) =\left\{x \in \vec{x}^{(i)}\Big| x+1 \in \vec{x}^{(j)}, \i>j \right\},
Z_{( a , m ) | ( a^{\prime}, m^{\prime})}( q ) = \sum_{( a^{\prime \prime}, m^{\prime \prime})}\hat{N}_{( a , m )^{*}\; ( a^{\prime}, m^{\prime})}{}^{( a^{\prime \prime}, m^{\prime \prime})}\hat{\chi}_{( a^{\prime \prime}, m^{\prime \prime})}( q )
\prod_{j\in J_i}\left(n\sum_{p\mid j}\frac1p\right) \le \left(\frac n{e^{\alpha^{i-1}}}\right)^{\#J_i}=\exp(\#J_i\log n-\#J_i\alpha^{i-2}) =\exp(\#J_i\log n-{(1+o(1))n(\alpha-1)/e^\gamma})
\left|- \sum\limits_{n=1}^{\infty}\kappa_{n,0}^{(q)}e^{-ns_0}\cos(n\eta)\right| \leq \mathcal{C}k^{\alpha_q}\sum\limits_{n=1}^{\infty}e^{-s_0n}n^{\alpha_q-1}\leq \mathcal{C}_{\alpha_q}' k^{\alpha_q}\int_{0}^{\infty}e^{-s_0 \theta}d\theta \leq \mathcal{C}_{\alpha_q, s_0}' k^{\alpha_q}.
\alpha_{X,Y}^{}(u;M\otimes N) \leq \left\|(x_{j})_{j=1}^{n}\right\|_{X(M)}\cdot \left\|(y_{j})_{j=1}^{n}\right\|_{Y(N)}&\leq\left\|(x_{j})_{j=1}^{n}\right\|_{X(E)}\cdot \left\|(y_{j})_{j=1}^{n}\right\|_{Y(F)}\leq \alpha_{X,Y}^{}(u; E\otimes F) + \eta.
{\bf P}\bigl[\beta_{M,N}(a,b)=1\bigr] =\exp\Bigl(-\int\limits_0^\infty e^{-b_0 t}\frac{\prod\limits_{j=1}^N (1-e^{-b_j t})}{\prod\limits_{i=1}^M (1-e^{-a_i t})}\frac{dt}{t}\Bigr), & = \exp\bigl(-(\mathcal{S}_N \log\Gamma_M)(0\,|a,\,b)\bigr).
{\big\| \mathcal{F}^{-1}\big[m_{\iota_1\iota_2\iota_3}(\xi,\eta,\sigma)\cdot \varphi_k(\xi)\varphi_{k_1}(\xi-\eta)\varphi_{k_2}(\eta-\sigma)\varphi_{k_3}(\sigma)\big] \big\|}_{L^1}\lesssim 2^{k/2}2^{\max(k_1,k_2,k_3)}
(\prod_{s=i_{2}}^{j_{2}}x_{s})^{m_{i_{2},j_{2}+1}}\cdots (\prod_{s=i_{1}}^{n}x_{s})^{m_{i_{1},n+1}}=q^{-(j_{2}-i_{1})m_{i_{1},n+1}m_{i_{2},j_{2}+1}}\cdots,
M \int_{T_3(\epsilon)}^{t-T_2(\epsilon)}f'(x(u)) \,du = \int_{T_3(\epsilon)}^{t-T_2(\epsilon)}\frac{f'(x(u))}{f(x(u))}\frac{Mf(x(u))}{x'(u)}x'(u)\,du &\geq \int_{T_3(\epsilon)}^{t-T_2(\epsilon)}\frac{f'(x(u))}{f(x(u))}x'(u)\,du &=\log f(x(t-T_2(\epsilon)))-\log f(x(T_3(\epsilon))) &>\log f(x(t))+\log (1-\epsilon) -\log f(x(T_3(\epsilon))).
\lim_{h\to 0}\mathbb{E}\left[\left(\frac{M(x+h)-M(x)}{h}- \frac{d}{dx}M(x) \right)^2 \right] & = \lim_{h \to 0}\mathbb{E}\left[ \int \left(\frac{\mu(s,x+h)-\mu(s,x)}{h}-\nu(s,x)\right)^2d[B]_s \right] = 0.
D_1\eta_2^{\alpha_1}=-\pi^{\alpha_1}+\frac 12C_{\;\;\beta_1\gamma_1}^{\alpha_1}\eta_1^{\beta_1}\eta_2^{\gamma_1}+\frac12C_{\;\;\beta_1\gamma_3}^{\alpha_1}\left( \eta_2^{\beta_1}\eta_1^{\gamma_3}+\eta_1^{\beta_1}\eta_2^{\gamma_3}\right) ,
P_\nu(1-2\beta)={_2}F_1\left( \left.\begin{array}{c}\nu+1,-\nu 1\end{array}\right|\beta \right)={}-\frac{\sin(\nu\pi)}{\pi}\int\mathbb D_{-\nu-1,\beta}V,{_2}F_1\left( \left.\begin{array}{c}\nu,-\nu 1\end{array}\right|\beta \right)={}-\frac{\sin(\nu\pi)}{\pi}\int(1-\beta V)\mathbb D_{-\nu-1,\beta}V.
|S_6|^2 \ll N \log N \left( \sum_{h=1}^H \sum_{M \leq m, m+h \leq M_1}|b(m)| \, |b(m+h)| \frac{X}{qh\log^{13}X}+ \sum_{M \leq m \leq M_1}|b(m)|^2 \right) \ll \frac{X_4 X \log^3 u}{\log^{11}X}+ X_4 \log \frac{X_4}{M}\log^3 u \ll \frac{X^2}{\log^6 X}.
\Theta={\mathcal P_2^*}\mathcal P_1\mbox{on}\,\,\tilde \Gamma \quad\mbox{and}\quad\tilde \Theta={\mathcal C_2^*}\mathcal C_1\mbox{on}\,\,\tilde \Gamma\mbox{and}\Theta,\tilde \Theta\in L^2(\Omega)
D_{i}( x , y ) = \int_{0}^{1}d \gamma \chi_{i}( \xi , \gamma ) \int{\frac{d^{4}p}{( 2 \pi )^{4}}}\cos [ p ( x - y ) ] \ln \left|{\frac{( 1 - \gamma^{2}) p^{2}+ 4 m^{2}}{\mu^{2}}}\right| ,
Y_{t}+\sum_{j=1}^{\infty}\Pi_{j}(-d_{t})u_{t-j}+\sum_{j=1}^{\infty}\sum_{k^{\prime}=j+1}^{\infty}\Pi_{j}(-d_{t})\Psi_{k^{\prime}-j}(t-j)u_{t-k^{\prime}}=u_{t}.
\tilde{g}_j(x,t) = \tilde{g}(x,t) \forall (x,t) \in \Omega \times [0,j], &\tilde{g}_j(x,t) = \tilde{g}(x,j) + D_t \tilde{g}(x,j)(t-j) \forall (x,t) \in \Omega \times (j,\infty)
\frac{T_{\nu}}{T_{\gamma}}=\frac{T_{1}}{T_{2}}=(\frac{g_{2}}{g_{1}})^{\frac{1}{3}}=(\frac{2}{2+\frac{7}{4}+\frac{7}{4}})^{\frac{1}{3}}=(\frac{4}{11})^{\frac{1}{3}}\approx0.714.
\frac{1}{2 \pi}c^{( b )}: e^{\mp i 2 \sqrt{\pi}\sqrt{\frac{\pi}{\pi + g N}}U_{1 b}\Phi^{( 1 )}( x )}:_{M^{( 1 )}}\prod_{I = 2}^{N}: e^{\mp i 2 \sqrt{\pi}U_{I b}\Phi^{( I )}( x )}:_{M^{( I )}}e^{\pm i \frac{\theta}{N}},
v(x)-P_\epsilon(x) = v(\lambda_1 x_1 + \lambda_2 x_2) - P_\epsilon(\lambda_1 x_1 + \lambda_2 x_2) \leq \lambda_1(v(x_1)-P_\epsilon(x_1)) + \lambda_2(v(x_2)-P_\epsilon(x_2)) &< \lambda_2(v(x_2)-P_\epsilon(x_2)) \leq v(z_\epsilon)-P_\epsilon(z_\epsilon).
S^{\delta_n^k+1+u-\alpha_{k-u}}\left (\alpha \right )= \alpha \iff \begin{array}{c}\{\alpha_{j+(k-u)}= \alpha_j+\alpha_{k-u}\}_{j=1}^{u}\mbox{and}\ \ \delta_{n}^k=\alpha_{1+u}+\alpha_{k-u}\mbox{and}\ \ \{\alpha_{j+1+u}= \alpha_{j}+ \alpha_{1+u}\}_{j=1}^{k-u-1}. \end{array}
A_2(n) = \begin{cases}p^{-3n/2}(-1)^n \chi_{\psi}(p^n)(1-p^{-1})p^{n}n > 0, p^{3n/2}(-1)^n \chi_{\psi}(p^n)(1-p^{-1})(1+p-p^{1-n}) n \leq 0. \end{cases}
\alpha_n'(a^2,x,q^2)=\frac{q^{n^2-n}(-x)^n(1-a^2q^{4n})(a^2/x;q^2)_n}{(1-a^2)(q^2x;q^2)_n}\sum_{0\le j\le n}\frac{(1-aq^{2j-1})(a)_{j-1}(x;q^2)_j}{q^{j(j-1)/2}x^j(q)_j(a^2/x;q^2)_j},
\varphi (g_1 g^{-1}_3) \varphi (g_1 g_3) \overset{(\ref{eq:var-noncomm})}{=}\varphi (g_1) \varphi (g^{-1}_3) \varphi (g_3) \varphi (g_1) \overset{{\bf A2}}{=}\varphi (g_1) \varphi (g_1) \overset{{\bf A3}}{=}\varphi (g^{2}_1) \,.
V_1(t)=2V_1^+(t)\le 2\int_{\eta_{k-1}}^{b(t)}v_1^{-p'}\le 2\int_{\eta_{k-1}}^{b(x)}v_1^{-p'}\le 2V_1(x)=4V_1^-(x), \eta_{k-1}\le t\le x,
\tilde Y_t^{b}= \ 0-\int_{t}^{T}g^b(t,\xi_s^a,\gamma_s)-D_s+(m^b(t,\tilde Y_s^b +\xi_s^a,\xi_s^a)+\tilde\lambda_s )\tilde Y_s^b+\eta^b(t,\tilde Z_s^b+\gamma_s,\gamma_s)\cdot \tilde Z_s^b ds & -\int_{t}^{T}\tilde Z_s^{b}\cdot dW_s, ~t\in[0, T],
\Psi ( r , \theta , t ) = \sqrt{{\frac{i}{r}}}\, \,{\frac{1}{\sqrt{2 \pi k_{0}}}}\, \,{\frac{\sin{\frac{\pi}{\alpha}}}{\cos{\frac{\pi}{\alpha}}+ \cos \theta}}\, \, e^{-{\frac{b^{2}}{4 \xi^{2}}}}\, \, \Psi_{\mathrm{{f r e e}}}( r , \alpha^{2}t ) ,
{\cal L}^{(0)}~=~ \frac{\kappa}{2 \pi}\epsilon^{mn}A_n \dot{A}_m + \pi_a \dot{z}_a + \pi^*_a \dot{z}^*_a + \Omega^{(0)}_1 \dot{\alpha}-{\cal H}^{(0)}, ~~~m, n=1,2 ,
e^{\beta Y^{b,n}_\rho}= \ 1-\int_\rho^T \beta e^{\beta Y_s^{b,n}}\left( g^b(s,Y_s^{b,n}+\xi_s^a,Z_s^{b,n}+\gamma_s)-D_s +\lambda_s^n \frac{1-e^{-\alpha Y_s^{b,n}}}{\alpha}\right)ds -\int_\rho^T \beta e^{\beta Y_s^{b,n}}Z_s^{b,n}\cdot dW_s -\frac{\beta^2}{2}\int_\rho^T e^{\beta Y_s^{b,n}}\|Z_s^{b,n}\|^2 ds.
\Phi^{\infty}=i\frac{\eta}{\sqrt{2}}\vec{p}\cdot\vec{\sigma}, \qquad\vec{p}=\left(\begin{array}{c}\sin\tau\cos N\phi \sin^2\tau\sin N\phi + \cos^2 \tau \sin\tau\cos\tau(\sin N\phi -1)\end{array}\right)
\iint_Q f_\epsilon(u^\tau) \partial_x I(u^\tau) \partial_x \varphi = -\iint_Q f_\epsilon(u^\tau) I(u^\tau) \partial^2_{xx}\varphi - \iint_Q n (u^\tau)^{n-1}\partial_x u^\tau I(u^\tau) \partial_x \varphi .
\min_{\mathbf{\Phi}_{g}}~ \tilde{f}_g(\mathbf{\Phi}_{g}) = \mathsf{Tr}(\mathbf{\Phi}_{g}\mathbf{Y}_{g,g}\mathbf{\Phi}_{g}^H\mathbf{Z}_{g}- 2\Re\{\mathbf{\Phi}_{g}\tilde{\mathbf{X}}_{g}\}) \mathrm{s.t.}~ \mathbf{\Phi}_{g}^H\mathbf{\Phi}_{g}= \mathbf{I}_{\bar{M}}, \forall g \in \mathcal{G}.
\mathcal{L}^{\mathrm{CLIP}}= \min_{\overline{\theta}}\mathbb{E}_{t}\Bigg[\mathrm{\frac{\pi_{\overline{\theta}^{*}}(S_t, A_t)}{\pi_{\overline{\theta}}(S_t, A_t)}\hat{A}_{t}, clip\Bigg(\frac{\pi_{\overline{\theta}^{*}}(S_t, A_t)}{\pi_{\overline{\theta}}(S_t, A_t)}, 1-\epsilon, \epsilon\Bigg)\hat{A}_{t}}\Bigg],
~\mathbf{P}\Big( \{U^*_{n}(\theta_{n},c^{*}_{n})= \emptyset \}\cap \{\mathfrak W^*(c_{\pi^*})\neq \emptyset \}\Big) \le ~ \mathbf{P}\Big( \{U^*_{n}(\theta_{n},c^{*}_{n})= \emptyset \}\cap \{\mathfrak W^{*,-\delta}(0)\neq \emptyset \}\Big) +\mathbf{P}\Big(\{\mathfrak W^{*,-\delta}(0)= \emptyset \}\cap \{\mathfrak W^*(0)\neq \emptyset \}\Big),
\nabla(a.[\omega]) = \nabla(f.d_{/}\xi) = \nabla(d_{/}(f.\xi)) - \nabla(d_{/}f\wedge \xi) & = d_{/}f\wedge \frac{\partial (f\xi)}{\partial \lambda}- \frac{\partial f}{\partial \lambda}.f.d_{/}\xi - \frac{\partial f}{\partial \lambda}.d_{/}f\wedge \xi - \nabla(b.[\omega]) & = a.\nabla([\omega]) - b.\nabla([\omega]).
\left| f \right|_{C^{\alpha}(E)}:= \sup_{\substack{x_1, x_2 \in E x_1 \neq x_2}}\frac{\left| f(x_2) - f(x_1)\right|}{\left| x_2 - x_1 \right|^{\alpha}}, \left\| f \right\|_{L^{\infty}(E)}= \sup_{x \in E}\left| f(x) \right| , \left\| f \right\|_{C^{\alpha}(E)}:= \left| f \right|_{C^{\alpha}(E)}+ \left\| f \right\|_{L^{\infty}(E)}.
\Delta h = \partial_r^2 h + \frac{d-1}{r}\partial_r h + \frac{1}{r^2}\Delta_{0}h,h \in C^2(B), \sum_{i,j=1}^d x_i x_j \partial_i \partial_j h(x) \Big|_{x=r\xi}=r^2 \partial_r^2 h (r\xi), h \in C^2(B),0<r \le 1, \xi \in S^{d-1},
\overline{{\hat\sigma}_{12}^{(\Lambda,N)}(\tilde{\boldsymbol{\phi}})}:=\frac{1}{(2\pi)^2}\iint_{[0,2\pi]\times[0,2\pi]}d\phi_1d\phi_2\;{\hat\sigma}_{12}^{(\Lambda,N)}(\tilde{\boldsymbol{\phi}})=\frac{1}{2\pi}\frac{p}{q}
\dot{z}_{y}=A\dot{u}_{y}-z_{y}(\beta|\dot{u}_{x}z_{x}|+\gamma\dot{u}_{x}z_{x}+\beta|\dot{u}_{y}z_{y}|+\gamma\dot{u}_{y}z_{y})\times(z_{x}^{2}+z_{y}^{2})^{\frac{n-2}{2}}
O = \left( \begin{array}{cccc}x_{3}0 \sqrt{1-x_{3}^{2}}0 0 1 0 0 -x_{1}0 \frac{x_{1}x_{3}}{\sqrt{1-x_{3}^{2}}}\frac{x_{2}}{\sqrt{1-x_{3}^{2}}}x_{2}0 -\frac{x_{2}x_{3}}{\sqrt{1-x_{3}^{2}}}\frac{x_{1}}{\sqrt{1-x_{3}^{2}}}\end{array}\right).
\begin{aligned}\int_{C^j-\zeta(j)}^{C^j}\frac{1}{u}\mathrm{d}u = \log\left(\frac{C^j}{C^j-\zeta(j)}\right) & = \log(C) - \log\left( 1 + (C-1) (1 - \tau_A(C,j)) \right).\end{aligned}
\intop_{0}^{a\left(x,t\right)}2\pi r\rho\frac{\partial u}{\partial t}dr+\intop_{0}^{a\left(x,t\right)}2\pi r\frac{\partial P}{\partial x}dr=\intop_{0}^{a\left(x,t\right)}2\pi r\mu\left(\frac{\partial^{2}u}{\partial r^{2}}+\frac{1}{r}\frac{\partial u}{\partial r}\right)dr.
\textstyle\psi_{i,j}^G=\begin{cases}\left(\prod_{\ell=2}^j \alpha_{1,\ell}\right) (-\partial_y+\partial_z){\rm if}\ 1=i<j, x\partial_x+y\partial_y+z\partial_z -\frac 1 3(x+y+z)\overline \partial{\rm if}\ (i,j)=(2,3),\end{cases}
\sigma_{i j}( x^{-}, y^{-}; x^{+}) = \int \! \frac{d P^{-}}{4 \pi}\frac{d p^{+}}{4 \pi}\frac{d k^{+}}{4 \pi}e^{- \frac{i}{2}P^{-}x^{+}}e^{- \frac{i}{2}( p^{+}x^{--}k^{+}y^{-})}~ \sigma_{i j}( p^{+}, k^{+}; P^{-}) ,
\sigma_{m+1}(x_0,x_1, \ldots,x_{2m},x_{2m+1})=\sigma_m(x_0,x_1) +\sigma_m(x_2,x_3,\ldots,x_{2m},x_{2m+1}) &= x_0 x_1 + x_0 + x_2 x_3 + x_2 + \ldots + x_{2m}x_{2m+1}+ x_{2m}.
\xi^*_i(u) = c^2_{s,i}\Bigl ( -u_i + \sum_{k\in \mathcal{J}}P_{i,k}u_k \Bigr )^2 + \sum_{k\in \mathcal{J}}P_{ik}u_k^2 - \big(\sum_{k\in \mathcal{J}}P_{ik}u_k\big)^2 = c_{s,i}^2 (-u_i+w_{ij})^2 + \sum_{k\in \mathcal{J}}P_{ik}(u_k^2-u_k) + w_{ij}- w_{ij}^2.
[(u,\tau)]=(p+u_1,\tau),\,\,\,\,\,\,\xi=\frac{(1+\frac{(B+t_1F_2)t}{u_1})\xi_1+\frac{(A+t_1F_1)t}{u_1}}{-\frac{(C+t_1F_3)t}{u_1}\xi_1+1}=\frac{(1+\frac{B't}{u_1})\xi_1+\frac{A't}{u_1}}{-\frac{C't}{u_1}\xi_1+1}=\frac{(u_1+B't)\xi_1+A't}{-C't\xi_1+u_1}
\mathbb{E}[\zeta_\mathcal{K}] \stackrel{(a)}{=}\mathbb{E}\big[\Re\{\zeta_\mathcal{K}\}\big]=\sum_{i\in\mathcal{K}}\sqrt{\bar{p}/\beta_i}\mathbb{E}\big[|\hat{h}_i|]\stackrel{(b)}{=}\frac{\sqrt{\pi\bar{p}}}{2}\sum_{i\in\mathcal{K}}\sqrt{1+\frac{1}{N_1\bar{\gamma}_i}}\stackrel{(c)}{\approx}\frac{K\sqrt{\pi\bar{p}}}{2}\sqrt{1+\frac{1}{N_1\bar{\gamma}'}},
P_{s,\omega,N}\left( g \right) (x):= \sum_{I\in X_N^{(\omega)}}\eta^{(\omega)}(I) e^{2\pi s c(I,x)}g\circ f_I (x), \eta^{(\omega)}(I) := \prod_{n=1}^{|I|}\textbf{p}^{(\omega_n)}_{I_n}.
\phi_{\chi_{j}\otimes \chi_{\bar{j}}}^{\alpha}\; \star \; \phi_{\chi_{k}\otimes \bar{\chi}_{k}}^{\beta}\ = \ \sum C \left( \chi_{j}, \chi_{\bar{j}}, \alpha \bigm | \chi_{k}, \chi_{\bar{k}}, \beta \bigm | \chi_{l}, \chi_{\bar{l}}, \gamma \right) \, \phi_{\chi_{l}\otimes \chi_{\bar{l}}}^{\gamma}\ .
\begin{aligned}I_4=\iint \partial_Y\left(\frac{h^0_e}{u^0_e}\right)\nabla w_2\cdot \nabla \partial_Yw_1\lesssim\Vert \nabla \partial_Yw_1\Vert_{L^2}\Vert w_2\Vert_{H^1},\end{aligned}
B_{n}(\theta_0,\varepsilon)=\{\theta\in\Theta:\, D(p_{\theta_0}^{(n)}, p_{\theta}^{(n)})\leq n\,\varepsilon^2,\, V(p_{\theta_0}^{(n)}, p_{\theta}^{(n)})\leq n\,\varepsilon^2\}.
\big[\psi(\tilde{\mathbb{A}}/f\tilde{\mathbb{A}})\big]_{\tilde{\mathbb{A}}}=\det(Q) &=f + c(f)p_1t^df(z_1)+ c(f)p_2t^{2d}f(z_1)^2 + \dots +c(f)t^{r_0d}f(z_1)^{r_0}.
\begin{array}{l l l}{{d = 1 0 :}}&{{\Gamma_{M}}}&{{M = 1 , \cdots , 1 0 \, ,}}\\{{d = 1 1 :}}&{{{\tilde{\Gamma}}_{\tilde{M}}}}&{{{\tilde{M}}= 1 , \cdots , 1 0 , 1 1 \, ,}}\\{{d = 1 2 :}}&{{{\hat{\Gamma}}_{\hat{M}}}}&{{{\hat{M}}= 1 , \cdots , 1 0 , 1 1 , 1 2 \, .}}\\ \end{array}
\begin{pmatrix}\vec{\Upsilon}_{lm}\vec{\Psi}_{lm}\end{pmatrix}\to \left( \begin{array}{l}\vec{\varUpsilon}_{lm}= (2l+1)^{-1/2}(\sqrt{l}\vec{\Upsilon}_{lm}+ \sqrt{l+1}\vec{\Psi}_{lm}) \vec{\varPsi}_{lm}= (2l+1)^{-1/2}(-\sqrt{l+1}\vec{\Upsilon}_{lm}+ \sqrt{l}\vec{\Psi}_{lm}) \end{array}\right)
Y(\theta)=\prod_{l=1}^{\infty}\frac{\textstyle \Gamma(1-l+\epsilon n\gamma/36\pi+1/4+i\theta/2\pi) \Gamma(l-\epsilon n\gamma/36\pi-1/4-i\theta/2\pi)}{\textstyle \Gamma(-l+\epsilon n\gamma/36\pi+1/4-i\theta/2\pi) \Gamma(l+1-\epsilon n\gamma/36\pi-1/4+i\theta/2\pi)}\;.
\|f\|_{L^{p,N}(\mathbb{R}^{6})}=(\sum\limits_{\substack{\theta \in \mathcal{P}_{N^{-1}}}}\|f*\Xi_{\theta}\|_{L^p(\mathbb{R}^{6})}^p)^{\frac{1}{p}}, \|f\|_{L^{p,N}(w_{B_{N}})}=(\sum\limits_{\substack{\theta \in \mathcal{P}_{N^{-1}}}}\|f*\Xi_{N^{-1}}\|_{L^p(w_{B_{N}})}^p)^{\frac{1}{p}}
\begin{cases}{}^C D^\nu_{0^+}u \left( x,t \right) = H t^{2H-1}\frac{\partial^2}{\partial x^2}u \left( x,t \right), \nu \in (0,1), \: x \in \mathbb{R}, \: t >0, u \left( x,0 \right) = \delta \left( x \right), \end{cases}
\rho_3(\mathbf{c}) \Sigma \rho_2(A) \Sigma^{-1}\rho_1(a) = \left( \begin{array}{ccc}a \iota(\alpha_{11}) \iota(\alpha_{12}) \sigma^{-1}a C_1 a \sigma \iota(\alpha_{21}) \sigma \iota(\alpha_{22}) \sigma^{-1}a C_2 0 0 a I_2\end{array}\right),
\lim\limits_{b\to a}\left(\frac{b}{a}\right)_{n}\sum_{k=0}^{n-1}\left(\frac{q^k}{1-\frac{bq^k}{a}}\right) =\lim\limits_{b\to a}\left[\left(\frac{b}{a}\right)_n\frac{1}{1-\frac{b}{a}}+ \left(\frac{b}{a}\right)_n\left(\frac{q}{1-\frac{bq}{a}}+\cdots+\frac{q^{n-1}}{1-\frac{bq^{n-1}}{a}}\right)\right] &=(q)_{n-1}.