harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
(\ldots, x_1, x_0)(w_n) \Phi = (\ldots, x_1, x_0p_n) \left((u_n) \Psi \circ \delta \circ (s_n)\Psi \right) = (\ldots, x_1, x_0p_n,z_1, z_2, \ldots, z_k) \delta \circ (s_n)\Psi.
f = \sum_K a_{ij}^K f_{ij,K}, \mbox{where}f_{ij,K}\in \mathbb{C}[\{{}_{(\alpha)}a_{st}: (s,t)\not=(i,j)\}] \mbox{and}a_{ij}^K := \prod_{\alpha=1}^{r+1}{}_{(\alpha)}a_{ij}^{k_{\alpha}}.
\begin{aligned}G'(u)(v-u)+\langle \lambda, B(v-u)\rangle + r\langle Bu-\eta,B(v-u)\rangle \geq 0, \quad\forall{v\in V},\, u\in V, F'(\eta)(q-\eta)-\langle \lambda, q-\eta\rangle+ r\langle\eta-Bu,q-\eta\rangle \geq 0, \forall q\in H,\, \eta\in H,\end{aligned}
{\bf a}^{\mu \nu}( x ) = \frac{1}{( 2 \pi )^{3}}\int \frac{d^{3}p}{\sqrt{2 p^{o}}}\left[{\bf a}^{\mu \nu}( p ) e^{i p x}+{\bf a}^{\dag \mu \nu}( p ) e^{- i p x}\right] ,
\mathcal{P}\approx1-\eta_{\mathcal{C}}\cdot\frac{\left( \Delta\mathcal{C}\right)^{2}}{\mathcal{C}_{}^{2}}\eta_{\mathcal{C}}\overset{}{=}\frac{8}{3}k_{\mathrm{o}}^{2}\left( 2k_{\mathrm{o}}^{2}+\sigma_{k_{\mathrm{o}}}^{2}\right) R_{\mathrm{o}}L^{3}.
h = \left( \begin{array}{c c}{1}{0}{X}{1}\end{array}\right) \left( \begin{array}{c c}{e^{- \frac{1}{2}\tilde{\Phi}}}{0}{0}{e^{\frac{1}{2}\tilde{\Phi}}}\end{array}\right) \left( \begin{array}{c c}{1}{Y}{0}{1}\end{array}\right) .
a_\kappa = \sum_{i,j\in \mathbb N\colon\atop{3(i+j)\leq \kappa}}(\kappa-3(i+j)+1),b_\kappa = \sum_{i,j\in \mathbb N\colon\atop{2(i+j)\leq \kappa}}\binom{\kappa-2(i+j)+3}{3}- a_\kappa, &c_\kappa = \frac16\left (\binom{\kappa+7}{7}-3b_\kappa - a_\kappa \right ).
B_{n,1}= -\frac{1}{2n}\sum \limits_{j = 1}^n \log \Big(1- \frac{G\left(\frac{j}{n}\right)+o(1)}{2\left(1 - \frac{j - 1}{n}\right) + o(1)}\Big) \xrightarrow{n \to \infty}-\frac{1}{2}\int \limits_0^1 \log \Big (1 - \frac{G(x)}{2(1-x)}\Big) \; dx~.
P_{D}=\mathrm{Pr}\left(\sigma_{\theta}^2|\mathbf{a}^H\mathbf{H}^H\mathbf{C}_w^{-1}\mathbf{y}|^2>\gamma^{'}|\mathcal{H}_1\right) P_{FA}=\mathrm{Pr}\left(\sigma_{\theta}^2|\mathbf{a}^H\mathbf{H}^H\mathbf{C}_w^{-1}\mathbf{y}|^2>\gamma^{'}|\mathcal{H}_0\right)\;.
F_{\gamma_{SCHCN}}(\gamma) =\mathcal{L}^{-1}\left[\frac{\mathcal{L}_{\gamma_{SCHCN}}(s)}{s}\right] &=\mathcal{L}^{-1}\left[\frac{\frac{N!}{N_c!N_c^{N-N_c}}\lambda_{0}\lambda_{eq}^N}{s(s+\lambda_{eq})^{N_c}(s+\lambda_{0}) \prod_{n=1}^{N-N_c}[s+(1+\frac{n}{N_c})\lambda_{eq}]}\right].
\int_{[t,T]}D^X_{s-}dX_s + \frac12 \int_{[t,T]}\Delta X_s \gamma_s dX_s = \frac12 \left( \gamma_T^{-1}(D^X_T)^2 - \gamma_t^{-1}d^2 - \int_{t}^T (D^X_s)^2\nu_s^2 d\left(\nu^{-2}_s\gamma_s^{-1}\right) \right)
64(n+ 4)\pi^{4}g_{n+4}-32q(11+4n)\pi^{3}g_{n+3}+ 4\left(32+54 n+29 n^2+5 n^3+8 \tilde{p}(3+2 n)+24 q^2+16 n q^2\right)\pi^{2}g_{n+2}- 4q\left(4 \tilde{p}(1+4 n)+n\left(4+11 n+5 n^2\right)\right) \pi g_{n+1}+ \left((n-1)\left(16 \tilde{p}^2+2 \tilde{p}n(5 n-2)+n^2\left(-2+n+n^2\right)\right)\right)g_n=0.
{\bf i}^2={\bf j}^2={\bf k}^2=-1,~{\bf i}{\bf j}={\bf k}=-{\bf j}{\bf i},~{\bf j}{\bf k}=-{\bf k}{\bf j}={\bf i},~ &\overline{h}=h_0-h_{1}{\bf i}-h_{2}{\bf j}-h_{3}{\bf k}~\,\,\,\,~|h|=\sqrt{h\overline{h}}.
d l^{2}\sim \biggl ( 1 + \frac{\psi_{i}^{\prime}}{r_{i}^{\prime}}\biggr )^{2 + 2 a}\biggl ( 1 + \frac{\chi_{i}^{\prime}}{r_{i}^{\prime}}\biggr )^{2 - 2 a}( d r_{i}^{\prime 2}+ r_{i}^{\prime 2}d \Omega^{2}) ,
\, \frac{N-4\sigma}{4\mu-2}= \frac{N+2-4\nu}{4\mu-2}+ 1 > \frac{N+2-4\left(\frac{N}{4}\right)}{4\mu-2}+ 1 > 1; \, \frac{N-4\sigma}{N+2-4\nu}= \frac{4\mu-2}{N+2-4\nu}+ 1 > 1;
W_{\wedge}= \left\{\begin{array}{c}{{L_{- 1}\quad L_{0}\quad L_{1}}}\\{{J_{- 2}\quad J_{- 1}\quad J_{0}\quad J_{1}\quad J_{2}}}\\{{\Lambda_{- 3}\quad \Lambda_{- 2}\quad \Lambda_{- 1}\quad \Lambda_{0}\quad \Lambda_{1}\quad \Lambda_{2}\quad \Lambda_{3}}}\\{{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots}}\\ \end{array}\right\}
\left\langle \Phi_{D}^{2}\right\rangle_{\phi}= \left\langle \Phi_{N D}^{2}\right\rangle_{\phi}= \left\langle \Phi_{D}^{2}\Phi_{N D}^{2}\right\rangle_{\phi}= \left\langle \Phi_{D}^{2}\Phi_{D}^{2}\right\rangle_{\phi}\equiv 0 .
Z_{\bold F_3}=\ker([P_0]\bold F_3\oplus [P_1]\bold F_3\oplus [P_t]\bold F_3\oplus [P_\infty]\bold F_3 \to \bold F_3) &a_0[P_0]+a_1[P_1]+a_t[P_t]+a_{\infty}[P_\infty] \mapsto a_0+a_1+a_t+a_{\infty}, H_{\bold F_3}=Z_{\bold F_3}/([P_0]-[P_1]-[P_t]+[P_\infty]).
\hat{\rm H}_{\pm}^s = \int_{-{\rm L}/2}^{{\rm L}/2}dx\hat{\cal H}_{\pm}^s(x)= \frac{1}{2}\hbar \int_{-{\rm L}/2}^{{\rm L}/2}dx (\psi_{\pm}^{\dagger s}d_{\pm}\psi_{\pm}^s- \psi_{\pm}^s d_{\pm}^{\star}\psi_{\pm}^{\dagger s}).
\mathbb{E}(|B_{E_t}- B_{E_s}|) = \mathbb{E}(|\int_{E_s}^{E_t}dB_z|) &=\int_0^{+\infty}\int_0^{+\infty}\mathbb{E}(|B_{r_1}- B_{r_2}|)h(r_1, t)h(r_2, s)dr_1 dr_2 &=\int_0^{+\infty}\int_0^{+\infty}\sqrt{r_1 - r_2}h(r_1, t)h(r_2, s)dr_1 dr_2 &=\mathbb{E}(\sqrt{E_t - E_s})\leq \sqrt{\mathbb{E}(E_t - E_s)}&\leq C(\beta, 1)^{\frac{1}{2}}|t - s|^{\beta/2},
\sum_{i,j=1}^{n}F_n(x_i - x_j) = \sum_{i,j=1}^{n}\sum_{|k| \leq n}\left(1 - \frac{|k|}{n}\right) e^{2\pi i k (x_i- x_j)}= \sum_{|k| \leq n}\left(1 - \frac{|k|}{n}\right) \sum_{i,j=1}^{n}e^{2\pi i k (x_i- x_j)}= \sum_{|k| \leq n}\left(1 - \frac{|k|}{n}\right) \left| \sum_{j=1}^{n}e^{2\pi i k x_j}\right|^2.
p_n(x)=a^{-n}(ab,ac,ad;q)_n\,\,{}_4\phi_3\left[{q^{-n}, abcdq^{n-1},ax,a/x\atop ab,ac,ad};q,q \right], &D p_n(x)= \left(q^{-n}+qbcdq^{n-1}-1-abcdq^{-1}\right)p_n(x).
\begin{cases}-\tilde\varphi_{f,k}''(x)-k^2\tilde\varphi_{f,k}(x)=f(x),\quad[0,2\pi], \tilde\varphi_{f,k}(0)=\tilde\varphi_{f,k}(2\pi), \tilde\varphi_{f,k}'(0)=\tilde\varphi_{f,k}'(2\pi).\end{cases}
N(H_r)=p^{s-(r+1)}\Big(1+(-1)^{\frac{(R_f-1)(p-1)}{4}}p^{-\frac{R_f-1}{2}}\varepsilon_{f}\sum_{(u,-\alpha)\in H_{r},y_1\in\Im L_f}\bar{\eta}(f(x_{u})\big)\Big).
\langle \theta_{\varphi^{\vee}}f , \theta_{\varphi^{\vee}}f \rangle = \frac{\mathrm{vol}\ [H]}{\kappa_{m,2n}}\, L(s_{m,2n}+1/2, \sigma, \chi) \, \prod_v Z^*_v(s_{m,2n},\varphi,f)
\frac{I_0}{P_i}= \sum_{j=1}^{N}\frac{P_j}{P_i}\sum_{X_k\in\Phi_j}H_{k0}\|X_k\|^{-\alpha}\stackrel{d}{=}\sum_{j=1}^{N}\sum_{X_k\in\Phi'_j}H_{k0}\|X_k\|^{-\alpha}&\stackrel{d}{=}\sum_{j=1}^{N}\eta_j^{\frac{\alpha}{2}}\left(\frac{P_j}{P_i}\right)\sum_{X_m\in\hat{\Phi}_j}H_{m0}\|X_m\|^{-\alpha},
[g]_{\mu \nu}= - \frac{1}{2}g^{\alpha \beta}\partial_{\alpha}\partial_{\beta}g_{\mu \nu}+ \nabla_{(\mu}\Gamma_{\nu)}+ g^{\alpha \beta}g^{\gamma \delta}[ \Gamma_{\alpha \gamma \mu}\Gamma_{\beta \delta \nu}+ \Gamma_{\alpha \gamma \mu}\Gamma_{\beta \nu \delta}+ \Gamma_{\alpha \gamma \nu}\Gamma_{\beta \mu \delta}],
F_j^i(y) =\partial_k f^i(f^{-1}(y))\sigma^k_j(f^{-1}(y)),y\in f(U), G^i(y) =\frac{1}{2}\partial_{jk}^2f^i(f^{-1}(y))\left(\, \sigma^j(f^{-1}(y))\cdot\sigma^k(f^{-1}(y)) \,\right),y\in f(U),
\sum_{i=1}^{n}(a_ia_i^*)^\frac{1}{2}\leq \sqrt{s}\langle x, x \rangle^\frac{1}{2}\iff \left(1-\frac{c_x}{2}\right) \sqrt{n}\leq \sqrt{s}\iff 1-\frac{c_x}{2}\leq \sqrt{\frac{s}{n}}.
\| (\tilde W-W) \|_{C^{m+2,\alpha}(\overline{B_{2R}}\setminus U)}+ \| (\tilde W-W) \|_{C^{m+2,\alpha}(\overline U)}\le C \| \partial_{\nu,\,{\rm out}}\tilde W\|_{C^{m+1,\alpha}(\partial U)}\le C \| \tilde W \|_{C^{m+2,\alpha}(\overline{B_{2R}}\setminus U)}\le C \|f\|_{C^{m,\alpha}(\overline{B_4})}.
Z[\vec{J}] = \exp{\{i S_{0}[\vec{J}]\}}\int D\delta\vec{q}\exp{\{i\int^{+\infty}_{-\infty}dt \left[\frac{1}{2}\delta\dot{\vec{q}}^{2}-\frac{1}{2}\delta\vec{q}^{T}\Sigma \delta\vec{q}+ \vec{J}^{T}\delta\vec{q}\right]\}}.
\|E\|_{F}^{2}=\|\widetilde{\Sigma}_{1}\widetilde{V}_{1}^{\ast}V_{1}-\widetilde{U}_{1}^{\ast}U_{1}\Sigma_{1}\|_{F}^{2}+\|\widetilde{\Sigma}_{1}\widetilde{V}_{1}^{\ast}V_{2}\|_{F}^{2}+\|\widetilde{U}_{2}^{\ast}U_{1}\Sigma_{1}\|_{F}^{2}.
A^{ij}=\overline{A^{ji}}^{t}, \quad\mbox{for all $i,j=1\dots,q$,}& B^{ij}=\overline{C^{ji}}^{t}, \quad\mbox{for all corresponding $i,j$,}& D^{ij}=\overline{D^{ji}}^{t},\quad\mbox{for all $i,j=1\dots,q'-q$.}
\frac{d^2 f}{d x^2}- \frac{(x^3-2)}{x (1-x^3)}\frac{d f}{d x}- \frac{l(l+3)}{x^2 (1-x^3)}f +\frac{\lambda^2}{x (1-x^3)^2}\left( 1 + \frac{\rho_0^3}{(\omega R)^3x^3}\right) f =0\,,
\zeta_{\Lambda'}(s) =\sum_{x\in\Lambda'\setminus\{0\}}\frac{1}{|x|^s}=\sum_{x \in \Lambda'\cap\Omega_\Lambda\setminus\{0\}}\sum_{v\in\Lambda}\frac{1}{|x+v|^s}+\sum_{v\in\Lambda\setminus\{0\}}\frac{1}{|v|^s}=\sum_{x \in \Lambda'\cap\Omega_\Lambda\setminus\{0\}}\zeta_\Lambda(s;x)+\zeta_\Lambda(s),
\widetilde{Z}_{d}\; = \; \int d{\bf z}\exp \left\{-{\frac{N}{2}}\left[{\frac{z^{2}}{\beta}}- d \left( \sqrt{1 + 4 z^{2}}- 1 - \ln \left({\frac{1}{2}}\sqrt{1 + 4 z^{2}}+{\frac{1}{2}}\right) \right) + \beta d \right] \right\}\; .
h(t) + u_4(t, 0) + \alpha u_3(t, 0) \chi_{[T_{opt}- \tau_1, T-\tau_1]}= \int_{T -\tau_2}^t g(t-s) u_4(s, 0) \, ds + \int_t^{T - \tau_1}f(s-t) u_4(s, 0) \, ds,
s_1 = \frac{s_2 \cdot{r}^{\top}A_{11}{r}+ s_4 \cdot{r}^{\top}A_{12}{r}+ m r_1 q_{12}}{{r}^{\top}Q_1{r}}+ O_g(\delta m^{1/2}), s_3 = \frac{s_2 \cdot{r}^{\top}A_{21}{r}+ s_4 \cdot{r}^{\top}A_{22}{r}+ m r_3 q_{12}}{{r}^{\top}Q_1{r}}+ O_g(\delta m^{1/2})
\begin{array}{l l}{{\bar{\bf s}X^{i *}= \psi^{i *}}}&{{{\bf s}X^{i *}= \xi^{i *}}}\\{{\bar{\bf s}\psi^{i}= - \frac{i}{2}\partial_{+}X^{i}}}&{{{\bf s}\psi^{i *}= - \frac{1}{2}F^{i *}}}\\{{\bar{\bf s}\xi^{i *}= \frac{1}{2}F^{i *}}}&{{{\bf s}\xi^{i}= - \frac{i}{2}\partial_{-}X^{i}}}\\{{\bar{\bf s}F^{i}= i \partial_{+}\xi^{i}}}&{{{\bf s}F^{i}= - i \partial_{-}\psi^{i}\ ,}}\\ \end{array}
i \left< a_{\mu}^{i}a_{\nu}^{j}A_{\rho}^{a}A_{\sigma}^{b}\right>_{\mathrm{b a r e}}= i g^{2}f^{a i c}f^{c j b}\left[ 2 g_{\mu \nu}g_{\rho \sigma}- \left( 1 - \frac 1 \alpha \right) ( g_{\mu \rho}g_{\nu \sigma}+ g_{\mu \sigma}g_{\nu \rho}) \right] .
\begin{aligned}\overline{16}^n_{209296}\to/<->/<1em> \overline{16}^n_{699643}16^n_{235548}\to/<->/<1em> 16^n_{635483}\overline{16}^n_{485898}\to/<->/<1em> 16^n_{543682}\overline{16}^n_{910482}\to/<->/<1em> 16^n_{919988}\end{aligned}
\mathbf{r}_p(\mathbf{x},\mathbf{y},\mathbf{s},\kappa,\tau)=\mathbf{Ax}-\tau \mathbf{b},\mathbf{r}_d(\mathbf{x},\mathbf{y},\mathbf{s},\kappa,\tau)={{\mathbf{A}}^{T}}\mathbf{y}+\mathbf{s}-\tau \mathbf{c}.
\beta_j = \; (\imath_{e_k}\varphi) \wedge (\imath_{e_j}\nabla_{e_k}\ast \varphi) - (\imath_{e_j}\imath_{e_k}\varphi) \wedge e^l \wedge (\imath_{e_k}\nabla_{e_l}\ast \varphi) &-(\imath_{e_j}\nabla_{e_k}\varphi) \wedge (\imath_{e_k}\ast \varphi) - e^l \wedge (\imath_{e_k}\nabla_{e_l}\varphi) \wedge (\imath_{e_j}\imath_{e_k}\ast \varphi).
(R_1^m)_{s,t}=-\int_s^t(\nabla_q\chi)(r,q_r^m,z_r^m)\cdot z_r^m dr &-\int_s^t (\nabla_z\chi)(r,q_r^m,z_r^m) \cdot \left[ (-\partial_r\psi(r,q_r^m)+\tilde F(r,q_r^m)-\nabla_qV(r,q_r^m))dr+\sigma(r,q^m_r) dW_r\right]
F = B_{\{1, 2\}}\cap B_{\{1, 2, 3\}}\cap B_{\{4, 5\}}\cap B_{\{4, 5, 6\}}\cap B_{\{1, 2, 3, 4, 5, 6\}}\cap B_{\{7, 8\}}\cap B_{\{7, 8, 9\}}\cap B_{\{10, 11\}}\cap B_{\{10, 11, 12\}}.
P_{Br}=\frac{16}{3}\frac{(\frac{e^{2}}{4\pi\epsilon_{0}})^{3}}{(m_{e}c^{2})^{\frac{3}{2}}\hbar}Z_{i}^{2}n_{i}n_{e}(k_{B}T_{e})^{\frac{1}{2}}G(y_{p})
\sum_{i=1}^m\theta_i\Big(N_{\lfloor nt_i\rfloor}(x)N_{\lfloor nt_i\rfloor}(y)-N_{\lfloor nt_{i-1}\rfloor}(x)N_{\lfloor nt_{i-1}\rfloor}(y)\Big)=\sum_{i,j=1}^m \theta_{\max(i,j)}d_{i,n}(x)d_{j,n}(y)
X_1 \, \colon \; -w^2 \; = \; (x^2+y^2)^2+y^3z-yz^3, X_2 \, \colon \; -w^2 \; = \; x^4+y^3z-yz^3, X_3 \, \colon \; \gamma w^2 \; = \; x^4+y^4+z^4.
\prod_{B}\frac{\xi(\varepsilon_b z + m_b)}{\xi(z+m_b +j_b)}\leq C^{|B|}\prod_{b: m_b = 1}q(z)^{-1}\cdot \log(5+|t|) \cdot \begin{cases}(1+|t|)^{-0.02}, \mbox{some $j_b \neq 0$,}(1+|t|)^{0.01/n^2}, \mbox{all $j_b=0$.}\end{cases}
\int_{\mathbb{R}^{n+1}_+\setminus B_{\rho_{i_j}}^+}{y^a|\nabla U_{k^\star}|^2\,dX}+\int_{\mathbb{R}^n\setminus\{B_{\rho_{i_j}}\cap\{y=0\}\}}{(U_{k^\star})_+^{2^*_s}(x,0)\,dx}\geq \eta_0,
B_{t}(m,\ell^{2n+1}d)-\ell^{\lambda-1}\left(\frac{(-1)^{\lambda}m}{\ell}\right)B_{t}(m,\ell^{2n-1}d) = B_{t}(m,\ell^{2n}(\ell d))-\ell^{\lambda-1}\left(\frac{(-1)^{\lambda}m}{\ell}\right)B_{t}(m,\ell^{2n-2}(\ell d)) = \ell^{(k-2)n}\, B_{t}(\ell^{2n}m,\ell d)\equiv 0\pmod{\ell^{(k-2)n}}.
\hspace{5 . 0 c m}\operatorname{e x p}\left\{- i \frac{2 \pi}{e L}\vec{\cal D}\vec{n}\right\}\operatorname{e x p}\left\{i \frac{2 \pi}{L}\int d^{3}x \tilde{\rho}_{\mathrm{m a t t}}^{3}\vec{n}\vec{x}\right\}\ ,
\phi_{\vec{u}}(\vec{t})= \frac{2^{N/2}\Gamma(N/2+1/2)}{E_N^{N-1}\sqrt{\pi}\;\Gamma(N/2)}\frac{1}{S_{N-1}}\int_0^{\sqrt{2E_N}}\int_\Omega e^{\frac{i}{N}\vec{t}^T A_N^{1/2}\hat{\vec{n}}r}\left( E_N - \frac{r^2}{2}\ \right)^{N/2-1}r^{N-1}d\hat{\vec{n}}dr.
\begin{array}{@{}l@{}}{\displaystyle |\sum_{k=1}^{m}J_2^{k,1}| \leq C\big\{h^{2r+2}+(\Delta t)^{4}\big\}+C\|\theta_{\psi}^{m}\|_{\mathcal{L}^2}^{2}+ C\Delta t \sum_{k=1}^{m-1}{\|\theta_{\psi}^{k}\|_{\mathcal{L}^2}^{2}}.}\end{array}
\partial_t v_j =\frac{1}{r^{N-1}\nu(r)}\partial_r(r^{N-1}\nu(r)\partial_r v_j) =\frac{1}{r^{N+k-1}r^{-k}\nu(r)}\partial_r(r^{N+k-1}r^{-k}\nu(r)\partial_r v_j), r>0,\,\,t>0,
E_{02}=\frac{5}{6}E_{01}\circ E_{01}-\frac{5}{2}E_{00}, E_{03}=\frac{5}{2}E_{01}\circ E_{01}\circ E_{01}-\frac{27}{2}E_{01}-6 E_{10}, E_{11}=E_{10}\circ E_{01}-\frac{4}{3}E_{01}\circ E_{01}+4E_{00}.
\dfrac{d^{k}}{dx^{k}}\phi_{-\varepsilon ,\lambda}\left( x\right) =-\frac{\beta_{1}}{\sqrt{\lambda}}\dfrac{d^{k}}{dx^{k}}\left( \sin \sqrt{\lambda}\left( x-a\right) \right) +O\left( \left( \sqrt{\lambda}\right)^{k-2}e^{\left\vert t\right\vert \left( x-a\right)}\right) ,
Q_{\rho}^{\nu}:=\{x\in\mathbb{R}^{n}:~|x\cdot\nu_{i}|<\rho/2~,~i=1,\ldots,n\}~,\quad\hat{Q}_{\rho}^{\nu}:=\mathbb{R}^{n}\setminus Q_{\rho}^{\nu},
\begin{array}{rl}\dot{x}= 1-x^2, \dot{y}= y^2+(2\beta-A-B)xy+(2\gamma+A-B)y &+\left((\beta^2+\beta-n(n+1)-(1+n+\beta)(A+B)\right)x^2 &+\left((\beta-\gamma)A-(\gamma+\beta)B+2\gamma\beta\right)x &+(n+1-\gamma)B+(n+1+\gamma)A+n(n+1)+\gamma^2-\beta .\end{array}
W_{\rm lr}(t) = W_{\rm lr}(t_1) + W_{\rm lfs}(t_1,t_2) + W_{\rm rfs}(t_2,t_1) + W_{\rm mm}(t_2,t_1,t_2) + W_{\rm lr}(t_2) \leq W_{\rm lr}(t_1) + W_{\rm mm}(t_1,t_1,t_2) + W_{\rm mm}(t_2,t_1,t_1) + W_{\rm mm}(t_2,t_1,t_2) + W_{\rm lr}(t_2)
\dim_{H}\left(\left\{x\in Y \mid \overline{\lim}\left\lvert \frac{1}{N}\sum_{n=0}^{N-1}f\left(u_{p(n)}.x\right)\right\rvert > 0 \right\}\right) = \dim_{H}\left(\left\{x\in Y \mid \overline{\lim}\left\lvert B_{N}f(x)\right\rvert > 0\right\}\right) \leq 3-\frac{\delta}{d},
\begin{array}{l l}{{y_{0}= 1}}&{{y_{4}= i b x_{1}}}\\{{y_{1}= - ( J_{1}^{2}+ J_{2}^{2}+ \frac{1}{4}+ 2 \alpha / x_{3}^{2})}}&{{y_{5}= i b x_{2}}}\\{{y_{2}= - 2 J_{3}}}&{{y_{6}= - \frac{1}{2}i b \{x_{3}, J_{1}\}}}\\{{y_{3}= b^{2}x_{3}^{2}}}&{{y_{7}= - \frac{1}{2}i b \{x_{3}, J_{2}\}\quad ,}}\\ \end{array}
\lambda_{j|v}^{r,s}(z_j,s) =[ -\Lambda_{M_j}^{r,s}(g_j^{v-1}(z_j,s),h^{v-1}(s))]_v+[\sum_{p,q=1}^n\Lambda_{N_j}^{p,q}(z_j,s)\frac{\partial{g_j^r}^{v-1}}{\partial z_j^{p}}\frac{\partial{g_j^s}^{v-1}}{\partial z_j^{q}}]_v
\left|\langle T(f)_b, \varphi \rangle - \langle T(f)(\cdot), \varphi \rangle \right| = \left| \lim_{r \nearrow 1}\int_0^{2\pi}T(f)(re^{i\theta})\varphi(\theta) \,d\theta - \int_0^{2\pi}T(f)(e^{i\theta}) \varphi(\theta) \,d\theta \right| &\leq C_\varphi \lim_{r\nearrow 1}\int_0^{2\pi}|T(f)(re^{i\theta}) - T(f)(e^{i\theta})| \,d\theta = 0,
p_{(W',S')}(t) =\frac{t^4}{t^4}\cdot \frac{2p_1(t^{-1})p_3(t^{-1})}{2p_1(t^{-1})p_3(t^{-1}) - 2np_3(t^{-1}) +n(n-1)}&=\frac{2p_1(t)p_3(t)}{2p_1(t)p_3(t) - 2ntp_3(t) +t^4 n(n-1)}
\varphi^{\prime}(a_{i})=-\frac{A_{i}e^{-\kappa a_{i}}(1+\kappa a_{i})}{a_{i}^{2}}=-\frac{1}{4\pi\epsilon_{r}\epsilon_{0}}\frac{z_{i}q}{a_{i}^{2}}=\varphi_{sp}^{\prime}(a_{i}),
\left\lbrace\begin{array}{lll}w''-\Delta w ={\cal L}\theta \hbox{in}(0,T) \times\Omega, \partial_\nu w={\cal R}\theta \hbox{on}(0,T) \times\Gamma, t=0: w=0,w'=\theta \hbox{in}~ \Omega \end{array}\right.
\sum_{m \geq 1}\frac{A(1, m)}{m}H_{m p, p}= L(1, \tilde f) \frac{2}{\pi}\int_0^\infty k(t) \tanh(\pi t) t \, d t +{}&\qquad{}+ O \Bigl ( p^{\frac{3}{7}- \varepsilon}\int_0^\infty k(t) \tanh(\pi t) t^{\frac{1}{7}+ \varepsilon}\, d t \Bigr ).
x_{n+1}^{-2s}w(x',x_{n+1})=x_{n+1}^{1-2s}\int_{0}^{1}\partial_{n+1}w(x',tx_{n+1})\,dt=\int_{0}^{1}(tx_{n+1})^{1-2s}\partial_{n+1}w(x',tx_{n+1})t^{2s-1}\,dt.
\alpha := d_1H_1 - \sum_{q=1}^{d_1^2}E_{1,q}, \beta_{p,1}:= d_{p,2}H_p + \sum_{q = 1}^{d_{p,2}^2}E_{p,q}(p=2,\ldots, m), \beta_{p,2}:= d_{p,3}H_p - \sum_{q = 1}^{d_{p,3}^2}E_{p,q}(p=2,\ldots, m).
K_{3 / 2}( \pi n_{1}| \vec{N}| ( 1 + T ) ) = \frac{1 + \pi n_{1}| \vec{N}| ( 1 + T )}{\pi n_{1}| \vec{N}| ( 1 + T )}\, ( 2 \pi n_{1}| \vec{N}| ( 1 + T ) ) )^{\frac{- 1}{2}}\, e^{- \pi n_{1}| \vec{N}| ( 1 + T )}. \, [ 5 p t ]
P\Big(- \sum_{j=1}^k \overline{U}_j \ge \epsilon c_o\bar h_4(n) \Big)\le \big(\epsilon c_o \bar h_4(n) \big)^{-2}\frac{C_1 k m^2}{(\log m)^4}\le \frac{2 C_1}{(\epsilon c_o)^2 k (\log_2 n)^2}\,.
E_{0}= \frac{\pi}{p^{+}}\left( \left( \frac{p^{I_{N}}}{\pi}\right)^{2}+ \left( \frac{m}{3}\right)^{2}( x^{i_{N}})^{2}+ \left( \frac{m}{6}\right)^{2}( x^{i_{N}^{\prime}})^{2}+ \frac{i}{\pi}\frac{m}{3}\psi_{0}\gamma^{4}\Omega \psi_{0}- \frac{i}{\pi}\frac{m}{6}\psi_{0}^{\prime}\gamma^{4}\Omega \psi_{0}^{\prime}\right) ~ .
\delta S ={\frac{1}{2}}\int_{{\cal M}^{2}}d^{2}\xi \sqrt{- g}\delta g^{m n}( \xi ) \left( \partial_{m}{\hat{X}}^{\underline{{m}}}\partial_{n}{\hat{X}}_{\underline{{m}}}-{\frac{1}{2}}g_{m n}g^{k l}\partial_{k}{\hat{X}}^{\underline{{m}}}\partial_{l}{\hat{X}}_{\underline{{m}}}\right) -
\binom{4n-2}{2n-1}\pm2\binom{2n-2}{n-1}= \frac{2^n(4n-3)(4n-5)\cdots(2n+1)}{(n-1)!}\pm\frac{2^n(2n-3)!!}{(n-1)!}= \frac{2^n}{(n-1)!}((4n-3)(4n-5)\cdots(2n+1)\pm(2n-3)!!).
g_sl_s^{D-p-1}~k_1.k_3~ k_2.k_4~\tilde P( \zeta ) \int d^{D-p-1}p_1~ \int d^{D-p-1}p_2~ \frac{q^2 e^{i ({\vec p_1}\cdot{\vec \epsilon_1}+{\vec p_2}\cdot{\vec \epsilon_2})}}{( q^2 + p_1^2) ( q^2 + p_2^2 )}~ \delta^{p+1}\left( \sum_i k_i\right) ~.
\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}d^2(\tilde{z}(t),z_*) =\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\big[d_W^2(\tilde{u}(t),u_*)+d_W^2(\tilde{v}(t),v_*)\big] &\leq \mathcal{H}(u_*)-\mathcal{H}(\tilde{u}(t))+\mathcal{H}(v_*)-\mathcal{H}(\tilde{v}(t)).
\begin{aligned}p_{\alpha(i)}(k,m,n) - p_{\alpha(i+1)}(k,m,n) = ~p_{v(\alpha(i))}\big(k,m-1,n-(m-1)Nw(\alpha(i)) - \alpha(i)\big) + p_{v(\alpha(i))}\big(k-1,m-1,n-(m-1)N(w(\alpha(i))-1) - \alpha(i)\big),\end{aligned}
\hat{\mathbb{E}}[\beta X]=\beta^+\hat{\mathbb{E}}[X]+\beta^-\hat{\mathbb{E}}[-X]=|\beta|\hat{\mathbb{E}}[X], \hat{\mathbb{E}}[\beta Y]=\beta^+\hat{\mathbb{E}}[Y]+\beta^-\hat{\mathbb{E}}[-Y]=|\beta|\hat{\mathbb{E}}[Y].
(\hat \lambda_i-\lambda_{k})P_{k}\hat P_i =P_{k}E \hat P_i, \frac{1}{\hat\lambda_i-\lambda_j}-\frac{1}{\lambda_{k}-\lambda_j}=\frac{\lambda_{k}-\hat\lambda_i}{(\hat\lambda_i-\lambda_j)(\lambda_{k}-\lambda_j)},
I_5= \delta T_4d(r^{6-n}) = -((T_4)_{ij}{r^{6-n}}_{,j})_{,i}= -(T_4)_{ij,i}(r^{6-n})_{,j}-(T_4)_{ij}(r^{6-n})_{,ji}= (n-6)\big[r^{4-n}(T_4)_{ij,i}x^j-(n-4)r^{2-n}(T_4)_{ij}x^ix^j+r^{4-n}{\rm tr}(T_4)\big] := (n-6)[I_1^{(5)}+I_2^{(5)}+I_3^{(5)}].
\begin{aligned}\alpha_{Lep}=\alpha(\delta,y^\delta):= &\max\{\alpha \in \Delta_q: \,\|x_{\alpha^\prime}^\delta-x_{\alpha}^\delta\| \le 2\, \Sigma(\alpha^\prime,\delta) \mbox{for all}\alpha^\prime\in\Delta_q\cap [\alpha_0, \alpha]\}.\end{aligned}
{\cal L}={\cal L}_{0}^{N = 2}+{\cal L}^{g r}+ \sum_{i = 0}^{k}t_{i}\int \phi_{i}^{( 2 )}+ \sum_{n = 1}^{\infty}\sum_{i = 0}^{k}t_{i , n}\sigma_{n}( \phi_{i})^{( 2 )}
{\cal L}= \frac{1}{2}\partial^{\mu}\phi_{1}\partial_{\mu}\phi_{1}- \alpha_{1}({\phi_{2}}^{2}+ \epsilon_{1}) ( 1 - c o s \phi_{1}) + \frac{1}{2}\partial^{\mu}\phi_{2}\partial_{\mu}\phi_{2}- \alpha_{2}({\phi_{1}}^{2}+ \epsilon_{2}) ( 1 - c o s \phi_{2})
\begin{array}{r c l}{{\delta C}}&{{=}}&{{\left( \Lambda^{( \cdot )}+ m \lambda \right) e^{B}\, ,}}\\{{}}&{{}}&{{}}\\{{G}}&{{=}}&{{d C - d B C + \frac{m}{2}e^{B}\, .}}\\{{}}&{{}}&{{}}\\ \end{array}
\mathbb K_{\alpha,\theta}^{\textrm{MB}}(x,y) := \frac{\theta}{(2\pi i)^2}\left( \frac{y}{x}\right)^{\alpha}\int_{C^{\delta}}ds \int_{\Sigma}dt \frac{x^{-\theta s -1}y^{\theta t}}{s-t}\frac{\Gamma(\theta s+\alpha+1)}{\Gamma(\theta t+\alpha+1)}\frac{\Gamma(s+1)}{\Gamma(t+1)}\frac{\sin \pi s}{\sin \pi t},
\begin{array}{lcl}\star_t\tau_3(t) = \frac{\sqrt{6}}{7y(t)^5}(-f^{1367}-f^{1457}-f^{2357}+f^{2467})+\frac{4\sqrt{6}}{21y(t)^5}(f^{1234}+f^{1256}+f^{3456}),\end{array}
\epsilon \left( C_{a}^{\alpha_{1}}\right) = \epsilon \left( l^{\alpha_{1}}\right) = \epsilon \left( f_{\alpha_{1}}\right) , \epsilon \left( p_{a}^{\alpha_{1}}\right) = \epsilon \left( f_{\alpha_{1}}\right) + 1 , a = 1 , 2 , 3 .
\theta\circ \langle \boldsymbol x, \boldsymbol x' \rangle = \theta\circ \langle \pi_2,\pi_1 \rangle\circ \langle \boldsymbol x', \boldsymbol x \rangle &= i\circ \theta \circ \langle \boldsymbol x', \boldsymbol x \rangle\;.
F_\pm (\theta,\phi)=\left( \begin{array}{l}f_{j}f_{j-1}\vdots f_{-j+1}f_{-j}\end{array}\right)_\pm \!\!\! (\theta,\phi)= v\left( \begin{array}{l}D^{j}_{j,j}D^{j}_{j-1,j}\vdots D^{j}_{-j+1,j}D^{j}_{-j,j}\end{array}\right) (\theta,\phi,\mp \phi),
+\sum_{l=t+1}^{r-2}q^{\theta(l>t+1)\sum_{\nu-r+t+1}^{\nu-r+l-1}n'_i}\prod_{\stackrel{i=1}{i\neq \nu-r+t}}^{\nu-2}\left[\begin{array}{c}n'_i+\tilde{n}_i-V_{i,r}+\theta(i>\nu-r+l)+\theta(t>0)\sum_{m=0}^{t-1}\delta_{i,\nu-r+m}n'_i-\delta_{i,\nu-r+l}\end{array}\right]_q \times
\int_{\Omega}\left(\nabla w_n \nabla v -\lambda_n m(x)w_n v -\lambda_n^{\frac{p-q}{2-q}}a(x)w_n^{p-1}v\right) -\int_{\partial \Omega}b(x)w_n^{q-1}v =0 \forall v \in H^1(\Omega)
2\Lambda_{ij}\L_{kj}\omega_{ki}=\sum_a - \Omega_{kj}(e_a)\wedge (\omega_{ki}\circ J_{ij})(e_a) + (\Omega_{kj}\circ J_{ij})(e_a) \wedge \omega_{ki}(e_a) + \langle J_{ij}, \omega_{ki}\rangle \Omega_{kj}.
\left\vert \frac{1}{n}\sum_{i=1}^{n}b(X_{i})b(X_{i})^{\prime}\{E[K_{h}(Y_{i}-\hat{\gamma}(X_{i}))|X_{i}]-f(0|X_{i})\}\right\vert_{\infty}& \leq C\frac{1}{n}\sum_{i=1}^{n}\left\vert \hat{\Delta}(X_{i})\right\vert+Ch^{2}=O_{p}(n^{-d_{\gamma}}+h^{2}),
\mathbb{E}^{0}({\bf 1}_{\tau_{x}>T}L_{T}|\mathcal{F}_{t}^{Q})= \mathbb{P}^{0}(\tau_{x}>T) + \int_{0}^{t}\mathbb{E}^{0}({\bf 1}_{\tau_{x>u}}L_{u}h(X_{u})G(T-u,x-X_{u})|\mathcal{F}_{u}^{Q})dQ_{u}.
i_1^* (Z_1 -Z_2 ) = g_1 (z)-g_2 (z), i_2^* (Z_1 -Z_2 ) = g_1 (b)-g_2 (b), i_{\Delta}^* (Z_1 -Z_2 ) = (0,\sqrt{a_0})+(0,-\sqrt{a_0})-\infty -w(\infty ),
\sum_{i=0}^{[\frac{2n+1-\nu}{2}]}{{2n+1-2i}\choose{\nu}}A_i = -q^{n-\nu+1/2}\sum_{i=0}^{[\frac{\nu}{2}]}{{2n+1-2i}\choose{2n+1-\nu}}A_i\qquad(\nu=0,1,\cdots, 2n+1),
d\varphi_{ri}\left( i\right) =\left. \frac{d}{dt}\right\vert_{r}\left(l\left( t\right) i,e^{\theta \left( t\right) i}\right) =\left( l^{\prime}\left( r\right) i,\theta^{\prime}\left( r\right) ie^{\theta \left(r\right) i}\right)
{\|T|_{H_0}x\|}^2 ={\left\|T\left(\displaystyle \sum_{n=1}^{\infty}s_{n}(t_n f_n+\sqrt{(1-{t_n}^2}g_n)\right)\right \|}^2 & = \displaystyle \sum_{n=1}^{\infty}{|s_{n}|}^2 \left({t_n}^2{a_n}^2+(1-{t_n}^2){b_n}^2\right) & > \sum_{n=1}^{\infty}{|s_{n}|}^2{c_n}^2 & >{a}^2.
(\sigma^{-1})^{*}h_{R}^{1(n)}(V,V)=h_{R}^{1(n)}((\sigma^{-1})_{*}V,(\sigma^{-1})_{*}V)=\eta|_{H_{R}^{1(n)}}((\sigma^{-1})_{*}V,(\sigma^{-1})_{*}V).
\lim_{t\rightarrow+\infty}P_\lambda\big(\eta_t(O)=1\big) =\lim_{t\rightarrow+\infty}P_\lambda\big(\xi_t(O)>0\big) &\geq\limsup_{t\rightarrow+\infty}\frac{(E\xi_t(O))^2}{E(\xi_t^2(O))}=\limsup_{t\rightarrow+\infty}\frac{1}{E(\xi_t^2(O))}&\geq \frac{\inf_{x\in \mathbb{Z}^d}K(x)}{K(O)}>0.
\hat{L}^{2}Y_{lm}(\theta,\phi)\equiv\{-\frac{1}{sin^{2}\theta}[sin\theta\frac{\partial}{\partial\theta}(sin\theta\frac{\partial}{\partial\theta})+\frac{\partial^{2}}{\partial\phi^{2}}]\}Y_{lm}(\theta,\phi)=l(l+1)Y_{lm}(\theta,\phi).
\sim T r \left[ \sigma^{3}{\cal T}^{a_{1}}. . .{\cal T}^{a_{n}}\right] \left( \partial^{k}\frac{1}{{{\partial \kern - 0 . 1 5 e m \raise 0 . 1 7 e x \llap{/}\kern 0 . 1 5 e m \relax}}_{f}^{m + ( n - m ) + k}}\right) .
\mathrm{(i)}~~\mathrm{E}[\mathbf{Y}|\boldsymbol{a}<\mathbf{Y}\leq \boldsymbol{b}]=\boldsymbol{\mu}+\frac{\mathbf{\Sigma}^{\frac{1}{2}}\boldsymbol{\delta}}{F_{\mathbf{Z}}(\boldsymbol{\xi_{a}},\boldsymbol{\xi_{b}})},
\mathcal{L}_{I_J}(s) &= \exp\Big(- 2 \pi \lambda_J \frac{s P_J R_{\tau}^{2 - \alpha}}{\alpha - 2}{}_2F_1(1, 1 - \frac{2}{\alpha}; 2 - \frac{2}{\alpha}; -\frac{s P_J}{R_{\tau}^{\alpha}})\Big) \times \int_{0}^{\infty}H(v, s) f(v) dv,
G(f) =(A+4)f(z)^2-4B(z) &=\left(\frac{(h(z+1)-h(z))^2}{h(z)h(z+1)}+4\right)\left(\frac{h(z)^2+Q(z)^2}{2h(z)Q(z)}\right)^2-4\left(\frac{(h(z+1)+h(z))^2}{4h(z)h(z+1)}\right) &=\frac{(h(z)+h(z+1))^2(h(z)-Q(z))^2(h(z)+Q(z))^2}{4h(z)^3h(z+1)Q(z)^2}.