harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
[X,\nabla_YZ] =\left.\frac{d\}{dt}\right|_{t=0}(\Phi_{-t})_*(\nabla_YZ)\ \textrm{(because of \eqref{eq:XYPhi})}=\left.\frac{d\}{dt}\right|_{t=0}\left(\nabla_{(\Phi_{-t})_*(Y)}(\Phi_{-t})_*(Z)\right) =\nabla_{[X,Y]}Z+\nabla_Y[X,Z].
f_m(k)\ =\ \frac{2(\partial_r J_m(kr)|_{r=R}H^{(1)}_{m-\beta}(kR)-\partial_r H^{(1)}_{m-\beta}(kr) |_{r=R}J_m(kR))}{H^{(2)}_{m-\beta}(kR)\partial_r H^{(1)}_{m-\beta}(kr) |_{r=R}-H^{(1)}_{m-\beta}(kR)\partial_r H^{(2)}_{m-\beta}(kr) |_{r=R}}\ ,
\frac{1}{\pi i}\int_{0}^{2\pi}\frac{f(\beta)z(\alpha)z(\beta)}{z(\alpha)-z(\beta)}\frac{z_{\beta}(\beta)}{(z(\beta))^{2}}d\beta = \frac{1}{\pi i}\int_{0}^{2\pi}\frac{f(\beta)z(\alpha)z_{\beta}(\beta)}{\left(z(\alpha)-z(\beta)\right)z(\beta)}d\beta=-zH\left(\frac{f}{z}\right).
\begin{array}{rcl}\left(\hat{A}_{3,p}\left(\check{A}_{p}\right)_{S}^{\left(+\right)}\hat{A}_{3,p}^{\top}\right) = \left(\hat{A}_{3,p}U\right) = \dim R\left(U\right)-\dim N\left(\hat{A}_{3,p}\right)\cap R\left(U\right).\end{array}
W_{cr}=({g^2 \over{4\pi}})^2\, 2\,C_F\,C_A ({\cal A})^2 \int_{0}^{1}d\,\xi \int_{0}^{1}d\,\eta\,log{|sin\rho(\xi-\eta)|\over{|sin\rho\xi|}}log{|sin\rho(\xi-\eta)|\over{|sin\rho\eta|}},
Q_{m}-\eta \, k^2 \, a^{5-h}\, \sum_{j=1 \atop j \neq m}^\aleph{\bf{P}}_0 \cdot \Upsilon_k(z_{m}, z_j) \cdot{Q}_j=i \, k \, a^{5-h}\;{\bf{P}}_0 \cdot H^{Inc}(z_{m}),
\mathcal{F}^{-1}\left[ \frac{1}{\xi^{2}+\lambda}\right] \left( x\right) =\frac{1}{2\sqrt{\lambda}}\mathrm{e}^{-\left\vert x\right\vert \sqrt{\lambda}},\;\;x\in \mathbb{R},\;\lambda \in \mathbb{C}\backslash \left( -\infty ,0\right] ,
\varphi_{\underline{d},\underline{d}''}(\varphi_{\underline{d}''',\underline{d}}(V_{\underline{d}'''}))=\varphi_{\underline{d}''',\underline{d}''}(V_{\underline{d}'''})=V_{\underline{d}''}^{X_{2}^{c},0}.
\sum_{\rho}\frac{1}{\rho}=\sum_{k=1}^{\infty}\frac{1}{\frac{1}{2}+i\tau_{k}}+\frac{1}{\frac{1}{2}-i\tau_{k}}=\sum_{k=1}^{\infty}\frac{1}{\frac{1}{4}+\tau_{k}^{2}},
T_{2,k}=\frac{B_{\tilde{n}}(0)}{M^{k}(M-N)^{\tilde{n}-k}}+\sum_{l=1}^{k-1}\binom{k}{l}\frac{B_{\tilde{n}}(0)}{M^{k}N^{\tilde{n}-l}(M-N)^{\tilde{n}-k+l}}&+\frac{B_{\tilde{n}}(0)}{M^kN^{\tilde{n}-k}}&=\biggl(\frac{1}{M^{k}(M-N)^{\tilde{n}-k}}+\frac{M^k-N^k-(M-N)^k}{M^{k}(N(M-N))^{\tilde{n}}}+\frac{1}{M^{k}N^{\tilde{n}-k}}\biggr)B_{\tilde{n}}(0)
\frac{d}{dt}\int_\Omega \gamma(v)u^2+2\int_\Omega |\nabla(u\gamma(v))|^2+\int_\Omega u^2\gamma''(v)|\nabla v|^2-\int_\Omega u^3\gamma'(v) &=-\int_\Omega u\gamma'(v)v-2\int_\Omega u\gamma'(v)\nabla u\cdot\nabla v.
\frac{\langle M^{2}\rangle}{N^{2}}=\frac{1}{N^{2}}\langle s_{x}^{2}+s_{y}^{2}\rangle=\frac{1}{N^{2}}\langle(\sum_{i=1}^{N}cos\theta_{i})^{2}+(\sum_{i=1}^{N}sin\theta_{i})^{2}\rangle
\widetilde\Delta_n u(j_1,j_2):= \frac{n^2}{4 \pi^2}\bigg( \frac{10}{3}u(j_1,j_2)-\frac{2}{3}\Big(u(j_1+1,j_2)+u(j_1,j_2+1) &+u(j_1-1,j_2)+u(j_1,j_2-1)\Big)-\frac{1}{6}\Big(u(j_1-1,j_2-1) &+u(j_1-1,j_2+1)+u(j_1+1,j_2-1)+u(j_1+1,j_2+1)\Big)\bigg).
\xi_m = -i d \theta_{\gamma_m}+ 2\pi i \cdot A + \pi i \cdot V \cdot ( \frac{1}{\zeta}d c - \zeta d \overline{c}) \xi_e = i d \theta_{\gamma_e}+ \pi R \cdot ( \frac{1}{\zeta}d c + \zeta d \overline{c})
{1\over N^2}\left<\Psi\right|{T_1}{T_2}\left| \Psi \right>={1 \over N^2}\left< \Omega \right|{T_1}{T_2}\left|\Omega\right> -{1 \over N^2}\int{\left[T_2,T\right]}\int{\left[T_1,T\right]}.
\mathcal{N}(u;f,\psi_h) := -\sum_{i,j}\int_{T_{ij}}f (u - v)\,\partial_v\psi_h\,{\rm d}v\,{\rm d}x \,\, - \sum_{i = 1}^{N_x}\sum_{j=0}^{N_v-1}\int_{I_i}\left((u - v) f[\![\psi_h]\!]\right)_{x,j+1/2}\,{\rm d}x \, ,
\partial_su(s,t)+J_t(\partial_tu(s,t)-X_{H^{(k)}}(u(s,t))) =\partial_sv(s,t-\rho(s))-\rho'(s)\partial_tv(s,t-\rho(s)) +J_t(v(s,t-\rho(s)))(\partial_tv(s,t-\rho(s))-X_{H^{(k)}}(v(s,t-\rho(s))))
\mathbf M^{*,0}_{\infty}(\phi):=\mathbf M^{\{U^*_0,0\}}_{\infty}(\phi)=\mathbf M^{\{U^*_0,0\}}_{c^*_{0}}(\phi)\cup\,\bigcup_{i=0}^{\infty}\mathbf M^{\{\emptyset,c^*_i\}}_{c^*_{i+1}}(\phi),
P_{\sigma^{-1}(0)}z_n-y=\sum_{i\in\sigma^{-1}(0)}x^{0*}_i(z_n)x^{0}_i- \sum_{i\in\sigma_n^{-1}(0)}x^{0*}_i(z_n) x^0_i=\sum_{i\in\sigma^{-1}(0)\cap(n,\infty)}x^{0*}_i(z_n)x^{0}_i.
\mathcal{R}_{f_1, f_2}(y, T) \ll \frac{1}{q^{1-\varepsilon}}+ \frac{T^{\varepsilon}}{T}(qT)^{1/2 + \theta}\Big(q + \frac{T}{Q}\Big)^{1/2}\Big( 1+ \frac{T}{qQ}\Big)^{1/2}&\ll \frac{1}{q^{1-\varepsilon}}+ T^{\varepsilon}(qT)^{\theta}\Big( \frac{q}{T^{1/2}}+ \frac{T^{1/2}}{Q}\Big).
{\bf r}( \lambda ) = \frac{\lambda^{\frac{3}{2}}+ \lambda^{- \frac{3}{2}}}{\lambda^{\frac{3}{2}}- \lambda^{- \frac{3}{2}}}\sum_{i = 1}^{3}H_{i}\otimes H_{i}+ \frac{2}{\lambda^{\frac{3}{2}}- \lambda^{- \frac{3}{2}}}\sum_{( i j )}\left( \lambda^{- \frac{1}{2}}E_{i j}\otimes{E}_{j i}+ \lambda^{\frac{1}{2}}E_{j i}\otimes{E}_{i j}\right) ,
{\cal H}_{\tilde F}(\Psi(t))=\Vert\dot\psi(t)\Vert^2+\Vert\nabla\psi_{reg}(t)\Vert^2+m^2\Vert\psi_{reg}(t)\Vert^2+\tilde U(\zeta(t))-{\cal G}(\zeta(t))=const,t\in [0,\tau].
t_{1}^{p^{+}- r^{-}}\int_{M}| D \mathrm{u}( z ) |^{p( z )}\,\, dv_{g}( z ) + t_{1}^{q^{+}- r^{-}}\int_{M}\mu ( z ) \,| D \mathrm{u}( z ) |^{q( z )}\,\, dv_{g}( z ) & - t_{1}^{-r^{-}- \gamma^{-}+ 1}\int_{M}g( z ) | \mathrm{u}( z ) |^{1 - \gamma ( z )}\,\, dv_{g}( z ) = \lambda \int_{M}| \mathrm{u}( z )|^{r( z )}\,\, dv_{g}( z ),
\|\delta_n(a)\|_{M,N}= \left\|\int_0^1e^{2\pi in\theta}\rho_\theta^{-1}\delta\rho_\theta(a)\,d\theta\right\|_{M,N}\le \int_0^1\|\rho_\theta^{-1}\delta\rho_\theta(a)\|_{M,N}\,d\theta \le C\|a\|_{M',N'}\,.
\mu\big(\begin{smallmatrix}b_{11}b_{12}b_{21}b_{22}\end{smallmatrix}\big)^\ast = \mu_2(\sqrt{L}a_{11})^\ast\mathcal F^\ast \mu_1(-La_{11}a_{12})^\ast \mathcal{F}\, \mu_1(a_{21}/a_{11})^\ast & = \mu_2(1/(\sqrt{L}a_{11}))\,\mathcal F^\ast \mu_1(La_{11}a_{12}) \mathcal{F}\, \mu_1(-a_{21}/a_{11}).
\mathrm{R e s}_{\bar{z}^{\prime}= \bar{z}}^{\Psi}\bar{\partial}^{\Psi}\! : \! \psi ( z^{\prime}) \! : \bar{\partial}^{\Psi}\! : \! \phi ( z ) \! : = - : \! ( \phi_{; \mu \eta}- i C_{\eta \nu \mu}\phi_{; \nu}) \psi^{; \eta}\bar{\partial}^{\Psi}\! X^{\mu}\! : + O ( \epsilon^{5}) .
\lambda_{d}=(\underbrace{a_{1},\dots,a_{1}}_{k_{1}},\underbrace{a_{2},\dots,a_{2}}_{k_{2}},\dots,\underbrace{a_{b},\dots a_{b}}_{k_{b}},\underbrace{0,\dots,0}_{z}, \quad\ \underbrace{-a_{b},\dots,-a_{b}}_{l_{b}},\dots,\underbrace{-a_{2},\dots,-a_{2}}_{l_{2}},\underbrace{-a_{1},\dots,-a_{1}}_{l_{1}})\in\mathbb{Z}^{v},
A = \frac{\Gamma \left( 1 - n + m \alpha_{-}^{2}\right)}{\Gamma \left( - n + ( 1 + m ) \alpha_{-}^{2}\right) \Gamma \left( 1 - \alpha_{-}^{2}\right)}\, C_{n m , 1 2}^{n , m + 1}A_{n , m + 1}^{\alpha}
X_{n+1}=\bullet|X_{[1:n]}\sim \frac{\alpha}{\alpha+n-1}c_0(\{\bullet-X_n+1\})+\frac{1}{\alpha+n-1}\sum\limits_{i=1}^{n-1}\delta_{X_{i+1}-X_i+(1-\delta_{X_i 0})}(\{\bullet\}).
L_s(g-g_M)(z) = \sum_{k=0}^M (-1)^k \frac{g^{(k)}(1)}{k!}\frac{2^{2s+k}e^{2\pi i a}}{\lambda^{2s+k}(1-z)^{2s}}\zeta\left( 2s+k, a, 1+\frac{2}{\lambda(1-z)}\right).
\sum_{j=1}^k (R-\rho)^{j}f_j(\rho) \leq \epsilon \sum_{j=1}^k (R-\rho)^j f_j(R) + \epsilon (R-\rho)^{k+\alpha}[\nabla^k\sigma]_{C^{\alpha}(B^{g_P}(p,R))}+ C_\epsilon \|\sigma\|_{L^\infty(B^{g_P}(p,R))}.
M = \frac{g_{YM}^2}{2V{\cal N}}(w^i+\Theta^{ij}n_j)^2 +\frac{V}{4g_{YM}^2{\cal N}}(m_{ij}+{\cal N}\Phi{ij})^2 +\frac{1}{\cal N}\sqrt{\frac{k_i^2}{\Sigma_i^2}},
\|v(t,\cdot)\|_{W^{\rho-1,\infty}_h}\le C_1 A\varepsilon \, , &\|\mathcal{L}v(t,\cdot)\|_{H^1_h}\le 5 B\varepsilon h^{-\sigma}\,, \|v(t,\cdot)\|_{H^s_h}\le B\varepsilon h^{-\sigma}\,,
S_{I I}\sim \frac{\sigma_{0}}{\pi}\left( \frac{\sigma_{0}}{2 \pi}\right)^{2 \nu - 1}\sum_{n \neq 0}^{}\left( \frac{r}{| n |}\right)^{2 \nu - 1}\frac{{\cos}^{2}\left( - \frac{2 \pi}{\sigma_{0}}| n | \tau_{0}- \frac{\pi}{2}\nu - \frac{\pi}{4}\right)}{{\cos}^{2}\left( - r \tau_{0}- \frac{\pi}{2}\nu - \frac{\pi}{4}\right)}~ | \phi_{n}( \sigma_{0}) |^{2}~ S_{I}
\frac{1}{s}=\sum_{i=1}^m \frac1{s_i}=\frac{1}{q_1}+ \frac{1}{q_2}+ \sum_{i=3}^m\frac1{p_i}\ge\sum_{i=1}^m\frac{1}{q_i}=\frac 1q>\frac1{r_{m+1}'},
u =\left(\frac{\eta_{1}\eta_{20}}{\eta_{4}\eta_{5}}\right)^{2},v=\left(\frac{\eta_{2}\eta_{5}\eta_{20}}{\eta_{1}\eta_{4}\eta_{10}}\right)^{2}k =\left(\frac{\eta_{4}\eta_{20}}{\eta_{2}\eta_{10}}\right)^{2},w =\left(\frac{\eta_{2}\eta_{20}}{\eta_{4}\eta_{10}}\right)^{3}.
H=\dot\rho\left(\dot s+s^2 \dot\rho \right), P=\dot s +2 s^2 \dot\rho-2 \nu s, begin{align*}4pt] D=\rho P-s \dot\rho, K=\rho^2 P+(1-2s \rho)\dot\rho+2\nu\rho.
S_{\mu_{T^*T}}\left(\psi_{\mu_{T^*T}}\left(-\frac{1}{s(|\lambda|,0 )^2}\right) \right) =-\frac{1}{s(|\lambda|,t)^2}\cdot-\frac{s(|\lambda|,t)^2}{|\lambda|^2}=\frac{1}{|\lambda|^2}
{\cal H}_{c a n}= \Pi^{-}\partial_{+}A_{-}+ \Pi^{i}\partial_{+}A_{i}-{\cal L}_{W e y l}= \frac 1 2 \Pi^{2}+ \frac 1 2 ( \partial_{i}A_{j})^{2}+ \Pi \partial_{i}A_{i}- A_{i}J^{i}- A_{-}J^{-}
\mu^{2}( t ) = \frac{\lambda_{\mathrm{R}}}{2}\tilde{I}_{1}^{\phi}( t ) - \frac{1}{2}a^{2}( t ) \hat{m}^{2}\left( 1 + \frac{\lambda_{\mathrm{R}}}{1 6 \pi^{2}}\bar{c}\right) + O ( \lambda_{\mathrm{R}}^{2}) \, ,
\lim_{t\rightarrow 0}\frac{\gamma_{n+1}([h_t])-\gamma_{n+1}(Q)}{t}= \frac{1}{(\sqrt{2\pi})^{n+1}}\int_{\mathbb{S}^n}g(\alpha_Q(u))e^{\frac{-r^2_Q(u)}{2}}r^{n+1}_Q(u)du &=\frac{1}{(\sqrt{2\pi})^{n+1}}\int_{\partial Q}g(\nu_Q(x))e^{\frac{-|x|^2}{2}}(x\cdot\nu_Q(x))d\mathcal{H}^n(x) &=\int_{\mathbb{S}^n}g\dS_{p,\gamma_{n+1},Q}.
D_uD_vD_wL=0\forall u,v,w \in 0\times(\mathbb{R}^n)^{\otimes 3}\times(\mathbb{R}^n)^{\otimes 4}\times (\mathbb{R}^n)^{\otimes m}\times(\mathbb{R}^n)^{\otimes m+1}\subset Y_n^m.
\tilde U_{N_sN_r}(\{\sigma_p\})=\int\limits_{C_{v_r}}\Psi_{\sigma,N_s}(t)\frac{d\theta dz}{2\pi i}P_{N_r}(t)+\int\limits_{\tilde C_r}\Psi_{\sigma,N_s}(t)\frac{d\theta dz}{2\pi i}P_{N_r}^{(a)}(t)
\phi_{k,n}(y_1,\cdots,y_k)=n^k(2\ln n)^{\frac{k}{2}}\times & \mathbb{P}(\xi^{(n)}(I_{n,k})=0,\ \xi^{(n)}(J_{n,k,j})>0,\ \forall\ 0\leq j\leq k).
T(n_i)+T(n_{i+1})+T(n_{i+1}) < \left(\frac{297}{81}+\frac{128}{243}\right) \cdot \frac{1}{X_0}&= \frac{1019}{243}\cdot \frac{1}{X_0}< 6 \cdot \frac{3}{4}\cdot \frac{1}{X_0}&= (k_i+k_{i+1}+k_{i+2}) \cdot \frac{3}{4}\cdot \frac{1}{X_0}.
\delta S = \frac{1}{4\pi}\,\mbox{Im}\,\, \left\{- \frac{i}{24}\int\!\! d^{12}z\D^{\alpha{\mathbf j}}L_{{\mathbf i}{\mathbf j}}D_\alpha^{\mathbf i}{\cal F}-3 \int\!\! d^8z_+ \sigma{\cal A}{\cal A}_D - \frac{i}{48}\int\!\! d^{12}z\ (H -{\bar H}){\bar{\cal A}}{\cal A}_D \right\}.
\int_{S^2(\tau,r')}|\Phi|^2 dVol_S^2 \leq \int_{r'}^\infty \int_{S^2(\tau,r)}\left(\frac{|\nabla_\rho \Phi|^2}{r^2}+\frac{|\Phi|^2}{r^2}\right)r^2dVol_{S^2}dr &\leq C\int_{\Sigma_\tau\cap\{r\geq r'\}}\frac{1}{(r')^p}e^p[\Phi]dVol_{\Sigma_\tau}&=\frac{C}{(r')^p}E^p[\Phi].
\sigma_{r , 3}= \sigma_{r , 1}^{*}; \sigma_{r , 4 s + 3}= \sigma_{r , 4 s + 1}^{*}; \quad | \sigma_{r , 4 s + 1}| > | \sigma_{r , 1}| \quad ( s = 1 \ldots \frac{r - 1}{2}) .
\begin{cases}\dot{y}_k = \frac{1}{N}\sum^N_{j = 1}K(y_k - x_j) + f_k(y)+B_k u, k = 1, \ldots, m \dot{x}_i = \frac{1}{N}\sum^N_{j = 1}K(x_i - x_j) + g(y)(x_i), \; \; i = 1, \ldots, N,\end{cases}
x =\frac{r(r^{2}s + 8rs- r - s)}{8r^{3}s-3r^{2}s +r-s}, \frac{1-x}{2 x}=\frac{1}{4\eta(\tau)^2 \eta(17\tau)^2}\left(\sum_{m,n=-\infty}^{\infty}{(e^{\pi i m}-e^{\pi i n})q^{\frac{1}{4}n^2+\frac{17}{4}m^2}}\right)^2.
\begin{aligned}\iint h^\epsilon \partial_x v_s \epsilon g^\epsilon \leq L^2 \epsilon \Vert v_{sx}\Vert_{L^\infty}\Vert h^\epsilon_x\Vert_{L^2}\Vert g^\epsilon_x\Vert_{L^2}= \epsilon L^2 \Vert v_{sx}\Vert_{L^\infty}\Vert g^\epsilon_y \Vert_{L^2}\Vert g^\epsilon_x\Vert_{L^2},\end{aligned}
P_{{\cal C}, 1}\triangleq 2^{-nR}\sum_{u^n(m) \in{\cal C}}Q_{V^n|U^n=u^n(m)}\mathbf{1}_{(V^n,u^n(m)) \in{\cal A}_{\epsilon}}, P_{{\cal C}, 2}\triangleq 2^{-nR}\sum_{u^n(m) \in{\cal C}}Q_{V^n|U^n=u^n(m)}\mathbf{1}_{(V^n,u^n(m)) \notin{\cal A}_{\epsilon}}.
R_{\overline q}M_zR_{\overline q}(z^n)=\overline q^{2n+1}z^n,R_{\overline q}M_zR_{\overline q}M_z^*(z^n)=\overline q^{2n-1}z^n &R_{\overline q}M_z^*R_{\overline q}(z^n)=\overline q^{2n-1}z^{n-1},R_{\overline q}M_z^*R_{\overline q}M_z(z^n)=\overline q^{2n+1}z^n.
K ( X^{1}, X^{2}, X^{3}) = g^{2}e^{x^{A}K^{A B}y^{B}}, \K^{A B}= g^{- 1}\left( \begin{matrix}{0}&{1}&{- 1}\\{- 1}&{0}&{1}\\{1}&{- 1}&{0}\\ \end{matrix}\right)^{A B},
\sum_{n=1}^{m-1}\frac{2m^6g(\frac{n}{m})}{\alpha(m,n)}\leq 2m^6\sum_{n=1}^{m-1}\frac{(\frac{n}{m}+1)^2}{n^4}\leq 2m^4\sum_{n=1}^{m-1}\frac{(n+m)^2}{n^4}\leq 2m^4\sum_{n=1}^{m-1}\frac{n^2+2nm+m^2}{n^4}\leq 2m^4\int_{1}^{m-1}\left(\frac{1}{t^2}+\frac{2m}{t^3}+\frac{m^2}{t^4}\right)dt\leq Cm^6.
\sum_{n=1}^{\infty}\intop_{0}^{\infty}\,y^{(s)-\frac{1}{2}}|J_{s-\frac{1}{2}}(2\pi xy)|\,e^{-2\pi ny}\,dy\leq C_{s}\,\sum_{n=1}^{\infty}\intop_{0}^{\infty}\,y^{(s)-1}\,e^{-2\pi ny}\,dy\leq D_{s}\,\sum_{n=1}^{\infty}\frac{1}{n^{(s)}}<\infty,
\big[[f, \tilde{p}_{i, j}], \tilde{p}_{k, l}\big] = \frac{1}{(i - 1)!k!}\big[\tilde{X}_{1, i - 1}(\eta^{(j - i + 2)/2}), \tilde{X}_{0, k}(\eta^{(l - k + 2)/2})\big] & - \frac{1}{(i - 2)!k!}\big[\tilde{X}_{0, i - 2}(\eta^{(j - i + 4)/2}), \tilde{X}_{0, k}(\eta^{(l - k + 2)/2})\big].
\tilde \Omega = \Omega + d^{\nabla^\omega}\Omega^3 \wedge \theta = \left(\Omega^1 + \nabla^\omega_{Z_1}\Omega^3 \right) dx \wedge \theta + \left( \Omega^1 + \nabla^\omega_{Z_2}\Omega^3\right) dy \wedge \theta & =: \Psi^1 dx \wedge \theta + \Psi^2 dy \wedge \theta,
dq_t= \tilde\gamma^{-1}(t,q_t)\left(-\partial_t\psi(t,q_t)-\nabla_qV(t,q_t))\right) dt & +\tilde S(t,q_t)dt+\tilde\gamma^{-1}(t,q_t)\sigma(t,q_t)\circ dW_t,
S^{(l)}(x_1) \; = \; \Big(S^{(1)}(x_1)\Big)^l \; \; , \; \;S^{(1)}(x_1) \; = \; \exp\left( \; i \pi \; \frac{x_1}{\sqrt{x_1^2 + \lambda^2}}\; \right) \; .
L_{a,n}^{XXZ}(\lambda )=\left( \begin{array}{cc}\left[ \lambda q^{s+S_{n}^{z}/2}-1/(\lambda q^{s+S_{n}^{z}/2})\right] /2 S_{n}^{-}S_{n}^{+}\left[ \lambda q^{s-S_{n}^{z}/2}-1/(\lambda q^{s-S_{n}^{z}/2})\right] /2\end{array}\right)
( K^{( l )}\: V \: K^{( r )}) ( x , y ) \; = \; \frac{1}{2^{l + r}}\left( \frac{\partial}{\partial_{u}}+ \frac{\partial}{\partial v}\right)^{l}\left( \frac{\partial}{\partial_{u}}- \frac{\partial}{\partial v}\right)^{r}\frac{\partial}{\partial u}\left. A_{u v}\right|_{u = 0 = v}\; .
{\bf{\cal I}}( E ) = \left( \begin{array}{c c}{{\int \frac{d^{3}k}{( 2 \pi )^{3}}\frac{1}{E^{+}- \frac{k^{2}}{2 \mu}}}}&{{\int \frac{d^{3}k}{( 2 \pi )^{3}}\frac{k^{2}}{E^{+}- \frac{k^{2}}{2 \mu}}}}\\{{\int \frac{d^{3}k}{( 2 \pi )^{3}}\frac{k^{2}}{E^{+}- \frac{k^{2}}{2 \mu}}}}&{{\int \frac{d^{3}k}{( 2 \pi )^{3}}\frac{k^{4}}{E^{+}- \frac{k^{2}}{2 \mu}}}}\\ \end{array}\right) .
\begin{aligned}\int_0^T \int_{B_{2R}^d}\int_{\mathbb{S}^{d-1}}\mathbf{1}_{\left(\tau,v_{s+k+1},\omega_{k+1}\right) \in \mathcal{B}^+_{III}}d\omega_{k+1}dv_{s+k+1}d\tau \leq & \leq C_{d,\alpha}(s+k-1) T R \eta^{d-1}\end{aligned}
\Delta(x_{1}\wedge\cdot\cdot\cdot\wedge x_{k})=\sum_{p=0}^{k}\sum_{\sigma\in Sh(p+1,k-p)}sgn(\sigma)(x_{\sigma(0)}\wedge\cdot\cdot\cdot\wedge x_{\sigma(p)})\otimes(x_{\sigma(p+1)}\wedge\cdot\cdot\cdot\wedge x_{\sigma(k)}).
T\Vert f'_h - f' \Vert_1 = \int_0^T |g'(t)| \, \mathrm{d}t = \sum_{k = 1}^M \int_{t_{k-1}}^{t_{k}}| g'(t)| \, \mathrm{d}t &= \sum_{k=1}^M \big| g(t_k) - g(t_{k-1}) \big|,
\arrowvert B_{1}\cup B_{2}\cup \dots \cup B_{k}\arrowvert = \sum_{m=1}^{k}(-1)^{m+1}\sum_{1\leq i_1 < i_2 < \dots <i_j \leq k}\arrowvert B_{i_1}\cap B_{i_2}\cap \dots \cap B_{i_m}\arrowvert \,.
\mathfrak{X}:=L^2([0,1]; x^{3/2}(1-x)^{N/2-1}dx), \mathfrak{X}^1 :=\{\phi\in\mathfrak{X}|\ \dot{D}\phi:=\sqrt{x(1-x)}\frac{d\phi}{dx}\in\mathfrak{X}\}, \mathfrak{X}^2 :=\{\phi\in \mathfrak{X}^1 |\ -\Lambda\phi\in\mathfrak{X}\},
\widehat{S}_{\mu}=\left\{\begin{array}{rclrcl}x_{1}^{2}+x_{2}^{2}+x_{3}^{2}= 0, \mu_{4}\; x_{1}^{2}+x_{2}^{2}+x_{4}^{2}= 0, \mu_{5}\; x_{1}^{2}+x_{2}^{2}+x_{5}^{2}= 0, \mu_{6}\; x_{1}^{2}+x_{2}^{2}+x_{6}^{2}= 0, \mu_{7}\; x_{1}^{2}+x_{2}^{2}+x_{7}^{2}= 0\end{array}\right\}\subset{\mathbb P}_{\mathbb C}^{6}.
{\cal G}_\omega (r,r',\theta)=\sum_{n=0}^\infty r^{1-D/2}\left [ b_nJ_{n-1+{D\over 2}}(|\omega| r)+c_n J_{-n+1-{D\over 2}}(|\omega| r)\right ] C_n^{(-1+D/2)}(z)\quad (r'<r<a).
\mathcal{J}^{\pm}(n^2_1n_2,m,q) = \displaystyle\int_{\mathbb{R}}\,V_1\left(\frac{\nu}{K}\right)\, \displaystyle\int_{\mathbb{R}}W(u)\,g(u,q)\,\eth_1^{\pm}(n^2_1n_2,u,q,p_1)\,\mathcal{I}^{\pm}(m,u,q,p_1,p_2)\,du\,d\nu.
D_1: (x_3 x_5-x_4^2)\begin{vmatrix}0 x_1 x_2 x_1 x_3 x_4 x_2 x_4 x_5 \end{vmatrix}=0, D_2: (x_2^2x_3^2-4x_1x_3^3-4x_2^3x_4+18x_1x_2x_3x_4-27x_4^2x_1^2)x_5=0.
M_{\Gamma}(t) = \frac{1}{4\pi i}\log \sideset{}{^{\star}}\prod_{\psi}\left(\frac{L(1+2it, \psi) L(1+2it, \overline{\psi})}{L(1-2it, \psi) L(1-2it, \overline{\psi})}\right)+O(t \log t).
f(x,y)+f(y,z) =f(x,z)+\bigg\langle \bigg(\frac{3}{2}-\|x\|\bigg)x-\bigg(\frac{3}{2}-\|y\|\bigg)y,y-z\bigg\rangle &=f(x,z)+\frac{3}{2}\langle x-y,y-z\rangle-\langle\|x\|x-y\|y\|,y-z\rangle &=f(x,z)+\frac{3}{4}[\|x-z\|^2-\|x-y\|^2-\|y-z\|^2] &\hskip6cm-\langle\|x\|x-y\|y\|,y-z\rangle.
A = \left[\begin{matrix}1+\delta \ \ \ \ 1 \ \ \ \frac{1}{2}1-\delta_1 \ \ \ \ (1-2\delta_1)I_{n-3}\end{matrix}\right], \Delta A = \left[\begin{matrix}0 \cdots 0 \epsilon \vdots \vdots 0 \vdots \vdots \vdots \vdots \vdots 0 \cdots 0\end{matrix}\right].
\psi(\delta) =-2p \int_{B}(U_{\delta,e_n}^{p-1}+U_{\delta,e_n}^{p-1})(\delta \partial_\delta U_{\delta,e_n})(\delta \partial_\delta U_{\delta,-e_n}) &+p\int_{B}(U_{\delta,e_n}^{p-1}(\delta \partial_\delta U_{\delta,e_n})-U_{\delta,-e_n}^{p-1}(\delta \partial_\delta U_{\delta,-e_n}))(\delta \partial_\delta(w_\delta-W_\delta)).
\begin{aligned}\bigg[-\frac{\partial^2}{\partial{\phi^i_j}^2}-(j-1)\cot \phi^i_j\frac{\partial}{\partial \phi^i_j}+G^i_j(\phi^i_j)+\frac{\alpha^i_{j-1}}{\sin^2 \phi^i_j}\bigg]h^i_j(\phi^i_j)=\alpha^i_{j}h^i_j(\phi^i_j), i=1,2,\cdots,N~~~j=1,2,\cdots,d_i-1 \lambda_i=\alpha^i_{d_i-1},~~~\alpha^i_0=0. \end{aligned}
(i) \a_1 < b_1 < b_2 < a_2 \Longrightarrow Q(A,B) = 2 \cdot 2 - 4 = 0, (ii) \b_1 < a_1 < b_2 < a_2 \Longrightarrow Q(A,B) = 2 \cdot 3 - 4 = 2.
H_{n m}^{\{{\bf R}\}}( 0 , 0 ) \approx \frac{( - 1 )^{\theta_{n m}}}{\sqrt{2 \pi}}\Gamma \left( \frac{n + m + 1}2 \right) R_{1 1}^{n / 2}R_{2 2}^{m / 2}\left( \frac{| R_{1 2}|}{\sqrt{R_{1 1}R_{2 2}}}+ 1 \right)^{\frac{n + m + 1}2},
\langle\nabla H(v), \pi^V(v)-v\rangle =3\alpha\left(\sum_i\frac{v_i^{2\alpha-1}(H_i(v))^2}{H(v)}-\sum_i v_i^{\alpha}H_i(v) \right)\;, &=\frac{3\alpha}{H(v)}\big[ v h^2(v)-\big(v h(v)\big)^2\big]\;,
\overline f = \Phi (\nu_i \nu_j)^{2t}\left(\sum_{s=0}^{t}\frac{\Phi^s}{(\nu_i \nu_j d)^{2s}}\right) (\phi_i \circ \phi_j) = (\nu_i \nu_j)^{2t+2}\left( \Gamma_{ij}^{(t+1)}- \delta_{ij}\right).
G^E(2\pi n T, k) = \pi \, C_{\cal{O}}{(2\pi T n)^2+k^2\over 4}\left[ \psi \left( 1+{n\over 2}-{i k \over 4\pi T}\right) + \psi \left( 1+{n\over 2}+{i k \over 4\pi T}\right)\right]\,.
D_{\mu \nu}^{0}( k )_{=}-{\frac{1}{k^{2}+ i \epsilon}}\bigl ( g_{\mu \nu}-{\frac{k_{\mu}\eta_{\nu}+ k_{\nu}\eta_{\mu}}{\eta \cdot k}}+ k_{\mu}k_{\nu}{\frac{\lambda k^{2}+ \eta^{2}}{( \eta \cdot k )^{2}}}\biggr ) ,
L_{\Gamma}(s, \varrho) = \prod_{\overline{\gamma}\in \overline{\Gamma}_{p}}\prod_{k=0}^{\infty}\det\left( 1-\varrho(\gamma)e^{-(s+k)\ell(\gamma)}\right) & = \prod_{\overline{\gamma}\in \overline{\Gamma}_{p}}\prod_{k=0}^{\infty}\det\left( 1-\varrho(\gamma)N_G(\gamma)^{-(s+k)}\right)
b_\varepsilon (\vartheta)[\mathbf{u},\mathbf{u}]=\varepsilon^{-2}b(\varepsilon ;\vartheta)[T_\varepsilon\mathbf{u},T_\varepsilon \mathbf{u}],\mathbf{u}\in H^1(\mathbb{R}^d;\mathbb{C}^n),
L{^{\pm}}_{CS}= \int_{\cal M}\varepsilon^{ijk}\bigg({^{\pm}}A_i^{AB}\partial_j{\^{\pm}}A_{kAB}+{\frac{2}{3}}{^{\pm}}A_{iA}^B{^{\pm}}A_{jB}^C{^{\pm}}A_{kC}^A \bigg),
r^S_i \triangleq f^{\langle 1, S\rangle}_i - f^{\langle 0, S\rangle}_i = R_i + \beta \sum_{j \in S}p_{ij}f^S_j - \beta f^S_i =\begin{cases}(1-\beta) f^S_i i \in S R_i + \beta \sum_{j \in S}p_{ij}f^S_j \end{cases}
\hat{\mathbb{E}}[|\varphi_m(X^x_{\tau+\cdot})-\varphi(X^x_{\tau+\cdot})|] \leq\hat{\mathbb{E}}[\sup_{0\leq t\leq T}|\varphi_m(X^x_{t+\cdot})-\varphi(X^x_{t+\cdot})|] =\hat{\mathbb{E}}^x_2[\sup_{0\leq t\leq T}|\varphi_m(B'_{t+\cdot})-\varphi(B'_{t+\cdot})|].
\mathsf{N}^X_{g, \ell_1 F_1 + \ell_2 F_2}=\begin{cases}12 \delta_{\ell_1 0}\frac{\sigma(\ell_2)}{\ell_2}+ 12 \delta_{\ell_2 0}\frac{\sigma(\ell_1)}{\ell_1}g = 1 0 g \neq 1.\end{cases}
f_n =[X^n Y^n] Y (1-P'_Y(XY,Y) )(1+\sum_{m\geq 1}(\frac{P(XY,Y)}{Y})^m) =[X^n Y^n] Y (1-P'_Y(XY,Y) )(\sum_{m\geq 1}(\frac{P(XY,Y)}{Y})^m) =\sum_{m\geq 1}[X^n Y^{m-1}] (1-P'_Y(X,Y))P(X,Y)^m.
\Pr\{N_0^{(k)}= k_0, N_1^{(k)}= k_1,\ldots,N_n^{(k)}= k_n\}= \frac{k!}{k_0!k_1!\cdots k_n!}p_0^{k_0}p_1^{k_1}\cdots p_n^{k_n}.
\mathrm{Re}\big(T_N(l)\big)=\frac{1}{4}\bigg(F_N(l)-2F_N(l-1)+F_N(l-2) &\qquad-F_N(2)+2F_N(1)\bigg) &=\frac{1}{4}\Big[\big(E_N(l)-E_N(l-1)\big)-\big(E_N(2)-E_N(1)\big)\Big]
\begin{gathered}\det\left(I_N+H\right)^{-1}\exp\left(W^*H\left(I_N+H\right)^{-1}W\right) =\sum_{K}\prod_{1\le m,n\le N}\frac{\left(h_{m,n}\right)^{k_{m,n}}}{k_{m,n}!}H_{r_1,c_1}\left(\overline{w_1},w_1\right)\dotsm H_{r_N,c_N}\left(\overline{w_N},w_N\right),\end{gathered}
J_{n}\left( \frac{r_{1}r_{2}}{t}\right) \sim \sqrt{\frac{2 t}{\pi r_{1}r_{2}}}\exp \left[ - i \left( \frac{r_{1}r_{2}}{t}+ \frac{n^{2}t}{2 r_{1}r_{2}}- \frac{n \pi}{2}\right) \right] .
\int_1^\infty \tau^{2s}\left| T_\Sigma(i\tau/2) - \sum_{l = 0}^k T_\Sigma^l (i\tau)^{-(2j-1+l)}\right| d\tau \lesssim\frac{1}{|\sin (2\pi s)|}\| (u,v) \|_{\dot H^s}^2\|(u,v)\|_{l^2_1 DU^2}^{2j-2}.
d(\mathcal{U}((\overline{w}_{\i})_{\i=1}^{\eta}),\mathcal{U}((\overline{v}_{i})_{i=1}^{\eta}))\leq~cK^{*}_{\eta}((\overline{w}_{\i})_{\i=1}^{\eta},(\overline{v}_{\i})_{\i=1}^{\eta})),
{f^{[1]}\left(x_0,x_{1}\right)}:=\begin{cases}\frac{f(x_0) - f(x_1)}{x_0 - x_1}, \text{if~$x_0\neq x_1$}\frac{f'_r(x_0)+f'_l(x_0)}{2}\end{cases}, \ \x_0, x_1\in\mathbb{R}.
\boldsymbol{D}^{\mu}u := \dfrac{1}{\Gamma (\sigma)}\dfrac{{\rm d}^n}{{\rm d}x^n}\int_{-\infty}^{x}(x-s)^{\sigma-1}u(s) \,{\rm d}s, \boldsymbol{D}^{\mu*}u := \dfrac{(-1)^n}{\Gamma(\sigma)}\dfrac{{\rm d}^n}{{\rm d}x^n}\int_{x}^{\infty}(s-x)^{\sigma-1}u(s) \,{\rm d}s.
\lim_{n\to\infty}\left\{\log \left[\frac{\lambda_n^j}{\lambda_n^k}\right]+\frac{|x_n^j-x_n^k|^2}{\lambda_n^j\lambda_n^k}+\frac{|t_n^j(\lambda_n^j)^2-t_n^k(\lambda_n^k)^2|}{\lambda_n^j\lambda_n^k}\right\}=\infty.
V_{\alpha}( x , \theta ) = \xi_{\alpha}( x ) - \frac{i}{2}( \gamma_{5}\gamma_{\mu}\theta )_{\alpha}\frac{A^{\mu}( x )}{g}+ \frac{1}{2}\theta_{\alpha}\phi ( x ) - \frac{1}{2}N ( x ) ( \gamma_{5}\theta )_{\alpha}- \frac{1}{2}\bar{\theta}\theta \sqrt{2}\Psi_{\alpha}( x ) ,
T^{\mu \nu}= \Sigma_{n}\frac{m_{n}}{\sqrt{g}}\int ( 1 - \alpha \phi ) U_{n}^{\mu}U_{n}^{\nu}\delta^{4}( x - x_{n}( \tau_{n}) ) d \tau_{n}+ \frac{1}{\beta}\left[ \nabla^{\mu}\phi \nabla^{\nu}\phi - \frac{1}{2}g^{\mu \nu}\left( \nabla_{\alpha}\phi \nabla^{\alpha}\phi - \Lambda^{2}e^{2 \phi}\right) \right] .
c^k_{\mathrm{b2s},3}\left( N_{\mathrm{scr}}\right) = \mathsf{E}\left\lbrace B N_{\mathrm{scr}}\log_2\left[ \frac{P_{\mathrm{bs}}}{BN_{\mathrm{bs}}\sigma^2_{\mathrm{sc}}}\right. \right. \left. \left. \,\,\, \times \left(\max \{\|\mathbf{g}_i \| \}_{i=1}^{N_{\mathrm{bs}}}\right)^{-2}+1 \right] \right\rbrace,
{E}^{(k)}_n(x;a,b,c)=\sum_{m=0}^n\sum_{j=0}^m\sum_{i=0}^j\frac{2(-1)^{m-j+i}}{j^k}\binom{j}{i}(x\ln c-(m-j+i+1)\ln a-(m-j+i+1)\ln b)^n.
\left(\overset{3}{R}^{(\epsilon)}_{\epsilon^\prime}ch^{N=3[M,m;\epsilon]}_{j,k;\epsilon^\prime}\right)(\tau,z) = \Psi^{[B;M, m;\epsilon]}_{j,k;\epsilon\prime}(\tau,z,-z,0),\,\,j,k\in \Omega^{[3;M]}_{\epsilon^\prime}.
C = e^{D'(0)}[2\pi(1+s_0)]^{-\frac{1}{2}}[A\Gamma(s_0+1)\zeta(s_0+1)]^\frac{1-2D(0)}{2s_0+2}\kappa =\frac{D(0)-1-\frac{1}{2}s_0}{1+s_0}\kappa_1 =\frac{s_0}{s_0+1}\min \left(\frac{-\sigma_0}{s_0}-\frac{\delta}{4}, \frac{1}{2}-\delta \right)
\begin{aligned}K(e_0,e_1) =g^{00}R^1_{010}=-\mu_1\mu_2, K(e_1,e_2) =g^{11}R^2_{121}=-\mu_2^2, K(e_0,e_3) =g^{00}R^3_{030}=-\mu_1^2,\end{aligned}
\xi_{L,E}^\omega(I) = Tr(\chi_{\Lambda_L}E_{\beta_L(H_L^\omega - E)}(I)) \eta_{p,E}^\omega(I) = Tr(\chi_{\Lambda_L}E_{\beta_L(H_{C_p}^\omega - E)}(I)), ~ p=1,\dots, N_L^d \zeta_{L,E}^\omega(I) = Tr(\chi_{\Lambda_L}E_{\beta_L(H^\omega - E)}(I)).
\int_0^T\langle\partial_t u_m,v\rangle\mathrm{dt}+\int_{Q_T}( \mathsf{A}\nabla u_m+T_m(u_m){\bf E}) \cdot \nabla v\mathrm{dx}\mathrm{dt}+ +\int_{\Sigma_T}b(u_m)u_mv\mathrm{ds}\mathrm{dt}=\int_{\Sigma_T}h v\mathrm{ds}\mathrm{dt},\qquad\forall v\in L^2(0,T;V_{2,\ell}),
\sum_{i=1}^{k_n}\sum_{j=1}^{\vert A \vert}\mu_{\theta^i\omega}(A\cap\sigma^{-j}A) =\sum_{i=1}^{k_n}\sum_{j=p_n}^{\vert A \vert}\mu_{\theta^i\omega}(A\cap\sigma^{-j}A) \le \sum_{i=1}^{k_n}c_2e^{-h_1 p_n}\mu_{\theta^i\omega}(A) &\le Ce^{-h_1 p_n}