harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
S_{\mathrm{D i r a c}}= \mathrm{\frac{i}{2 \p i}}\int d^{2}\! \sigma{\bar{\Psi}}\gamma^{a}\partial_{a}\Psi = \mathrm{\frac{i}{\p i}}\int d^{2}\! \sigma{\bar{\Psi}}_{L}{\bar{\partial}}\Psi_{L}- \mathrm{\frac{i}{\p i}}\int d^{2}\! \sigma{\bar{\Psi}}_{R}{\partial}\Psi_{R}\ ,
I =\frac{2B_t}{B}MN_s+\frac{2Q}{N}N_t+\left(\frac{B_t}{B}\frac{M^2+KB^2}N-\frac{B_s}{B}M+BB_tN\right)N_t &-\frac{B_t}{B}NM_s-\left(\frac{B_t}{B}\right)_sMN-Q_t.
\sum_{k \geq 0}\frac{\mathcal{B}^{k}z^{k}}{k!}+ \sum_{k \geq 0}\delta_{k-1}\frac{z^{k}}{k!}= \frac{z}{e^{z}-1}+ z = \frac{ze^{z}}{e^{z}-1}= e^{(\mathcal{B}+1)z}= \sum_{k \geq 0}\frac{( \mathcal{B}+1)^{k}z^{k}}{k!}
\int_{\mathbb{R}^3}\frac{|\widehat{u_{01}}(\xi)|}{|\xi|}d\xi+\int_{\mathbb{R}^3}\frac{|\widehat{h_{01}}(\xi)|}{|\xi|}d\xi + \int_0^t \int_{\mathbb{R}^3}\frac{|\widehat{F}(\tau,\xi)|}{|\xi|}d\xi d\tau+\int_0^t \int_{\mathbb{R}^3}\frac{|\widehat{G}(\tau,\xi)|}{|\xi|}d\xi d\tau \leq \delta_0.
v \partial_x f + G(M(x)) \partial_v f = \partial_v(\sigma\partial_v f + vf)\,, M(x) := \frac{\int_{\mathbb{T}_L}\!\mathrm{d}y \int_{\mathbb{R}}\!\mathrm{d}w\, f(y,w)\,\varphi(x-y)\, w}{\int_{\mathbb{T}_L}\!\mathrm{d}y \int_{\mathbb{R}}\!\mathrm{d}w\, f(y,w)\,\varphi(x-y)}\,.
\varphi_{q}(2 \cdot - \varrho ) = \sum_{k \in \mathbb{Z}}\sum_{r = 0}^{p}e_{k,r,\varrho}\varphi_{r}( \cdot - k ) + \sum_{k \in \mathbb{Z}}\sum_{r = 0}^{p}d_{k,r, \varrho}\psi_{r}( \cdot - k ) .
D = \det \left( \begin{array}{c c}{{g_{t t}}}&{{\lambda D_{t}\Phi^{J}\nonumber}}\\{{- \lambda D_{t}\Phi^{I}}}&{{g^{I J}+ i \lambda [ \Phi^{I}, \Phi^{J}] \nonumber}}\\ \end{array}\right) = \det \left( \begin{array}{c c c}{{D_{t t}}}&{{D_{t j}}}&{{D_{t b}}}\\{{D_{i t}}}&{{D_{i j}}}&{{D_{i b}}}\\{{D_{a t}}}&{{D_{a j}}}&{{D_{a b}}}\\ \end{array}\right) \ ,
< x_{f}^{\mu}| U_{\mathrm{p h y s i c a l}}( \tau_{f}, \tau_{i}) | x_{i}^{\mu}> = \int_{( \infty )}\frac{d^{D}p^{\mu}}{( 2 \pi )^{D}}\, e^{i \Delta x \cdot p}\int_{- \infty}^{\infty}\frac{d \gamma}{2 \pi}\, e^{- \frac{1}{2}i \gamma ( p^{2}+ m^{2})}\ .
\Pr\Big(\sum_{j=1}^rjZ_j=n\Big) = \frac{Q(x)}{2\pi}\left(\int_{|t|\leq t_0}+\int_{t_0<|t|\leq \pi}\right)\exp\left\{g_r(t,x)\right\}e^{-itn}dt &=:\frac{Q(x)}{2\pi}\big(K_1(n)+K_2(n)\big)
G^{(2)}_{\mathcal R}(x,y) = & \frac{e^{ik_{+}\vert x \vert}}{\sqrt{\vert x \vert}}\frac{e^{i\frac{\pi}4}}{\sqrt{8\pi k_{+}}}\left(\frac{2i\sin\theta_{\hat x}\mathcal S(\cos\theta_{\hat x},n)}{n^2-1}\right)e^{-ik_{+}\vert y \vert\cos(\theta_{\hat x}+\theta_{\hat y})}+ G^{(2)}_{\mathcal R,Res}(x,y),
A(\gamma) = \frac{1}{2}\int\limits_{s}^{t}{\gamma_t(\tau)^2 d \tau}+\int\limits_{s}^{t}{\Bigg( \gamma_t(\tau) \Big(\frac{\partial G}{\partial x}(\gamma(\tau),s)-\frac{\partial G}{\partial x}(\gamma(\tau),\tau) \Big) \Bigg) d \tau}+ \Big( G(\gamma(t),t)-G(\gamma(t),s) \Big)
C^{\alpha \beta}(x,y) = \left( \begin{array}{cc}\{\chi^{i j}(x) , \chi^{m n}(y) \}\{\chi^{i j}(x) , \phi^{m n}(y) \}\{\phi^{i j}(x) , \chi^{m n}(y) \}\{\phi^{i j}(x) , \phi^{m n}(y) \}\end{array}\right) ,
F_{\pm N}= Z_{F}^{- 1}\int D \bar{\chi}D \chi \, p r o d_{i = 1}^{N}\bar{\chi}_{\pm}\chi_{\pm}( w_{i}) \exp ( - \int d^{2}x \bar{\chi}( i \! \! \not \! \partial +{\not \! \! A}^{( \pm N )}) \chi )
v_{0}(r) =A\left( r/r_{0}\right)^{1/2}\left[ 1+c\ln\left(r/r_{0}\right) \right] ,r>R, & =A\left( R/r_{0}\right)^{-1/2}\left[ 1+c\ln\left( R/r_{0}\right)\right] \left( r/r_{0}\right) ,r<R.
BA_t(g_1,g_2)(\omega) = A_t(g_1(\cdot){\widetilde{g}_2(R_\omega\cdot)})(\omega) = \int_{\mathbb{S}^{n-1}}\delta(\omega\cdot \xi - t) g_1(\xi){\widetilde{g_2}(R_\omega(\xi))}\, d\sigma(\xi).
\int\limits_{R}^{\infty}r^{a}\mathrm{e}^{-b r^2}dr \leq \int\limits_{R}^{\infty}\mathrm{e}^{-b r^2 + a r}dr = \frac{\sqrt{\pi}\mathrm{e}^{\frac{a^2}{4b}}}{2}\mathrm{erfc}\left(\sqrt{b}R - \frac{a}{2\sqrt{b}}\right).
A(x)\left|I\right\rangle = \alpha(x)\prod_{k=1}^m\mathbf{f}(x,x_{k})|I\rangle +\sum_{j\in I}\left[\frac{{\mathcal{A}}^{I}_j}{x-x_{j}}-\frac{\tilde{\mathcal{D}}^{I}_j}{1+x+x_{j}}\right]B(x)\left|I\setminus\left\{j\right\}\right\rangle\,,
D(s, S_f \times \overline{S_g}) := \sum_{n \geq 1}\frac{S_f(n) \overline{S_g(n)}}{n^{s + k - 1}}= W(s; f,\overline{g}) + \frac{1}{2\pi i}\int_{(\gamma)}W(s - z; f,\overline{g}) \zeta(z) \frac{\Gamma(z) \Gamma(s - z + k - 1)}{\Gamma(s + k - 1)}\; dz,
\begin{array}{@{}l@{}}{\displaystyle |\sum_{k=1}^{m}V_5^{k,1}| + |\sum_{k=1}^{m}V_5^{k,2}| \leq C\big(h^{2r}+\tau^{4}\big) +C \|\theta_{\Psi}^{m}\|_{\mathcal{L}^2}^{2}}begin{align*}2mm]{\displaystyle \quad+ \frac{1}{32}\|\nabla\theta_{\Psi}^{m}\|_{\mathbf{L}^2}^{2}+C\tau \sum_{k=1}^{m}\|\nabla\theta_{\Psi}^{k}\|_{\mathbf{L}^2}^{2}.}\end{array}
\begin{aligned}u^\varepsilon (z_3) \leq \int_0^{t_3 + t_4}L \big( \xi^\varepsilon (s, z_1, -\gamma ), -\gamma (\xi^\varepsilon (s, x, -\gamma )) \big) e^{-\lambda s}\, ds + u^\varepsilon (z_4) e^{-\lambda (t_3 + t_4)}\leq C(1 + \lambda ) \cfrac{4 \sqrt{h}}{c_0 \mu}+ u^\varepsilon (z_4).\end{aligned}
P(\Pi) = \left\{y \in [1,m]^m : \begin{array}{rlc}\sum_{i \in E}y_i = \binom{m + 1}{2}\sum_{i \in S}y_i \leq \binom{m+1}{2}- \binom{m+1-|S|}{2}, \forall S \subseteq E: S \neq \emptyset \end{array}\right\}.
|U-\bar{U}| =\left|\int_0^1\frac{d}{d\tau}[A^{-1}(\tau A(U)+(1-\tau)A(\bar{U}))]\:d\tau\right| &\leq\left|\int_0^1\nabla_V(A^{-1})(\tau A(U)+(1-\tau)A(\bar{U}))\:d\tau\right|\:|A(U)-A(\bar{U})| &\leq C |A(U)-A(\bar{U})|,
\widehat{y}_{i}^{[K,L]}(x) = (1-q) \sum_{n=K}^{L}s^{\mu+1}\frac{P_{\lambda}(x, s)}{s-b_{i}}\frac{(\alpha_1 s, \alpha_2 s;q)_{\infty}}{(\beta_1 s, \beta_2 s;q)_{\infty}}\bigg|_{s= q^n \xi}, i=0,1,2, & b_0=0, b_1 = 1/\alpha_1, b_2 = 1/\alpha_2
\begin{aligned}\|A\pm B\| =\|(A^\ast\pm B^\ast)(A\pm B)\|^{\frac{1}{2}}&=\|A^\ast A\pm(A^\ast B+B^\ast A)-B^\ast B\|^{\frac{1}{2}}&=\|A^\ast A+B^\ast B\|^{\frac{1}{2}}=\max\{\|A\|,\|B\|\}&\leq 1.\end{aligned}
\textrm{esssup}_{(\omega,t,x)\in\Omega \times \mathcal{O}_t}u^{\pm}\leq \, C\left(\textrm{esssup}_{(\omega,t,x)\in\Omega \times \partial_p\mathcal{O}_t}u^{\pm}+A_p(f_0^{\pm},g_0;\mathcal{O}_t)+B_2(f_0^{\pm},g_0;\mathcal{O}_t)\right)
\begin{array}{rclcl}m_1x_1 + m_2x_2 + m_3x_3 + kp_1 = 0 m_1x_4 + m_2x_5 + m_3x_6 + kp_2 \equiv n_1x_1 + n_2x_2 + n_3x_3 + lp_1 = 0 & n_1x_4 + n_2x_5 + n_3x_6 + lp_2 = 0.\end{array}
\bar \Phi \mkern 7mu = \{s_1 = s'_1, \dotsc, s_\alpha = s'_\alpha\}\cup\{t_1 \le t'_1, \dotsc, t_\beta \le t'_\beta \}\Psi_1 = \{s_1 = s'_1, \dotsc, s_\alpha = s'_\alpha\}\Psi_2 = \{t_1 = t'_1, \dotsc, t_\gamma=t'_\gamma \}\Psi_3 = \{t_{\gamma+1}\le t'_{\gamma+1},\dotsc, t_\beta \le t'_\beta \}
Li_{C, \mathfrak{s}}({\bf z}):=\sum_{i_1>\cdots>i_r\geq0}\frac{z_1^{q^{i_1}}\cdots z_r^{q^{i_r}}}{L_{i_1}^{s_1}\cdots L_{i_r}^{s_r}}\in\mathbb{C}_{\infty}.
\Omega_R/dR \cong \mathbb{C}\omega_0 \oplus \bigoplus_{i=3}^{4}U_i^{\frac{(1-(-1)^k)n}{2k}}\oplus \bigoplus_{h=1}^{k-1}V_{h}^{\oplus \frac{(1-(-1)^h)n}{k}}.
\ell_0 = \frac{1}{2}\left [ \binom{\frac{d}{2}+ n-1}{n-1}+ \frac{1}{2}- \sqrt{\left [ \binom{\frac{d}{2}+ n-1}{n-1}+ \frac{1}{2}\right ]^2 - 2 \binom{d+n-1}{n-1}}\ \right ].
dq^\prime_t= \frac{1}{m}(p_t^\prime+\psi(t^*,q_t^\prime))dt, d(p_t^\prime)_i= -\frac{1}{m}\delta^{jk}(\gamma_{ij}(t^*,q_t^\prime)+ \partial_{q^i}\psi_j(t^*,q_t^\prime))(p_t^\prime+\psi(t^*,q_t^\prime))_kdt &+(-\partial_{q^i}V(t^*,q_t^\prime)+\tilde F_i(t^*,q_t^\prime,-p_t^\prime))dt+\sigma(t^*,q_t^\prime)\circ dW_t.
\gamma_{k,\ell}:=\frac{1}{1-e^{-c_{\ref{th-codegree}}d}}\left[\frac{p_\ell}{p_k}\, \mathbf{1}_{p_\ell<p_k}+ \Big( 1+\frac{p_\ell -p_k}{p_k-\lfloor 0.9d\rfloor}\Big)\, \mathbf{1}_{p_\ell\geq p_k}\right],
\partial_k \Gamma_k[\phi] = \lim_{\delta k\to 0}\frac{1}{2\delta k}\left[\ln\det\frac{\delta^2 \Gamma_k}{\delta \phi^{\Lambda}_{k}\delta \phi^{\Lambda}_{k}}-\ln\det\frac{\delta^2 \Gamma_k}{\delta \phi^{\Lambda}_{k-\delta k}\delta \phi^{\Lambda}_{k-\delta k}}\right],
\sum_{k=1}^n c_k f^{s}_k = \sum_{i=1}^n b_i h^{s}_i,{\rm where}b_i =\sqrt{J_{2s}(i)}\big(\sum_{k=1}^n c_kk^{-s}{\bf 1}_{i|k}\big), \i=1,\ldots, n, n\ge 1.
{\rm{det}}(\mathbf{M}) = \left(\prod_{q=0}^{n-1}\lambda_{q+1}^f \prod_{1 \leq i < q+1}(\lambda_{q+1}-\lambda_i)^{m+1}\right)^{m+1}= \left(\prod_{i=1}^n \lambda_i\right)^{f(m+1)}\left(\prod_{1 \leq i < j \leq n}(\lambda_j - \lambda_i)\right)^{(m+1)^2},
\nabla_{z_{\alpha}}^{0}\left(\begin{array}{c}\langle v| \hat V \end{array}\right)= \frac{1}{\theta_{\alpha}}\left(\begin{array}{c}\langle v| \widehat{\bar z}_{\alpha}-\lbrack \widehat{\bar z}_{\alpha}, \hat V \rbrack \end{array}\right) \, .
{\cal L}_{1 / m^{3}}^{( 2 + 1 ) s p i n}= \frac{\alpha}{7 2 0 m^{3}}\left[ 1 8 F^{\nu \lambda}F_{\nu \lambda , \mu}^{~ \mu}+ 7 F^{\nu \lambda , \mu}F_{\nu \lambda , \mu}- 2 F_{\lambda}^{\nu \lambda ,}F_{\nu \mu ,}^{\mu}\right] ,
\lambda_{j,\varepsilon}- \lambda_j = 4u_j^2(x^*) e^{\phi(x^*)}\varepsilon + \frac{4}{\pi}(H(x^*)-F^\perp(x^*)) u_j^2(x^*) e^{\phi(x^*)}\varepsilon^2\log\varepsilon + O(\varepsilon^2).
\begin{aligned}\big\|\nabla w\big\|_{L^2(\Omega)}\leq C\|g-v\|_{H^{\frac{1}{2}}(\partial\Omega)}\leq C\Big\{\|g\|_{H^{1}(\partial\Omega)}+\|v\|_{H^{1}(\partial\Omega)}\Big\}&\leq C\Big\{\|F\|_{L^q(\Omega)}+\|h\|_{W^{1,q}(\Omega)}+\|g\|_{H^1(\partial\Omega)}\Big\},\end{aligned}
\begin{array}{lll}\hat{q}_{i1}(x) = \displaystyle\frac{\tilde{a}_{i}(x)-\tilde{a}_1(x)-\left[\tilde{a}_{i}(x)-\tilde{a}_1(x)\right]_{d_{1i}(x)}}{d_{1i}(x)}&= \displaystyle q_{i1}(x)+\frac{e_{i}(x)-e_1(x)-\left[ e_{i}(x)-e_1(x)\right]_{d_{1i}(x)}}{d_{1i}(x)}\end{array}
\mathcal{Q}_\ell^k(z) = \begin{cases}\frac{(z^2-1)^{-k/2}}{(\ell-k)!}\left[\frac{1}{2}A^{(1)}_{k\ell}(z) \ln\left(\frac{z+1}{z-1}\right)+ A^{(2)}_{k\ell}(z)\right], \ell \geq k \frac{(-1)^k}{(z^2-1)^{k/2}}A^{(2)}_{k\ell}(z), \ell < k \end{cases}
M_k: \; \mathcal A_k \longrightarrow \mathcal W_k = \left\{0, \frac{1}{k}, \frac{2}{k}, \ldots, \frac{k-1}{k}, 1 \right\}\mathbf A_k(x_0) \longrightarrow M_k(\mathbf A_k(x_0)) = \frac{1}{k}|\mathbf A_k|_{0}^{k-1}
\lambda\int_{-\infty}^\infty \sin(t\lambda)\frac{\sin(t\sqrt{\smash[b]{\Delta_g}})}{\sqrt{\smash[b]{\Delta_g}}}\widehat{\chi}(t)\,dt=2\lambda\int_{0}^\infty \sin(t\lambda)\frac{\sin(t\sqrt{\smash[b]{\Delta_g}})}{\sqrt{\smash[b]{\Delta_g}}}\widehat{\chi}(t)\,dt.
\left( \left( \gamma^{(p-1)m_1}- \gamma^{(p-1)m_2}\right)^{\frac{p^n-1}{p-1}}\right)^{p^i-1}= \left( \gamma^{(p-1)(m_1+m_2)}\left( \gamma^{(p-1)m_1}- \gamma^{(p-1)m_2}\right)^{p^i-1}\right)^{\frac{p^n-1}{p-1}}&= \left( -1 \right)^{\frac{p^n-1}{p-1}}= - 1.
F^{i,j}=F(x^i, y^j) \ \ \ \ \ \ \ \ i,j\in\{1, 2, \star\}, G_x^{i,j}=\nabla_x F(x^i, y^j) \ \ \ \ \ \ \ \ i,j\in\{1, 2, \star\}, G_y^{i,j}=\nabla_y F(x^i, y^j) \ \ \ \ \ \ \ \ i,j\in\{1, 2, \star\}.
\left| \{x \in \Omega : \, \left| u_n(x,t) \right| > c \}\right| \le \frac{b}{\min\{\psi(c),\psi(-c)\}}\, , b:= \limsup_{n\to\infty}\int_{\Omega}\psi(u_{0n}(x)) \, dx < \infty \, ,
E=- \frac{3\xi^2}{64 a}\left [\sum_{l=1}^{\infty}\nu^0-\frac{9}{128}\sum_{l=1}^{\infty}\frac{1}{\nu^2}+\frac{423}{16384}\sum_{l=1}^{\infty}\frac{1}{\nu^4}+{\cal O}\left (\frac{1}{\nu^6}\right )\right ]{.}
\|R\| \geq |R(\hat{x}_L,\hat{x}_R)| = \bigl|\hat{T}(\hat{x}_L,\hat{x}_R)\bigr| + \eta \bigl|\hat{T}(\hat{x}_L,\hat{x}_R)\bigr|^2> \left(1-\frac{\eta^3}{2^4}\right) + \eta\left(1-\frac{\eta^3}{2^4}\right)^2.
\Vert\Omega\Vert^2:=\Omega_{i_1...i_m}^{j_1...j_n}\Omega_{j_1...j_n}^{i_1...i_m}=g^{j_1l_1}\cdots g^{j_nl_n}g_{i_1k_1}\cdots g_{i_mk_m}\Omega_{l_1...l_n}^{k_1...k_m}\Omega_{j_1...j_n}^{i_1...i_m}.
y_{k,l}^2 = 1, y_{i,j}\, y_{k,\, l}= y_{k,\, l}\, y_{i,j}\Leftrightarrow (y_{i,j}\, y_{k,\, l})^2 = 1, y_{i,k}\, y_{k,j}\, y_{i,k}= y_{k,j}\, y_{i,k}\, y_{k,j}\Leftrightarrow (y_{i,k}\, y_{k,j})^3 = 1,
\liminf_{j\rightarrow+\infty}\int_{\Pi^{\xi_{k+1}}}\mathcal{F}_{\varepsilon_{j}}^{\xi_{k+1}}((u_{j})_{z}^{\xi_{k+1}},\Omega_{z}^{\xi_{k+1}})~d\mathcal{H}^{n-1}(z)=:M_{k+1}<+\infty~,
\frac{1}{2}\Big ( \partial_{\mu}\varphi^{( 1 )}\Big )^{2}\; + \; \frac{1}{2}\Big ( \partial_{\mu}\varphi^{( 2 )}\Big )^{2}\; + \; \frac{1}{2}\Big ( \varphi^{( 1 )}\Big )^{2}\frac{e^{2}N}{\pi + g N}\; .
\sum_{n=-n_0}^{\infty}\frac{a_n^{(1)}e^{-n\pi i/2}}{\left( x-x_0\right)^n}= \sum_{n=-n_0}^{\infty}\frac{(-1)^na_n^{(2)}e^{-n\pi i/2}}{\left( x-x_0\right)^n},
\displaystyle\int_{U}e^{-\tau{f(x)}}\psi(x)dx =\frac{e^{-\tau{f(x_0)}}}{\sqrt{\det({\rm Hess}(f)(x_0))}}\Big(\,\frac{2\pi}{\tau}\Big)^{n/2}\Big(\psi(x_0) +\Vert \psi\Vert_{C^{0, 1}(\overline{U})}O(\tau^{-1/2})\Big)
\frac{2s_{N_n}\left( (1+r)^{d_N+1}- (1-r)^{d_N+1}\right)}{r(d_N+1)}- \frac{k\left( (1+r)^{d_N+2}- (1-r)^{d_N+2}\right)}{r^2(d_N+1)(d_N+2)}+ 2c=0.
S = \int d^{5}x \sqrt{- G}\left[{\frac{c^{4}}{1 6 \pi G_{5}}}{\cal R}- \Lambda \right] + \int d^{4}x \sqrt{- g}\left[{\cal L}_{\mathrm{m a t}}+{\cal L}_{\psi}+ e^{- 2 \psi}{\cal L}_{\mathrm{e m}}- \sigma \right] ,
(\boldsymbol{\tau}\cdot n_{E_{mn}})|_{E_{mn}}= (\lambda_{mn}(\nabla \eta)|_{T_m}+ (1-\lambda_{mn}) (\nabla \eta)|_{T_n})\cdot n_{E_{mn}}, (\boldsymbol{\rho}\cdot n_{E_{mn}})|_{E_{mn}}= (\lambda_{mn}(\nabla \zeta)|_{T_m}+ (1-\lambda_{mn}) (\nabla \zeta)|_{T_n})\cdot n_{E_{mn}},
\|\mathcal{L}_X\mathcal{L}_Y-\mathcal{L}_Y\mathcal{L}_X\|_2 =\max_{\|Z\|_F=1}\|\mathcal{L}_X\mathcal{L}_YZ-\mathcal{L}_Y\mathcal{L}_XZ\|_F &=\max_{\|Z\|_F=1}\frac{\|(XY-YX)Z-Z(XY-YX)\|_F}{4}&\le \frac{\|XY-YX\|_F}{2}&\le \|X\circ Y-\mu I\|_F,
{\cal L}= -{\frac{1}{4}}F_{\mu \nu}F^{\mu \nu}+{\frac{\kappa}{4}}\epsilon^{\mu \nu \alpha}A_{\mu}F_{\nu \alpha}+{\frac{1}{2}}| D_{\mu}\Phi |^{2}- \frac{m^{2}}{2}| \Phi |^{2},
\|\boldsymbol{u}\|_{l^2(t_0,T;\boldsymbol{A})}^2 :=\Delta t\sum_{n=1}^N\|\boldsymbol{u}^n\|_{\boldsymbol{A}}^2, \|p\|_{l^2(t_0,T;B)}^2:=\Delta t\sum_{n=1}^N\|p^n\|_B^2, \|\boldsymbol{u}\|_{l^\infty(t_0,T;\boldsymbol{A})}:=\max_{1\le n\le N}\|\boldsymbol{u}^n\|_{\boldsymbol{A}}, \|p\|_{l^\infty(t_0,T;B)}:=\max_{1\le n\le N}\|p^n\|_B.
P_{2}(\zeta) = 2, P_{3}(\zeta) = 6, P_{4}(\zeta) = 2\zeta+22, P_{5}(\zeta) = 30\zeta+90, P_{6}(\zeta) = 16\zeta^2 +310 \zeta +394, P_{7}(\zeta) =504 \zeta^2+2730 \zeta+1806, P_{8}(\zeta) = 360 \zeta^3+9422 \zeta^2+21980 \zeta +8558, P_{9}(\zeta) = 18264 \zeta^3+135954 \zeta^2+167076 \zeta +41586.
ds^{2}=-(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}})dt^{2}+(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}})^{-1}dr^{2}+r^{2}d\theta^{2}+r^{2}sin^{2}\theta d\phi^{2}.
\sigma(\rho)=\frac{\cosh^{4/5}\left[{5 \over 4}k (\rho-\rho_0) \right]}{\cosh^{4/5}\left[{5 \over 4}k\rho_0 \right]}~~~,~~~\gamma(\rho)=A\frac{\sinh^{\alpha}\left[{5 \over 4}k (\rho-\rho_0) \right]}{\cosh^{6/5}\left[{5 \over 4}k (\rho-\rho_0) \right]}
\frac{d^2\tilde F_0}{d\tilde z^2}+\tilde z\frac{d\tilde F_0}{d\tilde z}+(1-\frac{1}{4\tilde z^2}+\frac{\mu^{2/(7-p)}}{(7-p)^2 \tilde z^{2+2/(7-p)}})\tilde F_0=0
\sum_{i \in \Lambda^{\circ}_J}{r}_{i}\geq \frac{1}{2}m \Big( \bigcup_{i \in \Lambda^{\circ}_J}(b_i - r_i, b_i + r_i) \Big) = \frac{1}{2}m \Big( \bigcup_{i \in \Lambda_J}(b_i - r_i, b_i + r_i) \Big) &\geq \frac{1}{2}m \Big( \bigcup_{i \in \Lambda^{\circ}_J}(c_i, b_i) \Big)= \frac{1}{4}m(J).
T(S;\nu^{-1})f :=[(\zeta\circ\nu^{-1})\cdot S((\zeta\circ\nu^{-1})\cdot(f\circ\nu^{-1}))]\circ\nu =\nu^*\circ (\zeta\circ\nu^{-1})\cdot S\circ \nu^{-1,*}\zeta\cdot f.
\displaystyle\sum\limits_{n = 0}^{n_{12}+ j}Q_{n}^{\mu_1}(x) Q_{n_{12}+j-n}^{\mu_2}(y) \langle n, n_{12}+j-n |n_{12},j\rangle = \displaystyle\sum\limits_{n = 0}^{j+1}Q_{n}^{\mu_1}(x) Q_{1+ j-n}^{\mu_2}(y) \langle n, 1+j-n | 1,j\rangle Q_{n_{12}-1}^{\mu_{12}}(z)
\langle f_+^\lambda(\log f_+)^m\varphi, \psi\rangle = \int_{\{x \in U \mid f(x) > 0\}}\varphi(x)f(x)^\lambda(\log f(x))^m \varphi(x)\psi(x)\,dx (\forall \psi \in C_0^\infty(U)),
\tilde{T}_{\alpha}(a)=\left( e_{\alpha}(\tau(a)), \rho^{i}_{\alpha}\left.\frac{\partial}{\partial x^{i}}\right|_{a}\right), \tilde{V}_{\alpha}(a)=\left( 0, \left.\frac{\partial}{\partial y^{\alpha}}\right|_{a}\right), a\in E
\begin{aligned}\iint v^\epsilon \partial_y u_s u^\epsilon \leq L \Vert v^\epsilon_y\Vert_{L^2}\Vert y\partial_y u_s\Vert_{L^\infty}\Vert u^\epsilon_x\Vert_{L^2}&= L \Vert y\partial_y u_s\Vert_{L^\infty}\Vert v^\epsilon_y\Vert_{L^2}^2,\end{aligned}
\epsilon(q,0)=1-V_{q}\sum_{k,i}\frac{-q_{i}\frac{\partial f}{\partial k_{i}}}{-\frac{\hbar^{2}k\cdot q}{m}}=1-V_{q}\sum_{k,i}\frac{q_{i}\frac{\partial f}{\partial k_{i}}}{\frac{\hbar^{2}k\cdot q}{m}}
v^{\prime}* v^{\prime \prime}= ( b_{i}^{\prime}+ b_{i}^{\prime \prime}, l_{\pm ( i , i + 1 )}^{\prime}e^{- ( h / 2 ) b_{i + 1 ; n - 1}^{\prime \prime}}+ e^{- ( h / 2 ) b_{i ; n - 1}^{\prime}}\; l_{\pm ( i , i + 1 )}^{\prime \prime}) .
Z(\bar{v},x,t) = - A(x) \sin(\omega_1 t) -B(x) \sin(\omega_2 t) + \frac{A(x)^3}{3}\sin^3(\omega_1 t) + \frac{B(x)^3}{3}\sin^3(\omega_2 t) &+ A(x)\bar{v}^2 \sin(\omega_1 t) +B(x)\bar{v}^2 \sin(\omega_2 t) + A(x)^2 B(x)\sin^2(\omega_1 t) \sin(\omega_2 t) &+ A(x) B(x)^2 \sin(\omega_1 t) \sin^2(\omega_2 t)- \partial_x^2 J_0(x,t) +v_0^2 J_0.
{\mathcal L}_{m}\; = \; \frac{1}{2}m_{e l}^{2}\, A_{0}( x , \tau ) A_{0}( x , \tau ) \; + \; \frac{1}{2}m_{m a g}^{2}\, A_{j}( x , \tau ) \delta_{j k}^{\perp}A_{k}( x , \tau )
\left( k_{3}-k_{2}\right)_{\lambda}T_{\lambda \mu \nu}^{AVV}\left(k_{1},k_{2},k_{3};m\right) =-2miT_{\mu \nu}^{PVV}\left(k_{1},k_{2},k_{3};m\right) +T_{\mu \nu}^{AV}\left( k_{1},k_{2};m\right)-T_{\nu \mu}^{AV}\left( k_{3},k_{1};m\right) ,
p_{\tau,h}^{k}{}_{|I_n}(t) = \sum_{j=0}^r P^{j,k}_{n,h}\varphi_{n,j}(t)\,, \vec z_{\tau,h}^{k}{}_{|I_n}(t) = \sum_{j=0}^r \vec Z^{j,k}_{n,h}\varphi_{n,j}(t)\,,
\prod_{\substack{m,n \geq 1 m \leq n ; \, (m,n)=1}}\left( \frac{1}{1-z^n}\right)^{\frac{m^4}{n^5}}= \sqrt[3]{\frac{1}{1-z}}\; \exp\left\{\frac{z(7-2z)}{10(1-z)^2}- \frac{1}{30}Li_3(z) \right\},
R^e_n(s_{v_0})(w_{i}) =s_{w(i-e,i-1)}s_{w(i-e+1, i)}(w_i)=s_{w(i-e,i-1)}(w_i-w(i-e+1, i)) &=w_i+(w(i-e,i-1),w_i)w(i-e,i-1) &~~~~-w(i-e+1, i)-(w(i-e+1, i),w(i-e,i-1))w(i-e, i+1) &=w_i+w(i-e,i-1)-w(i-e+1, i) &=w_{i-e}.
\limsup_{\eta\rightarrow0}\limsup_{\epsilon\rightarrow0}\limsup_{N\rightarrow\infty}\frac{1}{N}\log\mathbb{P}_{N}\left[\frac{1}{N}\sum_{i\in I_{c_{1}}^{N}}\left|\int_{0}^{\eta}\rho_{\epsilon,i}^{(c_{2})}(x^{N}(t))dt-A_{i,c_{2}}^{N}(\eta)\right|>\delta\right]=-\infty.
P(R_j \in \{t_1,\dots, t_c\}, \; X_j \in \{s_1,\dots,s_d\}) & \le P(R_j \in \{t_1,\dots, t_c\})=\frac{c}{n}&=P(R_j \in \{1,\dots,c\}) &=P(R_j \in \{1,\dots,c\}, X_j \in \{1,\dots,d\})\;\;\mbox{(if}\; d\ge c),
\delta \psi_{M}= D_{M}\varepsilon +{\frac{1}{6 4}}e^{3 \phi / 4}( \Gamma_{M}^{M_{1}M_{2}}- 1 4 \delta_{M}^{M_{1}}\Gamma^{M_{2}}) \Gamma^{1 1}\varepsilon F_{M_{1}M_{2}}
\begin{array}{rcl}\dim Gor(H)-\dim Gor(H') = \frac12(3h_r+h_{r-1}-\sum_{i=0}^eh_ip_i)-\frac12(3h'_r+h'_{r-1}-\sum_{i=0}^eh'_ip'_i) = h_{r+1}-2h_{r+3}+h_{r+4}+1 \end{array}
\|P_{l}T^{\,j}P_{l}\,x\|\ge\Bigl\| \sum_{k\,\in\,[b_{l},b_{l+1})\backslash I_j}\Bigl(\prod_{s=k+1}^{b_{l+1}-1}w_{s}\Bigr)x_{k}e_{k}\,\Bigr\|\ge\Bigl\| \sum_{k\,\in\,I_{j_{0}}}\Bigl(\prod_{s=k+1}^{b_{l+1}-1}w_{s}\Bigr)x_{k}e_{k}\,\Bigr\|\ge\dfrac{X_{l}}{2}\cdot
\lim_{V \to \infty}\omega_{\beta,\mu,\Lambda}(\frac{b_{{0}}^{*}b_{{0}}}{V})= \lim_{\lambda \to +0}\lim_{V \to \infty}\omega_{\beta,\mu,\Lambda,\lambda}(\eta(b_{{0}}^{*})\eta(b_{{0}})) \ forall \phi \in [0,2\pi) \ .
B'_1T\le \mathrm{exp}\left (2^{12n+4}(\alpha(n+1)+1)^{4n}\delta^{-2n}d^{2n+2}\Delta^{n(2n+2)}\right) &\times 4(m+1)s(2e(\alpha(n+1)+1)\delta^{-1})^{(m+1)s-1}\log (4C)\cdot\log\log (4C)=A_1.
\varphi_j^\pm(s,z) =2z\gamma_j^\pm(s,\theta(x-h\gamma_j^\pm(s,z),x))-z\theta(x-h\gamma_j^\pm(s,z),x+h(z-\gamma_j^\pm(s,z))) &=2\gamma_j^\pm(s,z)\theta(x-h\gamma_j^\pm(s,z),x)-z\theta(x-h\gamma_j^\pm(s,z),x+h(z-\gamma_j^\pm(s,z))) ,
\delta_{\lambda \mu}=\langle P_{\lambda}(x;\zeta),{Q'}^{(r)}_{\mu}(x;\zeta)\rangle_{0}=b_{\lambda}(\zeta)^{-1}\sum_{\rho\in CP_{r,n}}Q_{\rho}^{\lambda}{Q'}_{\rho}^{\mu}z_{\rho}.
\tilde U \tilde S_2 = U^{-2}S_2S_1S_1^*+U^{-2}S_1^2S_2^* & = U^{-2}S_2US_2S_1^*+U^{-2}US_2US_2S_2^* & = U^{-2}U^2S_2S_2S_1^*+U^{-2}UU^2S_2S_2S_2^* & = S_2^2S_1^*+US_2^2S_2^* & = S_2^2S_1^*+S_1S_2S_2^*=\tilde S_1
\sigma_{j}^{a}= \mathbf{1}_{V}^{\otimes ( j - 1 )}\otimes \sigma_{}^{a}\otimes \mathbf{1}_{V}^{\otimes ( N - j )}, \qquad{\cal V}= V^{\otimes N}= \underbrace{V \otimes V \otimes \ldots \otimes V}_{\mathrm{~ N ~ f a c t o r s}},
|(q+\xi_1)^2 - E - i \eta_1 |^2 - |(q+\xi_2)^2 - E - i \eta_2 |^2 = 2 (q^2 - E) ( d_1 - d_2 ) + d_1^2 - d_2^2 + \eta_1^2 - \eta_2^2 \geq \frac{1}{4}\eta^2 \geq 0,
\sum_{\mathbf{r},r_0}\frac{y_{\mathbf{r},r_0}^2}{\phi_{\omega*}(rr_0)^2}\sum_{\substack{\mathbf{s},s_0 ss_0=rr_0}}\prod_{p|rr_0}|S_p(\mathbf{r},\mathbf{s},r_0,s_0)|\le \sum_{\mathbf{r},r_0}y_{\mathbf{r},r_0}^2\prod_{p|rr_0}\frac{p+\omega^*(p)-2}{(p-\omega^*(p))^2}=\sum_{\mathbf{r},r_0}\frac{y_{\mathbf{r},r_0}^2}{\prod_{p|rr_0}(p+O(k))}.
r_{\mu}(t)^{2}= \mu^{2}v(t)^{2}+ v'(t)^{2}\geq \biggl{(}\dfrac{8 k \Vert b \Vert_{L^{1}(\mathopen{[}0,T\mathclose{]})}}{\pi}\biggr{)}^{2}, t \in \mathopen{[}0,kT\mathclose{]}.
L^{\prime}[ C ] = - \frac{1}{2 \pi}\oint_{C}d u^{\mu}\, \partial_{\mu}\theta^{\prime}= - \frac{1}{2 \pi}\oint_{C}d u^{\mu}\, \partial_{\mu}\left[ \theta + \chi \right] = L [ C ] - \frac{1}{2 \pi}\oint_{C}d u^{\mu}\, \partial_{\mu}\chi ,
U^+_1(x,t) = f^+_0\int\limits_{-\infty}^{\eta}\! \mathrm{Ai}(\theta)\, d\theta +\sum\limits_{n=1}^{N-1}f^+_n (3t)^{-n/3}\left[ \int\limits_{1}^{\infty}\theta^{-n}\mathrm{Ai}(\eta-\theta)d\theta +B^+_n(\eta) \right]+
l_{2j,2N+1}(z) =\frac{(z-e_{2N})(z-e_{2j+1})}{(e_{2j}-e_{2N})(e_{2j}-e_{2j+1})}\prod_{i=0,i\not=j}^{N-1}\frac{(z^2-e_i^2)}{(e_j-e_i)}&=\frac{(z-e_{2N})(z+e_{2j})(e_{2j}+e_{2N})}{2e_{2j}(e_j-e_N)}l_{j,N}(z^2).
{\mathbf E}( w)=\int_{\mathbb{R}^2}\frac{1}{2}|\nabla w|^2+ \int_{\mathbb{R}^2}\frac{1}{4}\Big[\,A_+(| w^+|^2-{t^+}^2)^2+A_-(| w^-|^2-{t^-}^2)^2 begin{align*}1mm] \qquad\quad+2B(| w^+|^2-{t^+}^2)(| w^-|^2-{t^-}^2)\,\Big],
\left[\begin{array}{cc}\alpha I+A B^{T}-C \alpha I \end{array}\right]{\bf x}^{(k+1)}=\left[\begin{array}{cc}\alpha I-A -B^{T}C \alpha I \end{array}\right]{\bf x}^{(k)}+2\left[ \begin{array}{c}b q \end{array}\right].
\sum_{a=b}^{p'}\binom{a}b^2(H_a-H_{a-b}) =\sum_{a=b}^{p-1}\binom{a}b^2\sum_{j=0}^{b-1}\frac{1}{a-j}=\sum_{a=1}^{p-1}\binom{a}b^2\sum_{j=0,j\not=a}^{b-1}\frac{1}{a-j}&=\frac1{b!^2}\sum_{a=1}^{p-1}\sum_{i=1}^{2b-1}\alpha_i\,a^i=\sum_{i=1}^{2b-1}\frac{\alpha_i}{b!^2}\sum_{a=1}^{p-1}a^i\equiv_p 0,
1+\frac{a_{1}\cdot\cdot\cdot a_{p}}{b_{1}\cdot\cdot\cdot b_{q}}\frac{z}{1!}+\frac{a_{1}(a_{1}+1)\cdot\cdot\cdot a_{p}(a_{p}+1)}{b_{1}(b_{1}+1)\cdot\cdot\cdot b_{q}(b_{q}+1)}\frac{z^{2}}{2!}+\cdot\cdot\cdot
\begin{aligned}\rho\bigl(f\bigr)(s) = \operatorname{Ad}(u(s)) \Bigl( \operatorname{diag}\bigl[\overbrace{c(f(0)), \dots, c(f(0))}^a, f(t_1(s)), \dots, f(t_k(s)), \underbrace{c(f(1)), \dots, c(f(1))}_b \bigr] \Bigr), \end{aligned}
\chi_{n}( \xi_{1}) \equiv \langle \xi_{1}| n \rangle = \frac{( - i )^{n}}{\pi^{1 \! / \! 4}}\frac{1}{\sqrt{2^{n}n !}}H_{n}( \xi_{1}) e^{- \frac{1}{2}\xi_{1}^{2}}\hspace{0 . 1 5 c m},
B_5=\begin{cases}\{10\}\mbox{if}m_1=H(V_1) \{2,3\}\mbox{if}R_2<H(X_{2,1}), R_3<H(X_{3,1}), m_1<H(V_1) \{2,8\}\mbox{if}R_2<H(X_{2,1}), R_3\geq H(X_{3,1}), m_1<H(V_1) \{3,6\}\mbox{if}R_2\geq H(X_{2,1}), R_3<H(X_{3,1}), m_1<H(V_1) \end{cases},
\overline{{mdim}}_M(T,-(cg+1+\overline{{mdim}}_M(T,X,d)),d) = \overline{{mdim}}_M(T,-cg,d)-1-\overline{{mdim}}_M(T,X,d) \leq \overline{{mdim}}_M(T,0,d)-1-\overline{{mdim}}_M(T,X,d)=-1<0,
\sup_{x}|\hat{\gamma}_{2\ell}(x)-\beta_{L}^{\prime}b(x)| \leq\max_{j}\sup_{x}\left\vert b_{j}(x)\right\vert \sum_{j=1}^{p}\left\vert \hat{\beta}_{j}-\beta_{Lj}\right\vert =O_{p}((\varepsilon_{n}^{2})^{-1/(2\xi_{1}+1)}r_{2}) & =O_{p}(n^{-d_{1}(2\xi_{1}-1)/(2\xi_{1}+1)}\ln(n)).
Y_{m}^{S_{n}}(\mathrm{d}x):=\frac{\sqrt{\pi}}{4}\sum_{j=1}^{16^{m}}e^{\pi\lambda\Phi^{S_{n},\tilde{S}_{n,m}}(x)+\frac{4\lambda^{2}}{\gamma}s_{S_{n+m,j}}(x)}1_{S_{n+m,j}}(x)\mathrm{d}x
\psi_0(\vec r)\,=\,\sum_{n=-\infty}^{\infty}T_{n}^{0}(r, \phi)\,=\,\sum_{n=-\infty}^{\infty}\int_{-\infty}^{\infty}d\lambda (-i)^{|\lambda|}J_{|\lambda|}(kr) e^{i\lambda (\phi + \pi - \theta + 2n\pi)}.
{1\over 2}R^{(+)}_{ab,cd}\lambda_+{}^a \lambda_+{}^b\lambda_-{}^c\lambda_-{}^d ={1\over 4}R^{(+)}_{ab,cd}\lambda_+{}^a \lambda_+{}^b\lambda_-{}^c\lambda_-{}^d +{1\over 4}R^{(-)}_{ab,cd}\lambda_-{}^a \lambda_-{}^b\lambda_+{}^c\lambda_+{}^d \ .
n_{+}\big( r,\textup{\textbf{w}}p \textup{\textbf{w}}^{\ast}\big) = n_{+}\big( r,p \textup{\textbf{w}}^{\ast}\textup{\textbf{w}}p \big) = n_{+}\Big( r,p \textbf{\textup{W}}\big( H_+^{-1}\big) p \Big), r > 0.