harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
t^{1/2}\lVert \nabla \tilde{V}_m(t)\rVert_{L^\infty_H L^p_z}\le C_2\biggl( 2 H_m(t) \tilde{K}_{m-1}(t)+2 \tilde{H}_{m-1}(t) K_{m-1}(t)+K_m(t)\tilde{K}_{m-1}(t) &+K_{m-1}(t)\tilde{K}_{m-1}(t)+2 R t^{1/2}\tilde{K}_{m-1}\biggr)
\sigma_x\sigma_y=\sigma_{\sigma_x(y)}\sigma_{\gamma_y(x)}\gamma_y\gamma_x=\gamma_{\gamma_y(x)}\gamma_{\sigma_x(y)}\gamma_{\sigma_{\gamma_y(x)}(z)}(\sigma_x(y))=\sigma_{\gamma_{\sigma_y(z)}(x)}(\gamma_z(y))
i\partial_{t}\epsilon_{t}^{\mathbb T}(x) = \left( -\frac{1}{2}\Delta+ U*|\epsilon_{t}^{\mathbb T}|^{2}(x) + U*2\Re{\epsilon^{\mathbb T}_{t}}^{*}(x) \right) \epsilon_{t}(x) + U*\left( |\epsilon^{\mathbb T}_{t}|^{2}(x) + 2\Re{\epsilon^{\mathbb T}_{t}}^{*}(x) \right).
\pm\int_{(B^c_\varepsilon(x))^\pm}\frac{\xi \cdot (x - y)}{\, |x-y|^{N+2s}\,}\,dy =-\int_\varepsilon^{+\infty}\frac{d\rho}{\, \rho^{2s}\,}\int_{\partial B_1^\pm}(\pm\xi \cdot w) \, d{\mathcal H}^{N - 1}
\begin{cases}\rho_{n-j}(u_1,\ldots,u_{2n})=0, \rho_{n-j-1}(u_2,\ldots,u_{2n-1})=0, \rho_{n-j}(u_1,\ldots,u_{2n-1})=0, \rho_{n-j}(u_2,\ldots,u_{2n})=0, \end{cases}1\leq j\leq w.
\hat{B}\; = \; \chi_{L}\: ( \hat{A}_{R}\! \! \! \! \! \! \! \slash \; \: - m \: \hat{Y}_{R}) \: + \: \chi_{R}\: ( \hat{A}_{L}\! \! \! \! \! \! \! \slash \; \; - m \: \hat{Y}_{L}) \; \; \; ,
II=\int_{0}^{\pi}sin^{3}\theta d\theta=-\int_{0}^{\pi}sin^{2}\theta d(cos\theta)=\int_{0}^{\pi}(cos^{2}\theta-1)d(cos\theta)=\frac{cos^{3}\theta}{3}|_{0}^{\pi}-cos\theta|_{0}^{\pi}=\frac{4}{3}
\left( \Psi_{m}[ \{x_{i}\}] , \Psi_{n}[ \{y_{i}\}] \right) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
X= \sum_{k=0}^{g-1}(P^{d})^{k+2}RT^{k}T^{\pi}-P^{d}\sum_{j=0}^{n-1}R^{j+1}\sum_{h=0}^{l-1}(Q^d)^{h+j+1}S^{h}, T^{d}= P^\pi\sum_{j=0}^{n-1}\sum_{k=0}^{j}P^kR^{j-k}\sum_{h=0}^{l-1}(Q^d)^{h+j+1}S^{h},
\left\lbrace \begin{array}{lllll}\zeta = \left(\frac{\alpha_{m+1}}{\alpha_1}\right)^{\frac{1}{\gamma_{m+1}-\gamma_1}}, \alpha_i = \alpha_1\zeta^{\gamma_i-\gamma_1}1\leqslant i \leqslant r, \alpha_1 \sum\limits_{i=1}^r \zeta^{\gamma_i-\gamma_1}= 1, \sum_{i=1}^r \gamma_i \zeta^{\gamma_i-\gamma_1}= 0. \end{array}\right.
\widetilde{\psi}_{k}^{( n^{2})}(x;N,M,\lambda_{1}) =[ E_{n}(N,M) -E_{k}(N,M) ] \psi_{k}( x;N,M) \hphantom{\widetilde{\psi}_{k}^{( n^{2})}(x;N,M,\lambda_{1}) =}{}+(1-z)^{( 3N+5/2) /2}(1+z)^{( 3M+5/2) /2}\frac{P_{n,k}^{(N,M)}(z) P_{n}^{(N,M)}(z)}{\lambda_{1}+Q_{n}^{( N,M)}(z)},
|\Delta_h^{m}(\psi f,t)| \leq \sum\limits_{j=0}^m\Big|\binom{m}{j}\psi(t+jh)\Big|\cdot |f(t + jh)| \leq c_{m,\psi}c_{m,a}\max\{1,|bh|^a\}\sup\limits_{|\xi|\leq mh}\frac{|f(t-\xi)|}{(1+|b\xi|)^a}\leq C_{m,a,\psi}\max\{1,|bh|^a\}P_{b,a}f(t)\,.
\langle \partial_t \bar{u}, z \rangle_{V^*,V}+ \int_\Omega \nabla \bar{\mu}\cdot \nabla z \,dx=0, (\bar{\mu}, z)_H = (\nabla \bar{u}, \nabla z)_{H}-\langle \partial_{\boldsymbol{\nu}}\bar{u}, z_{|_\Gamma}\rangle_{Z_\Gamma^*,Z_\Gamma}+ (\bar{\xi},z)_H + \bigl( \pi (u_1)-\pi(u_2) -\bar{f},z \bigr)_{\!H}
\lim_{N_1,N_2\to+\infty}\frac 1{N_1N_2}\sum_{n_1\in[N_1], n_2\in[N_2]}\Psi(\tau_1^{p_1n_1}\tau_2^{p_2n_2}\cdot e_X)\, \overline{\Psi(\tau_1^{p'_1n_1}\tau_2^{p'_2n_2}\cdot e_X)}=0
M_i(x) = x^{-(m_i-m_{i-1})}\left(M_{i-1}(x) - \frac{f^{(m_i)}(0)}{m_i!}x^{m_i-m_{i-1}}\right) = \frac{M_{i-1}(x)}{x^{m_i-m_{i-1}}}-\frac{f^{(m_i)}(0)}{m_i!}.
\mathbb{E}\big[ \mathfrak{m}^{(p,p)}\big]= \mathbb{E}\big[(O_\prec(\hat{\Pi}^2)+O_\prec(\Psi^2 \hat{\Pi}))\mathfrak{m}^{(p-1,p)}\big]+\mathbb{E}\big[O_\prec(\Psi^2\hat{\Pi}^2) \mathfrak{m}^{(p-2,p)}\big] &\qquad+\mathbb{E}\big[O_\prec(\Psi^2\hat{\Pi}^2) \mathfrak{m}^{(p-1,p-1)}\big].
\psi(\zeta,\xi,\xi_1)=\frac{(\zeta-\xi)(\xi_1-\overline{\xi})+\i(\zeta-\overline{\xi})(\xi_1-\xi)}{(\zeta-\overline{\xi})(\xi_1-\xi)+\i(\zeta-\xi)(\xi_1-\overline{\xi})}, \phi(w)=\frac{w-\i}{\i w-1}.
\varphi(s,r) = \frac{(2\pi)^{(k_1-k_2)^2}}{2^{2k_1k_2}k_1!k_2!}\, \frac{\det\left[J_0(s_{p1}r_{p'1})\right]_{p,p'=1,\ldots,k_1}\det\left[J_0(s_{q2}r_{q'2})\right]_{q,q'=1,\ldots,k_2}}{B_{k_1k_2}(s^2)B_{k_1k_2}(r^2)}\ ,
{{\mathrm{\boldmath ~ \s i g m a ~}}\cdot{\bf p}}{\frac{f ( r )}{r}}{\phi^{l}}_{j m}= -{\frac{i}{r}}({\frac{d f}{d r}}+{\frac{\kappa f}{r}}){{\mathrm{\boldmath ~ \s i g m a ~}}\cdot{\bf n}}{\phi^{l}}_{j m},
\cos F(\delta)=\dfrac{\left\langle A^{1/2}\dfrac{x-x_0}{|x-x_0|}, A^{1/2}\dfrac{e}{|e|}\right\rangle}{\left| A^{1/2}\dfrac{x-x_0}{|x-x_0|}\right|\,\left| A^{1/2}\dfrac{e}{|e|}\right|},
\begin{array}{c}{( 3 2 1 ) , ( 4 7 1 ) , ( 5 7 2 ) , ( 6 7 3 ) , ( 5 6 1 ) , ( 6 4 2 ) , ( 4 5 3 ) ,}\\{\fbox{2}\fbox{1}5 \fbox{3}6 4 7 .}\\ \end{array}
f^{-1}(\omega)\triangle g^{-1}(\omega)= \Big(\bigcup_{x\in \omega_c}[x-1,x+1] \Big )\setminus \Big(\bigcup_{x\in \omega}(x-1,x+1) \Big )\subset \bigcup_{x\in \omega_c}\{x-1,x+1\}
\partial_{\mu}\hat{J}_{V}{}^{\mu}= \lim_{y \rightarrow x}{\frac{i e^{2}}{2}}\int_{x}^{y}d w ( \hat{E}( w ) \hat{\Pi}_{+}( y , x ) + \hat{\Pi}_{+}( y , x ) \hat{E}( w ) ) \; .
U_{1 2}= V_{1 2}- \{P_{0}\! - \! S ,{\frac{V_{1 2}}{2 ( P_{0}\! - \! h_{3})}}\}+{\frac{V_{1 2}( P_{0}+ S - 2 h_{3}) V_{1 2}}{4 ( P_{0}- h_{3})^{2}}}
\begin{array}{ll}{}~~~[D,P_{\mu}]=-iP_{\mu}\,,~~~~ ~~~~[D,K_{\mu}]=iK_{\mu}\,,{}{}{}~~~[D,Q^{a}]=-i\textstyle{\frac{1}{2}}Q^{a}\,,~~~~ ~~~~{[D,S^{a}]=i\textstyle{\frac{1}{2}}S^{a}}\,,{}{}\multicolumn{2}{c}{[D,D]=[D,M_{\mu\nu}]=[D,A_{a}{}^{b}]=0\,.}\end{array}
\|\lambda_{y}\|^2 = \int_{\mathbb{R}^3_+}\mathrm{d}k \,\frac{\chi^2_{\Lambda}(k)}{\pi^2|k|}\sum_{\gamma=1,2}\left\{\mathbf{e}^{(1)\,2}_{\gamma}(k) \cos^2(k_1 y) + (\mathbf{e}^{(2)\,2}_{\gamma}(k) +\mathbf{e}^{(3)\,2}_{\gamma}(k)) \sin^2(k_1 y) \right\},
M=\int d^{3}x\epsilon^{abc}\{A_{c}^{k},\sqrt{V}_{\epsilon}\}F_{ab}^{k}(x)\int d^{3}y\chi_{\epsilon}(x,y)\epsilon^{a^{\prime}b^{\prime}c^{\prime}}\{A_{c^{\prime}}^{k^{\prime}},\sqrt{V}_{\epsilon}\}F_{a^{\prime}b^{\prime}}^{k^{\prime}}(y)
\bigg( \sum_{k=-\infty}^{M-1}\bigg( \int_0^{x_k}f^r v \bigg)^{\frac{q}{r}}\int_{x_k}^{x_{k+1}}u(t) dt \bigg)^{\frac{1}{q}}\leq \mathfrak{C'}\bigg( \sum_{k=-\infty}^{M-1}2^k \bigg( \int_{x_k}^{x_{k+1}}f \bigg)^p \bigg)^{\frac{1}{p}}
\boldsymbol{\delta}_\emptyset^\star\mathbf{W}_{\mathbf{g}}^\star\mathbf{X}\mathbf{W}_{\mathbf{g}}\boldsymbol{\delta}_\emptyset=\delta_\emptyset^\ast \boldsymbol{F}_\nu\mathbf{W}_{\mathbf{g}^\bullet}^\star\mathbf{X}\mathbf{W}_{\mathbf{g}^\bullet}\boldsymbol{F}_\nu^\star \delta_\emptyset,
{\cal L}_{W e y l}^{n m}= \partial_{+}{A}_{i}\left( \partial_{-}{A}_{i}- \partial_{i}A_{-}\right) + \frac 1 2 \left( \partial_{+}{A}_{-}\right)^{2}- \frac 1 4 \left( \partial_{i}{A}_{j}- \partial_{j}{A}_{i}\right)^{2}+{A}_{-}{J}^{-}+{A}_{i}{J}^{i}\; ,
\vec{w}_{C}= \sum_{A \in \binom{C}{k-2}}\beta_{A}= (k-1-r_{C}) \cdot (-1)^{r_{C}}r_{C}! (k-2-r_{C})! + r_{C}\cdot (-1)^{r_{C}-1}(r_{C}-1)! (k-1-r_{C})! = 0.
X_3(y_1, y_2, y_3)= \frac{1}{\sqrt{N}}\int dx_1dx_2 \Big\{&\quad\u(y_1, x_1)v_N(x_1-x_2)\phi(x_2)\bar c(y_2, x_1) c(x_2, y_3) &+ u(y_1, x_1)v_N(x_1-x_2)\phi(x_2) u(y_2, x_2) c(x_1, y_3) \Big\}
\frac{i g^{4}\epsilon^{2}R}{1 6 \pi^{2}n^{3}}\int \frac{d^{d - 2}q_{i}}{( 2 \pi )^{d - 2}}( 1 + \epsilon^{2}q_{i}q_{i}R^{2}/ n^{2})^{- 3 / 2}\ .
\langle h,x_U \rangle =A_Uh=A_I\pi(U)^{-1}h=\langle \pi(U)^{-1}h,x_U \rangle=\langle h,\pi(U)x_U \rangle, &\langle h,\tau_U \rangle =\Psi_Uh=\Psi_I\pi(U)^{-1}h=\langle \pi(U)^{-1}h,\tau_U \rangle=\langle h,\pi(U)\tau_U \rangle,
\begin{aligned}T =(\mu+\nu)\bar{g}(\cdot,X)\otimes\bar{g}(\cdot,X)+\nu\bar{g}=\sec^2(\xi(x))\left[\nu(\xi(x)) x^*g_0+\mu(\xi(x))\left(e^{-\xi(x)\sqrt{n-1}}\right)^2dt^2\right], \end{aligned}
s_{i}^{2}= 1 i=1,2,...n s_{i}s_{i+1}s_{i}=s_{i+1}s_{i}s_{i+1}i=1,2,...n-1 s_{n-1}s_{n}s_{n-1}s_{n}= s_{n}s_{n-1}s_{n}s_{n-1}
\tilde{\Xi}_\pm D_aA\varphi =\tilde{\Xi}_\pm D_a\gamma^{(2n+1)}(P_{\rm F}-D_aP_{\rm F}D_a)\varphi &=-\tilde{\Xi}_\pm\gamma^{(2n+1)}D_a(P_{\rm F}-D_aP_{\rm F}D_a)\varphi &=\tilde{\Xi}_\pm\gamma^{(2n+1)}(P_{\rm F}-D_aP_{\rm F}D_a)D_a\varphi=A\tilde{\Xi}_\pm D_a\varphi,
\omega_1^{-1}(h)=\lambda_2 \nu, \omega_2^{-1}(h)=\lambda_2, \omega_3^{-1}(h)=\lambda_1\lambda_2\nu^2, \omega_4^{-1}(h)=\lambda_1\lambda_2\nu\ \text{for $h=\operatorname{diag}(\lambda_1,\lambda_2,\nu) \in T$.}
\sigma\left(\sum_{i=1}^{n}c_{i}x_{i}\right) \geq \rho\left(\varphi\left(\sum_{i=1}^{n}c_{i}x_{i}\right)\right) = \rho\left(\sum_{i=1}^{n}c_{i}\varphi(x_{i}) \right) & = \rho\left(\sum_{i=1}^{n}c_{i}y_{i}\right) \geq\delta\sum_{i=1}^{n}\lvert c_{i}\rvert.
n_{uv}=(\kappa t-\tau b)_{v}=(\kappa_{v}-\lambda_{u}\tau)t+(\kappa\lambda\tau+\tau\gamma)n+(\lambda_{u}\kappa-\tau_{v})b, n_{vu}=(\lambda\tau t+\gamma b)_{u}=(\lambda_{u}\tau+\lambda\tau_{u})t+(\kappa\lambda\tau+\tau\gamma)n+(\gamma_{u})b.
h_0 \partial_{t}v^{(0)}_0 =h_0 \partial^2_{x_2 x_2}v^{(0)}_0 - h_0 k_0\big(v^{(0)}_0\big) - 2 \delta_{\alpha_0,1}\kappa_0\big(v^{(0)}_0\big) + 2 \delta_{\beta_0,1}g^{(0)}_0
g = \sum_{m,n}\mathbf 1_{A_n}\Big(\sum_m \nu(B_{nm})\|x_{nm}^*\|_{X^*}^{q'}\Big)^{\frac{p'}{q'}-1}\ \mathbf 1_{B_{nm}}\x_{nm}\|x_{nm}^*\|_{X^*}^{q'-1},
\nabla^\lambda_X(\lambda Y \xi)- \lambda Y \nabla^\lambda_X\xi=\nabla_X(\lambda Y \xi)+\frac{1}{4\lambda}(Z X-XZ)(\lambda Y \xi) &\phantom{\lambda Y \nabla^\lambda_X\xi=}-\lambda Y \left(\nabla_X\xi+\frac{1}{4\lambda}(Z X-XZ)\xi\right) = \left(\nabla^{LC}_X(\lambda Y)\right)\xi+\frac14(ZXY-XZY-YZX+YXZ)\xi = \lambda\left(\nabla^{LC,\lambda}_X(Y)\right)\xi\,,
\sigma^{0 k}\equiv i \alpha^{k}= i \,\left( \begin{array}{cc}0 \sigma^k \sigma^k 0 \end{array}\right)\; ; \; \sigma^{i j}= \epsilon_{ijk}\,\left( \begin{array}{cc}\sigma^k 0 0 \sigma^k \end{array}\right) \; ,
\begin{array}{rl}\displaystyle \Delta_{\pm}=\displaystyle\frac{c}{24}\pm \left( I^{\pm}_{H}-2I^{\pm}_{S}-I^{\pm}_{C}-I^{\pm}_{W}\right) +\displaystyle\frac{1}{2\pi}\Sigma_{\pm}\mp \displaystyle\frac{1}{2\pi}\displaystyle\int\limits^{\infty}_{-\infty}\displaystyle\frac{dx}{2\pi}\varphi^{,}_{\pm}(x){\cal Q}_{\pm}(x)\: ,\end{array}
\overline \varphi_{m_1,\dots,m_k}^{(j)}(X,\dots ,X) &\ = \gamma^k\int_{{\cal K}_0}\dots\int_{{\cal K}_0}\int_{SO_d}\varphi^{(j)}_{m_1,\dots,m_k}(\vartheta K_{1},K_2,\dots,K_{k})\,\nu( d\vartheta)\,{\mathbb Q}(d K_1)\cdots \,{\mathbb Q}(d K_k)
\alpha =1^{c+d}a1^c \beta=1^{c+d+1}a1^{c-1}\alpha =(a-1)1^{c-1}21^{c+d}\beta=(a-1)1^{c+d}21^{c-1}\alpha =1^{c-1}21^{c+d-1}a \beta=1^{c+d}21^{c-2}a \alpha =1^{c-1}a1^{c+d-1}2 \beta=1^{c+d}a1^{c-2}2 \alpha =1^d21^{c-1}a1^{c-1}\beta=1^d21^{c-2}a1^c
P_{\infty,0\#}F_{\infty\#}N_\infty(fdg_1\wedge df_2)=N_\infty(f\circ P_{\infty,0}\circ F_\infty d(g_1\circ P_{\infty,0}\circ F_\infty)\wedge d(g_2\circ P_{\infty,0}\circ F_\infty));
\mathbf{B}^i_{t}(x_1)-\mathbf{B}^i_{t}(x_2)= rA_{i1}^{}+ rA_{i1}^{}+rA_{i1}^{}+rA_{i1}^{}+r\varepsilon_1 A_{i2}+r(\mathbf{e}_1.\eta_t^\varepsilon(x_1))A_{i1}^{},
\chi(\tau,\bar{\tau})=\left[\sum_{s}\Lambda_{s}(\tau,\bar{\tau})\right]\theta_{3}^{'}\left(0\left|\frac{i\beta^{2}}{2\pi^{2}\tau_{2}}\right.\right)-\tau^{-\frac{d-2}{2}}_{2}\frac{2\bar{\theta}_{2}^{4}}{\bar{\eta}^{12}}(z_{s}+z_{c})\theta_{2}\left(0\left|\frac{2i\beta^{2}}{\pi^{2}\tau_{2}}\right.\right)
\begin{cases}\phi_{xx}+(c+(\chi_2V_2-\chi_1 V_1)_x)\phi_x+(a+(\chi_2\lambda_2V_2-\chi_1\lambda_1V_1)(x))\phi\cr -((b+\chi_2\mu_2-\chi_1\mu_1)U(x))\phi=\lambda\phi,x_\epsilon<x<x_\epsilon+L\cr\phi(x_\epsilon)=\phi(x_\epsilon+L)=0.\end{cases}
\mathbb{E}\left[\varphi(t\wedge \tau_0,W(t \wedge \tau_0))]=\mathbb{E}[\int_{s}^{t\wedge \tau_0}\left(\partial_t \varphi(u,W(u))+\frac{1}{2}\partial^2_y \varphi(u,W(u)) \right. \right. \left. \left. +\varphi(u,W(u))\phi''(u)W(u)\right)\left(\int_{s}^{u}\phi''(\omega)W(\omega)d\omega\right)du \right]
\tilde{g}(t,\,x)= \tilde{f}(t,\,G_{\rho}(t,\,x)) J(t,\,x)= \int_{\mathbb{T}}\frac{\sum_{c=1}^{m}\rho_{c}(t,\,y)f_{x}^{(c)}(t,\,y)}{\lambda+\rho(t,\,y)}\nu(y-x)dy.
P_{{\bf b}_s, d_s}\biggl\{g_0(t)+g_1(t)\biggl(\sum_{n=0}^{\infty}\Bigl(\prod_{l=1}^{d_1-1}(\theta^{q^{l+n}}-t)^{c_1(l)q^{d-l}}\alpha_1^{q^{n+d}}\Bigr)\biggr)+ &\cdots+g_s(t)\biggl(\sum_{n=0}^{\infty}\Bigl(\prod_{l=1}^{d_s-1}(\theta^{q^{l+n}}-t)^{c_s(l)q^{d-l}}\alpha_s^{q^{n+d}}\Bigr)\biggr)\biggr\}=0.
\frac{d}{dt}=\frac{d\tau}{dt}\frac{d}{d\tau}=\frac{1}{t_{c}}\frac{d}{d\tau}\Rightarrow\frac{d^{n}}{dt^{n}}=(\frac{d}{dt})^{n}=(\frac{1}{t_{c}}\frac{d}{d\tau})^{n}=\frac{1}{t_{c}^{n}}\frac{d^{n}}{d\tau^{n}}.
\partial_{i}\hat{n}\cdot \partial_{i}\hat{n}=\left( \frac{2 k^{2}}{\lambda^{2k}}\right) |z -z_{0}|^{2k-2}\frac{1}{\left[1+\frac{1}{4 \lambda^{2k}}|z-z_{0}|^{2 k}\right]^{2}}
V_{0}= \textstyle{\left[ -{\frac{1}{2}}\left( -{\frac{1}{2}}-{\frac{1}{2}}-{\frac{1}{2}}\right)^{3}| | -{\frac{1}{2}}\left( -{\frac{1}{2}}-{\frac{1}{2}}-{\frac{1}{2}}\right)^{3}\right]}~ .
(\mathcal{M}+\mathcal{D})u^{r-1}=0 \implies \mathcal{M}u^{r-1}+\mathcal{D}u^{r-1}=0 \implies \mathcal{M}u^{r-1}= -\mathcal{D}u^{r-1}<0\; (\leq 0),
{}(\frac{1}{n})^{\frac{1}{1 + \alpha}- \frac{\bar{\beta}}{2 \bar{\beta}+ d}\frac{d-2}{\bar{\beta_3}}}& = (\frac{1}{n})^{\frac{\bar{\beta}(2 \bar{\beta_3}+ d - 2)}{\bar{\beta_3}(2 \bar{\beta}+ d)}- \frac{\bar{\beta}(d-2)}{\bar{\beta_3}(2 \bar{\beta}+ d)}}& = (\frac{1}{n})^{\frac{2 \bar{\beta}}{2 \bar{\beta}+ d}},
\begin{array}{rcl}Z^n_m Z^n_m\propto Z^n_m~, & S_m w S_m\propto S_m~,~~~ \forall w \in A_{2m}~, & Y_m w Y_m\propto Y_m~,~~~ \forall w \in A_n~, & Y_m Y_m\not=0~.\end{array}
[h_{i},h_{j}]=0, [h_{i}, e_{j}] =a_{ij}e_{j}, [h_{i}, f_{j}] =-a_{ij}f_{j}, &[e_{i},f_{j}]=\delta_{ij}\frac{q^{h_{i}}-q^{-h_{i}}}{q-q^{-1}}, &[e_{i},[e_{i},[e_{i},e_{j}]_{q^{2}}]]_{q^{-2}}=[f_{i},[f_{i},[f_{i},f_{j}]_{q^{-2}}]]_{q^{2}}=0i \ne j ,
-t_{11}\cdot s_{12}+ t_{12}= -\frac{(1+|v|^2)\cdot v\cdot (1+\overline{v}^2)}{|v|^2(\overline{v}-v)}\cdot\frac{v(1-\overline{v}^2)}{2|v|^2}+{2}\frac{(1+|v|^2)(1+\overline{v}^2)}{(1-\overline{v}^2)(v-\overline{v})}&= \frac{(1+|v|^2)(1+\overline{v}^2)^3}{2(v-\overline{v})\overline{v}^2(1-\overline{v}^2)}\ne 0,
S_N-S_{N,1}=\sum^N_{n=1}\sum^{m_0}_{j=1}\left( \mathbb{E}\big[K(X_n)|\mathcal{F}_{n-j}\big]-\mathbb{E}\big[K(X_n)|\mathcal{F}_{n-j-1}\big]\right):=\sum^N_{n=1}\sum^{m_0}_{j=1}\mathcal{P}_{n,n-j}
\phi =\left[ \begin{array}{ccc}a_{11}a_{12}0 a_{21}a_{22}0 a_{31}a_{32}a_{11}a_{22}+a_{21}a_{12}\end{array}\right] :a_{11}^{2}a_{22}^{2}-a_{12}^{2}a_{21}^{2}\neq 0\mbox{and}a_{11}a_{21}=a_{12}a_{22}=0.
\left( \begin{array}{c}{{\mathrm{{\small ~ n u m b e r ~ o f ~ d i f f e r e n t}~}}}\\{{\mathrm{{\small ~ W e y l ~ p a r t i c l e s}~}}}\\ \end{array}\right)^{2}= \left( \begin{array}{c}{{\mathrm{{\small ~ n u m b e r ~ o f ~}~}}}\\{{\mathrm{{\small ~ g a u g e ~ b o s o n s}~}}}\\ \end{array}\right)
\sum_{w\in\Lambda^*}e^{-\frac{\pi^2|w|^2}{t}}t^{-\frac{d}{2}}=\pi^{-d/2}\sum_{v\in\Lambda}e^{-|v|^2 t}=\pi^{-d/2}+\pi^{-d/2}e^{-l_0^2 t}\sum_{v\in\Lambda\setminus\{0\}}e^{-(|v|^2-l_0^2)t}=\pi^{-d/2}+O(e^{-l_0^2 t}),
G_{\mu \nu}^{0 A}( k , \epsilon = 0 ) = -{\frac{1}{k^{2}}}\biggl ( g_{\mu \nu}+{\frac{( \lambda k^{2}+ \eta^{2})}{( \eta \cdot k )^{2}}}k_{\mu}k_{\nu}-{\frac{k_{[ \mu}\eta_{\nu ]_{+}}}{\eta \cdot k}}\biggr ) .
\sup_{f_{+}+f_{-}\in\mathcal{F}_{RDT,p}(C)}2(f_{+}(0)-f_{-}(0)) \sum_{i=1}^{n}\frac{f_{-}(x_{i})^{2}}{\sigma^{2}(x_{i})}+ \sum_{i=1}^{n}\frac{f_{+}(x_{i})^{2}}{\sigma^{2}(x_{i})}\leq \delta^{2}/4.
g_{1}(\beta_1) =\sum_{j=3}^{n}l_j\Big[q_j \frac{\psi(\beta_1 q_2)}{q_2}\Big(\frac{\psi'(\beta_2 q_1)\psi'(\beta_1 q_j)+\psi'(\beta_2 q_j)\psi'(\beta_1q_1)}{\Delta}\Big) - \frac{q_j \psi(\beta_1 q_1)}{q_1}\Big(\frac{\psi'(\beta_1 q_2)\psi'(\beta_2 q_j)+\psi'(\beta_2 q_2)\psi'(\beta_1 q_j)}{\Delta}\Big) +\psi(\beta_1 q_j)\Big],
\langle Q'(u,v), (u,v)\rangle =p\|(u,v)\|^p-q\int_\Omega (\lambda |u|^{q}+\mu |v|^{q})dx -2(\alpha+\beta)\int_\Omega |u|^{\alpha}|v|^\beta dx &=(p-1)\|(u,v)\|^p-(q-1)\int_\Omega (\lambda |u|^{q}+\mu|v|^q)dx-2(\alpha+\beta-1) \int_\Omega |u|^{\alpha}|v|^\beta dx=\varphi_{u,v}''(1)\neq 0,
\int_I L(r, u ,u') \, dr = \int_I ((u_1 - v_1)^2 + (u_2^3 - t)^2) (u_2')^6 + \epsilon \left( (u_1')^2 + (u_2')^{\sigma}\right) \, dr & > \int_I ( u_2^3 - t)^2 (u_2')^6 + \epsilon \left( (v_1')^2 + (u_2')^{\sigma}\right) \, dr & = \int_I L(r, \hat{u}, \hat{u}')\, dr,
U_1(d_1)=\begin{pmatrix}1 d_1 0 0 0 1 0 0 0 0 1 -d_1 0 0 0 1\end{pmatrix}, U_2(d_2)=\begin{pmatrix}1 0 d_2 0 0 1 0 -d_2 0 0 1 0 0 0 0 1\end{pmatrix}, U(d_1,d_2) = U_1(d_1) U_2(d_2),
{F_{s+1}^{\pm}}\left( \xi \right) =-{\dfrac{1}{2}}\frac{d{F_{s}^{\pm}}\left( \xi \right)}{d\xi}-{\dfrac{1}{2}}\sum\limits_{j=0}^{s}{F_{j}^{\pm}\left( \xi \right) F_{s-j}^{\pm}\left( \xi \right)}\ \left({s=1,2,3,\cdots}\right) .
\psi_{k}( z , \bar{z}) ={\frac{1}{\sqrt 2}}\left[ \left({\frac{f_{n , m}( z )}{\overline{{f}}_{n , m}( z )}}\right)^{k}+ \left({\frac{\overline{{{f}}}_{n , m}( z )}{\f_{n , m}( z )}}\right)^{k}\right] , \qquad k = 0 , 1 , 2 , \ldots ,
I=\int_{0}^{\frac{\pi}{2}}\frac{1}{\sqrt{a^{2}cos^{2}(\theta)+b^{2}sin^{2}(\theta)}}d\theta=\int_{0}^{\frac{\pi}{2}}\frac{1}{AGM(a,b)}d\theta=\frac{\pi}{2AGM(a,b)}
\psi (u) \geq \frac{1}{2p}\|u\|^p_{W_0^{s,p}(M)}-C C_{1}(q,p)\|u\|^q_{W_0^{s,p}(M)}= \|u\|^p_{W_0^{s,p}(M)}(\frac{1}{2p}-C C_{1}(q,p)\|u\|^{q-p}_{W_0^{s,p}(M)}) = b^{p}(\frac{1}{2p}-C C_{1}(q,p)b^{q-p})=:a.
v^4(z) =|(v^2\varphi)(z)|^2\le\int_0^{2r}|\partial_x(v^2\varphi)(x+s,y)|ds \cdot\int_0^{2r}|\partial_y(v^2\varphi)(x,y+t)|dt \le\int_{\{s;(s,y)\in B\}}|\partial_x(v^2\varphi)(s,y)|ds \cdot\int_{\{t;(x,t)\in B\}}|\partial_y(v^2\varphi)(x,t)|dt,
U^e_t(\phi)= \int_0^t \sum_{(e,\tilde{e})}\,2C^n_{\tilde{e},e}\big(\phi_s(x^e_1)-\phi_s(x^{\tilde{e}}_1)\big)\,\big(u^n_{s-}(x^{\tilde{e}}_1) - u^n_{s-}(x^e_1)\big)\,ds V^e_t(\phi)= \,-\int_0^t \alpha_e\big(u^n_{s-}(x^e_2) - u^n_{s-}(x^e_1)\big)\,\nabla_L\phi_s(x^e_1)\;ds.
{1 \over 2\pi}\int_0^{2\pi}d\varphi \, \left[ R(r,\varphi) \, K_1(R(r,\varphi) ) \right]^2 = \sum_{n=-\infty}^{\infty}\left[ r (I_n \, K_n )'(r) \right]^2 \equiv F(r) ,
\vec{\chi}( x ) = \left( \begin{array}{c}{\vec{\chi}_{\mathrm{v e c}}( x )}\\{\vec{\chi}_{\mathrm{g r}}( x )}\\ \end{array}\right) = \left( \begin{array}{c}{A ( x ) \vec{n}}\\{B ( x ) \vec{p}}\\ \end{array}\right) \ , \qquad \partial_{i}\partial_{i}\vec{\chi}( x ) = 0 \ ,
\hat{G}^{\hat{\alpha}_{1}\ldots\hat{\alpha}_{4}}= \left(f^{-1}\right)^{\hat{\alpha}_{1}\ldots\hat{\alpha}_{4}}{}_{\hat{\beta}_{1}\ldots\hat{\beta}_{4}}\left(\hat{B}^{\hat{\beta}_{1}\ldots\hat{\beta}_{4}}-{}^{\star}\hat{B}^{\hat{\beta}_{1}\ldots\hat{\beta}_{7}}\hat{A}_{\hat{\beta}_{5}\hat{\beta}_{6}\hat{\beta}_{7}}\right)\, ,
\overline{L}_{-1}\big(\rho_0u^p\bar{u}^q\big)=\left(\sum\limits_{j\ge1}jp_ju_j^{-1}u_{j-1}u^p\bar{u}^q\right)+\left(\bigg(2\sum\limits_{j\ge1}q_j-2\bigg)\bar{u}_1u^p\bar{u}^q\right) \hphantom{\overline{L}_{-1}\big(\rho_0u^p\bar{u}^q\big)=}{}-\left(\sum\limits_{j\ge1}(j+2)q_j\bar{u}_j^{-1}\bar{u}_{j+1}u^p\bar{u}^q\right).
\frac{\partial w^\pm}{\partial{x_1}}=\frac{1}{2}\Big({U^\pm}'-\frac{U^\pm}{r}\Big)e^{2i\theta}\,+\,\frac{1}{2}\Big({U^\pm}'+\frac{U^\pm}{r}\Big), begin{align*}1mm] \frac{\partial w^\pm}{\partial{x_2}}=\frac{-i}{2}\Big({U^\pm}'-\frac{U^\pm}{r}\Big)e^{2i\theta}\,+\,\frac{i}{2}\Big({U^\pm}'+\frac{U^\pm}{r}\Big).
\begin{aligned}E\tau^{N,i}\approx e^{N\min\big(I_i(r_i),\thinspace I_i(r_{i+1})\big)},\ 1\le i\le l-1; &E\tau^{N,l,l}\approx e^{NI_l(r_l)};\ \E\tau^{N,0}\approx e^{NI_0(r_1)}. \end{aligned}
\vec{y}( \omega ) ={\frac{- e \vec{E}^{e}( \omega ) / \omega^{2}}{m_{0}+{\frac{2}{3}}e^{2}\int{\frac{d \vec{k}| f ( \vec{k}) |^{2}}{( 2 \pi )^{3}( k^{2}- \omega^{2})}}}}{~ ~}.
P_{3,c}^{\mathrm{PNC}}(g_{r1},g_{r2})= \frac{\left(2^{R_{3,c}^{\mathrm{PNC}}}-1\right)\left( 1+P_{3,p}^{\mathrm{PNC}}\min(g_{r1},g_{r2}) \right)}{\min (g_{r1},g_{r2})}.
\sum_{k=n}^{\infty}\frac{\Gamma(k+\alpha +1)}{x^k}\equiv \frac{\Gamma(n+\alpha+1)}{x^n}\; \Lambda_{n+\alpha}(-x) + \frac{2 \pi i\,x^{s+1}\,e^{-x}}{\Gamma(n+\alpha+1)}\; S \;\;,
\widehat{\gamma}_{1}:=\left( \sum\limits_{i=1}^{k}\frac{\mathbf{F}_{n}\left( X_{n-i+1:n}\right)}{C_{n}\left( X_{n-i+1:n}\right)}\right)^{-1}\sum_{i=1}^{k}\frac{\mathbf{F}_{n}\left( X_{n-i+1:n}\right)}{C_{n}\left( X_{n-i+1:n}\right)}\log \frac{X_{n-i+1:n}}{X_{n-k:n}},
\begin{array}{lcr}q_{n-1}^{n-1}=q_n^n=-2,\quad q_{n-1}^{n-2}=q_n^{n-2}=1 q_{n-2}^{n-1}=q_{n-2}^{n}=q_{n+3}^{n-1}=q_{n+1}^{n}=q^n_{n+2}=1 q_{n}^{n-1}=q_{n-1}^n=q_{n+1}^{n-1}=q_{n+1}^{n}=q_{n+2}^{n-1}=0. \end{array}
\mathfrak{I}_2(\omega) ={}\inf \{S_\omega ({\bf u})- \tfrac{d-2s_c}{2d}V ({\bf u}) \ |\{\bf u}\neq0, \, V({\bf u}) \le 0\}\le{}\inf \{S_\omega ({\bf u})- \tfrac{d-2s_c}{2d}V ({\bf u}) \ |\{\bf u}\neq0, \, V({\bf u}) = 0\}= \inf \{S_\omega ({\bf u}) \ |\{\bf u}\neq0, \, V({\bf u}) = 0\}.
H_{T}= \int d x^{1}\mathrm{T r}\left\{- \omega_{0}( D_{1}\pi_{\omega_{1}}- [ \pi_{\phi}, \phi ] ) - ( \phi^{2}+ \{\pi_{\omega_{1}}, \phi \}) B + \lambda_{\omega_{0}}\pi_{\omega_{0}}+ \lambda_{B}\pi_{B}\right\}.
\sum\limits_{j=0}^{2}\varepsilon^{\frac{j}{2}}\left\vert \lambda\right\vert^{1-\frac{j}{2}}\left\Vert u_{1}^{\left( j\right)}\right\Vert_{X}+\left\Vert Au_{1}\right\Vert_{X}\leq C\left\Vert f\right\Vert_{X}.
\sum_{1\leq k < \ell \leq m}N_{k,\ell}= 2\!\!\!\sum_{1\leq k < \ell \leq m}3^{\ell - k - 1}= 2\!\sum_{1 < \ell \leq m}\, \sum_{1\leq i < \ell}3^{i - 1}\leq \sum_{1 < \ell \leq m}3^{\ell - 1}\leq 3^{m}.
\frac{{\partial{{\mathcal B}_i^{DF}}\left({{W_i}}\right)}}{{{W_i}}}&= \frac{{{X_i}}}{{2{Y_i}{Z_i}}}\left[{1 - \frac{{{X_i}{W_i}+{X_i}-{Y_i}{Z_i}+{Z_i}}}{{\sqrt{{{\left({{X_i}{W_i}+{X_i}-{Y_i}{Z_i}+{Z_i}}\right)}^2}+ 4{Y_i}Z_i^2}}}}\right].
\xi_{0}( t ) = \alpha^{- 1}\left[ 1 + \frac{1}{2 \kappa^{2}\cos^{4}{t}}- \frac{1}{\kappa^{2}\cos{t}}( c_{1}\sin{( 2 \kappa \sin{t})}+ c_{2}\cos{( 2 \kappa \sin{t})}) \right] .
\dot{\bar{z}}= \bar{f}(\bar{z},e,v,w) \dot{\bar{\eta}}= M \bar{\eta}+M c_{1}e-\sum_{i=1}^{r}c_{i}\bar{g}_{i}(\bar{z},e,v,w) \dot{\xi}= A_{c}\xi+B\Psi\bar{\eta}+G(\bar{z},e,v,w)+B\bar{u}+B\Psi\tilde{\eta}
\displaystyle\int_{\mathbb{R}_+^{N+1}}y^{1-2\alpha}|\nabla v_{\varepsilon,z}|^2dxdy=\displaystyle\int_{\mathbb{R}_+^{N+1}}y^{1-2\alpha}|\nabla \omega_\varepsilon|^2dxdy+O(\varepsilon^{N-2\alpha}),
C^{(w)}(t,x)~=~E^{(w)}_1(t)-(c_1-1)\cdot \left[{\dot{\sigma}^{(w)}(t)t\over 2}-(w(t,x))-w(t,0-)) \right]\cdot\phi_x\left(x,{\sigma^{(w)}(t)t\over 2}\right),
c_1^{11}(A) = \bar{a}(1+\bar{b}) \int_Y r_B b_{11}\partial_1 w_A = -2\bar{a}(1+\bar{b}) \int_Y (\partial_1 w_A)(\partial_{22}^2 w_B), c_2^{11}(A) = \bar{a}(1+\bar{b}) \int_Y r_B b_{22}\partial_2 w_A = 2\bar{a}(1+\bar{b}) \int_Y (\partial_2 w_A)(\partial_{11}^2 w_B),
V^\nu({\bf x},t;\mbox{{\boldmath$\varepsilon$}})=-\frac{2\delta^\nu_k\varepsilon^k}{i\pi^2{\mbox{{\boldmath$\varepsilon$}}}^4}\ ;\V^\nu_5({\bf x},t;\mbox{{\boldmath$\varepsilon$}})=\frac{e\varepsilon_i}{4i\pi^2{\mbox{{\boldmath$\varepsilon$}}}^2}{\tilde{F_{\stackrel{{\scriptstyle{l}}}{{\scriptstyle{r}}}}}}^{\nu i}({\bf x},t).
sec(U, V) = \hat{s}ec(U, V) - \lambda^2 \mid \nabla (f + \log \lambda )\mid^2 + \frac{\lambda^4}{4}\mid grad_{\mathcal{H}}\frac{1}{\lambda^2}\mid^2 -\lambda^2g\left(\nabla f, \nabla_{\mathcal{H}}\frac{1}{\lambda^2}\right).
[\varphi_{I(T_{k}\oplus T_{k}^{\ast})}]\left(\frac{[\varphi_{\Omega_{w}^{-1}(T_{+})}]}{[\varphi_{I(T_{+})}]}+\frac{[\varphi_{\Omega_{w}^{-1}(T_{-})}]}{[\varphi_{I(T_{-})}]}\right)=[\varphi_{T'_{+}}]+[\varphi_{T'_{-}}].
d_{k}\big((A,r),(B,s)\big) =2^{k}\cdot\min\big(1-r+{\scriptstyle\frac12}+{\scriptstyle\frac12}-s, r+s \big) &=2^{k}\cdot\min\big(2-r -s, r+s \big) &=2^{k}\begin{cases}2-r-s r+s \end{cases}
P(\frac{1}{b}\sum_{i=1}^{b}\bar T_i < \eta) \le e^{\lambda^2 b \eta/2}E[e^{-\lambda^2 \bar{T}_1/2}]^b \le \Big( c_\delta e^{\lambda^2 \eta/2-\lambda (1-\delta)}\Big)^b \,.
D(a^2) =(2 \alpha_{1}+ \alpha_{3})a^2 + (\alpha_{2}+\gamma\alpha_{3})a^3 D(a^3) =(\alpha_{2}+ \gamma \alpha_{3})a^2 + [3 \alpha_{1}+ \gamma\alpha_{2}+ (\gamma^{2}+1) \alpha_{3}]a^3 D(aa^3) = [4 \alpha_1+\gamma \alpha_2+(\gamma^2+1)\alpha_3]a^2+[4 \gamma \alpha_1+(\gamma^2+1)\alpha_2+(\gamma^3+2\gamma)\alpha_3]a^3
\lambda(K_{t}^{\ast}) \geq \frac{t+1}{2t}+ 2\left(\frac{t+1}{2t}\right)\sum_{i=2}^{t}\frac{1}{2t}+ \sum_{2\leq i<j\leq t}\left(\frac{1}{2t}\right)^2 = \frac{t+1}{2t}+ 2\left(\frac{t+1}{2t}\right)\left(\frac{t-1}{2t}\right)+2\binom{t-1}{2}\left(\frac{1}{2t}\right)^2 = \frac{5}{4}-\frac{1}{4t}> \alpha.
P^{\mu}({\cal A}) = \frac{4}{3}\left[{\cal A}_{\nu}\partial_{\rho}{\cal A}_{\sigma}+ \partial_{\rho}{\cal A}_{\sigma}{\cal A}_{\nu}+ \frac{3}{2}{\cal A}_{\nu}{\cal A}_{\rho}{\cal A}_{\sigma}\right] \epsilon^{\mu \nu \rho \sigma}