harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\chi_{u}^{-}(j,\mathcal{A}) := \max \bigg( e_{f(u)}, ~ \max_{i\in \Theta (j, \mathcal{A}), ~ i \leq f(u)}\tilde{e}_{j, i}\bigg) -B_{u}, \chi_{u}^{+}(j,\mathcal{A}) := E_{u}- \min \bigg( \ell_{l(u)}, ~ \min_{i\in \Theta (j,\mathcal{A}), ~ i \geq l({u})}\tilde{\ell}_{j, i}\bigg).
\Psi_{1_{\chi|_{1=2}}}(T_1T_2) = \Phi_{1_{\chi|_{1=2}}}(T_1T_2) = \Phi_{1_\chi}(T_1, T_2) = \Psi_{1_\chi}(T_1, T_2) + \Psi_{0_\chi}(T_1, T_2)
\hat{\mathbf w}_{4}^{Q}= g_{1}{\mathbf w}_{4}^{Q}- \frac 1{r_{Z}^{V}}\partial{\frac{\delta}{\delta Z}}\sin \Theta^{V}- \frac 1{r_{A}^{V}}\partial{\frac{\delta}{\delta A}}\cos \Theta^{V}
L^{'}_{F}=\max\bigg\{\max_{1\leq j \leq n}\bigg(\bigg( KL_{\nu_{j}}+KL_{q}\kappa_{b}\gamma b\bigg) \bigg(1+\frac{M^{2}K^{2}b}{\delta^{j}}\bigg)\bigg), &\qquad\qquad\bigg(1+\frac{M^{2}K^{2}b}{\delta^{0}}\bigg)KL_{q}\kappa_{b}\gamma b,\ \max_{1\leq j \leq n}L_{\nu_{j}}\bigg\}.
\frac{\partial r_{t}}{\partial{t}}=\sum_{n=1}^N\left(\frac{1}{2}\frac{\partial^2}{\partial\zeta_n^2}+\mu_n\frac{\partial}{\partial\zeta_n}\right)r_{t}. \frac{\partial r_{t}}{\partial\zeta_i}=0,\qquad \zeta_i=\zeta_{i-1},\ 2\leq i\leq N
Q = \bar Q = \bar H = S = \bar S = q = \bar q = \bar r = 0, R(s) = e^{-\mu(s-t)}, \bar R(s) = \beta e^{-\mu(s-t)}, r(s) = -\alpha e^{-\mu(s-t)}, H = e^{-\mu(T-t)}.
\chi^1_{\dot q\dot p}=\chi^1\delta_{\dot q\dot p}+\chi^{1ij}\tilde\gamma^{ij}_{\dot q\dot p}+\chi^{1ijkl}_{\dot q\dot p},\ \chi^2_{qp}=\chi^2\delta_{qp}+\chi^{2ij}\gamma^{ij}_{qp}+\chi^{2ijkl}\gamma^{ijkl}_{qp}.
\int |u-G_{h/2}\ast u|^2dx = \int \left| \int G_{h/2}(z) (u(x)-u(x-z))\,dz\right|^2 dx \lesssim \int \int G_{h/2}(z) \big|u(x)-u(x-z)\big|^2dz\,dx \lesssim h E_{h}(u^h,\zeta) \lesssim h|\log h| \to 0.
\lambda_j(\mu)=\min_{\substack{V\subset H^2(\Omega){\rm dim}V=j}}\max_{\substack{v\in V \frac{\partial v}{\partial\nu}\ne 0}}\frac{\mathcal{Q}_{\mu,D}(v,v)}{\int_{\partial\Omega}\left(\frac{\partial v}{\partial\nu}\right)^2d\sigma},
[e_1, e_1]=\alpha_1e_3+\alpha_2e_4+\alpha_3e_5, [e_1, e_2]=\alpha_4e_3+ \alpha_5e_4+ \alpha_6e_5, [e_2, e_1]=\beta_1e_3+\beta_2 e_4+ \beta_3e_5, [e_2, e_2]=\beta_4e_3+\beta_5e_4+\beta_6e_5, [e_1, e_3]=\gamma_1e_4+\gamma_2e_4, [e_2, e_3]=\gamma_3e_4+\gamma_4e_5.
c_{i}=-\frac{1}{\left\vert B_{4\Lambda}\right\vert}\int_{B_{4\Lambda}}X_{i}u\left( x\right) dxi=1,2,...,q c_{0}=-\frac{1}{\left\vert B_{4}\right\vert}\left( \int_{B_{4}}u\left(x\right) dx+\sum_{i=1}^{q}c_{i}\int_{B_{4}}x_{i}dx\right) .
\Phi(z)=\sum_{\pm}\sum^{\pm}w_0\left(\frac{4\pi\sqrt{2nb}}{zC}\right)W_{\pm}\left(\frac{m_1z^2}{32\pi^2b},\frac{m_1Hz^2}{32\pi^2nb}\right)V^{\pm}\left(\frac{m_2z^2}{32\pi^2b},\frac{m_2H'z^2}{32\pi^2nb}\right)e^{\pm i\sqrt{\frac{m_1}{2b}}z}e^{\pm i\sqrt{\frac{m_2}{2b}}z}.
p_{A}(a)q_{R|A}(r|a) =\sum_bp_{A,B}(a,b)p_{R|B}(r|b) \!=\!\epsilon\, p_{A}(a)p_{R|B}(r|\mathtt e)\!+\!(1\!-\!\epsilon)p_{A}(a)p_{R|B}(r|a) &\geq\! \epsilon\, p_{A}(a)p_{R|B}(r|\mathtt e).
t\mapsto\exp(tD)=\begin{pmatrix}e^t 0 0 0 e^{-\frac{t}{2}}\cos t\delta e^{-\frac{t}{2}}\sin t\delta 0 -e^{-\frac{t}{2}}\sin t\delta e^{-\frac{t}{2}}\cos t\delta\end{pmatrix}\,.
- \frac{1}{8}\left[ \left( \nabla^{\mu}\nabla_{\mu}+ \frac{1}{1 0}\nabla^{\gamma}\nabla_{\gamma}- 7 2 \right) h_{2}^{I}+ \nabla^{\gamma}\nabla_{\gamma}h^{\prime I}- 1 9 2 \sqrt{2}\nabla^{\gamma}\nabla_{\gamma}b^{I}\right] Y^{I}= 0 ,
P_{n}(\lambda)= (\lambda^2-4)\left[(\lambda^2-4)P_{n-2}-2(\lambda+2)(\lambda^2-4)^{n-2}\right]-2(\lambda+2)(\lambda^2-4)^{n-1}P_{n}(\lambda)= (\lambda^2-4)^2P_{n-2}-2\times 2(\lambda+2)(\lambda^2-4)^{n-1}
\langle v_1,v_1\rangle=3\alpha^2,\quad\langle v_1,v_2\rangle=\frac{1}{3}\langle\alpha^\perp,\alpha^\perp\rangle=\alpha^2,\quad \langle v_2,v_2\rangle=\frac{1}{9}(\langle\alpha^\perp,\alpha^\perp\rangle+\langle\beta^\perp,\beta^\perp\rangle)=\frac{\alpha^2+\beta^2}{3}.
g_i \cdot x P_{i+1}x^{-1}g_i^{-1}= g_{i+1}xx_{i+1}P_{i+1}x_{i+1}^{-1}x^{-1}g_{i+1}^{-1}= g_{i+1}xP_{i+1}x^{-1}g_{i+1}^{-1},
P_\pm u\left[x\right]= (2\pi)^{-d}\int_{\mathbb{T}^d}\sum_{y\in\mathbb{Z}^d}L^k\left(e^{i(x-y)\cdot\xi}\right) p_\pm(y,\xi)u\left[y\right]d\xi = (2\pi)^{-d}\int_{\mathbb{T}^d}\sum_{y\in\mathbb{Z}^d}e^{i(x-y)\cdot\xi}(L^*)^k \left(p_\pm(y,\xi)\right)u\left[y\right]d\xi
\begin{array}{l}\underset{\epsilon \rightarrow 0}{\lim}\left\vert \left( \breve{F}_{\epsilon}(M)-\breve{F}_{\epsilon}(m)\right) -\left( MF_{\epsilon}(M)-mF_{\epsilon}(m)-\int_{m}^{M}F_{\epsilon}(y)dy\right) \right\vert =0.\end{array}
- [ 1 - ({\frac{r_{0}}{r}})^{D - 3}] d t^{2}+ [ 1 - ({\frac{r_{0}}{r}})^{D - 3}]^{- 1}d r^{2}+ r^{2}d \Omega_{D - 2}+ d z^{2}+ ( d x_{1}^{2}+ d x_{2}^{2}+ \ldots + d x_{p}^{2})
\ \|s(t_1)-s(t_2)\|_{L^p(\Omega)}= \|s_i(t_1)-s_i(t_2)-A_i(A_i+A_e)^{-1}(s_i(t_1)-s_i(t_2)+s_e(t_1)-s_e(t_2))\|_{L^p(\Omega)}\le \|s_i(t_1)-s_i(t_2)\|_{L^p(\Omega)}+C(\|s_i(t_1)-s_i(t_2)\|_{L^p(\Omega)}+\|s_e(t_1)-s_e(t_2)\|_{L^p(\Omega)}) \le C|t_1-t_2|^\nu.
\left\{\sum_{i=1}^k Q_i^{\Pi_1}(t)-kn(N)\right\}_{t\geq 0}\leq_{st}\left\{\sum_{i=1}^k Q_i^{\Pi_2}(t)\right\}_{t\geq 0}\leq_{st}\left\{\sum_{i=1}^k Q_i^{\Pi_1}(t)\right\}_{t\geq 0},
||X||_{s,r,q,\mathbf{a}}:=\sup_{(y,z,\bar{z})\in D(r)}||(\sum_{k,i,\alpha,\beta}e^{\mathbf{a}|\pi(k,\alpha,\beta;v)|}|X_{k,i,\alpha,\beta}^{(\mathbf{v})}|e^{|k|s}|y|^i|z^{\alpha}||\bar{z}^{\beta}|)_{\mathbf{v}\in V}||_{s,r,q},
D'^2\cdot\omega_0 =(D+\epsilon_0\Delta)^2\cdot\omega_0 &= D\cdot(D+\epsilon_0\Delta)\cdot\omega_0+\epsilon_0(D+\epsilon_0\Delta)\cdot\Delta\cdot\omega_0 =\underbrace{D^2\cdot\omega_0}_{>0}+\epsilon_0\underbrace{(D\cdot\omega_0\cdot\Delta)}_{\>0}+\epsilon_0\underbrace{(D'\cdot\omega_0\cdot\Delta)}_{\>0}>0.
\mathbb{I}\{\hat{z}_i^{(s+1)}\neq z_i\}\le \sum_{h=1}^{k}\mathbb{I}\left\{W_{i}(T_h) \ge \frac{\epsilon(a-b)n_h}{n}\right\}+ \mathbb{I}\left\{W_{i}(T_{z_i}) \le -\frac{\epsilon(a-b)n_{z_i}^*}{n}\right\}.
B_m(x) =x\big(L(x)-1\big)A_{m-1}(x) &+\sum_{d\geq1}\left(\frac{x^{d+2}}{(1-x)^d}L(x)+\left(x^{d+1}+\sum_{p=1}^{d-1}\frac{x^{d+2}}{(1-x)^p}\right)\big(L(x)-1/(1-x)\big)\right)A_{m-1}(x) &=x\big(L(x)-1\big)A_{m-1}(x)+\frac{x^3}{(1-2x)^2}A_{m-1}(x),
\lambda_g \theta_f x =\lambda_g\left(\sum_{h\in G}f_h(x)\delta_h\right)=\sum_{h\in G}f_h(x)\delta_{gh}=\sum_{u \in G}f_{g^{-1}u}(x)\delta_u &=\sum_{u \in G}f(\pi_{u^{-1}g}x)\delta_u=\sum_{u \in G}(f\pi_{u^{-1}})(\pi_gx)\delta_u=\theta_f(\pi_gx)\in \theta_f(\mathcal{X}), \forall x \in \mathcal{X}.
\mathbf{w}_{\mathrm{opt}}=\mathbf{R}_{\mathbf{rr}}^{-1}\mathbf{R}_{\mathbf{r}x}=\frac{\mathcal{E}_\mathrm{s}}{1+\mathcal{E}_\mathrm{s}\|\mathbf{h}\|^2}\mathbf{h}.
P_{n+1}(x, x_{n+1}) \leq \tfrac{1}{6}\left((n+3)x_{n+1}^{3}- 3x_{n+1}+ (n-1)\sqrt{\tfrac{n+2}{n}}|x|^{3}\right) &=\tfrac{1}{6}\left((n+3)x_{n+1}^{3}- 3x_{n+1}+ (n-1)\sqrt{\tfrac{n+2}{n}}(1 - x_{n+1}^{2})^{\tfrac{3}{2}}\right) \leq \tfrac{n}{6},
3\eta^2 + 1 +2\gamma (\sqrt{3}\eta - 1) \ge 3\eta^2 + 1 +2\cdot \frac{1+\sqrt{3}}{\sqrt{3}}\cdot (\sqrt{3}\eta - 1) = 3(\eta - \frac 13) \cdot (\eta + \frac{2+\sqrt{3}}{\sqrt{3}}) \ge C(\eta_0).
v_{n+1}=\sum_{k=1}^{n+1}\tau_{k-1}\left( \lambda^{n+1-k}-\nu^{n+1-k}\right) =\sum_{k=1}^{n}\tau_{k-1}\left( \lambda^{n-k+1}-\nu^{n-k+1}\right) , v_{n+2}=\sum_{k=1}^{n+1}\tau_{k-1}\left( \lambda^{n-k+2}-\nu^{n-k+2}\right) =\left( \lambda-\nu\right) \tau_{n}+\sum_{k=1}^{n}\tau_{k-1}\left( \lambda^{n-k+2}-\nu^{n-k+2}\right) .
\frac{1}{r!}\Big(\frac{d}{dx}\Big)^{r}\bigg[\frac{x^{r}}{(1-x)^{2}}F_{p,\lambda}\Big(\frac{x}{1-x}\Big)\bigg] = \frac{1}{r!}\Big(\frac{d}{dx}\Big)^{r}\bigg[\sum_{n=1}^{\infty}\Big((1)_{p,\lambda}+(2)_{p,\lambda}+\cdots+(n)_{p,\lambda}\Big)\bigg]x^{n+r}&=\sum_{n=1}^{\infty}\Big((1)_{p,\lambda}+(2)_{p,\lambda}+\cdots+(n)_{p,\lambda}\Big)\binom{n+r}{r}x^{n}.
K_{\mathrm{GUE}}(\pi;u,v)= \int_{-i\infty}^{i\infty}\frac{dz}{2i\pi}\int_{\mathcal{C}_{\{\pi_{1},\ldots,\pi_m\}}}\frac{dw}{2i\pi}\frac{1}{z-w}e^{\frac{1}{2}z^2 -\frac{1}{2}w^2}e^{vz-uw}\prod_{k = 1}^{m}\frac{z-\pi_{k}}{w - \pi_{k}}
\frac{m_{1}+m_{2}+m_{3}}{(\sqrt{m_{1}}+\sqrt{m_{2}}+\sqrt{m_{3}})^{2}}=1-\frac{2(x_{1}x_{2}+x_{2}x_{3}+x_{3}x_{1})}{(x_{1}+x_{2}+x_{3})^{2}}=1-\frac{3n^{2}}{(3n)^{2}}=\frac{2}{3}
S ({\cal M}) = \left( \begin{array}{c c}{1}&{0}\\{b}&{1}\\ \end{array}\right) \left( \begin{array}{c c}{\det G^{- 1 / 2}}&{0}\\{0}&{\det G^{1 / 2}G^{- 1}}\\ \end{array}\right) \left( \begin{array}{c c}{1}&{b^{T}}\\{0}&{1}\\ \end{array}\right) .
\Sigma= \bigl\{(y_1,y_2;a,b,c)\mid y_1,y_2\in S,1\leq a,b,c\leq n,[a,c,b] \sqsubseteq \Gamma(y_1), [a, b] \sqsubseteq \Gamma(y_2), (c)\subseteq \Gamma(y_2),\mbox{the orbit that contains $[a,c,b]$ in $\Gamma(y_1)$ is not a cycle}\mbox{and the orbit that contains $[a,b]$ in $\Gamma(y_2)$ is not a cycle}\bigr\}.
\hat{H}_{JC}=\hbar\omega\hat{a}^{\dagger}\hat{a}+\hbar\omega\hat{b}^{\dagger}\hat{b}+\frac{\hbar\Omega}{2}(\hat{a}\hat{b}^{\dagger}+\hat{a}^{\dagger}\hat{b}+\hat{a}\hat{b}+\hat{a}^{\dagger}\hat{b}^{\dagger}),
r\frac{d}{dr}\left[{}_2\!F_{1}\left(\lambda,\lambda+1;2\lambda+1;\tilde{A}\right)\right]=\frac{\lambda(\lambda+1)}{2\lambda+1}{}_2\!F_{1}\left(\lambda+1,\lambda+2;2\lambda+2;\tilde{A}\right)r\frac{d}{dr}\tilde{A},
\dim H_{m}= \ell_{m}+ \ell_{m+1}- \dim C_{m+1}= k_{m-1}- \dim C^{m-1}+ \dim C_{m}& + k_{m}- \dim C^{m}+ \dim C_{m+1}- \dim C_{m+1}= k_{m-1}+ k_{m}- \dim C^{m-1}( \; \dim C^j= \dim C_j \;) = \dim H^{m}\;.
\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,
p^2\sum_{k=\frac{p-1}2}^{p-1}\frac{(1)_k^2}{(\frac32)_k^2}\cdot(-1)^k\equiv \frac{p^2(1)_{p-1}^2}{(\frac32)_{p-1}^2}\cdot\big(\Psi(0)+p\cdot\Psi'(0)\big) \equiv (1-2pH_{\frac{p-1}2})\cdot\Psi(0)(1+2pH_{\frac{p-1}2})\equiv\Psi(0)\pmod{p^2}.
d\tau=\sqrt{(1-\frac{2m}{r})dt^{2}-\frac{1}{c^{2}}(1-\frac{2m}{r})^{-1}dr^{2}-\frac{r^{2}}{c^{2}}d\phi^{2}-\frac{r^{2}}{c^{2}}sin^{2}(\phi)d\theta^{2}},
\sum_{j=1}^{2}(-1)^{j-1}S_{j}\le...\le\sum_{j=1}^{n-4}(-1)^{j-1}S_{j}\le\sum_{j=1}^{n-2}(-1)^{j-1}S_{j}\le\sum_{j=1}^{n}(-1)^{j-1}S_{j}=\mathbb{P}(\bigcup_{i=1}^{n}A_{i})
\sum_{k=0}^n W(n,k)\:\hat{c}_k^{q}(-r)=\int_{0}^{1}\sum_{k=0}^n\:W(n,k)(-x-r|q)_k\:dx=\int_{0}^{1}(-1)^n\,x^n\:dx=\frac{(-1)^n}{n+1}
\left( \left( \left( b_{n + 1}[ d_{n}> < d_{n}] \right) + \left( b_{n + 1}[ d^{\prime \prime}> < d^{\prime \prime}] \right) \right) \left/ \bigoplus_{d^{\prime}\in R_{d}^{\prime}}b_{n + 1}[ d_{n}^{\prime}> < d_{n}^{\prime}] \right. \right)
\tfrac{\tfrac{2\sqrt{\beta/\sigma}}{2\sqrt{\beta/\sigma}+1}}{2-\tfrac{2\sqrt{\beta/\sigma}}{2\sqrt{\beta/\sigma}+1}}=\tfrac{2\sqrt{\beta/\sigma}}{2+2\sqrt{\beta/\sigma}}=\tfrac{\sqrt{\beta/\sigma}}{1+\sqrt{\beta/\sigma}}.
\left(\Pi_{\lambda_1,\lambda_2}g\right)^{1-\lambda_1,\lambda_2}_{i-\frac{1}{2},j+\frac{1}{2}}= g^{1-\lambda_1,\lambda_2}_{i-\frac{1}{2},j+\frac{1}{2}}, \mbox{if}\left(i,j\right) \in (\Bbbk_2^+,\Bbbk_1^-)
\frac{(x_{1}+su)^{2}}{a^{2}}+\frac{(y_{1}+sv)^{2}}{b^{2}}=1\Rightarrow2s(\frac{x_{1}u}{a^{2}}+\frac{y_{1}v}{b^{2}})+s^{2}(\frac{u^{2}}{a^{2}}+\frac{v^{2}}{b^{2}})=0.
\int_U \varphi(x) \bar{\nu}(x) dx \geq \bigg(\frac{R + T}{R}\bigg)^n \cdot \int_{Q_1}\varphi_{+}(x) \bar{\nu}(x) \dx - C(n) \cdot (T/R) \cdot (\sup_U \varphi) \cdot ||\bar{\nu}||_{L^\infty(Q_1)}\cdot |Q_1| & - \omega_\varphi(\delta) \cdot ||\bar{\nu}||_{L^\infty(Q_1)}\cdot |Q_1|.
\left [{d^2 \over dx^2}+{3 \over x}{d \over dx}-{{2 s^2 (x_+^2 -x_-^2)}\over{x^4 (\cosh 2 \delta + \cosh 2 \gamma )}}-{{m^2 l^2}\over x^2}\right ] \Phi_s^{\rm far}=0.
S = \int d^4x \frac{1}{2}\epsilon^{\mu\nu\lambda\rho}\left( B^a_{\mu\nu}(x)R^a_{\lambda\rho}(x) +\phi^{ab}(x)(B^a_{\mu\nu}B^b_{\lambda\rho}-\frac{1}{3}\delta^{ab}B^c_{\mu\nu}B^c_{\lambda\rho})\right)\ .
\frac{1}{\hat{\Pi /}- M + i 0}= \int_{0}^{\infty}d T \int d \chi \exp \left[ - \frac{i}{2 \kappa_{0}}( M^{2}T + M \chi ) \right] \exp \left[ \frac{i}{2 \kappa_{0}}( \hat{\Pi /}^{2}T + \hat{\Pi /}\chi ) \right]
\mathbb{P\left[A_{UE}\right]}= \int_0^\infty \int_0^{\min\left(X_{21}, M^{\frac{1}{\alpha\mu}}\left(\frac{N}{P}\right)^{\frac{2-\mu}{\alpha\mu}}X_{21}^{\frac{2}{\mu}}\right)}{\left(2\pi\lambda_b\right)^2 x_{11}x_{21}\mathrm{e}^{-\pi\lambda_b x_{21}^2}\mathrm{d}x_{11}\mathrm{d}x_{21}}.
F_{r{\tilde{\theta}}}^{1}= - \, \frac{1}{2 \lambda}\,{\dot{a}}~ ~ , ~ ~ F_{r{\tilde{\varphi}}}^{2}= \frac{1}{2 \lambda}\,{\sin{\widetilde{\theta}}\,{\dot{a}}}~ ~ , ~ ~ F_{{\tilde{\theta}}{\tilde{\varphi}}}^{3}= \frac{1}{2 \lambda}\,{\sin{\widetilde{\theta}}}\left( 1 - a^{2}\right) ~ ~ .
{J_{\varepsilon}(W_{\varepsilon,\xi})=\frac{1}{2}\int_{\mathbb{R}^{n}}c(\xi)|\nabla V^{\xi}(z)|^{2}+a(\xi)\left( V^{\xi}\right)^{2}(z)dz-\frac{1}{p}\int_{\mathbb{R}^{n}}b(\xi)\left( V^{\xi}\right)^{p}(z)dz+O(\varepsilon)}
S_{HG}=\int d^{4}x\sqrt{-g^{L}}M^{3}g_{L}^{\mu\nu}\partial_{5}G_{\mu\nu}\Big|_{y=L}-\int d^{4}x\sqrt{-g^{0}}M^{3}g_{0}^{\mu\nu}\partial_{5}G_{\mu\nu}\Big|_{y=0}\ .
{{P}_{SISO}}\approx A \prod\limits_{\begin{smallmatrix}j=1 j\ne k\end{smallmatrix}}^{p+q}{\Gamma \left({{b}_{j}}-{{b}_{k}}\right)}{{\left( \frac{2}{\sqrt{{\bar{\gamma}}}}\right)}^{p{{\gamma}_{2}}{{b}_{k}}}}\frac{\Gamma \left( \left( 1+p{{\gamma}_{2}}{{b}_{k}}\right)/2 \right)}{2\sqrt{\pi}p{{\gamma}_{2}}{{b}_{k}}}
o_k(1) \underset{\eqref{pscomau}}{=}[u_k-v]_{s,p}^p-\mu\int_\Omega \frac{|u_k-v|^{p^*_\alpha}}{|x|^{\alpha}}\, dx \geq [u_k-v]_{s,p}^p-\mu S_\alpha^{-\frac{p^*_\alpha}{p}}[u_k-v]_{s,p}^{p^*_\alpha}= [u_k-v]_{s,p}^p\left(1-\mu S_\alpha^{-\frac{p^*_\alpha}{p}}[u_k-v]_{s,p}^{p^*_\alpha-p}\right)\geq \omega_1[u_k-v]_{s,p}^p,
(M_r{V}^x)_S= &=\sum_{a,b}\left [\binom{a}{2}+\binom{b}{2}\right ]\binom{|S|-S(x)}{a-S(x)}\binom{r-|S|-(1-S(x))}{b}(r-2|T|) &=\sum_{a,b}\left [\binom{a}{2}+\binom{b}{2}\right ]\binom{|S|-S(x)}{a-S(x)}\binom{r-|S|-(1-S(x))}{b}(2|S|-r+2b-2a) &=P_1(x)+P_2(x)+P_3(x)+P_4(x)+P_5(x)+P_6(x),
4K_1^2+4M+8C_4M^{-\frac{p-2}{2}}K_2^p+16C_4M^{-\frac{p-2}{2}}K_2^p\frac{\alpha^2}{\varrho^2}&\quad\leq4K_1^2+4M+24C_4M^{-\frac{p-2}{2}}K_2^p\frac{\alpha^2}{\varrho^2}&\quad=4K_1^2+4K_2^2(1+6C_4)\alpha^{\frac{4}{p}}\varrho^{-\frac{4}{p}}\leq4K^2(1+6C_4)\alpha^{\frac{4}{p}}\varrho^{-\frac{4}{p}}.
\chi_{h,0}(\o):=\phi(M_{UV}^{-2}h^2|1-e^{i\frac{\o}{h}}|^2), &\chi_{h,l}(\o):=\phi(M_{UV}^{-2}M^{-2l}h^2|1-e^{i\frac{\o}{h}}|^2)-\phi(M_{UV}^{-2}M^{-2(l-1)}h^2|1-e^{i\frac{\o}{h}}|^2), &(l\in\{1,2,\cdots,N_h\}).
\begin{array}{rr}\psi_k(t) = -\rho \sum_{m=1}^N \left[\frac{1}{2}\;\overline{n_m^2(t)}\;\overline{\psi_m(t)}+ \frac{1}{2\pi i}\int_{L_m}\overline{n_m^2(t)}\;\overline{\psi_m(t)}E_1(t-z) \textrm{d}t \right]+C, t \in L_k \;(k=1,2,\ldots, N).\end{array}
P^{(-x_{j})}(z) = E_{-x_{j}}(z) \Phi_{\mathrm{HG}}(\sqrt{r}f_{-x_{j}}(z);\beta_{j})(s_{j}s_{j+1})^{-\frac{\sigma_{3}}{4}}e^{-\sqrt{r}g(z)\sigma_{3}}e^{\frac{\pi i \alpha}{2}\theta(z)\sigma_{3}},
|| v^*_\mu - v^0_{\tilde \mu}+ (\mu - \tilde \mu) v^1_{\tilde \mu}||_{\rho''}\le C|\mu - \tilde\mu|^2 &|| \lambda^*_\mu - \lambda^0_{\tilde \mu}+ (\mu - \tilde \mu) \lambda^1_{\tilde \mu}||_{\rho''}\le C|\mu - \tilde\mu|^2.
dX^{(n)}_i(t)=\sqrt{2((X_i^{(n)})^2(t)+1)}dW^{(n)}_i(t)+\left[\left(2-2n-2\Re(s)\right)X_i^{(n)}(t)+2\Im(s)+\sum_{j\ne i}^{}\frac{2((X_i^{(n)}(t))^2+1)}{X_i^{(n)}(t)-X_j^{(n)}(t)}\right]dt,
\begin{array}{l}{{\vert 0 , z \rangle_{- k}= \displaystyle \exp_{q^{2}}( - \frac{1}{2}| q^{- 2}z | ) \sum_{m = 0}^{\infty}\frac{( - q^{- 2}z )^{m}}{[ m ]_{q^{2}}!}({\cal D}_{- k})^{m}\vert 0 \rangle_{- k}~ ,}}\\{{{\cal X}^{- k}\vert 0 , z \rangle_{- k}= z \vert 0 , z \rangle_{- k}~ ,}}\\ \end{array}
\begin{cases}\bar h_1L_1\leq \bar h_2+\bar h_3 L_2,\,\,\, \bar p_1L_2\leq \bar p_2+\bar p_3 L_1 \underbar h_1l_1\geq \underbar h_2+\underbar h_3 l_2,\,\,\, \underbar p_1l_2\geq \underbar p_2+\underbar p_3 l_1. \end{cases}
c_{x,x+1}(\eta)= [ \eta(x-1) + \eta(x+1) + \eta(x) + \eta(x+2) ] \big( \eta(x) [1 - \eta(x+1)] + \eta(x+1) [ 1 - \eta(x) ] \big) = \xi_{x,x+1}(\eta) [ \eta(x-1) + \eta(x+2) + 1].
\sum_{j=0}^{\infty}V_{1,j}^{\mu}= \sum_{j=0}^{\infty}\int_{\mathcal{V}_{\alpha,\theta}}V_{1,j}\mu(\mathrm{d}V) = \int_{\mathcal{V}_{\alpha,\theta}}\sum_{j=0}^{\infty}V_{1,j}\mu(\mathrm{d}V) = \int_{\mathcal{V}_{\alpha,\theta}}1 \mu(\mathrm{d}V) = 1.
\begin{aligned}\frac{dE}{dt}=\begin{cases}F_+(\tau) - B E(\tau), E > 0, F_-(\tau), E<0, F_->0, \zeta F_-(\tau)/\left(\zeta - E(\tau)\right), E<0, F_- \le 0,\end{cases}\frac{d \tau}{dt}=1.\end{aligned}
(E_{1,2},0,0)t =e_{1,1,1}\wedge e_{1,2,1}\wedge e_{1,1,2}+e_{1,1,1}\wedge e_{1,1,2}\wedge e_{1,2,1}=0, (0,E_{1,2},0)t =e_{1,1,1}\wedge e_{2,1,1}\wedge e_{1,1,2}+e_{1,1,1}\wedge e_{1,1,2}\wedge e_{2,1,1}=0, (0,0,E_{1,2})t =e_{1,1,1}\wedge e_{2,1,1}\wedge e_{1,2,1}+e_{1,1,1}\wedge e_{1,2,1}\wedge e_{2,1,1}=0.
\big(\varepsilon_{n, b}(x,t),\psi(x)\big):=\int_{\mathbb{R}^{n}}\varepsilon_{n, b}(x,t)\psi(x)dx=\psi(0)\int_{\mathbb{R}^{n}}\varepsilon_{n, b}(x,t)dx &+\int_{\mathbb{R}^{n}}\varepsilon_{n, b}(x,t)\big(\psi(x)-\psi(0)\big)dx \to \big(\delta(x),\psi(x)\big):=\psi(0),~~t \to 0+.
\rho :=\left[\Gamma^2 -\frac{1}{\kappa^2}\Big(\sum_{i\in I\smallsetminus J}\frac{1+\alpha_i}{1-\alpha_i}-(1-|J|)\min_{i\in I\smallsetminus J}\frac{1+\alpha_i}{1-\alpha_i}\Big)^{-1}\Big(\frac{1 +\varepsilon}{1 -\varepsilon}\Big)^{|J|}\right]_+^{\frac{1}{2(m-1+|J|)}}<1.
\Vert (\mathcal{A}_{D,\varepsilon}-\zeta I)^{-1}-(\mathcal{A}_D^0-\zeta I)^{-1}\Vert_{L_2\rightarrow L_2}\leqslant 2C_2\max \lbrace 1; 8c_*^{-2}\rbrace \varepsilon &\leqslant C_{12}' \left(\vert \zeta \vert^{-1/2}\varepsilon +\varepsilon^2\right), \zeta \in \gamma ,\; \vert \zeta \vert \leqslant \check{c},
\int_\Omega|\mathbf{b}|^4dx+4\nu\iint_{Q_t}|\mathbf{b}|^2|\mathbf{b}_x|^2dxds+8\nu\iint_{Q_t}|\mathbf{b}\cdot \mathbf{b}_x|^2dxds &=\int_\Omega|\mathbf{b}_0|^4dx+4\iint_{Q_t}\mathbf{w}_x \cdot(|\mathbf{b}|^2\mathbf{b})dxds-4\iint_{Q_t}(u \mathbf{b})_x\cdot(|\mathbf{b}|^2\mathbf{b})dxds.
\frac{m_{B^{\prime}}}{m_{B}}= \frac{\rho_{B}V^{\prime}}{\rho_{B}\cdot V}= \frac{S h^{\prime}}{S h}= \frac{h^{\prime}}{h}= \frac{0 . 2 7 5 m}{0 . 3 7 5 m}= \frac{1 1}{2 5}
0=2\rho^q(\rho_j\Psi_{ikq}-\rho_i\Psi_{jkq}-\rho_k\Psi_{qij})+\rho^q\rho_q\Psi_{kij}+2\rho_k(\rho_i\Psi_j-\rho_j\Psi_i)\;; &0=2\rho^q\rho^r\Psi_{qir}-4\rho_i(\rho^q\Psi_q)+\rho^q\rho_q\Psi_i\;.
f_{Z|H_1}(z | H_1) =\frac{e^{\frac{\sigma_{\mathrm{w}}^2+\sigma_{\mathrm{a}}^2}{\zeta}}}{\pi^{n}}\int_{\sigma_{\mathrm{w}}^2+\sigma_{\mathrm{a}}^2}^{\infty}\left ( \frac{1}{v}\right)^{n}e^{-\frac{z}{v}}e^{-\frac{v}{\zeta}}dv.
T^{\underline\alpha}={1\over{288}}F_{\underline b_1...\underline b_4}E^{\underline a}\left(\Gamma_{\underline a}^{~\underline b_1...\underline b_4}-8\delta_{\underline a}^{[\underline b_1}\Gamma^{\underline b_2...\underline b_4]}\right)^{\underline\alpha}_{~\underline\beta}E^{\underline\beta},
+ \underbrace{\sum_{l=1}^{M_{T}}\sum_{m=1}^{M_{R}}\sum_{i=1}^{Q}\Phi_{R,m}(\varphi_{i}(k)) \Phi^{*}_{T,l}(\varphi_{i}(k)) \beta_{i}(k) \, \frac{e^{j(\theta_{R,m}(k)+\theta_{T,l}(k))}}{\sqrt{M_{T}M_{R}}}}_{\mbox{{\fontsize{0.35cm}{1em}\selectfont Scattered component}}\,\,\, V(k)}.
{[}\, [ \, C_{k m}\, , \, \sigma_{k l}\, ] \, , \, \sigma_{m n}\, ] ~ ~ = ~ ~ - \, [ \, [ \, \sigma_{k n}\, , \, \sigma_{m n}\, ] \, , \, \sigma_{k l}\, ] ~ ~ = ~ ~ [ \, [ \, C_{l n}\, , \, \sigma_{k l}\, ] \, , \, \sigma_{m n}\, ] ~ . ~ ~ ~
\sum_{j=1}^{m}\| A_{t_j}^\varphi (F,G)-A_{t_{j-1}}^\varphi (F,G) \|_{\textup{L}^2(\mathbb{R}^2)}^2\leq C_\varphi^2 \,\|F\|_{\textup{L}^4(\mathbb{R}^2)}^2 \|G\|_{\textup{L}^4(\mathbb{R}^2)}^2,
{d\vec{n}_k \over dt}= 4\nu_0^2 i\sum_{n=k\pm1}^{}e^{-\Delta_{kn}}\left[e^{i(\widetilde{\delta}_n -\widetilde{\delta}_k)}\vec{n}_n -R_{kn}\vec{n}_k +e^{i(\widetilde{\delta}_n - \widetilde{\delta}_k)}\vec{n}_n+R_{kn}^*\vec{n}_k \right] + C_k\vec{n}_k .
\rho_{\varepsilon}(|u-v|) \triangleq\rho\left(\mathcal{A}_{\varepsilon}(u)\bigcap\mathcal{A}_{\varepsilon}(v)\right)= \begin{cases}\log(1/|u-v|) \, , 1+\log(1/\varepsilon)-|u-v|/\varepsilon \, \text{if$|u-v|<\varepsilon$},\end{cases}
\widetilde{X_n}^i (\, \{e_1,\ldots,e_k\}) = u_n^*X_n^iu_n (\, \{e_1,\ldots,e_k\}) &\subset u_n^*X_n^i (V_{k-1}^{X_n}) &\subset u_n^* (V_k^{X_n}) &\subset \, \{e_1,\ldots,e_{\alpha(k,d)}\}.
\mathcal{B}(x,y)={\begin{cases}1{}x < y, \ \x,y \in \mathbb{N}\cup\{0\}1/2{}x=y\ne 0 \x,y \in \mathbb{N}\cup\{0\}0{}x>y \{}x=y=0, x,y\in \mathbb{N}\cup\{0\}\end{cases}}
\varphi_{1}= \frac{(u_{\xi_{1}}^{p}-u_{\xi_{2}}^{p})^{+}}{u_{\xi_{1}}^{p-1}}\, , \varphi_{2}= \frac{(u_{\xi_{1}}^{p}-u_{\xi_{2}}^{p})^{+}}{u_{\xi_{2}}^{p-1}}\, ,
\frac{1}{k}(u^l_h,\bar{u}_h)+ (\nabla u^{l}_h, \nabla\bar{u}_h)+ (u^{l-1}_h{\boldsymbol \sigma}^{l}_h,\nabla\bar{u}_h)+(u^{l}_h{\boldsymbol \sigma}^{l-1}_h,\nabla\bar{u}_h)=0, \ \forall\bar{u}_h\in U_h,
\begin{array}{lll}{0\leq t\leq \frac{1}{2}}: x^*(t)=\displaystyle{t+\frac{1+3e}{2(1-e)},}u^*(t)=1, begin{align*}.10in]{\frac{1}{2}\leq t\leq 1}: x^*(t)=\displaystyle{\frac{e^t + e^{2-t}}{\sqrt{e}(1-e)},}\displaystyle{u^*(t)=\frac{e^t - e^{2-t}}{\sqrt{e}(1-e)}.}\end{array}
S = \int d t \int_{0}^{L}d x \left\{\partial_{0}A^{1}\pi_{1}+ \frac{1}{2}i \psi^{\dagger}\partial_{0}\psi - \frac{1}{2}i \partial_{0}\psi^{\dagger}\psi -{\cal H}- A^{0}\phi + \partial_{1}( A^{0}\pi_{1}) \right\}\ ,
\log \, Z\left(\beta \right)=-\frac{\beta}{2r}\sum_{\ell =0}^{\infty}D_{\ell}M_{\ell}\left(\mu ,\chi \right)-\sum_{\ell =0}^{\infty}D_{\ell}\log \left(1-e^{-\beta M_{\ell}\left(\mu ,\chi \right)/r}\right),
Z[\lambda,T]=\frac{\int dA_\mu e^{-\int_0^{1/T}F_{\mu\nu}^2/4+\lambda\int dx e^{ie\int_0^{1/T}d\tau A_0(\tau,x) +{\rm c.c.}}}}{\int dA_\mu e^{-\int_0^{1/T}F_{\mu\nu}^2/4}}
\sum_{t=1}^T\frac{\gamma_0}{\gamma_{t}^2}\le\sum_{t=1}^T\frac{1}{\gamma_{t}}=\frac{1}{\gamma_{0}}\sum_{t=1}^T\frac{1}{t^c}\le\frac{1}{\gamma_{0}(1-c)}T^{1-c}.
\hat{\mathit{B}}(u,v)(k)=i\sum_{j\in \mathbb{Z}^2\setminus\{0\}}[(\hat{u}(k-j)\cdot j)\hat{v}(j)-\frac{(\hat{u}(k-j)\cdot j)(\hat{v}(j) \cdot k)}{k \cdot k}k], \forall k\in\mathbb{Z}^2\setminus\{0\}.
\hat \psi_j^{(K)}= \mathop{\max}\limits_{1 \le{k_j}\le K}\sum\limits_{i ={k_j}}^K{\hat \phi_{ji}^{\left( 4 \right)}}= \max \left\{{\hat \psi_j^{(K - 1)},0}\right\}+ \hat \phi_{jK}^{\left( 4 \right)}\;\; \;\; \hat \psi_j^{(0)}=0,
\begin{aligned}Y(s)= \frac{\hat{p}}{\bar{y}}\hat G_y(s)\hat G_h(s)U_h(s)+\frac{\hat{g}}{\bar{y}}\hat G_y(s)U_b(s) &+\hat G_y(s)\hat G_d(s)\frac{\hat d_{yz}}{\bar{y}}D_{yz}(s) \end{aligned}
P_{k_{0}}Q_{>k_{0}+20}(\chi \varphi)= P_{k_{0}}Q_{> k_{0}+20}(\chi_{< k_{0}+10}\varphi_{[k_{0}-5, k_{0}+5]}) +P_{k_{0}}Q_{>k_{0}+20}(\chi_{[k_{0}-5, k_{0}+5]}\varphi_{< k_{0}-5}) &+ \sum_{k_{1}\geq k_{0}+10}\sum_{k_{2}\in [k_{1}-5, k_{1}+5]}P_{k_{0}}Q_{> k_{0}+20}(\chi_{k_{1}}\varphi_{k_{2}}).
\sum_{n=1}^\infty |v_n|^2 \le 4 \sum_{n=1}^\infty \left[ (\gamma_{n+2}^-)^2 (\gamma_{n+1}^-)^2 + (\gamma_n^-)^2 (\gamma_{n-1}^+)^2 + (\gamma_n^+)^2 (\gamma_{n+1}^-)^2 + (\gamma_{n-2}^+)^2 (\gamma_{n-1}^+)^2 \right] |v_n|^2.
\left(D_{0+}^{\alpha,\alpha',\beta,\beta',\gamma}f\right)(x)=\left(\frac{d}{dx}\right)^{[\operatorname{Re}(\gamma)]+1}\left(I_{0+}^{-\alpha',-\alpha,-\beta'+[\operatorname{Re}(\gamma)]+1,-\beta,-\gamma+[\operatorname{Re}(\gamma)]+1}f\right)(x)
{\Phi}(s,z,\omega)=\bar{c}(z)[\vert{Y}^{M_{{\mathcal{\P}}}}_{s-}\vert_{1,l}+\vert{X}^{M_{{\mathcal{P}}}}_{s-}\vert_{1,l}^{2l}+\vert{Y}^{M_{{\mathcal{P}}}}_{s-}\vert_{l-1}^{2l}],
\int \frac{k(qx)}{( -x^2;q^2)_\infty}\Big( D_q u(x)+ u(x) u(qx)-\frac{q^n x}{(1-q)}u(qx)+\frac{[n]_q}{(1-q)}\Big)y(x) d_qx = & \frac{k(x)}{( -x^2;q^2)_\infty}\Big ( y(x)u(x) - D_{q^{-1}}y(x)\Big).
\big(I+(A^{\dagger})^{\ast}VE_{S_{A}}V^{\ast}A^{\dagger}\big)^{-1}AA^{\dagger}=AA^{\dagger}\big(I+(A^{\dagger})^{\ast}VE_{S_{A}}V^{\ast}A^{\dagger}\big)^{-1}, \big(I+A^{\dagger}UF_{S_{A}}U^{\ast}(A^{\dagger})^{\ast}\big)^{-1}A^{\dagger}A =A^{\dagger}A\big(I+A^{\dagger}UF_{S_{A}}U^{\ast}(A^{\dagger})^{\ast}\big)^{-1}.
\frac{1}{\rho}w(x)- 2R\left(\frac{x-a}{|x-a|^2},0 \right)= -R \left( \frac{2\rho^2 (x-a)}{(\rho^2+|x-a|^2)|x-a|^2}, \frac{2\rho}{\rho^2+|x-a|^2}\right).
\frac{d^2}{dt^2}\phi(\gamma(t))={}\gamma^\prime(t) \gamma^\prime(t) \phi ={}\tilde \nabla^2 \phi(\gamma^\prime (t),\gamma^\prime (t))+ (\tilde \nabla_{\gamma^\prime(t)}\gamma^\prime(t))(\phi) ={}\tilde\nabla^2 \phi(\gamma^\prime (t),\gamma^\prime (t))\geq c \Vert \gamma^\prime(t)\Vert^2
\frac{1}{l}\Biggl( \int^{t+l}_{t}L_{f}^{r}(s+\tau)\, ds\Biggr)^{1/r}\Biggl( \int^{t+l}_{t}\bigl\|x(s+\tau)-x(s)\bigr\|^{p}\, dt\Biggr)^{1/p}\leq M l^{(1/p)+(1/r)-1}\|L_{f}\|_{W^{r}}=l^{(1/q)-1}\|L_{f}\|_{W^{r}}\leq \|L_{f}\|_{W^{r}},t\in I.
\sum_{i\in I}(\operatorname{id}\otimes\varphi)\bigl((1\otimes c^*p_i)\Delta(q_i^*d)\bigr)\longrightarrow \,(\operatorname{id}\otimes\varphi)\bigl((1\otimes c^*)[(\tilde{x}^*\otimes1)E](\Delta d)\bigr) &=\tilde{x}^*\,(\operatorname{id}\otimes\varphi)\bigl((1\otimes c^*)(\Delta d)\bigr).