harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{cases}\partial_t u_1 - d_1\Delta u_1 = -u_1u_2 + u_3u_4, x\in\Omega, \; t>0, \partial_t u_2 - d_2\Delta u_2 = -u_1u_2 + u_3u_4, x\in\Omega, \; t>0, \partial_t u_3 - d_3\Delta u_3 = +u_1u_2 - u_3u_4, x\in\Omega, \; t>0, \partial_t u_4 = +u_1u_2 - u_3u_4, x\in\Omega, \; t>0, \end{cases}
\sum_{n=1}^{N}\frac{(-1)^{n-1}\left(\frac{c}{d}\right)_{n}d^n q^{\frac{n(n+1)}{2}}}{(q)_n (q)_{N-n}(cq)_{n}}\left(n + \sum_{k=1}^n \frac{q^k}{1- q^k}\right) =\frac{c}{(c-d)(q)_N}\left(1-\frac{(dq)_N}{(cq)_N}\right) +\frac{1}{(cq)_N}\sum_{k=1}^N \frac{\left(\frac{cq}{d}\right)_k (dq)^k(dq)_{N-k}}{(q)_k (1-q^k)(q)_{N-k}}
|(u - w_i)(t)| \leq |u(t) - w(x_n)| + |w(x_n) - w_i(t)| = |u(t) - w (x_n)| + |w_i(x_n)- w_i (t)| & \leq 3 \theta_n |g_n(t)| + 2 \theta_n |g_n(t) | & = 5 \theta_n |g_n (t)|.
u(\cdot,0)= B_{(P,Q)}\left(u(\cdot,0)\,e^{-\int_0^a(\mu(P,Q,\cdot,r)+\lambda)dr}\right) + B_{(P,Q)}\left(\int_0^a e^{-\int_x^a(\mu(P,Q,\cdot,r)+\lambda)dr}f(\cdot,x)\, dx \right).
\begin{cases}(\forall i\in\{1,\ldots,m\})C_i(x_{1,n},\ldots,x_{m,n}) \to C_i(\overline{x}_{1},\ldots,\overline{x}_m) (\forall k\in\{1,\ldots,s\})D_{k}^{-1}v_{k,n}\to D_{k}^{-1}\overline{v}_k. \end{cases}
\left ( \begin{array}{c}\psi_{0}(x) \vdots \psi_{p}(x) \end{array}\right ) = \sum_{k \in \mathbb{Z}}B_{k}\left ( \begin{array}{c}\varphi_{0}(2x - k) \vdots \varphi_{p}(2x - k) \end{array}\right ) .
_{s+1}\mathcal{F}_s(\alpha)^{q^d}=\sum_{n\geq 0}\sum_{H\geq h\geq 1}f_h(t)\theta^{hq^{n+d}}\alpha^{q^{n+d}}=\sum_{H\geq h\geq 1}f_h(t)\sum_{n\geq 0}\theta^{hq^{n+d}}\alpha^{q^{n+d}}.
\mathrm{t r}( S_{( i j )})^{4}\propto C^{\mu \nu \lambda \rho \alpha_{1}. . . \alpha_{8}}\xi_{\alpha_{1}}^{i j}\cdots \xi_{\alpha_{8}}^{i j}V_{\mu \nu \lambda \rho}^{i j}/ ( x^{i j})^{1 6},
E \Big[ \prod_{i=1}^p |S_{s_i}-\tilde{S}_{t_i}|_+^{-2}\Big]\le C^{2p}\prod_{i=1}^p |s_i-s_{i-1}|_+^{-1/2}\prod_{j=1}^p |t_{\sigma(j)}-t_{\sigma(j-1)}|_+^{-1/2}\,.
S_{{\mathrm{eff}}}= \frac 12 \int d\tau \left[\dot{\vartheta}^2 - \frac{(\pi_u - \pi_v)^2}{4\sinh^2\frac{\vartheta}{2}}+ \frac{(\pi_u + \pi_v)^2}{4\cosh^2\frac{\vartheta}{2}}\right]\,,
n=3,d=4, b_1=2h,b_2=-2j,b_3=1; &\alpha_1=\frac{r+\sqrt{ab}}{2},\alpha_2=\frac{s+\sqrt{ac}}{2},\alpha_3=\frac{\sqrt{c}(\sqrt{a}+\sqrt{b})}{\sqrt{b}(\sqrt{a}+\sqrt{c})}.
M = 4 \pi \int_{0}^{\infty}d r r^{2}\{\frac{F_{\pi}^{2}}{8}[ F^{2}+ 2 \frac{s i n^{2}F}{r^{2}}] + \frac{1}{2 e^{2}}\frac{s i n^{2}F}{r^{2}}[ \frac{s i n^{2}F}{r^{2}}+ 2 F^{2}] \}
L\left(\tfrac{1}{2},\pi\otimes\chi\right)=\sum_{n=1}^\infty \frac{\lambda_\pi(n,1)\chi(n)}{\sqrt{n}}V\left(\frac{nX}{q^{\frac{3}{2}}}\right)+\frac{g(\chi)^3}{q^{\frac{3}{2}}}\sum_{n=1}^\infty \frac{\lambda_{\tilde\pi}(n,1)\bar\chi(n)}{\sqrt{n}}V\left(\frac{n}{Xq^{\frac{3}{2}}}\right)
\begin{array}{lcl}\bar R(X,Y)Z = \langle Y,Z\rangle X-\langle X,Z\rangle Y+2\langle Z,E_3\rangle(\langle X,E_3\rangle Y-\langle Y,E_3\rangle X) & +2(\langle X,Z\rangle\langle Y,E_3\rangle-\langle Y,Z\rangle\langle X,E_3\rangle)E_3,\end{array}
\begin{array}{llllll}F_{i,j}(\Theta_1,\cdots,\Theta_{l+q+1}) \equiv \sum\limits_{k=1}^{l+q}\alpha_{ij}^k\Theta_k+q_{ij}(\Theta_1,\cdots,\Theta_{l+q+1}) (\mbox{mod}_{m_i+m_j+1}U^{}(\ggg,e)),\end{array}
t_{z_1}^{2n}t_{\alpha_1}^{-2n}t_{y^\prime_1}^{2n}t_{\beta_1}^{-2n}= [t_{z_1}^{2n}t_{\alpha_1}^{-2n}, f_2], t_{x_1}^{2n}t_{\beta_1}^{-2n}t_{z^\prime_1}^{2n}t_{\gamma_1}^{-2n}= [t_{x_1}^{2n}t_{\beta_1}^{-2n}, f_3],
\Xi:\mathbb{F}_{\ell}[\lambda]^{(\eta+1)m\times (\epsilon+1)n}\longrightarrow \mathbb{F}_{d}[\lambda]^{m\times n}M(\lambda) \longrightarrow \Xi[M](\lambda)=(\Lambda_\eta(\lambda^\ell)^T\otimes I_m)M(\lambda)(\Lambda_{\epsilon}(\lambda^\ell)\otimes I_n).
\left(\int_D|F_N-(1-b_{t_0}(\Psi))f|^2e^{-\phi}\right)^{1/2}\geq \left(\int_D|F_N|^2e^{-\phi}\right)^{1/2}-\left(\int_D|(1-b_{t_0}(\Psi))f|^2e^{-\phi}\right)^{1/2}\geq \left(\int_D|F_N|^2e^{-\phi}\right)^{1/2}-\left(\int_{\{\Psi<-t_0\}}|f|^2e^{-\phi}\right)^{1/2}.
e\cdot\frac{d}{dt}\bigg|_{t=0}J =\frac{d}{dt}\bigg|_{t=0}e\cdot\int_{\mathbb T}\sigma g ~ d\theta=\int_{\mathbb T}(e\cdot\sigma)\partial_{t}g_{t}(\theta,0) ~ d\theta =\int_{\mathbb T}\langle\nabla(e\cdot\sigma),\nabla\varphi\rangle\rho ~ d\theta =e\cdot\int_{\mathbb T}\nabla\varphi\rho ~ d\theta,
\| \rho^n \|_{\mathcal{C}_h}= \| \mathcal{P}_h \, u(t_n)- u(t_n)^I \|_{\mathcal{C}_h}\leq C\, h^2 \left( |u_0|_{H^2(\Omega)}+ |u_t|_{H^1(0, t_n, H^2(\Omega))}\right).
\pi(x)=\rho(x_7h_7+x_8h_8+x_9h_9)=\begin{pmatrix}0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_9 -x_7 0 0 0 0 x_8 -x_9 0 0 0 0 0 0 x_9 -x_7 0 0 0 0 x_8 -x_9 \end{pmatrix}.
- \frac{i}{2}\sum_{n}g_{n}\oint_{\gamma_{n}}( \phi_{1}+ \alpha_{1}\phi_{B}) \frac{d z}{z - z_{n}}- \frac{i}{2}g_{\infty}\oint_{\gamma_{\infty}}( \phi_{1}+ \alpha_{1}\phi_{B}) \frac{d z}{z}- \alpha_{1}\int \phi_{1}\partial_{z}\partial_{\bar{z}}\phi_{B}\frac{i d z \wedge d \bar{z}}{2}.
(\partial_1 b)(p_1, p_2) := p_1 + \frac{p_1 p_2}{1-p_2}\Phi_s + \frac{(p_1^2 + p_2^2) p_1 p_2}{1-p_2}\frac{\Phi_s^\prime}{s^2}(\partial_2 b)(p_1, p_2) := p_2 + \frac{p_2^2}{1-p_2}\Phi_s + \frac{1}{2}\frac{p_1^2 +p_2^2}{(1-p_2)^2}\Phi_s + \frac{(p_1^2+p_2^2) p_2^2}{1-p_2}\frac{\Phi_s^\prime}{s^2}.
F_{i+\frac{1}{2}}^{m}= \max (\vartheta_{}^{i+1},0)\Big[ \widetilde{m}_{i}+ \frac{\Delta x}{2}(m_{x})_{i}\Big] + \min (\vartheta_{m}^{i+1},0)\Big[ \widetilde{m}_{i}- \frac{\Delta x}{2}(m_{x})^{i}\Big],
4^{n+1}(l-1)^2 = (2lw-(l-1))^2+(2lx-(l-1))^2+2(2ly-(l-1))^2+2(2lz-(l-1))^2 = (2lw-(l-1))^2+(2lx-(l-1))^2+(2l(y+z)-2(l-1))^2+(2l(y-z))^2
\chi_1(s) = \sum_{m \leq s^{-1/2}}-\phi'(ms) &= Ku\Gamma(u+1) \sum_{m \leq s^{-1/2}}(ms)^{-u-1}(1+O((ms)^{\epsilon/2})) & = Ku\Gamma(u+1) s^{-u-1}\left( \sum_{m \leq s^{-1/2}}\frac{1}{m^{u+1}}+ O(s^{\epsilon/2}) \sum_{m \leq s^{-1/2}}\frac{1}{m^{u+1 - \epsilon/2}}\right) & = Ku\Gamma(u+1) s^{-u-1}\zeta(u+1)(1+O(s^{\epsilon/2})),
\frac{\Psi_{N_\varepsilon}'(u_\varepsilon)}{2}~ =u_\varepsilon H(u_\varepsilon) \varphi_{N_\varepsilon}(u_\varepsilon^2)+u_\varepsilon (1+g(u_\varepsilon))\frac{u_\varepsilon^{2N_\varepsilon}}{N_\varepsilon !}+O\left(\gamma_\varepsilon^3 \right) &\le (1+o(1))\gamma_\varepsilon\varphi_{N_\varepsilon-1}(\gamma_\varepsilon^2)\,.
\|\omega_z\|_{q/2,u^o}= \bigg(z^{-q}\int_0^z u^o(t)\,dt+\int_z^\infty u^o(t)\,\frac{dt}{t^q}\bigg)^{2/q}\le\bigg(q'z^{-q}\int_0^zu^o(t)\,dt\bigg)^{2/q}.
\hat{V}^{C}_{\sigma}(C_{1},C_{2}) = \left ( \frac{1}{2\pi(\sigma')^2}\right ) \frac{1}{N}\sum\limits_{i=1}^N exp \left( - \frac{(x_{i}-y_{i})^2 + (z_{i}-s_{i})^2}{2(\sigma')^2}\right )
\left\langle \left\langle (g\otimes s)\xi_{1},(g\otimes s) \xi_{2}\right\rangle \right\rangle =s\overline{s}\tau (\xi_{2})\tau (g)g \xi_{1}=\tau (\xi_{2})\xi_{1}= \left\langle \left\langle \xi_{1}, \xi_{2}\right\rangle \right\rangle , g\otimes s \in Spin_{(n+m)}^{\mathbb{C}}\subset \mathbb{C}l_{(n+m)},
\frac{dI}{dt}=-2\frac{\partial E}{\partial \varphi}=\frac12\sqrt{1-I^2}\sin(\varphi) &=\pm\frac12\sqrt{-4I^2+(8|p|^2-5)I^2+4|p|^2(1-|p|^2)}&=\pm\sqrt{(a-I^2)(b+I^2)}\ ,
\phi = \left( \begin{array}{l}{\phi^{+}}{\phi^{0}}\end{array}\right) = \frac{1}{\sqrt{2}}\left( \begin{array}{l}{\chi^{2}( x ) + i \chi^{1}( x )}{v + \psi ( x ) - i \chi^{3}( x )}\end{array}\right) ,
\beta(\widehat{x}_0) = \zeta\left(e^{i \widehat{p}_k\otimes\widehat{x}_k}(1\otimes\widehat{x}_0)e^{- i \widehat{p}_k\otimes \widehat{x}_k}\right) = \zeta\left(1\otimes\widehat{x}_0 - \frac{1}{\kappa}\widehat{p}_k\otimes\widehat{x}_k\right)= \widehat{x}_0 - \frac{1}{\kappa}\widehat{p}_k\widehat{x}_k.
(1-u^{2})^{\epsilon- \nu}f_{S}(u) = (1-u^{2(k+1)})^{\epsilon- \nu}f_{X}(u^{k+1}), (1-R_{S}^{2})^{\epsilon- \nu}R_{S}f_{S}^{'}(R_{S}) = (k+1)(1-R_{X}^{2})^{\epsilon- \nu}R_{X}f_{X}^{'}(R_{X}).
\frac{\left| z\left(\left(\frac{1}{2}-\sigma\right) + j\omega\right)\right|}{\left|z\left(\left(\frac{1}{2}+\sigma\right) - j\omega \right)\right|}= \frac{\left|j P^{\frac{1}{2}}\right|}{\left|G\left(\overline{\chi},P\right)\right|}\frac{\left|\left(\frac{3}{2}-\sigma\right) + j\omega\right|}{\left|\left(\frac{3}{2}+\sigma\right) - j\omega\right|}
N=\int_{t_{\rm i}}^{t_{\rm f}}dt H\simeq \frac{-1}{M_{Pl}^2\sqrt{1+4\alpha\Lambda_b/3M^5}}\int_{\phi_{\rm i}}^{\phi_{\rm f}}d\phi{V\over V'}\left[ 1+{V \over 2\Lambda_2^+}\right]\,.
\limsup_{n\to\infty}{\log m_n(x,y) \over -\log n}\geq \limsup_{k\to\infty}{\log m_{n_k}(x,y) \over -\log n_k}\geq \lim_{k\to\infty}{\log r_k \over -\log W_{B(y,r_k)}(x)}=1.
\frac{\sum_{i = 1}^n b^2_{i,n}}{\sum_{i = 1}^n a^2_{i,n}}- 1 = o_{\mathbb{P}_n}(1) + 2 \frac{\sum_{i = 1}^n a_{i,n}[b_{i,n}- a_{i,n}]}{\sum_{i = 1}^n a^2_{i,n}},
dS^{-1}_{mk}(t)=S^{-1}_{mk}(t) d V_{mm}(t)+ \sum_{j: j \not=m}S^{-1}_{j k}(t) dV_{m j}=- S^{-1}_{mk}(t) d U_{mm}(t)- \sum_{j: j \not=m}S^{-1}_{j k}(t) dU_{m j},
A + B = - \frac{s \left( 2 \alpha_{-}^{2}\right)}{s \left( \alpha_{-}^{2}\right)}\frac{\Gamma \left( 2 - 2 \alpha_{-}^{2}\right)}{\Gamma^{2}\left( 1 - \alpha_{-}^{2}\right)}A_{n m}^{\alpha}A_{1 2}^{\alpha}
\left(\bar{X}^{(N)}(t);t \ge 0\right)\overset{def}{=}\left(\mathfrak{r}_N\left(X^{(N)}\right)(t);t \ge 0\right)=\left(\alpha_i^{\pm}\left(X^{(N)};t\right),\gamma_1\left(X^{(N)};t\right),\delta\left(X^{(N)};t\right), i\in \mathbb{N};t\ge 0\right),
E_{ADM}=\left.{\Delta}\left[{\lambda}\sqrt{\kappa}({\psi}_0-{\psi}_1) +\sqrt{\kappa}({\psi}_0-{\psi}_1){\phi}^{\prime}-\frac{\sqrt{\kappa}}{2}({\psi}_0^{\prime}-{\psi}_1^{\prime})\right]\right|^{\infty}_{-\infty}
N_j(u_1,u_2,\ldots,u_m)=\sum_{k_1=0}^{\infty}\sum_{k_2=0}^{\infty}\ldots\sum_{k_m=0}^{\infty}\frac{\partial^{k_1+k_2\ldots+k_m}N_j(0,0,\ldots,0)}{\partial^{k_1}u_1\partial^{k_2}u_2\ldots\partial^{k_m}u_m}\prod_{j=1}^{m}\frac{u_j^{k_j}}{k_j!},
\nabla_{i}= g_{0}^{- 1}h_{1}^{- 1}( \partial_{i}- A_{i}\partial_{2}) ~ , ~ ~ \nabla_{2}= g_{0}^{- 1}h_{2}^{- 1}\partial_{2}~ , ~ ~ \nabla_{p}= g_{0}^{- 1}h_{3}^{- 1}( \partial_{p}- A_{p}\partial_{2}) .
\int_{-\infty}^{\infty}\int_0^1 \left| \alpha \left(\frac{\partial u}{\partial s}, X^{\omega_s}_H \right)\right| dtds = \int_{\mathcal I}\int_0^1 \left| \alpha \left(\frac{\partial u}{\partial s}, X^{\omega_s}_H \right)\right| dtds \leq |\alpha| \cdot C' \int_{\mathcal I}\int_0^1 \left| \frac{\partial u}{\partial s}\right|_{g_s}dtds
f\left( x\right) \backsim \frac{a_{0}\left( f\right)}{2}+\sum\limits_{k=1}^{\infty}\left( a_{k}\left( f\right) \cos kx+b_{k}\left(f\right) \sin kx\right) =:\sum\limits_{k=0}^{\infty}A_{k}\left( x,f\right)
{e}_{\hat{0}}^{\;\;0}{\bar{e}}_{\alpha}^{\;\; i}\left[- \frac{1}{2}\frac{\partial\Gamma^{(2)}}{\partial v^{i}}\right] ={e}_{\hat{0}}^{\;\;0}{\bar{e}}_{\alpha}^{\;\; i}\left[B^{v(2)}_{i}\right] +{e}_{\alpha}^{\;\;\mu}{e}_{\hat{0}}^{\;\;\nu}K^{g}_{\mu\nu},
n_{-ty+\frac{x}{|x|^2}}r_x e^{-(2\log |x|)H_0}\bar{n}_{\frac{x}{|x|^2}}&=s n_{\frac{z}{|z|^2}}r_z e^{-(2\log |z|)H_0}\bar{n}_{\frac{z}{|z|^2}}r_x e^{-(2\log |x|)H_0}\bar{n}_{\frac{x}{|x|^2}}&\in s n_{\frac{z}{|z|^2}}r_z r_x e^{-(2\log |z|+2\log |x|)H_0}\bar{N}.
\operatorname E\int_\mathbb S\mathcal X^2(v,t)\mu(\mathrm{d}v) =\int_\mathbb S\frac{\sigma^2(v)c(v)}{[1-d(v)][3-2d(v)]}\cdot t^{3-2d(v)}\mu(\mathrm dv) &\le\max\{t,t^2\}\Bigl[\int_\mathbb S\frac{\sigma^2(v)}{[1-d(v)]^2}\mu(\mathrm dv)+\int_\mathbb S\frac{\sigma^2(v)}{[1-d(v)][2d(v)-1]}\mu(\mathrm dv)\Bigr]
\prod_{k=1}^{\nu(n)}(\det D\tilde{f}_{\lambda})(\tilde{z}^{(n)}_k(\lambda)) =\prod_{k=1}^{\nu(n)}\prod_{j=1}^{2d-2}(\tilde{z}^{(n)}_k(\lambda)\wedge\tilde{c}_j(\lambda)) =\prod_{j=1}^{2d-2}\tilde{H}_n^{(j)}(\lambda),
v_m(x):= \begin{cases}\sqrt{\frac{\log m}{16 \pi^2}}+ \frac{1}{\sqrt{16 \pi^2 \log m}}(1 - m|x|^2), \ \mbox{if}\ 0 \leq |x| \ \leq \frac{1}{\sqrt{m}}, \frac{1}{\sqrt{4 \pi^2 \log m}}\log \frac{1}{|x|}, \mbox{if}\ \frac{1}{\sqrt{m}}< |x| \leq 1, \xi_m(x), \mbox{if}\ |x| \geq 1, \end{cases}
|\lambda \cap S_\mu(m)| = |\pi_1(\sigma) \cap S_\mu(m)| = |\sigma \cap (S_\mu(m) \times T_\mu)|, |\lambda \cap T_\mu(m)| = |\pi_2(\sigma) \cap T_\mu(m)| = |\sigma \cap (S_\mu \times T_\mu(m))|, |\lambda| = |\pi_1(\sigma)| + |\pi_2(\sigma)| = 2|\sigma|.
dX^{\epsilon, i}(t) = \Bigl (A + \sum\nolimits_{j \neq i}B_j \gamma_j\Bigr)X^{\epsilon, i}(t) dt + B_i u_i(t)dt + \sqrt{\epsilon}\sigma\bigl(X^{\epsilon, i}(t)\bigr) dW(t), X^{\epsilon,i}(0)=x_0, i =1, 2, \ldots, n,
\begin{array}{r c l}{{d s^{2}}}&{{=}}&{{\left( \sigma^{- 1}\eta_{i j}- \sigma^{- 2}\eta_{i k}\eta_{j l}x^{k}x^{l}\right) d x^{i}d x^{j}}}\\{{}}&{{=}}&{{- \displaystyle \frac{d x J ( I - x^{\prime}x J )^{- 1}d x^{\prime}}{1 - x J x^{\prime}}~ ,}}\\ \end{array}
\hat{P}^{\hat{a}\hat{b}\hat{c}}\equiv \hat{\cal H}^{\hat{a}\hat{b}\hat{d}}\hat{C}^{\hat{c}}_{\hat{d}}={1 \over K}\, \left[ (1+{1 \over 12}\hat{{\cal H}}^2) \, \hat{\cal H}^{\hat{a}\hat{b}\hat{c}}-\,{1\over 4}(\hat{{\cal H}}^3)^{\hat{a}\hat{b}\hat{c}}\right] \, .
\dots \rightarrow \bigoplus_{j\in \widetilde{\Gamma}}\bigoplus_{l=1}^{v_{ij}}P_j^{(d_l^{ij})}\rightarrow \bigoplus_{j \in \widetilde{\Gamma}}\bigoplus_{l=1}^{u_{ij}}P_j^{(d_l^{ij})}\rightarrow P_i^{(0)}\rightarrow s_i^{(0)}\rightarrow 0
H_{5}={\frac{P}{| \vec{z}- \vec{z}_{0}|^{3 - n}}}, H_{W}= 1 +{\frac{Q_{W}}{[ x^{2}+{\frac{4 P}{( n - 1 )^{2}}}| \vec{z}- \vec{z}_{0}|^{n - 1}]^{\frac{n + 1}{n - 1}}}}.
\sum_{\pi:k\in\{\pi(1),\dotsc,\pi(p)\}}f(z_{\pi(1)},\dotsc,z_{\pi(p)})g(z_{\pi(p+1)},\dotsc,z_{\pi(p+q)}) &=\sum_{j=1}^p \sum_{\pi:\pi(j)=k}f(z_{\pi(1)},\dotsc,z_{\pi(j-1)},z_k,z_{\pi(j+1)},\dotsc,z_{\pi(p)})g(z_{\pi(p+1)},\dotsc,z_{\pi(p+q)}) &=p\sum_{\pi:\pi(1)=k}f(z_k,z_{\pi(2)},\dotsc,z_{\pi(p)})g(z_{\pi(p+1)},\dotsc,z_{\pi(p+q)})
\mathbb P \big( \{W= \sum v_n e^{inx}\in \mathcal W | \sigma_{n_i}\in (a_i, b_i)\subset [-1/2, 1/2] \mbox{for}i = 1, \dots N ,\mbox{otherwise}w_n = 0\}\big) = \prod_{i=1}^N (b_i -a_i).
\begin{aligned}m_{jj}:=\frac{H(y_{i_j},\,y_{i_j})}{\bigl(K(y_{i_j})\bigr)^{\frac{n-2}{2}}},\,\,\forall\,\,1\leq j\leq s &{}m_{jk}:=-\frac{G(y_{i_j},\,y_{i_k})}{\bigl(K(y_{i_j})K(y_{i_k})\bigr)^{\frac{n-2}{4}}},\forall\,\,k\neq j,\,\,1\leq k,\,j\leq s. .\end{aligned}
\hat{\cal W}_{\mu}\to \hat{\cal W}_{\mu}+ i [\alpha^a T^a, \hat{\cal W}_{\mu}]+ i [\beta^b T^b (\hat{C}_+ - \hat{C}_-), \hat{\cal W}_{\mu}]- \partial_{\mu}\alpha^a T^a- \partial_{\mu}\beta^b T^b (\hat{C}_+ - \hat{C}_-).
H_{2 r}( x ) = ( - 1 )^{r}2^{2 r}r ! L_{r}^{( - 1 / 2 )}( x^{2}) , \quad H_{2 r + 1}( x ) = ( - 1 )^{r}2^{2 r + 1}r ! x L_{r}^{( 1 / 2 )}( x^{2}) .
\begin{array}{l}|w_j|^p|u_j|^q-|w_{j-1}|^p|u_{j-1}|^q =(|w_j|^p-|w_{j-1}|^p)|u_j|^q+|w_{j-1}|^p(|u_j|^q-|u_{j-1}|^q)\end{array}
\Psi(x,u) = \prod_{i_0 \in \mathbb J^{0}_k(x)}\mbox{\bf{1}}_{\{u \in \mathbb J^0_{k,j_i}(x)\}}\mathbb P\Bigg( \big( \bigcap_{i_{-}\in \mathbb J^{-}_{k}(x)}A_{i_{-},k}\big) \cap \big( \bigcap_{i_{+}\in \mathbb J^{+}_{k}(x)}A_{i_{+},k}\big) \Bigg).
\nu (x,y,z) := \dfrac{1}{y^{\lambda -1}},~\forall (x,y,z)\in \Omega:=\{(x,y,z)\in \mathbb{R}^{3}: y\neq 0 \}, H(x,y,z) := xy^{\lambda},~\forall (x,y,z)\in \Omega, C(x,y,z) :=\dfrac{1}{2}\left(x^2 +y^2 +z^2 \right),~\forall (x,y,z)\in \Omega.
f[F,g][F,h] = \begin{bmatrix}f^{+}f^{-}\end{bmatrix}\begin{bmatrix}d^{-}g d^{+}g \end{bmatrix}\begin{bmatrix}d^{-}h d^{+}h \end{bmatrix}= \begin{bmatrix}f^{+}d^{-}g f^{-}d^{+}g \end{bmatrix}\begin{bmatrix}d^{-}h d^{+}h \end{bmatrix}= \begin{bmatrix}f^{+}d^{-}gd^{+}h f^{-}d^{+}gd^{-}h\end{bmatrix}.
\tilde\mu(z) = 2(1-|z|^2)^2 \sum_{n=0}^\infty (n+1)\lambda_n |z|^{2n}&= 2(1-|z|^2) \left( \lambda_0 + \sum_{n=1}^\infty \big((n+1)\lambda_n-n\lambda_{n-1}\big)|z|^{2n}\right).
\sigma_{1}(\alpha(w)+\operatorname{ker}\beta \cap \operatorname{im}\alpha) =\sigma_{1}(\alpha(x_{1})+\alpha(x_{2}) + \operatorname{ker}\beta \cap \operatorname{im}\alpha) &=\sigma_{1}(\alpha(x_{2})+\operatorname{ker}\beta \cap \operatorname{im}\alpha)
\mathbf K(t)=\int_0^1 \frac{\mathrm{d}s}{\sqrt{(1-s^2)(1-s^2 t^2)}}\mathbf E(t)=\int_0^1 \sqrt{\frac{1- s^2 t^2}{1-s^2}}\,\mathrm{d}s, 0\leq s\leq 1.
f^{\ast}(\eta ,t ) =\int_{0}^{\infty}(1-e^{-\eta x})\nu_{t}^{\ast}(dx) & =\lambda^{\alpha (t)}\sum_{k=1}^{\infty}\binom{\alpha (t)}{k}(-1)^{k+1}\int_{0}^{\infty}(1-e^{-\eta x})\delta_{k}(dx) & =\lambda^{\alpha (t)}\sum_{k=1}^{\infty}\binom{\alpha (t)}{k}(-1)^{k+1}(1-e^{-\eta k}) & =\lambda^{\alpha (t)}(1-e^{-\eta})^{\alpha (t)}.
M(d\mid x,k) \propto M(d)\frac{\mathrm{d}P_{\mathrm{S}(\cdot\mid d,k)}}{\mathrm{d}P_{0,\varepsilon^{2}}}(x) \propto M(d)\prod_{i=1}^{d}(1+(d/i)^{2k+1})^{-1/2}\exp\bigg{\{}-\frac{x^{2}_{i}}{2\varepsilon^{2}(1+(d/i)^{2k+1})}+\frac{x^{2}_{i}}{2\varepsilon^{2}}\bigg{\}}.
A_1^{-(k+1)}= \Big( \frac{\alpha}{4e(1+\alpha)}\Big)^{\alpha k}\frac{1}{2^{k}}\cdot \frac{1}{A_1 (1+\eta)^{k}}\leq \Big( \frac{\alpha}{4e(1+\alpha)}\Big)^{2\phi A \lambda + 8}\frac{1}{2^{k+1}}\cdot \frac{2}{A_1 (1+\eta)^{k}}
a_{0,2}(\ell,k) =-1(k>n_1)c_2(\ell,k) a_{0,3}(\ell,k) =1(\ell\le n_1)c_3(\ell,k) b_2(\ell,k) =-1(k>n_1+1)c_2(\ell,k) b_3(\ell,k) =1(\ell\le n_1+1)c_3(\ell,k) a_2^\ast(\ell) =c_2(\ell,n_1) a_3^\ast(k) =c_3(n_1+1,k),
\frac{\mathbf{E}_\rho^t(S')- \mathbf{E}_\rho^t(S)}{t}=\frac{\sum_{i=1}^n\mathcal{E}_\rho^t(S',h_i^t\circ f_i^t)- \mathbf{E}_\rho^t(S)}{t}> \frac{\sum_{i=1}^n\mathcal{E}_\rho^s(S',h_i^s\circ f_i^t)- \mathbf{E}_\rho^s(S)}{s}> \sum_{i=1}^n\mathcal{E}_\rho(S',\pi\circ \tilde{f}_i^t)- \mathbf{E}_\rho(S).
\varphi_{n;\textrm{out}}=P_{\delta}(W_{n,m;\textrm{out}})\varphi_n \varphi_{n;\textrm{in}}=P_{\delta}(W_{n,m;\textrm{in}})\varphi_n \varphi_{n;\textrm{sur}}=P_{\delta}(W_{n,m;\textrm{sur}})\varphi_n
\frac{(M_{1}+\cdots+M_{\ell}-2)!}{(M_{1}-1)!M_{2}!\cdots M_{i}!\cdots M_{\ell}!}\left(M_{1}-1+M_{2}+\cdots+M_{\ell}\right)=\frac{(M_{1}+\cdots+M_{\ell}-1)!}{(M_{1}-1)!M_{2}!\cdots M_{i}!\cdots M_{\ell}!}.
U_{p,0}(u,x)\cap U_{p,p}(u,x)=\bigcup_{\kappa=0}^{\infty}\left(U^{\kappa+1}_{p,0}(u,x)\cap U^\kappa_{p,p}(u,x)\right)\cup\{\zeta\}&=\bigcup_{\kappa=0}^{\infty}\{X_0>u, X_p>u, \ldots, X_{(\kappa+1)p}>u, X_{(\kappa+2)p}\leq u,m_{u}(\{X_p,\ldots,X_{(\kappa+1)p}\})>x\}\cup\{\zeta\}
e^{-C_\varphi/h}\|\langle x \rangle^{-s}\mathbf{1}_{\le M}v \|^2_{L^2}+ \|\langle x \rangle^{-s}\mathbf{1}_{\ge M}v \|^2_{L^2}\le \frac{C}{h^2}\|\langle x \rangle^{s}(P(h) - E \pm i\varepsilon)v \|^2_{L^2}+ \frac{C\varepsilon}{h}\| v \|^2_{L^2},
(r, \mathcal{O}_{(0,0), \rho}, R):= \sup \{|y| : (x, y) \in \left(\partial B^k_{\rho}(0) \times B^{n-k}_{R}(0)\right) \cap \mathcal{O}_{(0,0), \rho}\}M(r, \mathcal{O}_{(0,0), \rho}, R) := \sup \{u_{(0,0), \rho}(x, y) : (x, y) \in \left(\partial B^k_{\rho}(0) \times B^{n-k}_{R}(0) \right) \cap \mathcal{O}_{(0,0), \rho}\}.
\langle \phi^{-1}(-Ad_{g_1^{-1}}X), Ad_{g_1^{-1}}Y\rangle_1 + \langle \phi^{-1}(X), Y\rangle_1 = \langle -Ad_{g_1^{-1}}X, Ad_{g_1^{-1}}Y\rangle_0 + \langle X,Y\rangle_0 = -\langle X,Y\rangle_0 + \langle X,Y\rangle_0 = 0
d_1 :=\frac{S_1\mu}{R_1}\left(S_1 \frac{g_*^3}{3}+S_1\mu\left(\frac{f_*^3}{3}+ f_*^2g_*+f_*g_*^2\right)\right), begin{align*}5pt]d_3 :=-z(\mu f_*+g_*)\Gamma_*\sigma^\prime(\Gamma_*)+zD.
\mathbb{E}\left[{\int}_{t}^{t+\delta}|Z_{s}^{1,v}|^{2}ds\right] =|Y_{t}^{1,v}|^{2}+\mathbb{E}\left[ \left({\int}_{t}^{t+\delta}F_{1}(s,X_{s}^{v},Y_{s}^{1,v},Z_{s}^{1,v},v_{s})ds\right)^{2}\right] & \leq2\mathbb{E}\left[ \left({\int}_{t}^{t+\delta}|F_{1}(s,X_{s}^{v},Y_{s}^{1,v},Z_{s}^{1,v},v_{s})|ds\right)^{2}\right] .
(Gu)(t)= T(t)u_{0}+\chi\int_{0}^{t}T(t-s)\nabla\cdot ( u(s) \nabla (\Delta-I)^{-1}u(s))ds & +(1+a)\int_{0}^{t}T(t-s)u(s)ds-b\int_{0}^{t}T(t-s)u^{2}(s)ds.
\varphi_*^{{\mathbb{P}}}(\mathcal{L}(Y)) = [[y_{1-}, y_{1+}]],\;\;\;\;\;\;y_{1\pm}:= \frac{a_{\pm}q^* - b_{\pm}p^*}{a_{\pm}p - b_{\pm}q}=\mkern1mu \frac{\mkern.5muq^*\mkern-2.5mu}{p}\mkern-5mu\left(1 - \frac{b_{\pm}}{q^*(a_{\pm}p-b_{\pm}q)}\right).
{\displaystyle{\delta^2 \Gamma_{m}[0,\Psi,\bar\Psi]\over\delta\Psi (x^\prime)\ \delta\bar\Psi (x)}}Bigg\vert_{\bar\Psi = \Psi = 0}\ =\S_{m}^{-1}(x-x^\prime ) \hspace{0.3cm}.\hspace{0.3cm}
\ddot{\phi}+ \frac{3 \dot{a}}{a}\dot{\phi}= \eta \dot{\phi}^{2}\frac{f^{\prime}}{1 + 2 \eta f}+ \eta \frac{\Lambda}{2 \pi G}\frac{f^{\prime}( 1 + 2 \eta f )}{( 1 + 4 \eta f )^{2}}
\Omega_{\beta_1,\bar{\beta}_2}= \frac{\tanh(\beta_2-\beta_1)}{2}\Big{\{}\frac{1}{\sinh(2\beta_1)}-\frac{1}{\sinh(2\beta_2)}\Big{\}}\chi([\theta(e_{\beta_2}),e_{\beta_1}]) +\frac{\tanh(\beta_1+\beta_2)}{2}\Big{\{}\frac{1}{\sinh(2\beta_2)}+\frac{1}{\sinh(2\beta_1)}\Big{\}}\chi([\theta(e_{\beta_2}),\sigma(e_{\beta_1})])
\partial^\mu \left( \partial^\nu \tilde{\partial}_{+}\bar{\phi}\,\,\varepsilon^{\prime}\sigma_{\nu \mu}\tilde{\partial}_{-}\psi\,\,\right) +\tilde{\partial}_{+}\left( \tilde{\partial}_{-}\bar{V}_{+}\,\bar{\varepsilon}_{\dot{\alpha}}\bar{\sigma}_\mu^{\dot{\alpha}\beta}\partial^\mu \tilde{\partial}_{-}\psi_\beta \right) .
A^+ = U \Lambda^+ U^T = \begin{pmatrix}U_1 U_2 \end{pmatrix}\begin{pmatrix}0 0 0 \Lambda_2^{-1}\end{pmatrix}\begin{pmatrix}U_1^T U_2^T \end{pmatrix}= U_2 \Lambda_2^{-1}U_2^T.
\sum^n_{\ell=0}t^{-\ell}\Bigr(\mathrm{Tr\,}\widetilde\alpha^+_\ell(x)-\mathrm{Tr\,}\widetilde\alpha^-_\ell(x)\Bigr)dv_X(x) &=\frac{1}{2\pi}\left[\mathrm{Td_b\,}(\nabla^{T^{1,0}X},T^{1,0}X)\wedge e^{-m\frac{d\omega_0}{2\pi}}\wedge\omega_0\right]_{2n+1}(x)+O(t)
t_{j}= \frac{\tan ( \frac{\pi}{4}- \frac{\pi j}{2 H})}{2 \cos^{2}( \frac{\pi j}{2 H})}\prod_{i = 0}^{j - 1}\frac{\tan ( \frac{\pi ( j - 1 - 2 i + w_{+})}{2 H}) \tan ( \frac{\pi ( j - 1 - 2 i + w_{-})}{2 H})}{\tan^{2}( \frac{\pi ( i + 1 )}{2 H})}.
S_2(...) = \sum_{m=1}^\infty \lambda_f (m) e\left( -\frac{(a+bq)m}{p_1q}\right) v_2(n) &= \frac{1}{p_1q}\frac{\eta_f(p_2)}{\sqrt{p_2}}\sum_{m=1}^\infty \lambda_{f}( m) e\left(-m \frac{\overline{(a+bq) p_2}}{p_1q}\right) \mathcal{V}_2\left( \frac{m}{p_2 (p_1q)^2}\right),
\begin{array}{c c c c c}{{\{L = ( 0 0 0 0 0 ) , M = 5 \}}}&{{+}}&{{\{L = ( 0 0 0 0 0 ) , M = 7 \}}}&{{=}}&{{\{L = ( 0 0 0 0 1 ) , M = 6 \}\nonumber}}\\{{( 1 , 0 , 0 , 0 )}}&{{+}}&{{( - 4 , - 1 , - 8 , 5 )}}&{{=}}&{{( - 3 , - 1 , - 8 , 5 )}}\\ \end{array}
\mathbb{R}^n = \bigcup_{i=1}^{2^n}A_i(\mathbb{R}^n_+),\;{\rm int}\, \big( A_k(\mathbb{R}^n_+) \big) \cap{\rm int}\, \big( A_l(\mathbb{R}^n_+) \big) = \emptyset,\; \mu( c + A_i(\mathbb{R}^n_+)) = \frac{1}{2^n}\mu( \mathbb{R}^n )
\underline{M}:=\frac{(b_{\inf}-\chi\mu)a_{\inf}-\chi\mu a_{\sup}}{(b_{\sup}-\chi\mu)(b_{\inf}-\chi\mu)-(\chi\mu)^2}>\frac{a_{\inf}-\frac{\chi\mu a_{\sup}}{b_{\inf}-\chi\mu}}{b_{\sup}-\chi\mu}
\Psi(\omega;\lambda)= \Phi(\omega;\lambda) -{1 \over4\lambda}\sum_k \left[(\phi^{(\lambda)}_{\omega,k},\partial_\lambda H^2(\lambda)\phi_{\omega,k}^{(\lambda)}) +(\phi^{(\lambda)}_{\omega,k},\partial_\lambda H^2(\lambda)\phi_{\omega,k}^{(\lambda)})^{*}\right],
(\frac{1}{\alpha}- \Delta t \delta_{xx}) u^*_{i,j}= (\frac{1}{\alpha}+ \Delta t \delta_{yy}) u^k_{i,j}+ \frac{\Delta t}{\alpha}f^{k+1}_{i,j}, (\frac{1}{\alpha}- \Delta t \delta_{yy}) u^{k+1}_{i,j}= \frac{1}{\alpha}u^*_{i,j}- \Delta t \delta_{yy}u^k_{i,j}.
Q_{B R S T}= \oint d z \left\{c ( -{\frac{1}{2}}\partial X \partial X +{\frac{1}{2}}\psi \partial \psi -{\frac{1}{2}}\partial \beta \gamma -{\frac{3}{2}}\beta \partial \gamma ) + c \partial c b +{\frac{1}{2}}\gamma \psi \partial X -{\frac{1}{4}}\gamma^{2}b \right\},
\beta_{0}=\frac{r_{1}^{T}r_{1}}{r_{0}^{T}r_{0}}\approx\frac{[\begin{matrix}-0.2810&0.7492\end{matrix}][\begin{matrix}-0.2810\\ 0.7492\end{matrix}]}{[\begin{matrix}-8&-3\end{matrix}][\begin{matrix}-8\\ -3\end{matrix}]}=0.0088.
p(\lambda) =\Delta_1^{1}f+(\lambda-\mu_1)\Delta_1^{2}f+(\lambda-\mu_2)(\lambda-\mu_1)\Delta_1^{3}f &+(\lambda-\mu_3)(\lambda-\mu_2)(\lambda-\mu_1)\Delta_1^{4}f+\dots &+(\lambda-\mu_{n-1})(\lambda-\mu_{n-2})\dots(\lambda-\mu_1)\Delta_1^{n}f.
\partial_{0}J_{1}- \partial_{x}J_{0}= \frac{1}{2}( \partial_{0}L_{1}\, - \partial_{x}L_{0}) + \frac{1}{2}( \partial_{0}L_{0}- \partial_{x}L_{1}) g + \, \frac{1}{2}( L_{0}^{2}- L_{1}^{2}) g ,
\frac{dx_\mu}{ds\,}\frac{d^2 x_\mu}{ds^2\,}\,=\,0; \qquad\frac{dx_\mu}{ds\,}\frac{d^3 x_\mu}{ds^3\,}\,=\,k_1^2;\qquad\frac{d^2 x_\mu}{ds^2\,}\frac{d^3 x_\mu}{ds^3\,}\,=\,-\,k_1\,\frac{dk_1}{ds\,}\,{.}
\exists !\,f\in L^p({\mathbb{R}}^n)\mbox{such that, for all}t> 0, \; u(t, \cdot)=\Gamma(t,0)f \mbox{in}L^2_{\rm loc}({\mathbb{R}}^n); &u \mbox{is a global weak solution of \eqref{eq1}with}\tilde N(u)\in L^p({\mathbb{R}}^n).
\hat{s}_{1}=[\begin{matrix}0&0&0\\ 0&0&-i\\ 0&i&0\end{matrix}],\hat{s}_{2}=[\begin{matrix}0&0&i\\ 0&0&0\\ -i&0&0\end{matrix}],\hat{s}_{3}=[\begin{matrix}0&-i&0\\i&0&0\\ 0&0&0\end{matrix}],
\nabla_{\rm tr}^*\nabla_{\rm tr}\phi =-\sum_a \nabla_{E_a}\nabla_{E_a}\phi +\nabla_{\kappa_B^\sharp}\phi, A_Y(\phi)=\mathcal{L}_Y\phi -\nabla_Y\phi,Y\in TM, F(\phi)=\sum_{a,b}\theta^a \wedge E_b\lrcorner R^\nabla(E_b, E_a)\phi.
I = \int d \mu{\cal L}= \int d^4 x\widetilde{D}^{i_1 j_1, \ldots ,i_p j_p}_{k_1 l_1, \ldots k_q l_q}L_{i_1 j_1, \ldots, i_p j_p}^{k_1 l_1, \ldots, k_q l_q}