harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left( [ \delta_{\varepsilon_{1}}, [ \delta_{\varepsilon_{2}}, \delta_{\varepsilon_{3}}] ] + [ \delta_{\varepsilon_{3}}, [ \delta_{\varepsilon_{1}}, \delta_{\varepsilon_{2}}] ] + [ \delta_{\varepsilon_{2}}, [ \delta_{\varepsilon_{3}}, \delta_{\varepsilon_{1}}] ] \right) q^{r}= 0 ,
M_{4}(\Gamma_{0}(20))=\mbox{span}_{\mathbb{C}}\left\{Q(q),\,Q(q^2),\,Q(q^4),\,Q(q^5),\,Q(q^{10}),\,Q(q^{20}),\,z^{2},\atop\frac{z^{2}}{u},\,z^{2}v,\,z^{2}k^{2},\,z^{2}kw\right\}.
{\cal Z}=\left<\int{\cal D}\varphi\exp\left\{-A\int d^4x\left[\frac12(\partial_\mu\varphi)^2-2\zeta A^{-1}\cos\left(\frac{\pi\varphi}{\sqrt{c}}\right)+h\varphi\right]\right\}\right>_{\frac12\int d^4xh^2}.
- \int_{\mathbb{R}^2_+}\partial_y^2 g_{n}\, \langle y\rangle^{2\ell'}g_{n}dx dy = \|\partial_y g_{n}\|_{L^2_{\ell'}(\mathbb{R}^2_+)}^2 + \int_{\mathbb{R}^2_+}\partial_y g_{n}(\langle y\rangle^{2\ell'})' g_{n}dx dy & \ge \frac{3}{4}\| \partial_y g_{n}\|_{L^2_{\ell'}(\mathbb{R}^2_+)}^2 - 4 \| g_{n}\|_{L^2(\mathbb{R}^2_+)}^2,
\begin{aligned}I_{35}\lesssim ~{}\iint_{D_t}\left( (1+\left|\underline{u}\right|^2)^{1+\delta}| \underline{L}\partial_x\tilde{\Lambda}|\right)\left( |\partial_x\tilde{\Lambda}||L\phi|| \underline{L}\phi|+|L\partial_x\phi ||\underline{L}\phi|+ |L\phi| |\underline{L}\partial_x\phi|\right) =: ~{}I_{35,1}+I_{35,2}+I_{35,3}.\end{aligned}
D_{P}= \beta \left( \left| k \right|^{2}- 4 p_{0}^{2}\right) \left( \left| k \right|^{2}- 4 \xi p_{0}^{2}\right) \quad , \quad \widetilde{D}_{P}= 2 \beta \left( \left| k \right|^{2}- 4 \xi p_{0}^{2}\right)
d\left(\sum_{j=1}^{n-1}\phi_j(x)i^*_{\partial M}\left(\star d\tau\wedge dx^j\right)\right)=(-1)^{n-2}\star_{\partial M}\star_{\partial M}\sum_{j=1}^{n-1}d\left(\phi_j(x)\cdot \star_{\partial M}dx^j\right)=
\lambda_2=\Gamma\frac{\int_{\Omega}{\log{(p)}dr}\int_{\Omega}{h^2dr}-\int_{\Omega}{hdr}\int_{\Omega}{\log{(p)}hdr}}{\mathrm{mes}(\Omega)\int_{\Omega}{h^2dr}-\left(\int_{\Omega}{hdr}\right)^2}.
0 =\|\sum_{i\in\sigma}\Lambda^*g_i+\sum_{i\in\sigma^c}\Gamma^*g_i\|^2=\lim_{j\rightarrow\infty}\|\sum_{i\in\sigma}\Lambda^*g_i+\sum_{i\in\sigma^c}\Gamma^*g_i\|^2 &\ge \lim_{j\rightarrow\infty}A\big(\sum_{i \in \sigma_j}|a_i|^2+\sum_{i \in \sigma^c_j}|a_i|^2\big)
\mathrm{I}(\xi_{1}^{x},\mathcal{T}_{x}) =\int_{G^{\mathbb{N}}}\log\frac{d\left(\omega_{1}\lambda\right)_{x}}{d\lambda_{x}}(\psi_{x}(\omega))d\mathbb{P}_{\mu}^{\pi(x)}(\omega) =\int_{G}\int_{Z_{x}}\log\frac{d(g\lambda)_{x}}{d\lambda_{x}}(z)d(g\lambda)_{x}(z)\varphi_{g}(\pi(x))d\mu(g).
\nabla^{\Gamma_0}_{v_m}s =\left[p,\mathbf{d}f(p)(u_p)+[\Gamma_0(u_p),f(p)]\right]_G, \nabla^{\Gamma}_{v_m}s =\left[p,\mathbf{d}f(p)(u_p)+[\Gamma(u_p),f(p)]\right]_G, [\sigma_2,s](v_m) =\left[p,\left[\Gamma_0\left(\operatorname{Hor}^\Gamma_p(v_m)\right),f(p)\right]\right]_G,
\left\{\widehat{A}(TX){\rm exp}(\frac{c}{2}){\rm ch}(\widetilde{T_CM}-\widetilde{L_C}+8+W_i+W_j)\right\}^{(14)}&=A\left\{e^{\frac{1}{24}A}\widehat{A}(TX){\rm exp}(\frac{c}{2})-\frac{e^{\frac{1}{24}A}-1}{A}\widehat{A}(TX){\rm exp}(\frac{c}{2}){\rm ch}(\widetilde{T_CM}-\widetilde{L_C}+8+W_i+W_j)\right\}^{(10)}.
B(w_n^j\cos(n \cdot x),\,w_n^j\cos(n \cdot x)) =-\Pi(w_n^j\cdot n)\cos(n \cdot x)\sin(n \cdot x)w_n^j=0, B(w_n^j\sin(n \cdot x),\,w_m^j\sin(n \cdot x)) =\Pi(w_n^j\cdot n)\sin(n \cdot x)\cos(n \cdot x)w_n^j=0,
\frac{e^{-(\sqrt{\mu}-\sqrt{\lambda})^{2}}}{(\lambda+\mu)^{2}}-\frac{e^{-(\lambda+\mu)}}{2\sqrt{\lambda\mu}}-\frac{e^{-(\lambda+\mu)}}{4\lambda\mu}\le P(X-Y\ge0)\le e^{-(\sqrt{\mu}-\sqrt{\lambda})^{2}}
-\lvert r\rvert + \lvert\log(1 + r)\rvert \leq \lvert r - \log(1 + r)\rvert = \biggl\lvert\frac{r^2}{2}- \frac{r^3}{3}+ \frac{r^4}{4}- \dotsb\biggr\rvert\leq \frac{\lvert r\rvert^2}{2}+ \frac{\lvert r\rvert^3}{2}+ \frac{\lvert r\rvert^4}{2}+ \dotsb= \frac{\lvert r\rvert^2}{2}\frac{1}{1 - \lvert r\rvert};
\lambda_\mu z_\nu z_\mu - \lambda_\nu z_\mu z_\nu = (-1)^{\mu' + \nu'}\left( \lambda_{\nu'}z_{\mu'}z_{\nu'}- \lambda_{\mu'}z_{\nu'}z_{\mu'}\right) &\lambda_\mu z_\mu z_\nu - \lambda_\nu z_\nu z_\mu = (-1)^{\mu' + \nu'}\left( \lambda_{\mu'}z_{\mu'}z_{\nu'}- \lambda_{\nu'}z_{\nu'}z_{\mu'}\right) .
\Bigg|\int_{S_{k,0,T}}\int_{(\mathbb{R}^d)^{k+1}}\Biggl(\prod_{i=1}^kh(y_i) \Biggr)f(z)J_{2k}dz \prod_{j=1}^k dy_j \prod_{i=1}^k ds_{i}\Bigg| &\leq3 C_T \frac{1}{(k-1)!}\|h\|^k\|f\|T^{k}\frac{\log n}{n}.
\widetilde{\Delta}_{q}^{\alpha}\,b(x,\xi) := v_{\xi}^{-1}(x)\int_{\Omega}\widetilde{K}_{B_{q^{\alpha}}}(x,y)v_{\xi}(y)dy{}& = v_{\xi}^{-1}(x)\int_{\Omega}\widetilde{q}^{\alpha}(x,y)\widetilde{K}_{B}(x,y)v_{\xi}(y)dy,
\sum\limits_{p+q=k}h^{p,q}_{(2)}(X_j) = \sum\limits_{p+q=k}{\rm dim}_\mathbf{G_j}(\mathcal{H}^{p,q}_{(2)}(X_j)) \rightarrow\sum\limits_{p+q=k}{\rm dim}_\mathbf{G}(\mathcal{H}^{p,q}_{(2)}(\widetilde{X})) = \sum\limits_{p+q=k}h^{p,q}_{(2)}(\widetilde{X}),
G(z) = z^{2\gamma-p}\left(\sum_{h\geq 0}z^h \sum_{s=0}^{\lfloor\frac{h}{2}\rfloor}N_{\mathcal L}(h-2s,\ell) \binom{s+n-2}{n-2}- \Upsilon_{p-2\gamma-1}^{(\ell)}(z) \right) &= z^{2\gamma-p}\left(\frac{\vartheta_{\mathcal L}^{(\ell)}(z)}{(1-z^2)^{n-1}}-\Upsilon_{p-2\gamma-1}^{(\ell)}(z) \right).
m_{1/2,\lambda}(x|\mu) = \frac{\lambda^\mu}{2\Gamma(\mu)\sqrt{\pi x^3}}\int_0^\infty y^\mu \, e^{-y^2/4x-\lambda y}\, dy = \sqrt{\frac{2}{\pi}}\, \mu\lambda^\mu\, (2x)^{\mu/2-1}\, e^{\lambda^2 x/2}\, D_{-\mu-1}(\lambda\sqrt{2x}\,)
c_1 := \max_{h\in(0,1]}\,\,\,\, 2 h - h^2 \frac{-\kappa}{1-\kappa}= 2 - \frac{-\kappa}{1-\kappa}c_2 := \max_{h\in[1,\bar{h}(\kappa)]}2 h - h^3 \frac{-\kappa}{2-(1+\kappa)h}\geq 2 - \frac{\kappa}{2-(1+\kappa)}= c_1
\frac{\partial\overline{u_{i}}}{\partial t}+\overline{u_{j}}\frac{\partial\overline{u_{i}}}{\partial x_{j}}=-\frac{1}{\rho}\frac{\partial\overline{p}}{\partial x_{i}}+\nu\frac{\partial^{2}\overline{u_{i}}}{\partial x_{j}\partial x_{j}}-\frac{\partial\tau_{ij}}{\partial x_{j}}.
\gamma_{x}=\left[\begin{array}{cc}0 0 1 0\end{array}\right]\otimes 1,\;\;\gamma_{y}=\left[\begin{array}{cc}1 0 0 -1\end{array}\right]\otimes\left[\begin{array}{cc}0 0 1 0\end{array}\right], \;\;\gamma_{\bar{x}}=\gamma_x^T,\;\; \gamma_{\bar{y}}=\gamma_y^T
\sum_{i=1}^{q}\int_0^T \alpha (t) \theta_{i}(t) \frac{d}{dt}f_{i}(u(t)) dt \leq \liminf_n \sum_{i=1}^{q}\int_0^T \alpha (t) (T\circ \theta_{n})_i(t) \frac{d}{dt}f_{i}(u(t)) dt.
\begin{aligned}h(c(y_k)+\nabla c(y_k)(\hat x-y_{k})) \leq (h\circ c)(\hat x) +\frac{r}{2}\|\hat x-y_k\|^2 &\leq a_kh(c(x))+(1-a_k)h(c(x_{k-1})) &+\rho a_k(1-a_k)\|x-x_{k-1}\|^2+\frac{ra_k^2}{2}\|x-v_{k-1}\|^2.\end{aligned}
h_{1i}(1)-h_{2i}(1)=\sum_{j=1}^2 \frac{1}{j!}\left(h_{1i}^{(j)}(0)-h_{2i}^{(j)}(0)\right)+\frac{1}{3!}\left(h_{1i}^{(3)}(\tau_1)-h_{2i}^{(3)}(\tau_2)\right),
\varsigma(m_{N_n}, m_{N_{n + 1}}, \Phi(x), \Phi(y)) = \sum_{N_n \leq i < N_{n + 1}}\varsigma(m_i, m_{i+ 1}, \Phi(x), \Phi(y)) > \sum_{N_n \leq i < n_{n + 1}}\left(\frac{1}{i + 1}- 2^{-i - 2}\right) > 3^{n + 2}
\left|\varphi^d_{m_n}(\lambda;x) - 1 - \sum_{k=1}^{d}\lambda^kG^k_{m_n}(x) \right| \leq (-1)^{d}\lambda^{d+1}\left(\int_{0}^{x}G^{d}_{m_n}(y)dm_n(y)\right)x\mathrm{e}^{\lambda (m_n \bullet s)(x)},
[A\xi_1\frac{\partial}{\partial \xi_1}\wedge \frac{\partial}{\partial u_1}, c_0\frac{1}{u_1}\frac{\partial}{\partial u_1}+c_1\frac{\xi_1}{u_1}\frac{\partial}{\partial \xi_1}]=-Ac_0\frac{\xi_1}{u_1^2}\frac{\partial}{\partial \xi_1}\wedge \frac{\partial}{\partial u_1}\equiv 0, \,\,\,\,\,c_0,c_1\in \mathbb{C}.
\|h-S_{y,\tau}S_{x,\tau}^{-1}h\| =\left\|\sum_{j \in \mathbb{J}}\langle h,x_j \rangle S_{x,\tau}^{-1}\tau_j-\sum_{j \in \mathbb{J}}\langle h,y_j \rangle S_{x,\tau}^{-1}\tau_j \right\|=\left\|\sum_{j \in \mathbb{J}}\langle h,x_j-y_j \rangle S_{x,\tau}^{-1}\tau_j\right\| &\leq\sum_{j \in \mathbb{J}}\|h\|\|x_j-y_j\|\| S_{x,\tau}^{-1}\tau_j\|=\beta \|h\|.
U^{-1}(\varphi )m^{*}U(\varphi )e_{a}=U^{-1}(\varphi )(\eta (\varphi ,a){\overline{m(\varphi (a))}}e_{\varphi (a)})=\eta (\varphi ,a){\overline{m(\varphi (a))}}U^{-1}(\varphi )e_{\varphi (a)}.
x-\sigma_{\mathrm{res}(\alpha)}(x) = (x- \sigma_{\beta}(x))+ \sigma_{\beta}(x- \sigma_{\alpha}(x)) + (\sigma_{\beta}\circ \sigma_{\alpha})(x- \sigma_{\beta}(x)) =b\beta+a(\alpha+\beta)+b\alpha =(a+b)(\alpha+\beta),
\left\| fl\left(B^{(k+1)}\right) - B^{(k+1)}\right\|_{1,2}\le \left\| fl \left(B^{(1)}\right) - B^{(1)}\right\|_{1,2}\, (1+u)^{k}& + u \, \sum_{i=1}^{k}\left\| B^{(i)}\right\|_{1,2}\, (1+u)^{k-i}+ 3 \, u \, \sum_{i = 1}^{k}\left\| B^{(i)}\right\|_{1,2}\left\| L_i^T \right\|_1 \, (1+u)^{k-i}+ O(u^2).
\#\mathcal{A}'_q =\#\{a\in\mathcal{A}:\,q|a,\,(a,10)=1\}&=\sum_{\substack{a\in\mathcal{A}q|a}}\sum_{d|(10,a)}\mu(d) &=\sum_{d|10}\mu(d)\sum_{a\in\mathcal{A}}\Bigl(\frac{1}{d q}\sum_{0\le b<d q}e\Bigl(\frac{a b}{d q}\Bigr)\Bigr) &=\sum_{d|10}\frac{\mu(d)}{d q}\sum_{0\le b< d q}S_{\mathcal{A}}\Bigl(\frac{b}{d q}\Bigr).
\hat{\Psi}_{k,n,2}(u) =\frac{1}{f_0(x_0)}\left(\hat{f}_n(u)-\sum_{j=0}^{k-1}\frac{f_0^{(j)}(x_0)}{j!}(u-x_0)^j\right) &\quad+\frac{r}{g_0(x_0)}\left(\hat{g}_n(u)-g_0(x_0)-g_0'(x_0)(u-x_0)\right).
S^{a b}( x ) S^{T b c}( x ) = S^{a b}( x ) S^{c b}( x ) = \frac{1}{4}\tau_{i j}^{b}\tau_{k l}^{b}( U ( x ) \tau^{a}U^{\dagger}( x ) )_{i j}( U ( x ) \tau^{c}U^{\dagger}( x ) )_{k l}= \frac{1}{2}t r [ \tau^{a}\tau^{c}] = \delta^{a c}
S^{env,0}_{s,t}= \ln(\beta(t,q_t)/\beta(s,q_s))-\int_s^t (\beta^{-1}\partial_r\beta)(r,q_r)dr &+(\beta V)(s,q_s)-(\beta V)(t,q_t)+\int_s^t \partial_r(\beta V)(r,q_r) dr &+\int_s^t ( \beta \tilde F+V\nabla_q\beta)(r,q_r)\circ dq_r.
\mathbf{v}^T \mathbf{S}^\mathrm{c}\mathbf{v}\geq \alpha C^{-2}C_\mathrm{P}^{-2}\|v\|_{L^2(D)}^2 = \alpha C^{-2}C_\mathrm{P}^{-2}\mathbf{v}^T\mathbf{M}\mathbf{v}\geq cH^{d}\|\mathbf{v}\|_2^2.
x = \phantom{.}e_{1,2}(r_{1,2}) e_{2,3}(r_{2,3}) \cdot \ldots \cdot e_{n-1,n}(r_{n-1,n}) \cdot \cdot e_{1,3}(r_{1,3}) \cdot \ldots \cdot e_{n-2,n}(r_{n-2,n}) \cdot \ldots \ldots \cdot e_{1,n}(r_{1,n}).
\frac{\lambda_0(w)\sum_{k,m}B_{k,m}-\sum_{k,m}A_{k,m}(w)}{(\sum_{k,m}B_{k,m})-\frac{A_{0,1}(w)}{\lambda_0(w)}}=\frac{\sum_{k,m}B_{k,m}(\lambda_0(w)-\lambda_k(w))}{\sum_{k,m}B_{k,m}}& =\lambda_0(w)-\lambda_{h(l)}(w) & \to \lambda_0(w)-\lambda_\ast(w),
c = \frac 1{2k_0}h_1', \hskip 2ex d = -\frac1{2k_0}h_2', \hskip 2ex e = \frac1{k_0}\left(\frac1{2k_0}h_2'\right)'+\frac1{2k_0}(k_2h_1-k_1h_2).
V^{(n+1)}(t) = \int_0^t f(t - s) V^{(n+1)}(s) + \int_t^T g(t -s) V^{(n+1)}(s) \, ds + \sum_{k=0}^{n}(-1)^{n-k+1}g^{(n-k)}(T-t) V^{(k)}(T).
M_2 = \frac{\left(1+\rho^2\right)^2}{\sqrt{1+3\rho^2 + 4 \rho^4}\sqrt{1-\rho^2}}\le \frac{\left(1+\rho^2\right)^2}{1+ \rho^2 -\rho^4}\le 1 + \rho^2 + 2\rho^4,
\varphi(\xi)=\frac{C_n}{(1+\xi^2)^{n/2+(1-\beta)/2}},~~u=\frac 1{y^n}\varphi\left(\frac ry\right)=\frac{C_ny^{1-\beta}}{(y^2+r^2)^{n/2+(1-\beta)/2}}.
F(x) = \frac{1}{1+\theta}\log^{\theta+1}\left(x+ e^{\theta +1}\right) - \frac{1}{1+\theta}\log^{\theta+1}\left( 1+ e^{\theta + 1}\right) + \int_{\log \left( 1+ e^{\theta + 1}\right)}^{\log \left(x+ e^{\theta + 1}\right)}\frac{w^\theta}{e^w-1}dw, x > 1.
\int_{0}^{\infty}sn(t)e^{-tz}dt=\frac{1}{1^{2}(1+k^{2})+z^{2}-}\frac{1\cdot2^{2}\cdot3k^{2}}{3^{2}(1+k^{2})+z^{2}-}\frac{3\cdot4^{2}\cdot5k^{2}}{5^{2}(1+k^{2})+z^{2}-}\cdot\cdot\cdot
(\tilde{a}_{ij})=\frac{1}{\tilde{\lambda}^2}\left(\begin{matrix}1-\mu^2x^2 -\mu^2xy 0 -\mu^2xy 1-\mu^2y^2 0 0 0 \tilde{\lambda}\end{matrix}\right),(\tilde{b}_i)=-\frac{1}{\tilde{\lambda}}\widetilde{W},
I ( x ) = \frac{1}{2 \pi}\, \Bigl ( g_{0}\,{\sqcap \! \! \! \! \sqcup}\, \tilde{\phi}- 2 i e g_{0}\, A_{\mu}\partial_{\mu}\tilde{\phi}+ e \, \epsilon_{\mu \nu}\, \partial_{\mu}A_{\nu}\Bigr ) \, .
{\cal D}D_{n , \mp 1 / 2}^{( j )}D_{. , \pm 1 / 2}^{( 1 / 2 )}= - [ D_{n , n^{\prime}}^{( j )}( J_{\alpha}^{( j )})_{n^{\prime}, \mp 1 / 2}] D_{. , \mu^{\prime}}^{( 1 / 2 )}[ \frac{1}{2}\sigma_{3}, [ \frac{1}{2}\sigma_{3}, \sigma_{\alpha}] ]_{\mu^{\prime}, \pm 1 / 2}
{\rm lim}_{L\rightarrow +\infty}\frac{k_{\gamma,\Sigma_1}^{L}}{\sqrt{L}}=\frac{|\frac{d}{dt}(\omega(\dot{\gamma}(t)))|}{\left[\overline{q}{\dot{\gamma}_3}-\frac{\sqrt{2}}{2}\overline{p}\left(-e^{-\gamma_3}\dot{\gamma_1}+e^{\gamma_3}\dot{\gamma_2}\right)\right]^2},~~if ~~\omega(\dot{\gamma}(t))= 0~~and~~\frac{d}{dt}(\omega(\dot{\gamma}(t)))\neq 0.
\int_{\mathcal{V}\times \mathcal{V}}\varphi_\lambda \, d \widetilde{\pi}= \int_{\mathcal{V}}g \, d \gamma - \int_{\mathcal{S}_{1}\times \mathcal{S}_{1}}\lambda_1 c_{1}\, d \nu_{1}-\int_{\mathcal{S}_{2}\times \mathcal{S}_{2}}\lambda_2 c_{2}\, d \nu_{2}< \infty,
h(x) \le \Bigl(g_4(x)+\frac{2}{6^5}(2^5x-13)(101-3^5x) &+\frac{2}{6^6}(2^6x-26)(303-3^6x)\Bigr)\Bigr|_{x=\frac{27}{65}}+ \frac1{10\cdot 6^6}- \eta &=-\frac{1906349521544493873261549971920349}{425623699615692261466802423622110256000}<0,
K^t =e^{-itH/\hbar}=e^{-\frac{it}{\hbar}(\frac{P^2}{2m}+\frac{1}{2}m\omega^2Q^2)}&=e^{-\frac{im\omega\tan(t\omega/2)}{2\hbar}Q^2}e^{-\frac{i\sin(t\omega)}{2\omega m\hbar}P^2}e^{-\frac{im\omega\tan(t\omega/2)}{2\hbar}Q^2}.
\begin{array}{l}\int_{\Omega}(\nabla u\nabla ((\theta_{\delta}\circ u)\varphi )+u(\theta_{\delta}\circ u)\varphi )\,\mathrm{d}x=\lambda \int_{\Omega}|u^{+}-\phi_{1}|(\theta_{\delta}\circ u)\varphi \,\mathrm{d}x.\end{array}
\left[\frac{\frac{2\epsilon Z\alpha}{\hbar c}}{1+2\sqrt{\left(\ell+\frac{1}{2}\right)-(Z\alpha)^2}+2n}\right]^2=\frac{m_0^2c^4-\epsilon^2}{\hbar^2c^2}\Leftrightarrow \frac{\epsilon Z\alpha}{m_0^2c^4-\epsilon^2}=n'+\frac{1}{2}+\sqrt{\left(\ell+\frac{1}{2}\right)-(Z\alpha)^2},
C_{\Phi ,\mathcal{J}}:=\max_{N\geq 0}\left\{-A_{\Phi}N^{P}+B_{\Phi ,\mathcal{J}}^{\prime}N^{p}\right\}\leq \left( A_{\Phi}\right)^{-\frac{p}{P-p}}\left( B_{\Phi ,\mathcal{J}}^{\prime}\right)^{\frac{P}{P-p}}.
Dv(\hat x) = \frac{1}{\epsilon}a(z)(\hat x-\hat y) + \varphi(z,v(z))\beta(z) - \delta Dd(\hat x) + 2\delta (\hat x-z), Du(\hat y) = \frac{1}{\epsilon}a(z)(\hat x-\hat y) + \varphi(z,v(z))\beta(z) + \delta Dd(\hat y),
U(t)x := \begin{cases}\frac{1}{\Gamma(\alpha)}\left( \frac{t}{2}\right)^{2\alpha}\int\limits_0^\infty r^{-\alpha}\exp^{-\frac{t^2}{4r}}T(r) x \, \frac{dr}{r}t > 0, x t = 0. \end{cases}
z_{b_1}+ \frac{-w_{21}}{W}z'_{21}+ \frac{w_{22}}{W}z'_{11}= 0 z_{w_2}= z'_{21}= z'_{22}z_{b_2}+ \frac{w_{11}}{W}z'_{22}+ \frac{-w_{12}}{W}z'_{12}= 0 z_{w_1}= z'_{11}= z'_{12}
\begin{aligned}f\left( M+m-\bar{a}\right) =f\left( \sum\limits_{i=1}^{n}{{{w}_{i}}\left( M+m-\frac{\bar{a}+{{a}_{i}}}{2}\right)}\right) \le \sum\limits_{i=1}^{n}{{{w}_{i}}f\left( M+m-\frac{\bar{a}+{{a}_{i}}}{2}\right)}\end{aligned}
\phantom{={}}\int_0^1 \frac{f(y)}{(1-xy)^m}y^k\, dy &= \int_0^1 \frac{f(y)(1-y)^{a}}{(1-xy)^{b}}\frac{(1-y)^{-a}}{(1-xy)^{m-b}}y^k dy \le \left[ \int_0^1 \frac{|f(y)|^p (1-y)^{ap}}{(1-xy)^{bp}}y^k dy\right]^{1/p}\times \left[ \int_0^1 \frac{(1-y)^{-aq}}{(1-xy)^{(m-b)q}}y^k dy\right]^{1/q},
\vert\mathcal R_1^{(d)}(\rho_z) \vert\leq \begin{cases}C|\Lambda|\rho_z^{2}\delta^{2}K_{\ell}^{-1}, d=2, C|\Lambda| \rho_z^{2}a \big(\rho_z a^{3}\big)^{\frac 1 2}\log(\rho_z) K_{\ell}^{-1}, d=3.\end{cases}
[\begin{matrix}0&2\\ 0&1\end{matrix}]=[\begin{matrix}1&1\\ 0&1\end{matrix}][\begin{matrix}0&1\\ 0&1\end{matrix}]\ne[\begin{matrix}0&1\\ 0&1\end{matrix}][\begin{matrix}1&1\\ 0&1\end{matrix}]=[\begin{matrix}0&1\\ 0&1\end{matrix}]
\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}.
\frac{\rho u_{0}^{2}}{L}(\frac{\partial}{\partial\tilde{t}}+\tilde{u}\cdot\tilde{\nabla})\tilde{u}=-\frac{\eta u_{0}}{L^{2}}(\tilde{\nabla}\tilde{p}+\tilde{\nabla}^{2}\tilde{u}),\tilde{\nabla}\cdot\tilde{u}=0
(\varphi^f T^{P_1}_w)(X) = \varphi^f(T^{P_1}_wX) & = \varphi(f(T^{P_1}_w)f(X)) & = (-1)^{\ell(w) - \ell_{P_1}(w)}\varphi (T_w^{P_1}f(X)) & = (\varphi T_w^{P_1})(f(X)) = (\varphi T_w^{P_1})^f(X).
m=1 \texttt{c[k]}= (k+1)\texttt{c[k+1]}(1+\eta_2) m=2 \texttt{c[k]}= (k+1)(k+2)\texttt{c[k+2]}(1+\eta_3) \dots m \texttt{c[k]}= (k+1)(k+2)\cdots (k+m)\texttt{c[k+m]}(1+\eta_{m+1})
T_{N}=\sum^N_{n=1}\sum^{\infty}_{j=1}\Big( \mathbb{E}\big[K(X_n+a_j\widetilde{\varepsilon}_{n-j})|\varepsilon_{n-j}\big]-\mathbb{E}\big[K(X_n+a_j\widetilde{\varepsilon}_{n-j})\big]\Big),
k_{s}^n = \nu_{s}^*( (P_s^*)^n 1) = \mathbb{E}\int_{\mathcal{S}}\frac{1}{\| g_n^{\mathrm T}\ldots g_1^{\mathrm T}v \|^{-s}}\nu_{s}^*(dv) \geq c_s \mathbb{E}\|g_1 \ldots g_n\|^s= c_s \mathbb{E}\|g_n \ldots g_1\|^s,
\overline{{v}}^{s , \pm}( \mathbf{p}) v^{r , \mp}( \mathbf{p}) = \varepsilon_{s}\delta_{s r}\frac m{p_{0}}, \left( \overline{{v}}^{s , \pm}( \mathbf{p}) = v^{* s , \pm}( \mathbf{p}) \Gamma_{4}\overline{{\Gamma}}_{4}\right) ,
G(x,y) =\frac{\Gamma(\frac{d-2}{2})e^{i(h(x)-h(y))\cdot k_0}}{2\pi^{d/2}\sqrt{\det H}|H^{-1/2}(h(x)-h(y))|^{d-2}}\cdot \frac{\phi_{k_0}(x)\overline{\phi_{k_0}(y)}}{\|\phi_{k_0}\|_{L^2(M)}^2}& \times \left(1+O\left(d_X(x,y)^{-1}\right)\right)+O\left(d_{X}(x,y)^{1-d}\right),
\widehat{I}^{[\beta]}= \left.\left\langle \frac{x^{\beta}}{m}~\right|~ m\in \mathcal{G}(I)\right\rangle = \left\langle \frac{x^{\beta}}{m_1}, \frac{x^{\beta}}{m_2}, \ldots, \frac{x^{\beta}}{m_p}\right\rangle~~\mbox{where $x^\beta = x_1^{\beta_1}\cdots x_n^{\beta_n}$.}
v_{4,s}(t,r) = \frac{-1}{2\pi}\int_{B_{s-t}(0)}\frac{dA(y)}{\sqrt{(s-t)^{2}-|y|^{2}}}\frac{v_{4,c}(s,|x+y|)\left((x+y)\cdot \widehat{x}\right)}{|x+y|}
\begin{array}{c}{\bf P}_A=\bar p_A\ ,\quad{\bf M}^{AB}=x^A\bar p^B-x^B\bar p^A+S^{AB}\ , S^{AB}=2i\{Z_i^A\partial^{iB}-\overline{Z}_i^A\overline{\partial}^{iB}-(\Sigma^{AB})_b^{\a}(\psi_a\partial^b+\overline{\psi}_a\overline{\partial}^b)\}\Phi\end{array}
l_a^+ = \langle e_1-a\,f_2, \e_2+a\,f_1 \rangle, l_b^- = \langle e_1+b\,e_2, \ -f_2 +b\,f_1 \rangle, l_a^+ \cap l_b^- = \langle \phi_{ab}\rangle, \ \ \mbox{for}\phi_{ab}\coloneqq e_1+b\,e_2-a\,f_2+ab\,f_1,
:{\hat{\psi}_{m}}^{\dagger}\hat{\psi}_{n}: ~ = ~{\hat{\psi}_{m}}^{\dagger}\hat{\psi}_{n}{}~ ~ \mathrm{i f}~ n > 0 ~ ~ :{\hat{\psi}_{m}}^{\dagger}\hat{\psi}_{n}: ~ = ~ - \hat{\psi}_{n}{\hat{\psi}_{m}}^{\dagger}~ ~ \mathrm{i f}~ n \le 0
\Delta_G =\left\{\widehat{x}_i-\widehat{x}_{i+1}\,, 1\leq i\leq n-1\,,\; 2\,\widehat{x}_{n}\right\}\,, \intertext{and}\Delta_K =\left\{\widehat{x}_i-\widehat{x}_{i+1}\,, 1\leq i\leq n-1\right\}\,,
T_n = \left\{\left(\frac{1}{p_1}, \frac{1}{q_1}\right) \in \left(0,\frac{1}{2}\right] \times \left(0, \frac{1}{2}\right) \left| \frac{2}{p_1}+ \frac{n-1}{q_1}\geq \frac{n-1}{2}\right.\right\}.
\begin{aligned}\mathbb{E}\int_0^\infty e^{-\beta t}\Big(|\hat{m}|^2+|\hat{\tilde{m}}|^2+||\hat{n}||^2+|\hat{q}|^2\Big)dt\leq \frac{1}{\delta}\mathbb{E}\int_0^\infty e^{-\beta t}|\kappa^i_t-\kappa^j_t|^2dt\xrightarrow{i,j\rightarrow\infty}0.\end{aligned}
\begin{aligned}\leq \mathbb{E}\int_t^\infty e^{-\beta s}\big(e^{-K_0s}|y_n-y|^2+e^{-K_1s}|z_n-z|^2+e^{-K_2s}|\tilde{z}_n-\tilde{z}|^2 &+e^{-K_3s}||\gamma_n-\gamma||^2\big)ds\rightarrow0,\ \n\rightarrow\infty,\end{aligned}
\begin{cases}u_t+(-\Delta)^{d}u=|x|^{\alpha}|u|^{p}+\zeta(t){\mathbf w}(x) \ \quad\mbox{for}(x,t)\in\mathbb{R}^{N}\times(0,\infty), u(x,0)=u_0(x),x\in \mathbb{R}^{N},\end{cases}
z^q_{\mathfrak{m}}(\mathcal{O}, n)^{pc}= z^q_{\mathfrak{m}}(\mathcal{O}, n) \mbox{and}z^q_{\widehat{\mathfrak{m}}}(\widehat{\mathcal{O}}, n)^{pc}= z^q_{\widehat{\mathfrak{m}}}(\widehat{\mathcal{O}}, n).
({\gamma}^{\mu}i{\partial}_{\mu}-m_2){\psi}_2(x,t)- \frac{1}{2}e_2 b(t){\gamma}^1{\psi}_2(x,t)+ e_2 \int_{0}^{\rm L}dy D(x,y|{\rm L}) J^0(y,t){\gamma}^0{\psi}_2(x,t) =0.
\frac{\partial^k u^{\alpha}}{\partial (x^1)^k}=f^{\alpha}\left(x,n,[u]_{x^1},\left[\frac{\partial^1 u}{\partial (x^1)^1}\right]_{x^1},\left[\frac{\partial^2 u}{\partial (x^1)^2}\right]_{x^1},\ldots,\left[\frac{\partial^{k-1}u}{\partial (x^1)^{k-1}}\right]_{x^1}\right),
\psi_{k+1}^*\colon \Omega_{k+1}\to \Omega_{k,k+1},\mspace{20mu}\begin{cases}x_{i,k+1}\mapsto w_{i,k}+ \xi_{k+1}w_{i-1,k}, begin{align*}1.5ex]s_{i,k+1}\mapsto \sigma_{i,k+1}. \end{cases}
- \mathrm{e x p}\left[ 2 \pi i \frac{P}{Q}( h_{n_{2}, e_{2}}- h_{e_{1}, n_{1}}) \right] ( \delta_{- e_{2}, e_{1}+ p e}\delta_{1 - n_{2}, n_{1}+ p n}- \delta_{- e_{2}, e_{1}+ p e}\delta_{1 - n_{2}, n_{1}+ p n - p})
\overline{Q}_{k,a}(x;q) =\overline{Q}_{k,a-2}(x;q)+(xq)^{a-1}\overline{Q}_{k,k-a+1}(xq;q)+(xq)^{a-2}\overline{Q}_{k,k-a+2}(xq;q) &\qquad+(xq)^a\overline{Q}_{k,k-a}(xq;q)+(xq)^{a-1}\overline{Q}_{k,k-a+1}(xq;q) &\quad=\overline{Q}_{k,0}(x;q)+\sum_{i=1}^a(xq)^i\overline{Q}_{k,k-i}(xq;q)+\sum_{i=0}^{a-1}(xq)^i\overline{Q}_{k,k-i}(xq;q).
\frac{\partial a_{4}}{\partial c}|_{c=0}=a_{3}[\frac{(3+c+\alpha)(3+c+\beta)}{(4+c)(1+c)}(\frac{1}{\alpha+3+c}+\frac{1}{\beta+3+c}-\frac{1}{4+c}-\frac{1}{1+c})]_{c=0}
\tilde{\mu}(\theta_t):=\left[\frac{\sum_i (\theta_t^i\delta^i)^{\frac{1}{\gamma}}}{\sum_i (\theta_t^i)^{\frac{1}{\gamma}}}\right]^\gamma=\left[\textstyle\sum_i a^i(\theta_t^i)\,(\delta^i)^{\frac{1}{\gamma}}\right]^\gamma.
h(1) = \| \log|1-e(\varphi)| \|_{L^{1}_\varphi}= -2 \int_{0}^{1}\min( \log|1-e(\varphi)|,0)\, d\varphi &= -2 \int_{-\frac16}^{\frac16}\log|1-e(\varphi)| \, d\varphi = -4 \int_{0}^{\frac16}\log(2\sin(\pi \varphi))\, d\varphi < 13/20.
\begin{array}{rcl}e=\displaystyle\frac{1}{q-q^{-1}}x_1(a_1Z-a_1^{-1}Z^{-1})X~, begin{align*}3mm] f=\displaystyle\frac{1}{q-q^{-1}}x_1^{-1}(a_2Z^{-1}-a_2^{-1}Z)X^{-1}~, begin{align*}3mm] t=\displaystyle\frac{a_1}{a_2}Z^2~,\end{array}
L_{1 1}^{( \pm )}= q^{\mp \left( \begin{array}{c c}{{1}}&{{0}}\\{{0}}&{{0}}\\ \end{array}\right)}, L_{2 2}^{( \pm )}= q^{\mp \left( \begin{array}{c c}{{0}}&{{0}}\\{{0}}&{{1}}\\ \end{array}\right)},
a_{1}=1, b_{0}=0 , a_{3k+0}=-2(4k - 1)^2kx^3, b_{3k+0}= -16k^2x^2 - 4kx + 1 , a_{3k+1}=-2(4k + 1)^2kx^3, b_{3k+1}= -2(3k + 1)x + 1 , a_{3k+2}=-(2k + 1)^2x^2; b_{3k+2}= -2(3k + 2)x + 1 .
\begin{aligned}P_c (\Psi_t)P_c^T(\Psi_t)(x) =\begin{bmatrix}I_n+O(t) \Upsilon_c^T+ O(t) \Upsilon_c +O(t) \frac{1}{2t}\cdot \begin{pmatrix}\begin{bmatrix}I_{\frac{n(n-1)}{2}}0 0 3 \cdot \Xi(\frac{1}{3})-\frac{n+2}{n}\cdot J_n\end{bmatrix}+O(t)\end{pmatrix}+\Gamma_c(x)\end{bmatrix}, \end{aligned}
\frac{d}{dt}T(t)=e^{-bt}\left( \frac{ae^{bt}}{\beta + Ce^{bt}}- \frac{a}{C}\log\left(\frac{\beta+Ce^{bt}}{\beta+Ce^{bt_1}}\right) - b \left(T(t_1)-T_e+\frac{a}{b}\right)e^{bt_1}\right)
E_i(P,X)= ( ( M^s(\widetilde{Y}_i) - \overline{M}^s(F) )\nabla u^s(X) )\cdot(P-\overline{X}_i), ~ ~P\in b_i\cap T^{s'}, X\in T_{non}^s, & F_i(P,X)= \left( (\overline{M}^s(F) - I)\nabla u^s(X) \right) \cdot (\widetilde{Y}_i-\overline{X}_i), ~ ~P\in b_i\cap T^{s'}, X\in T_{non}^s,
\Psi(z) = \begin{cases}\Psi_0(z) = 1 - F_1(z), z \in \mathfrak{R}_0, \Psi_1(z) = F_1(z) - F_2(z), z \in \mathfrak{R}_1, \Psi_2(z) = F_2(z), z \in \mathfrak{R}_2, \end{cases}
\langle b_m\cdot w,\nabla\cdot(w|w|^{q-2})\rangle = \langle b_m\cdot w,\Delta u |w|^{q-2}\rangle+(q-2)\langle b_m\cdot w,|w|^{q-3}w\cdot\nabla|w|\rangle &:= S_1+S_2.
Q^n_1(t) = Q^n_1(0) + A\left(\lambda_n n t\right) - D_1\left(\int_0^t\left(Q^n_1(s) - Q^n_2(s)\right)ds\right) - U^n_1(t), Q^n_i(t) = Q^n_i(0) + U^n_{i-1}(t) - D_i\left(\int_0^t\left(Q^n_i(s) -Q^n_{i+1}(s)\right) ds\right) - U^n_i(t), i \ge 2,
A=[\begin{matrix}0&0&0&1&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1\\ 3n^{2}&0&0&0&2n&0\\ 0&0&0&-2n&0&0\\ 0&0&-n^{2}&0&0&0\end{matrix}].
\langle u_i, \hat{u}_j\rangle =\frac{\langle B^{\ell}\chi_i, (B^*)^{\ell}\check{\chi}_j\rangle}{\nu_i^{2\ell}\nu_j^{2\ell+2}}=\frac{\langle B^{2\ell}\chi_i, \check{\chi}_j\rangle}{\nu_i^{2\ell}\nu_j^{2\ell+2}}.
\sum_{i_1=1}^A \cdots \sum_{i_k=1}^A 2^{kd/2}\epsilon^{- d/2}\prod_{\beta = 1}^k \zeta_{i_\beta}^d = 2^{kd/2}\epsilon^{- d/2}\sum_{i_1=1}^A \zeta_{i_1}^d \sum_{i_2=1}^A \zeta_{i_2}^d \cdots \sum_{i_k=1}^A \zeta_{i_k}^d = \epsilon^{-d/2}2^{kd/2}B_u^k
\sin \phi_1 = \sin (\phi_2 + \phi_3) = \sin \phi_2 \cos \phi_3 + \cos \phi_2 \sin \phi_3 \sin \phi_1 = \frac{2q(1-p^2)}{(1+q^2)(1+p^2)}+ \frac{2p(1-q^2)}{(1+q^2)(1+p^2)}
\frac{1}i \tilde s = (r-r_{ps})f(r)\xi + s_1(r,\xi, \theta,\Theta,\Phi,\Psi)+s_0(r,\xi, \theta,\Theta,\Phi,\Psi) \tau)+ \gamma(\tau, r,\xi, \theta,\Theta,\Phi,\Psi) p
k_\delta(x)= \begin{cases}1-\frac{|x|}{\delta},\ |x|\leq \delta 0,\ \delta<|x|<1-\delta \frac{P(X_1=0)}{2}\cdot (\frac{|x|}{\delta}+1-\frac{1}{\delta}), 1-\delta\leq |x|\leq 1 \frac{P(X_1=0)}{2},\ |x|\geq 1. \end{cases}
l_\sigma(u ; 0,0) = (\gamma \circ c_\sigma)( b_1 b_2) = \gamma( B a_1 a_2 B^{-1}) = (B;0,0) (v^{-1}u^{-1};1,1) (v;0,-1) (B^{-1},0,0) & = (B v^{-1}u^{-1}\theta(1,1)(v) \theta(1,0)(B)^{-1}; 1,0) = ( B v^{-1}u^{-1}B v u^{-2}B^{-1}; 1,0) = (B u^{-1}B^{-1}; 1,0).
\Theta_{}:=\sum_{i=1}^{k-1}\Theta_{h_i}^2+\sum_{i=1}^{w}\Theta_{x_i}\Theta_{x^*_i}+\sum_{i=1}^{w}\Theta_{x^*_i}\Theta_{x_i}+\sum_{i=1}^{\ell}\Theta_{y_i}\Theta_{y^*_i}-\sum_{i=1}^{\ell}\Theta_{y^*_i}\Theta_{y_i},