harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{\partial^2v_m}{\partial r^2}-( -\Delta r+\frac{2m}{r}+2\rho^{\prime})\frac{\partial v_m}{\partial r}+A_r v_m+(\frac{m(m+1)}{r^2}+\frac{m}{r}(2\rho^{\prime}-\Delta r)-V_0-V_1-V_2+\lambda)v_m=0.
U^{\ast}=u_{0}+\grave{u}_{1}t+\grave{u}_{2}t^{2}+... U_{\varphi}^{\ast}=\mu_{0}+\mu_{1}t+\mu_{2}t^{2}+... U_{\theta}^{\ast}=\eta_{0}+\eta_{1}t+\eta_{2}t^{2}+....
\frac{\partial\psi_w^{-1}}{\partial z}(z) = \frac{1}{2\pi i}\frac{z-\zeta^-(w)}{z-\zeta^+(w)}\frac{(z-\zeta^-(w)) - (z-\zeta^+(w))}{(z-\zeta^-(w))^2}& = \frac{1}{2\pi i}\frac{\zeta^+(w)-\zeta^-(w)}{(z-\zeta^+(w))(z-\zeta^-(w))}.
P(y^{n}_{[0:2]}|x^{n},f_{[0:2]}) = \frac{P(y^{n}_{[0:2]},x^{n},f_{[0:2]})}{\sum_{y^{n}_{[0:2]}\in \mathcal{Y}^{n}_{[0:2]}}P(y^{n}_{[0:2]},x^{n},f_{[0:2]})}=\frac{\phi(x^{n})p(y^{n}_{[0:2]}|x^{n})P(f_{[0:2]}|y^{n}_{[0:2]})}{\sum_{y^{n}_{[0:2]}\in \mathcal{Y}^{n}_{[0:2]}}\phi(x^{n})p(y^{n}_{[0:2]}|x^{n})P(f_{[0:2]}|y^{n}_{[0:2]})},
\left\{0\right\}=\mathfrak{g}_{\mathbb{C}}^{\left( 0\right)}\subset\mathfrak{g}_{\mathbb{C}}^{\left( 1\right)}\cdots\subset\mathfrak{g}_{\mathbb{C}}^{\left( n-1\right)}\subset\mathfrak{g}_{\mathbb{C}}^{\left( n\right)}=\mathfrak{g}_{\mathbb{C}}.
\displaystyle\sum_{l'=1}^{p-q}\left(\varphi_{il'}^{\star\star}\left(Z\right)\right)^{(2)}\overline{\left(\varphi_{il'}^{\star\star}\left(Z\right)\right)^{(2)}}\equiv 0,\quad\mbox{for all $i=1,\dots,q$,}
\tan\theta=\frac{-f_{uv}\pm\sqrt{f_{uv}^2-f_{uu}f_{vv}}}{f_{vv}}\bigg|_{u=v=0}\; , \;\;\; \;\;\; \cot\theta=\frac{-f_{uv}\pm\sqrt{f_{uv}^2-f_{uu}f_{vv}}}{f_{uu}}\bigg|_{u=v=0}\; .
A d ( \tilde{g})^{T}= \left( \begin{matrix}{{1}}&{{0}}&{{0}}&{{- \beta^{- 1}\rho e^{- \sigma}}}\\{{- \rho}}&{{e^{\sigma}}}&{{{\beta}^{- 1}{\rho}}}&{{\beta^{- 1}\rho^{2}e^{- \sigma}}}\\{{0}}&{{0}}&{{1}}&{{{\rho}{e^{- \sigma}}}}\\{{0}}&{{0}}&{{0}}&{{e^{- \sigma}}}\\ \end{matrix}\right)
E_1 = \{v \in TM \,|\, \theta(v) = 0, \, dp'_1(v) = 0,\dots, dp'_n(v) = 0 \}, E_2 = \{u \in TM \,|\, \theta(u) = 0, \, dx'_1(u) = 0,\dots, dx'_n(u) = 0 \},
f_{\tilde{\rho}}(x_+) - f_{\tilde{\rho}}(x_-) = \tilde{\rho}\left(\frac{z-m}{z-y}(m-y)^{\tilde{\rho}-1}-\frac{m-y}{z-y}(z-m)^{\tilde{\rho}-1}\right)(x_+-x_-) + o(x_+-x_-) &\quad+ (m-x_-)^{\tilde{\rho}}-(x_+-m)^{\tilde{\rho}}.
\begin{cases}R^{(2)}=R^2-(l-2)(1-\delta)R-d(l-1)(1-\delta)I. R^{(k+1)}=R^{(k)}R-(l-2)(1-\delta)R^{(k)}-(l-1)(1-\delta)(d-1+\delta)R^{(k-1)}.\end{cases}
T_{\xi}\tilde{\cal K}_{{\bf e}, -{\bf e}}^{min}({\mathbb S}^{2k+1})=T_{\xi}{\cal K}_{{\bf e}, -{\bf e}}^{min}({\mathbb S}^{2k+1})\cap T_{\xi}\tilde{\cal K}_{{\bf e}, -{\bf e}}({\mathbb S}^{2k+1})(\xi \in \tilde{\cal K}_{{\bf e}, -{\bf e}}^{min}({\mathbb S}^{2k+1})).
(VII)' \hspace{7mm}\Psi_{\bf D}=\left(\begin{array}{c}a_1 +{\rm j}b_1 a_2 +{\rm j}b_2 a_3 +{\rm j}b_3 a_4 +{\rm j}b_4\end{array}\right) \hspace{5mm}\leftrightarrow \hspace{5mm}\Psi^{SR}_{\bf C}=\left(\begin{array}{c}-b_1+{\rm i}b_3 -b_2+{\rm i}b_4 -a_1+{\rm i}a_3 -a_2+{\rm i}a_4\end{array}\right)
\prod_{S}\chi(S)\geq \left(\prod_{S:Q\cap S=\emptyset}\chi(S)\right)\left(\prod_{S:q_1\in S,q_2\notin S}\left(\sum_{i=1}^m\chi(S,i)\chi(S^*,i)\right)\right)\left(\prod_{S:Q\subseteq S}\sum_{i=1}^m \chi(S,i) \right).
\bar F_t(q) = e^{q}\left(\frac{q}{q+1}\int_t^\infty e^{-q\tau(s)}ds\right) = e^{q}\left(\frac{q}{q+1}\right)\left((1-t)e^{-q}+ \int_1^\infty e^{-qs}ds\right) = \frac{t}{q+1}+(1-t).
\begin{aligned}\left|\xi_{n,j}(a)-\mu_{n,j}(a)\right| \sim_C |Df_{a}^{n}(\xi_{0,j}(a))||\xi_{0,j}(a)-\mu_{0,j}(a)| &=|Df_{a}^{n}(\xi_{0,j}(a))||x_{j}(a)|\end{aligned}
{\Psi}_{\mu}^{( 1 )}({\alpha},{\bf n}, t ,{\bf x}) ={\frac{1}{{( 2 \pi )}^{3 / 2}}}\sum_{\mu^{'}= - 1}^{1}\int \frac{d{\bf p}}{p_{0}}\,{\xi}_{\bf p \mu \mu^{'}}^{( 1 )}({\alpha},{\bf n}) \, \exp [ - i ( p \cdot{x}) ]{\Psi}_{\mu^{'}}^{( 1 )}({\bf p}) .
Q_{0,1}\left(\frac{x}{|x|}\right)=q_*:=|{\bf S}^{N-1}|^{-\frac{1}{2}},Q_{1,i}\left(\frac{x}{|x|}\right)=q_N\frac{x_i}{|x|}\quad\mbox{with}q_N=N^{\frac{1}{2}}q_*,
\frac{Cf(t)^2}{|X_t + x|^{2\theta - 2}}= \frac{Cf(t)^{1/q + 1}}{r^{1/r}}\cdot\frac{f(t)^{1/r}r^{1/r}}{|X_t + x|^{2\theta - 2}}\leq \frac{C^q f(t)^{q + 1}}{q r^{q/r}}+ \frac{f(t)}{|X_t + x|^{(2\theta - 2)r}}.
(T^{(1)}_{n \times p})e_i= \sum_{j=0}^{n-1}a^{(1)}_{j}\chi_{[1, n]}(i+j) e_{i+j}+ \sum_{j=1}^{p-1}a^{(1)}_{-j}\chi_{[1, n]}(i-j) e_{i-j}= \sum_{j=-(p-1)}^{n-1}a^{(1)}_{j}\chi_{[1, n]}(i+j) e_{i+j}.
\Sigma^{+}=W^{+}=\{e_{1}-e_{2},\e_{2}-e_{3},\e_{1}-e_{3}\}, &\Pi=\{\alpha_{1}=e_{1}-e_{2}, \alpha_{2}=e_{2}-e_{3}\},\ \tilde{\alpha}=\alpha_{1}+\alpha_{2}, &m:=m(\lambda )= n(\lambda ) (\lambda \in \tilde{\Sigma}).
\partial_{t}^{k}\theta \in C^{0}\big([0, T], H^{s + 1 - k}(\Omega) \cap H^{1}_{0}(\Omega)\big) k = 0, 1, \dots, s - 2, \partial_{t}^{s - 1}\theta \in C^{0}\big([0, T], H^{1}_{0}(\Omega)\big) \cap L^{2}\big(0, T; H^{2}(\Omega) \cap H^{1}_{0}(\Omega)\big) \partial_{t}^{s}\theta \in L^{2}\big(0, T; L^{2}(\Omega)\big).
\nabla^{2}U=-\frac{\partial^{2}}{{\partial x}^{2}}(M_{x}B_{x}+M_{y}B_{y}+M_{z}B_{z})-\frac{\partial^{2}}{{\partial y}^{2}}(M_{x}B_{x}+M_{y}B_{y}+M_{z}B_{z})-\frac{\partial^{2}}{{\partial z}^{2}}(M_{x}B_{x}+M_{y}B_{y}+M_{z}B_{z})
\begin{array}{c}[(2,\chi_{1}^{1}) \oplus (-1,\chi_{2}^{1})]\otimes ((a+2i)-(k'-1),\chi_2^{k'-1}) =((a+2(i+1))-(k'-1),\chi_2^{k'-1}) \oplus ((a+2i)-k',\chi_1^{2}) \oplus((a+2i)-k',\chi_1^{4}) \oplus ((a+2(i-1))-(k'-2),\chi_2^{k'-2})\end{array}
Y^{j,n}_t =\xi^j+\int_t^Tf^j(r,Y^n_r)\,dr+\int_t^T\,dV^j_r &\quad+\int_t^Tn(Y^{j,n}_r-H^j(r,Y^n_r))^-\,dr-\int_t^T\,dM^{j,n}_r.
\left(\mathcal{D}f\right)(x) =(a-bx)\frac{\partial f(x)}{\partial x}+\frac{1}{2}\sigma^{2}x\frac{\partial^{2}f(x)}{\partial x^{2}}, \left(\mathcal{J}f\right)(x) =\int_{0}^{\infty}\left(f(x+z)-f(x)\right)\nu(\mathrm{d}z),
\varphi^{(3,3)}(\xi)= I_{0}+I_{1}+\sum_{j=0}^{\infty}I_{j+2}\xi^{j}+(1-\xi)\left(I_{0,1}^{2}+I_{1,2}^{2}\right)+(1-\xi)\sum_{j=0}^{\infty}I_{j+2,3}^{2}\xi^{j}+(1-\xi)^{2}\left(I_{0,1}^{3}+I_{1,2}^{3}\right)+(1-\xi)^{2}\sum_{j=0}^{\infty}I_{j+2,3}^{3}\xi^{j}.
ds^{2}= \frac{\Delta -\alpha^{2}\sin^{2}\theta}{\Sigma}\left( dt -\omega d\varphi \right)^{2}-\Sigma \left(\frac{d\hat{\rho}^{2}}{\Delta}+d\theta^{2}+\frac{\Delta\sin^{2}\theta d\varphi^{2}}{\Delta-\alpha^{2}\sin^{2}\theta}\right)\, ,
\sum_{r=1}^{n-1}\left( \frac{\partial^2 \Omega_r}{\partial x_j \partial x_k}\frac{\partial^2 \Omega_k}{\partial x_r \partial x_i}- \frac{\partial^2 \Omega_r}{\partial x_i \partial x_j}\frac{\partial^2 \Omega_k}{\partial x_r \partial x_k}\right)
\begin{array}{l l}{{g_{_L}( \widehat{\mathbf{f}}_{\mu}, \widehat{\mathbf{f}}_{\nu}) = g_{\mu \nu}}}\\{{g_{_L}( \widehat{\mathbf{f}}_{\mu}, \mathbf{e}_{k}) = 0}}\\{{g_{_L}( \mathbf{e}_{k}, \mathbf{e}_{l}) = g_{0 \, k l}\, ,}}\\ \end{array}
d\beta_t = \sqrt{\epsilon}\mathfrak{M}^{-1}(u_t,\beta_t)\big\langle P^{-1}(\beta_t)\Phi'(\beta_t),P^{-1}(\beta_t)G(u_t)dW_t\big\rangle + F.V.T = \sqrt{\epsilon}\mathfrak{M}^{-1}(u_t,\beta_t) \Phi'(\beta_t)^{\top}P^{-\top}(\beta_t)P^{-1}(\beta_t)G(u_t)dW_t + F.V.T,
\frac{1}{2}( S^{-}- ( S^{-})^{\dagger}) = \int_{M}\left( d^{4}x \frac{i}{2}\partial_{\mu}({\tilde{\overline{{\Psi}}}}_{L}\gamma^{\mu}{\tilde{\Psi}}_{L}) - \frac{i}{4}{\tilde{\overline{{\Psi}}}}_{L}\gamma^{A}{\tilde{\Psi}}_{L}e^{- 1}\epsilon_{A B C D}T^{B}\wedge e^{C}\wedge e^{D}\right) .
u \mbox{is a global weak solution of \eqref{eq1}in}L^\infty((0,\infty);L^p({\mathbb{R}}^n)) ; begin{align*}4pt] \exists !\,u_0\in L^p({\mathbb{R}}^n) \mbox{such that}u(t,\cdot)=\Gamma(t,0)u_0\mbox{in}L^p({\mathbb{R}}^n)\mbox{for all}t>0 ; begin{align*}4pt] u \mbox{is a global weak solution of \eqref{eq1}in}X^p.
\sum_{k=0}^{\infty}P_{\phi_k}(a) =\frac{1}{(2\pi)^n}\int_{\mathbb R^n}\!\int_{\mathbb R^n}\!e^{-is\cdot\xi}\sum_{k=0}^{\infty}\phi_k(\xi)\alpha_s(a)\,\mathrm ds\,\mathrm d\xi &=\frac{1}{(2\pi)^n}\int_{\mathbb R^n}\!\int_{\mathbb R^n}\!e^{-is\cdot\xi}\phi(\xi)\alpha_s(a)\,\mathrm ds\,\mathrm d\xi,
\left( \begin{array}{c}{\hat{\alpha}( k )}\\{\hat{\beta}( k )}\\ \end{array}\right) = \left( \begin{array}{c}{{a}( k )}\\{i}\\ \end{array}\right) \hat{\xi}( k ) + \left( \begin{array}{c}{- i{b}( k )}\\{1}\\ \end{array}\right) \hat{\eta}( k )
K \wedge K = \frac{1}{8}\frac{\bar{\kappa}^4}{\lambda^4}(\hat{I}_4)^2 \epsilon(x^{11})^2 + 36 dC' \wedge dC' + \frac{6}{\sqrt{2}}\frac{\bar{\kappa}^2}{\lambda^2}\epsilon(x^{11}) \hat{I}_4 \wedge dC'
|u_1|_{x_k}= A , |v_1|_{x_k}= A' , |u_2|_{x_k}= B , |v_2|_{x_k}= B', |u_1|_{x_l}= C , |v_1|_{x_l}= C' , |u_2|_{x_l}= D , |v_2|_{x_l}= D'.
M_L=\#\left\{\alpha : \ |\alpha|\leq C\delta L\right\}=\sum_{m\leq C\delta L}\frac{\Gamma(n+m)}{\Gamma(n)m!}\leq C\delta L\frac{\Gamma(n+C\delta L)}{\Gamma(n)(C\delta L)!}\simeq C^n\delta^nL^n.
J_{2|\alpha_i|}= \tilde{S}^T_{\gamma_i}J_{2n}\tilde{S}_{\gamma_i}= \sum_{k=1}^r \tilde{S}_{\gamma_k\gamma_i}^T J_{2|\alpha_k|}\tilde{S}_{\gamma_k\gamma_i}= \tilde{S}_{\gamma_i\gamma_i}^T J_{2|\alpha_i|}\tilde{S}_{\gamma_i\gamma_i}+ \sum_{k\neq i, k=1}^r \tilde{S}_{\gamma_k\gamma_i}^T J_{2|\alpha_k|}\tilde{S}_{\gamma_k\gamma_i}.
\sum_{e,f=1}^{8}\big\langle \gamma_1, \gamma_2, T_e \big\rangle^Y g^{ef}\big\langle \gamma_3, \gamma_4, T_f \big\rangle^Y=\sum_{e,f=1}^{8}\big\langle \gamma_1, \gamma_4, T_e \big\rangle^Y g^{ef}\big\langle \gamma_2, \gamma_3, T_f \big\rangle^Y.
(-1)^{n}\sum_{j \in \mathbb{Z}, \, j \neq 0}W_{n,m}^{(\beta, \alpha)}(2 \pi \imath j;j)= (-1)^{m-1}\sum_{j \in \mathbb{Z}, \, j \neq 0}W_{n,m}^{(\alpha,\beta)}(2 \pi \imath j;j).
\widetilde{S^{K\textnormal{th}}}= S^{K\textnormal{th}}\left.\right._\circ \left.\right. \prod_{j=1}^r \phi_j^{\ell_q(Q_j)}\widetilde{T^{K\textnormal{th}}}= \prod_{j=1}^r \phi_j^{-\ell_q(Q_j)}\left.\right._\circ \left.\right. T^{K\textnormal{th}}
\langle \xi_{\tilde K}\nabla w,\xi_{\tilde K}\nabla w\rangle=\frac{(1+\delta)^2}{1+2\delta}\left(\langle \nabla u_{q,r}, \nabla \nu\rangle-(1+u_{q,r})^{2\delta+1}\langle\nabla u_{q,r}, \nabla(\xi_{\tilde K})^2\rangle\right).
g(t,x,z)=f_{\mu,A}^{1-m}(x)z^{m-1}\frac{A+A^t}{2}(x-\mu )\ ,\textrm{and}\Lambda(t,x,z)=f_{\mu,A}^{1-m}(x)z^{m-1}Tr\left(\frac{A+A^t}{2}\right).
J_{ex}=\int\Phi_{a}^{*}(\vec{r}_{1})\Phi_{b}^{*}(\vec{r}_{2})(\frac{1}{R_{ab}}+\frac{1}{r_{12}}-\frac{1}{r_{a1}}-\frac{1}{r_{b2}})\Phi_{b}(\vec{r}_{1})\Phi_{a}(\vec{r}_{2})d^{3}r_{1}d^{3}r_{2}
\begin{array}{l}{\partial X ( z ) \sigma_{\vartheta}( w , \bar{w}) \sim ( z - w )^{- ( 1 - \vartheta )}\tau_{\vartheta}( w , \bar{w}) ,}{\partial \bar{X}( z ) \sigma_{\vartheta}( w , \bar{w}) \sim ( z - w )^{- \vartheta}\tau_{\vartheta}^{\prime}( w , \bar{w}) ,}\end{array}
\sum_{0 \le \beta \le \frac{\alpha}{2}}\frac{1}{p^{2\beta}}= \bigg(1 - \frac{1}{p^2}\bigg)^{-1}\bigg(1 - \frac{p^{\alpha \bmod 2}}{p^{\alpha + 2}}\bigg) = \bigg(1 - \frac{1}{p}\bigg)^{-1}\bigg(1 + \frac{1}{p}\bigg)^{-1}+ O\bigg(\frac{1}{x}\bigg),
|e'''(t)| \leq C \sup_{x \geq t}|RHS_{3}(x)| &\leq \frac{C}{t^{3}\log^{b+1}(t)}+ \frac{C \sup_{x \geq t}\left(x^{3/2}|e'''(x)|\right)}{t^{3/2}\sqrt{\log(\log(t))}}
\psi_\omega(T_xT_y^*) =\big(e_{0,e}\otimes \pi_{\omega}(S)^{*t}\xi_\omega\,|\,e_{0,e}\otimes \pi_{\omega}(S)^{*s}\xi_\omega\big) &=\big(\pi_{\omega}(S)^{*t}\xi_\omega\,|\,\pi_{\omega}(S)^{*s}\xi_\omega\big) &=\big(\pi_{\omega}(S^sS^{*t})\xi_\omega\,|\,\xi_\omega\big) &=\omega(S^sS^{*t}),
\begin{aligned}\phi_X(d\vec{1}) = \int\cos (d\sum_{n=1}^n y_j)\mathrm{d}\mu_X(y)+i\int\sin (d\sum_{n=1}^n y_j)\mathrm{d}\mu_X(y) = \int\cos (d\sum_{n=1}^n y_j)\mathrm{d}\mu_X(y) &= \int\left(\cos(dy_1) \cos (d\sum_{j=2}^n y_j)-\sin(dy_1)\sin(d\sum_{j=2}^ny_j)\right)\mathrm{d}\mu_X(y). \end{aligned}
\hat{{\cal L}}_{1}( \hat{F}) = \frac{\sqrt{\det G}}{G_{s}}\left[ \hat{T}_{3}+ ( 2 \pi \alpha^{\prime})^{2}\left( - \frac{1}{4}\hat{J}_{1}+ 2 \hat{J}_{2}+ \hat{J}_{3}\right) + O ( \alpha^{4}) \right] ,
\theta^{r-1}a_{i}=\sum_{j=1}^{n}q_{rij}a_{j}=\sum_{j=1}^{n}q_{rij}\sum_{k=1}^{n}k_{jk}\theta^{k-1}=\sum_{k=1}^{n}\big(\sum_{j=1}^{n}q_{rij}k_{jk}\Big)\theta^{k-1}.
\chi_{n,0}(\cos(\alpha_{1}),...,\cos(\alpha_{n})|\rho)\allowbreak=\allowbreak\sum_{k\geq0}\rho^{k}\prod_{j=1}^{n}T_{k}(\cos(\alpha_{j}))\allowbreak= \allowbreak\frac{1}{2^{n}}\sum_{i_{1}\in\{-1,1\}}...\sum_{i_{n}\in\{-1,1\}}\frac{(1-\rho\cos(\sum_{k=1}^{n}i_{k}\alpha_{k}))}{(1-2\rho\cos(\sum_{k=1}^{n}i_{k}\alpha_{k})+\rho^{2})},
M = \left[ \begin{array}{c c c c c c c c c c c c c c c c}0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 \end{array}\right]\,.
\left(-1+\frac{1}{n}\right)e_{i,g}^{n-1}+\frac{1}{n}e^{n-1}_{i,g+1}+\frac{1}{n}e_{i,g+2}^{n-2}-e_{i,g+1}^{n-2}&+\left(1-\frac{1}{n}\right)\left[\frac{1}{n^2}e_{i,g+1}^{n-3}+\frac{1}{n}e_{i,g+2}^{n-3}\right] -e_{i,g+2}^{n-3}+\left(-1+\frac{1}{n}\right)\sum_{\tau=3}^{n-2}e_{i,g+\tau}^{n-1-\tau}.
[T_\alpha,T^\dagger_\beta]=\bar{U}^\gamma_{\alpha\beta}T_\gamma+T^\dagger_\gamma\bar{U}^{\dagger\gamma}_{\beta\alpha}+\bar{U}^\gamma_{\alpha\delta}\bar{U}^{\dagger\delta}_{\beta\gamma}(-1)^{\varepsilon_\gamma\varepsilon_\delta}.
2\rho_{0}^{3}v_{0}v_{1}+3\rho_{0}^{2}v_{0}^{2}\rho_{1}-4v_{0}\mathfrak{S}_{1}+\frac{2\omega}{K_{0}}[v_{0}\rho_{1}+\rho_{0}v_{1}-\frac{K_{1}}{K_{0}}\rho_{0}v_{0}^{2}]=0
\begin{aligned}J(x)\cong\begin{cases}\frac{1}{4x^2}+\frac{a_0-\frac{1}{2}}{x}+a_2-a_1-\frac{5}{4}, \qquad x\rightarrow0, -\frac{3}{4(x-1)^2}+\frac{a_1+\frac{1}{2}}{x-1}-a_1^2-a_1-\frac{1}{4}, x\rightarrow1, a_2+a_3 x, x\rightarrow\infty,\end{cases}\end{aligned}
K_1 =\sup_{j\in\mathbb{Z}}2^{-j}\|[\Delta_{j},u_{1}\cdot\nabla]\delta u\|_{L^1_{t}L^2}, K_2 = \sup_{j\in\mathbb{Z}}2^{-j}\|\Delta_{j}(\delta u\cdot\nabla u_{2})\|_{L^1_{t}L^2}, &K_3 = \sup_{j\in\mathbb{Z}}2^{-j}\|\Delta_{j}(B_{1}\cdot\nabla\delta B)\|_{L^1_{t}L^2}, K_4 =\sup_{j\in\mathbb{Z}}2^{-j}\|\Delta_{j}(\delta B\cdot\nabla B_{2})\|_{L^1_{t}L^2}.
V^{2 - 2 h}\sum_{e n}[ \mathrm{e x p}( i \pi e / k ) - \mathrm{e x p}( - i \pi e / k ) ]^{2 h - 2}= V^{- 2 h}\left( \begin{array}{c}{{h - 1}}\\{{2 h - 2}}\\ \end{array}\right)
\dot{p}= -\Gamma\log{p(t,r)}+ \frac{\Gamma\int_{\Omega}\log{p(t,r)dr}}{\mathrm{mes}(\Omega)}=-\Gamma\left(\log{p(t,r)}-\frac{\int_{\Omega}\log{p(t,r)dr}}{\mathrm{mes}(\Omega)}\right).
\mathcal{H}= \int d^{\, 2}\, x \, \left[ \, \frac{1}{2}\, \left( \xi \,_{5}^{2}+ \xi \,_{6}^{2}\right) + \frac{1}{2}\, \left( \partial_{1}\, \xi_{3}- \partial_{2}\, \xi_{2}\right)^{\, 2}+ \frac{1}{2}\, m^{2}\, \left( \xi \,_{2}^{2}+ \xi \,_{3}^{2}\right) + m \left( \xi \,_{2}\, \xi \,_{6}- \xi \,_{3}\, \xi \,_{5}\right) \right] ;
P_{q,z}\cdot \varphi_{q,z}^* J^{K\textnormal{th,eq}}(t,z,Q) = \sum_{i=0}^N \left( \Lambda_i^{-\ell_q(Q)}\sum_{d \geq 0}\frac{1}{z^{d(N+1)}}\frac{(1-q)^{d(N+1)}Q^d}{\left(q\Lambda_0 \Lambda_i^{-1}, \dots q, \dots ,q\Lambda_N \Lambda_i^{-1};q \right)_d}\right) \Psi_i
\left\lbrace\begin{array}{ll}u_r'' -\Delta u_r + \sum_{s=1}^d\alpha_{rs}u_s =0 \hbox{in}(0, +\infty)\times\Omega, u_r =0 \hbox{on}(0, +\infty)\times \Gamma_0, \partial_\nu u_r +\sum_{s=1}^d\beta_{rs}u_s =0 \hbox{on}(0, +\infty)\times\Gamma_1\end{array}\right.
\tilde x\frac{\lambda_{j-2}}{\lambda_{j-2}-\lambda_{j-1}}=\tilde x\Big(1+\frac{\lambda_{j-1}}{\lambda_{j-2}-\lambda_{j-1}}\Big)\leq 2\tilde{x}\frac{\lambda_{j-1}}{g_{j-1}}\leq 2z_2\frac{\lambda_{j-1}}{g_{j-1}}.
V_\lambda (t) = t + L_\lambda^0 (t) ( \frac{\lambda}{f(\lambda)}-1) = t + R_\lambda (t) \frac{\lambda- f(\lambda)}{(\sigma \sqrt\lambda - \mu) f(\lambda)}= t + R_\lambda (t) \frac{1- f(\lambda)/ \lambda}{(\sigma - \mu/ \sqrt\lambda) f(\lambda) /\sqrt\lambda}.
z^{A}= \left( A_{i}^{a}, B_{a}^{0 i}, B_{a}^{i j}, \pi_{i j}^{a}, \varphi_{a}, \pi^{a}, \varphi_{a}^{\left( 1 \right)}, \pi^{( 1 ) a}, \varphi_{a}^{\left( 2 \right)}, \pi^{( 2 ) a}\right) .
\hspace{- 0 . 9 5 i n}\lambda = 4 \; \mathrm{f o r}\; \left| \vec{p}\right| < \bar{m}/ 2 , \quad \lambda = 3 \ \; \mathrm{f o r}\; \bar{m}/ 2 < \left| \vec{p}\right| < \bar{m}, \quad \lambda = 3 \; \mathrm{o r}\; 4 \; \mathrm{f o r}\; \left| \vec{p}\right| > \bar{m}.
\tilde{A}_0 : = \sup_{t > 0}\bigg( \int_0^t [U_1^*]^q (\tau) w(\tau)\,d\tau \bigg)^{\frac{1}{q}}[V_1^*]^{- \frac{1}{p}}(t), \tilde{A}_1 : = \sup_{t > 0}W_*^{\frac{1}{q}}(t) \bigg( \int_0^t [U_1^*]^{p'}(\tau) [V_1^*]^{- (2 + p')}(\tau) V_*^{-2}(\tau)v(\tau)\,d\tau \bigg)^{\frac{1}{p'}};
\vartheta(\varepsilon j)\eta(\varepsilon j)-\vartheta(\varepsilon i)\eta(\varepsilon i) = \eta(\varepsilon j) \left(\vartheta(\varepsilon j)-\vartheta(\varepsilon i) \right) + \vartheta(\varepsilon i) \left(\eta(\varepsilon j) -\eta(\varepsilon i) \right).
L2_H \lesssim \| \nabla_L C \|_{L^2 H^N}\| \nabla_L U_0^2 \|_{L^2 H^N}\lesssim \frac{C_1 \epsilon}{\nu^{3/2}}\frac{C_0 \epsilon}{\sqrt \nu}= \frac{C_0}{C_1}\left( \frac{C_1 \epsilon}{\nu}\right)^2 .
|W_t\!-\!W_\tau|^2=|W_t|^2+|W_\tau|^2\pm2|W_t||W_\tau|\cos\overline\beta=\varepsilon_1^2\sin^2\overline\beta+(|W_t|\!\pm\!\varepsilon_1\cos\overline\beta)^2\ge\varepsilon_1^2\sin^2\overline\beta\ge\frac{4}{\pi^2}\varepsilon_1^2\overline\beta^2
\begin{array}{ll}\forall \epsilon >0~\forall r>0~\forall x\in B_r~\exists R_0~\forall R\geq R_0 ~E^x\left(f^{\epsilon,R}\left(\sum_{i=1}^nc_iX^{\epsilon,R}_i(t)\right) \right)\leq E^x\left(f^{\epsilon,R}\left(\sum_{i=1}^nc_iY^{\epsilon,R}_i(t)\right) \right)\end{array}
\begin{array}{ll}1-2\cos\left(\frac{4\pi s}{p+1}\right) = 1-2\left(1-\frac{2\pi^2s^2}{(p+1)^2}+ O\left(\frac{r^4}{p^4}\right)\right) = -1 + \frac{4\pi^2s^2}{(p+1)^2}+O\left(\frac{r^4}{p^4}\right) &= -1 + \frac{4\pi^2s^2}{p^2}\left(1+O\left(\frac{1}{p}\right)\right)+O\big(\frac{r^4}{p^4}\big) & = -1 +\frac{4\pi^2s^2}{p^2}+ O\big(\frac{s^2}{p^3}\big).\end{array}
b^*_{k,n}:= 2n I\left(\frac{k}{n}\right) - \tanh^{-1}\left(\frac{k}{n}\right) = \sum_{j=1}^{\infty}\left(\frac{k}{n}\right)^{2j-1}\frac{1}{2j-1}\left(\frac{k}{j}-1\right) .
(\widetilde{X}_1 \widetilde{X}_2 \widetilde{X}_c)^2 \widetilde{C^\prime}_4 \widetilde{C}_4 \widetilde{C}_1^2 \widetilde{W}_1 \widetilde{W}_2 \widetilde{W}_c \widetilde{W_1^\prime}\widetilde{W_2^\prime}\widetilde{W_c^\prime}= \partial^2.
{\rm Tutte}\big(K_{\underbrace{n,\ldots,n}_{r}}^{({\boldsymbol{\ell}})},1,1\big) = k^{n}(r-1)^{n-1}((r-1) \ell + k )^{r-2}((r-2) \ell +k )^{(r-1)(n-1)}n^{nr-1}.
p^{(m)}_{1,k_1,0}(x,t) = \sum_{k=1}^{\frac{m-1}{2}}\frac{c_{m,k}(-1)^k}{(2\pi)^{\frac{m-1}{2}}}\frac{1}{|t|^{m-1-k}}p^{(1)}_{1,k_1,k}(x,|t|)
\sum_{l=1}^{h(m,n)}C_{m}^{l}(F^{0})(\lambda)(C_{m}^{l}(F^{0})(\lambda))^{*}=d_{m}^{0}(\lambda )^{\top}\left\{(\beta_{mk}^{2}+\gamma_{m_{1}k}(\lambda )+\gamma_{m_{2}k}(\lambda ))\delta_k^l \right\}_{k,l=1}^{\infty}\overline{d_{m}^{0}(\lambda )},
\left|\frac{n_{cs}\ \tilde{u}N}{M}(\ell-m)+(p-k)\right| &\le \left|\frac{n_{cs}\ \tilde{u}N}{M}(\ell-m)\right|+|(p-k)| &\le \frac{n_{cs}\ \tilde{u}N (G-1)}{M}+(N_1-1) &\le N-N_1-N_1+1= N-1
\langle \phi' | \phi \rangle =\int e^{iS_c}{\cal D}\{\eta\}{\cal D}\{a\}{\cal D}\{c\}{\cal D}\{\bar{c}\}e^{i\int\eta\hat{M}\eta dx}e^{i\left\{\frac{1}{2}\left[\int\Psi_0\eta dx\right]^2 +\bar{c}c\int\left(\Psi_0^2-\Psi_0\frac{d\eta}{dx}\right)dx \right\}}.
\|U_{C}^{(s-1)k+q}\|=\|(1-\frac{q}{k}) U_C^{(s-1)k}+\frac{q}{k}U_C^{sk}\| \leq (1-\frac{q}{k})\|U_C^{(s-1)k}\|+\frac{q}{k}\|U_C^{sk}\| \leq \bar{C},
\Pi_\infty^{(s, v)}(z_1, z_2) = \frac{1}{2 \prod_{i = 1}^m \sqrt{(v_i + z_1) (v_i + z_2)}}\int_{0}^1 \frac{A_{II}^{(s,v)}(\kappa, z_1) \cdot A_{II}^{(s,v)}(\kappa, z_2)}{\big[ C_{II}^{(s,v)}(\kappa)\big]^2}\kappa d \kappa.
\mathcal{D}= \{2X_1 \stackrel{D_1}{\longrightarrow}X_1, X_1 + X_2 \stackrel{D_2}{\longrightarrow}X_1, X_1 + X_2 \stackrel{D_3}{\longrightarrow}X_2, X_2 + X_4 \stackrel{D_4}{\longrightarrow}X_2\}.
=\begin{cases}1.0 \times b^{-(exponent-1)}b \geq 5, X=0 (b-1).1_b \times b^{-exponent}b \geq 3, X=1 1.01_2 \times b^{-exponent}b=2, X=2 (b-1).\underbrace{00\ldots00}_{\mathclap{}}(b-1)(b-1)_b \times b^{-exponent}b=2, X \geq 3; \b \geq 3, X \geq 2 \end{cases}
F_1|_{\Pi_{y,t}}= u s + z^3 + \delta_1 z^2 x^2 + \delta_2 z x^4, \F_2|_{\Pi_{y,t}}= u^2 + s^2 z + \varepsilon_1 z^3 x + \varepsilon_2 z^2 x^3 + \varepsilon_3 z x^5,
\begin{aligned}\mathfrak{S}_{\delta, 3, \flat}^{\Lambda*\mathbb{1}}\ll_{\varepsilon}\big((x L^3 M^4 N^3)^{1/6}+ M (LN)^{1/2}+ M^{1/2}LN+ (x^{-1}DH')^{1/2}\big) x^{\varepsilon}& \ll_{\varepsilon}\big((x^2 D^7 L^{-1})^{1/12}+ (D^5L)^{1/6}+ (x^{-1}DH')^{1/2}\big) x^{\varepsilon},\end{aligned}
\max_{j}\left\lVert\overset{3}{\mathbb{P}}\left(\tilde{E}^T_j\right)\right\rVert^2\lesssim\begin{cases}a^4 \mbox{if}t<\frac{2}{3}, a^{12-2h}\aleph d^{-6}\mbox{if}\frac{2}{3}\leq t<\frac{5}{7}.\end{cases}
A_{T i}^{a}({\bf{r}}) = \sum_{{\bf{k}}; s = 1 , 2}\frac{\epsilon_{i}^{s}({\bf{k}})}{\sqrt{2 k}}[ a_{s}^{a}({\bf{k}}) e^{+ i{\bf{k}}\cdot{\bf{r}}}+ a_{s}^{a \dagger}({\bf{k}}) e^{- i{\bf{k}}\cdot{\bf{r}}}] ,
q\cdot N_d =q^2+q\phi(b)+\frac{\phi(-b)}{q-1}\sum_{m=0}^{q-2}G_{-m}G_{md}G_{-m(d-1)+\frac{q-1}{2}}G_{\frac{q-1}{2}}T^{m}\left(\frac{b^{d-1}}{a^d}\right).
w(t,x) = \frac{1}{2^{\sqrt{\delta}}}\int_0^t \int_{x-t+b}^{x+t-b}\left[u_0''(y)+bu_1''(y)-\left(\frac{\nu^2}{(1+b)^2}u_0(y)+\frac{\mu}{1+b}u_1(y)+\frac{\nu^2\, b}{(1+b)^2}u_1(y)\right)\right] E(t,x;b,y;\mu,\nu^2) \, \mathrm{d}y\, \mathrm{d}b \doteq I_1+I_2+I_3+I_4+I_5.
=e^{-\lambda t}\sum_{k=0}^\infty I_{2k+2}\Bigl(\frac{2\lambda}{c_1+c_2}\sqrt{(c_1t-\beta)(c_2t+\beta)}\Bigr)\Biggl[ \Biggl(\sqrt{\frac{c_2t+\beta}{c_1t-\beta}}\Biggr)^{2k+2}-\Biggl(\frac{c_2}{c_1}\sqrt{\frac{c_1t-\beta}{c_2t+\beta}}\Biggr)^{2k+2}\: \Biggr]
\everymath{\displaystyle}\begin{array}{rcl}\epsilon^{i j}\hat D_i \frac{\delta F}{\delta E^{a j}}= -2{e^a}_j \left( \partial_i \frac{\delta F}{\delta G_{i j}}+ \Gamma^j_{i k}\frac{\delta F}{\delta G_{i j}}\right) \equiv -2{e^a}_j \nabla_i \frac{\delta F}{\delta G_{i j}}. \end{array}
\widetilde{(LU)_j}= \sum_{k=1}^{n+1}\sum_{|\beta|=0}^{s_j+t_k}\widetilde{b_{jk}^{\beta}}(\phi(x))^{-|\beta|}\partial_z^{\beta}\widetilde{U^k}&=\sum_{k=1}^{n+1}\sum_{|\beta|=0}^{s_j+t_k}(\phi(x))^{-4+t_k-|\beta|}\widetilde{b_{jk}^{\beta}}\partial_z^{\beta}((\phi(x))^{4-t_k}\widetilde{U^k}).
\left|\nabla_y u(x)\right|^2=\left|\frac{\partial x}{\partial y}\nabla_x u(x)\right|^2=\frac{|x|^2}{|y|^2}\left(\left|\nabla_{\mathbb{S}^{n-1}}u(x)\right|^2+\left(\frac{|y|}{|x|}\phi'(|y|)\right)^2\left|\nabla_r u(x)\right|^2\right).
A_{1}= \left({\frac{\omega - m \Omega_{H}}{4 \pi T_{H}}}\right)^{2}, ~ ~ B_{1}= -{\frac{\rho_{-}^{2}}{\rho_{+}^{2}}}\left({\frac{\omega - m \Omega_{H}\rho_{+}^{2}/ \rho_{-}^{2}}{4 \pi T_{H}^{\mathrm{B T Z}}}}\right)^{2}.
B(t)=\frac{\sum_{i=0}^{n}(\begin{matrix}n\\i\end{matrix})t^{i}(1-t)^{n-i}P_{i}w_{i}}{\sum_{i=0}^{n}(\begin{matrix}n\\i\end{matrix})t^{i}(1-t)^{n-i}w_{i}}.
|\Phi^{+}\rangle=\frac{1}{\sqrt{d}}\sum_{i=0}^{d-1}|i\rangle\otimes|i\rangle=\frac{1}{\sqrt{d}}(|0\rangle\otimes|0\rangle+\cdot\cdot\cdot+|d-1\rangle\otimes|d-1\rangle)
0 =\widetilde{H}'_{n+1}+2 \widetilde{H}_n+2x\widetilde{H}'_n+\widetilde{H}''_n &= -2(n+1)\widetilde{H}_n+2\widetilde{H}_n+2x\widetilde{H}_n'+\widetilde{H}''_n &= \widetilde{H}''_n+2x\widetilde{H}'_n-2n\widetilde{H}_n.
\begin{aligned}\frac{\partial^2R_1}{\partial z^2}-p^2R_1=2\langle p,\frac{\partial R_0}{\partial x}\rangle, \ -D(x)\leq z\leq0\cr R_1|_{z=0}=0,\frac{\partial R_1}{\partial z}|_{z=-D(x)}=\langle\nabla D,p\rangle B_0|_{z=-D(x)}\cr\end{aligned}
\frac{\left(q;q\right)_{n}L_{n}^{(\alpha)}\left(abq^{-n-\alpha};q\right)}{\left(-ab\right)^{n}\left(\frac{q^{\alpha+1}}{a};q\right)_{n}}=\sum_{k=0}^{n}\frac{\left(q^{-n};q\right)_{k}q^{\binom{k+1}{2}}}{\left(\frac{q^{\alpha+1}}{a};q\right)_{k}\left(ab\right)^{k}}L_{k}^{(\alpha+n-k)}\left(b;q\right).
\int_2^\infty \frac{e^{- i \xi x}}{x}dx=\int_{2\xi}^\infty\frac{e^{- i u}}{u}du=\int_{2\xi}^1\frac{e^{- i u}}{u}du+\int_{1}^\infty\frac{e^{- i u}}{u}du=O(|\log \xi |+1)\,.
q ( a , r ) = - \frac{1}{2 \pi a^{3}r^{n}S_{D}}\sum_{l = 0}^{\infty}D_{l}\int_{m a}^{\infty}d z \, z^{3}\frac{\bar{K}_{\nu}( z )}{\bar{I}_{\nu}( z )}\frac{F_{\nu}^{( q )}\left[ I_{\nu}( z r / a ) \right]}{\sqrt{z^{2}- m^{2}a^{2}}}, \quad q = \varepsilon , \, p , \, p_{\perp}, \quad r < a ,
\widetilde{\Delta}_h\varphi_{\nu,\ell}=\lambda_{\nu}\varphi_{\nu,\ell},\langle\varphi_{\nu,\ell},\varphi_{\nu,\ell'}\rangle_{L^2(Y)}=\delta_{\ell,\ell'}= \begin{cases}1, \ell=\ell' 0, \ell\neq \ell'.\end{cases}
\arraycolsep=1pt\left.\begin{array}{lll}(-\Delta)^s v+ v^p=0\ \ \rm{in}\ \, B_1\setminus\{0\}begin{align*}2mm] \phantom{(-\Delta)^s + u^p}\displaystyle u=h \rm{in}\ \, B_1^c.\end{array}\right.
\Big|\Big(\Omega^{5k}\setminus \big( \bigcup\nolimits^{n'}_{t=1}Y_t \cup \bigcup\nolimits^{n}_{t=n'+1}X_t \cup \bigcup\nolimits_{p \in \hat{J}}Q^\lambda(p) \big) \Big) \setminus U\Big| \le C_u k \Vert U\Vert_*