harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\overline{_{}}= \frac{\sqrt{2}}{\pi^{1/4}}|m|^{3/4}\sum_{\mathfrak{a}^{\prime}\in \mathcal{E}^{\prime}_{4k}}\int_{|1/4+t^2|\leq T}\frac{1}{1/4+t^2}\rho_{E_{\mathfrak{a}^{\prime}}(.,1/2+it/2),\infty}(m)\langle \theta*W,E_{\mathfrak{a}^{\prime}}(.,1/2+it/2) \rangle dt.
{\cal L}= \frac{1}{2}\left( \partial_{\mu}\chi - \frac{e}{\sqrt{\pi}}A_{\mu}\right)^{2}- \frac{1}{4}F_{\mu \nu}F^{\mu \nu}+ ( D_{\mu}\phi ) ( D^{\mu}\phi )^{*}- \lambda ( | \phi |^{2}- c^{2})^{2}
K_n\subset E_n,\mathbb{P}(K_n)\geq \mathbb{P}(E_{n+1})>0 \mbox{and}\overset{n}{\underset{m=1}{\cap}}K_m\subset \overset{n}{\underset{m=1}{\cap}}E_m,\mbox{for all}n \in \mathbb{N}.
R(z) = \frac{\left[a_{21}a_{32}b_3 - (a_{21}b_2 + a_{31}b_3 ) \gamma + b_1 \gamma^2 \right]z^3}{(1-\gamma z)^3}+ \frac{\left[ (a_{21}b_2 + a_{31}b_3 + a_{32}b_3) - (2b_1+b_2+b_3)\gamma + \gamma^2 \right] z^2 + \left[b_1 + b_2 + b_3 - 2\gamma \right] z + 1}{(1-\gamma z)^3}
|\bar{\mathbf{y}}^{i,\rho}(v)| =|\mathbf{y}^{i,\rho}(v)-\mathbf{y}^{i,\rho}(v_0)+\mathbf{y}^{i,\rho}(v_0)-\mathbf{y}^{m^0,\rho}(v_0)| &\leq\frac{C_{v}}{C_{\eta}-C_v}|v-v_0|+\frac{1}{q^{\min}}\left(K_f+\frac{C_vC_{\eta}C_z}{(C_{\eta}-C_v)^2}\right).
w\varpi_i(g \cdot t^{-\lambda_w})=w\varpi_i(g)\cdot w\varpi_i (t^{-\lambda_w})=w\varpi_i(g)\cdot t^{-(w\varpi_i,\ \lambda_w)}=w\varpi_i(g) t^{-A_{w\varpi_i}}=\gamma(g) z^{-A_\gamma}\gamma=w\varpi_i.
\partial J(u)v= \partial J(\alpha^{i}\varphi_{i}+v)v=\partial J(\alpha^{i}\varphi_{i})v+\partial^{2}J(\alpha^{i}\varphi_{i})v^{2}+o(\Vert v \Vert^{2}) \geq \partial J(\alpha^{i}\varphi_{i})v+\gamma \Vert v\Vert^{2}+o(\Vert v \Vert^{2}).
\int_{|z|\leq R}|z|^{-\beta q}dz =\int_{|z|\leq 1}|z|^{-\beta q}dz+\int_{1<|z|\leq R}|z|^{-\beta q}dz & \leq C \int_{0}^{1}r^{-\beta q+n-1}dr+C\max\{1,R^{-\beta q}\}\int_{1<|x|\leq R}dx & \stackrel{\beta<\frac{n}{q}}\leq C+CR^{-\beta q+n}e^{2|\rho|R}& \leq C(1+R^{-\beta q+n}e^{2|\rho|R}).
g_{\mu \nu}^{( 6 )}= \left( \begin{array}{c c}{{g_{\alpha \beta}^{( 5 )}+ e^{\psi_{1}}A_{\alpha}A_{\beta}}}&{{e^{\psi_{1}}A_{\alpha}}}\\{{e^{\psi_{1}}A_{\beta}}}&{{e^{\psi_{1}}}}\\ \end{array}\right) ,
\left\lbrace\begin{aligned}d\eta_q^{\epsilon,{\bf h}}(t) =\frac{\eta_p^{\epsilon,{\bf h}}(t)}{\epsilon}dt d\eta_p^{\epsilon,{\bf h}}(t) =-\frac{\eta_p^{\epsilon,{\bf h}}(t)}{\epsilon^2}dt+\frac{1}{\epsilon}Df(q^\epsilon(t)).\eta_q^{\epsilon,{\bf h}}(t)dt+\frac{1}{\epsilon}D\sigma(q^\epsilon(t)).\eta_q^{\epsilon,{\bf h}}(t)d\beta(t),\end{aligned}\right.
T_{\it 3}= m_{1}dz^1 \wedge d\bar z^2 \wedge d\bar z^3 +m_{2}d\bar z^1 \wedge dz^2 \wedge d\bar z^3 +m_{3}d\bar z^1 \wedge d\bar z^2 \wedge dz^3 +m_{4}dz^1 \wedge dz^2 \wedge dz^3 \ .
\sum_{n=0}^{\infty}\left(\frac{z^nq^{n^2+n}}{(zq;q^2)_{n+1}}\right)^2=\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}b_\omega^3(m,n)z^mq^n=\sum_{n=0}^{\infty}\frac{z^nq^n}{(zq;q^2)_{n+1}},
H_t(u,p) =\begin{cases}\sum^{N}_{k=1}\sum^4_{\mu=1}\beta^\mu_{kx}(u)\,p_{x\mu k}+\,\sum^{N}_{k=1}\sum^4_{\mu=1}\, \beta^\mu_{ky}(u)\,p_{y\mu k}, t\neq n\,T,\,n\in\, \mathbb{Z}, 0, t= n\,T,\,n\in\, \mathbb{Z}. \end{cases}
\frac{\xi^T D^2 \xi}{\tau_p k_s}- \xi^T D \xi = \sum_{i=1}^{k_s}\left[ \frac{\lambda_i^2}{\tau_p k_s}- \lambda_i\right] \xi_i^2 &\ge \left[ \frac{n^2 \rho_{2k}^2}{\tau_p k_s}- n \rho_{2k}\right] \|\beta^*_{S^* \backslash \widehat{S}_s}\|_2^2 \,,
x_1 = f_1(b,y) = (\phi_0 \circ \xi_1)(b,y), u_1 = g_1(b,y) = (\psi_0 \circ \xi_1)(b,y), x_2 = f_2(b,y) = (\phi_B \circ \xi_2)(b,y), u_2 = g_2(b,y) = (\psi_B \circ \xi_2)(b,y).
E_{\mu,\xi}=\mbox{span}\left\{P\psi^h_{\mu,\xi}: h=0,1,\cdots,n,\i=1,\cdots,k\right\}, & E_{\mu,\xi}^\perp =\left\{\phi\in H^1_0(\Omega): \langle \phi,P\psi^h_{\mu,\xi}\rangle =0: h=0,1,\cdots,n,\i=1,\cdots,k\right\}.
\liminf_{\alpha\to\infty}I(\alpha) \geqslant\int_{0}^{\infty}\liminf_{\alpha\to\infty}\exp\left\lbrace \frac{-\alpha w}{\alpha+w}\right\rbrace \frac{w^{-\kappa}}{\left(1+w/\alpha\right)^{2}}\mathrm{d}w =\int_{0}^{\infty}\exp\left\lbrace -w\right\rbrace w^{-\kappa}\mathrm{d}w=\Gamma(1-\kappa)>0.
\Pi^{\mu_{1}\cdot\cdot\cdot\mu_{l}\nu_{1}\cdot\cdot\cdot\nu_{l}}(p)=\sum_{m=-l}^{l}[e_{m}^{\mu_{1}\cdot\cdot\cdot\mu_{l}}(p)][e_{m}^{\nu_{1}\cdot\cdot\cdot\nu_{l}}(p)]^{*}
\psi_2 \circ \varphi_2 = (\psi_1 \sqcup (\psi_2 \setminus \psi_1)) \circ (\varphi_1 \sqcup (\varphi_2 \setminus \varphi_1)) = (\psi_1 \circ \varphi_1) \sqcup ((\psi_2 \setminus \psi_1) \circ (\varphi_2 \setminus \varphi_1)).
\sum_{N_{r}^{\prime}}\sum_{p = 1 , 2}\int_{C_{p}^{( r )}}\chi_{N_{r}^{\prime}}( t ) d t \left[ E_{p , N_{r}}^{( - 2 )}( Y_{p , N_{r}^{\prime}}( t ) ) - \partial_{q_{N_{r}}}Y_{p , N_{r}^{\prime}}( t ) + ( - 1 )^{e ( N_{r}^{\prime}) e ( N_{r})}\partial_{q_{N_{r}^{\prime}}}Y_{p , N_{r}}( t ) \right] = 0
\phi ( \theta , \varphi ) = a \left\{\cos^{2}\frac{\theta}{2}| 2 \lambda_{\phi}> - \frac{\sqrt{2}}{2}\sin \theta e^{- i \varphi}| 2 \lambda_{\phi}- \alpha > + \sin^{2}\frac{\theta}{2}e^{- 2 i \varphi}| 2 \lambda_{\phi}- 2 \alpha > \, . \right\}
\kappa^2 g^2 \alpha' \left(\frac{1}{2}(\frac{11}{3^2})^2-26 \cdot \frac{1}{2}(\frac{5}{3^2 \sqrt{2}})^2 \right) \phi^4=-\kappa^2 g^2 \alpha' \left(\frac{34}{27}\right) \phi^4.
\begin{aligned}\gamma \,\mathbf T_{\partial_j}\,K=-\Lambda^{-1}\,T_{\partial_j r/2\|\partial r\|}\,, \mathbf T_{\partial_j}= -V\, T_\Lambda^{-1/2}T_{\partial_j r/2\|\partial r\|}T_\Lambda^{-1/2}\,V^*. \end{aligned}
(\omega^0)' =\frac{\partial x^0}{\partial u}du=\partial' x^0, (\omega^0)'' =\frac{\partial x^0}{\partial v}dv=\partial'' x^0, (\omega^1)' =e^{-\mu_1x^0}\partial' x^1,\ (\omega^2)'=e^{-\mu_2x^0}\partial' x^2, (\omega^1)'' =e^{-\mu_1x^0}\partial'' x^1,\ (\omega^2)''=e^{-\mu_2x^0}\partial'' x^2.
R_{Tot}=\frac{R_{S}}{2\pi}[ln(\frac{r_{o}}{r_{i}})+\frac{L_{T}}{r_{i}}\frac{I_{0}(r_{i}/L_{T})}{I_{1}(r_{i}/L_{T})}+\frac{L_{T}}{r_{o}}\frac{K_{0}(r_{o}/L_{T})}{K_{1}(r_{o}/L_{T})}]
Z^N Y^{n_{\mu,i}+ N+1}\cdot v_{+}=\left(\sum_{\gamma \in \Gamma/\Gamma_q}\,\left( \gamma \cdot (x_i[f])\right)_q\right)^NY^{n_{\mu,i}+N+ 1}\cdot v_{+}&=(x_i[f_q])^NY^{n_{\mu,i}+ N+1}\cdot v_{+}.
\lim_{j\rightarrow+\infty}\left(\sup_{D_j}e^{-u(-v_{t_0,B}(\Psi))}\right)\int_{D_j}\frac{1}{B}\mathbb{I}_{\{-t_{0}-B<\Psi<-t_{0}\}}|f|^2e^{-\Psi}\le \left(\sup_{D}e^{-u(-v_{t_0,B}(\Psi))}\right)\int_{D}\frac{1}{B}\mathbb{I}_{\{-t_{0}-B<\Psi<-t_{0}\}}|f|^2e^{-\Psi}< +\infty
\langle \tau_p(\nu_{\gamma}\beta_1\nu_{\delta})\breve{\mathbf g}_p, \breve{\mathbf g}_p \rangle = p^{1/2}\xi_1(p)^2 \int_{K_{00}}\mathbf g_p(h \nu_{\gamma}\beta_1\nu_{\delta}\varpi_p) \underline{\chi}_p^{-1}(h) dh = p \underline{\chi}_p(\gamma)\mathrm{vol}(K_{00}) = \frac{\underline{\chi}_p(\gamma)}{p+1}.
| 1 \rangle , \, \, J_{+}( \vec{k}) | 1 \rangle , \, \, J_{+}( \vec{k}) J_{+}( \vec{k}^{\prime}) | 1 \rangle \, \, \mathrm{f o r}\, \, \vec{k}\not = \vec{k}^{\prime}, \, \, ( J_{+}( \vec{k}) )^{2}| 1 \rangle , \, \, \cdots
\epsilon^{2s-1}\int_{\Omega}{\int_{\Omega}{\frac{\overline{u}(x)-\overline{u}(y)}{|x-y|^{n+2s}}(v(x)-v(y))dx dy}}+\frac{1}{\epsilon}\int_{\Omega}{\overline{W}'(\overline{u})v\,dx}=0.
{\bf W}_{aa}=-\sum_{c\neq a}{\mbox{\boldmath $\beta$}}_a^*\cdot{\mbox{\boldmath $\beta$}}_c^*{\bf w}_{ac},\,\,\,\,{\bf W}_{ab}={\mbox{\boldmath $\beta$}}_a^*\cdot{\mbox{\boldmath $\beta$}}_b^*{\bf w}_{ab}\quad\hbox{\hskip 1cm if $a\neq b$}.
\nu_{\tau}((s,t]\times J) = \nu((\tau_s, \tau_t]\times J) = \nu((0, \tau_t]\times J) - \nu((0, \tau_s]\times J) \leq\nu((0, \tau_t]\times J) - \nu((0, \tau_s]\times J) + (\tau_t - \tau_s) = (\nu((0, \tau_t]\times J) + \tau_t) - (\nu((0, \tau_s]\times J) + \tau_s) = t-s.
CFS=max_{S_{k}}[\frac{r_{cf_{1}}+r_{cf_{2}}+\cdot\cdot\cdot+r_{cf_{k}}}{\sqrt{k+2(r_{f_{1}f_{2}}+\cdot\cdot\cdot+r_{f_{i}f_{j}}+\cdot\cdot\cdot+r_{f_{k}f_{k-1}})}}].
\left.\begin{array}{ll}\displaystyle \dfrac{6}{N}\left(\dfrac{a_2}{2m_1}\int_{\mathbb{R}^N}|\nabla \xi|^2dx+\dfrac{a_1}{4m_2}\int_{\mathbb{R}^N}|\nabla \eta|^2dx\right)-\dfrac{3}{2}a_1a_2\int_{\mathbb{R}^N}\eta \xi^2dx=0.\end{array}\right.
{\cal L}_{g,4A}= -\frac{1}{4}\sum_{n,m,l,k\neq(0,0,0)}(\sum_{j=0}^{N}a_{jn}a_{jm}a_{jl}a_{jk}) g_L f^{abc}g_L f^{ade}\tilde{A}_{\mu}^{n}\tilde{A}_{\nu}^{m}\tilde{A}^{l~\mu}\tilde{A}^{k~\nu},
{\widetilde{\cal C}}_T^\prime=2b{\widehat c}\sum_{\alpha,\beta}2B^2 \left(q_\alpha^2+q_\beta^2\right)\Delta_2(X_{\alpha\beta}^2)\Big|_\sigma -8d^2 \sum_{\alpha,\beta}2B^2 q_\alpha q_\beta \Delta_2(X_{\alpha\beta}^2)\Big|_{C_{\mu\nu}}~.
\left\langle f,g\right\rangle_{(-\Delta )^{2\alpha-d/2}}=\int_{\mathbb{R}^{d}}\frac{1}{\nu (d-4\alpha)}\int_{\mathbb{R}^{d}}\frac{1}{\|\mathbf{x}-\mathbf{y}\|^{4\alpha}}f(\mathbf{y})\overline{g(\mathbf{x})}d\mathbf{y}d\mathbf{x},\forall f,g \in \mathcal{S}(\mathbb{R}^{d}),
R_{k}\!\Big(\!\boldsymbol{a}, \boldsymbol{v}_{k},\boldsymbol{\Theta},\boldsymbol{\rho}\!\Big)\!\!=\!\!B\log_{2}\!\Big(\!1+\gamma_{k}\Big(\boldsymbol{a}, \boldsymbol{v}_{k},\boldsymbol{\Theta},\boldsymbol{\rho}\Big)\!\Big), \ \ \forall k \!\in \!\mathcal{K},
[d, s, x] = [y_1 \dots y_k, s, x] = \sum_{i=1}^k y_1 \dots y_{i-1}[y_i, s, x] y_{i+1}\dots y_k + \ \sum_{1 \le i < i' \le k}y_1 \dots y_{i-1}\bigl( [y_i, s] y_{i+1}\dots y_{i' - 1}[y_{i'}, x] + [y_i, x] y_{i+1}\dots y_{i' - 1}[y_{i'}, s] \bigr) y_{i' +1}\dots y_k
v_{\mathrm{nc}}:=\sum_{{\ell}=1}^{j}\xi_{\ell}\phi_{\mathrm{nc}}({\ell}), v_{\mathrm{pw}}:=\sum_{{\ell}=1}^{j}\xi_{\ell}\phi_{\mathrm{pw}}({\ell}), w_{\mathrm{pw}}:=\sum_{{\ell}=1}^{j}\xi_{\ell}\lambda_h(\ell)\phi_{\mathrm{pw}}({\ell}).
|f(x)-f(y)|\leqslant C_n \cdot\frac{(\Vert Q\Vert_1)^{\frac{1}{q}}}{\log^{\frac{1}{n}}\left(1+\frac{r_0}{2|x-y|}\right)}, \,\,r_0 =d(K, \partial{\mathbb B}^n),
\phi_0(t) =x_0t^{-\frac{1}{4}},t\in(0,l], \phi_n(t)=3t^{-\frac{1}{4}}+ \int_{0}^{t}\frac{(t-s)^{-\frac{1}{2}}}{\Gamma(\frac{1}{2})}s^{-\frac{1}{3}}[1+(\phi_{n-1}(s))^{\frac{4}{3}}]ds,t\in(0,l],n=1,2,\cdots.
{d\tilde{n}^{(\lambda)}(\omega) \over d\omega}={dn^{(\lambda)}(\omega) \over d\omega}-{1 \over 4\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]~~~,
\begin{aligned}\ln_{e_1}\eta =\bigg(\tilde{a}\,e_1(\tilde{b})-\tilde{b}\,e_1(\tilde{a})+\frac{\tilde{b}\lambda_x-\tilde{a}\lambda_y}{\lambda^2}\bigg)\,e_1-r\,e_2, \ln_{e_2}\eta =(\tilde{a}\,e_2(\tilde{b})-\tilde{b}\,e_2(\tilde{a})-r)\,e_1.\end{aligned}
\begin{pmatrix}\ell_1 \ell_2\end{pmatrix}\begin{pmatrix}-K 1 1 -M \end{pmatrix}^{-1}\begin{pmatrix}\ell^t_1 \ell^t_2\end{pmatrix}= \begin{pmatrix}\ell_1 \ell_2\end{pmatrix}\begin{pmatrix}-K 1 1 -M \end{pmatrix}^{-1}\begin{pmatrix}m_1^t m_2^t\end{pmatrix}.
w_k(x)-w_k(y) =v_k(x)\,\eta(x)+\Psi_a(x)\,(1-\eta(x))-v_k(y)\,\eta(y)-\Psi_a(y)\,(1-\eta(y)) &=\eta(x) \Big(v_k(x)-v_k(y)\Big)+(1-\eta(x))\Big(\Psi_a(x)-\Psi_a(y)\Big) &+\Big(\eta(x)-\eta(y)\Big)\,\Big(v_k(y)-\Psi_a(y)\Big).
I_{2}= f(x) \int_{-\delta}^{+\delta}(1-v^{2})^{n}\dv + \int_{-\delta}^{+\delta}\bigl(f(x+v)-f(x)\bigr) (1-v^{2})^{n}\dv & = 2 f(x) (J_{n}-J_{n}^{*}) + \int_{-\delta}^{+\delta}\bigl(f(x+v)-f(x)\bigr) (1-v^{2})^{n}\dv.
F(s) = \left[\frac{1 + \sin \theta}{2\sin\theta}\log \frac{1 + \sin \theta}{2 \sin \theta}- \frac{1 - \sin \theta}{2 \sin\theta}\log \frac{1 - \sin \theta}{2 \sin \theta}\right], \theta(s) =2\sin^{-1}\left(\frac{1}{2s}\right)
\left(D_{0+}^{\alpha,\beta,\gamma}f\right)(x)=\left(\frac{d}{dx}\right)^{[\operatorname{Re}(\alpha)]+1}\left(I_{0+}^{-\alpha+[\operatorname{Re}(\alpha)]+1,-\beta-[\operatorname{Re}(\alpha)]-1,\alpha+\gamma-[\operatorname{Re}(\alpha)]-1}f\right)(x)
e\left( G\right) =\binom{n}{3}-\sum_{i=1}^{k}\binom{n_{i}}{3}=\binom{n}{3}-\sum_{i=1}^{k}f\left( n_{i}\right) \leq\binom{n}{3}-kf\binom{n/k}{3}& =\binom{n}{3}-k\binom{n/k}{3}.
\bigl<n;t|i\hbar\frac{\partial}{\partial t}|n;t\bigr>=\int d q \bigl<n;t|q\bigr>\Bigl(i\hbar\frac{\partial}{\partial t}\Bigr)\bigl<q|n;t\bigr>=(n+\frac{1}{2})\frac{\hbar}{2\sqrt{\kappa}}\frac{\dot{B}(t)A(t)-B(t)\dot{A}(t)}{A(t)},
B_{n,\mu,0}(1)\Lambda_p=\sum_{j=0}^k \lambda_jB_{n,\mu,j}(1)+\sum_{j=1}^k \lambda_jS_{n,\mu,j}(1) \coloneqq T(n, \mu)+\sum_{j=1}^k \lambda_jS_{n,\mu,j}(1).
\Psi_{\kappa}^{( R )}= \frac 1{\pi \sqrt{2}}\exp \left( \frac{\pi \kappa}2 - i \frac{\kappa}2 \ln \left( \frac{x_{+}}{- x_{-}}\right) \right) K_{i \kappa}\left( m \sqrt{- x_{-}x_{+}}\right) .
{\cal F}_{\epsilon}^{i}( \vec{x}, t ) = \frac{1}{2 \pi}\int d^{2}\vec{p}\mathrm{e}^{i \vec{p}\vec{x}}\tilde{\cal F}_{\epsilon}^{i}( \vec{p}, t ) , \qquad \overline{{\tilde{\cal F}}}{}_{\epsilon}^{i}( \vec{p}, t ) = \tilde{\cal F}_{\epsilon}^{i}( - \vec{p}, t ) ,
\log|{\mathbf{\Gamma}_2}|+{\rm Tr}({\mathbf{\Gamma}_2}^{-1}(\sigma^{2}{\bf I}_2+{\bf G}_{13}{\bf Q}_3{\bf G}_{13}^{T})-{\rm Tr}({\bf I}_2) &= \log|\sigma^{2}{\bf I}_2+{\bf G}_{13}{\bf Q}_3{\bf G}_{13}^{T}|+\epsilon.
g(\omega)= \exp\Big(\int_0^t\frac{m_1(t)-m_2(t)}{\sigma^2(t)}d\omega_t-\frac{1}{2}\int_0^t\frac{m_1^2(t)-m_2^2(t)}{\sigma^2(t)}dt\Big) = \exp\Big(\int_0^t\frac{m_1(t)-m_2(t)}{\sigma^2(t)}(d\omega_t-m_2(t)dt)-\frac{1}{2}\int_0^t\frac{(m_1(t)-m_2(t))^2}{\sigma^2(t)}dt\Big).
\max_{w_\mathcal{I}\in \mathcal{W}_\mathcal{I}}\Pr\{\hat W_\mathcal{I}\ne w_\mathcal{I}| W_\mathcal{I}=w_\mathcal{I}\}= \max_{w_\mathcal{I}\in \mathcal{W}_\mathcal{I}}\Pr\{\cup_{i=1}^2\{Y_i^n \notin \mathcal{D}_i^{(n)}(w_i)\}| W_\mathcal{I}=w_\mathcal{I}\}.
\frac{\partial g_{ab}}{\partial x^{c}}=\frac{\partial}{\partial x^{c}}\langle\frac{\partial\vec{\Psi}}{\partial x^{a}},\frac{\partial\vec{\Psi}}{\partial x^{b}}\rangle=\langle\frac{\partial^{2}\vec{\Psi}}{\partial x^{c}\partial x^{a}},\frac{\partial\vec{\Psi}}{\partial x^{b}}\rangle+\langle\frac{\partial\vec{\Psi}}{\partial x^{a}},\frac{\partial^{2}\vec{\Psi}}{\partial x^{c}\partial x^{b}}\rangle
w (\lambda\cdot \zeta_k(a_1)) = \frac{(-1)^{n-1}}{n-1}w \zeta_k x_\lambda \zeta_2(a_1) & = \frac{(-1)^{n-1}(-1)^{n-k}}{n-1}\zeta_{n+1-k}x_\lambda \zeta_2(a_1) & = (-1)^{n-k}\lambda \cdot \zeta_{n+1-k}(a_1) & = \lambda \cdot ((-1)^{n-k}\zeta_{n+1-k}(a_1)) & = \lambda \cdot (w \zeta_k(a_1))
\left\{\nabla_{\bf r^{\prime}}^{2}+ \frac{2 M}{\hbar^{2}}\left[ E - V ({\bf r^{\prime}}) \right] \right\}{\mathcal G}_{D}({\bf r^{\prime \prime}},{\bf r^{\prime}}; E ) = \delta ({\bf r^{\prime \prime}}-{\bf r^{\prime}}) \; ,
\left(1-\frac{\beta}{\tilde{\beta}}\right)\,\Omega(x_n) \leq-\beta\,E^\dagger(x_n)+\Omega(x_n)-\Omega^\dagger+r_n\,\|F(x_n)-y^\dagger\|+\Omega^\dagger-\frac{\beta\,\tilde{c}}{\tilde{\beta}}&=-D_\beta(r_n)+\Omega^\dagger-\frac{\beta\,\tilde{c}}{\tilde{\beta}}&\leq\Omega^\dagger-\frac{\beta\,\tilde{c}}{\tilde{\beta}}.
\begin{aligned}\Pr(T_k = t) = \Pr( \{X_{t-1}= k-1\}\cap \{X_t = k \}) = \Pr(X_{t-1}= k-1) \cdot \Pr(X_t = k \, \vert \, X_{t-1}= k-1 ) = \Pr(X_{t-1}= k-1) \cdot p_{k-1}^{},\end{aligned}
-\int_0^T(\theta, \phi_t)\, dt-\int_0^T(u\theta, \nabla\phi)\, dt+\kappa\int_0^T(\Lambda^{\frac{1}{2}}\theta, \Lambda^{\frac{1}{2}}\phi)\, dt=(\theta_0, \phi(x,0))+\int_0^T(f, \phi)\, dt.
A\cdot B=(\begin{matrix}A^{0}&A^{1}&A^{2}&A^{3}\end{matrix})(\begin{matrix}-1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{matrix})(\begin{matrix}B^{0}\\B^{1}\\B^{2}\\B^{3}\end{matrix})
(m-n)\binom{m+n+1}{k}=\sum_{i=0}^k(m-n-k+2i)\binom{m+1}{k-i}\binom{n+1}{i}, \sum_{i=0}^k\binom{m+1}{k-i}\binom{n+1}{i}\binom{m-k+i+1}{3}\delta_{m+n-k,0}=\binom{m+1}{3}\delta_{m+n,0}\delta_{k,0}
\begin{aligned}a^+=-\frac{\log R_1}{\log\left(\sqrt{\frac{R_2}{R_1}}\right)}, a^-=-\frac{\log R_2}{\log\left(\sqrt{\frac{R_1}{R_2}}\right)}, b^+=\frac{1}{\log\left(\sqrt{\frac{R_2}{R_1}}\right)}, b^-=\frac{1}{\log\left(\sqrt{\frac{R_1}{R_2}}\right)}.\end{aligned}
T(f_1,f_2,f_3)=\frac{1}{N^2}\sum\limits_{\substack{n_1,n_2,n_3\leqslant N \vert n_1 - \sqrt{2}n_2 + (-1+\sqrt{2})n_3\vert\leqslant \frac{1}{2}}}f_1(n_1)f_2(n_2)f_3(n_3).
\delta E_k (\rho_k,n_k) = \frac{n_k^2}{2 \rho_k}\frac{\Gamma\left( 2-{1 \over 2n_k}\right)}{\Gamma\left({1 \over 2n_k}\right)}+ \frac{n_{k-1}^2}{2 \rho_k}\left( \frac{\rho_k}{\rho_{k-1}}\right)^{2n_{k-1}}\frac{\Gamma\left({4n_{k-1}-1 \over 2n_k}\right)}{\Gamma\left({1 \over 2n_k}\right)}
\sigma_{lb}(T): =\{\lambda \in \mathbb{C}: \lambda I -T \thinspace \mbox{is not lower semi-Browder}\}, \sigma_{ub}(T): =\{\lambda \in \mathbb{C}: \lambda I -T \thinspace \mbox{is not upper semi-Browder}\}, \sigma_b(T): =\{\lambda \in \mathbb{C}: \lambda I -T \thinspace \mbox{is not Browder}\}, \thinspace \mbox{respectively}.
a(pmI)a(p^rI)-\chi(p^2)p^{2k-3}a(mI)a(p^{r-1}I) &\quad+\epsilon\chi(p)p^{k-2}a(mI) a\begin{pmatrix}p^{r-1}m &p^{r+1}m\end{pmatrix}&+\epsilon\chi(p)p^{k-2}a(mI)\sum_{\substack{1\le u<p/2 u^2\not\equiv-1\,(p)}}a\left(p^rm\begin{pmatrix}(1+u^2)/p u u p\end{pmatrix}\right).
( C{\cal F}^{( 2 )}+ D ) \partial_{a}^{2}\widetilde{\cal F}( \tilde{a}) - C{\cal F}^{( 3 )}\partial_{a}\widetilde{\cal F}( \tilde{a}) - ( A{\cal F}^{( 2 )}+ B ) ( C{\cal F}^{( 2 )}+ D )^{2}= 0 ,
\partial_{x^{2}}f^{k}=\partial_{x}f^{k}{\displaystyle\sum\limits_{j=0}^{k-1}}\frac{F_{x^{2}}^{j}\left( F_{y}^{j}\right)^{2}-2F_{xy}^{j}F_{x}^{j}F_{y}^{j}+F_{y^{2}}^{j}\left( F_{x}^{j}\right)^{2}}{F_{x}^{j}\left(F_{y}^{j}\right)^{2}}\partial_{x}f^{j}.
TS_{N}={\textstyle\sum\limits_{n\geq1}}TP_{N=ne}S_{n}={\textstyle\sum\limits_{n\geq1}}TS_{n}.TP_{N=ne}e={\textstyle\sum\limits_{n\geq1}}nTx.TP_{N=ne}e & =Tx.{\textstyle\sum\limits_{n\geq1}}nTP_{N=ne}e=Tx.TN.
\Psi_{F}[r_{i}]=NA\Psi[r_{i}]=N\sum_{\pi\in S_{N}}(-1)^{\pi}\prod_{i=1}^{N}\psi_{\alpha_{\pi(i)}}(r_{i})=N\sum_{\pi\in S_{N}}(-1)^{\pi}\psi_{\alpha_{\pi(1)}}\otimes\psi_{\alpha_{\pi(2)}}\otimes\cdot\cdot\cdot\otimes\psi_{\alpha_{\pi(N)}}.
\emptyset =\mathcal{X}^{\Delta}_0\subset X=\mathcal{X}^{\Delta}_1\subset \mathcal{X}^{\Delta}_2\subset\cdots \subset \mathcal{X}^{\Delta}_k\subset \mathcal{X}^{\Delta}_{k+1}\subset\cdots \subset \bigcup_{k= 0}^{\infty}\mathcal{X}^{\Delta}_k=\mathcal{X}^{\Delta}.
\begin{array}{lll}\noindent (1-\lambda_2)b_2+2(1-\lambda_2)a_2+\lambda_3b_2a_1-\lambda_3b_2a_2-\lambda_3a_2^2 +\lambda_3a_1a_2\ & \ =(\lambda_3-\lambda_2)(1-\lambda_2),\end{array}
g_{\,\,j}^{i}=\mathbf{E}^{i}\cdot\mathbf{E}_{j}=\delta_{\,\,j}^{i}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,g_{i}^{\,\,j}=\mathbf{E}_{i}\cdot\mathbf{E}^{j}=\delta_{i}^{\,\,j}
\sum_{n\geq0}(\sum_{i\geq0}R_iq_{n-i})w^n=\sum_{i\geq0}R_iw^i\cdot\sum_{j\geq0}q_jw^j &=\exp\Big(\sum_{m\nmid n\geq1}\frac{w^{n}p_{n}}{n}(1-\xi_m^n)c^n\Big)\exp\Big(\sum_{m\nmid n\geq1}\frac{w^n p_{n}}{n\epsilon_n}\Big)=\exp\Big(\sum_{m\nmid n\geq1}\frac{w^np_{n}}{n\epsilon_n}q^n\Big) &=\sum_{n\geq0}q_n(qw)^n.
\sum_{\substack{1\leq \tilde{\jmath}<\infty p|\tilde{\jmath}\Rightarrow p|j}}\frac{(j,\tilde{\jmath})}{\tilde{\jmath}}=\prod_{p|j}\sum_{\nu=0}^{\infty}\frac{p^{\min\{j_p,\nu\}}}{p^{\nu}}= \prod_{p|j}\bigg(j_p + \frac{1}{1-\frac{1}{p}}\bigg) \ll j^{\varepsilon}
Z_{EH}={1\over 4}{1\over (\eta\bar\eta)^4}\int dp_xdp_y \,\,q^{p_x\bar p_x + p_y\bar p_y}\sum_i\left|{\theta_i\over \eta}\right|^4 +{1\over 4}\sum_{i,j}\left|{\theta_i\over\theta_j}\right|^4\,\, .
_{2}\phi_{4}\left(\sqrt{a},-\sqrt{a};i\sqrt{b}q^{1/4},-i\sqrt{b}q^{1/4},\sqrt{az},-\sqrt{az};\sqrt{q},-bz\sqrt{q}\right)=\frac{\left(z;q\right)_{\infty}}{\left(az;q\right)_{\infty}}\sum_{n=0}^{\infty}\frac{\left(a;q\right)_{n}S_{n}\left(b;q\right)z^{n}}{\left(-bq^{1/2};q\right)_{n}}.
g_1+g_3+g_5+g_7 =g_1+g_4+g_6+g_7 =g_2+g_3+g_6+g_7 =g_2+g_4+g_5+g_7 =g_1+g_2+g_3+g_4 =g_1+g_2+g_5+g_6 =g_3+g_4+g_5+g_6.
\mathbb{P}\Bigl( \sup_{(s,t)\in\mathbf{E}}X(s,t)> u \Bigr) =S a_1a_2\mathcal{P}_{\alpha_2}^{ba_2^{-\alpha_2}}\mathcal{H}_{\alpha_1}u^{{2}/{\alpha_1}}\Psi(u) \bigl(1+o(1)\bigr);
( \hat{\gamma}_{\hat{m}\hat{n}})_{\alpha}{}^{\beta}( \hat{\gamma}^{\hat{m}\hat{n}})_{\gamma}{}^{\delta}= 2 \delta_{\alpha}{}^{\beta}\delta_{\gamma}{}^{\delta}- 8 \delta_{\alpha}{}^{\delta}\delta_{\gamma}{}^{\beta}\, , \qquad ( \hat{\gamma}_{\hat{m}\hat{n}})_{\alpha}{}^{\beta}( \hat{\gamma}^{\hat{p}\hat{q}})_{\beta}{}^{\alpha}= - 8 \delta_{[ \hat{m}}{}^{\hat{p}}\delta_{\hat{n}]}{}^{\hat{q}}\, .
\|\hat{F}_2^l\|_{0,r}\le\|\hat{F}_2^{l+1}\|_{0,r}+\|T_2^{l+1}\|_{0,r}+2^4c_0^{\frac{4-2}{2}}\|\hat{F}_4^{l+1}\|_{0,r}&+2^4 c_0^{\frac{4-2}{2}}\|T_4^{l+1}\|_{0,r}+\sum_{m=6}^N2^m c_0^{\frac{m-2}{2}}\|J_m^{l+1}\|_{0,r}.
[\begin{matrix}c_{2}c_{3}&-c_{3}s_{2}c_{1}+s_{3}s_{1}&c_{3}s_{2}s_{1}+s_{3}c_{1}\\s_{2}&c_{1}c_{2}&-c_{2}s_{1}\\ -s_{3}c_{2}&s_{3}s_{2}c_{1}+c_{3}s_{1}&-s_{3}s_{2}s_{1}+c_{3}c_{1}\end{matrix}]
\begin{cases}u_{tt}- \displaystyle \left(x^\alpha u_{x}\right)_x=f, (t,x)\in Q, u(t,1)=0, (0,T), \begin{cases}u (t,0)=0, \alpha \in (0,1), & (x^\alpha u_{x})(t,0)=0, \alpha \in [1,2),\end{cases}t\in (0,T), u (0,x)=u_0(x) u_{t}(0,x)=u_1(x) x\in (0,1),\end{cases}
(M_0 \cdot M_1 M_0 M_1)^r = M_0 (M_1 M_0)^{2r-1}M_1, (M_0 \cdot M_1 M_0 M_1)^r M_0 = M_0 (M_1 M_0)^{2r}, (M_1 M_0 M_1 \cdot M_0)^r =(M_1 M_0)^{2r}, (M_1 M_0 M_1 \cdot M_0)^r M_1 M_0 M_1 =(M_1 M_0)^{2r+1}M_1 ,
\sum_{n\leqslant N}r(n)^2\omega(n) =\prod_p\big(1+r(p)^2\omega(p)\big)-\sum_{n>N}r(n)^2\omega(n) &=\big(1+O\big(N^{-C/(\log\log N)^3}\big)\big)\prod_p\big(1+r(p)^2\omega(p)\big),
y_{\mathbf{s}}=y_{\mathbf{t}}+O\Bigl(T_k Y_{\mathbf{s}}\sum_{i=1}^k\frac{\log{[r_i,s_i]/s_i}}{\log{R}}\Bigr)=y_{\mathbf{t}}+O\Bigl(T_k Y_{\mathbf{s}}\frac{\log{A}}{\log{R}}\Bigr).
\lim_{\rho\to\infty}\gamma_k^\mathrm{B}[\iota]= \frac{M\beta_{k}}{\sum_{i=1}^{K}\beta_{i}+\sum_{i=1}^{K}\beta_{i}}\lim_{\rho\to\infty}\gamma_k^\mathrm{C}[\iota]= \frac{M\beta_{k}}{\sum_{i=1}^{K}\beta_{i}}
\int_0^{\infty}g^T(x,v) \mathrm{d}v = \int_0^{\infty}\int_0^1 \frac{e^{-(1-y)/\kappa |v|}}{\kappa |v| (1-e^{-1/\kappa|v|})}\rho_g(x+y) \left( \alpha\mathcal{M}_{T(y+x)}+ (1-\alpha) \mathcal{M}_{\tau(y+x)}\right) \mathrm{d}y \mathrm{d}v = \int_0^1 [ \alpha \mathcal{K}_1(T,y)+ (1-\alpha) \mathcal{K}_1(\tau,y) ] \rho_g (x+y) \mathrm{d}y.
\hat{\tilde{sec}}(U, V)= \frac{\tilde{g}(\hat{\tilde{R}}(U,V)V, U)}{\tilde{g}(U, U) \tilde{g}(V, V)- \tilde{g}(U, V)^2}= \frac{1}{\lambda^2}g(\hat{\tilde{R}}(U,V)V, U).
I_{1,n}(t) \le C_0\Vert{\mathbf F}(t)\Vert_{L^2(\mathbb R^2_+)}^2+C_1\left\{I(t)+I_0(t)+I_{\sigma}(t)+I_2(t)\right\}&=C_0\Vert{\mathbf F}(t)\Vert_{L^2(\mathbb R^2_+)}^2+C_1 I_{1,\ast}(t)\,.
\frac{\omega^{-1}(2b;\mathcal{F}_{RDT,p}(C))^{2}}{4}=\inf_{f_{+}-f_{-}\in\mathcal{F}_{RDT,p}(C)}\sum_{i=1}^n\left( \frac{f_+^2(x_i)}{\sigma^2(x_i)}+ \frac{f_-^2(x_i)}{\sigma^2(x_i)}\right) 2(f_+(0)-f_-(0))\ge 2b
\theta^1 :=du-u_xdx-u_ydy,\quad\theta^2:=du_v-v_{x}dx-v_{y}dy, \theta^3 :=du_x-F_1dx-u_{xy}dy,\theta^4:=du_y-u_{xy}d x-u_{yy}dy \theta^5 :=dv_x-F_2dx-v_{xy}dy,\quad\theta^6:=dv_y-v_{xy}d x-v_{yy}dy.
L_j :={}_{r_k^{j-1}}(L_1), L_j^\prime :={}_{r_k^{j-1}}(L_1^\prime), \delta_j := r_k^{j-1}(\delta_1), \delta_j^\prime := r_k^{j-1}(\delta_1^\prime), &X_{i,j}:={}_{r_k^{j-1}}(X_i), Y_{i,j}:={}_{r_k^{j-1}}(Y_i),
W_{n-1}\cdot \exp\left (\alpha_1 U_{n-1}^2\right ) = W_{n-2}\cdot\exp\Bigg ( \alpha_1\left(Z_{n-1}+\left (a+ \frac{s}{2\alpha_1}\right )U_{n-2}\right)^2 + &\left (a^2\alpha_1 - s\eta\right )U_{n-2}^2 - \alpha_1\left(a+\frac{s}{2\alpha_1}\right)^2 U_{n-2}^2\Bigg).
\hat{\mathbb{E}}_{\tau+}[\phi(B_{\tau+t_1}-B_{\tau},\cdots,B_{\tau+t_m}-B_{\tau})]=\hat{\mathbb{E}}[\phi(B_{\tau+t_1}-B_{\tau},\cdots,B_{\tau+t_m}-B_{\tau})]=\hat{\mathbb{E}}[\phi(B_{t_1},\cdots,B_{t_m})].
\widetilde{F}_{0}^{(\rho)}(z)=\begin{cases}1-a^{(\rho)}z^{-1}+O\left(z^{-2}\right), \ \mbox{if}\ \rho\not\equiv p \mod (p+1), z-a^{(\rho)}+O\left(z^{-1}\right), \ \mbox{if}\ \rho\equiv p \mod (p+1).\end{cases}
\langle \Omega, e^{-B_{\tau^\mathrm{eff}}}\mathcal{G}_N^\mathrm{eff}e^{B_{\tau^\mathrm{eff}}}\Omega\rangle \le\; 4\pi \mathfrak{a}(N-1)+e_\Lambda \mathfrak{a}^2 &-\frac{1}{2}\sum_{p\in \Lambda_+^*}\left[ p^2+8\pi\mathfrak{a}-\sqrt{|p|^4+ 16\pi \mathfrak{a}p^2}-\frac{(8\pi \mathfrak{a})^2}{2p^2}\right] &+CN^{-\nu/2}.
\{k \tau \}- \{F_n \tau \}= \{(F_{a_1}+ \ldots + F_{a_s}) \tau \}- \{F_n \tau \}&\geq \tau^{a_1}- \tau^{a_2}- \ldots - \tau^{a_s}- \tau^n &> \tau^{a_1}- \tau^{a_1+2}- \tau^{a_1+4}- \ldots & = \tau^{a_1}\left( 1 - \frac{\tau^2}{1-\tau^2}\right) & > 0
\frac{\partial Y}{\partial x}=\left( \frac{A_3}{x^2}+\frac{A_2}{x}+A_1+A_0 x \right)Y , \frac{\partial Y}{\partial t_1}=\left( x B_{11}+B_{10}\right)Y ,\frac{\partial Y}{\partial t_2}=\left( -\frac{A_3}{t_2x}+B_{20}\right)Y ,
\widetilde{\mathcal{L}}(U) := M_1UA + (B_1 + \gamma M_3)UM + M_2UB_2 - C_2UC_1 - C_2^TUC_1^T; &\widetilde{\mathcal{R}}(U) := M_3FM; &\widehat{\mathcal{L}}(U) := M_0D_1^{-1}UA + (A_1+\gamma M_0D_1)UM_0 + M_0D_1^{-1}D_2^2UA_2 - CD_2UCD_3 - D_2C^TUD_3C^T; &\widehat{\mathcal{R}}(U) := M_0D_1FM_0;