harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\lambda_{n}^{n-1}(\widehat{\lambda}_{n}-1)+\widehat{\lambda}_{n}^{2}\frac{\lambda_{n}^{n-1}-\widehat{\lambda}_{n}^{n-1}}{\lambda_{n}-\widehat{\lambda}_{n}}=\lambda_{n}^{n-1}(\widehat{\lambda}_{n}-1)+\widehat{\lambda}_{n}^{2}(\lambda_{n}^{n-2}+\lambda_{n}^{n-3}\widehat{\lambda}_{n}+\cdots+\widehat{\lambda}_{n}^{n-2})=0.
\begin{aligned}\sqrt{\eta_2}\, R_\eta \in L^\infty(0,T; H^{2s+1}(\mathbb T^d_{\ell})), \sqrt{\eta_2}\, \Delta^{s+1}R_\eta \in L^2(0,T; L^2(\mathbb T^d_{\ell})), \eta_1^{\frac{1}{\alpha}}\, R_\eta^{-1}\in L^\infty(0,T; L^{\alpha}(\mathbb T^d_{\ell})), \sqrt{\eta_1}\, \nabla R_\eta^{-\frac{\alpha}{2}}\in L^2(0,T; L^2(\mathbb T^d_{\ell})).\end{aligned}
( \Delta X )_{j}= ( e^{\textstyle \alpha^{i}H_{i}^{\prime}}X^{\prime \prime}+ e^{\textstyle \beta^{i}H_{i}^{\prime \prime}}X^{\prime})_{j}, \Delta H_{i}= H_{i}^{\prime}+ H_{i}^{\prime \prime}, \alpha^{i}\equiv \beta^{i}+ \gamma^{i}.
{\cal F}_{0}= - \frac{\mathrm{l n}Z_{0}}{\beta V}= \frac{1}{\beta V}\sum_{n}\sum_{\bf p}\mathrm{l n}\left( \frac{\beta M^{2}}{\rho}\right) + 2 T \int \frac{d^{2}p}{( 2 \pi )^{2}}\left[ \beta \omega + l n ( 1 - e^{- \beta \omega}) \right] \; ,
f(u) = -\left(C_iD_i+C_gD_g\right)u(1-u)^2 - C_gD_g u(1-u), D(u) = D_i\left(1-4u+3u^2\right) + D_g\left(4u-3u^2\right), g(u) = \lambda_g u (1-u) +\left[\lambda_i-\lambda_g- \left(k_i - k_g\right) \right] u(1-u)^2 -k_g u.
\beta = \sum_{i=0}^\infty \sum_{j=0}^{j_i}(\beta_{i,j}^+ + \beta_{i,j}^-), \;\,\beta_{i,j}^+ = i\partial\bar\partial(r^{d_{i,j}^+}\phi_{i,j}), \;\,\beta_{i,j}^- = i\partial\bar\partial(r^{d_{i}^-}\phi_{i,j}),
\left (\begin{array}{cc}\Re\left(\frac{\nu_{2k-1}}{|\nu_{2k-1}|}\right) -\Im\left(\frac{\nu_{2k-1}}{|\nu_{2k-1}|}\right) \Im\left(\frac{\nu_{2k-1}}{|\nu_{2k-1}|}\right) \Re\left(\frac{\nu_{2k-1}}{|\nu_{2k-1}|}\right)\end{array}\right)\ ,k=1,\dots, s,
(B_q u_j)(m)= \sum_{n=1}^q S(m,n;q) u_j(n) = \sum_{n=1}^q \sum_{\substack{k=1 \gcd(k,q)=1}}^q \exp \Bigl(\frac{2 \pi i}{q}(mk+nk^* ) \Bigr) \exp \Bigl(\frac{2 \pi i jn}{q}\Bigr) = \sum_{\substack{k=1 \gcd(k,q)=1}}^q \exp \Bigl(\frac{2 \pi i mk}{q}\Bigr) \sum_{n=1}^q \exp \Bigl(\frac{2 \pi i n}{q}(k^*+j) \Bigr).
d \mu = \frac{V \kappa^{2}}{2 \pi^{2}\hbar^{3}( a + 1 )^{2}}\left( 1 - e^{- ( a + 1 ) P_{0}/ \kappa}\right)^{2}e^{- ( 4 a + 1 ) P_{0}/ \kappa}d P_{0}.
\sum_{{k=1,k\neq j}}^n\dfrac{1}{z_{\nu,k}-z_{\nu,j}}= \dfrac{-z_{\nu,j}K_\nu''(z_{\nu,j})-2z_{\nu,j}K_{\nu+1}'(z_{\nu,j})+2(z_{\nu,j}+2\nu-1)K_\nu'(z_{\nu,j})- 2K_{\nu+1}(z_{\nu,j})}{-2z_{\nu,j}K_\nu'(z_{\nu,j})},
|f_{3}|\leq\chi_{E'_{3}}\chi_{(Dil_{2^{\mu}}(I_{P'}\times I_{P''}))^{c}}=\chi_{E'_{3}}\{\chi_{(2^{\mu}I_{P'})^{c}}+\chi_{(2^{\mu}I_{P''})^{c}}-\chi_{(2^{\mu}I_{P'})^{c}}\chi_{(2^{\mu}I_{P''})^{c}}\},
\mbox{\boldmath$\Psi$\unboldmath}=\left( \begin{array}{c}\Psi_{+1}(x) \Psi_{-1}(x)\end{array}\right) \;,\;\;\Psi_{\zeta}(x)=\left( \begin{array}{c}\chi_{\zeta}(x) \varphi_{\zeta}(x)\end{array}\right) \;,
\prod_{n \le N}\left( 1-R^{n}\right)^{\pi(n)}= \dfrac{e^{-\gamma}}{C_{X}}\cdot \dfrac{1}{N}\exp \left( O \left( \dfrac{1}{N}\right) \right)= \dfrac{e^{-\gamma}}{C_{X}}\cdot \dfrac{1}{N}\left( 1+ O \left( \dfrac{1}{N}\right) \right),
{\partial \qquad \over \partial Q_I(t)}= \int_0^{\Delta t}ds \left[{\partial Q(s,t) \over \partial Q_I(t)}\right]{\delta \qquad \over \delta Q(s,t)}= \int_0^{\Delta t}ds{s^{I-1}\over (I-1)!}{\delta \qquad \over \delta Q(s,t)}\; ,
\|\Delta^2u\|_{W^{-2,q}(\Omega)}=\sup_{0\neq\varphi\in W^{2,q'}_0(\Omega)}\dfrac{\bigg|\int_\Omega g(x)|u|^{p-1}u\varphi\bigg|}{\|\varphi\|_{W^{2,q'}_0(\Omega)}}\leq C(p,q,\Omega)\|g\|_1\|u\|_{H^2(\Omega)}^p,
M=(\begin{matrix}z_{1}&w_{1}&x_{1}&x_{2}&x_{3}&y_{1}&y_{2}\end{matrix})=(\begin{matrix}0&1&-1&0&0&-2&1\\ 0&3&0&0&1&0&0\\ -1&1&1&1&0&2&0\\ -2&0&-1&0&0&-2&0\\ 1&0&0&0&0&0&0\\ 0&1&0&0&0&0&0\\ 0&0&0&-1&0&-1&0\end{matrix}),
\phi\bigl(\textup{Tr}_{M^{\pi_n}}^{\mathbf{S}}(B)\bigr)=\textup{Tr}_{M^{\pi_n}}\bigl((\textup{id}_{M^{\pi_n}}\otimes\phi)B\bigr)\qquad\forall\, \phi\in\mathbf{S}^*,
\renewcommand{\arraystretch}{1 . 4}T_{i}^{( 2 )}( s ) = \left( \begin{array}{c c}{\rho_{i}/ s + O ( 1 )}&{O \left( s^{( k - n - 1 ) / 2}\right)}\\{O \left( s^{( k - n - 1 ) / 2}\right)}&{T_{i}^{( 1 )}( 1 ) + O ( s )}\\ \end{array}\right) , \quad i = 1 , 2 , 3 .
d \bar{s}^2 = \frac{r^2}{1-r^2}\{d \tau^2 + \sigma_{ij}d \xi^i d \xi^j \}\equiv e^{2 \psi}\{d \tau^2 + \sigma_{ij}d \xi^i d \xi^j \}, d \tilde{s}^2 = r^2 \{d \tilde{\tau}^2 + \sigma_{ij}d \xi^i d \xi^j \}\equiv e^{2 \tilde{\psi}}\{d \tilde{\tau}^2 + \sigma_{ij}d \xi^i d \xi^j\}.
\left( \frac{-c\alpha^2}{3}+ d\right) y = p, \mbox{\so\}p\in (y),\mbox{\and}\alpha - \alpha^{p^m}= \alpha \left( 1 - \alpha^2 (-18)^{\frac{p^m-3}{4}}\right) \equiv \alpha\left( 1 + \frac{3d}{c}(-18)^{\frac{p^m-3}{4}}\right) \mod (y) \equiv 0 \mod (y),
\nu_t\left(x\right) \log \frac{\nu_t\left(x\right)}{e^{-U^{\pi}\left(x\right)}}= \frac{\nu_t\left(x\right)}{e^{-U^{\pi}\left(x\right)}}e^{-U^{\pi}\left(x\right)}\log \frac{\nu_t\left(x\right)}{e^{-U^{\pi}\left(x\right)}}\geq -\frac 1e e^{-U^{\pi}\left(x\right)}=: -f_2 \left(x\right),
F^{ij}\nabla_m\nabla_m h_{ij}= F^{ij}\nabla_m \nabla_i h_{mj}&= F^{ij}\left[\nabla_i\nabla_m h_{mj}+R_{miml}h_{lj}+R_{mij l}h_{lm}\right] &= F^{ij}\nabla_i\nabla_jh_{mm}+F^{ij}h_{li}h_{lj}h_{mm}-F^{ij}h_{ij}|A|^2,
\int_{0}^{t}q_T \bigl(\xi_T(s) \bigr)\,ds=\varPhi_T \bigl(\xi_T(t) \bigr)- \varPhi_T(x_0)- \int_{0}^{t}\varPhi'_T \bigl(\xi_T(s) \bigr) \,dW_T(s)-\alpha_T(t),
- e_A E_t \frac{\kappa}{r}+ \frac{e_A}{r}\, (\vec{\Theta}\gamma^{ts}\vec{\Theta}) E_s + \frac{E_t}{2r}\, \left( \vec{\Theta}_{\hat{\alpha}A}( \vec{\Theta}_{\hat{\alpha}}\cdot \vec{e}) -( \vec{\Theta}_{\hat{\alpha}}\cdot \vec{e}) \vec{\Theta}_{\hat{\alpha}A}\right) \, .
\|x_k-x_{k+1}\|\leq \frac{1+\theta}{1-\theta}\left[ \frac{1+\theta}{2}\frac{L}{1-\sqrt{2bL}}\,\|x_{k-1}-x_k\|+\theta\,\frac{1+\sqrt{2bL}}{1-\sqrt{2bL}}\right]\|x_{k-1}-x_k\|, k= 0,1, \ldots .
\frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\vec{v})=0, &\rho\Big(\frac{\partial\vec{v}}{\partial t}+(\vec{v}\cdot\nabla)\vec{v}\Big)+\nabla P=-\rho\nabla\Phi, &\triangle\Phi=4\pi\mathsf{G}\rho.
\sum_{j \in T^c}g_j u_j = \sum_{j \in \hat S}g_j u_j + \sum_{j \notin \hat S \cup T}g_j u_j \le \sum_{j\in \hat S}|g_j| | u_j| + g_s^*\sum_{j\notin \hat S \cup T}|u_j|.
\Bigr[(Q_{m}{\varepsilon}_{ln}Q_{l}^{*}Q_{n}^{*}),(Q_{m}{\varepsilon}_{ln}Q_{l}^{*}Q_{n}^{*}Q_{a}Q_{a}^{*}),(Q_{a}Q_{a}^{*}Q_{b}Q_{b}^{*}Q_{m}^{*})\Bigl]
\left(\mathbf{1}_{n\times\cdots\times n}-\boldsymbol{\Delta}\right)\circ\mbox{Prod}\left(\mathbf{X},\mathbf{X}^{\top^{\left(m-1\right)}},\cdots,\mathbf{X}^{\top^{2}},\mathbf{X}^{\top}\right)=\mathbf{0}_{n\times\cdots\times n},
\mathbb{E}\left(|I| \; \big| \; Z=1 \right) \le \epsilon n^{1 - \alpha}= c_1 \left(\frac{1 + \ln(1+ n^{\alpha})}{n^{\alpha}}\right)^{1/2}n^{1 - \alpha}\leq Cn^{1 -{3 \alpha}/{2}}(\ln n^\alpha)^{1/2}.
\forall\:0\le t<n,\quad\left(\mathbf{1}_{n\times\cdots\times n}-\boldsymbol{\Delta}\right)\circ\mbox{Prod}_{\boldsymbol{\Delta}^{(t)}}\left(\mathbf{X},\mathbf{X}^{\top^{\left(m-1\right)}},\cdots,\mathbf{X}^{\top^{2}},\mathbf{X}^{\top}\right)=
\bigg| \int_{\partial\mathfrak D}\sum^3_{k,\ell =1}\mathfrak Z_{jk}(y,x) u_k (y) u_{\ell}(y) n_{\ell}(y) d 0_y\bigg| &\le c(\delta,S) (|x|S_{\tau}(x))^{-1}\|u_k(y)u_{\ell}(y)\|_1 &\le c(\delta,S) (|y|S_{\tau}(y))^{-1}\|u\|^2_{2-1/p,\partial\Omega}
R_{i j k}:\qquad0 = x_{i j}f_k - x_{i k}f_j + x_{j k}f_i &\qquad+ \sum_{r=1}^{i-1}y_r \Phi_{r i j k}- \sum_{r=i+1}^{j-1}y_r \Phi_{i r j k}+ \sum_{r=j+1}^{k-1}y_r \Phi_{i j r k}- \sum_{r=k+1}^{n}y_r \Phi_{i j k r}
\sup_{0 \not= (z,q) \in (H^{1,\frac{1}{2}}_{0,per}(Q_T))^2}\frac{\mathcal{F}_{(\eta,\zeta)}(z,q)}{|(z,q)|_{1,\frac{1}{2}}}\leq \, C_F \, \|\mathcal{R}_1(\eta,\zeta,\boldsymbol{\tau})\| + \|\mathcal{R}_2(\eta,\boldsymbol{\tau})\| + C_F \, \|\mathcal{R}_3(\eta,\zeta,\boldsymbol{\rho})\| + \|\mathcal{R}_4(\zeta,\boldsymbol{\rho})\|
[b,R_j](f)(g)-\big([b,R_j]f\big)_B =J_1(g)-\big(J_1\big)_B+J_2(g,\tilde g)-\big(J_2(\cdot,\tilde g)\big)_B \quad+\left(J_3(\cdot, g)\right)_B+J_4(g,\tilde g).
E_{m\bar a\bar b}\equiv G_{m\bar a\bar b}-\mu\epsilon^{mnp}\partial_{n}h_{p\bar a\bar b}+ \frac{\mu}{2}[\epsilon^{pna}\partial_{n}v_{b}+\epsilon^{pnb}\partial_{n}v_{a}-\frac{2}{3}\eta_{ab}\epsilon^{pmn}\partial_{m}v_{n}] = 0,
\begin{cases}D_{A}\psi_=0 F_{A}^{+}= \frac{r}{2}(q(\psi,\psi)- i\Omega) - \frac{1}{2}F^+_{A_{K^{-1}}}- \frac{i}{2}\wp_4^+ *d*b -2^{-\frac{1}{2}}r^{\frac{1}{2}}(\eta^* \psi^{\xi}-{\psi^{\xi}}^* \eta)=0, \end{cases}
\Psi(\tau):=\sum_{n\geq 0}q^{n(n+1)/2}=e^{-\frac{2\pi i\tau}{8}}\cdot\dfrac{\eta(2\tau)^2}{\eta(\tau)}\quad\tilde{\theta}(\tau):=\sum_{n=-\infty}^\infty (-1)^n q^{n^2}=\dfrac{\eta(\tau)^2}{\eta(2\tau)}.
\exp\left(\bar p_1 \frac{\mu''(s)}{2 \mu'(s)}\right) \mu'(s)^{\bar\rho}\,\nabla\, \mu'(s)^{-\bar\rho}\exp\left( \! -\bar p_1 \frac{\mu''(s)}{2 \mu'(s)}\right) = d + \bar p_{-1}ds + \sum_{i\in \bar E}\tilde{\bar v}_i(s) \bar p_i ds
\bar Y_t = \bar Y_n+\int_{t\wedge\tau}^{n\wedge\tau}\bigl(f\bigl(s,Y^m_s,Z^m_s,V^m_s \bigr)-f\bigl(s,Y^n_s,Z^n_s,V^n_s \bigr) \bigr) ds - \int_{t\wedge\tau}^{n\wedge\tau}\bar Z_s dW_s - \int_{t\wedge\tau}^{n\wedge\tau}\int_U \widehat V_s(e) \widehat\pi(de,ds) - \bar M_{n\wedge\tau}+ \bar M_{t\wedge \tau}.
\Psi(P_{S}, P_{D}) = \min\left(1, \frac{\lambda_S}{P_S}\right)\min\left(1, \frac{\lambda_D}{P_D}\right) & = \min \left( 1, \frac{\lambda_{S}}{P_{S}}, \frac{\lambda_{D}}{P_{D}}, \frac{\lambda_{S}\lambda_{D}}{P_{S}P_{D}}\right),
A+\frac{\sqrt{ab}}{a}B=\frac{\alpha}{t^\alpha}\mathrm{e}^{-\frac{\alpha(x+y)}{t^\alpha}}\bigg(\frac yx\bigg)^{\frac{\sqrt{ab}-1}{2}}I_{\sqrt{ab}-1}\bigg(\frac{2\alpha\sqrt{xy}}{t^\alpha}\bigg),
L_i u := -\varepsilon u_i'' + b_{ii}u_i' + a_{ii}u_i = f_i + \sum_{\genfrac{}{}{0pt}{}{j=1}{j\ne i}}^m \left[ (b_{ij}u_j)' - (b_{ij}'+a_{ij})u_j \right], u_i(0) = u_i(1) = 0.
\forall j_{1},j_{2}\in\mathcal{L}^{(r-1)},\beta_{1},\beta_{2}\in(\mathcal{L}^{(r-1)})^{\mathtt{C}}:\, h((\mathcal{L}^{(r-1)})_{\gamma_{r}\omega}^{j_{1}j_{2}})=h((\mathcal{L}^{(r-1)})_{\beta_{1}\beta_{2}}^{l_{r}m})=0
(e^{-t[H_{j,A}^m-m+V]}g)(x) = E^{\lambda^m}\bigg[ e^{-S^{m}_{j,A,V}(x,t; X)}g(x+X(t)) \bigg], j=1,2, (e^{-t[H_{3,A}^m-m+V]}g)(x) = E^{\mu\times \nu^m}\bigg[e^{-S_{3,A,V}(x, t; B, T)}g\Big(x+B\big(T(t)\big)\Big) \bigg].
D_1h(X,Y,Z)= \frac{1}{\sqrt{3}}(\nabla_X h(Y,Z)+\nabla_Y h(Z,X)+\nabla_Z h(X,Y)), D_2h(X,Y,Z)= \frac{1}{\sqrt{2}}(\nabla_X h(Y,Z)-\nabla_Y h(Z,X)).
\lambda(p^\mu)\,\equiv\left(\begin{array}{c}\left(\zeta_\lambda\,\Theta_{[j]}\right)\,\phi^\ast_{_L}(p^\mu) \phi_{_L}(p^\mu) \end{array}\right)\,\,,\quad\rho(p^\mu)\,\equiv\left(\begin{array}{c}\phi_{_R}(p^\mu) \left(\zeta_\rho\,\Theta_{[j]}\right)^\ast\,\phi^\ast_{_R}(p^\mu)\end{array}\right)\,\,\quad,
\mathbb{P}(\mathbf{c}, t) \Leftrightarrow \neg \left(\mathbb{P}_{b_{11}= 0}\vee \mathbb{P}_{b_{11}= 1}\vee\mathbb{P}_{b_{10}= 0}\vee\mathbb{P}_{b_{10}= 1}\vee\mathbb{P}_{b_{2}= t-1}\vee\mathbb{P}_{b_{3}= t-1}\right).
\begin{array}{l}{{A_{\mu}^{\prime}= U A_{\mu}U^{- 1}+ \frac i g U \partial_{\mu}U^{- 1},}}\\{{Q_{\mu}^{\prime}= U Q_{\mu}U^{- 1}, \, \, \, J_{\mu}^{\prime}= U J_{\mu}U^{- 1},}}\\{{C_{\mu}^{\prime}= U C U^{- 1}, \, \, \, \overline{{{\xi}}}^{\prime}= U \overline{{{\xi}}}U^{- 1},}}\\{{\overline{{{C}}}^{\prime}= U \overline{{{C}}}U^{- 1}, \, \, \, \xi^{\prime}= U \xi U^{- 1}.}}\\ \end{array}
R_{\phi_0 (x,u,v)}^{-1}R_{[u\backslash (xv)]}=I\Leftrightarrow \phi_0 (x,u,v)=[u\backslash (xv)] (u\backslash([(uv)/(u\backslash (xv))]v))=[u\backslash (xv)]\Leftrightarrow x\backslash (uv)=u\backslash (xv).
z_i = [v_{i-2}, v_{i-1}, ((v_{i-2},v_{i-1}) \cap (v_i, v_{i+1})), ((v_{i-2},v_{i-1}) \cap (v_{i+1}, v_{i+2}))], & w_i = [((v_{i-2},v_{i-1}) \cap (v_{i+1}, v_{i+2})), ((v_{i-1},v_i) \cap (v_{i+1}, v_{i+2})),v_{i+1}, v_{i+2}]
I_{\textup{nested}, 1}= \Big\| 1_{Q_0}\Big( \mathop{\sum_{R \in \mathcal{D}_{\textup{good}}:\, R \subset Q^*}}_{2^{-\kappa}< \ell(R) < 2^{s-r}}1_{R}\int_{\ell(R)/2}^{\ell(R)}\Big| \mathop{\sum_{\ell=r+1}^{s-\textup{gen(R)}}}\theta_t (1_{R^{(\ell)}\setminus R^{(\ell-1)}}\Delta_{R^{(\ell)}}f) \Big|^2\, \frac{dt}{t}\Big)^{1/2}\Big\|_{L^q(\mu)}
f_{R}(r) := \frac{f_\infty(r)}{f_\infty(R)}, f_\infty(r) := \begin{cases}\ln \left( \frac{\tanh(r/2)}{\tanh(a/2)}\right) : d = 2, 1 - \frac{\tanh a}{\tanh r}: d = 3, \end{cases}
u^{\dagger}.A.v=\left(\begin{array}{c}u_{+}u_{-}\end{array}\right)^{\dagger}.(DP)^{\dagger}.\left( \begin{array}{cc}1 0 0 -1 \end{array}\right).(DP).\left(\begin{array}{c}v_{+}v_{-}\end{array}\right)=0,
c- 0.25\cdot3^2b^3=r_{3j+1}^2-4-2.25(r_j^2-4)^3= =8((2-\sqrt{3})^{4j+2}+(2+\sqrt{3})^{4j+2})-34((2-\sqrt{3})^{2j+1}+(2+\sqrt{3})^{2j+1})+\frac{128}{3}>0
\left|\Delta\rho_\eta^\varepsilon-\overline{\Delta_\Gamma\eta}-k_d^{-1}\Bigl(\overline{V_\Gamma H\eta}\Bigr)-\Bigl(\bar{\zeta}_1-\bar{\zeta}_0\Bigr)\right| \leq c\varepsilon\sum_{\xi=\eta,\zeta_0,\zeta_1}\left(|\bar{\xi}|+\Bigl|\overline{\nabla_\Gamma\xi}\Bigr|+\Bigl|\overline{\nabla_\Gamma^2\xi}\Bigr|\right)
l_{0}^{o}+ l_{1}^{o}+ l_{2}^{o}+ l_{3}^{o}= 2 l_{0}- 1, l_{0}^{o}+ l_{1}^{o}- l_{2}^{o}- l_{3}^{o}= 2 l_{1}+ 1, & l_{0}^{o}- l_{1}^{o}+ l_{2}^{o}- l_{3}^{o}= 2 l_{2}+ 1, l_{0}^{o}- l_{1}^{o}- l_{2}^{o}+ l_{3}^{o}= 2 l_{3}+ 1.
|h''_{\alpha}(x)| \leq \sum_{i=0}^{n}\Bigl| h_r'(x-t_{i+1}\alpha) p_r(t_{i+1}) - h_l'(x-t_{i}\alpha) p_l(t_{i}) \Bigr| \leq \frac{C\alpha}{4Pn}\cdot 2Pn = \frac{C\alpha}{2}.
\begin{aligned}u(x+h)+u(x-h)-2u(x) \leq C\left(1 + \frac{1}{\mathrm{dist}\left(x, \partial \Omega\right)}\right)|h|^2 \leq \frac{C}{\mathrm{dist}\left(x, \partial \Omega \right)}|h|^2,\end{aligned}
V ( | z |_{3}) - V ( z_{t r}) = \frac{b^{2}}{3 6}\frac{| z_{t r}|^{4}}{( 1 - | z_{t r}|^{2})^{2}}\left( - \frac{x^{2}}{3}+ \frac{2 x^{3}}{2 7}+ \ldots \right) = V ( z_{t r}) \left( - \frac{x}{3}+ \frac{2 x^{2}}{2 7}+ \ldots \right)
g = \alpha_1(n_1g_1)\alpha_2(n_2g_2)\alpha_3(n_1g_1)^{-1}\alpha_4(n_2g_2)^{-1}&= \alpha_1(n_1)\alpha_1(g_1)\alpha_2(n_2)\alpha_2(g_2)\alpha_3(g_1)^{-1}\alpha_3(n_1)^{-1}\alpha_4(g_2)^{-1}\alpha_4(n_2)^{-1}.
\alpha_{I J}= \left( \begin{matrix}{{0}}&{{1}}&{{0}}&{{0}}\\{{- 1}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{1}}\\{{0}}&{{0}}&{{- 1}}&{{0}}\\ \end{matrix}\right) \ ; \qquad \beta_{I J}= \left( \begin{matrix}{{0}}&{{- 1}}&{{0}}&{{0}}\\{{1}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{1}}\\{{0}}&{{0}}&{{- 1}}&{{0}}\\ \end{matrix}\right) \ .
t_1 = \vcenter{\offinterlineskip\small\hrule\hbox{\vrule height 9 pt depth 4 pt\hbox to 13pt{\hfil\(2\)\hfil}\vrule\hbox to 13pt{\hfil\(3\)\hfil}\vrule\hbox to 30pt{\hfil\(\cdots\)\hfil}\vrule\hbox to 25pt{\hfil\(n-1\)\hfil}\vrule\hbox to 13pt{\hfil\(n\)\hfil}\vrule}\hrule}\,.
P_{\lambda}(x, s) = (x/s)^{\lambda}\frac{(x/s;q)_{\infty}}{\big(q^{-\lambda}x/s ;q\big)_{\infty}}, y(s)=s^{\mu '}\frac{(q/(\beta s);q)_{\infty}}{(q/(\alpha s);q)_{\infty}}
\frac{\widetilde{W}}{x}\sum_{n \leq x/\widetilde{W}}\Big(f(\widetilde{W}n+A) - S_f(x;\widetilde{W},A)\Big) F(g(n)\Gamma) = o_{G/\Gamma}\bigg(\frac{1}{\log x}\frac{\widetilde{W}}{\phi(\widetilde{W})}\prod_{\substack{p \leq x,~ p \nmid \widetilde W}}\left(1+ \frac{|f(p)|}{p}\right)\bigg)
\frac 1{L^2}\sum_{k\le I}b_k^2k^{1+2\alpha}= \frac 1{L^2}\sum_{k\le I}b_k^2k^{2\beta}k^{1+2\alpha-2\beta}\le \frac 1{L^2}\|b_0\|^2_{2,\beta}I^{1+2\alpha-2\beta},
\frac{p+1}{p}\cdot \frac{\Lambda}{a_{*}}\geq \frac{\Lambda}{a_{*}}\|(-\Delta)^{\frac{1}{4}}W\|_{L^{2}}^2 + \frac{1}{\Lambda^p}\int_{\mathbb{R}^{3}}V(x)|W(x)|^{2}{\rm d}x & = \frac{\Lambda b}{a_{*}}\|(-\Delta)^{\frac{1}{4}}\mathcal{Q}\|_{L^{2}}^2 + \frac{1}{\Lambda^p b^p}\int_{\mathbb{R}^{3}}V(x-x_0)|\mathcal{Q}(x)|^{2}{\rm d}x.
q_{\infty}\left(\mathcal F_1^{c'},\mathcal F_2^{c'},\mathbf T^{'}\right)-q_{\infty}^{*}=\left(a_{n_1}-a_{n_3}\right)\left(f_{2,K_2^c,\infty}(T_{n_3}^{*})-f_{1,K_1^c+F_1^{b*},\infty}\right)> 0,
R_{\mu_k}(w) \approx \frac{r_A}{N}\sum_{j=0}^{N-1}\xi_N^{j}\cdot d_j^{(k)}\cdot J_{[a_k, b_k]}'(\xi_N^j) \cdot \frac{J_{[a_k, b_k]}(\xi_N^j) - 1/c_j^{(k)}}{c_j^{(k)}- w}, k \in \{1,2\}.
V_{1}=1,V_2=\frac{1}{2},V_3=\frac{3}{4}\sum_{n=0}^\infty\left[ \frac{1}{(3n+1)^2}-\frac{1}{(3n+2)^2}\right],V_4=\sum_{n=0}^\infty\frac{1}{(2n+1)^3}
P_{{\cal S}}(A)= \sum_{j=1,2}P_{{\cal S}_j/{\cal S}}P_{{\cal S}_j}(A)+2 \sqrt{P_{{\cal S}_1/{\cal S}}P_{{\cal S}_1}(A)P_{{\cal S}_2/{\cal S}}P_{{\cal S}_2}(A)}\lambda(A;{\cal S},{\cal S}_j)\;.
\langle [O_{3}^{(1)}](x) [O_{3}^{(2)}](y)\rangle +b_{1}\langle [O_{3}^{(1)}](x) [O_{3}^{(1)}](y)\rangle +a_{1}\langle [O_{3}^{(2)}](x) [O_{3}^{(2)}](y)\rangle=O(N^{-4})
F\begin{pmatrix}g_{1,1}g_{1,2}g_{1,3}g_{2,1}g_{2,2}g_{2,3}g_{3,1}g_{3,2}g_{3,3}\end{pmatrix}=\begin{pmatrix}\frac{g_{1,2}}{\sqrt{-3}}\frac{g_{1,3}}{\sqrt{-3}}\frac{g_{2,1}}{\sqrt{-3}}\frac{g_{3,1}}{\sqrt{-3}}\end{pmatrix}.
2u^3t+18u^2t^2+45ut^3+36t^4+4u^3+52u^2t+193ut^2 & +207t^3+32u^2+260ut+431t^2+108u+382t+120 & =4u^3t+24u^2t^2+54ut^3+36t^4+2u^3+56u^2t+217ut^2 & +216t^3+22u^2+255ut+449t^2+80u+375t+100
\int_{0}^{\infty}cn(t)e^{-tz}dt=\frac{1}{z+}\frac{1^{2}}{z+}\frac{2^{2}k^{2}}{z+}\frac{3^{2}}{z+}\frac{4^{2}k^{2}}{z+}\frac{5^{2}}{z+}\cdot\cdot\cdot
\left(\hat{F}_j\right)^{-1}\int_{\hat{F}_j}\frac{\partial \hat{v}_1}{\partial \eta}~\hat{s}:=P_0\hat{u}_1\vert_{\hat{F}_j}~~,~~~~~~\left(\hat{F}_j\right)^{-1}\int_{\hat{F}_j}\frac{\partial \hat{v}_2}{\partial \eta}~\hat{s}:=P_0\hat{u}_2\vert_{\hat{F}_j}.
\frac{1}{\lambda\ell}\log_2\left[\frac{\ell}{n}\sum_{t=0}^{n/\ell-1}2^{\lambda L(y_{t\ell+1}^{t\ell+\ell})}\right]\ge\hat{H}_\lambda^\ell(x^n)-\frac{\gamma(s,\ell)}{\ell},
{\cal L}^{\mathrm{g a u g e}}\ = \ -{\textstyle{\frac{i}{8}}}\left({\cal N}_{I J}F_{\mu \nu}^{+ I}F^{+ \mu \nu J}\- \ \bar{\cal N}_{I J}F_{\mu \nu}^{- I}F^{- \mu \nu J}\right) ,
\tilde{\hat P}_n(x) =\frac{1}{2^{n+1}}\begin{bmatrix}1 -1 1 1\end{bmatrix}\begin{bmatrix}U_n(x)- U_{n-1}(x) 0 0 U_n(x)+ U_{n-1}(x)\end{bmatrix}\begin{bmatrix}1 1 -1 1\end{bmatrix}&=\frac{1}{2^{n}}\begin{bmatrix}U_n(x) -U_{n-1}(x) -U_{n-1}(x) U_n(x)\end{bmatrix}.
(\delta^{n}\omega)(X_{1},X_{2}, . . ., X_{n+1}) =\sum_{i=1}^{n}{[X_{i},\omega(X_{1}, . . ., \widehat{X_{i}}, . . ., X_{n+1})]_{L}}&\,+(-1)^{n+1}[\omega(X_{1}, . . ., X_{n}),X_{n+1}]_{R}&\,+\sum_{1\leq i<j\leq n+1}{(-1)^{i}\omega(X_{1}, . . ., \widehat{X_{i}}, . . ., X_{j-1},[X_{i},X_{j}], . . ., X_{n+1})},
B \leq \int_{|u-t|>2^{2s}\ell(Q)^{2s}}\frac{|P_s*(\phi\mu)(x,t) - P_s*(\phi\mu)(y,t)|}{|u-t|^{2-\frac1{2s}}}\,du &+\int_{|u-t|>2^{2s}\ell(Q)^{2s}}\frac{|P_s*(\phi\mu)(x,u) - P_s*(\phi\mu)(y,u)|}{|u-t|^{2-\frac1{2s}}}\,du=B_1+B_2.
P_{\gamma,\xi}^+d\theta_{\xi}(\gamma) h P_{\gamma,\xi}^+=P_{\gamma,\xi}^+h_{\xi}P_{\gamma,\xi}^+\,\,\,\mathrm{and}\,\,P_{\gamma,\xi}^- d\theta_{\xi}(\gamma) h P_{\gamma,\xi}^-=0,
\tilde{z}_{\pm}=-Q_{\pm}^{-1}\displaystyle\displaystyle\frac{\partial}{\partial \overline{z}_{\pm}}-z_{\pm}~,~~~\tilde{p}_{\pm}=\displaystyle\displaystyle\frac{\partial}{\partial z_{\pm}}-Q_{\pm}^{-1}\displaystyle\frac{\partial p_{\pm}}{\partial z_{\pm}}\displaystyle\displaystyle\frac{\partial}{\partial \overline{z}_{\pm}}~.
P_{T}f(u,\xi)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}P_{V}f(u^{\prime},\xi^{\prime})P_{V}\phi_{\gamma(u,\xi)}(u^{\prime},\xi^{\prime})du^{\prime}d\xi^{\prime}
\gamma^{8}= \left( \begin{array}{c c}{{0}}&{{1}}\\{{1}}&{{0}}\\ \end{array}\right) , \gamma^{a}= \left( \begin{array}{c c}{{0}}&{{\lambda^{i}}}\\{{- \lambda^{i}}}&{{0}}\\ \end{array}\right) , i = 1 , . . . . , 7 ,
\frac{\triangle_{V_s}w}{w}- \frac12 = \frac{\triangle r}{r + A}- \frac{dr (V_s)}{r + A}- \frac12 = \frac{\triangle r}{r + A}- \frac{dr(V_s + \frac12 r \partial_r)}{r + A}+ \frac{r}{2(r + A)}- \frac12 .
X_x(\sqrt{2r}\cos\theta_0,\sqrt{2r}\sin\theta_0,z)\sin\theta_0 -X_y(\sqrt{2r}\cos\theta_0,\sqrt{2r}\sin\theta_0,z)\cos\theta_0 &=\sqrt{2r}\left(\delta^{-1}\alpha+cz-\delta^p\tilde G(r,z)\right).
\Psi (\mathbf{x})=\dfrac{1}{\vert T\vert}\mathcal{E}(\mathbf{x},2)=-\dfrac{1}{\pi^{2}\vert T\vert}\sum_{\mathbf{k}\in \mathbb{Z}^{n}\setminus \mathbf{\lbrace 0\rbrace}}\dfrac{e^{i\pi\mathbf{x}.\mathbf{k}}}{\Vert\mathbf{k}\Vert^{2}}.
w^\beta_\epsilon w^\beta_\epsilon = (\theta^\beta_\epsilon)^2 c_\beta t_\beta + c_{\alpha+\beta}t_{\alpha+\beta}+ 2\theta^\beta_\epsilon b_{\beta, \alpha+\gamma}e^\alpha = \left( (\theta^\beta_\epsilon)^2 c_\beta + c_{\alpha+\beta}\right) t_\beta + c_{\alpha+\beta}t_\alpha + 2\theta^\beta_\epsilon b_{\beta, \alpha+\gamma}e^\alpha
P_{U_n}(w) =P_{U_n}(P_D(g))+P_{U_n}(R^Df(\cdot,w)+R^D\mu) &=P_{D}(g)+(R^Df(\cdot,w)-\Pi_{U_n}(R^Df(\cdot,w)))+(R^D\mu-\Pi_{U_n}(R^D\mu))\mbox{q.e.},
\mathcal{P}_{A,B}(\mu) =\lim_{n \rightarrow \infty}\left(\frac{\int \varphi_{n}^{1}d \mu}{\int \psi_{n}^{1}d \mu}, \ldots, \frac{\int \varphi_{n}^{d}d \mu}{\int \psi_{n}^{d}d \mu}\right).
\psi^B_{i,2n+1-i}(x_u\pm x_v) &= \left( \prod_{\ell=i+1}^n (x_u-x_\ell)(x_u + x_\ell) \right) x_u \pm \left( \prod_{\ell=i+1}^n (x_v-x_\ell)(x_v + x_\ell) \right) x_v.
\frac{1}{2}\left( \int_{\Omega}|\nabla u_{0}|^{2}-\int_{\Omega}a(x)u_{0}^{2}\right) =\lim I_{q_{n}}(u_{n})\leq\lim I_{q_{n}}(\phi) & =\frac{1}{2}\left( \int_{\Omega}|\nabla\phi|^{2}-\int_{\Omega}a(x)\phi^{2}\right) <0,
\left[T_{\mu\nu}\right]_P(x)=\frac{C_P}{2}\frac{1}{\sqrt{|g(x)|}}g_{\mu\rho}(x)g_{\nu\lambda}(x)\int_{-\infty}^{+\infty}d\xi\frac{1}{\beta(\xi)}\frac{dy^\rho(\xi)}{d\xi}\frac{dy^\lambda(\xi)}{d\xi}\delta^{(11)}(x-y(\xi))
\langle \Upsilon \rangle_{\Sigma}\ \longrightarrow \, l a n g l e \Upsilon \rangle_{\Sigma}+ \delta \! \langle \Upsilon \rangle_{\Sigma}, \hspace{1 e m}\delta \! \langle \Upsilon ( z_{0}) \rangle_{\Sigma}= \frac{1}{\pi}\int_{\Sigma}\langle \Psi ( z ) \Upsilon ( z_{0}) \rangle_{\Sigma}\, d^{2}\! z .
d\log(\{1+aS^iT^j, S\}) =\frac{d(1+aS^iT^j)}{1+aS^iT^j}\wedge\frac{dS}{S}&=-aj\sum_{k=0}^\infty(-a)^kS^{(k+1)i-1}T^{(k+1)j-1}dS\wedge dT.
\mbox{$G$-regular:}\theta_L(G) \le \theta_z(G) \le -|V(G)|{\lambda_n\over \lambda_1 - \lambda_n}\mbox{... and $G$-edge trans.:}\quad\theta_L(G) =\theta_z(G) = -|V(G)|{\lambda_n\over \lambda_1 - \lambda_n}\,,
\begin{cases}u\in K\,, \noalign{\medskip}\displaystyle{\int_{\Omega}\bigl[a(x,u,\nabla u)\cdot\nabla (v-u) +b(x,u,\nabla u)\,(v-u)\bigr]\,dx \geq 0}\noalign{\medskip}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\,.\end{cases}
- ~ \frac{( \mu - 1 ) ( 2 \mu - 3 )}{2 ( \mu - 2 )}\left[ \hat{\Phi}( \mu ) ~ + ~ \hat{\Psi}^{2}( \mu ) ~ - ~ \frac{8 ( 2 \mu - 3 )}{( \mu - 1 ) ( \mu - 2 ) \eta_{1}}~ - ~ \frac{2}{( \mu - 2 )^{2}}\right] \eta_{1}^{2}
S \ : \ \tau \ \rightarrow \ \tilde{\tau}= - \frac{1}{\tau}\ , \A_{\bar{z}}^{+ +}( 0 , 0 ) \ \rightarrow \ \tilde{A}_{\bar{z}}^{+ +}( 0 , 0 ) = - \bar{\tau}A_{\bar{z}}^{+ +}( 0 , 0 ) \ ,
|q_{i,l+1}- q_{i,l+1}'| \leq | \varphi(\bar{q}_l) - \varphi_t(\bar{q}_l') | \leq | \varphi(\bar{q}_l) - \varphi(\bar{q}_l') | + | \varphi_t(\bar{q}'_l) - \varphi(\bar{q}_l') | & \leq c\d(\bar{q}_l, \bar{q}_l') + \log_t M\leq (cP_l(c)+1) \log_t M =P_{l+1}(c) \log_t M.
B_i= \{\omega:\|u_i-u_i'\|\leq q \|u_{i-1}-u_{i-1}'\|\}\boldsymbol{A}_{0}=B_{m}^c \boldsymbol{A}_k = \big(\cap_{i=m}^{k+m-1}B_{i}\big)\cap B_{k+m}^c, \forall 1\leq k\leq n-1, \boldsymbol{A}_{n}=\cap_{i=m}^{n+m-1}B_i.
\langle 0 | (X^1(\sigma) - \overline{X^1})^2 | 0 \rangle\ =\ \sum_{m = 1}^\infty \frac{1}{m^2}\langle 0 | (\alpha_m \alpha_{-m}+ \tilde\alpha_m \tilde\alpha_{-m})| 0 \rangle\ = 2 \sum_{m = 1}^\infty \frac{1}{m}.
\mathbf{A}_{i}-\mathbf{A}_{j}=\psi(\pi,\mathbf{a},\mathbf{d}_{i})-\psi(\pi,\mathbf{b},\mathbf{d}_{j}) &=\psi(\pi,\mathbf{a}-\mathbf{b},\mathbf{d}_{i}-\mathbf{d}_{j}).
\begin{array}{c}{U_{0}( z ) = \frac 1 2 ( u_{0}+ u_{- 1}) + \frac 1{\sqrt{2}}( u_{0}- u_{- 1}) z_{p e r}}\\{V_{0}( z ) = \frac 1 2 ( v_{0}+ v_{- 1}) + \frac 1{\sqrt{2}}( v_{0}- v_{- 1}) z_{p e r},}\\ \end{array}
A(x,p) \star B(x,p) = e^{\kappa(\partial_x\partial_{\widetilde{p}}{~}-{~}\partial_p\partial_{\widetilde{x}})}A(x,p)B({\widetilde{x}},{\widetilde{p}}) \Big |_{{\widetilde{x}=x},{\widetilde{p}= p}}.
-D_{\infty}^{\beta}\tilde u(t,x)= \ \int_0^{\infty}(\tilde u(t-r,x)-\tilde u(t,x))\,\nu(r)dr = \ \int_0^{t}(\tilde u(t-r,x)-\tilde u(t,x))\,\nu(r)dr+\int_t^{\infty}\phi(t-r,x)\,\nu(r)dr &\quad-\tilde u(t,x))\int_t^\infty\nu(r)dr \pm \phi(0,x)\int_t^\infty\,\nu(r)dr = \ -D_{0}^{\beta}\tilde u(t,x)+f_\phi(t,x).
A_{0}( x ) = \left[ \begin{array}{c}{{\frac{\pi T}g \exp \left\{\frac g{\sqrt{\pi}}( x - x_{0}) \right\}, \ \ \x \leq x_{0}}}\\{{\frac{\pi T}g \left[ 2 - \exp \left\{\frac g{\sqrt{\pi}}( x_{0}- x ) \right\}\right] , \ \ \x \geq x_{0}}}\\ \end{array}\right.