harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
R_{i j k l}=\frac{1}{(n-1)(n-2)}R\left(g_{i k}-\partial_i f \partial_k f\right)\left(g_{j l}-\partial_j f \partial_l f\right) -\frac{1}{(n-1)(n-2)}R\left(g_{i l}-\partial_i f \partial_l f\right)\left(g_{j k}-\partial_j f \partial_k f\right) +O\left(r^{-1-8\delta}\right).
0\leq\lim_{\stackrel{\hbox{\scriptsize $z\!\in\!\Omega$}}{z\rightarrow\theta}}\frac{\zeta(z)-\phi_k(z)}{{\delta(z)}^s}=\lim_{\stackrel{\hbox{\scriptsize $z\!\in\!\Omega$}}{y\rightarrow\theta}}\int_\Omega\frac{G_\Omega(y,z)}{{\delta(z)}^s}\left(1-\psi_k(y)\right)\;dy\leq\int_\Omega M_\Omega(y,\theta)\left(1-\psi_k(y)\right)\;dy\xrightarrow{k\rightarrow+\infty}0,
f(t) = - (1-\gamma) \log \frac{1}{ps^2}+ 1 + 2 \sqrt{\frac{p}{s^2}}(1-s^2)-2p+ps^2 \le - (1-\gamma) \log \frac{1}{ps^2}+ 1 + 2 \sqrt{\frac{p}{s^2}}\le- \left(1-\gamma - \frac{\epsilon}{48}\right) \log \frac{1}{ps^2}\, ,
\begin{array}{rl}\underset{\mathbf{h},\boldsymbol{\theta},\boldsymbol{\phi}=\mathbf{F}e^{j\boldsymbol{\theta}}}{}\left\|\mathbf{r}-\boldsymbol{\Phi}_{\boldsymbol{\phi}}\mathbf{S}\check{\mathbf{F}}\mathbf{h}\right\|^2,\end{array}
\ell_0 \ge \frac{r_1 (M_1+7)+r_2 (M_1+4)+r_3 (M_1+1)+r_1^2+r_2^2+r_3^2-r_1r_2-r_1r_3-r_2r_3}{3 (M_1+4)}= \frac{r_1 (6-\tfrac{3}{2}k)+r_2 (3-\tfrac{3}{2}k)+r_3 (-\tfrac{3}{2}k)+r_1^2+r_2^2+r_3^2-r_1r_2-r_1r_3-r_2r_3}{3 (3-\tfrac{3}{2}k)},
\int_{\tau_{b_i}(V)}\left(e^{\alpha u^2}- 1 \right) \dv_g = \int_{V}\left(e^{\alpha (u \circ \tau_{b_i})^2}- 1 \right) \dv_g, \leq C \int_{V}\left(e^{q_2 \left(\sqrt{\frac{\alpha}{q_2}}u \circ \tau_{b_i}\right)^2}- 1 \right) \dx, \leq C \int_{U}|u \circ \tau_{b_i}|^2 \dv_g , \leq C \int_{\tau_{b_i}(U)}u^2 \dv_g .
\|E_{\rho,\sigma}((0,t))\xi_\sigma\|^2 =\|JE_{\sigma,\rho}((t^{-1},+\infty))JJ\Delta_{\sigma,\rho}^{1/2}\xi_\rho\|^2=\|E_{\sigma,\rho}((t^{-1},+\infty))\Delta_{\sigma,\rho}^{1/2}\xi_\rho\|^2 &=\int_{(t^{-1},+\infty)}s\,d\|E_{\sigma,\rho}(s)\xi_\rho\|^2=\int_{(0,t)}s^{-1}\,d\|E_{\sigma,\rho}(s^{-1})\xi_\rho\|^2,
F_1(u)=\frac{b}{k+1}u^{k+1}+cu^{k-1}u_x^2-a(k-2)u^{k-3}u_x^4, F_2(u)=[k(k+2)-8a-b-c(k+1)]u^{k-2}u_x^3-3a(k-2)u^{k-3}u_x^3u_{xx}.
\begin{cases}\partial_t\boldsymbol \zeta + \boldsymbol u_1\cdot \nabla \boldsymbol \zeta + \boldsymbol \zeta \cdot \nabla \boldsymbol u_2 + \nu(-\Delta)^\alpha \boldsymbol \zeta= - \nabla \pi + \eta \boldsymbol e_2, \partial_t \eta + \boldsymbol u_1\cdot \nabla \eta + \boldsymbol \zeta \cdot \nabla \theta_2+ \kappa(-\Delta)^\beta \eta= 0.\end{cases}
x_j^Tz^*_j = x_{j0}z_{j0}+ \bar x^T\bar z^*_j = x_{j0}z_{j0}+ \bar x^T\bar z^*_j \stackrel{r_j(z_j^*)<0}{>}x_{j0}\|\bar z_j^*\| + \bar x^T\bar z^*_j = 0
\begin{array}{ll}m\dot{u}^\alpha+\displaystyle\frac{1}{c^2}S^{\alpha\sigma}\ddot{u}_{\sigma}=f^\alpha begin{align*}8pt] \dot{S}^{\alpha\beta}- \displaystyle\frac{1}{c^2}S^{\alpha\sigma}\dot{u}_{\sigma}u^\beta + \displaystyle\frac{1}{c^2}S^{\beta\sigma}\dot{u}_{\sigma}u^\alpha =0 \end{array}
\left(\partial_{0,\nu}\left(1+\kappa*\right)\left(\begin{pmatrix}\frac{1}{2}0 0 \frac{1}{2}\end{pmatrix}+\partial_{0,\nu}^{-1}\begin{pmatrix}\frac{1}{2}0 0 \frac{1}{2}\end{pmatrix}\right)+\begin{pmatrix}0 \partial_{1}\partial_{1,c}0\end{pmatrix}\right)\begin{pmatrix}u v\end{pmatrix}=\begin{pmatrix}f g\end{pmatrix}.
A_n \leq R(1^n,1^n_*)+R(1^n_*,3^n_*)+R(3^n_*,3^n) &= 5\lambda(2A_{n-1}-B_{n-1})+\left[(10\lambda B_{n-1}+2(5\lambda)^n)^{-1}+(15\lambda B_{n-1}+3(5\lambda)^n)^{-1}\right]^{-1}&= 10\lambda A_{n-1}+\lambda B_{n-1}+\frac 65(5\lambda)^n.
\times \exp \left( \frac{1}{2}\sum_{b = 1}^{N}\sum_{j = 1}^{m_{b}}\sum_{j^{\prime}= 1}^{m_{b}}( 1 - \delta_{j j^{\prime}}) \ln ( y_{j}^{( b )}- y_{j^{\prime}}^{( b )})^{2}\right) \; .
T^1_\eta(t) := \eta^2 \int_0^t \int_\Omega \nabla(af_1+bg_1)\cdot \frac{\nabla f_1}{(f_1+\eta)^2}\, \mathrm{d}x \mathrm{d}s begin{align*}1ex] +\eta^2 \cfrac{b}{c}\int_0^t \int_\Omega \nabla(cf_1+dg_1)\cdot \frac{\nabla g_1}{(g_1+\eta)^2}\, \mathrm{d}x \mathrm{d}s
{\bf P}\left( \bigcup_{i\in \Upsilon_{1}}I_{i,n}\right) =\sum_{i\in\Upsilon_{1}}p_{i,n}\leq \frac{4\,||K^{2}||\,R_{n}(E_{n},E_{n})}{\sigma^{2}}\rightarrow 0,\mbox{as}n\rightarrow \infty .
\underset{[0,T]}{\sup}\int\limits_{-\infty}^{\infty}e^{a(t) x_{+}^{5/4}}|u(x,t)|^2\,dx\le c^{*}=c^{*}(a_0; \|u_0\|_{10,2}; \|e^{a_0 x_{+}^{5/4}}u_0\|_2;T)
(1+\delta)\|T^\ast \omega\|_{\varphi_1}^2 + \|S\omega\|_{\varphi_3}^2 \ge \int_{\Omega_\varepsilon}\left(\sum_{j,k=1}^n \frac{\partial^2\varphi}{\partial z_j\partial \bar{z}_k}\omega_j\bar{\omega}_k-(1+1/\delta)|\partial \psi|^2 |\omega|^2\right) e^{-\varphi}
I_{12}= \int_{t_q}^T \log{|F(b+it,\chi)|}dt= \int_{t_q}^T \left( \left( \frac{1}{2}- b \right) \log{\frac{qt}{2\pi}}+ O\left( \frac{1}{t^2}\right) \right) dt &= \left( \frac{1}{2}- b \right) \left( T\log{\frac{qT}{2\pi}}- T - t_q\log{\frac{qt_q}{2\pi}}+ t_q \right) + O(1).
\Psi (BK) =\ 6\cdot \Psi(H_{n-2}\setminus \{a,f\}) + 5\cdot \Psi(H_{n-2}\setminus \{c,f\})+12\cdot \Psi(H_{n-2}\setminus c) &+5\cdot \Psi(H_{n-2}\setminus \{e,h\})+3\cdot \Psi(H_{n-2}\setminus \{c,h\})+8\cdot \Psi(H_{n-2}\setminus \{a,e\}) &+5\cdot \Psi(H_{n-2}\setminus \{a,c\})+7\cdot \Psi(H_{n-2}\setminus\{c,e\}).
T_{2,r}=-\frac{\theta_3(q)^4}{2}-\frac{\theta_4(q)^4}{2}=-\frac{1}{2}\frac{\eta(q^2)^{20}}{\eta(q)^8\eta(q^4)^8}-\frac{1}{2}\frac{\eta(q)^8}{\eta(q^2)^4}
\Bigl|\sum_n \psi_N(n) e_q(f(n))\Bigr| \leq \frac{1}{K}\sum_n \left|\sum_{k=1}^K \psi_N(n+kr) e_{s}( \overline{r}f(n+kr) )\right| \ll \frac{N^{1/2}}{K}\left(\sum_n \left|\sum_{k=1}^K \psi_N(n+kr) e_{s}( \overline{r}f(n+kr) )\right|^2\right)^{1/2},
\gamma^{0}= \left( \begin{array}{c c}{{0}}&{{{- i}}}\\{{i}}&{{{0}}}\\ \end{array}\right) , \quad \gamma^{1}= \left( \begin{array}{c c}{{0}}&{{{i}}}\\{{i}}&{{{0}}}\\ \end{array}\right) , \quad \gamma^{2}= \left( \begin{array}{c c}{{i}}&{{{0}}}\\{{0}}&{{{- i}}}\\ \end{array}\right) .
Z_{t}^{\nu}=\int_{0}^{t}\int_{\mathbf{R}_{0}^{d}}\chi_{\alpha}\left(y\right) y\tilde{J}\left( ds,dy\right) +\int_{0}^{t}\int_{\mathbf{R}_{0}^{d}}\left( 1-\chi_{\alpha}\left( y\right) \right) yJ\left(ds,dy\right) ,~t\geq 0,
\phi_1 := \phi_{7AB}(\tau, z) = 2 \zeta^{\pm 1}- 1 + (-\zeta^{\pm 2}+ 5 \zeta^{\pm 1}- 8) q + (2 \zeta^{\pm 3}- 8 \zeta^{\pm 2}+ 17 \zeta^{\pm 1}- 22) q^2 + O(q^3)
U_{ni}= u_n + c_i \Delta t \varphi_{1}( c_i \Delta t J_n)F(u_n) + \Delta t \sum_{j=2}^{i-1}a_{ij}(\Delta t J_n) D_{nj}, u_{n+1}= u_n + h\varphi_{1}(\Delta t J_n)F(u_n) + \Delta t \sum_{i=2}^{s}b_{i}(\Delta t J_n) D_{ni}
\sum_{\sigma \in W(\Phi(C_n))}{(-1)^{\ell(\sigma)}x^{L_{\phi(C_n)}(\sigma)}}= (1-x^{\left\lceil\frac{n}{2}\right\rceil})\prod_{i=1}^{\left\lceil\frac{n}{2}\right\rceil}(1-x^{2i})^2.
|\hat u|^2 \ = \ |\widehat u(\xi_1,0)|^2 \chi_{D'_{\eta}(\xi_1)}(\xi') + |\hat u - \widehat u(\xi_1,0) \chi_{D'_{\eta}(\xi_1)}(\xi')|^2 + 2 \Re \Big[(\hat u - \widehat u(\xi_1,0)) \widehat u(\xi_1,0) \chi_{D'_{\eta}(\xi_1)}(\xi') \Big],
\frac{1}{N(\lambda+1)}\sum_{j=1}^{N}\left[\phi_{\epsilon}(x_{j}^{N}(\tau_{i,\epsilon}^{+})-x_{i}^{N}(0))-\phi_{\epsilon}(x_{j}^{N}(0)-x_{i}^{N}(0))\right] \le(\kappa-1)\epsilon+\frac{\kappa+1}{\lambda+1}\epsilon:=-\frac{\gamma}{\lambda+1}\epsilon.
U\left( t,s\right) \varphi =\varphi-i\int_{s}^{t}\bar{Q}\left(\tau\right) U\left( \tau,s\right) \varphi d\tau U\left( t,s\right) \varphi =\varphi+i\int_{t}^{s}U\left( t,\sigma\right) \bar{Q}\left( \sigma\right) \varphi d\sigma
\nabla_{(t,z)}G(t,x,j) = G(t, x_{t-},j_{t-}+\delta_{t,z}(\cdot))-G(t,j_{t-}), &=\psi(t,x_{t-},(j_{t-}+\delta_{t,z}(\cdot))_{t-})- \psi(t, x_{t-},j_{t-}), &=\psi(t, x_{t-},j_{t-}) - \psi(t, x_{t-},j_{t-}) = 0.
\gamma_t=\gamma_0\delta^{n_t},\ \beta_t=\frac{\gamma_{t-1}(1-\kappa_t)}{\gamma_t(1-\kappa_{t-1})}\left(1+\frac{2\mu\gamma_{t-1}}{1-\kappa_{t-1}}\right)^{-1},\ \alpha_t=\frac{\kappa_{t-1}\gamma_{t}\beta_t}{\gamma_{t-1}},
\varphi(h_s, h_{-t}) = \varphi(C_s, C_{-t}) = 0,\varphi(h_s, C_{-t}) = \delta_{st}\frac{q^s[s]}{s(q-q^{-1})},\varphi(C_s,h_{-t}) = \delta_{st}\frac{q^{-s}[s]}{s(q-q^{-1})}, & \varphi(E_m, F_{-n}) = \delta_{mn}(q-q^{-1}), \varphi(F_s,E_{-t}) = \delta_{st}(q^{-1}-q).
\int \frac{d^{2 \sigma}Q}{( 2 \pi )^{2 \sigma}}\frac{Q_{\mu}}{( Q^{2}+ \vec{Q}^{2})}\ = 0 \, , \qquad \int \frac{d^{2 \sigma}K}{( 2 \pi )^{2 \sigma}}\frac{K_{\mu}}{( K^{2}+ \vec{K}^{2})}\ = 0 \, ,
\psi_{\alpha}={1\over 2\pi i}\int_{C_{+}}dte^{-ix\cosh t}{e^{(1-\alpha) t}\over{e^{t}+e^{i\theta}}}+{1\over 2\pi i}\int_{C_{-}}dte^{-ix\cosh t}{e^{(1-\alpha) t}\over{e^{t}+e^{i\theta}}},
R \geq a_{l_{1}}+\sum_{j=2}^{n}a_{l_{j}}\geq \varepsilon\sum_{j=l_{1}+1}^{\infty}a_{j}+\sum_{j=2}^{n}a_{l_{j}}& \geq (1+\varepsilon )\sum_{j=2}^{n}a_{l_{j}}\geq (1+\epsilon)^{n-1}a_{l_{n}}\geq (1+\epsilon)^{n-1}a_{k-1}.
S_q(a,n) =\bigg\{\frac{1}{(1-q)^2}-\frac{n(1-a)a^{(n-1)/2}}{(1-q/a)(1-aq)(1-a^n)}\bigg\}begin{align*}5pt] \quad\times\frac{(1-aq^n)(a-q^n)}{(1-a)^2}-\frac{1}{q(q^{n-1};q^2)_2}.
{\rm Hilb}({\cal A}_3,t)=(1,2,3,1),{\rm Hilb}({\cal A}_4,t)=(1,3,6,10,11,6,1),{\rm Hilb}({\cal A}_5,t)=(1,4,10,20,35,51,64,60,35,10,1),{\rm Hilb}({\cal A}_6,t)=(1,5,15,35,70,126,204,300,405,490,511,424,245,85,15,1).
\eta^{\prime \prime}(x) = m^2_1(e^{m_2}-e^{m_3}) e^{m_1 x}+ m^2_2 (e^{m_3}-e^{m_1}) e^{m_2 x}+ m_3^2 ( e^{m_1}- e^{m_2}) e^{m_3 x}.
w_j(X,Y,t,\tilde X, \tilde Y,\tilde t) = v(X,Y,t) - v(\tilde X, \tilde Y, \tilde t) - w(Y,t) - \bigl(\frac{j^4}{4}|X-\tilde X|^4+\frac{j^4}{4}|Y-\tilde Y|^4+\frac j2|t-\tilde t|^2\bigr ).
\left(\overline{u}+\overline{\rho(u)}\right)\left(\overline{N}(^{\ast}b)(n)\right) =\left(-\underline{u}_{1}\pi_{1}p\xi_{be_{k}}(n)-\pi_{1}^{1}\rho_{2}\gamma \xi_{be_{k}}(n),-u_{2}\gamma \xi_{be_{k}}(n),0\right) &=(-\pi_{1}^{1}u_{1}p\xi_{be_{k}}(n)-\pi_{1}^{1}\rho_{2}\gamma \xi_{be_{k}}(n), -u_{2}\gamma \xi_{be_{k}}(n),0)
\frac{\|e_{m+1}\|_{\mathcal G_q^{\omega}}}{\|e_{m+1}\|_{\mathcal{PG}_q^{\omega}}}\le\frac{1}{\omega(1)\left(\sum_{n=1}^{m}\frac{1}{n}\right)^{\frac{1}{q}}}\xrightarrow[m\to \infty]{}0.
\zeta ({m_i},{k_i},{\alpha_{ie}}) = \int_0^\infty{\frac{1}{{\Gamma ({k_i})}}{{(\frac{{{k_i}}}{{{\alpha_{ie}}}})}^{{k_i}}}y_{ie}^{{m_i}+{k_i}- 1}}&\quad\quad\times \exp ( - \frac{{{k_i}{y_{ie}}}}{{{\alpha_{ie}}}})d{y_{ie}}.
\eta_{mix}(P,T)=(\frac{T_{cmix}}{T_{cz}})^{-1/6}\cdot(\frac{P_{cmix}}{P_{cz}})^{2/3}\cdot(\frac{M_{mix}}{M_{z}})^{1/2}\cdot\frac{\alpha_{cmix}}{\alpha_{cz}}\cdot\eta_{z}(P_{z},T_{z})
d s^{2}= G_{M N}d x^{M}d x^{N}= e^{2 \phi}g_{\mu \nu}d x^{\mu}d x^{\nu}+ \sum_{i = 1}^{n}e^{2 \phi_{i}}d s_{i}^{2}, \qquad \mu = 0 , . . . , d ,
\lvert\eqref{eq:8}\rvert \lesssim y^{-\gamma}\sum_{n=l+1}^{\infty}\alpha_{n}e^{-\lambda_{n}( s- s_{0})}e^{-\lambda_{l}s_{0}}e^{-\kappa s_{0}}= e^{-\kappa s_{0}}e^{-\lambda_{l}s}y^{-\gamma}\sum_{n=l+1}^{\infty}\alpha_{n}e^{-(\lambda_{n}-\lambda_{l})( s- s_{0})}
\lim_{n \to \infty}\frac{P_{n + p(p+1) ,k}(z)}{P_{n,k}(z)}=\prod_{\rho =0}^{p(p+1)-1}\widetilde{F}_{k}^{(\rho)}(z),z\in\mathbb{C}\setminus[a_{k},b_{k}], k=0,\ldots,p-1,
\begin{array}{l}{{L G_{2 X}= X_{R}^{- 1 / 3}X_{L}^{1 / 3}L G_{2}X_{L}^{- 1 / 3}X_{R}^{1 / 3}}}\\{{G_{2 X}= X_{R}^{- 1 / 3}X_{L}^{1 / 3}G_{2}X_{L}^{- 1 / 3}X_{R}^{1 / 3}.}}\\ \end{array}
\frac{\partial^{3}\bar{t}}{\partial x^{3}}|_{P_{0}}= - \frac{\partial^{2}\bar{\alpha}}{\partial \bar{t}^{2}}|_{P_{0}}\sinh ( \chi ) \cosh^{2}( \chi ) = - \frac{R}{2}\sinh ( \chi ) \cosh^{2}( \chi )
I_{C}=\frac{I_{CBO}+\alpha I_{B}}{1-\alpha-(\frac{V_{CE}-V_{BE}(I_{B})}{BV_{CBO}})^{n}}\cong\frac{I_{CBO}+\alpha I_{B}}{1-\alpha-(\frac{V_{CE}}{BV_{CBO}})^{n}}
\Vert g \Vert_{C^{1,\bar 1}(\bar \Omega \times I)}:= \max \left\{\Vert g \Vert, \Vert \nabla g \Vert, \Vert \Delta g \Vert, \Vert g_t\Vert, \Vert g_{t t}\Vert \right\}\leq A,
e^{-t\widetilde{\Box}^+_{b,m}}(x,x)\sim \Bigr(\sum\limits^{p}_{s=1}e^{\frac{2\pi(s-1)}{p}mi}\Bigr)\sum^\infty_{j=0}t^{-n+j}\widetilde\alpha^+_{n-j}(x)\mod O\Bigr(t^{-n}e^{-\frac{\varepsilon_0\hat d(x,X_{\mathrm{sing\,}})^2}{t}}\Bigr)
\begin{aligned}\mathbb{E}\left[\|\bar{y}^{k+1}-\bar{g}^{k+1}\|^2|\mathcal{F}^k\right] &= (1-2\alpha^k+(\alpha^k)^2)\| \bar{y}^k-\bar{g}^k\|^2 +(\gamma_2^k)^2 \mathbb{E}\left[\|\bar{\xi}_w^k\|^2\right]\end{aligned}
\varpi \partial_i(\partial_j(u_k w_\ell))=\partial_i(\partial_j(\varpi u_k w_\ell))-\partial_i((\partial_j\varpi)u_kw_\ell)-\partial_j((\partial_i\varpi)u_kw_\ell)-(\partial_i\partial_j\varpi)u_kw_\ell.
\mathcal{G}(u;x,p)\* \mathcal{G}(v;x,p)={\frac{1}{2}}\int \!dw~\exp \left( -{\frac{1}{2}}\left( e^{u+v-w}+e^{u-v+w}+e^{-u+v+w}\right) \right) ~ \mathcal{G}(w;x,p).
\begin{cases}\dot{y}_{_{4i+1}}=y_{_{4i+2}}\dot{y}_{_{4i+2}}=-y_{_{4i+1}}-2y_{_{4i+1}}y_{_{4i+3}}\dot{y}_{_{4i+3}}=y_{_{4i+4}}\dot{y}_{_{4i+4}}=-y_{_{4i+3}}-y^2_{_{4i+1}}+y^2_{_{4i+3}}\, , \end{cases}
Av= \begin{bmatrix*}[r]a_0 v_0 + a_1v_1 + \cdots + a_{n-1}v_{n-1}a_0 v_1 + \cdots + a_{n-2}v_{n-1}\ddots a_0 v_{n-1}\end{bmatrix*}\in \mathbb{C}^n.
\mathbb{P}_{n}^{(N)}(f)=\sum_{j=1}^{m_{N}}\frac{P(A_{j}^{(N)})}{\mathbb{P}_{n}^{(N-1)}(A_{j}^{(N)})}\mathbb{P}_{n}^{(N-1)}(f1_{A_{j}^{(N)}}),
B(z) = \psi_{n}(x;N,M) \psi_{n-1}(x;N+1,M+1) /4n \hphantom{B(z)}{}=\frac{(1-z)^{N+1}(1+z)^{M+1}}{4n}P_{n}^{( N,M)}(z) P_{n-1}^{(N+1,M+1)}(z) .
{1\over 2}\bar \theta ( 1- \Gamma^5){\slash \hskip -7pt{\Pi_{\bar z}}}\,( 1-\Gamma^5)\partial_z \theta ={1\over 2}\bar \theta ( 1- \Gamma^5) \left( \gamma_0\Pi_{\bar z}{}^0 + \sum_{a=5}^{a=9}\gamma_a \Pi_{\bar z}{}^a \right)\partial_z \theta \ .
|x_0:x_1:x_2| = \begin{cases}x_0\cdot|x_0:x_1:x_2| = |x_0^2:x_0x_1:x_0^2x_2| =|y_0:y_1:y_0y_3|; x_1\cdot|x_0:x_1:x_2| = |x_0x_1:x_1^2:x_1^2x_2| =|y_1:y_2:y_2y_3|, \end{cases}
F^{{\mu}{\nu}{\sigma}}= - G^{{\mu}{\nu}{\sigma}}+ \frac{1}{({\partial}_{\lambda}a ) ({\partial}^{\lambda}a )}\left[{\partial}^{[{\mu}}a{\cal G}^{{\nu}{\sigma}]{\rho}}{\partial}_{\rho}a + \frac{1}{3 !}{\epsilon}^{{\mu}{\nu}{\sigma}{\rho}{\eta}{\delta}}{\partial}_{[{\rho}}a{\cal G}_{{\eta}{\delta}]{\theta}}{\partial}^{\theta}a \right] .
\sum_{J}\frac{(-i)^{|J|}}{J !}\partial_{\xi}^{J}q \, \partial_{x^\prime}^{J}q - \sum_{J}\frac{(-i)^{|J|}}{J !}\partial_{\xi}^{J}b \, \partial_{x^\prime}^{J}q - \frac{\partial q}{\partial x_n}+ c = 0.
X_{N}^{( K )}( a ) \ = \ \left( \frac{\alpha}{\pi}\right)^{K^{2}}\, \int d [ \sigma ] \exp \left( - \alpha{\mathrm{t r}\,}( \sigma - a ) ( \sigma^{\dagger}- a ) \right) \, \mathrm{d e t}^{N}\sigma \sigma^{\dagger}\: ,
|P\nabla f|^2 \le \frac{\delta}{c_2}(a\Pi P\nabla f)\cdot (P\nabla f) = \frac{\delta}{c_2}(\Pi^Ta\Pi\nabla f)\cdot (P\nabla f) = \frac{\delta}{c_2}(a\Pi \nabla f)\cdot (\Pi\nabla f)\,.
dq^\prime_t= (\phi_* b_+)(t^*,q^\prime_t)dt-(\phi_*b_-)(t^*,q^\prime_t)dt+ (\hat{\phi}_*(\tilde\gamma^{-1}\sigma))(t^*,q^\prime_t)\circ d\tilde W_t = b^+(t^*,q^\prime_t)dt-(-b_-)(t^*,q^\prime_t)dt+ (\hat{\phi}_*(\tilde\gamma^{-1}\sigma))(t^*,q^\prime_t)\circ d\tilde W_t = b(t^*,q_t^\prime)dt+ (\hat{\phi}_*(\tilde\gamma^{-1}\sigma))(t^*,q^\prime_t)\circ d\tilde W_t
( 1 - \eta^{2})^{- M / 2}e^{-{\frac{\eta^{2}}{1 - \eta^{2}}}x^{2}}H_{M - 1}({\frac{x}{\sqrt{1 - \eta^{2}}}}) = \sum_{n = 0}^{\infty}{\frac{( - \eta^{2}/ 4 )^{n}}{n !}}H_{2 n + M - 1}( x ) .
(\frac{\partial}{\partial z_{\nu}})dz_{\lambda}=\delta_{\nu\lambda},(\frac{\partial}{\partial\overline{z}_{\nu}})dz_{\lambda}=0,(\frac{\partial}{\partial z_{\nu}})d\overline{z}_{\lambda}=0,(\frac{\partial}{\partial\overline{z}_{\nu}})d\overline{z}_{\lambda}=\delta_{\nu\lambda}
\begin{cases}w = (D_t + \sqrt{1+D_x^2}) u & \bar{w}= -(D_t - \sqrt{1+D_x^2})u\end{cases}\, , \begin{cases}u = \langle D_x \rangle^{-1}(\frac{w + \bar{w}}{2}) & D_t u = \frac{w-\bar{w}}{2}\end{cases}\,.
g(t,\theta,x)^{\gamma_0 - 1}= \Big\| \| (\phi^{-1}f)(\xi_t)^{\gamma_0 - 1}e^{-J_g(t,r,\xi)}\|_{\Pi_x^{(\phi)};\frac{1}{\gamma_0 - 1}}\Big\|_{\mathbf 1_{0\leq r\leq \theta}dr;\frac{1}{\gamma_0 - 1}},t\geq 0, \theta \geq 0, x\in E.
\hat{G}^{\hat{\alpha}_{1}\ldots \hat{\alpha}_{4}}= \left( f^{- 1}\right)^{\hat{\alpha}_{1}\ldots \hat{\alpha}_{4}}{}_{\hat{\beta}_{1}\ldots \hat{\beta}_{4}}\left( \hat{B}^{\hat{\beta}_{1}\ldots \hat{\beta}_{4}}-{}^{\star}\hat{B}^{\hat{\beta}_{1}\ldots \hat{\beta}_{7}}\hat{A}_{\hat{\beta}_{5}\hat{\beta}_{6}\hat{\beta}_{7}}\right) \, ,
M_{r}=\sum_{\pi \in \mathcal{P}_2(2 r)}\int_{[0,1] \times [-m,m]^{|SP(\pi)|}\times[-\lambda,1]^{|DP(\pi)|}}\prod_{s=1}^{r}\chi_{[0,\lambda]}\left(x_0+ \sum_{i=1}^{2s-1}\epsilon_\pi(i) x_{\pi(i)}\right) & \times \chi_{[0,1]}\left( x_0+ \sum_{i=1}^{2s}\epsilon_\pi(i) x_{\pi(i)}\right) \prod_{l=0}^{r}\mathrm{~d}x_l,
\sigma^2(n,k)= \frac{2\lambda^4}{(\alpha^2+\lambda^2)^2}\frac{k(k-1)(N-2k)}{(n-1)(N-n-1)}\approx \frac{2\lambda^4}{(\alpha^2+\lambda^2)^2}\frac{k^2(N-2k)}{n(N-n)}\ .
\begin{bmatrix}1 -q_m -a_{m-1}-a_{m-2}\ldots -a_1 -a_0 1 -q_{m-1}0 \ldots 0 \ddots \ddots \ddots \vdots 1 -q_2 0 1 -q_1 1\end{bmatrix}s = \begin{bmatrix}0 0 \vdots 0 0 1\end{bmatrix}
\rho_0^{\mbox{\tiny{bos}}}(\omega , k) = \frac{\pi k (\omega^2-k^2)^2}{32}\left[1 +\frac{2 T}{k}\log \left( \frac{1- e^{-\frac{\omega + k}{2T}}}{1- e^{-\frac{\omega + k}{2T}}}\right)\right]\,.
\mathrm{rk}(L'_{2,0}) =n-1,\ \mathrm{rk}(L'_{2,1})=2, \mathrm{rk}(L'_{v,0}) =n,\ \mathrm{rk}(L'_{v,1})=1(v=p_1,\dots,p_k), \mathrm{rk}(K_{\ell,v,0}) =n-1,\ \mathrm{rk}(K_{\ell,v,1})=1 (v=2,p_1,\dots,p_k).
u_{w\lambda}=f_{i_{1}}^{(\langle h_{i_{1}}, s_{i_{2}}\cdots s_{i_{\ell}}\lambda\rangle)}\cdots f_{i_{\ell-1}}^{(\langle h_{i_{\ell-1}}, s_{i_{\ell}}\lambda\rangle)}f_{i_{\ell}}^{(\langle h_{i_{\ell}}, \lambda\rangle)}.u_{\lambda}
J_{1}\le C(c_{1},c_{2},C_{1},C_{2},n) \biggl(\sum\limits_{s=1}^{\infty}\sum\limits_{\alpha \in \widetilde{A}^{s}}\widehat{\gamma}^{s}_{\alpha}\int\limits_{\widetilde{Q}^{s}_{\alpha}}|\varphi^{s}_{\alpha}-\varphi(x)|\,dx + \sum\limits_{m \in \mathbb{Z}^{n}}\widehat{\gamma}_{0,m}\|\varphi|L_{1}(Q_{0,m})\|\biggr).
\rho^{-m_{1}}b_{j_{1}}p^{j_{2}+j_{3}}N_{j_{2}}N_{j_{3}}-\rho^{-m_{2}}b_{j_{2}}p^{j_{1}+j_{3}}N_{j_{1}}N_{j_{3}}=\rho^{-m_{3}}b_{j_{3}}p^{j_{1}+j_{2}}N_{j_{1}}N_{j_{2}}.
\begin{aligned}n_1 = -8m(t-1)(m^2-9t^4+24t^3+24t^2-96t+48)^{-1}, n_2 =-\{m^2-(6t^2-8t+8)m+3(t+2)(3t-2)(t-2)^2\}& \times (m^2-9t^4+24t^3+24t^2-96t+48)^{-1},\end{aligned}
D=-\sum_{\substack{i,j=1, i\not=j}}^n\partial_y\Bigg[\bigg(\frac{\tau_i(\mathcal{L}B_{ij,0})}{x_{ij}^3}\bigg)(\varphi_{ij}-1)\Bigg]-\sum_{\substack{i,j=1, i\not=j}}^n\partial_y\bigg(\frac{\tau_i([|D|,\varphi_{ij}]B_{ij,0})}{x_{ij}^3}\bigg).
{\cal Z}_{_0}= \int D \overline{{{\xi}}}D \xi \exp \left\{i \int d^{2}x \left[ - i \overline{{{\xi}}}( x ) \partial \! \! \! / \partial \! \! \! /^{\dagger}\partial \! \! \! / \xi ( x ) + m \overline{{{\xi}}}( x ) \partial \! \! \! / \partial \! \! \! / \xi ( x ) \right] \right\}.
(\partial_{\alpha}[\hat \Psi_{\alpha}\circ F_{\alpha, n}^{-1}])' =\psi' \left(\frac{z_{n}+1}{2}\right)z_{n}'\partial_{\alpha}z_{n}+ \psi \left(\frac{z_{n}+1}{2}\right)\partial_{\alpha}z_{n}' &+\sum_{i=1}^{n-1}\psi' \left(z_{i}\right)z_{i}'\partial_{\alpha}z_i + \psi \left(z_i \right)\partial_{\alpha}z_i',
\displaystyle{F^{+I}_{\mu \nu}\longrightarrow \tilde F^{+I}_{\mu \nu}=U^I_J\, F^{+J}_{\mu \nu}+Z^{IJ}\,G^{+}_{\mu \nu J}\;\;G^{+}_{\mu \nu I}\longrightarrow \tilde G^+_{\mu \nu I}= V_I^J\;G^{+}_{\mu \nu J}+ W_{IJ}\,F^{+J}_{\mu \nu}},
\Sigma_4=\sum\limits_{1\leq|m|\leq M}\frac{\Theta_m}{2\pi i m}\,, &\Theta_m=\sum\limits_{D\leq d<2D}e\left(\frac{\sqrt{X}m}{d}\right)\sum\limits_{n\in \mathcal{N}(d)}e\left(-\frac{nm}{d}\right)\,, &\Sigma_5=\sum\limits_{D\leq d<2D}\sum\limits_{n\in \mathcal{N}(d)}f_{M}\left(\frac{\sqrt{X}-n}{d}\right)\,.
I_{\mu}^{\ \nu}\equiv \left[ \delta_{\mu}^{\nu}+ \phi^{- 1}\phi_{\mu \alpha}h^{\alpha \nu}\right] , \quad I^{- 1}\_{\mu}^{\ \alpha}I_{\alpha}^{\ \nu}= \delta_{\mu}^{\nu}, \quad I \equiv d e t I_{\mu}^{\ \nu}.
\begin{array}{c}\sum_{0\le p \le 1-a_{ij}}(-1)^{p}\left[ \begin{array}{clcr}1-a_{ij}p \end{array}\right]_{Q_{i}}e_{i}^{1-a_{ij}-p}e_{j}e_{i}^{p}= 0 \sum_{0\le p \le 1-a_{ij}}(-1)^{p}\left[ \begin{array}{clcr}1-a_{ij}&p \end{array}\right]_{Q_{i}}f_{i}^{1-a_{ij}-p}f_{j}f_{i}^{p}=0 \end{array}
\begin{aligned}\widetilde{M}_{k}(\alpha,\theta,E,0) =M_k(\alpha,\theta,E,0)+\sum_{i=0}^{k-1}M_{k-i-1}(\alpha,\theta,E,i+1)\begin{pmatrix}g(i) 0 0 0 \end{pmatrix}\widetilde{M}_{i}(\alpha,\theta,E,0). \end{aligned}
\left[\begin{array}{cc}0 -\left({2\pi n\over m_l-m_r}\right)^2 b_i b_i 0\end{array}\right],\;\;\;\;\;\left[\begin{array}{cc}0 -\left({2\pi n\over m_l-m_r}\right)^2 b_i^* b_i^* 0\end{array}\right]
\left(\begin{matrix}x=0 x=t_1 x=t_2 x=\infty \, \left( \frac12 \right) \begin{matrix}0 \theta^0 \end{matrix}\begin{matrix}0 \theta^{t_1}\end{matrix}\begin{matrix}0 \theta^{t_2}\end{matrix}\overbrace{\begin{matrix}1 \theta^\infty_1/2 -1 \theta^\infty_1/2 \end{matrix}}\ \end{matrix}\right) ,
V(P, I) = \sum_{\{n_{i, k}\}}\left(\sum_{j \geq 0}\frac{\hbar^j}{j!}\partial^j_P\right)\prod_{i, k}\frac{\hbar^{(i - 1)n_{i, k}}}{n_{i, k}!}I_{i, k}^{n_{i, k}}= \sum_{\{n_{i, k}\}}\sum_{j \geq 0}C(\{n_{i, k}\}, j) \hbar^{p(\{n_{i, k}\}, j)}\partial^j_P\prod_{i, k}I_{i, k}^{n_{i, k}}
\begin{aligned}u_{0,\boldsymbol{\lambda},\boldsymbol{x}}\equiv u_{\lambda_{1,0},\lambda_{2,0},..\lambda_{m,0}, x_{1,0},x_{2,0},..x_{m,0}}&\equiv \sum_{j=1}^{m}\chi(x-x_{j,0})u_{\lambda_{j,0}}(x-x_{j,0}).\end{aligned}
\delta_\epsilon q^n=\epsilon\,\left\{q^n,\phi\right\}=\epsilon\,\frac{\partial H}{\partial p_n}\ \ ,\ \ \delta_\epsilon p_n=\epsilon\,\left\{p^n,\phi\right\}=-\epsilon\,\frac{\partial H}{\partial q^n}\ \ ,\ \ \delta_\epsilon\lambda=\dot{\epsilon}\ ,
(J(f)(t))(u) =\sum_{j=1}^{\infty}\Phi(t)e_j(\cdot)\langle u, e_j\rangle =\sum_{j=1}^{\infty}\Phi(t)\big(\langle u, e_j\rangle e_j\big)(\cdot) =\Phi(t)\left( u\right)(\cdot),
u_{k,n}^{[j],-}(y) := y^{-\frac{k}{2}}\frac{\partial^j}{\partial s^j}W_{\mathrm{sgn}(n)\frac{k}{2}, s-\frac{1}{2}}(4\pi |n|y)\bigg|_{s=\frac{k}{2}}, u_{k,n}^{[j],+}(y) := y^{-\frac{k}{2}}\frac{\partial^j}{\partial s^j}\mathcal{M}^+_{\mathrm{sgn}(n)\frac{k}{2}, s-\frac{1}{2}}(4\pi |n|y)\bigg|_{s=\frac{k}{2}},
A^{S_1G_1S_2G_2}(z)=\frac{1}{z^{4/3}}\begin{pmatrix}-t^{1/3}0 0 0 -\omega t^{1/3}0 0 0 -\omega^2 t^{1/3}\end{pmatrix}\hphantom{A^{S_1G_1S_2G_2}(z)=}{}+\frac{1}{z}\begin{pmatrix}\theta^\infty_1/3-2/3 * * * \theta^\infty_1/3-2/3 * * * \theta^\infty_1/3-2/3 \end{pmatrix}+\cdots.
p_{i,j}^{(t+1)}= \begin{dcases}0^{t+1}\mbox{if}i=j=0; \frac{j}{n}p_{i,j}^{(t)}\,+\, \frac{n-j+1}{n}p_{i,j-1}^{(t)}\mbox{if}\; 0 < i \le j \le n j < t; 0 \mbox{otherwise}; \end{dcases}
(1-q)P_{n}(0;q)P_{m}(0;q)-\!\!\sum_{k\in\mathbb{Z}\setminus\{0\}}\frac{\mathcal{S}_{q}(qs_{k}(q))}{s_{k}(q)\mathcal{S}_{q}'(s_{k}(q))}P_{n}\left(q^{1/2}s_{k}(q);q\right)P_{m}\left(q^{1/2}s_{k}(q);q\right)=\delta_{mn}
\frac 12 \frac{d}{dt}\sum_{i=1}^N\frac{\|v_i\|_2^2}{u_{i,\infty}}+ \frac \beta 2 \sum_{i=1}^{N}\frac{\|v_i\|_2^2}{u_{i,\infty}}+ \frac \beta 2 \sum_{i=1}^{N}\|v_i\|_{H^1(\Omega)}^2 \leq C\sum_{i=1}^{N}(\|v_i\|_{\mu+1}^{\mu+1}+ \|v_i\|_{2+{\delta}}^{2+{\delta}}).
\begin{aligned}\lambda\sp+(A) =\lim_{n\to\infty}\frac{1}{n}\int\log\|A\sp n(y)\|d\mu(y) &=\lim_{n\to\infty}\frac{1}{n}\int\log\|H\sp n(y)\|d\mu(y) &=\lambda\sp+(H).\end{aligned}
L(x) \stackrel{T^\ast_s}{\longmapsto}D_s L(x) D_s^{-1}\stackrel{\eqref{eq:DLD}}{=}Q_1'\cdots Q_m' P^k \stackrel{e^\ast_u}{\longmapsto}Q_{u+1}' \cdots Q_m' P^k Q_1' \cdots Q_u'.
|P_{U}| \leq \left| P(0) + P'(0) + \frac{1}{2}\, P''(0) \right| + C \, (1+|x|^6)\, |U|_2^3 = \frac{1}{2}\, \left| P''(0) \right| + C \, (1+|x|^6)\, |U|_2^3 \, .
\frac{1}{p}\cdot \frac{d}{dt}\int_{\Omega}u^{-p}(t,x)= -(p+1)\int_{\Omega}u^{-p-2}(t,x)|\nabla u(t,x)|^2 + (p+1) \chi \int_{\Omega}\frac{u^{-p-1}(t,x)}{v(t,x)}\nabla u (t,x)\cdot \nabla v(t,x) &- \int_{\Omega}a(t,x) u^{-p}(t,x)+ \int_{\Omega}b(t,x) u^{-p+1}(t,x)\forall\, t>s.
\left( \int_{B^{c}(0,R)}\frac{u(x)}{|x|^{(N-\ell)q}}dx \right)^{1/q}= |\mathbb{S}^{N-1}|^{1/q}\int_{R}^{\infty}r^{-\beta q-(N-\ell)q+N-1}dr=C R^{\alpha},
i\,\lim_{r\rightarrow 0}\left(Mr\right)^{\nu +1}g_n(r) \sin\left(\frac{\pi}{4}+\frac{\Theta}{2}\right)=\lim_{r\rightarrow 0}\left(Mr\right)^{-\nu}f_n(r) \cos\left(\frac{\pi}{4}+\frac{\Theta}{2}\right)
F_a(q) = \frac{f(-q^2,-q^3)}{f^2(-q,-q^4)}f^2(-q^5)-4\cos^2(2n\pi/5)q^{1/5}\frac{f^2(-q^5)}{f(-q,-q^4)}+2\cos(4n\pi/5)q^{2/5}\frac{f^2(-q^5)}{f(-q^2,-q^3)}-2\cos(2n\pi/5)q^{3/5}\frac{f(-q,-q^4)}{f^2(-q^2,-q^3)}f^2(-q^5).
E[X]=\sum_{i=1}^{\infty}x_{i}p_{i}=2\cdot\frac{1}{2}+4\cdot\frac{1}{4}+8\cdot\frac{1}{8}+16\cdot\frac{1}{16}+\cdot\cdot\cdot=1+1+1+1+\cdot\cdot\cdot.
\left (\mathcal{K}_m' (\varphi_m) v \right )^{(\beta)}(s) = \tilde{h}\sum_{j=1}^m \sum_{i=1}^\rho w_i \; \frac{\partial^{\beta+1}\kappa}{\partial s^\beta\partial u}(s, \zeta_i^j, \varphi_m (\zeta_i^j)) v (\zeta_i^j), \;\;\; s \in [a, b].