harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{aligned}\mu_c^{\pm}(Y_1,Y_2,\dots,Y_m)=\mu_c^{\pm}(R^{-1}Y_1,R^{-1}Y_2,\dots,R^{-1}Y_m), \nu_c^{\pm}(Y_1,Y_2,\dots,Y_m)=\nu_c^{\pm}(R^{-1}Y_1,R^{-1}Y_2,\dots,R^{-1}Y_m).\end{aligned}
\otimes^L_n(F_1,\dots,F_n)X = \sum_{(U,V)\in \mathrm{mFil}_nX}{L(U_1\backslash V_1) \otimes \dots \otimes L(U_{n-1}\backslash V_{n-1}}) \phantom{\otimes^L_n(F_1,\dots,F_n)X =}{}\otimes F_1 ( U_1\backslash V_0 ) \otimes \dots \otimes F_n ( U_n\backslash V_{n-1}).
\sum_{n=0}^{\infty}\frac{H_{n}(x)H_{n}(y)}{n!}(\frac{u}{2})^{n}=\frac{1}{\sqrt{1-u^{2}}}e^{\frac{2u}{1+u}xy-\frac{u^{2}}{1-u^{2}}(x-y)^{2}}.
n\int\limits_T \ell( x_\infty^n(t) - y^n_\infty) d\mu(t) + \ell( y_\infty^n - x_0) + \sum\limits_{i=1}^{\infty}\eta_i \ell( y_\infty^n- y^n_i ) = \inf\limits_{u \in X}\left( n\int\limits_T \ell( x_\infty^n(t) - u) d\mu(t) + \ell( u - x_0) + \sum\limits_{i=1}^{\infty}\eta_i \ell( y_\infty^n- y^n_i )\right).
{}B(t)\bar{\bf z}^{(\epsilon)}(t)=B(t){\bf z}^{(\epsilon)}(t)\leq{\bf c}(t)+\mbox{\boldmath $\epsilon$}+\int_{0}^{t}K(t,s){\bf z}^{(\epsilon)}(s)ds={\bf c}(t)+\mbox{\boldmath $\epsilon$}+\int_{0}^{t}K(t,s)\bar{\bf z}^{(\epsilon)}(s)ds.
a_{k+1}=a_k\, (1+4z_k)^2 + 8\,b_k\,\frac{z_k(1-z_k)(1+4z_k)^2}{1-22z_k-4z_k^2}, b_{k+1}=5\,b_k\,(1+4z_k)^2\frac{1+2z_k-4z_k^2}{1-22z_k-4z_k^2}
\hat{P}_{\lambda_{h,\beta}}={\mathop{Sym}}_{\mathfrak{S}_{d_{\beta}}}\left(\prod_{s=1}^{d_{\beta}}p_{\beta,r_{\beta}(h,s)}(w_{\beta,s}) \prod_{1\leq s<r\leq d_{\beta}}\Big(1+\frac{(\beta,\beta)\cdot \hbar}{2(w_{\beta,s}-w_{\beta,r})}\Big)\right).
\begin{array}{c}|0\rangle \quad \bar{R}(\lambda ,\bar{\eta},\Theta ) M\cos (\nu_{0})\end{array}\dots \begin{array}{c}|n\rangle \quad \bar{R}(\lambda ,\bar{\eta},\Theta )a_{n}(\eta ,\theta ) M\cos (\nu_{n})\end{array}\quad .
\left\langle \mathbf{K}\left[1-e^{-h}\right](e_n),e_n \right\rangle =\sum_{i=1}^{+\infty}\int_{E\times E}\lambda_i \varphi_i(x)\varphi_i(y)e_n(x)e_n(y)dxdy =\sum_{i=1}^{+\infty}\lambda_i \left(\int_{E}\varphi_i(x)e_n(x)dx\right)^2 =\sum_{i=1}^{+\infty}\lambda_i \left\langle \varphi_i,e_n \right\rangle^2
ae_{r}a^{-1}= \left(\sum_{j=1}^{n}f_{\sigma(j)}b_{j}e_{j}\right)e_{r}\left(\sum_{l=1}^{n}e_{l}c_{l}f_{\sigma(l)}\right)= (f_{\sigma(r)}b_{r}e_{r}) e_{r}(e_{r}c_{r}f_{\sigma(r)})= f_{\sigma(r)},
H_{nm}^{\{{\bf R}\}}(0,0)=\frac{m!n!(-1)^{\frac{m+n}2}}{2^{\frac{|m-n|}2}\left(\frac{m+n}2\right)!}\left[R_{11}^nR_{22}^m\left(r^2-1\right)^{\mu_{mn}}\right]^{\frac 12}P_{\mu_{mn}}^{(\frac{|m-n|}2,\frac{|m-n|}2)}\left(\frac r{\sqrt{r^2-1}}\right),
x =\sum_{j=1}^n[\sum_{i=1}^m E_N\{E_M(x{\mu_j}^*){\lambda_i}^*\}\lambda_i]\mu_j = \sum_{i=1}^m\sum_{j=1}^nE_N\{E_M(x{\mu_j}^*{\lambda_i}^*)\}\lambda_i \mu_j.
\left( \partial_{\mu}1 \otimes 1 + \frac{1}{4}\epsilon_{\alpha \beta \delta}\omega_{\mu}^{\alpha \beta}1 \otimes \gamma^{\delta}+ \frac{i}{2}A_{\mu}^{c}\sigma_{c}\otimes 1 - \frac{1}{2 \ell}e_{\mu \alpha}1 \otimes \gamma^{\alpha}\right) \varepsilon_{r}= 0 \, ,
\lim_{\varepsilon\to{0}^{+}}\frac{f(\mathbf{x}_{0}+\varepsilon \mathbf{x})}{\varepsilon^{\alpha}L(\varepsilon)}=g(\mathbf{x}) \ \ \ \mbox{strongly in}\mathcal{K}_{M,r}'(\mathbb{R}^{n}) \ (\mbox{resp.}\mathcal{S}'_{r}(\mathbb{R}^{n})).
\log T_{\mathfrak{P}_{\Gamma}}(X;E_\rho)=\log T_{X_0}(X;E_\rho)+\sum_{P_j\in\mathfrak{P}_{\Gamma}}\log(t_{P_j})\left(\sum_{k=0}^n\frac{(-1)^k\dim(\sigma_{\rho,k})\lambda_{\rho,k}}{2}\right).
k_3(\langle H\rangle, a) &=\frac{1}{2\times 4^5}\sum_{x\neq 0,a}\sum_{y\neq 0,a,x}[(2+h\chi_4(a-x)+\overline{h}\overline{\chi_4}(a-x)) &\times (2+h\chi_4(a-y)+\overline{h}\overline{\chi_4}(a-y))(2+h\chi_4(x-y)+\overline{h}\overline{\chi_4}(x-y)) &\times (2+h\chi_4(x)+\overline{h}\overline{\chi_4}(x))(2+h\chi_4(y)+\overline{h}\overline{\chi_4}(y))].
\left.T^{(2)}_{eff}\right.^2 = \cosh(2\rho)\,\cosh^2\left(\frac{\chi}{\sqrt 3}+\sigma\right)\pm \sqrt{\sinh^2(2\rho)\,\cosh^4\left(\frac{\chi}{\sqrt 3}+\sigma\right)+\sinh^4\left(\frac{\chi}{\sqrt 3}+\sigma\right)}\quad.
G(n,k)=\frac{n^2 2^{4 k-16 n+10}\binom{2 n}{n}^3 \binom{n}{k}\binom{k+n}{n-k}\binom{k+2 n-1}{n-1}\binom{2 k+4 n-2}{k+2 n-1}}{\binom{2 k}{k}^2}.
\Gamma ( A^{( 0 )}) \, = \, - \log \left| \frac{\vartheta ( \alpha , \tau )}{\vartheta ( 0 , \tau )}\right|^{2}\, + \, \frac{e^{2}}{2 \pi}\beta L{A}_{\mu}^{( 0 )}{A}^{( 0 ) \mu}
\begin{aligned}\lim_{t\to\infty}\textnormal{Prob}\bigl(Z_{t+\theta t}=Z_t \; \forall \theta \in [0,s] \bigr) = \lim_{t\to\infty}\textnormal{Prob}\bigl(Z_{t+st}=Z_t\bigr) & = \textnormal{Prob}\bigl( \overline{Z}_{1+s}= \overline{Z}_1 \bigr) = \textnormal{Prob}\left( \Theta > s \right),\end{aligned}
\tilde{\gamma}_{q}(\nu)=\frac{1}{\tau_{c}}|\frac{\tilde{E}_{q}(\nu)}{E_{q,s}}|^{2}=\frac{1}{\tau_{c}}\frac{1}{(2\tau_{c})^{-2}+4\pi^{2}(\nu-\nu_{q})^{2}},
(-1)^mP_k^{(m)}(\epsilon) = \frac{1}{m!}\sum_{l=1}^k\binom{k}{l}(-\epsilon)^l(2l)_m & = \frac{1}{m!}\sum_{l=1}^{k}\binom{k}{l}(-1)^l \kappa^{2l}\frac{(2l+m-1)!}{(2l-1)!}& = \frac{\kappa}{m!}\partial_{w}^{m}\left\{\sum_{l=1}^{k}\binom{k}{l}(-1)^l(w)^{2l+m-1}\right\}_{|w=\kappa}& = \frac{\kappa}{m!}\partial_{w}^{m}[w^{m-1}(1-w^2)^{k}]_{|w=\kappa}.
\begin{matrix}S&\Rightarrow^{L}&w_{1}A\chi_{1}&\Rightarrow&w_{1}\nu\chi_{1}&\Rightarrow^{*}&w_{1}x\\S&\Rightarrow^{L}&w_{2}A\chi_{2}&\Rightarrow&w_{2}\omega\chi_{2}&\Rightarrow^{*}&w_{2}y,\\ \end{matrix}
\Gamma_{1}^{( I )}\; = \; \frac{1}{2}\mathrm{T r}\ln \left[ \delta ( x_{1}- x_{2}) - 2 g^{2}\int_{y}\Delta_{\varphi}( x_{1}, y ){\bar{\chi}}( y ){\mathcal D}^{- 1}( y , x_{2}) \chi ( x_{2}) \right] \; .
\frac{1}{I_{{q'}^{2}}+\mathcal{L}\left(W',Z'\right)}\otimes W'-\overline{\frac{1}{I_{{q'}^{2}}+\mathcal{L}\left(W',Z'\right)}\otimes W'}^{t}=2\sqrt{-1}\left(V\otimes\left( Z'-A\otimes W \right)\right)\overline{V\otimes\left( Z'-A\otimes W \right)}^{t},
m_2 + O\left(\frac{3^n\log{n}}{d_1}+ \frac{m_1\log{n}}{d_2}\right) = \frac{\ell_n(n)}{1-4d_2/n}+ O\left(\frac{3^n\log{n}}{d_1}+ \frac{\ell_n(n)\log{n}}{d_2}\right) &\leq \ell_n(n) + \frac{8d_2\ell_n(n)}{n}+ O\left(\frac{3^n\log{n}}{d_1}+ \frac{\ell_n(n)\log{n}}{d_2}\right) & = \left(1+O\left(\sqrt{\log{n}/n}\right)\right)\ell_n(n)
\mathbb{H}^{3}_{1}(-c^2)_{+}= \{(u^0,u^1,u^2,u^3)\in\mathbb{H}^{3}_1(-c^2): \c(u^{1}+u^{2})>0\}; \mathbb{H}^{3}_{1}(-c^2)_{0}=\{(u^0,u^1,u^2,u^3)\in\mathbb{H}^{3}_1(-c^2): \u^{1}+u^{2}=0\}; \mathbb{H}^{3}_{1}(-c^2)_{-}=\{(u^0,u^1,u^2,u^3)\in\mathbb{H}^{3}_1(-c^2): \c(u^{1}+u^{2})<0\}.
0\leqslant \varphi(0)\leqslant\frac{\varepsilon}{h^*_0}=\frac{\varepsilon}{h_0+\varepsilon/l},\ \ 0\leqslant \varphi(1)\leqslant\frac{\varepsilon(1-h_1)}{(h_0+\varepsilon/l)^2},
\frac{d}{dz}\left\lbrace{}^{\Psi}\Phi_{p}^{(\alpha,\beta)}(b;c;z)\right\rbrace =\frac{(b)}{(c)}\sum_{n=0}^{\infty}\frac{{}^{\Psi}B_{p}^{(\alpha,\beta)}(b+n+1,c-b)}{B(b+1,c-b)}\frac{z^{n}}{n!}&=\frac{(b)}{(c)}\left[{}^{\Psi}\Phi_{p}^{(\alpha,\beta)}(b+1;c+1;z)\right].
(\tilde{A}_1)_j^{j'}(l):=-\frac{(2 c_2\, j\, (j-j')^2+6\,c_3\, j) \sqrt{\lvert j-j'\rvert \xi_{j-j'}}\{(\omega-\overline{\omega})\cdot l+(m_3-1) (j'^3-j^3) \}}{(\omega\cdot l+m_3 (j'^3-j^3))(\overline{\omega}\cdot l+j'^3-j^3)}
\mathrm{hofib}(\mathcal N^{\geq i}A\Omega\{i\}/p^n\xrightarrow{\varphi_i-1}A\Omega\{i\}/p^n) = \tau^{\leq i}\mathrm{hofib}(\tau^{\leq i}\mathcal N^{\geq i}A\Omega\{i\}/p^n\xrightarrow{\varphi_i-1}\tau^{\leq i}A\Omega\{i\}/p^n)\ .
N^{n,5}_\tau:= \frac 1n \sum_{j=0}^{\lfloor n\tau \rfloor}(\sigma'\sigma)^2(X_{\frac j n}) \int_{0}^{1\wedge(n \tau -j )}\gamma_{\frac{x+j}n}\sum_{k=j+1}^{\infty}[ (x + k)^\alpha - k^{\alpha}]^2 dx=0.
\left\| \nabla\phi(t,\cdot) \right\|_{H_{s,\delta+1}}^{2}\leq C_{e}^2 \left\| \rho(t,\cdot)\right\|_{H_{s-1,\delta +2}}^{2}\leq C_{e}^2 C_{n}^2 c_{K,\gamma}^{2\beta}\left\|w(t,\cdot)\right\|_{H_{s,\delta}}^{2\beta}\leq C_{e}^2C_{n}^2c_{K,\gamma}^{2\beta}(2M_0)^{2\beta}.
\mathbf{u}^{\operatorname*{mult}}=\left( \mathbf{u}_{j}^{\operatorname*{mult}}\right)_{j=1}^{n_{\Omega}}=\left( \left( u_{\operatorname*{D};j}^{\operatorname*{mult}},u_{\operatorname*{N};j}^{\operatorname*{mult}}\right) \right)_{j=1}^{n_{\Omega}}\in\mathbb{X}\left( \mathcal{P}_{\Omega}\right)
\sum_{n=0}^{\infty}\frac{2^{n}}{3^{n+1}}=\frac{1}{3}+\frac{2}{9}+\frac{4}{27}+\frac{8}{81}+\cdot\cdot\cdot=\frac{1}{3}(\frac{1}{1-\frac{2}{3}})=1.
e^{i(t-a_{j+1})\Delta}w(a_{j+1})= e^{it\Delta}w(0)+i\int_{0}^{a_{j+1}}e^{i(t-s)\Delta}(F(\tilde{u}+w)-F(u))ds -i\int_{0}^{a_{j+1}}e^{i(t-s)\Delta}eds,
J^\mu=\left[\left(1 - \frac{\partial_{z}^2}{M_r^2}\right)\left(\not\!\partial \phi+iF\right)\right] \gamma^\mu \psi +\frac{1}{M_r^2}\,\delta^\mu_1\left\{-\partial_z (\!\not\!\partial\phi \not\!\partial\psi) +iF\, \partial_z\!\not\!\partial\psi -i \,(\partial_z F)\!\not\!\partial\psi\right\}\;.
\begin{aligned}\int_{[e_n, b_n+\varepsilon_n)}\left(h^+_n-u_2\right)^2dm =\int_{(b_n, b_n+\varepsilon_n)}\left(h^+_n-u_2\right)^2dm &\leq 2\int_{(b_n, b_n+\varepsilon_n)}\left(h^+_n-h\right)^2dm+2\int_{(b_n, b_n+\varepsilon_n)}\left(h-u_2\right)^2dm &<\frac{2\epsilon}{M}+2\int_{(b_n, b_n+\varepsilon_n)}\left(h-u_2\right)^2dm.\end{aligned}
d C_{4}- C_{2}\wedge d b_{2}= \frac{i}{6}\bar{L}\tau_{3}\tau_{1}\hat{L}^{3}L + \frac{1}{3 0}( \varepsilon^{a_{1}. . . a_{5}}L^{a_{1}}\wedge \cdots \wedge L^{a_{5}}+ \varepsilon^{a_{1}^{\prime}. . . a_{5}^{\prime}}L^{a_{1}^{\prime}}\wedge \cdots \wedge L^{a_{5}^{\prime}}) .
Y(T,U) = \int_{\cal F}\frac{d^2\tau}{\Im \tau}\Gamma_{2,2}(T,U) \left(-6\left[{\overline{\Omega}}_2^{\phantom 2}- \frac{1}{8\pi \Im\tau}\right]\frac{\overline{\Omega}}{\overline{\eta}^{24}}- \frac{\overline{j}}{8}+126\right)\ ,
\sum_{b = 1}^{N}\Big ( a_{\mu}^{( b )}, j_{\mu}^{( b )}\Big ) = \sum_{I = 1}^{N}\sum_{b = 1}^{N}\Big ( U_{I b}A_{\mu}^{( I )}, j_{\mu}^{( b )}\Big ) : = \sum_{I = 1}^{N}\Big ( A_{\mu}^{( I )}, J_{\mu}^{( I )}\Big ) ,
{\cal M}_{a b}^{o g}= \left( \begin{array}{c c c c}{{0}}&{{\zeta_{W}^{o}M_{W}^{o 2}}}&{{0}}&{{0}}\\{{\zeta_{W}^{o}M_{W}^{o 2}}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{\zeta_{Z}^{o}M_{Z}^{o 2}}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{M_{g A}^{o 2}}}\\ \end{array}\right)
\kappa_{X^+}(\zeta,t) = \int_{0}^\infty \left( \sum_{k=1}^{\lfloor t \rfloor}\bigg( \kappa_X \left( e^{\xi}\eta\left(k, \xi\right) \zeta \right) -\kappa_X \big( \eta\left(k, \xi\right) \zeta \big) \bigg) + \kappa_X \big( \eta\left(\lfloor t \rfloor, \xi\right) \zeta \big) \right)\pi(d \xi),
|\widehat{T}_{\varepsilon}h(x) | = \bigg|\int_{\{y:|y-x|>\varepsilon\}}\widehat{K}(x,y) h(y)\, dy\bigg| \leq \bigg|\int_{\{y: |y-x_k|>\varepsilon\}}\widehat{K}(x,y)h(y)\, dy\bigg| + \int_V | \widehat{K}(x,y) h(y)|\, dy.
\mathcal P(\tilde Y_i,\tilde Y_j)=F_{\mathcal M}\;w^T_{i,j}\,\tilde X_i^{\dag}\,\tilde X_j\; F_{\mathcal M},\mathcal M=F_{\tilde X_i}\;w_{i,j},w_{i,j}:=w(Y_i,Y_j),
\phi(0) = \psi(0,0)=0, &\phi^{(1)}(0) = \phi^{(3)}(0) = \phi^{(5)}(0)=0, &\phi^{(2)}(0) = \partial_1^2\psi(0,0) = \beta, \phi^{(4)}(0) = \partial_1^4\psi(0,0) = -2\beta^3 + 24 c.
\frac{|f_k|_{\infty,p+1}}{|f_k|_{\infty,p}^r}=\frac{a_{k,p+1}N_{k,p+1}^{p+1}}{a_{k,p}^rN_{k,p}^{pr}}=\frac{a_{k,p+1}N_{k,p+1}^{p}}{a_{k,p}}\cdot\frac{1}{a_{k,p}^{r-1}}\cdot\frac{N_{k,p+1}}{N_{k,p}^{pr}}=N_{k,p}^p\cdot\frac{1}{a_{k,p}^{r-1}}\cdot\frac{N_{k,p+1}}{N_{k,p}^{pr}}&\geq\frac{N_{k,p+1}}{N_{k,p}^{pr}}\xrightarrow[k\to\infty]{}\infty,
:\hat{b}^{\dagger}\hat{b}\hat{b}\hat{b}^{\dagger}\hat{b}\hat{b}^{\dagger}\hat{b}:=\hat{b}^{\dagger}\hat{b}^{\dagger}\hat{b}^{\dagger}\hat{b}\hat{b}\hat{b}\hat{b}=(\hat{b}^{\dagger})^{3}\hat{b}^{4}.
\frac{d}{d t}\mathfrak{F}\le\bigg(-\frac{1}{2}+\frac{\chi \lVert v_0\rVert_{L^\infty(\Omega)}^{1-\gamma}}{2}\bigg)\int_\Omega \frac{\lvert \nabla u \rvert^2}{u}+\frac{1}{2}\int_\Omega u^{\beta+1}\; \textrm{on}\; (0,T_{max}),
v^k =\frac{1}{c_k}(x^k-x^{k+1})+\frac{\alpha_k}{c_k}(x^k-x^{k-1}) y^k =x^{k+1}\varepsilon_k^1 =\frac{\|x^{k+1}-x^k\|^2}{4\gamma}\varepsilon_k^2 =\frac{\alpha_k}{\gamma}\|x^k-x^{k-1}\|^2 \varepsilon_k =\varepsilon_k^1+\varepsilon_k^2.
L_{_{( 2 )}}( q , \dot{q}, \alpha , \dot{\lambda}_{_{( 1 )}}, \dot{\lambda}_{_{( 2 )}}) = L_{_{( 2 )}}( Q_{_{( 2 )}}) = \Delta_{_{( 2 )}}^{^A}\dot{Q}_{_{( 2 )}}^{^A}- W ( Q_{_{( 1 )}}) \; ,
\nabla_q (h_{pi}P_{\nu j \nu \nu \nu}) = (\nabla_i h_{pq}) P_{\nu j \nu\nu\nu}+ h_{pi}(D_q P_{\nu j \nu \nu \nu}-h_{qj}P_{\nu \nu\nu\nu\nu}) & +h_{pi}h_{qk}(P_{kj\nu\nu\nu}+P_{\nu jk\nu\nu}+ P_{\nu j \nu k \nu}+ P_{\nu j \nu \nu k}).
p_1 r_3 + p_3 r_1 + q_1 r_4 + q_3 r_2 = \frac{a}{c t_d}\;, p_2 r_3 + p_4 r_1 + q_2 r_4 + q_4 r_2 = \frac{a}{c t_{(\ell-1)d}}\;.
\Gamma^{( 1 )}={\frac{3 1}{1 8 0}}\ln ( \mu^{2}a^{2}) +{\frac{7}{4 5}}\ln ( 2 ) +{\frac{1}{2}}\ln ( \pi ) +{\frac{6 9 3 1}{3 0 2 4 0}}-{\frac{1}{3}}\zeta_{R}^{\prime}( - 3 ) +{\frac{1 1}{6}}\zeta_{R}^{\prime}( - 1 ) .
\begin{aligned}\bar{p}'(r) =\frac{1}{\pi r}\int_{0}^{\pi}[\omega_{\theta}+uv_{\theta}-(v+b)u_{\theta}]d\theta &=\frac{1}{\pi r}\int_{0}^{\pi}[2uv_{\theta}-(vu_{\theta}+uv_{\theta})]d\theta &=\frac{2}{\pi r}\int_{0}^{\pi}(u-\bar{u})v_{\theta}d\theta.\end{aligned}
\lim_{N\to\infty}\varphi_N\left(P\left({\bf X}_N+{\bf Y}_N,{\bf X}_N+{\bf X}_N{\bf Y}_N^{-1}{\bf X}_N\right)\right)=\lim_{N\to\infty}\varphi_N\left(Q\left({\bf X}_N,{\bf Y}_N^{-1},{\bf Y}_N\right)\right).
E_0^{(r;[-L_1,L_2])}T_a=\begin{cases}\frac{v_{+,a;-L_1,L_2}(0)}{u_{+,a;-L_1,L_2}(0)}, \ 0<a\le L_2; \frac{v_{-,a;-L_1,L_2}(0)}{u_{-,a;-L_1,L_2}(0)}, \ -L_1\le a<0.\end{cases}
a_l := \max\{h : h \in E_{p,1}^l, 1 \le p \le m \},a'_l := \min\{h : h \in E_{p,1}^l, 1 \le p \le m \}, e_l := \max\{\delta_p^l - a_l \cdot t_{p,1}^l : 1 \le p \le m \mbox{for which}a_l \in E_{p,1}^l \}\mbox{and}e'_l := \min\{\delta_p^l - a_l \cdot t_{p,1}^l : 1 \le p \le m \mbox{for which}a_l \in E_{p,1}^l \}.
[y_1,y_2^3]_c =y_{12}y_2^2+\chi_2(g_1)y_2y_{12}y_2+\chi_{2}(g_1)^2y_2^2y_{12}&=\chi_2(g_{12})y_2y_{12}y_2+\lambda_{122}(1+\xi)y_2+\chi_{2}(g_1)^2(1+\xi)y_2^2y_{12}&=\lambda_{122}(1+\xi+\xi^2)y_2+\chi_{2}(g_1)^2(1+\xi+\xi^2)y_2^2y_{12}=0.
\left\{\omega \in \Omega : \varphi_{t_1+t_0}(\cdot,B(x_0,r_0)) \subset \bar{B}\left( \varphi_{t_1+t_0}(\cdot,x_0),\frac{r_0}{2}\right) \right\}& \quad\subset \left\{\omega \in \Omega : \varphi_{t_1+t_0}(\cdot,B(x_0,r_0)) \subset B \left( z_m, \frac{2}{3}r_0 \right) \textrm{for some}m \in \mathbb{N}\right\}.
I_{h,T}\mathbf{u}(X)-\mathbf{u}(X) = \sum_{i\in\mathcal{I}^{s'}}\Phi_{i,T}(X) (\mathbf{E}_i^s(X) + \mathbf{F}_i^s(X))+\sum_{i\in\mathcal{I}}\Phi_{i,T}(X) \mathbf{R}_i^s(X), ~s = \pm,
\varphi(r,t)=\frac{1}{4\pi\epsilon_{0}}(\frac{q}{|r-r_{s}|(1-\beta_{s}\cdot(r-r_{s})/|r-r_{s}|)})_{t_{r}}=\frac{1}{4\pi\epsilon_{0}}(\frac{q}{(1-n_{s}\cdot\beta_{s})|r-r_{s}|})_{t_{r}}
\left( g_{2}^{\lambda_2}\cdots g_{\ell}^{\lambda_\ell}\right)' = \left(g_{2}^{\lambda_2}\right)'\cdots g_{\ell}^{\lambda_\ell}+ \cdots + g_{2}^{\lambda_2}\cdots \left(g_{\ell}^{\lambda_\ell}\right)',
W ( \phi , \chi , \varphi ) ={\textstyle{\frac{1}{4}}}e^{2 \varphi / \sqrt{7}}\left( m_{2}\sinh ( \phi ) + m_{3}\cosh ( \phi ) + m_{1}e^{\phi}\chi -{\textstyle{\frac{1}{2}}}( m_{2}- m_{3}) e^{\phi}\chi^{2}\right) \, .
S_\Psi = \int d^6 x \sqrt{^6g}\bar{\Psi}i \Gamma^A D_A \Psi\simeq \epsilon^2 \int_0^\infty dz \phi^7 (\phi ')^{3/2}\int d^4 x\sqrt{-\eta}\bar{\psi}i \gamma^\nu\partial_\nu \psi .
{{P}_{}}=\sum\limits_{n=1}^{N}{\prod\limits_{k=1,k\ne n}^{N}{\int_{0}^{\infty}{{{f}_{{{I}_{n}}}}\left({{I}_{\max}}\right)}}}{{F}_{{{I}_{k}}}}\left({{I}_{\max}}\right) &\times \operatorname{Q}\left({{I}_{\max}}\sqrt{\frac{{\bar{\gamma}}}{2N}}\right)d{{I}_{\max}}
\begin{array}{l l}{{m \dot{u}^{\alpha}+ \displaystyle \frac{1}{c^{2}}S^{\alpha \sigma}\ddot{u}_{\sigma}= f^{\alpha}}}\\{{\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}
S_{1}^{C S}( \beta , \alpha = 1 , y ) = - \left[ \mathrm{T r}\left( \hat{\rho}^{H}( \beta ) \ln \hat{\rho}^{H}( \beta ) \right) - \mathrm{T r}\left( \hat{\rho}^{R}( 2 \pi \mu ) \ln \hat{\rho}^{R}( 2 \pi \mu ) \right) \right] .
\pi_{A}( x ) = \frac{\partial{\cal L}( x )}{\partial \left( \partial_{0}\Psi_{A}( x ) \right)}= i \left( \overline{{\Psi}}( x ) \Gamma_{4}\right)_{A}= i \left( \Psi^{+}( x ) \overline{{\Gamma}}_{4}\right)_{A}.
f''_n=\int_Q\Bigl[p(p-1)t^{p-2}g_n^{\frac p2}+p^2t^{p-1}g_n^{\tfrac{p-2}{2}}\partial_t g_n+\tfrac{p}2t^p\bigl(\bigl(\tfrac{p}2-1\bigr)g_n^{\frac{p-4}{2}}(\partial_t g_n)^2+ g_n^{\frac{p-2}{2}}\partial_{tt}g_n\bigr)\Bigr ]\,dx\,.
{\Theta_J}{res_J^{M}}([x_K^{M}]) ={\Theta_J}({res_J^{M}}([x_K^{M}])) &={\Theta_J}(\sum_{d\in{{W_M}\bigcap{X_{JK}}}}[{x_{J\cap{d(K)}}^J}]) &=\sum_{d\in{{W_M}\bigcap{X_{JK}}}}{\Theta_J}([{x_{J\cap{d(K)}}^J}]) &=\sum_{d\in{{W_M}\bigcap{X_{JK}}}}{W_{J}/W_{J\cap{d(K)}}}.
\gamma^1_{ik}\alpha_{jl}^0+\alpha_{ik}^0\gamma^1_{jl}-\gamma^1_{il}\alpha_{jk}^0-\alpha_{il}^0\gamma^1_{jk}= T_{ijkl}+ \frac{1}{|A_0|}(\alpha_{ik}^0 \alpha_{jl}^0-\alpha_{il}^0 \alpha_{jk}^0) (\overline{\overline{\alpha}}_1^{\,0}T_{23}-\overline{\overline{\alpha}}_2^{\,0}T_{13}+\overline{\overline{\alpha}}_3^{\,0}T_{12})^2.
CH_{alg}^d(X):=\{Z\in CH^d(X):Z\sim_{alg}0\}, CH_{hom}^d(X):=\{Z\in CH^d(X):Z\sim_{hom}0\},{\rm NS}_{alg}^d(X):=CH^d(X)/CH_{alg}^d(X),\;{\rm NS}_{hom}^d(X):=CH^d(X)/CH_{hom}^d(X),{\rm Griff}^d(X):=CH_{hom}^d(X)/CH_{alg}^d(X).
\int_M\left(|\widetilde \nabla^g\psi|^2-|\widetilde D^g\psi|^2+\frac{R_g-\varepsilon n(n-1)}{4}|\psi|^2\right)dM=\int_{\Sigma}\left\langle{\widetilde D}^\gamma\psi-\frac{H}{2}\psi,\psi\right\rangle d\Sigma,
I_{\mathrm{b2u}}\left( k \right)= \frac{P_{\mathrm{bs}}a_{\mathrm{b2u},k}}{N_{\mathrm{bs}}}\left(\max \{\|\mathcal{H}^{\dagger}_{\mathrm{b2s},i}\| \}_{i=1}^{N_{\mathrm{bs}}}\right)^{-2}\| \mathbf{H}_{\mathrm{b2u},k}\mathcal{H}^{\dagger}_{\mathrm{b2s}}\|_\mathrm{F}^2
\frac{\partial}{\partial \lambda_i}\frac{\pi_\mathrm{S}(\lambda)}{\pi_\mathrm{P}(\lambda)}= \left( - \frac{d}{2}+1 \right) \left( \frac{\lambda_1}{s_1}+ \dotsb + \frac{\lambda_d}{s_d}\right)^{-\frac{d}{2}}\frac{1}{s_i}
\nabla^a\left( R_{ab}+ \nabla_a\nabla_b f +\frac{1}{2}g_{ab}\right) - (\nabla^a f)\left(R_{ab}+ \nabla_a\nabla_b f + \frac{1}{2}g_{ab}\right) = & \frac{1}{2}\nabla_{b}\left( R + 2\Delta f - |\nabla f|^2 - f \right).
\frac{(p(x,y|z))^a}{(p(x|z)p(y|z))^{a-1}}\left(\log \frac{p(x,y|z)}{p(x|z)p(y|z)}\right)^2 = (p(x,y|z))^{a-(a-1)(k_1+k_2)}(\log (p(x,y|z))^{1-(k_1+k_2)})^2 & \stackrel{}{\le}(p(x,y|z))^{1/2}(\log (p(x,y|z))^{-1})^2 & = 4(p(x,y|z))^{1/2}(\log (p(x,y|z))^{-1/2})^2 & \stackrel{}{\le}16 e^{-2}\log e,
\mathcal{T}^1_{\neq \neq}= \int B Q^3 B \left( (U_{\neq}^1)_{Hi}(\partial_X Q^3_{\neq})_{Lo}\right) dV dt + \int B Q^3 B \left( (U_{\neq}^1)_{Lo}(\partial_X Q^3_{\neq})_{Hi}\right) dV dt = \mathcal{T}^1_{\neq \neq;HL}+ \mathcal{T}^1_{\neq \neq;LH}.
e^{\alpha}_y(z) =\exp (\sum_{n\ge 1}y_n z^n)\exp (-\sum_{n\ge 1}\frac{\partial}{n\partial y_n}z^{-n})e^{\alpha}z^{\partial_{\alpha}}, e^{-\alpha}_y(z) =\exp (-\sum_{n\ge 1}y_n z^n)\exp (\sum_{n\ge 1}\frac{\partial}{n\partial y_n}z^{-n})e^{-\alpha}z^{-\partial_{\alpha}},
(\frac{z-a}{z-b})^{k}(\frac{z-c}{z-b})^{l}P\{\begin{matrix}a&b&c&\\ \alpha&\beta&\gamma&z\\ \alpha^{\prime}&\beta^{\prime}&\gamma^{\prime}&\end{matrix}\}=P\{\begin{matrix}a&b&c&\\ \alpha+k&\beta-k-l&\gamma+l&z\\ \alpha^{\prime}+k&\beta^{\prime}-k-l&\gamma^{\prime}+l&\end{matrix}\}
T=\frac{1}{2}m_{1}v_{1}\cdot v_{1}+\frac{1}{2}m_{2}v_{2}\cdot v_{2}=\frac{1}{2}(m_{1}+m_{2})L_{1}^{2}\dot{\theta}_{1}^{2}+\frac{1}{2}m_{2}L_{2}^{2}\dot{\theta}_{2}^{2}+m_{2}L_{1}L_{2}cos(\theta_{2}-\theta_{1})\dot{\theta}_{1}\dot{\theta}_{2}.
L^{(s,t)}:=\left(\begin{array}{cccc}1 e_1^{(s,t)}1 &\ddots \ddots & e_{m-1}^{(s,t)}1\end{array}\right),R^{(s,t)}:=\left(\begin{array}{cccc}q_1^{(s,t)}1 &q_2^{(s,t)}\ddots & \ddots 1 & q_m^{(s,t)}\end{array}\right).
c(E) = 1 + c_1 + |n|\cdot pt + \sum_{0\leq i<j \leq s}c_1(L_i)c_1(L_j) & = 1 + (s+1)c_1(L_s) + \sum_{i=1}^s i \beta^i & + |n|\cdot pt + \frac{s}{2 (s+1)}c_1^2 - \sum_{1\leq i<j\leq s}\frac{i(s+1-j)}{s+1}\beta^i\beta^j - \sum_{1\leq i\leq s}\frac{i(s+1-i)}{2(s+1)}(\beta^i)^2\,,
\overline{{g}}_{\mu \nu}= \left[ \begin{array}{c c}{1 + l ( x^{0})^{2}\overline{{\lambda}}( x^{0}, x^{1})^{2}}&{l ( x^{0})^{2}}\\{l ( x^{0})^{2}\overline{{\lambda}}( x^{0}, x^{1})^{2}}&{l ( x^{0})^{2}}\\ \end{array}\right] .
{\cal M}= \int \frac{d^{D}\! q}{( q - p )^{2}( q \! \cdot \! n )}\longrightarrow^{\! \! \! \! \! \! \! \! \! \! \! \! \! \! M L}\int \frac{d^{D}\! q ( q \! \cdot \! n^{*})}{( q - p )^{2}\left[ ( q \! \cdot \! n ) ( q \! \cdot \! n^{*}) + i \epsilon \right]}.
\mu_{p,\Lambda}^{\eta}(A) = \sum_{\omega \in A}\mu_{p,\Lambda}^{\eta}(\omega) =\sum_{\omega \in A}\gamma_{\beta,\Lambda}^{c_\eta}(c_{\eta, \Lambda}(\omega))= \sum_{\sigma \in B}\gamma_{\beta,\Lambda}^{c_\eta}(\sigma)= \gamma_{\beta,\Lambda}^{c_\eta}(B).
\langle \partial_t(\chi_{\varepsilon}+ \ell \varphi_{\varepsilon}) (t), z \rangle_{V^*,V}+ \int_{\Omega}\nabla \chi_{\varepsilon}(t) \cdot \nabla z+ \langle \zeta_{\varepsilon}(t), z \rangle_{V^*,V}= \langle f_{\varepsilon}(t), z \rangle_{V^*,V}\ \ \textrm{for all $z \in V$,}
\sum_{n=0}^{p-1}R_{\mathbf{s}_{n}}(\tau) = \sum_{i\in \Omega_{1}}\sum_{n=0}^{p-1}\omega_{q}^{f_{n,i}-f_{n,j}}+ \sum_{i\in \Omega_{2}}\sum_{n=0}^{p-1}\omega_{q}^{f_{n,i}-f_{n,j}}&=p\sum_{i\in \Omega_{1}}\omega_{q}^{f_{0,i}-f_{0,j}},
-\sum_{j=1}^2 \frac{\lambda_j}{\pi}\frac{1}{(0-z_j(t))^2}D_tZ(0)=-\frac{\lambda i}{\pi}\frac{x}{x^2+y^2}\frac{\lambda}{\pi}\frac{4xyi}{(x^2+y^2)^2}=\frac{4\lambda^2 x^2y}{\pi^2 (x^2+y^2)^3}
{\cal C}(\bar{x};q;P) \, = \, \left\{\, v \, \in \,{\cal T}(\bar{x};P) \, \mid \, \exists \lambda \, \in \, \Lambda(\bar{x})\mbox{such that}A_{\, i \bullet}v \, = \, 0 \mbox{for all}i \, \in \, \mbox{supp}(\lambda) \, \right\},
Q(z) =V(-z,1-\xi_{2,k},\ldots,1-\xi_{k,k})=V(1-\xi_{2,k},\ldots,1-\xi_{k,k})\prod_{l=2}^{k}(1-\xi_{l,k}+z) &=V(1-\xi_{2,k},\ldots,1-\xi_{k,k})R(z+1),
A = 2a_{0}^{(2)}(b_{0}(0))b_{0}^{(1)}(0) + 2\partial_{2}a_{0}^{(1)}(b_{0}(0)) - a_{0}^{(2)}(0) B = 2a_{0}^{(1)}(b_{0}(0)) - 2a_{0}^{(2)}(0)a_{0}(0) - 2a_{0}^{(1)}(0)^{2}C = -3a_{0}(0)\left(a_{0}(0)a_{0}^{(2)}(0) + 2a_{0}^{(1)}(0)^{2}\right)
\begin{array}{l l l}{{h_{b}( z , T , x^{1 0}) = \tilde{h}_{b + N^{\prime}/ 2}( \bar{z}, - 1 / \bar{T}, - x^{1 0})}}&{{,}}&{{\tilde{h}_{b}( z , T , x^{1 0}) = h_{b + N^{\prime}/ 2}( \bar{z}, - 1 / \bar{T}, - x^{1 0})}}\\{{h_{c}( z , T , x^{1 0}) = \tilde{h}_{c}( \bar{z}, - 1 / \bar{T}, - x^{1 0})}}&{{,}}&{{}}\\ \end{array}
\int_{0}^{T}\int_{\Omega_{\varepsilon}}-u_{\varepsilon}(x,t)v(x)\partial_{t}c(t)+A\left( \frac{x}{\varepsilon},\frac{t}{\varepsilon^{2}}\right)\nabla u_{\varepsilon}(x,t)\cdot \nabla v(x)c(t)dxdt =\int_{0}^{T}\int_{\Omega_{\varepsilon}}f_{\varepsilon}(x,t)v(x)c(t)dxdt,v\in V_{\varepsilon},c\in D(0,T)
\left( \begin{matrix}{{a_{D 1}}}\\{{a_{D 2}}}\\{{a_{1}}}\\{{a_{2}}}\\ \end{matrix}\right) \to M_{1}\left( \begin{matrix}{{a_{D 1}}}\\{{a_{D 2}}}\\{{a_{1}}}\\{{a_{2}}}\\ \end{matrix}\right) , \qquad M_{1}={\tilde{M}}_{1}^{4}, \quad \,{\tilde{M}}_{1}= \left( \begin{matrix}{{1}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{1}}&{{0}}&{{0}}\\{{- 1}}&{{0}}&{{1}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{1}}\\ \end{matrix}\right)
{\cal L}_{\eta}| \Phi_{p h}\rangle = ( \eta \hat{\partial}+ \frac{1}{2}\hat{\partial}^{I}\eta^{J}M^{I J}+ \frac{1}{2}\eta^{I}M^{z I}) | \Phi_{p h}\rangle - \hat{\partial}^{+}\eta^{I}\alpha^{I}( \bar{\alpha}^{-}| \Phi \rangle ) \Bigr |_{\alpha^{+}= 0}^{\vphantom{5 p t}}\, ,
e^{-t\Delta_V}=\frac{1}{2\sqrt{\pi t}}\int_{\mathbb{R}}\cos(s\sqrt{\Delta_V})e^{-\frac{s^2}{4t}}ds &=\frac{1}{2\sqrt{\pi t}}\int_{\mathbb{R}}\cos(s\sqrt{\Delta_V})(1-\psi(s))e^{-\frac{s^2}{4t}}ds &\ \ \ \ +\frac{1}{2\sqrt{\pi t}}\int_{\mathbb{R}}(1+\Delta_V)^{-m}\cos(s\sqrt{\Delta_V})(1-\frac{d^2}{ds^2})^m(\psi(s)e^{-\frac{s^2}{4t}})ds.
\pi_+(V_0) = v_1 \wedge v_{-1}- v_{-1}\wedge v_1 - q (q - q^{-1}) v_0 \wedge v_0 = 2 v_1 \wedge v_{-1}+ (q - q^{-1})^2 v_1 \wedge v_{-1}= (q^2 + q^{-2}) v_1 \wedge v_{-1}.
M_{B,C}(X) = b_2^3X^4 + \big(-2c_0b_2^3 + (c_1b_1 + 2c_2b_0)b_2^2 + (-c_2b_1^2 - 3b_0b_1)b_2 + b_1^3\big)X^2+{}\quad+ c_0^2b_2^3 + \big(-c_0c_1b_1 + (-2c_0c_2 + c_1^2)b_0\big)b_2^2 + \big(c_0c_2b_1^2 + (-c_1c_2 + 3c_0)b_0b_1+{}\quad+(c_2^2 - 2c_1)b_0^2\big)b_2 + (-c_0b_1^3 + c_1b_0b_1^2 - c_2b_0^2b_1 + b_0^3).
U=U_\mu dx^\mu= \frac{x^0+x^1}{1+\eta_{\mu\nu}x^\mu x^\nu}[x^0(\textsf{a}x^2+\textsf{b}x^3+\textsf{c})dx^0-x^1(\textsf{a}x^2+\textsf{b}x^3+\textsf{c})dx^1 &+(\textsf{a}(1+\eta_{\mu\nu}x^\mu x^\nu)-x^2(\textsf{a}x^2+\textsf{b}x^3+\textsf{c}))dx^2 &+(\textsf{a}(1+\eta_{\mu\nu}x^\mu x^\nu)-x^3(\textsf{a}x^2+\textsf{b}x^3+\textsf{c}))dx^3].