harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{aligned}R(\mathbf{F}_R, \mathbf{F}_B, \mathbf{W}_R, \mathbf{W}_B) = \log_2 \left( \left| \mathbf{I}_{N_s}+ \frac{P}{N_s}\mathbf{R}_n^{-1}\tilde{\mathbf{H}}\tilde{\mathbf{H}}^H \right| \right),\end{aligned}
t_{\Omega \Sigma}^{( 1 )}{}^{\Pi \Gamma}{}_{\Delta \Lambda}= \delta_{[ \Delta}^{[ \Pi}\delta_{\Lambda ] [ \Omega}\delta_{\Sigma ]}^{\Gamma ]}, \qquad t_{\Omega \Sigma}^{( 2 )}{}^{\Pi \Gamma \Delta \Lambda}= 0 .
\psi_\delta(t):=\begin{cases}0, \,\,\,t \in (t_0-\rho^p, t_1), \frac{1}{\delta}(t-t_1), \,\,\,t \in [t_1, t_1+\delta), 1, \,\,\,t \in [t_1+\delta, t_0).\end{cases}
(x_1, x_2, \dots, x_n) \sigma \tau = (x_n, x_1, \dots, x_{n-1})\tau & = ((x_1, x_2, \dots, x_n )\mu, (x_2, x_3, \dots, x_1)\mu, \dots, (x_n, x_1, \dots, x_{n-1})\mu) & = ((x_2, x_3, \dots, x_1)\mu, (x_3, x_4, \dots, x_2) \mu, \dots, (x_1, x_2, \dots, x_n )\mu ) \sigma & = (x_1, x_2, \dots, x_n) \tau \sigma.
\mathcal A:=\!\left\{\left(0,2\right),\left(p,p+2\right),\left(p-1,p-1\right),\left(-1,-1\right),\left(p+1,p-1\right),\left(1,-1\right)\right\}\!, \varepsilon_1(\boldsymbol{A}):= \varepsilon\left(\frac{A_1-A_2}{2p},\frac{A_2}{p}\right).
C^k_{n,(\alpha,\beta),\rho}= [t^n] t^\rho C_{k,(\alpha-1,k-2)}(t) \kern+1pt C_k(t)^{\beta + (\rho-1)(k-1)}= [t^{n-\rho}]C_{k,(\alpha-1,k-2)}(t) \kern+1pt C_k(t)^{\beta+(\rho-1)(k-1)}.
\begin{array}{r l}{E_{8}^{+}=}&{\{\pm e_{a}\}}\\{\cup}&{\{( \pm e_{a}\pm e_{b}\pm e_{c}\pm e_{d}) / 2 : a , b , c , d \mathrm{d i s t i n c t}, e_{a}( e_{b}( e_{c}e_{d}) ) = \pm 1 \},}\\{}&{a , b , c , d \in \{0 , . . . , 7 \}.}\\ \end{array}
\sin^{2}{\psi}= \frac{1}{2}\pm \frac{1}{4}\frac{r_{1}^{2}+ r_{5}^{2}- 2 r_{K}^{2}}{\sqrt{r_{1}^{4}+ r_{5}^{4}+ r_{K}^{4}- r_{1}^{2}r_{5}^{2}- r_{1}^{2}r_{K}^{2}- r_{5}^{2}r_{K}^{2}}}.
\|f\|_{A^2_\alpha}= \left(\sum_{n=0}^\infty \frac{1}{c_\alpha(n)}\left|\sum_{n_1+n_2=n}a_{n_1}b_{n_2}\right|^2\right)^\frac{1}{2}&\leq \left(\sum_{n=0}^\infty \sum_{n_1+n_2=n}\frac{|a_{n_1}|^2}{c_{\alpha_1}(n_1)}\frac{|b_{n_2}|^2}{c_{\alpha_2}(n_2)}\right)^\frac{1}{2}= \|g\|_{A^2_{\alpha_1}}\|h\|_{A^2_{\alpha_2}},
\|(z-T)^{-1}\|\leq\frac{1}{|z|}+\sum\limits_{k=1}^{\infty}\frac{\|T^k\|}{|z|^{k+1}}=\frac{1}{|z|}+\sum\limits_{k=1}^{\infty}\frac{\prod\limits_{j=1}^{k}w_{j}}{|z|^{k+1}}.
\delta{\cal E}_{0}= - i \frac{3 \hbar c}{2}\int d^{3}\vec{r}\varphi^{\dagger}\epsilon_{0}\left[ 1 - \frac{\hbar^{2}}{4 m^{2}c^{2}}( \vec{\sigma}\cdot \vec{\nabla})^{2}\right] \varphi \,{.}
\begin{minipage}[c]{35pc}$d(s)\ =$ greatest integer $n\geq 0$ suchthat $s\in x^n(S)$ if this exists,or $\infty$ if $s\in\bigcap_{n\geq 0}x^n(S).$ (In statements such as~\eqref{d.dxs}and~\eqref{d.ds-1}below, we will understand $\infty+1=\infty=\infty-1.)$\end{minipage}
\Big|u\Big|_{\mathrm{H}^{\mathrm{s}}\Big(\mathbb{R};\mathrm{L}^2(\partial\Omega)\Big)}^2 := \int_\mathbb{R}\int_\mathbb{R}\dfrac{\Vert u(\cdot,t)-u(\cdot,\tau)\Vert_{\mathrm{L}^2(\partial\Omega)}^2}{|t-\tau|^{1+2\mathrm{s}}}dt d\tau.
\left( \ln \left( 1+Z_{\geq m_+}^{(\varepsilon')}\right) \right)^k = \left( \ln(1+Z_{\geq m_+}) \right)^k + \sum_{\ell=1}^{k}\binom{k}{\ell}\ln(1+Z_{\geq m_+})^{k-\ell}\ln(W_{m_+})^\ell.
\frac{1}{sin\theta}\frac{\partial}{\partial\theta}(sin\theta\frac{\partial Y_{l}^{m}}{\partial\theta})+\frac{1}{sin^{2}\theta}\frac{\partial^{2}Y_{l}^{m}}{\partial\phi^{2}}=-l(l+1)Y_{l}^{m}(\theta,\phi)
J^-_{jk}= \int_0^M\int_0^M(\eta-\eta')_+\phi_{N+j}(\eta)\phi_k(\eta')\,d\eta d\eta', J^+_{jk}= \int_0^M\int_0^M(\eta'-\eta)_+\phi_{N+j}(\eta)\phi_k(\eta')\,d\eta d\eta'.
\left\| \begin{bmatrix}u v \theta \end{bmatrix}\right\|_{L^2(\Omega)\times H^{-2}(\Omega)\times H^{-2}(\Omega)}=\|u\|_{L^2(\Omega)}+\|v\|_{H^{-2}(\Omega)}+\|\theta\|_{H^{-2}(\Omega)}&=\|u\|_{L^2(\Omega)}+\|\Lambda^{-\frac{1}{2}}v\|_{L^{2}(\Omega)}+\|\Lambda^{-\frac{1}{2}}\theta\|_{L^{2}(\Omega)}.
h_F(q_{12}) = ( \frac{a_1 d_1 - b_1 c_1}{b_1 d_1}\cdot J_1, -\frac{a_1 d_1 - b_1 c_1}{b_1 d_1}\cdot J )_F = ( \frac{a_1 d_1 - b_1 c_1}{b_1 d_1}, J )_F = ( \frac{\nu_1}{a_1 b_1 + c_1 d_1 J_1}, J )_F.
2C_1\int_{\theta = 0}^{\pi}\left(\frac{\rho_{1}}{\tau_{1}}+ \frac{\alpha\rho_{0}\rho_{1}}{2\tau_{1}}(1 +c_{0}\cos\theta)\right)\sin\theta e^{\frac{1}{\lambda_{1}}\cos\theta\cos\phi}\sinh\left(\frac{1}{\lambda_{1}}\sin\theta\sin\phi\right) h_{0}(\cos\theta)d\theta &-\frac{\alpha c_{1}}{2\tau_{0}}\beta_{0}\rho_{0}\rho_{1}\sin\phi = 0.
{\cal L}= \frac{1}{2}D_{+}\rho \, D_{-}\rho + \frac{1}{2}D_{+}\psi \, D_{-}\psi - \frac{1}{2}\mathrm{t h}^{2}\rho D_{+}\omega \, D_{-}\omega + \mathrm{s h}\rho \, e^{\psi}.
\int\frac{[\mu_{+}^{\prime}+\mu_{-}^{\prime}]}{v^{2}}dv =2\int_{v>0}\frac{[\mu_{+}^{\prime}+\mu_{-}^{\prime}]}{v^{2}}dv & \geq c_{0}\left( \frac{1}{2\sigma_{0}}-\frac{1}{a}\right) +\int_{v>a}\frac{[\mu_{+}^{\prime}+\mu_{-}^{\prime}]}{v^{2}}dv & >0,
C_1 = 1/\left[ (S_{\max}-S_{\min}) - \frac{\sqrt{\pi}}{2}\frac{1-\beta}{\beta}\alpha\left( \left( \frac{S_{\max}-K}{\alpha}\right) - \left( \frac{S_{\min}-K}{\alpha}\right) \right) \right], C_2 = \left[ \frac{\sqrt{\pi}}{2}\frac{1-\beta}{\beta}\alpha\left( \frac{S_{\min}-K}{\alpha}\right) - S_{\min}\right]C_1.
\frac{Cl_{2}(\theta)}{\theta}=3-log[|\theta|(1-\frac{\theta^{2}}{4\pi^{2}})]-\frac{2\pi}{\theta}log(\frac{2\pi+\theta}{2\pi-\theta})+\sum_{n=1}^{\infty}\frac{\zeta(2n)-1}{n(2n+1)}(\frac{\theta}{2\pi})^{2n}.
\dim S^0_U \leq\dim({\pi'_{1}}^{-1}(p)\cap S_U^0) +\dim \pi'_1(S_U^0) \leq &\leq\dim{\pi'_{1}}^{-1}(p) +\dim G_s \overset{(\ref{E:L}), (\ref{E:D})}{=}&=N- \frac{kt(k-1)}2+\frac{st(s-1)}2 + (k-s)(n-k+s)= &=s^2 \left( \frac t 2 -1 \right)+s\left(2k-n-\frac t 2 \right) -\frac{kt(k-1)}2 +k(n-k) +N.
\frac{\mu^{s + 1}}{2}\left( \frac{2 \pi}{L}\right)^{- s}\sum_{n = - \infty}^{\infty}\frac{1}{\Gamma \left( \frac{s}{2}\right)}\int_{0}^{\infty}d t t^{( \frac{s}{2}- 1 )}e^{- ( n^{2}+ ( \frac{m L}{2 \pi})^{2}) t}\rfloor_{s = - 1}=
\begin{array}{c}y_i=h_y(x_i)+u_i, z_i=h_z(x_i)+v_i,\end{array}\begin{pmatrix}u_i v_i \end{pmatrix}\sim\mathcal{N}\left(0,\Sigma(x_i)\right),\, h_y\in\mathcal{H}_y, \, h_z\in\mathcal{H}_z,
d(s_n,s_m) = d(s_n-s_m,0)=d\Big(\sum_{k=m+1}^n ky_{\beta_k},0\Big) \leq \sum_{k=m+1}^n d\big(ky_{\beta_k},0\big) \leq \sum_{k=m+1}^n \frac{1}{2^k}\to 0, n,m \to \infty.
\ell_{ij}u_{jk}+u_{ij}\ell_{jk}+\beta_{ij}u_{jk}x_i +u_{ij}\beta_{jk}x_j =u_{jk}\ell_{ik}+\ell_{ik}u_{ij}+u_{jk}\beta_{ik}x_i+\beta_{ij}x_i u_{ij},
(S\otimes_{\mathcal{O}}\mathcal{O}G)(\Delta_{\varphi_x}(Q)) =S(\Delta_{\varphi_x}(Q))\otimes_k (\mathcal{O}G)(\Delta_{\varphi_x}(Q)) &=\bigoplus_{t\in [C_G(Q)/C_N(Q), z=xt]}(S(\Delta_{\varphi_x}(Q))\otimes_k \bar{N}_{\mathcal{O}Nz}^{\varphi_x}(Q)).
\frac{1}{|B_{r}|}\int_{B_{r}}|X_{i}X_{j}u\left( x\right) -(X_{i}X_{j}u)_{B_{r}}|dx & \leq\frac{c}{k}\sum_{i,j=1}^{q}\frac{1}{|B_{kr}|}\int_{B_{kr}}|X_{i}X_{j}u\left( x\right) |dx+ck^{2+Q/p}\left( \frac{1}{|B_{kr}|}\int_{B_{kr}}|\overline{L}u(x)|^{p}dx\right)^{1/p}\ & \equiv A+B.
\int_{{\Bbb R}^2}\phi (x_1,x_2,x_3)dx_1 dx_2=\int_{{\Bbb R}^2}\phi(x_1,x_2,x_3)dx_2 dx_3=\int_{{\Bbb R}^2}\phi (x_1,x_2,x_3)dx_3 dx_1=0.
\Psi = \sum_{i \bullet}c_{\bullet}\mathrm{\Large e^{i{\bf k \cdot r}_{i}}}a_{i}^{+}\left| O \right\rangle + \sum_{i \circ}c_{\circ}\mathrm{\Large e^{i{\bf k \cdot r}_{i}}}a_{i}^{+}\left| O \right\rangle
\big\| \|\textbf{1}_{[-s_k,-s_{k-1}]}(P_{\leq M}u)\|_{V^p_{\Delta_x}}\big\|_{l^{p}}=\big\{\sum_{k=1}^{K}\|\mathbf{1}_{[-s_k,-s_{k-1})}P_{\leq M}u\|^{p}_{V^p_{\Delta_x}}\big\}^{\frac{1}{p}}\leq \|P_{\leq M}u\|_{V^p_{\Delta_x}}.
\hat{R}F(v_{0}\otimes v_{1}) =[2]^{1/2}(q^{-4}v_{1}\otimes v_{-1}+\hat{R}(v_{0}\otimes v_{0})), F\hat{R}(v_{0}\otimes v_{1}) =[2]^{1/2}(v_{0}\otimes v_{0}+v_{1}\otimes v_{-1}).
I(s, v) = 3 \int_0^{(1-s-v)/6}\log \frac{\sin ( \pi (\varphi+1/6 ) )}{\sin ( \pi (-\varphi+1/6 ) )}d \varphi+ 6 \int_0^{(1-s+v)/6}\log \frac{\sin(\pi(\varphi+1/6))}{\sin ( \pi (-\varphi+1/6 ) )}d \varphi &+ 6 \int_0^{(1-s)/3}\log \frac{\cos(\pi(\varphi+1/6))}{\cos(\pi(-\varphi+1/6))}d \varphi\mbox{for $(x, v) \in \overline{\Lambda}_2$}.
e_R[\Phi] ({t_2}) - e_R[\Phi]({t_1}) = \int_{\{\vert{\bf x^\prime}\vert \leq R\}}\left(v( t_1,{\bf x^\prime})- v(t_2,{\bf x^\prime})\right)\cdot (1,{\bf 0}) \,d{\bf x^\prime}&-\int_{\{\vert{\bf x^\prime}\vert = R\}\times [t_1, t_2]}\dot\Phi \nabla^\prime\Phi\cdot \frac{{\bf x^\prime}}{\vert{\bf x^\prime}\vert}\,d\sigma_{cyl}\,,-\infty< t_1< t_2<0\,,
\sum_{m=0}^\infty \lambda^m B_m^{(k)}\left(\frac{x}{\lambda}\right) \frac{t^m}{m!}= \sum_{m=0}^\infty \sum_{n=0}^m D_{n,\xi}^{(k)}(x|\lambda) \xi^{-n}s_2 (m,n) \frac{t^m}{m!}.
{\bf E}_L \left( \prod_{j=1}^n \hat{V}_{L,\omega}(p_j - p_{j+1}) \right)= \sum_{A \in \mathcal{A}_n}\prod_{a \in A}\left\{m_{|a|}\delta_{*,L}\left( \sum_{l \in a}(p_l - p_{l+1}) \right) \prod_{l \in a}\hat{B}_\#(p_l - p_{l+1}) \right\}.
v-u\in V,v-u=\begin{cases}0 \mbox{a.e.~in}\Omega\setminus N, w- u <0 \mbox{a.e.~in}N, \end{cases}\nabla (v-u)=\begin{cases}0 \mbox{a.e.~in}\Omega\setminus N, \nabla (w-u) \mbox{a.e.~in}N. \end{cases}
\sum_{a=2}^{n-5}\sum_{c=a+1}^{n-4}\sum_{d=c+2}^{n-2}(c-a)(d-c-1)=\sum_{a=2}^{n-5}\sum_{c=a+1}^{n-4}(c-a)\binom{n-2-c}{2}=\sum_{c=3}^{n-4}\binom{n-2-c}{2}\sum_{a=2}^{c-1}(c-a) &=\sum_{c=3}^{n-4}\binom{n-2-c}{2}\binom{c-1}{2}=\binom{n-2}{5}
\omega_t= \omega_M+ dd^c \phi_t = \omega_M+ \frac{1}{|\log |t||}\sum_{1\leq i,j\leq m}\frac{\partial^2 u}{\partial x_i\partial x_j}d\log |F_i|_{h^{\otimes d_i}}\wedge d^c \log |F_j|_{h^{\otimes d_j}}& -\sum_1^m \frac{\partial u}{\partial x_i}dd^c \log |F_i|_{h^{\otimes d_i}}.
\vert \Delta_i \vert =\frac{\vert G_\alpha \vert}{\vert G_{\alpha\beta}\vert}=\frac{2(m!)^2}{2((m-i)!i!)^2}=\left(\frac{m!}{(m-i)!i!}\right)^2=2^{-\lfloor \frac{2i}{m}\rfloor}\left(\frac{m!}{(m-i)!i!}\right)^2.
o(t)=\tau_t(a_1\cdots a_n) +\sum_{r=1}^n (-1)^r \sum_{1\leq k_1<\ldots<k_r\leq n}\left(\prod_{i=1}^{r}\tau_t(a_{k_i}) \right)\tau_t(a_1\cdots \hat{a}_{k_1}\cdots \hat{a}_{k_r}\cdots a_n)
\varphi(e_{1};\tau) =432g_{2}e_{1}^{4}+864g_{3}e_{1}^{3}-72g_{2}^{2}e_{1}^{2}-216g_{2}g_{3}e_{1}-g_{2}^{3}-108g_{3}^{2}& =64(2\alpha-\beta)^{3}>0.
\frac 1{N(Z-W)}= N(W)^{-1}\cdot \sum_{l,m,n}t^l_{m\,\underline{n}}(Z) \cdot t^l_{n\,\underline{m}}(W^{-1}),\begin{matrix}l=0,\frac 12, 1, \frac 32,\dots, m,n = -l, -l+1, \dots, l, \end{matrix}
\sum_{\mu , \nu = 0}^{3}\left( \hat{A}_{\mu}\hat{A}_{\nu}\right) \left( \hat{A}^{\mu}\hat{A}^{\nu}\right) - \sum_{\nu = 0}^{3}\left( \left( \hat{A}_{\mu}\hat{A}_{\nu}\right) \hat{A}^{\mu}\right) \hat{A}^{\nu}= 3 \lambda \sum_{\nu = 0}\hat{A}_{\nu}\hat{A}^{\nu}
p\widetilde{\mathbb{S}}_{\lambda^i}p\widetilde{\mathbb{S}}_{\lambda^j}=\sum_{\lambda^\ell\succeq \lambda^i,\lambda^j}\sum_{P : \delta P=\Delta_{\sigma_r\lambda^i~\sigma_r\lambda^j}^{\sigma_r\lambda^\ell}}\sigma_r wt(P)p\widetilde{\mathbb{S}}_{\lambda^\ell},
\min_{0{}\leq{}i{}\leq{}N}\|\nabla f(x_i)\|^2{}\leq{}\frac{2L\,\big(f(x_0)-f_*\big)}{1 + \sum\limits_{i=0}^{N-1}\left[2 h_i - \frac{h_i^2}{2}\max(1, h_i) \right]}
\gamma_1T_1 + \gamma_2T_2 = \gamma_1 \widetilde{M}_L^{-1}\cdot Q_1(v, w)\cdot \widetilde{M}_R^{-1}+ \gamma_2 \widetilde{M}_L^{-1}\cdot Q_2(v, w)\cdot \widetilde{M}_R^{-1}= &= \widetilde{M}_L^{-1}\cdot Q(v, w)\cdot \widetilde{M}_R^{-1}.
T_{a b}{}^{c}\; = \; - \left[ i_{k_{( m )}}\hat{C}\right]_{a b}Q^{m n}k_{( n )}^{c}\hspace{. 2 c m}, \hspace{. 2 c m}2 K_{a b c}\; = \; - T_{a b c}+ T_{b c a}- T_{c a b}\; , \; \check{\omega}\; = \; \omega + K \; .
V = \left( \begin{array}{c}{X^{\Lambda}( \phi )}\\{F_{\Sigma}( \phi ) = \frac{\partial}{\partial X^{\Sigma}}\ln \mathrm{N}( X ) \}\\ \end{array}\right) \quad \quad ; \quad \quad \Lambda , \Sigma = 1 , \dots , n_{V}= 1 + n
\phi_{0}^{( - 1 )}={\frac{1}{{J_{H}^{- 1}}^{\prime}}}, \qquad \phi_{1}^{( - 1 )}={\frac{J_{H}^{- 1}}{{J_{H}^{- 1}}^{\prime}}}, \qquad \phi_{2}^{( - 1 )}={\frac{\left( J_{H}^{- 1}\right)^{2}}{{J_{H}^{- 1}}^{\prime}}}.
P_{\alpha\beta}(x')=\frac{4\mu}{3\mu+\lambda}\frac{\delta_{\alpha\beta}}{\omega_{n-1}}\frac{1}{(|x'|^2+1)^{\frac{n}{2}}}+\frac{\mu+\lambda}{3\mu+\lambda}\frac{2n}{\omega_{n-1}}\frac{(x',1)_\alpha(x',1)_\beta}{(|x'|^2+1)^{\frac{n+2}{2}}},
M\models \forall \; x \; \theta_{m}(x, \overline{b})\leftrightarrow \theta_{n}(x, \overline{a}) \leftrightarrow \theta_{n}(x, \overline{a}^{\prime})\leftrightarrow \theta_{m^{\prime}}(x, \overline{b}^{\prime}) & c=g_{m}(\overline{b}) & c^{\prime}= g_{m^{\prime}}(\overline{b}^{\prime})
{}^{M\!L\!\!}\textrm{F}_{p}(a,b;c;z)= \frac{1}{B(b,c-b)}\int_{0}^{\infty}u^{b-1}(1+u)^{a-c}(1+u(1-z))^{-a}E_{\alpha, \beta}^{\gamma}\left(-p\left(2+u+\frac{1}{u}\right)\right)du,
\lambda^{3}+ 7 \sqrt{\frac{W ( z_{o})}{6}}\lambda^{2}+ ( 2 W ( z_{o}) + z_{o}^{2}W^{\prime \prime}( z_{o}) ) \lambda + 4 \sqrt{\frac{W ( z_{o})}{6}}z_{o}^{2}W^{\prime \prime}( z_{o}) = 0 \hspace{5 m m}\ ,
\alpha_{\pm}\bigl(\mathcal{E}^{\prime}(n)\bigr) =\mathcal{E}^{\prime}(n\mp 1)-\mathcal{E}^{\prime}(n), R_{-1}\bigl(\mathcal{E}^{\prime}(n)\bigr) R_0\bigl(\mathcal{E}^{\prime}(n)\bigr)^{-1}=A^{(-)}_n+C^{(-)}_n, a^{(+)}\phi^{(-)}_n(x)=C^{(-)}_n\phi^{(-)}_{n-1}(x), \ \a^{(-)}\phi^{(-)}_n(x)=A^{(-)}_n\phi^{(-)}_{n+1}(x),
O_A^{(b^{-1})}=\rho O_A^{(1)}+\frac{\rho-\rho^{-1}}{\big(Q^2-Q^{-2}\big)^2}O_B^{(1)}O_B^{(1)}O_A^{(1)},\qquad\textrm{where}\rho=\mathrm{pexp}\left(-\frac{Q^8-Q^{-8}}{(1+p^{-1})(1+s^{-1})(1+ps)}\right).
\frac{c}{x}\bigg(-\tau_t+\frac{1-\alpha}{t}\tau\bigg)+mx^k\phi_2-t^{1-\alpha}\xi_{1t}-2\phi_{1x}+\xi_{1xx}+\frac{c}{x^2}\xi_1+\frac{c}{x}\xi_{1x}-nx^k\eta_2=0,
\begin{array}{l l l}{J}&{\rightarrow}&{J -{\frac{i m \pi}{2}}}\\{J_{1}}&{\rightarrow}&{J_{1}+{\frac{i m \pi}{4}}}\\{\varepsilon}&{\rightarrow}&{\varepsilon}\\{A}&{\rightarrow}&{A +{\frac{i m \pi}{4}}}\\ \end{array}
\sum_{D \in \mathcal{D}_{v,d}(132,213)}y^dx^{des(\sigma(D))}= 1 + \sum\limits_{d=1}^\infty y^d(1+x)^{d-1}&= 1 + \frac{1}{1+x}\left( -1 + \frac{1}{1-y(1+x)}\right) &= \frac{x}{1+x}+ \frac{1}{(1+x)(1-y(1+x))}&= \frac{x(1-y(1+x))+1}{(1+x)(1-y(1+x))}&= \frac{1-yx}{1-y(1+x).}
\phi(x) = \frac{2V\left(\lambda' - \delta' x\right)}{2\kappa' x(n-x) +\lambda' +\delta' x};\varphi(x) = \frac{2V}{2\kappa' x(n-x) +\lambda' +\delta' x},{\cal I}(x) = \exp\left\{\int_0^x \phi(z)dz \right\}.
G ( x , y ; B ) = \frac{1}{2 \pi}\frac{1}{( x - y )^{2}}\left( \begin{array}{c c}{{0}}&{{e^{- [ \chi ( x ) - \chi ( y ) ]}\; \overline{{{( \tilde{x}- \tilde{y})}}}}}\\{{e^{+ [ \chi ( x ) - \chi ( y ) ]}\; ( \tilde{x}- \tilde{y})}}&{{0}}\\ \end{array}\right) \; .
D_{{\cal F}}^{( N , \alpha )}( \tau ; \Delta ) = \frac{2 \pi}{\alpha}\int_{0}^{\infty}\frac{d s}{( 4 \pi i s )^{N / 2}}e^{- i ( \tau^{2}- 2 r^{2}) / 4 s}\sum_{n = - \infty}^{\infty}I_{2 \pi | n | / \alpha}\left( r^{2}/ 2 i s \right) e^{i 2 \pi n \Delta / \alpha}.
w_j(X,Y,t,\tilde X,\tilde Y,\tilde t) :=u(X,Y,t)-\phi(\tilde X,\tilde Y,\tilde 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 ),
\begin{array}{l}\displaystyle \# \{\sum_{\substack{j\in J\setminus J_{k}}}X(u_{j})(t_{j+1}-t_{j})|\forall J_{k}=\{i_1,\cdots,i_k\}\subset J, 1\leq k\leq K(n)=n^r, \ \ \ \ \ \ \forall r\in(0,1), \displaystyle \lim_{\substack{n\rightarrow +\infty}}\displaystyle \frac{K(n)}{n}=0 \}\end{array}
S_{d u a l}= \int d^{6}x \left[ -{\frac 1 6}G_{{\mu}{\nu}{\sigma}}G^{{\mu}{\nu}{\sigma}}+ \frac{1}{2 ({\partial}_{\lambda}a ) ({\partial}^{\lambda}a )}{\partial}^{\mu}a{\cal G}_{{\mu}{\nu}{\sigma}}{\cal G}^{{\nu}{\sigma}{\rho}}{\partial}_{\rho}a + A_{{\nu}{\sigma}}{\partial}_{{\mu}}G^{{\mu}{\nu}{\sigma}}\right] .
(2n+5)(2n+4) f_4^2 f^n_{2n-5}= \big( (n+3)(n+4) f_3^2 - 3(2n+4) f_2 f_4 \big) f^n_{2n-3}\phantom{=}\ + \big( 2(n+4) f_2 f_3 + (n-2)(2n+4) f_1 f_4 \big) f^n_{2n-2}\phantom{=}\ + \big(\mathop-(n-1)(n+4) f_1 f_3 + (2n-1)(2n+4) f_0 f_4 \big) f^n_{2n-1}\phantom{=}\ - 2n(n+4) f_0 f_3 f^n_{2n}.
\frac{1+\sum_{i=0}^hk_1(i)}{k_1(0)+r}a^{\widehat{Q}^{+r}}+\frac{1+\sum_{i=0}^hk_2(i)}{k_2(0)+s}b^{\widehat{Q}^{+s}}= 1 + \sum_{j=1}^h\left(\frac{k_1(j)}{k_1(0)+k_1(j)}a^{\widehat{Q}_j}+\frac{k_2(j)}{k_2(0)+k_2(j)}b^{\widehat{Q}_j}\right).
G_{m}( D ; \alpha_{1}, . . . , \alpha_{m}) = [ \prod_{i = 1}^{m}\Gamma ( \alpha_{i}) ]^{- 1}( \prod_{i = 1}^{m}\int_{0}^{\infty}d t_{i}t_{i}^{\alpha_{i}- 1}) T^{- \frac{D}{2}}e^{- \frac{1}{T}\sum_{i < j}t_{i}t_{j}x_{i j}^{2}}
V\left(\sum_{j\in J}\Lambda_{\psi}(p_j)\otimes q_j^*\zeta\right)\longrightarrow\, (\Lambda_{\psi}\otimes\Lambda_{\psi})\bigl(E(\operatorname{id}\otimes\operatorname{id}\otimes\omega_{d^*\xi',c^*\zeta'})(W_{13})(1\otimes a)\bigr) &=E\bigl(\xi\otimes\Lambda_{\psi}(a)\bigr)=E(\xi\otimes\zeta).
{\cal L}_{i}= 2^{- i}\delta_{c_{1}d_{1}\cdots c_{i}d_{i}}^{a_{1}b_{1}\cdots a_{i}b_{i}}R_{\ \a_{1}b_{1}}^{c_{1}d_{1}}\cdots R_{\ \a_{i}b_{i}}^{c_{i}d_{i}}.
{\cal L}= - \frac{\epsilon_{0}}{4}F_{\mu \nu}F^{\mu \nu}- \frac{\epsilon_{0}}{4}G_{\mu \nu}G^{\mu \nu}- \frac{1}{c}{j_{e}}_{\mu}A^{\mu}- \frac{1}{c}{j_{m}}_{\mu}W^{\mu}+ (gauge fixing terms) \, .
\mathcal{K}:=\exp\bigg \{M(\log(M(1+\|V\|_{1})N_{0}^2 e^{M_{1}{(1-\tau)}/{\tau}}\rho^{-2}\Theta^{1/\tau}))+M\|V\|_{1}^{1/2s}\bigg\}&\leq \exp\bigg \{M\log( M N_{0}^2\rho^{-2}\Theta^{1/\tau}\big)+M \log(1+\|V\|_{1})+M M_{1}(1-\tau)/\tau+M\|V\|_{1}^{1/2s}\bigg\}.
d\mathbf{Y}_t = \mathbf{F}_t^{\ast}d\mathbf{a}_1+\mathbf{F}_t d\mathbf{a}_2 + d\mathbf{a}_3 + \mathbf{G}_t \left(d\mathbf{b}_1+ d\mathbf{b}_2+d\mathbf{b}_3+d\mathbf{b}_4\right),
\xi^{0}=f^{0}(\tilde{x}^{1},\tilde{x}^{2},\tilde{x}^{3}),\xi^{\alpha}=\int g^{\alpha\beta}(\tilde{x})\frac{\partial f^{0}(\tilde{x}^{1},\tilde{x}^{2},\tilde{x}^{3})}{\partial\tilde{x}^{\beta}}d\tilde{x}^{0}+f^{\alpha}(\tilde{x}^{1},\tilde{x}^{2},\tilde{x}^{3}),
\begin{cases}-\sigma_t - \sigma_x - v_{x}=f (0,T)\times (0,1), -v_{t}- v_{xx}- v_x - \sigma_x = g (0,T)\times (0,1), \sigma(t,0)=\sigma(t,1) t\in(0,T), v(t,0)=v(t,1)=0 t\in(0,T), \sigma(T,x)= \sigma_T(x),\ \v(T,x)= v_T(x) x\in(0,1).\end{cases}
\begin{array}{c}{{A^{\left( * \right)}( p ) = \sum_{m}v ( p , m ) a^{*}( p , m )}}\\{{B^{\left( * \right)}( p , ) = \sum_{m}v ( p , m ) b^{*}( p , m )}}\\{{A ( p ) = \sum_{m}u ( p , m ) a ( p , m )}}\\{{B ( p ) = \sum_{m}u ( p , m ) b ( p , m )}}\\ \end{array}
\left\Vert e^{\gamma \left\vert x\right\vert^{p}}\tilde{u}_{k}\left(x,0\right) \right\Vert_{X}=\left\Vert e^{\gamma \left( \frac{\alpha}{\beta}\right)^{p/2}\left\vert x\right\vert^{p}}\upsilon \left( x,0\right)\right\Vert_{X}=a_{0},
\delta \eta_{\alpha_{2 k + 1}}^{*}= \Phi_{\alpha_{2 k}}^{*}Z_{\alpha_{2 k + 1}}^{\alpha_{2 k}}+ \Phi_{\alpha_{2 k + 2}}^{*}A_{\alpha_{2 k + 1}}^{\alpha_{2 k + 2}}, k = 0 , \cdots , b ,
x^{t+1}(\gamma)=(I+\gamma B)^{-1}\left(x^t+\alpha(\gamma)(x^t-x^{t-1})-\gamma \left(F(x^t)+\beta(\gamma)(F(x^t)-F(x^{t-1}))\right)\right) \forall \gamma >0, &L_{t+1}(\gamma)= \|F(x^{t+1}(\gamma))-F(x^{t})- \kappa_t \gamma^{-1}(x^{t+1}(\gamma)-x^t)\|/\|x^{t+1}(\gamma)-x^t\| \forall \gamma >0,
\begin{cases}dp(t,x,z)= -[A_{\hat{\pi}(t,z)}p(t,x,z)+xq(t,x,z)] dt + q(t,x,z) dR(t);0\leq t \leq T, p(T,x,z)=U(x)\mathbb{E}_{\tilde{P}}[\delta_{Z}(z)|\mathcal{R}_T],\end{cases}
\mu_i^{-\frac{1}{2}}\pi_i(x)=-4\pi\alpha_3\,H_{\lambda}(x,\zeta_i)+\mu_i\,\mathcal{D}_0\Bigl(\frac{x-\zeta_i}{\mu_i}\Bigr)+\mu_i^{2-\sigma}\,\theta(\mu_i,x,\zeta_i)
h'_0=-\left\{e^{\frac{1}{24}A_3}\widehat{A}(TM)\right\}^{(12)}&h'_1=\left\{e^{\frac{1}{24}A_3}\widehat{A}(TM){\rm ch}(\widetilde{T_CM}+120)\right\}^{(12)}&h'_2=\left\{-e^{\frac{1}{24}A_3}\widehat{A}(TM){\rm ch}(3712+81\widetilde{T_CM}+\wedge^2\widetilde{T_CM}+W_i)+e^{\frac{1}{24}A_3}A_3\widehat{A}(TM)\right\}^{(12)}
\begin{array}{ll}s_{m}^{-1}E^{2}_{pq}= s_{m}^{-1}H_{p}(O_{n-k,n-k}; H_{q}(L_{k}) ) & & \cong H_{p}(O_{n-k,n-k}; s_{m}^{-1}H_{q}(L_{k})) \end{array}\Rightarrow s_{m}^{-1}H_{p+q}(T_{k}),
p_{\mathrm{S}_0}(t_\mathrm{p}|n) = \begin{cases}\delta\left(t_{\mathrm{s},1}\right) n=0 \delta\left(t_{\mathrm{w},1}(n)\right) \end{cases}p_{\mathrm{S}_1}(t_\mathrm{p}|n,q) =\delta\left(t_\mathrm{c}+ t_{\mathrm{w},1}(n+ q )\right),
g_{ij}=\begin{pmatrix}1+\lambda (u^2)^2 \lambda(- u^1 u^2+u^3) \lambda u^2 \lambda(- u^1 u^2+u^3) 4+\lambda(u^1)^2 -2\lambda u^1 \lambda u^2 -2\lambda u^1 \lambda\end{pmatrix}, w_{12}=\frac{\lambda\sqrt{\lambda}u^1}{\sqrt{-\det g}},w_{23}=0,w_{31}=-\frac{\lambda\sqrt{\lambda}}{\sqrt{-\det g}},
R_{k,g}^{Ndp}=\left(1-\frac{K}{T}\right)\log_2\left(1+\frac{p_d\beta_g^k\gamma_{k,g}\left|\mathbb{E}\left[\frac{f_{k,g}}{\hat{f}_{k,g}}\right]\right|^2}{p_d\beta_g^k \mathrm{Var}\left[\frac{f_{k,g}}{\hat{f}_{k,g}}\right]+\mathbb{E}\left[\left|\frac{I_{k,g}}{\hat{f}_{k,g}}\right|^2\right]+\mathbb{E}\left[\left|\frac{1}{\hat{f}_{k,g}}\right|^2\right]}\right)
V_{-\textstyle{1\over 2}; \chi}= e^{-\textstyle{\phi\over 2}}\prod_{j=1}^3 \sigma_{1-\nu}^je^{i(\textstyle{1\over 2}-\nu_j)H_j}e^{ik_\mu X^\mu}\, , \ \ \V_{-\textstyle{1\over 2};\chi^*}= e^{-\textstyle{\phi\over 2}}\prod_{j=1}^3 \sigma_\nu^j e^{-i(\textstyle{1\over 2}-\nu_j)H_j}e^{ik_\mu X^\mu}\, .
\phi(x_{[k]}) = \sum_{i \in [r_1]}\beta^{(1)}_{i, 1}(x_I) \gamma_i(x_{[k] \setminus I}) + \sum_{i \in [r_2]}\beta^{(2)}_{i, 1}(x_{I^{(2)}_{i, 1}}) \dots \beta^{(2)}_{i, d^{(2)}_i}(x_{I^{(2)}_{i, d^{(2)}_i}}) + \sum_{i \in [r_3]}\beta^{(3)}_{i, 1}(x_{I^{(3)}_{i, 1}}) \dots \beta^{(3)}_{i, d^{(3)}_i}(x_{I^{(3)}_{i, d^{(3)}_i}}).
\prod_{i=1}^k as_\lambda (V_i) \leq (2\pi)^{k n\lambda}\, \left( \int_{\mathbb{R}^{n}}\rho^{\frac{1}{k}}\Bigl( \frac{k(k-1)}{2}\, |u|^2 \Bigr) du\right)^{k(1-2 \lambda)}.
\mathbb{P}_x(\tau^{\{0\}}>t) \sim v_{{\rm sign}(x)}(\theta) |x|^{\theta}t^{-\theta/\alpha}\sum_{j=\pm1}\pi^\theta_{j}\frac{\mathbf{E}_{0,j}[I^{\theta/\alpha-1}]}{\mu_{\theta}|\alpha-\theta|v_j(\theta)}, t\to\infty,
F^{(1)}= - i \int_{x_0}^x dx_j A^{(1)}_j - \frac{i}{2}\int_{x_0}^x dx_j\, \epsilon^{ml}k_m \,\partial_l A^{(0)}_j + \frac{i}{2}\int_{x_0}^x dx_j\, \epsilon^{ml}A^{(0)}_m \,\partial_l A^{(0)}_j.
\|z\|^{\ell}_{\dot B^{s}_{p,r}}\stackrel{\mathrm{def}}{=}\|z^{\ell}\|_{\dot B^{s}_{p,r}}\ \hbox{and}\ \|z\|^{h}_{\dot B^{s}_{p,r}}\stackrel{\mathrm{def}}{=}\|z^{h}\|_{\dot B^{s}_{p,r}}.
\begin{aligned}\mathbf{y}=\mathbf{h}_1x_1+\mathbf{h}_2x_2+\mathbf{n}=\sqrt{P_{1,1}}\mathbf{h}_1s_{1,1}+\sqrt{P_{1,2}}\mathbf{h}_1s_{1,2}+\sqrt{P_{2}}\mathbf{h}_2s_{2}+\mathbf{n},\end{aligned}
X(\delta)+\omega_{\mu, L, \varepsilon}(t) = \omega_{\mu, Le^{\delta/\mu}, \varepsilon}(t), X(\delta)+\omega_{\mu, L, \varepsilon}(t) = \omega_{\mu, L, \varepsilon e^{-\delta/\mu}}(t e^{-\delta/\mu}), X(\delta)+\omega_{\mu, L, \varepsilon}(t) = \omega_{\mu-\delta, L,\varepsilon}(t) + \bar{\omega}_{\delta, e L, \varepsilon}(t).
\begin{aligned}\|\delta u\|_{L^\infty_{t}\dot{B}_{2,\infty}^{-1}}+\nu\|\delta u\|_{\tilde{L}^1_{t}\dot{B}_{2,\infty}^1}\le C\|(u_{1},u_{2})\|_{L^1_{t}\dot{B}_{2,1}^2}\|\delta u\|_{L^\infty_{t}\dot{B}_{2,\infty}^{-1}}+C\int_{0}^{t}\|(B_{1},B_{2})\|_{\dot{B}_{2,1}^1}\|\delta B\|_{\dot{B}_{2,\infty}^0}d\tau.\end{aligned}
I_1(\ell_1,\ell_2)-I_2(\ell_1,\ell_2) &\leq\Delta\nu p^k_{j,+}(t-\Delta \ell_2) &\quad\; \cdot \sum_{n=1}^\infty \sum_{m=1}^\infty \sum_{i=1}^m i P(Z_0(\Delta\ell_2)=m,Z_0(\Delta \ell_1)=n,X_{\ell_1,\Delta}= 1,B_{\ell_1\Delta}(\Delta\ell_2)=i).
\begin{array}{l}{{( \mathrm{{B o s e}\oplus{F e r m i \o s c i l l a t o r s})_{L}^{\left( l_{L}\right)}}}}\\{{\times ( \mathrm{{B o s e}\oplus{F e r m i \o s c i l l a t o r s})_{R}^{\left( l_{R}\right)}}}}\\{{\times \, \, | v a c , \, \, p^{\mu}; \vec{m}, \vec{n}>}}\\ \end{array}
\left\langle (J_U\circ C_f\circ J^{-1}_V)(J_V(g)),\delta_U(x)\right\rangle =\left\langle J_U(C_f(g)),\delta_U(x)\right\rangle=\left\langle J_U(g\circ f),\delta_U(x)\right\rangle &=g(f(x))=\left\langle J_V(g),\delta_V(f(x))\right\rangle .
2 \int V \mbox{Re}\Big( \bar{u}(r \partial_r - n/2) (\delta A^2 + 1)^{-1}u \Big) dx \stackrel{\delta \downarrow 0}{\longrightarrow}2 \int V \mbox{Re}\Big( \bar{u}(r \partial_r - n/2) u \Big) dx = \int V \nabla \cdot x ( |u|^2 ) dx .
\left| \left(u_x(x)\right)^{p}\right|= |\langle u_x, p\rangle p|= |\langle u_x, p- u\rangle|\leq 3\varepsilon |u_x(x)|, \left| \left(u_t(x)\right)^{p}\right|\leq 3\varepsilon |u_t(x)| \; \textrm{and}\; \left| \left(u(x)- p\right)^{p}\right|\leq \frac{3\varepsilon}{2}\left|u(x)- p\right|.
\begin{array}{l l}{{\delta_{( \xi )}^{S}M^{\mu \nu}={\cal L}_{\xi}( \sqrt{g}g^{\mu \nu}) ,}}&{{}}\\{{\delta_{( \xi )}^{S}L^{\mu \nu}={\cal L}_{\xi}\lbrack M^{\mu \nu}- s ( \sqrt{g}g^{\mu \nu}) \rbrack .}}\\ \end{array}
\Big\{x \in \mathbb{I}: \log a_1(x) \geq b + \varepsilon, \cdots, \log a_n(x) \geq b + \varepsilon\Big\}\supseteq\Big\{x \in \mathbb{I}:a_1(x) = b_\varepsilon, \cdots, a_n(x)= b_\varepsilon\Big\},
[ a^{m}, a^{\dagger n}] = \sum_{g = 0}^{\infty}\frac{1}{2^{2 g}( 2 g + 1 ) !}\prod_{k = 0}^{2 g}( m - k ) ( n - k ) ( a^{m - 2 g - 1}a^{\dagger n - 2 g - 1})_{W}.