harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
E_{3}(f):=-\frac{1}{\pi i}\int_{0}^{2\pi}\frac{f(\beta)\left(\frac{\epsilon(\beta)}{z(\beta)}-\frac{\epsilon(\alpha)}{z(\alpha)}\right)^{2}(z(\alpha)z(\beta))^{2}}{(z(\beta)-z(\alpha))|z(\beta)-z(\alpha)|^{2}}\partial_{\beta}\left(\frac{1}{z(\beta)}\right)d\beta.
\left(x^{j}f(x)\right)^{(j+n)}=\sum_{l=0}^{n+j}\binom{n+j}{l}\left( x^j\right)^{(l)}f^{(n+j-l)}(x) &=\sum_{l=0}^{j}\binom{n+j}{l}\frac{j!}{(j-l)!}x^{j-l}f^{(n+j-l)}(x) &=\sum_{m=0}^{j}\binom{n+j}{j-m}\frac{j!}{m!}x^{m}f^{(n+m)}(x).
d_X(Z_s^0,Z_s^1) = \min\left(\inf_{1\leq i \neq j \leq s}|x_i^0 - x_j^1|,\inf_{1\leq i \leq s \; : \;(x_i^0,v_i^0) \neq (x_i^1,v_i^1)}|x_i^0 - x_i^1| \right)
F_{\Phi}(X) = \frac{q^{1-g}h}{\zeta(2)(1-q^{-1})(q^2-1)}q^{2X}- \sum_{j=1}^{2g}\frac{Z(q \gamma_j^{-1})\gamma_j}{Z'(\gamma_j^{-1})}\frac{\gamma_j^X}{\gamma_j - 1}+ \varepsilon_{\Phi}(X),
g_n(k)=\frac{k!}{\langle n \rangle_k \alpha^k}\prod_{j=1}^n (\alpha d_j +1) \sum_{i_1<\ldots<i_k}\prod_{j=1}^k \frac{\alpha d_{i_j}}{\alpha d_{i_j}+1}\prod_{\ell\in [n]\setminus\{i_1,\ldots,i_k\}}\frac 1{\alpha d_\ell +1}.
1) \; 0 \leq \langle\mathbf{z},\mathbf{z}\rangle= x_1^2+y_1^2+ \cdots +x_n^2+y_n^2-x_{n+1}^2-y_{n+1}^2 2) \; 0 \geq f(\mathbf{z})={\displaystyle \sum_{j=1}^{n}}4(x_jy_{n+1}-x_{n+1}y_{j})^{2}-\sum_{1\leq j<k \leq n}4(x_jy_{k}-y_{k}x_{j})^{2}.
\frac{n}{a_nQ'(a_n)}=\frac{2}{\pi}\int_0^1\frac{tQ'(a_nt)}{Q'(a_n)\sqrt{1-t^2}}\, dt &\rightarrow \frac{2}{\pi}\int_0^1 \frac{t^{\alpha}}{\sqrt{1-t^2}}\, dt=B_{\alpha}.
2(m+1)\xi_j+v_{q_0}(\xi_j)+v_{q_1}(\xi_j)+\sum_{\substack{1\leq k\leq n k\neq j}}\Bigl( v_q(\xi_j+\xi_k)+ v_q(\xi_j-\xi_k)\Bigr)=2\pi (\varrho_{\texttt{b};j}+\lambda_j) ,
\int_{\gamma}e^{-t\mu}(-\mu)^{-n}\log(-\mu)d\mu =-\frac{d}{dn}\int_{\gamma}e^{-t\mu}(-\mu)^{-n}d\mu=\frac{d}{dn}\left(\frac{2\pi i}{\Gamma(n)}t^{n-1}\right) &=2\pi i\frac{t^{n-1}\log t\Gamma(n)-t^{n-1}\Gamma(n)'}{\Gamma(n)^2}&=\frac{2\pi i}{\Gamma(n)}t^{n-1}\log t-\frac{2\pi i}{\Gamma(n)}\frac{\Gamma'(n)}{\Gamma(n)}t^{n-1}.
0 \geq \nabla_k\nabla_k \|\nabla w\|^2 &=2\left(\langle \nabla_k\nabla_k\nabla w,\nabla w\rangle + \|\nabla_k\nabla w\|^2\right) &\geq 2\langle \nabla_k\nabla_k\nabla w,\nabla w\rangle &=2\|\nabla w\| \langle \nabla_k\nabla_k\nabla w,E_n\rangle.
A_{s_i}^{\mathbf{z}'}= \frac{1-vz_i^n z_{i+1}^{-n}}{1-z_i^n z_{i+1}^{-n}}I_{V_{z^n_1}}\otimes \cdots \otimes \tau \tilde R^{\gamma}(z_i^n z_{i+1}^{-n}) \otimes \cdots \otimes I_{V_{z^n_r}}.
C_1^{S^{\star}}=\begin{cases}\,\,\int_{-\beta^{\star}\eta^{\star}}^{\eta^{\star}-\beta^{\star}\eta^{\star}}{\gamma p(\gamma)d\gamma}, \eta^{\star}> 0, \,\,\int_{\eta^{\star}-\beta^{\star}\eta^{\star}}^{-\beta^{\star}\eta^{\star}}{\gamma p(\gamma)d\gamma}, \eta^{\star}< 0\end{cases}
E_{1}^{rs}=E(Y^{r+1})\otimes_{K_0}H^s_{rig}(Y^{r+1}/{\mathfrak S}^{0})\Longrightarrow H^{r+s}_{rig}(Y/{\mathfrak S}^{0},E) E_{1}^{rs}=E(Y^{r+1})\otimes_KH^s_{rig}(Y^{r+1}/{\mathfrak S}^{\pi})\Longrightarrow H^{r+s}_{rig}(Y/{\mathfrak S}^{\pi},E).
\frac{\partial Y}{\partial x}=\left(\frac{A_3}{x^2}+\frac{A_2}{x}+A_1+A_0 x \right)Y ,\frac{\partial Y}{\partial t_1}=-\frac{A_3}{t_1x}Y, \frac{\partial Y}{\partial t_2}=(E_2 x+B_1)Y.
\mathop{\lim}\limits_{\varepsilon \to \varepsilon_0}\mathop{\sup}\limits_{(x,j) \in K\times S}P\left ({\|{y^\varepsilon \left({t, s, \xi,j}\right)-y^{\varepsilon_0}\left({t,s,\xi,j}\right)}\|_H \ge \eta}\right ) = 0.
\bigcup_{j=1}^k (\alpha_j^-\cup \beta_j^-)\subset N^-, \ \ \bigcup_{j=1}^k (\alpha_j^+\cup \beta_j^+)\subset N^+,\ \ (\alpha_j^-\cup \beta_j^-)\cap (\alpha_i^-\cup \beta_i^-)=\emptyset \ \ \i\neq j.
\frac{1+ d(y)}{2}= \frac{W_{Y|U}(y|0)}{W_{Y|U}(y|0)+W_{Y|U}(y|1)}= W_{U|Y}(0|y), \frac{1 - d(y)}{2}= \frac{W_{Y|U}(y|1)}{W_{Y|U}(y|0)+W_{Y|U}(y|1)}= W_{U|Y}(1|y).
H_n^{\alpha_n}H_{n-1}^{\alpha_{n-1}- \alpha_n}\cdots H_2^{\alpha_2 - \alpha_3}H_1^{\alpha_1 - \alpha_2}=\left( \frac{H_1}{H_0}\right)^{\alpha_1}\left( \frac{H_2}{H_1}\right)^{\alpha_2}\cdots \left( \frac{H_n}{H_{n-1}}\right)^{\alpha_n}
\begin{array}{llll}e^I = S^I_J\theta^J, \hbox{i.e.}e^a:= \theta^a\hbox{and}e^i = S^i_j\theta^j p_I = (S^{-1})^J_I\pi_J \hbox{i.e.}p_a:= \pi_a\hbox{and}p_i = (S^{-1})^j_i\pi_j \omega^{IJ}= S^I_KS^J_L\varphi^{KL}- \hbox{d}S^I_KS^J_L\textsf{h}^{KL}\Omega^{IJ}= S^I_KS^J_L\Phi^{KL}\end{array}
b(\upsilon)+c(\upsilon) =b_\upsilon(i,j')+b_\upsilon(1,n)+\sum\limits_{(s,t)\in\sigma^-_{(i,j)(i',j')}}b_\upsilon(s,t)+\sum\limits_{(s,t)\in\sigma^-_{(i,j)(i',j')}}c_\upsilon^\ell(s,t) &=2(k+2)+b(\sigma)+2r+c(\sigma)=b(\sigma)+c(\sigma)+4+2(k+r),
\max_{j<\beta N}\left\{\Pr_{v\sim \mu_t}[|v H|=j]\cdot 2^{2\epsilon N\log(1-\frac{2j}{N})}\right\}\leq 2^{-\frac{\epsilon N}{\ln2}}\cdot 2^{\epsilon N\log(\frac{2\epsilon}{1-\eta})}\leq 2^{-h(\epsilon)N}\cdot 2^{\epsilon N\log(\frac{2}{1-\eta})},
{H}_{int}= \bar{\psi}* e \gamma^\mu \bar{B}_\mu \left[1 + \frac{e}{2}\gamma^{+}\frac{1}{i \partial_{-}- e\bar{B}_{-}}e \gamma^{\nu}\bar{B}_{\nu}\right] \ast \psi =\bar{\psi}\ast{\cal V}[\bar{B}_\mu] \ast \psi.
\psi^m_{\pm}=u^{m(n)}\varphi_{\pm (n)}=\frac12 u^{m[+2]}\varphi^{[-2]}_{\pm}+\frac12 u^{m[-2]}\varphi^{[+2]}_{\pm}-u^{(i)}_m\varphi^{(i)}_{\pm}.
\begin{aligned}l(\mathcal Y^{[0]},0,N+1) \le l(\mathcal Y,0,N+1)\le l(\mathcal Y^{[N+1]},0,N+1), l^*(\mathcal Y^{[N+1]},0,N+1) \le l^*(\mathcal Y,0,N+1)\le l^*(\mathcal Y^{[0]},0,N+1),\end{aligned}
\sum_{n\in E}1 \leq\sum_{n\in E}\Phi_3(\frac{C}{u(A_n)}).\mu(A_n) &=\sum_{n\in E}\int_{A_n}\Phi_3(\frac{C}{u(A_n)})d\mu &\leq\int_{X}\Phi_3(\frac{C}{u(A_n)})d\mu<\infty.
I^{q}_{2,1}=\sum_{k=-\infty}^{k_0}\int_{2^{k}\leq|x|\leq 2^{k+1}}\left(\int_{B(0,2|x|)\setminus B\left(0,\frac{|x|}{2}\right)}|G_{\xi, \sigma}(y^{-1}x)u(y)|dy\right)^{q}\frac{dx}{|x|^{\beta q}},
\big(a^+(f_1)\dotsm a^+(f_n)a^-(g_1)\dotsm a^-(g_m)\big)\diamond\big(a^+(\varphi_1)\dotsm a^+(\varphi_k)a^-(\psi_1)\dotsm a^-(\psi_l)\big) &\qquad:=q^{km}\,a^+(f_1)\dotsm a^+(f_n)a^+(\varphi_1)\dotsm a^+(\varphi_k) &\qquad\qquad\qquad\qquad\times a^-(g_1)\dotsm a^-(g_m)a^-(\psi_1)\dotsm a^-(\psi_l),
hAh^{-1}= A, hA = Ah; hBh^{-1}= B, hB = Bh; hI = I, (h-1_{V}) I = 0; hA'h^{-1}= A', hA' = A'h; hB'h^{-1}= B',hB' = B'h; hFh'^{-1}= F
N_{j}( t ) \; = \; \det \{2 G_{j}( t ) \}^{- 1 / 4}\; \; , N_{1 2}( t ) \; = \; \det \{1 - G_{1}( t ) G_{1 2}( t ) G_{2}( t ) G_{1 2}( t ) \}^{1 / 4}\; \; ,
\begin{aligned}r_N=r\cos \theta_{N-1}r_{N-1}=r\sin \theta_{N-1}\cos \theta_{N-2}\cdots \cdots r_2=r\sin \theta_{N-1}\sin \theta_{N-2}\cdots \sin \theta_2 \cos \theta_1 r_1=r\sin \theta_{N-1}\sin \theta_{N-2}\cdots \sin \theta_2 \sin \theta_1.\end{aligned}
a^{(1)}= \frac{r^{\frac{1-r}{1+r}}(r+1)^2}{4}&a^{(2)}= \frac{(r-q)^{\frac{1-r+q}{1+r-q}}(r-q+1)^2}{4}&a^{(3)}= \frac{\theta^{\frac{1-3\theta}{1+\theta}}(1+\theta)^2}{4},
\overset{\cdot}{\vec{r}}_{n}\cdot \vec{r}_{n}=0~,~~~\overset{\cdot}{\vec{r}}_{n}\cdot \overset{\cdot}{\vec{r}}_{n}=\dot{\theta}_{n}^{2},~~~\left( \vec{r}_{n}\wedge \overset{\cdot}{\vec{r}}_{n}\right) \cdot \hat{z}=\dot{\theta}_{n}~,
T_1 \, \Theta(z+\iota(P-A_2)) \, \Theta(z+\iota(A_1-O)) + T_2 \, \Theta(z+\iota(P-O)) \, \Theta(z+\iota(A_1-A_2))\, = T_3 \, \Theta(z+\iota(P+A_1-A_2-O)) \, \Theta(z) \,
\Bigg[: e^{- i 2 \sqrt{\pi}\sqrt{\frac{\pi}{\pi\!+\!gN}}U_{1 b}\Phi^{(1)}(x)}:_{M^{(1)}}\; \prod_{I=2}^N: e^{- i 2 \sqrt{\pi}U_{I b}\Phi^{(I)}(x)}:_{M^{(I)}}e^{+ i \frac{\theta}{N}}
\varphi_{0}=\frac{\partial S}{\partial L}=\frac{\partial S_{\varphi}}{\partial L}+\frac{\partial S_{r}}{\partial L}=\varphi-\frac{L}{\sqrt{2m}}\int^{r}\frac{dr}{r^{2}\sqrt{E_{tot}-U(r)-\frac{L^{2}}{2mr^{2}}}}
\mathbb{V}:=\varinjlim \left( S_2^{new}(U_n,\widetilde{\psi},F/\mathcal{O})^{ord}\oplus S_k^{new}(\Gamma_1(Np^n\ell),\psi,F/\mathcal{O})^{\left(\frac{-}{\ell}\right),ord}\right),
\Big(\Gamma \big(1 + \frac{1}{2g({\bf R})}\big)\Big)^{-g({\bf R})}\leq \Big (\frac{1}{2eg({\bf R})}\Big)^{-\frac{1}{2}}=\Big(\frac{2edg({\bf R})}{d}\Big)^{\frac{1}{2}},
K_{h_1}\circ K_{h_2}((x_1,x_2), A_1\times A_2)=\int_{X_1}\int_{X_2}K_{h_2}^2(y_2,A_2) \delta_{y_1}(A_1)\> d\omega_{X_2}(y_2)\> K_{h_1}^1(x_1,dy_1) &=K_{h_1}^1(x_1,A_1)\cdot \omega_{X_2}(A_2) = K_{h_1}((x_1,x_2), A_1\times A_2)
\alpha_{0}( \sigma , \bar{\sigma}) = \frac{1}{2}\left( \mathrm{a r t a n h}\frac{\sigma}{4}+ \bar{\sigma}\right) \mathrm{a n d}\bar{\alpha}_{0}( \sigma , \bar{\sigma}) = \frac{1}{2}\left( \mathrm{a r t a n h}\frac{\sigma}{4}- \bar{\sigma}\right)
L := \langle x_0^{d_0 +1}, x_0^{d_0}x_1^{d_1 +1}, x_0^{d_0}x_1^{d_1}x_2^{d_2 +1}, \ldots, x_0^{d_0}x_1^{d_1}\cdots x_{k-1}^{d_{k-1}+1}, x_0^{d_0}x_1^{d_1}\cdots x_k^{d_k}\rangle.
\Psi(0)=\frac{(1-\varrho_a)_a(\epsilon-\varrho_a)_a}{a!\cdot(\epsilon)_a}\cdot{}_4F_3\bigg[\begin{matrix}\varrho_a -a \delta-\beta \delta-\gamma &\delta \varrho_a-a 1+\varrho_a-a-\epsilon\end{matrix}\bigg|\,1\bigg].
\Gamma_{s}( P^{a}) ={\textstyle \frac{i R_{5}}{3 2}}\sigma^{2}\Gamma^{0 1 2 3 4}\Gamma^{a}\, , \, \, \, \, \Rightarrow \Gamma^{a}={\textstyle \frac{3 2 i}{R_{5}}}\mathcal{S}\Gamma_{s}( P^{a}) \, , \hspace{1 c m}\mathcal{S}= \sigma^{2}\Gamma^{0 1 2 3 4}\, ,
\Psi_E^{n\leq \ell-1}\left(r,\theta\right)= \left(\begin{array}{l}B_n^+ J_{\ell+a-n}\left(k r\right) e^{i n \theta}i\frac{k}{E-m}B_n^+ J_{\ell+a-n-1}\left(k r\right)e^{i\left(n+1\right) \theta}B_n^- J_{\ell+a-n}\left(k r\right)e^{i n \theta}i\frac{k}{E+m}B_n^- J_{\ell+a-n-1}\left(kr\right) e^{i\left(n+1\right) \theta}\end{array}\right)
\widetilde{\mathcal{J}}(\delta)=\big\{(\mathcal{X},\lambda,\mathcal{Y})\in \mathbb{X}\times\mathbb{R}^{p}\times\mathbb{X}\ |\ \Psi\big(\mathcal{X},\lambda,\mathcal{Y}\big)\in \delta+\mathcal{N}_{\mathbb{X}\times\mathbb{R}^p\times K^{\circ}}(\mathcal{X},\lambda,\mathcal{Y})\big\}.
\sum_{I \in J_r^{-1}(K)}f = \sum_{I \in J_r^{-1}(K)}(x-1)^{n-r}(y-1)^{\#I-r}={c(K) \choose c(K)-n+r}(x-1)^{n-r}\sum_{j=0}^{\overline{c}(K)}{\bar{c}(K) \choose j}(y-1)^j ={c(K) \choose c(K)-n+r}(x-1)^{n-r}y^{\bar{c}(K)}.
T_{\mu \nu}= ( - g )^{\frac{1}{4}}\left\{- \frac{1}{4}\gamma_{\mu \nu}\left[ \gamma^{\rho \sigma}\left( g_{\rho \sigma}- g^{\kappa \tau}F_{\rho \kappa}F_{\tau \sigma}\right) - ( p - 3 ) \Lambda \right] + g_{\mu \nu}- g^{\rho \sigma}F_{\mu \rho}F_{\sigma \nu}\right\}.
\alpha Af(\bar{\eta})=\alpha f(\bar{\eta})\sum_{x\in\bar{\eta}}\frac{B\bar{g}(x)}{\bar{g}(x)}+\alpha f(\bar{\eta})\sum_{\{x,x'\}\subset\bar{\eta}}r(x,x')(\frac 12\frac{\bar{g}(x')}{\bar{g}(x)}+\frac 12\frac{\bar{g}(x)}{\bar{g}(x')}-1),
\pi(x) - \textrm{li}(x) = \frac{\theta(x)}{\log x}+ \int_{2}^{x}\frac{\theta(t)}{t \log^{2}t}\, dt - \int_{2}^{x}\frac{dt}{\log t}- \textrm{li}(2) &= \frac{\theta(x)-x}{\log x}+ \frac{2}{\log 2}+ \int_{2}^{x}\frac{\theta(t)-t}{t \log^{2}t}\, dt - \textrm{li}(2).
\sigma_{impl}^{n}=\alpha\frac{F_{0}-K}{D(\zeta)}\{1+[\frac{2\gamma_{2}-\gamma_{1}^{2}}{24}(\frac{\sigma_{0}C(F_{mid})}{\alpha})^{2}+\frac{\rho\gamma_{1}}{4}\frac{\sigma_{0}C(F_{mid})}{\alpha}+\frac{2-3\rho^{2}}{24}]\epsilon\}.
E\{R(\psi)\}=\eta^{1/2}\{\lambda^{1r}\lambda^{st}\lambda_{rs,t}+{\textstyle{1\over 2}}\lambda^{1r}\tau^{st}\lambda_{rs,t}+{\textstyle{1\over 2}}\lambda^{1r}\lambda^{st}\lambda_{rst}+{\textstyle{1\over 3}}\lambda^{1r}\tau^{st}\lambda_{rst}\}+O(n^{-1}).
X^m(\sigma^{\pm2},\eta^{\pm})=X^m(\sigma^{\pm2})+\frac{i}{\sqrt{8}}\eta^+\psi^m_+(\sigma^{\pm2})+\frac{i}{\sqrt{8}}\eta^-\psi^m_-(\sigma^{\pm2})+i\eta^+\eta^-F^m(\sigma^{\pm2}),
R_{\mathrm{avg}}(\boldsymbol{\rho},\boldsymbol{\alpha}_a,\boldsymbol{\alpha}_b,\boldsymbol{\gamma},\mathbf{d}_b)=\frac{1}{N}\sum_{i=1}^{N}R(\rho_i, \alpha_{a_i},\alpha_{b_i}, \gamma_i,d_{b_i})
T^{( n )}{}_{\mu \nu}= G^{( n )}{}_{\mu}{}^{\rho_{1}\cdots \rho_{n - 1}}G^{( n )}{}_{\nu \rho_{1}\cdots \rho_{n - 1}}-{\textstyle \frac{1}{2 n}}g_{\mu \nu}G^{( n ) \, 2}\, ,
N(w_1,w_2) =N\left(\prod_i S^{a_{ki}}_i T^{b_{ki}}_i,w_2\right) &=\sum a_{1i}N(S_i,w_2)+\sum b_{1i}N(T_i,w_2) &=\sum a_{1i}a_{2k}N(S_i,S_k)+\sum a_{1i}b_{2k}N(S_i,T_k) &+\sum b_{1i}a_{2k}N(T_i,S_k)+\sum b_{1i}b_{2k}N(T_i,T_k),
\begin{aligned}\lim_{n\to\infty}k\sum_{l=1}^k\binom{c_nkn}l(k)_l\sum_{j=n-N+1}^{n-1}\frac{(k(j-1))_{c_nkn-l}}{(kn)_{c_nkn}}\frac 1{k(n-j+1)-l}= &k\sum_{l=1}^k\binom kl(\frac c{1-c})^l\sum_{s=1}^{N-1}(1-c)^{k(s+1)}\frac 1{k(s+1)-l}.\end{aligned}
\partial_{t}u(t, x) = \partial_{i}\big(a_{ij}(x) \partial_{j}u(t, x)\big) (t, x) \in (0, T) \times \Omega, u(t, x) = 0 (t, x) \in (0, T) \times \partial \Omega, u(0, x) = u^{0}(x) x \in \Omega.
\int_{\mathbb{D}}\frac{|h(w)|^p}{|w|^{pk_p}}\leq\liminf_{k\rightarrow\infty}\int_{\mathbb{D}}\frac{|g_{z_{j_k}}(w)|^p}{|w|^{pk_p}}=\frac{1}{K_{p,\varphi_p}(0)}.
\lim_{\varepsilon\to 0^{+}}\frac{(q_{\frac{1}{\varepsilon}}f)(\mathbf{x}_{0}+\varepsilon \mathbf{x})}{\varepsilon^{\alpha}L(\varepsilon)}= g(\mathbf{x}) \ \ \ \mbox{weakly$^{\ast}$ in}\mathcal{K}_{M,r}'(\mathbb{R}^{n}) \ (\mbox{resp.}\mathcal{S}'_{r}(\mathbb{R}^{n}))
\mu_{\lambda \rho}(C_{\bf 0}\in\Lambda({\bf k})\mid ~ \Gamma_{\bf 0})= \lambda^{|{\bf k}| -1}|{\bf k}|\frac{p_0^{k_0}p_\alpha^{k_\alpha}p_\beta^{k_\beta}}{k_0!k_\alpha!k_\beta!}F_\lambda ({\bf k}),
d s^{2}= \alpha^{\prime}\left[ \frac{u^{2}}{\tilde{R}^{2}}( \cos^{2}\varphi ( - \tilde{f}d x_{0}^{2}+ d x_{1}^{2}) + \cos^{- 2}\theta ( d x_{2}^{2}+ d x_{3}^{2}) ) + \frac{\tilde{R}^{2}}{u^{2}}( \tilde{f}^{- 1}d u^{2}+ u^{2}d \Omega^{2}) \right] ,
Y(\mathcal{I})_{\beta\omega}^{mj}=-Y(\mathcal{I})_{\beta\alpha}^{mi}\cdot Y(\mathcal{I})_{\alpha\omega}^{mi}\cdot Y(\mathcal{I})_{\beta\alpha}^{ij}\cdot Y(\mathcal{I})_{\alpha\omega}^{ij}\in\mathbb{C},(j,\beta)\in\mathfrak{c}.
(A^{(V)})^{\hat{b}}_{\mbox{}\hat{c}}\equiv[\omega^{a}J^{(V)}_{a}+E^{a}\tilde{P}^{(V)}_{a}]^{\hat{b}}_{\mbox{}\hat{c}}=\left( \begin{array}{cccc}0 \omega^{2}-\omega^{1}E^{0}\omega^{2}0 -\omega^{0}E^{1}-\omega^{1}\omega^{0}0 E^{2}-E^{0}E^{1}E^{2}0 \end{array}\right).
\left[ \bar{Z}_{a},\bar{Z}_{b}\right] =\frac{1}{2}\left[ P_{a}-Z_{a},P_{b}-Z_{b}\right] , & =\frac{1}{2}\left( \left[ P_{a},P_{b}\right] -\left[ P_{a},Z_{b}\right] -\left[ Z_{a},P_{b}\right] +\left[ Z_{a},Z_{b}\right]\right) , & =\frac{1}{2}\left( J_{ab}-Z_{ab}+Z_{ba}-J_{ab}\right) , & =-Z_{ab}.
\sum_{k=0}^2\|u\|_{n+1,\gamma;\Omega_k}&\le C(\varepsilon)\Big(\sum_{k=0}^2\sum_{j=1}^2\|f_j\|_{n,\gamma;\hat{\Omega}_k}+\sum_{k=0}^2\|u\|_{L_\infty(\hat{\Omega}_k)}\Big)+C_1(l^\gamma+\varepsilon)\sum_{k=0}^2[u]_{n+1,\gamma;\hat{\Omega}_k},
\frac{1}{({M_{i}}^{2}-{p_{1}}^{2}) ({M_{i}}^{2}-{p_{2}}^{2})}= \int_{0}^{1}d \xi \frac{1}{{[{M_{i}}^{2}-{p_{1}}^{2}- ({p_{2}}^{2}-{p_{1}}^{2}) \xi ]}^{2}}
\int\frac{x^2dG_{y_2}(x)}{(1+x\underline{m}(z))^2}=m'_{y_2}(-\underline{m}(z))=-\frac{1-y_2}{y_2}\cdot\frac{1}{(\underline{m}(z))^2}+\frac{1}{y_2}\cdot\underline{m}'_{y_2}(-\underline{m}(z)).
\begin{aligned}(u_pI_p^{[k-1]}:I^{[\ell]})\ =\ (u_pI_p^{[k-1]}:I_p^{[\ell]})\cap(u_pI_p^{[\ell-1]}:u_pI_p^{[\ell-1]})\cap(u_pI_p^{[k-1]}:J) &=\u_pI_p^{[k-1]},\end{aligned}
L = \frac{1}{2kz}\left[A_\mu( \partial^2 \eta^{\mu\nu}-z\partial_z(\frac{1}{z}\partial_z) \eta^{\mu\nu}-\partial^\mu\partial^\nu) A_\nu + 2A_5\partial_z \partial^\mu A_\mu -A_5 \partial^2 A_5\right].
\Vert u(t)\Vert_{q_0}\leq \left( \frac{C}{M^{m_1-m_2}}\, t + \Vert u_0\Vert_{q_0}^{1-m_1}\right)^{-\frac{1}{m_1-1}}\forall t \ge 0 \, , C := \frac{4 \, c \, q_0}{|\Omega|^{\frac{m_1-1}{q_0}}\, C_P^{\ast 2}\, (m_1-1) \left( m_1+q_0-1 \right)^2}\, .
\left\{A , B \right\}= \left( \frac{\partial A}{\partial q^{i}}\frac{\partial B}{\partial p_{i}}+ \frac{\partial A}{\partial p_{i}}\frac{\partial B}{\partial q^{i}}\right) + ( - 1 )^{\sigma ( A )}\left( \frac{\partial A}{\partial \theta^{i}}\frac{\partial B}{\partial \pi_{i}}+ \frac{\partial A}{\partial \pi^{i}}\frac{\partial B}{\partial \theta_{i}}\right) \ .
\left\| \frac{1}{r}\left(\sum_{i=1}^r{h(x_i)\rho_{(x_is)^{-1}}(f)}\right) - \frac{1}{r}\left(\sum_{i=1}^r{h(y_i)\rho_{(y_it)^{-1}}(f)}\right)\right\|_{L_p(m_{XAS})}\leq \frac{1}{6}\epsilon \|\nu\|\sup{\left\{\left\|\rho_{(as')^{-1}}(f)\right\|_{L_p(m_{XAS})}: s' \in S, a \in A\right\}}\leq \frac{1}{6}\epsilon \|\nu\|\|f\|_{L_p(m_{XAS})}
f(x) =a_0x^n+a_1x^{n-1}+\cdots+a_{n-1}x+a_n, g(x) =b_0x^m+b_1x^{m-1}+\cdots+b_{m-1}x^x+b_m, u(x) =u_0x^{m-1}+u_1x^{m-2}+\cdots+u_{m-2}x+u_{m-1}, v(x) =v_0x^{n-1}+v_1x^{n-2}+\cdots+v_{n-2}x+v_{n-1}.
\operatorname{Tail}(v ; x_{0}, r) :=\left(r^{sq}\int_{\mathbb{R}^{N}\backslash B_{r}\left(x_{0}\right)}\frac{|v(x)|^{q-1}}{\left|x-x_{0}\right|^{N+s q}}d x\right)^{\frac{1}{q-1}}r>0, x_0\in\mathbb{R}^N.
\begin{array}{c}{\displaystyle{\frac{e^{\displaystyle{- \frac{i}{2}\pi \left( 1 + s \right)}}}{{\nu}^{1 + s}8 \pi}}\left( \frac{8{\nu}^{2}}{1 - s}+ \frac{4 \nu}{s}+ \frac{1 - 4{\nu}^{2}}{1 + s}\right)}\\ \end{array}
v(t) = e^{t A_p}a + \int_0^t e^{(t-s)A_p}\left(P_p f(s) + F_p(v(s),\zeta(s))\right) ds, \zeta(t) = e^{t \Delta_{\zeta}}b + \int_0^t e^{(t-s)\Delta_{\zeta}}\left(g(s) + G_q(v(s),\zeta(s))\right) ds.
S = \frac{i}{8 \pi \alpha^{\prime}}\int d \tau d \sigma ( \tilde{\psi}\hat{\partial}\psi - \phi \hat{\partial}\phi - \rho \hat{\partial}\rho - \eta \hat{\partial}\eta + \theta \hat{\partial}\theta + \xi \hat{\partial}\xi + \chi \hat{\partial}\chi + \sigma_{i}\hat{\partial}\sigma_{i}+ \sigma_{i j}\hat{\partial}\sigma_{i j}) .
\mathbb{P}\left(\tilde Y_2< y\right) =1-\tilde G(y)\stackrel{(a)}{\sim}\pi\frac{\lambda}{2}r^2\frac{\delta}{1-\delta}y, y\rightarrow 0 \mathbb{P}\left(\tilde Y_1< y\right) =1-U(y)\sim 1-\tilde G(y),y\rightarrow 0.
\Psi_{\Lambda}({\bf A}) = \exp ({\frac{i}{2 \Lambda}}\int d^{3}x \mathrm{S T r}({\bf A}_{a}\partial_{b}{\bf A}_{c}+{\frac{\sqrt{2}i}{3}}{\bf A}_{a}{\bf A}_{b}{\bf A}_{c}) \tilde{\epsilon}^{a b c}
h(\theta) =\frac{A(e^{i\theta})f(\theta)-\sum\limits_{j=0}^{\infty}(\bold{B}^{-1}\bold{R}\bold{a})_je^{ij\theta}}{f(\theta)+g(\theta)}= &=A(e^{i\theta})-\frac{A(e^{i\theta})g(\theta)+\sum\limits_{j=0}^{\infty}(\bold{B}^{-1}\bold{R}\bold{a})_je^{ij\theta}}{f(\theta)+g(\theta)},
WF^{-\frac12-\epsilon+\tau}(\omega^{+}|_{\cal{Q}})=WF^{-\frac12-\epsilon+\tau}(\rho^*\omega^{+}|_{\cal{Q}})=\rho^*WF^{-\frac12-\epsilon+\tau}(\omega^{+}|_{\cal{Q}})
\tilde{\phi}( \lambda ) \sim i \sqrt{\frac{4 \lambda}{\pi}}e^{- 2 \lambda}K_{\nu}( - 2 \lambda ) \sim \sum_{n \geq 0}( - 1 )^{n}\frac{\Gamma ( \nu + n + 1 / 2 )}{4^{n}n ! \Gamma ( \nu - n + 1 / 2 )}\lambda^{- n}, \ \ \lambda \rightarrow \infty ,
\begin{array}{rcl}V_e/m^2 = \displaystyle +[D(k_1+p)D_m(k_1)+D(k_1)D_m(k_1+p)] & \displaystyle \times[D(k_2+p)D_m(k_2)+D(k_2)D_m(k_2+p)]K_{12}& \displaystyle +D_m(k_2)[D(k_2+p)-D(k_2-p)]D_m(k_1+k_2)D(k_1+k_2+p)K_1, \end{array}
\rho_{L}( q ) = \frac{1}{q^{L - 1}}\left[ \ 1 + \sum_{1 \leq i < j \leq L}( q - 1 )^{j - i}B_{i j}\ + \ \sum_{1 \leq i < j < k < l \leq L}( q - 1 )^{j - i + l - k}\B_{i , j , k , l}^{( 2 )}+ \cdots \right] \,
\hat{H}=\hat{H}_{A}(\hat{a}^{+},\hat{a}^{-})+V(\hat{a}^{+},\hat{a}^{-})\sum_{j}c_{j}\hat{x}_{j}+\sum_{j}[\frac{\hat{p}^{2}}{2}+\frac{1}{2}\hat{x}_{j}^{2}]
|\kappa,\Omega\rangle(z)=\sqrt{A_1(\kappa)}\,\sum_{m=1}^{\infty}V_{m-1}^{(1)}(\kappa)\,\frac{z^m}{m}\equiv\sum_{m=1}^{\infty}\langle\kappa,1|m-1,1\rangle\,\frac{z^m}{\sqrt{m}}.
R_i^{(\alpha,\beta)}(x):= \frac{(\alpha+1)_i}{i!}\sum_{j=0}^{i}\frac{(-i)_j(i+\alpha+\beta+1)_j}{j!(\alpha+1)_j}(1-x)^j (i=0,1,\ldots,n;\;\alpha,\beta > -1),
\mathfrak{g}_{\mathbb{C}}(\Lambda)=_\mathbb{C}\{e_{{\varphi_i}}, e_{{-\varphi_i}}, h_{{\varphi_i}}\}_{i=1}^r\simeq (\mathfrak{sl}_2(\mathbb{C}))^r, \quad \mathfrak{g}_{\mathbb{R}}(\Lambda)=_\mathbb{R}\{x_{{\varphi_i}}, y_{{\varphi_i}}, \sqrt{-1}h_{{\varphi_i}}\}_{i=1}^r\simeq (\mathfrak{sl}_2(\mathbb{R}))^r.
f_n(t) =f_{0,n}+\int_0^t [Q^{+}(s,f_{n-1}(s))-Q^{-}(s,f_{n-1}(s))]ds &+ \int_0^t \biggl[ a\left( \left\| \Lambda f_{n-1}(s)\right\|+\int_0^s\Delta (\tau,f_{n-2}(\tau))d\tau\right) \Lambda f_{n-1}(s) &-a(\left\|\Lambda f_0\right\| )\Lambda f_n(s)\biggr]ds,
u (t) =\check{f}(\hat{\xi}(t_{k}),e(t_{k}))+\Psi \eta (t_{k}) \dot{\hat{\xi}}(t) =A_{o}\hat{\xi}(t)+\lambda e(t_{k})+B\check{f}(\hat{\xi}(t_{k}),e(t_{k})) \dot{\eta}(t) =M \eta (t)+N u (t),~t\in[t_{k},t_{k+1}),~k\in\mathbb{S}
\Theta_{2\kappa +1,2\lambda}^{\circlearrowleft}= \Theta_{2\kappa,2\lambda}^\circlearrowleft + \sum_{r\geq 1}\frac{(\lambda-r+1)r(r+1)(2r+1)}{(\kappa +2+r)(N+1)(N+ 2)^2}\binom{N+2}{\kappa+1}\binom{N+2}{\kappa-r+1}\binom{N+2}{\kappa+r+1}=\frac{\binom{N+1}{\kappa}\binom{N +1}{\kappa}\binom{N+1}{\kappa +1}}{(N+1)}
\pi_n: \prod_{i=0}^{\infty}\frac{1+x_iu}{1-x_iu}\mapsto \prod_{i=1}^{n}\frac{1+ t_iu}{1 + \pmb{z}_i u}=\prod_{i=1}^{n}\frac{1-\pmb{z}_i u}{1-t_iu}.
\left| \frac{1}{N}\sum_{n=0}^{N-1}a_n \right|^2 \leq \frac{N+H}{N^2(H+1)}\sum_{n=0}^{N-1}|a_n|^2 &+ \frac{2(N+H)}{N^2(H+1)^2}\sum_{h=1}^H (H + 1 - h) Re \left( \sum_{n=0}^{N-h-1}a_n \overline{a}_{n+h}\right).
\mathbf{a}_{\mathrm{d}}(\gamma_z) =[1 \quad\exp(j\gamma_z)\quad\cdots\quad\exp(j(R-1)\gamma_z)]^T \mathbf{a}_{\tau}(\eta_z) =[1 \quad\exp(-j\eta_z)\quad\cdots\quad\exp(-j(U-1)\eta_z)]^T
\left\{\begin{array}{ll}-\Delta_p u(z)=\hat{\lambda}|u(z)|^{p-2}u(z) \mbox{in}\ \Omega, \frac{\partial u}{\partial n_p}+\beta(z)|u|^{p-2}u=0 \mbox{on}\ \partial\Omega.\end{array}\right\}
\overline{\left [\begin{matrix}a_{11}1 a_{21}a_{22}\end{matrix}\right ]}=\left [\begin{matrix}a_{11}1 a_{21}a_{22}\end{matrix}\right ]+\left (v^{a_{11}- a_{22}-1}\ \overline{[a_{11}+1]}- v^{-a_{11}+ a_{22}+1}[a_{11}+1]\right )\left [\begin{matrix}a_{11}+1 0 a_{21}-1 a_{22}+1\end{matrix}\right ].
\hat{H}^1=\sum^N_{i=0}\frac{\vec{p}^2_{1i}}{2m_i}=H^1_s+H^1_c=\frac{\vec{p}_{1s}^2}{2M_{N+1}}+\sum_{j=2}^{N+1}\frac{\vec{\pi}_{1j}^2}{2\mu_{1j}}
a(n) - 2\sum_{m= 1}^{\left\lfloor \sqrt{n}\right\rfloor}(-1)^{m-1}a_k(n-m^2) = a(n) - 2\sum_{m= 1}^{\left\lfloor \sqrt{\frac{n}{2}}\right\rfloor}(-1)^{m-1}a_k(n-2m^2),
f(s) =\left(c_{1}\lambda_{1}^{\alpha_1}+c_2\lambda_{2}^{\alpha_2}\right)+\left(c_{1}\alpha_{1}\lambda^{\alpha_{1}-1}+c_{2}\alpha_{2}\lambda^{\alpha_{2}-1}\right)s+... & =f(0)+\sum_{k=0}^{\infty}{a_{k}s^{k+\beta}},
- \Delta w + \frac{5}{3}\left({u_{1}}^{7/3}- u_{2}^{7/3}\right) - \phi_{1}u_{1}+ \phi_{2}u_{2}= 0, &- \Delta \psi + a_{1}^{2}\psi = 4 \pi \left( u_{2}^{2}- u_{1}^{2}\right) + R.
u_x(\xi) = \log \frac{df_*\rho_x}{d\rho_{F(x)}}(f(\xi)) = \exp(B(F(x), \overrightarrow{x\xi}, f(\xi))) u_y(\xi) = \log \frac{df_*\rho_y}{d\rho_{F(y)}}(f(\xi)) = \exp(B(F(y), \overrightarrow{y\xi}, f(\xi)))
w_0 (\sigma) = C_0\exp\Big\{-\Big[\frac{\tau_m}{2}\Big]^{1/2}e^{i\frac{\pi}{12}}\sigma^2\Big\}, \lambda_1 = \sqrt{2\tau_m}e^{i\tau_m\frac{\pi}{12}}+\beta\tau_{m,2}\,,
\frac{\partial \gamma_1^{B_1}}{\partial p_1^{B_1}}=\frac{1}{p_1^{B_1}}\frac{1-\gamma^*\gamma_1^{B_1}}{\gamma^*\gamma_1^{B_1}}(\gamma_1^{B_1})^2\gamma^*=\frac{1}{p_1^{B_1}}(\gamma_1^{B_1}-\gamma^*(\gamma_1^{B_1})^2).
\langle 0|T_{ik}(x)|0\rangle =\lim_{x^{\prime}\rightarrow x}\partial_{i}\partial_{k}^{\prime}\langle 0|\varphi (x)\varphi (x^{\prime})|0\rangle +\left[ \left( \xi -\frac{1}{4}\right) g_{ik}\nabla_{l}\nabla^{l}-\xi \nabla_{i}\nabla_{k}-\xi R_{ik}\right] \langle 0|\varphi^{2}(x)|0\rangle ,
U(t,\xi):=\left(\begin{array}{c}i\langle\xi\rangle \widehat{u}(t,\xi) \partial_{t}\widehat{u}(t,\xi)\end{array}\right), U_{0}(\xi):=\left(\begin{array}{c}i \langle\xi\rangle\widehat{u}_{0}(\xi) \widehat{u}_{1}(\xi)\end{array}\right),
P(X)=Q(X^{-1}-x^{-1})=\sum_{i=0}^{p-1}(-1)^ix^{-i}Q_i+\sum_{j=1}^{p-1}\left(\sum_{i=j}^{p-1}(-1)^{i-j}{i\choose j}x^{-p-i+j}Q_i\right)X^{p-j}.
L_{\mathrm{e f f}}={\frac{f^{2}}{2 \, g_{Y M}^{2}}}\left[ 1 + \sum_{L , m = 1}^{\infty}c_{L m}\left({\frac{g_{Y M}^{2}f^{2}}{x^{7 - p}}}\right)^{L}~ \left({\frac{f^{2}}{x^{4}}}\right)^{m - L}\right]
e^{a}e^{b}\hat{\Omega}_{b , a}^{~ ~ i}= 0 \quad \rightarrow \quad \hat{\Omega}_{b , a}^{~ ~ i}= \hat{\Omega}_{a , b}^{~ i}={\nabla}_{b}E_{a}^{~ \underline{b}}u_{\underline{b}}^{~ i},
\varrho:=\lim_{n\to\infty}\, \frac{1}{n}\log\int_{\mathcal{X}^n}\prod_{m=0}^{n-1}(C(x_m)c^{x_{m+1}})\,\kappa(x_0,\mathrm{d}x_1)\,\kappa(x_1,\mathrm{d}x_2)\,\ldots\,\kappa(x_{n-1},\mathrm{d}x_n)
C = - \frac{4i}{g}\frac{1}{I_2}[B,F] + \Omega \Bigl( \frac{1}{2}a^1 \lambda_1 + \frac{1}{2}a^2 \lambda_2 + \frac{1}{2}a^3 \lambda_3 + \frac{1}{2}a^8 \lambda_8 \Bigr) \Omega^{\dagger}.
\log \nu_t^{(n)}(x) = e^{-\frac{\sigma^2}{2}t}\log \nu_0(x) &- \int_0^t \frac{\sigma^2}{2}e^{-\frac{\sigma^2}{2}(t-s)}\left( \frac{2}{\sigma^2}\frac{\delta F}{\delta \nu}(\nu_s^{(n-1)}, \mu_s^{(n-1)}, x) - \log \pi(x) - \operatorname{D_{KL}}(\nu_s^{(n-1)}|\pi) \right) \mathrm{d}s,
\frac{d \tau}{d \alpha}\langle \int_{\tilde{c}}\frac{d z}{2 \pi i}( \frac{d z}{d w}) J^{--}( z ) ~ \tilde{G}_{0}^{+}\Phi_{D}( \alpha^{- 1}) ~ G_{0}^{+}\Phi_{A}( - \alpha^{- 1}) ~ G_{0}^{+}\Phi_{B}( - \alpha ) ~ \tilde{G}_{0}^{+}\Phi_{C}( \alpha ) \rangle