harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
x_1 = \Phi_1(r,\phi,\theta_1,\ldots,\theta_{d-2}) := r \cos\phi \prod_{k=1}^{d-2}\sin\theta_k, x_2 = \Phi_2(r,\phi,\theta_1,\ldots,\theta_{d-2}) := r \sin\phi \prod_{k=1}^{d-2}\sin\theta_k, x_i = \Phi_i(r,\phi,\theta_1,\ldots,\theta_{d-2}) := r \cos\theta_{i-2}\prod_{k=i-1}^{d-2}\sin\theta_k,\,3\leqslant i\leqslant d.
\begin{cases}1<p<\infty \mbox{when}\ \n=1, \displaystyle{1<p\leqslant \max\left\{p_0(n),1+\frac{2}{n},\frac{1}{2(n-1)}\left(1+\sqrt{4n^2-3}\,\right)\right\}}\mbox{when}\ \n\geqslant 2,\end{cases}
L_{j,\psi_\omega}a_{k_1,k_2,\omega}= i^{-j}\sum_{\mu=0}^{2j}\frac{(\psi_\omega''(0)^{-1}\partial, \partial )^{\mu+j}[ (\psi_\omega-P_{2,0}\psi_\omega )^\mu a_{k_1,k_2,\omega}](0)}{2^{\mu+j}\mu !(\mu+j) !}.
g ( x_{+}) = \left( \begin{array}{c c}{\cos ( \omega x_{+})}&{- \sin ( \omega x_{+})}\\{\sin ( \omega x_{+})}&{\cos ( \omega x_{+})}\\ \end{array}\right) \quad ; \qquad J ( x_{+}) = \left( \begin{array}{c c}{0}&{- \omega}\\{\omega}&{0}\\ \end{array}\right) .
P_{\mu,x}^{n}\left(L_{x}g^{-1},A\right)=\int_{G}{\bf 1}_{A}\left(x,L_{x}g^{-1}s\right)\frac{\varphi_{g^{-1}s}(\pi(x))}{\varphi_{g^{-1}}(\pi(x))}d\mu^{(n)}(s),
A_{m}(\omega,\gamma)=\frac{4}{\sqrt{3\pi}}\frac{1}{|m|!}\frac{\gamma^{2}}{1+\gamma^{2}}\sum_{n>v}^{\infty}e^{-(n-v)\alpha(\gamma)}w_{m}(\sqrt{\frac{2\gamma}{\sqrt{1+\gamma^{2}}}(n-v)})
\mu^\ast(Z|C\sqcap( V|A\sqcup W|B ))=\mu^\ast(Z|C\sqcap V|A\sqcap W|B )+\mu^\ast(Z|C\sqcap V|A\sqcap W|B^\sqsubset)+\mu^\ast(Z|C\sqcap V|A^\sqsubset\sqcap W|B ).
|2v-v'-v'_*|^2 = |v - v'|^2 + |v - v'_*|^2 = |\bar{v}- \bar{v}'_*|^2 + s^2, \frac{|v-v'|}{|2v-v'-v'_*|}= \frac{s}{\sqrt{|\bar{v}- \bar{v}'_*|^2 + s^2}}.
\begin{aligned}\omega N_{1}<p_{1}\leq N_{1},\p_{2},p_{3}\sim U_{1},\p_{4},p_{5}\sim V_{1}, &\omega N_{2}<p_{6}\leq N_{2},\p_{7},p_{8}\sim U_{2},\p_{9},p_{10}\sim V_{2},\v_{j}\leq L\end{aligned}
[\bar{\xi}_{0,0}, \bar{x}^{\pm}_{0,s}]=0,\ [\bar{\xi}_{0,1}, \bar{x}^{\pm}_{0,s}]=0,\ [\bar{\xi}_{0,2}, \bar{x}^{\pm}_{0,s}]=\mp 2\beta^2\bar{x}^{\pm}_{0,s},
\left[ \hat X_{2,q},\;\hat P_{2,q}\right] = i\hbar - (1-q^2)\left(\frac{i\hbar}{2(\lambda_1 +\lambda_2)^2}\right) \left(\lambda_1 K_2 \tilde A_2^\dagger \tilde A_2 + \lambda_2 K_1 \tilde A_1^\dagger \tilde A_1 \right);
D_xf_B(S_A) = f_B(S_A+D_xS_A)-f_B(S_A) & = \sum_{k=1}^{D_xS_A}\big(f_B(S_A+k)-f_B(S_A+k-1)\big) & = \sum_{k=1}^{D_xS_A}\Delta f_B(S_A+k-1) & = \Delta f_B(S_A) D_xS_A + \sum_{k=2}^{D_xS_A}\big(\Delta f_B(S_A+k-1)-\Delta f_B(S_A)\big).
U_{j}=\frac{1}{\alpha}\left[\left(\frac{a-1}{a-b}\right)(X.Y)^{a-2}-\left(\frac{b-1}{a-b}\right)(X.Y)^{b-2}\right]\times N &V_{j}=\frac{1}{\alpha}\left[\left(\frac{a-1}{a-b}\right)(X.Y)^{a-2}-\left(\frac{b-1}{a-b}\right)(X.Y)^{b-2}\right]\times M
\begin{aligned}(\tilde{\Delta}_E(\tilde{h}_1+\tilde{h}_2+\tilde{h}_3),\tilde{h}_1+\tilde{h}_2+\tilde{h}_3)_{L^2(\tilde{g})}\geq C(n)\left(\left\|\tilde{h}_1\right\|^2_{L^2(\tilde{g})}+\left\|\tilde{h}_2\right\|^2_{L^2(\tilde{g})}+\left\|\tilde{h}_3\right\|^2_{L^2(\tilde{g})}\right) &=C(n)\left\|\tilde{h}\right\|^2_{L^2(\tilde{g})}.\end{aligned}
R^{(2)}(d) =-\psi^{(1)}(d)+(-\psi(d)+\log(r-s))^{2}, R^{(3)}(d) =-\psi^{(2)}(d)-3(-\psi(d)+\log(r-s))\psi^{(1)}(d)+(-\psi(d)+\log(r-s))^{3},
\nabla_x H(t,x,\psi,\operatorname{p}^1\sharp\nu,u) = \psi\nabla_xf(t,x,\operatorname{p}^1\sharp\nu,u)-\nabla_xf_0(t,x,\operatorname{p}^1\sharp\nu,u), \nabla_\psi H(t,x,\psi,\operatorname{p}^1\sharp\nu,u) =f(t,x,\operatorname{p}^1\sharp\nu,u),
\begin{aligned}\pi^{a_{\psi_0}+1}P_{\psi_0}+ IP_{\psi_0}\subset Q_{\psi_0}\subset\pi^{a_{\psi_0}}P_{\psi_0}+ IP_{\psi_0}, Q_{\psi}= \pi^{a_{\psi}}P_{\psi}+ IP_{\psi}, \psi \neq \psi_0. \end{aligned}
(0~0~0~0~0~0~0~-\frac{7}{2}) = [{\bf 1}_B \oplus{\bf \overline{8}}_F \oplus{\bf \overline{28}}_B \oplus{\bf \overline{56}}_F \oplus{\bf 70}_B \oplus{\bf 56}_F \oplus{\bf 28}_B \oplus{\bf 8}_F \oplus{\bf 1}_B].
\left \{{\partial^2 \over{\partial r^2}}+{5 \over r}{\partial \over{\partial r}}+\tilde\omega^2 \left (1 +{{\tilde R_{\pi\over 2}^4}\over r^4}\right ) -{{2 R_{\pi\over 2}^8}\over{r^{10}f^2}}\right \}\hat \phi = 0.
G ( x , x^{\prime}) = \frac 1{( 2 \pi )^{3}}\sum_{n = - \infty}^{\infty}e^{i n ( \theta - \theta^{\prime})}\int_{- \infty}^{\infty}d k e^{i k ( z - z^{\prime})}\int_{- \infty}^{\infty}d \omega e^{i \omega ( \tau - \tau^{\prime})}g_{n}( r , r^{\prime}) \ .
\{\tilde\Theta_\alpha (q,\phi ) ,\tilde\Theta_\beta (q,\phi )\}=0~~;~~\tilde\Theta_\alpha (q,\phi )=\Theta_\alpha (q)+\Sigma_{n=1}^\infty \tilde\Theta^{(n)}_\alpha (q,\phi )~~;~~\tilde\Theta^{(n)}\approx O(\phi^n)
\begin{cases}x_{k+1}=A_1x_k+A_2\mathcal{E}x_k+Bu^F_k \ \ \ \ \ \ \ \ \ \ +(C_1x_k+C_2\mathcal{E}x_k+Du^F_k)w_k, x_0=\xi\in{\mathcal R}^n, \k\in{\mathcal N}_{T-1},\end{cases}
g_{2/3, 3/4}^2 (w_2) = \frac{n_0}{4 k_{\ell}}\log\left( 1 + \frac{w_2 P'}{3}\right) - \frac{n_0}{8 k_{\ell}}\log \left( 1 + \frac{2 w_2 P'}{3}\right) - \frac{3 \ell}{4 k_{\ell}}H_2 \left(\frac{w_2}{\ell}\right) .
f^1 =\frac{1}{m}(q\varepsilon^{12}+ r \varepsilon^{13}),~~~f^2 = \frac{1}{m}(-p\varepsilon^{12}+ r \varepsilon^{23}), ~~~f^3 = \frac{-1}{m}( p\varepsilon^{13}+ q \varepsilon^{23})
\underline h(t):=(K_1t+\theta)^{1/({\gamma}-1)},\ \t\geq 0, &\underline U(t,x):=K_2\frac{\underline h(t)-|x|}{\underline h(t)}{\Theta}, \ \ \t\geq 0,\x\in [-\underline h(t), \underline h(t)],
\begin{array}{l}\partial^{\mathbf{A}}_c p{_a}^{jc}= \partial_cp{_a}^{jc}+ \mathbf{A}{^k}_c c^j_{k\ell}p{_a}^{\ell c}\partial^{\mathbf{A}}_cp{_i}^{jc}= \partial_c p{_i}^{jc}- \mathbf{A}{^k}_c c^\ell_{ki}p{_\ell}^{jc}+ \mathbf{A}{^k}_c c^j_{k\ell}p{_i}^{\ell c}\end{array}
\bar J_{i}(u_i,f) =\{\mathbb E}\int_0^T \left\{| x_i-(\Gamma m_i +\eta)|^2_Q+u_i^T R u_i -\frac{1}{\gamma}|f(t)|^2 \right\}dt+{\mathbb E}x_i^T(T)Hx_i(T).
x'(t) =s-dx(t)-\frac{\beta x(t)v(t)}{(1+ax(t))(1+b v(t))}, y'(t) = \frac{\beta x(t-\tau)v(t)e^{-s\tau}}{(1+ax(t-\tau))(1+b v(t-\tau))}-ay(t)-py(t)z(t), v'(t) = ky(t)-u v(t), z'(t) = cy(t)z(t)-b z(t).
\begin{aligned}\frac{d}{dt}M(v_\lambda) = \ 2 \Re ( \partial_t v_\lambda \cdot \bar v_\lambda) = \ 2 \Im (\partial_x g_{[<\lambda]}\partial_x v_\lambda \cdot \bar v_\lambda) + 2 \Im ( f_\lambda \bar u_\lambda) = \ - 2 \partial_x [g_{[<\lambda]}\Im (\partial_x u_\lambda \cdot \bar u_\lambda)] + 2 \Im ( f_\lambda \bar u_\lambda)\end{aligned}
\rho(\alpha,\beta)=\frac{1}{Z}\sum_{n=0}^{\infty}e^{-n\hbar\beta\omega}D(\alpha)|n\rangle\langle n|D^{\dagger}(\alpha)=\frac{1}{Z}\sum_{n=0}^{\infty}e^{-n\hbar\beta\omega}|\alpha,n\rangle\langle\alpha,n|,
\int_{0}^{t}X\mathcal{Z}_{i,j}d\tilde{W}_{k,m}= \Big( \frac{\emph{q}_{m,m}^{k,k}\emph{q}_{j,j}^{i,i}}{\lambda_{j}}\Big)^{\frac{1}{2}}\mathcal{O}(\varepsilon^{-r}),
\int d^{3}x \; \frac{1}{2 \pi q_{I}}\varepsilon^{\mu \nu \lambda}\partial_{\mu}\bar{\omega}_{I}\partial_{\nu}H_{\lambda}^{I}= \frac{1}{q_{I}}\int d^{3}x \; K_{I}^{\mu}H_{\mu}^{I}\; \; ,
v_0 =z_{(0)}, \v_1=\frac{1}{2}(s_2z_{(0)}+ cx_{(0)}), \v_{m+2}=s_2v_{m+1}-v_m w_0 =z_{(1)}, \w_1=\frac{1}{2}(tz_{(1)}+ cy_{(1)}), \w_{n+2}=tw_{n+1}-w_n.
V_{y}={\frac{{x^{7}}}{3}}- 3 \,{x^{6}}\,{y^{2}}+ 5 \,{x^{5}}\,{y^{4}}- 6 \,{x^{3}}\,{y^{5}}+{\frac{7 \, x \,{y^{6}}}{3}}= 0
W(X,Y)Z =R(X,Y)Z+YL(X,Z)-XL(Y,Z)+ \stackrel{*}{L}^{\sharp}(Y)g(X,Z)-\stackrel{*}{L}^{\sharp}(X)g(Y,Z) & =K\{Xg(Y,Z)-Yg(X,Z)\}+\frac{K}{2}Yg(X,Z)-\frac{K}{2}Xg(Y,Z)+ & +\frac{K}{2}Yg(X,Z)-\frac{K}{2}Xg(Y,Z) & =0
h(va) = h(v)a; h \left( v \sum_{i=0}^{p-1}\vartheta^{i}* \mu_{g_{i}}\right) = h(v) \sum_{i=0}^{p-1}\vartheta^{i}* \mu_{g_{i}}; \intertext{by using (\ref{derivacionAccion}),}h \left( \sum_{i=0}^{p-1}r^i(v) \mu_{g_{i}}\right) = \sum_{i=0}^{p-1}(r')^i h(v) \mu_{g_{i}}.
\omega^{(1)}= u_\beta x^\beta, \omega^{(2)}= D^{\ell_2}(u-u_0) \mbox{with}\, 0 \le \ell_2 \le \ell+4, \omega^{(j)}\in \left\{1, u_0, D^{\ell_j}(u-u_0)\right\}\mbox{with}\, 0 \le \ell_j \le \frac{\ell+4}{2}\, \mbox{for}\, j \ge 3.
\| \Xi\| =\left\Vert T (T^{1/2}C T^{1/2}+ \lambda I)^{-2}T\right\Vert &= \sup_{i}\frac{\mu_i^2}{(\mu_i\xi_i+ \lambda)^2}\lesssim \sup_i \frac{i^{-2t}}{(i^{-(t+c)}+\lambda)^2}= \left[ \sup_i \frac{i^{-t}}{i^{-(t+c)}+\lambda)}\right]^2 &\leq \left[ \lambda^{\tfrac{t -(t+c)}{t+c}}\right]^2 = \lambda^{-\tfrac{-2c}{t+c}},
V ={x^{7}}\,{y^{7}}-{x^{5}}\,{y^{5}}\, \left({x^{3}}+{y^{3}}\right) +{x^{3}}\,{y^{3}}\, \left({x^{6}}+{y^{6}}\right) + x \, y \, \left({x^{9}}+{y^{9}}\right) \,
\omega^{(2)}= \left(\omega^{(2)}- \sum_{\beta \le i_2 < \alpha^\prime+N-m-1-\beta}\omega^{(2)}_{i_2}x^{i_2}\right) + \sum_{\beta \le i_2 < \alpha^\prime+N-m-1-\beta}\omega^{(2)}_{i_2}x^{i_2},
S=\int d^6 x \sqrt{g}\left[{1\over 2}(\partial\phi)^2 +{1\over 2}m^2 \phi^2 +{g_4\over 4!}\phi^4+\cdots\right]+ \int d^4 x \sum_{n=0}^\infty{\lambda_{2n}\over (2n)!}\phi^{2n}
\mathcal{I}(t_n u_n) \geq \mathcal{I}\left(\kappa \frac{u_n}{\|u_n\|}\right) = \mathcal{I}(\kappa w_n) &\geq (\kappa)^{p_2^-}\frac{1}{2}\rho(w_n)-\int_{\Omega}F(x,\kappa w_n)dx &=2\mu. \frac{1}{2}\rho(w_n)-\int_{\Omega}F(x,\kappa w_n)dx = \mu + o_n(1).
(1^{\alpha_0},h_{i_1}^{\alpha_1},h_{i_2}^{\alpha_2},\ldots,h_{i_k}^{\alpha_k}) := (\underbrace{1,\ldots,1}_{\alpha_0},\underbrace{h_{i_1},\ldots,h_{i_1}}_{\alpha_1},\ldots,\underbrace{h_{i_k},\ldots,h_{i_k}}_{\alpha_k}).
\chi_{-\frac{1}{2}}^n|0\rangle = H^{\gamma}_{(-n)}\dots H^{\gamma}_{(-2)}H^{\gamma}_{(-1)}|0\rangle \chi_{-\frac{3}{2}}^n|0\rangle = H^{\beta}_{(-n)}\dots H^{\beta}_{(-2)}H^{\beta}_{(-1)}|0\rangle.
U_5(FZ t^i)=F\Big((\sum_{j=\left\lceil \frac{i+1}{5}\right\rceil}^\infty y_0(i,j)t^j)+\rho(\sum_{j=\left\lceil \frac{i+1}{5}\right\rceil}^\infty y_1(i,j)t^j) \Big),
\tau(w) = \tau(x_{j_1}^* x_{i_2}\cdots x_{j_{k-1}}^* x_{i_k}\, d_i^* d_i) &= \tau(x_{j_1}^* x_{i_2}\cdots x_{j_{k-1}}^* x_{i_k}) - \tau(x_{j_1}^* x_{i_2}\cdots x_{j_{k-1}}^* x_{i_k}\, x_i^* x_i) &= 0 - \tau(x_{i_1}x_{j_1}^* \cdots x_{i_k}x_{j_k}^*) = 0 \; ,
k^{-2}\sup_{B(p,k^{-1})}|\overline{\partial}\chi_{k}|^2e^{-k\varphi}\lesssim \frac{1}{k(\log k)^2}\sup_{B(p, k^{-1})}\large|\chi'(\frac{\sqrt{k}d_{X}}{\log k})\large|^2 e^{-k\pi (d_{X})^2}= O(k^{-\infty}),
|f_{\omega,1}^{n+1}(z,w)| \le\begin{cases}|f_{\omega,1}^n(z,w)| \textrm{if}f_n=f; |f_{\omega,1}^n(z,w)|\left(1+|f^n_{\omega,2}(z,w)|\right) \textrm{if}f_n=g;\end{cases}|f_{\omega,2}^{n+1}(z,w)| =\begin{cases}\frac{1}{2}|f_{\omega,2}^n(z,w)| \textrm{if}f_n=f; |f_{\omega,2}^n(z,w)| \textrm{if}f_n=g.\end{cases}
J_k(\varphi)\,:=\,\frac{1}{p}\int_{\mathbb{R}^{2N}}\frac{|\varphi(x)-\varphi(y)|^p}{|x-y|^{N+ps}}\,dx\,dy-\int_{\mathbb{R}^{N}}f(x)\Phi_{k}(\varphi)\,dx\qquad\varphi\in W^{s,p}_0(\Omega)\,.
C(\Delta) :=\sup_{\xi\in\Delta}|c(\xi)|=\max_{s\in\{0,\infty\}}\left(\limsup_{t\to s}|c(t)|\right), D(\Delta) :=\sup_{\xi\in\Delta}|d(\xi)|=\max_{s\in\{0,\infty\}}\left(\limsup_{t\to s}|d(t)|\right).
\varphi(\widehat{F})(k_1,k_2)(x_1,(a_2,x_2)) = \widehat{F}(k_1,k_2)(\kappa_{x_1,(a_2,x_2)}) = \widehat{F}(k_1,k_2)({}^{x_1}a_2 F(x_1,x_2)) = \widehat{F}(k_1,k_2)({}^{x_1}a_2) \ \widehat{F}(k_1,k_2)(F(x_1,x_2)).
\mathrm{div}_g X_{(q)}= \frac{1}{2}(2q-1)! \ \mathrm{div}_g \left( T_{(2q-1)\beta}\cdot e_{\beta}^{\top}\right) = \frac{1}{2}(2q-1)! \big[ \mathrm{div}_g \left( T_{(2q-1)\beta}\right) \cdot e_{\beta}^{\top}+ T_{(2q-1)\beta}\cdot \nabla e_{\beta}^{\top}\big].
\eta^{1/2}[(\breve\xi^{rst}-\xi^{rst})l_{rs}l_t-(\breve\xi^{rs}-\xi^{rs})l_rl_s-\{(\breve\xi^{tuv}-\xi^{tuv})\lambda_{tu,v}+(\breve\xi^{tu}-\xi^{tu})\lambda_{tu}\}\tau^{rs}l_rl_s]=O_p(n^{-1}).
\begin{aligned}E(\rho_2) \leq (\sigma\gamma_0)\delta^{1/\sigma}\rho_2^{1/\sigma}\cdot\rho_2^{1-1/\sigma}\ln \rho_2+C(\alpha,\sigma)\rho_2^\alpha\ln\rho_2 &= (\sigma\gamma_0)\delta^{1/\sigma}\rho_2\ln \rho_2+C(\alpha,\sigma)\rho_2^\alpha\ln\rho_2\,.\end{aligned}
\widetilde \Pi^{\prime}= \left( \begin{matrix}{\widetilde F_{0}^{\prime}}\\{\widetilde F_{1}^{\prime}}\\{\widetilde X^{0}}\\{\widetilde X^{1}}\\ \end{matrix}\right) = \left( \begin{matrix}{\widetilde X_{0}}\\{\widetilde F_{1}}\\{- \widetilde F_{0}}\\{\widetilde X^{1}}\\ \end{matrix}\right) \ .
\sum_{n= \frac{1}{2}}^{R/2}~\frac{\triangle k_n^+}{k_n^+}~ f( k^+_n ) = \sum_{n= \frac{1}{2}}^{R/2}~\frac{\triangle k_n^+}{k_n^+}~ e^{- \sigma^2 / ( \Lambda' k_n^+ )}
g^{2}\, \Phi_i^2 ={m(z=h_i;\mu)\over a'(z=h_i)}={\displaystyle\prod_{j=1}^{N-1}(h_{i}-\mu_{j})\over\displaystyle\prod_{j=1,j\neq i}^{N}(h_{i}-h_{j})}\,\cdotp
w_1^\eta\Big( \int w_2^x\big(h_N\|z_2\|) K(z_2)dz_2\Big)+ \int w_1^x(h_N\|z_1 \|) K(z_1)dz_1 &\; = w_1^\eta \left(\ell h_N^{\alpha_2}\int \|z_2\|^{\alpha_2}K(z_2)dz_2\right) + \ell h_N^{\alpha_1}\int\|z_1 \|^{\alpha_1}K(z_1)dz_1 &\; = \ell \left(\ell h_N^{\alpha_2}m_{\alpha_2}(K)\right)^{\alpha_3}+ \ell h_N^{\alpha_1}m_{\alpha_1}(K)
\delta_{c_1}\circ \delta_{c_2}= \alpha_1\delta_{c_1}+ \alpha_2 \delta_{c_2}+ \alpha_3\delta_{c_3}+ \alpha_4\delta_{c_4}, &\delta_{c_2}\circ \delta_{c_1}= \alpha_1\delta_{c_1}+ \alpha_2 \delta_{c_2}+ \alpha_4\delta_{c_3}+ \alpha_3\delta_{c_4},
f(sq) \ = \ \frac{(md+2)(d+1)}{2}+\frac{(ms+2)(s+1)}{2}-\frac{(m\delta+2)(\delta+1)}{2}-\frac{(m\sigma+2)(\sigma+1)}{2}& \ = \ \frac{1}{2}m(d(d+1)+s(s+1)-\delta(\delta+1)-\sigma(\sigma+1)) & \ = \ \frac{1}{2}(d^2+s^2-\delta^2-\sigma^2)m & \ = \ (\sigma \delta - sd)m.
E_{_{a}}^{\mu}= \left( \begin{array}{c c c}{1}&{0}&{0}\\{0}&{\frac{\beta}{\rho}\cos \phi}&{- \frac{1}{\rho}\frac{\alpha}{\sin \alpha \theta}\sin \phi}\\{0}&{\frac{\beta}{\rho}\sin \phi}&{\frac{1}{\rho}\frac{\alpha}{\sin \alpha \theta}\cos \phi}\\ \end{array}\right) .
K = M_1 \otimes M_2 K(u,v) = M_1(r_1, s_1) M_2(r_2, s_2)u, v, r_i, s_i \in \mathbb{Z}_p u \equiv r_1 p + r_2 v \equiv s_1 p + s_2 (F_p \otimes F_p) (u,v) = \omega_p^{r_1 s_1}\omega_p^{r_2 s_2}
\Delta(A) = \left\{\sum_{i=0}^n t_i a_i : t_i \geq 0 i=0,1,\ldots, n \sum_{i=0}^n t_i = 1 \right\}& = a_0 + \left\{\sum_{i=1}^n t_i (a_i - a_0) : t_i \geq 0 i= 1,\ldots, n \sum_{i=1}^n t_i \leq 1 \right\}.
\lim_{\xi \to \{\xi_\beta^2\}^+}\phi_{\beta,\gamma}'(\xi)=\lim_{\xi \to \{\xi_\beta^2\}^+}\frac{z\left(\phi_{\beta,\gamma}(\xi)\right)}{D\left(\phi_{\beta,\gamma}(\xi)\right)}=\lim_{s\to \beta^-}\frac{z(s)}{D(s)}=\lim_{t\to \gamma^+}\frac{w(t)}{D_3(t)},
f'(\lambda) = \lambda \|\nabla \Theta\|^2_{L^2}-\frac{3(p-1)}{2(p+1)}\lambda^{\frac{3(p-1)}{2}-1}\|\Theta\|^{p+1}_{L^{p+1}}+ \frac{3(q-1)}{2(q+1)}\lambda^{\frac{3(q-1)}{2}-1}\|\Theta\|^{q+1}_{L^{q+1}}= \frac{G(\Theta_\lambda)}{\lambda}
\|(-A_n)^{\frac{1}{2}}(F_n(a)-F_n(b))\|^2_{l_n^2}=\frac{n}{\pi}\sum_{j=1}^n|f(a_j)-f(b_j)-f(a_{j-1})+f(b_{j-1})|^2 \le\frac{2n}{\pi}\sum_{j=1}^n|a_j^3-b_j^3-a_{j-1}^3+b_{j-1}^3|^2+2\|(-A_n)^{\frac{1}{2}}(a-b)\|^2_{l_n^2}.
{\frac{2}{g^{2}}}\sin \alpha_{j}+{\frac{4 \sin \alpha_{j}}{m^{4}+ 4 m^{2}+ 4 \cos^{2}({\frac{\alpha_{j}}{2}})}}= \sum_{k \neq j}\cot \left|{\frac{\alpha_{j}- \alpha_{k}}{2}}\right| .
-z(z^{-1})'' = - \frac{9}{4}{\cal H}^2 + \frac{9}{2}{\cal H}' + \frac{\varphi'''}{\varphi'}- 2 \biggl(\frac{\varphi''}{\varphi'}\biggr)^2 - 2 \frac{{\cal H}'}{\cal H}\frac{\varphi''}{\varphi'}- 3{\cal H}\frac{\varphi''}{\varphi'}-\frac{{\cal H}''}{\cal H}.
\prod_{\substack{\gcd(j_1,j_2,j_3,j_4,k)=1 j_1,j_2,j_3,j_4<k j_1,j_2,j_3,j_4\geq 1; k \geq 2}}\left(\frac{1}{1-y^{j_1+j_2+j_3+j_4}z^k}\right)^{\frac{1}{k}},
\mathrm{usp}_S(T^{-1},[f])= \mathrm{usp}_S(T^{-1},[\sum_{v\in I}\alpha(v)\phi v]) \leq \sup_{v\in I}\mathrm{usp}_S(T^{-1},[\alpha(v)\phi v]) = \sup_{v\in I}\mathrm{usp}_S(T^{-1},[\phi v]).
(\vec{B}\cdot\vec{x})^2+(\vec{C}\cdot\vec{x})^2+(\vec{D}\cdot\vec{x})^2 =(\alpha b)^2+(\alpha c_1+\beta c_2)^2+(\gamma d)^2 &=(b^2+c_1^2-d^2)\alpha^2+(c_2^2-d^2)\beta^2+2c_1c_2\alpha\beta+d^2.
\mathrm{~ e}^{- S_{\mathrm{e f f}}[ A_{0}]}\equiv \int [ d \vec{a}] [ d \vec{\pi}] \mathrm{~ e}^{\int_{0}^{1 / T}d \tau \int_{S^{2}}\left\{i \vec{\pi}\cdot \dot{\vec{a}}- \frac{1}{2}( \pi_{i}- \frac{\kappa}{2}\epsilon_{i j}a_{j})^{2}- \frac{1}{2}( B_{M}^{2}+ b^{2}) \right\}+ i \frac{\kappa}{T}\int_{S^{2}}A_{0}\left( B_{M}+ b / 2 \right)}
C_k=\lim\limits_{r\to+\infty}\frac{f_k}{u_0}=\lim\limits_{r\to+\infty}\langle \frac{u(r,\theta)}{u_0(r)}, \varphi_k \rangle &=\langle \lim\limits_{r\to+\infty}\frac{\bar u(r,\theta)}{u_0(r)}, \varphi_k \rangle=\lim\limits_{r\to + \infty}\frac{\bar f_k}{u_0}=\bar C_k.
F(\eta_{E_0,F_0},\eta_{E_0,E_1},\eta_{E_1,F_1},\eta_{F_1,F_0},\eta_{E_0,E_2},\eta_{E_2,F_2},\eta_{F_2,F_0},\eta_{E_0,E_3},\eta_{E_3,F_3},\eta_{E_3,F_0}) = F_1(X_0,X_1,X_2,X_3),F_1(x,y,z,t)=F(x,y,y,y,z,z,z,t,t,t).
V(z) = (I_2-Az)^{-1}(I-Bz)T^{-1}&= (\underbrace{\left( z-\alpha_+ \right) \left( z-\alpha_- \right)}_{=a(z)})^{-1}\underbrace{(w, \overline{w}) \begin{pmatrix}z - \alpha_- 0 0 z- \alpha_+\end{pmatrix}(w, \overline{w})^{-1}(I-Bz)T^{-1}}_{=b(z)}= a^{-1}(z)b(z).
\left\Vert z\right\Vert_{X^{\alpha/2}}=\left\Vert z\right\Vert_{L^{2}\left(\Omega\right)}+[z]_{n,\alpha}=\left\Vert z\right\Vert_{L^{2}\left(\Omega\right)}+\left( \int\limits_{Q}\frac{\left\vert z\left( x\right)-z\left( y\right) \right\vert^{2}}{\left\vert x-y\right\vert^{n+\alpha}}dxdy\right)^{1/2}.
\phi_1-\frac{\phi_1^7}{7}+3 \phi_1^5 \phi_2^2- 5 \phi_1^3 \phi_2^4+ \phi_1 \phi_2^6=\gamma;\hspace{0.4cm}\phi_2 - \phi_1^6 \phi_2 +5 \phi_1^4 \phi_2^3 -3\phi_1^2 \phi_2^5+\frac{\phi_2^7}{7}=\gamma_\perp
V_{1/2}(x) := -\frac{1}{\pi}\left( \frac{1}{\sqrt{1+x^2}}- \frac{2|x|\,\mathrm{arcsinh}|x|}{1+x^2}\right) \mbox{and}u_{1/2}(x) :=\frac{1}{\sqrt{1+x^2}},
G_{F}( x , y ) = e^{-{\frac{i e^{2}}{2}}\int d^{2}z_{1}\int d^{2}z_{2}j^{\mu}( z_{1}; x , y ) G_{\mu \nu}( z_{1}, z_{2}) j^{\nu}( z_{2}; x , y )}S_{F}( x , y ) ,
U = \hat{U}_{1 2}\cdot \cdot \cdot \hat{U}_{1 n}\hat{U}_{2 3}\cdot \cdot \cdot \hat{U}_{2 n}\cdot \cdot \cdot \hat{U}_{i , i + 1}\cdot \cdot \cdot \hat{U}_{i n}\cdot \cdot \cdot \hat{U}_{n - 1 , n},
p^{\nu_2}(2,t)=-\frac{\lambda_1}{\lambda_2}\sum_{k=1}^{\infty}(-1)^k\underset{\Lambda^{k}_{2}}{\sum}\frac{\lambda_1^{k_1}\lambda_2^{k_2}t^{k_1\nu_1+k_2\nu_2}}{\Gamma\left(k_1\nu_1+k_2\nu_2+1\right)},
\mathbb{E}[||U||_p^p] = \int_{\Omega}||U_{\omega}||_p^p\mathbb{P}(d\omega) = \int_{L^p(\mathcal{D})}||f||_p^p\mu(df) = \int_{L^p(\mathcal{D})}||f||_p^{2\times p/2}\mu(df) \geq \bigg(\int_{L^p(\mathcal{D})}||f||_p^{2}\mu(df)\bigg)^{p/2}\geq \nu(\mathcal{E}_k)^{p/2}
\begin{aligned}H(t,x,y,z,\tilde{z},\gamma,\mathcal{Z},u;p,q,m,\tilde{m},n,\tilde{M}):=g(x,y,z,\tilde{z},\gamma,u)p+\tilde{b}(x,u)q+\sigma(x,u)m &+\tilde{\sigma}(x,u)\tilde{m}+\int_{\mathcal{E}}l(x,u,e)n_{e}\nu(de)+e^{-\frac{\beta}{2}t}h(x,u)\tilde{M}\mathcal{Z}+f(x,y,z,\tilde{z},\gamma,u)\mathcal{Z}.\end{aligned}
\cos(\tau \mathcal{A}_\varepsilon^{1/2}) = T_{\varepsilon}^* \cos(\varepsilon^{-1}\tau \mathcal{A}^{1/2}) T_{\varepsilon},\mathcal{A}_\varepsilon^{-1/2}\sin(\tau \mathcal{A}_\varepsilon^{1/2}) = \varepsilon \, T_{\varepsilon}^* \mathcal{A}^{-1/2}\sin(\varepsilon^{-1}\tau \mathcal{A}^{1/2}) T_{\varepsilon}.
\frac{1}{r}\frac{d}{dr}[r\frac{dv}{dr}]=\frac{1}{r}[r(\frac{dv}{dr})^{\prime}+(r)^{\prime}\frac{dv}{dr}]=\frac{1}{r}[r\frac{d^{2}v}{dr^{2}}+\frac{dv}{dr}]=\frac{d^{2}v}{dr^{2}}+\frac{1}{r}\frac{dv}{dr}.
E_{i - 1}\triangleright E_{\xi_{i}}= E_{i - 1}E_{\xi_{i}}- K_{i - 1}E_{\xi_{i}}K_{i - 1}^{-1}E_{i - 1}& = E_{i - 1}E_{\xi_{i}}- q^{-1}E_{\xi_{i}}E_{i - 1}= - E_{\xi_{i - 1}}.
k_4=\frac{(1-k_1)+\sqrt{(1-k_1)^2-4C}}{2},\ \k_5=\frac{1}{2}+\frac{k_1}{2}-\frac{k_2}{2k_3}+\sqrt{\left[\frac{1}{2}+\frac{k_1}{2}-\frac{k_2}{2k_3}\right]^2-\left[\frac{A}{k_3^2}+\frac{B}{k_3}+C\right]},
{\cal A}= 6 4 \cdot 9 6 \pi^{3}\lim_{\rho \rightarrow 0}\rho^{4}\int_{0}^{r}x_{0}^{3}d x_{0}\int_{0}^{\pi}{\frac{\sin^{2}\theta \, \, d \theta}{( x^{2}+ x_{0}^{2}+ \rho^{2}- 2 x x_{0}\cos \theta )^{4}}}.
\begin{cases}U_i''+c_i U_i'+f(U_i)=0,\; U_i(-\infty) = p_{i-1}, \ \U_i(+\infty) = p_{i}\quad\ \ (i=1,2,\ldots,n), c_1 \leq c_2 \leq \cdots \leq c_n,\end{cases}
\hat{\phi}( x^{0}\, , \, x^{1}) \, = \, \frac{1}{\sqrt{2 \pi}}\left( \frac{\pi}{R}\right) \sum_{n \geq 0}\frac{1}{\sqrt{\omega_{n + \frac{1}{2}}}}\left[ \hat{\Lambda}_{n}e^{- i \omega_{n + \frac{1}{2}}( x^{0}+ x^{1})}\, + \, \hat{\Lambda}_{n}^{\dagger}e^{i \omega_{n + \frac{1}{2}}( x^{0}+ x^{1})}\right] ,
A[< 0]^{\perp_{f}}= \{Y \in \mathcal{R}(A) \mid \mathrm{Hom}_{\mathrm{D}(A)}(A[< 0],X) = 0\}&=\{Y \in \mathcal{R}(A) \mid \mathrm{H}^{i}\mathrm{RHom}_{A}(A,X) = 0~\textrm{for}~i>0\}&=\{Y \in \mathcal{R}(A) \mid \mathrm{H}^{i}X = 0~\mathrm{for}~i>0\}.
\int_{\overline{M}_{g,n}}\frac{c(E^{*})}{(1-k_{1}\psi_{1})\cdot\cdot\cdot(1-k_{n}\psi_{n})}=\int_{\overline{M}_{1,1}}\frac{1-\lambda_{1}}{1-k_{1}\psi_{1}}=[\int_{\overline{M}_{1,1}}\psi_{1}]k_{1}-[\int_{\overline{M}_{1,1}}\lambda_{1}].
\varphi_n (g_1,\ldots,g_s)^T=((g_1(\omega),\ldots,g_s(\omega))^T/{m^n}) \bmod 1= =(A\ (f_1(\omega),\ldots,f_s(\omega))^T/{m^n}) \bmod 1=(A\ \varphi_n (f_1,\ldots,f_s)^T)\bmod 1.
\frac{\Gamma(\frac{k+1}{2})\Gamma(\lfloor \frac k2 \rfloor +1)}{\Gamma(\frac{k}{2})\Gamma(\lfloor \frac k2 \rfloor +\frac 32)}=\frac{\Gamma(\frac{k}{2}+\frac 12)^2}{\Gamma(\frac{k}{2})\Gamma(\frac{k}2 +1)}\le 1.
\int^{T}_{0}\int_{\mathbb{R}^{3}}\left(z(\partial_{t}\varphi+\Delta\varphi+u^{c,\gamma}\cdot\nabla\varphi)+(w_{1}\otimes w_{2})\cdot\nabla\varphi-(z\cdot\nabla)u^{c,\gamma}\varphi\right)dxdt=0.
B(\theta) =2\pi \frac{d}{dE}{\rm Ai}^{(k)}(E \Omega ){\rm Ai}^{(k)}(E \Omega^{-1}),{\rm e}^{-A(\theta)}= -4\pi \cos \frac{\pi}{k+2}\frac{d}{dE}{\rm Ai}^{(k)}(E)^2.
\left[\begin{array}{cc}S T \end{array}\right]\left[\begin{array}{cc}0 I I 0 \end{array}\right]\left[\begin{array}{c}S^H T^H \end{array}\right]= \left[\begin{array}{cc}U V \end{array}\right] \left[\begin{array}{cc}\Phi_{11}I \Phi_{21}I \Phi_{12}I \Phi_{22}I \end{array}\right]\left[\begin{array}{c}U^H V^H \end{array}\right] = 0
x(P_1)= \, 4 (5 + t)^2 (11 + t)^2 (79 + 8 t + t^2)^2, x(P_2)= \, (5 + t) (11 + t) (79 + 8 t + t^2)^3, x(P_3)= \, \frac{64 (4 + t)^2 (5 + t)^2 (11 + t)^2 (79 + 8 t + t^2)^2}{(29 + 6 t + t^2)^2}.
|D_kf(x)-D_kf(y)| =\bigg|\int_{\Bbb R^N}(D_k(x,z)-D_k(y,z))f(z)d\omega(z)\bigg| \leqslant \|f\|_{L^\infty}\bigg(\int_{\|x-z\|\leqslant r^{4-k}}+\int_{\|y-z\|\leqslant r^{4-k}}\bigg)\frac{(r^k\|x-y\|)^s}{V_k(x)+V_k(z)}d\omega(z),
D_{1}=\frac{1}{2}\left. \frac{\partial}{\partial y_{1}}(\tilde{\gamma}(y))^{2}\right\vert_{y=0}\tilde{c}(0)\int_{\mathbb{R}^{n}}|\nabla_{z}\left(U(\sqrt{\tilde{A}(0)}z)\right) |^{2}dz+O(\varepsilon)
\begin{aligned}\bigg|-l\int_0^\infty \langle y\rangle^{2l-2}2y(\nu u_\beta \partial_y u_\beta+\kappa h_\beta\partial_yh_\beta)\mathrm{d}y\bigg| &\leq \frac{\nu}{4}\Vert \partial_y u_\beta\Vert_{L^2_l}^2+\frac{\kappa}{4}\Vert \partial_y h_\beta\Vert_{L^2_l}^2+Cs_\beta(x).\end{aligned}
U^*_{n}(\theta_{n},c_{n}^{*}) \equiv\big\{\lambda\in B^d_{n,\rho}: p^\prime \lambda = 0 \cap u^*_{n,j,\theta_n}(\lambda)\le c_{n}^{*}, \: \forall j=1,\dots,J\big\}, \mathfrak W^*(c_{\pi^*}) \equiv \big\{\lambda\in \mathfrak B^d_\rho: p^\prime \lambda = 0 \cap \mathfrak{w}^*_{j}(\lambda)\le c_{\pi^*},~\forall j=1,\dots,J\big\}.
\textup{Tr}(\Delta_n+z^2)^{-\alpha}= n^{-2\alpha+2}\int_0^1 \int_0^1 f(x, y,1,z/n)^{-\alpha}dx dy &+ 2\, I^{n}_{y}\circ E^{n}_{x,M}f^{-\alpha}+ E^{n}_{y,M}\circ E^{n}_{x,M}f^{-\alpha},
(-\overline{v}_1)dF(\overline{x})(z)=-\overline{v}_1(-z_1+\|z_2\|_q) =\overline{v}_1 z_1-\overline{v}_1\|z_2\|_q \ge\overline{v}_1 z_1+ \|\overline{v}_2\|_p\|z_2\|_q \geq \langle \overline{v},z\rangle.
v_i(t,x)= \int_0^1g(t,x, y)h_i(y)dy +\int_0^t \int_0^1 g(t-s,x,y)b(u_i(s,y;h_i))dsdy &+\int_0^t\int_0^1g(t-s, x, y)\sigma(u_i(s, y;h_i))W(dsdy).
\int x^{m}sin(x^{n})dx=\frac{x^{m+n+1}}{m+n+1}_{1}F_{2}(\begin{matrix}\frac{1}{2}+\frac{m+1}{2n}\\ \frac{3}{2}+\frac{m+1}{2n},\frac{3}{2}\end{matrix}|-\frac{x^{2n}}{4}).
d\Big(F_\beta\big(P^{\mathcal{P}(\textbf{x}, h),\mathcal{P}(\textbf{x}, h)}(\textbf{y}'_{\mathcal{P}(\textbf{x}, h)})\big),\,e\Big)\leq c(\beta)\sum_{i=2}^md(f_i',f'_{i-1})^\kappa.
\begin{aligned}\left\langle\frac{\partial\varphi}{\partial u},\frac{\partial\varphi}{\partial u}\right\rangle =\left\langle\frac{\partial \varphi}{\partial v},\frac{\partial\varphi}{\partial v}\right\rangle=0, \left\langle\frac{\partial\varphi}{\partial u},\frac{\partial\varphi}{\partial v}\right\rangle =\frac{1}{2}e^\omega.\end{aligned}
(^{c}_{0}{D}_{t}\, u, v ) \, + \, a\big(l(u)\big) \, (\nabla u, \nabla v) = \big(f, v\big), \forall v \in H^1_0(\Omega), \big(u(\cdot, 0), v\big) = \big(u_0 (\cdot), v\big), \mbox{in}\; \, \Omega.
\Delta^{+}( \omega ) = \ln \left( \frac{\Lambda^{2}}{m^{2}}\right) - \int_{- \Lambda}^{+ \Lambda}\frac{d{\bf k}}{[{\bf k}^{2}+ m_{\mathrm{\scriptsize e f f}}^{2}]^{1 / 2}}\frac{{\bf k}^{2}}{[{\bf k}^{2}+ m_{\mathrm{\scriptsize e f f}}^{2}- \omega^{2}/ 4 + i \epsilon ]},