harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
R_0(\l_0)=\lim_{n\to\infty}\frac{F_n^{(1)}(\l_n)}{F_n^{(2)}(\l_n)}=\lim_{n\to\infty}R_n(\l_n)\frac{\prod_{w\in W_n}(\l_n-w)}{\prod_{u\in U_n}(\l_n-u)}=\tau_0,
f_X(x)=\frac{\sqrt{2}}{\lambda\sqrt{\pi}}e^{-\frac{x^2}{2\lambda^2}}=\frac{1}{\lambda\sqrt{2\pi}}I^{1,0}_{0,1}\Bigg[\frac{x}{\lambda\sqrt{2}}\left|\begin{matrix}\left(0,\frac{1}{2},1\right) \end{matrix}\right.\Bigg],\ \x>0.
\Omega_{1 2}= Q \bar{K}_{2}Q \bar{K}_{2}Q = \left( \begin{array}{c c}{- u_{1}^{2}d_{1}}&{u_{1}^{3}}\\{- u_{1}u_{2}d_{1}}&{u_{1}^{2}u_{2}}\\ \end{array}\right)
\sum_{\substack{i+j+k=n i,j,k\ge 1}}F(i,j,k) \zeta(2i) \zeta(2j) \zeta(2k) = \sum_{k=0}^{\lfloor(d+1)/2\rfloor}K_{F,k}(n)\zeta(2k)\zeta(2n-2k),
\partial_a\hat{g}_{ij}= -4\gamma_n\alpha\frac{z^a}{\rho^{n-2}}\delta_{ij}+ \partial_a H^2_{ij}+\partial_a H^3_{ij}+O_3\left(\rho^{-(n-2)}\right), \partial_{ba}\hat{g}_{ij}=-4\gamma_n\alpha\left(\frac{\delta_{ab}}{\rho^{n-2}}-(n-2)\frac{z^bz^a}{\rho^{n}}\right)\delta_{ij}+ \partial_{ba}H^2_{ij}+\partial_{ba}H^3_{ij}+O_2\left(\rho^{-(n-1)}\right).
\tilde{a}_0 (t,\vec{k}) |\tilde{0}(t) \rangle = \tilde{U}(t) a_0 (\vec{k})\tilde{U}(t)^{-1}\tilde{U}(t) |0 \rangle = \tilde{U}(t) a_0 (\vec{k}) | 0 \rangle = 0 \, ,
\sum_{{\bf{q}}_{1}\neq{\bf{0}},{\bf{q}}}{\bar{n}}_{{\bf{k}}-{\bf{q}}/ 2 +{\bf{q}}_{1}}e^{i \mathrm{~}\theta ({\bf{k}}+{\bf{q}}_{1}/ 2 , -{\bf{q}}+{\bf{q}}_{1})}e^{- i \mathrm{~}\theta ({\bf{k}}-{\bf{q}}/ 2 +{\bf{q}}_{1}/ 2 ,{\bf{q}}_{1})}= 0
\lim_{x\to\infty}e^{\Lambda_u}\frac{2}{\Lambda_2^2}\cdot \frac{f(x)}{g_u(x)}= \lim_{x\to\infty}e^{\Lambda_u}\frac{2}{\Lambda_2^2}\cdot \frac{f(x)}{g_1(x)}\frac{g_1(x)}{g(x)}\frac{g(x)}{g_u(x)}=e^{\Lambda_u}\frac{2}{\Lambda_u}.
\left( \begin{array}{l}{{\dot{b}( t )}}\\{{\dot{c}^{\dagger}( t )}}\\ \end{array}\right) = \left( \begin{array}{l l}{{0}}&{{1}}\\{{1}}&{{0}}\\ \end{array}\right) \left( \begin{array}{l}{{b ( t )}}\\{{c^{\dagger}( t )}}\\ \end{array}\right)
\bigl\langle (f_{k,m})_{k=1, m=0}^{k=K, m=M}, (g_{k,m})_{k=1, m=0}^{k=K, m=M}\bigr\rangle \eqsim_{p, q}\bigl\|(f_{k,m})_{k=1, m=0}^{k=K, m=M}\bigr\|_{Q^{p'}_{q'}}.
I m \Sigma_{F}( Q ) \simeq \frac{g^{2}}{4 \pi}C_{F}T \gamma^{0}\frac{g_{\tau}^{( i )}( q_{0}, q )}{| g_{\tau}^{( i )}( q_{0}, q ) |}\ln \left( \frac{m_{T}}{\Gamma_{q}( Q )}\right) .
G(\Theta) = \|\nabla \Theta\|^2_{L^2}-\frac{3(p-1)}{2(p+1)}\|\Theta\|^{p+1}_{L^{p+1}}+\frac{3(q-1)}{2(q+1)}\|\Theta\|^{q+1}_{L^{q+1}}&\geq \frac{3p-7}{3p+1}\|\nabla \Theta\|^2_{L^2}+ \frac{9(q-1)(p-q)}{2(q+1)(3p+1)}\|\Theta\|^{q+1}_{L^{q+1}}&\geq \frac{3p-7}{3p+1}\|\nabla \Theta\|^2_{L^2}.
((\tilde \gamma+2\gamma I)^{-1})^i_j=\delta^{ik}(\tilde \gamma+2\gamma I)^{-1}_{kj}=\left( \begin{array}{ccc}3\gamma/(9\gamma^2+B_0^2) -B_0/(9\gamma^2+B_0^2) 0 B_0/(9\gamma^2+B_0^2) 3\gamma/(9\gamma^2+B_0^2) 0 0 0 1/(3\gamma) \end{array}\right).
S(w,z)=2\mathrm{i}\left(w_c^{(m)}\left((u,v)\right)w_c^{(m)}\left((x,y)\right)\right)^{1/2}\sum_{j=0}^{N-2}\bigg(\prod_{i=1}^m\frac{(L_i+j)!}{L_i!j!}\bigg)(\bar{z}-w)(w\bar{z})^j,
E_q(J_1,J_2)= \sum_{n_1,n_2 = 0}^{\infty}\frac{J_1^{n_1}J_2^{n_2}}{[n_1]_q![n_2]_q!}= \mathcal{N}^2(J_1,J_2);\; 0 \le J_i < \frac{1}{1-q^2}, i =1,2 .
b(x,\xi,u,v)=(\sigma\circ\mathcal{A}^{-1})(x,\xi,u,v), &\tilde b(x,\xi,u,v)=(\tilde\sigma\circ\mathcal{A}^{-1})(x,\xi,u,v), & c(x,\xi,u,v)=b(x,\xi,u,v)\tilde b(x,\xi,u,v).
-h'(s) = Ku\Gamma(u+1)\zeta(u+1) s^{-u-1}\zeta(u+1)(1+O(s^{\epsilon/2})) + O(s^{-u/2-1}) + O(s^{-1}) & = (1+O(s^{\epsilon/2}))Ku\Gamma(u+1)\zeta(u+1) s^{-u-1},
([T_{|u|^2},\chi_p]f|1) =(T_{|u|^2}\chi_pf-\chi_pT_{|u|^2}f|1) &=(\chi_pf|H_u^2(1))-(\chi_p|1)(H_u^2f|1) &=(H_u^2(\chi_pf)|1)-(\chi_p|1)(H_u^2f|1) &=(\chi_pf-(\chi_p|1)f|u)(u|1)\ ,
\bar{u}_j^{n+1}=\bar{u}_j^{n}-\lambda[h(\tilde{u}_{j+\frac{1}{2}}^{-},\tilde{u}_{j+\frac{1}{2}}^{+})-h(\tilde{u}_{j-\frac{1}{2}}^{-},\tilde{u}_{j-\frac{1}{2}}^{+}) ].
R(t) :=n!^{A-15}2^{18n}(2t+n)\frac{(t-n)_n^3(t+n+1)_{n}^3(t-n+\frac12)_{3n}^3}{(t)_{n+1}^{A}}&= n!^{A-15}2^{-3}(2t+n)\frac{(2t-2n)_{6n+1}^3}{(t)_{n+1}^{A+3}}
\Psi\;:=\; \left(\begin{array}{cccc}f_1^{}f_2 \cdots f_r \sigma^*f_1 \sigma^*f_2 \cdots \sigma^*f_r \vdots \vdots \vdots \sigma^{(r-1)*}f_1 \sigma^{(r-1)*}f_2 \cdots \sigma^{(r-1)*}f_r \end{array}\right)
\sigma^K_k = \frac{1 - \frac{\theta_k}{\theta_{k-1}}\frac{1-\theta_0}{1-\theta_k}}{1 + \frac{\theta_{k-1}}{\theta_0 \mu}}, k \in \{1, K-1\}\sigma^K_K = \frac{1 - \frac{\theta_{K-1}}{\theta_0}(1-\theta_0)}{1 + \frac{\theta_{K-1}}{\theta_0\mu}}.
\sum_{j = 1}^{N}\left( \frac{m}{2 \epsilon}\Delta{r_{j}}^{2}+ \frac{m}{2 \epsilon}{r_{j}}^{2}\left( 1 - \frac{1}{2}\frac{\Delta r_{j}}{r_{j}}\right)^{2}\Delta{\theta_{j}}^{2}\right) \; .
\mathcal{CP}_{n,k}( a)=\bigcup_{i_1+\dots+i_k=n}\bigcup_{j_1=1}^{a_{i_1}}\cdots\bigcup_{j_k=1}^{a_{i_k}}\{[\{1,\dots,i_1\}, j_1]\}\Cup\cdots\Cup\{[\{1,\dots,i_k\}, j_k]\}.
\begin{aligned}\left[ T_{ij},T_{jk}\right] = T_{ik}i \neq k, \left[ T_{ij},T_{k\ell}\right] = 1 i \neq \ell, j \neq k, & i,j,k \in \{1,\dots,n\}(T_{12}T_{21}^{-1}T_{12})^4 = 1.\end{aligned}
\mathcal{P}: \underset{\left\{\boldsymbol{v}_{2,p}\right\}_{p=1}^{P_{2}}}{\max}\lambda \textrm{tr}\left(\boldsymbol{v}_{2,p}\boldsymbol{v}_{2,p}^{H}\right)\leq P_{t}, \mathbf{J}_{\textrm{eff}}\left(\left\{\boldsymbol{v}_{2,p}\right\}_{p=1}^{P_{2}}\right)\succeq\lambda\mathbf{I},
R \le H(U_{2},U_3|X_{2},X_3)-H(U_2|X_2,Y_2)-H(U_3|X_3,Y_3) &\le H(U_2|X_2)+H(U_3|X_3)-H(U_2|X_2,Y_2)-H(U_3|X_3,Y_3) &=I(U_2;Y_2|X_2)+I(U_3;Y_3|X_3) &\le I(U_2,X_1;Y_2|X_2)+I(U_3,X_1;Y_3|X_3) &\le H(Y_2)-H(Y_2|X_1)+H(Y_3)-H(Y_3|X_1) &=I(X_1;Y_2)+I(X_1;Y_3) &\le 1-\varepsilon+1-H(\varepsilon)\approx 1.5101,
\begin{array}{l}{\sum\limits_{i=0}^{n-1}}a_{s_1s_2\cdots s_k}^{(i)}b_{s_1s_2\cdots s_k}^{(i)}=0, ~~~~~{\rm for ~ all ~}s_i, ~ 1 \leq s_i \leq m_i, 1 \leq i \leq k.\end{array}
\big( \left( D\left( y\right) ,z\right) +\left( y,D(z)\right) \big) x+\big( \left( D\left( x\right) ,z\right) +\left( x,D(z)\right) \big) y+\big( \left( D\left( x\right) ,y\right) +\left( x,D(y)\right) \big) z=0,
F_N(a,b;t)=\frac{(1-tq^N)(1-b)}{(1-bq^N)(1-t)}\sum_{n=0}^{N}\frac{(q^{-N})_n(atq/b)_n(q)_nq^n}{(tq)_n(q^{1-N}/b)_n(q)_n}.
\int_M u^2 \leq c\int_M |\nabla u|^2 + c\left(\bar{u}_0^2 + \sum_{\ell=1}^L (\bar{u}_\ell')^2 + \sum_{\ell=1}^L (\bar{u}_\ell'')^2 \right) &\leq c\int_M |\nabla u|^2 + c\left(\sum_{\ell=1}^L (\bar{u}_\ell' - \bar{u}_\ell'')^2 + \sum_{\ell=1}^L (\bar{u}_\ell'' - \bar{u}_0)^2 \right).
-u(t)=\int_{t}^{\infty}w(s)g(s)H^{\alpha-\gamma-1}(s)\Delta_\alpha s \leq \frac{1}{\alpha-\gamma+\theta}\int_{t}^{\infty}T_\alpha^\Delta \Big(-w(s)H^{\alpha-\gamma}(s)\Big)\Delta_\alpha s &=\frac{1}{\alpha-\gamma+\theta}w(t)H^{\alpha-\gamma}(t).
\sqrt{P}=C_1 e^{-\frac{GM m}{\Lambda K}r-\frac{\Lambda m}{K}\ln r+\frac{m\Lambda}{K}\ln \left(\Lambda^2 e^{\frac{2 r}{r_0}}-2 GM \textsf{Ei}\left(\frac{2r}{r_0}\right)r-C_2 \Lambda^2 r\right)}
\frac{\hat\rho^\star}{\frac1{\hat\gamma(\hat\rho^\star)}\frac{1-\hat\rho^\star}{1-(1-\hat\rho^\star)c}+\hat\rho^\star}= \frac{T_{\check\rho^\star}}{1-\check\rho^\star+T_{\check\rho^\star}}= \rho^\star.
\|\nabla f_3\|_{L^2}^2= -\langle f_3,\Delta f_3\rangle=-\langle \Delta^{-1}(f_1 f_2),\Delta f_3\rangle=-\langle (f_1 f_2), f_3\rangle = -\langle f_1 ,f_2f_3\rangle=-\langle f_1 ,P_0(f_2f_3)\rangle\leq \|f_1\|_{L^2}\|P_0(f_2f_3)\|_{L^2}.
\frac{1}{3}x^{3}ln|\frac{r+\sqrt{r^{2}-x^{2}}}{r-\sqrt{r^{2}-x^{2}}}|-\int\frac{x^{3}r(r-\sqrt{r^{2}-x^{2}})dx}{\sqrt{r^{2}-x^{2}}(\sqrt{r^{2}-x^{2}}+r)(r\sqrt{r^{2}-x^{2}}+\frac{x^{2}}{2}-r^{2})}
f_1 :=4x-z, f_2 :=4y-z, f_3 :=4x^2y-z(y-t)^2, f_4 :=4xy^2-z(x-t)^2, f_5 :=(x-y)^2-2(x+y)t+t^2, f_6 :=(x-y)^2-2(x+y)(z+t)+(z+t)^2.
P_c^{\textnormal{SIC}}=\sum_{k=1}^\infty (1-\eta)^{k-1}\eta p_k &\leq \sum_{k=1}^\infty (1-\eta)^{k-1}\eta \frac{(\pi\theta)^{-k/2}}{\Gamma(k/2+1)}&= \frac{\eta}{1-\eta}\left(\exp\Big( \frac{(1-\eta)^2}{\pi\theta}\Big)\left(1+\operatorname{erf}\Big(\frac{1-\eta}{\sqrt{\pi\theta}}\Big)\right)-1\right).
- \frac{i}{2}\left( \frac{n + 2}{n^{2}}\right) \Delta^{a b}\left[ n \tau_{1 \cdots n a b}+ \left( n - 1 \right) \tau_{1 \cdots a n b}+ \cdots + 0 \cdot \tau_{a 1 \cdots n b}\right] + \mathrm{c y c}_{1 \cdots n}
L_{1}(x(u,z)) =\left( \big( u^{2}-z^{2}\big) \prod_{j=1}^{N-2}x_{j}\right)^{-\gamma\kappa}\rho\big(z^{-\kappa},z^{\kappa}\big) \sum_{n=0}^{\infty}\alpha_{n}( x (u,0 ) ) z^{n},
\epsilon^+(z)=\frac{z}{2\pi}+\frac{i}{\pi}\sum_{n>0}\frac{e^{-in(z-i\varepsilon)}}{n},~~~~\epsilon^-(z)=\frac{z}{2\pi}+\frac{i}{\pi}\sum_{n<0}\frac{e^{-in(z+i\varepsilon)}}{n}.
\|w\|_{L^{2}(\mathbb{R}^{n}\times\{0\})}=\bigg[\int_{\mathbb{R}^{n}\times\{0\}}(\langle\xi\rangle^{2-2s}|\hat{w}|^{2})^{s}(\langle\xi\rangle^{-2s}|\hat{w}|^{2})^{1-s}\,d\xi\bigg]^{\frac{1}{2}}\le(\mu^{1-s}\|w\|_{H^{1-s}(\mathbb{R}^{n}\times\{0\})})^{s}(\mu^{-s}\|w\|_{H^{-s}(\mathbb{R}^{n}\times\{0\})})^{1-s}
\left( \tau_{b_g}\tau_{d_g}\ldots \tau_{b_1}\tau_{d_1}\tau_{d_1}\tau_{b_1}\ldots \tau_{d_g}\tau_{b_g}\right) \omega = \omega \left( \tau_{b_g}\tau_{d_g}\ldots \tau_{b_1}\tau_{d_1}\tau_{d_1}\tau_{b_1}\ldots \tau_{d_g}\tau_{b_g}\right)
E_1 E_2 \cdot E_1 = E_1 \cdot (-E_1 \wedge E_2) = 0, F_1 F_2 \cdot E_1 = F_1 \cdot 0 = 0, E_1 E_2 \cdot E_2 = E_1 \cdot 0 = 0, F_1 F_2 \cdot E_2 = F_1 \cdot (-2) = 0.
\frac{2^{k+1}}{t_0}\int_{T_{k-1}}^{t_0}\int_{\mathbb T^2}\theta_k^2(x,s)\,dxds \leq \frac{2^{k+1}}{t_0}\int_{T_{k-1}}^{t_0}\int_{\mathbb T^2}\theta_{k-1}^2(x,s)1_{\left\{\theta_k>0\right\}}\,dxds \leq \frac{2^{2k+1}}{t_0M}\int_{T_{k-1}}^{t_0}\int_{\mathbb T^2}|\theta_{k-1}|^3\,dxds \leq C\frac{2^{2k+1}}{\nu t_0M}U_{k-1}^{3/2}.
e^{\hat{\mu}}= \tilde{E}_{\alpha}^{\hat{\mu}}d \theta^{\alpha}+ \tilde{E}_{\nu}^{\hat{\mu}}d x^{\nu}+ \tilde{E}_{i}^{\hat{\mu}}d \Phi ( x , \theta )^{i}+ \tilde{E}_{\alpha^{\prime}}^{\hat{\mu}}d \Psi^{\alpha^{\prime}}( x , \theta ) ,
P_{n+j-1}(x_k) =\frac{(ab,ac,ad;q)_{n+j-1}}{(2a)^{n+j-1}(abcdq^{n+j-2};q)_{n+j-1}}&\quad\times\sum_{l\geq 0}\frac{(q^{1-j-n},abcdq^{n+j-2},a\xi q^{k-1},a\xi^{-1}q^{1-k};q)_l}{(q,ab,ac,ad;q)_l}\,q^l,
p(y|x, \theta, m) = \sum_{j=1}^m\frac{\alpha_j \exp\{-0.5||x-\mu_j^x||^2/\sigma^2 \}}{\sum_{i=1}^m \alpha_i \exp\{-0.5||x-\mu_i^x||^2/\sigma^2 \}}\phi_{\mu_j^y,\sigma}(y),
j(\gamma_1) =\prod_{s\in S\smallsetminus U_1}c_s\ \cdot\ \prod_{\iota=1}^\mu \tau_\iota^{(12)}\tau_\iota^{>2}\,; j(\gamma_2) =\prod_{s\in S\smallsetminus U_2}c_s\ \cdot\ \prod_{\iota=1}^\mu \tau_\iota^{(12)}\,; j(\gamma_3) =\prod_{s\in S\smallsetminus U_3}c_s\ \cdot\ \prod_{\iota=1}^\mu \tau_\iota^{>2}\,.
\frac{1}{N}D\bigl(p_{Y^N|X^N}(\,\cdot\,|x^N)\|q(\,\cdot\,)\bigr) =\frac{1}{N}\sum_{i=1}^{n_c(x^N)}T_{q,p_{Y|X}}(c_i(x^N))+o(1) &=t(q,p_{Y|X})\left(\frac{1}{N}\sum_{i=1}^{n_c(x^N)}l(c_i(x^N))\right)+o(1) &=t(q,p_{Y|X})+o(1).
|\partial_{t}E_{5}(t,r)| \leq \frac{C r}{t^{3}\log^{b+1}(t)}+\frac{C r \sup_{x \geq t}\left(\frac{|e'''(x)|x}{\lambda(x)^{3-2\alpha}}\right)}{t\log^{(3-2\alpha)b}(t)}
C_{b}=\frac{b}{(b-1)^{2}}-\sum_{n=1}^{\infty}(\frac{b^{2n}-b^{n}+1}{(b^{n}-1)^{2}}-\frac{b^{2n+1}-b^{n}+1}{(b^{n+1}-1)^{2}})b^{-\delta_{b}(n+1)}.
e^{x^{2}}pe^{-x^{2}}=p+1\left[ x^{2},p\right]+\frac{1}{2!}\left[ x^{2},\left[x^{2},p\right]\right]+\frac{1}{3!}\left[x^{2},\left[x^{2},\left[x^{2},p\right]\right]\right]+\cdots
\delta ( t ) = \frac{\delta ( 0 )}{1 - \frac{\delta ( 0 )}{E_{d}^{( 0 )}}\int_{0}^{t}\left( \partial^{0}\tilde{\phi}_{d \, ( \rightarrow )}^{( 1 )}\, \partial^{\xi}\tilde{\phi}_{d \, ( \rightarrow )}^{( 1 )}\right) \mid_{\xi = 1 + \delta}d t^{\prime}}.
R_{\xi,d,\kappa}:= \begin{cases}\Bigl(\frac{(\kappa/(2^d \xi))^{1-\kappa}- 1}{1-\kappa}\Bigr)^{1/d}\kappa \in (0,1) \log^{1/d}\Bigl(\frac{1}{2^d\xi}\Bigr) \kappa=1 \Bigl(\frac{1 - \{2^d\xi/(\kappa a_{d,\kappa}^d)\}^{\kappa-1}}{\kappa-1}\Bigr)^{1/d}\kappa \in (1,\infty). \end{cases}
\displaystyle{\langle \Psi^{\rho r}( z_{1}) \Psi^{\sigma s}( z_{2}) \rangle = C_{\Psi}\frac{I^{\rho \sigma}( \hat{{\cal X}}_{1 2}) I^{r s}( V ( z_{1 2}) )}{( \det{\cal X}_{1 2})^{\frac{1}{2}\eta}}}
\left( \begin{array}{c}{{\theta^{1}}}\\{{\theta^{2}}}\\ \end{array}\right) \Rightarrow \left( \begin{array}{c}{{\tilde{\theta}^{1}}}\\{{\tilde{\theta}^{2}}}\\ \end{array}\right) = \left( \begin{array}{c c}{{\cos \alpha}}&{{\sin \alpha}}\\{{- \sin \alpha}}&{{\cos \alpha}}\\ \end{array}\right) \; \left( \begin{array}{c}{{\theta^{1}}}\\{{\theta^{2}}}\\ \end{array}\right) \; ,
a_{j}^{B W}({\mathsf s}) = \frac{R_{Z}}{{\mathsf s}-{\mathsf s}_{R}}= \frac{R_{Z}}{{\mathsf s}- \bar{M}_{Z}^{2}+ i \bar{M}_{Z}\bar{\Gamma}_{Z}}= \frac{R_{Z}}{{\mathsf s}- ( M_{R}- i \frac{\Gamma_{R}}{2})^{2}}\, , \quad m_{0}^{2}\leq{\mathsf s}< \infty \, .
{\bf{A_0(v)}}=\left(\begin{array}{ccc}u 0 -\frac{R\theta}{a-q}\frac{2a\theta q}{Q}u 0 -\frac{2(P-q)q}{Q}0 u\end{array}\right),{\bf{B_0(v)}}=2q\left(\begin{array}{ccc}-\frac{R\theta}{P-q}0 0 0 -\frac{a\theta}{Q}\frac{P+q}{P-q}\frac{a\theta}{Q}0 \frac{2aq}{Q}-\frac{P-q}{Q}\end{array}\right),
F_{6}\big(3^{-2/3}t\big) = \frac{(q_2-1)}{2q_2}\exp\left(\frac{1}{3}\int_{t}^{\infty}\omega(s)ds -\frac{2}{3}\int_{t}^{\infty}\frac{u_s(s)}{u(s)}\frac{1+q_2(s)}{q_2(s)}ds\right),
{\bf T}e x p \biggl ( i e \sum_{a = 1}^{N}\int_{0}^{T}d t \,{\dot{\bf x}}_{a}\cdot{\hat{\bf A}}({\bf x}_{a}) \biggr ) = e x p ( i \omega ) \, e x p \biggl ( i e \sum_{a = 1}^{N}\int_{0}^{T}d t \,{\dot{\bf x}}_{a}\cdot{\hat{\bf A}}({\bf x}_{a}) \biggr ) \, ,
| \{c_{q i}^{a}\}, p_{q}, \cdots , p_{1}, I , k \rangle = \prod_{a = 1}^{4}\prod_{j = 1}^{N_{q}}( P_{q}^{a \dagger j})^{c_{q j}^{a}}| \{0 \}, p_{q}, \cdots , p_{1}, I , k \rangle ,
\langle E_{1}(l,m,t)E_{2}^{*}(l,m,t)\rangle=\langle A(l,m,t-\frac{R_{1}}{c})A^{*}(l,m,t-\frac{R_{2}}{c})\rangle\times\frac{e^{i\omega\frac{R_{1}}{c}}}{R_{1}}\times\frac{e^{-i\omega\frac{R_{2}}{c}}}{R_{2}}
+ \overline{{{\psi}}}_{+}e^{\mu}{}_{a}\gamma_{+}^{a}( i \partial_{\mu}+ i \omega_{+ \mu}+ q V_{\mu}) \psi_{+}+ \overline{{{\psi}}}_{-}e^{\mu}{}_{a}\gamma_{-}^{a}( i \partial_{\mu}+ i \omega_{- \mu}+ q V_{\mu}) \psi_{-}
\sum_{n_{1}, \, n_{2}}\, \! \! \!^{\prime}\, f ( n_{1}, \, n_{2}) = \sum_{n_{1}= 1}^{\infty}\ \sum_{n_{2}= - \infty}^{\infty}\f ( n_{1}, \, n_{2}) + \sum_{n_{2}= 0}^{\infty}\f ( 0 , \, n_{2})
V(u,-\mu,z) = zK_{-\mu}(uz)\left(\sum_{s=0}^{N-1}\frac{A_s(-\mu,z)}{u^{2s}}+g_2(u,z)\right) \hphantom{V(u,-\mu,z) =}{}-\frac{z}{u}K_{-\mu+1}(uz)\left(\sum_{s=0}^{N-1}\frac{B_s(-\mu,z)}{u^{2s}}+zh_2(u,z)\right).
L_{\mathrm{ChS}}^{(5)}=\alpha_{1}l^{2}\varepsilon_{abcde}R^{ab}R^{cd}e^{e}+\alpha_{3}\varepsilon_{abcde}\left( \frac{2}{3}R^{ab}e^{c}e^{d}e^{e}+2l^{2}k^{ab}R^{cd}T^{e}+l^{2}R^{ab}R^{cd}h^{e}\right) ,
\begin{array}{c}{{U_{0}( z ) = \frac 1 2 ( u_{0}+ u_{- 1}) + \frac 1{\sqrt{2}}( u_{0}- u_{- 1}) \, z_{p e r}}}\\{{V_{0}( z ) = \frac 1 2 ( v_{0}+ v_{- 1}) + \frac 1{\sqrt{2}}( v_{0}- v_{- 1}) \, z_{p e r},}}\\ \end{array}
\Gamma (s_1) \Gamma (s_2) \Phi_2 (s_1,s_2, a, z_1,z_2) = \int_0^\infty \frac{y^{s_2-1}}{e^y-z_2}\int_0^\infty \frac{x^{s_1-1}e^{(1-a)(x+y)}}{e^{x+y}-z_1}dx dy.
\begin{aligned}t a = a \delta, \delta a^* = a^* t, &aa^*=(t^4 - T^{\delta}_2t^2 - T_1^{\delta}t -T_0^{\delta})e_0, a^*a=\delta^4 - T^{\delta}_2\delta^2 - T_1^{\delta}\delta -T_0^{\delta}e_4, a_4a_4^*-\delta =\frac{t_1+2t_2+3t_3+4t_4}{4}e_4=-R_1-\frac{t}{5}e_4, &a_4^*a_4 +B + C +R_1= 0, B^2+R_1B+S_2=0, &C^3-R_1C^2+R_2C-R_3 = 0.\end{aligned}
\sum_{j=1}^{n_{S}}\sum_{b_{j}=0}^{a_{j}}\sum_{\beta_{j}}x_{b_{j}}b_{j}=\sum_{h=1}^{n_{P}}\sum_{d_{h}=0}^{c_{h}}\sum_{\gamma_{h}}u_{\gamma_{h}}y_{d_{h}}d_{h}.
\rho(L_{0}) \, =\, -\frac{h(z)}{h'(z)}\partial + b(z) \, , \rho(L_i) \, =\, h(z)^i \left(\rho(L_0) +i \lambda \right) P(\rho(L_0) +\lambda+1 ,-i) \, , \ \i\neq 0,
\Theta_{[ 1 / 2 ]}= \left[ \begin{array}{c c c}{0}&{{}}&{- 1}\\{1}&{{}}&{0}\\ \end{array}\right] , \Theta_{[ 1 ]}= \left[ \begin{array}{c c c c c}{0}&{{}}&{0}&{{}}&{1}\\{0}&{{}}&{- 1}&{{}}&{0}\\{1}&{{}}&{0}&{{}}&{0}\\ \end{array}\right] \quad .
Y_{2h}^{(a,b)}=\frac{b+\sqrt{ab}}{\sqrt{ab}}\left( \frac{r+\sqrt{ab}}{2}\right)^{2h}+\frac{-b+\sqrt{ab}}{\sqrt{ab}}\left( \frac{r-\sqrt{ab}}{2}\right)^{2h}, &Y_{2m}^{(b,d)}=\frac{\varepsilon b+\sqrt{bd}}{\sqrt{bd}}\left( \frac{y+\sqrt{bd}}{2}\right)^{2m}+\frac{-\varepsilon b+\sqrt{bd}}{\sqrt{bd}}\left( \frac{y-\sqrt{bd}}{2}\right)^{2m}.
X_d=2ig\sum_{\alpha\in\Delta}x_{d}(\alpha\cdot q, \xi)E_{d}(\alpha),\quad Y_d=ig\sum_{\alpha\in\Delta}y_{d}(\alpha\cdot q, \xi)E_{d}(\alpha),\quad E_{d}(\alpha)_{\beta \gamma}=\delta_{\beta-\gamma,2\alpha}.
P(d[x_1,x_3]dx_2) + P([dx_3,x_1]dx_2) + P(dx_2[dx_1,x_3]) = P(d[x_1,x_3]dx_2) + P([dx_3,x_1]dx_2) + P([dx_1,x_3]dx_2) + P([dx_2,[dx_1,x_3]]) = P((d[x_1,x_3] + [dx_3,x_1] + [x_3,dx_1])dx_2) + P([dx_2,[dx_1,x_3]]) = P((d[x_1,x_3] + d[x_3,x_1])dx_2) + P([dx_2,[dx_1,x_3]]) = P([dx_2,[dx_1,x_3]]) .
(\Omega_\psi)^{n+1}=\epsilon f (\omega_X)^{n+1}, M\times \left(1, \frac{3}{2}\right) \psi|_{M\times \{r=1\}}=\phi_0, \psi|_{M\times \{r=3/2\}}=\psi_1+4\log(3/2)
\displaystyle \left\lvert D\left(\lambda\mapsto (\pi_\lambda(g)e,e)\right)_{\lambda=0}\right\rvert \ll \sigma(g)^{\deg(D)}\int_K \delta_P(m_P(kg))^{1/2}\lvert(\tau(m_P(kg))e(k_P(kg)),e(k))\rvert dk
\mathbf{C}_{{\rm SO}(V,g)}(2)=\mathbf{S}^{2}_{1}+\mathbf{S}^{2}_{2}+\mathbf{S}^{2}_{3},\mathbf{C}_{{\rm SO}(U,\eta)}(2)=\mathbf{V}^{2}_{1}+\mathbf{V}^{2}_{2}+\mathbf{V}^{2}_{3}.
P_1\partial([0,1,2]\times[0,1,2]) = P_1([1,2]\times[1,2]-[0,2]\times[0,2]+[0,1]\times[0,1]) = [1,1,2]\times[1,2,2]-[0,0,2]\times[0,2,2]+[0,0,1]\times[0,1,1].
\partial_t f+v\cdot \nabla_x f+{\rm{div}}_v((u-v)f)-\nabla_x\Phi\cdot\nabla_vf-\Delta_vf=0, &-\Delta\Phi=\int_{\mathbb{R}^3}f\,dv-c(x), &\partial_t\rho+{\rm{div}}_x(\rho u)=0, &\partial_t(\rho u)+{\rm{div}}(\rho u\otimes u)+\nabla\rho^{\gamma}+\delta\nabla\rho^{\beta}={\rm{div}}\mathbb{S}(\nabla u)-\int_{\mathbb{R}^3}(u-v)f\,dv,
\mathrm{G L}( \gamma_{1}, \gamma_{2}) ={\textstyle{\frac{1}{4 \pi}}}\oint_{\gamma_{1}}d s \oint_{\gamma_{2}}d t \, d o t{\gamma_{1}}^{a}( s ) \dot{\gamma_{2}}^{b}( t ) \epsilon_{a b c}{\frac{( \gamma_{1}^{c}( s ) - \gamma_{2}^{c}( t ) )}{| \gamma_{1}( s ) - \gamma_{2}( t ) |^{3}}}
\Lambda_{\psi}((\operatorname{id}\otimes\omega_{\Lambda(a),\Lambda(b)})(\Delta(r^*y)) =\Lambda_{\psi}\bigl((\operatorname{id}\otimes\varphi)[(1\otimes b^*)\Delta(r^*y)(1\otimes a)]\bigr) &=\Lambda_{\psi}\bigl((\operatorname{id}\otimes\varphi)[\Delta(r^*y)(1\otimes s)]\bigr).
\|w\|^2 =(p+q-1)\lambda\int_{-1}^1f(x)|w_1(x, 0)|^p|w_2(x,0)|^q &+\int_{-1}^1 \left\{(\alpha+\beta+2w_1^2+2w_2^2)(\alpha+\beta-1+2w_1^2+2w_2^2)+4(w_1^2+w_2^2)\right\}|w_1|^\alpha|w_2|^\beta e^{w_1^2+w_2^2}.
{\cal L}_{N J L}= \bar{q}[ i \gamma^{\mu}\partial_{\mu}- g ( \sigma + i \vec{\pi}\cdot \vec{\tau}\gamma_{5}) ] q -{\frac{\mu^{2}}{2}}( \sigma^{2}+ \vec{\pi}^{2}) +{\frac{M_{0}\mu^{2}}{g}}\sigma ,
\begin{array}{rl}\rho_s^* = \displaystyle \inf_{y}L_y(f), \textnormal{subject to}\displaystyle y_0 = 1, & \displaystyle{\bf M}_{\lfloor s/2 \rfloor}(y) \succeq 0, & \displaystyle{\bf M}_{\lfloor (s-\deg(g_j))/2 \rfloor}(yg_j) \succeq 0, \forall j \in \{1,\dots, m\}, \end{array}
R^2(\omega)=1+\frac{1}{N}\sum\limits_{j=1}^{N}\frac{\|f^{(j)}(0)\|^2}{L}\int_{-\infty}^{0}e^{Lu-2O_{u}^{(j)}}e^{\frac{2}{N}\sum\limits_{k=1}^{N}\int_{u}^{0}O_{\tau}^{(k)}d\tau}du.
\left(\mathbf{1}_{n\times\cdots\times n}-\boldsymbol{\Delta}\right)\circ\mbox{Prod}\left(\mathbf{X}^{(1)},\mathbf{X}^{(2)},\cdots,\mathbf{X}^{(m)}\right)=\mathbf{0}_{n\times\cdots\times n},
H^{\prime}=\frac 12\int d^3x\left(B_a^{ij}F_{ij}^a-A_i^aA_a^i+\left( \left( D^i\right)_{\;\;b}^aB_{0i}^b\right) \left( D_j\right)_a^{\;\;c}B_c^{0j}\right) \equiv\int d^3x\,h^{\prime},
\begin{aligned}x_i^{k+1}-{x'_i}^{k+1}= (1-\gamma_1^k |R_{ii}|)(x_i^k-{x'_i}^k)-\lambda^k(y_i^k-{y'_i}^k), y_i^{k+1}-{y'_i}^{k+1}= (1-\alpha^k-\gamma^k_2|C_{ii}|)(y_i^k-{y'_i}^k) +(g_i^{k+1}-{g'_i}^{k+1})- (1-\alpha^k)(g_i^k-{g'_i}^k), \end{aligned}
\langle u^-,v^-\rangle_{*\mathcal R^-}=\int_0^\infty \Big[\,A_-\big({t^-}^2-{U^-}^2\big)-B\big({t^+}^2-{U^+}^2\big)\,\Big](u^-_1v^-_1+u^-_2v^-_2)r\,{\mathrm d}r.
E\left[\int_s^t (\nabla_z\chi)(r,q_r^m,z_r^m) \cdot (-\nabla_q V(r,q_r^m)-\partial_r\psi(r,q_r^m)+\tilde F(r,q_r^m))dr\right] = E\left[\int_s^t (-\nabla_q V(r,q_r)-\partial_r\psi(r,q_r)+\tilde F(r,q_r))\cdot \left(\int (\nabla_z\chi)(r,q_r,z) h(r,q_r,z)dz\right)dr\right]+O(m^{1/2})
\left( \begin{array}{l l}{\quad \tilde{d}}&{- \tilde{b}/ q}\\{- q \tilde{c}}&{\quad \tilde{a}}\\ \end{array}\right) = \left( \begin{array}{l l}{a^{*}}&{c^{*}}\\{b^{*}}&{d^{*}}\\ \end{array}\right) .
\begin{aligned}\|\nabla^2 u_1\|_{L^p(\mathbb{R}^d)}+\|\nabla p_1\|_{L^p(\mathbb{R}^d)}\leq C\|F\|_{L^p(\mathbb{R}^d)}, \|\nabla^2 u_2\|_{L^p(\mathbb{R}^d)}+\|\nabla p_2\|_{L^p(\mathbb{R}^d)}\leq C\|\Delta\psi\|_{L^p(\mathbb{R}^d)}\leq C\|\nabla h\|_{L^p(\mathbb{R}^d)},\end{aligned}
\begin{aligned}\| \{\Phi_{1,t,r_1}(\widehat u_0)(\xi,\eta)\}^{\lor}\|_2 =\|\Phi_{1,t,r_1}(\widehat u_0)\|_2 \leq c_{r_1}(1+|t|) ( \|\widehat u_0\|_2 + \| \,(1+\xi^2 + \eta^2)^{r_1}\,\widehat u_0\|_2 \leq c_{r_1}(1+|t|) (\|u_0\|_2 + \|\,D_x^{s}u_0\|_2 + \|\,D_y^{s}u_0\|_2). \end{aligned}
\tilde{\tilde{\delta}}_{s u p e r}\psi = -{\frac{1}{2^{1 / 4}g}}\epsilon_{1}\partial_{-}A_{+}+ 2^{1 / 4}\epsilon_{2}\partial_{-}\phi + 2^{3 / 4}g \epsilon_{1}\left[{\frac{1}{\partial_{-}}}\psi , \psi \right]
a_1 \frac{d b_1}{b_1}\wedge \frac{d b_2}{b_2}+a_2 \frac{d b_1}{b_1}\wedge \frac{d b_3}{b_3}+a_3 \frac{d b_2}{b_2}\wedge \frac{d b_3}{b_3}= a_1' \frac{d b_1}{b_1}\wedge \frac{d b_2}{b_2}+a_2' \frac{d b_1}{b_1}\wedge \frac{d b_3}{b_3}+ \frac{d z_1}{z_1}\wedge \frac{d z_2}{z_2}
N | \psi_{n}\rangle = ( \nu_{0}+ n ) | \psi_{n}\rangle , \quad a^{\dagger}| \psi_{n}\rangle = | \psi_{n + 1}\rangle , \quad a | \psi_{n}\rangle = \lambda_{n}| \psi_{n - 1}\rangle , \quad \lambda_{n}= [ n + \nu_{0}]_{q}- [ \nu_{0}]_{q}
\lim_{u\to g(0)}\frac{g^{-1}(u-\log(x)q(u))}{g^{-1}(u)}=x \lim_{u\to g(0)}\frac{g(g^{-1}(u)x)-u}{q(u)}=\lim_{u\to g(0)}\frac{g\left(g^{-1}(u)\frac{g^{-1}(u-\log(x)q(u))}{g^{-1}(u)}\right)-u}{q(u)}=-\log(x)
B = \Omega\bigg( \ln \frac{\mathsf{Num}_\Psi \cdot d}{\epsilon}\bigg)^{d/2}; \ \delta = \bigg( \frac{\epsilon \cdot \eta^{1/2d}}{d \cdot B \cdot \sqrt{\mathsf{Num}_\Psi}\cdot \mathsf{Coeff}_\Psi^{1/d}}\bigg)^d; \c = \frac{\mathsf{Num_\Psi}}{\delta \cdot \sqrt{\epsilon}}.
\begin{array}{l}{\hat{\partial}_{i}\hat{\partial}_{j}= q^{- 1}\hat{\cal R}_{j i}^{l k}\hat{\partial}_{k}\hat{\partial}_{l},}\\{\hat{\partial}_{i}\partial_{j}= q \hat{\cal R}_{j i}^{l k}\partial_{k}\hat{\partial}_{l}.}\\ \end{array}
\Delta_{R \mu \alpha}( k ) = \frac{1}{k^{2}- m^{2}+ i \epsilon k_{0}}\left[ g_{\mu \alpha}- \frac{k_{\mu}k_{\alpha}}{m^{2}}- i m \varepsilon_{\mu \alpha \rho}\frac{k^{\rho}}{m^{2}}\right]
\bigsqcup_{c_1 + c_2 = e}\mathcal{M}_{c_1}( \tilde{P_X}, n, \mu_X^{+}) \times \mathcal{M}_{c_2}( \tilde{P_Y}, n, \mu_Y^{+}) \xrightarrow{\cong}\bigsqcup_{\tilde{P}\in S}\mathcal{M}_c ( \tilde{P}, n, \mu^{+}).
\left[(z^2-1)\frac{d^2}{dz^2}+2a\left(\frac{1-bz}{b-z}\right)\left((z-c)\frac{d}{dz}-1\right)\right]\hat{P}_n^{(\alpha,\beta)}(z)=(n-1)(\alpha+\beta+n)\hat{P}_n^{(\alpha,\beta)}(z)
W^m(p_0) \cap f^{-1}(W^m(p_1)) \subseteq \bigcup \big\{W^{m+1}(x) \,\big|\, x\in f^{-1}(p_1),x\in\overline{W}^m(p_0) \big\}= \bigcup\limits_{x\in E_m(p_0;p_1)}W^{m+1}(x)
\int_{2x_0+v}^u (\phi_u\cdot \phi_t)\phi(u_0, u_0- 2x_0) du_0 = \int_{t-d}^{t+d}(\phi_u\cdot \phi_t)\phi(s, x_0) ds = \frac{1}{2}\int_{t-d}^{t+d}(\phi_x\cdot \phi_t)\phi(s, x_0) ds + \frac{1}{2}\int_{t-d}^{t+d}|\phi_t|^2\phi(s, x_0) ds.
E(u,{\cal P},T) = \int_\Omega \frac{1}{2}Q(e(\nabla T^T \nabla u)) +{\cal H}^1\Big(J_u \setminus \bigcup\nolimits_j\partial^* P_j\Big) +{\cal H}^1\Big(\bigcup\nolimits_j \partial^* P_j \cap \Omega\Big),
\mathcal{M}^{r}(t)\geq \mathcal{M}^{r}(\delta)+(r-1)\theta_{1}\varrho_{0}^{1-\sigma}\int_{\delta}^{t}(\mathcal{M}^{r}(s))^{\sigma+1}ds,0<\delta<t<t_{0}<T_{\mathrm{gel}}.
\|w(t)\|_{L^{2}}^{2}+\int^{t}_{s}\|\nabla\otimes w(\tau)\|_{L^{2}}^{2}d\tau\leq\|w(s)\|_{L^{2}}^{2},\quad\lim\limits_{t\rightarrow\infty}\|w(t)\|_{L^{2}}=0,
\begin{array}{ll}(a_1+a_2)\gamma=0, (n-2k-2)a_1\gamma=0, begin{align*}1mm]\delta_{1}=-a_1a_{n-2k}, a_1b_{2}=(-1)^{k-1}(a_2+1)^k\theta_{n}, begin{align*}1mm]\theta_{n-i}=(-1)^{i}(a_2+1)^i\theta_{n}, 0\leq i\leq k-1.\end{array}
\mathcal{F}\big[R_nf\big](\xi)=\frac{1}{\pi(2\pi)^{n+1}}\int_{\mathbb{R}^{n+1}}M_{n+1}(\xi;\underline{\eta}) \widehat{f}(\xi-\eta_1-\ldots-\eta_{n+1}) &\times\widehat{h}(\eta_1)\cdot\ldots\cdot\widehat{h}(\eta_{n+1})\, d\underline{\eta}.
v\left( x,x_{0}\right) \mid_{x\in \Gamma ,x_{0}\in \left[ 0,1\right]}=\widetilde{g}_{0}\left( x,x_{0}\right) ,\partial_{n}v\left( x,x_{0}\right)\mid_{x\in \Gamma ,x_{0}\in \left[ 0,1\right]}=\widetilde{g}_{1}\left(x,x_{0}\right) ,