harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
+ \sum_{j=1}^{d}(A_{0}(A_{0}^{-1}J_{\varepsilon}A_{j}(J_{\varepsilon}(\tilde{\textbf{u}}^{\varepsilon}+\bar{\textbf{u}})))' J_{\varepsilon}\textbf{w}^{\varepsilon}\partial_{x_{j}}J_{\varepsilon}\tilde{\textbf{u}}^{\varepsilon}, \textbf{w}^{\varepsilon})_{0}.
\partial \gamma_f = -\frac23 \sum_{x \in E_N}\xi_f(x) ([g_x \rho] + [g_{x \tau}\rho]+[g_{x \tau^2}\rho]) & = -\frac23 \sum_{x \in E_N}(\xi_f(x)+\xi_f(x \tau)+\xi_f(x \tau^2)) [g_x \rho] = 0.
\tfrac{1}{(g!)^2}\int_{X^g}\log\|\theta_{\alpha_X}\|(P_1+\dots+P_g-Q-\alpha_X)((\gamma|_{X^{g+1}})\circ s_Q)^*\nu^g = \tfrac{1}{(g!)^2}\int_{pr_{g+1}|_{X^{g+1}}}\log\|\theta_{\alpha_X}\|(P_1+\dots+P_g-P_{g+1}-\alpha_X)(\gamma|_{X^{g+1}})^*\nu^g,
t_\delta A_{n-1}= \rho_0(t) A_{n-1}\rho_{n-1}(t)^{-1}, A_i = \rho_{i+1}(t) A_i \rho_i(t)^{-1}(i\neq n-1), B_i = \rho_i(t) B_i \rho_i(t)^{-1}, t_i^{-1}a_i = \rho_i(t) a_i, t_\delta t_0 b_{n-1}= b_{n-1}\rho_{n-1}(t)^{-1}, t_{i+1}b_i = b_i \rho_i(t)^{-1}(i\neq n-1),
\begin{pmatrix}t_0^{2s}t_0^{2s+2}... t_0^{2r}t_2^{2s}t_2^{2s+2}... t_2^{2r}\vdots \vdots \vdots \vdots t_{2p-2}^{2s}t_{2p-2}^{2s+2}... t_{2p-2}^{2r}\end{pmatrix}\begin{pmatrix}b_s b_{s+1}\vdots b_{r}\end{pmatrix}=\begin{pmatrix}a_0 a_1 \vdots a_{p-1}\end{pmatrix}.
(\partial_t - \Delta) \left( |\tilde{A}|^2 - \frac{1}{2}\tilde{H}^2 \right) = 2|A|^2 \left( |\tilde{A}|^2 - \frac{1}{2}\tilde{H}^2 \right) & -2 \left( |\nabla \tilde{A}|^2 -\frac{1}{2}|\nabla \tilde{H}|^2 \right) + O(1+|A|^3).
\mathcal{E}_k(u):=\displaystyle{\int_0^L}\left\{\frac{1}{2}\left( R_k f^2+S_k\mu(f+g)^2\right)+\mu\Phi(\Gamma)\right\}\,dx ,\mbox{for}k=1, begin{align*}10pt] \mathcal{E}_k(u):=\displaystyle{\int_0^L}\left\{-\frac{1}{2}\left( R_k |\partial_xf|^2+S_k\mu|\partial_x(f+g)|^2\right)+\mu\Phi(\Gamma)\right\}\,dx ,\mbox{for}k=3
\sum_{k=0}^{n-1}\frac{(q^{d+r};q^d)_k^{d-r-1}(q^r;q^d)_k^{r}(q^{r-d};q^d)_kq^{dk}}{(q^d;q^d)_k^d}\equiv 0\pmod{\Phi_n(q)^2}.
\lim_{s\to 1}\gamma_{n,1}^{()}(s) =\prod_{j=0}^{\frac{n-5}{4}}\frac{\Gamma(\tfrac{1}{2}-j)}{\Gamma(-\tfrac{1}{2}-j)}\cdot\left(\lim_{t\to 0}\frac{\Gamma(t-j)}{\Gamma(-t-1-j)}\right) & =\frac{\Gamma(\tfrac{1}{2})}{\Gamma\left(-\tfrac{n-3}{4}\right)}\cdot \left(\tfrac{n-1}{4}\right)!.
\lim_{\sigma\to0}\mathcal{F}_\sigma =F(T,\tilde{x},u(T,\tilde{x}),\nabla_x\varphi(\tilde{x},\tilde{y}),\nabla_x^2\varphi(\tilde{x},\tilde{y})) &-F(T,\tilde{y},v(T,\tilde{y}),-\nabla_y\varphi(\tilde{x},\tilde{y}),-\nabla_y^2\varphi(\tilde{x},\tilde{y}))).
E_3(s) = (p(p-1)^2x^5)\left(\frac{\splitfrac{p^7x^6 - p^7x^5 - p^6x^4 + p^5x^5 + p^6x^3}{- p^5x^4 - p^4x^3 + p^4x^2 - p^3x^3 - p^2x^2 + p^2x + 1}}{(1-x)(1-p^2x^2)(1-p^3x^3)(1-px^2)}\right).
\prod_{i=1}^n \left( \ell^{3 \sum_{j \neq i}f^{(i,j)}_\ell + 3(n-2)v}\right) = 2^{3n(n-2)v}\prod_{i=1}^n \prod_{j \neq i}\ell^{3 f_\ell^{(i,j)}}.
|Dw|^2\log|Dw|^2\frac{\partial\varphi}{\partial\gamma}(x_0)\geq (\alpha_0-|\psi|_{C^0(\overline\Omega\times[-M_0, M_0])}-C_0)|Dw|^2\log|Dw|^2 = |Dw|^2\log|Dw|^2 > 0.
\int_{\mathbf{R}^{d+1}}g(t,x)\mathcal{G}f(t,x)dxdt =\int_{\mathbf{R}}\left(\int_{\mathbf{R}^{d}}f(-s,-y)\mathcal{P}\bar{g}(s,y)dy\right)ds &\leq \int_{\mathbf{R}}\|f(-s,\cdot)\|_{L_{p}}\|\mathcal{P}\bar{g}(s)\|_{L_{p'}}ds &\leq N\|f\|_{L_{q}(\mathbf{R},L_{p})}\|g\|_{L_{q'}(\mathbf{R},L_{p'})}
\zeta(s)=\sum_{n\ge1}n^{-s}=\frac{1}{\Gamma(s)}\int_{0}^{+\infty}(\sum_{n\ge1}e^{-nt})t^{s-1}dt=\frac{1}{\Gamma(s)}\int_{0}^{\infty}t^{s-1}\frac{F_{\zeta}(t)}{t}dt.
G_3^{\mathfrak H/\overline{\varGamma}_{0}(2)}\left(\frac{1+i}{2},\frac{i}{\sqrt{2}}\right)={}-\log2, G_3^{\mathfrak H/\overline{\varGamma}_{0}(3)}\left(\frac{3+i\sqrt{3}}{6},\frac{i}{\sqrt{3}}\right)={}2\log2-\frac{3\log 3}{2}.
{\mathbb E}(\|U_i\|_H^2) = \sum_{j=(i-1)q +1}^{iq}{\mathbb E}(\|Y_j\|_H^2) +2\sum_{j=(i-1)q +1}^{iq}\sum_{\ell=(i-1)q +1}^{j-1}{\mathbb E}(<Y_j, Y_\ell>)\, ,
W= \prod_{a=1}^{p-1}\left( \frac{n^{\frac{1}{l}}}{L_a}\right)\int_0^{\infty}\frac{ds}{s}\left ( \frac{\pi}{2s}\right )^{\frac{p-1}{2}}e^{-r^2s}\frac{4\sinh^2(\omega_ls\sin{\frac{\phi}{2}}) - \sinh^2(2\omega_ls\sin{\frac{\phi}{2}})}{\cos{\frac{\phi}{2}}~\sinh(2\omega_ls\sin{\frac{\phi}{2}})}\,.
\bar{\Phi}= 2 \sqrt 2 r , \bar{F}_{t r}= Q_{E}e^{- 2 \sqrt 2 r}, \bar{T}= 0 ,{}\bar{G}_{\mu \nu}= \left( \begin{array}{c c}{- f}&{0}\\{0}&{f^{- 1}}\\ \end{array}\right) ,
f_{n,u}(\alpha,\beta,\underline{X},\underline{Y})=f(\alpha,\beta,\underline{X},\underline{Y}) =\displaystyle c(t_{i,s})\prod_{k\geq 3}\prod_{s{{=0}}}^{{r-1}}\prod_{1\leq i,j \leq r}(t_{k,s}-X_i)(t_{k,s}-Y_j) & \cdot \displaystyle \prod_{s={{0}}}^{{r-1}}(X_sY_s)^u[(X_s-\alpha)(X_s-\beta)(Y_s-\alpha)(Y_s-\beta)]^{rn}\enspace,
1_{N_B}(x_{\eta}^{-1}x_{\zeta})\beta_{i,j}(x_{\eta}^{-1}x_{\zeta})\frac{1}{|K|}\int_{K}{\rm d}k\,\phi(k x_{\eta}g_{B}^{-1}x^{-1}_{\eta}x)=1_{N_{B}}(x_{\eta}^{-1}x_{\zeta})\beta_{ij}(x_{\eta}^{-1}x_{\zeta})\frac{1}{|K|}\int_{K}{\rm d}k\, \phi(kx)
Q_{\mu_{1}\dots \mu_{p}}^{\mathrm{R}}\equiv{\frac{Q_{p}^{\mathrm{R}}}{v_{p}}}\int_{{\cal M}_{p}}d X^{\mu_{1}}\wedge \cdots d X^{\mu_{p}}\ , \quad \mathrm{w i t h}\quad Q_{p}^{\mathrm{R}}={\frac{1}{\mathrm{V o l}\, S^{8 - p}}}\int_{S^{8 - p}}\! \!{}^{*}G^{( p + 2 )}\ .
S={1\over{16\pi G_{10}}}\int d^{10}x\sqrt{-G^{dual}}\left[e^{-{6\over 7}\phi}({\cal R}^{dual}+{{16}\over{49}}\partial_M\phi\partial^M\phi)-{1\over 4}e^{{6\over 7}\phi}F_{MN}F^{MN}\right].
{\rm Re}\, \chi^{\rm D}_T(\omega)=-\frac{\hbar A}{\pi^2}\int\int_{R_4}d\omega_{\rm in}dk_{\|}k_{\|}\sqrt{k_{\|}^2-(\omega+\omega_{\rm in})^2}\sqrt{\omega_{\rm in}^2-k_{\|}^2}\;{\overline n}(-\omega_{\rm in})
\sum_{k=1}^{\infty}\left(\frac{k}{2}\right)_{r}\frac{6^{r}}{k\pi^{k}24^{\frac{k}{2}}\left(n+1\right)^{\frac{k}{2}+r}\left(1-\frac{25}{24(n+1)}\right)^{\frac{k}{2}+r}}\leq \frac{a_{2}+a_{3}}{(n+1)^{r+\frac{1}{2}}},
f(r, \theta)=f\Big(\frac{r}{\rho_L(\theta)}\rho_L(\theta), \theta\Big) \geq \frac{r}{\rho_L(\theta)}f\Big(\rho_L(\theta), \theta\Big)+ f(0, \theta) \left(1-\frac{r}{\rho_L(\theta)}\right) &\geq f(0, \theta) \Big(1-\frac{r}{\rho_L(\theta)}\Big).
\langle\chi,\chi\rangle_{\mathcal{OP}_n(\Gamma_{*})}=1, \ \ \langle\chi,\chi\rangle_{\mathcal{SP}^1_n(\Gamma_{*})}=0 \hbox{~for~}\chi \hbox{~a~double~spin}, \langle\chi,\chi\rangle_{\mathcal{OP}_n(\Gamma_{*})}=\langle\chi,\chi\rangle_{\mathcal{SP}^1_n(\Gamma_{*})}=\frac{1}{2}\ \ \hbox{for~}\chi \hbox{~an~associate~spin}.
{\Theta}_{-}^{\; \; +}= \frac{i}{2}R ( \bar{\psi}{\gamma}^{+}{\partial}_{-}{\psi}-{\partial}_{-}\bar{\psi}{\gamma}^{+}{\psi}) = \frac{1}{2}({\partial}_{-}{\lambda},{\partial}^{+}{\lambda})_{+}.
\delta L ( q , \dot{q}) = \frac{1}{2}\dot{q}^{\mu}( \delta q^{\alpha}\partial_{\alpha}g_{\mu \nu}+ g_{\mu \alpha}\partial_{\nu}\delta q^{\alpha}+ g \_{\alpha \nu}\partial_{\mu}\delta q^{\alpha}) q^{\nu}+ + \frac{1}{4}\left[ \left[ \partial_{\alpha}\delta q^{\mu}, \dot{q}^{\alpha}\right] g_{\mu \nu}, \dot{q}^{\nu}\right] - \delta q^{\alpha}\partial_{\alpha}U_{q}( q ) \, .
\tilde{F}^{j}:=\max \{||\tilde{f}^{0}||_{\mathbf{B}_{\sigma}},||\tilde{f}^{1}||_{\mathbf{B}_{\sigma}},\ldots ,||\tilde{f}^{j-1}||_{\mathbf{B}_{\sigma}}\}, j=1,2,\ldots,[[T/\Delta t]].
2\int_{\Omega}\left\langle\frac{uT}{r^{a+b}},\widetilde{\nabla}u\right\rangle_TdV_{\widetilde{g}}=-\int_{\Omega}\frac{|u|^2}{r^{a+b+1}}\left(r\widetilde{\Delta}r-(a+b)\right)dV_{\widetilde{g}}.
\tilde{I}[u] \coloneqq I_{+}[u] -\psi(\|u\|_E^p) I_{-}[u] &=\frac{1}{p}\|u\|_{D^{s,p}}^p+ \frac{1}{p}\int_{\mathbb{R}^N}V(x)|u|^p dx &\phantom{=}-\frac{1}{2\cdot p_{r;s}^{\uparrow *}}\psi(\|u\|_E^p)\int_{\mathbb{R}^N}(K\ast g(u))g(u)dx-\varepsilon_W \psi(\|u\|_E^p)\int_{\mathbb{R}^N}W(x)f(u)dx.
\lim_{\mu_j\to 0}\,\frac{1}{\mu_j}\int_{\mu_j}^{2\mu_j}\int_{|x|<4t}\,\left(\partial_tu+\frac{x}{t}\cdot\nabla u+\left(\frac{d}{2}-1\right)\frac{u}{t}\right)^2\,dx \,dt=0.
F_2 = B \cdot (C^{4}- 8 B C^{2}- 3 C^{3}+ 16 B^{2}- 20 B C + 3 C^{2}+ B - C)^{-1}F_3 = B F_4 = C F_5 = C - B F_6 = C^{2}- B + C F_7 = C^{3}- B^{2}+ B C F_8 = B C^{2}- 2 B^{2}+ 3 B C - C^{2}
A_{\mu\nu}(x) ~\sim~ \frac{B_A}{(x^2)^{\beta_A}}\left[ \eta_{\mu\nu}~+~\frac{2\beta_A(1-\rho)}{(2-2\beta_A+\rho)}\frac{x_\mu x_\nu}{x^2}~+~i \vartheta h(\beta_A) \frac{\epsilon_{\mu\nu\sigma}x^\sigma}{(x^2)^{1/2}}\right]
\begin{array}{cccc}A = \left[ \begin{smallmatrix}0 1 A_{21}A_{22}\end{smallmatrix}\right], B = \left[ \begin{smallmatrix}0 b \end{smallmatrix}\right], E_1 = \left[ \begin{smallmatrix}0 e_1 \end{smallmatrix}\right], E_2 = \left[ \begin{smallmatrix}0 e_2 \end{smallmatrix}\right].\end{array}
\begin{aligned}\mbox{there are positive constants}\,\,\, M_0\,\, \mbox{and}\,\, \sigma\,\, \mbox{such that}& \,\,\,\,\,\,\,\displaystyle\max_{(x,s)\in \mathop D\limits^\circ\times[-M_0,M_0]}|f(x,s)|\leq \mu_0^2\frac{M_0}{(\sigma+1)}\frac{\lambda_1}{2},\end{aligned}
u^\varepsilon(t,x)=\sup_{x'\in\overline{\Omega}}\{u(t,x')-\varepsilon^{-1}|x-x'|^2\}\quad &v_\varepsilon(t,x)=\inf_{x'\in\overline{\Omega}}\{v(t,x')+\varepsilon^{-1}|x-x'|^2\},
T_1(s) = \prod_p (1 - b_p p^{-2s}+ (2b_p-2) p^{-3s}-b_p p^{-4s}+ p^{-6s}) T_2(s) = \prod_p \left(1 - \sum_{i,j=1}^3 \alpha_{i,p}^{-1}\alpha_{j,p}^{-1}p^{-2s}+ \frac{2b_p - 2}{p^{3s}}- \sum_{i,j=1}^3 \alpha_{i,p}\alpha_{j,p}p^{-4s}+ p^{-6s}\right) b_p = \sum_{i=1}^3 \sum_{j=1}^3 \frac{\alpha_{i,p}}{\alpha_{j,p}}.
\sigma'_j =\sigma'_j(\delta_5,\delta_3,\delta_2,\delta_1,\sigma_{3},\sigma_{2},\alpha_1)-\sigma'_j(\delta_5,\delta_3,\delta_2,0,\sigma_{3},\sigma_{2},\alpha_1)+\sigma'_j(\delta_5,\delta_3,\delta_2,0,\sigma_{3},\sigma_{2},\alpha_1) &=K_{2j}\delta_1+\delta_j+\sigma_j,j=2,3,
\phi_{\alpha}(\lambda )=\frac{\prod_{i=2}^{n}\varphi_{\alpha_{i}}(\lambda_{i})}{\prod_{i=1}^{n}(1+\beta t_{i})^{n-i}},\beta^{\alpha_{i}}\varphi_{\alpha_{i}}(\lambda_{i})=\Gamma_{i-1-\alpha_{i}}(i-1,n+\lambda_{i}-1)\;.
\Omega_{\gamma}=tr\{dP\,i\,u^{-1}\,du-\frac{1}{2\gamma}(e^{i\gamma P}u^{-1}\,du\,e^{-i\gamma P}\,u^{-1}\,du+L_{+}^{-1}dL_{+}L_{-}^{-1}dL_{-})\}.
\tilde{g}(\theta,\theta)=\rho(z)|\vartheta|^2, \rho=\sum_{k=1}^{n}|\partial_{z}w_k\circ w_i^{-1}|^2, \alpha^{*}\tilde{g}'(\theta,\theta)=\tilde{g}'(\theta',\theta')=\rho'(z')|\vartheta'|^2, \rho'=\sum_{k=1}^{n}|\partial_{z'}w'_k\circ w_i^{'-1}|^2.
\begin{array}{rcl}\displaystyle\sum_{k}K^j_k (z) \phi^k (z) |\mbox{vac}\rangle_i = \nu^{(i)}(z) \phi^j (-z)|\mbox{vac}\rangle_i ,{}_i\langle \mbox{vac}| \displaystyle\sum_{k}\phi_k (z) K_j^k (z) = \nu^{(i)}(z){}_i\langle \mbox{vac}| \phi_j (-z), \end{array}
u_{a+1}^* y_i u_k = q_1 (u_{a+1}^{(t)})^* y_i^{(t)}u_k u_j^* y_i u_{a+1}= u_j^* y_i^{(t)}u_{a+1}^{(t)}q_1 u_{a+1}^* y_i u_{a+1}= q_1(u_{a+1}^{(t)})^* y_i^{(t)}u_{a+1}^{(t)}q_1
\gamma_{a_{2 k}}\equiv Z_{\; \; a_{2 k}}^{a_{2 k - 1}}\pi_{a_{2 k - 1}}+ A_{a_{2 k}}^{\; \; a_{2 k + 1}}\pi_{a_{2 k + 1}}\approx 0 , \; k = 1 , \cdots , \Gamma .
CB.{h}^{n}\equiv a_{n+1}h^{n+1}+a_{n-1}h^{n-1}+\cdots+a_{1}h a_{n+1}=2(n+1) k =1,3,\ldots,k-2\ (k-1) a_{n-k}=\frac{1}{2}\bigg(-\mu{n\choose k}4^{k}+2{n \choose k+1}4^{k+1}+{n\choose k+2}4^{k+2}\bigg).
\begin{cases}r\varphi_r=\lambda(r,\theta) r^2+\frac{r^2-1}{r^2+1}\varphi, \forall r\geq0 \varphi(0,\theta)=\varphi_0(\theta), \ \ \varphi_r(0,\theta)=0, \forall \theta\in{\mathbb{S}}^{n-1}. \end{cases}
q_j \|f_j\|\;\|\tfrac{1}{A}f_j + \epsilon_{2}^{j}h_j \| =q_j\sqrt{\frac{1}{A^2}\|f_j\|^4 + (\epsilon_{2}^{j})^2\|f_j\|^2\|h_j\|^2 + \frac{2}{A}\epsilon_{2}^{j}\|f_i\|^2 Re( \langle f_j, h_j \rangle)}\;\;<\; q_j\frac{1}{A}\|f_i\|^2 < max_{i=1}^N \frac{q_i}{A}\|f_i\|^2 .
G^{(n)}_\sigma(\zeta_1,\cdots,\zeta_{2n})=\sum_{\varepsilon_j =\pm}v_{\varepsilon_1}\otimes \cdots \otimes v_{\varepsilon_{2n}}G^{(n)}_\sigma(\zeta_1,\cdots,\zeta_{2n})^{\varepsilon_1 \cdots \varepsilon_{2n}},
p_2\left(g_1X(w_{i_1,j_1})\times g_2X(w_{i_2,j_2}) \right) = g_1^\bot \cdot p_2(X(w_{i_1,j_1})) \times g_2^\bot \cdot p_2(X(w_{i_2,j_2})) \subset \P^{n-1}\times \P^{n-1}&= (g_1^\bot \cdot w_0L_{n-j_1+1}) \times (g_2^\bot \cdot w_0 L_{n-j_2+1}) &=L_{g_1}\times L_{g_2},
t_1 = \dfrac{(c-1)^2}{4}\, t_2 = \dfrac{(c+1)^2}{4}\, , S_q = \dfrac{(1-c^2+4a^2)\lambda_0 + 2a(c^2 + 1)\lambda_1}{4a^2(\lambda_0+a\lambda_1)}\, , P_q = -\dfrac{(c-1)(c+1)(4a\lambda_0+(1-c^2))}{16a^3(\lambda_0+a\lambda_1)}\; .
[y_{(1\,l)},y_{(2\,l)}]_c =[y_{(1\,l)},[y_2,y_{(3\,l)}]_c]_c=[[y_{(1\,l)},y_2]_c,y_{(3\,l)}]_c &=\lambda_{122}(1-\xi^2)\chi_2(g_{(3\,l)})(1-\chi_{(3\,l)}(g_{122}g_{(3\,l)}))y_{(3\,l)}^2 &=\lambda_{122}(1-\xi^2)\chi_2(g_{(3\,l)})(1-\xi^2)y_{(3\,l)}^2=-3\lambda_{122}\xi^2\chi_2(g_{(3\,l)})y_{(3\,l)}^2,
\begin{aligned}\int_{A}2\langle \nabla_{x}V_{t}(x,0),X(x)\rangle+2V_{t}(x,0)\mathrm{div}(X(x))dx &\qquad=2\int_{A}\mathrm{div}(V_{t}(x,0)X(x))dx=2\int_{\partial A}V_{t}(x,0)\langle X(x),N(x)\rangle dx &\qquad=2\int_{\partial A}\int_{\partial A}G(x,t)\langle X(x),N(x)\rangle\langle X(y),N(y)\rangle dxdy.\end{aligned}
N_{n,0}= \left( 2^n \left\lfloor \frac{n}{2}\right\rfloor ! \, (\nu)_{\lceil \frac{n}{2}\rceil}\right)^{-1/2}, N_{2m-1,1}= \frac{1}{\sqrt{2^{2m}(m-1)!\, (\nu)_{m+1}}}, N_{2m,1}= \frac{1}{\sqrt{2^{2m+1}m!\, (\nu)_{m+1}}}.
\begin{matrix}0&1&2&3&4\\ 1&2&3&4&0\\ 2&3&4&0&1\\ 3&4&0&1&2\\ 4&0&1&2&3\end{matrix}\begin{matrix}0&1&2&3&4\\ 1&0&3&4&2\\ 2&3&4&0&1\\ 3&4&1&2&0\\ 4&2&0&1&3\end{matrix}
\mathcal{D}(p_{_1}||p_{_0}) = N \times \mathcal{D}(p_{_1}(y[i])||p_{_0}(y[i])), \mathcal{D}(p_{_0}||p_{_1}) = N \times \mathcal{D}(p_{_0}(y[i])||p_{_1}(y[i])),
\check{R}(u)=I-\frac{u}{k-u}T = \begin{bmatrix}1 0 0 0 0 \frac{k}{k-u}\frac{-u}{k-u}0 0 \frac{-u}{k-u}\frac{k}{k-u}0 0 0 0 1 \end{bmatrix}
\begin{cases}\partial_t u+u\partial_x u+v\partial_yu+\partial_xP-\partial_y^2u=0, \partial_xu+\partial_yv=0,{\mbox in}\quad\Omega, (u, v)|_{y=0}=0, \lim\limits_{y\to+\infty}u=U(t,x),\end{cases}
\psi (\theta, x) \ = \ \inf_{v_2 \in{\mathcal M}_2}E^{v_1, v_2}_x \Big[e^{\theta \int^{\infty}_0e^{-\alpha t}r_2(X(t), \hat v_1(\theta e^{-\alpha t}, X(t)), v_2(t, X(t)))dt}\Big](:=\psi^{\hat v_1}_{\alpha,2}(\theta, x)).
e^{\textstyle - J \alpha_{-}\Phi ( \sigma , \tau )}={\frac{1}{\sqrt{\varpi}}}\sum_{M = - J}^{J}( - 1 )^{J - M}e^{i h ( J - M )}\xi_{M}^{( J )}( x_{+}){\overline{\xi}_{- M}^{( J )}}( x_{-}) \sqrt{\varpi}
\sqrt{\det \left( \frac{\hat{{\cal G}}_{\perp}^{( 0 )}}{\hat{{\cal G}}_{\perp}^{( 1 )}}\right)}= \frac{{\cal N}_{\perp}}{8 i \sin \left( \frac{| \omega_{-}( \kappa^{2}) | \beta_{T}\mu}{2}\right) \sinh \left( \frac{\omega_{+}( \kappa^{2}) \beta_{T}\mu}{2}\right)}e^{- h_{\perp}( \beta_{T}\mu , \kappa^{2})}
j_\delta ' (u_\delta)h =J_y (y_\delta, u_\delta )S_\delta '(u_\delta;h)+J_u (y_\delta, u_\delta)h+(u-\overline{u},h) =(y_\delta -z_d ,S_\delta '(u_\delta;h)) + \alpha (u,h)+(u-\overline{u},h)
\sum_{\lambda = i}^{m + i-1}= ~(-1)^i ~ h ( a^{m+n-1}_{1, i-1},~ a_i^{m+n-2}\cdot f^{n-1},~ a^{m+n-1}_{i+m+1, j}, ~ g^m, ~ a^{m+n-1}_{j+n+1, m+n+p-1}) &~ + (-1)^{m+i-1}~ h (a^{m+n-1}_{1, i-1}, ~ f^{n-1}\cdot a^{m+n-2}_{i+m},~ a^{m+n-1}_{i+m+1, j}, ~ g^m, ~ a^{m+n-1}_{j+n+1, m+n+p-1}).
= \frac{\left\langle \Psi \left| H^{S c h w .}- H^{F l a t}\right| \Psi \right\rangle}{\left\langle \Psi | \Psi \right\rangle}+ \frac{\left\langle \Psi \left| H_{q u a s i l o c a l}\right| \Psi \right\rangle}{\left\langle \Psi | \Psi \right\rangle}.
(\tilde{f}\circ w)(s_{\lambda},n)= \sum_{\gamma\in\Pi_{\lambda}}K_{\lambda,\gamma}\tilde{f}(e^{w(\gamma)},n) = \sum_{\gamma\in\Pi_{\lambda}}K_{\lambda,w^{-1}(\gamma)}\tilde{f}(e^{\gamma},n) = \sum_{\gamma\in\Pi_{\lambda}}K_{\lambda,\gamma}\tilde{f}(e^{\gamma},n)=\tilde{f}(s_{\lambda},n).
\frac{(2/\delta^2_k)|B| 2^{\gamma_k}}{2^{\hat{n}_k}}\leq \frac{2^{C(x_k \mid b) + \gamma_k + \log(2/\delta^2_k) + 1}}{2^{\hat{n}_k}}= 2^{C(x_k \mid b) -(n_k + \eta_k - \gamma_k - \log(2/\delta^2_k))}.
E=h\nu=E_{i}-E_{f}=\frac{m_{e}q_{e}^{4}}{8h^{2}\epsilon_{0}^{2}}(\frac{1}{1^{2}}-\frac{1}{2^{2}})(Z-1)^{2}\approx\frac{3}{4}(Z-1)^{2}\times13.6eV.
{\not \! \! D}^{2}= ( D_{\mu})^{2}- \frac{i}{4}[ \gamma^{\mu}, \gamma^{\nu}] F_{\mu \nu}= ( D_{\mu})^{2}+ \left( \begin{array}{c c}{( E + B ) \sigma^{3}}&{0}\\{0}&{( - E + B ) \sigma^{3}}\\ \end{array}\right)
g_{\zeta_i(1)}\left( \tilde{\chi}_{\sigma_{i+1}}^{-1}\circ \tilde{\chi}_{\sigma_i}\right) = \delta\!\left( s_{i,i+1}\mu_{i,i+1}e^{\pi i\left( \frac{k_{i+1}}{3}- \frac{[k_i + \eta_i]_3}{3}+ \frac{1}{2}\right)}\right) \tilde{\alpha}\tilde{t}_i\tilde{\alpha}^{-1},
\mathcal{G}_0 =\{u_0\in H_0^1(\Omega):T_{\max}(u_0)=\infty \mathrm{and}u(t)\to 0, \textrm{as}\t\to \infty \}, &\mathcal{B}=\{u_0\in H_0^1(\Omega): T_{\max}(u_0)<\infty \}, &\mathcal{N}:=\{u\in H_0^1(\Omega)\backslash \{0\}:I(u)=0\}.
g_V(0) = \int_{0}^{\infty}\frac{(1-\theta^2)(w_2^2x^2-2\theta w_1w_2x+w_1^2)x}{2\pi\big(w_2^2x^2-2\theta w_1w_2x+w_1^2\big)^{3/2}\bigl(x^2+2\theta x+1\big)^{3/2}}dx &+\int_{0}^{\infty}\frac{(1-\theta^2)(w_2^2x^2-2\theta w_1w_2x+w_1^2)x}{2\pi\big(w_2^2 x^2-2\theta w_1w_2x+w_1^2\bigr)^{3/2}\bigl(x^2-2\theta x+1\big)^{3/2}}dx,
\frac{2}{3}|\lambda| \|f\|_{L^2}^2\leq{}\left|\lambda \int_{\mathbb R}g'(x) U(x)\,\mathrm dx\right|=\left|\int_{\mathbb R}g'(x)\bigl(f'(x)\overline{f(x)}+ V(x)\bigr)\,\mathrm dx\right| \leq{}\left(16\cdot \sqrt{3}+12\right)\|q_-\|_{L^1}^{2}\|f\|_{L^2}^2.
b_{jl}(t)=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}u(t,y)\overline{\hat \psi_{jl}(y)}dy=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}u(t,y)2^{-j/2}\exp\Bigl\{\Bigr. i\frac{y}{2^j}l \Bigl\}\Bigr. \overline{\hat \psi \left(\frac{y}{2^j}\right)}dy,
R_{\alpha{\bar{m}}\nu{\bar{n}}}= \nabla_{\bar{m}}\partial_{\bar{n}}H_{\alpha \nu}= \partial_{\bar{m}}\partial_{\bar{n}}H_{\alpha \nu}- \Gamma_{\bar{m}\bar{n}}^{\bar{k}}\partial_{\bar{k}}H_{\alpha \nu}\, ,
\gamma_{1,i}^{(k)}+ \sum_{j=2}^d v_{j,i}^{(k)}- \gamma_{1,i}^{(k+1)}= \prod_{j=2}^d \bigl( 1 - \epsilon_{j,i}^{(k-1)}\bigr)^{-1}+ \sum_{j=2}^d v_{j,i}^{(k)}- \prod_{j=2}^d \bigl( 1 - \epsilon_{j,i}^{(k-1)}- v_{j,i}^{(k)}\bigr)^{-1}.
P_{n}\left( x\right) =\frac{1}{\sqrt{h_{n}h_{n+1}}}\left( \begin{array}{llll}\gamma_{0}\gamma_{1}... \gamma_{n}\gamma_{1}\gamma_{2}... \gamma_{n+1}... ... ... ... \gamma_{n-1}\gamma_{n}... \gamma_{2n-1}1 x ... x^{n}\end{array}\right) ,
[-] \circ ( L\otimes [-] ) \big( \big(L\otimes \overline{\gamma}_{L,L}\big) \overline{a}_{L,L,L}\big(\overline{\gamma}_{L,L}\otimes L\big) \overline{a}_{L,L,L}^{-1}\big) ( x\otimes ( y\otimes z ) ) \qquad{}=\big[ d^{-1}y,\big[ c^{-1}d^{-1}z,cd^{2}x\big] \big],
X \triangleq \beta_{2,2}^{-1}\bigl(\tau=1,b_0=1,\,b_1=1+\alpha,\,b_2=1+\alpha\bigr), Y \triangleq \beta_{1,0}^{-1}(\tau=1, b_0=2\alpha+2), Y' \triangleq \beta_{1,0}^{-1}\bigl(\tau=1, b_0=1\bigr)
F_1 = \begin{pmatrix}0 0 0 1-q^{\mu}B_1-1+q^{\lambda}B_2 0 0 0 \end{pmatrix}\!,F_2 = \begin{pmatrix}0 0 0 0 0 0 1-q^{\mu}B_1 B_2-1+q^{\lambda}\end{pmatrix}, & F_{\infty}= \begin{pmatrix}q^{\mu}-B_1 -B_2 q^{\mu}-1 1-B_1 -B_2 q^{\mu}-1 -B_1 1-B_2 \end{pmatrix}\! .
\frac{b_{p,q}(x)-b_{p,q}(x_n)}{x-x_n}=\frac{\pi_q(0^m1\overline{b_{n,m+2}b_{n,m+3}\cdots})}{\pi_p(0^m1\overline{b_{n,m+2}b_{n,m+3}\cdots})}&\ge \frac{\pi_q(0^m10^{\infty})}{\pi_p(0^m1^{\infty})}=\frac{p-1}{p}\left(\frac{p}{q}\right)^{m+1}
\begin{array}{c c c}{{\alpha_{1}\to \alpha_{1}+ \omega_{1}\alpha_{2},}}&{{\ \ \}}&{{\beta_{1}\to c ~ \beta_{1}+ \omega_{2}\beta_{2}}}\\{{\alpha_{2}\to c ~ \alpha_{2}\phantom{\omega_{2}c \alpha_{2}},}}&{{\ \ \}}&{{\beta_{2}\to \beta_{2}\phantom{n n c + \omega_{2}( \beta_{2}+ 1 )}.}}\\ \end{array}
\rho^{\flat}(\mathcal{X}; \hat{z}) = -\mathsf{C}^{T}\hat{x}+ \mathsf{A}^{T}\hat{p}, \rho^{\sharp}(\mathcal{X}; \hat{z}) = -\mathsf{D}^{T}\hat{x}+ \mathsf{B}^{T}\hat{p},
P^R_{out}(\tau_R) =\frac{2^{M+1}(K^R_h+1)^M(K_g+1)^M(K_d+1)}{(2M+2)!\ \Omega_h^{M}\Omega^M_g\Omega^\chi_d}(\beta^R)^{-2M}\cdot e^{-MK^R_h-MK_g-K^R_d}(\tau_R)^{M+1}P_{BS}^{-(M+1)},
\frac{1}{2}\epsilon^{a b c d}\gamma_{5}\gamma_{b}e_{c}^{\mu}e_{d}^{\nu}\left( e_{\mu}^{e}p_{e}+ \frac{1}{2}i \nabla_{\mu}\right) W_{\nu \rho}= \hat{X}_{\mu \rho}^{( 3 / 2 ) \nu \lambda}W_{\nu \lambda}
+\cos\psi\cot\eta\cdot\left(\frac{F[q_{0}](\frac{i}{\mu_{0}})w_{2}[q_{0}](\frac{i}{\mu_{0}})w_{3}[q_{0}](\frac{i}{\mu_{0}})}{w_{1}[q_{0}](\frac{i}{\mu_{0}})}\right)^{-\frac{1}{2}}\gamma^{0}\gamma^{1}\gamma^{0}\partial_{\psi}\left(-\gamma^{0}u_{n}(\frac{1}{\mu})\right)\mid_{\mu=\mu_{0}}
| D^\alpha(A_j \partial_j H)-A_j \partial_jD^\alpha|_r\leq C(|\nabla v|_{\infty}|\nabla^2 H|_r+|\nabla^2 v|_r |\nabla H|_{\infty}) \leq C\|\nabla v\|_2 \|H\|_{H^2 \cap W^{2,q}}.
4 f ( k , z ) ={\frac{\alpha}{\pi}}\int_{- 1}^{1}d z^{\prime}\int_{0}^{1}d \gamma \; A ( \theta , \theta^{\prime}, \gamma ) \left\{\gamma^{3 / 2}f ( \gamma k , z^{\prime}) + \gamma^{- 5 / 2}f ({\frac{k}{\gamma}}, z^{\prime}) \right\}
{\frac{\Gamma_{t \rightarrow f}}{\Gamma_{f \rightarrow t}}}= e^{S_{E}( \phi_{t}) - S_{E}( \phi_{f})}= \exp \left[ -{\frac{3 M_{\mathrm{P l}}^{4}}{8}}\left({\frac{1}{V ( \phi_{t})}}-{\frac{1}{V ( \phi_{f})}}\right) \right]
\langle S(b\otimes c_1\otimes a\otimes c_2),L\otimes M\otimes N\rangle =\langle ( c_1\otimes a\otimes c_2),S^*(L\otimes M\otimes N)[b]\rangle &=\langle ( c_1\otimes a\otimes c_2),(L\otimes M(b)\otimes N)\rangle &=\langle ( c_1\otimes (b\otimes a)\otimes c_2),(L\otimes M\otimes N)\rangle
\Phi(r)-\Phi(0^+) \phantom{A}\geq \Phi(r)\,e^{\bar{C_3}r^{1-\frac{n}{\Theta}}}+ C_4\,\int_0^r \left(t^{-\frac{n}{\Theta}}+\frac{\omega(t)}{t}\right)e^{\bar{C_3}t^{1-\frac{n}{\Theta}}}\,dt -\Phi(0^+) - c\,\left(r^{1-\frac{n}{\Theta}}+\int_0^r\frac{\omega(t)}{t}\,dt\right).
\lim_{\beta \downarrow 0}1 + \frac{1-\beta}{p\beta}( 1+ (q-1)e^{-2s}) &=1 - \frac{1+ (q-1)e^{-2s}}{p}+\frac{1+ (q-1)e^{-2s}}{p}\lim_{\beta \downarrow 0}\frac{1}{\beta}&=- \infty.
o_\lambda(\mathbf{x}^{\pm};-1)=\frac{\det(x^{\lambda_j+n-j+\frac{1}{2}}_i+x^{-(\lambda_j+n-j+\frac{1}{2})}_i)^{n}_{i,j=1}}{\det(x^{n-j+\frac{1}{2}}_i+x^{-(n-j+\frac{1}{2})}_i)^{n}_{i,j=1}}
p(x_1, x_2, x_3, x_4) = \sum_{x_0}p(x_0) \cdot p(x_1) \cdot p(x_2 \mid x_1, x_0) \cdot p(x_3 \mid x_2) \cdot p(x_4 \mid x_3, x_0) &= p(x_1) \cdot p(x_3 \mid x_2) \cdot \underbrace{\sum_{x_0}p(x_0) \cdot p(x_2 \mid x_1, x_0) \cdot p(x_4 \mid x_3, x_0)}_{p^*(x_2, x_4 \mid x_1, x_3)}.
H(X) =\tfrac{1}{4(g!)^{2}}\int_{X^{g}}\log\|\theta\|(2P_{1}+P_{2}+\dots+P_{g-1}-P_g)\Psi^{*}\nu^{g}&=\tfrac{1}{(g!)^{2}}\int_{X^{g}}\log\|\theta\|(2P_{1}+P_{2}+\dots+P_{g-1}-P_{g})\Phi^{*}\nu^{g},
\Delta(L_i) =L_i\otimes L_i \varepsilon(L_i) = 1 S(L_i) = L_i^{-1}, \Delta(F_i) =L_i^{-1}\otimes F_i + F_i \otimes 1 \varepsilon(F_i) = 0 S(F_i) = -L_i F_i.
\left\{\begin{array}{c}k_1^0,\ldots,k_s^0\geq0 begin{align*}.2cm]k_i^j,\ell_i^j\geq 1 \end{array}\ \Bigg|\ \begin{array}{c}k_1^0+\cdots+k_1^{u_1}=k_1 \vdots k_s^0+\cdots+k_s^{u_s}=k_s \end{array}, \begin{array}{c}\ell_1^1+\cdots+\ell_1^{v_1}=\ell_1 \vdots \ell_t^1+\cdots+\ell_t^{v_t}=\ell_t \end{array}\right\}
\frac{d\tau}{d\alpha}\langle\int_{\tilde c}\frac{dz}{2\pi i}(\frac{dz}{dw}) 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
\mathbb{E}|V_k| \leq \mathbb{E}\left| e^{\lambda F_m}-e^{\lambda F_m^{(k)}}\right| = 2 \mathbb{E}\left( e^{\lambda F_m^{(k)}}-e^{\lambda F_m}\right)_+ &= 2 \mathbb{E}\left( e^{\lambda F_m}-e^{\lambda F_m^{(k)}}\right)_+ \ .
\ell_{j}\left( s\right) \left( \mathsf{N}_{j}\left( s\right) f,w\right) =\left\langle f,\overline{w}\right\rangle_{\mathbb{R}^{3}}\qquad\forall f\in H^{-1}\left( \mathbb{R}^{3}\right) ,\quad\forall w\in H^{1}\left( \mathbb{R}^{3}\right) .
\|f\|_{1,'}^2 =\sum_{\nu=0}^{r-1}\int^1_0 |f^{(\nu)}(y)|^2 \,{\rm d}y, \|f\|_{1,\pitchfork'}^2 =\sum_{\nu=0}^{r-1}|f^{(\nu)}(a)|^2, \|f\|_{1,'}^2 =\sum_{\nu=0}^{r-1}\left|\int_0^1 f^{(\nu)}(y) \,{\rm d}y \right|^2 .
P_{J,K}\circ T_{J,K}^{-1}=P_{J,K}\circ \mathcal{D}_{J,K}^{-1}\circ\theta^{-1}\circ \mathcal{D}_{J,K}=P_{K,J}\circ\theta^{-2}\circ \mathcal{D}_{J,K}&=P_{K,J}\circ\theta^{-1}\circ \mathcal{D}_{J,K}=P_{K,J}\circ\mathcal{D}_{J,K}^{-1}\circ\theta=P_{J,K},
\sqrt{k_{n}(x)\,k_{n}(y)}=h_{n}^{-1}\left\| K^{2}\right\| \,\sqrt{f(x)\,f(y)}\,\exp \left( \frac{A\,\theta \varepsilon_{n}}{\left\| K^{2}\right\|}\left( f^{-1}(x)+f^{-1}(y)\right) \right) .
m_1( \tau \ge n ) = 2 m ( [ \tfrac12, \hat{b}_{n-1}] ) \le 2 ( \hat{b}_n - \tfrac12 ) \le 2^{\frac{1}{\beta}( \frac{1}{\alpha^2}+ \frac{2}{\alpha})}n^{- \frac{1}{\alpha \beta}}\le C(B) n^{- \frac{1}{\alpha_u \beta_u}}.
f_{\mu \nu}^{\xi}( x ) = \partial_{\mu}[ n^{A}( x ){\cal A}_{\nu}^{A}( x ) ] - \partial_{\nu}[ n^{A}( x ){\cal A}_{\mu}^{A}( x ) ] -{\frac{1}{g}}f^{A B C}n^{A}( x ) \partial_{\mu}n^{B}( x ) \partial_{\nu}n^{C}( x ) ,
(P_V - \mu^2)B(\mu) = 1-\chi_1 - [P_0,\chi_0]R_0(\mu)(1-\chi_1) \phantom{=}+ (\chi_1 + \chi_2(\lambda^2 - \mu^2)R_V(\lambda)\chi_1+T)[P_0,\chi_0]R_0(\mu)(1-\chi_1) = (1-\chi_1) + (\chi_2(\lambda^2 - \mu^2)R_V(\lambda)+T)[P_0,\chi_0]R_0(\mu)(1-\chi_1).