harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sum_{i=1}^K\frac{nR_i}{r_i}\leq \sum_{i=1}^K\frac{I(W_i;\textbf{Y}_i^n|\Omega^n,W_{i+1},\ldots,W_K)}{r_i}\leq \frac{h(\textbf{Y}_K^n|\Omega^n)}{r_K}- \underbrace{\frac{h(\textbf{Y}_1^n|\Omega^n,W_{1},\ldots,W_K)}{r_1}}_{n\log (2\pi e)}+
P_r^{\perp}S_r(E)P_r = & - \sum_{k\geq 2}(-1)^k\frac{1}{2\pi i}\oint_{\gamma_r}\biggl[\frac{1}{(\mu_r-\eta)^2}\sum_{s=2}^{k}(\tilde R_{\Sigma}(\eta)E)^{s-1}P_rE (R_{\Sigma}(\eta)E)^{k-s}P_r + & \frac{1}{\mu_r-\eta}(\tilde R_{\Sigma}(\eta)E)^k P_r\biggr]d\eta.
M_{ij}=\max\{p_{ij}(x_i^{*},y_j^{*}),u_{i\mp\frac{1}{2},\beta}^{\pm},u_{\beta,j\mp\frac{1}{2}}^{\pm}\}, m_{ij}=\min\{p_{ij}(x_i^{*},y_j^{*}),u_{i\mp\frac{1}{2},\beta}^{\pm},u_{\beta,j\mp\frac{1}{2}}^{\pm}\},
\tilde{p}^{\nu_n}(n,s)=\int_0^{\infty}p^{\nu_n}(n,t)e^{-st}\,\mathrm{d}t=\frac{s^{\nu_1-1}\prod_{k=1}^{n-1}\lambda_k}{\prod_{k=1}^n(s^{\nu_k}+\lambda_k)},\ \s>0.
\Vert u \Vert_{C^\kappa([0,T];\mathbb{U})}:=\vert u(0)\vert_\mathbb{U}+ \sup \Bigl\{\frac{\vert u(t)-u(s)\vert_\mathbb{U}}{\vert t -s \vert^\kappa}: s<t \in [0,T]\Bigr\}<\infty.
\int_0^{+\infty}\!\!\!x\bar{u}u_tdx + \int_0^{+\infty}\!\!\!x\bar{u}u_{xx}dx=\alpha \int_0^{+\infty}\!\!\!x|u|^2vdx+ \beta \int_0^{+\infty}\!\!\!x|u|^4dx.
D H_i(h)[h_1] = D \tilde H_X( G_X(h) )[ D G_X(h)[h_1]], D^2 H_i (h)[h_1,h_2] = D^2 \tilde H_X(G_X(h))[D G_X(h)[h_1],D G_X(h)[h_2] ] &+ D \tilde H_X(G_X(h))[D^2 G_X(h)[h_1,h_2]].
\left|F_n^\ast(\ell) + F_n^\ast(k) - 1\right| = \left|F_n^\ast(\ell) - F^\ast(\ell) + (1 - F^\ast(k)) + F_n^\ast(k) - 1\right| \le \left|F_n^\ast(\ell) - F^\ast(\ell)\right| + \left|F_n^\ast(k) - F^\ast(k)\right|,
\frac{1}{\mu_{r,k}}\left(1-e^{-\mu_{r,k}}\frac{\mu_{r,k}^{k-1}}{(k-1)!f_{k-1}(\mu_{r,k})}\right)=\frac{1}{\mu_{r,k}}\left(1-\frac{1}{(k-1)(r-1)}\right).
\widetilde{\Pi}^{\varepsilon}_{\vec{\mathbb{P}}}(f_{1},f_{2})(x):=\sum_{\vec{P}\in\vec{\mathbb{P}}}\frac{\widetilde{C}^{\varepsilon}_{Q_{\vec{P}}}}{|I_{\vec{P}}|^{\frac{1}{2}}}\langle f_{1},\Phi^{1}_{\vec{P}_{1}}\rangle\langle f_{2},\Phi^{2}_{\vec{P}_{2}}\rangle\Phi^{3}_{\vec{P}_{3}}(x),
\| \phi_X(b) \|^2 = \| \phi_1(b) \|^2 \geq \| \phi_m(b)\|^2 \geq \| \phi_m(b) \zeta \|^2 = \| \langle \phi_m(b) \zeta, \phi_m(b) \zeta \rangle \| &= \| \langle \zeta, \phi_m(b^*b) \zeta \rangle \| = \| \langle \zeta, \phi_m(a) \zeta \rangle \| > \| a \| - \epsilon = \| b^* b \| - \epsilon = \| b \|^2 - \epsilon.
\left| 2 \Gamma_{B}( m^{2}- m_{n}^{2}- m_{n^{\prime}}^{2}) - \Gamma_{D}m_{n}^{2}- \Gamma_{D}^{y}m_{n^{\prime}}^{2}\right| \ll \Gamma_{D}m_{n}^{2}+ \Gamma_{D}^{y}m_{n^{\prime}}^{2}.
\langle\xi'(0),\omega\rangle=- \frac{p\mathcal{A}(u,\omega_1)+p\mathcal{A}(v,\omega_2)-K_{\lambda,\mu}(z,\omega)-2\displaystyle\int_\Omega(\alpha|u|^{\alpha-2}u\omega_1|v|^\beta+\beta|u|^\alpha|v|^{\beta-2}v\omega_2)dx}{(p-q)\|(u,v)\|^p-2(\alpha+\beta-q) \displaystyle\int_{\Omega}|u|^{\alpha}|u|^{\beta}dx},
B_{2l-1,2j-1}=-2\sum_{k=1}^\infty\frac{\alpha^k{{2l+2j-3}\choose{2j-2}}}{(kR_0)^{2l+2j-2}},B_{2l,2j}=2\sum_{k=1}^\infty\frac{(-\alpha)^k{{2l+2j-1}\choose{2j-1}}}{(kR_0)^{2l+2j}},
F(z+\omega_2\,|\,\overline\omega_1,\omega_2)F(z\,|\,\overline\omega_1,-\omega_2)=\exp\Bigg(\int_{C}\frac{e^{(z+\omega_2)s}}{(e^{\overline\omega_1 s}-1)(e^{\omega_2 s}-1)}\frac{ds}{s}+ \int_{c\cdot C}\frac{e^{zs}}{(e^{\overline\omega_1 s}-1)(e^{-\omega_2 s}-1)}\frac{ds}{s}\Bigg).
(E\otimes F)\otimes^{\frac{1}{2}}(E\otimes F) =\frac{1}{2}\bigl((E\otimes F)\otimes(E\otimes F)\bigr) &\cong \frac{1}{2}\bigl((E\otimes F\otimes E)\otimes F) &=(E\otimes F\otimes E)\otimes^{\frac{1}{2}}F,
P_{X}(z)=\sum_{\gamma\in\mathcal{P}(\Gamma_{X})}g_{\mathbb{H}}(z,\gamma z)=\sum_{\gamma\in\mathcal{P}(\Gamma_{X})}4\pi\int_{0}^{\infty}K_{\mathbb{H}}(t;z,\gamma z)dt, &E_{X}(z)=\sum_{\gamma\in\mathcal{E}(\Gamma_{X})}g_{\mathbb{H}}(z,\gamma z)=\sum_{\gamma\in\mathcal{E}(\Gamma_{X})}4\pi\int_{0}^{\infty}K_{\mathbb{H}}(t;z,\gamma z)dt.
\frac{1}{h}\int_{1/h}^{+\infty}\bigg\vert\vert \sin(t)\vert -\frac{2}{\pi}\bigg\vert \frac{dt}{t^2}\geq \pi(m-1)\sum_{k=m}^{+\infty}\frac{c}{(k+1)^2}\geq \frac\pi c(m-1)\int_{m}^{+\infty}\frac{1}{(x+2)^2}dx=\frac{\pi c(m-1)}{(m+2)}\nrightarrow 0
\int^{T}_0\int_{\Omega\times\mathbb{R}^3}f\big(\partial_t\varphi+v\cdot\nabla_x\varphi+(u-v)\cdot\nabla_v\varphi+L[f]\cdot\nabla_v\varphi+\Delta_v\varphi\big)\,dxdvdt &+\int_{\Omega\times\mathbb{R}^3}f_0\varphi(0,x,v)\;dxdv=\int^{T}_{0}\int_{\Sigma^-}\big(v\cdot \nu(x)\big)g\varphi\,d\sigma(x)\,dvdt,
t_{f}- t_{i}= \int_{r_{i}}^{r_{f}}\, d u \, \frac{\pm 1}{\sqrt{2 \left[ E_{r}- V ( u ) - \frac{L^{2}}{2 u^{2}}\right]}}\ \ , \ \L = \frac{\varphi_{f}- \varphi_{i}}{g^{2}( t_{f}- t_{i}) + \int_{t_{i}}^{t_{f}}\, \frac{d t}{r^{2}( t )}}\ \ \ ,
\partial \|d_i^*(x)\|-{\lambda_{d_i}^*(x)}\ni 0, \lambda |I-\mathbf{C}|-\Lambda_{\mathbf{W}}+\partial \delta_\Delta(\mathbf{W}) \ni\mathbf{0}, \int_\Omega \partial |I(x)-c_i^*|w^*_i(x)dx \ni 0, \nabla^T\lambda_{d_i}(x)+\lambda_{w_i}(x) =0, \nabla u_i(x)-d_i(x) =0, u_i(x)-w_i(x) =0.
\sup_{t\in[0,T)}\|B(t)\|_{L^2}\geq \ \left(\sum_{q\geq0}e^{-2\mu\lambda_q^\alpha t}\|B_q(0)\|^2_{L^2}\right)^{\frac12}&-C\left(\sup_{t\in[0,T)}\|B(t)\|_{L^2}\right) \left(\sup_{t\in(0,T)}t^{\frac{m-(\frac52-\alpha)}{\alpha}}\|B(t)\|_{H^m}\right)
\begin{aligned}|F^{ij}(\nabla_\beta u)_{ij}| \leq \, |\nabla_\beta f^{1/k}| + C \left(f^{1/k}+ b_\alpha \sum_{i=1}^n F^{ii}\right) \leq \, C m^{1/k}m^{-1/2(k-1)}+ C b_\alpha \sum_{i=1}^n F^{ii}\leq \, C b_\alpha^{-1/2}m^{1/k}+ C b_\alpha \sum_{i=1}^n F^{ii}\end{aligned}
|p_N| = \frac{1}{\sin(\theta)}|a_1p_1\sin(\theta)+a_2p_2\sin(2\theta)+\ldots+a_{N-1}p_{N-1}\sin((N-1)\theta)| &=|a_1p_1U_1+a_2p_2U_2+\ldots+a_{N-1}p_{N-1}U_{N-1}| &\leq|a_1U_1^2+a_2U_2^2+\ldots+a_{N-1}U_{N-1}^2| + 2C\sum_{i=1}^{N-1}|a_iU_i| &\leq\frac{C}{N}+ 2C\frac{C}{N^3}N^2 = \frac{C'}{N}.
\mathcal{I}_0(1^{\prime}) = \tilde{\tau}_{\ge 2}^{\leq n-1}R \wedge \tilde{\tau}_{\ge n-1}^{\leq n-1}R, \mathcal{I}_0(0) = \tilde{\tau}_{\ge 1}^{\leq n-1}R \wedge \tilde{\tau}_{\ge n-1}^{\leq n-1}R, \mathcal{I}_0(1) = \tilde{\tau}_{\ge 2}^{\leq n-1}R \wedge \tilde{\tau}_{\ge n-2}^{\leq n-1}R.
Z = \frac{1}{q+1}\cdot \left( \sum_{i=0}^{\infty}q^{-2i}+ \sum_{i=1}^{\infty}q^{-2i}+ \sum_{i=0}^{\infty}q^{-2i-1}+ \sum_{i=1}^{\infty}q^{-2i+1}\right) = \frac{1}{q+1}\cdot \frac{1 + q^{-2}+ q^{-1}+ q^{-1}}{1 - q^{-2}}= \frac{1}{q-1}
(j-1)^2+\cdots+2^2+1^2+1^2+2^2+\cdots+(r_i -j)^2 = \frac{j(j-1)(2j-1)}{6}+\frac{(r_i -j)(r_i -j+1)(2r_i-2j+1)}{6}= \frac{1}{6}[2j^3-3j^2+j+2r^3_i+3r^2_i-6r^2_ij+6r_ij^2-6r_ij+r_i-2j^3+3j^2-j] = \frac{1}{6}\left [2r^3_i+3r^2_i+r_i-6r^2_ij+6r_ij^2-6r_ij\right ]
\frac{1}{10}\tilde{u}_{j-\frac{1}{2}}+\frac{6}{10}\tilde{u}_{j+\frac{1}{2}}+\frac{3}{10}\tilde{u}_{j+\frac{3}{2}}=\frac{10}{30}\bar{u}_{j}+\frac{19}{30}\bar{u}_{j+1}+\frac{1}{30}\bar{u}_{j+2}.
\overline{\Gamma}_5=-\varepsilon^{\widetilde{0},0}-\varepsilon^{0,\widetilde{0}}-\varepsilon^{\widetilde{\mu},\mu}-\varepsilon^{{\mu},\widetilde{\mu}}-\frac 12 e_{\mu \nu \rho\omega}\varepsilon^{[\mu \nu],[\rho \omega ]}.
\delta g = -2u\,h, \delta g^{-1}= 2\,u\,\hat{h}\delta(dS) = -2Hu \, dS, \delta (2H) = \Delta{u}+2u(2H^2-K+2k_0), \delta K = 2H\Delta{u}- \langle h, \,u \rangle + 2HKu,
\mathbf{h}_{kl}= \sqrt{\frac{M}{N_{\mathrm{cl},k}N_{\mathrm{ray},k}}}\sum_{i=1}^{N_{\mathrm{cl},k}}\sum_{j=1}^{N_{\mathrm{ray},k}}\alpha_{ijkl}\mathbf{a}\left(\phi_{ijk},\theta_{ijk}\right),
q_{n_1,n_2}= n_1n_2 -\sum_p\lfloor n_1/p \rfloor \lfloor n_2/p\rfloor +\sum_{p_1\le p_2}\lfloor n_1/(p_1p_2)\rfloor\lfloor n_2/(p_1p_2)\rfloor - ... = \sum_{k=1}^{n_1}\nu(k)\lfloor n_1/k\rfloor\lfloor n_2/k\rfloor
\left\Vert \chi\right\Vert_{n}^{2}=\left\langle \tilde{w},\tilde{w}\right\rangle_{L^{2}}+\left\langle \tilde{w}_{z},\tilde{w}_{z}\right\rangle_{L^{2}}+\left\langle \tilde{p},\tilde{p}\right\rangle_{L^{2}}\,.
\overline{\not\! \partial}=\gamma^a \delta_a^\mu\partial_\mu\,,\qquad \not\! \partial=\gamma^\mu \partial_\mu =a^{-1}\overline{\not\! \partial}\,,\qquad \overline{\gamma}_\mu =\delta_\mu^a\gamma_a \,,\qquad \overline{\gamma}^\mu =\delta^\mu_a\gamma^a\,.
I_h z_R(t)= I_h z_0 +\int_0^t i\Delta I_h z_R(s)-i \nu I_h z_R(s) -\epsilon I_h (\gamma z_R(s) -\mu \overline{z_R}(s))\d s -\tfrac{1}{2}\int_0^t I_h (z_R(s)F_\Phi )\d s+i\kappa \int_0^t \theta_R(|z_R|_{X_s})I_h(|z_R(s)|^2z_R(s))\d s -i\int_0^t I_h(z_R(s)\d W(s)).
\frac{-v_m(\lambda,r)}{\sin(|m+\alpha|\, \pi)\, W_m(\lambda)}\, J_{-|m+\alpha|}(\sqrt{\lambda}\, r') =\frac{-v_m(0,r)\, \lambda^{-\frac{|m+\alpha|}{2}}}{\sin(|m+\alpha|\, \pi)\, W_m(\lambda)}\, \lambda^{\frac{|m+\alpha|}{2}}\, J_{-|m+\alpha|}(\sqrt{\lambda}\, r') = G_{m,0}(r,r') + \frac{\mathcal{O}(\lambda)}{1+|m|}\ , \lambda\to 0+
\frac{d\rho_{0}}{d\Lambda}= \frac{\Lambda\,T^2}{2\pi}\,g\left(\frac{\Lambda}{T}\right)\left[{\frac{6\,\lambda_{{4}}+12\,\rho_{{0}}\lambda_{{6}}}{{\Lambda}^{2}+4\,\rho_{{0}}\lambda_{{4}}}}\right]\frac{1}{2\lambda_{4}}\; ,
\mathcal{E}_{a}^{\rm H}(u) = \|(-\Delta+m^{2})^{\frac{1}{4}}u\|_{L^{2}}^2 + \int_{\mathbb{R}^{3}}V(x)|u(x)|^{2}{\rm d}x -\frac{a}{2}\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\frac{|u(x)|^{2}|u(y)|^{2}}{|x-y|}{\rm d}x{\rm d}y.
|x_{s_{l+1}}- y_{s_{l+1}}| < (3/5)^{s_{l+1}/2 + 1/2^{n+1}(s_{l+1}- s_l + 2K_n + 20)}|x_0 - y_0| = &< (3/5)^{s_{l+1}/2 + 20K_n}|x_0 - y_0|
\left({\frac{\Delta X^{\mu}}{d \tau}}\right) \left({\frac{\partial}{\partial \xi^{c}}}|{\frac{\Delta X^{\mu}}{\partial \xi^{c}}}|^{- 1}\left[{\frac{\Delta X^{\nu}}{d \tau}}\right] \right) + \left({\frac{\Delta X^{\mu}}{d \tau}}\right) \Gamma_{\mu \lambda}^{\nu}\left({\frac{\Delta X^{\lambda}}{d \tau}}\right) = 0 \quad .
\tilde{R}^{a b}= R^{a b}- K_{c}^{a}\wedge K^{c b}+ \frac{1}{2}C_{\; \cdot c}^{a b}\tilde{T}^{c}+ \frac{1}{4}\gamma_{J}^{I}\gamma_{L}^{K}\theta_{I}^{a}\theta_{K}^{b}A^{J}\wedge A^{L}+ i \left( D K^{a b}- \frac{1}{4}\gamma_{J}^{I}( \theta_{I}^{a}e^{b}+ \theta_{I}^{b}e^{a}) \wedge A^{J}\right) .
- [ 8 Q_{0}^{2}/ r ] \{\Gamma [ 7 / 4 ] / 3 - \Gamma [ 3 / 4 , r^{4}] / 4 - ( \Gamma [ 7 / 4 ] - \Gamma [ 3 / 4 , 2 r^{4}] ) / 2^{3 / 4}\}
1- \overline{\Psi(z_l,x_{1i},x_{2i},x_{3i})}\Psi(z_k,x_{1j},x_{2j},x_{3j}) &=(1-\overline{z_l}z_k)\overline{\gamma(\lambda_i,z_l)}\gamma(\lambda_j,z_k)+(1-\overline{\lambda_i}\lambda_j)\eta(\lambda_i,z_l)^*\frac{I-F(\lambda_i)^*F(\lambda_j)}{1-\overline{\lambda_i}\lambda_j}\eta(\lambda_j,z_k).
\sum_{i=1}^{3}|\nabla J_i|^2 + c\,J_3^2\,V_2(x)=\sum_{i=1,2}\frac{|\nabla J_i|^2}{1-J_i^2}+ c\,J_3^2\,V_2(x)\le \frac{k+2c}{r_0^2}.
X(k_{1},k_{2},k_{3})=\sum_{n_{1}=1}^{N-1}\sum_{n_{2}=1}^{N-1}\sum_{n_{3}=1}^{N-1}\tilde{x}(n_{1},n_{2},n_{3})cos(\varphi k_{1})cos(\varphi k_{2})cos(\varphi k_{3})
E\max_{1\leq k\leq n}\left\vert \sum\nolimits_{j=1}^{k}a_{j}E^{j}X\right\vert^{p}\ll E\max_{1\leq k\leq n}|s_{k}E^{k}X|^{p}+\sum\nolimits_{i=1}^{n-1}E|s_{i}P^{i}(X)|^{p}.
+\epsilon^{2}\underset{\mathbb{R}^{2}}{\int}(V\star(\partial_{t}\mathfrak{U}))\mathfrak{U}(t,x)dx+\frac{\epsilon}{2N}\stackrel[i=1]{N}{\sum}\mathrm{trace}(u^{k}(t,x_{i})\lor x_{i}^{k}R_{N}(t)).
\frac{G(z)}{A(z)}R(z) = \bigg(\sum_n \frac{c_n\mu_n^{1/2}}{z-t_n}\bigg) \cdot\bigg(\sum_n \frac{G(t_n) \bar c_n}{A'(t_n)\mu_n^{1/2}(z-t_n)}\bigg)- \frac{G(z)}{A(z)}\sum_n \frac{|c_n|^2}{z-t_n}.
\displaystyle\int_{\Omega}p\left|\nabla u\right|^{2}\geq 2\pi p_{0}\left(\sum_{i=1}^{m}d_{i}^{2}\right) \left( \log \dfrac{R}{R_{0}}- I\left(\dfrac{R}{R_{0}}\right)\right)+2\pi p_{0}a^{2}\displaystyle\sum_{i\neq j}d_{i}d_{j}\log \frac{R}{|a_{i}-a_{j}|}-C,
\left( \frac{\partial}{\partial u}\right)^{n_{1}}a_{\mu}^{m_{1}}\left( \frac{\partial}{\partial \bar{u}}\right)^{n_{2}}\bar{a}_{\mu}^{m_{2}}= \left( \frac{\partial}{\partial u}\right)^{n_{1}}\left( \frac{\partial}{\partial \bar{u}}\right)^{n_{2}}a_{\mu}^{m_{1}}\bar{a}_{\mu}^{m_{2}},
\phi_a(x,t):=\sum_i\mathbf{1}_{I_i}(x)\sum_j\mathbf{1}_{[(j-1)\Delta t,j\Delta t)}(t)\left[\frac{a_{i,j}-a_{i,j-1}}{\Delta t}t+j\cdot a_{i,j-1}-(j-1)\cdot a_{i,j}\right].
V_n ={(1+3\beta_n) (1+\gamma_n) \Delta^2 +{24 (\sqrt{p \log{n}}+ \log{n}) \over n-1}\over \sqrt{4(1-{1\over n}-2\beta_n) (1-\gamma_n) \Delta^2 +{16p \over n-1}(1-\theta_n)}}.
T_{L}\ = \ - 3 \omega e^{- b_{L}d}\coth \left[ \omega ( x_{s}\! - \! L ) \right] \left( \frac{\tanh \left[ \frac{\omega}{2}( x_{s}\! - \! L ) \right]}{\tanh \left( \frac{\omega x_{s}}{2}\right)}\right)^{\cosh^{- 1}\left[ \omega ( x_{s}\! - \! L ) \right]}\ .
m''(a ) = \int_{M \setminus \{p \}}G_a P_a G_a''\, dv^{g}& = -2 \int_{M \setminus \{p \}}\varphi G_a' G_a dv^g & = 2 \int_{M \setminus \{p \}}G_a' P_a G_a'\, dv^g \geq 0.
\left. S_{n , m}\, ( x_{1}\, q_{1}, . . . . . . , x_{n}\, q_{n}; \, p_{1}. . . . . . p_{m}) = \langle{\cal D}( x_{1}\, q_{1}, . . . . . . , x_{n}\, q_{n}) \, \psi_{p_{1}}. . . . . . . \psi_{p_{m}}\rangle \right.
f(u,v)+\langle G(u,v), (x-u, y-v)\rangle=\begin{cases}-x \textrm{if}(u,v)\in E_1, u<0 x \textrm{if}(u,v)\in E_1, u>0 -x \textrm{if}(u,v)\in E_2, u<0, 1\leq v\leq j+1 x \textrm{if}(u,v)\in E_2, u>0, 1\leq v\leq j+1 2(y-j-1) \textrm{if}(u,v)\in E_2, v>j+1,\end{cases}
U_{2k,2a-1}(x;q)=U_{2k,2a}(x;q)-U^{2a}_{2k,2a}(x;q) &=(-xq;q^2)_\infty\overline{Q}_{k,a}(x^2;q^2)-xq(-xq^3;q^2)_\infty\overline{Q}_{k,a}(x^2;q^2)+xq(-xq^3;q^2)_\infty\overline{Q}_{k,a-1}(x^2;q^2) &=(-xq^3;q^2)_\infty\overline{Q}_{k,a}(x^2;q^2)+xq(-xq^3;q^2)_\infty\overline{Q}_{k,a-1}(x^2;q^2).
V_n(x_n)=y_n(x_n) \mbox{and}\;\;\; V_j(x_j)=\max\left\{y_j(x_j),\frac{1}{j+1}\sum_{i=1}^{j+1}V_{j+1}(i)\right\}\;\;\mbox{for}\;\;j=n-1,n-2,\dots,1.
\langle( \psi )'(u),v\rangle = \int_{M}|u(x)|^{p-2}u(x)v(x)d\mu_g(x) & + \iint_{M\times M}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))(v(x)-v(y))}{(d_g(x,y))^{N+ps}}d\mu_g(x)d\mu_g(y)\cdot
y_k^{\star}=\frac{\sqrt{\left(1+\eta_k\right)\left|\left(\boldsymbol{v}_{k}\right)^{\mathrm{H}}\!\boldsymbol{g}_{k}\right|^{2}a_{k}{E}_{k}}}{\sum_{l \in\mathcal{K}}\left|\left(\boldsymbol{v}_{k}\right)^{\mathrm{H}}\!\boldsymbol{g}_{l}\right|^{2}a_{l}{E}_{l}+ L\sigma^2\left\|\boldsymbol{v}_k\right\|^2}, \forall k \in \mathcal{K},
\Gamma_{ij}^{1}=\begin{cases}0, (i,j)\in \{(1,1), (1,3), (3,1), (3,3)\}, \tfrac14{x_2}L, (i,j)\in \{(1,2),(2,1)\}, -\tfrac12{x_1}L, (i,j)= (2,2), \tfrac12{L}, (i,j)\in \{(2,3),(3,2)\},\end{cases}
\| X_s^{I, R_1}- \hat X_s \|_H^{p-2}\| \hat X_s - X_s^{I, R_2}\|_{H_\delta}^2(1 + \|\hat X_s\|_{H_\gamma}^c + \| X_s^{I, R_2}\|_{H_\gamma}^c )^2 &\leq\tfrac{p-2}{p}\| X_s^{I, R_1}- \hat X_s \|_H^p+\tfrac{2}{p}\| \hat X_s - X_s^{I, R_2}\|_{H_\delta}^p(1 + \|\hat X_s\|_{H_\gamma}^c + \| X_s^{I, R_2}\|_{H_\gamma}^c )^p.
{\mathbb{E}}({\widetilde{R}}_{1})=0\,{\mathbb{E}}({\widetilde{R}}_{2})=\frac{1}{2}\sum_{j,\ell=1}^{p}{\mathbb{E}}\big ({\mathbb{E}}(d_{iu}^{(j)}d_{iu}^{(\ell)}|{\mathcal{F}}_{i,u-1}^{X})\partial_{j}\partial_{\ell}s_{iu}({\mathbf{0}})\big )\,.
\frac{\Gamma(t)}{\Gamma(s)}= \frac{\sigma}{\tau}\Big( 1 + C_{\sigma,\tau}\big (| \sigma| +|\tau|\big) \sqrt{\frac{N}{M + 1}}\Big), \frac{1}{s - t}= \frac{1}{\tau - \sigma}\sqrt{\frac{M + 1}{N}},
< T_{1}T_{2}T_{3}> = \frac{- 8 J_{1}J_{2}J_{3}}{( 2 m_{2}- p_{2}) ( 2 m_{3}- p_{3})}\frac{2}{\pi}( 2 \mu )^{J_{1}+ J_{2}+ J_{3}- 1}\mid \log \mu \mid ,
f^{(N)}_{\mathcal{A},t}(x)=\int_{\mathcal{A}}^{}e^{-\lambda_{N,s}t}\frac{\Delta_N(y)}{\Delta_N(x)}\det\left(p^{(N),s}_t(x_i,y_j)\right)^N_{i,j=1}dy>0 \ ,\ \forall x \in \mathring{W}^N.
\psi\left(-\sum_{i=1}^{n}k_{i},\sum_{i=1}^{n}s_{i}\right)\left[\mathcal{T}^{*}\left(\mathcal{T}^{*}\right)^{-1}\overline{f}\left(\mathbf{k},\mathbf{s}\right)\right]=\overline{h}\left(\mathbf{k},\mathbf{s}\right).
L^{- 1}: = \overline{{{L}}}= \{\overline{{{L}}}_{M}^{A}\}= \left( \begin{array}{c c}{{\overline{{{L}}}_{\alpha}^{a}}}&{{\overline{{{L}}}_{\alpha}^{i}}}\\{{0}}&{{\overline{{{L}}}_{\alpha}^{i}}}\\ \end{array}\right) \, , \quad \overline{{{L}}}_{\mu}^{a}= 0 \, ,
m+1=\underbrace{n_{1}+\cdots+n_{1}}_{M_{1}\;\mathrm{times}}+\underbrace{n_{2}+\cdots+n_{2}}_{M_{2}\;\mathrm{times}}+\cdots+\underbrace{n_{\ell}+\cdots+ n_{\ell}}_{M_{\ell}\;\mathrm{times}},
y_{m}=\arg \min_{y\in \mathbb{R}^{N}}\left\Vert C_{m}C_{m}^{T}y-b\right\Vert=W_{m+1}\arg \min_{t\in \mathbb{R}^{m+1}}\left\Vert H_{m}H_{m}^{T}t-\Vert b\Vert e_{1}\right\Vert =W_{m+1}t_{m}.
\mathcal{K}_{old}P_{old}= \ \mathcal{K}_{old}(I+(s^{2}_{new}-s^{2}_{old})\mathcal{K}_{old}^{-1}M+(s_{new}-s_{old})\mathcal{K}_{old}^{-1}D) \ \cdot (I + (s^{2}_{new}-s^{2}_{old})\mathcal{K}_{old}^{-1}M + (s_{new}-s_{old})\mathcal{K}_{old}^{-1}D)^{-1}P_{old}= \ \mathcal{K}_{new}P_{new},
( \rho_1 + a A) \tilde\varphi + a A^\frac12\tilde\psi = h_1, &a A^\frac12\tilde\varphi + ( \rho_2 + a + bA)\tilde\psi - \delta A^\gamma\theta = h_2, &\delta A^\gamma \tilde\psi +( \rho_3 + c A)\theta = h_3,
H^0\left(G_\Sigma,D^*_{\rho_{\pmb{1},2}}\right)^\vee = H^0\left(G_\Sigma, \hat{R[[\Gamma]]}(\tau^{-1}\varkappa \kappa) \right)^\vee \cong \frac{R[[\Gamma]]}{\left(\tau^{-1}\varkappa \kappa(\gamma_0)-1\right)}.
\prod_{\alpha > 0}\frac{[ \langle w_{i}( \mu ) + w_{i}( \rho ) , \alpha \rangle ]}{[ \langle \rho , \alpha \rangle ]}= - \prod_{\alpha > 0}\frac{[ \langle \mu + \rho , \alpha \rangle ]}{[ \langle \rho , \alpha \rangle ]},
g_{l}^{\left(j\right)}=\frac{1}{2\chi}\left\{\left(\mathcal{D}_{l,m}^{(j)},\mathcal{D}_{l,m}^{(j)}\right)\left\vert \left(\mathcal{B}_{l,m}^{(j)},\mathcal{D}_{l,m}^{(j)}\right) \right\vert^{-2}+\Re\left[\gamma_{l,m}^{(j)}\right] \right\}.
{\psi_{x,y,-1}}={\psi_{x,y,1}}+ \sqrt{{{\left({{\psi_{x + 1,y,0}}-{\psi_{x - 1,y,0}}}\right)}^2}+{{\left({{\psi_{x,y + 1,0}}-{\psi_{x,y - 1,0}}}\right)}^2}}\tan \left({\frac{\pi}{2}- \theta}\right),
\Omega_0= (-1)^{m+1}{2m \choose m}&\Omega_1= -2+ (-1)^{m+1}{2m \choose m}&\Omega_2=4m+ (-1)^{m+1}{2m \choose m}\intertext{and}\Omega_{\Phi(n)-1}= 2 &\Omega_{\Phi(n)-2}=-2n+2 &\Omega_{\Phi(n)-3}= n^2-3n+2 .
\begin{cases}\rho_{2n-j-2}(v_1,u_2,\ldots,v_{2n-1},u_{2n},v_{2n})=0, \rho_{2n-j-1}(u_1,v_1,\ldots,u_{2n},v_{2n})=0, \rho_{2n-j-2}(v_1,u_2,\ldots,v_{2n-1},u_{2n})=0, \rho_{2n-j-1}(u_1,v_1,\ldots,u_{2n-1},v_{2n-1},u_{2n})=0, \end{cases}0\leq j\leq s.
\|(-\Delta)^{\frac{1}{4}}Q\|_{L^{2}}^{2}= \|Q\|_{L^{2}}^{2}= \frac{a_{*}}{2}\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\frac{|Q(x)|^{2}|Q(y)|^{2}}{|x-y|}{\rm d}x{\rm d}y=1.
(1+i |\nabla|^{-1}\partial_t)(u_{\lambda_1}v_{\lambda_2})={}\big( (1+i |\nabla|^{-1}\partial_t) u_{\lambda_1}\big) v_{\lambda_2}-i |\nabla|^{-1}\mathcal{M}( |\nabla|^{-1}\partial_tu_{\lambda_1},v_{\lambda_2}) &{}+i|\nabla|^{-1}( u_{\lambda_1}\partial_t v_{\lambda_2})-i|\nabla|^{-1}( |\nabla|^{-1}\partial_t u_{\lambda_1}|\nabla| v_{\lambda_2}).
d(o,x)\int_{\varepsilon D^2}^{b D^2}\sqrt{\frac{d}{et}}h_{t/2}(o,o) dt \leq d(o,x)(b-\varepsilon)D^2\sqrt{\frac{d}{e\varepsilon D^2}}h_{\varepsilon D^2/2}(o,o) \leq C' \frac{d(o,x)}{D}\frac{D^2}{n}\leq \varepsilon \frac{D^2}{n}.
C_1 \max_{j=1,\ldots,N}|p_j(\mathcal A)|^{\frac{1}{d_j}}\leq \inf_{M \in G}||\rho_M \mathcal A|| \leq C_2 \max_{j=1,\ldots,N}|p_j(\mathcal A)|^{\frac{1}{d_j}}\mbox{for all}\mathcal A \in V.
\omega = \partial_{z}\log{\frac{E ( z , P_{+})}{E ( z , P_{-})}}- 2 \pi i \sum_{j , k = 1}^{h}\left( \mathrm{I m}\, \int_{P_{-}}^{P_{+}}\omega_{j}\right){\Omega_{j k}^{( 2 )}}^{- 1}\omega_{k}( z ) ,
\begin{multlined}C_n=2^{n-4}(n-4)^2 \dfrac{\omega_{n-1}}{\omega_n}\int_0^\infty \dfrac{r^{\frac{n-2}{2}}\ln (1+r)}{(1+r)^n}dr +\dfrac{(n-4)^2}{8(n-2)}(1-\dfrac{1}{2}\ln \sqrt{n(n-4)(n^2-4)}).\end{multlined}
\lim_{V \to \infty}\omega^{'}\left\{\left(\frac{1}{V}\,\int_{\Lambda}d{x}\ \tau_{{x}}(A)\right)^{2}-\left(\frac{1}{V}\, \int_{\Lambda}d{x}\ \omega^{'}(\tau_{{x}}(A))\right)^{2}\right\}= 0 \ .
{\cal H}_{\mathrm{TCS}}(s,w,E_{\mathrm{cut}})=\left\{\left| \Psi \right\rangle :\: \left( L_{0}-\bar{L}_{0}\right) \left| \Psi \right\rangle =s\left| \Psi \right\rangle ,\: Q_{k}\left| \Psi \right\rangle =w\left| \Psi \right\rangle ,H_{\mathrm{CFT}}\left| \Psi \right\rangle \leq E_{\mathrm{cut}}\left| \Psi \right\rangle \right\}\: ,
(r^{n-1}(g^m)')'=-\alpha r^{n-1}h_1(r)\le 0\qquad\forall \xi_0<r<R_0 \Rightarrow\quad r^{n-1}(g^m)'(r)\le\xi_0^{n-1}(g^m)'(\xi_0)<0\quad\,\forall \xi_0<r<R_0 \Rightarrow\quad g'(r)<0\qquad\qquad\qquad\qquad\qquad\,\,\,\forall \xi_0\le r<R_0.
\begin{cases}c_0 \leq u^k+u_s,v^k+v_s,\phi^k+\phi_s \leq c_1, (u^k+u_s)^2 - g(\phi^k+\phi_s)\geq c_2^2,\,(v^k+v_s)^2 - g(\phi^k+\phi_s)\geq c_2^2;\end{cases}
q^0 = 1, q^{\nu_1 N}q^{\nu_2 N}= q^{(\nu_1 + \nu_2)N}, q^{\nu N}b^\dagger q^{-\nu N}= q^\nu b^\dagger, q^{\nu N}b q^{-\nu N}= q^{-\nu}b, b^\dagger b = \frac{q^N - q^{- N}}{q - q^{-1}}, b b^\dagger = \frac{q q^N - q^{-1}q^{- N}}{q - q^{-1}}.
B = \biggl(\sum_{l=0}^{n-1}\kappa^{(l)}\tilde{W}D^{(2l+1)}(\frac{{\gamma}^{st}}{\Delta})+\sum_{l=0}^{n}\kappa^{(l) \prime}W D^{(2l)}(\frac{{\gamma}^{st}}{\Delta})\biggr)W^{-1}.
\begin{aligned}\sum_{j=0}^{n}(\lambda-\delta(x_j))x_j =\sum_{j=0}^{n-1}(\lambda-\delta(x_j))x_j + (\lambda-\delta(x_n)) x_n \geq (1+\delta(x_{n-1}))x_n+ (\lambda-\delta(x_n)) x_n = (1+\lambda+\delta(x_{n-1})-\delta(x_n)) x_n \geq(1+\lambda)x_n= (1+\delta(x_n))x_{n+1}.\end{aligned}
\widetilde{\sigma}_{ji}=(2\,\mu^*+\varkappa) ( u_{i,j}+u_{j,i})+\varkappa \, \epsilon_{jim}\Big(\underbrace{\frac{1}{2}\epsilon_{mks}u_{s,k}}_{r_m}-\vartheta_m\Big)+\lambda \, u_{k,k}\delta_{ji}, *m_{ji}=\beta \,\vartheta_{j,i}+\gamma\, \vartheta_{i,j}+ \alpha \, \vartheta_{k,k}\delta_{ji}\, .
\alpha(d,d, x_3, x_4, x_5) = \sum_{i \in [q], j \in [t]}\lambda_{ij}\tau_i(d, d, x_3) \tilde{\gamma}_j(x_4, x_5) + \sum_{i \in [q], j \in [t]}\lambda'_{ij}\tau_i(d, d, x_4) \tilde{\gamma}_j(x_3, x_5) + \sum_{i \in [q], j \in [t]}\lambda''_{ij}\tau_i(d, d, x_5) \tilde{\gamma}_j(x_3, x_4)
\widehat{R}_{\nu}(S)=\left( \frac{1}{2\pi\hbar}\right)^{n}\frac{i^{\nu}}{\sqrt{|\det(S-I)|}}\int_{\mathbb{R}^{2n}}e^{\frac{i}{2\hbar}\Phi(S)z^{2}}\widehat{T}(z)dz.
\lim_{m\rightarrow+\infty}\frac{x_{1}^{\left(t_{0}+m\right)}}{\varpi\circ W\left(z^{\left(t_{0}+m\right)}\right)}=\begin{cases}\gamma_1 \mbox{if}\gamma_{2}=0,\delta_{1}\neq0, \frac{\gamma_1x_{1}^{(t_{0})}}{x_{1}^{\left(t_{0}\right)}+x_{2}^{\left(t_{0}\right)}}\mbox{if}\gamma_{2}=\delta_{1}=0, \qquad0 \mbox{if}\gamma_{2}\neq0,\delta_{1}=0,\end{cases}
f^{*}H\equiv \sum_{j=1}^{r}\sum_{k=1}^{r}c_{j}a_{kj}D_{k}=\left< A\left( \begin{array}{c}c_{1}c_{2}\vdots c_{r}\end{array}\right),\left(\begin{array}{c}D_{1}D_{2}\vdots D_{r}\end{array}\right)\right>=\left<A\vec{c},\vec{D}\right>.
\lbrace S_i,S_j\rbrace = \varepsilon_{ij}^{\ \k}S_k,\ lbrace R_i,S_j\rbrace = \varepsilon_{ij}^{\ \k}R_k,\ lbrace R_i,R_j\rbrace = -d(M)\varepsilon_{ij}^{\ \k}S_k.
{\widetilde{\cal C}}_{T}^{\prime}= 2 b{\widehat c}\sum_{\alpha , \beta}2 B^{2}\left( q_{\alpha}^{2}+ q_{\beta}^{2}\right) \Delta_{2}( X_{\alpha \beta}^{2}) \Big |_{\sigma}- 8 d^{2}\sum_{\alpha , \beta}2 B^{2}q_{\alpha}q_{\beta}\Delta_{2}( X_{\alpha \beta}^{2}) \Big |_{C_{\mu \nu}}~ .
T^{*3(j-l-2)}x = \sum_{i=0}^{3l+6}\alpha_{i}e_{i}, T^{*3(j-r-l-2)}y = \sum_{i=0}^{3l+6}\beta_{i}e_{i}\quad T^{*3(n-2j+r-l-3)}z = \sum_{i=0}^{3l+6}\gamma_{i}e_{i}.
{\mathcal{F}^{N}}[ g ] ={\frac{1}{1 6 \pi}}\int{d^{2}}\xi^{\prime}{d^{2}}\xi^{\prime \prime}\sqrt{g ( \xi^{\prime})}{J^{N}}( \xi^{\prime}) G ( \xi^{\prime}, \xi^{\prime \prime}) \sqrt{g ( \xi^{\prime \prime})}{J^{N}}( \xi^{\prime \prime})
\varphi (g^{n}_1 g_2) \overset{{\bf A3}}{=}\varphi (g_2) \varphi (g^{n}_1) \overset{{\bf A3}}{=}\varphi (g_2) \varphi \big( g^{j}_1 \big) \varphi \big( g^{i}_1 \big) \overset{(\ref{eq:9})}{=}\varphi \big( g_2 g^{j}_1 \big) \varphi \big( g^{i}_1 \big) \,,
\dot{Q}_{B}= \frac{q^{2}\dot{c}( p^{2}- m^{2})}{1 + q^{2}}( 1 - q^{2}) \quad \mathrm{a n d}\quad \dot{Q}_{r}= \frac{d}{d \tau}( \lambda e ) \frac{p^{2}- m^{2}}{1 + q^{2}}( 1 - q^{2}) .
\psi\left( m ,z\right) = N_m \sqrt{\left| z\right| +\frac{1}{k}}\left[ Y_2 \left( m\left| z\right| +\frac{1}{k}\right) +\frac{4k^2}{\pi m^2}J_2\left( m\left( \left| z\right| +\frac{1}{k}\right)\right)\right] ,
\sqrt{\|u(t)\|_{L^2}^2 + \|v(t)\|_{L^2}^2}\leq e^{-\gamma t}\sqrt{\|u_0\|_{L^2}^2 + \|v_0\|_{L^2}^2}+ 2\frac{\|f \|_{L^2}+ \|g\|_{L^2}}{\gamma}(1 - e^{-\gamma t}) \leq C = C(\|u_0\|_{L^2}, \|v_0\|_{L^2}, (\|f\|_{L^2}+ \|g\|_{L^2}), \gamma).
\frac{|B_{[H]}|}{|[H]|}= \sum_{H \le K \le G}\mu(H,K) q^{n/|K|}&= q^{n/m}- |S_H| q^{n/p_H}+ \sum_{K \notin S_H}\mu(H,K) q^{n/|K|}&= q^{n/m}- (|S_H| - \sum_{K \notin S_H}\mu(H,K) q^{n/|K| - n/p_H}) q^{n/p_H}.
S^2=\Delta\left( Y^\prime -\frac{2C_2+\frac{1}{\Delta}( a C_3 + \frac{\tilde{d}\, C_4}{D-2}) r^{\tilde{d}}}{r(C_2+C_1r^{2\tilde{d}})}\right)^2 +\frac{K}{(C_2+C_1r^{2\tilde{d}})^2}r^{2\tilde{d}-2},
\begin{aligned}\bigl|[V(t)w(t)]_{x=0}\bigr| \lesssim t^{-\frac14}\|w(t)\|_{H^1}, \|V(t)w(t)\|_{L^\infty}\lesssim \|w(t)\|_{L^\infty}+ t^{-\frac14}\|w(t)\|_{H^1}, \|V(t)w(t)\|_{\dot H^1}\lesssim \|w(t)\|_{H^1}.\end{aligned}