harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
M_{jk}:= E_{j+2,k+2}-E_{k+2,j+2}, 1\leq j<k\leq n, X_j := E_{j+2,1}-E_{j+2,2}+E_{1,j+2}+E_{2,j+2}, 1\leq j\leq n, \overline{X}_j := E_{j+2,1}+E_{j+2,2}+E_{1,j+2}-E_{2,j+2}, 1\leq j\leq n.
(zxe_{1}+x^{2}e_{2})[X+1] =zxp_{1}[X+1]+\frac{x^{2}}{2}(p_{1}^{2}-p_{2})[X+1] =zx(p_{1}+1)+\frac{x^{2}}{2}((p_{1}+1)^{2}-(p_{2}+1)) =zx(p_{1}+1)+\frac{x^{2}}{2}(p_{1}^{2}+2p_{1}-p_{2}).
\sum_{k=0}^{n-1}\frac{(q^{d+r};q^d)_k^{d-r}(q^r;q^d)_k^{r-1}(q^{r-d};q^d)_kq^{dk}}{(q^d;q^d)_k^d}&=\frac{[d]}{[r]}\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}-\frac{[d-r]}{q^{-r}[r]}\sum_{k=0}^{n-1}\frac{(q^{d+r};q^d)_k^{d-r-1}(q^r;q^d)_k^{r+1}q^{dk}}{(q^d;q^d)_k^d}.
L(\zeta_{i_3+1}-\zeta_{i_1+1})\frac{\zeta_j -\zeta_{i_2+1}}{\zeta_j -\zeta_{j+1}}+ L(\zeta_{i_1+1}-\zeta_{i_2+1})\frac{\zeta_j -\zeta_{i_3+1}}{\zeta_j -\zeta_{j+1}}= -L(\zeta_{i_2+1}-\zeta_{i_3+1})\frac{\zeta_j -\zeta_{i_1+1}}{\zeta_j -\zeta_{j+1}},
K_{u v}^{x}k_{\mathbf{I}}^{u}k_{\mathbf{J}}^{v}-{\frac{\lambda}{2}}\varepsilon^{x y z}{\cal P}_{\mathbf{I}}^{y}{\cal P}_{\mathbf{J}}^{z}={\frac{1}{2}}f_{\phantom{\mathbf{K}}\mathbf{I}\mathbf{J}}^{\mathbf{K}}{\cal P}_{\mathbf{K}}^{x}
y_{jk}(x,\l) = \left(\delta_{jk}-i b_j \int_{a_{jk}}^x e^{i (b_k-b_j) \l (t-x)}\left(q_{jk}(t) +\sum_{l=1}^n q_{jl}(t)\rho_{lk}(t,\l)\right)dt\right)e^{i b_k \l x}.
(\begin{matrix}1&1&1&0&0&0&0\\ 1&0&0&1&1&0&0\\ 1&0&0&0&0&1&1\\ 0&1&0&1&0&1&0\\ 0&1&0&0&1&0&1\\ 0&0&1&1&0&0&1\\ 0&0&1&0&1&1&0\end{matrix}).
\varepsilon_\mu(k,0)=\left(\begin{array}{c}1 0 0 0\end{array}\right)\,\,,\,\,\varepsilon_\mu(k,1)=\left(\begin{array}{c}0 1 0 0\end{array}\right)\,\,,\,\,\varepsilon_\mu(k,2)=\left(\begin{array}{c}0 0 1 0\end{array}\right)\,\,,\,\,\varepsilon_\mu(k,3)=\left(\begin{array}{c}0 0 0 1\end{array}\right)\,\,.
\mathcal{F}\left[ |x|^{-\beta}\right] (\xi )=2\Gamma (1-\beta )\sin \frac{\beta \pi}{2}\frac{1}{|\xi |^{1-\beta}}, \mathcal{F}\left[ \mathcal{E}_{x}^{\beta}u\left( x\right) \right] (\xi )=\mathrm{i}\frac{\xi}{|\xi |^{1-\beta}}\sin \frac{\beta \pi}{2}\hat{u}(\xi).
\Big[ \sum_{l=M}^{L-M}{\sum_{P\in\mathcal{Q}(l)}{\int_{P}{\int_{\Omega}{\Big( \sum_{n=1}^{L-l}{2^{-2{s}{t_n}}\sum_{Q\in\mathcal{V}_n(l,P)}{\theta_Q(\omega)\big|{\Gamma}_{{t_l}}\ast\chi_Q(x)\big|^2}}\Big)^{{q}/{2}}}d\lambda}dx}}\Big]^{{1}/{q}}.
b_{\alpha}(\varphi,\psi) = \int_{\mathcal{D}\times\mathcal{D}}k(x,y)\partial^{\alpha}\varphi(x)\partial^{\alpha}\psi(y)dxdy = \int_{\mathcal{D}\times\mathcal{D}}\partial^{\alpha,\alpha}k(x,y)\varphi(x)\psi(y)dxdy = \langle \mathcal{E}_k^{\alpha}\varphi,\psi\rangle_{L^2}
\theta(n_s^{-1}(\lambda))T_{n_s}- T_{n_s}\theta(n_s^{-2}(\lambda)) = \sum_{k = 0}^{\langle \nu(\lambda),\alpha\rangle - 1}\theta(n_s^{-1}(\lambda)\mu_{n_s}(k))c_{n_s,k}& = \sum_{k = 1}^{\langle \nu(\lambda),\alpha\rangle}c_{n_s,k}\theta(\mu_{n_s}(-k)n_s^{-2}(\lambda)).
\begin{array}{rcl}\psi^j = h^{j+1}\circ d^j + d^{j-1}\circ h^j h^j((s \otimes t) \cdot p) = (s \otimes t) \cdot h^j(p) - (1 \otimes \partial)(s \otimes t) \cdot k^j(p)\,. \end{array}
\left(S^\textnormal{coh}\right)^{-1}(\tau,z)(T_a) = \sum_{j=0}^N \textbf{g}\left( e^{\tau_2/z}T_j + \sum_{\substack{d \in H_2(X;\mathbb{Z})-\{0\}l \geq 0}}\sum_k \frac{1}{l!}e^{\tau_2(d)}\left \langle \frac{e^{\tau_2/z}T_j}{-z+\psi}, T_k, \tau', \dots, \tau' \right \rangle^\textnormal{coh}_{0,l+2,d}T^k , T_a \right) T^j
\mathbf{Oper}^{\mathsf{alg}}_{\mathcal G}:= \mathbf{PSh}(\widetilde{\mathbb G}_{\mathcal G}\rightrightarrows\widetilde{\mathbb E}_{\mathcal G})\times_{\mathbf{Cat}^{/\widetilde{\mathbb E}_{\mathcal G}}}\mathbf{Oper}'_{\mathcal G}\ ,
\|x+y\|^2 =\|t\tfrac{1}{t}x+(1-t)\tfrac{1}{1-t}y\|^2\leq t\|\tfrac{1}{t}x\|^2+(1-t)\tfrac{1}{1-t}\|y\|^2 &=\tfrac{1}{t}\|x\|^2+\tfrac{1}{1-t}\|y\|^2.
\int^{Y(z)}_{Y(w)}\frac{dt}{P(t)}=-\frac{2}{\sqrt[3]{4}}\left(B\left(m^{*}(2z)^2;\frac{1}{6},\frac{2}{3}\right)-B\left(m^{*}(2w)^2;\frac{1}{6},\frac{2}{3}\right)\right)
\inf_{w \in \mathcal X}R(w) = C_m^{-1}> 0, \inf_{w \in \mathcal X}\|w\|_{L^m(\Omega)}\geq (\lambda_m C_m^2)^{-1/(m-2)}> 0, R(w) \leq (\lambda_m C_m^2)^{1/(m-2)}\|w\|_{H^1_0(\Omega)}\mbox{for all}\w \in \mathcal X.
V(x,i)=\sup_{D,R}\mathrm{E}_{x,i}\left [ \int_{0}^{e_{\lambda_{i}}}e^{-\int_{0}^{t}\delta_{Y_s}\mathrm{d}s}\mathrm{d}D_{t}-\phi\int_{0}^{e_{\lambda_{i}}}e^{-\int_{0}^{t}\delta_{Y_s}\mathrm{d}s}\mathrm{d}R_{t}+e^{-\int_0^{e_{\lambda_{i}}}\delta_{Y_s}ds}V(U_{e_{\lambda_{i}}}, Y_{e_{\lambda_{i}}})\right],
x\frac{\mathrm{d}y}{\mathrm{d}x}=xy-2v(y-1)^2+2\Theta(y-1), &x\frac{{\rm d}v}{{\rm d}x}=yv^2-\frac{1}{y}(v+\Theta)^2, &x\frac{{\rm d}}{{\rm d}x}{\rm ln}u =-2v-\Theta+yv+\frac{1}{y}(v+\Theta).
[\begin{matrix}x_{1}\\x_{2}^{*}\end{matrix}]=[\begin{matrix}h_{1}&-h_{2}\\h_{2}^{*}&h_{1}^{*}\end{matrix}][\begin{matrix}S_{1}\\S_{2}^{*}\end{matrix}]+[\begin{matrix}n_{1}\\n_{2}^{*}\end{matrix}]
\begin{aligned}C_{\bullet}(W^{u}_{Z}, F^{*})=\bigoplus_{x\in B}F^{*}_{x}\otimes o^{u}_{x}C_{i}(W^{u}_{Z}, F^{*})=\bigoplus_{x\in B^{i}}F^{*}_{x}\otimes o^{u}_{x},\end{aligned}
k_1^\mu T_{\rho \sigma \mu \nu}^{(1)abcd}(p_1,p_2;k_1,k_2)=C_1k_{2\nu}S_{\rho \sigma}^{(1)}(q_1)+C_1\Gamma_{\sigma \nu \rho}^{(1)}(p_2,k_2,q_1)
\|\mathbf{E}_{}^T(\boldsymbol{\alpha}^{k+1}- \boldsymbol{\alpha}^{\star})\|^2 = \|\nabla f(\mathbf{x}^{k+1}) - \nabla f(\mathbf{x}^{\star}) + \frac{\rho (1-\eta)}{2}\mathbf{E}_{}^T\mathbf{E}_{}(\mathbf{x}^{k+1}- \mathbf{x}^{\star}) + \mathbf{M}(\mathbf{x}^{k+1}- \mathbf{x}^k)\|^2,
\det A^{\widetilde{K}}(\widetilde{G})=\begin{vmatrix}M_1 iM_2\cr iM_2 M_1\end{vmatrix}=\begin{vmatrix}M_1+iM_2 iM_2\cr M_1+iM_2 M_1\end{vmatrix}=\begin{vmatrix}M_1+iM_2 iM_2\cr 0 M_1-iM_2\end{vmatrix}=\left|\det A^{K,\omega}(G)\right|^2\,.
\qquad\qquad\frac{(\frac{1}{4})_k(\frac{3}{4})_k}{(1)_{k}^2}=\frac{\binom{2k}{k}\binom{4k}{2k}}{64^k}, begin{align*}1mm] H_k(-\tfrac{1}{4})+H_k(-\tfrac{3}{4})=4H_{4k}-2H_{2k},
\hat{R}_{i j , k l}^{\epsilon}= \epsilon_{j s}^{t}\hat{R}_{i s , k t}( \epsilon^{t})_{t l}^{- 1}\quad ( \hat{R}^{\epsilon}= ({\bf 1}\otimes \epsilon^{t}) \hat{R}({\bf 1}\otimes ( \epsilon^{- 1})^{t}) ) ,
{\ddot{a}_0\over a_0}=-{{16\pi^2G^2_5}\over{9c^6}}(2\varrho^2_{\rm tot}c^4+\sigma\varrho_{\rm tot}c^2+3\sigma\wp_{\rm tot}+3\wp_{\rm tot}\varrho_{\rm tot}c^2)-c^2{a^{\prime\prime}_{R\,0}\over a_0}+{{16\pi^2G^2_5}\over{9c^6}}\sigma^2,
\begin{aligned}\langle DX_{i},DX_{j}\rangle_{\mathcal{H}}=\mathbb{E}[X_{j}X_{k}] =\frac{1}{2}\left[\left(k-j+1\right)^{2H}+\left(k-j-1\right)^{2H}-2\left(k-j\right)^{2H}\right].\end{aligned}
{C(1,1)\over2^{5/2}}=\int_0^1\frac{dx}{\sqrt{3-x^2}}\,\left(\log\frac34+\log\frac{3+x}{2+x}-\frac{x^2}{4-x^2}\log\frac{4}{2+x}+\frac{x}{2+x}\log\frac{3+x}{3}\right)
\mathcal{P}_c(\theta_c, \tau) &= \sum_{k \in \mathcal{K}}\mathcal{A}_k \mathbb{E}_{D_{\mathbf{b}_0, \mathbf{u}_0}}[\mathbb{P}(_U \geq \theta_c | \mathbf{b}_0 \in \Phi_k, D_{\mathbf{b}_0, \mathbf{u}_0}= r)].
M(t)\left[p(t)+(1-p(t))\frac{|\mathcal{A}_i\setminus\mathcal{A}_{3-i}|}{M(t)-H(t)}\right]= |\mathcal{A}_1\cap\mathcal{A}_2|+|\mathcal{A}_i\setminus\mathcal{A}_{3-i}|=|\mathcal{A}_i|, i= 1,2.
\mu_{k}^{\prime \prime}+ \biggl \{k^{2}\biggl [ \frac{\epsilon_{1}}{1 + \mathrm{e}^{x}}+ \frac{\epsilon_{3}\mathrm{e}^{x}}{( 1 + \mathrm{e}^{x})^{2}}\biggr ] - ( 1 - 6 \xi ) \frac{\beta ( \beta + 1 )}{\eta^{2}}\biggr \}\mu_{k}= 0 \, ,
\sum_{\substack{h=1 h\neq\frac{d}{2}}}^{d-1}\lfloor\langle \frac{-hp^i}{d}\rangle+\frac{mp^i}{q-1}\rfloor=\sum_{\substack{h=1 h\neq\frac{d}{2}}}^{d-1}\lfloor\langle \frac{h}{d}\rangle+\frac{mp^i}{q-1}\rfloor,
\langle u(t),\varphi \rangle - \langle f,\varphi\rangle & + \mu\int_0^t \langle u,\varphi \rangle ds + \int_0^t \bigl \langle b \cdot \nabla u,\varphi \bigr\rangle ds - \sigma \int_0^t \langle u,\nabla \varphi \rangle dB_s + \frac{\sigma^2}{2}\int_0^t \bigl\langle u, \Delta \varphi\bigr\rangle ds =0 .
(w,v)_{\mathcal{T}_h}:= \sum_{K\in\mathcal{T}_h}(w,v)_K = \sum_{K\in\mathcal{T}_h}\int_K w v , \left\langle \zeta,\rho\right\rangle_{\partial\mathcal{T}_h}:= \sum_{K\in\mathcal{T}_h}\left\langle \zeta,\rho\right\rangle_{\partial K}= \sum_{K\in\mathcal{T}_h}\int_{\partial K}\zeta\rho.
G_i^p(m)=\begin{cases}2^{{m\choose 2}-i(p-1)m+\frac{i^2p(p-1)}{2}+\frac{i(p-1)}{2}}\cdot\frac{m!}{(p-1)(m-pi)!p^ii!}\textrm{if}p>2, 2^{{m\choose 2}-im+i^2+i}\cdot\frac{m!}{(m-2i)!2^ii!}\textrm{if}p=2.\end{cases}
S=2\pi\, \frac{\sqrt{(y^2+x_0^2+x_1^2)(y^2+x_0^2+x_2^2)(y^2+x_0^2+x_3^2)}}{y^2+x_0^2}- 2\pi i \frac{x_0 x_1 x_2 x_3}{y (y^2 + x_0^2)}\, .
m_{0}(l) =\sum_{(i_1,k_1,\ldots,i_l,k_l)\in I_{0}}2^{(k_1+\ldots+k_{l})r}d_{i_1}^r\cdots d_{i_l}^r\Pr(B_{i_1,k_1,\ldots,i_l,k_l}) &\leq\left(\frac{{4}C_6r\sigma_{r,X}(rl)}{t}\right)^{r(l-1)}\sum_{(i_1,k_1)\in P_1I_0}2^{k_1r}d_{i_1}^r\Pr(B_{i_1,k_1}).
\begin{array}{l}[{H}_{\pi_\alpha}]:={\dot u}^{\alpha}(x)- \frac{\partial{H}}{\partial \pi_{\alpha}}, begin{align*}2mm][{H}_{u^\alpha}]:={\dot \pi}_{\alpha}(x)+\frac{\partial{H}}{\partial u^{\alpha}}-\nabla_a (\frac{\partial{H}}{\partial(\nabla_a u^{\alpha})}).\end{array}
U(A\otimes 1+1\otimes B)U^* =A+B^{(0)}\oplus \bigoplus_{n=1}^\infty( A\otimes 1+1\otimes B^{(n)}) U A\otimes B U^* =A \otimes B= B^{(0)}A \oplus \bigoplus_{n=1}^\infty A\otimes B^{(n)}.
g_{\mu \nu}= \left( \begin{array}{c c c c}{{- N^{2}}}&{{}}&{{0}}&{{}}\\{{}}&{{}}&{{}}&{{}}\\{{0}}&{{}}&{{\frac{1}{8}e^{2 \alpha}\chi_{i A B}\chi_{j}^{A B}}}&{{}}\\{{}}&{{}}&{{}}&{{}}\\ \end{array}\right) = \left( \begin{array}{c c c c}{{- N^{2}}}&{{}}&{{0}}&{{}}\\{{}}&{{}}&{{}}&{{}}\\{{0}}&{{}}&{{e^{2 \alpha}\chi_{i}^{I}\chi_{I j}}}&{{}}\\{{}}&{{}}&{{}}&{{}}\\ \end{array}\right) ,
\sum_{\substack{p\leq n p\notin\mathcal{S}}}\beta_p\log p \leq \sum_{\substack{p \leq n p\in\mathcal{S}}}\log(2n^2+l)+\dfrac{1}{\lambda}\sum_{i=1}^{r}e_i\log p_i+\sum_{n < p < 2n}\dfrac{\alpha_p}{\lambda}\log p.
b_{\pm}(\eta,z) ~=~ \bar{b}~ \exp \left \{i k g_{\pm}(\eta) ~-~ i k z~\mp~ i \left[ \frac{p^2}{2}( h \eta ~+~ h_{0}) z^2 ) ~+~ O(p^4) \right] ~+~ O\left(\frac 1 k \right) \right \}.
0 \leq v_\nu (x_0, t_0, \xi_0) = u_{\xi_0 \xi_0 \nu}+2u_{i\xi_0}D_j{\xi_0}^i\nu^j- a^l u_{l\nu}- D_\nu a^l u_{l}- b_\nu + 2u_i u_{i\nu}+ K_1 (x \cdot \nu) \leq u_{\xi_0 \xi_0 \nu}+ C_{14}\leq -2 \kappa_{min}u_{\xi_0 \xi_0}+ C_{13}(1+ | u_{\nu \nu}|) + C_{14}.
\displaystyle{\lim_{n\rightarrow\infty}Y_{l_{1}}^{m_{1}}\star_{\vec{c}}^{n}Y_{l_{2}}^{m_{2}}}\ = & ={\displaystyle\sum\limits_{\substack{l=|l_{1}-l_{2}| l\equiv l_{1}+l_{2}(\operatorname{mod}2)}}^{l_{1}+l_{2}}}\sqrt{\frac{(2l_{1}+1)(2l_{2}+1)}{2l+1}}\C_{m_{1},m_{2},m}^{l_{1},l_{2},l}C_{0,0,0}^{l_{1},l_{2},l}\displaystyle{\frac{l}{l_{1}l_{2}}}\Y_{l}^{m}
\left[ \begin{array}{c c}{{1}}&{{0}}\\{{0}}&{{1}}\\ \end{array}\right] , \hspace{2 m m}\left[ \begin{array}{c c}{{0}}&{{1}}\\{{1}}&{{0}}\\ \end{array}\right] , \hspace{2 m m}\left[ \begin{array}{c c}{{0}}&{{i}}\\{{- i}}&{{0}}\\ \end{array}\right] , \hspace{2 m m}\left[ \begin{array}{c c}{{1}}&{{0}}\\{{0}}&{{- 1}}\\ \end{array}\right] .
(c^2+d^2+e^2 + a^2+b^2+ab)^2 - ((a-b)^2c^2+(2b+a)^2 d^2 + (2a+b)^2 e^2) = c^2(a+b)^2 + (c^2+ab)^2 + d^2a^2+ (d^2-b^2-ab)^2 +e^2b^2+(e^2-a^2-ab)^2.
\sum_{j = 1}^{n}{\frac{1 - y \operatorname{c o s}{\frac{( 2 j - 1 ) \pi}{n}}}{( y - \operatorname{c o s}{\frac{( 2 j - 1 ) \pi}{n}})^{2}}}\geq 0 , \quad \mathrm{f o r}\y \geq 1 .
a(z)= \sum_{j=0}^{d+\epsilon}\frac{a_j(\dot{z})}{|z|^{d+\alpha+j}}+O(\frac{1}{|z|^{2d+\alpha+1+\epsilon}}), |z|\to\infty, ~~~ \alpha\in (0,2), a_j\in C^{d+1-j+\epsilon}(S^{d-1}),
\frac{\partial^2 w}{\partial t\partial s}(t,s) = \langle (1,\frac{\partial}{\partial t}\mathbb{E}_{\mathbb{P}}[S(X)|\mathcal{F}_t](\omega)), (1,\frac{\partial}{\partial s}\mathbb{E}_{\mathbb{Q}}[S(Y)|\mathcal{G}_s](\omega^\prime)) \rangle_{\mathbb{R}\oplus \mathcal{H}^1}w(t,s).
\log \prod_{i=1}^\infty(1-q^{-i-1}) = \sum_{i=1}^\infty \log\left( 1-q^{-i-1}\right) \geq \sum_{i=1}^\infty -2q^{-i-1}= -\frac{2}{q(q-1)}\geq - \frac{1}{10}
Q^{-}\sim \sqrt{g^{2}N}\times \frac{K}{\sqrt{N}}\times \sum_{n , l = 1}^{\infty}\left( \frac{1}{n}+ \frac{1}{l}\right) \times \left( \Gamma^{\dagger}( l + n ) \Gamma ( l ) \Gamma ( n ) + \Gamma^{\dagger}( n ) \Gamma^{\dagger}( l ) \Gamma ( n + l ) \right) ,
\int_\Omega|\nabla_Hv|^4dxdydz=-\int_\Omega\nabla_H\cdot(|\nabla_Hv|^2\nabla_Hv)vdxdydz \leq C\int_\Omega|\nabla_Hv|^2|v||\nabla^2_Hv|dxdydz\leq C\|\nabla_Hv\|_4^2\|v\|_6\|\nabla_H^2v\|_3
G^{2}=(\begin{matrix}1&0&1&0\\ 0&1&0&1\\ 0&1&1&0\\ 1&0&1&1\end{matrix})G^{3}=(\begin{matrix}0&1&0&1\\ 1&1&2&0\\ 1&0&1&1\\ 0&2&1&1\end{matrix})
C^{\pi}_t(y_{t-J}^{t-1})= \sup_{\big\{r_i(x_i|y_{i-M}^{i-1},y_i):~i=t,t+1,\ldots,n\big\}}{\bf E}^{\pi}\bigg\{\sum_{i=t}^n\log\Big(\frac{r_i(X_i|y_{i-M}^{i-1},Y_i)}{{\pi}_{i}(X_i|y_{i-J}^{i-1})}\Big)\Big{|}Y_{t-J}^{t-1}=y_{t-J}^{t-1}\bigg\},~t\in\mathbb{N}_0^n
g \simeq \tilde{\Lambda}^{2}( - A^{2}\eta^{2}d s^{2}+ d \eta^{2}+ d \rho^{2}) + \tilde{\Lambda}^{- 2}\rho^{2}d \phi^{2}, \qquad \tilde{\Lambda}\equiv 1 + \frac{B^{2}\rho^{2}}{4},
\int_{-\infty}^\infty \frac{dx}{\sqrt{\pi}}\, e^{i\lambda x^2}=\lim_{\epsilon \rightarrow 0}\int_{-\infty}^\infty \frac{dx}{\sqrt{\pi}}\, e^{i\lambda x^2}\,e^{-\epsilon x^2}=\frac{1}{|\sqrt{\lambda}|}\exp \left( \frac{i\pi}{4}\mbox{sgn}\, \lambda\right),
\frac{\partial Q}{\partial t}=\Delta(1+\frac{\alpha'}{2}R)Q+2\langle\nabla Q,\nabla L\rangle+2\big|\nabla\nabla L+\frac{1}{2}\big(R+\frac{\alpha'}{4}R^2-r\big)g\big|^2+\big(\frac{\alpha'}{4}R^2+r\big)Q-\frac{\partial r}{\partial t}.
c_1^{11}(A) = \bar{\gamma}\left( c_1^{11}(A^1) + \bar{a}_{11}^1 \int_Y r^1 A^1 e_1\cdot \nabla w\right) = s\,\bar{a}_{11}^1 \int_Y r^1 A^1 e_1\cdot \nabla q \neq 0,
\forall \, (t,x) \in \mathfrak{C}(a,b)^c \left| \int_{-N}^N U(p) \, e^{i t \psi(p)}\, dp \right| \leqslant \left| \int_{-N}^{0}\dots \right| + \left| \int_0^N \dots \right| & \leqslant \big(c_{-N}^c(u_0,f) + c_{+N}^c(u_0,f) \big) \, t^{-\mu}& =: c_1^c(u_0,f) \, t^{-\mu}\; ,
\mathcal{Li}^{\ast}_{K,\mathfrak{s}}({\bf z}):=\sum_{i_1\geq i_2\geq \cdots\geq i_r>0}\frac{z_1^{q^{i_1}}z_2^{q^{i_2}}\cdots z_r^{q^{i_r}}}{(\theta^{q^{i_1}}-t)^{s_1}(\theta^{q^{i_2}}-t)^{s_2}\cdots (\theta^{q^{i_r}}-t)^{s_r}}\in\mathbb{T}
\begin{array}{r c l}\psi'(t,x) = e^{-ie\epsilon(t,x)}\psi(t,x)=e^{-i\frac{2\pi k}{L}x}\psi(t,x)\ ,{A^1}'(t,x) = A^1(t,x)-\partial_1\epsilon(t,x)=A^1(t,x)-\frac{2\pi k}{eL}\ , \pi'_1(t,x) = \pi_1(t,x)\ ,\{A^0}'(t,x) = A^0(t,x)+\partial_0\epsilon(t,x)=A^0(t,x)\ .\end{array}
\frac{1}{c_{n}-c_{n-1}}- \frac{1}{c_n}\ = \ \frac{c_{n-1}}{c_n^{2}-c_{n}c_{n-1}}&\ = \ \frac{c_{n-1}}{c_{n-1}(c_{n+1}-c_{n})+s c_{n}}\ <\ \frac{1}{c_{n+1}-c_{n}}.
a = \dfrac{(r-p)i-q j}{r(r-p-q)_p}, b = \dfrac{(r-q)j-p i}{r(r-p-q)_q}, a' = \dfrac{(r-p)i'-q j'}{r(r-p-q)_p}, b' = \dfrac{(r-q)j'-p i'}{r(r-p-q)_q}.
\rho_{\{j\}}(\Delta) = \frac{1}{2}\log \left ( \frac{\sigma_{{j}}^2 + \sum \limits_{i \neq j}r_{ij}^2 \sigma_{i}^2}{\Delta - \Delta_{\min, \{j\}}}\right ) = \frac{1}{2}\log \left ( \frac{\sum \limits_{i=1}^{m}\sigma_i^2 - \Delta_{\min, \{j\}}}{\Delta - \Delta_{\min, \{j\}}}\right )
\frac{dJ}{dS_{th}}=\lambda+\int\limits_{0}^{R}\left(\frac{\left(K(T,r)-\ln{(1+S_{th})}\right)\sum\limits_{i\in \psi}r^\alpha d_i^{-\alpha}\left ( \underset{j \in \psi\setminus i}\prod{(1+S_{th}r^\alpha d_j^{-\alpha}})\right)}{\left(\underset{j \in \psi}\prod{(1+S_{th}r^\alpha d_j^{-\alpha})}\right)^2}\right)f_R(r)r=0
\lambda^{2 n}H , E_{\alpha}, \lambda^{2 n}E_{\pm \alpha}, F_{\alpha_{1}}, \lambda^{2 n}F_{\pm \alpha_{1}}, F_{\alpha_{2}}, \lambda^{2 n}F_{\alpha_{2}}, \lambda^{2 n}F_{- \alpha_{2}}, \lambda H^{\prime}, \lambda^{2 n + 1}H^{\prime}, ~ ~ n > 0
\begin{cases}-\Delta u_j=f(u_j)+\lambda u_j \ \ \ \ \mathbb R^N, u_j(x)>0,\ \lim_{|x|\to\infty}u_j(x)=0,i=1,2, \cdots, \ell \sum_{i=1}^{\ell}|u_j|_2^2=\alpha. \end{cases}
F_{total}=F_{1}+\frac{F_{2}-1}{G_{1}}+\frac{F_{3}-1}{G_{1}G_{2}}+\frac{F_{4}-1}{G_{1}G_{2}G_{3}}+\cdot\cdot\cdot+\frac{F_{n}-1}{G_{1}G_{2}\cdot\cdot\cdot G_{n-1}}
\left\| R(\epsilon;t-s u_{(n)})\cdots R(\epsilon;t-su_{(1)}) -\left(\tilde{R}^n - \epsilon\sum_{i=0}^{n-1}s u_{(i+1)}\tilde{R}^{n-i-1}\tilde{R}^{(1)}\tilde{R}^{i}\right) \right\| = O(\epsilon^2 s^3)
C = - s A^{L^{\prime}}\; \; , \; \; \hspace{1 e m}\bar{{\cal P}}= - i s A_{0}\; \; , \; \; \hspace{1 e m}\mathrm{\boldmath ~ \p i ~}^{L}= \frac{\nabla}{\nabla^{2}}( s{\cal P}+ \rho^{( 2 )}) \; \; , \; \; \hspace{1 e m}\pi_{0}= i s \bar{C}\; \; ,
\frac{du}{dt}(t_0)=\displaystyle\sum_{i=0}^{k}\alpha_{k,i}\left(u(t_i)+u(t_{i+1})\right)+ \mathcal{O}\left(\displaystyle\sum_{i=0}^{k}\alpha_{k,i}\left((\Delta t_i)^{k+1}+(\Delta t_{i+1})^{k+1}\right)\right)
\frac{h}{N}|T_0^{l,(n)}| &\le (c_0 M^{a_1(l+1)})^{-n+1}(c_0 M^{-a_2(l+1)})^{n-1}\Bigg(\sum_{m=4}^N2^{2m}(c_0 M^{a_1(l+1)})^{\frac{m}{2}}\|J_{m}^{l+1}\|_{l,0}\Bigg)^n &\le M^{(a_1+a_2)(l+1)}(c M^{a_4(l+1)}\alpha^{-4})^n.
lim_{x\rightarrow0^{+}}f_{\kappa}(x)=(1+\kappa\nu)(2\kappa)^{\nu}\frac{\Gamma(\frac{1}{2\kappa}+\frac{\nu}{2})}{\Gamma(\frac{1}{2\kappa}-\frac{\nu}{2})}\frac{\alpha\beta^{\nu}}{\Gamma(\nu)}x^{\alpha\nu-1}
\quad\det V_{n}-I_{0}H_{n-1}\left(\frac{B_{2k+5}(r+1)}{2k+5}\right) =H_{n}\left(\frac{B_{2k+1}(r+1)}{2k+1}\right)-\left(r+\frac{1}{2}\right)H_{n-1}\left(\frac{B_{2k+5}(r+1)}{2k+5}\right),
\left|N(f(x), r)-\mu_{k+m-r}{p \choose r}p^{k-r}\right| \leq \sum_{j=r}^{k+m}{j \choose r}{m\sqrt{p}+1+j \choose m\sqrt{p}+1}{m-1 \choose k+m-j}p^{\frac{k+m-j}2}\leq m\cdot \max_{r\leq j\leq k+m}E_j,
g^{(n)}_{\mbox{\footnotesize\boldmath $a$}}(z,v) =\sum_{\mbox{\footnotesize\boldmath $\alpha $}\in G_m}\omega^{2\langle \mbox{\footnotesize\boldmath $\alpha $},\mbox{\footnotesize\boldmath $a$}\rangle}u^{(n)}_{2\mbox{\footnotesize\boldmath $\alpha $}}(z,v)u^{(n)}_{2(\mbox{\footnotesize\boldmath $a$}+\mbox{\footnotesize\boldmath $\alpha $})}(-z,v).
z(\sigma,\bar{\sigma},\varphi)=-\frac{\sqrt{2}\bar{\sigma}}{c_0}+\frac{4}{c_0}\cos(\varphi/2)\,\,,\quad\bar{z}(\sigma,\bar{\sigma},\varphi)=\frac{\sigma}{\sqrt{2}}+\frac{4}{c_0}\cos(\varphi/2)
C_{NE}=\mathrm{Cone}\left\{\begin{aligned}(-1,\;\;\,0,\;\;\,0,1,0,0,0,1,-1), (\;\;\,0,-1,\,\;\;0,1,-1,1,0,0,0), (\;\;\,0,\;\;\,0,-1,0,0,1,-1,1,0),\end{aligned}\;\begin{aligned}(-1,\;\;\,0,\;\;\,0,0,1,-1,1,0,0) (\;\;\,0,-1,\,\;\;0,0,0,0,1,-1,1) (\;\;\,0,\;\;\,0,-1,-1,1,0,0,0,1)\end{aligned}\right\}.
R^{(k_s)}\cdot \left( \prod_{i=1}^{k_s}(1-\beta T_i)^{-\lambda_i}c_{\lambda_i}(i)\right) = R^{(k_s)}\cdot \left( \prod_{i=1}^{k_s}(1-\beta T_i)^{-\lambda_i}c'_{\lambda_i}(i)\right),
H(\hat{P}_B) = \sum_{i = 1}^S-\hat{p}_{B,i}\ln \hat{p}_{B,i}& = H(P_B) + \sum_{i =1}^S (p_{B,i}- \hat{p}_{B,i}) \ln p_{B,i}- \sum_{i = 1}^S \hat{p}_{B,i}\ln \frac{\hat{p}_{B,i}}{p_{B,i}}.
p(t,v,g)=\frac{1}{V_F}\sum_{k=-\infty}^{+\infty}\exp\left(-k^2(\frac{2\pi}{V_F})^2D(t)+i\frac{2\pi}{V_F}kv+i\frac{2\pi}{V_F}kg\right)(p_{t,k}*\bar{G}_{t,k})(g).
-\frac{\Gamma_{\rm b}\left(a=-a_l\right)}{{\cal E}_l}={\rm{\cal A}}_l=\frac{\left({\cal C}_{66}-{\cal C}_{11}\right)\left({\cal C}_{33}-a_l{\cal C}_{44}\right)+\left({\cal C}_{13}+{\cal C}_{44}\right)^2}{{\rm E}_l}
\left[ \begin{array}{c}{{A_{j m}^{( 1 )}}}\\{{A_{j m}^{( 2 )}}}\\{{A_{j m}^{( 3 )}}}\\ \end{array}\right] = R \cdot \left[ \begin{array}{c}{{a_{j m}^{( 1 )}}}\\{{a_{j m}^{( 2 )}}}\\{{a_{j m}^{( 3 )}}}\\ \end{array}\right] ,
\log M + N_1 - N_0 \le 2n \binom{2(k-t)}{k-t}- \frac{1}{n}\binom{n-k-1}{k-t}+ n &\le \left( 3n - \frac{1}{n}\left( \frac{n-k-1}{2(k-t)}\right)^{k-t}\right) \binom{2(k-t)}{k-t}&\le \left( 3n - \frac{n^2}{64 (k-t)^3}\left( \frac{t(k-t) + \eta_{k,t}}{2(k-t)}\right)^{k-t-3}\right) \binom{2(k-t)}{k-t}.
{\cal N}_M[{\cal H}_{\vec{x}}]={\frac{1}{2}}\partial_t\phi_x\partial_t\phi_x+{\frac{1}{2}}\partial_{\vec{x}}\phi_{\vec{x}}\partial_{\vec{x}}\phi_{\vec{x}}+{\cal N}_M[V(\phi_{\vec{x}})] -{\frac{1}{2}}I_0[M^2]+{\frac{1}{4}}M^2 I_1[M^2] \;,
\tilde{A}_1({\mathbf U}^{\pm},\Psi^{\pm})=\frac{1}{\partial_1\Phi^{\pm}}\Big(A_1({\mathbf U}^{\pm})-A_0({\mathbf U}^{\pm})\partial_t\Psi^{\pm}-A_2({\mathbf U}^{\pm})\partial_2\Psi^{\pm}\Big),
K (x,y,\Psi_{x,y}^{-1}(u) ) \stackrel{\eqref{sec.two_2:eq_defKernelK}}{\geq}\sum_{k = 1}^p |F_{n+k}(x,y,\Psi_{x,y}^{-1}(u)) |^{{1}/{\sigma^*_k}}\stackrel{\eqref{sec.two_2:def_mapPsixy}}{=}\sum_{k = 1}^p |\Psi_{x,y}(\Psi_{x,y}^{-1}(u) ) |^{{1}/{\sigma^*_k}}= \sum_{k = 1}^p|u_k|^{{1}/{\sigma^*_k}}{=}N(u).
{}_3F_2\bigg[\begin{matrix}\alpha \beta \gamma &1 \delta\end{matrix}\bigg|\,1\bigg]=\frac{\Gamma(\delta)\Gamma(1+\delta-\alpha-\beta-\gamma)}{\Gamma(\delta-\alpha)\Gamma(1+\delta-\beta-\gamma)}\cdot{}_3F_2\bigg[\begin{matrix}\alpha 1-\beta 1-\gamma &1 1+\delta-\beta-\gamma\end{matrix}\bigg|\,1\bigg].
q_k:= \int_{0}^{\infty}e^{-\gamma_k \sigma_k t}|\varphi'(t)|^2 + \sigma_k^2 \Bigl(\lambda \beta_k^2 e^{-\gamma_k \sigma_k t}|\varphi(t)|^2 - \bigl \langle U_k(\sigma_k t) \varphi(t),\varphi(t)\bigr \rangle\Bigr)\,dt <0
(\partial_t \phi)(\tilde X_j, \tilde Y_j, \tilde t_j) = \partial_{\tilde t}\Psi(P_j) = - \partial_t \Psi(P_j), (\nabla_X \phi)(\tilde X_j, \tilde Y_j, \tilde t_j) = \nabla_{\tilde X}\Psi(P_j) = - \nabla_X\Psi(P_j), (\nabla_Y \phi)(\tilde X_j, \tilde Y_j, \tilde t_j) = \nabla_{\tilde Y}\Psi(P_j) = - \nabla_Y\Psi(P_j).
8^2 \int_{C/\ell}^{\pi -C/\ell}\frac{\gamma_4^2}{\ell^8}\sin \phi d \phi = 8^2 h^2_0(0) \frac{1}{\ell}\int_{C/\ell}^{\pi -C/\ell}\cos^2 \psi_{0,\ell}d \phi+O(\ell^{-2})=\frac{2 \cdot 32}{\ell}+O(\ell^{-2}),
{\frac{1}{g_{1}^{2}}}+{\frac{1}{g_{2}^{2}}}={\frac{1}{4 \pi g_{s}}}; \qquad{\frac{1}{g_{1}^{2}}}-{\frac{1}{g_{2}^{2}}}={\frac{1}{4 \pi^{2}g_{s}}}\left( \frac{1}{2 \pi \alpha^{\prime}}\int_{S^{2}}B_{( 2 )}- \pi \right) .
\sum_{k=1}^{n}\frac{q^k}{1-q^k}-\sum_{k=1}^{n-1}\frac{xq^k}{1-xq^k}=\frac{x}{1-x}-\frac{1}{(x)_n}\sum_{k=1}^{n}\left[\begin{matrix}n k\end{matrix}\right]\frac{\left(\frac{q}{x}\right)_k(x)_{n-k}x^k}{1-q^k}.
\tau_{W}\left( q_{\zeta_{1}}q_{\zeta_{2}}\right) =\left( 1+\varphi_{W}\left( \zeta_{1}\right) \right) \left( 1+\varphi_{W}\left( \zeta_{2}\right) \right) \dfrac{m_{W}\left( \zeta_{1}\right) -m_{W}\left(\zeta_{2}\right)}{\zeta_{1}-\zeta_{2}}
-\int_0^L (\psi'+ \psi''')y_0 dx -\frac{1}{2}\int_0^L \psi' y^2_0 dx + \int_0^L \Big(\dot m_1(0)\varphi_1(x) + \dot m_2(0)\varphi_2(x) +\frac{\partial g}{\partial m_1}( \mathbf{m}^0) \dot m_1(0) + \frac{\partial g}{\partial m_2}( \mathbf{m}^0)\dot m_2(0)\Big)\psi dx = 0.
E[\hat\beta_l (\alpha_{l,{{n}_t}}- \alpha_{l,{\hat{n}_t}}e^{-j\omega})] = \frac{\sqrt{\pi}}{2}E(\hat\beta_l), &Var[\hat\beta_l (\alpha_{l,{{n}_t}}- \alpha_{l,{\hat{n}_t}}e^{-j\omega})] =2-\frac{\pi}{4}E^2(\hat\beta_l).
c_{f_{\lambda}, v_{w_0\lambda}}^{\lambda}.|(0)\rangle_{{\bf i}}=c_{f_{\lambda}, v_{s_{i_1}\lambda}}^{\lambda}.|(0)\rangle_{i_1}\otimes \cdots \otimes c_{f_{s_{i_{l(w_0)-1}}\cdots s_{i_1}\lambda}, v_{w_0\lambda}}^{\lambda}.|(0)\rangle_{i_{l(w_0)}}&=|(0)\rangle_{i_1}\otimes \cdots \otimes |(0)\rangle_{i_{l(w_0)}}=|(0)\rangle_{{\bf i}},
||\alpha_{a_{n}}(x)|| =||\alpha_{b_{n}c_{n}}(x)|| \leq ||\alpha_{b_{n}}(\alpha_{a}(x))|| + ||\alpha_{b_{n}}(\alpha_{c_{n}}(x)-\alpha_{a}(x))|| \leq ||\alpha_{b_{n}}(\alpha_{a}(x))|| + ||\alpha_{c_{n}}(x)-\alpha_{a}(x)||.
\sum_{\substack{x \leq n < x+q^\delta (n,q) = 1}}1= \sum_{d | q}\left(\mu(d) \frac{q^\delta}{d}+ O(1)\right)= q^\delta \sum_{d|q}\frac{\mu(d)}{d}+ O(\tau(q)) = q^\delta \frac{\varphi(q)}{q}+ O(\tau(q)),
f_a(t) = \int_{-1}^{i\infty}[u, R(t,\cdot)^s],f_b(t) = -\int_{1}^{i\infty}[u, R(t,\cdot)^s],\intertext{and}f_c(t) = \int_{0}^{i\infty}[u, R(t,\cdot)^s].
\int_{\Omega}\rho |u_{tt}|^2 x+\frac{1}{2}\frac{d}{dt}\int_{\Omega}\Big(\mu|\nabla u_t|^2+(\lambda+\mu)(u_t)^2\Big) x = \int_{\Omega}\Big(\big(-\nabla P_t-(\rho u\cdot \nabla u)_t-\rho_tu_t+(H\times H)_t\big)\cdot u_{tt}\Big)x=\frac{d}{dt}\Phi_1(t)+ \Phi_2(t),
f(x)\;=\;\bigl(x^2\,-\,1\bigr)x,\; g(x)\;=\;(x^2\,-\,1)\log\Bigl\vert\dfrac{1+x}{1-x}\Bigr\vert,\;\phi_\epsilon(x)\;=\;\dfrac 2{\epsilon^{2(1-\kappa)}}f(x)\,-\,\dfrac{\dot{\kappa}}{K(\epsilon)}g(x).
h_1 = -\Delta\rho_0 - \frac{1}{\epsilon^2}(1-|\nabla\tilde{\chi}_0|^2-\rho_0^2)\rho_0 H_2 = \frac{1}{\epsilon}\rho_0^2\nabla\tilde\chi_0 - \sigma_0\nabla\tilde\phi_0 h_3 = \sigma_0\Delta\tilde\phi_0- \frac{1}{\epsilon^2}\rho_0^2\tilde{\phi}_0 \,.
\psi\left( \frac{\mathrm{d}}{\mathrm{d}t}(Ru(t))\right) =\int_{\Omega}\lim_{s\rightarrow t}A\left( \frac{Ru(s)-Ru(t)}{s-t}\right) \mathrm{d}x & =\lim_{s\rightarrow t}\int_{\Omega}A\left( \frac{Ru(s)-Ru(t)}{s-t}\right)\mathrm{d}x
=\frac{(a_{1}-a_{2})pq}{a_{1}pq-Q(q-p)}\left(\frac{Q(q-p)-a_{2}pq}{a_{1}pq-Q(q-p)}\right)^{\frac{a_{2}pq-Q(q-p)}{(a_{1}-a_{2})p^{2}}}\left(\frac{a_{1}pq-Q(q-p)}{(a_{1}-a_{2})(q-p)}d\right)^{\frac{p-q}{p}}.