harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\{\eta_{A B}\}= \left( \begin{array}{c c}{{g_{a b}}}&{{0}}\\{{0}}&{{g_{i j}}}\\ \end{array}\right) \, , \quad \{\eta^{A B}\}= \left( \begin{array}{c c}{{g^{a b}}}&{{0}}\\{{0}}&{{g^{i j}}}\\ \end{array}\right) \, ,
\sum_{k=0}^{n-1}\frac{(q^{d-1};q^d)_k^dq^{dk}}{(q^d;q^d)_k^d}\equiv \frac{(q^d;q^d)_{(d-1)(n-1)/d}q^{(d-1)(n-1)(d+n-1)/(2d)}}{(q^d;q^d)_{(n-1)/d}^{d-1}(-1)^{(d-1)(n-1)/d}}\pmod{\Phi_n(q)^2}.
U^{M}\left( t,s\right) \varphi =\varphi-i\int_{s}^{t}Q_{M}\left(\tau\right) U^{M}\left( \tau,s\right) \varphi d\tau\text{and}U^{M}\left( t,s\right) \varphi =\varphi+i\int_{t}^{s}U^{M}\left(t,\sigma\right) Q_{M}\left( \sigma\right) \varphi d\sigma
\Phi = W\circ \Phi\circ W^{-1}&=(W\circ t_{\gamma_{-1}}\circ W^{-1})\circ (W\circ t_{\beta}\circ W^{-1})\circ (W\circ t_{\gamma_{1}}\circ W^{-1}) &=t_{W(\gamma_{-1})}\circ t_{W(\beta)}\circ t_{W(\gamma_{1})}.
\begin{array}{rcccl}S_{\pm\vec{a_1}}\psi (\vec{r}) = e^{\pm\frac{i}{\hbar}a_1(\hat{p}_1+\frac{e}{c}By)}\psi (\vec{r}) = \psi (\vec{r}\pm\vec{a_1})e^{\pm 2\pi i\alpha y/a_2}, S_{\pm\vec{a_2}}\psi (\vec{r}) = e^{\pm\frac{i}{\hbar}a_2\hat{p}_2}\psi (\vec{r}) = \psi (\vec{r}\pm\vec{a_2}) ,\end{array}
I_{\mu}(z) \sim\left(\frac{z}{2}\right)^\mu/\Gamma(\mu+1),z\to 0, \mu>-1, I_{\mu}(z) = \frac{e^z}{\sqrt{2 \pi z}}\left(1+O\left(\frac{1}{z}\right)\right),z\to \infty,|\arg z|<\frac{\pi}{2}.
\sum_{i,j\geq 0}\phi_i(m\rq_{1,j}(x,y))t^{i+j}=\sum_{i,j,k\geq 0}m_{1,i}(\phi_j(x),\phi_k(y))t^{i+j+k}, \sum_{i,j\geq 0}\phi_i(m\rq_{2,j}(x,y))t^{i+j}=\sum_{i,j,k\geq 0}m_{2,i}(\phi_j(x),\phi_k(y))t^{i+j+k}.
\varphi_a(x,\xi+2\pi m) =\varphi_a(x,\xi)+2\pi x\cdot m, m\in\mathbb{Z}^d, |\partial_x^\alpha\partial_\xi^\beta\left[\varphi_a(x,\xi)-x\cdot\xi\right]| \leq C_{\alpha\beta,a}\langle x\rangle^{1-\varepsilon-|\alpha|}, |{}^t\nabla_x\nabla_\xi\varphi_a(x,\xi)-I| <\frac{1}{2}
\phi_3^{(brst)}= \frac{1}{2}(p^2+\gamma )+\gamma\gamma^{ij}[\partial_jc^3 \mu_i] +(\partial_ic^i)\mu_3+\frac{3}{2}\partial_ic^3\partial_jc^3\mu_j\mu_i+\partial_i(c^i\mu_3).
\rho_W(P_t \nu_1, P_t \nu_2) = \rho_W( P_\epsilon \mathcal{P}_n \nu_1, P_\epsilon \mathcal{P}_n \nu_2)\leq C' \rho_W(\mathcal{P}_n \nu_1, \mathcal{P}_n \nu_2)\leq C C' \delta^n \rho_W(\nu_1, \nu_2).
R(\kappa) =\sum_{j=-2}^{\beta-6}\kappa^jG_j+\mathcal O(\kappa^{\beta-5}); G_j =G_{0,j}-\sum_{\genfrac{}{}{0pt}{}{j_1\ge 0,j_2\ge -2,j_3\ge 0}{j_1+j_2+j_3=j}}G_{0,j_1}vC_{j_2}v^*G_{0,j_3}
B(\beta_0)=\dfrac{|u_c|L^{2}_{2M}(2N_{2M})^{\frac{\delta-3\mu+5}{\xi}}}{\bigg[\exp\bigg(\tfrac{2\beta_0^{\xi}N_{2M}}{L_{2m}\xi}\bigg)L_{2m}^\frac{\delta-3\mu+5}{\xi}(\xi)^{\tfrac{1-\delta-\nu}{\xi}}\bigg[\Gamma\bigg(\tfrac{3-\nu-\mu}{\xi}\bigg)-\gamma\bigg(\tfrac{3-\nu-\mu}{\xi},\tfrac{2N_{2M}\beta_0^{\xi}}{L_{2m}\xi}\bigg)\bigg]\bigg]^2}>0.
M_{\rho}=\begin{pmatrix}0 0 \dots 0 -a_0 1 0 \dots 0 -a_1 0 1 \dots 0 -a_2 \vdots \vdots \ddots \vdots \vdots 0 0 \dots 1 -a_{m-1}\end{pmatrix}M_{\omega}=\begin{pmatrix}0 0 \dots 0 1 \end{pmatrix}.
I_{WZ}=\int d^{4}x[\partial_{\mu}\phi^{\dagger}\partial^{\mu}\phi-i\chi\sigma^{\mu}\partial_{\mu}\overline{\chi}-|\frac{\partial W}{\partial\phi}|^{2}-\frac{1}{2}\frac{\partial^{2}W}{\partial\phi^{2}}\chi\chi-\frac{1}{2}\frac{\partial^{2}W^{\dagger}}{\partial\phi^{\dagger2}}\overline{\chi}\overline{\chi}]
P_{\mathrm{d}}=\frac{1}{P\left({\mathcal{H}}_{1}\right)}\sum_{i=1}^{N}\sum_{m=0}^{N}\sum_{k=\max\left( 0,m+i -N \right)}^{\min\left( m, i \right)}P\left({\mathcal{H}}_{1,i},m,k \right) \times P\left({\mathcal{H}}_{1}|{\mathcal{H}}_{1,i},m,k \right).
M_{k,\ell}(\theta) = 2\cos(\tfrac{(k-\ell)\theta}{2}) +2\cos(\tfrac{k \theta}{2})(2i\sin(\tfrac{\theta}{2}))^{-\ell}+ 2\cos(\tfrac{\ell \theta}{2})(2i\sin(\tfrac{\theta}{2}))^{-k},
\lambda_{2 N}( \alpha ) ={\frac{1 + \sum_{m = 0}^{N - 1}\left({\frac{i \alpha}{2}}\right)^{2 m + 1}{\frac{( 2 N + 2 m ) !}{( 2 m + 1 ) ! ( 2 N - 2 m - 2 ) !}}}{\sum_{m = 0}^{N - 1}\left({\frac{i \alpha}{2}}\right)^{2 m}{\frac{( 2 N + 2 m - 1 ) !}{( 2 m ) ! ( 2 N - 2 m - 1 ) !}}- i \alpha ( 4 N + 1 )}}\times \lambda_{0}( \alpha )
\hat{d}_{\Pi}({\hat{\pi}_1},{\hat{\pi}_2}):= |\tanh ({\varsigma_{{\pi_1}}}) - \tanh ({\varsigma_{{\pi_2}}})| \vee \mathop{\sup}\limits_{t \ge{\varsigma_{{\pi_1}}}\wedge{\varsigma_{{\pi_2}}}}\left|{\frac{{\tanh (\hat{\pi}_{1}(t ))}}{{1 + |t|}}- \frac{{\tanh (\hat{\pi}_{2}(t ))}}{{1 + |t|}}}\right|,
I(t_1u_1+t_2u_2+t_3u_3) =\sum_{i=1}^3I(t_iu_i)+\frac{\lambda}{2}\sum_{i<j}t_{i}^{2}t_{j}^{2}\int_{\mathbb R^3}\phi_iu_j^2dx I'(u)[u_i] =I'(u_i)[u_i]+\lambda\sum_{j\neq i}\int_{\mathbb R^3}\phi_ju_i^2dx,
{\cal{A}}_{n o n r e l a t}^{+ + c l a s s}= i \frac{e^{2}}{m \theta}\frac{\vec{s}\wedge \vec{q}}{\vec{q}^{2}}+ \frac{e^{4}}{4 \pi m \theta^{2}}\ln \Biggl ( \frac{- \Lambda^{2}}{\vec{p}^{2}}\Biggr )
\zeta_T(\nu|\beta)={\nu \varrho^{-2\nu}\over \pi i}\int_{\mu}^{\infty}d\omega \int_{C_+}dz{2z \over (z^2+\omega^2)^{\nu+1}}\ln\left(1-e^{i\beta z}\right)\breve{\Psi}(\omega;z),
\sum_{n=0}^{\infty}\Delta_{k}(n)q^{n}=\dfrac{(q^{2};q^{2})_{\infty}(q^{2k+1};q^{2k+1})_{\infty}}{(q;q)_{\infty}^3(q^{2(2k+1)};q^{2(2k+1)})_{\infty}}.
=\ln\frac{k_0}{4}-\sum\limits_{k=4}^{k_0-1}\int\limits_{\frac{1}{k}-2^{-4k-1}}^{\frac{1}{k}}\,\frac{dr}{r}\geqslant\ln\frac{k_0}{4}-\sum\limits_{k=4}^{k_0-1}\frac{2^{-4k-1}}{\frac{1}{k}-2^{-4k-1}}\geqslant\ln\frac{k_0}{4}-c
f^{\Delta_i}_{t_i}(t) = \lim \limits_{\substack{s_i \to t_i s_i \neq \sigma_i(t_i)}}\frac{f(t_1, \ldots, t_{i-1}, \sigma_i(t_i), t_{t+1}, \ldots, t_n) - f(t_1, \ldots, t_{i-1}, s_i, t_{t+1}, \ldots, t_n)}{\sigma_i(t_i) - s_i},
\nu^{3}\sum_{i \ne j \ne k}\sum_{l \ne m}\bigg [ \epsilon_{l}( y_{l m}K_{l m}^{\mathrm{t o t}}) , \xi_{i}( y_{i j}K_{i j}^{\mathrm{t o t}}) ( y_{j k}K_{j k}^{\mathrm{t o t}}) \bigg ] \, .
{\bf Z({\bf x})}=\frac{1}{{\bf x}}+\sum_{q\neq 0}\{\frac{1}{{\bf x-q}}+\frac{1}{{\bf q}}+\frac{1}{{\bf q}}{\bf x}\frac{1}{{\bf q}}+\frac{1}{{\bf q}}{\bf x}\frac{1}{{\bf q}}{\bf x}\frac{1}{{\bf q}}+\frac{1}{{\bf q}}{\bf x}\frac{1}{{\bf q}}{\bf x}\frac{1}{{\bf q}}{\bf x}\frac{1}{{\bf q}}\},
\sum_{n=1}^{N}\left[\begin{matrix}N n\end{matrix}\right]_{q^2}\frac{(-1)^{n-1}z^nq^{n^2}(q^2;q^2)_{n}}{(zq;q^2)_n(1-zq^{2n})}&=\sum_{n=1}^{N}\left[\begin{matrix}N n\end{matrix}\right]_{q^2}\left(\frac{(q;q)_{2n-2}z^{2n-1}q^{n(2n-1)}}{(zq;q)_{2n-1}}+\frac{(q;q)_{2n-1}z^{2n}q^{n(2n+1)}}{(zq;q)_{2n}}\right)\frac{(q^{2};q^2)_n}{(zq^{2N+1};q^2)_{n}}.
\prod_{d\mid n}\binom{kd-1}{\lfloor d/3\rfloor}^{\mu(n/d)}\equiv (-1)^{\phi_3(n)}\left\{\frac{1}{2}(27^{k\phi(n)}+1)+k(\frac{1}{2}k-\frac{1}{3})n^{2}A_{3}(n)\right\}\pmod{n^3};
\alpha_{n}^{(N)}(f)=\alpha_{n}^{(0)}(\phi_{(1)}\circ...\circ\phi_{(N)}f)+\digamma_{n}^{(N)}(f), \digamma_{n}^{(N)}(f)=\sum_{k=1}^{N}\Gamma_{n}^{(k)}(\phi_{(k+1)}\circ...\circ\phi_{(N)}f).
\frac{1}{exp\psi(x)}\sim\frac{1}{x}+\frac{1}{2\cdot x^{2}}+\frac{5}{4\cdot3!\cdot x^{3}}+\frac{3}{2\cdot4!\cdot x^{4}}+\frac{47}{48\cdot5!\cdot x^{5}}-\frac{5}{16\cdot6!\cdot x^{6}}+\cdot\cdot\cdot
| \Phi \rangle = \sum_{r , s}b_{\overline{\jmath}_{1}. . . \overline{\jmath}_{s}}^{i_{1}. . . i_{r}}\lambda_{i_{1}}\dots \lambda_{i_{r}}\psi^{\overline{\jmath}_{1}}\dots \psi^{\overline{\jmath}_{s}}| 0 \rangle ,
i\left\{Y_{l_{1}}^{m_{1}},Y_{l_{2}}^{m_{2}}\right\}={\displaystyle\sum\limits_{_{\substack{l=|l_{1}-l_{2}|+1 l\equiv l_{1}+l_{2}-1}}}^{l_{1}+l_{2}-1}}\sqrt{\frac{(2l_{1}+1)(2l_{2}+1)}{2l+1}}\C_{m_{1},m_{2},m}^{l_{1},\l_{2},\l}P({l_{1},l_{2},l})\Y_{l}^{m}
\phi(x)=\phi\big((\theta\rtimes\pi)(b)\big) =\sum_s\phi(\theta(b_s)\pi_s)=\phi(\theta(b_e)) &=\phi(\theta\circ\mathbb E(b))=\phi\big(\mathbb E_\theta\circ(\theta\rtimes\pi)(b)\big)=\phi(\mathbb E_\theta(x)),
\lbrack F_{_{(1)}}]^{-1}=\left(\begin{array}{ccc}f^{-1}-f^{-1}M_{_{(1)}}R_{_{(1)}}^{-1}0 -R_{_{(1)}}^{-1}M_{_{(1)}}^{^T}f^{-1}R_{_{(1)}}^{-1}\omega_{_{(1)}}R_{_{(1)}}^{-1}-R_{_{(1)}}0 R_{_{(1)}}0\end{array}\right) \;,
\widetilde{\textbf{c}}_{q,2n-2}^{(n)}=\frac{\binom{n}{q-1}}{(n!)^2}P_{2n}(q)\textbf{c}_{q,2n-2}^{(n)}=\frac{\binom{n}{q-1}}{(n!)^2}Q_{2n}(q),
\frac{1}{3^{s}}(1-\frac{1}{2^{s}})\zeta(s)=\frac{1}{3^{s}}+\frac{1}{9^{s}}+\frac{1}{15^{s}}+\frac{1}{21^{s}}+\frac{1}{27^{s}}+\frac{1}{33^{s}}+...
\frac{U}{V}\sim\frac{1}{R}\beta^{\prime}(\frac{x}{R};\alpha_{1},\alpha_{2})=\frac{(\frac{x}{R})^{\alpha_{1}-1}}{(1+\frac{x}{R})^{\alpha_{1}+\alpha_{2}}}\cdot\frac{1}{RB(\alpha_{1},\alpha_{2})},x\ge0
\mathtt{a}^{\ell_{+}}\square_{l}=\mathtt{a}^{\ell_{+}}\left(\frac{\partial\;}{\partial\mathtt{a}}\right)^{\ell_{+}}-\mathtt{a}^{\ell}\cdot\mathtt{a}^{\ell_{-}}\left(\frac{\partial\;}{\partial\mathtt{a}}\right)^{\ell_{-}},
{\vert O_{k}\rangle}_{\nu \bar{\nu}}= \epsilon^{l_{1}. . l_{k - 2}}\, \epsilon^{t_{1}\cdots t_{k - 2}}\, ( \Gamma_{A_{1}B_{1}}^{z}\, \nu_{l_{1}}^{A_{1}}\, \nu_{t_{1}}^{B_{1}}) \cdots ( \Gamma_{A_{k - 2}B_{k - 2}}^{z}\, \nu_{l_{k - 2}}^{A_{k - 2}}\, \nu_{t_{k - 2}}^{B_{k - 2}}) \, \vert \Phi \rangle
C^{0+ \alpha}(\mathbb R^d)= \{h \in C_b^{0,0}(\mathbb R^d) : \|h\|_{C^{0+\alpha}(\mathbb R^d)}<\infty \}&C^{1+\alpha}(\mathbb R^d) =\{h \in C_b^{1,0}(\mathbb R^d) : \|h\|_{C^{1+\alpha}(\mathbb R^d)}<\infty \},
\begin{array}{rlrl}a = \dfrac{p \{(r-p)i-q j\}}{r(r-p-q)}, b = \dfrac{q \{(r-q)j-p i\}}{r(r-p-q)}, begin{align*}4mm] a' = \dfrac{p \{(r-p)i'-q j'\}}{r(r-p-q)}, b' = \dfrac{q \{(r-q)j'-p i'\}}{r(r-p-q)}. \end{array}
\begin{aligned}P_b= \Pr(-2\Re\{y\sqrt{P_s}\zeta\eta\}+|\sqrt{P_s}\zeta\eta|^2 &>-2\Re\{y\sqrt{P_s}\zeta\hat \eta\}+|\sqrt{P_s}\zeta\hat \eta|^2) = \Pr\left(2\Re\{y\sqrt{P_s}\zeta(\hat\eta-\eta)\}\right. &+\left.|\sqrt{P_s}\zeta\eta|^2 -|\sqrt{P_s}\zeta \hat \eta|^2>0\right).\end{aligned}
\Big( \int_{\mathbb T}\left\| \sum_{k=0}^N x_k \left(cz \right)^k \right\|^r_X dz \Big)^{1/r}\le \Big( \int_{\mathbb T}\left\| \sum_{k=0}^N x_k z^k \right\|^s_X dz \Big)^{1/s}
\begin{cases}\phi^m_t+\mathbf{w}^m\cdot \nabla\phi^m+\gamma (-\Delta \phi^m+W'(\phi^m)) =0,\ (0, T)\times \Omega, \phi^m=\phi_b, \ (0,T)\times\Gamma, \phi^m|_{t=0}=\phi_0,\x\in \Omega,\end{cases}
A^i_{(1)}= -2\sigma_0S^i_0 - \sigma_0W_0\bar{k}^i + \sigma_0k_0W^i + 2\sigma_0\sigma_1S_0W^i, A^i_{(2)}= -k_0y^i + \frac{1}{2}h_{00}\bar{k}^i + \sigma_1R_{00}b^i, A^i_{(3)}= -4\sigma_1R_{00}y^i + 2\sigma_1h_{00}W_r\bar{k}^ry^i + \frac{\sigma_1}{2}h_{00}^{\frac{m+3}{2}}W_r\bar{k}^rb^i.
\mu|x_{1},...,x_{n}\sim N(\frac{\frac{\sigma^{2}}{n}\mu_{0}+\sigma_{0}^{2}\overline{x}}{\frac{\sigma^{2}}{n}+\sigma_{0}^{2}},(\frac{n}{\sigma^{2}}+\frac{1}{\sigma_{0}^{2}})^{-1})
\langle \nabla_i\nabla_j \Psi_t-\nabla_l\Psi_t\Gamma_{ij}^l, \nabla_m\nabla_b \Psi_t-\nabla_l\Psi_t\Gamma_{mb}^l\rangle =\frac{1}{2t}O(t)+\sum_l\Gamma_{ij}^l\Gamma_{mb}^l.
d\lambda^{(n)}_i(t) = 2 \sqrt{\lambda^{(n)}_i(t)}dB^{(n)}_i(t) + 2 \Big(|n - p| + 1\Big) dt + \sum_{j \neq i}\frac{4\lambda^{(n)}_i(t)}{\lambda^{(n)}_i(t) - \lambda^{(n)}_j(t)}dt
\begin{aligned}v_{ij}= (1 - \phi_u d) u_{ij}- \phi_{uu}d\, u_i u_j- (\phi_u d_j + \phi_{ju}d) u_i - (\phi_u d_i + \phi_{iu}d) u_j & - \phi_{ij}d - \phi d_{ij}- \phi_i d_j - \phi_j d_i,\end{aligned}
S^{\prime \prime}\, = \,{\frac{1}{4 \pi}}\int d^{2}x \Big [ \partial_{+}\, ( \Phi - \Lambda ) \partial_{-}\, ( \Phi - \Lambda ) \, - \, ( 2 \Phi - \Lambda \, ) ( \partial_{-}A_{+}\, - \, \p a r t i a l_{+}A_{-}) \Big ]
d U-\sum_{i=1}^n\sum_{j=1}^n a_{ij}(x,t)D_{ij}Udt+\sum_{i=1}^n b_i(x,t)D_iUdt+h(x,t)Udt=U\sum_{k=0}^mh_kdB_k,~~(x,t)\in Q_T,
\sum_{{k=1,k\neq j}}^n\dfrac{1}{z_{\nu,k}^2-z_{\nu,j}^2}=\dfrac{1-z_{\nu,j}-\nu}{2z_{\nu,j}^2}-\frac{1}{2z_{\nu,j}}\sum_{k=1}^n\dfrac{1}{z_{\nu,k}+z_{\nu,j}},
X_+(t)X_+'(t) = \sum_{i:\gamma^+_i>\gamma_*}\gamma^+_i|\xi_i^+(t)|^2+\sum_{i:\gamma^-_i>\gamma_*}\gamma^-_i|\xi_i^-(t)|^2 + \sum_{i:\gamma^+_i>\gamma_*}\xi_i^+(t)\mathcal{E}^+_i(t) +\sum_{i:\gamma^-_i>\gamma_*}\xi_i^-(t)\mathcal{E}^-_i(t) \geq (\gamma_*+\varepsilon_1) X^2_+(t) +X_+(t)Y_+(t).
\mu(B(x_\circ,\rho)) \le R^{\frac{n+2-\alpha}2}|S\cap B(x_\circ,\rho)|\le R^{\frac{n+2-\alpha}2}R^{-1}\rho^{n}\le \rho^{-(n-\alpha)}\rho^{n}= \rho^\alpha.
\lambda_j(0)=\min_{\substack{V\subset H_c^2(\Omega){\rm dim}V=j}}\max_{\substack{v\in V \frac{\partial v}{\partial\nu}\ne 0}}\frac{\mathcal{Q}_{\sigma}(v,v)}{\int_{\partial\Omega}\left(\frac{\partial v}{\partial\nu}\right)^2d\sigma},
\|u - g\|_{L^2(\Gamma_D)}^2 = \|u\|_{L^2(\Gamma_D)}^2 + \|g\|_{L^2(\Gamma_D)}^2 -2\int_{\Gamma_D}ug\,d\mathcal{H}^{N-1}\ge \frac{1}{2}\|u\|_{L^2(\Gamma_D)}^2 - 7\|g\|_{L^2(\Gamma_D)}^2,
\left\| g(x)-h(x)\right\| =\lim_{n\rightarrow\infty}|2|^{n}\left\| g(2^{n+1}x)-h(2^{n+1}x)\right\| &\leq\lim_{k\rightarrow\infty}|2|^{n}\max\{\left\| g(2^{n+1}x)-h(2^{n+1}x)\right\|, \left\| g(2^{n+1}x)-h(2^{n+1}x)\right\|\}&\leq\lim_{j\rightarrow\infty}\lim_{n\rightarrow\infty}\max\{|2|^{k}\mu(0, 2^{k+1}x);j\leq k<n+j\}&=0
\sum_{k=1}^{\infty}\frac{B_{2k}}{2k(2k-1)z^{2k-1}}\equiv - \frac{1}{2\pi i}\int_{c-i\infty \atop{{\rm Max}[0<c={\rm Re}\,s<1}}^{c+i\infty}\; \frac{z^{-s}\, \zeta(-s)}{\sin(s\pi)}\; ds \;\;.
G_o(K;\vec{x},\phi;\vec{x'},\phi') = -\frac{1}{(2\pi)^4}\sum_{n=-\infty}^{n=+\infty}\int d^3p\frac{e^{i\vec{p}.(\vec{x}-\vec{x'})}e^{in(\phi-\phi')}}{[p^2 + n^2/R^2 - K^2-i \epsilon]},
[\begin{matrix}M_{11}\\M_{22}\\M_{12}\end{matrix}]=-\frac{2h^{3}E}{3(1-\nu^{2})}[\begin{matrix}1&\nu&0\\ \nu&1&0\\ 0&0&1-\nu\end{matrix}][\begin{matrix}w_{,11}^{0}\\w_{,22}^{0}\\w_{,12}^{0}\end{matrix}]
(-1)^n\sum_{J \subseteq I}(-1)^{|J|}(1+t)^{s(\lambda;\mu)+1-|J|}x^{|\lambda|-|\mu|}=(-1)^n(1+t)^{s(\lambda;\mu)+1-|\mathcal{I}|}x^{|\lambda|-|\mu|}\left(\sum_{J \subseteq I}(-1)^{|J|}(1+t)^{|\mathcal{I}|-|J|}\right) &=(-1)^nt^{|\mathcal{I}|}(1+t)^{s(\lambda;\mu)+1-|\mathcal{I}|}x^{|\lambda|-|\mu|}.
0 \equiv 1 - (-18)^{p^m-1}\mod p = (1 - (-18)^{\frac{p^m-1}{2}})(1 + (-18)^{\frac{p^m-1}{2}}) \equiv (1 + (-18)^{\frac{p^m-1}{2}})(1 + \frac{3d}{c}(-18)^{\frac{p^m-3}{4}})(1 - \frac{3d}{c}(-18)^{\frac{p^m-3}{4}}) \mod p,
\begin{array}{l}u^{m_t}(s,y) = \int_{\mathcal{C}_t^d}K(y-X_s(\omega)) \exp\left(\int_0^s \Lambda(r,X_r(\omega),u^{m_t}(r,X_r(\omega))) dr \right)m_t(d\omega) \ \forall s\in [0,t]\ .\end{array}
F_1(a;b,b';c;x;y) = \sum_{m=0}^\infty \sum_{n=0}^\infty \frac{(a)_{m+n}(b)_m (b')_n}{m! n! (c)_{m+n}}x^m y^n F_2(a;b,b';c,c';x,y) = \sum_{m=0}^\infty \sum_{n=0}^\infty\frac{(a)_{m+n}(b)_m (b')_n}{m! n! (c)_{m+n}(c')_{n+m}}x^m y^n,
H = \frac{| 1 - \Theta{\cal B}|}{2}\left\{\frac{b_{1}^{2}}{2}\tilde{p}_{1}^{2}+ \frac{b_{2}^{2}}{2}\tilde{p}_{2}^{2}+{\vec{b}}_{1}\cdot{\vec{b}}_{2}( \tilde{p}_{1}\tilde{p}_{2}+ \tilde{p}_{2}\tilde{p}_{1}) \right\}~ ,
d_2 = c_1 d_3 = c_2 d_4 = c_3 + c_4 \hat{G}_1 d_5 = c_5 + c_6 \hat{G}_1 d_6 = c_7 + c_8 \hat{G}_1 + c_9 \hat{G}_2 + c_{10}\hat{G}_3 d_7 = c_{11}+ c_{12}\hat{G}_1^2 + c_{13}\hat{G}_1 + c_{14}\hat{G}_2 +c_{15}\hat{G}_3 + c_{16}\hat{G}_4.
\frac{n^{2}-1}{n^{2}+2}(1/\overline{\rho})=a_{0}+a_{1}\overline{\rho}+a_{2}\overline{T}+a_{3}\overline{\lambda}^{2}\overline{T}+\frac{a_{4}}{\overline{\lambda}^{2}}+\frac{a_{5}}{\overline{\lambda}^{2}-\overline{\lambda}_{UV}^{2}}+\frac{a_{6}}{\overline{\lambda}^{2}-\overline{\lambda}_{IR}^{2}}+a_{7}\overline{\rho}^{2}
B{u}_i^\prime + (A+BQ)u_i \,= -H_i(u^\prime+Qu),\i=1,2, (B+H_1)\widetilde{u}\,^\prime + \bigl(A+(B+H_1)Q\bigr)\widetilde{u}\,= -H_2(u^\prime_++Qu_+), B\overline{\overline{u}}\,^\prime + (A+BQ)\overline{\overline{u}}\,= -H_1({u}_2'+Q{u}_2)-H_2({u}_1'+Q{u}_1).
\begin{aligned}\overline{Z}^{p,q}_r = \mathrm{ker}\big(d_{r-1}:E^{p,q}_{r-1}\to E^{p+r-1,q-r+2}_{r-1}\bigskip), \overline{B}^{p,q}_r =\mathrm{im}\big(d_{r-1}:E^{p-r+1,q+r-2}_{r-1}\to E^{p,q}_{r-1}\big).\end{aligned}
\alpha_{2}(k)=(\begin{matrix}2k\\k\end{matrix})\sum_{j=0}^{k}(\begin{matrix}k\\j\end{matrix})\sum_{m=0}^{j}(\begin{matrix}j\\m\end{matrix})^{3}=(\begin{matrix}2k\\k\end{matrix})\sum_{j=0}^{k}(\begin{matrix}k\\j\end{matrix})^{2}(\begin{matrix}2j\\j\end{matrix})=1,6,90,1860,44730,...
V_{,k}\eta^{k}+V\xi_{,t}+\xi V_{,t}=-f_{,t}\frac{\partial\eta^{i}}{\partial t}g_{ij}-\xi_{,j}V =f_{,j}L_{\eta}g_{ij}=2\left( \frac{1}{2}\frac{\partial\xi}{\partial t}\right)g_{ij}\frac{\partial\xi}{\partial x^{k}}=0.
\tilde w={\hat T_k}w{\hat T_k}^{-1}= -e^{-\lambda_0^{(k)}}+2\lambda_1^{(k)}(\hat L_0-\lambda_{-1}^{(k)}w) -\lambda_1^{(k)2}e^{\lambda_0^{(k)}}(\hat L_1+2\lambda_{-1}^{(k)}\hat L_0-{\lambda_{-1}^{(k)2}}w),
\nu_{\mathbf{f}}\left( \left \{x\in \mathbb{R}^{\mathbb{N}}:\int_{\mathbb{R}^{\mathbb{N}}}\pi^{\star}d\nu_{\mathbf{f}}\leq \lim_{n}\frac{1}{n}\sum_{k=1}^{n}\pi\left( \tau^{k-1}\left( x\right) \right) =\pi^{\star}\left( x\right) \leq\int_{\mathbb{R}^{\mathbb{N}}}\pi^{\star}d\bar{\nu}_{\mathbf{f}}\right \}\right) =1.
T_{2k+1,2a+1}(x;q)-T_{2k+1,2a-1}(x;q) &=(1+xq)[(xq)^{2a+1}\overline{T}_{2k+1,2k-2a}(xq;q)+(xq)^{2a-1}\overline{T}_{2k+1,2k-2a+2}(xq;q)].
a_2 =\frac{-a_{11}a_4a_6+4a_{11}a_5a_7-3a_{10}a_6a_7-a_{10}a_4a_8+a_6^2a_8+2a_5a_8^2-2a_7a_8a_9}{2a_{11}a_9-a_{10}^2}, begin{align*}5pt] a_4 =\frac{-a_{11}a_5a_6+a_{10}a_6^2+2a_{11}a_3a_7-a_{10}a_5a_8+a_3a_8^2-2a_{10}a_7a_9+3a_6a_8a_9}{2(2a_{11}a_9-a_{10}^2)}, begin{align*}5pt] a_5 =\frac{a_{11}a_3a_6+a_{10}a_3a_8-a_{10}a_6a_9-2a_8a_9^2}{2a_{11}a_9-a_{10}^2}.
\phi_\ell\left( h \begin{bmatrix}0 \frac{1}{m}I 0 \Omega \end{bmatrix}\right)= \begin{bmatrix}\tfrac{1}{\ell!}I h \phi_{\ell + 1}( h \Omega ) 0 \phi_\ell(h\Omega) \end{bmatrix},\ell\ge 0.
\begin{aligned}\frac{\delta}{2}\frac{d}{dt}\| \nabla N_{q}\|_{L^{2}(\Omega)}^{2}+ \frac{1}{2}\| q \|_{L^{2}(\Omega)}^{2}+ \delta \| \nabla q \|_{L^{2}(\Omega)}^{2}& \leq C \big (1 + \sum_{i=p,d}\big ( \| \nabla \mu_{i}\|_{L^{2}(\Omega)}^{2}+ \| \varphi_{i}\|_{L^{2}(\Omega)}^{2}\big ) \big ).\end{aligned}
\frac{\big(\sum_{j=1}^{l(n-1)}\lambda(j)\big)\sigma(l(n))}{(l(n-1)+1)!\lambda(l(n))}\leq \frac 12,\frac{\big(\sum_{j=1}^{l(n-1)}\lambda(j)\big)\sigma(l(n))}{(l(n-1)+1)!}\leq \frac{\lambda(l(n))}{2},
DJ_{\varepsilon_\alpha}(u_0-W_{\delta_{\varepsilon_\alpha}(t_\alpha),\xi_\alpha}+\phi_{\delta_{\varepsilon_\alpha}(t_\alpha),\xi_\alpha})=\sum_{i=0}^n \lambda_{i,\alpha}\left\langle Z_{i,\alpha}, .\right\rangle_{P_g}.
\int_{G}H(|u_j(x)-b|) \, dx \ge C s_j^n H(|s_j^{- \frac{n-p}{p}}-b|) + C s_j^n H(|-s_j^{- \frac{n-p}{p}}-b|) \ge C s_j^n H(|s_j^{- \frac{n-p}{p}}|).
\uppercase\expandafter{\romannumeral1}+\uppercase\expandafter{\romannumeral4}=\uppercase\expandafter{\romannumeral2}-\uppercase\expandafter{\romannumeral3}-\uppercase\expandafter{\romannumeral5}\leq c\mu\lambda_1|Q^{(4)}\backslash E(\lambda_1)|+cs^{-1}\int_{Q^{(3)}}|u-u_{Q^{(1)}}|^2dz
2E(\frac{1}{2}\sqrt{2})-K(\frac{1}{2}\sqrt{2})-2E[arccos(x);\frac{1}{2}\sqrt{2}]+F[arccos(x);\frac{1}{2}\sqrt{2}]=\int_{0}^{1}\frac{\sqrt{2}x^{3}y^{2}}{\sqrt{1-x^{4}y^{4}}}dy
U_{1}(\phi_{0},a_{1},...,a_{d})=\mu^{D}\sqrt{a_{1}\cdots a_{d}}\sum_{s=1}^{\infty}\frac{(-1)^{s}}{2s}g^{s}\phi_{0}^{2s}\sum_{n_{1},...,n_{d}=-\infty}^{+\infty}\int \frac{d^{D-d}q}{(a_{1}n_{1}^{2}+\cdots +a_{d}n_{d}^{2}+c^{2}+{\bf q}^{2})^{s}}.
\min_{\mathrm{P}_1, \ldots, \mathrm{P}_M}\,\, \sum_{i=1}^{M}\mathrm{P}_i \mathrm{P_T}(\Hat{\theta}) \leq \Hat{\mathrm{T}}0 \leq \mathrm{P}_i \leq \Hat{\mathrm{P}}\quad\quad\quad\quad\, .
\left| \sum_{m=-M}^M \frac{1}{c-m}\right| \le \frac{1}{|c|}+ \sum_{m=1}^M \frac{2 |c|}{|c^2 - m^2|}\le \frac{1}{|c|}+ \sum_{1 \le m < |c|-1}\frac{2|c|}{(|c|-m)(|c|+m)}+ \frac{3}{\left\| c \right\|}+ \sum_{m>|c|+1}\frac{2 |c|}{m^2 - c^2}= O \left( \frac{\log (|c|+1)}{\left\| c \right\|}\right) .
{V}^{( 0 )}={\frac{m}{2}}\left[ p_{i}( t ) p^{i}( t ) + \omega^{2}q_{i}( t ) q^{i}( t ) \right] + e A_{0}( t , \vec{q}) + 2 \theta \int d^{2}x \partial^{i}\Pi_{i}( t , \vec{x}) A_{0}( t , \vec{x}) .
\sum_{j = 0}^k{\sum_{s = 0}^{k - j}{( - 1)^{n(j + s)}\binom kj\binom{k-j}sF_n^{j + s}G_{m - 2nk + (n + 1)j + (n - 1)s}}}= ( - 1)^{nk}G_m\,,
\left[\varphi^*_{(d,-d)}{\mathcal{Z}}^{ct}_2\right] - \left[\overline{\mathcal{DR}}_2(d)\right] = \frac{2d^2-1}{4}\left(\psi_1+\psi_2 \right)\left(\delta_{1,2}-\delta_{1,1}\right)-\delta_2\left(\delta_2+\frac{7}{10}\delta_{1,2}\right).
\frac{Q(a_ns)}{n}= \frac{1+o(1)}{B_\alpha}\int_0^s \frac{a_nQ'(a_nx)}{a_nQ'(a_n)}\,dx = \frac{1+o(1)}{B_\alpha}\int_0^s x^{\alpha-1}(1+o(1))\,dx = \frac{s^\alpha}{\alpha B_\alpha}(1+o(1)) = \gamma_\alpha s^\alpha (1+o(1)) \mbox{as}n\to\infty.
i{\cal G}_2(p)=-{2ie^2B^2p_{\bot}^2\over (p^2-m^2)^4}D+2 e^3B^3 \left[{1 \over (p^2-m^2)^4}+{4 p_{\bot}^2 \over (p^2-m^2)^5}\right] \bar D ,
I_2 =2\ell \mu{}^1 + \frac{2\delta^2\ell^3}{3}\int_{-1}^1 |\rho_{\ell}W_1'' + W_1'\rho'_\ell |^2 \leq 2\ell \mu{}^1 + \frac{2\delta^2\ell^3}{3}(C_1 + C_2\ell^{-2\alpha}),
\eta^{2}\cdot \tau^{2}{\cal{G}}: ={\cal{Y}}_{[ p_{1}, . . . , p_{N}]}\circ \left({\frac{1}{4}}\sum_{i , j , k , l = 1}^{N}\left( \eta^{( i j )}\eta^{( k l )}+ 2 \eta^{( i k )}\eta^{( l j )}\right) \tau^{( i j )}\tau^{( k l )}\right){\cal{G}}
G^{({\cal Q})}( x_{1}, x_{2}, x_{3}, x_{4}) = \langle{\cal Q}^{i_{1}j_{1}}( x_{1}){\cal Q}^{i_{2}j_{2}}( x_{2}){\cal Q}^{i_{3}j_{3}}( x_{3}){\cal Q}^{i_{4}j_{4}}( x_{4}) \rangle
\widetilde{m}_{n_2}^{(i_2)}= e_2^{\prime}\left(e_2^{-1}\cdot m_{n_2}^{(i_2)}\right), \textit{where}e_2^{\prime}\textit{is the overlap of}b^{(i_1)}\textit{and}\widetilde{m}_{n_2}^{(i_2)}.
\int_s^t -\nabla_qV(r,q_r)\beta(r,q_r)\circ dq_r = (\beta V)(s,q_s)-(\beta V)(t,q_t)+\int_s^t \partial_r(\beta V)(r,q_r) dr+\int_s^t \nabla_q\beta(r,q_r) V(r,q_r)\circ dq_r
\omega_j(t) = \frac1{\Gamma(1-\beta)}\int_{t_j}^{\min(t,t_{j+1})}(t-\tau)^{-\beta}\,d\tau &= \frac1{\Gamma(2-\beta)}\left((t-t_j)^{1-\beta}-(t-\min(t,t_{j+1}))^{1-\beta}\right)
A^{(i)}_j :={\mathcal O}_{\tilde{D}_i\times T}[W^{(i)}_j] \big/ \big( (W^{(i)}_j)^{r^{(i)}_j}-t^{r^{(i)}_j}-z_i \big) A^{(i)}_{j,k}:=A^{(i)}_j\Big/\Big( z_{i,1}\cdots z_{i,m_i-1}\big(W^{(i)}_j-\zeta_{r^{(i)}_j}^k t\big) \Big),
N[\mathfrak{e},{\widetilde{V}}]\ \leq \ \mathcal{L}_{\widetilde{V}}[(1-{\varepsilon})\eta (\mathfrak{e})]\ =\ \#\big\{x\in \Gamma \,\big|\V(x)\geq(1-{\varepsilon})\eta (\mathfrak{e})\big\}.
v_d^{(0)}= \frac{2^{d(d + 1)/2}}{d!}S_d(1,1,1/2), v_d^{(1)}= 2^{(d - 1)(d - 2)/2-2}\sum_{j=0}^{d-2}\sum_{k=0}^{d-2-j}\frac{(-1)^{d-k}2^{2d-2-2k-j}}{j!k!(d-2-j-k)!}B_{d-2}(d-2-k,d-2-k-j) &\int_{z=0}^{1}\int_{y=-2\sqrt{z}}^{2\sqrt{z}}y^j(y+z+1)^k\:dy\:dz
[[x_1, x_2, \ldots, x_n], y_{1},\ldots , y_{n-1}] = \sum_{i=1}^{n}[[x_1, x_{2}, \ldots , x_{i-1},y_{1}, x_{i+1}, \ldots , x_{n}],x_{i},y_{2},\ldots , y_{n-1}]
\theta(y_0,y_1) = \frac{1}{s_*}(s_0^p + s_1^p)^{\frac 1 p},{\rm prd}_{\theta}(x_0,s_0,x_1,s_1) := \left(x_0,\frac{s_0}{\theta},x_1,\frac{s_1}{\theta}\right).
\hat{H}= \frac{1}{2}\hat{p}_{0}^{2}+ \frac{\omega_{0}^{2}}{2}\hat{q}_{0}^{2}+ \frac{1}{2}\sum_{k = 1}^{N}\left[ \hat{p}_{k}^{2}+ \omega_{k}^{2}\hat{q}_{k}^{2}+ 2 \lambda_{k}( t ) \hat{q}_{0}\hat{q}_{k}\right] .
\frac{\partial^2}{\partial a^2}\zeta'_B(0;a,1,1)-2\left(\frac 1{12}-\zeta'_R(-1)\right)\frac 1{a^3}=\int_0^\infty \frac{\partial^2}{\partial a^2}F(at)\frac 1{t(e^{t}-1)}\,dt.
LI_n=\max_{0=k_0\leq k_1\leq \dots\leq k_m=n}\sum_{i=1}^m \left(N^i_{k_i}-N^i_{k_{i-1}}\right)=\max_{\substack{\lambda\in \Lambda_d}}\sum_{i=1}^m \left(n p_i\lambda_i+\sqrt{n}\widetilde{V^{n}_i}(\lambda)\right),
( \tilde{b}\delta-b^{\prime}) f(\mathcal{Y}_1,\ldots,\mathcal{Y}_{n-1},\mathcal{V}_n)f(\chi_1,\dots,\chi_n)+f(\chi_1,\dots,\chi_n)b^{\prime}f(\mathcal{Y}_1,\ldots,\mathcal{Y}_{n-1},\mathcal{V}_n) =0
\mbox{Var}_{\mu_z}(f) = \int_{\Sigma_z}f^2\,d\mu_z - \Big(\int_{\Sigma_z}f\,d\mu_z\Big)^2 \le \frac{1}{\beta\rho_0}\int_{\Sigma_z}|P\nabla f|^2 d\mu_z\,,
\sum_{i=0}^n\sum_{j=0}^i((s+1)i(i-1) + (n-i)i - (s+1)(j(j-1) + (i-j)(i-j-1)))\frac{x_{i-j}}{(i-j)!}\frac{x_j}{j!}\sum_{\ell=0}^n (s+\ell)!\frac{B_{n-i, \ell}(x_1, x_2, \ldots)}{(n-i)!}= 0
d s^{2}= e^{2 A ( r )}[ - 4 d x^{+}d x^{-}+{\cal H}d x^{+ \, 2}] + e^{2 B ( r )}[ d r^{2}+ r^{2}d \Omega_{n}^{2}] + \sum_{i}e^{2 C_{i}( r )}d y_{i}^{2}
\zeta_c^d q \, \phi_{k,m}^{+}(c,d;\tau) = \frac{1}{2\pi i}\int_{|s|=r}\alpha^+_{m,c}(s) \frac{e^s \zeta_c^d q}{1-e^s \zeta_c^d q}ds = -\frac{1}{2\pi i}\int_{|s|=r}\alpha^+_{m,c}(s) \sum_{n\geq 0}\left(e^s \zeta_c^d q\right)^{-n}ds,
{J^z}_{[\kappa, \lambda]}(z, \bar{z}) = 2 \left( 2\omega_{\bar{z}z}- \frac{1}{\mu}p \right) \lambda_{\bar{z}}- 2 \left( 2\lambda_{\bar{z}z}- \frac{1}{\mu}\kappa \right) v_{\bar{z}},
r \partial_r (\xi^4) = r 4 \xi^3 \xi_r = 4 \xi^3 ( r \partial_r ( \tfrac{H}{r}) ) = 4 \xi^3 (D_r H - 3 \tfrac{H}{r}) = 4 \xi^3 D_r H - 12 \xi^3 \tfrac{H}{r}.
R_{\sf tt}= -\frac{d-1}{a}\left( \frac{d^2a}{d{\sf t}^2}\right) \;, ~~~R_{ij}=\left[ a\left(\frac{d^2a}{d{\sf t}^2}\right)+(d-2)\left(\frac{da}{d\sf t}\right)^2+(d-2)k\right]\gamma_{ij}\;.
W=\frac{1}{2}\sum_{i=1}^{3}\sum_{j=1}^{3}\sigma_{ij}\epsilon_{ij}=\frac{1}{2}(\sigma_{x}\epsilon_{x}+\sigma_{y}\epsilon_{y}+\sigma_{z}\epsilon_{z}+2\sigma_{xy}\epsilon_{xy}+2\sigma_{yz}\epsilon_{yz}+2\sigma_{xz}\epsilon_{xz})