harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
H^{\beta}(z) \to e^{-\alpha}_y(z) =\exp (-\sum_{n\ge 1}y_n z^n)\exp (\sum_{n\ge 1}\frac{\partial}{n\partial y_n}z^{-n})e^{-\alpha}z^{-h^y_0}H^{\gamma}(z) \to \partial_z e^{\alpha}_y(z) =:h^y (z)\exp (\sum_{n\ge 1}y_n z^n)\exp (-\sum_{n\ge 1}\frac{\partial}{n\partial y_n}z^{-n})e^{\alpha}z^{h^y_0}:.
\gamma_{n+2}=\frac{\gamma_{n}\left( \gamma_{n}+\gamma_{n-1}-z^{2}-n+\frac{1}{2}\right) +\gamma_{n+1}\left( n+\frac{3}{2}-\gamma_{n+1}+z^{2}\right)-\frac{1}{2}z^{2}}{\gamma_{n+1}}.
c_k(n):=c_k\big(n;X_n,{\boldsymbol{g}}^{(n)}\big)=\sum_{{1 \le i_1 < i_2 < \cdots < i_s < n \atop j_1 \ge 1,\ldots, j_s \ge 1}\atop m:=\sum(j_a+1) \le n}\prod_{a=1}^{s}g_{i_a}[j_a]e_{k-m}\big(X_{[1,n]{\setminus}\bigcup\limits_{a=1}^{s}[i_a,i_a+j_a ]}\big),
\sum_{\alpha \subset (k^{n})}\frac{\Pi (y_{\alpha})}{\Pi (t_{\lambda})}\frac{G_{\lambda^{\vee}}(y_{\alpha}|\ominus t^{\prime})G_{\mu}(y_{\alpha}|\ominus t)}{e(y_{\alpha},y_{\alpha})}=\delta_{\lambda \mu}\;,
\begin{cases}\boldsymbol u(0,x,y)=\boldsymbol u_0(x,y),\quad\theta(0,x,y)=\theta_0(x,y):=T_0(x,y)-S(x,y), \boldsymbol u(t,x,y)\cdot \boldsymbol n=0,\quad\theta(t,x,y)=0,(x,y)\in\partial\Omega.\end{cases}
(a^{\dagger}_{\oplus}(v)f)(k) =a^{\dagger}_+(v)f(k) (a_{\oplus}(v)f)(k) =a_+(v)f(k) (\varphi_{\oplus}(v)f)(k) =\varphi_+(v)f(k) (g(d\Gamma_{\oplus}(\omega)+m)f)(k) =g(d\Gamma_+(\omega)+m(k))f(k).
\alpha^a=\sqrt{\frac{m}{2}}\,\left[\hat{A}^a+\frac{i}{m}\hat{\pi}^a\right]\ \ ,\ \ \{\alpha^a}^\dagger=\sqrt{\frac{m}{2}}\,\left[\hat{A}^a-\frac{i}{m}\hat{\pi}^a\right]\ \ ,
Q_k^{(s,t)}:=\frac{H_{k-1}^{(s,t)}H_{k}^{(s,t+1)}}{H_{k}^{(s,t)}H_{k-1}^{(s,t+1)}}, &E_k^{(s,t)}:=\frac{H_{k+1}^{(s,t)}H_{k-1}^{(s,t+1)}}{H_{k}^{(s,t)}H_{k}^{(s,t+1)}}.
f^{-1}[T_t] = \omega^*\cap\left( \bigcup_{A\in\mathcal A_s}\hat{A}(t)\setminus\bigcup\left\{\hat{B}:B\in\mathcal A(t,A)\right\}\cup \bigcup_{t^\prime\in succ_T(t)}f^{-1}[T_{t^\prime}] \right),
\mathbb{E}\Big(\Big|\sum_{m=1}^n X_m\Big|^p\Big)\le K_p\max\Bigg\{\Big(\sum_{m=1}^n\mathbb{E}\big(|X_m|^2\big)\Big)^{p/2},\sum_{m=1}^n \mathbb{E}(|X_m|^p)\Bigg\}.
N(\phi) = \sum_{j \in J}\frac{\frac{\alpha_j}{N^j}}{|x-y|}\leq \sum_{j\in J}\frac{\frac{\alpha_j}{N^j}}{\frac{\alpha_{j_0}}{N^{(j_0-1)k + 1}}}= \frac{N^{(j_0-1)k +1}}{\alpha_{j_0}}\sum_{j\in J}\frac{\alpha_j}{N^j}\leq \frac{N^{(j_0-1)k +1}}{\alpha_{j_0}}< \infty,
\int_{S_R}\langle B^{-1}_{I_F^{m\psi_k},\omega}f_F,f_F\rangle_{h\otimes I_F,\omega}e^{-m\psi_k}dV_\omega=\int_{S_R}\frac{1}{m}\langle B^{-1}_{I_F^{\psi_k},\omega}f_F,f_F\rangle_{h\otimes I_F,\omega}e^{-m\psi_k}dV_\omega<+\infty,
fg = \ \delta_i(e)(e-1) + \biggl(\sum_{j=1}^{n}\delta_i(e)(e-1)x_j\biggr) + \sigma_i(e)x_i(e-1) + \sum_{j=1}^{n}\sigma_i(e)x_i(e-1)x_j = \ \delta_i(e)(e-1) + \biggl(\sum_{j=1}^{n}\delta_i(e)(e-1)x_j\biggr) + \sigma_i(e)[\sigma_i(e-1)x_i + \delta_i(e-1)] + \ \sum_{j=1}^{n}\sigma_i(e)[\sigma_i(e-1)x_i + \delta_i(e-1)]x_j.
f_\varepsilon\big(\mathcal{L}^n(\{\tilde{u}_\varepsilon>0\}\cap\mathbb{R}^n\cap\Omega)\big)-f_\varepsilon\big(\mathcal{L}^n(\{u_\varepsilon>0\}\cap\mathbb{R}^n\cap\Omega)\big)=-\frac{1}{\varepsilon}V.
\|A(\upsilon+\tau)\| = \| A(\upsilon) A^{-1}(\upsilon) A(\upsilon+\tau) \| \leq \|A(\upsilon)\| \cdot \| [\phi_{\upsilon}(\tau)]^T \| \smash{\overset{\eqref{eq:PhiHigherDerivativeEstimate}}{\leq}}\|A(\upsilon)\| \cdot v_0(\tau),
S_{ij}= R_{ij}- h R_{i\gamma}R_{\gamma j}+ h^2 R_{i\gamma}R_{\gamma\gamma}R_{\gamma j}\dots (-1)^k h^k R_{i\gamma}R^{k-1}_{\gamma\gamma}S_{\gamma j}= R_{ij}- h R_{i\gamma}R_{\gamma j}\left( \sum_{l=0}^k\left(-h R_{\gamma \gamma}\right)^l \right) + (-1)^k h^k R_{i\gamma}R^{k-1}_{\gamma\gamma}S_{\gamma j}
I_x = \frac{\hat c_1 (\mathcal G) \sqrt{2 \pi}}{\hat \sigma}+ \frac{2 \hat c_1 (\mathcal G) \sqrt{2 \pi}}{\hat \sigma}\lim_{M\rightarrow \infty}\lim_{\delta \rightarrow 0}\Big\{\sum_{k=1}^{\lfloor \frac{M}{\sqrt \delta}\rfloor}d_{k,x,\delta}+ \sum_{k={\lfloor \frac{M}{\sqrt \delta}\rfloor}}^{\infty}d_{k,x,\delta}\Big\}.
\frac{d}{dx}\left[xF\left(1,m;1+m;x^{\frac{1}{m}}\right)\right]=F\left(1,m;1+m;x^{\frac{1}{m}}\right)+\frac{x^{\frac{1}{m}}}{1+m}F\left(2,1+m;2+m;x^{\frac{1}{m}}\right).
\varphi_j(W) = \frac{b_j b_{j+1}}{c_{j-1}c_j}\frac{P_{j-2}(\psi)}{P_j(\psi)}, \varphi_j(W') = \frac{b_j b_{j+1}}{c_{j-1}c_j}\frac{P_{j-2}(\psi')}{P_j(\psi')}(2 \le j \le D-2).
\sum_{i=0}^n \Delta(f) = f\,\Big|_0^{n+1}= f(n+1)-f(0) \Delta (fg)(n) = f(n+1) \Delta g(n) + g(n)\Delta f(n) \sum_{i=0}^n g(i)\Delta f(i) = (fg)\Big|_0^{n+1}- \sum_{i=0}^n f(i+1)\Delta g(i)
\mathrm{Re}\Big[ a_{m_0 n_0}(i\alpha -1)^{m_0}(P(z_2))^{m_0}\big(z_2^{n_0}+o(|z_2|^{n_0})\big)+ b_{k_0 l_0}(i\alpha -1)^{k_0}(z_2^{l_0}+o(|z_2|^{l_0}) &\times (P(z_2))^{k_0}P'(z_2) \Big ]=0
G(t) = \prod_{j=1}^{\frac{n}{2}}(t - \lambda_{2j}) + \frac{4}{n + 3}\sum_{j=1}^{\frac{n}{2}}\sin^{2}\left(\frac{2j \pi}{n + 3}\right) (\lambda_{2j}- \lambda_{n+2}) \prod_{\substack{m=1 m \neq j}}^{\frac{n}{2}}(t - \lambda_{2m}).
\sum_{i=2n-1}^{2n}\sum_{j=1}^{2n-2}\left| D_iD_j u(p) - D_iD_j u(q) \right| \leq C \left( d^{\frac{1}{\beta}-1}\sup_{B_\beta(0,1)}|u| + \int_0^d \frac{\omega( r)}{r}dr + d^{\frac{1}{\beta}-1}\int_d^1 \frac{\omega( r)}{r^{1/\beta}}dr \right),
\int d^{2 K N}w = 2^{N K - K ( K + 1 ) / 2}\int \, \big | \det_{K}W \big |^{N - K}\, d^{K^{2}}W \, \Big \{\prod_{a = 1}^{K}d^{2 N - 2 a + 1}\hat{\Omega}_{a}\Big \}\ .
d ( \bar\lambda ) = (e_i , 0) , 1 \le i \le k d ( \bar\lambda ) = ( 0, e_i ) , 1 \le i \le l ; d ( \bar\mu ) = ( e_j , 0) , 1 \le j \le k d ( \bar\mu ) = ( 0, e_j ) , 1 \le j \le l .
\P(\exists v\in \Lambda_n: v \mbox{is undetermined}) \leq \sum_{v\in \Lambda_n}\sum_{1\leq \ell \leq r}\sum_{I_1, \ldots, I_\ell}\sum_{B'_{I_1}, \ldots, B'_{I_\ell}}\P(\sigma|_{B_{I_j}(v)}= \sigma|_{B'_{I_j}}\mbox{for}1\leq j\leq \ell) &\leq n^d r \binom{r-1}{\ell-1}n^{d\ell}q^{-(r-1)^{d}(1 + \ell)}= o(1)\,,
\rho\overline{u}_{j}\frac{\partial\overline{u}_{i}}{\partial x_{j}}=\rho\overline{f_{i}}+\frac{\partial}{\partial x_{j}}[-\overline{p}\delta_{ij}+2\mu\overline{S}_{ij}-\rho\overline{u_{i}^{\prime}u_{j}^{\prime}}].
\ll \Bigl(\sum_{p>2k^2}\frac{\omega(p)^2\log{p}}{\phi_{\omega}(p)^2}\Bigr)\Bigl(\sum_{(A',WB)=1}\frac{\omega(A')^2}{\phi_{\omega}(A')^2}\Bigr)\Bigl(\sum_{\substack{\mathbf{r}'\in\mathcal{D}'_k r_m=1}}\frac{\phi(r') Y_{\mathbf{r}'}^2}{\phi_\omega(r')^2}\Bigr).
(\sqrt{a}-\sqrt{b})^2-a\nabla_{\nu}b =b\nabla_{\nu}a-2\sqrt{ab}-K(h^{\frac{1}{2^n}},2)^{-r_n}a\sharp_{\nu}b+\sum_{k=0}^{n-1}r_{k}\big[\big(a^{\frac{m_k}{2^k}}b^{1-\frac{m_k}{2^k}}\big)^{\frac{1}{2}}-\big(a^{\frac{m_k+1}{2^k}}b^{1-\frac{m_k+1}{2^k}}\big)^{\frac{1}{2}}\big]^{2}.
\langle n_1 + 1, \mu_1,\epsilon_1, n_2, \mu_2,\epsilon_2 | 0,\mu_{12},\epsilon_{12}\rangle = - \frac{(-1)^{n_2}}{\epsilon_2}\sqrt{\frac{[n_2 + 1]_{\mu_2}}{[n_1+1]_{\mu_1}}}\langle n_1, \mu_1,\epsilon_1, n_2+1, \mu_2,\epsilon_2 | 0,\mu_{12},\epsilon_{12}\rangle.
d^-_G(v, W_{i-1}) \ge |W_{i-1}| + |W_i| - 2n/9 - \gamma n \ge |W_{i-1}| - (\gamma + \lambda) n, d^+_G(v, W_{i+1}) \ge |W_{i+1}| - (\gamma + \lambda) n d^-_G(v, W_{i-1}), d^+_G(v, W_{i+1}) \ge (2/9 - 2\lambda - \gamma)n.
\hat{Z}_{88}^{10}= \hat{Z}_{88}^{\overline{10}}= \frac{(det \Im \Omega)^2}{4}\biggl\vert \Theta \left[ \begin{array}{ll}1/2 1/2 1/2 1/2\end{array}\right] (0 \vert \Omega ) \biggr\vert^8
{\cal Z}^{( N_{f})}( \{\mu \}) = \det \left( \begin{array}{l l}{{A ( \{\mu_{f}\})}}&{{A ( \{- \mu_{f}\})}}\\{{A ( \{- \mu_{f}\})}}&{{A ( \{\mu_{f}\})}}\\ \end{array}\right) \Delta ( \{\mu \})^{- 1}, A_{i j}= ( \mu_{i})^{j - 1}e^{\mu_{i}}.
0 < Q^{2}< \frac{1 6}{3 \omega_{4}^{2}}\, \biggl [ - \frac{\ell^{2}}{3}\left( \omega_{4}M + \frac{2 \ell^{2}}{9}\right) + \frac{2 \ell}{3 \sqrt{3}}\left( \omega_{4}M + \frac{\ell^{2}}{3}\right)^{3 / 2}\biggr ] \, ,
|\begin{matrix}3&4&8\\ 5&2&2\\ 8&7&0\end{matrix}|=3|\begin{matrix}2&2\\ 7&0\end{matrix}|-4|\begin{matrix}5&2\\ 8&0\end{matrix}|+8|\begin{matrix}5&2\\ 8&7\end{matrix}|=3(-14)-4(-16)+8(35-16)=-42+64+152=174
\begin{tabular}[c]{cc}$-\Delta u\left( s,t\right) =f\left( s,t,u\left( s,t\right) \right) ,$ $\quad\left( s,t\right) \in\Omega$ $u\left( s,t\right) =0,$ $\quad\left( s,t\right) \in\partial\Omega$\end{tabular}
x^{-x}e^{x}\Gamma(x)=\int_{\mathbb{R}}e^{-x\tau^{2}/2}(1-\tau/3+\tau^{2}/12+4a_{4}\tau^{3}+O(\tau^{4}))d\tau=\sqrt{2\pi}(x^{-1/2}+x^{-3/2}/12)+O(x^{-5/2})
{\cal I}= \frac{1}{4 \pi}\int \left( \frac{(|R_z|^2 - |R_{{\bar z}}|^2)(1+ |z|^2)^2}{(1+|R|^2)^2}\right)^2 \frac{2i~{\rm d}z{\rm d}{\bar z}}{(1+ |z|^2)^2}.
\begin{aligned}3^{l_1+1}x^{r + 18}+ a(r)x^{15}+ a(r) x^{33}+3^{2l_1+1}x^{2r+9}+ 2 \cdot 3^{l_1+1}a(r) x^{24+r}+ 2 \cdot 3^{l_1+1}b(r) x^{42 + r}\in \langle F \rangle . \end{aligned}
P(q_1,\ldots,q_n) =(q_n-a_n)*P_n(q_1,\ldots,q_n)+(q_{n-1}-a_{n-1})*P_{n-1}(q_1,\ldots,q_{n-1})+ &+\cdots+(q_{2}-a_{2})*P_{2}(q_1,q_2)+R_{2}(q_1)
\begin{cases}u_{xx}(0,t)=h_0(t),\,\,u_x(L,t)=h_1(t),\,\,u_{xx}(L,t)=h_2(t), v_{xx}(0,t)=g_0(t),\,\,v_x(L,t)=g_1(t),\,\,v_{xx}(L,t)=g_2(t),\end{cases}
| \frac1{N}\sum_{i=1}^{N}(\gamma (E\cap E_i)-\gamma (E)\gamma (E_i)) | = | \int_{E}(\frac1{N}\sum_{i=1}^{N}({\mathcal X}_{E_i}-\gamma (E_i)))d\gamma | < \frac1N \sqrt{\sum_{i=1}^{N}\gamma (E_i)}.
\frac{\beta}{2}\int_{x\neq y}\log |x-y|\,dL_N(x)\, dL_N(y)-tF_0^a(L_N,\tau_B^N)-\sum_k\int W_k \,dL_k^N+J^a_t(\mu_{1,t}^*,\ldots,\mu_{d,t}^*) &=\frac{\beta}{2}\int_{x\neq y}\log |x-y|\,d[L_N-\mu^*](x)\,d[L_N-\mu^*](y)+R(L_N-\mu^*) &= \frac{\beta}{2}\int\log|x-y|\,d[\bar L_N-\mu^*](x)\,d[\bar L_N-\mu^*](y)+R(L_N-\mu^*)+O(\log N/N),
(X, \delta_X f \wedge \delta_X (D^2 g(\nu_{X}Df , \cdot) - D^2 f(\nu_{X}Dg , \cdot) )) &= D^2 g(\nu_{X}Df , \nu_{X}Dh) - D^2 f(\nu_{X}Dg , \nu_{X}Dh) +\circlearrowright,
W_{z}\left( \begin{array}{c c}{{a + N \bar{\epsilon}_{j}}}&{{b + N \bar{\epsilon}_{j}}}\\{{d + N \bar{\epsilon}_{j}}}&{{c + N \bar{\epsilon}_{j}}}\\ \end{array}\right) = W_{z}\left( \begin{array}{c c}{{a}}&{{b}}\\{{d}}&{{c}}\\ \end{array}\right) .
\left\Vert \frac{1}{M}{\textstyle\sum_{j=0}^{M-1}}\widehat{T}^{j}u-\frac{1}{M}{\textstyle\sum_{j=0}^{M-1}}\widehat{T}^{j}u^{\ast}\right\Vert_{1}\longrightarrow0\begin{array}[c]{c}M\rightarrow\infty u,u^{\ast}\in\mathcal{U}\end{array}
n^{-1}\tilde{l}^{GEL}_{\gamma}(\theta) = \sum_{k,l}\bar{\psi}^k \omega^{kl}\bar{\psi}^l + \frac{2}{3}\sum_{k,l,m}\bar{\psi}^k \bar{\psi}^l \bar{\psi}^m \omega^{kl}\omega^{km}\omega^{lm}\alpha_{klm}\,,
\sharp S' \leq \begin{cases}2q^{\frac{2l+t}{2}}\frac{2l+t}{2}< \lfloor\frac{r-1}{2}\rfloor, 2q^{\lfloor\frac{r-1}{2}\rfloor}\frac{2l+t}{2}\geq \lfloor\frac{r-1}{2}\rfloor, 0 ,\end{cases}
\Phi^n\circ \Lambda^\alpha\circ i_G^\alpha(s) =\Phi^n\circ i_G^{\alpha,r}(s) =i_A^{\beta,r}(u_s)i_G^{\beta,r}(s) &=\Lambda^\beta\bigl(i_A^\beta(u_s)i_G^\beta(s)\bigr) =\Lambda^\beta\circ\Upsilon\circ i_G^\alpha(s).
C_4 =| L_X^M+4\pi |^{-\sigma_0}(\frac{4\pi}{L^m_X}+1)^{C_2 -\sigma_0 +\frac{1}{2}}, C_5 =\delta^{-s_0+\sigma_0}| L_X^M+\frac{4\pi}{\delta}|^{-\sigma_0}(\frac{4\pi}{\delta L^m_X}+1)^{C_2 -\sigma_0 +\frac{1}{2}}
\phi_1 := \phi_{8A}(\tau, z) = 2 \zeta^{\pm 1}- 2 + (-2 \zeta^{\pm 2}+ 8 \zeta^{\pm 1}- 12) q + (2 \zeta^{\pm 3}- 12 \zeta^{\pm 2}+ 30 \zeta^{\pm 1}- 40) q^2 + O(q^3)
\begin{aligned}z_h^{(0)}=z^0, & z_h^{(n+1)}\in \underset{\mathcal{K}\times \mathcal{K}}{\mbox{Argmin}}\left\{\frac{1}{2h}d^2\big(\cdot,z_h^{(n)}\big)+\mathcal E(\cdot)\right\}\end{aligned}
\int_{0}^{\pi}\int_{2}^{3}\rho^{2}cos\varphi d\rho d\varphi=\int_{0}^{\pi}cos\varphi d\varphi[\frac{\rho^{3}}{3}]_{2}^{3}=[sin\varphi]_{0}^{\pi}(9-\frac{8}{3})=0.
\mathfrak{h}_j(\varphi) = \lim_{k\to\infty}Q(0,k)^{-1}\circ P_{U^{(k)}_j}\circ \mathcal{M}^{(k)}\circ S(k)(\varphi) &= \sum_{l \geq 0}Q(0,l)^{-1}\circ P_{U^{(l)}_j}\circ\big(\mathcal{M}^{(l)}\circ S(l)-Z(l)\circ\mathcal{M}^{(l-1)}\circ S(l-1)\big)(\varphi)
J_2= -2\int \overline{u}^{k}\cdot \nabla u^{k}\cdot \overline{u}^{k+1}\leq C|\nabla u^k|_\infty | \overline{u}^{k}|_2 | \overline{u}^{k+1}|_2\leq \frac{C}{\nu}|\nabla u^k|^2_\infty | \overline{u}^{k+1}|^2+\nu | \overline{u}^{k}|^2_2.
w^\xi_\eta(\rho) :=\sum_{\{\alpha,\beta;\epsilon\}\in\vec{\mathcal{\triangle}}^\xi_\eta}\left(e^{\frac{1}{2}(\ell_{\alpha}(\rho)+\ell_{\beta}(\rho))}+(-1)^{\alpha\cdot\beta}\right)^{-1}&=1-\sum_{\{\alpha,\beta;\epsilon\}\in\vec{\mathcal{\triangle}}^\eta_\xi}\left(e^{\frac{1}{2}(\ell_{\alpha}(\rho)+\ell_{\beta}(\rho))}+(-1)^{\alpha\cdot\beta}\right)^{-1}
\arraycolsep=5pt\def\arraystretch{1.4}\begin{array}{rl}\xi(R)=6M(s,t)+1=5, s\in\left[\frac{4}{9},\frac{5}{9}\right], \t \in\left[\frac{4}{9},\frac{5}{9}\right]; \xi(R)=6M(s,t)+1>5, s\notin\left[\frac{4}{9},\frac{5}{9}\right], \t \notin\left[\frac{4}{9},\frac{5}{9}\right]. \end{array}
\frac{\partial}{\partial z^j}(\log{\rm tr}_{\omega_{\epsilon}}\omega_{\varphi_{\epsilon}}+\Psi_{\epsilon,\rho}-B \varphi_{\epsilon})=\frac{1}{{\rm tr}_{\omega_{\epsilon}}\omega_{\varphi_{\epsilon}}}\varphi_{\epsilon i \bar{i}j}+\Psi_{\epsilon, \rho,j}-B\varphi_{\epsilon j}.
\psi_-=\pi_*\left(-\frac{1}{||X||}\iota_X\phi\right), &\omega\;\;\;\;\begin{cases}\frac{1}{6}\omega^3=\pi_*(\frac{1}{||X||}\iota_X\varepsilon), \sigma=\frac{1}{2}\omega^2=\pi_*(\phi+\frac{1}{||X||}\iota_X\phi\wedge\eta),\end{cases}
\mu_a = - \eta_i \rho^i_a, \rho^j_a b_{ji}+ \rho^j_a \partial_j \eta_i + \eta_j \partial_i \rho^j_a+ \Gamma^b_{ai}\mu_b = 0, \rho^i_a \partial_i \mu_b - C^c_{ab}\mu_c - \rho^i_b \Gamma^c_{ai}\mu_c = 0.
\zeta_{0}^{( p )}( \nu ) ={\frac{1}{\pi \Gamma ( \nu )}}\int_{0}^{\infty}d x \int_{0}^{1}d t ~ t^{\nu - 1}e^{- t x^{2}}{\frac{1}{( 4 \pi t )^{( D - 1 ) / 2}}}\sum_{n = 0}^{\infty}\breve{a}_{n}( x ) t^{n}.
a_1= c_2 c_3 (c_4-1)+1, a_2=\alpha_{21}c_3 c_4+ (c_1-\alpha_{21}-1)(c_3-1)+ c_3, a_3= c_1c_4+ (c_1-\alpha_{21}-1)(c_2- 1)(c_4-1)-c_4+1, a_4=c_1 c_2(c_3-1)+ \alpha_{21}(c_2-1)+ c_2.
\gamma_0 = \frac{1}{(4\pi)^\frac{n}{2}\Gamma\left(1+\frac{n}{2}\right)}, \gamma_1 = \frac{1}{2(4\pi)^\frac{n}{2}\Gamma\left(2+\frac{n}{2}\right)}\intertext{and}\gamma_2 = \frac{1}{4(4\pi)^\frac{n}{2}\Gamma\left(3+\frac{n}{2}\right)},
\lim_{N\to\infty}\Bigg|\frac1N\sum_{n=1}^N F(S_X^n(x,\ldots,x)) G(S_Y^n(y,\ldots,y)) &-\Bigg(\frac1N\sum_{n=1}^N F(S_X^n(x,\ldots,x))\Bigg) \Bigg(\frac1N\sum_{n=1}^NG(S_Y^n(y,\ldots,y)) \Bigg)\Bigg|=0.
\sum_{t = t_{k,j}^- -1}^{t_{k,j}^+}d_n(t) = Q_n^f( t_{k,j}^+ +1 ) - Q_n^f( t_{k,j}^- ) - \sum_{t = t_{k,j}^-}^{t_{k,j}^+}a_n^f(t).
\max_{1\leq j,k \leq n, j\neq k}|f_j(\tau_k)f_k(\tau_j)|^m\geq \frac{\frac{1}{(G_{f,\tau}^{\circ^m})}\left(\sum_{j=1}^nf_j(\tau_j)^m\right)^2-\sum_{j=1}^n|f_j(\tau_j)|^{2m}}{n^2-n},
\begin{array}{ll}\sum\limits_{k_1=0}^{k-1}\int\limits_0^{\infty}<y>^{2\ell}|\partial_t^{k_1}\alpha|^2\,\mathrm{d}y\leq C(T), begin{align*}13pt]\sum\limits_{k_1=0}^{k-2}\int\limits_0^{\infty}<y>^{2\ell}|\partial_t^{k_1}\alpha_1|^2\,\mathrm{d}y\leq C(T).\end{array}
\alpha:=-\frac{1}{2}(n^{a}\overline{\delta}l_{a}-\overline{m}^{a}\overline{\delta}m_{a})=-\frac{1}{2}(n^{a}\overline{m}^{b}\nabla_{b}l_{a}-\overline{m}^{a}\overline{m}^{b}\nabla_{b}m_{a}).
\left(c_{f_{\lambda_0}, v_{w_0\lambda_0}}^{\lambda_0}.\varphi(G^{\mathrm{low}}(b_0))\right).|(0) \rangle_{\bf i}=\sum_{{\bf d}\in (\mathbb{Z}_{\geqq 0})^{l(w_0)}}{_{{\bf i}}\zeta'}_{{\bf d}}^{G^{\mathrm{low}}(b_0)}|({\bf d})\rangle_{{\bf i}},
\theta_{p}^0 = dp^0, \theta_{p}^i = d p^i, \quad\theta_x = dx, \theta_y^0 = dy^0 - p^0 dx, \theta_y^i = dy^i - p^i dx, \theta_{z}^i = dz^i - p^0 dy^i - p^i dy^0 + p^0 p^i dx.
\mathbb{P}\left\{v^T\cdot C_t \cdot v=0,\ \v\neq 0 \right\}&=\mathbb{P}\left\{v^TD^jY_t=0,\ 1\leq i\leq d,\ \v\neq 0 \right\}&=\mathbb{P}\left\{\int_{0}^{t}v^TJ_{t\leftarrow s}V_j(Y_s)d D^jX^j_s=0,\ 1\leq i\leq d,\ \v\neq 0 \right\}.
\frac{\partial f_0}{\partial t}=\, \Delta f_0 + \frac{4H}{|H|^2}\left\langle\nabla |H|, \nabla f_0\right\rangle - \frac{2}{|H|^2}\left[|\nabla h|^2 - \left(\frac{1}{n}+ f_0\right) |\nabla H|^2\right] begin{align*}5pt] +\frac{2}{|H|^2}\left[R_1 - \left(\frac{1}{n}+f_0\right) R_2\right] - 4nK f_0.
\mathcal{M}^2=2\,(\alpha+\beta)\left[\begin{array}{cccc}b_2^2 -b_1\, b_2 -b_2\, a_2 a_1\, b_2 -b_1\,b_2 b_1^2 b_1\, a_2 -b_1\, a_1 -b_2\, a_2 b_1\, a_2 a_2^2 -a_1\, a_2 a_1\, b_2 -b_1\, a_1 -a_1\, a_2 a_1^2\end{array}\right]
\eta ( s , t ) = \frac{1}{B_{k_{1}k_{1}}( s^{2})}\, \mathrm{d e t}\left[ \frac{1}{s_{p 1}^{2}+ s_{q 2}^{2}}\left( 1 - \exp \left( - \frac{s_{p 1}^{2}+ s_{q 2}^{2}}{2 t}\right) \right) \right]_{p , q = 1 , \ldots , k_{1}}\ .
\limsup\limits_{t\rightarrow\infty}\|P_\delta(W_{n,m;\textrm{far}}) e^{-itH}\psi - P_\delta(W_{n,m;\textrm{far}})e^{-itH_0}\varphi\| \leq \limsup\limits_{t\rightarrow\infty}\|P_\delta(W_{n,m;\textrm{far}}) \|_{\textrm{op}}\|e^{-itH}\psi-e^{-itH_0}\varphi\|=0
\{(k_1, \ldots, k_d) \in [s]^d ~|~ (j_\ell, k_\ell) \in Q, 1 \leq \ell \leq d\}= \{(k_1, \ldots, k_d) \in [s]^d ~|~ (j'_{\sigma(\ell)}, k_\ell) \in Q, 1 \leq \ell \leq d\}.
g_{\mu \nu}\left( x \right) = \eta_{m n}e_{\mu}^{m}\left( x \right) e_{\nu}^{n}\left( x \right) , \quad g^{\mu \nu}\left( x \right) = \eta^{m n}e_{m}^{\mu}\left( x \right) e_{n}^{\nu}\left( x \right) ,
0 = \frac{1}{a}\left(\frac{a'}{p}\right)' + 2n \frac{1}{b}\left(\frac{b'}{p}\right)' \frac{1}{p}\left(\frac{a'}{p}\right)' = 2n\left(\frac{a^3}{b^4}- \frac{a'b'}{bp^2}\right) \frac{1}{p}\left(\frac{b'}{p}\right)' = \frac{2n+2}{b}-2\frac{a^2}{b^3}- \frac{a'b'}{ap^2}- (2n-1)\frac{1}{b}\left(\frac{b'}{p}\right)^2.
\Pi_{\mu \nu}^{({\rm tadpole})}= \frac{1}{2}(2 e^2\delta_{\mu \nu}) \frac{1}{\beta}\sum_n \int \frac{d^2k}{(2 \pi)^2}\frac{1}{k_0^2 + \omega_{\sigma}^2}= e^2 \delta_{\mu \nu}\frac{\beta}{(2 \pi)^2}\sum_n \int\frac{d^2k}{(2 \pi)^2}\,\frac{1}{n^2 + (\frac{\beta \omega_{\sigma}}{2\pi})^2},
ab+c+1 =\frac{\gamma^2+\gamma+1}{\gamma+1}\cdot \frac{\gamma^2+1}{\gamma^3+\gamma^2+\gamma}+\frac{\gamma+1}{\gamma^3+\gamma^2+\gamma}+1= \frac{(\gamma^3+1)+(\gamma+1)+(\gamma^3+\gamma^2+\gamma)}{\gamma^3+\gamma^2+\gamma}&=\frac{\gamma^2}{\gamma^3+\gamma^2+\gamma}\ne0
f(z) =f(\varphi_{y}(s)) &=f\left(\varphi_{x_i}(s)- \sum_{j\in J}u_j\varphi_{x_j}(s)\right) &=f(\varphi_{x_i}(s))- \sum_{j\in J}u_jf(\varphi_{x_j}(s)) &=\psi_i(s)-\sum_{j\in J}u_j\psi_j(s) &=0
\int_{\mathcal{V}}\phi(v) \mathcal{L}\phi(v)\,{\rm d}\mu(v)=\frac{1}{2}\iint_{\mathcal{V}\times\mathcal{V}}\Big( \phi(v) - \phi(w) \Big)^2\,{\rm d}\mu(w)\,{\rm d}\mu(v)\ge 0.
\zeta ( \Omega^{s_{1}- 1}\omega_{p_{1}}\Omega^{s_{2}- 1}\omega_{p_{2}}\ldots \Omega^{s_{k}- 1}\omega_{p_{k}}) : = \zeta \left( \begin{array}{r l c l}{{s_{1},}}&{{s_{2}}}&{{\ldots}}&{{s_{k}}}\\{{\lambda^{p_{1}},}}&{{\lambda^{p_{2}- p_{1}}}}&{{\ldots}}&{{\lambda^{p_{k}- p_{k - 1}}}}\\ \end{array}\right)
\int_{0}^{L}\! \! d x_{4}\xi_{M}^{+}( x_{4}) \xi_{M^{\prime}}^{+}( x_{4}) = \int_{0}^{L}\! \! d x_{4}\xi_{M}^{-}( x_{4}) \xi_{M^{\prime}}^{-}( x_{4}) = \delta_{M M^{\prime}}.
\Psi(|v'|^2) = \Big (Y(\theta) + Z(\theta)\cos \varphi \Big ) \rho(Y(\theta) + Z(\theta)\cos \varphi) & \le \Big (Y(\theta) + Z(\theta)\cos \varphi \Big )\Big( \rho(Y(\theta)) + \rho'(Y(\theta) )Z(\theta) \cos \varphi \Big),
< 0 | T_{\mu \nu}( x ) | 0 > = \mathop{{\sum}^{\prime}}_{\gamma}{\frac{\Delta_{\gamma}( x , x )^{1 / 2}}{4 \pi^{2}\sigma_{\gamma}( x , x )^{2}}}\; \left( t_{\mu \nu}( x ; \gamma ) + s_{\mu \nu}( x ; \gamma ) \right) + O ( \sigma_{\gamma}( x , x )^{- 3 / 2}) .
\sum_{i\neq1}A_i(T)=\sum_{i\neq 1}\frac{Q_1(g,T,\lambda_i,\lambda_c)}{T^2(2\lambda_i^{11}-8\lambda_i^9\lambda_c^2+12\lambda_i^7\lambda_c^4-8\lambda_i^5\lambda_c^6+2\lambda_i^3\lambda_c^8)}.
\mathbb{E}\left\Vert pr_n \left( \xi \right) \right\Vert^2 + \mathbb{E}\left\Vert N^2 \, pr_n \left( \xi \right) \right\Vert^2\leq\mathbb{E}\left\Vert \xi \right\Vert^2 + n^2 \mathbb{E}\sum_{k=0}^n \sum_{\eta = \pm}\left\vert \langle e_{k}\otimes e_{\eta}, \xi \rangle \right\vert^2\leq 1 + n^2 ,
D_{b-}^{\alpha}f\left( t\right) =\frac{\left( -1\right)^{\alpha}}{\Gamma\left( 1-\alpha \right)}\left( \frac{f\left( t\right)}{\left( b-t\right)^{\alpha}}+\alpha \int_{t}^{b}\frac{f\left( t\right) -f\left( s\right)}{\left( s-t\right)^{\alpha +1}}ds\right)\,.
f (r, Z_0) = - 2 i \int_0^{\omega - \alpha}d p \, \frac{p e^{- i p r}}{(\omega^2 - p^2)^{3 / 2}}\cosh (\sqrt{\omega^2 - p^2}\, Z_0) + i \sqrt{\frac{2}{\omega \alpha}}\, e^{- i \omega r}- \pi Z_0 e^{- i \omega r}.
\mathcal{D}_{i}p(x) :=\frac{\partial}{\partial x_{i}}p(x) +\kappa\sum_{j\neq i}\tau((i,j)) \frac{p(x) -p(x(i,j))}{x_{i}-x_{j}}, \mathcal{U}_{i}p(x) :=\mathcal{D}_{i}( x_{i}p(x)) -\kappa\sum_{j=1}^{i-1}\tau( (i,j)) p(x(i,j)) .
\frac{d}{dt}\int u^2 \varphi_{\delta}(x)\,dx + \epsilon \int (\partial_x^2 u)^2 \partial_x\varphi_{\delta}(x)\,dx &\le c_0\beta^5 \int u^2\varphi_{\delta}(x)\,dx +|a\int \partial_x^2 u\partial_x^3u u\varphi_{\delta}(x)\, dx|.
\langle \psi_1^{a_1}\gamma_1,\dots,\psi_k^{a_k}\gamma_k,\psi_{k+1}\rangle_{g,k+1,\beta}=(2g-2+k)\langle\psi_1^{a_1}\gamma_1,\dots,\psi_k^{a_k}\gamma_k\rangle_{g,k,\beta}.
P^\eta( X_{t_{n+1}}^{\vec v}-X_r^{\vec v}< c_{n+1}t_{n+1}+r) &\leq P^\eta\Big(\sum_{k=1}^{t_n-1}Z_k\leq c_{n+1}t_{n+1}+r\Big)+\sum_{\substack{y\in B_{t_{n+1}}r\leq s \leq t_{n+1}}}\P^\eta\big(y\not\in G(\eta(s),\phi_{t_n})\big)\,.
Q(v): =f\,[k\,v^{2}+\delta \,\acute{}\,v+\delta^{2}(k+\lambda)](v^{2}+\delta^{2})^{\frac{n+1}{2}}& +f^{2}(v^{2}+\delta^{2})^{n+2}-\delta^{2}=0\forall \;u\in I,\;v\in\mathbb{R}.
\Lambda_{n,2}^{(1,3)}=\frac{16}{n^4\delta^2_n(2)\delta^2_n(3)}\sum_{r \ne r'}^d \sum_{q_1,q_2=1}^{r-1}\sum_{q_3,q_4=1}^{r'-1}\sum_{p_1=1}^{q_1-1}\sum_{p_2=1}^{q_2-1}A(p_1, q_1, p_2, q_2, q_3, q_4)
(a)u_1'=u_1=0 \quad\frac{u_2'}{u_2}>0 \qquad\qquad(b)\frac{u_1'}{u_1}>0 \quad \frac{1}{u_1'}e^\frac{u_2'}{u_1'}=\frac{1}{u_1}e^\frac{u_2}{u_1}=:u\,.
\max_{z_n^{\rm R}\le 0,z_n^{\rm I}\le0}\ln \left(\frac{1}{2}-\frac{1}{2}{\rm{erf}}\big(z_n^{\rm R}+\sqrt{P/2}\big)\right) + \ln\left(\frac{1}{2}-\frac{1}{2}{\rm{erf}}\big(z_n^{\rm I}+\sqrt{P/2}\big)\right) \mathrm{s.t.}~{z_n^{\rm R}}^2+{z_n^{\rm I}}^2 = B_n.
S_{(a)}[J^{\mu},\varphi ,\theta ,{\tilde\alpha}^r,\beta_r;{}^4g_{\mu\nu}]=\int d^4x \Big[ |J|a({{|J|}\over{\sqrt{{}^4g}}},\partial_{\mu}\theta J^{\mu}) -J^{\mu}(\partial_{\mu}\varphi +\beta_r\partial_{\mu}{\tilde \alpha}^r )\Big]
\frac{\dim \mathsf{V}^\lambda}{\dim \mathsf{W}_M^\lambda}= \frac{d!}{M^d}\Omega_{\frac{1}{M}}^{-1}(\lambda)\quad\frac{\dim \mathsf{V}^\lambda}{\dim \mathsf{W}_N^\lambda}= \frac{d!}{N^d}\Omega_{\frac{1}{N}}^{-1}(\lambda),
\; + \{x_{i}^{\pm}( l ) x_{i}^{\pm}( k ) x_{j}^{\pm}( m ) - ( q + q^{- 1}) x_{i}^{\pm}( l ) x_{j}^{\pm}( m ) x_{i}^{\pm}( k ) + x_{j}^{\pm}( m ) x_{i}^{\pm}( l ) x_{i}^{\pm}( k ) \}
(f_k)_r = 2(w'(z)z^{l_k + \frac{1}{2}}+ l_kz^{l_k - \frac{1}{2}}w(z)), (f_k)_{rr}= 4w''(z)z^{l_k+1}+ 2(4l_k + 1)z^{l_k}w'(z) + l_k(4l_k - 2)z^{l_k -1}w(z).
\int_{\mathbb{R}^{N}}|(-\Delta)^{\frac{\alpha}{2}}u_{0}|^2\,dx+\int_{\mathbb{R}^{N}}V(x)u_0^2\,dx=\eta\int_{\mathbb{R}^{N}}k(x)f(u_0)u_0\,dx+\eta\lambda\int_{\mathbb{R}^{N}}|u_{0}|^{2_{\alpha}^{*}}\,dx.
g_{S_{N}}(s) ={\textstyle\sum\limits_{n\geq0}}TP_{N=ne}s^{S_{n}}={\textstyle\sum\limits_{n\geq0}}T\left( s^{S_{n}}.P_{N=ne}e\right) & ={\textstyle\sum\limits_{n\geq0}}Ts^{S_{n}}.TP_{N=ne}e={\textstyle\sum\limits_{n\geq0}}\left( Ts^{x}\right)^{n}.TP_{N=ne}e=\widetilde{g}_{N}\left( g_{x}(s)\right) .
{\;}_p F_q \left({a_1,\ldots, a_p \atop b_1,\ldots,b_q}\Big{|}z \right)=\sum_{k=0}^\infty \frac{(a_1)_k\cdots (a_p)_k}{(b_1)_k \cdots (b_q)_k}\frac{z^k}{k!}
\frac{\partial \mathcal{V}_{L}}{\partial z}(x_k)= (1+\alpha-x_k){\sum_{i=1}^{n}\frac{1}{x_k-w_i}}+ x_k\sum_{i=1}^{n}\frac{R_n^{\prime \prime}(w_i)}{R_n^{\prime}(w_i)(x_k-w_i)}= 0.
\sum_{j=1}^k (-1)^{k-j}\binom{k}{j}\left( \|x\|^{2j}-1 \right) \quad=\sum_{j=0}^k (-1)^{k-j}\binom{k}{j}\left( \|x\|^{2j}-1 \right) &=\left( \|x\|^2 - 1 \right)^k = \left( 1 - \|x\|^2 \right)^k
\lim_{x \to 0}P_n(x)=(-1)^n\frac{(b^2-a^2)^{\mu+\nu+1}\Gamma(\nu)}{a^\mu b^{2\nu}\Gamma(\mu+\nu+1)n!}{\;}_2 F_1 \left({-n, \nu \atop \mu+\nu+1}\Big{|}1-\frac{a^2}{b^2}\right)
s^2\mathcal{Q}_0 =\hat{\Theta}_s^{i+1}(\psi_i-\psi)+\sum_{j=0}^i\Theta_s^j(s\mathfrak{c}_{i+1}^g-s\mathfrak{h}_{i+1}^g), s^2\mathcal{Q}_1 =\hat{\Theta}_s^{i+1}(s\mathfrak{e}_{i+1})+\sum_{j=0}^i\Theta_s^j(s\mathfrak{e}_{i+1}), s^2\mathcal{Q}_2 =-\hat{\Theta}_s^{i+1}(s\mathfrak{c}^g_{i+1}-s\mathfrak{h}^g_{i+1}).
c( \{e_2,e_5\}) = c( \{e_3,e_4\})= 1 c( \{e_4,e_5\}) = c( \{e_1,e_3\}) = 2 c( \{e_1,e_5\}) = c( \{e_2,e_4\})= 3 c( \{e_1,e_2\}) = c( \{e_3,e_5\}) = 4 c( \{e_1,e_4\}) = c( \{e_2,e_3\}) = 5.
\sum_{s=1}^{n-1}\frac{q^{s(n-s)+n}}{y^{s}z^{n-s}}+\frac{q^{m^2+3m+1}}{y^{m}z^{m}}\cdot \frac{J_2^3j(-1;q^2)j(yz;q^2)}{j(y;q^2)j(-y;q^2)j(z;q^2)j(-z;q^2)}