harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
a_d(0) = 1, a_d(1) = d, a_d(2) = d^2+d, a_d(3) = d^3+\frac{3}{2}d^2+\frac{1}{2}d, a_d(4) = d^4+2d^3+2d^2+d, a_d(5) = d^5+\frac{5}{2}d^4 + \frac{7}{2}d^3 + 2d^2, a_d(6) = d^6+3d^5+\frac{21}{4}d^4+\frac{9}{2}d^3+\frac{9}{4}d^2+d.
\Gamma_{{\rm RRS}\vert k}= \max\limits_{r \in \mathcal{C}}{\Gamma_{r, \mathrm{D}}}= \max\limits_{r \in \mathcal{C}}\frac{\gamma_{r, \mathrm{D}}}{1+\sum_{l=1}^{L}\gamma_{\mathrm{I}_{l, \mathrm{D}}}},
\left( \begin{array}{c}{{z^{\prime}}}\\{{\bar{z}^{\prime}}}\\ \end{array}\right) = \left( \begin{array}{c c}{{u}}&{{0}}\\{{0}}&{{\bar{u}}}\\ \end{array}\right) \left( \begin{array}{c}{{z}}\\{{\bar{z}}}\\ \end{array}\right) \ .
J_1=-T_{\lambda^{(1)}}\left(B\dot\eta-T_B\dot\eta-T_{\dot\eta}B\right), J_2=T_{\lambda^{(0)}}T_B\dot\eta+[T_{\dot\eta}, T_{\lambda^{(1)}}]B+T_{\dot\eta}F(\eta, B)+(\dot\eta-T_{\dot\eta})\div V-F(\eta, B\dot\eta)-T_{\dot\eta}R.
\begin{array}{rl}\sigma_i\displaystyle\frac{\rm d}{{\rm d}t}\theta_i h_i ={\cal L}_i(h,\boldsymbol{v})+\sigma_i\mathfrak{M}(t,h), \sigma_i\displaystyle\frac{\rm d}{{\rm d}t}\theta_i h_i v_{a\,i}={\cal J}_{a\,i}(h,\boldsymbol{v})+{\cal S}_{a\,i}(h,w)-{\cal K}(h)|\boldsymbol{v}_i| v_{a\,i}. \end{array}
\lim_{n \to \infty}n^{1 + 4m^2 - 2m - 4ms -2s}\Big[ D_n^{(n)}(x_1^{(n)}) D_{n-1}^{(n)}(x_2^{(n)}) - D_n^{(n)}(x_2^{(n)}) D_{n-1}^{(n)}(x_1^{(n)})\Big] = \frac{2^{4ms +2s}(z_1z_2)^{-\frac{s}{2}}}{(w_1 \cdots w_m)^{2+2s}}\Big( A_I^{(s, w)}(1, z_2) B_I^{(s,w)}(1, z_1) - A_I^{(s, w)}(1, z_1) B_I^{(s,w)}(1, z_2) \Big).
\frac{1}{h}\left(\frac{d^n\phi_{ij}}{d\xi^n}(\xi+h)-\frac{d^n\phi_{ij}}{d\xi^n}(\xi)\right)=\frac{(-1)^{n+1}}{-1}\frac{1}{-h}\left(\frac{d^n\phi_{ji}}{d\xi^n}(-\xi-h)-\frac{d^n\phi_{ji}}{d\xi^n}(-\xi)\right).
\varphi_{\Omega^1}\circ \psi_{\Omega^1}(df) = \varphi_{\Omega^1}\circ \psi_{\Omega^1}(\partial_{\mathrm{inv}}(f) g_{LT}dZ) = (\varphi_{L}\circ \psi_{L}(\partial_\mathrm{inv}(f))g_{LT}dZ = \partial_\mathrm{inv}(\varphi_{L}\circ \psi_{L}(f))g_{LT}dZ = d(\varphi_{L}\circ \psi_{L}(f)) \ .
[\begin{matrix}1&1&1&1\\x_{1}&x_{1}&x_{2}&x_{2}\\y_{1}&y_{2}&y_{1}&y_{2}\\x_{1}y_{1}&x_{1}y_{2}&x_{2}y_{1}&x_{2}y_{2}\end{matrix}][\begin{matrix}w_{11}\\w_{12}\\w_{21}\\w_{22}\end{matrix}]=[\begin{matrix}1\\x\\y\\xy\end{matrix}],
w(r, X^{s,x; u,\gamma}_r)=w(r, X^{s,x; u, \gamma [r,\eta]}_r)\geq \mathbb{E}[w(t, X^{s,x; u, \gamma [r,\eta],}_t)|\mathcal{F}_r] -(t-r)\varphi (t-r)\a.s.
\mathbb{P}^{(2)}_{\ell}:= \mathbb{P}\bigg[ \sup_{\substack{X_{\ell-1}< n\leqslant X_{\ell}1 \leqslant j\leqslant J}}V^{(2)}(n,y_j)> \frac{T_1(\ell)}{\ell^{K/2}}\bigg]
{\rm Prob}(a\leq x_1<b,x\in S^2) \ = \ \frac{\int_a^b\frac{1}{\sqrt{1-x_1^2}}S_{1}(\sqrt{1-x_1^2}) dx_1}{S_2(1)}&\ = \ \frac{\int_a^b2\pi dx_1}{4\pi}&\ = \ \frac{b-a}{2},
\lambda_n=\int_{-\infty}^{+\infty}\frac{e^{-\theta^2+2m\theta}}{(zq^{1/2}e^{-2ik\theta};q)_{\infty}}\frac{h_n(-qe^{2mki},q^{1/2}e^{2ki \theta})}{(q;q)_n}d\theta.
K^{\Omega}(t)=\frac{1}{\left| \Omega \right|}\sum_{x,y\in \Omega}\delta_{x^{y},x^{-1}}\prod_{\xi \in \mathcal{O}_{-}(x,y)}K\left( \frac{\lambda_{\xi}^{2}t}{\left| \xi \right|}\right) \prod_{\xi \in \mathcal{O}_{+}(x,y)}Z\left( \frac{\left| \xi \right|}{2\lambda_{\xi}^{2}it}+\frac{\kappa_{\xi}}{\lambda_{\xi}}\right)
a_{n}=\lim_{\varepsilon\downarrow0}\frac{G_{\mu_{1}}^{(n)}(\alpha+i\varepsilon)}{n!}=\lim_{\varepsilon\downarrow0}(-1)^{n}\int_{\mathbb{R}}\frac{d\mu_{1}(t)}{(\alpha-t+i\varepsilon)^{n+1}}=(-1)^{n}\int_{\mathbb{R}}\frac{d\mu_{1}(t)}{(\alpha-t)^{n+1}},
\|(f_n)_{n\geq 0}\|_{Q_q^p}:= \Bigl( \mathbb E \Bigl\|\Bigl(\sum_{n=0}^{\infty}|f_n|^2\Bigr)^{\frac 12}\Bigr\|^{p}_{L^q(S)}\Bigr)^{\frac 1p}<\infty.
v_{jj}= V(t_{jj}) = V \left( \frac{p_{jj}\cdot 1 + \sum_{i \in S, i \ne j}p_{ji}(t_{ij}+ 1)}{1 - \sum_{i \not \in S}p_{ji}}\right) = \frac{\sum_{i\in S, i \ne j}p_{ji}^2 v_{ij}}{\left( 1- \sum_{i \not \in S}p_{ij}\right)^2}
\widetilde{f}^{jk}\,_{p}\widetilde{f}^{pi}\,_{m}=\widetilde{f}^{ji}\,_{p}\widetilde{f}^{pk}\,_{m}+\widetilde{f}^{ki}\,_{p}\widetilde{f}^{jp}\,_{m}, & \widetilde{f}^{jk}\,_{p}\widetilde{f}^{ip}\,_{m}=\widetilde{f}^{ij}\,_{p}\widetilde{f}^{pk}\,_{m}+\widetilde{f}^{ik}\,_{p}\widetilde{f}^{jp}\,_{m},
-2\sum_{n\ge 0}\sum_{m\ge 1}(-1)^{m}m^\ell q^{m(2n+1)}=-4\sum_{m\ge 1}\frac{(2m)^\ell q^{2m}}{1-q^{2m}}+2\sum_{m\ge 1}\frac{m^\ell q^m}{1-q^{2m}}.
U_{g_{1}}U_{g_{2}}\Psi_{\mathbf n}^{\kappa}( \mathbf r ) = \operatorname{e x p}( i \alpha_{2}( \mathbf r ; g_{1}, g_{2}) ) U_{g_{1}g_{2}}\Psi_{\mathbf n}^{\kappa}( \mathbf r )
[y_1, y_1]=\theta_1y_5, [y_1, y_2]=y_3, [y_2, y_1]=-y_3+\theta_2y_4+\theta_3y_5, [y_2, y_2]=\theta_4y_4+\theta_5y_5, [y_1, y_3]=y_4=-[y_3, y_1], [y_2, y_3]=y_5=-[y_3, y_2].
\begin{aligned}\tilde{p}_{k}=\alpha p_k+\left(1-\alpha\right)p_{k+1},~\tilde{p}_{k+1}=\left(1-\alpha\right) p_k+\alpha p_{k+1}, \tilde{q}_{k}=\alpha q_k+\left(1-\alpha\right)q_{k+1},~\tilde{q}_{k+1}=\left(1-\alpha\right) q_k+\alpha q_{k+1},\end{aligned}
\begin{aligned}\limsup_{N\rightarrow \infty}\underset{\substack{0 \leq t \leq T_L Z_s \in \mathbb{R}^{2ds}}}{\sup}\left| f_N^{(s)}(t,Z_s) - f_t^{\otimes s}(Z_s)\right|\mathbf{1}_{Z_s \in \mathcal{K}_s \cap \mathcal{U}_s^{\eta(\varepsilon)}}\mathbf{1}_{E_s (Z_s) \leq 2 R^2}= 0 \end{aligned}
\tilde \varphi_i(z)=\varphi_i(\bar x_i)^{-1}\varphi_i(\varphi_i(\bar x_i)^{-(p_i-1)/4}z+\bar x_i)\leq 1\mbox{for}|z| \le \frac12 \varphi_i(\bar x_i)^{(p_i-1)/4}.
\sum_{n=1}^\infty\P_0\{X_n\in(x,x+\Delta]\}=\sum_{n=1}^\infty\int_x^{x+\Delta}V(y)\P\{S_n\in dy,\tau_0>n\}&\sim V(x)\sum_{n=1}^\infty\P\{S_n\in(x,x+\Delta],\tau_0>n\}=V(x)u(x).
\max \left( \frac{d-1}{d}s_c, \frac{s_c}{2}, s_c - \frac{d}{2(d+1)}\right)=\begin{cases}\frac{s_c}{2}, d=1, \frac{d-1}{d}s_c, d\ge 2\ \m=2, 3, s_c - \frac{d}{2(d+1)}, d\ge 2\ \m\ge 4.\end{cases}
(L(z_{n+1}-z_{n}), z_{n+1})_{H}=(L^{1/2}(z_{n+1}-z_{n}), L^{1/2}z_{n+1})_{H}=\frac{1}{2}\|L^{1/2}z_{n+1}\|_{H}^2 - \frac{1}{2}\|L^{1/2}z_{n}\|_{H}^2 + \frac{1}{2}\|L^{1/2}(z_{n+1}- z_{n})\|_{H}^2,
\int_{\P^2}H^2=\sum_{j=0}^2\frac{i_j^*H^2}{e(i_j^*T\P^2)}=\frac{\alpha_0^2}{(\alpha_0-\alpha_1)(\alpha_0-\alpha_2)}+\frac{\alpha_1^2}{(\alpha_1-\alpha_0)(\alpha_1-\alpha_2)}+\frac{\alpha_2^2}{(\alpha_2-\alpha_0)(\alpha_2-\alpha_1)}=\dots=1,
\left(f*{{\rm P}}^l\right)\left(x_1,x_2\right)=\int_{\widehat{G^2}}{\Phi}_1({\alpha}^{-l}_i\left(\omega_i\right))\cr{\Phi}_2({\alpha}^{-l}_j\left(\omega_j\right)){\mathcal{F}}_r\left(f\right)\left(\omega_i,\omega_j\right)\omega_j{(x}_2)\omega_i(x_1){d^2}_{{\mu}_{\widehat{G^2}}}\left(\omega_i,\omega_j\right)
\int_{\mathbb{R}^n}e^{-|\eta|^2}|\eta|^{a}d\eta = C_n \int_0^{\infty}e^{-r^2}r^{a+n-1}dr= \frac{C_n}{2}\int_0^{\infty}e^{-x}x^{\frac{a+n}{2}-1}dx = \frac{C_n}{2}\ \Gamma\left(\frac{n+a}{2}\right),
D_{j}\geqslant\frac{C_0^{p+1}M((pq)^{1/2}-1/2)}{B_0\widetilde{B}_0^{2p}}(pq)^{-\frac{3j}{2}-(j-1)p+1}\,D_{j-2}^{pq}:= E_0(pq)^{-\frac{3j}{2}-(j-1)p+1}\,D_{j-2}^{pq}, Q_{j}\geqslant\frac{C_0^{q+1}M^q((pq)^{1/2}-1/2)^q}{B_0^q\widetilde{B}_0^2}(pq)^{-j-\frac{3(j-1)q}{2}+q}\,Q_{j-2}^{pq}:=\widetilde{E}_0(pq)^{-j-\frac{3(j-1)q}{2}+q}\,Q_{j-2}^{pq}.
\inf_{0 \not = u \in H^1 (\mathbb{R}^N)}\frac{\| u \|_2^{(N+\gamma)- (N-2)p}\| \nabla u \|_2^{Np -(N+\gamma)}}{\int_{\mathbb{R}^N}\left( |\cdot|^{-(N-\gamma)}* |u|^p \right) |u|^p dx}
(H-\epsilon)u_{k+1}=\sum_{l=2}^{k}C_{k-l+1}\frac{\partial_{\epsilon}^{l-2}u}{(l-2)!}+\sum_{l=2}^{k}D_{k-l+1}\frac{\partial_{\epsilon}^{l-2}v}{(l-2)!}+\frac{\partial_\epsilon^{k-1}u}{(k-1)!},
\begin{aligned}\gamma_{1}(y,t) = -\frac{1}{24}\frac{\partial^{3}}{\partial y^{3}}p_{0}(y,t) = \left(\frac{2}{\pi}\right)^{1/2}\left(\frac{8}{3}y^{3}-2y\right)e^{-2y^{2}}\end{aligned}
\sharp T(\varepsilon_1,\varepsilon_2,B)=\frac{6(\varepsilon_1-\varepsilon_2)}{\pi^2 (\alpha^\prime)^2}B^{2-\frac{1}{r}}+O_{\varepsilon_i,\sigma}(B^{1-\frac{1}{2r}+\sigma(\frac{3}{4}+\frac{1}{8r})}).
\langle 0 | \mathrm{T r}\lambda^{2}| 0 \rangle^{2}= \frac{2^{1 0}\pi^{4}}{5}e^{- 8 \pi^{2}/ g^{2}}\frac{M_{\mathrm{P V}}^{6}}{g^{4}}= \frac{1 4 4}{5}\Lambda_{G}^{6},
\iint_{Q_r^+}|v(x,t)-v_{Q_r^+}|^2\,dx\,dt\le 4\iint_{Q_r^+}|v(x,t)-v^{\theta}_{Q_{2r}^+}|^2\,dx\,dt &\,\,\le 16\iint_{Q_r^+}|v(x,t)-v^{\theta}_{B_{2r}}(t)|^2\,dx\,dt+16\iint_{Q_r^+}|v^{\theta}_{B_{2r}}(t)-v^{\theta}_{Q_{2r}^+}|^2\,dx\,dt.
K_{N}(z_{1},z_{2}) = \frac{1}{N}\sum_{k=1}^{N}\frac{N-k}{N}\frac{s(z_{1})s(z_{2})}{1-\frac{N-k}{N}s(z_{1})s(z_{2})}+\frac{1}{N}\sum_{k=1}^{N}\frac{E_{N,k}}{1-\frac{N-k}{N}s(z_{1})s(z_{2})}+ O_{L^1}(d_{N}/N)
\Pi_{M,-}^\mathrm{pol}\equiv \!\!\! \sum_{\sigma \in S_M \times S_{N-M}}\!\!\!\!\!\!\!\!\! \mathrm{sgn}(\sigma) \, s_\sigma \, , \bigl(\Pi_{M,-}^\mathrm{pol}\bigr)^2 = \binom{N}{M}\, \Pi_{M,-}^\mathrm{pol}\, ,
f_{i}(\alpha) ={f}_{i}^0(\alpha) + \sum_{j=1}^k \ \sum_{\{p: j \in S_p\}}({f}_{j}^0)|_{C_{p,S_p}}(\alpha) &={f}_{i}^0(\alpha) + \sum_{j \in S_q}({f}_{j}^0) (\alpha) &= f(\gamma_p)
{\rm DM}_t(\varphi_1,\varphi_2):= \left<{<\!\!<\!\hat{W}^{n}\!>\!\!>}_{t}(\varphi_{1}), \varphi_{2}\right> = \frac{1}{n}\int_{0}^{t}\left<u_{z},\varphi_{1}\varphi_{2}\Psi(\cdot,\overline{\gamma}(z))\right> dz,
\begin{aligned}\bigg |\lim_{t\to\infty}\sum_{T(n)<|\Im(\rho)| \le t}\Re\left(1-\left(\frac{\rho}{\rho-\tau}\right)^n \right) \bigg | &\quad<\sum_{h=0}^{\infty}\sum_{|\Im(\rho)| \in (2^hT(n),2^{h+1}T(n)]}\frac{3n\tau^2+(ne\tau)^2}{3\Im(\rho)^2}.\end{aligned}
\zeta_{D}^{c}( 2 , x^{2}) = - \frac{\alpha \, R}{8 \, x^{3}}+ \frac{3 \, \alpha}{5 1 2 \, R \, x^{5}}+ \frac{5 5 5}{2^{1 7}}\, \frac{\alpha}{R^{3}\, x^{7}}+{\cal O}( x^{- 9}) \,{.}
\kappa^t ( x_0, x_n) \ge \sum_{i = 1}^n \frac{d ( x_{i - 1}, x_i)}{d ( x_0, x_n)}\kappa^t ( x_{i - 1}, x_i) = \frac{1}{d ( x_0, x_n)}\sum_{i = 1}^n \kappa^t ( x_{i - 1}, x_i)
\begin{aligned}0 \geq \sum_p\frac{F^{kk}}{12K}(|u_{pk}|^2 + |u_{p\bar k}|^2) +\frac{1}{2}\psi'' F^{kk}|u_k|^2 + \psi' F^{kk}u_{k\bar k}&- 2\psi'^2\delta^{-1}F^{11}K - C\big[F^{kk}\lambda_1^{-1}|g_{1\bar 1k}| + \mathcal{F}\big] &+ C_\epsilon\psi'F^{kk}|u_k|^2 + \epsilon C\psi'\mathcal{F}. \end{aligned}
\bar{\lambda}_i^{(\ell)}= \frac{\bigl(a^{2g_i}\bar{\lambda}_{i}^{(\ell-g_i)}+ (1-a^{2g_i})\lambda_i\bigr)}{\rho\bigl(a^{2g_i}\bar{\lambda}_{i}^{(\ell-g_i)}+ (1-a^{2g_i})\lambda_i\bigr) + 1},
{e}_{\hat{0}}^{\; \; 0}{\bar{e}}_{\alpha}^{\; \; i}\left[ - \frac{1}{2}\frac{\partial \Gamma^{( 2 )}}{\partial v^{i}}\right] ={e}_{\hat{0}}^{\; \; 0}{\bar{e}}_{\alpha}^{\; \; i}\left[ B_{i}^{v ( 2 )}\right] +{e}_{\alpha}^{\; \; \mu}{e}_{\hat{0}}^{\; \; \nu}K_{\mu \nu}^{g},
f i t ( k ) = - 2 . 1 2 5 8 5 + 1 . 3 3 3 3 3 \log ( k ) - 0 . 2 3 8 4 8 k^{({1 . 3 3 3 3})}+ 0 . 0 5 3 8 2 k^{({2 . 6 6 6 6})}- 0 . 0 3 3 3 5 k^{({2 . 6 6 6 6})}\log ( k ) .
A_{31}= \left(2-e^{h\sigma}\left((h(bd-a_\sigma)+1)(h\sigma+2)+h\sigma\right)\right) \;, A_{32}= he^{h\sigma}(h(bd-a_\sigma)+2) \;, B_3 = -\left(e^{h\sigma}\left((h(bd-a_\sigma)+1)(h\sigma+2)+h\sigma\right)+2\right) \;.
\Big| \frac1n \sum_{j=0}^{n-1}\phi (F_j( x )) - \frac1n \sum_{j=0}^{n-1}\phi (f^j( h^{-1}(x) )) \Big| = \Big| \frac1n \sum_{j=0}^{n-1}\phi_j (f^j( h^{-1}(x) )) - \frac1n \sum_{j=0}^{n-1}\phi (f^j( h^{-1}(x) )) \Big| \le \frac1n \sum_{j=0}^{n-1}\| \phi_j -\phi\|_{C^0}
\int_{0}^{1}d \alpha_{r}\; \cdots \tilde{U}_{| \tau_{r}= ( 1 - \alpha_{1}) \cdots ( 1 - \alpha_{r - 1}) ( \alpha_{r}- \alpha_{r}^{2}) \tau}\; \leq \; \sup_{\alpha_{r}\in [ 0 , 1 ]}\; \cdots \left( ( 1 - \alpha_{1}) \cdots ( 1 - \alpha_{r - 1}) \tau \right)^{- 1}\; g ,
\alpha + \sum_{i = 1}^{3}\alpha_{i}= 0 , \alpha \mu^{2}+ \sum_{i = 1}^{3}\alpha_{i}\mu_{i}^{2}= 0 , \sum_{i = 1}^{3}\alpha_{i}\mu_{i}^{2}\ln ( \mu_{i}^{2}/ \mu^{2}) = 0 ,
\cdots \otimes \Phi ( \zeta_{B}) \otimes \Phi ( \zeta_{B}) \otimes \cdots = \cdots \otimes ( \cdots \widehat{R}( \zeta_{B}) \widehat{R}( \zeta_{B}) ) \otimes ( \cdots \widehat{R}( \zeta_{B}) \widehat{R}( \zeta_{B}) ) \otimes \cdots ,
q(x) = -\frac{1}{\pi i}\oint_{\Gamma_N}{\Bigg(\tilde \varphi'(x, \mu)\varphi(x, \mu) + \tilde \varphi(x, \mu)\varphi'(x, \mu)\Bigg)\hat M(\mu)}d\mu - 2 \sum_{k=N+1}^\infty \sum_{j=0}^1 (-1)^j\alpha_{kj}(\tilde \varphi'_{k,j}(x)\varphi_{k,j}(x) + \tilde \varphi_{k,j}(x)\varphi'_{k,j}(x)).
d\left(S_\omega(x), S_{\omega^\prime}(y)\right) = d\left(c x + a(\omega), c y+ a(\omega^\prime)\right) = \sum_{n=1}^\infty 2^{-n}\left|c x_n + a(\omega)_n - c y_n - a(\omega^\prime)_n\right| \leq c d(x, y) + d\left(a(\omega), a(\omega^\prime)\right).
{\bf E}[Y(\tau)^q] =\Gamma(1-q/\tau), \Gamma_{2}(\tau-q\,|\,\tau) =\frac{\tau^{(\tau-q)/\tau-1/2}}{\sqrt{2\pi}}{\bf E}[Y(\tau)^q]\,\Gamma_2\bigl(1+\tau-q\,|\,\tau\bigr).
\delta_2 = \frac{16 (C_0+1)}{|\log \delta_1|}\bigg ( |\alpha|^{-\frac{1}{2}}\delta_1 + \big |\log |\alpha| \big | \, \big ( |\alpha| + \| F \|_{L^\infty_{2}(\Omega)}\big ) \, + \| F_{12}- F_{21}\|_{L^1 (\Omega)}\bigg )\,,
k \langle r_1^{-1}T_1, \dots, r_n^{-1}T_n \rangle := \{\sum_{i_1, \dots, i_n}a_{i_1, \dots, i_n}T_1^{i_1}\dots T_n^{i_n}\in k [[T_1, \dots T_n ]]| \a_{i_1, \dots, i_n}r_1^{i_1}\dots r_n^{i_n}\to 0 \},
\zeta_T(\nu|\beta)={\varrho^{-2\nu}\over \pi i}\int_{\mu}^{\infty}d\omega \int_{C_+}dz\ln\left(1-e^{i\beta z}\right)\left[{2\nu z \breve{\Phi}(\omega;z) \over(z^2+\omega^2)^{\nu+1}}-{\partial_z\breve{\Phi}(\omega;z) \over(z^2+\omega^2)^{\nu}}\right]~~.
\ker(D|_{L^2(M\setminus\Sigma;\mathcal{S}\otimes \mathcal{I})})^0:=\big\{\mathfrak{u}\in \ker(D|_{L^2(M\setminus\Sigma;\mathcal{S}\otimes\mathcal{I})}) &\mbox{with its leading coefficients in}\ell^2\times\ell^2\big\}.
R^4 = t^{\mu_1\nu_1\mu_2\nu_2\mu_3\nu_3 \mu_4\nu_4}t^{\rho_1\sigma_1\rho_2\sigma_2\rho_3\sigma_3\rho_4\sigma_4}R_{\mu_1\nu_1\rho_1\sigma_1}R_{\mu_2\nu_2\rho_2\sigma_2}R_{\mu_3\nu_3\rho_3\sigma_3}R_{\mu_4\nu_4\rho_4\sigma_4}.
\lefteqn{\Phi(N+P,x) - \Phi(N,x)}={}\bar{F}(S(x)+T(N+P,x),x) - \bar{F}(S(x)+T(N,x),x) ={}\bar{F}(S(x)+ T(N,x) + T(P,x),x) - \bar{F}(S(x)+T(N,x),x).
\tilde{g}_{s}^{\prime}= \left( \frac{\tilde{R}}{R}\right)^{1 / 4}\frac{R^{1 / 2}l_{p}^{3 / 2}}{R_{8}R_{9}}\ , \ \tilde{l}_{s}^{\prime}= \left( \frac{\tilde{R}}{R}\right)^{1 / 4}\frac{l_{p}^{3 / 2}}{R^{1 / 2}},
\Delta(H_1\star P_1)\star \Delta(H_2\star P_2)=\Delta(H_1)\star \Delta(P_1)\star \Delta(H_2)\star \Delta(P_2)=\Delta(H_1)\star \Delta(P_1 \star H_2)\star \Delta(P_2) = \Delta(H_1)\star \Delta(H_2\star H_2(P_1))\star \Delta(P_2)=\Delta(H_1)\star \Delta(H_2)\star \Delta(H_2(P_1))\star \Delta(P_2).
\mathcal{W}_\beta(r_0^{d}T_{r_0}\#\mu_\infty)=r_0^{2d}\mathcal{W}_\beta(T_{r_0}\#\mu_\infty)=r_0^{2d}r_0^\beta\mathcal{W}_\beta(\mu_\infty)=r_0^{2d+\beta}\mathcal{W}_\beta(\mu_\infty).
X_N(t) = \{\phi_1(\cdot,t),\dots,\phi_N(\cdot,t)\}, X_N = \{\, \sum_{j=1}^N u_j(t)\phi_j(x,t)~|~ u_j \in H^1(0,T; \Bbb{R}), ~u_j(0)=0, ~1 \leq j \leq N\,\}.
\int_{B_{H_0}}\widetilde u(\nabla E_0)\,dy = \int_\Omega \widetilde u (\nabla u)\,dx = \int_\Omega \nabla u \cdot x - u \, dx = \int_\Omega{\rm div}(u x) -n u -u \, dx =\int_{\partial \Omega}u x\cdot \nu \, d\mathcal H^{n-1}(x) - (n+1) \int_\Omega u\, dx = - (n+1) \int_\Omega u\, dx.
\left[ ( \lambda z - \mu j)(1-z) + (c-j)\gamma z + j\xi (1-z) \right]P_{j}(z) - \xi z (1-z)P^{'}_{j}(z) = (c-j+1)\gamma z P_{j-1}(z) + j \mu (1-z)\sum_{n=0}^{j}z^{n}p_{j,n}- \mu z \sum_{n=1}^{j}n z^{n}p_{j,n}+ \mu \sum_{n=1}^{j+1}n z^{n}p_{j+1,n}+j\xi (1-z) \sum_{n=0}^{j}z^{n}p_{j,n},
\partial_{\tau}r_j^2 = 2 r_1r_2r_3 \sin(\theta_1+\theta_2+\theta_3), j=1,2,3, \left. \begin{array}{cc}r_1 \partial_{\tau}\theta_1 = r_2r_3\cos(\theta_1+\theta_2+\theta_3), r_2 \partial_{\tau}\theta_2 = r_1r_3\cos(\theta_1+\theta_2+\theta_3), r_3 \partial_{\tau}\theta_3 = r_1r_2\cos(\theta_1+\theta_2+\theta_3), \end{array}\right.
T_1=\frac{-N_f/2+N_c}{2 \pi i},\ \T_0=-\frac{1}{4 \pi i}\sum_{i=1}^{N_f}m_i,\ \ \frac{\partial F}{\partial T_1}=2u-\frac{1}{2}\sum_{i=1}^{N_f}m_i^2
\tilde{X}^{(K)}_{jk}B_k^{\dagger}v = \begin{bmatrix}\tilde{X}_{jk}A_{1k}^{\dagger}v \tilde{X}_{jk}A_{2k}^{\dagger}v \vdots \tilde{X}_{jk}A_{Kk}^{\dagger}v \end{bmatrix}, B_k^{\dagger}v = \begin{bmatrix}A_{1k}^{\dagger}v A_{2k}^{\dagger}v \vdots A_{Kk}^{\dagger}v \end{bmatrix}.
2\sqrt{6}\cos \frac{11\pi}{36}\ +\ 6\cos \frac{10\pi}{36}\ - \ \left(3\sqrt{2}+\sqrt{6}\right)\cos \frac{\pi}{36}= 0 begin{align*}1.2ex](\sqrt{3}-\sqrt{7})\cos \frac{\pi}{42}\ -\ 2\sqrt{7}\cos \frac{25\pi}{42}\ -\ 8\cos \frac{\pi}{42}\cos \frac{25\pi}{42}\ =\ 3.
z^TKz = \left\langle \sum_{i=1}^{n-1}p_i v_i,\sum_{i=1}^{n-1}p_i Kv_i\right\rangle=\left\langle \sum_{i=1}^{n-1}p_i v_i,\sum_{i=1}^{n-1}p_i \lambda_i v_i\right\rangle &=\sum_{i=1}^{n-1}\lambda_i p_{i}^{2}<max(\lambda_i)\sum_{i=1}^{n-1}p_{i}^{2}<0.
\begin{aligned}\lim_{\theta\to\infty}-\frac{2b_n+2a(\theta)^2}{\theta}= \frac{N-1+2\Delta}{2}, \lim_{\theta\to\infty}\frac{4 u_n}{\theta^2}= \frac{n(N+1-n)(2n-N-1-2\Delta)(2n-N-1+2\Delta)}{4(2n-N)(2n-N-2)}, \end{aligned}
\delta \left({1\over 2}\int_{M_{10}}*F_2' \wedge F_2'\right)=\int_{M_{10}}*F_2'\wedge m \delta B_2 =\mu_8 \int_{M_{10}}d*F_2' \wedge \Lambda_1=\mu_8 \mu_0 \int_{W_1}\hat{\Lambda}_1,
dV\epsilon_{\alpha\beta\gamma\delta}\Phi_{\alpha}\frac{\partial{(\Phi_{\beta},\Phi_{\gamma},\Phi_{\delta})}}{\partial(x,y,z)}=3 d\eta\hspace{0.5mm}d\xi\hspace{0.5mm}d\phi\hspace{0.5mm}mn \frac{d}{d\eta}\biggl(\frac{1}{1+f^2(\eta)}\biggr),
I_{m+\nu}^{(2)}(z;q)=\frac{(z/2)^{\nu}}{(q;q)_{\infty}}\sum_{n=0}^{\infty}\frac{q^{n^{2}+n\nu}}{(q;q)_{n}}\left(\frac{z^{2}}{4}\right)^{n}\left(q^{n+\nu+1};q\right)_{\infty}\frac{\left(zq^{n}/2\right)^{m}}{(q^{n+\nu+1};q)_{m}}.
\| f\|_{C^{k,\alpha}_{-q}(\mathbb{R}^3\setminus B)}= \sum_{|I| \le k}\sup_{x\in \mathbb{R}^3\setminus B}\left| |x|^{| I| + q}(\partial^{I}f)(x) \right| + \sum_{|I| = k}\left[ |x|^{k + q+\alpha}(\partial^{I}f)(x) \right]_{\alpha, \mathbb{R}^3\setminus B}.
2 i \z_{1 2}^{- 1}z_{1 3}^{- 3 / 4}z_{1 4}^{- 3 / 4}z_{1 5}^{- 1 / 2}z_{2 3}^{- 3 / 4}z_{2 4}^{- 3 / 4}z_{2 5}^{- 1 / 2}z_{3 4}^{- 1}z_{3 5}^{- 1 / 2}z_{4 5}^{- 1 / 2}\cdot ( z_{1 4}z_{2 3}- z_{1 3}z_{2 4})
\mathop{S}_{\r T}^{\d = 2}\mathrm{\LARGE \sl S}_{\n r T}^{\d = 2}={\frac{1}{\lambda}}\int d^{3}x \varepsilon^{\mu \nu \rho}E_{\mu}^{\underline{{\alpha}}}T_{\nu \rho}^{\underline{{\beta}}}\eta_{\alpha \beta}.
\frac{dc}{d\tau}=\mu h\frac{b+c}{1+c}-\frac{\Gamma}{K_1}\frac{c}{K+c}+\Lambda\theta=F_1, \frac{d\theta}{d\tau}=\epsilon(-\theta+T(c))=\epsilon R_2, \frac{dh}{d\tau}=\epsilon\left(\frac{1}{1+c^2}-h\right)=\epsilon R_3.
\bar{D}(r_s)=\frac{D(r_s)}{L(s)}=\frac{\sum_{i=0}^{K-1-s}\binom{K}{s+1+i}(N-1)^i N}{\binom{K-2}{s-1}+\sum_{i=0}^{K-1-s}\binom{K-1}{s}(N-1)^iN}.
\begin{array}{c}{{a^{-}a^{+}- q a^{+}a^{-}~ = ~ q^{- N}}}\\{{a^{-}a^{+}- q^{- 1}a^{+}a^{-}~ = ~ q^{N}}}\\{{q^{N}a^{+}q^{- N}~ = ~ q a^{+}}}\\{{q^{N}a^{-}q^{- N}~ = ~ q^{- 1}a^{-}}}\\{{q^{N}q^{- N}~ = ~ q^{- N}q^{N}~ = 1 ~}}\\ \end{array}
P_\lambda(\lambda)=c\det\begin{bmatrix}0 \left[f_{j_1}(v_1,\lambda)\right]_{j_1=1,...,p_1}\cdots \left[f_{j_K}(v_K,\lambda)\right]_{j_K=1,...,p_K}\left[g_i(\lambda)\right]_{i=1,..,p}\left[h_{i_1,j_1}(v_1)\right]_{\substack{i_1=1,...,p j_1=1,...,p_1}}\cdots \left[h_{i_K,j_K}(v_K)\right]_{\substack{i_K=1,...,p j_K=1,...,p_K}}\end{bmatrix}.
\begin{array}{cllcll}{[A^i_m,A^j_n]}=\; m \delta^{ij}\delta_{m,-n}\;\;\; ,\;\;\; [C_m,C_n] =\; m \delta_{m,-n}\;\;\; , begin{align*}6pt]\{B^i_r,B^j_s \}=\; \delta{ij}\delta_{r,-s}\;\;\; ,\;\;\; \{D_r,D_s \}=\; \delta_{r,-s}\;\;\; ,\end{array}
\rho_t(x,y) = \frac{t^{-d/\alpha}p_1^{\Gamma}(t^{-1/\alpha}x,t^{-1/\alpha}y)}{t^{2\beta/\alpha}M_{\Gamma}(t^{-1/\alpha}x)M_{\Gamma}(t^{-1/\alpha}y)}= t^{-(d+2\beta)/\alpha}\rho_1 (t^{-1/\alpha}x,t^{-1/\alpha}y).
{\cal B}= \left( \begin{matrix}{{H_{i j}}}&{{}}&{{0}}\\{{0}}&{{}}&{{- k^{- 2}}}\\ \end{matrix}\right) \left( \begin{matrix}{{\delta_{k j}}}&{{}}&{{H_{k l}^{- 1}U_{l}^{-}}}\\{{k^{2}U_{j}^{+}}}&{{}}&{{1}}\\ \end{matrix}\right) \, ,
\frac{1}{2}q^{N+1}\left\{\sum_{n=1}^{\infty}nq^{n-1}(-q^{n})_{N}- \sum_{n=1}^{\infty}nq^n (-q^{n+1})_{N}\right\}&=\frac{1}{2}q^{N+1}\left\{\sum_{n=1}^{\infty}nq^{n-1}(-q^{n})_{N}- \sum_{n=1}^{\infty}(n+1)q^n (-q^{n+1})_{N}+ \sum_{n=1}^{\infty}q^n(-q^{n+1})_N\right\}&=\frac{1}{2}q^{N+1}\sum_{n=0}^{\infty}q^n(-q^{n+1})_N.
G_c(\gamma_1) = \frac{\sqrt{J_1}}{E_q(J_1,J_2)}\left( F_q(J_1,J_2,\gamma_1) + F_q(J_1,J_2,-\gamma_1)\right) = \frac{2\sqrt{J_1}}{E_q(J_1,J_2)}\sum_{n_1,n_2 =0}^\infty \frac{J_1^{n_1}J_2^{n_2}}{[n_1]_q![n_2]_q!}\cos(\gamma_1q^{2[n_1]_q});
z^{1+s(p-1)}f(sp)-zf(0)= z^{1+s(p-1)}\cdot\big(f(sp)-f(0)\big)+\big(z^{1+s(p-1)}-z\big)\cdot f(0) \equiv z^{1+s(p-1)}\cdot spf'(0)+s(z^{p}-z)\cdot f(0) \equiv z^p\cdot s(f(p)-f(0))+s(z^{p}-z)\cdot f(0)=s\big(z^pf(p)-zf(0)\big)\pmod{p^2}.
\sum_{n=1}^{N}\mu\left( z_{n}\right) =\sum_{n=1}^{N}\frac{1}{d\left(z_{n}\right)}=\sum_{n=1}^{d\left( z_{0}\right)^{2}}\frac{1}{d\left(z_{0}\right) +n+d\left( z_{0}\right)^{2}}& \approx\log\frac{d\left( z_{0}\right) +2d\left( z_{0}\right)^{2}}{d\left( z_{0}\right) +1+d\left( z_{0}\right)^{2}}\approx\log2,
\vec{\gamma}=\left(\begin{array}{cc}0 \vec{\sigma}-\vec{\sigma}0 \end{array}\right)\; ,\;\gamma_5=\left(\begin{array}{cc}1 0 0 -1\end{array}\right) \; ,\;\gamma_0= \left(\begin{array}{cc}0 -1 -1 0 \end{array}\right)
(|A u_c - (A u_c)_{Q_{\varkappa, c \varkappa}(z_0)}|^2)^{1/2}&= (\nu r)^{-2}\big(|A \overline{u}_{c}- (A \overline{u}_{c})_{Q_{\varkappa/(2\nu r), c \varkappa/(2\nu r)}}|^2\big)^{1/2}_{Q_{\varkappa/(2\nu r), c \varkappa/(2\nu r)}}, A = (-\Delta_x)^{1/3}, D^2_v,
\deg\left(_\infty\left(\frac{\Delta}{\Delta_{p^n,12}}\right)\right)=p^{n-1}(p+1)-1 &\deg\left(_\infty\left(\frac{\Delta_p^n}{\Delta_{p^n,12}}\right)\right)=p^{n-1}(p+1-p)=p^{n-1}.
q(0,3) =d_3^{-1}d_{29}, q(0,5) =d_5^{-1}d_{27}, q(0,7) =d_7^{-1}d_{25}, q(0,9) =d_9^{-1}d_{23}, q(0,11) =d_{11}^{-1}d_{21}, q(0,13) =d_{13}^{-1}d_{19}, q(0,15) =d_{15}^{-1}d_{17},
\psi(z)\approx lnz-\frac{1}{2z}-\frac{1}{12z^{2}}+\frac{1}{120z^{4}}-\frac{1}{252z^{6}}+\frac{1}{240z^{8}}-\frac{1}{132z^{10}}+\frac{691}{32760z^{12}}-\frac{1}{12z^{14}}+\cdot\cdot\cdot.
\begin{aligned}\begin{aligned}\|g^k\|^2 =m^2\|\nabla f(x^k)-\nabla f(x^*)+\nabla f(x^*)\|^2 &\leq 2m^2\|\nabla f(x^k)-\nabla f(x^*)\|^2+2m^2\|\nabla f(x^*)\|^2 &\le 2L^2\|x^k- x^*\|^2+2m^2\|\nabla f(x^*)\|^2\end{aligned}\end{aligned}
2n^2\exp\Bigl[-\Omega\bigl(n^{1/2}\bigr)\Bigr]+2n^2\exp\Bigl[-\Omega\bigl( nc^{\frac{1}{1-s}}\bigr)\Bigr] &=2n^2\exp\Bigl[-\Omega\bigl(n^{1/2}\bigr)\Bigr]+2n^2\exp\Bigl[-\Omega\Bigl( \log^{\frac{\alpha}{1-s}}n\Bigr)\Bigr] &< \frac{1}{n^\beta},
\mathrm{d}x\left( t\right) =f\left( t,x_{t},r\left( t\right) ,u\left(t\right) \right) \mathrm{d}t+g\left( t,x_{t},r\left( t\right) ,u\left(t\right) \right) \mathrm{d}w\left( t\right) ,t\in J,
\tilde{b}^{~[A_1 \cdots A_{F-k}]}_{~[l_1 \cdots l_{F+4+k}]}=\epsilon_{i_1 \cdots i_{2F+8}}x_{i_1}^{A_1}\cdots x_{i_{F-k}}^{A_{F-k}}\bar{C}_{l_1i_{F+1-k}}\cdots \bar{C}_{l_{k+F+4}i_{2F+4}}W_{i_{2F+5}i_{2F+6}}W_{i_{2F+7}i_{2F+8}}.
T^{L}= \left( \begin{matrix}{{1}}&{{0}}\\{{0}}&{{0}}\\ \end{matrix}\right) , T^{R}= \left( \begin{matrix}{{0}}&{{0}}\\{{0}}&{{1}}\\ \end{matrix}\right) , Q = \left( \begin{matrix}{{0}}&{{1}}\\{{0}}&{{0}}\\ \end{matrix}\right) , Q^{\dagger}= \left( \begin{matrix}{{0}}&{{0}}\\{{1}}&{{0}}\\ \end{matrix}\right) .
\frac d{dt}\| u(t)\|_2^2 = \|u(t)\|_{p+1}^{p+1}\leq C \|u(t)\|_{H^1(\mathbb R)}^{p+1}, \frac d{dt}\| \nabla u(t)\|_2^2 = \mathrm{Re}\int_{\mathbb R}\nabla (|u(t,x)|^{p-1}u(t,x)) \cdot \overline{\nabla u(t,x)}dx\leq C \|u(t)\|_{H^1(\mathbb R)}^{p+1},
\lim_{k\rightarrow \infty}\mathcal{L}_k(\mu_1, \overline{D}_1)-\mathcal{L}_k(\mu_0,\overline{D}_0)=\frac{1}{\mathrm{vol(D)}}\bigl(\mathcal{E}({\overline{D}_1}_X)-\mathcal{E}({\overline{D}_0}_X)\bigr)=\mathcal{E}_{\rm{eq}}({\overline{D}_1})-\mathcal{E}_{\rm{eq}}({\overline{D}_0}).
\Gamma\big(\big[(1/2) + \varepsilon s \pm z^{1/2}\big]\big/2\big)\Gamma\big(\big[(3/2) - \varepsilon s \mp z^{1/2}\big]\big/2\big) & =\dfrac{\pi}{\sin\big(\pi\big[(1/2) + \varepsilon \pm z^{1/2}\big]\big/2\big)}, \varepsilon \in \{-1,1\}.
\begin{array}{rll}F(f_{0}^{-1/2}f_{1},f_{0}^{-1/2}f_{2}) = F(f_{0}^{-1/2}f_{2},f_{0}^{-1/2}f_{1})\;, \mbox{and}\;\;\;F(f_{0}^{-1/2}f_{1},f_{0}^{-1/2}f_{2}) = 0\;\;\;\mbox{at}{\rm P}_{\rm JB}\;.\end{array}
S_{D B I}= - \frac{T_{p}}{\kappa}\int_{V_{p + 1}}d^{p + 1}\xi \mathrm{e}^{- \frac{\kappa ( 3 - p )}{2 \sqrt{2}}\phi}\sqrt{- \det \left[ g + \sqrt{2}\kappa A +{\hat{F}}\mathrm{e}^{- \frac{\kappa}{\sqrt{2}}\phi}\right]}.
l^+(z) := \sum_{j=1}^p l_j z^j, l^-(z) := \sum_{j=-q}^{-1}l_j z^j, & l^{+,\circ}(z) := \sum_{j=0}^p l_j z^j, l^{-,\circ}(z) := \sum_{j=-q}^0 l_j z^j, l^{\circ}(z) := l_0.
((r,x)\cdot (s,y)) \cdot ((r,x)\cdot (t,z)) &= (\alpha_{x,y}(r,s),x\cdot y)\cdot (\alpha_{x,z}(r,t),x\cdot z) &=(\alpha_{x\cdot y,x\cdot z}(\alpha_{x,y}(r,s),\alpha_{x,z}(r,t)),(x\cdot y)\cdot (x\cdot z))