harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int_{B^n(1)}Dv(\boldsymbol{\nabla}_{F_a}u){\text d}V_{F_a}(x)=\lambda\int_{B^n(1)}\kappa(x)g(u(x))v(x){\text d}V_{F_a}(x),\ \forall v\in C_0^\infty(B^n(1)).
C_*(\mathcal{G})=(0\rightarrow \pi_2(K_{\mathcal{G}})\rightarrow C_2(\tilde{K_{\mathcal{G}}})\xrightarrow{\partial_2}C_1(\tilde{K_{\mathcal{G}}}) \xrightarrow{\partial_1}C_0(\tilde{K_{\mathcal{G}}}) \xrightarrow{\partial_0}\mathbb{Z}\rightarrow 0),
2\kappa^{2}e^{-1}L = R +4m^2e^{(2\lambda_{1}+2\lambda_{2})}- 5\partial_{\mu}(\lambda_{1}+\lambda_{2})^2 - \partial_{\mu}(\lambda_{1}-\lambda_{2})^2 - e^{-4\lambda_{1}}{F_{\mu\nu}^{(1)}}^{2}- e^{-4\lambda_{2}}{F_{\mu\nu}^{(2)}}^{2}.
\begin{array}{l}\L_0 \psi_0^- =\lambda_0^{T}\psi_0^- \mbox{in}\mathbb R_-\,, \L_0 \psi_0^+ =\lambda_0^{T}\psi_0^+ \mbox{in}\mathbb R_+\,, \partial_\tau \psi_0^- (\sigma,0)= \partial_\tau \psi_0^+ (\sigma,0)\,,\partial_\tau \psi_0^+ (\sigma,0) = \kappa \, \bigl( \psi_0^+(\sigma,0) - \psi_0^-(\sigma,0)\bigr) \,.\end{array}
(\widetilde{B}_{t_1},\widetilde{B}_{t_2}-\widetilde{B}_{t_{1}},\cdots ,\widetilde{B}_{t_n}-\widetilde{B}_{t_{n-1}})\overset{d}{=}(B_{t_1},{B}_{t_2}-{B}_{t_{1}},\cdots ,B_{t_n}-B_{t_{n-1}}).
B_1+B_2 =\frac{2N(2-p)-4\tau}{p}\int_{|x|<R}|x|^{-\tau}|u|^p(I_\alpha*|\cdot|^{-\tau}|u|^{p})dx &+O\bigg(\int_{|x|>R}|x|^{-\tau}|u|^{p}(I_\alpha*|\cdot|^{-\tau}|u|^p)dx\bigg) &=\frac{2(N(2-p)-2\tau)}p\mathcal{P}[u]+O\bigg(\int_{|x|>R}|x|^{-\tau}|u|^{p}(I_\alpha*|\cdot|^{-\tau}|u|^p)dx\bigg).
T_3 := \sum_{k=1}^d \sum_{i=k+1}^d \sum_{j=i+1}^d \frac{\cos^2(x_j/2)(\sin(x_i/2)-\sin(x_k/2)) - \cos^2(x_i/2)(\sin(x_j/2)-\sin(x_k/2))}{Y}+\sum_{k=1}^d \sum_{i=k+1}^d \sum_{j=i+1}^d \frac{\cos^2(x_k/2)(\sin(x_j/2)-\sin(x_i/2))}{Y},
|\alpha_3| \leq 2^{10}A^{-1}K^{-\frac{1}{2}}N_1, |\beta_3| \sim K^{-1}N_1, (\mathfrak{R}_1^{A}+ \mathfrak{R}_2^{A}) \cap \mathfrak{R}_3^{A}\not= \emptyset, \mathfrak{R}_1^{A}\cap \mathfrak{D}_{j_1}^A \not= \emptyset, \mathfrak{R}_2^{A}\cap \mathfrak{D}_{j}^A \not= \emptyset.
S=\sum_{n=0}^{\infty}\frac{(-i)^{n}}{n!}\int_{-\infty}^{\infty}dx_{1}^{4}\cdot\cdot\cdot\int_{-\infty}^{\infty}dx_{n}^{4}T[H(t_{1})\cdot\cdot\cdot H(t_{n})].
r^{l}(\begin{matrix}Y_{l}^{m}\\Y_{l}^{-m}\end{matrix})=[\frac{2l+1}{4\pi}]^{1/2}\overline{\Pi}_{l}^{m}(z)(\begin{matrix}(-1)^{m}(A_{m}+iB_{m})\\ (A_{m}-iB_{m})\end{matrix}),m>0.
f_{n}^{\left( \alpha \right)}\left( \textbf{A}+x\right) =f_{n}^{\left( \alpha +1 \right)}\left( x\right), \left(\alpha +1\right)\left( \textbf{A}+x\right) f_{n}^{\left( \alpha \right)}\left( \textbf{A}+x\right) =f_{n+1}^{\left( \alpha +1\right)}\left( x\right) +\alpha xf_{n}^{\left( \alpha +1\right)}\left( x\right).
\partial_x B =0\,, &\partial_t B + 2B \partial_x A + A \partial_xB-2\sigma+ 2B =0\,, & \partial_tA + AB \partial_xA-G(M)+A=0\,, &-\frac{\int_{\mathbb{T}}\mathrm{d}x\, B^{-1/2}\partial_t B}{2 \int_{\mathbb{T}}\mathrm{d}x\, \sqrt B}+ \frac{\sigma}{B}-1=0\,.
[\omega_0^{(k,l)}: \omega_1^{(k,l)}] = [a(\xi_0^k\xi_1^l + \xi_0^l \xi_1^k) + \xi_0^l \xi_1^k : a(\xi_0^k\xi_1^l + \xi_0^l \xi_1^k) + \xi_0^k \xi_1^l]
(k\k+1\cdots k+m)=\left(\begin{array}{ccccccccccccc}x_{1}\cdots x_{k-1}x_{k}\cdots x_{k+m-1}x_{k+m}x_{1}\cdots x_{k-1}x_{k+1}\cdots x_{k+m}x_{k}\end{array}\right. \left.\begin{array}{ccc}x_{k+m+1}\cdots x_{r}x_{k+m+1}\cdots x_{r}\end{array}\right)
\frac{\det D ( \varphi )}{\det D ( 0 )}= \lim_{\Lambda \to \infty}\left[ \frac{\langle + \mid \varphi + \rangle}{\mid \langle + \mid \varphi + \rangle \mid}\frac{\langle \varphi + \mid \varphi - \rangle}{\langle + \mid - \rangle}\frac{\langle \varphi - \mid - \rangle}{\mid \langle \varphi - \mid - \rangle \mid}\right]
\begin{gathered}\left [\eta \cdot \nabla, \Gamma \right ] \left (\tau \circ X^{-1}\right ) = \eta(t) \cdot \nabla \Gamma \left (\tau \circ X^{-1}\right ) - \Gamma \left (\eta \cdot \nabla \left (\tau \circ X^{-1}\right ) \right )\end{gathered}
\begin{array}[c]{c}Ric_{\alpha\overline{\beta}}f^{\alpha}-i(n+2)\mu A_{\overline{\alpha}\overline{\beta}},^{\overline{\beta}}-i(n+1)A_{\overline{\alpha}\overline{\beta}}f^{\overline{\beta}}+2Kf_{\overline{\beta}}=0.\end{array}
{\hat{\cal K}}^{( 2 )}= 2 \partial{\hat{\omega}}^{( 1 )}+ l_{p}^{- 2}D{\hat{X}}^{\hat{\mu}}D{\hat{X}}^{\hat{\nu}}( i_{\hat{k}}{\hat{C}})_{{\hat{\mu}}{\hat{\nu}}}- m l_{p}^{2}\partial{\hat{\omega}}^{( 0 )}{\hat{b}}.
{\mathcal D}^{(1/2,1/2)}\otimes{\mathcal D}^{(1/2,1/2)}={\mathcal D}^{(1,1)}\oplus{\mathcal D}^{(1,0)}\oplus{\mathcal D}^{(0,1)}\oplus{\mathcal D}^{(0,0)}\, .
K_{\alpha}^{s}=\{(v,w)\in \mathbb{R}^{k}\oplus \mathbb{R}^{d-k}: \Vert w\Vert< \alpha \Vert v \Vert\}; K_{\alpha}^{u}=\{(v,w)\in \mathbb{R}^{k}\oplus \mathbb{R}^{d-k}: \Vert v\Vert< \alpha \Vert w\Vert\}.
\mathcal{D}_z^{2a}\left(\frac{\zeta^d}{1-\zeta^\ell}\right)=-\ell^{2a}\left(\frac{(2a)!}{\left(2\pi i\ell z\right)^{2a+1}}+\sum_{b\geq 0}\frac{\left(2\pi i\ell z\right)^{b}B_{2a+b+1}\left(\frac{d}{\ell}\right)}{b!(2a+b+1)}\right).
\begin{array}{c}{\nabla_{\rho}\nabla^{\rho}\Lambda^{\mu_{2}\ldots \mu_{s}}+ m^{2}\Lambda^{\mu_{2}\ldots \mu_{s}}= 0 ,}\\{\mathrm{w i t h}m^{2}= ( s - 1 ) ( s + d - 3 )}\\ \end{array}
\langle\delta_x^{(v)},(1+\Delta_V)^{-m}\cos(s\sqrt{\Delta_V})\delta_y^{(w)}\rangle=\langle(1+\Delta_V)^{-\frac{m}{2}}\delta_x^{(v)},\cos(s\sqrt{\Delta_V})(1+\Delta_V)^{-\frac{m}{2}}\delta_y^{(w)}\rangle.
{\cal M}^{og}_{ab}=\left(\begin{array}{cccc}0 \zeta_W^o M_{W}^{o2}0 0 \zeta_W^oM_{W}^{o2}0 0 0 0 0 \zeta_Z^o M_{Z}^{o2}0 0 0 0 M_{gA}^{o2}\end{array}\right)
|C|=\frac{1}{3}\sum_{j=1}^3\sum_{i=1}^q |C^j_i|=\frac{1}{3}\sum_{j=1}^3\sum_{i=1}^q(q-a^j_i)= q^2-\frac{1}{3}\sum_{j=1}^3\sum_{i=1}^qa^j_i.
\begin{array}{l}{{{\cal X}^{0}\vert n , z \rangle_{0}= q^{n}z \vert n , z \rangle_{0}~ ,}}\\{{{\cal D}_{0}\vert n , z \rangle_{0}= \displaystyle \frac{q^{1 - n}}{q - 1}z^{- 1}\vert n - 1 , z \rangle_{0}~ ,}}\\{{\mu_{0}\vert n , z \rangle_{0}= \vert n - 1 , z \rangle_{0}~ ,}}\\ \end{array}
\bar{F}_{\bar{L}^d_h}= \bar{R}_{h}^{e-}\circ F_{\bar{L}_h^d}\circ (\bar{R}_{h}^{e-})^{-1}, \; \; \bar{F}_{{\mathbb{\bar{L}}}^{e}_h}= \bar{R}_{h}^{e-}\circ F_{{\mathbb{\bar{L}}}^{e}_h}\circ (\bar{R}_{h}^{e-})^{-1}.
\begin{aligned}f_n =\sum_{k=0}^{n}\sum_{l=n}^{\infty}\lambda p^{-2l}f_k+\sum_{k=n+1}^{\infty}\sum_{l=k}^{\infty}\lambda p^{-2l}f_k &=\frac{\lambda}{(1-p^{-2})}\left[p^{-2n}\sum_{k=0}^n f_k+\sum_{k=n+1}^{\infty}p^{-2k}f_k\right].\end{aligned}
(C_{ij}^{ij}+C_{ji}^{ji})e_{ii}-(C_{ij}^{ij}+C_{ji}^{ji})e_{jj}=(C_{ii}^{ii}-C_{ii}^{jj})e_{ii}+(C_{jj}^{ii}-C_{jj}^{jj})e_{jj}&+\sum_{x\neq i,j}C_{xx}^{ii}e_{xx}-\sum_{y\neq i,j}C_{yy}^{jj}e_{yy}.
P(N_{t+h}-N_t\ge 2|Z_t = i,U_t=u) =\sum_{n=0}^\infty P(N_{t+h}-N_t\ge 2|Z_t = i,U_t=u,N_t=n)g(n) =\sum_{n=0}^\infty P(t<T_{n+1}<T_{n+2}\le t+h|Z_t = i,U_t=u,N_t=n)g(n).
W_{\rho}(\mu, \nu):=\inf_{\pi \in \mathcal C(\mu, \nu)}\int_{(\mathbb R^d\times\mathcal M) \times (\mathbb R^d\times\mathcal M)}\rho((x,i), (y,j)) \pi(dx\times \{i\}, dy\times \{j\})=\inf E \rho(X, Y),
\begin{array}{rcl}a_{i,\kappa}= \frac{\kappa}{n-2}+\frac{n-i}{n}-\frac{1}{n(n-2)}\,\,\, (i=1,2,\dots,n), b_{i,\kappa}= \frac{\kappa}{n-2}+\frac{3-i}{2}-\frac{1}{2(n-2)}\,\,\, (i=1,2), b_{i,\kappa}= \frac{\kappa}{n-2}+\frac{i-2}{n-2}+\frac{\delta_{i}}{n-2}\,\,\, (i=3,\dots,n-1),\end{array}
- \frac{1}{2 \pi i ( u - v )}\left[ \left( e^{- i [ \theta_{p}( u ) - \theta_{p}( v ) + \mathrm{a r g}\Gamma ( 1 + i u ) - \mathrm{a r g}\Gamma ( 1 + i v ) ]}- 1 \right){\left( \frac{a}{2}\right)}^{i ( u - v )}\right.
\begin{aligned}\mathcal{A}[\overline{u}](x) =\lambda-\mathcal{L}[u](ax)+ a^{(N+1)m}|Du|^m(ax)-f(ax) & \qquad+\mathcal{L}[u](ax) -\mathcal{L}[\overline{u}](x)+f(ax)-f(x) \geq f(ax)-f(x)+\mathcal{L}[u](ax)-\mathcal{L}[\overline{u}](x). \end{aligned}
w_0 = s_{\alpha_4}s_{\alpha_7}s_{\alpha_2 +\alpha_4 +\alpha_5}s_{\alpha_3 + \alpha_4 + \alpha_5}\times s_{\scriptscriptstyle{d6}, max}s_{\alpha_2 + \alpha_3 + \alpha_4}s_{\alpha_{max}}s_{\scriptscriptstyle{e7}, max}.
\bar{T}^{D}_{\Delta,k}[V](x_k, r_k):= \min_{(u,r^{D,\prime})\in F^D_k\left(x_k, r_k\right)}\, \biggl\{c(x_k,u) +\sum_{x^{\prime}\in S}\, Q(x^{\prime}|x_k,u) \,V(x^{\prime}, r^{D,\prime}(x^{\prime})) \biggr\},
m(x) \cdot s = n_{B}^{-1}\left( \frac{x}{|x|}\right) \cdot n_{B^{\perp}}^{-1}\left( \frac{(z-x)^{\perp}}{|z-x|}\right) = n_{B}^{-1}\left( \frac{x}{|x|}\right) \cdot n_{B}^{-1}\left( \frac{(z-x)}{|z-x|}\right)^{\perp}.
\frac{1}{N}\sum_{i=1}^{N}\pi_{i,N}(1-\pi_{i,N}) =\frac{1}{N}\sum_{i=1}^{N}\pi_{i,N}-\frac{1}{N}\sum_{i=1}^{N}\pi_{i,N}^{2}&\overset{P(as)}{\rightarrow}\frac{\theta Ew_{\theta}(X_{1})}{Ew(X_{1})}-\frac{\theta^{2}Ew_{\theta}^{2}(X_{1})}{[Ew(X_{1})]^{2}}.
g_{a,b,c}(x,y,z_1,z_0;q) :=\sum_{t=0}^{a-1}(-y)^tq^{c\binom{t}{2}}\Theta(q^{bt}x;q^a)m\Big (-q^{a\binom{b+1}{2}-c\binom{a+1}{2}-tD}\frac{(-y)^a}{(-x)^b},z_0;q^{aD}\Big ) &\ \ \ \ \ +\sum_{t=0}^{c-1}(-x)^tq^{a\binom{t}{2}}\Theta(q^{bt}y;q^c)m\Big (-q^{c\binom{b+1}{2}-a\binom{c+1}{2}-tD}\frac{(-x)^c}{(-y)^b},z_1;q^{cD}\Big ).
T_\Lambda\mathcal{T}\{g_i\}_{i\in\mathbb{N}}=\sum_{i\in\mathbb{N}}\Lambda_{i+1}^*g_i=\sum_{i\in\mathbb{N}}T^*\Lambda_i^*g_i=T^*\Big(\sum_{i\in\mathbb{N}}\Lambda_i^*g_i\Big)=0.
\frac{\mathcal{C}_{\mathcal{M}^{}}\left[ \mathcal{D}_{\theta}^{}\left( \tau;\rho\right) \right]}{\mathcal{C}_{\mathcal{M}^{}}\left[ \mathcal{D}_{\theta}^{}\left( \tau;0\right) \right]}=\sqrt{\frac{1-\rho}{1+\rho}}.
I_2 =\sum_{j=-y}^{-1}\zeta(j)^p \zeta(j+y)^p=2\zeta(y-1)^p+\sum_{j=-y+2}^{-y/2}\zeta(j)^p \zeta(j+y)^p+\sum_{j=-y/2+1}^{-2}\zeta(j)^p \zeta(j+y)^p &\leq 2\zeta(y-1)^p+12 \zeta(y) &\leq 14 \zeta(y-1)^p.
\bar{H}(y_l | \bar{x}^{B(l)})= -\frac{1}{K}\log \int_{\frac{1}{K}\sum_{i \in B(l)}x_i = y_l}\exp \left(- H(x^{B(l)}| \bar{x}^{B(l)})\right) \mathcal{L}^{K-1}(dx^{B(l)}).
\frac{1}{7}\times ( \frac{7 7}{8}) = \frac{1 1}{8}- ( \frac{1 3}{8}) = 0 . 2 5 \times ( \frac{1}{4 0}) = \frac{1}{1 0}+ ( \frac{9}{1 0}) = \frac{9}{7}\div \frac{9}{7}
\widehat\Psi_-^{-1}(z)\widehat\Psi_+(z)=\Psi_0^{-1}(z)\Psi(z)=\begin{pmatrix}1 f_2(z) 0 1\end{pmatrix}e^{\left(-\frac{s}{z}\right)^k}z^{\frac{\alpha}{2}\sigma_3}H_j,
\ddot{h}=\left( \frac{\left( 2+r\right) r}{\left( 1+r\right)^{2}}\right)^{2}h^{\prime \prime}+\left( \frac{2}{\left( 1+r\right)^{3}}+\frac{2+r}{\left( 1+r\right)^{2}}\right) h^{\prime}-\kappa^{2}f^{2}\left( 1+r^{2}h \right) ,
\tilde W^T \tilde \Lambda \tilde W= \begin{bmatrix}W - \epsilon \Lambda^{-1}T^T \tilde F \,\ - \epsilon \tilde F_{\tau}\end{bmatrix}^T\begin{bmatrix}\Lambda 0 0 -1/u_n \end{bmatrix}\begin{bmatrix}W - \epsilon \Lambda^{-1}T^T \tilde F \,\ \ - \epsilon \tilde F_{\tau}\end{bmatrix}.
\sqrt{\eta^2+\frac{2p+2}{\Lambda_{m,n}}}-\sqrt{\eta^2+2m+2}=\sqrt{\eta^2+2m+2+\frac{a}{(\Lambda_{m,n})^2}}-\sqrt{\eta^2+2m+2}&=\sqrt{\eta^2+2m+2}\left(\sqrt{1+\frac{a}{(\eta^2+2m+2)(\Lambda_{m,n})^2}}-1\right).
\ell^{(k-2)n}B_{t}(\ell^{2n}m,d) =C_{n}(d)=C_{0}(\ell^{2n}d)-\ell^{\lambda-1}\left(\frac{(-1)^{\lambda}m}{\ell}\right)C_{0}(\ell^{2n-2}d) &=B_{t}(m,\ell^{2n}d)-\ell^{\lambda-1}\left(\frac{(-1)^{\lambda}m}{\ell}\right)B_{t}(m,\ell^{2n-2}d).
I_1=C_{0}\int_{B_{r_1}(x_1)}\frac{v^p (y)}{|x_1-y|^2}\mathrm{d}y\leq C_1r_1^{n-2}N^p=C_2M^{1-\frac{2}{n}}N^{\frac{2p}{n}}q^{\frac{2}{n}-1}.
M_{q_1^* - q_0^*}\lesssim M_{u_1^* - u_0^*}+ \|u_1^* - u_0^*\|_{\alpha}+ \| a^*_1 - a^*_0\|_{\alpha}(\displaystyle\max_{i=0,1}\|u_i^*\|_{\alpha}+ N_0^{int}) + \delta N_0^{int}+ \delta N_0,
2 \left( 1 -{\frac{1}{1 2}}N \, e^{2 \Phi}\right) \partial_{-}\Sigma - 4 \left( 1 -{\frac{1}{2 4}}N \, e^{2 \Phi}\right) \partial_{-}\Phi \, \Sigma = \left( 1 -{\frac{1}{2 4}}N \, e^{2 \Phi}\right) \lambda^{2}e^{2 \rho}\; ,
Q_1(Q_0(x_1y_1z_1))=-x_2^{\ell}y_2 z_1+x_2^\ell y_1z_2+ x_2y_2^\ell z_1-x_2y_1z_2-x_1y_2^\ell z_2+x_1 y_2 z_2^\ell \not=0.
(z \cdot \overline{r}) \cdot \overline{e}= \overline{\overline{z}\cdot r}\cdot \overline{e}= \overline{(\overline{z}r) \cdot e}&= \overline{\overline{z}( r\cdot e)}= z \overline{r \cdot e}&= z (\overline{r}\cdot \overline{e}).
R_{k, \beta}(z)=\begin{cases}\prod_{p}\left(1-\frac{1}{p}+\frac{1}{p}\left(\frac{p}{p-1}\right)^z \right) k , \prod_{p}\left(1-\frac{1}{p}+\frac{1}{p}\left(\frac{p}{p-1}\right)^z \alpha_k(p)^{-z}\right) k . \end{cases}
\bar{r}_{\mathcal{A}}^{\mathrm{eff}}(\alpha) =\frac{1}{|\mathcal{A}|}\int_{\mathcal{A}}\, r^{\mathrm{eff}}(\mathbf{x}_{*},\alpha)\,\mathrm{d}\mathbf{x}_{*}\in[0,\bar{r}_{\mathcal{A}}^{\mathrm{ref}}].
\frac{d+\varepsilon\sqrt{ad}}{\sqrt{ad}}\left(\frac{x+\sqrt{ad}}{2}\right)^{2l}- \frac{d+\sqrt{ad}}{\sqrt{ad}}<W_{2l}^{(a,d)}=W_{2m}^{(b,d)}< &<\frac{d+\varepsilon\sqrt{bd}}{\sqrt{bd}}\left(\frac{y+\sqrt{bd}}{2}\right)^{2m}.
\partial_\Sigma^* \Psi(f) \star_H \partial_\Sigma^* \Psi(g) = \partial_\Sigma^* \Psi(f) \cdot \partial_\Sigma^* \Psi(g) + \hbar \left\langle (\partial_\Sigma \otimes \partial_\Sigma)[\tfrac{i}{2}E + H], f \otimes g \right\rangle_{\Sigma^2}.
^{( 3 )}R = - 2 \, \frac{\left( \theta^{\prime}\left( 1 - \frac{3 M}{2 r}\right) - \frac{1}{r \tan \theta}\right)^{2}+ \left( \frac{\sqrt{3}M}{2 r}\theta^{\prime}\right)^{2}}{1 + r^{2}( \theta^{\prime})^{2}\left( 1 - \frac{2 M}{r}\right)}
\int_{B}U_{\delta,e_n}U_{\delta,-e_n}^p = \alpha_n^{p+1}\int_{B}\frac{\delta^{\frac{n-2}{2}}}{(\delta^2+|x-e_n|^2)^{\frac{n-2}{2}}}\left(\frac{\delta^{\frac{n-2}{2}}}{(\delta^2+|x+e_n|^2)^{\frac{n-2}{2}}}\right)^\frac{n+2}{n-2}\, dx &= \alpha_n^{p+1}\int_{B}\frac{\delta^{n}}{(\delta^2+|x-e_n|^2)^{\frac{n-2}{2}}(\delta^2+|x+e_n|^2)^{\frac{n+2}{2}}}\, dx.
\Gamma(z):D\subset L^2(\mathbb{R}^3)\longrightarrow L^2(\mathbb{R}^3) \Gamma(z):=\Gamma^\lambda- (\lambda+z)\mspace{1.5mu}G(\bar{z})^\ast G(-\lambda),\lambda\!>\!\lambda_0,\,z\in\rho(\mathcal{H}_0)
\begin{array}{l c l}{C ( \Lambda ) E_{- \alpha}( \mu ) \omega}&{=}&{\frac{2}{\Lambda - \mu}\left( E_{- \alpha}( \Lambda ) h_{\alpha}( \mu ) \right.}\\{}&{}&{\left. - E_{- \alpha}( \mu ) h_{\alpha}( \Lambda ) \right) \omega + E_{- \alpha}( \Lambda ) C ( \Lambda ) \omega}\\ \end{array}
\|\tilde G(v)\|^2_{L_{HS}(\mathcal H;W^{-g,2})}= \sum_{k=1}^\infty \|\tilde G(v)h_k\|^2_{W^{-g,2}}= \sum_{k=1}^\infty \| (G(v)h_k)\|^2_{W^{-g,2}}&\le \sum_{k=1}^\infty \|G(v)h_k\|^2_{H^{1-g,2}}=\|G(v)\|^2_{L_{HS}(\mathcal H;H^{1-g,2})},
\left(\prod\limits_{j=0}^{s-1}(1-\zeta_p^{2mb^j})\right)^{4k}=\left(\prod_{i=0}^{s-1}\zeta_p^{mb^i}\right)^{4k}\prod\limits_{i=0}^{s-1}\left(\zeta_p^{-mb^i}-\zeta_p^{mb^i}\right)^{4k}\geq 0,
\Omega_{x^0}-\phi_{sz}=[\varepsilon_1]_{x^0}+[\varepsilon_2]_{x^0}-z[\varepsilon_2]_{x^0z}-[\varepsilon_4]_{sz}=0, \Omega_r-r\phi_{ss}=[\varepsilon_1]_r+[\varepsilon_3]_r-s[\varepsilon_3]_{rs}-r[\varepsilon_3+\varepsilon_4]_{ss}=0.
\phi_{P(n)}(t,y) = \left( 1 - \frac{1}{2}i y - \frac{1}{2(n-1)}\sum_{i=1}^{n-1}t_i\right)^{-\lambda}= 2^{\lambda}\left( 2 - i y - \frac{1}{n-1}\sum_{i=1}^{n-1}t_i\right)^{-\lambda}
\frac{v_{in}-v_{out}(\frac{Z_{2}}{Z_{4}}+1)}{Z_{1}}=\frac{v_{out}(\frac{Z_{2}}{Z_{4}}+1)-v_{out}}{Z_{3}}+\frac{v_{out}(\frac{Z_{2}}{Z_{4}}+1)-v_{out}}{Z_{2}}.
\int_{-L+1}^{L-1}\int_{-L}^L \Vert (t,1,0) - (s,0,1) \Vert^p \,ds\,dt = 2\,{\frac{2\,{p}^{2}L + 4 \left( 2\,L-1 \right)^{p}{L}^{2}- 4 \left( 2\,L-1 \right)^{p}L + \left( 2\,L-1 \right)^{p}+ 4\,p L - 1 - 2\,{p}^{2}-4\,p}{{p}^{2}+3\,p+2}}.
\|g\|_{1}\geq \int_{\triangle \setminus \triangle_{\delta}}gdU(x) = \int_{\triangle \setminus \triangle_{\delta}}g^{+}dU(x) + \int_{\triangle \setminus \triangle_{\delta}}g^{-}dU(x) \geq (3/20) \cdot \|g\|_{1}.
D_{m~e}^{~~n~f}= \biggl( R - 2\Lambda \biggr) \biggl( \delta^n_m\delta^f_e - \delta^f_m \delta^n_e \biggr) + 2R^{fn}_{~~me}+ \frac{1}{2}\biggl(R_m^{~f}\delta^n_e - R^n_{~m}\delta^f_e + R_e^{~n}\delta^f_m - R^f_{~e}\delta^n_m \biggr)
\Phi_{+}^{+ Y^{\prime}}( x , \theta^{+}, u ) = \rho_{+}^{+ Y^{\prime}}( x , u ) + \theta_{+}^{+ A^{\prime}}\phi_{A^{\prime}}^{Y^{\prime}}( x , u ) + i ( \theta_{+}^{+})^{2}\chi_{-}^{- Y^{\prime}}( x , u ) \; ,
s_n(G)\left(\sum^{s-1}_{i=0}l(g_{1,i})\otimes...\otimes l(g_{n,i})+\mathcal Q_n(G)\right):= \overline\bigotimes^{s-1}_{i=0}[\langle1,-g_{1,i}\rangle]\overline\otimes...\overline\otimes[\langle1,-g_{n,i}\rangle]\overline\otimes I^{n+1}(G).
[ e_{\{m \}, i}^{( n )}\otimes^{n}1_{N}] [ e_{\{m^{\prime}\}, j}^{( n )}\otimes^{n}1_{N}] = \delta_{i , j}\delta ( \{m \}, \{m^{\prime}\}) \frac{n !}{f_{\{m \}}^{( n )}}e_{\{m \}, i}^{( n )}\; ,
\|u\|^p_{W^{k,p}_\delta(M)}:=\sum_{j=0}^{k}\left\vert\left\vert\rho^{-\delta-\frac{n}{p}+j}\left\vert\hat\nabla^j u\right\vert_{\hat g}\right\vert\right\vert^p_{L^p(M,\hat g)}
< T_{\gamma}> = \frac{\sum_{n = - \infty}^{\infty}B_{n}^{S_{1}}B_{n + \nu}^{S_{2}}}{\sum_{n = - \infty}^{\infty}B_{n}^{V}}= \frac{\sum_{n = - \infty}^{\infty}e^{- S_{1}e^{2}n^{2}/ 2}e^{- S_{2}e^{2}( n + \nu )^{2}/ 2}}{\sum_{n = - \infty}^{\infty}e^{- V e^{2}n^{2}/ 2}}.
\begin{aligned}\frac{1}{\tau^k}[f(x)-f(x^{k+1})]-\frac{1}{\tau^{k-1}}[f(x)-f(x^{k})]+(x-\bar{x}^{k+1})^T[-A^T\lambda^k &+\tau^k\beta^kA^T(A\bar{x}^{k+1}-b)+\tau^k\beta^k(D-A^TA)(\bar{x}^{k+1}-\bar{x}^k) &+(1-\tau^k) \beta^kA^T(A{x}^{k}-b) ]\geq0,~\forall x.\end{aligned}
-{\mathbb E}\tilde x_i^T(T)H x_i(T)={\mathbb E}\int_0^T \left\{(\tilde x_i -\Gamma \tilde m_i)^TQ [x_i-(\Gamma m_i+\eta)] + y_i^T B \tilde u_i+ \gamma (y+y_i)^T \tilde p_i\right\}dt.
E_{e,l}\!\!=\!\!\eta_{e,l}(1\!\!-\!\!\rho_{e,l})\!\bigg[\textrm{tr}(\mathbf{H}_{e,l}^{H}\mathbf{q}\mathbf{q}^{H}\mathbf{H}_{e,l})\!\!+\!\!\textrm{tr}(\mathbf{H}_{e,l}^{H}\mathbf{V}\mathbf{H}_{e,l}\!\!+\!\!N_{E}\sigma_{ea,l}^{2})\bigg]
\phi_y(\alpha) = \inf\left\{\left(\sum_{n=1}^N\phi(\alpha_n)^y\right)^{1/y}:(\alpha_1,\alpha_2,\ldots)\in \tau^{-1}(\alpha)\right\}& = \inf\left\{\left(\sum_{n=1}^N\phi(\alpha_n)^y\right)^{\frac{x}{y}\cdot\frac{1}{x}}:(\alpha_1,\alpha_2,\ldots)\in \tau^{-1}(\alpha)\right\}.
\psi_{1}\left( r \right) = \bar{n}^{3 / 4}\psi_{\lambda - \frac{1}{2}, 1 - \mu}\left( \rho \right) \, , \; \psi_{2}\left( r \right) = \bar{n}^{1 / 4}\psi_{\lambda , \mu}\left( \rho \right) \, , \; 2 \bar{n}= 2 \lambda + \mu - 1 \, .
\left({}^{c}D_{-}^{\alpha,\alpha',\beta,\beta',\gamma}f\right)(x)=\left(-1\right)^{[\operatorname{Re}(\gamma)]+1}\left(I_{-}^{-\alpha',-\alpha,-\beta',-\beta+[\operatorname{Re}(\gamma)]+1,-\gamma+[\operatorname{Re}(\gamma)]+1}f^{({[\operatorname{Re}(\gamma)]+1})}\right)(x),
\Sigma_{f}(x+1)=\sum_{b=\pm1}\sum_{k=1}^{\lfloor\frac{\sqrt{24x+1}-b}{6}\rfloor}(-1)^{k+1}\Sigma_{f}(n+1-\frac{k(3k+b)}{2})+\sum_{n=0}^{x}\sum_{k=1}^{n+1}s_{n+1,k}f(k).
\quad\ \|\mathcal{V}[v+\eta] - \mathcal{V}[v] - \mathcal{V}^{BD}[v;\eta]\|_{L^r(0,T)}\qquad\qquad\qquad\qquad\qquad\le \rho_{v,p,r}(\|\eta\|_{\infty}) (\|\eta'\|_{L^p(0,T)}+ |\eta(0)|) .
\ |\mathcal H(k,d,2)|-|\mathcal H(k,d,l)| = \ \sum_{i=d+2}^{2d}\sum_{j=d+2}^{i}\binom{n-j}{k-d-1}-\sum_{i=d+2}^l\binom{n-d-i+1}{k-d}-(d-1)\binom{n-d-l+1}{k-d-l+2}< \ \frac{d(d-1)}{2}\binom{n-d-2}{k-d-1}-\binom{n-d-l+1}{k-d}.
(1-2tc(t))^2 = 1-4t (1-2tc(t))^{-1}= (1-4t)^{-\frac{1}{2}}= \sum_{n=0}^\infty \binom{-\frac{1}{2}}{n}(-4t)^n &= \sum_{n=0}^{\infty}\binom{2n}{n}t^n = \sum_{n=0}^\infty (n+1)C_nt^n,
\{u : \mathbb{R}\rightarrow \mathfrak{g}\mid u \in H^0, \u(t + 2 \pi) = \sigma (u(t)) , \ \rho(u(-t)) = - u(t)\}\cong \ \{u : [0, 2 \pi] \rightarrow \mathfrak{g}\mid u \in H^0, \ \rho(\sigma^{-1}u(2\pi-t)) = - u(t)\}\cong \ \{u : [0, \pi] \rightarrow \mathfrak{g}\mid u \in H^0\}.
b_{1,0}(s) = \prod_{j = 1}^d (s_1 + \dfrac{d + 1}{2}- \dfrac{j - 1}{2}) \prod_{k = 1}^{n - d}(s_1 + s_2 + \dfrac{n + 1}{2}- \dfrac{k - 1}{2}), b_{0,1}(s) = \prod_{j = 1}^d (s_2 + \dfrac{d + 1}{2}- \dfrac{j - 1}{2}) (s_2 + \dfrac{n}{2}- \dfrac{j - 1}{2}) \prod_{k = 1}^{n - d}(s_1 + s_2 + \dfrac{n + 1}{2}- \dfrac{k - 1}{2}) .
n = y_n n + y_nz_n^2C''(z_n)\delta_n + o(C''(z_n)\delta_n) + mc_mS_n &= n + n\frac{L-c_mS}{C(z_n)}+\delta_n\bigg( z_n^2C''(z_n)- n\frac{z_nC'(z_n)}{C(z_n)}\bigg) + o(C''(z_n)\delta_n) + mc_mS_n.
\kappa_1 =\langle \frac{de_1(s)}{ds}, e_2(s)\rangle &=-2\lambda \sum_{j=1}^n \Big\{( A_j\sin(2\lambda s)- B_j \cos(2\lambda s))^2+( A_j \cos(2\lambda s)+ B_j \sin(2\lambda s))^2 \Big\}&=-2\lambda,
\det\nabla(\phi_m\circ\cdots\circ \phi_1) =\det\nabla(\phi_m\circ\cdots\circ \phi_i)(\phi_{i-1}\circ\cdots\circ \phi_1)\cdot \det\nabla (\phi_{i-1}\circ\cdots\phi_1) &\geq \det\nabla(\phi_m\circ\cdots\circ \phi_i)(\phi_{i-1}\circ\cdots\circ \phi_1)\cdot 2f \chi_{A_{i-1}\setminus A_i}&\geq f \chi_{A_{i-1}\setminus A_i},
a_0(\ell,k) =a_{0,1}(\ell,k)-a_{0,2}(\ell,k)-a_{0,3}(\ell,k) b(\ell,k) =-b_1(\ell,k)+ b_2(\ell,k)+b_3(\ell,k) A_0^\ast(\ell,k) =-(\delta_{k,n_1+1}-a_3^\ast(k))(\delta_{\ell,n_1}-a_2^\ast(\ell)).
M_\ell(t)=\begin{cases}0, t=0,\cr\frac{1}{\ell}e^{-\frac{q}{(\ell t)^{2/q}}}, t\in \Big(0,\frac{1}{\ell}\left(\frac{2q}{q+2}\right)^{q/2}\Big),\cr\frac{(q+2)^{1+q/2}}{2^{q/2}q^{1+ q/2}}e^{-\frac{q}{2}}t-\frac{2}{eq\ell}e^{-\frac{q}{2}}, t\geq\frac{1}{\ell}\left(\frac{2q}{q+2}\right)^{q/2}.\end{cases}
D_1 := \{x_3 + x_4 = 0,\ 2x_1^2 + 2x_2^2 + 2x_1 x_2 - 1 = 0 \}D_2 := \{x_3 - x_4 = 0,\ 2x_1^2 + 2x_2^2 + 2x_1 x_2 - 1 = 0 \}
Y^{0}_{q},\,Y^{\bf i}_{q},\,Y^{\bf j}_{q},\,Y^{\bf k}_{q}, &[(d\pi_{GM})(\tilde{Y}^{0}),\,(d\pi_{GM})(\tilde{Y}^{\bf i})]_{q}=[Y^{0},\,Y^{\bf i}]_{q}, &[(d\pi_{GM})(\tilde{Y}^{0}),\,(d\pi_{GM})(\tilde{Y}^{\bf j})]_{q}=[Y^{0},\,Y^{\bf j}]_{q}, &[(d\pi_{GM})(\tilde{Y}^{0}),\,(d\pi_{GM})(\tilde{Y}^{\bf k})]_{q}=[Y^{0},\,Y^{\bf k}]_{q}
||[D^{\alpha}_{\ast},\mathcal{A}_1]\partial_1{\mathbf V}||^2_{L^2(\Omega_t)}\lesssim\sum_{0<\beta\leq\alpha}||D^{\beta}_{\ast}\mathcal{A}_1D^{\alpha-\beta}_{\ast}(\partial_1{\mathbf V})||^2_{L^2(\Omega_t)}\,.
{\chi_{\overline{R}}(x_1,x_2)\over |\overline{R}|}\iint_{\overline{R}}f_1(y_1,y_2)dy_1dy_2+\{\chi_{\overline{R}}(x_1,x_2)\over |\overline{R}|}\iint_{\overline{R}}f_2(y_1,y_2)dy_1dy_2=0.
\alpha(n;k_1,k_2,\ldots,k_n) = (-1)^{n-1}\alpha(n; k_2,k_3,\ldots,k_n,k_1-n), \alpha(n;k_1,k_2,\ldots,k_n) = \alpha(n;k_1+c,k_2+c,\ldots,k_n+c), c \in \mathbb{Z}.
\sum_{k=3}^n (qk^2-(6q-q')k) \frac{n!}{(n-k)!(2k)}\leq \sum_{k=3}^n (qk+q')\frac{n!}{(n-k)!(2)}&= \frac{n!}{2}\sum_{k=3}^n \frac{qk + q'}{(n-k)!}&\leq(qn+q'n)\frac{n!}{2}\sum_{j=1}^\infty \frac{1}{j!}&\leq \left(\frac{q+q'}{2}\right)(n)n!e.
e_{\mu}^{a}= e^{\frac{\rho ( x )}{2}}\delta_{\mu}^{a}, \quad e_{a}^{\mu}= e^{- \frac{\rho ( x )}{2}}\delta_{a}^{\mu}, \quad \omega_{m , a b}= \frac{1}{2}( \eta_{a m}\partial_{b}\rho ( x ) - \eta_{b m}\partial_{a}\rho ( x ) ) , \quad \omega_{c , a b}= e_{c}^{m}\omega_{m , a b}= e^{- \frac{\rho ( x )}{2}}\omega_{c , a b}.
A'(r)= 2h^{-3}h'\partial^2_\theta+\frac12\frac{(2h'+2rh''h'+2rh''+r^2h'''h')h+(2rh'+r^2h'')h'}{h^2}&-\frac14\frac{2(h+rh')(2h'+rh''h')h^2+2hh'(h+rh')^2}{h^4}.
\varepsilon\frac{d\Delta_{2}(t,\varepsilon)}{dt}=-{\mathcal{A}}_{2}^{T}(\varepsilon)\Delta_{1}(t,\varepsilon) -{\mathcal{A}}_{4}^{T}(\varepsilon)\Delta_{2}(t,\varepsilon)+ \Gamma_{2}(t,\varepsilon),
E = (t_{1}+ c) \partial_{1}+\partial_{2}& E = (t_{1}+c) \partial_{1}& E = (t_{1}+c) \partial_{1}+{c}_{0}t_{2}\partial_{2}, E = (t_{1}+c) \partial_{1}+ t_{2}^{r}(1 + c_{1}t_{2}^{r-1})\partial_{2},
y^* = \left( 2 -\mu^2-\frac{1}{\tau}-1+\mu^2 + \frac{1}{\tau}- \frac{\mu^2}{\tau}\right) y^* + \frac{\mu^2}{\tau}T y^* = \lim_{n\rightarrow +\infty}\mathbb{E}\left[ y[n] \right] = T.
={\frac{1}{Z}}\sum_{n = 0}^{\infty}\big ( -{\frac{1}{4 \pi \alpha^{\prime}}}\big ){\frac{1}{n !}}\int_{{\cal H}_{n}^{\varepsilon}}d^{d}z d^{d}\bar{z}\langle \Phi_{1}\Phi_{2}\dots \Phi_{N}\prod_{a = 1}^{n}B_{i_{a}j_{a}}J^{i_{a}}( z_{a}) \bar{J}^{i_{a}}( \bar{z}_{a}) \rangle
M(s,z) = \int_{0}^{\infty}\theta_2(z,t)t^{s}dt/t=\frac{1}{2\sqrt{k}}\int_{0}^{\infty}\sum_{h_1,h_2 \in \mathbb{Z}}\exp\big( -\frac{4\pi t^2}{y}\big| \frac{h_1}{4\sqrt{k}}+\sqrt{k}zh_2\big|^2\big) t^{s+1}dt/t.
{\frac{1}{\beta_{c}}}= b ( d - 2 ) ( 1 - \kappa ){\int}_{- 1}^{1}\; d \xi \; H_{c}( \xi ) \;{\frac{\sqrt{1 - \xi^{2}}}{\sqrt{( \xi + \frac{3 + \kappa}{1 - \kappa}) ( \xi + \frac{1 + 3 \kappa}{1 - \kappa})}}}\; .
\mathcal{G}_{n+1}(u,t) =e^t\left(\frac{\partial}{\partial u}\mathcal{G}_n(u,t)+n\mathcal{G}_n(u,t)\right) & =e^t\left(e^{-nu}\sum_{m= 2}^\infty \frac{(m+n-1)!}{(m-2)!}e^{-mt}(1-e^{-u})^{m-2}e^{-u}\right).
\begin{aligned}B_{\beta,I}^*B_{\beta,I}= A_{\beta,I}^*A_I, B_{\beta,I}^*y_\beta = A_{\beta,I}^*y, B_{\beta,I}^*B_{\beta,I}= A_{\beta,I}^*A_I, B_{\beta,I}^*y_\beta = A_{\beta,I}^*y. \end{aligned}
\frac 1 y \int_0^{2 \pi}\int_{0}^{1}x^{- 2 \rho + 1}d x d \phi + \frac 1 y \int_0^{2 \pi}\int_{1}^{4 y^{2 \delta}}\sqrt x d x d \phi = O \big( y^{3 \delta - 1}\big).