harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{ll}{\sf I}^{\sf i}_{{\sf p},{\rm ab}}=\langle\underline{\sf p},{\rm a}|{\sf I}^{\sf i}|\underline{\sf p},{\rm b}\rangle{\sf i}=1,2,3 begin{align*}2mm]{\sf C}^{\sf c}_{{\sf p},{\rm ab}}=\langle\underline{\sf p},{\rm a}|{\sf C}^{\sf c}|\underline{\sf p},{\rm b}\rangle{\sf c}=1,2, ..., 8\end{array}\hskip7cm
T_{a,\alpha q}^{\underline\alpha}E_{\underline\alpha,\beta r}=T_{a,\alpha q}^bE^{\underline\alpha}_bE_{\underline\alpha,\beta r}-(\nabla_{\alpha q}E^{\underline\alpha}_a+\omega_{\alpha q,a}^{~b}E^{\underline\alpha}_b)E_{\underline\alpha,\beta r}-(\nabla_aE^{\underline\alpha}_{\alpha q})E_{\underline\alpha,\beta r}.
U\left({{\bf y},{\bf 0}}\right) \equiv P\exp \left({ig\int\limits_{\bf 0}^{\bf y}{dz^i}A_i^a \left( z \right)T^a}\right) = P\left({1 + ig\int\limits_{\bf 0}^{\bf y}{dz^i}A_i^a\left( z \right)T^a + ...}\right) .
\langle H_y \rangle_{u_{\alpha}\otimes \Phi_y^{(0)}}= \Big(e_{\alpha}+ \alpha \|\lambda_y\|^2 -\frac{\alpha^2}{L^3}+ O\Big( \frac{\alpha^2}{L^5}\Big) + O\big( \alpha^2 L e^{-L/2}\big) \Big) |\Phi_y^{(0)}|^2 ,
\int_{S'_{d}}\big|X^{\beta}(\varphi,w)\big|^{2n+1}\,d\mu_{\beta}(w) =0, \int_{S'_{d}}\big|X^{\beta}(\varphi,w)\big|^{2n}\,d\mu_{\beta}(w) =\frac{(2n)!}{2^{n}\Gamma(\beta n+1)}|\varphi|_{0}^{2n}.
I = - \tau_{D p}\int d^{p + 1}\sigma \e^{- \phi}\sqrt{- \det \left( G_{\alpha \beta}+ \left[ 2 \pi F_{\alpha \beta}- B_{\alpha \beta}\right] \right)}- \tau_{D p}\int e^{2 \pi F - B}\wedge \bigoplus_{n}C_{( n )}\quad ,
r=r^{\prime}[1-\frac{1}{2r^{\prime2}}[2(x_{0}x^{\prime}+y_{0}y^{\prime})-(x^{\prime2}+y^{\prime2})]+\frac{1}{8r^{\prime4}}[2(x_{0}x^{\prime}+y_{0}y^{\prime})-(x^{\prime2}+y^{\prime2})]^{2}+\cdot\cdot\cdot],
\Bigl| \xi - \frac{s}{b^{|W|}(b^{|UV|}- 1)}\Bigr| \le \frac{1}{b^{|W (UV)^{t+1}U|}}= \frac{1}{b^{|WUV| + |(UV)^{t}U|}}&= \frac{1}{b^{|WUV|}b^{n |WUV| / (r(n,{\mathbf x}) - n)}}.
\quad\left(H(0)\right)^{t}\equiv \begin{pmatrix}0 0 \dots 0 b_{1}0 \dots 0 0 0 \dots 0 0 b_{2}\dots 0 \vdots \vdots \ddots \vdots \vdots \vdots \ddots \vdots 0 0 \dots 0 0 0 \dots b_{q}\end{pmatrix},\quad\mbox{where $b_{1},b_{2},\dots,b_{q}\in[0,1)$.}
\sum_{\epsilon_{l}< 0}u_{\epsilon_{l}l}\left( r \right) u_{\epsilon_{l}l}\left( r^{\prime}\right) + \int_{0}^{\infty}\frac{\mathrm{d}k}{2 \pi}\, u_{k l}\left( r \right) u_{k l}\left( r^{\prime}\right) = \delta \left( r - r^{\prime}\right) \, ,
dq_t= \tilde\gamma^{-1}(t,q_t)\left(-\partial_t\psi(t,q_t)-\nabla_qV(t,q_t)+\tilde F(t,q_t,\psi(t,q_t))\right) dt & +\tilde S(t,q_t)dt+\tilde\gamma^{-1}(t,q_t)\sigma(t,q_t)\circ dW_t,
\hat S[\hat B]=\int d^{3}x [-\frac{1}{8\pi}\epsilon^{\mu\nu\lambda}(\hat B_{\mu}*\partial_{\nu}\hat B_{\lambda}+\frac{2i}{3}\hat B_{\mu}*\hat B_{\nu}*\hat B_{\lambda})+\frac{1}{2g^{2}}\hat B^{\mu}\hat B_{\mu}] +O(\frac{1}{m}).
g_{BTZ}= -\frac{(r^2-r_{+}^2)(r^2-r_{-}^2)}{r^2}dt^2+\frac{r^2}{(r^2-r_{+}^2)(r^2-r_{-}^2)}dr^2 + r^2(d\phi -\frac{r_{+}r_{-}}{r^2}dt)^2
A_1(z)=(1-zbq)(1-zbq^2)(q^{-n}-q^\gamma)(q^{-n}-q^{\gamma+1})(abq^{2n}-1)(abq^{2n-1}-1), B_1(z)=bq^{n+\gamma}(1-zbq^2)(1+q)(aq^n-1)(1-q^{-n})(q^{-n}-q^\gamma)(abq^{2n-1}-1), C_1(z)=b^2q^{2n+2\gamma}(aq^n-1)(aq^{n-1}-1)(1-q^{-n})(1-q^{-n+1}).
x^{(2)}=[\begin{matrix}0&-1/2\\ -5/7&0\\ \end{matrix}][\begin{matrix}5.0\\ 8/7\\ \end{matrix}]+[\begin{matrix}11/2\\ 13/7\\ \end{matrix}]=[\begin{matrix}69/14\\ -12/7\\ \end{matrix}]\approx[\begin{matrix}4.929\\ -1.714\\ \end{matrix}].
\exists\ \{\nu_{j}\}_{j\in I(\bar{x})}\geq 0\ \forall y\in \ell_p \sum_{k=1}^{+\infty}|\bar{x}_k-x_k|^{p-2}{(\bar{x}_k-x_k)}y_k=-\sum_{i\in I(\bar{x})}\nu_i \langle f_i \ |\y \rangle,
=(\underbrace{\beta_{1},\dots,\beta_{1}}_{u_{1}},\dots,\underbrace{\beta_{m},\dots\beta_{m}}_{u_{m}},\underbrace{0,\dots,0}_{x};\underbrace{\gamma_{1},\dots,\gamma_{1}}_{r_{1}},\dots,\underbrace{\gamma_{m},\dots,\gamma_{m}}_{r_{m}},\underbrace{0,\dots,0}_{y}),
\int_s^t[\sum_{i=1}^m f^{i,j}_-\circ \tilde{c}^i(r)+g^j_-(r)]^+dr \leq\tilde{c}^j(t)-\tilde{c}^j(s) &\leq \int_s^t[\sum_{i=1}^m f^{i,j}\circ\tilde{c}^i(r)+g^j(r)]^+dr
{1\over |B|}\int_B\big|[b,R_j](f)(g)-\big([b,R_j]f\big)_B\big|dg \leq{2\over |B|}\int_B|J_1(g)|dg+{2\over |B|}\int_B\left|J_2(g,\tilde g)\right|dg +{1\over |B|}\int_B\left|\left(J_3(\cdot, g)\right)_B\right|dg \quad+{1\over |B|}\int_B\left|J_4(g,\tilde g)\right|dg =:2L_1+2L_2(\tilde g)+L_3+L_4(\tilde g).
WP(\lambda)=n!\sum_{k=0}^{l(\lambda)}(-1)^{l(\lambda)-k}\binom{n+k}{k}\sum_{\substack{\nu^1,\dots,\nu^k \nu^1+\nu^2+\cdots+\nu^k=\lambda \nu^i \neq0}}\dfrac{\prod_{j=1}^{l(\lambda)}\binom{m_j(\lambda)}{m_j(\nu^1),m_j(\nu^2)\dots,m_j(\nu^k)}}{\prod_{i=1}^{k}(|\nu^i|+1)!}.
-j^{tL(\frac{\cdot}{t})}(tH'(\rho_{-}))=\rho_{-}+\int_{\rho_{-}}^{\infty}\frac{1-p(\rho_{-},\rho,t)}{\sqrt{2\pi t(H'(\rho)-H'(\rho_{-}))^3}}H''(\rho) d\rho
C_{-2,n}\int\limits_0^Te^{(\lambda_n+q_{k,n})s}ds+C_{-1,n}\int\limits_0^Te^{(\overline{\lambda}_n+q_{k,n})s}ds+\sum\limits_{j=0}^{N-1}C_{j,n}\int\limits_0^Te^{(q_{j,n}+q_{k,n})s}ds=b_{k,n},k=0,1,2,...,N-1.
\|u\|_{C^{\alpha}_{WF}(\Omega)}:= \sup_{(s,z)\in\Omega}|u(s,z)| + \sup_{\stackrel{(s_1,z_1)\neq(s_2,z_2)}{(s_i,z_i)\in\Omega, i=1,2}}\frac{|u(s_1,z_1)-u(s_2,z_2)|}{\left(\sqrt{|s_1-s_2|}+\rho(z_1,z_2)\right)^{\alpha}}.
\begin{cases}A =(LR \times \overline{L}R) \cdot (L^{2}R \times \overline{L}R), B =(LR \times \overline{L}R) \cdot (L^{2}R \times LR), C =(LR \times \overline{L}R) \cdot (LR \times \overline{L}R).\end{cases}
\stackrel{( 0 )}{S}= S_{0}\left[ A_{\alpha \beta}^{( \sigma )}\right] + \int d^{D}x \left( A_{( \sigma )}^{* \alpha \beta}\partial_{\left[ \alpha \right.}\eta_{\left. \beta \right]}^{( \sigma )}+ \eta_{( \sigma )}^{* \alpha}\partial_{\alpha}C^{( \sigma )}\right) ,
( q^{x}\cdot \Theta_{, s s}\Theta_{, r r s}+ q^{- x}\cdot \Theta_{, r r}\Theta_{, s s s}) - ( q^{x}+ q^{- x}) \Theta_{, r s}\Theta_{, r s s}+ \Theta_{, r \alpha s}+ \Theta_{, s \beta s}= 0 .
\widetilde{t}_{j,k+1}^{\left[\widetilde{p}-1,q-1\right]}=\widetilde{t}_{j,k}^{\left[\widetilde{p}-1,q-1\right]}+{q-1 \over \widetilde{p}-2}\,\widetilde{t}_{j,k+1}^{\left[\widetilde{p}-2,q-2\right]}\,\left(t_{j,k+\widetilde{p}-1}-t_{j,k+1}\right).
\Big(2 x^1_i \partial_{x^1_i}^2 + 2(|p - n + 1| + 1) \partial_{x^1_i}\Big)^* (x^1_i)^{n - p - 1}= (2(n - p)(n - p - 1) - 2(n - p - 1)(n - p)) (x^1_i)^{n - p - 2}= 0.
\varphi_i(t)= C(t,0)x_0+S(t,0)y_0+\int_0^t S(t,s)f_i(s)ds+\int_0^t S(t,s)BB^{*}S^{*}(b,t) R(a,\Upsilon_0^b)\Big[x_b-C(b,0)x_0 &-S(b,0)y_0-\int_0^b S(b,\eta)f_i(\eta)d\eta\Big](s) ds,i=1,2.
w(\delta_{1}) = S(\delta_{1})(1 - \chi(\frac{10 x}{\delta_{1}}))(u_{0}, u_{1}) + \int_{0}^{\frac{\delta_{1}}{10}}S(\delta_{1}- \tau) (1 - \chi(\frac{10x}{\delta_{1}})) F(u(\tau)) d\tau + \int_{\frac{\delta_{1}}{10}}^{\delta_{1}}S(\delta_{1}- \tau) F(u(\tau)) d\tau,
\begin{array}{l}f_0=1,\mbox{and}f_k=z\sum\limits_{\ell=0}^{k-1}f_\ell+z\sum\limits_{\ell=0}^{k-1}g_\ell +z\sum\limits_{\ell=0}^{k-1}h_\ell, k\geq 1, g_k=z f_{k+1}+zg_{k+1}+zh_{k+1},k\geq 0, h_k=zf_k+zg_k+zh_k,k\geq 0. \end{array}
\begin{aligned}a( P_h\vec{v}_h , \vec{w}_h ) + b( \vec{w}_h ,p_h) = a( \vec{v}_h, \vec{w}_h ), \forall\vec{w}_h\in V_h, b(P_h\vec{v}_h ,q_h) =0, \forall q_h\in Q_h.\end{aligned}
\frac{dM_1}{dv}=-\frac{2}{v^3}\int_0^v\left(1-e^{-\int_{v'}^vA(v'')dv''}\right)v'^2dv'-\frac{A(v)}{v^2}\int_0^ve^{-\int_{v'}^vA(v'')dv''}v'^2dv'.
\int_{X}f(x)d\lambda(x) =\int_{X_{0}}\mathcal{E}_{s}f(e,e,x_{0})d\lambda(x_{0})=\int_{X'}\phi d\lambda' =\int_{X'}\phi\left(x'\right)du.\lambda'(x')=\int_{X_{0}}\mathcal{E}_{s}f(u,e,x_{0})d\lambda(x_{0})=\int_{X}f(x)du.\lambda(x),
v_m^+(x,t)=\frac{3aL^3}{2\pi}\sum_{j=0}^{2}\int_{0}^{\infty}e^{ia\rho^3L^3t}\frac{\Delta_{j,m}^+(\rho)}{\Delta^+(\rho)}e^{\lambda_j^+(\rho)x}\hat{h}^+_m(\rho)\rho^2d\rho v_m^-(x,t)=\overline{v_m^+(x,t)},
u^k = \left[ S^0 \subset S^{k\sigma}\to \Sigma^{k\sigma}E_{C_2}\simeq \Sigma^{|k\sigma|}\Sigma^{k\sigma - |k\sigma|}E_{C_2}\to \Sigma^{|k\sigma|}\widehat{E}_{C_2}\right] \in \widehat{E}^{C_2}_{-|k\sigma|}
S_{}\phantom{}_{b_1 b_1}^{b_1 b_1}(\zeta) = S_{}\phantom{}_{b_1 b_2}^{b_2 b_1}\left(\zeta + \frac{i\pi \nu}2\right) S_{}\phantom{}_{b_1 b_1}^{b_1 b_1}\left(\zeta - i\pi (1 - \nu)\right)
V_{0}( \sigma ) = \frac{\sigma^{2}}{2 \lambda_{0}}- \int_{0}^{\sigma}d s \int_{- \infty}^{\infty}\frac{d \omega}{2 \pi}\frac{s}{( 4 \pi )^{\frac{D - 1}{2}}}( \omega^{2}+ s^{2})^{\frac{D - 3}{2}}\Gamma ( - \frac{D - 3}{2}) \mathrm{t r}\mathrm{\boldmath 1}.
B_{\bar{b}}:= \left\{\bigg(x_{\bar{a}}: \bar{a}\in \bigcup_{I \subseteq [k]}\prod_{i \in I}Q_i \bigg) \in B : \left( x_{\bar{b}_I}: I \subseteq [k] \right) \in A \setminus \left( S_+ \setminus S_- \right) \right\}.
{\cal A T}_{q}^{- 1}~ = ~ \left( \begin{matrix}{{1}}&{{0}}&{{0}}&{{0}}\\{{1}}&{{1}}&{{0}}&{{0}}\\{{- 5}}&{{3}}&{{1}}&{{- 1}}\\{{8}}&{{5}}&{{0}}&{{1}}\\ \end{matrix}\right)
G(u_\mu, u_\nu)=\beta_{\mu\nu}|u_\nu|^{p+1}|u_\mu|^{p-1}u_\mu+\displaystyle{\sum_{\substack{\mu=1}}^N}\lambda_{\mu\nu}|u_\nu|^{p+1}|u_\mu|^{p-1}u_\mu
\Delta^h_{R_{02}}=\left[ \begin{matrix}3 5 -3 5 11 -5 -3 -5 3\end{matrix}\right],\quad\Delta^h_P=\left[ \begin{matrix}4 8 0 8 16 0 0 0 8\end{matrix}\right],\Delta^0_{\mathcal{R}}=\left[ \begin{matrix}0 -2 -6 -2 0 -12 -6 -12 0\end{matrix}\right]
\left\Vert u\right\Vert_{H_{0}^{s}(\Omega)}^{2}=\left\Vert u\right\Vert_{L^{2}(\Omega)}^{2}+\iint_{Q}\frac{|u(x)-u(y)|^{2}}{|x-y|^{d+2s}}\mathrm{d}x\mathrm{d}y & =\left\Vert u\right\Vert_{L^{2}(\Omega)}^{2}+[u]_{\mathbb{R}^{d},s}^{2}
, \ldots , \left( \begin{array}{c c c c c c c c c c c}{{0}}&{{\cdots}}&{{0}}&{{1}}&{{0}}&{{\cdots}}&{{0}}&{{0}}&{{0}}&{{\cdots}}&{{0}}\\{{0}}&{{\cdots}}&{{0}}&{{0}}&{{0}}&{{\cdots}}&{{0}}&{{1}}&{{0}}&{{\cdots}}&{{0}}\\ \end{array}\right)
\sum_{w \in \{I, w_4, w_5\}}K_{w_6}^{+-}(y; w(\mu)) = \frac{1}{24\pi^2}\sum_{w \in \{I, w_4, w_5\}}\Bigl( \mathcal{J}_2(y; w(\mu)) + \mathcal{J}_3(y; w(\mu)) + \mathcal{J}_4(y; w(\mu)) \Bigr) ;
(\mathcal F'_{\mathbf H}f\,(i\lambda),w\otimes1) =(2\pi)^{\ell/2}|W|\int_{A}f(a)\, \mathbf G(-i\lambda,w^{-1}a)\,da &=(2\pi)^{\ell/2}|W|\,\mathcal F_{\mathbf H}f\,(\lambda,w)\qquad\forall w\in W
\tau_{a}\big(\varphi_{\mathrm{MH}}, (x,a), (y,b)\bigr) = \Bigr(z,\min\bigl(\{1, 2, 3\}\setminus \{a,b\}\bigr)\Bigr), \tau^{a}\big(\varphi_{\mathrm{MH}}, (x,a), (y,b)\bigr) = \Bigr(z,\max\bigl(\{1, 2, 3\}\setminus \{a,b\}\bigr)\Bigr).
\int_{\mathbb{R}^{2N}}\frac{|u(x) - u(y)|^{p-2}\, (u(x) - u(y))\, (\varphi(x) - \varphi(y))}{|x - y|^{N+s\,p}}\, dx\, dy=\int_{\Omega}\frac{f(x)}{u^\gamma}\,\varphi\, dx,
T(x;C)=\big\{z\in X~|~f\big(\Pi_{X_2}(z)\big)\in T\big(f\big(\Pi_{X_2}(x)\big);C_f\big)\big\}, &N(x;C)=\big\{x^*\in Z^*~|~g(x^*)\in N\big(f\big(\Pi_{X_2}(x)\big);C_f\big)\big\}.
0\le \zeta_{j,N}\le \sum_{{\ell=1}}^{\infty}\frac{d^{2\ell+1}}{(\ell!)^2}{(1+x_j)}^\ell \left(1-x_j- \frac 1 N\right)^\ell &\le \sum_{{\ell=1}}^{\infty}\frac{d^{2\ell+1}}{(\ell!)^2}{(1-x_j^2 )}^\ell \,.
i G_{a b}( x - y ) = \frac 1{i^{2}}\, \frac{\delta^{2}\ln Z [ \phi_{1}, \phi_{2}]}{\delta \phi_{a}( x ) \, \delta \phi_{b}( y )}= i \left( \begin{array}{c c}{{G_{1 1}}}&{{- G_{1 2}}}\\{{- G_{2 1}}}&{{G_{2 2}}}\\ \end{array}\right) \ .
\frac{\partial P_{\rm{tr},1}(h_{\beta,1})}{\partial h_{\beta,1}}= 0 \Leftrightarrow \int_0^1 \Big\{2h_{\beta,1}\big( \eta_1 + P_{0}(r, h_{\beta,1})(\eta_0-\eta_1) \big) + & \big(r^2 + h_{\beta,1}^2\big) \big( \eta_1 + \frac{\partial P_{0}(r, h_{\beta,1})}{\partial h_{\beta,1}}(\eta_0-\eta_1) \big) \Big\}r{\rm{d}}r = 0,
Z_t = X_t + \frac{e^{\beta_t}}{\frac{d}{dt}e^{\beta_t}}\dot X_t , \frac{d}{dt}\nabla h(Z_t) = \frac{\frac{d}{dt}e^{\beta_t}}{e^{\beta_t}}\left( \nabla h(X_t) - \nabla h(Z_t) - \frac{1}{\mu}\nabla f(X_t)\right),
\begin{cases}t_i = \alpha_i+\dots+\alpha_{n-2}+\frac{1}{2}(\alpha_{n-1}+\alpha_n) , \, (1\leq i\leq n-2) t_{n-1}= \frac{1}{2}(\alpha_{n-1}+\alpha_{n}), t_n = \frac{1}{2}(-\alpha_{n-1}+\alpha_{n}) . \end{cases}
\pi^2_{\rm co}:N\otimes_{\rm max}N^\circ\rightarrow \mathcal{B}(L^2(N,\tau)\otimes \overline{L^2(N,\tau)}) \pi^2_{\rm co}(x\otimes y^o)(\xi\otimes\overline{\eta}) :=x\xi\otimes\overline{\eta y}x, y\in N\, ,\xi, \eta\in L^2(N,\tau)\,
g_{\alpha,T}(\underline{m}) = \int_{\mathcal{T}^k}\prod_{j=1}^{k}\gamma_j^{m_j}\,d \mu_{\alpha,T}(\underline{\gamma}) &= \frac{1}{T}\int_{0}^{T}\left\{(n_1+\alpha)^{m_1}\cdots(n_k+\alpha)^{m_k}\right\}^{-i \tau}\,d \tau
\rho^{2}( n^{2}\rho^{2}+ 8 s i n^{2}n F ) F^{\prime \prime}+ 2 n^{2}\rho^{3}F^{\prime}+ \frac{1}{n}s i n 2 n F ( 4 n^{2}\rho^{2}F^{2}- 4 s i n^{2}n F - n^{2}\rho^{2}) = 0
\int_{M_{\delta}}d p p^{\alpha}\ \leq \ \frac{1}{\delta^{2}}\int_{0}^{\infty}d p p^{\alpha}\left| \chi \left( p \right) - \phi_{0}\right|^{2}\ < \ \frac{q_{0}^{2}\varepsilon \left( q_{0}\right)}{\delta^{2}}\ .
\sum_{u\in E^{0,0}}\sum_{i=1}^{k_u}n_i^u a_{X_i^u}(w,u) = \sum_{u\in E^{0,0}}\sum_{j=1}^{k_u}m_j^u a_{X_j^u}(w,u) (\forall w\in E^{0,1}) .
w\left({{\eta_1},...,{\eta_d}}\right) = \mathop{\lim}\limits_{T \to \infty}\frac 1T \ln{\left \langle E_{1}^{\eta_1-\eta_2}E_{2}^{\eta_2-\eta_3}...E_{d-1}^{\eta_{d-1}-\eta_d}E_{d}^{\eta_d}\right \rangle}
\sum_{n =1}^{\infty}\frac{(b/a)_n a^n}{(1- c q^n) (b)_n}= \sum_{m=0}^{\infty}\frac{(b/c)_mc^m}{(b)_m}\left(\frac{aq^m}{1-aq^m}-\frac{bq^m}{1-bq^m}\right).
\frac{d^{n-1-l}\hat{\rho}}{dv^{n-1-l}}=P_{l-1}+\frac{v^{l-1}}{2}\left\{(-1)^l\frac{(n-l-1)!}{(n-1)!}\hat{I}_{n,n-1}-\sum_{m=1}^l\frac{(-1)^m}{(m-1)!(l-m)!(n-m)}\hat{I}_{n,m-1}\right\}.
\rho_u\rho_u\Phi^v=\rho_u(\Phi^u R_u\Phi^v)=\Phi^v\lambda_u^2=\rho_u\rho_w\Phi^u=\rho_u(\Phi^w)R_{vu}(\Phi^u)=\Phi^vR_v(\lambda_u^2)
\begin{cases}\lambda_1 = \sqrt{\frac{s_{+}(a)s_{+}(c)}{s_{+}(b)}}\lambda_2 = \sqrt{\frac{s_{+}(a)s_{+}(b)}{s_{+}(c)}}\lambda_3 = \sqrt{\frac{s_{+}(b)s_{+}(c)}{s_{+}(a)}}\end{cases}
X=\left(\begin{array}{ccc}\alpha_{1}x_{3}\overline{x}_{2}\overline{x}_{3}\alpha_{2}x_{1}x_{2}\overline{x}_{1}\alpha_{3}\end{array}\right) ,~\alpha_{1},\alpha_{2},\alpha_{3}\in \mathbb{R},~x_{1},x_{2},x_{3}\in \mathbb{A}\mathbb{A}_{s},
\delta K_{\mu \nu}= \frac{\epsilon}{2}\delta g_{\perp \perp}K_{\mu \nu}^{( 0 )}- \delta \Gamma_{\perp \mu \nu}- \bar{\nabla}_{\mu}\bar{\nabla}_{\nu}\delta Z_{\perp}+ \epsilon \delta Z_{\perp}( K_{\mu}^{( 0 ) \rho}K_{\rho \nu}^{( 0 )}- R_{\perp \mu \perp \nu \perp}^{( 0 )}) + \bar{\cal L}_{\delta Z_{\parallel}}K_{\mu \nu}^{( 0 )},
\left|\int_0^t\langle f,w\rangle d\tau\right| &\leq\frac{1}{4}\int_0^t\|\nabla w\|_2^2 d\tau+C\int_0^t\frac{d\tau}{1+\tau}+C\int_0^{T_\varepsilon}\tau^{-1/2}d\tau +2\varepsilon\int_{T_\varepsilon}^t\tau^{-1/2}d\tau
{}_{N}\! \langle \varrho_{n}^{\prime}, \bar{\varrho}_{n}^{\prime}| \varrho_{n}, \bar{\varrho}_{n}\rangle_{N}= \prod_{\mu = 0}^{p}\delta ( \mathrm{R e}\varrho_{n \mu}^{\prime}- \mathrm{R e}\varrho_{n \mu}) \delta ( \mathrm{I m}\varrho_{n \mu}^{\prime}- \mathrm{I m}\varrho_{n \mu}) ~ .
\sum\limits_n\big|\langle d_n,f(z)\rangle\langle g(z), e_n\rangle\big| \leq \ \Big(\sum\limits_n \big|\langle d_n,f(z)\rangle\big|^2\Big)^{\frac{1}{2}}\Big(\sum\limits_n \big|\langle g(z),e_n\rangle\big|^2\Big)^{\frac{1}{2}}begin{align*}0.2cm]= \ \|f(z)\|\, \|g(z)\|.
\frac{d}{dt}f(K_u^2)H_uu =[B_u,f(K_u^2)]H_uu+f(K_u^2)([B_u,H_u]u+i\bar{J}Qu)+ f(K_u^2)H_u(B_uu-iJH_uu) &=B_uf(K_u^2)H_uu+i\bar{J}Qf(K_u^2)u+i\bar{J}f(K_u^2)H_u^2u &=B_uw_k^K+i\bar{J}Qu_k^K+i\bar{J}f(K_u^2)(K_u^2u+Qu) &=B_uw_k^K+i\bar{J}(2Q+\sigma_k^2)u_k^K,
\frac{d\tau^{2}}{dt^{2}}=\frac{1}{c^{2}}\eta_{\alpha\beta}\frac{dx^{\alpha}}{dt}\frac{dx^{\beta}}{dt}=1-\frac{1}{c^{2}}\frac{dr^{2}}{dt^{2}}=\frac{1}{\gamma(\dot{r})^{2}}
J_i(\pi)= \int_{\mathbb{K}^\infty}\sum_{n=1}^\infty \beta^{n-1}r^+_i(s_n,a_n)dP_{\mu}^\pi+\int_{\mathbb{K}^\infty}\sum_{n=1}^\infty \beta^{n-1}r^-_i(s_n,a_n)dP_{\mu}^\pi.
\int_{I_{\bar{l}}}{y^a|\nabla U_k|^2\,dX}+\int_{I_{\bar{l}}}{y^a|U_k|^{2\gamma}\,dX}+\int_{I_{\bar{l}}\cap\{y=0\}}{|U_k(x,0)|^{2^*_s}\,dx}\leq \varepsilon^\alpha .
f_{[ c d ] [ e f ]}^{[ a b ]}={\frac{1}{2}}\bigg ( \eta_{c e}\delta_{d}^{a}\delta_{f}^{b}- \eta_{c f}\delta_{d}^{a}\delta_{e}^{b}+ \eta_{d f}\delta_{c}^{a}\delta_{e}^{b}- \eta_{d e}\delta_{c}^{a}\delta_{f}^{b}- ( a \leftrightarrow b ) \bigg ) .
\int_{\Sigma}\sqrt{W}[ - \overline{{{\Psi}}}\Gamma_{-}\Gamma_{r}\mathcal{D}_{r}\Psi + \overline{{{\Psi}}}\Gamma_{-}\Gamma_{m}\{X^{m}, \Psi \}+ \Lambda \{\overline{{{\Psi}}}\Gamma_{-}, \Psi \}] ,
h_{\ell}^{\prime \prime}( i t , r ) = - 2 t \xi_{\ell}( t r ) h_{\ell}^{\prime}( i t , r ) + \left[ \sigma ( r ) - 2 t^{2}\left. \frac{d \xi_{\ell}( \tau )}{d \tau}\right|_{\tau = t r}\right] h_{\ell}( i t , r ) \, .
B_n \subset \{Z_n(x) \geq -\frac{\beta \log n}{\sqrt{2}}+ \sigma \sqrt{2\log n}f(x) \geq C e^{(3\sigma+1)\sqrt{\log n}}n^{-\beta}\}& \cap \{Z_n(x) < -\frac{\beta \log n}{\sqrt{2}}+ \sigma \sqrt{2\log n}f(x) \leq c e^{(\sigma-1)\sqrt{\log n}}n^{-\beta}\}.
h_2 = \begin{cases}(k/\sqrt{|C_1|}) f_{\nu-1}/f_\nu + C_4 x^{-n/2}/f_\nu , n\leq 2 -(k/\sqrt{|C_1|}) f_{\nu+1}/f_\nu + C_4 x^{-n/2}/f_\nu , n\geq 2\end{cases}.
|StarFC(S)| = \prod_{i=1}^n |StarFC(S_{Q_i \cap S})| \cdot \prod_{M \not=N}|StarFC(R_M)| = \prod_{i=1}^n |StarFC(T_{Q_i})| \cdot \prod_{M \ne N}|StarFC(R_M)| = |StarFC(T)| \cdot \prod_{M \ne N}|StarFC(R_M)| > |StarFC(R_N)| \cdot \prod_{M \ne N}|StarFC(R_M)| = |StarFC(R)|.
\left[ \begin{array}{c c}{{a_{0}}}&{{1}}\\{{1}}&{{0}}\\ \end{array}\right] \left[ \begin{array}{c c}{{a_{1}}}&{{1}}\\{{1}}&{{0}}\\ \end{array}\right] \cdots \left[ \begin{array}{c c}{{a_{i}}}&{{1}}\\{{1}}&{{0}}\\ \end{array}\right] = \left[ \begin{array}{c c}{{p_{i}}}&{{p_{i - 1}}}\\{{q_{i}}}&{{q_{i - 1}}}\\ \end{array}\right] .
\wp(p_{C}^{(k)}(\tau)|\tau)=\frac{(\frac{g_{2}(\tau)}{2}-3e_{k}(\tau)^{2})(C\eta_{1}(\tau)-\eta_{2}(\tau))+\frac{g_{2}(\tau)}{4}e_{k}(\tau)(C-\tau)}{3e_{k}(\tau)(C\eta_{1}(\tau)-\eta_{2}(\tau))+(\frac{g_{2}(\tau)}{2}-3e_{k}(\tau)^{2})(C-\tau)}
V^{[n]}_{z_1z_1z_1}- V^{[n]}_{z_1}-U_{z_2}=0, & V^{[j]}_{z_1z_1}-V^{[j]}+U\,\Omega^{(j+1)}=0, j=1\dots n-1, & \left(\Omega^{[1]}\,U\right)_{z_1}+\frac{U}{2n}-z_2U_{z_2}=0, & \Omega^{[j]}_{z_1}=UV^{[j]}_{z_1}, j=1\dots n.
D_n(x+1|a_1,\dots,a_r)-D_n(x|a_1,\dots,a_r) =n D_{n-1}(x|a_1,\dots,a_r)\,, \widehat D_n(x+1|a_1,\dots,a_r)-\widehat D_n(x|a_1,\dots,a_r) =n\widehat D_{n-1}(x|a_1,\dots,a_r)\,.
\begin{aligned}D^*_N(M_1x,\ldots,M_Nx) \:\:\: = \:\:\: \sup_{\beta\in [0,1)^d}\left|\frac{\sum_{n=1}^Nf_{\beta}(M_nx)}{N}\right| \leq \:\:\: \max_{{\beta}_L\in B_L}\left|\frac{\sum_{n=1}^Nf_{\beta_L}(M_nx)}{N}\right|+\sup_{\beta\in [0,1)^d}\left|\frac{\sum_{n=1}^Nf_{\beta_L,\beta}(M_nx)}{N}\right|.\end{aligned}
\begin{matrix}p(w,b|log\mu,)=(\frac{\mu}{2\pi})^{\frac{n_{f}}{2}}exp(-\frac{\mu}{2}w^{T}w)\frac{1}{\sqrt{2\pi\sigma_{b}}}exp(-\frac{b^{2}}{2\sigma_{b}})\\ \propto(\frac{\mu}{2\pi})^{\frac{n_{f}}{2}}exp(-\frac{\mu}{2}w^{T}w)\end{matrix}.
e^{- S_{b}^{f r e e}( m ; \Lambda , \lambda )}= \int{\cal D}A{\cal D}\psi{\cal D}\overline{{\psi}}\quad J ( A ) \quad e^{\overline{{\psi}}\gamma^{\mu}\partial_{\mu}\psi + m \overline{{\psi}}e^{- 2 i \varphi \gamma_{5}}\psi + ( A , d \lambda + i \ast d \Lambda )}.
\begin{array}{lcl}\mbox{At PU-Rx:}\,\,\, r_{p}(k) = h_{p}(k) x_{p}(k) + h_{sp}(k) x_{s}(k) + n_{p}(k), \mbox{At SU-Rx:}\,\,\, r_{s}(k) = h_{s}(k) x_{s}(k) + h_{ps}(k) x_{p}(k) + n_{s}(k), \end{array}
k \int_0^L \omega u_1 u_2 = \int_0^L t \mu_1( 1 - u_1) u_1 + \int_0^L (1-t) \frac{\mu_1}{\alpha^2}\left(\alpha - (\alpha u_{1}- d u_2)^+\right)(\alpha u_{1}- d u_2)^+.
\cfrac{t^2 - \sigma_i^2}{\mu^2}=\cfrac{2\sigma_i}{\sigma_i^o -\sigma_i}\rightarrow \cfrac{2t^*}{\zeta_i-t^*}\;\; i=1,\ldots,\mathbf k^*, \; \cfrac{\mu^2}{t^2 - \sigma_i^2}\rightarrow \cfrac{0}{(t^*)^2 - (\zeta_i)^2}=0 \; \, i=\mathbf k^*+1,\ldots,m.
\delta[B_\mu \dot y^\mu+B_\alpha\dot\psi^\alpha+\lambda^\alpha\tilde\chi_\alpha-\tilde H]={d\over{dt}}\{[B_\mu f^{\mu\nu}{{\partial\tilde\chi_\alpha}\over{\partial y^\nu}}+B_\beta \omega^{\beta\rho}{{\partial\tilde\chi_\alpha}\over{\partial \psi^\rho}}+\tilde\chi_\alpha]\epsilon^\alpha\}\,,
M^{-(a_1+a_2+a_4)l+ra_3l}\sum_{m=2}^Nc_0^{\frac{m}{2}}M^{\frac{a_1 l}{2}m}\alpha^m\Bigg(\|F_m^l\|_{l,r}+\sum_{n=2}^{\infty}\|T_m^{l,(n)}\|_{l,r}\Bigg) &\le c (M^{-a_1+a_2+a_4}+M^{-2a_1+a_2+a_4}\alpha^{-2}).
F_X(x)=1-M\int_x^\infty\mathrm{e}^{\beta t}t^\nu K_{\nu}(\alpha t)\,\mathrm{d}t=1-M\sum_{k=0}^\infty\frac{\beta^k}{k!}\int_x^\infty t^{\nu+k}K_\nu(\alpha t)\,\mathrm{d}t.
\left\{e^{\frac{1}{24}A}\widehat{A}(TX){\rm exp}(\frac{c}{2}){\rm ch}(\widetilde{T_CM}-\widetilde{L_C}-16+W_i+W_j)-e^{\frac{1}{24}A}A\widehat{A}(TX){\rm exp}(\frac{c}{2})\right\}^{(14)}&=-24\left\{e^{\frac{1}{24}A}\widehat{A}(TX){\rm exp}(\frac{c}{2})\right\}^{(14)}.
H = \left. \left( \frac{\partial_{r}L}{\partial \dot{q}}\dot{q}- L \right) \right|_{\frac{\partial_{r}L}{\partial \dot{q}}= P}, \, \, \, \, \, \, q = ( x , e , \chi , \psi^{n}) , \; P = ( p , P_{e}, P_{\chi}, P_{n}) \; .
Y^{\tau_1,i}_0 := \log(\varepsilon) \geq Z^i_{\tau_1}, Y^{\tau_1,i}_{n+1}-Y^{\tau_1,i}_n :=\begin{cases}Z^i_{\tau_1+n+1}-Z^i_{\tau_1+n}, Z^i_{\tau_1+n+1}-Z^i_{\tau_1+n}\geq -d -d, \end{cases}
\sup_{\frac{1}{8(1+|x|)^2}<s<1}K_{23}(t,x,y)|_{t=\log\left(\frac{1+s}{1-s}\right)}\lesssim\sup_{\frac{1}{8(1+|x|)^2}<s<1}\frac{|x|}{s^{(n-1)/2}}e^{|y|^2-|x|^2}& \lesssim(1+|x|)^ne^{|y|^2-|x|^2}, (x,y)\in N^c.
Q^+_i =2y_i\sqrt{w_i(1+\gamma_i)|h_{i,i}|^2p_i}-y^2_i\left(\sum_{j=1}^{L}|h_{i,j}|^2p_j+\sigma^2_i\right), Q_k^- ={2\tilde{y}_k\sqrt{\sum_{j\ne k}|\tilde{h}_{k,j}|^2 p_j+\tilde{\sigma}^2_k}-\tilde{y}_k^2w_k(1-\tilde{\gamma}_k)|\tilde{h}_{k,k}|^2p_k}.
\mathcal{U}_{k}(\lambda) = \big\{\mathbf{P}_k \in \mathcal{U}_k : c_{k-1}(\mathbf P_k) = 2k-2-\lambda \big\}, \mathcal{W}_{k}(\lambda) = \big\{\mathbf{P}_k \in \mathcal{W}_k : c_{k-1}(\mathbf P_k) = 2k-2-\lambda \big\},
\liminf_{t\to\infty}\frac 1 t \int_0^t a_{\max}(N-S(\bar t))d\bar t \geq\liminf_{t\to\infty}\frac 1 t \int_0^t a(\xi_{\bar t})(N-S(\bar t))d\bar t &\geq\liminf_{t\to\infty}\frac 1 t \int_0^t a(\xi_{\bar t})(N-b(\xi_{\bar t}))d\bar t=\lambda.
|L_{\mathcal{G}}(u)|/n^2 \le \binom{t}{2}\left(\frac{1}{t+2}+ \epsilon\right)^2 + t\left(\frac{1}{t+2}+ \epsilon\right)\left(\frac{1}{t+2}+ 2\epsilon\right) & \le \frac{t(t+1)}{2(t+2)^2}+ t\epsilon + \frac{t(t+3)}{2}\epsilon^2\le \frac{t(t+1)}{2(t+2)^2}+ 2t\epsilon.
B_j =-K_4+2\lambda_j C_j =\lambda_jJ_3 D_j =K_2+2J_2\lambda_j E_j =\lambda_jJ_1 F_{\varepsilon,j}(t) =-K_0+J_0\lambda_j-L_0\lambda_j^2+\lambda_j^3-\hat{c}_nv_{\varepsilon}^{\frac{12}{n-6}}(t)
R_k=\frac{4}{\sqrt{5}}[(\alpha(\begin{bmatrix}1 0\end{bmatrix},t))(\alpha(\begin{bmatrix}-1 2\end{bmatrix},t))-(\alpha(\begin{bmatrix}-1 0\end{bmatrix},t))(\alpha(\begin{bmatrix}1 2\end{bmatrix},t))]=0,
\{g(z,\bar z),J_0(z_1)\}=\frac{1}{2}\,(t_0\,g(z,\bar z))\,\delta(z-z_1),~~~~\{g(z,\bar z),\bar J_0(\bar z_1)\}=\frac{1}{2}\,(g(z,\bar z)\,t_0)\,\delta(\bar z-\bar z_1),
\Sigma ( \tau , \xi , p_{t}, \theta ) = - \frac{g E \cos \theta}{8 \pi^{3}}\ln \left[ 1 - \exp \left( -{\frac{2 \pi p_{t}^{2}}{g E \cos \theta}}\right) \right] ({\frac{\alpha}{\pi}})^{1 / 2}\exp ( - \alpha \xi^{2})
\int_0^{2\pi}\frac{d\phi_1}{2\pi}f(\phi_1,\ldots,\phi_{N_f},\theta)\sum_{\nu=-\infty}^\infty e^{i\nu(\theta-\phi_1\ldots-\phi_{N_f})}=f(\phi_2+\ldots+\phi_{N_f}-\theta,\phi_2,\ldots,\phi_{N_f},\theta)\;,
\phi_{\varepsilon ,\lambda}\left( \theta_{-\varepsilon}\right) =\delta^{-1}\phi_{-\varepsilon ,\lambda}\left( \theta_{-\varepsilon}-\right) ,\phi_{\varepsilon ,\lambda}^{\prime}\left( \theta_{-\varepsilon}\right) =\delta^{-1}\phi_{-\varepsilon ,\lambda}^{\prime}\left( \theta_{-\varepsilon}-\right) ,
H^{\wedge Q!^{(i+1)(i+2)/2}}= \big(H^{\wedge Q!^{i(i+1)/2}}\big)^{\wedge Q!^{i+1}}\subseteq \big(H_{i+1}(H \cap \Gamma) \big)^{\wedge Q!^{i+1}}\subseteq H_{i+1}^{\wedge Q!^{i+1}}(H \cap \Gamma)^{\wedge Q!^{i+1}}H_{i+2}\subseteq H_{i+2}(H_1 \cap \Gamma).
P_{G_{2}}^{\Delta_{S}}( 3 | x ) = P_{D_{4}}^{{\cal R}}( x ) / x^{2}\quad ({\cal R}={\bf V},{\bf S},{\bf \bar{S}}) , \quad P_{G_{2}}^{\Delta_{L}}( 3 | x ) \left( P_{G_{2}}^{\Delta_{S}}( 3 | x ) \right)^{3}= P_{D_{4}}^{\Delta}( x ) .
P_j^E \ni m^T g = z^\ell \left(\prod_{i=j+1}^n x_i^{a_i + T}\right) \left(\prod_{i=1}^j x_i^{a_i}\right) &= z^\ell \left(\prod_{i=1}^n x_i \right)^{T'}\left( \prod_{i=j+1}^n x_i^{a_i + T- T'}\right) \left(\prod_{i=1}^j x_i^{a_i-T'}\right) &= \left(z^{D \cdot T' + \ell}\prod_{i=1}^j x_i^{a_i-T'}\right) \left( \prod_{i=j+1}^n x_i^{a_i + T- T'}\right),
\int_{S^2}(k^{(3)}- \frac{1}{4}- h^{(3)}) dS^2 =-\frac{3}{4}\int_{S^2}W_0^2 d S^2-\frac{1}{60}\int_{S^2}|\alpha|^2 dS^2 + \frac{11}{45}\int_{S^2}|\beta|^2 dS^2.
\begin{aligned}Y_1(\Omega_1)=h^1_1(\phi^1_1)\cdot \prod_{a=2}^{d_1-1}(\sin \phi^1_a)^{c_{a-1}+\frac{1}{2}-\frac{a-1}{2}}P^{(c_{a-1},-1/2)}_{J_a}(\cos 2\phi^1_a) c_a=\sum_{s=1}^{a}J_s+\frac{a-1}{2}+A+J_1,\end{aligned}