harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{{\mathbf{\dot F}}_k}=\sqrt{\gamma}\left[{{\mathbf{U}}_k^{[ 1]}{\mathbf{V}}_k^{[ 1]}, \ldots ,{\mathbf{U}}_k^{[ J]}{\mathbf{V}}_k^{[ J]}}\right]\in\mathbb{C}^{M\times d_\Sigma}
W(N,n) = \frac{1}{2\pi i}\, \oint_{|z|=\rho}\, dz \, \bigl( f(z) \bigr)^N \, z^{-n-1}= \frac{1}{2\pi i}\, \oint_{|z|=\rho}\frac{dz}{z}\, \exp \{N ( \ln f(z) - \mu \ln z ) \}\,,
\mathbf{R}\mathbf{v}^{0}=\mathbf{R}\frac{(\sum\limits_{j=1}^{N}\mathbf{R}_{.j})^{H}}{\mathbf{S(R)}}=\mathbf{R}(\mathbf{1}^{H}_{N\times1}\mathbf{R})^{H}/\mathbf{S(R)}&=\mathbf{R}\mathbf{R}^{H}\mathbf{1}_{N\times1}=\mathbf{1}_{N\times1}
\begin{array}{l c r}{{d s^{2}={\frac{1}{2}}K_{\rho \sigma}[ d \phi_{\rho}d \phi_{\sigma}+ a_{\rho}a_{\sigma}\sum_{i = 1}^{3}d \Omega_{\rho , i}d \Omega_{\sigma , i}] ;}}\\{{H = - 2 a_{\rho}d \Omega_{\rho},}}\\ \end{array}
\begin{array}{lll}\psi(t) \displaystyle \approx \tilde{\psi}(t)=\sum\limits_{i=1}^N \mu_{i}\,\tau_{i}^{-1}\,e^{-t/\tau_{i}}\approx \sum\limits_{i=1}^N \mu_{i}\,\tau_{i}^{-1}\,\left(1 - \frac{t}{\tau_{i}}\right)\mathop{\sim}\limits_{t\rightarrow 0^+}1 - \frac{t}{\tau_{eq}}\mathop{\sim}\limits_{t\rightarrow 0^+}e^{-t/\tau_{eq}},\end{array}
\sum_{i=1}^{\ell-1}\sum_{j=0}^{2^{i-1}k_1}\binom{2^{i-1}k_1^2}{j}\left(\frac{1+k_1+\cdots+k_1^i}{nr_n^d v_d(1-\epsilon)-(1+k_1+\cdots+k_1^i)}\right)^{2^{i-1}k_1^2-j}& \le \sum_{i=1}^{\ell-1}2^{i-1}k_1 2^{2^{i-1}k_1^2}\left(\frac{k_1^{\ell}}{nr_n^d v_d(1-\epsilon)-k_1^{\ell}}\right)^{2^{i-1}k_1^2-2^ik_1}.
\frac{q^{n\left(n+\nu\right)/2}h_{n}\left(i\cosh\left(\log q^{\nu/2}\right)\vert q\right)}{i^{n-\nu}\left(q;q\right)_{\infty}}=q^{\nu\left(\nu+1/2\right)}J_{\nu}^{(2)}\left(\frac{2i}{q^{\nu+\left(n+1\right)/2}};q\right)
x(x^{\lambda}\cdot x\phi^{-1})=x\phi^{-1}\Rightarrow (yx)(x^{\lambda}\cdot x\phi^{-1})=(y\cdot x\phi^{-1}) \Rightarrow (x\cdot yx)(x^{\lambda}\cdot x\phi^{-1})=x(y\cdot x\phi^{-1})\Rightarrow R_{x^\lambda}\cdot x\phi^{-1}=L^{-1}_{x}R^{-1}_{x}R_{x\phi^{-1}}L_{x}.
\sum_{n = - \infty}^{+ \infty}z^{n}q^{n^{2}}= \prod_{n = 0}^{\infty}\left( 1 - q^{2 n + 2}\right) \left( 1 + z q^{2 n + 1}\right) \left( 1 + z^{- 1}q^{2 n + 1}\right) ,
|(P_-v_s,E_-(-s)^*v)| &=|(P_-v_s,E_-(-s)^*(\chi_{\{|x|\geq R\}}+\chi_{\{|x|<R\}})v)| &\leq \|E_-(-s) P_- v_s\| \|\chi_{\{|x|\geq R\}}v\| + \|P_- v_s\| \|\chi_{\{|x|< R\}}E_-(-s)\| \| v\| &\leq C_v (\|\chi_{\{|x|\geq R\}}v\| + \|\chi_{\{|x|< R\}}E_-(-s)\|)
B_{M N}= ( - )^{M ( N + \underline{{{N}}})}\partial_{N}z^{\underline{{{N}}}}\partial_{M}z^{\underline{{{M}}}}B_{\underline{{{M N}}}}\; , E_{\pm}^{\underline{{{A}}}}= E_{\pm}^{M}\partial_{M}z^{\underline{{{N}}}}E_{\underline{{{N}}}}^{\underline{{{A}}}}\; ,
|E(G)| = |E(G \setminus \{v_r\})| + d(v_r) &\leq \operatorname{ex}(K_{n_1,\ldots,n_r-1},kK_r) + d(v_r) &=h_k(n_1,\ldots,n_r-1)+d(v_r) & = \sum_{1\leq i<j\leq r}n_in_j - \sum_{i = 1}^{r-1}n_i - n_1n_2 + n_2(k-1) + d(v_r) &\leq \sum_{1\leq i<j\leq r}n_in_j - n_1n_2 + n_2(k-1) &= h_k(n_1,n_2,\ldots,n_r).
\deg_{x_n}\left(\sum_{i=1}^n a_i(x) v_{i,n+1-i_0}(x) \right) =\deg_{x_n}(a_{i_0}(x) v_{i_0, n+1-i_0}(x))>0 =\deg_{x_n}(c_{n+1-i_0}(x')).
\|\nu_{n}^{r(h)}\cdot h -\nu_{n}^{s(h)}\|_{1}\le \|\nu_{n}^{r(h)}\cdot h -\mu_{n}^{z}\cdot h\|_{1}+\| \mu_{n}^{z}\cdot h - \mu_{n}^{\dot z}\| + \|\mu_{n}^{\dot z}- \nu_{n}^{s(h)}\| = \|\nu_{n}^{\dot z'}-\mu_{n}^{z'}\|_{1}+0+\|\mu_{n}^{z'}- \nu_{n}^{\dot z'}\| <\epsilon.
\beta: \mathbb{C}^3\times \mathbb{C}^3 \to \mathbb{C}^3, \left(\begin{bmatrix}s_1 s_2 s_3\end{bmatrix},\begin{bmatrix}t_1 t_2 t_3\end{bmatrix}\right)\mapsto\begin{bmatrix*}[r]s_1t_2+s_2t_3 -s_2t_1+s_3t_2 -s_1t_1-s_3t_3\end{bmatrix*},
Z_{\mathrm{F}}[ A ] = \int \! D{\bar{\psi}}D \psi e^{- \int \! d^{2}x \, [ \psi_{1}^{\dagger}i \partial \psi_{1}+ \psi_{2}^{\dagger}i{\bar{\partial}}\psi_{2}- \psi_{1}^{\dagger}A \psi_{1}- \psi_{2}^{\dagger}{\bar{A}}\psi_{2}]}\; ,
u(t,X_t) =u(0,X_0)+\int_0^t (u_t+b\cdot \nabla u+\frac{1}{2}\Delta u)(s,X_s)ds+\int_0^t \nabla u(s,X_s)dB_s &= u(0,X_0)-\int_0^t b(s,X_s)ds+\int_0^t \nabla u(s,X_s)dB_s &=u(0,X_0)-X_t+X_0+B_t+\int_0^t \nabla u(s,X_s)dB_s.
\bar{s}:={\sup\left\{s\mid \lambda_{\rm min}(X+s\Delta X)\ge 0,s\ge 0\right\}}&=\begin{cases}-\displaystyle{\frac{1}{\lambda_{\rm min}(X^{-1}\Delta X)}}\ \mbox{if}\lambda_{\rm min}(X^{-1}\Delta X)<0{\infty}\ \mbox{otherwise}\end{cases}
E_{1}= e^{2 x \cos \rho + 2 i t \sin \rho + c}= E_{2}^{*}, \quad E_{3}= e^{2 x + d}; \quad A_{1 2}= - \tan^{2}\rho , \ \A_{1 3}= A_{2 3}= - \tan^{2}( \rho / 2 ) .
E(a^3b^4h^2m^{\ast}\tilde{m}\overline{l_1}^2,a^3b^2hz\tilde{m}^2\overline{l_1}^2;d) \ll d^{1/2+\varepsilon}\left(a^3b^2hz\tilde{m}^2\overline{l_1}^2,d\right)^{1/2}=d^{1/2+\varepsilon}(z,d)^{1/2}.
(N^{(\lambda)}-k) ((\ell+1) p)^{N^{(\lambda)}-k}\big\langle u_0^{(\lambda)}\big| P_{\ell+1}^k H^k \big|u_0^{(\lambda)}\big\rangle = k!(N^{(\lambda)}-k)! (p (\ell+1))^{N^{(\lambda)}}.
\lVert af\rVert_S =\lVert \sum_{v\in I}a\alpha(v)\#v\rVert_S &=\max\{\lvert v\rvert_S\mid v\in I, a\alpha(v)\neq 0\}&=\max\{\lvert v\rvert_S\mid v\in I, \alpha(v)\neq 0\}&=\lVert f\rVert_S &=\lvert a\rvert_0\lVert f\rVert_S.
H(n,j_{1},...,j_{m}):=\left(\frac{m^{2}}{4n^{2}}-\frac{m}{2n}\right)\sum_{i=1}^{m}j_{i}^{2}+\left(\frac{m^{2}}{6n^{2}}-\frac{m^{3}}{6n^{3}}\right)\sum_{{i}=1}^{m}j_{i}^{3}-\frac{m^{3}}{3n^{3}}\sum_{i=1}^{m}j_{i}^{4}+O(n^{-1}),
\begin{array}{rclll}\Lambda^{(s)t}_{\pi,i}A_i^{(s)}\Lambda_{\pi,i}^{(s)}= B^{(s)}, \ \ \ \ \ \ \ (\Lambda^{(s)}_{\pi,i}-1)_{kj}- (\Lambda^{(s)}_{\pi,j}-1)_{ki}= \pi\cdot L^{k(s)}_{\pi, ij}(\Lambda^{(s)}_{\pi}) = p \cdot L^{k(s)}_{p,ij}(\Lambda^{(s)}_{\pi}).\end{array}
\langle \tilde{\pi}(g)\eta^l_x, \eta^l_y\rangle =\sum_{g'\in D^y}\langle (\tilde{\pi}(g)\eta^l_x)(g'), \eta^l_y(g')\rangle = \, \frac{1}{\, |A_1|^{1/2}|A_2|^{1/2}\,}\sum_{g'\in D_0}\langle \pi(h(g'))v^m_{s(h(g'))}, v^m_{s(g')}\rangle,
I_{2}= - \frac{\frac{n-2}{2}u_{a_{j}}( a_{i})\gamma_{n}}{(\frac{\lambda_{i}}{\lambda_{j}}+ \lambda_{i}\lambda_{j}\gamma_{n}G_{a_{j}}^{\frac{2}{2-n}}( a_{i}) )^{\frac{n}{2}}}\int_{\mathcal{A}}\frac{\nabla G^{\frac{2}{2-n}}_{a_{j}}( a_{i})\lambda_{j}x}{(1+r^{2})^{\frac{n+2}{2}}}
\|\lambda\nabla\pi\|_{\tilde L^1_t(B^s_{p,r})}\leq C \lambda^\ast\|\nabla\pi\|_{\tilde L^1_t(B^s_{p,r})},\hbox{if}s>-\min\{\frac{d}{p},\frac{d}{p'}\}, \hbox{with}r=1 \hbox{if}s=\frac{d}{p}.
\begin{array}{c}Z_{S,s,t}Z_{S,t,u}\bar{S}^{u}=Z_{S,s,t}\bar{S}^{u}_{t}= \bar{S}^{u}_{s}=Z_{S,s,u}\bar{S}^{u}\mbox{and}Z_{S,s,t}Z_{S,t,s}\bar{S}^{s}=Z_{S,s,t}\bar{S}^{t}_{s}=\bar{S}^{s}.\end{array}
[\begin{matrix}s_{x}cos\theta&-s_{y}sin\theta&t_{x}s_{x}cos\theta-t_{y}s_{y}sin\theta+t_{x}^{\prime}\\s_{x}sin\theta&s_{y}cos\theta&t_{x}s_{x}sin\theta+t_{y}s_{y}cos\theta+t_{y}^{\prime}\\ 0&0&1\end{matrix}]
L_{\mu \nu}^{- 1}( \tau , \tau^{\prime}) - \frac{{\cal F}_{\mu}^{\alpha}}{2}\int_{0}^{1}\epsilon ( \tau - \tau^{\prime \prime}) L_{\alpha \nu}^{- 1}( \tau^{\prime \prime}, \tau^{\prime}) d \tau^{\prime \prime}= \eta_{\mu \nu}\delta ( \tau - \tau^{\prime})
{\cal{S}}= \frac{1}{16 \pi}\int d^4 x \sqrt{-\tilde{g}}[ \tilde{\Phi}\tilde{R}- \frac{\omega(\tilde{\Phi})}{\tilde{\Phi}}\tilde{g}^{\mu \nu}\partial_{\mu}\tilde{\Phi}\partial_{\nu}\tilde{\Phi}] +{\cal{S}}_{m}[\psi_m, \tilde{g}_{\mu \nu}]
(\operatorname{id}\otimes\varphi)\bigl((c\otimes1)\Delta((\theta\otimes\operatorname{id})[(y\otimes1)(\Delta a)])\bigr) =(\operatorname{id}\otimes\varphi)\bigl((\theta\otimes\operatorname{id}\otimes\operatorname{id})[(y\otimes c\otimes1)(E\otimes1)\Delta_{13}(a)]\bigr).
\Omega_i = \frac{dz_i}{z_i}+ \left( \lambda\frac{df_i}{f_i}+ \frac{df_i}{f_i^{k+1}}\right)\ \ \ \ \ \ \Omega_j = \frac{dz_j}{z_j}+ \left( \lambda\frac{df_j}{f_j}+ \frac{df_j}{f_j^{k+1}}\right).
\begin{aligned}\frac{V_{\ell}(x_i;f)}{x_i}\ll \ell \log \ell \Bigg( \sum_{k=1}^3\sup_{\substack{0\leqslant j\leqslant J}}\lambda^{(k)}_{\ell}(x_i,y_j;f)\Bigg)+ L^{(12)}_{\ell}(x_i;f) + L^{(2)}_{\ell}(x_i;f) \,\,\,\,\,\,\,\, + \frac{W_{\ell}(x_i;f)}{x_i}.\end{aligned}
\frac{m\cdot m^{-2}}{m^{-3}}=m\cdot m^{-2}\cdot\frac{1}{m^{-3}}=m\cdot\frac{1}{m^{2}}\cdot m^{3}=\frac{m\cdot m^{3}}{m^{2}}=\frac{m^{4}}{m^{2}}=m^{2}
|\gamma_l|\leqslant N^2\big|\langle\nabla_{x_2}\psi_N,\,g_\beta(x_1-x_2)\,\mathbf{A}(x_1) \widehat{r}\,\psi_N \rangle\big| &+N^2\big|\langle\psi_N,\,g_\beta(x_1-x_2)\,\mathbf{A}(x_1)\nabla_{x_2}\widehat{r}\,\psi_N \rangle\big|.
\psi_1^h = u^h - u - x_2\Big(\frac{R_{21}^h}{h}- v_2'\Big) - x_3\Big(\frac{R_{31}^h}{h}- v_3'\Big) + \beta_1^h\,, \psi_i^h = \frac{1}{h}(v_i^h - v_i) + (w^h - w)x_i^\bot + \beta_i^h\,,i=2,3\,.
\sigma^\mathcal{A}(x)= \Psi_{\mathcal{A}}\left(\left[j^1s(x)\right]\right)=\Psi_{\mathcal{A}}\left(\left[\dot s_t(m),j^1s_t(m)\right]\right) = \sigma_x^{-1}[\dot s_t(m)]+\left[s_t(m),\mathcal{A}_M\!\cdot\!j^1s_t(m)\right] = :\left(\sigma^1_t(m),\sigma^2_t(m)\right)\in\left(\operatorname{Ad}P_M\right)_m\times_X L\left(T_mM,\left(\operatorname{Ad}P_M\right)_m\right).
\forall \psi\in{\cal P}^n_{u,v}([a,b]),\,\, \hbox{if}\,\, \psi^{(n-2)}(s_1)=\psi^{(n-1)}(s_1)=0, s_1\in(a,b), &\hbox{then}\,\, \psi^{(n-2)}(s)=0, s\in[a,b].
\Gamma_{l,s}^{\,\alpha,\,\alpha_{0}}(p)=\underbrace{\Gamma_{l,s}^{\,\alpha,\,\alpha_{0}}(0)}_{\rm{as\in\ (\ref{gao}})}+\frac{1}{2}\sum_{\mu,\nu=0}^{3}\,p_{\mu}\,p_{\nu}\,\partial_{\mu}\,\partial_{\nu}\,\Gamma_{l,s}^{\,\alpha,\,\alpha_{0}}(0)
S:= \begin{pmatrix}c'(0) b'(0) a'(0) 0 0 c'(0) b'(0) a'(0) -b'(0) -2(a'(0)-c'(0)) b'(0) 0 0 -b'(0) -2(a'(0)-c'(0)) b'(0)\end{pmatrix}.
U^+\cap T_i(U^+)= \{u\in U^+\mid\tau(u,U^-f_i)=\{0\}\}, U^-\cap T_i(U^-)= \{u\in U^-\mid\tau(U^+e_i,u)=\{0\}\}, U^+\cap T_i^{-1}(U^+)= \{u\in U^+\mid\tau(u,f_iU^-)=\{0\}\}, U^-\cap T_i^{-1}(U^-)= \{u\in U^-\mid\tau(e_iU^+,u)=\{0\}\}.
\int_\sigma^\infty{{d\sigma'}\over{\sqrt{1+4{\sigma'}^3}}}={1\over 3}\int_0^\infty{{X^{-2/3}dX}\over{\sqrt{1+4X}}}-{1\over 3}\int_0^{\sigma^3}{{X^{-2/3}dX}\over{\sqrt{1+4X}}}=ig_3^{1\over 6}(\eta-\eta_\infty),
\int \Bigg(e^{-{\vec y\cdot \vec y\over 2\hbar}}\Big(\int g(\vec x) e^{f(\vec x)+\vec x\cdot\vec y \over \hbar}d^n\vec x\Big)\Bigg)e^{\vec y\cdot \vec z\over \hbar}d^n\vec y=(2\pi\hbar)^{n/2}e^{{\vec z\cdot\vec z\over 2\hbar}}\int e^{{1\over \hbar}(f(\vec x)+{\vec x\cdot \vec x\over 2}+\vec x\cdot \vec z)}d^n\vec x
\int_{t}^{t+1}ds |K_{1}(s-t,\lambda(t))-\frac{\lambda(t)^{2}}{4(1+s-t)}| \leq \int_{t}^{t+1}ds K_{1}(s-t,\lambda(t) + \int_{t}^{t+1}ds \frac{\lambda(t)^{2}}{4(1+s-t)}&\leq C \lambda(t)^{2}
{\rm Hilb}\big(3T_6^{0}(3,3)^{ab},t\big)=(1,9,36,75,78,31)=(1+t)^5+4t(1+t)^4 \hphantom{{\rm Hilb}\big(3T_6^{0}(3,3)^{ab},t\big)=}{}+10t^{2}(1+t)^3+11t^{3}(1+t)^2+5t^{4}(1+t),
E_j^{(b)}=e^{D\sigma (z_j)}\int_{z_1}^{z_2}\langle T_0^0\rangle^{(b)}_{{\mathrm{ren}}}e^{-(D+1)\sigma (z)}dz=-\frac{J_D(B_1,B_2) e^{D\sigma (z_j)}}{2^D\pi^{D/2}\Gamma (D/2+1)a^D}.
\begin{aligned}(x\sqrt{x}- y \sqrt{y})^2 ~\leq~ \frac{2y^2}{(\sqrt{x}+ \sqrt{y})^2}(x-y)^2 + 2b(x-y)^2 begin{align*}.5em] \leq ~ 2 y (x-y)^2 + 2b(x-y)^2,\end{aligned}
\begin{aligned}Q_a^2Q_b + Q_aQ_bQ_a + Q_bQ_a^2 + 4Q_{a,b}^2Q_a +4Q_aQ_{a,b}^2 + 4Q_{a,b}Q_aQ_{a,b}= Q_{a^2b}+ 4Q_{a(ab)}+ 4Q_{a^2b,a(ab)}+ 4 Q_{a^3,b(ab)}+ 2Q_{a^3,b^2a}\end{aligned}
\int_{0}^{2 \pi}d \varphi \int_{0}^{r ( \varphi )}{\frac{r d r}{2 ( \rho_{i}^{2}- r^{2})}}={\frac{1}{4}}\int_{0}^{2 \pi}d \varphi \ln{\frac{\rho_{i}^{2}}{\rho_{i}^{2}- r^{2}( \varphi )}}.
- \Re\Big\{\frac{L'}{L}(s,\chi)\Big\}= \frac{1}{2}\log D_{\chi}+\Re\Big\{\frac{\delta(\chi)}{s-1}+ \frac{\delta(\chi)}{s}\Big\}- \sum_{\rho}\Re\Big\{\frac{1}{s-\rho}\Big\}+ \Re\Big\{\frac{\gamma_{\chi}'}{\gamma_{\chi}}(s)\Big\}.
K_{\nu}( z ) = \frac{2^{\nu}\Gamma ( \nu + 1 / 2 )}{\sqrt{\pi}z^{\nu}}\int_{0}^{\infty}{\frac{\cos z t \, d t}{( t^{2}+ 1 )^{\nu + 1 / 2}}}, \quad \sum_{k = - \infty}^{+ \infty}e^{i k z}= 2 \pi \sum_{n = - \infty}^{+ \infty}\delta ( z - 2 \pi n ) ,
\big\langle \alpha_p(\omega),\mu \big\rangle_{Z}=\big\langle \omega,\mu \big\rangle_{Z_{1}},\big\langle \beta_p(\mu),\nu \big\rangle_{Z_{2}}=\big\langle \mu,\nu \big\rangle_{Z_{2}},\big\langle \delta_p(\nu),\omega' \big\rangle_{Z_1}=\big\langle \nu,i_{e_{\mathfrak{n}}}\omega' \big\rangle_Y.
\begin{aligned}u_y =-\left(u\omega_{[1]}\right)_x, (v_{[i]})_{xxx}-(v_{[i]})_x =-\left(u\omega_{[i+1]}\right)_x, u_t =(v_{[n]})_{xxx}-(v_{[n]})_x=\delta_x,\end{aligned}i=1,\dots, n-1,
{\dot{y}}_{( \alpha )}= \{H , \{y}_{( \alpha )}\}= \{E + U , \{y}_{( \alpha )}\}={\frac{p_{( \alpha )}}{E}}-{\frac{s \epsilon_{( \alpha ) ( \beta ) ( \gamma )}p_{( \gamma )}}{E^{3}}}{\frac{\partial U}{\partial{y}_{( \beta )}}},
C_{D_{induced}}=\frac{C_{L}^{2}}{\pi AR}(1+\frac{\sum_{n=2}^{\infty}nA_{n}^{2}}{A_{1}^{2}})-\frac{\pi AR\overline{p}}{2}A_{2}=\frac{C_{L}^{2}}{\pi ARe}-\frac{\pi AR\overline{p}}{2}A_{2}
\begin{aligned}\frac{\partial}{\partial t}R_{ijkl}= \Delta R_{ijkl}+2R_{imjn}R^{\,\,\, m\,\, n}_{k \,\,\,\, l}-2R_{imjn}R^{\,\,\, m\,\, n}_{l \,\,\,\, k}&+2R_{imkn}R^{\,\,\, m\,\, n}_{j \,\,\,\, l}-2R_{imln}R^{\,\,\, m\,\, n}_{j \,\,\,\, k}&-R^{m}_{l}R_{ijkm}+R^{m}_{k}R_{ijlm}-R^{m}_{j}R_{klim}+R^m_i R_{kljm}\, ,\end{aligned}
\frac{1}{2}\partial_{t}\lVert \Lambda^{1+r_{1}}b\rVert_{L^{2}}^{2}+ \lVert \Lambda^{2}b\rVert_{L^{2}}^{2}= -\int (u\cdot\nabla) b\cdot\Lambda^{2+2r_{1}}b + \int (b\cdot\nabla) u \cdot \Lambda^{2+2r_{1}}b := III_{1}+ III_{2}
l_i(\zeta_1,\ldots,\zeta_s,y) = y + h\sum_{j=1}^s a_{ij}\zeta_j + h\sum_{j=1}^s a_{ij}v(l_j(\zeta_1,\ldots,\zeta_s,y)).\intertext{Now let $d$ be an arbitrary number. Then we have for each $i$,}l_i(w(k_1),\ldots,w(k_s),y) = y + h e_i + h\sum_{j=1}^s a_{ij}d + h\sum_{j=1}^s a_{ij}v(l_j(w(k_1),\ldots,w(k_s),y)),
g^{\beta \gamma}\xi_{,\beta \gamma}^{\alpha}-\xi_{,\gamma}^{\alpha}\Gamma^{\gamma}+\xi^{\gamma}\Gamma_{,\gamma}^{\alpha}-\xi_{,\gamma}^{\alpha}C^{\gamma}+\xi^{\gamma}C_{,\gamma}^{\alpha}-2g^{\beta \alpha}a_{,\beta}+\left( a-\lambda \right) \Gamma^{\alpha}+\left( a-\lambda\right) C^{\alpha}-\xi_{,t}^{\alpha}=0.
\begin{aligned}a_2 = (fu-gv)(-gu+3fv)\{(3f^2+4fg-3g^2)u^2+ & (12f^2-24fg+4g^2)uv+(-27f^2+12fg+3g^2)v^2\}, a_3 = 2(3f^2+g^2)(gu^2+6fuv-3gv^2)(fu^2-2guv-3fv^2).\end{aligned}
\dim_H(\Lambda \cap W^{(u)}) = \underline{\dim}_B (\Lambda \cap W^{(u)}) = \overline{\dim}_B(\Lambda \cap W^{(u)}) = r^{(u)}, \dim_H(\Lambda \cap W^{(s)}) = \underline{\dim}_B (\Lambda \cap W^{(s)}) = \overline{\dim}_B(\Lambda \cap W^{(s)}) = r^{(s)},
\frac{F}{V}= ( - 1 )^{f}\frac{1}{\beta}\int_{0}^{\infty}\frac{d p^{-}}{\sqrt{2}\pi}\int \frac{d^{D - 2}p_{T}}{( 2 \pi )^{D - 2}}\ln \left( 1 - ( - 1 )^{f}e^{- \frac{\beta}{\sqrt{2}}\left( \frac{{\vec{p}_{T}}^{~ 2}+ M^{2}}{2 p^{-}}+ p^{-}\right)}\right)
|B>_{(\frac{1}{2},\frac{1}{2})}=\sum\limits_{k_p}\left((\tilde A_{\mu\nu}(k_p)\psi^\mu_{-1/2}\psi^\nu_{-1/2}+\tilde B(k_p)(\bar\beta_{-\frac{1}{2}}\gamma_{-\frac{1}{2}}-\beta_{-\frac{1}{2}}\bar \gamma_{-\frac{1}{2}})\right)|\Omega, k>\ ,
SINR^n_{1,n}= \rho |\mathbf{v}_{1,n}^H\mathbf{H}_{1,1}\mathbf{p}_1 |^2 \alpha_{1,n}^2 = \rho |\mathbf{v}_{1,n}^H\mathbf{H}_{1,1}\mathbf{p}_1 |^2 \left(1- \frac{\epsilon_{1,k}\left( |\mathbf{v}_{1,k}^H\mathbf{h}_{1,1k}|^2 +\frac{1}{\rho}\right)}{|\mathbf{v}_{1,k}^H\mathbf{h}_{1,1k}|^2 (1+\epsilon_{1,k})}\right),
\lim\limits_{N \rightarrow\infty}\frac{\mathcal{R}_2}{N}=\lim\limits_{N \rightarrow\infty}\frac{MN-1}{2K}m\tan^2\frac{\pi}{2m}+\lim\limits_{N \rightarrow\infty}\frac{Q({{\textbf{{w}}}},0)}{N},
g_{\theta\theta^{\prime}}\left( u,\nabla P\right) =\left( a_{\theta}-a_{\theta^{\prime}}\right) \nabla P+a_{\theta^{\prime}}\operatorname{div}\left( b_{\theta^{\prime}}D\left( u\right) \right) -a_{\theta}\operatorname{div}\left( b_{\theta}D\left( u\right) \right) .
\Vert K\Vert_{HS}^2 =\int_{S_2}\int_{S_1}\left(\frac{p(x_1,x_2)}{p_1(x_1)p_2(x_2)}\right)^2p_1(x_1)p_2(x_2)d\nu_1(x_1)d\nu_2(x_2) &=\int_{S_2}\int_{S_1}\frac{(p(x_1,x_2))^2}{p_1(x_1)p_2(x_2)}d\nu_1(x_1)d\nu_2(x_2)<\infty.
\mathcal{E}(t) \le \frac{1}{N-\widetilde{C}}\mathcal{F}(t) \le\frac{1}{N-\widetilde{C}}\mathcal{E}(0)e^{-\widehat{C}/(N+\widetilde{C})}t = C \mathcal{E}(0) e^{-c_0t}.
-2 g(U,V) X(f) = 2g(H , X) g(U,V) = g(\nabla_U V , X) + g(\nabla_V U ,X) & = - g(\nabla_U X , V) - g(\nabla_V X , U) & = g([ X, U ] , V ) - g(\nabla_X U , V) + g([ X , V ] , U ) - g(\nabla_X V , U) & = - g(\nabla_X U , V) - g(\nabla_X V , U) & = - X (g(U,V)).
V_{\rm eff}(\phi_R) = \pi^2\zeta^2\phi^4_R \left( \ln{{\phi^2_R}\over{v^2_R}}-{{1}\over{2}}\right) +{{1}\over{4}}(\zeta-1)m^2_h\phi^2_R(1-{{\phi^2_R}\over{2v^2_R}})
\int_{t_{v-1}}^{t_{v}}K_{ij}(s,t)\cdot\widehat{w}_{i}(s)ds-\left (t_{v}-t_{v-1}\right )K_{ij}^{(v,u)}w_{i}^{(v)}& =\int_{t_{v-1}}^{t_{v}}\left (K_{ij}(s,t)-K_{ij}^{(v,u)}\right )w_{i}^{(v)}ds\leq\bar{\epsilon}\left (t_{v}-t_{v-1}\right )w_{i}^{(v)}.
\P\left\{|H(s+\varepsilon\,,0)-H(s\,,0)| \ge \frac12\| H(s+\varepsilon\,,0)-H(s\,,0)\|_1 \right\}&\hskip2.4in\ge \frac{\| H(s+\varepsilon\,,0)-H(s\,,0)\|_1^2}{4\| H(s+\varepsilon\,,0)-H(s\,,0)\|_2^2}= \frac{1}{2\pi},
\int\limits_0^\infty{{x^{s - 1}}}\left({\frac{1}{{{{\left({1 - x}\right)}^a}}}- \frac{1}{{{{\left({1 + x}\right)}^a}}}}\right)dx = 2{\left({- 1}\right)^{\frac{{s + 1}}{2}}}\frac{{\Gamma \left({a - s}\right)\Gamma \left( s \right)}}{{\Gamma \left( a \right)}}\sin \frac{{\pi s}}{2}
{}_{1}\phi_{0}\biggl(\genfrac..{0pt}{}{a}{\mbox{--}};q,\,x\biggr) = \sum_{i = 0}^{\infty}\frac{(a)_{i}}{(q)_{i}}x^{i}= \frac{(a x)_{\infty}}{(x)_{\infty}}, \lvert x \rvert < 1.
f(x) = \left(\prod_{i = 0}^N \frac{1}{\log^{(i)}(x)}\right)^2\left(\frac{1}{\log^{(N + 1)}(x)}\right)^{2 + \epsilon}&= \frac{1}{x^2\log^2(x)\cdots \left(\log^{(N)}(x)\right)^2 \left(\log^{(N + 1)}(x)\right)^{2 + \epsilon}}
\psi_{\mu\vert A}\, \leftrightarrow \, \mbox{$ \begin{array}{|c|}\hline\stackrel{}{A}\cr\hline\end{array}$}\, \equiv \,{\bf 8}\quad ; \quad \chi_{ABC}\,\leftrightarrow \,\mbox{$ \begin{array}{|c|}\hline\stackrel{}{A}\cr\hline\stackrel{}{B}\cr\hline\stackrel{}{C}\cr\hline\end{array}$}\, \equiv \,{\bf 56}
\Vert \zeta^{n}\Vert = \left\vert \sum_{j=2}^{k}\zeta_{j}^{n}\right\vert \geq \left\vert\zeta_{2}^{n}\left(2 \vert\rm{cos}(n\psi_{2})\vert-\sum_{j=4}^{k}r_{j}^{n}\right)\right\vert\geq \left\vert\zeta_{2}\right\vert^{n}\left(2 \rm{cos}(n\psi_{2})-(k-3)r_{4}^{n}\right), n\geq n_{0}.
0 = \langle S_{\tilde{h}^{\pi}_{i,j}}(T_{\tau}), w\rangle_t= \langle S_{N(v)}(T_\tau), w\rangle_t= \langle T_\tau, N(v)w\rangle_t &= \langle \tau, P(N(v)w)\rangle_a = \langle \tau, vP(w)\rangle_a
h[k]=\Re\{h[k]\}+j \Im\{h[k]\}, \Re\{h[k]\}=\sqrt{\frac{2}{N_1}}\sum \limits_{n=1}^{N_1}\cos(2\pi f_{D}k \cos(a_n)+\phi_n) \Im\{h[k]\}=\sqrt{\frac{2}{N_1}}\sum \limits_{n=1}^{N_1}\cos(2\pi f_{D}k \sin(a_n)+\psi_n) a_n=\frac{2\pi n-\pi +\theta}{4N_1},n=1,2,\cdots,N_1
\frac{I(U;Y|V)}{I(U;X|V)}\leq \max_{v}\frac{I(U;Y|V=v)}{I(U;X|V=v)}&\quad\overset{(a)}{\leq}s^*_r(X;Y) &\quad\leq \max_{\substack{q_{XY}:\: q_{XY}=q_{X}p_{Y|X}}}s_q^*(X;Y) &\quad=\eta\big(p_{Y|X}\big)
\lambda_1^{p^{A_1}-1}=\lambda_2^{p^{A_2}-1},\ \ \ \ (\lambda_1-1)^{p^{B_1}-1}=(\lambda_2-1)^{p^{B_2}-1},\ \ \ \ \left(\frac{\lambda_1}{\lambda_1-1}\right)^{p^{C_1}-1}=\left(\frac{\lambda_2}{\lambda_2-1}\right)^{p^{C_2}-1}.
\phi = \phi_{e}+ C_{2,0}\Delta x^{2}+ C_{3,0}\Delta x^{3}+ \dots + C_{0,2}\Delta t^{2}+ C_{0,3}\Delta t^{3}+ \dots + C_{2,2}\Delta t^{2}\Delta x^{2}+ C_{2,3}\Delta x^{2}\Delta t^{3}+ \dots.
\Pi_{\alpha}\Big(D^{(n)}_{\alpha}(\theta,\,\theta^\ast) \geq \frac{M}{1-\alpha}\, n\,\varepsilon_n^2\ \Big| \, X^{(n)}\Big)\leq e^{-M\, n \, \varepsilon_n^2/2}e^{M\, n \, \varepsilon_n^2/4}= e^{-M\, n \, \varepsilon_n^2/4}\to 0,\quad\mbox{as $n\to\infty$}.
G^{C}_{\sigma}(C_1 - C_2 )= \frac{1}{2\pi\sigma^2}exp \left ( -\frac{(C_{1}- C_{2}) (C_{1}- C_{2})^{*}}{2\sigma^2}\right ) = \frac{1}{2\pi\sigma^2}exp \left ( -\frac{(x_{n}- y_{n})^{2}+ (z_{n}- s_{n})^{2}}{2\sigma^2}\right )
\Pi^\alpha = \mathcal{P}_+ d \theta^\alpha \, , \qquad\Pi^\mu = dx^\mu +{1\over2}(C\Gamma^\mu)_{\alpha\beta}\theta^\alpha \Pi^\beta\, , \qquad\Pi^{\mu_1\dots \mu_5}= d \varphi^{\mu_1\dots \mu_5}+{1\over2}(C\Gamma^{\mu_1\dots \mu_5})_{\alpha\beta}\theta^\alpha\Pi^\beta\, .
b_{\scriptscriptstyle{l - 1}}a^{l}= - \pi^{- 2}( - 1 )^{m}i^{l}e^{i \delta_{\scriptscriptstyle{l}}}\Delta_{\scriptscriptstyle{l}}( 1 + \mathrm{O}( \Delta_{\scriptscriptstyle{l}}^{2}) ) .
| \phi_{s}\rangle \equiv \zeta^{i_{1}}\ldots \zeta^{i_{s}}\phi^{i_{1}\ldots i_{s}}, \qquad | \phi_{s - 1}\rangle \equiv \zeta^{i_{1}}\ldots \zeta^{i_{s - 1}}\phi^{i_{1}\ldots i_{s - 1}}.
{\lambda_{\ell}^{2}\frac 1 2 A_{33,\ell}=\lambda_{\ell}^{2}\left[ \frac{3 \cdot 2^7}{\pi^2}\frac{1}{\ell^2}\log \ell+O(\ell^{-2}) \right]= \frac{3 \cdot 2^7}{\pi^2}\ell^2 \log \ell +O(\ell^{2}).}
\int \varphi(x)\, d\mu(x) =\int \varphi\circ f^n(x)\, d\mu(x)=\int\left(\int_{\xi(x)}\varphi\circ f^n(y)\, d\mu_{x}(y)\right) d\mu(x) &=\int\left(\int_{\xi(f^{-n}(x))}\varphi\circ f^n(y)\, d\mu_{f^{-n}(x)}(y)\right) d\mu(x) &=\int\left(\int_{\xi_n(x)}\varphi(y)\, d\big(f^n_{\ast}\mu_{f^{-n}(x)}\big)(y)\right)d\mu(x).
\int_{B_{2}}\left( \kappa_{2}u_{\mu \nu}^{0}( x ) T^{\mu \nu}+ \kappa_{2}\sum_{n = 1}^{\infty}\frac{\psi_{n}( R ) e^{2 k R}}{N_{0}}u_{\mu \nu}^{n}( x ) T^{\mu \nu}- \frac{\kappa_{1}}{\sqrt{3}}\varphi T_{\mu}^{\mu}\right) d z .
\int{\cal D}x_{d}^{I}{\cal D}\psi_{\alpha}{\cal D}\bar{\psi}_{\dot{\alpha}}( \prod_{i = 1}^{2 g_{S}- 2 + p}V_{i}) ( \prod_{a = 1}^{p}\phi_{a}) e^{- S ( x_{d}^{I}, \psi_{\alpha}, \bar{\psi}_{\dot{\alpha}})}= < \prod_{a = 1}^{p}\phi_{a}( p_{a}) >
\overline{A}^{\mu}(\overline{q})=A^{\nu}(q)\,\overline{a}^{\mu}_{\nu}(q)=\overline{a}^{\mu}_{\nu}(q)A^{\nu}(q), \quad\overline{A}_{\mu}(\overline{q})=A_{\nu}(q)\,{a}_{\mu}^{\nu}(q)={a}_{\mu}^{\nu}(q)A_{\nu}(q)
\Theta(g)_{i\bar{j}k\bar{l}}=-\frac{\partial^{2}g_{k\bar{l}}}{\partial z_i\partial \bar{z}_j}+g^{p\bar{q}}\frac{\partial g_{p\bar{l}}}{\partial \bar{z}_j}\frac{\partial g_{k\bar{q}}}{\partial z_i}.
\begin{cases}g_1(c,\delta):=\quad\delta^2(4-c^2)^2(2c^2-(36-13c^2)\delta)+2c^2\delta^2; g_2(c,\delta):=-8c\delta(4-c^2)^2(1+\delta)(1-|\delta|^2); g_3(c,\delta):=-8(4-c^2)^2(8+|\delta|^2)(1-|\delta|^2); v(c,\delta,\eta):=\quad72\delta(4-c^2)^2(1-|\delta|^2)(1-|\eta|^2).\end{cases}
\text{LHS of (\ref{eqn:final})}= \sum_{m=3}^4 (-1)^m \binom{m}{3}\binom{4}{m}= - \binom{3}{3}\binom{4}{3}+ \binom{4}{3}\binom{4}{4}= - \binom{4}{3}+ + \binom{4}{3}= 0
t^l_{-l\,\underline{-l}}(Z) = (z_{11})^{2l}, t^l_{-l\,\underline{l}}(Z) = (z_{12})^{2l}, t^l_{l\,\underline{-l}}(Z) = (z_{21})^{2l}, t^l_{l\,\underline{l}}(Z) = (z_{22})^{2l}.
a^{3124}=a^{3214},a_{1}^{3124}=a_{1}^{3214},a_{2}^{3124}=a_{2}^{3214},a_{3}^{3124}=a_{3}^{3214},a_{4}^{3124}=a_{4}^{3214}, a_{12}^{3124}=a_{12}^{3214},a_{13}^{3124}=a_{13}^{3214},a_{14}^{3124}=a_{14}^{3214},a_{23}^{3124}=a_{23}^{3214},a_{24}^{3124}=a_{24}^{3214}, a_{123}^{3124}=a_{123}^{3214},a_{124}^{3124}=a_{124}^{3214}
\theta_{\alpha \beta}={\frac{1}{2 \kappa^{2}}}{\partial^{i}}\left[ -{\partial_{i}}{h_{\alpha \beta}}+{\eta_{\alpha \beta}}\left({\partial_{i}}{h_{\gamma}^{\gamma}}+{\partial_{i}}{h_{j}^{j}}-{\partial_{j}}{h_{i}^{j}}\right) \right] ,
b_{0,xx}=-2\int\rho_{x}b_{-1}dx -4\int\rho b_{-1,x}dx + C_1 = -2\rho b_{-1}-2\int\rho b_{-1,x}dx +C_{1}, & b_{0,x}=-2\int\biggl(\rho b_{-1}+\int\rho b_{-1,x}dx\biggr)dx+C_{1}x+C_{2}:= f(x)+C_{1}x+C_{2}, & b_{0}=\int f(x) dx+\frac{1}{2}C_{1}x^{2}+C_{2}x+C_{3}.
A(t) := \tfrac{(3t-1)(t+1)^3\log(t+1)}{16t^3}+ \tfrac{(3t+1)(-t+1)^3\log(2t+1)}{32t^3}+ \tfrac{(-t+1)(185t^4+698t^3-217t^2-160t+6)}{64t(3t+1)^2(t+3)}
\mathcal{T}\left(t_{0},a\left(t_{0}\right)\right)<\mathcal{T}\left(t_{0},\frac{2\sqrt{2}}{3}a_{0}\right)=\frac{3\sqrt{2}}{2}\;\frac{\left(1-\tanh^{2}\left(\frac{1}{2|a_{0}|}\right)\right)^{\frac{1}{2}}}{1-2\tanh\left(\frac{1}{2|a_{0}|}\right)}.
Z_N(\boldsymbol{\lambda})=\frac{\prod_{j=1}^N [a(\lambda_j)b(\lambda_j)]^N}{\prod_{n=0}^{N-1}n! \prod_{1\leq j<k\leq N}d(\lambda_k,\lambda_j)}\det\left[\partial_{\lambda_k}^{j-1}\varphi(\lambda_k)\right]_{j,k=1,\dots,N}.
\sup_{0<|h|<h_0}\left\|\frac{\delta_{h}\nabla (u\,\eta)}{|h|^{\Gamma-1}}\right\|^p_{L^p(\mathbb{R}^N)}\le \frac{C}{(2-\Gamma)^p\,(\Gamma-1)^p}\left(\frac{R}{r}\right)^N\,\left(\frac{R}{h_0}\right)^{1+p}\,\left[(1-s)\,\mathcal{M}_\gamma+h_0^{-\Gamma\,p}\,\|u\|^p_{L^p(B_{R+h_0})}\right].
\sigma_{\mathfrak{e}}i =\mathfrak{e}\mathrm{and}\sigma_{\mathfrak{e}}^{-1}\Gamma_{X,\mathfrak{e}}\sigma_{\mathfrak{e}}= \langle\gamma_{i}\rangle,\mathrm{where}\,\,\, \gamma_{i}=\left(\begin{array}{ccc}\cos(\pi\slash m_{\mathfrak{e}}) \sin(\pi\slash m_{\mathfrak{e}}) -\sin(\pi\slash m_{\mathfrak{e}}) \cos(\pi\slash m_{\mathfrak{e}})\end{array}\right).
\varphi (s) = (t^2 -s^2 )^{\ell+k+N - \frac{1}{2}}\, g_{( \ell+k , N)}(t^2 -s^2 ), \; \;\varphi (s) = \{Z(s) \}^{\ell - \frac{1}{2}}\, F_N ( Z(s) ),