harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
d s^{2}= ( \rho^{2}- \rho_{+}^{2}) \left( - ( - \frac{\rho_{+}^{2}}{l^{2}}R^{2}- k ) d T^{2}+ \frac{d R^{2}}{- \frac{\rho_{+}^{2}}{l^{2}}R^{2}- k}\right) + \frac{d \rho^{2}}{\frac{\rho^{2}- \rho_{+}^{2}}{l^{2}}}+ \rho^{2}d \phi^{2}
\Psi ( p = 0 ) = \frac{c_{1}}{\sqrt 2}\left( \begin{array}{c}{\omega + 1}\\{\omega - 1}\\{0}\\{0}\\ \end{array}\right) + \frac{c_{2}}{\sqrt 2}\left( \begin{array}{c}{0}\\{0}\\{\omega - 1}\\{\omega + 1}\\ \end{array}\right) + \frac{c_{3}}{\sqrt 8}\left( \begin{array}{c}{\omega - 1}\\{\omega + 1}\\{\lambda ( \omega + 1 )}\\{\lambda ( \omega - 1 )}\\ \end{array}\right) ,
<\theta^a_1,\vec Y_1|e^{-lH}|\theta^a_2,\vec Y_2>_B = \frac{V(B_1) V(B_2)}{\mbox{Vol}(T_6)}= \prod_a \frac{|\bar n^{(1)}_a \bar n^{(2)}_{a+1}- \bar n^{(1)}_{a+1}\bar n^{(2)}_a|}{\sin |\theta_a|}
\begin{aligned}\mathfrak{S}_{\delta, \flat}^{\tau_k}(H', M, N) \ll \frac{x^{\varepsilon}}{H'}\sum_{h\sim H'}\sum_{m\sim M}\Big\{\Big(\frac{xh}{mN^2}\Big)^{1/2}N^{1/2}+ \Big(\frac{xh}{mN^2}\Big)^{-1}\Big\}\noalign{\vskip 0,5mm}\ll \big((x H' M N^{-1})^{1/2}+ x^{-1}D^2\big) x^{\varepsilon}.\end{aligned}
\frac{\epsilon'((m_{\beta}-m_{\alpha-1})+2r(m_N))+r\log{C}}{m_{\beta}-m_{\alpha-1}}= \epsilon'+\frac{\epsilon'2r(m_N)}{(m_{\beta}-m_{\alpha-1})}+\frac{r\log{C}}{(m_{\beta}-m_{\alpha-1})}\leq \epsilon'+\epsilon'6Nr+\frac{r\log{C}}{(m_{\alpha}-m_{\alpha-1})}\leq (2+6Nr)\epsilon'=\epsilon.
k ={\frac{u_{i}}{2}}\Big [{\frac{{\cal E}+ \sqrt{{\cal E}^{2}+ \mu_{i}^{2}}}{K + \sqrt{K^{2}+ \mu_{i}^{2}}}}-{\frac{K + \sqrt{K^{2}+ \mu_{i}^{2}}}{{\cal E}+ \sqrt{{\cal E}^{2}+ \mu_{i}^{2}}}}\Big ] ,
-\frac{\sqrt{N}}{4\pi}(\theta_{1/2},\theta_{1/2}) = 4\sum_{n\geq 0}a(-n)\sigma_{1}(n)-4\sum_{\substack{b > 0 b \equiv 0(N)}}b\sum_{n>0}a(-bn) - 2\sum_{\substack{b>0 b \not \equiv 0 (N)}}b\sum_{n>0}a(-bn) &= 4\sigma_{1}(0) + 2\sum_{n>0}a(-n)(\sigma_{1}(n)-N\sigma_{1}(n/N)) = -\frac{1+N}{12}.
\alpha_{i}=Y_{i}/[W_{i}(\nu_{i}^{\prime\prime}-\nu_{i}^{\prime})]and\alpha_{T}=\frac{\int_{T_{ref}}^{T}c_{p}dT}{\sum_{i=1}^{N}h_{i}^{0}W_{i}(\nu_{i}^{\prime}-\nu_{i}^{\prime\prime})}
{\phi_\alpha}^{-1}([1^n.w1^n]_{\hat{X_\alpha}}) =\bigcap_{i=-n}^{\vert w \vert +n-1}{\hat{U}}^{-i}A_{(1^nw1^n)_{i+n+1}}\textrm{and}B(\tilde{\mathfrak{t}}_n(\alpha)) = \bigcap_{j=-n}^{n-1}{\hat{U}}^{-j}A_1.
\renewcommand{\arraystretch}{1 . 2}\begin{array}{r c l}{{\chi_{\mathcal{H}_{\frac 1 2 , 0}}( \tau )}}&{{=}}&{{| \Lambda_{0 , 2}( \tau ) |^{2}= \chi_{\mathcal{V}_{- 1 / 8 , - 1 / 8}}( \tau ) \, ,}}\\{{\chi_{\mathcal{H}_{\frac 1 2 , \frac 1 2}}( \tau )}}&{{=}}&{{| \Lambda_{2 , 2}( \tau ) |^{2}= \chi_{\mathcal{V}_{3 / 8 , 3 / 8}}( \tau ) \, ,}}\\ \end{array}
S_\mathrm{CS}=-\mathrm{i}\int \mathrm{d}^3 x \epsilon^{\mu\rho\nu}\left(\frac{1}{2}A_\mu^a \partial_\rho A_\nu^a+\frac{1}{3!}gf^{abc}A_\mu^a A_\rho^b A_\nu^c\right),
\sum_{k=1}^M \frac{2^{-k}r}{\varphi(2^{-k}r)}V(6 \cdot 2^{-k}r) \le \sum_{k=1}^M \frac{2^{-k}r}{V(2^{-k}r)}= \sum_{k=1}^M \int_{2^{-k}r}^{2^{-k+1}r}\frac{1}{V(2^{-k}r)}ds &\le \int_0^r \frac{1}{V(s)}ds \le c_5 \frac{r}{V(r)},
\lambda^{2^{n-1}}+ \sum\limits_{0\leq k\leq n-2}c_{n-1-k}^{2^{k+1}}\lambda^{2^k}= \lambda^{2^{n-1}}+ \sum\limits_{0\leq k\leq n-2}\alpha^{2^{n-1}-2^k}b_{n-1-k}^{2^{k+1}}\lambda^{2^k},
\hat{P}= - \frac{2 \lambda^{2}}{4 !}\left( \begin{array}{c c}{3{\varphi_{1}}^{2}+{\varphi_{2}}^{2}}&{2{\varphi_{1}}{\varphi_{2}}}\\{2{\varphi_{1}}{\varphi_{2}}}&{3{\varphi_{2}}^{2}+{\varphi_{1}}^{2}}\\ \end{array}\right) ,
Q_{\beta}= \{\pm(3,2), \pm(1,1), \pm(2,2), \pm(2,1), \pm(1,0), \pm(3,1), \pm(0,1), &\ \pm(2,0), \pm(1,-1), \pm(3,3), \pm(1,2), \pm(2,3), \pm(0,2), (0,0)\}
\mathrm{P}_{k,q,p}= \mathrm{P}\left( \log(1+SINR_{k,q,p}^n)<R_{k,q,n}, \forall n\in \{1, \cdots, p\}\right) = \mathrm{P}\left([ \mathbf{C}_{k,q,p}]_{q,q}>\rho \xi^*_{k,q,p}\right),
\langle \phi | x^{m l}\rangle = N^{1 / 2}( n ) ( \beta / 2 \pi )^{n / 4}\int_{I_{n}}d^{n}\rho ( 1 - \beta \rho^{2}/ 2 )^{1 / 2 + \sqrt{1 / 4 + n / 8}}e^{i x \cdot \rho / \hbar}\phi ( \rho )
\mathbf P_{\delta_x}[\eta_t X_t(f)|\|X_t\|\neq 0]= \frac{\mathbf P_{\delta_x}[\eta_t X_t(f) \mathbf 1_{\|X_t\|\neq 0}]}{\mathbf P_{\delta_x}(\|X_t\|\neq 0)}= \frac{\eta_t}{\mathbf P_{\delta_x}(\|X_t\|\neq 0)}P^\beta_t f(x)\stackrel[t\to \infty]{}{\sim}\langle f,\phi^* \rangle_m.
1 - \mathbb{E}[ \phi_\alpha(T_\tau)] \leq \frac{(T_\tau)}{( \frac{4m}{9n}z_{1 - \alpha}- \|\Theta_\tau\|_2^2 )^2}\;\leq\frac{\zeta_c^{h^T_\tau}B \sum_{c = 1}^4 n^{-c}}{( \frac{4m}{9n}z_{1 - \alpha})^2 - \frac{8m}{9n}z_{1 - \alpha}\|\Theta_\tau\|_2^2 + \|\Theta_\tau\|_2^4},
\begin{cases}dx_i^o =( Ax_i^o +BR^{-1}B^Ty_i^o +G m_i^o+\gamma p_i^o) dt +D dW_i, \dot{m_i^o}= (A + G) m_i^o +B\bar u +\gamma p_i^o, \dot{p_i^o}= -(A+G)^T p_i^o -(I-\Gamma)^T Q [\mathbb{E}x_i^o- (\Gamma m_i^o+\eta)], dy_i^o=\left\{-A^T y_i^o +Q[x_i^o-(\Gamma m_i^o+\eta)]\right\}dt +\zeta_i^o dW_i, \end{cases}
\tilde{f}^{j}-f(t_{j})=U^{\Delta t}(\tilde{f}^{j-1}-f(t_{j-1}))+\int_{t_{j-1}}^{t_{j}}[J^{\pi}(U^{\Delta t}\tilde{f}^{j-1})-U^{t_{j}-u}J(f(u))]du
R_{12}(\frac{z_1}{z_2}) \, R_{13}(\frac{z_1}{z_3}) \, R_{23}(\frac{z_2}{z_3})=R_{23}(\frac{z_2}{z_3}) \, R_{13}(\frac{z_1}{z_3}) \, R_{12}(\frac{z_1}{z_2}) \,.
c(E-F;u)=\frac{\prod_{i=1}^e(1+x_i u)}{\prod_{j=1}^f(1+y_j u)}=\sum_{p=0}^\infty\sum_{j=0}^p(-1)^je_{p-j}(x_1,\ldots,x_a)h_j(y_1,\ldots,y_b)\cdot u^p,
{\bf T}^{(2)}_1(E)=\xi'_0 \frac{\bar{{\mathbf Q}}(E q^{\frac{5}{4}})}{\bar{{\mathbf Q}}(E q^{\frac{1}{4}})}+\xi'_1 \frac{\bar{{\mathbf Q}}(E q^{-\frac{3}{4}})}{\bar{{\mathbf Q}}(E q^{\frac{1}{4}})}\frac{{\mathbf Q}(E q^{\frac{3}{4}})}{{\mathbf Q}(E q^{-\frac{1}{4}})}+\xi'_2 \frac{{\mathbf Q}(Eq^{-\frac{5}{4}})}{{\mathbf Q}(E q^{-\frac{1}{4}})}
\left(P e^{-ig\int_{x_{0}}^{x_{0}+\beta}dt'\,A_{0}(t',\vec{x})}\right)\rightarrow U^{-1}(x_{0},\vec{x})\left(P e^{-ig\int_{x_{0}}^{x_{0}+\beta}dt'\,A_{0}(t',\vec{x})}\right)U(x_{0},\vec{x})
A = \int . . . \int d k_{1}. . . d k_{r}A_{I}( q_{j}, k_{i}) \prod_{n = 1}^{r}\frac{W_{n}( k_{n}^{2}, q )}{k_{n}^{2}+ m_{n}^{2}}A_{I I}( q_{j}^{\prime}, k_{i}) \delta^{4}( q - k_{1}- . . . - k_{r}) ,
\Phi_1(f_n) = \frac{1}{2}\int |\nabla f_n|^2 + \frac{1}{2}\int |f_n|^2 - \frac{p}{2}\int \left( |\cdot|^{-(N-\gamma)}*(Q^{p-1}f_n) \right)(Q^{p-1}f_n) &- \frac{p-1}{2}\int \left( |\cdot|^{-(N-\gamma)}*Q^{p}\right) Q^{p-2}|f_n|^2 = o_n(1),
\left. \begin{array}{l l}\frac{\partial V(\textbf{x})}{\partial \textbf{x}_E}\!\!\!\! = \frac{\nu_E}{\Lambda}[x_E-x^*(\textbf{x}), \y_E - y^*(\textbf{x}), \z_E- z^*(\textbf{x}) ].\end{array}\right.
\varepsilon_{1,t}^N(\vec{f})=\int_0^t \frac{1}{N\gamma_N}\sum_{i=1}^N\sum_{j=1}^N\lambda_1\left(\frac{i}{N}\right)\lambda_2\left(\frac{j}{N}\right)\times &\left(\mathbb{E}\left(S_s^N(i)I_s^N(j)\right)-\mathbb{E}\left(S_s^N(i)\right)\mathbb{E}\left(I_s^N(j)\right)\right)\left(-f_1\left(\frac{i}{N}\right)+f_2\left(\frac{i}{N}\right)\right)ds
\underline{\tilde{c}}= \frac{1}{j_{0,N+1}^2}\mathbf{J}_0(\, \underline{\smash{j}}_0\,\underline{\smash{j}}_0^\intercal/j_{0,N+1}) D_{\underline{v}}\mathbf{J}_0(\, \underline{\smash{j}}_0\,\underline{\smash{j}}_0^\intercal/j_{0,N+1}) D_{\underline{v}}\underline{c}.
1+\sum_{i=1}^{+\infty}\frac{f^*(p^i)+g^*(p^i)}{p^i}=1+\frac{1/(p+1)(p-2)+(p-1)/p(p-2)}{p}+\frac{p-1}{p-2}\sum_{i\ge 2}\frac{1}{p^{2i}}&=1+\frac{1}{p(p+1)(p-2)}+\frac{p-1}{p-2}\frac{1}{p^2-1}&=\frac{(p-1)^2}{p(p-2)}&=\left (1-\frac{1}{(p-1)^2}\right )^{-1}.
- \Bigl [ \left[ Q ( \xi ) , Q ( \eta ) \right] , b ( k , \sigma ) \Bigr ] = \Bigl [ \left[ Q ( \eta ) , b ( k , \sigma ) \right] , Q ( \xi ) \Bigr ] + \Bigl [ \left[ b ( k , \sigma ) , Q ( \xi ) \right] , Q ( \eta ) \Bigr ]
0= \langle \tau_{H}, \alpha \rangle \left\{m_{1}\left(\frac{3}{2}- \left(\cot \frac{\vartheta}{2}\right)^{2}\right)+m_{1}\left(\frac{3}{2}- \left( \tan \frac{\vartheta}{2}\right)^{2}\right)\right. &\left. +4n\left(\frac{3}{2}- (\tan \vartheta )^{2}\right)+4m_{2}\left(\frac{3}{2}- (\cot \vartheta )^{2}\right)\right\}
u_{k,i}(x,t):=[e^{-tL_V}\varphi^{1,i}](|x|)=[e^{-tL_V^k}\phi^{k,i}](|x|)\,Q_{k,i}\left(\frac{x}{|x|}\right), u(x,t)=u_0(x,t)+\sum_{k=1}^{m-1}\sum_{i=1}^{\ell_k}u_{k,i}(x,t)+R_m(x,t),
\sup_{v \in L^{\widetilde A}(\Omega )}\frac{\int_\Omega uv\,dx}{\|v\|_{L^{\widetilde A}(\Omega )}}= \sup_{\varphi \in C^\infty_0(\Omega )}\frac{\int_\Omega u\varphi\,dx}{\|\varphi\|_{L^{\widetilde A}(\Omega )}},
L^\mathcal{L}(t,\varphi\circ\psi,\dot\varphi\circ\psi,\psi^*\Gamma) =\int_M\mathbf{i}_{\partial_t}\mathcal{L}\left(t,\dot\varphi(\psi(p)),\operatorname{Hor}_{\varphi(\psi(p))}^{(\varphi\circ\psi)_*\psi^*\Gamma}\right) &=\int_M\mathbf{i}_{\partial_t}\mathcal{L}\left(t,\dot\varphi(\psi(p)),\operatorname{Hor}_{\varphi(\psi(p))}^{\varphi_*\Gamma}\right).
d(\eta^\perp \phi) (X,Y) &=-g(\nabla_X\phi\xi^\perp,Y)+g(\nabla_Y\phi\xi^\perp,X) &=-g((\theta A-A\theta)X,Y)+2\sum^3_{a=1}\{\eta_a(\phi AX)\eta_a(\phi Y)-\eta_a(\phi AY)\eta_a(\phi X)\}.
e^{-\int_{\tau_{j+1}}^u G_j(v) \: dv}X_j(u) = X_j(\tau_{j+1}) + \int_{\tau_{j+1}}^u \mu X_{j-1}(w) e^{-\int_{\tau_{j+1}}^w G_j(v) \: dv}\: dw + Z_j''(u) &\leq 1 + \frac{s}{\mu}+ \frac{\delta s}{3 \mu}+ Z_j''(u).
\sum_{i\in\mathcal{I}_{n}^{\sigma}}(\mu_n^{\sigma}(i))^q \prod_{\ell=1}^n \big( \lambda_i^{(\sigma_{\ell})}\big)^{T_{\ell}^{\boldsymbol{\mu},\sigma}(q)- T_{\ell-1}^{\boldsymbol{\mu},\sigma}(q)}=1.
\left<\xi_i(\sigma){\cal O}\left[\vec\xi{\,}\right]\right>_{\vec\xi}=\int\limits_0^1 d\sigma^\prime G\left(\sigma-\sigma^\prime\right)\left<\frac{\delta{\cal O}\left[\vec\xi{\,}\right]}{\delta\xi_i\left(\sigma^\prime\right)}\right>_{\vec\xi},
\lbrack\partial_{V},\widetilde{JX}]=\frac{1}{2}[\tilde{V}-i\widetilde{JV},\widetilde{JX}]=\frac{1}{2}[\widetilde{JV}-i\widetilde{JJV},\tilde{X}]=\frac{1}{2}[\widetilde{JV}+i\tilde{V},\tilde{X}]=i[\partial_{V},\tilde{X}].
V_m\left(\mathbb{R}^{m+k}\right)\simeq_{\mathbb{Q}}\begin{cases}\prod_{\frac{k}{2}+1\leq\ell\leq \lfloor\frac{m+k-1}{2}\rfloor}K\left(\mathbb{Q},4\ell-1\right)\times K\left(\mathbb{Q},m+k-1\right)\times S^k \prod_{\frac{k}{2}+1\leq\ell\leq \lfloor\frac{m+k-1}{2}\rfloor}K\left(\mathbb{Q},4\ell-1\right)\times S^k \end{cases}
F(v_p)= \frac{|v_p|_R^2}{g_R(W_p,v_p)+\sqrt{\Lambda(p) |v_p|_R^2+ g_R(v_p,W_p)^2}}= \frac{|v_p|_R^2}{g_R(W_p,v_p)+\sqrt{h(v_p,v_p)}},
R_{f}( z , z^{\prime}; L ) = \exp \left\{\frac{1}{2}[ R_{b}( z , z ) + R_{b}( z^{\prime}, z^{\prime}) ] - R_{b}( z , z^{\prime}) \right\}\frac{\Theta [ l_{1}, l_{2}] ( J | \omega^{( r )})}{\Theta [ l_{1}, l_{2}] ( 0 | \omega^{( r )})}
E_{(x,T)}\,\, u(X_\sigma)\!\!\int_\sigma^T\! \langle R_\tau^\dag \dot{z}_\tau,dW_\tau\rangle=E_{(x,T)}u(X_\sigma)\,\,E_{(x,T)}\!\!\int_\sigma^T\! \langle R_\tau^\dag \dot{z}_\tau,dW_\tau\rangle=0,
\zeta= 81\frac{8{\lambda_1'}^2 \lambda_3^2 - 2 \lambda_1 \lambda_1'' \lambda_3^2 + \lambda_1^2 \lambda_3 \lambda_3'' - 7\lambda_1 \lambda_1' \lambda_3 \lambda_3' +\lambda_1^2{\lambda_3'}^2}{\lambda_1^5 \lambda_3}\,n_1+ 162 \frac{\lambda_1 \lambda_3' -2 \lambda_3 \lambda_1'}{\lambda_1^4}\,n_1'.
\sum_{p \,\leq\, x}(\log g_{u}(p-1))^{\lambda}=\sum_{\substack{d \, \le \, z (d,\,a_{2})\,=\,1}}\rho_{\lambda}(d)\pi(x;\l(d),1)+\sum_{\substack{d \, \ge \, z (d,\,a_{2})\,=\,1}}\rho_{\lambda}(d)\pi(x;\l(d),1) &=E'_{1}+E'_{2}.
\begin{aligned}\bigg|\sum_{\substack{x<n\le y \gcd(n,M/g_{1})>1}}\chi(n)\bigg|\hskip 0.1cm\le \hskip 0.1cm 2^{\omega(M/g_{1})-1}\left(\dfrac{1}{3\log 3}\sqrt{k}\log k+\dfrac{13}{2}\sqrt{k}\right).\end{aligned}
p^{(\delta)}_{n\delta}\le \int \rho^{(\delta)}_{n\delta}(x) \frac{1-e^{-\delta f^*}}{\delta}dx=\frac{1-e^{-\delta f^*}}{\delta}\le f^* \mbox{and}\bar \rho_{(n-1) \delta}^{(\delta)}\le R_{n-1},
\{p_2 ,\Psi\}(x_0, \xi) = H_{p_2}(\Psi)(x_0, \xi) = \frac{d}{ds}\Psi\circ x_s(x_0 ,\xi)|_{s=0}, \{p_2, \{p_2 ,\Psi\}\}(x_0, \xi) = H_{p_2}\big( H_{p_2}(\Psi)\big) (x_0, \xi) = \frac{d^2}{ds^2}\Psi\circ x_s(x_0 ,\xi)|_{s=0}.
\frac{{\lambda}^2}{16 \pi^2}\,\left({{\sigma}_c}\,{{{\phi}_c}}^2 + \frac{{{\sigma}_c}\,{{{\phi}_c}}^2}{2\,\epsilon}+ \frac{{{{\phi}_c}}^4}{4}+ \frac{{{{\phi}_c}}^4}{4\,\epsilon}+{{{\sigma}_c}}^2 + \frac{{{{\sigma}_c}}^2}{4\,\epsilon}\right)
\frac{(1+y)^k}{y^{k}}\frac{-1}{(r-t-1)!}\sum_{l=0}^{r-t-1}\frac{(t+l)!}{t!}\frac{(r-l-1)!}{t!}\frac{y^{r-t-1-l}}{(1+y)^{r-l}}= \frac{-1}{(r-t-1)!(t!)^2}\sum_{l=0}^{r-t-1}(t+l)!(r-l-1)!y^{r-t-1-l-k}\sum_{j=0}^{\infty}\binom{-r+l+k}{j}y^j.
r_{\varepsilon,\sigma}^2(\mathcal{E}_a,\Psi_{D^\dagger,M},\beta) \leq \inf_{D \in \mathbb{N}}\left[\tilde C(\alpha,\beta) \varepsilon^2 \sqrt{\sum_{j=1}^{D \wedge M_1}b_j^{-4}}+ (7+4\sqrt{x_{\alpha/2}}) \left[\sigma^2 \ln^{3/2}(1/\sigma) \vee a^{-2}_{D \wedge M_0}\right] \right],
S_{int}=\frac{g}{4!(2\pi)^d}\int dp_1dp_2dp_3dp_4\, e^{-i p_1\wedge p_2-i p_3\wedge p_4}\varphi(p_1)\varphi(p_2)\varphi(p_3)\varphi(p_4)\delta(p_1+p_2+p_3+p_4).
{\cal P}^{a \mu}= \alpha_{2}\sqrt{- \gamma}\left[ \left( K_{i}K^{i}\gamma^{a b}- 2 K_{i}K^{a b i}- 2{\cal G}^{a b}\right) e^{\mu}{}_{b}+ 2 ( \widetilde \nabla^{a}K^{i}) n^{\mu}{}_{i}\right] .
<\beta \circ \rho_{\beta}^*(f,\gamma),m> =<(f,\gamma),\rho_{\beta}\circ \beta(m)> &=<(f,\gamma),(\alpha \otimes \beta) \circ \rho_{\beta}(m)> &=<\rho_{\beta}^* \circ (\alpha \otimes \beta)(f,\gamma),m > &=<\rho_{\beta}^*(\alpha(f),\beta(\gamma)),m>
(-1)^i h_i - (-1)^i H(x_n) = (-1)^i H(Y^\varepsilon (\tau^\varepsilon (x_n), x_n)) - (-1)^i H(x_n) = \nu \int_0^{\tau^\varepsilon (x_n)}|DH(Y^\varepsilon (t, x_n))| \, dt \geq \nu c_0 \tau^\varepsilon (x_n),
C\left(\int_\Omega \overline{\xi_0}^2 w^{qj}\,dx\right)^{2/q}\,\leq \lambda_1\int_\Omega \overline{\chi}_1 \overline{\xi_0}w^{2j-1}\,dx+ \widetilde{C}j\int_\Omega \overline{\xi_0}^2 w^{2j}\,dx &\, \leq Cj\int_\Omega \overline{\xi_0}^2 w^{2j}\,dx.
f(\alpha) = \beta_\alpha(a_1^1, \dots, a_1^k, \dots, a_n^1, \dots, a_n^k) &= \beta_\alpha(f(z_1^1), \dots, f(z_1^k), \dots, f(z_n^1), \dots, f(z_n^k)) & = f(\beta_\alpha)
\chi^1_{D_m}(q) =\prod_{i=1}^m(q-2i+1)+m\prod_{i=1}^{m-1}(q-2i+1)=(q-m+1)\prod_{i=1}^{m-1}(q-2i+1), \chi^2_{D_m}(q) =\prod_{i=1}^{m}(q-2i)+2m\prod_{i=1}^{m-1}(q-2i)+m(m-1)\prod_{i=1}^{m-2}(q-2i) &=\left((q-2m)(q-2m+2)+2m(q-2m+2)+m(m-1)\right)\prod_{i=1}^{m-2}(q-2i) &=\left(q^2-2(m-1)q+m(m-1)\right)\prod_{i=1}^{m-2}(q-2i).
\frac{d s^{2}}{l_{s}^{2}}= \frac{u^{7 - p}}{R^{7 - p}}\Big [ - f d \tilde{t}^{2}+ ( d \tilde{x}^{1})^{2}\Big ] + ( d \tilde{x}^{2})^{2}+ \cdots + ( d \tilde{x}^{p})^{2}+ f^{- 1}d u^{2}+ u^{2}d \Omega_{8 - p}^{2}
\begin{cases}\frac{\partial u(x,t)}{\partial t}=dD^{\alpha}_{\theta}u(x,t)-e\frac{\partial u(x,t)}{\partial x}, \ \ \ \t>0, \ \ 0< x <\pi,\ \ \ 1< \alpha \leq2, u(0,t)=0, \u(\pi,t)=0 , \ \ \ \t>0, u(x,0)=sin(x), \ 0\leqslant x \leqslant \pi.\end{cases}
\theta_{A}(f)(x)=e^{\frac{(N-2)s}{2}}\left|\frac{x}{|x|}-a |x|\right|^{2-N}f\left( b+\frac{e^sP_c(x-a|x|^2)}{1-2\langle a,x\rangle +|a|^2|x|^2}\right).
\psi^{-1}\, (Y_t) = x + L_t + \lambda \int_0^t D \psi^{-1}\, (Y_s) u( \psi^{-1}(Y_s))ds - \int_0^t D \psi^{-1}\, (Y_s)ds \int_{\{|z| > 1 \}}\big([\psi (\psi^{-1}(Y_{s-}) \, + z) - Y_{s-}] - z \big) \nu(dz) + \int_0^t ds\int_{\{|z| \le 1 \}}\big [ z - D \psi^{-1}\, (Y_s) [\psi (\psi^{-1}(Y_{s-}) \, + z) - Y_{s-}] \big] \nu(dz).
\cdot \overrightarrow{b}> = \frac{\overrightarrow{a}\cdot \overrightarrow{b}^{2}}{\vert \overrightarrow{a}/ / \overrightarrow{b}\vert}= \frac{- \frac{1}{2}\vert \overrightarrow{b}\vert^{2}}{\vert \overrightarrow{a}/ / \overrightarrow{b}\vert}= - \frac{1}{2 .}
\tilde{p}(t,z)=\tilde{p}(0,z) \exp(-\int_0^t \frac{b_0(s,z)}{\sigma_0(s,z)}dB(s) - \frac{1}{2}\int_0^t (\frac{b_0(s,z)}{\sigma_0(s,z)})^{2}ds),
\sum_{n=1}^{[Nt]}E[(W^{(u)}_{N,i,n})(W^{(u)}_{N,i',n})]=E\big[(\sum_{n=1}^{[Nt]}W^{(u)}_{N,i,n})(\sum_{n=1}^{[Nt]}W^{(u)}_{N,i',n})\big].
p^{-1}E_2\lesssim C \int_0^{\infty}s^{p-3}\|g_{s}\|_2^2\,ds &=\int_0^{\infty}s^{p-3}\int_{\Theta^{s}}|g_{s}|^2\,d\gamma\,d\mu\,ds+\int_0^{\infty}s^{p-3}\int_{(\Theta^{s})^c}|g_{s}|^2\,d\gamma\,d\mu\,ds:= E_{2,1}+E_{2,2}.
p_{\hat{S}^m_1,\hat{S}^m_2|S^m}(\hat{s}^m_1,\hat{s}^m_2|s^m)=\sum\limits_{x^{\rho m},y^{\rho m}_1,y^{\rho m}_2}p_{X^{\rho m}|S^m}(x^{\rho m}|s^m)p_{Y^{\rho m}_1,Y^{\rho m}_2|X^{\rho m}}(y^{\rho m}_1,y^{\rho m}_2|x^{\rho m})\prod\limits_{i=1}^2p_{\hat{S}^m_i|Y^{\rho m}_i}(\hat{s}^m_i|y^{\rho m}_i)
u_1 = 2\exp\left( i\frac{z}{2{\hbar}c}\right){\rm Re}\left[ \frac{1}{{\Gamma}(1+i{\beta})}\exp\left( i\frac{z}{2{\hbar}c}\right) (\frac{iz}{{\hbar}c})^{-1+i{\beta}}G(1-i{\beta}, -i{\beta}; \frac{iz}{{\hbar}c}) \right] .
R_{\; B}^{A}\; = \; \left( \begin{array}{c c}{{\gamma^{\mu \nu}\, R_{\mu \nu}^{( 1 )}\;_{\; B}^{A}}}&{{k \, \gamma^{\mu}\, \gamma^{5}\, e^{- \sigma}\, P_{\mu}\;_{\; B}^{A}}}\\{{k^{*}\, \gamma^{\mu}\, \gamma^{5}\, e^{- \sigma}\, Q_{\mu}\;_{\; B}^{A}\quad}}&{{\gamma^{\mu \nu}\, R_{\mu \nu \; B}^{( 2 ) \, A}\, + \, ( k^{*}k - 1 ) \, L_{\; B}^{A}}}\\ \end{array}\right) \; .
F\left(s_{j}-s_{k}\right) =\int_{\mathbb{R}}e^{-is_{j}\lambda}e^{is_{k}\lambda}d\mu\left(\lambda\right) =\int_{\mathbb{R}}e^{-i\left(s_{j}-s_{k}\right)\lambda}d\mu\left(\lambda\right)=G_{\mu}\left(s_{j}-s_{k}\right)
\omega_{g,n+2}(p_0,q_0,J)-\omega_{g,n+2}(q_0,p_0,J) =\frac{1}{2\pi i}\oint_{C_{{\rm Rec}}^p}\cdot\frac{\eta^{p}(p_0)\cdot\Delta\omega_{0,2}(p,q_0)-\eta^{p}(q_0)\cdot\Delta\omega_{0,2}(p,p_0)}{4\,\omega_{0,1}(p)}\cdot \hat R_{g,n+1}(p,J).
d_{i}^{n}p^{6}d_{j}^{n}\rightarrow d_{i}^{n}p^{5}d_{j}^{n+1}\rightarrow d_{i}^{n+1}p^{4}d_{j}^{n+1}\rightarrow d_{i}^{n+1}p^{5}d_{j}^{n}\rightarrow d_{i}^{n}p^{6}d_{j}^{n}
C\star({\bf 1}^\epsilon_0 \|C^3\|_{L^1}-{\bf 1}^\epsilon_0 \|C_\infty^3\|_{L^1})\star S = \big(\|C^3\|_{L^1}- \|C^3_\infty\|_{L^1}\big)C\star S &=: -\gamma_t C\star S,
\delta (c\cup d) = \{\delta_H (c_H\cup d_H)\}& = \{\delta_H(c_H)\cup d_H + (-1)^{|c_H|}c_H\cup \delta_H(d_H)\}~~\mbox{(by the non-equivariant case)}& = \{\delta_H(c_H)\cup d_H\}+ (-1)^{|c_H|}\{c_H\cup \delta_H(d_H)\}& = \delta (c)\cup d + (-1)^{|c|}c \cup \delta (d)~~\mbox{( since $(-1)^{|c_H|}= (-1)^{|c|}$)}.
dpdq=\frac{\partial(p,q)}{\partial(\theta,\varphi)}d\theta d\varphi=(\frac{\partial p}{\partial\theta}\frac{\partial q}{\partial\varphi}-\frac{\partial p}{\partial\varphi}\frac{\partial q}{\partial\theta})d\theta d\varphi=n^{2}cos\theta sin\theta d\theta d\varphi=n^{2}cos\theta d\Omega,
\begin{aligned}dE(u)v = \Bigl(\int_0^1 v(s) ds, \int_0^1 \Bigl(x(s)-\frac{x}{2}\Bigr) \wedge v(s) ds \Bigr) & =\Bigl(\int_0^1 v(s) ds, \int_0^1 z\wedge \Bigl( s-\frac{1}{2}\Bigr) v(s) ds \Bigr)\end{aligned}
W_\phi\left(wz^1\cdots z^{C+1}\right) = W_\phi\left(W_\phi(wz^1\cdots z^{C})z^{C+1}\right)=W_\phi\left(W_\phi\left(w^1w^2\cdots w^{C+1}z^1\cdots z^{C}\right)z^{C+1}\right).
\begin{array}{r c l}{Q}&{=}&{\sqrt{\frac{2 5 - D}{6}}}\\{\alpha_{\pm}}&{=}&{\frac{Q}{2}\pm \sqrt{\frac{Q^{2}}{4}- 1}= \frac{1}{\sqrt{2 4}}\left( \sqrt{2 5 - D}\pm \sqrt{1 - D}\right) \ .}\\ \end{array}
q_{R}(x,\xi,\tau)=\frac{p_{\Phi}+\overline{p_{\Phi}}}{2}(x,\xi,\tau)=\Re p_{\Phi}(x,\xi,\tau) = p_2(x, \xi ) - \tau^2 p_2(x, d\Phi(x)) , &q_{I}(x,\xi,\tau)=\frac{p_{\Phi}-\overline{p_{\Phi}}}{2i}(x,\xi,\tau)=\Im p_{\Phi}(x,\xi,\tau)=2 \tau \widetilde{p_2}(x, \xi, d\Phi(x)) .
Z ( \kappa ) = \int \! [ d x ( t ) ] \, \exp \left[ - \int \! d \tau \, \Biggl ({\frac{1}{2}}\Bigl ({\frac{d x}{d \tau}}\Bigr )^{2}- x^{2}+{\frac{\kappa x^{4}}{2 \sqrt{2}}}+{\frac{1}{\sqrt{2}\kappa}}\Biggr ) \right] \, ,
G_r f(r) = \frac{1}{3g^2(r)}\partial^2_{rr}f(r) - \frac{1}{3g^2(r)}\left(\frac{\partial_r g(r)}{g(r)}-\frac{1}{r}+\frac{2\partial_r U(r)}{mv^2(r)}\right)\partial_r f(r).
m_{\tilde{G}}={\frac{1}{3}}\langle \, | M | \, \rangle = \frac{\, 1}{\, 4 \,}\, \langle \, | \sum_{a}b_{a}^{\prime}u_{a}- 4 e^{K / 2}W | \, \rangle = \frac{\, 1}{\, 4 \,}\, \langle \, | \sum_{a}b_{a}u_{a}| \, \rangle \, \approx \,{\frac{1}{4}}b_{+}\langle \, \rho_{+}\, \rangle .
\hat x_{12}=x_{12}+{2i\over (12)}\left[(1^-2) \theta^+_1 \bar\theta^+_1 - (12^-) \theta^+_2 \bar\theta^+_2 + \theta^+_1 \bar\theta^+_2 + \theta^+_2 \bar\theta^+_1\right] \ .
u_{2k-1}^{(s,t)}=\frac{q_k^{(s,t)}}{\kappa^{(t)}}\frac{H_{k-1}^{(s,t+1)}H_{k-1}^{(s+1,t)}}{H_{k-1}^{(s,t)}H_{k-1}^{(s+1,t+1)}}.
\Pr [ E_{k,s}^{\rm c}\cap \{X_{\tau_k}> x \}] \leq \sup_{y >x}\Pr [ E_{k,s}^{\rm c}\mid X_{\tau_k}=y , \, \eta_{\tau_k}= \eta^\star_{k,0}= 0 ] = \sup_{y > x}\Pr [ E_{s}^{\rm c}\mid X_0 =y , \, \eta_0 = \eta^\star_0 = 0 ] .
\left.\frac{dE(t_b+1)}{dE_b}\right|_{E_b=E_b^*}= e^{-B(t_f-t_m)}\ \left( \frac{\zeta}{\zeta-E_b^*}\right)\ \frac{F_-(t_f)}{F_+(t_f)}\ \frac{F_+(t_m)}{F_-(t_m)}.
\left|r\frac{d}{dr}\left[C(z,w)\right]\right|\lesssim \frac{1}{|1-z\overline{w}|^{2\lambda+2}}\ln\left(\frac{|1-z\bar{w}|^2}{|1-zw|^2}+2\right)+\frac{1}{|1-z\overline{w}|^{2\lambda+1}}\frac{1}{|1-zw|}\lesssim \frac{|1-z\overline{w}|^{-1}|1-zw|^{-1}}{(|1-z\overline{w}|+|1-zw|)^{2\lambda}},z=re^{i\theta}.
v_r^{\alpha}(X_l)\ge v_r^{\alpha}(R-P_lQ_l)-ri-v^{\alpha}(R_i)=c_l^{\alpha}, v_r^{\alpha}(Y_l)\ge v_r^{\alpha}(R-P_lQ_l)=c_l^{\alpha}+ri+v^{\alpha}(R_i).
G=\prod_{n=1}^{\infty}(\frac{4n-1}{4n}\cdot\frac{4n+2}{4n+1})=(\frac{3}{4}\cdot\frac{6}{5})(\frac{7}{8}\cdot\frac{10}{9})(\frac{11}{12}\cdot\frac{14}{13})\cdot\cdot\cdot
\delta_tv^{n-\frac{1}{2}}+ Av^{n-\frac{1}{2}}+ \sum\limits_{j=1}^m I^{(\alpha_j)}B_jv^{n-\frac{1}{2}}- \kappa v^{n-\frac{1}{2}}= g^{n-\frac{1}{2}}+ \mathcal{R}^{n-1/2},
[\begin{matrix}c_{3}c_{1}-s_{3}s_{2}s_{1}&-s_{3}c_{2}&c_{3}s_{1}+s_{3}s_{2}c_{1}\\s_{3}c_{1}+c_{3}s_{2}s_{1}&c_{3}c_{2}&s_{3}s_{1}-c_{3}s_{2}c_{1}\\ -c_{2}s_{1}&s_{2}&c_{2}c_{1}\end{matrix}]
h(Y_k|Y_1,\ldots,Y_{k-1}) :=\lim_{n\to\infty}\frac{H(Y_k^1,\ldots,Y_k^n|Y_1,\ldots,Y_{k-1})}{n}, h(Y_k|X_0,Y_1,\ldots,Y_{k-1}) :=\lim_{n\to\infty}\frac{H(Y_k^1,\ldots,Y_k^n|X_0,Y_1,\ldots,Y_{k-1})}{n},
\begin{array}{r l}{{}}&{{< q_{f}| \hat{U}( t_{f}, t_{i}) | q_{i}> =}}\\{{}}&{{}}\\{{}}&{{= \int_{- \infty}^{\infty}\prod_{\alpha = 1}^{N - 1}d q_{\alpha}\, \prod_{\alpha = 0}^{N - 1}d p_{\alpha}\, \prod_{\alpha = 0}^{N - 1}\left[ < q_{\alpha + 1}| p_{\alpha}> < p_{\alpha}| e^{- \frac{i}{\hbar}\epsilon \hat{H}_{0}}| q_{\alpha}> \right] \ .}}\\ \end{array}
\left( \nabla_T \nabla_{s_d}\phi \right)_{|\Sigma_p}= \left( \nabla_{s_d}\nabla_{T}\phi + R^{S}(T,s_d) \phi + \nabla_{[T,s_d]}\phi \right)_{|\Sigma_p}& = R^S(T,s_d) \phi_{|\Sigma_p},
\sum_{n \le x}(-1)^{n-1}\frac1{f(n)}= D_f \left( \left(\frac{2}{S_{1/f}(1)}-1\right) (\log x + E_f)+ 2(\log 2)\frac{S'_{1/f}(1)}{S_{1/f}(1)^2}\right) + O\left(T_{1/f}(x) \right),
\nabla^{2}\equiv -{\frac{1}{\sqrt g}}\partial_{i}\sqrt g g^{i j}\partial_{j}={\frac{1}{\sqrt g}}\delta_{i j}\partial_{i}\partial_{j}= ({\frac{1 + \vec{x}^{2}}{2}})^{2}( \partial_{1}^{2}+ \partial_{2}^{2}) \, .
\psi^{-1}(u_{h_g})=\psi^{-1}(u_h)=\psi^{-1}(z^{-1}), &\gamma_{\psi'}^{-1}(\det c_{h_g})=(\det c_h,\det c_{h_0})_2\gamma_{\psi'}^{-1}(\det c_h)=(z,\det c_{h_0})_2\gamma_{\psi'}^{-1}(z^{-1}), &\delta^{1/2}_{B_n}(b_g)=|z|^{-1}, &\chi(\mathfrak{s}(b_g))=|z|^{2s_{k+1}}=|z|^{-2s_{k}}.
R_k = E \left\{\log_2 \left(1 + \frac{\rho N}{1+\frac{\rho}{N}\sum_{j=1, j \neq k}^K|\mathbf{h}_k^H \mathbf{h}_j|^2}\right)\right\}&\geq \log_2 \left(1 + \frac{1}{\frac{1}{\rho N}+(K-1)E \left\{\frac{1}{N^2}|\mathbf{h}_k^H \mathbf{h}_j|^2\right\}}\right)\triangleq R_L
S = \frac{1}{2}\int d\tau tr\biggl\{\biggl(D_0 A^I\biggr)^2- \frac{1}{2}\biggl[A^I,A^J\biggr]~\biggl[A^J,A^I\biggr]+i{\bar\lambda}D_0\lambda+i{\bar\lambda}{\gamma}^I~\biggl[A^{I,}\lambda\biggr]\biggr\}
\|f - f_0\|_n \leq \|f - f_{a^0}\|_n + \|f_{a^0}- f_0\|_n \leq C J^{-1}\| Kf - Kf_n\|_n + \|f_{a^0}- f_0\|_n \leq C J^{-1}\| Kf - Kf_0\|_n + \|Kf_{a^0}- Kf_0\|_n + \|f_{a^0}- f_0\|_n.
\int_{B_1(0)}\left(|\nabla u_\kappa|^2+|\nabla v_\kappa|^2+\kappa u_\kappa^2v_\kappa^2\right)\mbox{div}Y-2DY(\nabla u_\kappa,\nabla u_\kappa)-2DY(\nabla v_\kappa,\nabla v_\kappa)=0.
\mathrm{Hom}_{DG_{T^{2n}_{J=T}}}(E_{(r,A,r',\mathcal{U},p,q)},E_{(s,B,s',\mathcal{V},u,v)}) &:=\Gamma (E_{(r,A,r',\mathcal{U},p,q)},E_{(s,B,s',\mathcal{V},u,v)})\bigotimes_{C^{\infty}(T^{2n}_{J=T})}\Omega^{0,*}(T^{2n}_{J=T}),
Y(R)=Z\left(\int_{\log R_1}^{\log R}e^{\theta(p-1) s}s^{-\kappa (p-1)}\,ds\right), 0<\rho<\rho_T=\int_{\log R_1}^{\log T}e^{\theta(p-1) s}s^{-\kappa (p-1)}\,ds.
\widetilde \chi^{( 3 )}\left( e_{i}, e_{j}\right) = \chi_{F}^{( 3 )}\left( e_{i}, e_{j}\right) + \sum_{k = 1}^{3 h - 3}a_{k}\oint_{\cal C}\Omega^{k + 1 - h_{0}}\left[ e_{i}, e_{j}\right] , \qquad h_{0}\equiv{\frac{3}{2}}h .
\left\{-\sum_{i=1}^3 \frac{\partial^2}{\partial x_i^2}+ \sum_{\stackrel{i,j=1}{i\neq j}}^3 \left({{\omega^2}\over{8}}\left( x_i-x_j \right)^2 +{2 \alpha\over{\left( x_i-x_j \right)^{2}}}+\Omega\left(\frac{x_i-x_j}{\sqrt{2}}\right) \right)\right\}\psi=E\psi~,