harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left(\frac{d}{d x}\right)^{n}(f_1(x)-f_1(x_0))^m\Big|_{x=x_0}&=\left(\frac{d}{d x}\right)^{n}\Bigg(\sum_{l=1}^n\frac{1}{l!}\left(\frac{d}{d y}\right)^{l}f_1(y)\Big|_{y=x_0}(x-x_0)^l\Bigg)^m\Bigg|_{x=x_0}&=n!\prod_{j=1}^m\Bigg(\sum_{l_j=1}^n\frac{1}{l_j!}\left(\frac{d}{d y}\right)^{l_j}f_1(y)\Big|_{y=x_0}\Bigg)1_{\sum_{j=1}^ml_j=n}.
\hat{T}_{\mu \nu}= - \frac{1}{2}\left( \hat{F}_{\mu \lambda}\star \hat{F}_{\nu}^{\, \lambda}+ \hat{F}_{\nu \lambda}\star \hat{F}_{\mu}^{\, \lambda}- \frac{1}{2}\, \eta_{\mu \nu}\, \hat{F}_{\lambda \rho}\star \hat{F}^{\lambda \rho}\right)
\nabla_{e_0}e_0=\frac{1}{2\phi^{3/2}}\sum_{i=1}^3 \frac{\partial\phi}{\partial x_i}e_i; \nabla_{e_i}e_0=-\frac{1}{2\phi^{3/2}}\sum_{j,k=1}^3 \epsilon_{ijk}\frac{\partial\phi}{\partial x_j}e_k;
\begin{aligned}\frac{(1+4\tau)r}{2\sqrt{2\tau}}\pm\frac u{2\sqrt\tau}=\sqrt{2N}+\frac{(T^2+R\pm U)N^{-1/6}}{\sqrt2}\mp\frac{TUN^{-1/2}}{\sqrt2}+o(N^{-1/2}), 2\tau r^2\pm\sqrt2ru =\frac N2+TN^{2/3}+\left(T^2+\frac R2\pm U\right)N^{1/3}+RT+T^3+o(1).\end{aligned}
D ( \alpha ,{\alpha^{\prime}}) ={\frac{\sum_{i , j}p_{i}^{\prime \prime}\langle \Phi_{i}| C_{\alpha}| \Psi_{j}\rangle \langle \Phi_{i}| C_{\alpha^{\prime}}| \Psi_{j}\rangle^{*}p_{j}^{\prime}}{\sum_{i , j}p_{i}^{\prime \prime}| \langle \Phi_{i}| C_{u}| \Psi_{j}\rangle |^{2}p_{j}^{\prime}}}.
\frac{1}{4}\gamma_{ab}\psi_A\overline{\psi}_B\gamma^{ab}\psi_C\epsilon^{AC}-\frac{1}{2}\gamma_{a}\psi_A\overline{\psi}_B\gamma^{a}\psi_C\epsilon^{AC}+3\psi_C\overline{\psi}_B\psi_A\epsilon^{BC}=0
S_1(t,q) = -\frac{1}{q^2}\sum_{j=0}^{n-1}j(j-1) \sum_{r \ge 1}r (-tq^j)^{r}&= -\frac{1}{q^2}\sum_{r \ge 1}r(-t)^r \sum_{j=0}^{n-1}j(j-1) q^{jr}\implies \omega_n^{2i}[t^r]S_1(t,\omega_n^i ) = - r(-1)^r \sum_{j=0}^{n-1}j(j-1) = -2\binom{n}{3}r(-1)^r .
\delta g_{m n}= - 4 i E_{\{m}^{\alpha}( \Gamma_{n \}})_{\alpha \beta}i_{\kappa}E^{\beta}, \qquad \delta \hat{H}^{( 3 )}= - i_{\kappa}d A^{( 3 )}, \qquad \delta_{\kappa}a ( y ) = 0 ,
0 =\det\left(M-\mu I_{n}+\sqrt{\frac{p\left(n+2\right)}{1-p}}ll^{\top}\right) =\det\left(M-\mu I_{n}\right)\det\left(I_{n}+G_{M}\left(\mu\right)\sqrt{\frac{p\left(n+2\right)}{1-p}}ll^{\top}\right) =\det\left(M-\mu I_{n}\right)\left(1+l^{\top}G_{M}\left(\mu\right)\sqrt{\frac{p\left(n+2\right)}{1-p}}l\right),
\widehat{F}'_n(\tau)-\Phi(\tau) =\widehat{Z}_1^n(\tau)+\widehat{Z}_2^n(\tau)+\frac{1}{k_n}\frac{\tau^2\Phi''(\tau)-\tau\Phi'(\tau)}{4}+o_p \biggl(\frac{1}{k_n}\biggr)
\begin{array}{rclcrcl}\psi_{-}\rightarrow e^{-i\theta}g^{-1}\psi_{-}\quad \quad \psi_{+}\rightarrow \psi_{+}\psi^{\dagger}_{-}\rightarrow \psi^{\dagger}_{-}g e^{i\theta}\quad\quad \psi^{\dagger}_{+}\rightarrow \psi^{\dagger}_{+}\end{array}\quad,
\alpha''_{2n+1}= \frac{1}{1-q}\left(q^{2n^2+2n}\sum_{j=0}^{n}q^{-2j^2-2j}\alpha_{2j+1}- q^{2n^2+4n+2}\sum_{j=0}^{n}q^{-2j^2}\alpha_{2j}\right),
\begin{array}{r c l r c l}{\chi_{1 2}}&{=}&{\chi_{3 4}}&{\equiv}&{\chi_{s},}\\{\chi_{1 4}}&{=}&{\chi_{2 3}}&{\equiv}&{\chi_{t},}\\{\chi_{1 3}}&{=}&{\chi_{2 4}}&{\equiv}&{\chi_{u}.}\\ \end{array}
\|e^{-tMz\partial_z}z_qu\|^2_{H_{\phi}}=\int_{\mathbb{C}^2}|v(e^{-tM}z)|^2|(e^{-tM}z)_{q}|^{2}e^{-2\phi(z)}L(dz) &=e^{2t\operatorname{Tr}M}\int_{\mathbb{C}^2}|v(z')|^2|z'_q|^2e^{-\phi(z')}e^{-2[\phi(e^{tM}z')-\phi(z')]}L(dz'){~.}
M_1(t)=\sum_{i=1}^n|\mu_i(t)|^2,M_2(t)=\sup_{1\leq i\leq n}|\dot{x}_i(t)-\mu_i(t)|, &M_3(t)=\sup_{1\leq i\leq n}\bigg|\dot{\mu}_i(t)+\sum^n_{j=1,j\not=i}\frac{a_{ij}}{x^3_{ij}(t)}+\sum^n_{\substack{k,j=1, j\not=i}}\frac{b_{ijk}\mu_k(t)}{x^3_{ij}(t)}\bigg|.
\left\vert\left\vert a-b\right\vert-\left\vert a-c\right\vert\right\vert \,\,=\,\, \left\vert\left\vert\left\vert b-a\right\vert -a\right\vert-c\right\vert & \,\,=\,\, \left\vert\left\vert b-\left\vert a-a\right\vert\right\vert-c\right\vert & \,\,=\,\, \left\vert \left\vert b-0\right\vert -c\right\vert & \,\,=\,\, \left\vert b-c\right\vert
\int \phi_1(n) L_{\tilde{w}_t}(\tilde{\psi}_t)( p^-_t(n) n z_0) J_1(n) \;d\mu_{z_0}^u =\int (\phi_1J_1)\circ u^+_t (n) L_{w_t}(F_t)( \tilde{p}^-_t(n) u^+_t(n) z_0) J_t^{-1}(n) \;d\mu_{z_t}^u =\int (\phi_1J_1)\circ u^+_t (n)\cdot L_{w_t}(F_t)(n z_t)\cdot J_t^{-1}(n) \;d\mu_{z_t}^u(n),
F_{q}^{(k)}(s) = \frac{1}{Q_{1}(2-q)}\left[ \displaystyle \sum_{n=0}^{\infty}\frac{1}{\left(\frac{3-2q}{1-q}\right)_{n}}\frac{\left(\pm \frac{\sigma_{\epsilon}}{1-q}\right)^{n}}{n!}(-1)^{k}\frac{\Gamma(n+k+1)}{s^{n+k+1}}\right].
T_{2}^{1}=6(-3a_{1}^{0}a_{2}^{2}c_{2}^{3}+3a_{2}^{1}a_{1}^{2}c_{2}^{2}+a_{1}^{0}a_{1}^{2}c_{1}^{1}+a_{1}^{0}a_{2}^{1}c_{2}^{4}-a_{2}^{0}a_{1}^{2}c_{2}^{3}-3c_{1}^{0}a_{2}^{2}a_{2}^{3}
d\left(T \{x_n^i\},T \{x_n^j\}\right) = d\left( \{x_n^{i+11}\}, \{x_n^{j+11}\}\right) &= \frac{10}{i+j+22}+1 &< \frac{10}{i+j}+1 &= d\left(\{x_n^i\},\{x_n^j\}\right).
( - \frac{1}{2}\frac{d^{2}}{{( d x^{1})}^{2}}+ \Lambda_{1}^{3}+ \frac{1}{2}{( \Lambda_{1}^{3})}^{2}( x^{1})^{2}+ \frac{1}{( x^{1})^{2}}( 8 ( m^{1})^{2}- \frac{1}{8}) ) Y_{1}^{1}( x^{1}) = 4 E_{I I}^{1}Y_{1}^{1}( x^{1}) ,
\check{\mathcal{B}}_\varepsilon =(\mathbf{D}-\mathbf{A}^\varepsilon )^*\check{g}^\varepsilon (\mathbf{D}-\mathbf{A}^\varepsilon )+\varepsilon^{-2}\check{v}^\varepsilon +\varepsilon^{-1}\widehat{v}^\varepsilon +\check{\mathcal{V}}^\varepsilon .
\sum_{i = 1}^{r}\Lambda ( H_{i}) \Lambda^{\prime}( H^{i}) = ( \Lambda_{0}, \Lambda_{0}^{\prime}) , \forall \Lambda = ( \Lambda_{0}, \kappa , \sigma ) , \Lambda^{\prime}= ( \Lambda_{0}^{\prime}, \kappa^{\prime}, \sigma^{\prime}) \in{\cal H}^{*}
{\cal V}_\sigma\left(\eta,\zeta,s\right)=1-\frac{1}{\eta}\sum_{n=0}^\infty\sum_{m=1}^\infty\left(-\eta\right)^n\zeta^m\,\theta\left(s+\sigma-n\sigma-m\right)\,F_{nm}\left(s+\sigma-n\sigma-m,\sigma\right)\,,
G(s) = \frac{\Gamma(2s - 1 + 2it)\Gamma(2s - 1 - 2it)}{\Gamma(2s-\tfrac12)}\sqrt{\pi}\frac{\Gamma(2s - \tfrac12)}{\Gamma(s+it)\Gamma(s-it)}&= \frac{2^{4s - 4}}{\sqrt{\pi}}\Gamma(s - \tfrac12 + it)\Gamma(s - \tfrac12 - it).
\sum_{\ell=0}^n B_{\Gamma,p}^{(\ell)}(z) = \frac{(-1)^{p-1}}{z^p}\sum_{g=0}^{\lfloor\frac{p-1}{2}\rfloor}z^{2g}(-1)^{g}\binom{n-1}{g}\; \sum_{h=0}^{\lfloor\frac{p-1}{2}\rfloor-g}z^{2h}\binom{h+n-2}{n-2}&= \frac{(-1)^{p-1}}{z^p}\sum_{m=0}^{\lfloor\frac{p-1}{2}\rfloor}z^{2m}\sum_{g=0}^{m}(-1)^{g}\binom{n-1}{g}\binom{m-g+n-2}{n-2}.
\sup_{f\in L^1(\Gamma_-,d\xi)\atop\|f\|_{L^1(\Gamma_-,d\xi)}=1}\Big|\int_{\Gamma_+}\phi(x,v)j_+K^{n+2}(I-K)^{-1}Jf(x,v) d\xi(x,v)\Big|\le C_{n+2}e^{(n+2)\|\tau\sigma_-\|_\infty}\|k\|_{\infty}^{n+2}\eta^{n-2},
L ~ = ~ \frac{1}{2}( \partial_{\mu}\phi^{a i})^{2}~ + ~ \frac{u}{4 !}\left( \phi^{a i}\phi^{a i}\right)^{2}~ + ~ \frac{v}{4 !}\left[ \left( \phi^{a i}\phi^{b i}\right)^{2}~ - ~ \phi^{a i}\phi^{a i}\phi^{b j}\phi^{b j}\right]
\tilde{a}_{\{1,2,3,4,6,7,8,9\}}(n) = \min_{n}\left(\tilde{a}_{\{1,9\}}(n), \tilde{a}_{\{2,8\}}(n), \tilde{a}_{\{3,7\}}(n), \tilde{a}_{\{4,6\}}(n) \right) \Rightarrow \tilde{a}_{\{1,2,3,4,6,7,8,9\}}(n) \geq \sqrt{5^n-1}.
\varepsilon_{u}= i \eta_{1 3}\gamma^{1 3}\varepsilon_{l}, \, v a r e p s i l o n_{u}= i \eta_{2 4}\gamma^{2 4}\varepsilon_{l}, \, v a r e p s i l o n_{u}= i \eta_{5 6}\gamma^{5 6}\varepsilon_{l}, \, v a r e p s i l o n_{u}= - \eta_{e}\eta_{p}\varepsilon_{l}
\| M \|_{r\ast}:= \max_{\|X \|_r \leq 1}\langle M,X \rangle = \max_{\sum_{i=1}^r s_i^2 \leq 1}\left[ \sum_{i=1}^{r}\sigma_i(M) s_i + s_r \sum_{i=r+1}^{n}\sigma_i(M) \right].
\big\|D^2 V_{s,t}(x) - D^2 V_{s,t}(y)\big\|_{L^q}\leq C_2 \sqrt{q}|t-s|^{1/(2 \varrho)}|x - y|^{\gamma}\sqrt{M}\sum_{n = 1}^{\infty}\| D^2 \sigma_n \|_{\mathcal{C}^{\gamma}}, \big\| D^2 V_{s,t}(x) \big\|_{L^q}\leq C_2 \sqrt{q}\frac{\sqrt{M}\kappa |t - s|^{1/(2\varrho)}}{1 + |x|^{\eta}},
2 \sum_{i = 1}^{N}\delta \alpha_{i}\lambda_{i}= \sum_{i = 1}^{N}\delta \alpha_{i}\lambda_{i}+ \sum_{i = 1}^{N}( \alpha_{i}- 1 ) ( \delta \lambda_{i}- \delta \lambda_{0}) = \delta \left[ \sum_{i = 1}^{N}( \alpha_{i}- 1 ) ( \lambda_{i}- \lambda_{0}) \right] .
Q ( \psi ) : = \frac{1}{4}F^{2}( \sin^{2}{\psi}+ \kappa^{2}\sin^{2}{F}) - \frac{1}{2}\sin^{2}{F}- \frac{1}{8}\kappa^{2}\frac{\sin^{4}{F}}{\sin^{2}{\psi}}, \quad \psi \in [ 0 , \pi ] .
F_{\mathcal{Y},a,b}(x) := \sum_{(i,j) \in [m] \times [n]}\overline{y}_{(i,j)}g_{(i,j)}x + \sum_{(q,p) \in \mathcal{Y}}\Delta y_{(q,p)}g_{(q,p)}(x) - \Delta y_{(a,b)}g_{(a,b)}(x).
\begin{aligned}\left|(u^2u_t)|_{x=0}\right|\leq\|u_t\|_{L^\infty}u^2|_{x=0}\leq\epsilon\left(\int_0^\infty\rho(r^2u_{xt})^2dx+2\int_0^\infty\rho^{-1}\frac{u_t^2}{r^2}dx\right)+C_\epsilon \|u^2r\|_{L^\infty}\left(\frac{dR}{dt}\right)^2.\end{aligned}
{\tilde{F}}_{\phi r}= -{\tilde{F}}_{r \phi}= - \frac{r}{1 + H^{2}r^{2}}\tan H t \ , \quad{\tilde{F}}_{t r}={\tilde{F}}_{r t}={\tilde{F}}_{t \phi}={\tilde{F}}_{\phi t}= 0
X = \frac{1}{2}\left[ -q_{N}(1 - q_{N}) + \sum_{i=1}^4 \Theta_{N}^{\gamma_i}(1 - \Theta_{N}^{\gamma_i}) - \sum_{\Gamma_{cut}}\Theta_{N}^{\gamma_+}(1 - \Theta_{N}^{\gamma_+}) \right].
V_{loss}^{'}=\sum_{s_i\in S_k^{(lf)}}\left\lfloor B_i\left(T_0\right)+\sum_{t=T_0+1}^N E_i^h(t)\right\rfloor+\sum_{s_i\in S_m-S_k^{(lf)}}\left\lfloor B_i\left(T_0\right)+\sum_{t=T_0+1}^N E_i^h(t)\right\rfloor \geq V_{loss}
A_{\mu}( x_{1}, t ) = c \int_{- \infty}^{\infty}d x_{1}^{\prime}\int_{- \infty}^{\infty}d t^{\prime}D^{c}( x_{1}- x_{1}^{\prime}; t - t^{\prime}) j_{\mu}^{t o t}( x_{1}^{\prime}, t^{\prime})
H(x_{1},x_{2}) = \chi([(0,0),(x_{1},x_{2})]^{c}) = \chi_{+}([(0,0),(x_{1},x_{2})]^{c}) - \chi_{-}([(0,0),(x_{1},x_{2})]^{c}) =: \int\limits_{[(0,0),(x_{1},x_{2})]^{c}}\!\!\!\!\!\! h_{+}(x_{1},x_{2}) \;\mathrm dx_{1}\mathrm dx_{2}- \int\limits_{[(0,0),(x_{1},x_{2})]^{c}}\!\!\!\!\!\! h_{-}(x_{1},x_{2})\;\mathrm dx_{1}\mathrm dx_{2}.
I = \int_M J\theta \wedge d \Omega^{n-1}= (n-1) \int_M J\theta \wedge \theta \wedge \Omega^{n-1}= -2(n-1) \int_M \sqrt{-1}\theta^{(1, 0)}\wedge \overline{\theta^{(1, 0)}}\wedge \Omega^{n-1}.
\alpha_2 = (y_1+y_2)^{n_2}y_2 &= (y_1^{n_2}+y_2^{n_2})y_2 ~(\mbox{as}~ \binom{n_2}{i}~\mbox{is even for}~ 0<i<n) &= y_2^{n_2+1}~( y_1^{n_2}=0, ~ \mbox{since}~ n_2 \geq n_1+1 ~\mbox{and}~ y_1^{n_1+1}= 0).
(b+a^3-k) x^{q+2}z + (2a^2 b) x^{q+1}z^2 + (b^2 a - 1) x^q z^3 + (-b - a^3 + k) x^3 z^q + (-2ab) x^2 z^{q+1}+ (-ab^2+1) x z^{q+2}= 0
\hat P_{2,q}= \frac{\lambda_2\sqrt{K_1}}{2(\lambda_1 +\lambda_2)}\left(\tilde A_1 + \tilde A_1^\dagger\right) + \frac{\lambda_1\sqrt{K_2}}{2(\lambda_1 +\lambda_2)}\left(\tilde A_2 + \tilde A_2^\dagger\right),
S^+_x = \sqrt{2S}a^*_x \left[ 1-\frac{a^*_x a_x}{2S}\right]^{1/2}, S^-_x = \sqrt{2S}\left[ 1-\frac{a^*_x a_x}{2S}\right]^{1/2}a_x, S^3_x = a^*_x a_x - S.
\Lambda_r\leq \Lambda_r (c_r\nu)^{-m}\Lambda_r^{m(\frac3r-1)}\|u_Q\|_r^m\leq(c_r\nu)^{-m}\Lambda_r^{m/2+1}\|u_Q\|_2^m \leq (c_r\nu)^{-m}\|u_Q\|_{\dot H^{1/2+1/m}}^m.
u^{(n)}(x)\quad=\begin{cases}\frac{\sum_{\ell=1}^\infty b^{(n)}_\ell(x_\ell)}{1+\sum_{\ell=1}^\infty b^{(n)}_\ell(x_\ell)}\sum_{\ell=1}^\infty b^{(n)}_\ell(x_\ell)<\infty\,, 1 \sum_{\ell=1}^\infty b^{(n)}_\ell(x_\ell)=\infty\,.\end{cases}
\begin{array}{c c c c}{{q_{2}=}}&{{q_{1}+ p_{1}+ p_{n}+ p_{n - 1}+}}&{{\ldots}}&{{+ p_{3}}}\\{{\vdots}}&{{\vdots}}&{{\ddots}}&{{\vdots}}\\{{q_{j}=}}&{{q_{1}+ p_{1}+ p_{n}+ p_{n - 1}+}}&{{\ldots}}&{{+ p_{j + 1}}}\\{{\vdots}}&{{\vdots}}&{{}}&{{}}\\{{q_{n}=}}&{{q_{1}+ p_{1}}}&{{}}&{{}}\\ \end{array}
L :=\{\ell \in \mathbb{N}_{>0}: \lceil \tfrac{i}{4}\rceil \leq \ell \leq i \}\neq \emptyset |L| < \infty K(\ell) := \{k \in \mathbb{N}_0 : k \leq \lceil \tfrac{i-1}{2}\rceil +1 \}3 \leq |K(\ell)| < \infty.
\Bigl ( \frac{d}{d u}+ 2 k \mathrm{s n}( u , k ) \Bigr ) \Bigl ( - \frac{d}{d u}+ 2 k \mathrm{s n}( u , k ) \Bigr ) \phi ( u ) = ( \frac{m_{F}}{\omega})^{2}\phi ( u )
\theta_1 (\tau ) = \sum_{n=-\infty}^{\infty}q^{(n+1/2)^2/2}, \quad\theta_2 (\tau ) = \sum_{n=-\infty}^{\infty}(-1)^n q^{n^2/2},\quad\theta_3 (\tau ) = \sum_{n=-\infty}^{\infty}q^{n^2/2}.
[M, N]_{\epsilon}=\sum_{N}N_{(1)}MN_{(2)}-\sum_{M}M_{(1)}NM_{(2)}= \begin{bmatrix}1 0 0 0\end{bmatrix}\begin{bmatrix}1 0 1 0\end{bmatrix}\begin{bmatrix}0 0 0 1\end{bmatrix}+\begin{bmatrix}0 0 1 0\end{bmatrix}\begin{bmatrix}0 1 0 1\end{bmatrix}\begin{bmatrix}0 0 1 0\end{bmatrix}= \begin{bmatrix}0 0 1 0\end{bmatrix}=E_{21}.
{\mathbb G}\left({k,n}\right) ={{{\mathbb S}{\mathbb T}\left({k,n}\right)}\mathord{\left/{\vphantom{{{\mathbb S}{\mathbb T}\left({k,n}\right)}{{\mathbb U}\left( k \right)}}}\right. \kern-\nulldelimiterspace}{{\mathbb U}\left( k \right)}}.
f(t\, x) =\int_{\mathbb{R}^N}\frac{\left|t^{-\beta}\,|x|^{-\beta}-|y|^{-\beta}\right|^{p-2}\, \left(t^{-\beta}\, |x|^{-\beta}-|y|^{-\beta}\right)}{|t\,x-y|^{N+r}}\, dy &=t^{-\beta\,(p-1)-N-r}\,\int_{\mathbb{R}^N}\frac{\left||x|^{-\beta}-|z|^{-\beta}\right|^{p-2}\,(|x|^{-\beta}-|z|^{-\beta})}{|x-z|^{N+r}}\, t^N\, dz &=t^{-\beta\,(p-1)-r}f(x),x\in\mathbb{R}^N,
d(x_k, \mathrm{Arg}\min f) \leq \|x_k-u_{k+1}\| \leq \|x_{k+1}-u_{k+1}\|+\|x_{k+1}-x_k\| = d(x_{k+1}, \mathrm{Arg}\min f)+ h\|\nabla f(x_k)\|, ~k\geq0.
\dot{Q}=\frac{T_{i}-T_{o}}{R_{i}+R_{1}+R_{2}+R_{o}}=\frac{T_{i}-T_{1}}{R_{i}}=\frac{T_{i}-T_{2}}{R_{i}+R_{1}}=\frac{T_{i}-T_{3}}{R_{i}+R_{1}+R_{2}}=\frac{T_{1}-T_{2}}{R_{1}}=\frac{T_{3}-T_{o}}{R_{0}}
F_a(f)' =\frac{f'e^{af}-f'e^{-af}}{e^{af}+e^{-af}}F_a(f)'' =\frac{f''e^{af}+a\left(f'\right)^2e^{af}-f''e^{-af}+a\left(-f'\right)^2e^{-af}}{e^{af}+e^{-af}}-\frac{a\left(f'e^{af}-f'e^{-af}\right)^2}{\left(e^{af}+e^{-af}\right)^2}&=\frac{f''e^{af}-f''e^{-af}}{e^{af}+e^{-af}}+\frac{4a\left(f'\right)^2}{\left(e^{af}+e^{-af}\right)^2}.
\int_X du \wedge d^c u \wedge \omega \wedge \Omega^{n-2}= \int_X dd^c(-u)^2 \wedge \omega \wedge\Omega^{n-2}- 2\int_X udd^cu \wedge \omega \wedge \Omega^{n-2}&=\int_X (-u)^2 dd^c[\omega \wedge \Omega^{n-2}] + 2\int_X (-u) \Omega_u \wedge \omega \wedge \Omega^{n-2}&- 2 \int_X (-u) \Omega\wedge \omega \wedge \Omega^{n-2},
2N^2\binom{2N}{N}^2\biggm|(-4096)^{N-1}\sum_{n=0}^{N-1}F(n,0)=\sum_{n=0}^{N-1}\left(20 n^2+8 n+1\right) \binom{2 n}{n}^5 (-4096)^{N- n-1},
\mathcal{I}\left(a, m, 0, \frac{\pi}{4}\right) = \frac{1}{(1+a)^{2}}\left\lbrace F_{1}\left(\frac{1}{2}, \frac{1}{2}- m, m, \frac{3}{2}, 1, \frac{1}{1+a}\right) - \frac{1}{\sqrt{2}}F_{1}\left(\frac{1}{2}, \frac{1}{2}- m, m, \frac{3}{2}, \frac{1}{2}, \frac{1}{2(1+a)}\right) \right\rbrace
\frac{\partial J_\varepsilon}{\partial v}= \sum_{j = 1}^2A_j U_{y^j, \lambda_j}+ \sum_{j = 1}^2B_j \frac{\partial U_{y^j, \lambda_j}}{\partial \lambda_j}+ \sum_{j = 1}^2\sum_{i = 1}^NC_{ji}\frac{\partial U_{y^j, \lambda_j}}{\partial y^j_i},
E\left[S(\theta)^{(r)}\right] = r \int_0^{\infty}t^{r-1}\left[1 - \left(1 - e^{-\theta t}\right) \left(1 - e^{-(1 - \theta) t / N}\right)^N \right] dt &= r \int_0^{\infty}t^{r-1}\left[1 - \left(1 - e^{-(1 - \theta) t / N}\right)^N \right] dt+ r \int_0^{\infty}t^{r-1}e^{-\theta t}\left(1 - e^{-(1 - \theta) t / N}\right)^N dt,
({\eta}_{\mu \nu}) = \left( \begin{array}{c c c c c}{1}&{0}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}&{0}\\{0}&{0}&{1}&{0}&{0}\\{0}&{0}&{0}&{0}&{- 1}\\{0}&{0}&{0}&{- 1}&{0}\\ \end{array}\right) .
\begin{array}{l}{b_{\alpha}^{2}= \lambda b_{\alpha}^{1}\quad , \quad E_{i k}= \left( \begin{array}{l l}{1}&{\lambda}\\{\lambda}&{\lambda^{2}}\\ \end{array}\right) E_{1 1}}\\{b_{\alpha}^{1}= - \frac{E_{1 1}}{| a |}( a_{\alpha}^{2}- \lambda a_{\alpha}^{1}) \quad , \quad | a | = \det a_{\alpha}^{i}\not = 0 .}\\ \end{array}
\displaystyle MAE=\displaystyle \frac{1}{N}\sum_{i=0}^{N}|[\sum_{\substack{j\in J\setminus J_{K(n)}}}B_{t_{j}}(B_{t_{j+1}}-B_{t_{j}})]-[\frac{1}{2}B_{T}^2 -\frac{1}{2}T]|.
\begin{aligned}K_b\wedge M_b\wedge K_m\wedge M_m>\widetilde J, &[N_b\wedge\lceil\!\lceil L_m\rceil\!\rceil\wedge(K_m-n-\alpha)]_+>\widetilde s, &[N_m\wedge\lceil\!\lceil L_b\rceil\!\rceil\wedge(K_b-n-\alpha)]_+>\widetilde J-n-\widetilde s.\end{aligned}
|\partial_{t}^{2}\left(-\lambda(t) \langle \left(\frac{\cos(2Q_{\frac{1}{\lambda(t)}}(r))-1}{r^{2}}\right)v_{4}\left(1-\chi_{\geq 1}(\frac{4r}{t})\right)\vert_{r=R\lambda(t)},\phi_{0}\rangle\right)| \leq \frac{C}{t^{4}\log^{2b+N-2}(t)}+ \frac{C \lambda(t)^{2-2\alpha}\sup_{x \geq t}\left(\frac{|e''''(x)| x}{\lambda(x)^{2-2\alpha}}\right)}{t \log^{b-3+N}(t)}
\partial_{x_n}^k \partial_{y_n}^l\partial_{\xi'}^\alpha \partial_{x'}^\beta a(x', 0, y', 0, 0,1)= (-1)^{m-|\alpha|}\partial_{x_n}^k \partial_{y_n}^l\partial_{\xi'}^\alpha \partial_{x'}^\beta a(x', 0, y', 0,0, -1)
\frac{\partial^2 (D' h_k)}{\partial (r_n)^2}= -\frac{1}{r_n}\frac{\partial (D' h_k)}{\partial r_n}- \frac{1}{\beta^2 (r_n)^2}\frac{\partial^2 (D' h_k) )}{\partial (\theta_n)^2}+ F.
e^{B}\left( \frac{1}{2}A^{\prime \prime}- \frac{1}{2}B^{\prime \prime}+ \frac{D_{1}}{4}{A^{\prime}}^{2}- \frac{1}{4}{B^{\prime}}^{2}- \frac{1}{4}( D_{1}- 1 ) A^{\prime}B^{\prime}\right) = \frac{\kappa^{2}}{2 a^{2}}( \frac{1}{e^{2}}{P^{\prime}}^{2}+ 2 v^{2}f^{2}P^{2}) ~ .
\|\Box_k^{\alpha_1}\|_{M_2}\lesssim 2^{jn(\alpha_2-\alpha_1)(1-1/p_2)}\|\Box_k^{\alpha_1}f\|_{L^{p_1}}\lesssim 2^{jn(\alpha_2-\alpha_1)(1-1/p_2)}\|f\|_{M_1}=2^{j\widetilde{A_2}(\mathbf{p},q,\alpha_1,\alpha_2)}\|f\|_{M_1}.
\int \prod_{\mu=0}^{p}\left(\frac{d\bar{\lambda}_{\mu}d\lambda_{\mu}}{2i}\right) \exp \left[ -\left(\bar{\lambda}_{\mu}+\alpha_{\mu}\right) M^{\mu\nu}\left(\lambda_{\nu}+\beta_{\nu}\right)\right] = \frac{\pi^{p+1}}{\det M^{\mu\nu}}~,
\frac{1}{2}\frac{d}{dt}\Big[ a(t)\frac{d}{dt}{\Vert u(t) - v(t)\Vert}^2 \Big] = \ \frac{d}{dt}\Big( a(u^{\prime}-v^{\prime}),u-v \Big) = \ \Big({\big( a(u^{\prime}-v^{\prime}) \big)}^{\prime}, u-v \Big) + a{\Vert u^{\prime}-v^{\prime}\Vert}^2 \ge \a{\Vert u^{\prime}- v^{\prime}\Vert}^2 \ge \ 0.
\| M \|_{r\ast}\; = \; \|\Sigma \|_{r\ast}\;= \; \max_{\|X \|_r \leq 1}\langle \Sigma,X \rangle \; = \; \max_{\sum_{i=1}^r\sigma_i^2(X)=1}\sum_{i=1}^{n}\sigma_i(M) \sigma_i(X) \;= \; \max_{\sum_{i=1}^r\sigma_i^2(X) \leq 1}\left[ \sum_{i=1}^{r}\sigma_i(M) \sigma_i(X) + \sigma_r(X) \sum_{i=r+1}^{n}\sigma_i(M) \right].
\begin{aligned}b_{s}^+(L) =\dfrac{\xi}{2\,N\, [s]}\, \left.\frac{d}{dq}\, \{1\}_q\left(\dfrac{V_{s}(L)}{[s]_q}- \dfrac{V_{2N-s}(L)}{[2N-s]_q}\right)\right|_{q=\xi}, b_{s}^-(L) =\dfrac{\xi}{4\,N\, [s]}\, \left.\frac{d}{dq}\, \{1\}_q\left(\dfrac{V_{2N+s}(L)}{[2N+s]_q}- \dfrac{V_{2N-s}(L)}{[2N-s]_q}\right)\right|_{q=\xi}.\end{aligned}
H =\int dx \left\{\frac{1}{2}\left[\xi_{5}^{2}+\xi_{6}^{2}+(\partial_{1}\xi_{3})^{2}-e^{2}\xi_{2}^{2}\right] +e\xi_{6}\xi_{2}+\xi_{1}(\partial_{1}\xi_{5}+e\partial_{1}\xi_{3})+\lambda \xi_{4}\right\}.
\Vert (x_{j})_{j=1}^{\infty}\Vert_{q,p}:=\sup_{(x_{n}^{\ast})_{n=1}^{\infty}\in B_{\ell_{p}^{w}(E^{\ast})}}\left( \sum_{j=1}^{\infty}\left( \sum_{n=1}^{\infty}|x_{n}^{\ast}(x_{j})|^{p}\right)^{{q}/{p}}\right)^{1/{q}}
(x+1)^{\beta}P_{m,n}^{(\alpha,\beta)}(x){}=(-1)^{m}\frac{(\beta+n)(\alpha+n-2m+1)}{(\alpha+n-m+1)}\int_{c}^x(t+1)^{\beta-1}P_m^{(-\alpha-1,\beta-1)}(t)P_{n-m}^{(\alpha+1,\beta-1)}(t)\,{\rm d}t
F_{{\mathcal V}(3,2)}= X_1X_2X_4+X_1X_3X_4+X_2X_3X_4+X_1X_2X_5+X_1X_3X_5+X_2X_3X_5 &+X_2X_4X_5+X_3X_4X_5+X_1X_2X_6+X_1X_3X_6+X_2X_3X_6+X_1X_4X_6 & +X_3X_4X_6+X_1X_5X_6+X_2X_5X_6+X_4X_5X_6+X_1X_2X_7+X_1X_3X_7 & +X_2X_3X_7+X_1X_4X_7+X_2X_4X_7+X_1X_5X_7+X_3X_5X_7+X_4X_5X_7 &+ X_2X_6X_7+X_3X_6X_7+X_4X_6X_7+X_5X_6X_7.
\Delta[nx_{cl},e_{0}]=\left(1+\frac{ge_{0}}{4}B_{\alpha\beta}\gamma^{[\alpha}\gamma^{\beta]}+\frac{g^{2}e_{0}^{2}}{16}\bar{B}_{\alpha\beta}\bar{B}^{*\alpha\beta}\gamma^{5}\right)\bar{\Lambda}(e_{0})+Q_{\alpha\beta}\gamma^{[\alpha}\gamma^{\beta]}
\langle \Lambda, \varphi\rangle =\lim_n\langle H'_q(u_n), \varphi\rangle=\lim_n\int_\Omega |u_n|^{q-2}u_n\, \varphi\, \frac{dx}{|x|^{\alpha}}=\int_\Omega\frac{|u|^{q-2}u\, \varphi}{|x|^{\alpha}}\, dx=\langle H'_q(u), \varphi\rangle.
D_{T}( 0 , x ) = \frac{T}{2 \pi}\sum_{n = - \infty}^{\infty}\int \frac{d k ~ e^{i k x}}{( 2 \pi n T )^{2}+ k^{2}+ \mu^{2}}= \frac{1}{4 \pi}\int d k e^{i k x}\frac{1}{\sqrt{k^{2}+ \mu^{2}}}\coth \frac{\sqrt{k^{2}+ \mu^{2}}}{2 T}
\sqrt{\rho}\partial_{x_i}\big((\sqrt{\rho})^{-1}u\big)=e^{-\frac{1}{4}\langle Q_{\infty}^{-1}x,x\rangle}\partial_{x_i}(e^{\frac{1}{4}\langle Q_{\infty}^{-1}x,x\rangle}u)=\Big(\partial_{x_i}+\frac{1}{2}(Q_{\infty}^{-1}x)_i\Big)u,
\lbrack{H , G_{r i}}\rbrack = 0 , \quad \lbrack{{\bf P}, Q_{r i}}\rbrack = 0 , \quad \lbrack{{\bf P}, G_{r i}}\rbrack = 0 , \quad \lbrack{G_{r 1}, N_{1}}\rbrack = - \frac{1}{4}Q_{r},
G_{k,\ell}(R,M)=\int_{R}^{2R}\int_{0}^\infty\big|(r\rho)^{-\frac{n-2}2}J_{\nu(k)}(r\rho)b^0_{k,\ell}(M\rho)\chi(\rho)\big|^2\frac{\mathrm{d}r\mathrm{d}\rho}{(1+\frac r M)^\beta}.
\begin{aligned}\mathcal{S}^n_t (\varphi) = \int_0^t S_n \mathcal{X}^n_s (\varphi ) ds = \frac{1}{2\sqrt{n}}\int_0^t \sum_{j \in \mathbb{Z}}g_n (\eta^n_j (s) ) \Delta^n \varphi^n_j (s) ds\end{aligned}
\lambda(g^{-1})\lambda(gh) = \sum_{\substack{A \ni g C \ni h^{-1}g^{-1}}}(A, g^{-1})(C,gh) = \sum_{\substack{C \ni h^{-1}C \ni h^{-1}g^{-1}}}(hC, g^{-1})(C,gh) = \sum_{\substack{C \ni h^{-1}C \ni h^{-1}g^{-1}}}(C,h).
\eta = \sum_{J = 1}^{\infty}\Big ( \bar{c}_{J}\psi_{J}+ c_{J}\bar{\psi}_{J}\Big ) , \qquad \tilde{\Pi}= \sum_{J = 1}^{\infty}\Big ({p}_{J}\psi_{J}+ \bar{p}_{J}\bar{\psi}_{J}\Big )
a(\kappa) =\tfrac{1}{\kappa}\bigl\{I_{Q\mathcal K}-Q(A(\kappa)+Q)^{-1}Q\bigr\}&=\sum_{j=0}^\infty (-1)^j\kappa^jQ\widetilde A_1(\kappa)\bigl[(A_0^\dagger+Q)\widetilde A_1(\kappa)\bigr]^{j}Q.
\frac{d}{dt}E_1(t)+\frac{Ca}{2}\frac{4\mu}{\gamma}\int_0^\infty\left(r^2(\rho^{\frac{\gamma}{2}})_x\right)^2dx=2\mu\int_0^\infty(\log\rho)_xru(u+\mu(\log\rho)_xr^2)dx+\mu(\rho r^2u_x)|_{x=0}R^2\frac{dR}{dt}.
V_{7}=-\varepsilon\int_{\mathbb{T}^{d}}(\mu^{1}-\mu^{2})(\mu^{2})\sum_{i=1}^{d}\left[\Theta_{p_{i}q}(t,x,\mu^{2},Dw^{2})\left(\frac{\partial(\mu^{1}-\mu^{2})}{\partial x_{i}}\right)\right]\dx,
K = i \int_{0}^{\infty}d \lambda^{\prime}\; s e^{i \lambda^{\prime}( \pi^{\prime \prime}- \pi^{\prime})}e^{- i [ s \lambda^{\prime}( \gamma^{\mu}P_{\mu}^{\prime}- m ) +{\frac{i}{s}}({\bar{\eta}}^{\prime \prime}-{\bar{\eta}}^{\prime}) ( \eta^{\prime \prime}- \eta^{\prime}) ]}\delta^{4}( P^{\prime \prime}- P^{\prime}) \, .
\exp(4it)P_4+P_0+\exp(-4it)P_{-4}=\tfrac{1}{2}(P(t)+P(t+\tfrac{\pi}{2})), P_0 =\tfrac{1}{3}(P(t)+P(t+\tfrac{\pi}{3})+P(t+\tfrac{2\pi}{3})), \exp(2it)P_2+\exp(-2it)P_{-2}=\tfrac{1}{2}(P(t)-P(t+\tfrac{\pi}{2})).
\hat{\cal R}_{mn}={\cal D}_{[m}\psi_{n]}+\frac{\kappa}{2}\gamma_{ab[m}\psi_{n]}v^{ab}+2i\kappa\gamma_{[m}\vec{\tau}\psi_{n]}\vec{t}-\frac{\kappa}{\sqrt{3}}\gamma^a\psi_{[n}\hat{F}_{m]a}
h_n^k =(-1)^k L^{2n-2k-2}z_n; h_n^k =c_{0n}\frac{z^{2k}}{(2k)!}+c_{1n}\frac{z^{2k-2}}{(2k-2)!}+\ldots+c_{kn}; h_n^k =0,k<0.
\left(\begin{matrix}x y\end{matrix}\right)A(\varepsilon_1)\cdots A(\varepsilon_{\nu-1})\left(\begin{matrix}1 1\end{matrix}\right)=\left(\begin{matrix}1 1\end{matrix}\right){}^t\!A(\varepsilon_1)\cdots{}^t\!A(\varepsilon_{\nu-1})\left(\begin{matrix}x y\end{matrix}\right)
\chi_{\{(x, z)\in W_\alpha: u>v\}}\omega^n_{\max(u, v)}\wedge \eta=\chi_{\{z\in V_\alpha: u>v\}}\omega^n_{\max(u, v)}\wedge dx &\chi_{\{(x, z)\in W_\alpha: u>v\}}\omega^n_u\wedge \eta=\chi_{\{z\in V_\alpha: u>v\}}\omega^n_{u}\wedge dx.
\|u_{l}v_{l}\|_{L^{p}(\Omega)}\le C \|u_{l}\|_{L^{p}(\Omega)}\|v_{l}\|_{L^{2}(\Omega)}^{\theta_{1}}\|v_{l}\|_{W^{2,p}(\Omega)}^{1-\theta_{1}},\qquad\theta_{1}=\frac{\frac{2}{d}-\frac{1}{p}}{\frac{2}{d}-\frac{1}{p}+\frac{1}{2}}\in(0,1),
I(\mu_E) = \int \left(\int \log\frac{1}{|x-t|}\,d\omega_{\Omega}(i,t) + \frac{1}{2}\log(1+x^2) \right)d\omega_{\Omega}(i,x) + \frac{1}{2}\int \log(1+t^2)\,d\omega_{\Omega}(i,t) = g_\Omega(i,\infty) + \frac{1}{2}\int \log(1+t^2)\,d\omega_{\Omega}(i,t).
4\sum_{j,k=1}^\infty \int_0^\infty m^s \int_{|x|>R}\partial^2_{jk}\varphi_R \partial_j \overline{u}_m(t) \partial_k u_m(t) dx dm = 4\int_0^\infty m^s \int_{|x|>R}\varphi''_R |\nabla u_m(t)|^2 dx dm.
\aligned\partial (\frak H_{(32)}^i \circ \frak H_{(ab)}^{i})= \frak N^{i+1}_{32}\circ \frak N^{i}_{2}\circ \frak H_{(ab)}^{i}\cup \frak H^{i}_{32}\circ \frak N^{i}_{b,21}&\cup\frak N^{i}_{3}\circ \frak N^{i}_{32}\circ \frak H_{(ab)}^{i}\cup \frak H^{i}_{32}\circ \frak N^{i}_{a,21}.\endaligned
\mathcal{C}_1=\mathcal{C}\bigcap \left\{(r_1,r_2): \tfrac{r_2}{r_1}\leq \tfrac{\tau_2}{\tau_1}\right\}, \mathcal{C}_2=\mathcal{C}\bigcap \left\{(r_1,r_2): \tfrac{r_2}{r_1}\geq \tfrac{\tau_2}{\tau_1}\right\}.
\left[ ( h r^{3}\partial_{r})^{2}+ \omega^{2}r^{6}f -{\frac{8 h r^{4}r_{\pm}^{4}}{( r^{2}+ r_{\pm}^{2})^{2}}}\left( 1 +{\frac{r_{0}^{2}}{r_{\pm}^{2}}}\right) \right] \tilde{\phi}_{\pm}= 0 ,
F \otimes \mathbb R_{n+}= \bigcup_{\epsilon \in \mathcal O^\times_+,\j \in J_n}\epsilon \overline{C(\mathbf v_j)}= \left(\coprod_{\epsilon \in \mathcal O^\times_+,\j \in J_n}\epsilon C(\mathbf v_j)\right) \coprod \left(\bigcup_{\epsilon \in \mathcal O^\times_+,\C \in F(D)}\epsilon C\right),