harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left[\begin{array}{ll}B^{(m)}_{n-m-1}B^{(m+1)}_{n-m-1}A^{(m)}_{n-m-1}A^{(m+1)}_{n-m-1}\end{array}\right]=\left[\begin{array}{ll}B^{(m-1)}_{n-m-1}B^{(m)}_{n-m-1}A^{(m-1)}_{n-m-1}A^{(m)}_{n-m-1}\end{array}\right]\left[\begin{array}{cc}0 1 1 a_{n}\end{array}\right]
S_3\psi_{l,m,j}=m\psi_{l,m,j}S_+\psi_{l,m,j}=\sqrt{l(l+1) m(m+1)}\psi_{l,m+1,j}S_-\psi_{l,m,j}=\sqrt{l(l+1)-m(m-1)}\psi_{l,m-1,j}S^2\psi_{l,m,j}=l(l+1)\psi_{l,m,j}
\Sigma(\xi-k)-\Sigma\left( \frac{\lvert \xi\lvert}{\lvert \xi-k\lvert}(\xi-k) \right) \geq -\omega\left( \frac{\lvert \xi\lvert}{\lvert \xi-k\lvert}(\xi-k)-\xi-k \right) &=-\omega\left( (\lvert \xi\lvert-\lvert \xi-k\lvert)\frac{\xi-k}{\lvert \xi-k\lvert}\right)
(q;q)_\infty\sum_{n=0}^\infty a_k(n)q^n=(q^k;q^k)_\infty\sum_{n=1}^{\infty}\frac{q^{kn}+ 2q^{2kn}+\dots+ (k-1)q^{(k-1)kn}}{1 + q^{kn}+ q^{2kn}+\dots+ q^{(k-1)kn}},
A_{1,0,0}+A_{2,0,0}+A_{3,0,0}+A_{4,0,0}=1 &z_1A_{1,0,0}+z_2A_{2,0,0}+z_3A_{3,0,0}+z_4A_{4,0,0}=a_{1,0,0}/\sqrt{q}&z_1^2A_{1,0,0}+z_2^2A_{2,0,0}+z_3^2A_{3,0,0}+z_4^2A_{4,0,0}=a_{2,0,0}/{q}&z_1^3A_{1,0,0}+z_2^3A_{2,0,0}+z_3^3A_{3,0,0}+z_4^3A_{4,0,0}=a_{3,0,0}/q\sqrt{q},
E[(\partial_x\phi_t(x))^2] = E[ \exp \{2 \int_0^t b'(r,\phi_r(x)) dr \}] = \tilde{E}[ \exp \{2 \int_0^t b'(r,\phi_r(x)) dr \}Z^{-1}] \leq \left( \tilde{E}[ \exp \{4 \int_0^t b'(r,\phi_r(x)) dr \}] \right)^{1/2}\left( \tilde{E}[ Z^{-2}] \right)^{1/2}.
S^{(m)}_2=\frac{X}{\varphi(V)\log X}\sum_{u_1,\ldots,u_k}\frac{(y^{(m)}_{u_1,\ldots,u_k})^2}{\prod_{i=1}^kg(u_i)}+O\left(\frac{(y^{(m)}_{max})^2\varphi(VP_f)^{k-1}X(\log X)^{k-2}}{\varphi(V)(VP_f)^{k-1}D_0}\right) &+O\left(\frac{y_{max}^2X}{\varphi(M)(\log X)^A}\right).
S_{5}= - \int d^{5}x \sqrt{- g}\, \left[ i \bar{\Psi}_{M}\gamma^{M N P}D_{N}\Psi_{P}+ i \frac{3}{2}\frac{a^{\prime}}{a}\bar{\Psi}_{M}\gamma^{M N}\Psi_{N}+ \left( i \bar{J}^{M}\Psi_{M}+ h . c . \right) \right] \, .
\psi_b(\mathbb{A}):= (\mathbb{P}(\mathbb{A}))= \begin{bmatrix}\psi_a(A) b\end{bmatrix}, \psi_a(A) := \frac{1}{2}\begin{bmatrix}a_{32}- a_{23}a_{13}- a_{31}a_{21}- a_{12}\end{bmatrix},
\mathcal{N}_2(h,\omega) = - [ |\partial_x| , \partial_x T_{\partial_x \o}] h - i T_{\partial_x^2 \o}|\partial_x|^{1/2}\omega + i [ |\partial_x|^{1/2}, T_{\partial_x \o}\partial_x] \omega & + |\partial_x| M_2(\omega,h) - i |\partial_x|^{1/2}Q_2(\omega,\omega) - i |\partial_x|^{1/2}T_{|\partial_x|^3 h}P_{\geq 1}h.
\lim_{t \rightarrow \pm \infty}\left\|\left(u(t) - S(t)(u_0^\pm, u_1^\pm), \partial_t u(t) - \partial_t S(t)(u_0^\pm, u_1^\pm)\right)\right\|_{\dot{H}^{\frac{1}{2}}\times \dot{H}^{-\frac{1}{2}}}= 0.
\log\left\{e^{1/2}\left(1-\frac1{m}\right)^{(m-1)/2}\right\}^m= \frac{m}2\left\{1-(m-1)\left(\frac1{m}+\frac1{2m^2}+o\left(\frac1{m^2}\right)\right) \right\}= \frac{1}{4}+o(1)
\lambda_{n,j}= \frac{t_p J_n^{\prime}(t_p) + in J_n(t_s)\sigma_j^{(n)}}{t_p H_n^{(1)\prime}(t_p) + in H_n^{(1)}(t_s)\sigma_j^{(n)}}, X_{n,j}= \frac{(1,\sigma_j^{(n)})^T}{\sqrt{1+|\sigma_j^{(n)}|^2}},
P_g\Big\{\big[v(t),w(t)\big];(v_0,w_0)\Big\}&=\frac{1}{Z}\exp\Bigg[-\frac{1}{2\sigma^2}\left(\begin{array}{c}v(t)-v_0 w(t)-w_0 \end{array}\right)^{\top}\Omega_{ij}^{-1}\left(\begin{array}{c}v(t)-v_0 w(t)-w_0 \end{array}\right)\Bigg],
(x_{1}+x_{2}+\cdot\cdot\cdot+x_{m})^{n}=\sum_{k_{1}+k_{2}+\cdot\cdot\cdot+k_{m}=n}(\begin{matrix}n\\k_{1},k_{2},...,k_{m}\end{matrix})x_{1}^{k_{1}}x_{2}^{k_{2}}\cdot\cdot\cdot x_{m}^{k_{m}},
\int_{I}|\widehat{\mu}(\lambda\gamma(t))|^{2}dt = \int_{I}|\widehat{\psi}\ast \widehat{\mu}(\lambda\gamma(t))|^{2}dt \lesssim \int_{\mathbb R^d}\int_{I}|\widehat{\psi}(\lambda\gamma(t)-\xi)| dt |\widehat{\mu}(\xi)|^{2}d\xi .
2^{(n-1)/2}\binom{(kn-1)/2}{(n-1)/2}=\prod_{r=1}^{(n-1)/2}\frac{kn-(2r-1)}{r}&=\prod_{r=1}^{(n-1)/2}\left(\frac{kn-r}{r}\cdot\frac{kn-(n-r)}{r}\cdot\frac{r}{kn-2r}\right) &=\prod_{d\mid n}\prod_{\substack{r=1 (r,n)=d}}^{(n-1)/2}\left(\frac{kn-r}{r}\cdot\frac{kn-(n-r)}{r}\cdot\frac{r}{kn-2r}\right) &=\prod_{d\mid n}S_{n/d}=\prod_{d\mid n}S_d.
\left\lbrace \begin{aligned}\dot{\boldsymbol{x}}(t) = A(t) \boldsymbol{x}(t) + B(t) \boldsymbol{u}(t) , \boldsymbol{y}(t) = C(t) \boldsymbol{x}(t) + D(t) \boldsymbol{u}(t) ,\end{aligned}\right.
\int_0^\infty \left\|\frac{d}{dt}\left(e^{itH}Je^{-itH_0}E_{H_0}(\Gamma)u \right)\right\| dt &=\int_0^\infty \left\|\frac{d}{dt}\left(e^{itH}J_ae^{-itH_0}u \right)\right\| dt &= \int_0^\infty \|e^{itH}(HJ_a-J_aH_0)e^{-itH_0}u\| dt &= \int_0^\infty \|(HJ_a-J_aH_0)e^{-itH_0}u\| dt
p_{k}\circ\varphi_{k}\circ\pi_{m_{k}}=p_{k}\circ\pi_{k}\circ\varphi =\pi_{k-1}\circ\varphi =\varphi_{k-1}\circ\pi_{m_{k-1}}=\varphi_{k-1}\circ p_{m_{k},m_{k-1}}\circ\pi_{m_{k}}
L^{-1}:=\overline{L}=\{\overline{L}^{A}_{M}\}=\left(\begin{array}{cc}\overline{L}^{a}_{\alpha}\overline{L}^{i}_{\alpha}0 \overline{L}^{i}_{\alpha}\end{array}\right)\,,\quad\overline{L}^{a}_{\mu}=0\,,
F_{B}= \bigg [ 2 \epsilon_{i j}^{1}\epsilon_{i l}^{2}p^{j}k^{l}- \epsilon_{i j}^{1}\epsilon_{i j}^{2}\vec{k}\cdot \vec{p}\bigg ]{\frac{\Gamma ( 1 - 2 k_{0}^{2}) \Gamma ( \vec{k}\cdot \vec{p}+ k_{0}^{2})}{\Gamma ( \vec{k}\cdot \vec{p}- k_{0}^{2}+ 1 )}}\ ,
\int \frac{d^{2}k}{( 2 \pi )^{2}}\frac{1}{( k^{2}+ M_{1}^{2}) ( k^{2}+ M_{2}^{2})}= \frac{1}{4 \pi}\frac{\log ( M_{1}^{2}/ M_{2}^{2})}{M_{1}^{2}- M_{2}^{2}}
\begin{aligned}Z(\sigma^{b,1}_{r}(v);(\alpha,\beta))= \sum_{\begin{subarray}{c}c^{'}_{1}r_1+\sum_{i=2}^{q}r_i=r c^{'}_{1}r_1, r_i\ge0\end{subarray}}Z^{*}_{(c^{'}_1r_1,\{r_i\}_{i=2}^{q})}(\tau(v);(\beta,\alpha))\end{aligned}
D_{{\cal F}}^{( N , \alpha )}( \Delta ) = \frac{- i \pi^{1 - N / 2}\Gamma ( 2 - N / 2 )}{2^{N - 2}\alpha}\sum_{n = - \infty}^{\infty}e^{i 2 \pi n \Delta / \alpha}\int_{0}^{\infty}d p \p^{N - 3}J_{2 \pi | n | / \alpha}^{2}( p r ) .
\frac{1}{\sum_{j = 1}^n \eta_j}\sum_{i = 1}^n \eta_i \left[\mathbb{E}_{X, Y}\{|X - \theta_i|\}- \mathbb{E}_X\{|X - \theta|\}\right] \leq \frac{2r}{1 - \varepsilon}\left(\frac{1+(e+1)^2}{\sqrt{n\alpha^2}}+ \varepsilon\right),
x_{1}^{1 2}+ x_{2}^{1 2}+ x_{3}^{6}+ x_{4}^{6}+ x_{5}^{2}+{\tilde{a}_{1}}a_{1}x_{1}^{6}x_{2}^{6}+{\tilde{a}_{2}}a_{2}x_{1}x_{2}x_{3}x_{4}x_{5}= 0 .
\sum(-1)^{f_\sigma}\cdots E_{ij}E_{kl}\cdots-\sum (-1)^{f_{\sigma'}}\cdots E_{kl}E_{ij}\cdots = \sum (-1)^{f_{\tilde\sigma}}\cdots E_{il}\cdots-\sum(-1)^{f_{\tilde\sigma'}}\cdots E_{kj}\cdots'
h_{d}(X_0, X_1, ..., X_k) \overset{}{=}\sum \limits_{\substack{c_0^{}+ c_1^{}+ \cdots + c_k^{}= d c_j^{}\geqslant 0}}X_{0}^{c_0^{}}\cdot X_{1}^{c_1^{}}\cdots X_{k}^{c_k^{}}\,,
\Phi^\ell(t,\bar t,\psi) = \sum_{n=0}^{\infty}a_n t^\ell_{0n}(t,\bar t,\psi) = \frac{e^{-i\ell\psi}}{(1-|t|^2)^\ell}\sum_{n=0}^{\infty}b_n t^n \equiv e^{-i\ell\psi}\varphi(t,\bar t)\,
\mu^2 = \frac{1}{(\deg\tau)^2}\tau\pi\psi\widehat{\tau}\tau\pi\psi\widehat{\tau}= \frac{1}{\deg\tau}\tau\pi\psi\pi\psi\widehat{\tau}= \frac{-dp}{\deg\tau}\tau\widehat{\tau}= -dp,
S_{\mu\nu}(m,n|a,b)= \sum_{p=1}^\infty \frac{p}{m}\frac{p}{n}S_{\mu\nu}(p)\oint \frac{dz}{2\pi i}z^{p-1}\left( \frac{b^*z+a^*}{az+b}\right)^m \oint \frac{d\bar z}{-2\pi i}{\bar z}^{p-1}\left( \frac{b\bar z+a}{a^*\bar z+b^*}\right)^n.
\operatorname*{l i m}_{| x_{1}| \rightarrow \infty}\int_{0}^{\infty}\frac{d \kappa^{2}}{| x_{1}|^{2}}\rho_{i i}\left( \frac{\kappa^{2}}{| x_{1}|^{2}}\right) G ( \kappa^{2}) = 0 ,
-C_1(1-\eta)^{\alpha_0}\leq\partial_\xi w+\partial_\tau w\leq \frac{(\delta b)^{\frac{1}{\alpha_0}}}{K}(1-\eta)on\xi=0 and \tau=0, & -C_1(1-\eta)^{\alpha_0}\leq\partial_\tau w \leq b\delta(1-\eta)^{\alpha_0}on\xi=0 and \tau=0,
Z_n(\lambda,T) := \sum_{k \geq 0, 0 \leq \ell \leq n}\frac{T^{nk}}{(nk)!}e^{- \lambda \ell}\mathrm{Vol}^{(n)}_{k,\ell}= \frac{1}{2}\sum_{\theta \in \{0,1\}}(-1)^{(\theta_1+1)(\theta_2+n+1)}\prod_{w^n = (-1)^{\theta_2}}(e^{Tw}- (-1)^{\theta_1}e^{- \lambda}).
\eta(\tau_n^{q-\ell}\tau_j, \tau_i) =\tau_n^q\tau_1\eta(\tau_n^{q-\ell+D(j)-D(i)}\tau_{j+1},\tau_{i+1})=(\tau_n^q\tau_1)^{n-j}\tau_n^{\ell-1+D(i)-D(j)+D(i+2,j+1)+D(i+1)}\tau_{i+n-j+1}=(\tau_n^q\tau_1)^{n-j}\tau_n^{\ell-1+D(i+1,j)+D(i)}\tau_{i+n-j+1},
\frac{\partial}{\partial\beta}\Big(\log \mid \frac{h_{\beta}(\beta, q_j,q_n)}{h_{\beta}(\beta, q_l,q_n)}\mid \Big) = \frac{h_{\beta}(\beta, q_l,q_n)}{h_{\beta}(\beta,q_j,q_n)}\cdot \frac{\partial}{\partial \beta}\Big(\frac{h_{\beta}(\beta,q_j,q_n)}{h_{\beta}(\beta,q_l,q_n)}\Big) \neq 0.
-\partial^2_{\xi_1\xi_1}V^{(i,m,j)}_2 =\Big(\partial^2_{x_2 x_2}v^{(i,m)}_0 - k_i\big(v^{(i,m)}_0\big)- \partial_{t}v^{(i,m)}_0\Big)\Big|_{x_1=\varepsilon(j+b_{i,m})},\ \ \xi_1\in I_{h_{i,m}}(b_{i,m}),
\begin{gathered}(f_{{W^0_V},i}^{t^2_m})^* (f_{{W^0_V},i}^{t^2_m})_* \omega^m_{X^{t^2_m}_{W^0_V,i}/W_{V,i}}\longrightarrow \omega^m_{X^{t^2_m}_{W^0_{V,i}}/ W^0_{V,i}}, (f^{t^2_m})^* f_*^{t^2_m}\omega^m_{U^{t^2_m}/V}\longrightarrow \omega^m_{U^{t^2_m}/V}\end{gathered}
\tilde{w}_s\cdot \Psi([\tilde{u},\tilde{v}])\cdot \tilde{w}_s^{-1}\overset{(1)}{=}\tilde{w}_s[\tilde{u},\tilde{v}]\tilde{w}_s^{-1}=[\widetilde w_s\tilde{u}\widetilde w_s^{-1}, \widetilde w_s \tilde{v}\widetilde w_s^{-1}].
L_M(w\,|\,a) = \int\limits_0^\infty \frac{dt}{t^{M+1}}\Bigl(e^{-wt}\,f(t) - \sum\limits_{k=0}^{M-1}\frac{t^k}{k!}\,B_{M,k}(w\,|\,a)- \frac{t^M\,e^{-t}}{M!}\, B_{M,M}(w\,|\,a)\Bigr).
a_{0,2}(\ell,k)=\frac 1{(2\pi i)^3}\int_{\Gamma_{D_2}}dw\int_{\gamma_{\tau_1}}d\zeta\int_{\gamma_{\tau_2}}d\omega\frac{G_{\Delta n,\Delta\mu,\Delta\xi}(w)}{\zeta^{k-n_1}G_{n_2+1-\ell,\Delta\mu,\Delta\xi}(\omega)(\zeta-w)(w-\omega)}.
\begin{aligned}P_f(X) = \nabla^{3}f(X,e_{b},e_{b})+\sum_{t=1}^{3}\nabla^{3}f(I_{t}X,e_{b},I_{t}e_{b})-4nSdf(X)+4nT^{0}(X,\nabla f) \hskip2.8in-\frac{8n(n-2)}{n-1}U(X,\nabla f), , P_f(X) = \nabla^{3}f(X,e_{b},e_{b})+\sum_{t=1}^{3}\nabla^{3}f(I_{t}X,e_{b},I_{t}e_{b})-4Sdf(X)+4T^{0}(X,\nabla f),\ .\end{aligned}
|\phi_j^*\psi_j-\phi_k^*\psi_k|=\Big|(\phi_j-\phi_k)^*\psi_j+\phi_k^*(\psi_j-\psi_k)\Big|\le \|\phi_j-\phi_k\|\,\|\psi_j\|+\|\phi_k\|\,\|\psi_j-\psi_k\|
\alpha_t(\psi_{j,\infty}(s_{\theta_{j}}^mi_{\theta_{j}}(f)s_{\theta_{j}}^{*n})) = e^{it(m-n)/N^{j}}\psi_{j,\infty}(s_{\theta_{j}}^mi_{\theta_{j}}(f)s_{\theta_{j}}^{*n}),
\hat\tau \ = \ \inf\Big\{t\geq0\colon K_t\,<\,\sup_{0\leq s\leq t}K_s\Big\}, \hat\tau' \ = \ \inf\Big\{t\geq\hat\tau\colon K_t\,=\,\sup_{0\leq s\leq t}K_s\Big\},
\int\frac{(|y|^{2})^{-u}}{|y|^{2}+q_{\mu}^{2}}d^{D}y=\pi^{D/2}(q_{\mu}^{2})^{-z-u}\frac{\Gamma(-u+D/2)\Gamma(1+u-D/2)}{\Gamma(D/2)}.
{\rm Hilb}\big(3T_{n+m}^{(0)}(K_{n,m})^{ab},t\big)= \sum_{k=0}^{n-1}(-1)^{k}(1+(n-k) t)^{m-1}\prod_{j=1}^{n-k}(1+j t){n \brace n-k}\hphantom{{\rm Hilb}\big(3T_{n+m}^{(0)}(K_{n,m})^{ab},t\big)}{}=t^{n+m-1}{\rm Tutte}\big(K_{n,m},1+t^{-1},0\big),
= \frac{px}{1-px}\sum_{r=1}^{\infty}(p^4x^3)^r \bigg( \frac{p^{-r-1}(1-p^{-r})}{1-p^{-1}}- (px)^{-r}\bigg( \frac{x^{r+1}(1-x^r)}{1-x}\bigg)\bigg)
\partial_{}^\gamma H_m = \sum_{\substack{m'<m m'\equiv m (\textrm{mod}R)}}\left(\partial_{}^{\gamma}g_{m'}-\sum_{0<\gamma' \leq \gamma}c_{\gamma',\gamma}\partial_{}^{\gamma'}G_{m'}\partial_{}^{\gamma-\gamma'}H_{m'}\right)\prod_{\substack{m'<m''<m m''\equiv m (\textrm{mod}R)}}(1-G_{m''}).
\sum_{n=0}^\infty B(n) \frac{x^n}{n!}\frac{d}{da}(a)_n = \sum_{n,k=0}^\infty B(n+k+1) \frac{x^{n+k+1}}{(n+k+1)!}(a)_{n+k+1}\frac{1}{a}\frac{(a)_{k}}{(a+1)_{k}}&= x \sum_{n,k=0}^\infty B(n+k+1) \frac{(1)_k (1)_n (a)_k (a+1)_{k+n}}{(a+1)_k (2)_{k+n}}\frac{x^n x^k}{k!n!}
\langle P_h^k(\cdot, s), V \rangle = \sum_j \langle P_h^k(\cdot, s), \mu_j \rangle \langle \mu_j, V \rangle + \sum_{\frac{1}{2}\leq \ell \leq k}\sum_j \langle P_h^k(\cdot, s), \mu_{j,\ell}\rangle \langle \mu_{j,\ell}, V \rangle.
d\mathfrak{p}= [\widetilde{a}_{12},\ \widetilde{b}_{12},\ \widetilde{c}_{12},\ \widetilde{d}_{12},\ \widetilde{e}_{12},\ , \widetilde{f}'_{12}] d\mathfrak{q}= [\widetilde{e}_{12},\ \widetilde{b}_{12},\ \widetilde{f}_{21},\ \widetilde{d}_{21},\ \widetilde{a}_{12},\ \widetilde{c}'_{21}].
\left\{\prod_{l_{i}^{( k )}}\left( \prod_{i = 0}^{l_{i}^{( k )}- 1}( \Theta_{j}- i ) \right) - \prod_{i = 1}^{| l_{0}^{( k )}|}( i - | l_{0}^{( k )}| - \Theta_{0}) \prod_{l_{i}^{( k )}, j \neq 0}\left( \prod_{i = 0}^{| l_{i}^{( k )}| - 1}( \Theta_{j}+ | l_{i}^{( k )}| - i ) \right) x_{k}\right\}\Pi ( x ) = 0
\int_{S^1}\varphi_n\,d\mu \;=\; \int_{S^1\setminus E_{n_0}}\varphi\,d\mu + \sum_{j=1}^{N}\sum_{k=n_0}^{n-1}\int_{I_k(c_j)\setminus I_{k+2}(c_j)}\varphi\,d\mu
\sum_{\substack{p \leq x^{1/2}p \nmid d}}\frac{(\lambda^2_f(p)-2)}{p^{1+2/\log x}}\frac{\log (x/p^2)}{\log x}=\sum_{\substack{p \leq x^{1/2}p \nmid d}}\frac{\lambda^2_f(p)-2}{p^{1+2/\log x}}- 2\sum_{\substack{p \leq x^{1/2}p \nmid d}}\frac{(\lambda^2_f(p)-2)}{p^{1+2/\log x}}\frac{\log p}{\log x}.
- \frac{e^{i \frac{\pi}{2}\mathrm{s i g n}\left( \frac{H}{P}\right)}}{\sqrt{8 K | H |}}\mathrm{s i g n}( P ) e^{- \frac{i \pi}{2 K}\phi}\sum_{j = 0}^{\infty}\frac{1}{j !}\left. \left( \frac{\pi}{2 i K}\frac{P}{H}\right)^{j}\left[ \partial_{\epsilon}^{( 2 j )}\frac{\prod_{i = 1}^{3}2 i \sin \frac{\epsilon}{p_{i}}}{2 i \sin \epsilon}\right] \right|_{\epsilon = 0}.
\sum_{\lambda}\frac{u^{|\lambda|}}{p^{n(\lambda)+|\lambda|}\prod_{i \geq 1}\prod_{j=1}^{\lfloor m_i(\lambda)/2 \rfloor}(1-1/p^{2j})}= (1-u/p)^{-1}\prod_{i \geq 3 \atop i \odd}(1-u^2/p^i)^{-1},
\widehat{M}= m_0^2 I\otimes D_1; &\widehat{A}= m_0(A\otimes D_1^{-1}+ I \otimes A_1 + A_2 \otimes D_2^{2}D_1^{-1}) + C D_3\otimes D_2 C + D_3 C \otimes C D_2.
N^{\prime}(m,m^{\prime}) = |<m,m^{\prime}>|^{2}=\left( \frac{1+\mathbf{n}\cdot\mathbf{n}^{\prime}}{2}\right)^{n}=\frac{4\pi}{n+1}\mathbb{U}_{\frac{1}{\vec{b}},\vec{b}}(\mathbf{n},\mathbf{n}^{\prime})
U_{i_1,j}= \left\{g+3i_1+5, \frac{1}{2}(2g-3j+6i_1+7),\frac{1}{2}(2g+3j-12i_1-23)\right\}_{g-1}=:\{\alpha_1,\beta_1,\gamma_1\}, U_{i_2,j}= \left\{g+3i_2+5, \frac{1}{2}(2g-3j+6i_2+7),\frac{1}{2}(2g+3j-12i_2-23)\right\}_{g}=:\{\alpha_2,\beta_2,\gamma_2\}.
\ell^{-1}\sum_{k=1}^{\ell}\xi^{-a_{j+k}}\xi^k \xi^{a_{j+k}}=\ell^{-1}\sum_{k=1}^{\ell}\xi^{a_{j+k}-a_{i+k}+k}=\xi^{a_j-a_i}{\ell}^{-1}\sum_{k=1}^{\ell}\xi^{k(j-i+1)}=\delta_{i,j+1}\xi^{a_j-a_i}.
\sum_{\alpha,\beta=1}^d n^\alpha \frac{\partial n^\beta}{\partial x^\alpha}\nu^\beta = \sum_{\alpha=1}^d n^\alpha\frac{\partial (n\cdot \nu)}{\partial x^\alpha}- \sum_{\alpha,\beta=1}^d \frac{\partial \nu^\beta}{\partial x^\alpha}n^\beta n^\alpha = - \sum_{\alpha,\beta=1}^d \frac{\partial \nu^\beta}{\partial x^\alpha}n^\beta n^\alpha.
P(x_{t}|x_{t-1},u_{t})=det(2\pi R_{t})^{-\frac{1}{2}}exp(-\frac{1}{2}(x_{t}-A_{t}x_{t-1}-B_{t}u_{t})^{T}R_{t}^{-1}(x_{t}-A_{t}x_{t-1}-B_{t}u_{t}))
\sum_{n\asymp x}\frac{c_j^{(II)}(n)\chi(n)}{n^s}=\int_{-x^{1+\varepsilon}}^{x^{1+\varepsilon}}\int_{-x^{1+\varepsilon}}^{x^{1+\varepsilon}}\int_{-x^{1+\varepsilon}}^{x^{1+\varepsilon}}E(s;\mathbf{t})F(s;\mathbf{t})P(s;\mathbf{t})K(\mathbf{t})dt_1 dt_2 dt_3,
\varphi_{v_0}(e_1,e_2,f_2) = (a_0,b_0) = (0,0), \varphi_{v_1}(e_1,e_2,f_2) = (a_1,b_1) = (-e_1,e_1+f_2), \varphi_{v_2}(e_1,e_2,f_2) = (a_2,b_2) = (e_2,e_1).
\tilde{f}_{p}(v,w) =((\tilde{f_{p}})_{1}(v,w),(\tilde{f_{p}})_{2}(v,w))=(\tilde{a}_{p}(v,w)+\tilde{F}_{p}(v) ,\tilde{b}_{p}(v,w) +\tilde{F}_{p}(w)),
u ( \rho ) = \left( C_{1}F \left( - \frac{e^{2}+ 1}{4}, - \frac{1}{2}, \rho^{2}\right) + C_{2}\rho^{3}F \left( - \frac{e^{2}- 5}{4}, \frac{5}{2}, \rho^{2}\right) \right) e^{- \frac{\rho^{2}}{2}}
[ \delta_{\eta}\, , \, \delta_{\omega}] \; = \; \int d^{3}x \, \Sigma_{\mu \nu}^{a b}( \theta ) \, \frac{\delta S_{i n v}}{\delta \theta_{\mu}^{a}( x )}\frac{\delta}{\delta \theta_{\nu}^{b}( x )}\; .
\mathcal{L}_{V_a}b - \mathcal{L}_{V_b}a = -2 d^\Lambda a \wedge d^\Lambda b + \iota_X(d^\Lambda a \wedge d b - d^\Lambda b \wedge da) + d \iota_X(d^\Lambda a \wedge b - d^\Lambda b \wedge a) + d d^\Lambda b \wedge \iota_X a - d d^\Lambda a \wedge \iota_X b
\sum_{n=0}^{\infty}p_{-6}\left(5^{j}n+\dfrac{3\times5^{j}+1}{4}\right)q^{n}=\dfrac{1}{q^{7}E_{25}^{6}}\sum_{l=1}^{\infty}b(j,l)T^{l}\zeta^{-(6l+6)}.
{\cal H}= \frac{1}{2}( \vec{E}^{a}\vec{E}^{a}+ \vec{B}^{a}\vec{B}^{a}) + \frac{1}{2}( \vec{\cal E}^{2}+ \vec{\cal B}^{2}) + \pi^{\dagger}\pi + ( \vec{D}\varphi )^{\dagger}( \vec{D}\varphi ) + V ( \varphi^{\dagger}\varphi )
H_{0}^{- ( 0 )}= \mu [ \frac{( \alpha - \imath )}{2{\alpha}}] \exp ( - i \frac{\partial}{{\partial}{\alpha}}) , \quad \widetilde{H}_{0}^{- ( 1 / 2 )}= \mu [ \frac{( \alpha - \frac{3 i}{2})}{2 ({\alpha}- \frac{i}{2})}] \exp ( - i \frac{\partial}{{\partial}{\alpha}}) ,
X^{(i)}_{k+1}= (1-h)X^{(i)}_{k}+h\sum_{j\in\mathcal{N}_i}\mathbf{W}_{ij}X^{(j)}_k -\alpha_{k+1}h \nabla u_i(X^{(i)}_k) dt+ \sqrt{2\sigma\alpha_{k+1}h}Z^{(i)}_{k+1}
F_{1}({\bf{k}}) = 1 - \sum_{i,{\bf{q}}}\frac{1 - n_{F}({\bf{k}}+{\bf{q}})}{({\tilde{\omega}}_{i}(-{\bf{q}}) + \frac{{\bf{k.q}}}{m}+ \epsilon_{{\bf{q}}})^{2}}g^{2}_{i}(-{\bf{q}})
\sum_{x\in\Theta_k^{n_1}}|\phi_E(x)|^2=1-\sum_{x\in\Lambda_{L_{k-1}}\setminus\Theta_k^{n_1}}|\phi_E(x)|^2\geq 1- CL^{\beta d}e^{-m'L^{\rho^{2n_1-1}}}\geq 1/2
\hat k = \underset{1 \leq k \leq{\frac{N}{2}}}{}\ \frac{1}{(N-k-1)^2}\sum_{i=k+1}^N(X_i-\bar{X}_{k,N})^2\ \ \ \ \bar{X}_{k,N}=\frac{1}{N-k}\sum_{i=k+1}^N X_i.
\Pi_{\alpha}\Big(D^{(n)}_{\alpha}(\theta,\,\theta^\ast) \geq \frac{M}{1-\alpha}\, n\,\varepsilon_n^2\ \Big| \, X^{(n)}\Big) = \frac{\int_{U_n}e^{-\alpha\, r_{n}(\theta,\,\theta^\ast)}\,\Pi(d\theta)}{\int_{\Theta}e^{-\alpha\, r_{n}(\theta,\,\theta^\ast)}\, \Pi(d\theta)},
I_{1}( B ) = - \frac{e B}{{\pi}^{\frac{3}{2}}\beta}\sum_{k = 1}^{\infty}\frac{( - 1 )^{k + 1}}{k}\int_{0}^{\infty}d x \; e^{- \frac{x^{2}}{4}- \frac{1}{x^{2}}\beta^{2}k^{2}m^{2}}\frac{1}{e^{\frac{2 \beta^{2}e B k^{2}}{x^{2}}}- 1}\; .
T_{\ell,1}\mapsto T_{\ell,1}-(z_\ell\cdot H_0+H_1)\prod_{\substack{2 \leq k\leq m k\neq \ell}}B_k, T_{\ell,j}\mapsto T_{\ell,j},\qquad\mbox{for}j=2,\ldots,n_\ell.
R_I = \sum_{\nu=0}^{\lfloor \frac{k-1}{2}\rfloor}e_{k-1-2\nu}(X_i \mid i\in I)Y^\nu S_I = \sum_{\nu=0}^{\lfloor \frac{k}{2}\rfloor}e_{k-2\nu}(X_i \mid i\in I)Y^\nu,
\sum_{\gamma}f_{\frac{1}{2}}(t-\gamma)\leq \frac{\widehat{h}(0)}{2\pi}\log t +\frac{1}{\pi}\sum_{n=2}^\infty\frac{\Lambda(n)}{\sqrt{n}}\left|\widehat{h}\left(\frac{\log n}{2\pi}\right)\right|.
M_{1}=[\begin{matrix}a&b\\c&d\end{matrix}],N_{1}=[\begin{matrix}e&f\\g&h\end{matrix}],M_{2}=[\begin{matrix}a&c\\e&g\end{matrix}],N_{2}=[\begin{matrix}b&d\\f&h\end{matrix}],M_{3}=[\begin{matrix}a&e\\b&f\end{matrix}],N_{3}=[\begin{matrix}c&g\\d&h\end{matrix}]
F(R_{\mathrm{CPF}},\gamma_i,\beta_i)=f_{\mathrm{CPF}}\bar{R}_{\mathrm{CPF}}-\beta_0 \left(\bar{P}_{\mathrm{CPF}}-P_{0}\right) &- \sum_{i=1}^M \beta_i \left(\bar{P}_{i}-P_{0}\right) - \sum_{i=1}^M \gamma_i \left( f_{\mathrm{CPF}}\bar{R}_{\mathrm{CPF}}-f_i\bar{R}_{i}\right) &-\alpha( f_{\mathrm{CPF}}+\sum_{i=1}^M f_i -1)
0 = (I-Q) F_{\Psi\Psi}(\Psi^q,p_0)[v_1, \psi_p(0,p_0)p'(0)] &+ (I-Q) F_{\Psi p}(\Psi^q,p_0)[v_1, p'(0)] &+ (I-Q) F_\Psi(\Psi^q,p_0)\psi_{vp}(0,p_0)[v_1,p'(0)],
\mathcal{I}_{s,c}^{\pm}(m)=\left(\frac{\mathcal{M}l^{3\epsilon n_l}}{m}\right)^{\frac{1}{4}}\min\left(M,\frac{BZl^{\frac{n_l}{2}}}{A}\right)e(\theta_{s,c}m)W_{s,c}(m)+O(l^{-100n_l}),
\int_{D_{\rho}}\frac{1}{2}|\mathrm{Ric}_{\tilde{g}}|^2_{\tilde{g}}dV_{\tilde{g}}=\int_{S_{\rho}}\tilde{g}(\tilde{\nabla}\Delta_{\tilde{g}}\Phi,\tilde{\nu})d\tilde{\omega}+ 2\int_{S_{\rho}}\mathrm{Ric}_{\tilde{g}}(\tilde{\nabla}\Phi,\tilde{\nu})d\tilde{\omega}.
m_{h}= \left( \begin{array}{c c c c}{{0}}&{{0}}&{{\xi_{1}}}&{{\xi_{4}}}\\{{0}}&{{0}}&{{\bar{\xi}_{4}}}&{{\bar{\xi}_{1}}}\\{{\xi_{1}}}&{{\bar{\xi}_{4}}}&{{\bar{\Phi}_{1 2}}}&{{\phi_{2}}}\\{{\xi_{4}}}&{{\bar{\xi}_{1}}}&{{\phi_{2}}}&{{\Phi_{1 2}}}\\ \end{array}\right) ,
\frac{\mathtt{g}_{a, \mu}'(1)}{\mathtt{g}_{a, \mu}(1)}= a\left(a^{a/2}-1\right)(\mu-1)+a^{a/2}-\sum_{n=1}^\infty \frac{2 a^{a/2}}{\mathtt{j}_{\mu,n}^2-1}&\geq a\left(a^{a/2}-1\right)(\nu-1)+a^{a/2}-\sum_{n=1}^\infty \frac{2 a^{a/2}}{\mathtt{j}_{\nu,n}^2-1}=\frac{\mathtt{g}_{a, \nu}'(1)}{\mathtt{g}_{a, \nu}(1)}.
c(u_0,f) := \frac{1}{2\pi}\, \frac{3}{\mu}\, \left\| \tilde{u}\right\|_{L^{\infty}(p_1,p_2)}+ \frac{1}{\pi}\left( 4 \left\| \tilde{u}\right\|_{L^{\infty}(p_1,p_2)}+ \left\| \tilde{u}' \right\|_{L^1(p_1,p_2)}\right) m^{-1}\; .
\hat{k}_2({\Delta}) =\sum_{j_1,j_2,j_3=1,2}\dfrac{(x_{1,1}x_{1,2}x_{2,1}x_{2,2}x_{3,1}x_{3,2})^4}{x_{1,j_1}x_{2,j_2}x_{3,j_3}}&=(x_{1,1}x_{1,2}x_{2,1}x_{2,2}x_{3,1}x_{3,2})^3(x_{1,1}+x_{1,2})(x_{2,1}+x_{2,2})(x_{3,1}+x_{3,2}).
\Phi_0 =\phi_0dvd\b{v}=\frac{\phi_0}{2}(dv\otimes d\b{v}+d\b{v}\otimes dv),\quad\dot{\Phi}_0=q dv^2+\o{q dv^2}, \ddot{\Phi}_0=\left(\frac{2|q|^2}{\phi^2_0}+2\alpha\right)\Phi_0.
(1-zz_0)^{-2\deg}f(v) = e^{-z(1-zz_0)L_{1}}e^{-(1-zz_0)^{-1}z_0L_{-1}}e^{zL_{1}}e^{z_0L_{-1}}f(v) = f(e^{-z(1-zz_0)L_{1}}e^{-(1-zz_0)^{-1}z_0L_{-1}}e^{zL_{1}}e^{z_0L_{-1}}v) = f((1-zz_0)^{-2\deg}v) = (1-zz_0)^{-2n}f(v),
x^2y\longrightarrow_{\mathcal{FJ}}x^2y-xf_{xy}= x^3+xy^2 \longrightarrow_{\mathcal{FJ}}x^3+xy^2-f_{x^3}= xy^2 \longrightarrow_{\mathcal{FJ}}xy^2-yf_{xy}= & x^2y+y^3 \longrightarrow_{\mathcal{FJ}}x^2y+y^3-f_{y^3}=x^2y \longrightarrow_{\mathcal{FJ}}x^2y-xf_{xy}\dots .
\frac{1}{u^{\frac{1}{r}}(Q)}\int_{Q}|b(x)-b_{Q}|dx \leq\frac{c}{u(Q)^{\frac{1}{r}}|Q|^{\frac{1}{r'}}}\int_{Q}|b(x)-b_{Q}|dx =c\left(\frac{1}{u(Q)}\int_{Q}|b(x)-b_{Q}|dx\right)^{\frac{1}{r}}\left(\frac{1}{|Q|}\int_{Q}|b(x)-b_{Q}|dx\right)^{\frac{1}{r'}},
e_{\overline{G}}(V_1) + e_{\overline{G}}(V_2) \geq d_{\overline{G}[V_i]}(v_i) + d_{\overline{G}[V_j]}(v_j)-1 = |V_i|-d_{V_i}(v_i) + |V_j| - d_{V_j}(v_j)-3 \geq D - d_{V_i}(v_i) + D - d_{V_j}(v_j) - 5\geq d_H(v_i) + d_H(v_j) - 5 &\geq e(H) - 5.
S_{(5)}= \frac{1}{2 \kappa^2_{(5)}}\int d^4 x \int dy \sqrt{-g_{(5)}}\left[ R^{(5)}+ 12k^2 \right] - \int d^4 x \left[\sqrt{-g_{(+)}}\lambda_{(+)}+\sqrt{-g_{(-)}}\lambda_{(-)}\right],
\frac{\mbox{d}\mu_{0,q^{1/4}}}{\mbox{d}x}=\left(\pi q^{1/4}{}_{2}\phi_{2}\left[\begin{matrix}0, q^{1/2}-q^{1/2}, -q \end{matrix}\ ;q, -q^{-1/2}x^{2}\right]\right)^{-1}\!.
\sum_{k=1}^\infty \sum_{m=1}^\infty \lambda_k(a^k_m)^2 - \sum_{k=1}^n \sum_{m=1}^\infty \lambda_m (a^k_m)^2= \sum_{k=1}^n \sum_{m=n+1}^\infty \lambda_k \left((a^m_k)^2 - (a^k_m)^2\right)+ \sum_{k=1}^n \sum_{m=n+1}^\infty (\lambda_k - \lambda_m)(a^k_m)^2.
D_2 \ =\ \frac{2}{9}\left(- 6 J_6^2 - 6J_5J_4 + 3J_3J_0 + 4 J_0^2 - 8J_6 + 3J_3 + 10 J_0 - 14 \right) A_9\ + \frac{8}{9}\left(3J_6J_5 + 9J_4J_1 + 4J_5\right) A_8 - \frac{2}{9}\left(12 J_5J_1 - 13 J_2J_0\right) A_7\ - - \frac{28}{9}J_6J_2 A_5 + \frac{28}{9}J_6J_1 A_3 \ ,
J_{\nu_{1}\nu_{2}\nu_{3}}^{( d )}= \frac{1}{\pi^{d}}\int \! \! \int{d^{d}k_{1}d^{d}k_{2}}P_{k_{1}, m_{1}}^{\nu_{1}}P_{k_{1}- k_{2}, m_{2}}^{\nu_{2}}P_{k_{2}, m_{3}}^{\nu_{3}}.
S=\int d^{4}x\sqrt{-g}[\frac{\mathcal{R}}{16\pi G}-\frac{1}{4\pi}\frac{1}{2}((\partial_{\mu}\phi)(\partial^{\mu}\phi)+(\frac{g\phi}{2})^{2}A_{\mu}A^{\mu}+V(\phi)+\xi\phi^{2}\mathcal{R})-\frac{1}{4\pi}\frac{1}{4}\mid F\mid^{2}].
\begin{cases}u_t=\Delta (ue^{-v}) x\in\Omega,\;t>0, v_t-\Delta v+v=u x\in\Omega,\;t>0, \partial_\nu u=\partial_\nu v=0, x\in\partial\Omega,\;t>0, u(x,0)=u_0(x),\;v(x,0)=v_0(x) x\in\Omega,\end{cases}
\operatorname{Pr}\Big(Y\cap(S+\epsilon B_2)=\emptyset\Big)\geq1-\frac{7}{2}\operatorname{exp}\bigg(-\frac{1}{2}\bigg(\frac{(1-\epsilon)a_M-\epsilon a_N-w(S)}{3+\epsilon+\epsilon a_N/a_M}\bigg)^2\bigg).
\|F\|_{W^{-s,p'}(\Omega)}=\min_{\varphi\in L^{p'}(\mathbb{R}^N\times\mathbb{R}^N)}\left\{\|\varphi\|_{L^{p'}(\mathbb{R}^N\times\mathbb{R}^N)}\, :\, R^*_{s,p}(\varphi)=F\mbox{in}\Omega\right\},
y^{(m)}_{r_1,\ldots,r_k}= \sum_{(u,VP_f\prod_ir_i)=1}\frac{\mu(u)^2}{\varphi(u)}F\bigl(\frac{\log r_1}{\log R},\ldots,\frac{\log r_{m-1}}{\log R},\frac{\log u}{\log R},\frac{\log r_{m+1}}{\log R},\ldots,\frac{\log r_k}{\log R}\bigr) &+O\left(\frac{F_{max}\varphi(VP_f)\log X}{VP_fD_0}\right).
\begin{array}{c}\dot{\bf x}={\rm i}\left({\bf z}_1-{\bf{\bar z}}_1\right), \dot{\bf z}_{i-1}=-{\rm i}\delta_{1.i-1}{\bf p}+k_{i-1}{\bf z}_{i}-k_{i-2}{\bf z}_{i-2}, \dot{\bf z}_{N}=-{\rm i}\delta_{1.N}{\bf p}+{\rm i}k_{N}{\bf z}_{N}-k_{N-1}{\bf z}_{N-1},{\dot{\bf p}}=0,\end{array}
\Omega_3 ={1\over 3R^2}\varepsilon_{\hat m\hat n\hat p\hat q}X^{\hat m}dX^{\hat n}dX^{\hat p}dX^{\hat q}\,,\qquad\tilde\Omega_3 ={1\over 3R^2}\varepsilon_{\hat m'\hat n'\hat p'\hat q'}X^{\hat m'}dX^{\hat n'}dX^{\hat p'}dX^{\hat q'}\,,
\|\sigma_2\|\, =\, \frac{1}{1\,+\,|z|^2}\ \,,\ \ \ \|\sigma_0\|\, =\, \frac{2\,|z|}{1\,+\,|z|^2}\ \,,\ \ \ \|\sigma_{-2}\|\, =\, \frac{|z|^2}{1\,+\,|z|^2}
H_{j}= r^{\frac{\sigma_{3}}{4}}G_{j}(-rx_{j};r\vec{x},\vec{s}) = R(-x_{j})E_{-x_{j}}(-x_{j})\begin{pmatrix}\Psi_{j,11}\Psi_{j,12}\Psi_{j,21}\Psi_{j,22}\end{pmatrix}.