harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
(d_{\omega}F)(\gamma) = -2\,\mbox{Re}\,\langle\langle \xi,\,\bar\partial^{\star}\omega\rangle\rangle_\omega + 2\,\mbox{Re}\,\int\limits_X \xi\wedge\rho_\omega^{2,\,0}\wedge\overline{\rho_\omega^{2,\,0}}\wedge\bar\partial\omega_{n-3}.
\sum_{j=1}^3\sum_k\int_{\mathbb{R}}(\mathbf{1}_{|k-k_0|\geq D k_0}+\mathbf{1}_{|\xi-\eta_0|\geq Dk_0})e^{|k-k_0|+|\xi-\eta_0|}|\widehat{g_j'}(t,k,\xi)|^2\,\mathrm{d}\xi\lesssim \varepsilon_1^2e^{-Dk_0/2}.
\lim_{n \to \infty}\left\| f_t \circ \left( 1_{A_1^{(n)}}* 1_{A_2^{(n)}}\right) \right\| = \| f_t \circ ( \phi_1 * \phi_2 ) \| , \lim_{n \to \infty}\left\| f_t \circ \left( 1_{A_1^{(n)}}^* * 1_{A_2^{(n)}}^* \right) \right\| = \| f_t \circ ( \phi_1^* * \phi_2^* ) \|
\left((a_1 \otimes a_2 \otimes \cdots \otimes a_n) \otimes \sigma\right) \left((b_1 \otimes b_2 \otimes \cdots \otimes b_n) \otimes \rho\right)=(a_1 b_{\sigma^{-1}(1)}\otimes a_2b_{\sigma^{-1}(2)}\otimes \cdots \otimes a_nb_{\sigma^{-1}(n)}) \otimes (\sigma \rho)
\!\!\!\!\!\!\!\!\!\!((1-t)\times_{p,\bar{\alpha}}f+_{p,\bar{\alpha}}t\times_{p,\bar{\alpha}}g) (z_1,\cdots, z_n) =\sup_{0<\lambda<1}\left( \sup_{z_i=M_{p,\alpha_i}^{(t,\lambda)}(x_i,y_i), 1\leq i\leq n}M_{p,\alpha_{n+1}}^{(t,\lambda)}(f(x_1,\cdots, x_n), g(y_1,\cdots, y_n))\right).
\Delta E_{\rm up}= \frac{1}{T^{(0)}}\frac{4e^{S^{(0)}}\tan^2 \frac{1}{2}(\phi_\alpha + \phi_\gamma) + 4 e^{S^{(0)}}}{4 e^{2S^{(0)}}\tan^2 \frac{1}{2}(\phi_\alpha + \phi_\gamma) - 1},
(q_{j}f)(\mathbf{x}):=\langle f(\mathbf{y}), q_{j}(\mathbf{x},\mathbf{y})\rangle=\int_{\mathbb R^{n}}f(\mathbf{y})q_{j}(\mathbf{x},\mathbf{y})d\mathbf{y}, \mathbf{x}\in \mathbb R^{n}.
\vec{v}( \vec{x}) \equiv \epsilon^{i j}\, \partial_{i}\hat{n}\times \partial_{j}\hat{n}\quad , \quad q ( \vec{x}) \equiv \epsilon^{i j}\, \epsilon^{a b c}\, \hat{n}^{a}\, \partial_{i}\hat{n}^{b}\, \partial_{j}\hat{n}^{c}
\sum_{n=2^{k-1}}^{2^k-1}1_{[-B,B]}\left( t + \sum_{i=0}^{n-1}g(T^i x) \right)\ge\beta \tau \sum_{n=0}^{2^k-1}1_{[-B,B]}\left( t + \sum_{i=0}^{n-1}g(T^i x) \right)
\textrm{II}_n\leq[\frac{(1+\lambda M^2)}{\lambda E\rho^2}]^n\frac{E\rho^2}{(E\rho)^2}P(\tau\geq n)=[\frac{(1+\lambda M^2)}{d\lambda E\rho^2}]^n\frac{E\rho^2}{(E\rho)^2}.
V ( r , u ) = - 2 ( 8 \pi^{2}\alpha^{\prime})^{- 1 / 2}\int_{0}^{\infty}d l e^{- r^{2}l / 2 \pi \alpha^{\prime}}{l}^{1 / 2}{\frac{\mathrm{t a n h}( u ) e^{- u^{2}l / \pi}}{\mathrm{S i n}( u l )}}[ 2 2 + 2 \mathrm{C o s}( 2 u l ) ] \quad .
S = \int d^{11}x{\cal E}\lbrace \frac{1}{4}R({\cal E}) - \frac{1}{48}{\cal F}^{2}+ \frac{2}{(12)^{4}}\epsilon^{x_{1}...x_{11}}{\cal F}_{x_{1}...x_{4}}{\cal F}_{x_{5}...x_{8}}{\cal A}_{x_{9}...x_{11}}\rbrace
\int |\nabla_{z_i}u_i|^2 \frac{|\Psi|^2}{|u_i|^2}\leqslant \int |\nabla_{z_i}u_i|^2 |\Psi_i|= \|\nabla u_i\|_{L^2}^2 \|\Psi_i\|_{L^2}^2 \leqslant \frac{1}{1-C\ell^2}\|\nabla u_i\|_{L^2}^2 \|\Psi\|_{L^2}^2.
g_t(\hat{x}_1+z,\hat{x}_1+z+t^{-1/d}(\hat{x}_2-\hat{x}_1+y_2),\hdots,\hat{x}_1+z+t^{-1/d}(\hat{x}_{m+1}-\hat{x}_1+y_{m+1})) & =g_1(\hat{x}_1,\hat{x}_2+y_2,\hdots,\hat{x}_{m+1}+y_{m+1})
\int_0^{2\pi}\frac{d(e^{\pm i N \chi}e^{\pm i N'(\cos^{-1}(p)\pm \cos^{-1}(q))})}{\pm i N \mp i N'(\frac{1}{\sqrt{1-p^2}}\frac{dp}{d\chi}\pm\frac{1}{\sqrt{1-q^2}}\frac{dq}{d\chi})}F(N,N',\chi)
S_{B u l k}= \frac{1}{4 \pi}\int d^{2}z \left( \frac{2}{\alpha^{\prime}}\partial X^{\mu}\partial X_{\mu}+ \psi^{\mu}\overline{{{\partial}}}\psi_{\mu}+ \tilde{\psi}^{\mu}\partial \tilde{\psi}_{\mu}\right) \ .
\sum_{h \in \mathcal{M}_m}\chi_f(h) = \sum_{\substack{h \in \mathcal{M}_m (h,f_1)=1}}\chi_D(h) = \sum_{B | f_1}\mu(B) \sum_{\substack{h \in \mathcal{M}_m B|h}}\chi_D(h) = \sum_{B | f_1}\mu(B) \chi_D(B) \sum_{h_1 \in \mathcal{M}_{m - d(B)}}\chi_D(h_1) \ll \sqrt{|f|},
\mathcal{L}= \int_0^D \frac{1}{2}\log\left(1+P(t)\right)dt -\int_0^D \lambda(t) \left(\int_0^{t}a e^{b \tau}P(\tau) d \tau - T_{\delta}e^{bt}\right) dt -\int_0^D \beta(t)\left( \int_{0}^{t}P(\tau) d\tau - \sum_{i=0}^{h(t)}E_i \right) dt
|\varphi(z) - \varphi(z')| = |\varphi(f_a(\xi)) - \varphi(f_{a'}(\xi'))| = |\varphi(f_a(|\xi|)) - \varphi(f_{a}(e^{-i\theta}|\xi'|))| = |\varphi(f_a(1)) - \varphi(f_{a}(e^{-i\theta})| = |\varphi(f_a(1)) - \varphi(f_{e^{i\theta}a}(1)| < \epsilon,
\Phi_{1,0}(\theta)= \Phi_{2,\bar{0}}(\theta)=\frac{\sin\frac{\pi}{3}}{\cosh \theta -\cos\frac{\pi}{3}}, \Phi_{1,\bar{0}}(\theta)= \Phi_{2,0}(\theta)=\frac{\sin\frac{\pi}{3}}{\cosh \theta +\cos\frac{\pi}{3}}.
{\bf D}^{\alpha}= \partial^{\alpha}+ i g [{\bf A}^{\alpha}, \cdot ] \quad \mathrm{{a n d}}\quad{\bf F}^{\alpha \beta}= \partial^{\alpha}{\bf A}^{\beta}- \partial^{\beta}{\bf A}^{\alpha}+ i g [{\bf A}^{\alpha},{\bf A}^{\beta}] .
T^{\tilde{D}}(u, v, w) - T^{D^{(1)}}(u, v, w) = & - \frac{1}{8}\sum_{(u,v,w)\, \mathrm{cyclic}}\left( \eta (\mathcal J v, u, w) + \eta ( w ,u,\mathcal J v) -\eta (v,u,\mathcal Jw) -\eta (\mathcal Jw,u,v)\right) = - \frac{1}{4}\sum_{(u,v,w)\, \mathrm{cyclic}}\left( \eta (\mathcal J u, w, v) + \eta ( u ,v,\mathcal J w) \right) ,
F_{d,\ell}(z;\tau) =\sum_{a\geq0}\mathcal{D}_z^{2a}\left(\frac{\zeta^d}{1-\zeta^\ell}\right)\frac{\left(2\pi i\tau\right)^a}{a!}, G_{d,\ell}(z;\tau) =2i\sum_{a\geq 0}\mathcal{D}_z^{2a}\left(\frac{\sin\left(2\pi dz\right)}{1-\zeta^\ell}\right)\frac{\left(2\pi i\tau\right)^a}{a!}.
\int_{B_{1}(x_{0},\widetilde{g})}|F_{A}|^{2}_{\widetilde{g}}dV_{\widetilde{g}}=(\sqrt{b})^{n-4}\int_{B_{\frac{1}{\sqrt{b}}}(x_{0})}|F_{A}|^{2}dV_{g}&\leq(\frac{1}{2})^{4-n}e^{\frac{a}{2}}\int_{B_{\frac{1}{2}}(x_{0})}|F_{A}|^{2}dV_{g}&\leq\varepsilon2^{n-4}e^{\frac{a}{2}}
\begin{aligned}\psi^{+}_{1}\left(\mathfrak{a}^{*}\right)=\frac{\sqrt{2}}{2}\Big(j_{1}^{*}(\mathfrak{a}^{*}|_{Z_{1}}) +j_{2}^{*}(\mathfrak{a}^{*}|_{Z'_{1}})\Big)\otimes 1_{\mathbb{C}^{+}}.\end{aligned}
\Gamma^{(2)}_{aa}(k) = \coth{{\beta k^0}\over 2}\Bigl[ \Gamma^{(2)}_{ar}(k)-\Gamma^{(2)}_{ra}(k)\Bigr]\, ,\qquad \Gamma^{(2)}_{ar}(k) = \Gamma^{(2)*}_{ra}(k)\, .
\left\langle \mathcal{C}^{( g )}( p ) \right\rangle = \left. \frac{\mathcal{C}^{( g )}\left( \frac{1}{i}\frac{\delta}{\delta J}\right) W_{0}( J )}{W_{0}( 0 )}\right\vert_{J = 0}= \bar{p}_{3}
\Delta_{t,h}(x) =\int_0^x \df_{t+h}^\star (\psi_{t+h}\nu) - \int_0^x \df_{t}^\star (\psi_{t}\nu) &= \underbrace{\int_0^x \df_{t+h}^\star (\psi_{t+h}\nu) - \int_0^x \df_{t+h}^\star (\psi_{t}\nu)}_{\Delta_1}&+\underbrace{\int_0^x \df_{t+h}^\star (\psi_{t}\nu) - \int_0^x \df_{t}^\star (\psi_{t}\nu)}_{\Delta_2}.
\tilde{p}_{n,\phi}^{d,i+1}=\frac{1}{2}\left({p}_{n,\phi}^{d,i}+ \lambda_{n,\phi}^{d,i}+ d_{n,\phi}+\sum_{m \in \mathcal{M}_{n,\phi}}e_m^i + \mu_{n,\phi}^i\right).
M=\frac{i}{\sqrt{Z}}\int d^{4}x_{1}e^{ip_{1}\cdot x_{1}}[(i{\partial/}_{x_{1}}+m)u_{p_{1}}^{s_{1}}]_{\alpha_{1}}\langle\beta out|\overline{\Psi}_{\alpha_{1}}(x_{1})|\alpha^{\prime}in\rangle.
\psi:C^{n-1}_c(G,V)\to C^n_c(G,V)^G, \psi(c)(x_0,\ldots,x_n) := x_0\cdot c(x_0^{-1}x_1,x_1^{-1}x_2,\ldots,x_{n-1}^{-1}x_n).
\int_{S^{n-1}}(-L_K z) z dV_K = - \frac{1}{n}\int_{S^{n-1}}z Q^{i,j}(h^2 z_i)_j d\theta = \frac{1}{n}\int_{S^{n-1}}Q^{i,j}h^2 z_i z_j d\theta.
P\check{X}_{i}:=e^{-v_{i}}P, P\check{T}_{i}:=P.s_{i}+\frac{(\alpha e^{v_{i}}+\beta)(\gamma e^{v_{i+1}}+\delta)}{e^{v_{i}}-e^{v_{i+1}}}\left(P-P.s_{i}\right),
u^* (x_0) = \lim_{t \to T}u(x_0, t) = (T- t_0 (x_0))^{- \frac{1}{p-1}}\lim_{\tau \to 1}U (x_0, 0, \tau) \sim (T- t_0 (x_0))^{- \frac{1}{p-1}}\left(\frac{(p - 1)^2}{4 p}K_0^2 \right)^{- \frac{1}{p-1}},
\langle\Upsilon_0(z_0)\cdots\Upsilon_n(z_n)\rangle_{\Sigma}^{\Psi}=\left\langle\Upsilon_0(z_0)\cdots\Upsilon_n(z_n)\exp\frac{1}{\pi}\int_{\Sigma}\Psi(z)\,d^2\!z\right\rangle_{\Sigma}.
\Phi_{x_0, \alpha}= \Bigl\{\varphi(t) \in C_{0T}([0,T], \mathbb{R}^d) \, \Bigl \vert \,\varphi(0) = x_0\,\, \,\, S_{0T}^{\mathcal{K}}(\varphi(t)) < \alpha \Bigl\}.
A'_1\cap B_2\cap H_{1,2}=\{e_1,\dots,e_{d/2}\},A'_2\cap B_1\cap H_{1,4}=\{f_1,\dots,f_{d/2}\}, A'_1\cap B_2\cap H_{3,4}=\{g_1,\dots,g_{d/2}\},A'_2\cap B_1\cap H_{2,3}=\{h_1,\dots,h_{d/2}\}.
rV_r(x) = \begin{cases}\mu(x_r^\ast)x_r^\ast+r(x-x_r^\ast), x\geq x_r^\ast, \mu(x_r^\ast)x_r^\ast-r\int_{x}^{x_r^\ast}\frac{\psi_r'(t)}{\psi_r'(x_r^\ast)}dt, x<x_r^\ast.\end{cases}
\dot{G}_{\pm}+ \frac{i}{2}\phi^{\pm}D_{\pm}^{2}G_{\pm}+ \frac{i}{4}D_{\pm}\phi^{\pm}D_{\pm}G_{\pm}+ \frac{3}{4}i D_{\pm}^{2}\phi^{\pm}G_{\pm}= \frac{\kappa_{0}}{2}D_{\pm}^{5}\phi^{\pm}.
J ( X_{\tau}, \alpha , \beta ) = \int_{X_{\tau}}d^{m + n}z \prod_{i = 1}^{m}z_{i}^{\alpha}( 1 - z_{i})^{\beta}\prod_{\hat{i}= \hat{1}}^{\hat{n}}z_{\hat{i}}^{\alpha^{\prime}}( 1 - z_{\hat{i}})^{\beta^{\prime}}\left[ \prod z_{i j}^{2 \rho}\prod z_{\hat{i}\hat{j}}^{2 \rho^{\prime}}\prod z_{i \hat{j}}^{2 \sigma}\right]_{\tau}.
\int_I \rho_3^2 |y_{xx}|^2 dx \leq C \left(\int_I \rho^2 |G|^2 dx + \int_\mathcal{O}\rho_1^2 |f|^2 dx + \int_I \rho_3^2 |y_t|^2dx \right. &+ \left. \int_I \rho_0^2|y|^2 dx + \int_I \rho_2^2|y_x|^2 dx + \sum_{i=1}^2 \int_I \rho_0^2 |p^i|^2 dx \right).
H_{\nu+m}^{(1)}(iy)=\left(\frac{2i}{y}\right)^{m}r_{m,\nu}\left(\frac{y^2}{4}\right)H_{\nu}^{(1)}(iy)+\left(\frac{2i}{y}\right)^{m-1}s_{m,\nu}\left(\frac{y^2}{4}\right)H_{\nu+1}^{(1)}(iy).
L=\sum_{i=1}^{n}m_{i}(r_{i}-R)\times\frac{d}{dt}(r_{i}-R)+(\sum_{i=1}^{n}m_{i})[R\times\frac{d}{dt}(r_{i}-R)+(r_{i}-R)\times v]+(\sum_{i=1}^{n}m_{i})R\times v
[ \tau \psi_{\beta}]_{2}= \left( 1 + \sum_{\alpha \in \Phi}( \alpha \beta / 2 k - ( \alpha \beta )^{2}/ 2 k^{2}+ N_{\alpha , \beta}N_{- \alpha , \alpha + \beta}) \right) \psi_{\beta}, \; ( \beta \in \Phi ) ,
\sum_{\substack{h_0,h_1,h_2\ge0 h_0+h_1+h_2=h h_0+2h_1+h_2=m}}\binom h{h_0,h_1,h_2}\sum_{n \leq x}\big( \log 2 \cdot F_{\omega_0}(n) \big)^{h_0}S_1^{h_1}S_2^{h_2}= \sum_{\substack{\beta\le B_h k_{h\beta}=m}}r_{h\beta}\prod_{j=1}^{\tilde k_{h\beta}}\mu(\omega_{q_{w(h,\beta, j)}}) \sum_{n \leq x}\prod_{i=1}^{k_{h\beta}}F_{\omega_{q_{v(h,\beta, i)}}}(n).
D_{\phi\phi\phi}^3\Phi(0, \mu^*_{k})[\phi^*_{k},\phi^*_{k},\phi^*_{k}]=-\frac{3}{2}\phi^*_{k}\left( \frac{1}{\mu^*_{k}}+\frac{1}{2(\mu^*_{k}-(2k)^{-r})}\right).
\left( T_\lambda^{\vec{v}}\right)_{i,j}=\displaystyle \beta_{-i+j}+ \sum_{k = j}^{n-1}\beta_k \sum_{r =0}^{\min \{k-j, n-i\}}\alpha_{n-j-r}\sum_{\mathclap{\substack{p_1\dots p_m m \leq k-j - r p_s \geq 1 \sum_{s=1}^m p_s = k-j - r}}}(-1)^{m+1}\alpha_{n-p_1}\dots \alpha_{n-p_m}.
\displaystyle\int_{\mathbb{R}^n}(- \Delta)^{\frac{\gamma}{2}}v(- \Delta)^{\frac{\gamma}{2}}wdx -(2^*_\gamma - 1)\displaystyle\int_{\mathbb{R}^n}U^{2^*_\gamma - 2}vwdx = 0,w\in F,
[\Theta_{i},\Theta_{j}]=0=[\bar\Theta_{i},\bar\Theta_{j}], \Theta_{i}\bar\Theta_{i}=\prod_{j \not= i}Q_{i,j}Q_{j,i}=\bar\Theta_{i}\Theta_{i},\prod_{i=1}^{n}\Theta_{i}=\prod_{1 \le i \not= j \le n}Q_{i,j}Q_{j,i}= \prod_{i=1}^{n}\bar\Theta_{i}.
(1-\dfrac{\Delta t}{2}\{(1-r_1)\delta_{\alpha_1 x}^{I^-}+r_1\delta_{\alpha_1 x}^{I^+}\}) I_{i,j}^{*}=(1+\dfrac{\Delta t}{2}\{(1-r_2)\delta_{\alpha_2 y}^{I^-}+r_2\delta_{\alpha_2 y}^{I^+}\})I_{i,j}^{n}+\dfrac{\Delta t}{2}\left \{\beta S_{i,j}^{n+1/2}V_{i,j}^{n+1/2}-(\mu+\gamma) I_{i,j}^{n+1/2}\right\}
\left(\mbox{LHS}_{\left(\ref{eq:u-7}\right)}=\right)\int_{0}^{1}\left(D_{F}^{*}h\right)\left(x\right)\varphi\left(x\right)dx=\int_{0}^{1}h\left(x\right)\varphi'\left(x\right)dx\left(=\mbox{RHS}_{\left(\ref{eq:u-7}\right)}\right)
\begin{array}{ll}16 = 2^4 \pm \sqrt{2}e_i, \, i = 1, \ldots, 8 224 = 2^4 \cdot 14 \frac{1}{\sqrt{2}}\sum_{i=1}^8 (\pm x_i) e_i, \; x \in \mathcal{H}_8 (x) = 4,\end{array}
1 = \sum_{l \in \bf{N}_{0}}\tilde{\epsilon}_{l}\frac{( ( l + 3 ) ! )^{2}}{( 2 l + 7 ) !}( v - 1 )^{l}_{2}F_{1}( l + 4 , l + 4 ; 2 l + 8 ; 1 - v )
\frac{|\mathbf{u}|^{N}}{N!}\Psi_{\mathbf{m}}^{(d)}(\mathbf{u}) = \sum_{\substack{|\mathbf{n}|-|\mathbf{m}|=N, \mathbf{n}\in \mathcal{P}}}\binom{\mathbf{n}}{\mathbf{m}}^{(d)}\Psi_{\mathbf{n}}^{(d)}(\mathbf{u}).
\mathbf{P}\left((\eta^N_n)_{n=1}^{N}=(x_n)_{n=1}^{N}\right)=\frac{1}{Z}\exp\left(-\sum_{k=0}^\infty\beta_kf_k\left((x_n)_{n=1}^{N}\right)\right)\mathbf{1}_{\{f_0\left((x_n)_{n=1}^{N}\right)<N/2\}},
\sum_{j=1}^{n}\frac{1}{\lambda_{j}}\left( \frac{z\mathbb{E}_{\alpha_{j},\beta_{j}}^{\prime}(z)}{\mathbb{E}_{\alpha_{j},\beta_{j}}(z)}\right) =\zeta (1-\delta )p(z)+\frac{(1-\delta )zp^{\prime}(z)}{\delta+(1-\delta )p(z)}+\sum_{j=1}^{n}\frac{1}{\lambda_{j}}-\zeta (1-\delta ).
\nabla g(x) \cdot S = \, n(\gamma(r) + \gamma(\tilde{r})) + r \gamma^{\prime}(r) (1-|S^{\perp}(\nabla r)|^2) &\, + \tilde{r}\gamma^{\prime}(\tilde{r}) \left(1-\left|S^{\perp}\left(i_x\left(\dfrac{\tilde{x}-a}{\tilde{r}}\right)\right)\right|^2\right) + \gamma(\tilde{r})\varepsilon_1(x,S) - \tilde{r}\gamma^{\prime}(\tilde{r})\varepsilon_2(x,S),
\begin{array}{c}{e_{1}e_{2}= e_{3}\ , \e_{2}e_{3}= e_{1}\ , \e_{3}e_{1}= e_{2}}{e_{3}e_{2}= - e_{1}\ , \e_{2}e_{1}= - e_{3}\ , \e_{1}e_{3}= - e_{2}\ .}\end{array}
\begin{aligned}D_t(u')+(S-\operatorname{id})\exp(u_{-1}-u) =\bold{E}(L), D_t(tu'-u)+(S-\operatorname{id})(t\exp(u_{-1}-u)) =t\bold{E}(L), D_t\left(\frac{(u')^2}{2}+\exp(u-u_1)\right)+(S-\operatorname{id})(u'\exp(u_{-1}-u)) =u'\bold{E}(L).\end{aligned}
u \left( t_{m n}^{( l )}\right) = \delta_{m , n}q^{2 ( m - k )}u \left( t_{k k}^{( l )}\right) , \quad v \left( t_{m n}^{( l )}\right) = \delta_{m , n}q^{2 ( k - m )}v \left( t_{k k}^{( l )}\right) .
\frac{\delta{\cal W}}{\delta\psi^\alpha}= \frac{\delta\Gamma}{\delta\psi^\alpha}- \frac{\delta\Gamma}{\delta\Sigma^m}\sigma^m_{,\alpha}(\varphi,\psi),\;\;\; \frac{\delta{\cal W}}{\delta\phi^*_A}= \frac{\delta\Gamma}{\delta\phi^*_A}\,,
< 0 | \hat{h}_{+ +}\omega \left( x_{1}\right) \cdots \omega \left( x_{N}\right) | 0 > ~ \sim ~ < \Delta | \left( L_{2}+ \lambda L_{1}^{2}\right) \omega \left( x_{1}\right) \cdots \omega \left( x_{N}\right) | 0 >
i \partial_t \Psi + \partial_{xx}\Psi + 2 |\Psi|^2 \Psi + \frac{1}{2}v^2 \Psi - \Re \Big( \Psi \big( 1 - 2 F(v, \overline{\Psi}) \big) \Big) \big( 1 - 2 F(v, \Psi) \big) = 0,
\sum_{i=1}^Ni^{1+2\alpha+2p}\xi_i^2 = \sum_{i=1}^Ni^{1+2\alpha+2p}\cdot i^{-2p}f_{0,i}^2 = \sum_{i=1}^Ni^{1+2\alpha-2\beta}f_{0,i}^2i^{2\beta}\leq N^{(1+2\alpha-2\beta)\vee 0}\|f_0\|^2_\beta.
K(L,1) =\sqrt{\pi}(n-2)!2^{n-3}\sum_{k=1}^\infty \left[ \frac{\Gamma(Lk+1/2)}{\Gamma(Lk+1)}\frac{\Gamma(2Lk-n+2)}{\Gamma(2Lk+1)}+ \frac{\Gamma(Lk-1/2)}{\Gamma(Lk)}\frac{\Gamma(2Lk-n)}{\Gamma(2Lk-1)}\right. & \left. -2 \frac{\Gamma(Lk+1/2)}{\Gamma(Lk+1)}\frac{\Gamma(2Lk-n+)}{\Gamma(2Lk)}\right]\ .
\tilde{t}'(v_0 \otimes v_0) = v_0 \otimes v_0 \tilde{t}'(v_1 \otimes v_1) = v_1 \otimes v_1 \tilde{t}'(v_0 \otimes v_1) = q^{-1}v_1 \otimes v_0 \tilde{t}'(v_1 \otimes v_0) = (1-q^{-2})v_1 \otimes v_0 + q^{-1}v_0 \otimes v_1.
lim_{\alpha\rightarrow0}u(r,t)=Re\{-\sum_{n=0}^{N}\frac{P_{n}^{\prime}}{4\mu}(R^{2}-r^{2})e^{in\omega t}\}=-\sum_{n=0}^{N}\frac{P_{n}^{\prime}}{4\mu}(R^{2}-r^{2})cos(n\omega t).
E_1(x)=\sum_{x\leq p\leq 2x}\frac{1}{p}\sum_{\substack{0< b< p gcd(m,p-1)=1}}e^{\frac{i2\pi b(\tau^{m}-v)}{p}}\cdot \frac{1}{p}\sum_{\substack{0\leq b< p gcd(m,p-1)=1}}e^{\frac{i2\pi b(\tau^{em}-v)}{p}}=0.
\sum_{k=0}^{n-1}T_k^{(2r)}\equiv \sum_{l=1}^{n-1}\frac{n^2(2n-3l)}{l^2}\binom{n-1}{l-1}^2\binom{2l-1}{l-1}\sum_{j=l}^{n-1}(2j+1)^{2r-1}(-1)^j\pmod{n^2}.
\sum_{i=1}^{n}\frac{1}{b_i}= \sum_{i=1}^{n-1}\frac{1}{b_i}+ \frac{1}{b_n}< \sum_{i=1}^{n-1}\frac{1}{a_i}+ \frac{1}{a_n}= \sum_{i=1}^{n}\frac{1}{a_i}.
T_i X_i = X_{i+1}T_i + 1 X_i T_i = T_i X_{i+1}+ 1 X_i X_j = X_j X_i T_i^2 =1 T_i T_j = T_j T_i |i-j|>1 T_i T_{i+1}T_i = T_{i+1}T_i T_{i+1}.
f_{T T T}= \frac{9 6 i}{\pi}\frac{( \Phi_{T}( T ) )^{2}}{\Phi ( T ) ( \Phi ( T ) - \Phi ( \frac{i}{\sqrt{2}}) )}\frac{\Phi^{\frac{1}{2}}( U )}{\Phi_{U}( U )}\frac{{( \Phi ( U ) - \Phi ( \frac{i}{\sqrt{2}}) )}^{\frac{1}{2}}}{( \Phi ( U ) - \Phi ( T ) )}.
\left( \frac{( 1 3 )}{( x_{1 3}^{2})^{\frac{D - 2}{2}}}\right)^{m - j - k}\left( \frac{( 2 3 )}{( x_{2 3}^{2})^{\frac{D - 2}{2}}}\right)^{n - j - k}\left[ \frac{( 1 3 )}{( x_{1 3}^{2})^{\frac{D - 2}{2}}}\ \frac{( 2 3^{-})}{( x_{2 3}^{2})^{\frac{D - 2}{2}}}- 1 \leftrightarrow 2 \right]^{j}\ .
\vert \Xi_{0 , p}[ \alpha , \beta ] \vert^{2}= \frac{\prod_{j = 3}^{p - 1}( - 1 )^{2 m_{\beta_{j}}}q^{4 i m_{\beta_{j}}\rho_{\beta_{j}}}}{{\psi}_{2}{}_{( \alpha_{1}, \alpha_{2}, \beta_{3})}\prod_{j = 3}^{p - 2}{\psi}_{2}{}_{( \beta_{j}, \alpha_{j}, \beta_{j + 1})}{\psi}_{2}{}_{( \beta_{p - 1}, \alpha_{p - 1}, \alpha_{p})}}\Xi_{0 , p}[ \alpha , \beta ]^{2}
G\in\mathcal{C}\Longleftrightarrow \alpha(G)=\frac{3}{4}\Longleftrightarrow \frac{1+n_2(G)+n_4(G)}{2^n}=\frac{3}{4}\Longleftrightarrow \frac{2^n-n_4(G)}{2^n}=\frac{3}{4}\Longleftrightarrow n_4(G)=2^{n-2}\Longleftrightarrow n_2(G)=2^{n-1}-1,
\dot{V}_1\mid_{(\ref{dyn:err:im})}=2\sum_{i=1}^ne_{vi}[\Upsilon_{2i}-\theta_i{\rm sgn}(e_{vi})]+2\sum_{i=1}^n{\bar \theta_i}\dot{\theta}_i \leq 2\sum_{i=1}^ne_{vi}[\Upsilon_{2i}-\Theta{\rm sgn}(e_{vi})]
f_{i,j+k}(L,L) + f_{j,i+k}(M,L) + f_{k,i+j}(N,L) - 2f_n(L) \leq \dim U_1 + j+\dim U_1) + k + \dim U_1) - 2(\dim U_1) = j + k + \dim U_1 = f_{i,j,k}(L,M,N).
[ d \Phi ] [ d \Phi^{\dagger}] [ d \zeta ] [ d \zeta^{\dagger}] = \Pi_{x , a}\frac{d \Phi_{a}( x ) d \Phi_{a}^{\dagger}( x )}{2 \pi i}d \zeta_{a}( x ) d \zeta_{a}^{\dagger}( x ) \, .
\int_{M}{\frac{\omega^{N}}{N !}}\exp \left( - \rho J_{X}\right) = \sum_{m_{c}}{\frac{\exp \left\{- \rho J_{X}\left( m_{c}\right) \right\}}{\rho^{N}D \left( m_{c}\right)}}\ , \d H \left( m_{c}\right) = 0 \ , \m_{c}\in M \ ,
\delta \tau_{j,k}:= B_{j,k}\tau_k+t_{j,k}-\tau_{j}-\frac{B_{j,k}A_{k}+A_{j}}{2}\left(y_k-y_{j}\right), \left|\delta \tau_{j,k}\right|< 2 c^{\tau}_JC_{|A|}2^d \sqrt{\epsilon_J}\quad.
h\wedge_L (g_1g_2) = (h_0 h') \wedge_L (g_1 h_0 g_2') = (h_0 h') \wedge_L (h_0 g_1 g_2') &= h_0 (h' \wedge_L (g_1 g_2')) = h_0 =h\wedge_L g_2.
- i \left( \frac{d}{\phi d \phi}\right) f_{3}^{( 6 )}\phi^{i}\phi^{j}\phi^{k}\phi^{l}v^{p}( \theta \gamma^{s i}\theta ) ( \theta \gamma^{p j}\theta ) ( \theta \gamma^{k n}\theta ) ( \theta \gamma^{l n}\theta ) ,
^{}_i = \frac{p^{}_i|(\mathbf{v}^{}_i)^{T}\mathbf{g}_i|^2}{\sum_{j \neq i}p^{}_j|(\mathbf{v}^{}_i)^{T}\mathbf{g}_j|^2 + \sigma^2\|\mathbf{v}^{}_i\|^2}, ~~ i =1,\cdots, K.
D^{\mu}\varphi_{a}= \partial^{\mu}\varphi_{a}+ g W_{a b}A^{b \mu}\varphi_{a}, \; \bar{F}_{\mu \nu}^{a}= \partial_{\left[ \mu \right.}A_{\left. \nu \right]}^{a}+ g \frac{\delta W_{b c}}{\delta \varphi_{a}}A_{\mu}^{b}A_{\nu}^{c}.
2\left \langle Tx,x \right \rangle =Tr(T(x\otimes x)+(x\otimes x)T) &=Tr(\psi(T) U(x\otimes x)U^{\ast}+U(x\otimes x)U^{\ast}\psi(T)) &=2\left \langle U^{\ast}\psi(T)U,x \right \rangle.
\phi_{S S}/ 2 = \tan^{- 1}\left( u \frac{\sinh ( 8 m \gamma x )}{\cosh ( 8 m \gamma u t )}\right) , \, \, \, \phi_{S \overline{{{S}}}}/ 2 = \tan^{- 1}\left( \frac{1}{u}\frac{\sinh ( 8 m \gamma u t )}{\cosh ( 8 m \gamma x )}\right) ,
\inf\{s\geq t:X_s=0\}\,\mathbb{Q}^\delta_{x^2}&~~\stackrel{\mathcal{L}}{=}\inf\{s\geq t:X_s+X_s'=0\}\,\mathbb{Q}_0^\delta(X)\times\mathbb{Q}_{x^2}^0(X') &~~\stackrel{\mathcal{L}}{=}\inf\big\{s\geq\max\{t,\,\tau_0'\}:X_s=0\big\}\,\mathbb{Q}_0^\delta(X)\times\mathbb{Q}_{x^2}^0(X').
\mathcal{L}=\mathcal{L}\left( Q,\dot{Q}\right) =\frac{1}{2}\left[\begin{array}[c]{l}\dot{Q}Q\end{array}\right]^{\mathrm{T}}M_{\mathrm{L}}\left[\begin{array}[c]{l}\dot{Q}Q\end{array}\right] ,M_{\mathrm{L}}=\left[\begin{array}[c]{ll}\alpha \theta \theta^{\mathrm{T}}-\eta\end{array}\right] ,
\frac{\partial \widehat{b}}{\partial \overline{k}}= \frac{\partial\widehat{b}}{\partial \overline{\alpha}}+\widehat{a}\widehat{b}_0, \frac{\partial \widehat{a}}{\partial{k}}= \widehat{b}\overline{\widehat{b}_0},~~~\widehat{b}_0=\widehat{b}(0,k,t), |\alpha|\leq A, ~~k\in \mathbb C,~~ 0\leq t\leq T.
\Phi (F) (Z) := \underset{t \rightarrow \infty}\lim F \left( \begin{smallmatrix}Z 0 0 i t \end{smallmatrix}\right) = \sum_{T \in \Lambda_{n-1}}a_F(\begin{psmallmatrix}T 0 0 0 \end{psmallmatrix})e(TZ).
E\left(\prod_{i=1}^{N_1}\prod_{j \in \Lambda_i}X_j^\lambda\right)\leq \ \prod_{i=1}^{N_1}\left(E\left(\prod_{j \in \Lambda_i}X_j^{\lambda N_1}\right)\right)^{\frac{1}{N_1}}= \left(\prod_{i=1}^{N_1}E\left(\prod_{j \in \Lambda_i}X_j^{t_0}\right)\right)^{\frac{1}{N_1}}.
\chi + k - 2 = -\frac 12 \sum_{j \neq l}\mathbb{E}[\mathcal{D}^{jl}(0) \mathcal{D}^{jl}(-J,-1)] = -\frac{k(k-1)}2 \mathbb{E}[\mathcal{D}^{jl}(0) \mathcal{D}^{jl}(-J,-1)]
W(X,Y)Z =R(X,Y)Z+\frac{1}{n(n-2)}\{Y[(n-1)Ric(X,Z)+\stackrel{*}{Ric}(X,Z)]- -X[(n-1)Ric(Y,Z)+\stackrel{*}{Ric}(Y,Z)]+[(n-1)\stackrel{*}{Ric}^\sharp (Y)+Ric^\sharp (Y)]g(X,Z)- -[(n-1)\stackrel{*}{Ric}^\sharp (X)+Ric^\sharp (X)]g(Y,Z)\}+\frac{\sigma}{(n-1)(n-2)}[Xg(Y,Z)-Yg(X,Z)]
d S^{2}={H_{6}}^{- 1 / 2}[ d u d v - \mu^{2}\sum_{i = 1}^{4}{z^{i}}^{2}d u^{2}+ \sum_{i = 1}^{4}d z^{i}d z^{i}+ d \tilde{x}^{2}] +{H_{6}}^{1 / 2}( d r^{2}+ r^{2}d{\Omega_{2}}^{2}) ,
m = 2 \pi R_{1}\times \tilde{R}_{7}\times{\frac{| a_{D}( u ) |}{\alpha}}\times{\frac{1}{g l_{s}^{4}}}={\frac{R_{1}}{R_{1 1}l_{s}^{2}}}{\frac{2 \pi | a_{D}( u ) |}{\alpha}}={\frac{1}{g_{s}^{\prime}l_{s}^{' 3}}}{\frac{2 \pi \tilde{R}_{7}| a ( u ) |}{\alpha}},
h \phi_{n}( \varphi ) = \lambda_{n}^{2}\phi_{n}( \varphi ) \Rightarrow{\tilde{h}}[ \frac{1}{\lambda_{n}}a \phi_{n}( \varphi ) ] = \lambda_{n}^{2}[ \frac{1}{\lambda_{n}}a \phi_{n}( \varphi ) ]
\tilde{H}_{v}- \tilde{E}_{v}= 2 \int_{\cal B}d \sigma_{\nu}\zeta_{\mu}\nabla_{\rho}\nabla_{\sigma}\left({\frac{\partial{\tilde{L}}_{v}}{\partial R_{\mu \sigma \rho \nu}}}+{\frac{\partial{\tilde{L}}_{v}}{\partial R_{\nu \sigma \rho \mu}}}\right) .
\Delta_{\nu}(x)>\frac{1}{\sqrt{\pi}}\frac{\Gamma\left(\nu+\frac{1}{2}\right)x^{2\nu}}{\Gamma(\nu+2)\Gamma(2\nu+1)}=\frac{\left(\frac{1}{4}x^{2}\right)^{\nu}}{\Gamma(\nu+1)\Gamma(\nu+2)}
\begin{cases}2a_1a_2-(x_1+x_{t_1})a_1-(x_1+x_{t_1})a_2+2x_1x_{t_1}=0, 2a_1a_2-(x_s+x_{t_2})a_1-(x_s+x_{t_2})a_2+2x_sx_{t_2}=0,\end{cases}
\sum_{k=0}^{n-1}\frac{\binom nk\binom{2k}k}{(2k+1)(-4)^k}=-\frac{\binom{2n}n}{(2n+1)(-4)^n}+\frac{4^n}{(2n+1)\binom{2n}n}\sum_{k=0}^n\frac{\binom{2k}k^2}{16^k}
\sum_{\lambda \in \mathcal{P}}t^{h_{\alpha,\beta}(\lambda)}q^{|\lambda|}=\prod_{i\geq 1 \atop{(\alpha+\beta) \nmid 1}}\frac{1}{(1-q^{i})}\prod_{i \geq 1}\frac{1}{(1-tq^{(\alpha+\beta)i})}.
\begin{aligned}\tilde{\mathcal{D}}_s = \left\{\left. Z_s = (X_s,V_s) \in\mathbb{R}^{ds}\times\mathbb{R}^{ds}\right| \; \forall 1\leq i < j \leq m-1, |x_i-x_j| > \varepsilon \right\}\end{aligned}
\bar{x}^{\mu}: = X^{\mu}\left( \pi / 2 \right) = x_{0}^{\mu}- \sum_{e}x_{e}^{\mu}w_{e}, \quad \langle \bar{x}| = \langle x_{0}| \exp \left( i \hat{p}\cdot \sum_{e}x_{e}w_{e}\right) .
R_{l}(r) =\frac{u_{l}(r)}{r^{( D_{0}- 1)/2}}\stackrel{(r \rightarrow 0 )}{\sim}\mbox{\boldmath\large $\left\{\right.$}\! \! \!r^{\alpha_{+,l}}\mbox{\boldmath\large $,$}\! r^{\alpha_{-,l}}\mbox{\boldmath\large $\left. \right\}$}\! \!\; ,
\mathbf A\tilde{\mathbf V}^{n+4}= 4\tilde{\mathbf V}^{n+3}- 3\tilde{\mathbf V}^{n+2}+ \frac{4}{3}\tilde{\mathbf V}^{n+1}- \frac{1}{4}\tilde{\mathbf V}^{n}+ k\mathbf b^{n+4}+ k\mathbf q(\tilde{\mathbf V}^{n+4}),
\prod_{j=1}^{\theta N - 1}{\big(z_rq^{\theta r - \theta + \tau_r^+ - \tau_r^-}- q^{\theta(N-1) - j}\big)}= T_{q, z_r}^{\theta r - \theta + \tau_r^+ - \tau_r^-}\prod_{j=1}^{\theta N - 1}{\big(z_r - q^{\theta(N - 1) - j}\big)}.
\phi = \frac{1}{4}P^2 - \frac{1}{2}(m_{1}^2 + m_{2}^2) +\frac{(m_{1}^2 - m_{2}^2)^2}{4P^2}+ q_\bot^2-\frac{\alpha \bigl(P^2 - (m_1 - \kappa m_2)^2\bigr)}{\epsilon P\cdot r}= 0,
\times\Bigg\{\sum_{\sigma\in Z_N}\frac{w(\omega u_2'u_3,\sigma+j_2+j_3)w(qu_2'',\sigma)s(\sigma,j_1)}{w(q^{-1}u_2',\sigma+j_2)w(u_2''u_3,\sigma+i_3)}\Bigg\}_0
J(t) = J^\ast+ MC\int_{T(\epsilon)}^{t-\tau}f'(x(u))\,du +M\int_{t-\tau}^t \int_0^{t-u}\tilde{M}(v)\,dv f'(x(u))\,du &\leq J^\ast+ MC\int_{T(\epsilon)}^{t-\tau}f'(x(u))\,du +M\int_{t-\tau}^t \int_0^{t-u}\tilde{M}(v)\,dv f'(x(u))\,du.
\Delta(L(mn-1, n), \omega) = -\frac{d(u)d(v)}{4(-1)^{\sigma_+(L)}}(u^2 v^{-2n}+ u^{-2}v^{2n})(u^{2m}v^{-2}+ u^{-2m}v^2) &= -\frac{d(u)d(v)}{4(-1)^{\sigma_+(L)}}((u v^{-n})^2 + (u^{-1}v^{n})^2)((u^{m}v^{-1})^{2}+ (u^{-m}v)^2) &= (-1)^{\sigma_+(L)+1}d(u)d(v).
\frac{\omega(\theta_j(\zeta_1),\gamma)}{\omega(\theta_j(\zeta_2),\gamma)}=\frac{1}{\beta_j(\zeta_1,\zeta_2)}\left(\frac{1-\bar\delta_j\zeta_2}{1-\bar\delta_j\zeta_1}\right)\frac{\omega(\zeta_1,\gamma)}{\omega(\zeta_2,\gamma)}.
[M^{(1)}]_{1,2}=\left<\psi_0,L_1\phi_1\right>_{L^2_{\rm per}([0,T])}\quad\textrm{and}[M^{(1)}]_{3,2}=\left<\psi_2,L_1\phi_1\right>_{L^2_{\rm per}([0,T])},
\begin{aligned}\sum\limits_{\ell=1}^n \Big( \frac{\partial B_{ij}(y)}{\partial y_\ell}B_{\ell k}(y) + \frac{\partial B_{jk}(y)}{\partial y_\ell}B_{\ell i}(y) + \frac{\partial B_{ki}(y)}{\partial y_\ell}B_{\ell j}(y) \Big) = 0 \textrm{for}i,j,k=1,...,n.\end{aligned}