harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
S_{\mathrm{e f f}}( A ) \; = \; \sigma_{H}\, S_{C S}( A ) \; = \; \frac{\sigma_{H}}{2}\int_{\Omega}d^{3}\xi \; \varepsilon^{\mu \nu \lambda}\, A_{\mu}( \xi ) \, \partial_{\nu}\, A_{\lambda}( \xi ) ~ ,
(f^1\otimes id) (s\gamma_i \otimes p_i) = (f^1\otimes id) (\gamma_i \otimes p_i s) f^1(s\gamma_i) \otimes p_i = f^1(\gamma_i) \otimes p_i s f^1(s\gamma_i) \otimes p_i = (f^1(\gamma_i) \otimes p_i) s.
\xi_{i,j}\left(\tau\right)= \frac{\sqrt{2}}{2}\left(-\tau_{i,2j-1}+\tau_{i,2j}\right) \mbox{and}\eta_{i,j}\left(\tau\right)=\frac{\sqrt{2}}{2}\left(\tau_{i,2j-1}+\tau_{i,2j}\right) \mbox{for}i,j\leq n.
n^d_t \le C_{d,T}(t)+2000\big( C_2^4+3( C_8+C_{5}^2) \big)\eta^4 d^4t^3\sum_{k=0}^{t-1}C_{d,T}(k) \exp\big(2000\eta^4 d^4\big( C_2^4 t^3+3( C_8+C_{5}^2) \big)t^3(t-k-1)\big).
r_k(\mathbf{x},\mathbf{y}):=\begin{bmatrix}\mathbf{x}\mathbf{y}\end{bmatrix}- \Pi_{D}(\begin{bmatrix}\mathbf{x}\mathbf{y}\end{bmatrix}- \begin{bmatrix}H\mathbf{x}+\mathbf{g}-A^T\mathbf{y}+\rho (\mathbf{x}-\mathbf{x}_k) A\mathbf{x}-\mathbf{b}+ \delta (\mathbf{y}-\mathbf{y}_k) \end{bmatrix}).
\varphi_{\tau(1)}(\tilde{\gamma}_1(0)) = x, \varphi_{\tau(i)}(\tilde{\gamma}_i(t_{i-1}))= \varphi_{\tau(i-1)}(\tilde{\gamma}_{i-1}(t_{i-1})), \varphi_{\tau(n)}(\tilde{\gamma}_n(1)) = y.
\mathfrak{R}_{\sigma_{3,6,2,0}}(q;\beta) = (1,6,11,16,{\bf 11})_{\beta}+q \beta^2 (10,20,35,34)_{\beta}+q^2 \beta^4 (5,14,{\bf 26})_{\beta}, \mathfrak{R}_{\sigma_{3,6,2,0}}(q;1) = (45,99,45)_{q}.
1= \alpha^{\frac{2n}{n-2}}c_{0}+\frac{n(n-6)}{(n-2)^{2}}\alpha^{\frac{4}{n-2}}\int \varphi_{a, \lambda}^{\frac{4}{n-2}}v^{2}+o_{\frac{1}{\lambda}}(1)+o(\Vert v \Vert^{2}),
L_1.\,c_1(X_1)^2 = (\mu^*L - E).(\mu^*c_1(X)-E)^2 &= L.\,c_1(X)^2 + \mu^*L.\,E^2 + 2\mu^*c_1(X).\,E^2 - E^3\quad({}\eqref{zero1}) &=L.\,c_1(X)^2 -L.C + 4(g_C-1)-\deg N_{C/X}.\quad({}\eqref{int3},\eqref{int4},\eqref{int1}).
\frac{\left(-1\right)^{\frac{5}{6}}x^{\frac{7}{2}}-\left(-1\right)^{\frac{1}{3}}x^{2}}{\left(x^{2}-x+1\right)^{\frac{1}{3}}\left(x+1\right)^{\frac{1}{3}}}-\frac{x^{6}+x\sqrt{-x}}{x^{3}+1}=0,
[XY]=[U_{X}U_{Y}][\begin{matrix}\Sigma_{X}&0\\ 0&\Sigma_{Y}\end{matrix}][\begin{matrix}V_{XX}&V_{XY}\\V_{YX}&V_{YY}\end{matrix}]^{*}=[U_{X}U_{Y}][\begin{matrix}\Sigma_{X}&0\\ 0&\Sigma_{Y}\end{matrix}][\begin{matrix}V_{XX}^{*}&V_{YX}^{*}\\V_{XY}^{*}&V_{YY}^{*}\end{matrix}]
|(f(z)-w)-(z-w)|=\frac{1}{2}|z|^{2}|f^{\prime\prime}(tz)|\le\frac{|z|^{2}}{1-t|z|}\le\frac{|z|^{2}}{1-|z|}=\frac{1}{6}<|z|-|w|\le|z-w|.
=\frac{r}{m}\Delta^k \left( \frac{r}{m}\right)^j + k \sum_{l=0}^k{k \choose l}(-1)^{k-l}\Big(l+\frac{r}{m}\Big)^j + k \sum_{l=0}^{k-1}{k -1\choose l}(-1)^{k-1-l}\Big(l+\frac{r}{m}\Big)^j &=\frac{r}{m}\Delta^k \left( \frac{r}{m}\right)^j + k \Delta^k \left( \frac{r}{m}\right)^j+ k \Delta^{k-1}\left( \frac{r}{m}\right)^j.
m_2(z)\equiv m_2^{(N)}(z):=\int \frac{1}{x-z}dF^{(N)}_{Y^*Y}(x)=\frac{1}{N}\sum_{i=1}^N (\mathcal G_2)_{ii}(z)=\frac{1}{N}\mathrm{Tr}\, \mathcal G_2(z).
U ( t , \vec{x}) = \left( P e^{- i g \int_{0}^{t}d t^{\prime}A_{0}( t^{\prime}, \vec{x})}\right) \left( P e^{i g \int_{0}^{\beta}d t^{\prime}A_{0}( t^{\prime}, \vec{x})}\right)^{{\frac{t}{\beta}}}
\check R_i(u)\check R_{i+1}(u+v)\check R_i(v) =\check R_{i+1}(v)\check R_i(u+v)\check R_{i+1}(u) i=1,\ldots,N-2 \check R_i(u)\check R_i(-u) =1 i=1,\ldots,N-1 \check R_i(u)\check R_j(v) =\check R_i(v)\check R_j(u) |i-j|>1
\begin{bmatrix}n+1 s \end{bmatrix}- \begin{bmatrix}n+1 s+1 \end{bmatrix}= J_s^{n+1}= J_{s-1}^n + J_s^n + J_{s-1}^n = \begin{bmatrix}n s-1 \end{bmatrix}- \begin{bmatrix}n s+2 \end{bmatrix}
- \log \left[ ( x )_{i \frac{\pi}{2}+ \theta}( x )_{i \frac{\pi}{2}- \theta}\right] \sim - 4 \sin \pi x \mathrm{e}^{- \left| \theta \right|}+ O \left( \mathrm{e}^{- 2 \left| \theta \right|}\right) \ .
\frac{\partial F_1}{\partial z_2}(z_1,z_2, \bar{z}_1 z_2 + \bar{z}_1^2)+(-z_2-\bar{z}_1)\frac{\partial F_1}{\partial w}(z_1,z_2, \bar{z}_1 z_2 + \bar{z}_1^2)= 0 .
a\Theta\wedge\omega_2-b\Theta\wedge\omega_1-c\omega_1\wedge\theta_{56}=0, &b\Theta\wedge\omega_2+a\Theta\wedge\omega_1+c\omega_2\wedge\theta_{56}=0, &c\Theta\wedge\omega_2+[a\omega_1+b\omega_2]\wedge\theta_{56}=0, &c\Theta\wedge\omega_1-[-b\omega_1+a\omega_2]\wedge\theta_{56}=0,
[\begin{matrix}-26&-3&-6&2&2&-1&0&0\\ 0&-2&-4&1&1&0&0&0\\ -3&1&5&-1&-1&0&0&0\\ -3&1&2&-1&0&0&0&0\\ 1&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\end{matrix}]
(x,t) = \big(x_0 + \xi\sqrt{T - t_0(x_0)},t_0(x_0) + \tau (T - t_0(x_0))\big), (\xi, \tau) \in \mathbb{R}^n \times \left[ - \frac{t_0(x_0)}{T - t_0(x_0)}, 1 \right).
f_k = k^{-1/2}(1+ \mathbf y^{-1/2}) \delta_a(k^{1/2}) = k^{-1}(1+ \mathbf y^{-1/2}) \frac{\mathbf y^{1/2}-1}{\mathbf y -1}( \delta_a (k) ) = k^{-1}\mathbf y^{-1/2}(\delta_a (k) ).
\mathbb{D}^m_n=\left\{(X_1,\ldots,X_m)\in M_n^m\: \left|\:\: \begin{array}{l}X_j X_k-X_k X_j=\mathbf{0}_n, X_jX_j^\ast-X_j^\ast X_j=\mathbf{0}_n, \|X_j\|\leq 1\end{array}1\leq j,k\leq m\right.\right\}
\mathcal{S}^{(int)}_k := \Big\{\mathbf{c}' \in \mathbb{R}^{n}: \mathbf{l}^{\mbox{\tiny \upshape (\itshape k\upshape)}}\leq \mathbf{c}' \leq \mathbf{u}^{\mbox{\tiny \upshape (\itshape k\upshape)}}\Big\},
\Phi^{\Delta}= \left( A_{\alpha \beta}^{( \sigma )}, \eta_{\alpha}^{( \sigma )}, C^{( \sigma )}\right) ; \Phi_{\Delta}^{*}= \left( A_{( \sigma )}^{* \alpha \beta}, \eta_{( \sigma )}^{* \alpha}, C_{( \sigma )}^{*}\right) ,
(\mathfrak{d}_1^ma)_n =\sum_{k=0}^m\tbinom{m}{k}(-1)^k \frac{(-1)^{n+k}}{(n+k+1)^\alpha}&=(-1)^n\sum_{k=0}^m\tbinom{m}{k}\frac{1}{(n+k+1)^\alpha}&\sim \frac{1}{(n+1)^\alpha}.
\tau_i \equiv \begin{cases}\theta_i + t_1 \delta_i i< j, \theta_i - t_1 \sum_{l=1}^{j-1}\delta_l i=j, \theta_i j<i<k; \theta_i - t_2 \sum_{l=k+1}^n \delta_l i=k, \theta_i + t_2 \delta_i i> k.\end{cases}
(G(M') \circ G(D)\circ G(M)\circ G(D'))^2 = (\mu_2 \circ d_2 \circ m_2\circ \delta_2)\circ (\mu_2 \circ d_2 \circ m_2\circ \delta_2) = (\mu_2 \circ d_2 \circ \mu_2 \circ d_2)\circ ( m_2\circ \delta_2 \circ m_2\circ \delta_2) = (\mu_2 \circ d_2 \circ m_2\circ \delta_2) = G(M') \circ G(D)\circ G(M)\circ G(D')
-d_{i}^{\varepsilon}\nabla u_{i}^{\varepsilon}\cdot\mbox{n}=\varepsilon\left(a_{i}^{\varepsilon}u_{i}^{\varepsilon}-b_{i}^{\varepsilon}v_{i}^{\varepsilon}\right)\quad\mbox{on}\;\left(0,T\right)\times\Gamma^{\varepsilon},
0 \leq E_t(\overline Q_t + P; B_R) - E_t(\overline Q_t; B_R) & = \widetilde{L}\int_{B_R}\nabla \overline Q_t \cdot\nabla P + \frac{\widetilde L}{2}\int_{B_R}|\nabla P|^2 &+ \frac{1}{2}\int_{B_R}(|\overline Q_t+P|^2-1)^2 -\frac 12 \int_{B_R}(|\overline Q_t|^2-1)^2 & + \frac{\lambda(t)}{t}\int_{B_R}\big[ h(\overline Q_t+P)-h(\overline Q_t) \big] \, dx.
\left( J_{\varepsilon}^{\prime}(W_{\varepsilon,\xi}+\phi_{\varepsilon,\xi})-J_{\varepsilon}^{\prime}(W_{\varepsilon,\xi})\right) \left[ \left.\frac{\partial}{\partial y_{h}}W_{\varepsilon,\xi(y)}\right\vert_{y=0}\right] =O(\varepsilon).
\chi_1 \cdot \chi_2 = \chi_1 \otimes \chi_2, \epsilon_1 \cdot \epsilon_2 = \epsilon_1 \otimes \epsilon_2 \chi_1 \cdot \epsilon_2 = \chi_1 \otimes \epsilon_2, E \cdot \chi_2 = \chi_2 \otimes E - D_{E}\chi_2.
\times \int \left( \prod_{i}^{}{\cal D}x^{( i )}( \xi ){\cal D}h_{\mu \nu}^{i}\right) \delta \left( \sum_{i}^{}\Sigma_{\mu \nu}^{i}\right) \exp \Biggl \{- \int d^{4}x \Biggl [ \frac{1}{2 4 \eta^{2}}\left( H_{\mu \nu \lambda}^{i}\right)^{2}+ \frac{N g_{m}^{2}}{8}\left( h_{\mu \nu}^{i}\right)^{2}- i \pi h_{\mu \nu}^{i}\Sigma_{\mu \nu}^{i{\,}( \alpha )}\Biggr ] \Biggr \}.
a\in \{u \in H_{per}^{2/p,p}(\Omega)^2\cap L^p_{\overline{\sigma}}(\Omega) \mid v\mid_{\Gamma_b}= 0 \}, b_{\tau}\in H_{per}^{2/{q_{\tau}},q_{\tau}}(\Omega), b_{\sigma}\in H_{per}^{2/{q_{\sigma}},q_{\sigma}}(\Omega).
\theta = [\mu_i,\sigma_{ij}]^\top_{i,j\in\{1,\ldots,d\}, i\le j},Z(\theta) = S(\theta)^{\frac{1}{\alpha -1}}N_{\theta,\alpha},h({\bf{x}})\equiv 1,w(\theta)=\big[w^{(1)}(\theta), w^{(2)}(\theta)\big]^\top,f({\bf{x}}) = \big[ f^{(1)}({\bf{x}}),f^{(2)}({\bf{x}})\big]^\top,
\lambda \int_U u^{\lambda-1}u_t^2 dx +\int_U K (|\nabla u+Z(u)|)(\nabla u+Z(u))\cdot (\nabla u+Z(u))_t dx &= \int_U K (|\nabla u+Z(u)|)(\nabla u+Z(u))\cdot (Z(u))_t dx + \int_\Gamma B u_t d\sigma + \int_U f u_t dx &\le C \int_U |\nabla u+Z(u)|^{1-a}|Z'(u)||u_t| dx + \int_\Gamma B u_t d\sigma + \int_U f u_t dx.
\tilde{L}_n(t)=\max\left\{0,\mathbb{F}_n(t)-a_n^{-1}({b_n+x_{\alpha}})\sqrt{{\mathbb{F}_n(t)(1-\mathbb{F}_n(t))}/{n}}\right\}, \tilde{U}_n(t)=\min\left\{1,\mathbb{F}_n(t)+a_n^{-1}({b_n+x_{\alpha}})\sqrt{{\mathbb{F}_n(t)(1-\mathbb{F}_n(t))}/{n}}\right\},
(\vec U,{\bf D_{x_i}}\vec V)=\vec U^T (I_n \otimes P_{\Omega})({\bf D_{x_i}}\vec V)= -({\bf D_{x_i}}\vec U, \vec V) + (E\vec U)^T (I_n \otimes P_{\partial\Omega}) N_{i}(E\vec V).
\begin{aligned}\min \limits_{w_{c},r}[ w_{c},r ] s.t.{\rm C_{1}:}\w_{c}\geq2 &{\rm C_{2}:}\ 0\textless r \textless 1 &{\rm C_{3}:}\R_{\rm FA}(\lambda^{\star}, w_{c}, r ) \leq \tau \end{aligned}
\left| \dfrac{z f^{\prime}(z)}{f(z)}- 1 \right|\leqq \frac{z- \sum\limits_{\nu=k+1}^{\infty}\nu a_{\nu}|z|^{\nu}}{z-\sum\limits_{\nu=2}^{\infty}a_{\nu}|z|^{\nu}}\leqq 1- \alpha(z\in \mathbb{U})
\mbox{Tr}\left( \mathcal{K}_{\alpha}^{2}\right)=\int_{\mathcal{D}}\int_{\mathcal{D}}\frac{1}{\|\mathbf{x}-\mathbf{y}\|^{2\alpha}}d\mathbf{x}d\mathbf{y}=[a_{d,1}^{\chi_{r}^{2}}(\mathcal{D})]^{2}<\infty.
\begin{aligned}\varphi(0, \lambda) = f_\downarrow(\lambda), \psi(0, \lambda) = f_\downarrow(\lambda), \chi(\pi, \lambda) = F_\downarrow(\lambda), \varphi'(0, \lambda) = -f_\uparrow(\lambda), \psi'(0, \lambda) = -f_\uparrow(\lambda) - \alpha f_\downarrow(\lambda), \chi'(\pi, \lambda) = F_\uparrow(\lambda).\end{aligned}
\begin{array}{|c|c|c|c|}\hline e^{\ell \rho}e^{\ell\phi}e^{\ell\chi}\hline \mbox{in the original theory}(\frac{1}{2}\ell (\ell-3),0) (-\frac{1}{2}\ell(\ell+2),0) (\frac{1}{2}\ell(\ell-1),0) \hline\mbox{in the twisted theory}(\frac{1}{2}\ell (\ell-1),0) (-\ell^2/2,0) (\frac{1}{2}\ell(\ell+1),0) \hline \end{array}
\left[ \!\!\begin{array}{c}u'_r v'_r \end{array}\!\!\right]= \frac{1}{R^2}\left[ \!\!\begin{array}{cc}u -v v u \end{array}\!\!\right] \left[ \!\!\begin{array}{c}R'_rR \Theta '_rR^2 \end{array}\!\!\right] .
d((x,i), (y,j)):= \sqrt{\mathcal\mathbb 1_{i \neq j}+ \hat V(x- y)},~x,y \in \mathbb R^d, ~i,j \in \mathcal M, &\mathcal P_d:=\left\{\mu \in \mathcal P(\mathbb R^d \times \mathcal M): \int_{\mathbb R^d \times \mathcal M}d((x,i), (0,1))\mu(dx\times \{i\})< \infty \right\}.
M_{}= 2P_1(1)P_3(1)\sideset{}{^+}\sum_{\chi(q)}\epsilon(\chi) L \left( \frac{1}{2},\overline{\chi}\right) - 2P_1(1)P_3(1) \sideset{}{^+}\sum_{\chi(q)}\sum_n \frac{\chi(n)}{n^{\frac{1}{2}}}F(n),
a_h(R_hu,v_h)=\frac{1}{2}\sum_{K\in \mathcal{T}_h}\int_K(a_5+a_7)\frac{\partial^3 v_h}{\partial x_1^3}\;\mathrm{d}x_1\mathrm{d}x_2+\frac{1}{2}\sum_{K\in \mathcal{T}_h}\int_K(a_6+a_8)\frac{\partial^3 v_h}{\partial x_2^3}\;\mathrm{d}x_1\mathrm{d}x_2.
\zeta(5) = \frac{16 \pi^2}{147}\zeta(3) + \frac{32 \pi^4}{49}\times \sum_{j \geq 0}\sum_{1 \leq i \leq 5}\binom{j+\frac{i}{2}}{\frac{i}{2}}^{-1}\frac{a_i}{i}\frac{\left(-\frac{1}{4}\right)^j}{j!}D_z^{(j)}\left[ -\frac{\pi \sqrt{z}}{2}\cot\left(\pi \sqrt{z}\right) \right] \Biggr\rvert_{z = \frac{1}{4}}
{B}_0=\left({\begin{array}{*{20}c}1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1 \end{array}}\right) \,\,\,\, \,\,\,\,{B}_1=\left({\begin{array}{*{20}c}-b 0 0 0 0 -a 0 0 0 0 b 0 0 0 0 a \end{array}}\right).
\left( \frac{\mathcal{\vartheta}_{1}\left( \nu \mid \theta \right)}{\nu \mathcal{\vartheta}_{1}^{^{\prime}}\left( 0 \mid \theta \right)}\right) = \operatorname{e x p}\left( - \sum_{k = 1}^{\infty}\frac{1}{2 k}\nu^{2 k}G_{2 k}( q^{2}) \right)
\left< T^{\mu}{}_{\nu}\right> = \frac{1}{r^{4}}\left( \begin{array}{c c c c}{- A}&{0}&{0}&{0}\\{0}&{- A}&{0}&{0}\\{0}&{0}&{3 A}&{\kappa B / r^{2}}\\{0}&{0}&{\kappa B}&{- A}\\ \end{array}\right) ,
\Pi^*(s)=\frac{b_0(s,Y(T_0))}{\sigma^2_0(s,Y(T_0))}+\frac{(Y(T_0)-Y(s))\beta(s)}{\sigma_0(s,Y(T_0))\|\beta\|^2_{[s,T_0]}}, 0 \leq s \leq T < T_0.
\int_0^1 h^n_{n-j}(t)\frac{t^s}{s!}dt=\frac{\sqrt{2n-2j+1}}{(s-n+j)!(s+1)_{n+1}}\prod_{m=0}^{\frac{j-1}{2}}(n-s-2m)\prod_{m=0}^{\frac{j-3}{2}}(n+s-2m).
\int_{r=r_0}|\varrho^{i+1}(d\mathcal{M}^{j})(\mathcal{M}^{j})^{-1}|^2i_{\vec{n}}dVol(M)\leq \gamma_{_T}^4(i+1)(\frac{\mathfrak{r}}{T^{i+1}})\kappa_3^4s^4.
S ={1 \over\sqrt{M^2}}\varepsilon_{ij}L^i \tilde{D}L^j ={L^*{}^2 \over\sqrt{M^2}}\varepsilon_{ij}\hat{k}^i \tilde{D}\hat{k}^j ={L^*{}^2 \over \sqrt{M^2}}\overline\kappa_2 \,.
(\widehat{T}_{i}f)(x) =-\beta\gamma f(x)+(1-\alpha\delta)f(s_{i}x)-\alpha\gamma\sum_{j=0}^{-a_{i}(x)-1}f(s_{i}x-ja^{\vee}_{i}+v_{i+1}) -(\alpha\delta+\beta\gamma)\sum_{j=1}^{-a_{i}(x)-1}f(s_{i}x-ja^{\vee}_{i})-\beta\delta\sum_{j=1}^{-a_{i}(x)}f(s_{i}x-ja^{\vee}_{i}-v_{i+1}).
S_{m}^{( 1 )}= S_{m}^{( 1 )}( \Psi_{1}, A^{2}( Q , R ) g_{\mu \nu}) \mathrm{a n d}S_{m}^{( 2 )}= S_{m}^{( 2 )}( \Psi_{2}, B^{2}( Q , R ) g_{\mu \nu}) ,
f_p^{\infty}(x,u,b,\xi )-f_p^{\infty}(x_{0},u,b,\xi ) \geq\lim_{k\rightarrow +\infty}\frac{f(x,u,t_k^{\frac{1}{p}}b,t_{k}\xi)-f(x_{0},u,t_k^{\frac{1}{p}}b,t_{k}\xi )}{t_{k}}&\geq -\varepsilon \lim_{k\rightarrow +\infty}\frac{1+t_k|b|^{p}+|t_{k}\xi |}{t_{k}}=-\varepsilon (|b|^p+|\xi |).
\Big|\Big(\frac{d}{d\lambda}\Big)^\alpha F^{(\iota)}(\lambda )\Big| = \Big|\Big(\frac{d}{d\lambda}\Big)^\alpha \int_{-2}^2 2^\iota F(\tau) \check \chi(2^\iota(\lambda-\tau))\, d\tau \Big| \lesssim_N 2^{\iota(\alpha+1)}\int_{-2}^2 \frac{|F(\tau)|}{(1+2^\iota|\lambda-\tau|)^N}\, d\tau.
\rho_{ij}\to 0, \frac{\rho_{ij}}{2\mu^2}=\cfrac{t^2 -\sigma_i^2}{2\mu^2}\times\cfrac{(t^2 - \sigma_j^2)}{2\mu^2}\rightarrow \frac{(t^*)^2}{(\zeta_i-t^*)(\zeta_j-t^*)};
\kappa_m\left(\sigma W Z \right) =\left(\frac{\sigma}{2}\right)^m [\kappa_m(Z_1^2)+(-1)^m\kappa_m(Z_2^2)] &=\left(\frac{\sigma}{2}\right)^m [2^{m-1}(m-1)!+(-1)^m2^{m-1}(m-1)!],
W (t_\lambda f) = W( f_\lambda ( f\circ \sigma^{d(\lambda)}))= f_\lambda ( f\circ \sigma^{d(\lambda)})\sqrt{d\mu_\pi}= ( f\circ \sigma^{d(\lambda)})\sqrt{| f_\lambda|^2d\mu_\pi}= ( f\circ \sigma^{d(\lambda)})\sqrt{d [\mu_\pi \circ (\sigma_\lambda)^{-1}]}= S_\lambda^{univ}(f\sqrt{d\mu_\pi})=S_\lambda^{univ}W(f)
g_{1}(\nabla^{^{M_1}}_{U}V,Z) =-g_{1}(\nabla^{^{M_1}}_{U}V,\phi^{2}Z)-g_{1}(\nabla^{^{M_1}}_{U}V,\omega \phi Z)+g_{1}(\mathcal{H}\nabla^{^{M_1}}_{U}\varphi V,\omega Z)
h_1(\partial_x u_1+\partial_y u_2)=-\partial_\tau\hat h_1+\hat h_1\partial_\xi\hat u_1-\hat u_1\partial_\xi\hat h_1+\nu\hat h_1\big(\partial_\eta(\hat h_1\partial_\eta\hat h_1)-(\partial_\eta\hat h_1)^2\big).
\Vert \pi(\mu)(F) - \mu \ast F\Vert_2 = \frac{1}{2\vert T \vert}\Vert \sum_{g \in T \cup T^{-1}}(\sqrt{\frac{\rho(g \, \cdot)}{\rho}}- 1)F(g^{-1}\, \cdot)\Vert_2 & \leqslant \frac{1}{4}\frac{1}{2\vert T \vert}\sum_{g \in T \cup T^{-1}}\Vert \sqrt{\frac{\rho(g \, \cdot )}{\rho}}F(g^{-1}\, \cdot )\Vert_2 = \frac{1}{4}\Vert F \Vert_2.
B_{4 , 6 , 8}= \left[ \begin{matrix}{{}}&{{}}&{{\vdots}}&{{}}\\{{Q_{1 , 2 , 3}}}&{{}}&{{\vdots}}&{{0}}\\{{\cdots}}&{{\cdots}}&{{\cdots}}&{{}}\\{{0}}&{{}}&{{\vdots}}&{{0}}\\ \end{matrix}\right] , \ \B_{5 , 7 , 9}= \left[ \begin{matrix}{{}}&{{}}&{{\vdots}}&{{}}\\{{P_{1 , 2 , 3}}}&{{}}&{{\vdots}}&{{0}}\\{{\cdots}}&{{\cdots}}&{{\cdots}}&{{}}\\{{0}}&{{}}&{{\vdots}}&{{0}}\\ \end{matrix}\right]
J(u)=\frac{1}{2}\int_{\mathbb{R}^{2}}|\nabla u|^{2}dx+\frac{1}{4}\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}\ln |x-y|u^{2}(x)u^{2}(y)dxdy-\int_{\mathbb{R}^{2}}F(u)dx
I_{\tilde{Q}}(X_1;X_2,Y) = I_{\tilde{Q}}(X_1;X_2)+I_{\tilde{Q}}(X_1;Y|X_2) &= I_{\tilde{Q}}(X_1;X_2)+H_W(Y|X_2) &\ \ \ \ -H_{\tilde{Q}}(Y|X_1,X_2), f(W) = -H_W(Y|X_1,X_2),
\widehat{W}={\widehat M}_{\infty}= \left( \begin{array}{c c c c c c}{{- 1}}&{{0}}&{{0}}&{{2}}&{{0}}&{{0}}\\{{0}}&{{1}}&{{0}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{1}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{- 1}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{0}}&{{1}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{0}}&{{0}}&{{1}}\\ \end{array}\right) \; .
H_0^1(\mathbb{T}):=H_S\oplus H_S^{\perp}, H_S:=\mbox{span}\{e^{\mathrm{i}\,j\,x}: j\in S \}, H_S^{\perp}:=\{u=\sum_{j\in S^c}u_j\,e^{\mathrm{i}\,j\,x}\in H_0^1(\mathbb{T}) \},
\sum_{1 \leq i \leq 2r}\sum_{\sigma \in S_{2r}}r_\sigma\delta_{j,\sigma{(i)}}= \sum_{\sigma \in S_{2r}}\sum_{\substack{j=\sigma{(i)}1 \leq i \leq 2r}}r_\sigma = \sum_{\sigma \in S_{2r}}r_\sigma =\frac{1}{2}.
{\cal D}^{(n)}_{q^{-m/2}}(q,t;p)=\prod_{1\le i\le n}\frac{1}{\theta_p(u x_i^{\pm 1})}{\cal D}^{(n)}_{q^{-(m-1)/2}}(q,t;p)D^{(n)}_q\big(uq^{-(m-1)/2},pq^{-(m-1)/2}/u;t;p\big),
f(t)-\sum_{k=0}^{p-1}b_k\frac{t^{kr}}{(kr)!}=-\frac{1}{2\pi i}\int_{\delta_{\beta_1}}e^{t/u}\left(\varphi(u)-\sum_{k=0}^{p-1}b_ku^{kr}\right)\,\frac{du}{u}&=-\frac{1}{2\pi i}\sum_{j=1}^3\int_{\delta_j}e^{t/u}\left(\varphi(u)-\sum_{k=0}^{p-1}b_ku^{kr}\right)\,\frac{du}{u}.
0 = x^2 z^2 v = (1-q^2(q+q^{-1})\nu_y + q^4 \nu_y^2) v = (1-q^{4N+2}(q+q^{-1})\lambda + q^{8N+4}\lambda^2)v = (1-q^{4N+3}\lambda)(1-q^{4N+1}\lambda) v.
\prod_{m=1}^\infty\prod_{n=1}^\infty\frac{\mu_m^{(+)}-\nu_{n}^{(+)}}{(2 m)^2-(2n-1)^2}=\prod_{m=1}^\infty\prod_{n=1}^\infty\frac{(2 m)^2-(2n-1)^2}{\mu_m^{(-)}-\nu_{n}^{(-)}}.
\frac{dC}{dt}=\frac{A}{B}=\frac{A^2}{\lambda_1\lambda_2}=\frac{1}{\lambda_1\lambda_2}\Bigg(\frac{\lambda_1^2\lambda_3^2}{C^2}\Bigg), C^2\frac{dC}{dt}=\frac{\lambda_1\lambda_3^2}{\lambda_2}.
\Omega^{2,+}(\mathbb{R}^4) = \langle \, l_1^+=\d x_0\d x_1 + \d x_2\d x_3 \, , \, l_2^+=\d x_0\d x_2 - \d x_1\d x_3 \, , \, l_3^+=\d x_0\d x_3 + \d x_1\d x_2 \, \rangle \Omega^{2,-}(\mathbb{R}^4) = \langle \, l_1^-=\d x_0\d x_1 - \d x_2\d x_3 \, , \, l_2^-=\d x_0\d x_2 + \d x_1\d x_3 \, , \, l_3^-=\d x_0\d x_3 - \d x_1\d x_2 \, \rangle
A_{2,n}^{j,\ell}(x)=\frac{(c-1)B_{1,n-1}^{j,\ell}(x)}{(c+1)(n-1)+\beta c},B_{2,n}^{j,\ell}(x)=A_{1,n-1}^{j,\ell}(x)+A_{2,n}^{j,\ell}(x)(1-x) .
\varphi(w)=\varphi_{1}w_{1}+\cdot\cdot\cdot+\varphi_{n}w_{n}=[\varphi_{1},...,\varphi_{n}][\begin{matrix}w_{1}\\ \vdots\\w_{n}\end{matrix}]=\overline{[\begin{matrix}\overline{\varphi_{1}}\\ \vdots\\ \overline{\varphi_{n}}\end{matrix}]}^{T}\vec{w}=\overline{\vec{f_{\varphi}}}^{T}\vec{w}=\langle f_{\varphi}|w\rangle,
\xi^{(m+n)}_{{l}}=\frac{\pi \bigl({l}+\frac{1}{2}(\epsilon_- +\tilde{\epsilon}_-) \bigr)}{m+n-1+ \frac{1}{2}(\epsilon_+ +\epsilon_- + \tilde{\epsilon}_+ +\tilde{\epsilon}_-)}(0\le{l}<m+n).
y_{(2k)m+2i-1,j}+ y_{(2k)m+2i-1,j+1}= x_{2i-1,j}+ x_{2i-1,j+1}\mbox{and}y_{(2k)m+2i,j}+ y_{(2k)m+2i,j+1}= x_{2i,j}+ x_{2i,j+1}.
\Im G_{R}= -\,{\cal C}(\epsilon, h_{L,R}) \sinh{\left({p_+\over 2 T_L}+{p_-\over 2 T_R}\right)}\left| \Gamma \left( h_L +{i p_+\over 2 \pi T_L}\right) \Gamma \left( h_R +{i p_-\over 2 \pi T_R}\right)\right|^2\,,
\mathbb{E}_{\theta_0,b}(T_{D,M}\mid X):=\mathbb{E}_{\theta_0,b}\left[\sum_{j=1}^{D \wedge M}\left(\frac{Y_j}{X_j}-\theta_{j,0}\right)^2 \mid X \right]=\sum_{j=1}^{D \wedge M}\left(\frac{b_j}{X_j}-1\right)^2\theta^2_{j,0}+\varepsilon^2 \sum_{j=1}^{D \wedge M}X_j^{-2}.
\frac{h}{2i}Op_h^w\big(\theta_h'(x)(x + p'(\xi))\big)v^{\Sigma}_{\Lambda}= \frac{h^{\frac{3}{2}}}{2i}Op_h^w\Big(\theta'_h(x)\gamma(\frac{x+p'(\xi)}{\sqrt{h}})(\frac{x+p'(\xi)}{\sqrt{h}})\Big)v^{\Sigma}+ h^{\frac{3}{2}}Op_h^w(r_0)v^{\Sigma}\,,
-a\mu(|D|^2 +\sigma^2)\nu= \frac{\gamma}{1-\gamma}G_1(\nu)+ \frac{\sqrt{\mu}}{\gamma}|D| \mathcal T\nu-\frac{\mu}{\gamma^2}|D|^2 \mathcal T_1v\equiv G_2(\nu)\in L^2(\mathbb R),
\sum_{T<\gamma\le 2T}\bigg| M\Big( \frac{1}{2}+i\gamma + \frac{2\pi i \kappa}{\log T},P \Big) \bigg|^4 \ll T\log T \sum_{n\le y^2}\frac{d(n)^2}{n}+ T \sum_{n\le y^2}\frac{(\Lambda*d)(n) \, d(n)}{n}&\ll T \log^5 T
\left\langle \left(\prod_{i=1}^{n-k-1}P_{X(U_{i},...,U_{k+1+i})}d_k d^*_k \right) \chi_{X(U_0,...,U_{k+1})}, \chi_{X(U_{n-k-1},...,U_{n})}\right\rangle = \left\langle \left(\prod_{i=1}^{n-k}P_{X(U_{i},...,U_{k+i})}d_k^* d_k \right) \chi_{X(U_0,...,U_{k})}, \chi_{X(U_{n-k},...,U_{n})}\right\rangle.
z=V\int \limits_0^{\infty}dp_ne^{-\frac 12\beta c \left (p_n+\frac{m^2c^2}{p_n}\right )}\prod_{k=1}^{n-1}\int\limits_{-\infty}^{\infty}dp_ke^{-\frac{\beta c}{2p_n}p_k^2}.
\frac{1}{\sqrt{-g}}j^\mu=\exp(-2\lambda\phi)g^{\mu\nu}\partial_\nu a+\frac{q^2}{16\pi^2 f_{PQ}}\epsilon^{\mu\nu\rho\sigma}(A_\nu^j\partial_\rho A_{\sigma j}+\frac{q}{3}f_{ijk}A_\nu^i A_\rho^j A_\sigma^k)
\langle \hat u_j,Eu_k\rangle =\sum_{l\geq 1}\langle \hat u_j,u_l\rangle\langle u_l,Eu_k\rangle &=\langle \hat u_j,u_j\rangle\langle u_j,Eu_k\rangle+\sum_{l\neq j}\langle \hat u_j,u_l\rangle\langle u_l,Eu_k\rangle &=\langle\hat u_j,u_j\rangle\sqrt{\lambda_k\lambda_j}\bar\eta_{kj}+\sum_{l\neq j}\langle\hat u_j,u_l\rangle\sqrt{\lambda_k\lambda_l}\bar\eta_{kl}
\delta_{\xi}g_{\mu \nu}= 2 \nabla_{( \mu}\xi_{\nu)},\ \ \delta_{\xi}\Gamma^{\tau}_{\rho \sigma}=\frac{1}{2}g^{\tau \nu}\left(\nabla_{\rho}\delta_{\xi}g_{\sigma \nu}+ \nabla_{\sigma}\delta_{\xi}g_{\rho \nu}- \nabla_{\nu}\delta_{\xi}g_{\rho \sigma}\right).
\tilde\rho_{(1)}^r(x) =\frac{x^{L/2-1}e^{-x}}{2\Gamma(L/2)}\left(1-x^{L/2+1}\overline{\gamma}(L/2+1,x)+\frac{(2x)^L}{B(L/2,1/2)}\overline{\gamma}(L+1,2x)\right),
\mathcal{H}:= L^{2}(G, \mathbb{R}^{k})/\{\mathbf{1}\}\equiv\Big\{\mathbf{u}\in L^{2}(G, \mathbb{R}^{k}) \,\big|\, \int_{G}\mathbf{u}\, \mathrm{d}\mathbf{x}= \mathbf{0}\Big\}, \mathcal{V}:= H^{1}(G, \mathbb{R}^{k}) \cap \mathcal{H}.
\left(q_{1}q_{2}q_{3}\right)'^{Q}_{1}=L_{1}\left( f^{Q}_{11}f^{Q}_{22}f^{Q}_{33}\right)\left( q_{1}q_{2}q_{3}\right)^{Q}=\left({\bf f}^{Q}_{11}\bar{{\bf f}}^{Q}_{11}q_{1}\right)q_{2}q_{3}=\left( q_{1}q_{2}q_{3}\right)^{Q}_{1}=inv,
\frac{\partial^2 \psi}{\partial i_1^2}(i_1,i_2) =\frac{8\,i_2\,e^{\frac{k}{2}\log^2\frac{i_1+R}{i_1-R}}}{\sqrt{i_1^2-4\,i_2}(i_1^2-4\,i_2)(-1) (\sqrt{i_1^2-4\,i_2}-i_1)(\sqrt{i_1^2-4\,i_2}+i_1)}\cdot \qquad\cdot\left(2\,k\,R\log^2\frac{i_1+R}{i_1-R}+2\,R -i_1\log\frac{i_1+R}{i_1-R}\right),
\frac{R_{31}}{-i-\omega(1,1)}u_1+\frac{R_{32}}{-i-\omega(2,2)}u_2=i(-R_{31}+iR_{32})=-iR_{31}-R_{32}=0; \frac{R_{41}}{-i-\omega(1,1)}u_1+\frac{R_{42}}{-i-\omega(2,2)}u_2=-iR_{41}-R_{42}=i+i=2i=-2u_4.
d^*=\underset{\mathbf{y}\in\ell_\infty, \lambda\in\mathbb{R}}{\max.}\lambda & \mathbf{y}^T d-\lambda\geq 0, &\quad\quad\quad\quad\quad\, \mathbf{y}^T c \geq 0\quad,\forall\; c\in C_g, &\quad\quad\quad\quad\quad\, \mathbf{y}_{0}=1.
{\hspace{1mm}^*}\hspace{-1mm}f_{\mu \nu}(x) = - \frac{e_0}{4 \pi \varepsilon_0 c}[ \partial_\mu \vec{n}(x) \wedge \partial_\nu \vec{n}(x) ] \vec{n}(x)=\left( \begin{array}{cccc}0 B_x B_y B_z -B_x 0 \frac{E_z}{c}\frac{-E_y}{c}-B_y \frac{-E_z}{c}0 \frac{E_x}{c}-B_z \frac{E_y}{c}\frac{-E_x}{c}0 \end{array}\right)
(D^a R_\ell)(t) = \frac1{\Gamma(1-a)}\frac d{dt}\int_0^t \frac{R_\ell(t-r)}{r^a}\, dr = \bigg[ \frac{R_\ell(0)}{\Gamma(1-a) t^a}+ \int_0^t \frac{R_\ell'(t-r)}{r^a}\, dr \bigg].
{\tilde{\delta}}\biggl [{\dot{\varphi}}^{a}- \omega^{a b}{\frac{\partial H}{\partial \varphi^{b}}}\biggr ] = \int{\cal D}\lambda_{a}\exp i \int \lambda_{a}\biggl [{\dot{\varphi}}^{a}- \omega^{a b}{\frac{\partial H}{\partial \varphi^{b}}}\biggr ] d t \nonumber
X^{y'}_{\tau_{n_k}}\geqslant \pi_1(y)+v\,(m_k-\pi_2(y))-(\beta+1)n_k'' \geqslant \pi_1(y')-n_k''+v\,(m_k-n_k''-n_k')-(\beta+1)n_k'' =\pi_1(y')+(v-\epsilon_k)\,n_k.
\sum_{l \geq 0}\sum_{\mid m \mid \leq l}\int_{1}^{\infty}d \lambda u_{\lambda , l , m}( r , \theta , \phi ) \overline{{u}}_{\lambda , l , m}( r^{\prime}, \theta^{\prime}, \phi^{\prime}) = \Omega ( k ) \delta^{3}( k - k^{\prime}) .
S(E,A)=\int d^{4}x(-i\frac{1}{2}(\frac{N}{\sqrt{q}})\tilde{E}_{I}^{a}\tilde{E}_{J}^{b}{\epsilon^{IJ}}_{MN}{F_{ab}}^{MN}-2Nn^{b}\tilde{E}_{I}^{a}n_{J}{F_{ab}}^{IJ})
P(C'_{k,n,j})= P(S_k+b_j\notin\{S_m\}_{m=0}^{k-1})\times P(\{S_m\}_{m=k+1}^{n}\cap \{S_k,S_k+b_j\}= \emptyset) = P(b_j\notin\{S_m\}_{m=1}^{k})\times P(\{S_m\}_{m=1}^{n-k}\cap \{0,b_j\}= \emptyset)
\frac{\partial}{\partial \nu}H^{\phi}_{\nu}(f) = \frac{{\rm log}\int_{\mathbb{R}}\phi(x) (f(x))^\nu{\rm d}x}{(1-\nu)^2}+ \frac{\int_{\mathbb{R}}\phi(x) (f(x))^\nu{\rm log}(f(x)){\rm d}x}{(1-\nu)\int_{\mathbb{R}}\phi(x) (f(x))^\nu{\rm d}x}= I_1 + I_2
|D_{m_p-1}(t)-D_{m_k-1}(t)|\leq \sum_{j=k+1}^\infty|D_{m_{j+1}-1}(t)-D_{m_j-1}(t)| =\sum_{j=k+1}^\infty|X_j(t)|\leq\sum_{j=k+1}^\infty\frac{1}{j^2}.
[(A_1,A_2,A_0,0),(0,0,0,\nu)] = \frac{d}{dt}\frac{d}{ds}\Big|_{t,s=0}(e^{tA_1},e^{tA_2},e^{tA_0},0)(I,I,I,s\nu)(e^{-tA_1},e^{-tA_2},e^{-tA_0},0)(I,I,I,-s\nu) = \frac{d}{dt}\frac{d}{ds}\Big|_{t,s=0}\Big(I,I,I,-s\nu+(e^{tA_0}\circ s\nu)(e^{-tA_1}\times e^{-tA_2})\Big) = \big(0,0,0,A_0\circ \nu-\nu\circ (A_1\times I)-\nu\circ (I\times A_2)\big).