harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{ll}\big(\nabla_h v^{1},\mathfrak{L}_{2,\tau}^{1}(\nabla_h v^{n})\big)+\sum\limits^{\mathcal{N}}_{n=2}\big(\nabla_h v^{n-\frac{1}{2}},\mathfrak{L}_{2,\tau}^{n}(\nabla_h v^{n})\big)\geq 0, \end{array}
\zeta_{\Omega({\bf x})}(s)= R(s)+{1\over \Gamma(s)}\int_{0}^{1}\tilde Z_{{\Omega({\bf x})}}(t) t^{s-1}dt +{1\over \Gamma(s)}\int_{1}^{\infty}Z_{{\Omega({\bf x})}}(t) t^{s-1}dt,
I(T/\lambda)=\int \frac{d^{2}k}{(2\pi)^{2}}\frac{1}{e^{|k_0|/T}-1}\delta(k^{2}-\lambda^{2}) =\frac{1}{(2\pi)^{2}}\left[\frac{\pi T}{2\lambda}+\frac{1}{2}\ln\frac{\lambda}{T}+\cdots\right]
A_{i, a}:=Y_{i, aq_i^{-1}}Y_{i, aq_i}\left(\prod_{j\colon c_{ji}=-1}Y_{j, a}^{-1}\right)\left(\prod_{j\colon c_{ji}=-2}Y_{j, aq^{-1}}^{-1}Y_{j, aq}^{-1}\right)\left(\prod_{j\colon c_{ji}=-3}Y_{j, aq^{-2}}^{-1}Y_{j, a}^{-1}Y_{j, aq^2}^{-1}\right).
b(x,y) = -\left(\sum_{z \in X}b(x,z)(1_{gy}(gx) - 1_{gy}(gz)) +c(x)1_{gy}(gx)\right) = -HT_{g^{-1}}1_{gy}(x) = -T_{g^{-1}}H1_{gy}(x) &= -\left(\sum_{z \in X}b(gx,z)(1_{gy}(gx) - 1_{gy}(z)) + c(gx)1_{gy}(gx)\right) &= b(gx,gy),
H=\sum_{n=1}^{\mathsf{N}}\left( \sigma_{n}^{x}\sigma_{n+1}^{x}+\sigma_{n}^{y}\sigma_{n+1}^{y}+\sigma_{n}^{z}\sigma_{n+1}^{z}\right) +\frac{\sigma_{\mathsf{N}}^{z}}{p}+\frac{\sigma_{1}^{z}+\xi \sigma_{1}^{x}}{q}.
V_{p}^{K}=\left\{X=(dL_{p})_{Id}(u)~\biggr|~u=\begin{pmatrix}x 0{0}0\end{pmatrix}, ~-\overline{x}=x\,\in\mathbb{H}~\right\}. \intertext{The orthogonal complement $H^{K}_{p}$ of $V^{K}_{p}$ is}H_{p}^K=\left\{X=(dL_{p})(u)~\biggr|~u=\begin{pmatrix}0 y -\overline{y}z\end{pmatrix},\,-\overline{z}=z, y\,\in\mathbb{H}\right\}.
\frac{\partial v^1}{\partial t}-\sum_{ij}a^{v^1}_{ij}\frac{\partial v^1}{\partial x_i\partial x_j}=0,~\frac{\partial v^2}{\partial t}-\sum_{ij}a^{v^2}_{ij}\frac{\partial v^2}{\partial x_i\partial x_j}=0,
\sum_{(i,j) \in N}\left(1 - \cos\left(2 \pi x_{(i,j)}\right) \right) (x_{(i,j)})_{(i,j) \in N}\in \left[0, \frac{1}{2}\right]^N, \forall\, (k,\ell) \in S, \; 4x_{(k,\ell)}+\sum_{\|(i,j)-(k,\ell)\|_1 = 1}x_{(i,j)}\geq |v_{(k,\ell)}|.
\sum_{i=1}^p(x_{i-1}-x_i)^2q_i(t, x,\xi)+\sum_{i=1}^p\xi_i^2r_i(t, x,\xi) +\big\{(x_p- \phi_p(x^{(p+1)}, \xi^{(p+1)}))^2+\psi_p(x^{(p+1)}, \xi^{(p+1)})\big\}q_{p+1}(t, x,\xi),
|N(a_i) \backslash N(a_{i-1})| = |N(a_i)|-|N(a_i) \cap N(a_{i-1})| &=|N(a_i)|-|N(a_{i-1})|+|N(a_{i-1})|- |N(a_i) \cap N(a_{i-1})| &=|N(a_i)|-|N(a_{i-1})|+|N(a_{i-1}) \backslash N(a_{i})| & < |N(a_i)|-|N(a_{i-1})| + s.
\frac{1}{\rho^d}\int_{B_\rho}\left|D^2v(y)-M_m\right|^pdy\, \le \dfrac{1}{r}\cdot \dfrac{1}{r^m}\int_{B_{r^m}}\left|D^2v(y)-M_m\right|^p{dy}&=\,\int_{B_{r^m}}\left|D^2v_m(x)\right|^pdx &\,\leq C,
\delta_{t}u^{1}_{ij}-\mu\Delta_{h}u^{1}_{ij}- \mathfrak{w}_{1,1}\Delta_{h}u^{1}_{ij}=b^{1}_{ij}+\frac{g(u^1_{ij})+g(u^{0}_{ij})}{2}&+(R1)^{1}_{ij}+(R2)^{1}_{ij}+(R3)^{1}_{ij}+(R4)^{1}_{ij},(x_i,y_j)\in\Omega_{h},
\kappa(\Omega) = \sup\Bigg\{\frac{\|\nabla u\|_{L^2(\Omega)}}{\|D(u)\|_{L^2(\Omega)}};~~~ u\in W^{1,2}(\Omega,\mathbb{R}^n), ~ u\cdot\vec n = 0 \mbox{on}\partial\Omega\Bigg\}.
(1-2\lambda c_1)||y_n^i-x_n||^2 \le||x_n-x^*||^2-||z_n^i-x^*||^2 =(||x_n-x^*||-||z^i_n-x^*||)(||x_n-x^*||+||z^i_n-x^*||) &\le ||x_n-z^i_n||(||x_n-x^*||+||z^i_n-x^*||)
\tilde{R}_{ij}-\frac{1}{n}\tilde{R}\tilde{g}_{ij}=R_{ij}-\frac{1}{n}Rg_{ij}-(n-2)(\nabla_{i}\nabla_{j}\varphi-\nabla_{i}\varphi\nabla_{j}\varphi)+\frac{(n-2)}{n}(\Delta\varphi-|d\varphi|^{2})g_{ij}
\frac{d}{dr}(r^{k+\frac{1}{2}}W^-_{k,l})=-r^{k+\frac{1}{2}}2l W^+_{k,l}-r^{k+\frac{1}{2}}P^+_{k,l}(f); &\frac{d}{dr}(r^{-k-\frac{1}{2}}W^+_{k,l})=-r^{-k-\frac{1}{2}}2l W^-_{k,l}-r^{-k-\frac{1}{2}}P^-_{k,l}(f)
D_{1 2}^{( l )}\stackrel{1}{F^{( l )}}\stackrel{2}{F^{( 1 )}}D_{1 2}^{( l ) \; - 1}= \left( \begin{array}{c c}{{F^{( l + 1 )}}}&{{0}}\\{{0}}&{{- F^{( l - 1 )}}}\\ \end{array}\right) .
\sum_{d_i\mid a_i,r_i\mid d_i}\frac{\mu(d_i)d_i}{\varphi(d_i)}=\frac{\mu(r_i)r_i}{\varphi(r_i)}\sum_{d_i\mid\frac{a_i}{r_i}}\frac{\mu(d_i)d_i}{\varphi(d_i)}&=\frac{\mu(r_i)r_i}{\varphi(r_i)}\prod_{p\mid\frac{a_i}{r_i}}\frac{-1}{p-1}&=\frac{\mu(r_i)r_i}{\varphi(r_i)}\frac{\mu(a_i/r_i)}{\varphi(a_i/r_i)}=\frac{\mu(a_i)r_i}{\varphi(a_i)}.
PA(y;\beta,\theta)=lim_{a\rightarrow-\infty}GG(y;a,\beta,p=-\theta/a)=lim_{a\rightarrow-\infty}(\frac{\theta y^{-\theta-1}e^{-(y/\beta)^{a}}}{\beta^{-\theta}(-\theta/a)\Gamma(-\theta/a)})=
\left( \partial_{t}-{\frac{l}{2 \pi \alpha^{\prime}}}\partial_{\theta}\right)^{2}X_{4}- \left({\frac{l \, \rho}{2 \pi \alpha^{\prime}}}\right)^{2}\left({\frac{1}{\rho^{2}}}\partial_{\theta}^{2}+ \partial_{z}^{2}\right) X_{4}+ O ( l^{3}) = 0 \, ,
\phi_{Kerr}=\frac{2\pi\overline{n}_{2}I_{0}\Delta z}{\lambda}exp(\frac{-2r^{2}}{w_{0}^{2}})\approx\frac{2\pi\overline{n}_{2}I_{0}\Delta z}{\lambda}(1-\frac{2r^{2}}{w_{0}^{2}})
\Gamma_{div}^{lg}= \frac{1}{4\pi \epsilon}\int d^d x\,\sqrt{-g}\,\left\{\frac{24-n}{6}R - \left[ 8-n \Phi'(\phi)^2\right] g^{\mu\nu}\partial_\mu \phi \partial_\nu \phi +2\Lambda e^{2\phi}\right\},
\frac{N^{(9+16m)/(5+8m)}\eta^{8/(5+8m)}}{N^{(15+26m)/(10+16m)}\eta^{5/(5+8m)}}= N^{(3+6m)/(10+16m)}p^{6m/k(5+8m)}&\ge N^{(3+6m)/(10+16m)}N^{-3m/(5+8m)}&= N^{3/(10+16m)}\, .
\begin{cases}B_0 = x \wedge M_{B}, A_{i}= x(B_{0}^{-1}A_{1}^{-1}\cdots B_{i-1}^{-1}) \wedge M_{A}, B_{i}= x(B_{0}^{-1}A_{1}^{-1}\cdots B_{i-1}^{-1}A_{i}^{-1}) \wedge M_{B}.\end{cases}
\sum_{g \geq 2}u^{2g+2}\sum_{d \geq 1}\mathsf{H}^{\textup{FLS}}_{g,(1,d)}q^d =\sum_{g \geq 2}u^{2g}f_g(q) = \left( q \frac{d}{dq}\right)^2 \frac{1}{4}K(z,\tau)^4 \,.
q^\epsilon(t)-q^0(t) =Q^\epsilon(t)-q^0(t)-P^\epsilon(t) &=q_0^\epsilon-q_0^0+\int_0^t \bigl(f(q^\epsilon(s))-f(q^0(s))\bigr)ds+\int_0^t \bigl(\sigma(q^\epsilon(s))-\sigma(q^0(s))\bigr)d\beta(s)-\epsilon p^\epsilon(t).
\int^{+\infty}_{A_{7,r}}\frac{t^{11/3}}{4\left(49+13 t^4+t^8\right)^{2/3}}dt=\frac{3}{4}\sqrt[3]{2k_{4r}}\cdot{}_2F_1\left(\frac{1}{3},\frac{1}{6};\frac{7}{6};k_{4r}^2\right).
Spl_X(f,\sigma):= \{p\in Spl_X(f)\mid\sum_{i=1}^nm_{j,i}r_{\sigma(i)}\equiv m_j\bmod p\,(1\le{}^\forall j\le t)\}, Spl_X(f,\sigma,\{k_j\}):= \{p\in Spl_X(f,\sigma)\mid\sum_{i=1}^n m_{j,i}r_{\sigma(i)}=m_j+k_jp\,(1\le{}^\forall j\le t)\},
\delta_{\epsilon}b = (\bar\epsilon_1 \Gamma_m d \theta_1 - \bar\epsilon_2\Gamma_m d\theta_2) dX^m +\bar\epsilon_1\Gamma_m d\theta_1 \bar\theta_1 \Gamma^m d\theta_1- \bar\epsilon_2\Gamma_m d\theta_2 \bar\theta_2 \Gamma^m d\theta_2.
\lambda_{\mathbf{d}}=\mu(d)d\sum_{\mathbf{d}|\mathbf{r}}\frac{y_{\mathbf{r}}}{\phi_\omega(r)},y_{\mathbf{r}}=\frac{\mathbf{1}_{\mathcal{D}_k}(\mathbf{r})W^kB^k}{\phi(W B)^k}\mathfrak{S}_{WB}(\mathcal{L})F\Bigl(\frac{\log{r_1}}{\log{R}},\dots,\frac{\log{r_k}}{\log{R}}\Bigr),
{\hat{\Pi}}_{{\hat{\mu}}{\hat{\nu}}}{\cal D}_{\hat{\imath}}{\hat{X}}^{\hat{\mu}}{\cal D}_{\hat{\jmath}}{\hat{X}}^{\hat{\nu}}={\hat{g}}_{{\hat{\mu}}{\hat{\nu}}}D_{\hat{\imath}}{\hat{X}}^{\hat{\mu}}D_{\hat{\jmath}}{\hat{X}}^{\hat{\nu}},
\int_{0}^{\infty}x^{\mu}\exp(-ax^{\xi})\cos(xy)dx=-y^{-\mu-1}\sum_{\ell=0}^{\infty}\left(-\frac{a}{y^{\xi}}\right)^{\ell}\frac{1}{\ell!}\Gamma(\mu+1+\xi\ell)~\sin\left\{\frac{\pi}{2}(\mu+\xi\ell)\right\}.
\Vert W \Vert_{\cal H}\le \Vert V'' \Vert_{\cal H}+ \big|\bigcup\nolimits^n_{i=1}\partial X'_i \setminus \bigcup\nolimits_j T_j^0\big|_{\cal H}+ \big|\bigcup\nolimits^n_{i=1}\partial X'_i \cap\bigcup\nolimits_j T_j^0\big|_{\cal H}= \Vert V'' \Vert_{\cal H}+ \Vert U \Vert_{\cal H}\le \Vert V' \Vert_{\cal H}+ (1+c\nu)\Vert V \Vert_{\cal H},
( 1 - \delta_{n , N}) m b_{n + 2}\varepsilon^{\mu \nu \sigma}e_{\nu |}{}^{\alpha ( 2 )}h_{\sigma | \alpha ( n + 2 )}^{n + 2}+ ( 1 - \delta_{n , N}) m e^{\mu | \alpha ( 2 )}\lambda_{\alpha ( n + 2 )}^{n + 2}= 0 .
\min_{v \in{\mathcal{K}_{\psi}(\Omega)}}\int_{\Omega}G(Dv) \, dx = \max_{\sigma \in S^{p'}_{-}(\Omega)}\left ([\![\sigma, D\psi]\!]_{u_0}(\overline{\Omega}) - \int_{\Omega}G^*(\sigma) \, dx \right )
\int_{-1}^{1}\frac{2(4x^{2}-4xy-1+\rho^{2})\sqrt{1-y^{2}}dy}{\pi((1-\rho^{2})^{2}-4xy\rho(1+\rho^{2})+4\rho^{2}(x^{2}+y^{2}))}=4x^{2}-1,
(xq)^{2a}\sum_{h=1}^{k-a}(xq^2)^{2h}U_{2k,2k-2h}(xq^2;q)+(xq)^{2a}\sum_{h=0}^{k-a-1}(xq^2)^{2h}U_{2k,2k-2h}(xq^2;q).
ds^{2}=(\frac{r^{2}}{b^{2}}-\frac{b^{n-2}}{r^{n-2}})d\tau^{2}+\frac{dr^{2}}{(\frac{r^{2}}{b^{2}}-\frac{b^{n-2}}{r^{n-2}})}+r^{2}\sum_{i=1}^{n-1}dx_{i}^{2}.
- ( \frac{m}{e})^{2}[ \partial_{\mu}\hat{P}_{\nu}^{a}- \partial_{\nu}\hat{P}_{\mu}^{a}+ \frac{m}{e}\epsilon^{a b c}\hat{P}_{\mu}^{b}\hat{P}_{\nu}^{c}] = m^{2}\tilde{W}_{\mu \nu}^{a}.
A = \frac{i}{2}\{\frac{1}{2}d \ln ( \frac{d \overline{{g}}_{2}}{d \overline{{z}}_{2}}\frac{d z_{2}}{d g_{2}}) + \frac{( g_{2}d \overline{{g}}_{2}- \overline{{g}}_{2}d g_{2})}{( 1 - g_{2}\overline{{g}}_{2})}\},
W_{ijkl,st}-W_{ijkl,ts}= \, W_{rjkl}W_{rist}+W_{irkl}W_{rjst}+W_{ijrl}W_{rkst}+W_{ijkr}W_{rlst}+ &+\frac{R}{12}\big(W_{sjkl}\delta_{it}-W_{tjkl}\delta_{is}+W_{iskl}\delta_{jt}-W_{itkl}\delta_{js}&+W_{ijsl}\delta_{kt}-W_{ijtl}\delta_{ks}+W_{ijks}\delta_{lt}-W_{ijkt}\delta_{ls}\big)\,,
\psi_N=\; \sum_{s=0}^N\;\sum_{d=-N+s,\,-N+s+2,\,\dots}^{\dots,\,N-s-2,\,N-s}u_1^{\otimes (N-s+d)/2}\otimes_\mathrm{sym}u_2^{\otimes (N-s-d)/2}\otimes_\mathrm{sym}\Phi_{s,d}
R_{1^{\prime}1^{\prime}}= - R_{0^{\prime}1^{\prime}0^{\prime}1^{\prime}}+ \sum_{k}R_{1^{\prime}k 1^{\prime}k}= R_{1 1}+ \sinh^{2}\alpha \sum_{k}( R_{0 k 0 k}+ R_{1 k 1 k}) ,
ds^2=\frac{1+u_t^2}{\sqrt{1+u_t^2+u_x^2}}(dt^2+dy^2)+\frac{1+u_x^2}{\sqrt{1+u_t^2+u_x^2}}(dx^2+dz^2)+\frac{2u_tu_x}{\sqrt{1+u_t^2+u_x^2}}(dtdx+dydz),
\int_{\mathcal{B}^+(0, R)}(y^{1-2s_2}\nabla w) \cdot \nabla ((x, y) \cdot \nabla w)\,dxdy &=-\frac{N-2s_2}{2}\int_{\mathcal{B}^+(0, R)}y^{1-2s_2}|\nabla w|^2\,dxdy+ \frac R 2 \int_{\partial^+\mathcal{B}^+(0, R)}y^{1-2s_2}|\nabla w|^2\,dS.
\tilde{\Lambda}_{L}^{\prime 4 ( N_{c}^{\prime}- 1 ) - 2 ( N_{f}- 1 + \tilde{N}_{c}- 1 )}= ( - \mu m )^{2}\tilde{\Lambda}^{\prime 4 ( N_{c}^{\prime}- 1 ) - 2 ( N_{f}+ \tilde{N}_{c})}.
|f(z)| \leq \left(\frac{1-\alpha}{1+\alpha}\right)^2\log{\frac{1+\alpha |z|}{1-|z|}}+\frac{2|z|}{(1-|z|)^2}-\left(\frac{1-\alpha}{1+\alpha}\right)\frac{2|z|}{1-|z|}&\leq \frac{(1+\alpha)|z|(1+|z|)}{(1+\alpha |z|)(1-|z|)^2}, z \in \Delta.
\displaystyle\sum\limits_{m_1,m_2}\langle m_1,l_1,m_2,l_2 | 0,l_{12}\rangle P_{m_1}^{l_1}(x) P_{m_2}^{l_2}(y) = \frac{x^{-k}\langle 0,l_1,k,l_2 | 0,l_{12}, \rangle}{\sqrt{k! (2l_2)_k}}{}_2 F_1\left(\begin{matrix}-k ,1-k-2l_2 2l_1 \end{matrix}; -x^2\right).
\sum_{j=1}^{|F_{n}|-1}(a_{j-1}b_{j+1}-a_{j+1}b_{j-1})=\sum_{j=1}^{|F_{n}|-1}\|\begin{matrix}a_{j-1}&a_{j+1}\\b_{j-1}&b_{j+1}\end{matrix}\|=3(|F_{n}|-1)-2n-1,
(-1)^{\mathrm{1}_{i \in \mathcal{O}}}\frac{T^{\mathrm{1}_{i \in \mathcal{B}}}}{n}\sum_{w^n = (-1)^{\theta_2}}\frac{w^{\mathrm{1}_{i \in \mathcal{B}}+h_i-h_j}e^{- (Tw+\theta_1 \pi i+ \lambda)[t_j - t_i]_i}}{1 - (-1)^{\theta_1}e^{-(Tw+\lambda)}}= \mathrm{1}_{x_i = x_j}- K_n^{\lambda,\theta,T}(x_i,x_j),
g \in \begin{cases}D_+ \mbox{if $\omega_{Z(\Pi_g)}$ is a positive multiple of $\omega(P)$ at a point of $Z(\Pi_{g,> 0})$,}D_- \mbox{if $\omega_{Z(\Pi_g)}$ is a negative multiple of $\omega(P)$ at a point of $Z(\Pi_{g,> 0})$.}\end{cases}
\psi_{n,k}(z)=\begin{cases}z\int_{\Gamma_{k-1}}\frac{\Psi_{n,k-1}(t)\,t^{k-1-\ell}}{z-t^{p+1}}\,d\sigma_{k-1}(t), \ell< k, begin{align*}1em]\int_{\Gamma_{k-1}}\frac{\Psi_{n,k-1}(t)\,t^{p+k-\ell}}{z-t^{p+1}}\,d\sigma_{k-1}(t), k\leq \ell,\end{cases}
\partial(w\overline{u}^{d-i})+\partial(ws_i\overline{u}^{d-i-1})+\partial(ws_is_{i+1}\overline{u}^{d-i})=\partial(w\overline{u}^{d-i-1})+\partial(ws_{i+1}\overline{u}^{d-i})+\partial(ws_{i+1}s_i\overline{u}^{d-i-1})
\mathfrak{Re}(\Omega) =R_{+}^{3}\theta^{1234}-R_{+}R_{-}^{2}\left(\theta^{1267}+\theta^{1537}+\theta^{1564}\right) &+\frac{r}{2}dr\wedge\left(R_{+}\left(\theta^{237}+\theta^{264}+\theta^{534}\right)-\frac{R_{-}^{2}}{R_{+}}\theta^{567}\right),
\frac{4}{\pi}\int_{0}^{1}\sum_{n = 0}^{\infty}(-1)^{n}x^{2n}\sqrt{1-x^{2}}\binom{-\frac{1}{2}}{n}f_{n}\ln\left(x\right) \, dx + \frac{1}{2}\sum_{n=0}^{\infty}\left( \frac{1}{16}\right)^{n}\frac{\binom{2 n}{n}^2 ( 2 \ln (2) ( n + 1) + 1)}{(n+1)^2}f_{n},
I_{(M2)}= -{1\over 4\pi^2}\int d^3\xi \bigg\{\sqrt{-\det (\partial_\alpha{\bar X}^M \partial_\beta{\bar X}_M)}+{1\over 6}\epsilon^{\alpha\beta\gamma}A_{MNR}\partial_\alpha{\bar X}^M \partial_\beta{\bar X}^N \partial_\gamma{\bar X}^R\bigg\},
\Delta \o_a=\o_a \biggl[\alpha \frac{N-K-2}{|x|^2}+ \frac{\alpha^2}{|x|^2}+ \frac{a(a-1)}{|x|^2 (\textrm{log}|x|)^2}+ a\frac{N-k-2}{|x|^2 log|x|}+\frac{2 a \nabla |x| \nabla \alpha}{|x|}+ \frac{2 a \alpha}{|x|^2 \textrm{log}|x|}+O\bigl(\textrm{log}|x| |x|^{-3/2}\bigl)\biggl].
S_{s o u r c e}^{( 3 )}= \int d^{4}x \left( e^{- \phi}\sqrt{\det ( G +{\cal F})}+{\frac{i}{4}}\chi{\cal F}*{\cal F}+{\frac{i}{2}}B^{R}*{\cal F}+{\frac{i}{2 4}}\epsilon^{\alpha \beta \gamma \delta}A_{\alpha \beta \gamma \delta}^{( 4 )}\right) ,
\exp \left( - \Gamma ( F_{\mu \nu}, a ) \right) = \sum_{k = - \infty}^{\infty}\left( \hat{\Gamma}_{k}^{( 1 )}( F_{\mu \nu}) \cos 2 \pi k a + \hat{\Gamma}_{k}^{( 2 )}( F_{\mu \nu}) \sin 2 \pi k a \right)
\tau^{- 1}( t ) \equiv \frac{d}{d t}\ln S ( t ) \approx \frac{\mathrm{T r}\dot{Y}\ln Y}{\mathrm{T r}Y \ln Y}= \frac{\int \tilde{d}k \dot{Y}_{k}\ln Y_{k}}{\int \tilde{d}k Y_{k}\ln Y_{k}},
[ E_{0}\, \pi ({F}_{+}) E_{0}, E_{0}\, \pi ({G}_{+}) E_{0}] = \lim_{T \rightarrow \infty}\! \mathrm{~ T^{- 1}~}\! \int_{0}^{T}\! d t \, E_{0}\, \pi ( [{F}_{+}, \alpha_{t}( G_{+}) ] ) E_{0}= 0 ,
\frac{1}{\big(6\sqrt{13}\sqrt{-18+5\sqrt{13}}\big)\pi}=\sum_{k=0}^\infty\frac{\left(\frac{1}{2}\right)_k^3}{(k!)^3}\left(k+\frac{1}{4}-\frac{7}{156}\sqrt{13}\right)\left(649-180\sqrt{13}\right)^k.
\mbox{d}\bar{X}(t) = \left\{A_0 \big( X(t-r) \big) - R \right\}\, \bar{X}(t) \, \mbox{d}t + A_1 \big( X(t-r) \big) \, \bar{X}(t) \, \mbox{d}W(t) & = A_1 \big( X(t-r) \big) \, \bar{X}(t) \, \big\{\mbox{d}W(t) - \Sigma (t) \, \mbox{d}t \big\},
\left[ \begin{array}{c}w_{2}\left( t\right) w_{3}\left( t\right) w_{0}\left( t\right) w_{1}\left( t\right)\end{array}\right] =\left[ \begin{array}{cccc}0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0\end{array}\right] \left[ \begin{array}{c}w_{0}\left( t\right) w_{1}\left( t\right) w_{2}\left( t\right) w_{3}\left( t\right)\end{array}\right]
\begin{aligned}s^t_{i'j'}= G^{t}_{i'j'}(z,\delta,w):= a_{i'\tau(i',t-1)}(\delta_{it-1}- \delta_{it}) &+ \sum_{j= \tau(i',t-1)+1}^{\tau(i',t)-1}(m^{t}_{i'j}- m^{t}_{i'j-1})z_{ij}+(m^t_{i'j'}- m^t_{i'\tau(i't)-1})w^t_{i'j'}. \end{aligned}
{Z}_{1}( x ) ={\frac{K_{1}( 1 - a x )}{( 1 - a ) ( 1 - x )}}\ln{\frac{1 - a x}{1 - x}}-{\frac{2 a ( 1 - a x )}{( 1 + 3 \beta^{2}) ( 1 - x )}}\int_{0}^{x}d y \left[{\frac{( 1 - y )^{\frac{2 \beta^{2}}{1 + \beta^{2}}}}{1 - a y}}\right] ,
= \Delta_p^{u_0}(u_{k}^{\delta}, u_{k+1}^{\delta}) - \mu \langle j_p(F(u_k^{\delta})-v^{\delta}), F'(u_k^{\delta})(u_k^{\delta}-u^{\dagger})\rangle + \beta_k\langle J_p(u^{\dagger}-u_0), u^{\dagger}-u_k^{\delta}\rangle - \beta_k \langle J_p(u^{\dagger}-u_0) - J_p(u_k^{\delta}-u_0), u^{\dagger}-u_k^{\delta}\rangle.
\frac{4 \pi}{L_{p}^{3 - d}\, ( d - 1 )}\, \sigma = \frac{\pm \dot{a}\, \alpha \, t^{\alpha - 1}+ ( a \, \, t^{\alpha})^{- 1}}{[ 1 - ( \dot{a}\, \, t^{\alpha})^{2}]^{1 / 2}}\, \, .
G_{(\sigma)}(g_s(z),z')=(-1)^{2l_2-1}(c_rz+d_r)^{-1}\left(G_{(\sigma)}(z,z')+\sum_{F_r=\mu_r,\nu_r}\tilde Y_{\sigma,F_r}(z)\Phi_{\sigma,{F_r}}^{(0)}(z')\right)
[\begin{matrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{matrix}][\begin{matrix}y_{0}\\y_{1}\\y_{2}\end{matrix}]=[\begin{matrix}ay_{0}+2cy_{1}+2by_{2}\\by_{0}+ay_{1}+2cy_{2}\\cy_{0}+by_{1}+ay_{2}\end{matrix}]
{\cal H}_r^{(d(n-1))}=\ 2\,\sum_{i<j}^n \bigg(\frac{m_i+m_j}{m_i m_j}\bigg) \rho_{ij}p^2_{\rho_{ij}}\ +\ 2\,\sum_{i \neq j,i\neq k, j< k}^n\,\frac{1}{m_i}\,(\rho_{ij}+ \rho_{ik}- \rho_{jk})p_{\rho_{ij}}p_{\rho_{ik}}\ +\ \Omega \ ,
\|f\|_{{CMO}^{q,\lambda}_{\omega}\left(\mathbb R^d \right)}=\sup\limits_{R>0}\left(\frac{1}{\omega\left(B(0,R)\right)^{1+\lambda q}}\int_{B(0,R)}|f(x)-f_{B(0,R),\omega}|^q\omega(x)dx\right)^{1/q}<\infty.
(\alpha^{2}_{opt})^{ev}=\sqrt{\frac{2\mid\beta\mid^{2}tanh\mid\beta\mid^{2}+2\mid\beta\mid^{2}cos2\theta_{1}+1}{2\mid\beta\mid^{2}tanh\mid\beta\mid^{2}-2\mid\beta\mid^{2}cos2\theta_{1}+1}}(\alpha^{2}_{opt})^{o},
\widehat V_{\varphi}(t,x^{(t)},m^{(t)})=\inf_{(u_t)_{t=0}^{T-1}\in \mathcal A}\mathbb{E}_{\varphi}\left[\sum_{s=t}^{T-1}f(s,X_s,\mu_s,u_s)+ F(X_T,\mu_T)| X^{(t)}=x^{(t)}, \mu^{(t)}=m^{(t)}\right],
[f(-M_x)][g(H)] =\big([f(0)]+[-M_x][f_1(-M_x)]\big)\big([g(0)]+[H][g_1(H)]\big) &=f(0)g(0)+g(0)[-M_x][f_1(-M_x)]+f(0)[H][g_1(H)] &=f(0)g(0)+g(0)[f(-M_x)-f(0)]+f(0)[g(H)-g(0)] &=g(0)[f(-M_x)]+f(0)[g(H)]-f(0)g(0).
d\theta^{\lambda\cdot Id}(\tilde{X}_{p},\tilde{Y}_{p})= =\tilde{X}_{p}(\theta^{\lambda\cdot Id}(\tilde{Y}))-\tilde{Y}_{p}(\theta^{\lambda\cdot Id}(\tilde{X}))-\theta^{\lambda\cdot Id}([\tilde{X},\tilde{Y}])(p) &=-\theta^{\lambda\cdot Id}([\tilde{X},\tilde{Y}])(p),
G_{2,3N-2}(a,b,q) =\sum_{l=0}^{\lfloor N/2-1 \rfloor}(q^3)^{N-2l-2 \choose 2}\genfrac{[}{]}{0pt}{}{N-1}{2l+1}_{q^3}(-aq^5;q^6)_l(-bq;q^6)_{l+1}.
X=\xi^{i}\frac{\partial}{\partial x^{i}}+\eta^{i}\frac{\partial}{\partial y^{i}}+\zeta_i\frac{\partial}{\partial p_i}\,\,\,{\rm and}\,\,\,\varphi=\alpha_idx^{i}+\beta_idy^{i}+\gamma^{i}dp_i.
\pi ( u ) = \frac{2 k - 1}{4 \pi}\int_{\tau = 0}^{\infty}d \tau \; \sinh \tau \int_{\theta = 0}^{2 \pi}d \theta \left[ \cosh \frac{\sigma}{2}- i \sinh \tau \cos \theta \sinh \frac{\sigma}{2}\right]^{- 2 k}
s =\sum_{i=0}^{r-1}\left\{\lfloor\frac{-mp^i}{q-1}\rfloor+\lfloor\frac{mdp^i}{q-1}\rfloor+\lfloor\frac{-m(d-1)p^i}{q-1}\rfloor\right\}&-\sum_{i=0}^{r-1}\left\{\lfloor\frac{-2mp^i}{q-1}\rfloor+\lfloor\frac{2mdp^i}{q-1}\rfloor+\lfloor\frac{-2m(d-1)p^i}{q-1}\rfloor\right\},
[\hat{S}_0,\hat{S}_+]=(\hat{C}_2-\hat{C}')\hat{J}^+ \qquad[\hat{S}_0,\hat{S}_-]=-(\hat{C}_2-\hat{C}')\hat{J}^-\qquad [\hat{S}_+,\hat{S}_-]=2(\hat{C}_2-\hat{C}')\hat{H}
\frac{1}{2}\sum_{\substack{pq\in E_\Gamma \alpha(pq)=\lambda\alpha_i}}\frac{g_{pq}}{h_{pq}}+\sum_{p\in V_2}\frac{f(p)}{c_p\prod_{e\in E^p}\alpha(e)}=\frac{g}{\alpha_1\cdots\alpha_N}.
\triangle_f (-f)^{-b}= b(\triangle_f f) (-f)^{-b-1}+ b(b+1) |\nabla f|^2 (-f)^{-b-2}&= -b (-f)^{-b}-12b (-f)^{-b-1}+ b(b+1)(-f-R+10) (-f)^{b-2}&\leq -b (-f)^{-b}+b(b-11) (-f)^{-b-1}+ 11b(b+1)(-f)^{b-2}< - b(-f)^{-b},
L(s, f \otimes \chi_d) = \sum_{n=1}^{\infty}\frac{\lambda_f(n)\chi_d(n)}{n^s}= \prod_{p\nmid d}\left(1 - \frac{\lambda_f (p) \chi_d(p)}{p^s}+ \frac{1}{p^{2s}}\right)^{-1}.
\sum_{h}W\left(\frac{h}{H}\right)\sum_{X\leq n\leq 2X}a(n)\tau(n+h)\tau(n-h) =H\widehat{W}(1)\sum_{X\leq n\leq 2X}a(n)\sum_{d}\frac{r_d(2n)}{d^2}(\log n+2\gamma-2\log d)^2 &+O\left(X^{\varepsilon}\left(\frac{H^2}{X^{1/2}}+HX^{1/4}+(XH)^{1/2}+\frac{X}{H^{1/2}}\right)\|a\|_2\right).
\psi_{b^{-1}a^{-1}ba}\stackrel{\epsilon=1}{=}\psi_{b^{-1}}\, \psi_{a^{-1}}\, \psi_b\, \psi_a\stackrel{\epsilon=1}{=}\psi_b^{-1}\, \psi_a^{-1}\, \psi_b\, \psi_a = 1.
S^{I}( \tau ) \; = \; \int_{- \infty}^{\infty}\Bigl [ \; -{\frac{1}{2}}\; \varepsilon \;{\frac{x^{2}}{\tau^{2}}}\, \left({\partial_{x}\phi}\right)^{2}\; + \;{\frac{\lambda}{4}}{\frac{1}{1 +{\frac{x^{2}}{\tau^{2}}}\; ( 1 - \varepsilon )}}\; \phi^{4}\; \Bigr ] \; d x .
\begin{cases}\partial_t \bigl( \frac{\tilde \omega}r\bigr) + (u^{(l)}\cdot \nabla) \bigl( \frac{\tilde \omega}r \bigr) =0, \tilde \omega \Bigr|_{t=0}= \omega_0=\omega_0^{(l)}+ \omega_0^{(h)}. \end{cases}
= \binom{2k+1}{k}\frac{c_1^k\,c_2^{k+1}}{(c_1+c_2)^{2k+1}}+ \Bigl(1-\frac{c_1}{c_2}\Bigr)\sum_{j=0}^{k-1}\binom{2k+1}{j}\frac{c_1^{j}\,c_2^{2k+1-j}}{(c_1+c_2)^{2k+1}}=
\tilde{f}(L_n,L_p)=a_{s,n,p}L_{s+n+p},\ \tilde{f}(L_n,G_p)=b_{s,n,p}G_{s+n+p},\ \tilde{f}(G_n,G_p)=c_{s,n,p}L_{s+n+p}, \hat{f}(1,L_p)=a_{s,p}'L_{s+p}\textrm{and}\hat{f}(1,G_p)=b_{s,p}'G_{s+p}.
\left|{\rm Tr}(\Gamma a_0\prod_{k=1}^{p-1}[F,a_k]F[f(s|D|),a_p])-s{\rm Tr}(\Gamma a_0\prod_{k=1}^{p-1}[F,a_k]F\delta(a_p)f'(s|D|))\right| &\leq\|\Gamma a_0\prod_{k=1}^{p-1}[F,a_k]F\|_{q,\infty}\,\|[f(s|D|),a_p]-s\delta(a_p)f'(s|D|)\|_{p,1}=O(s)
\sigma_e(\sum_{\{j:\zeta_j\in P_m\}}C_{\psi_j}) \setminus\{0\}= \left\{\lambda: \lambda^\ell \in \sigma_e\left(C_{\psi_{j_0}}C_{\psi_{j_1}}\cdots C_{\psi_{j_{\ell -1}}}\right)\right\}\setminus\{0\}.
h_{C,C_1,D,D_1,E,E_1,E_2}(x)= H_{C,C_1,D,D_1,E,E_1,E_2}(x) \indent + H_{C,C_1,D,D \cup \{0\},E,E_1,E_2}(x)h_{C,C,D,D,E,E,E}H_{C,C \cup \{0\},D,D_1 ,E,E_1,E_2}(x).
[z^n](T-T_h)(z) =[u^n]\sum_{k\ge 0}(1-u^2)^2u^{h+1+(h+2)k}(u^2+3u+1)^{n-1}=\sum_{k\ge 0}[u^{n-(h+1+(h+2)k)}] (1-2u^2+u^4)(u^2+3u+1)^{n-1}=\sum_{k\ge 0}\bigl([u^{n-(h+1+(h+2)k)}](u^2+3u+1)^{n-1}&\qquad\qquad- 2 [u^{n-(h+1+(h+2)k)-2}](u^2+3u+1)^{n-1}&\qquad\qquad+ [u^{n-(h+1+(h+2)k)-4}](u^2+3u+1)^{n-1}\bigr).
\langle A^{*a}_\mu(p) c^b(-p) \rangle^{(1)}= - \lim_{M_i \rightarrow\infty}\frac{g_0^2 c_v}{16 \pi^2}\left\{\frac{1}{2}\sum_{i=1}^N c_i\log\left(\frac{M_i^2}{\kappa^2}\right) +{\rm fin}\right\}i p_\mu\delta^{ab}
\begin{gathered}f_+(b,c,d) -f_+(a+b,c,d) +f_+(a,b+c,d) -f_+(a,b,c+d) +f_+(a,b,c) = 0 begin{align*}.5ex] \begin{aligned}f_+(a,b,c) -f_+(a,c,b) +f_+(c,a,b) = g_+(b,c) -g_+(a+b,c) +g_+(a,c) f_+(a,b,c) -f_+(b,a,c) +f_+(b,c,a) = -g_+(a,c) +g_+(a,b+c) -g_+(a,b) \end{aligned}begin{align*}.5ex] g_+(a,b) + g_+(b,a) = 0 \,. \end{gathered}
\int_{I^c}(|u|^{p})^{\frac{k}{p-1}}\, dx\leq \int_{I^c}\left(\frac{\|u\|_{L^p}^{p}}{2|x|}\right)^{\frac{k}{p-1}}=\frac{\|u\|_{L^p}^{\frac{pk}{p-1}}r_0^{1-\frac{k}{p-1}}(p-1)}{2^{\frac{k}{p-1}}(k+1-p)}.
A'= \left( \begin{matrix}0 \sqrt{b_0c_0}0 \sqrt{b_0c_0}0 \sqrt{b_1c_1}\ddots 0 \sqrt{b_1c_1}0 \ddots 0 \ddots \ddots \ddots \sqrt{b_{m}c_m}0 \sqrt{b_{m}c_m}0 \end{matrix}\right)
\frac{1}{\phi(2N+1)}\left(\frac{1}{|S_{m,N}|}\sum_{k\in S_{m,N}}|\xi^{m_0,N_0}_k|^p\right)^\frac1p \le \frac{C_2(2N+1)^\frac1p}{(2N_0+1)^{\frac1p}\phi(2N_0+1)}\left(\frac{|S_{m_0,N_0}|}{|S_{m_0,N}|}\right)^\frac1p &= \frac{C_2}{\phi(2N_0+1)}.
\Omega_1(s) = \left\{x \in \Omega ;\, \langle x \rangle^{2-\alpha}\le t_0 + s \right\}, \Omega_2(s) = \Omega \setminus \Omega_1(s) = \left\{x \in \Omega ;\, \langle x \rangle^{2-\alpha}> t_0 + s \right\}.
\tau^2_{s(\mu)+\alpha^\vee, \mu}= q^{-1}g(B(\alpha^\vee, \mu) - Q(\alpha^\vee)) \mathbf{z}^{-\alpha^\vee}\frac{1-\mathbf{z}^{n_\alpha \alpha^\vee}}{1-q^{-1}\mathbf{z}^{n_\alpha \alpha^\vee}},
N_{\ell, m}(\Gamma) = \bigoplus_{\substack{0\le \ell' \le \ell 0\le \ell'+m' \le \ell+m m'\ge 0}}\sum_{X \in \mathcal{X}^{\ell, m}_{\ell', m'}}X(M^{*}_{\ell',m'}(\Gamma)),
\partial_{\alpha,\beta}(e_{z_{0}}\cdots e_{z_{k+1}}) =\sum_{i=1}^{k}\left(\Delta_{z_{i},z_{i+1}}^{\alpha,\beta}-\Delta_{z_{i},z_{i-1}}^{\alpha,\beta}\right)e_{z_{0}}\cdots\widehat{e_{z_{i}}}\cdots e_{z_{k+1}}
P_m^* \mathcal Q(\ell, \omega) P_m &= \frac1\ell P_m^* \mathcal Q_0 P_m -\frac{2 \varkappa \cdot |\partial_3 \psi_{\varkappa (1,0),3}(0)|^2}{\sqrt{\Lambda -\omega}\cdot \sqrt{2 \zeta_1''(\varkappa)}}\cdot \int_{\{|\theta| = 1\}}P_m^* T_\ell^* P_{\varkappa \theta}T_\ell P_m \; \mathrm{d}\theta + P_m^* R(\ell,\omega) P_m .
{}_2F_1\bigg[\begin{matrix}-a \beta &\gamma\end{matrix}\bigg|\,1\bigg]_{c}={}_2F_1\bigg[\begin{matrix}-a \beta &\gamma\end{matrix}\bigg|\,1\bigg]=\frac{\Gamma(\gamma)\Gamma(\gamma+a-\beta)}{\Gamma(\gamma+a)\Gamma(\gamma-\beta)}.
Z_c= \lim_{L \rightarrow \infty}\int \exp \{i \int_{-L}^{L}dt\sum_{\alpha}-( \frac{i}{2}\frac{\partial \bar{c}^{\alpha}}{\partial t}+ m_{\alpha}\bar{c}^{\alpha})c^{\alpha}+ h.c.+\chi_{\alpha}(\bar{c}^{\alpha}+c^{\alpha}) \}d \bar{c}^{\alpha}dc^{\alpha}
\forall u, \: v, \: w \: \in W^P, \: \Delta(u,v,w) =1 \: \mathrm{if}\: (-1)^{\ell(w_0w) - \ell(w_0u) - \ell(w_0 v) + \int_{d_1(u,v,w) l_1 + d_2(u,v,w) l_2}c_1(T_X)}\geq 0 &=0 \mathrm{if}\: (-1)^{\ell(w_0w) - \ell(w_0u) - \ell(w_0 v) + \int_{d_1(u,v,w) l_1 + d_2(u,v,w) l_2}c_1(T_X)}< 0
{\vec{P}}= ({\overline{{q}}}+ 3 A^{I}p^{I}w ,{\overline{{p}}}_{2},{\overline{{p}}}_{3}\vert \vert q + ( 3 A^{I}p^{I}+ \alpha_{2}) w , p_{2}, p_{3}\vert p^{I}+ \alpha_{2}A^{I}) ~ ,
\binom{4n-2}{2n-1}= \frac{(4n-2)(4n-3)\cdots(2n+1)(2n)!}{(2n-1)^2(2n-2)^2\cdots(n+1)^2(n!)^2}=\frac{2^n(4n-3)!!}{(2n-1)!!(n-1)!}= \frac{2^n(4n-3)(4n-5)\cdots(2n+1)}{(n-1)!},
\aligned({\rm Im}\,{\bf w}_2)^t =A_2\cdot\big (\,{\sf t}^2_1\, \ldots \,{\sf t}^2_{k_2}\,\big)^t, & \ \ \vdots ({\rm Im}\,{\bf w}_{\rho})^t =A_{\rho}\cdot\big(\,{\sf t}^\rho_1\, \ldots \,{\sf t}^{\rho}_{k_{\rho}}\,\big )^t.\endaligned
\theta(\mathsf{P}_{j}^{i})=q^{-(2\rho,\lambda_{i}-\lambda_{j})}(q^{(2\rho,\lambda_{n}-\lambda_{m})}(\mathsf{M}_{m}^{n})_{j}^{i}+q^{-(2\rho,\lambda_{n}-\lambda_{m})}(\mathsf{M}_{n}^{m})_{j}^{i}).