harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
B^{\mu \nu}( s ) = - \frac{1}{3 2 \pi^{2}}\Lambda^{s / 4 \pi}( g^{\mu \nu}- \frac{q^{\mu}q^{\nu}}{q^{2}}) \int_{0}^{1}d x \int_{0}^{\Lambda}d m \; m^{- 1 - s / 4 \pi}\ln{( 1 + \frac{x ( 1 - x ) q^{2}}{m})}
\Bigl\lVert \sup_{j_0 \geq s+k_0}\Bigl\lvert \sum_{j = j_0}^{\infty}T_{\mathcal N^{\sharp}(j,s)}B_{j-s}\Bigr\rvert\, \Bigr\rVert_2 \lesssim 2^{-s/5}|I_0|^{1/2}.
0=\sum_{t_{1,\nu}^0,\cdots,t_{k-1,\nu}^0}\lambda_{\nu,\iota^0_\nu,{\bf t}_\nu^0,0}\prod_{i=1}^{k-1}(\lambda(h_i)+\delta(h_i))^{t_{i,\nu}^0}=\bigg(\sum_{t_{1,\nu}^0,\cdots,t_{k-1,\nu}^0}\lambda_{\nu,\iota^0_\nu,{\bf t}_\nu^0,0}\prod_{i=1}^{k-1}(h_i+\delta(h_i))^{t_{i,\nu}^0}\bigg)(\lambda),
{\displaystyle\sum\limits_{n=1}^{\infty}}(-1)^{n+1}\,(2n)^{2n-1}={\displaystyle\sum\limits_{n=1}^{\infty}}\left( \frac{(-1)^{n+1}(2n)^{2n-1}}{(2n)!}{\displaystyle\int\limits_{0}^{\infty}}x^{2n}e^{-x}dx\right) \ ={\displaystyle\int\limits_{0}^{\infty}}\ln\left( \frac{x}{\sqrt{W(-i\,x)W(i\,x)}}\right) e^{-x}dx=0.3233674316....
\sum_{\substack{q\leq X^{e_f-\delta}(q,MP_f)=1}}\;\max_{(a,qM)=1}\;\Big\lvert\psi(X;qM,a) - \frac{\psi(X)}{\varphi(qM)}\Big\rvert\ll\frac{X}{\varphi(M)}(\log X)^{-A}.
-\nu(C_{j_k}(\alpha(k)) = -\nu\left(\frac{B(k)}{2^{j_k(n-k+1)}j_k!}\right) = -(\nu(B(k) - \nu(2^{j_k(n-k+1)}j_k!)) = (n-k+1)j_k + j_k - s(j_k) - \nu(B(k)) = (n-k+1)j_k - s(j_k)
\sum_{y\neq 0,1}\chi_4(y)\chi_4(1-y)\sum_{x\neq 0,1,y}\overline{\chi_4}(x)\overline{\chi_4}(x-y) &=\sum_{y\neq 0,1}\chi_4(y)\chi_4(1-y)\left[\varphi(y)\rho-\overline{\chi_4}(y-1) \right] &=\rho\sum_y \overline{\chi_4}(y)\chi_4(1-y)-\sum_y \chi_4(y) &=-\rho+1.
\Delta^hf = h^{\bar{\gamma}\alpha}\left[\nabla_{\mathcal{X}_\alpha}\nabla_{\mathcal{X}_{\bar{\gamma}}}f - \mathcal{C}_\alpha\left(\nabla_{\mathcal{X}_{\bar{\gamma}}}f\right)\right] \Delta^vf = h^{\bar{\gamma}\alpha}\left[\nabla_{\mathcal{V}_\alpha}\nabla_{\mathcal{V}_{\bar{\gamma}}}f + C_\alpha\left(\nabla_{\mathcal{V}_{\bar{\gamma}}}f\right)\right]
\Delta = 2 5 6 \tilde{u}_{1}^{2}\tilde{u}_{2}^{2}+ 2 8 8 \Lambda_{1}^{2}\Lambda_{2}^{2}\tilde{u}_{1}\tilde{u}_{2}+ 2 5 6 \Lambda_{2}^{2}\tilde{u}_{1}^{3}- 2 5 6 \Lambda_{1}^{2}\tilde{u}_{2}^{3}- 2 7 \Lambda_{1}^{4}\Lambda_{2}^{4},
\int d t^{\prime}G ( 0 , t ; 0 , t^{\prime}) = \int \frac{d k}{2 \pi}\left( \frac{i}{- k^{2}- 4}\right) \left( \frac{2 i k \left( i k - 2 \coth 2 x_{0}\right)}{( i k )^{2}- 2 i k \sqrt{1 + \sigma_{0}}\sqrt{1 + \sigma_{1}}+ 2 ( \sigma_{0}+ \sigma_{1})}\right) .
\big\langle m \big| N^{(\lambda)}-k\big\rangle = \big\langle u_0^{(\lambda)}\big| P_{\ell+1}^m C^{N^{(\lambda)}-m}H^k P_{\ell-1}^{N^{(\lambda)}-k}\big|u_0^{(\lambda)}\big\rangle ,k < m.
\phi_{i n t}^{1 2}= \lambda_{1}\overline{{{\lambda}}}_{2}+ \lambda_{2}\overline{{{\lambda}}}_{1}+ \left( x - a_{1}\right) \overline{{{\left( x - a_{2}\right)}}}+ \left( x - a_{2}\right) \overline{{{\left( x - a_{1}\right)}}}\ ,
g^{- 1}= \left( \begin{array}{l l}{- 2 \mu_{2}- 2 \mu_{2}^{2}+ \frac 2 3 \mu_{3},}&{- \mu_{3}( 6 + \frac{1 6}{3}\mu_{2})}\\{- \mu_{3}( 6 + \frac{1 6}{3}\mu_{2}) ,}&{\frac 8 3 \mu_{2}^{2}\mu_{3}- 8 \mu_{3}^{2}}\\ \end{array}\right)
t_{\Omega \Sigma}^{( 2 )}{}^{\Pi \Gamma}{}^{\Delta \Lambda}+ t_{\Delta \Lambda}^{( 2 )}{}^{\Pi \Gamma}{}^{\Omega \Sigma}- t_{\Pi \Gamma}^{( 2 )}{}^{\Delta \Lambda}{}^{\Omega \Sigma}= 0 \, .
|(f_i \pm \varphi)(x)-(f_i\pm \varphi)(u)|= =\begin{cases}|a^x_i-(g^{+}_i(u)-\frac12\pm \frac12)|\leq 1\leq d(x,u) |(g^{+}_i(u)-\frac12)-(g^{+}_i(u)-\frac12\pm \frac12)|=\frac12 \leq d(x,u),\end{cases}
\sigma^{S}_{abs}= \frac{1}{8}A_{h}(\omega^{2}-m^{2})^{2}r_{e}\frac{(e^{\frac{\omega - m\phi_{K}}{T_{H}}}- 1)}{(e^{\frac{(\omega - m)}{T_{H}}}- 1)(e^{\frac{1}{2}\pi (\omega + m) r_{e}}- 1)}.
\Big\|\frac{1}{2}\big(x_{\ell}- Jx_{\ell}m\big) - \frac{1}{2}\big(x-Jxm\big)\Big\| = \frac{1}{2}\Big\|(x_{\ell}-x) - \big(Jx_{\ell}m -Jxm\big)\Big\| &\leq \frac{1}{2}\big\|x_{\ell}-x\big\| + \frac{1}{2}\big\|J(x_{\ell}-x)m\big\| &= \big\|x_{\ell}-x\big\| & \longrightarrow 0, \; \; \; \ell \to \infty.
H[f](\phi,\psi) : = \frac{1}{2}\Big\{\big\langle \nabla\phi, \nabla \langle \nabla f,\nabla\psi \rangle \big\rangle +\big\langle \nabla\psi, \nabla \langle \nabla f,\nabla\phi \rangle \big\rangle -\big\langle \nabla f, \nabla \langle \nabla\phi,\nabla\psi \rangle \big\rangle \Big\}.
\delta \tilde{T}_{\mu\nu}dy^{\mu}dy^{\nu}= \int d^3{\bf k}\left[ \tilde{t}_{00}Ydt^2 + 2\tilde{t}_{(L)0}V_{(L)i}dtdy^i + (\tilde{t}_{(LL)}T_{(LL)ij}+\tilde{t}_{(Y)}T_{(Y)ij})dy^idy^j\right].
\frac{\lambda_v^{K_{\ell}}}{\lambda_v^L}=\Bigl\{q_v^{(\dim M_v^{K_{\ell}}-m+1)/2}\cdot|M_v^{K_{\ell}}|^{-1}\cdot\prod_{i=1}^{m-1}(q_v^{2i}-1)\Bigr\}\Bigl\{q_v^{(\dim M_v^L-m)/2}\cdot|M_v^L|^{-1}\cdot\prod_{i=1}^{m}(q_v^{2i}-1)\Bigr\}^{-1}=\frac{q_v^{1/2}}{q_v^{n+1}-1}.
\varphi\!\left(\! \frac{p_1(x)}{\widetilde{+}_{\varphi}(p_1,\cdots\!,p_m)(x)},\cdots\!, \frac{p_m(x)}{\widetilde{+}_{\varphi}(p_1,\cdots\!,p_m)(x)}\!\right) = 1,
\frac{\partial^{2}}{\partial x_{2}^{2}}V ( x_{2}) \; = \; \phi_{0}( x_{2}) \phi_{0}^{\dagger}( x_{2}) \int \frac{d^{2}k}{2 \pi}\frac{- i \; k_{0}+ i e \; V_{0 , 0}}{( - i \; k_{0}+ i e \gamma_{0}V_{0 , 0})^{2}}\; .
\left| G_{[i]}\right| =|G_{i-1}|s_i p_i = \frac{i-1}{d+i-1-k}\prod_{j=1}^i s_j p_j, \left| W_{[i]}\right| = |W_i|\prod_{j=1}^{i-1}s_j p_j = p_i\prod_{j=1}^{i-1}s_j p_j= \frac{1}{d+i-k}\prod_{j=1}^i s_j p_j.
\left| \dfrac{\sqrt{s_n^2}}{\sqrt{\sigma_n^2}}- 1 \right| = \dfrac{|s_n^2 - \sigma_n^2|}{\sqrt{\sigma_n^2}\cdot (\sqrt{s_n^2}+\sqrt{\sigma_n^2})}\leq \dfrac{|s_n^2 - \sigma_n^2|}{\sigma_n^2},
\limsup_{n\to\infty}\frac{\log|r_n|}{n}=\limsup_{n\to\infty}\frac{\log|\hat{r}_n|}{n}=-\rho=-19.10095491\hdots, \lim_{n\to\infty}\frac{\log|q_n|}{n}=\kappa=27.86755317\hdots.
(t_{1}-t_{3})(t_{2}-t_{3})\gamma_{3}c_{1}=qc_{0}, (t_{1}-t_{3})(t_{2}-t_{3})(m+1)(m+\gamma_{3})c_{m+1}=-(m-1+\alpha)(m-1+\beta)c_{m-1}+[m\{(m-1+\gamma_{3})(t_{1}+t_{2}-2t_{3})+ (t_{2}-t_{3})\gamma_{1}+(t_{1}-t_{3})\gamma_{2}\}+q]c_{m}.
J_{1}( i , j , 0 , 0 ; m ) = C^{( 4 )}_{3}F_{2}\left( \left. \begin{matrix}{- i ,}&{- j ,}&{- i - j - \frac{D}{2},}\\{\quad \frac{- i - j}{2},}&{\quad \frac{1 - i - j}{2}}\\ \end{matrix}\right| \frac{t}{4 m^{2}}\right)
\mathbf{S}^{(3)}\mathbf{x}+ \mathbf{U}^{(3)}\left( \mathbf{S}^{(2)}\mathbf{y}^{(3)}+ \mathbf{U}^{(2)}\mathbf{K}^{(2)}\mathbf{y}^{(2)}\right) = \mathbf{b}
\delta (x;g) := \prod_{1\leq j\leq n}(e^{\frac{1}{2}x_j}- e^{-\frac{1}{2}x_j})^{g_S}(e^{x_j}- e^{-x_j})^{g_L}&\times \prod_{1\leq j <k\leq n}(e^{\frac{1}{2}(x_j+x_k)}- e^{-\frac{1}{2}(x_j+x_k)})^{g_M}(e^{\frac{1}{2}(x_j-x_k)}- e^{-\frac{1}{2}(x_j-x_k)})^{g_M}
V^{\emph{r}}_{\epsilon J}(\xi ;g) =\prod_{j\in J}w (\epsilon_j\xi_j)\prod_{\substack{j\in J k\not\in J}}v(\epsilon_j\xi_j+\xi_k) v(\epsilon_j\xi_j-\xi_k) & \times\prod_{\substack{j,j^\prime \in J j<j^\prime}}v(\epsilon_j\xi_j+\epsilon_{j^\prime}\xi_{j^\prime})v(\epsilon_j\xi_j+\epsilon_{j^\prime}\xi_{j^\prime}+1) ,
m^2 = \left( \begin{smallmatrix}1 0 1 1 0 1 1 1 1 1 1 0 1 1 0 1 \end{smallmatrix}\right) \left( \begin{smallmatrix}1 0 1 1 0 1 1 1 1 1 1 0 1 1 0 1 \end{smallmatrix}\right) = I_4.
I=\int dx \left( 4\Omega' \rho' +2\Lambda e^{2\rho}-2\Omega \psi^{'2}-\frac{1}{2}\Omega \psi_1^{'2}+e^{-\psi_1-2\rho}\Omega^2 A_2^{'2}+e^{\psi_1-2\rho}\Omega^2 A^{'2}\right)
|h(t)|^2= 2 \int_0^t \langle h(s), h'(s) \rangle \,ds \le 2 e^{\gamma t}\int_0^t e^{-\gamma s}\langle h(s), h'(s) \rangle \,ds &\le 2 e^{\gamma t}\|h\|_{L^2_\gamma}\|h\|_* \le \frac{4}{\gamma}e^{\gamma t}\|h\|_*^2
{\cal L}_{m}( g ) = \frac{1}{2^{m}}\sqrt{- g}\; \; \delta_{c_{1}d_{1}\cdots c_{m}d_{m}}^{a_{1}b_{1}\cdots a_{m}b_{m}}\; \; R_{a_{1}b_{1}}{}^{c_{1}d_{1}}\cdots R_{a_{m}b_{m}}{}^{c_{m}d_{m}}
\Phi_{g,n,\beta}((\iota_*\alpha)\boxtimes\gamma)=\sum_{\beta_1+\beta_2=\beta}\left( \Phi_{g_1,n_1+1,\beta_1}\otimes \Phi_{g_2,n_2+1,\beta_2}\right)(\alpha\boxtimes (\gamma\boxtimes\eta)).
\det\left[\frac{1}{a_i+b_j}\right]_{i,j=1}^n=\prod_{i=1}^n \frac{1}{a_i+b_i}\prod_{1\le i <j \le n}\frac{(a_i-a_j)(b_i-b_j)}{(a_i+b_j) (a_j+b_i)}
\|1_{\mathcal{D}}\|_{\mathcal{H}_{4\alpha-d}}^{2}=\int_{\mathcal{D}}\frac{1}{\nu (d-4\alpha)}\int_{\mathcal{D}}\frac{1}{\|\mathbf{x}-\mathbf{y}\|^{4\alpha}}d\mathbf{y}d\mathbf{x}=\frac{[a_{d,2}^{\chi_{r}^{2}}(\mathcal{D})]^{2}}{\nu (d-4\alpha )}.
(\partial_t u_*)(0,t(s))=O(e^{-\frac{d}{2}s-s}), (\partial_r F_N^0)(r(s),t(s))=O(e^{-\frac{d}{2}s-s}r(s)), (\partial_tF_N^0)(r(s),t(s))=F_N^1(r(s),t(s))=O(e^{-\frac{d}{2}s-2s}r(s)^2),
m_{r_j}\big((a,b)\big) = \int^b_a W_{r_j}(t)\,dt = \frac{r_j}{1 - e^{-r_j}}\int^b_a e^{-r_j t}\,dt = \frac{-1}{1 - e^{-r_j}}\big(e^{-r_jb}- e^{-r_j a}\big).
{e}_{\hat{0}}^{0}{\bar{e}}_{\alpha}^{i}\left[ \frac{1}{2}\frac{\partial \Gamma^{( 2 )}}{\partial w^{i}}\right] = -{e}_{\hat{0}}^{0}{\bar{e}}_{\alpha}^{i}\left[ B_{i}^{w ( 2 )}\right] +{e}_{\alpha}^{\mu}{e}_{\hat{0}}^{\nu}K_{\mu \nu}^{b},
\sum_{j=\lfloor n/2 \rfloor +1}^n E_{\pi^Q}^Q h_{n-j}(Z_j) \ = \ \sum_{k=0}^{n- \lfloor n/2 \rfloor -1}E_{\pi^Q}^Q \tilde g (Z_0) E_{Z_0}h (Z_k).
\Phi_{L.W.}^{(3,0)}= \Phi_{0,3}^{(3,0)}=V^{(3,0)}\hspace{10mm}\Phi_{1,0}^{(3,0)}= ( a_{22}^{*}a_{11}^{*^{3}}-a_{12}^{*}a_{11}^{*^{2}}) V^{(3,0)}
z_{\ell}\left( r\right) :=r\frac{\left( h_{\ell}^{\left( 1\right)}\right)^{\prime}\left( r\right)}{h_{\ell}^{\left( 1\right)}\left(r\right)}=-\frac{p_{\ell}\left( r^{-2}\right)}{q_{\ell}\left(r^{-2}\right)}+\operatorname*{i}\frac{r}{q_{\ell}\left( r^{-2}\right)},
f\,w^{n+1}[(4\delta^{2}+\delta \,\acute{}\,^{2})v^{4}+4\delta^{2}\, \delta \,\acute{}\, \lambda \,v^{3}+2\delta^{2}[2\delta^{2}(\lambda^{2}+1)-\delta \,\acute{}\,^{2}]v^{2}& -4\delta^{4\,}\delta \,\acute{}\, \lambda \,v+\delta^{4\,}\delta \,\acute{}\,^{2}]+4\delta^{2}v\,[k\,v^{3}+\delta^{2}(k-\lambda)v+\delta^{2}\delta \,\acute{}\,]=0.
\mathbf r_1 \otimes a\mathbf r_2 \otimes \mathbf r_3 = \mathbf r_1 \otimes \mathbf r_2 \otimes \mathbf r_3a , \mathbf r_1\mathbf r_2 \otimes \mathbf r_3 = 1 \otimes 1 , \mathbf r_2 \otimes \mathbf r_3\mathbf r_1 = 1 \otimes 1 .
\begin{array}{ll}\partial_{t}^{j_1}\partial_{x}^{j_2}(\partial_y \tilde{u}^0 - \beta \tilde{u}^0)|_{y=0}begin{align*}4pt]= \sum\limits_{j=0}^{k_0}\sum\limits_{m=0}^{j_1}C_{j_1}^m \frac{\partial_t^{j_1-m}t^j}{j!}\partial_{x}^{j_2}\Big(\partial_y(\partial_t^{j+m}\tilde{u}_0) - \beta \partial_t^{j+m}\tilde{u}_0 \Big)\Big|_{y=0}=0.\end{array}
A_1 =\sum_{1\leq j \leq r}y_j J l_j + \sum_{\alpha \in \Phi_{P^u}}(z_{\alpha}e_{-\alpha}+\theta(z_{\alpha}e_{-\alpha})) \in \mathfrak{k}A_2 =\sum_{1\leq j \leq r}x_j l_j \in \mathfrak{a}_1 A_3 =\sum_{\alpha \in \Phi_{P^u}}\theta(z_{\alpha}e_{-\alpha}) \in \sum_{\alpha\in \Phi_{P^u}}\mathbb{C}e_{\alpha}.
l_{11}=1-\frac{t}{s^2}+s^2 t-t^2+\frac{t^2}{s^4}-\frac{t^2}{s^2}+s^2 t^2-t^3-\frac{t^3}{s^2}l_{12}=\frac{t}{s^3}+s^3 t-\frac{t^2}{s}-st^2 l_{21}=\frac{t^2}{s^3}-\frac{2t^2}{s}-2 st^2+s^3 t^{2}+\frac{t^3}{s^3}-\frac{2t^3}{s}-2 st^3+s^3 t^3-\frac{t^4}{s}-st^4 l_{22}=1+\frac{t}{s^2}-s^{2}t-t^2+\frac{t^2}{s^2}-s^2 t^2+s^4t^2-t^3-s^2 t^3
\frac{1}{2\delta}\left|\int^{\delta}_{-\delta}\exp(ib(\psi^s_{q,(-\delta,\delta)}(\tau;\varphi)+\alpha \tau)) d\tau\right|= \frac{1}{2}\left|\int^{1}_{-1}\exp(i\delta b(\psi^s_{q',(-1,1)}(\tau;\varphi)+\alpha' \tau)) d\tau\right|
y^{\prime}_c = \stackrel{-}{(+)}\frac{{r_0}^{p-1}}{\sqrt{r^{2p-2}-{r_0}^{2p-2}}},\;\;\;\sqrt{1+{y^{\prime}_c}^2}=\stackrel{-}{(+)}\frac{r^{p-1}}{\sqrt{r^{2p-2}-r_0^{2p-2}}}
\frac{\partial C_{1}}{\partial t}=\frac{\partial}{\partial x}[\frac{C_{1}+C_{2}}{C}D_{1}\frac{\partial C_{1}}{\partial x}-\frac{C_{1}}{C}[D_{1}\frac{\partial C_{1}}{\partial x}-D_{2}\frac{\partial C_{1}}{\partial x}]].
g_{\tilde{r}\tilde{r}}(G^{\tilde{r}}_{\tilde{r}}+G^{\theta}_{\theta})=16\pi G\frac{m}{f}(L_M-g^{\tilde{r}\tilde{r}}\frac{\partial L_M}{\partial g^{\tilde{r}\tilde{r}}}-g^{\theta\theta}\frac{\partial L_M}{\partial g^{\theta\theta}})
l i m_{Q \to q}[{{\frac{Q^{\frac{k N_{B}}{2}}{( B^{-})}^{k}}{( [ k ] ! )^{\frac{1}{2}}}},{{\frac{{( B^{+})}^{k}Q^{{\frac{k N_{B}}{2}}}}{( [ k ] ! )^{\frac{1}{2}}}}}}] = 1
\sum_{n=0}^{\infty}p_{-2}\left(5^{2j}(5n+2)+\dfrac{11\times5^{2j}+1}{12}\right)q^{5n+2}= \dfrac{1}{q^{3}E_{25}^{2}}\sum_{l=1}^{\infty}a(2j,l)T^{l}\left(\sum_{k=1}^{\infty}m(6l+2,k)T^{-k}\right).
T_0 = \Gamma_{11}, \quad T_2 ={1\over 2}\psi^2 +{\cal F}\Gamma_{11}, \quad T_4 ={1 \over 24}\Gamma_{11}\psi^4 +{1 \over 2}{\cal F}\psi^2 +{1 \over 2}{\cal F}^2 \Gamma_{11}, \quad{\rm etc.}
F_{2 r + 1}^{M}; \quad | n_{0}, n_{1}, \ldots , n_{r}; \bar{n}_{1}, \ldots , \bar{n}_{r}\rangle , \qquad n_{0}+ n_{1}+ \cdots + n_{r}+ \bar{n}_{1}+ \cdots + \bar{n}_{r}= M ,
\|\boldsymbol{\sigma}-\boldsymbol{\sigma}_{\mathcal{T}}\|_{\mathcal{C}}^2=\int_{\Omega}(\boldsymbol{\sigma}-\boldsymbol{\sigma}_{\mathcal{T}}):\mathcal{K}(\psi)dx + a(\boldsymbol{\sigma}-\boldsymbol{\sigma}_{\mathcal{T}}, \boldsymbol{\varepsilon}^{\perp}(\boldsymbol{\phi})).
\chi_{n}(\xi_{1})\equiv\langle\xi_{1}|n\rangle=\frac{(-i)^{n}}{\pi^{1\!/\!4}}\frac{1}{\sqrt{2^{n}n!}}H_{n}(\xi_{1})e^{-\frac{1}{2}\xi_{1}^{2}}\hspace{0.15cm},
h^{2}(P,Q)=\frac{1}{2}\int\left(\sqrt{\frac{dP}{d\nu}}-\sqrt{\frac{dP}{d\nu}}\right)^{2}d\nu\quad\mbox{et}\quad\rho(P,Q)=\int\sqrt{\frac{dP}{d\nu}\frac{dQ}{d\nu}}d\nu.
\begin{array}{ll}<L^{\pm}_1,M_2>=R^{\pm}_{12}\;,\quad \qquad< \tilde{L}^{\pm}_1, \tilde{M}_2>=R^{\pm}_{12}\;, <L^{\pm}_1, \tilde{M}_2>=A_{12}\;, \quad \qquad<\tilde{L}^{\pm}_1, M_2>= \tilde{A}_{12}\;.\end{array}
(\min_{j}\beta_{j}^{2})g_{E}\leq g^{k}_{\beta,\epsilon}\leq g_{E},\g_{E}=\Sigma_{j=1}^{k}(ds_{j}^2+s_{j}^2d\theta_{j}^2)+\Sigma_{j=k+1}^{n}dz_{j}\otimes d\bar{z}_{j}.
\log \bigg(\frac{k_N a_N}{N}\bigg( \frac{s}{\mu}\bigg)^{2k_N/3}\bigg) = \log k_N + \log \bigg( \frac{1}{s}\bigg) + \log \log \bigg( \frac{s}{\mu}\bigg) - \log N + \frac{2 k_N}{3}\log \bigg( \frac{s}{\mu}\bigg) &= \log k_N + \log \bigg( \frac{1}{s}\bigg) + \log \log \bigg( \frac{s}{\mu}\bigg) - \frac{1}{3}\log N,
\mathbb{A}[{\bf a}] = \dfrac{1}{h}\left[\begin{matrix}a_1+a_2 -a_2 -a_2 a_2+a_3 -a_3 & \ddots \ddots \ddots & -a_{N-1}a_{N-1}+a_{N}-a_{N}& -a_{N}2a_{N}\end{matrix}\right].
E\max_{1\leq k\leq2^{r}}\left\vert s_{k}E^{k}X\right\vert^{p}\leq\sum\nolimits_{i=0}^{r-1}E\max_{2^{i}\leq k\leq2^{i+1}}|s_{k}E^{k}X|^{p}\leq\sum\nolimits_{i=0}^{r-1}\left( s_{2^{i+1}}^{\ast}\right)^{p}E\max_{2^{i}\leq k\leq2^{i+1}}|E^{k}X|^{p}\ll\sum\nolimits_{i=0}^{r-1}\left(s_{2^{i+1}}^{\ast}\right)^{p}E|E^{2^{i}}X|^{p}.
\boldsymbol{x}_{A}^{post}=\boldsymbol{V}_{A}^{post}\left((\boldsymbol{V}_{A}^{pri})^{-1}\boldsymbol{x}_{A}^{pri}+\frac{\mathbf{F}^{H}\boldsymbol{y}}{\sigma_{n}^{2}}\right)
\textrm{rank}(\mathbf{Q}_{s})=\textrm{rank}\{[1+\tilde{f}^{*}(t)][\mathbf{I}+\mathbf{Y}+\sum_{l=1}^{L}t^{-1}\mathbf{H}_{e,l}\mathbf{A}_{l}\mathbf{H}^{H}_{e,l}]^{-1}&(\sum_{k=1}^{K}\mu_{k}\mathbf{h}_{s,k}\mathbf{h}^{H}_{s,k})\mathbf{Q}_{s}\}\leq\textrm{rank}(\sum_{k=1}^{K}\mu_{k}\mathbf{h}_{s,k}\mathbf{h}^{H}_{s,k})\leq K.
J^{\mu \nu}= E g^{\mu \nu}, \j_{1}^{\mu}={\frac{\nabla^{\mu}\phi}{( \nabla \phi )^{2}}}, \j_{2}^{\mu}= j_{\cal R}^{\mu}+ V{\frac{\nabla^{\mu}\phi}{( \nabla \phi )^{2}}}, \j_{3}^{\mu}= E j_{1}^{\mu},
\Sigma J(W^1_n) \cong C \big( l_1:J(\widetilde{W}_n^1) \rightarrow \Sigma J(Z^1_n) \big), \Sigma J(W^2_n) \cong C \big( l_2:J(Z^2_n ) \rightarrow \Sigma J(\widetilde{W}^3_n) \big), \Sigma J( \widetilde{W}^2_n ) \cong C \big( l_3:J(Z^1_n) \rightarrow \Sigma J(W^2_n) \big).
G_{R}^{+}( x , x^{\prime}) = \langle 0_{R}\vert \varphi ( x ) \varphi ( x^{\prime}) \vert 0_{R}\rangle = \frac{1}{\pi^{2}}\int \frac{d{\mathbf k}}{( 2 \pi )^{D - 1}}\, e^{i{\mathbf k}({\mathbf x}-{\mathbf x^{\prime}})}\int_{0}^{\infty}d \omega \sinh ( \pi \omega ) e^{- i \omega ( \tau - \tau^{\prime})}K_{i \omega}( \lambda \xi ) K_{i \omega}( \lambda \xi^{\prime}) .
S^{(m)}_2=\sum_{\substack{d_1,\ldots,d_k e_1,\ldots,e_k}}\lambda_{d_1,\ldots,d_k}\lambda_{e_1,\ldots,e_k}\sum_{\substack{x\leq n<2x n\equiv \nu_0\pmod{W^\prime}[d_i,e_i]\mid nM+h_i}}\chi_\mathbb{P}(nM+h_m).
X^{k}= x^{k}+ \frac{i \pi_{r}}{m ( \omega + m )}\sum_{a = 1}^{N}[ \psi_{a}^{k}, \psi_{a}^{r}] , \theta_{a}^{k}= \psi_{a}^{k}+ \frac{\pi_{k}( \omega - m )}{{m \vec{\pi}}^{2}}\pi_{r}\psi_{a}^{r}.
p_n^{\epsilon,\Delta t}=\frac{1}{(1+\frac{\Delta t}{\epsilon^2})^n}p_0^{\epsilon}+\frac{\Delta t}{\epsilon}\sum_{k=0}^{n-1}\frac{1}{(1+\frac{\Delta t}{\epsilon^2})^{n-k}}f(q_k^{\epsilon,\Delta t})+\frac{1}{\epsilon}\sum_{k=0}^{n-1}\frac{1}{(1+\frac{\Delta t}{\epsilon^2})^{n-k}}\sigma(q_k^{\epsilon,\Delta t})\Delta\beta_k.
K_{\mathrm{d i r e c t}}( r , \phi ; r , \phi ; t ) ={\frac{1}{8 \pi^{2}\alpha t}}\int_{A + B}{\frac{d \delta}{1 - e^{-{\frac{i \delta}{\alpha}}}}}e^{-{\frac{r^{2}}{2 t}}( 1 - \cos \delta )}\quad - ( \mathrm{t h e ~ p o l e ~ a t ~ \d e l t a = 0 ~})
\nabla_{\dot{A}a}\overline{{W}}^{2}= 2 W^{\dot{C}\dot{D}}\nabla_{\dot{A}a}W_{\dot{C}\dot{D}}= 4 i W^{\dot{C}\dot{D}}W_{\dot{C}\dot{D}\dot{A}a}- \frac{4}{3}W_{\dot{A}\dot{B}}\Lambda_{a}^{\dot{B}}
\varphi = \begin{bmatrix}x_1^{d_1}x_1^{d_2}\cdots x_1^{d_n}x_2^{d_1}x_2^{d_2}\cdots x_2^{d_n}\vdots \vdots \ddots \vdots x_n^{d_1}x_n^{d_n}\cdots x_n^{d_n}g_1 g_2 \cdots g_n \end{bmatrix},
||u||_{L^p_tL^q_z(I_t \times \mathbb{R}^m\times \mathbb{T}^n)}=\left(\int_{I_t}\left(\int_{\mathbb{R}^m\times \mathbb{T}^n}|u(t,z)|^q \, dz \right)^{\frac{p}{q}}\, dt\right)^{\frac{1}{p}}.
\begin{bmatrix}\lambda P_d+P_{d-1}P_{d-2}\cdots P_0\end{bmatrix}(\Lambda_{d-1}(\lambda)\otimes I_n)= &(\Lambda_{d-1}(\lambda)^T\otimes I_m)\begin{bmatrix}\lambda P_d+P_{d-1}P_{d-2}\vdots P_0\end{bmatrix}=P(\lambda),
\left|1-e^{2\pi i\epsilon\frac{F^{-1}(t+y)-F^{-1}(y)}{w(y)}}\right| = \sqrt{2-2\cos\left(2\pi \epsilon\frac{F^{-1}(t+y)-F^{-1}(y)}{w(y)}\right)}\leq \sqrt{2-2\cos\left(2\pi \epsilon C\delta w(\delta)\right)},
\delta g_{\mathcal{H}_t}=-\frac{1}{\rho_t}\frac{\delta \mathcal{H}_t}{\delta\mu}\delta\mu_{\mathcal{H}_t}=\frac{1}{\iota_\Lambda^*\rho_t}\bigg(\frac{\delta \mathcal{H}_t}{\delta g}-\iota_\Lambda^*\dot{\rho}_t\,\mu+\iota_\Lambda^*\left\{\rho_t,\frac{1}{\rho_t}\frac{\delta \mathcal{H}_t}{\delta \mu}\right\}\,\mu\bigg).
D_{c}^{(n)}(\eta,{\eta}') = \frac{\Gamma(\frac{n}{2}-1)}{4\pi^{\frac{n}{2}}}\big{[}\frac{1}{|{\eta}-{\eta}'|^{n-2}}-\frac{1}{|\frac{a_{{\eta}'}}{a_B}{\eta}-\frac{a_B}{a_{{\eta}'}}{\eta}'|^{n-2}}\big{]}
[m\delta_{ij}(\partial_{t}+w_{\pm}^{k}\partial_{k}\pm\frac{1}{2}w_{2}^{kl}\partial_{l}\partial_{k})-qF_{ij}]w_{\pm}^{j}=\pm\frac{q}{2}w_{2}^{jk}\partial_{k}F_{ij}-q\partial_{t}A_{i}-\partial_{i}U
{\cal D}_{n^{\prime}n}^{q^{\prime}q}( \mathrm{\bf ~ k}) \; = \; ( - 1 )^{n^{\prime}- n}\Bigl \{{\cal D}_{n \, n^{\prime}}^{- q \, - q^{\prime}}( \mathrm{\bf ~ k}) \Bigr \}^{\ast}\; \; ,
\Omega_3\mid_{C_0}=\frac{dv_2\wedge dv_1}{f_{v_0}},~\Omega_3\mid_{C_1}=\frac{dv_0\wedge dv_2}{f_{v_1}},~\Omega_3\mid_{C_2}=\frac{dv_1\wedge dv_0}{f_{v_2}},
\mathcal{F}[b_{m,n}] (\tau) = \sum_{(m',n') \in \mathbb{Z}^2_{0}}\frac{1}{\alpha n^2+i (\tau-l_{m,n})}\int_{\mathbb{R}}\mathcal{F}[{\hat \theta}_{m',n'}] (\tau_1) \mathcal{F}[{\hat \beta}_{m-m, n-n'}] (\tau-\tau_1) d\tau_1
\Omega_{M_c\star}=\Omega_{\mathcal{C}^c\star}/\Omega_{\mathcal{C}\mathcal{C}^c\star}=\{[\omega]=\omega+\Omega_{\mathcal{C}\mathcal{C}^c\star}~|~ \omega\in\Omega_{\mathcal{C}^c\star}\}.
S = \frac{\hat{j}l_{c}t \phi_{0}^{3 / 2}}{2 \pi l_{s}^{2}}\int_{1}^{\sqrt{\Lambda / \phi_{0}}}\sqrt{\frac{\chi^{2}( \chi^{2}+ \hat{j}) ( 2 \chi^{2}+ \hat{j})^{2}}{( \chi^{2}- 1 ) ( \chi^{2}+ \hat{p}^{2})}}d \chi
\mathbf{L}:(\mathfrak{L}_p|_{\delta=0})^0 \rightarrow (\mathcal{T}_{a^+,a^-}\circ B)\oplus \ker(D|_{L^2_1}); \mathfrak{u}\mapsto (c,\mathfrak{v})\mbox{where}c=\frac{\bar{u}^+}{\bar{a}^+}=\frac{u^-}{a^-}.
F_{M N}= \partial_{M}A_{N}- \partial_{N}A_{M}+ \left[ A_{M}, A_{N}\right] , \quad R_{M N}^{a b}= \partial_{M}\omega_{N}^{a b}- \partial_{N}\omega_{M}^{a b}+ \left[ \omega_{M}, \omega_{N}\right]^{a b}.
c_{n}=\sum_{k=1}^{n}a_{k}\sum_{\pi\in P_{n,k}}(\begin{matrix}k\\ \pi_{1},\pi_{2},...,\pi_{n}\end{matrix})b_{1}^{\pi_{1}}b_{2}^{\pi_{2}}\cdot\cdot\cdot b_{n}^{\pi_{n}},
R_{(0)}(r) = C_{{\rm in}}e^{-i\frac{\omega \ell^2}{2 r_{H}}\ln \left(\frac{r-r_{H}}{r+r_{H}}\right)}+ C_{{\rm out}}e^{i\frac{\omega \ell^2}{2 r_{H}}\ln \left(\frac{r-r_{H}}{r+r_{H}}\right)}.
(\#S_1) \mu_{S_1}= \sum_{i \in S_1}(|x_i|^2 - \alpha_i) = \gamma - \alpha_{S_1}, (\#S_2) \mu_{S_2}= \sum_{i \in S_1}(-|y_i|^2 - \alpha_i) = -\delta - \alpha_{S_2},
\lim_{n\to\infty}\Psi_{\rho(n)}(u_n(x), u_n(y))=e^{-\Phi(\sqrt{\lambda}+\frac{x-y}{2\sqrt{\lambda}})e^{-y}-\Phi(\sqrt{\lambda}+\frac{y-x}{2\sqrt\lambda})e^{-x}}=:H_\lambda(x,y)
\arg{G_1(\sigma\pm i\tau,\chi)}= O\bigg( m^{1/2}(\log{\log{(q\tau)}}) &\quad\quad\times \left( m^{1/2}+ (\log{(q\tau)})^{1/10}+ \frac{(\log{(q\tau)})^{2(1-\sigma)}}{(\log{\log{(q\tau)}})^{3/2}}\right) \bigg).
\begin{array}{c}\Delta_A (f_s,0)= \left(\int_{\mathbb{R}}x^{2n}| f_s (x)|^2 dx\right)^{\frac{1}{2}}= |s|^{-1/2}\left(\int_{\mathbb{R}}x^{2n}| f_1 \left( s^{-1}x\right)|^2 dx \right)^{\frac{1}{2}}= =|s|^{-1/2}\left(\int_{\mathbb{R}}(sy)^{2n}| f_1 \left(y\right)|^2 |s|dy \right)^{\frac{1}{2}}= |s|^n \Delta_A (f_1,0)\end{array}
\lim_{N\to\infty}\limsup_{n\to\infty}k\sum_{l=1}^k\binom{c_nkn}l(k)_l\sum_{j=1}^{n-N}\frac{(k(j-1))_{c_nkn-l}}{(kn)_{c_nkn}}\frac 1{k(n-j+1)-l}=0.
T = \left( \begin{array}{c c c c}{0}&{x_{3}}&{- x_{2}}&{- x_{1}}\\{- x_{3}}&{0}&{x_{1}}&{- x_{2}}\\{x_{2}}&{- x_{1}}&{0}&{- x_{3}}\\{x_{1}}&{x_{2}}&{x_{3}}&{0}\\ \end{array}\right)
X_{\lfloor nt \rfloor}\Rightarrow \phi(t), \sum_{k= \lfloor n t \rfloor}^n \log \frac{X_k/x_k^o}{Y_k/y_k^o}\Rightarrow \frac{1}{\sqrt{\beta}}\int_{\theta(t)}^1 \frac{d b_u}{\sqrt{u}}
\Psi^{\mu}( z ) \Psi^{\mu}( w ) = \frac{\alpha^{\prime}}{2}\frac{1}{z - w}+ \cdots , \quad \tilde{\Psi}^{\mu}( z ) \tilde{\Psi}^{\mu}( w ) = \frac{\alpha^{\prime}}{2}\frac{1}{\bar{z}- \bar{w}}+ \cdots ,
E(A)E(B) =\sum_{p\geq 0}\frac{A^p}{m(p)}\sum_{p\geq 0}\frac{B^p}{m(p)}&= \sum_{p\geq 0}\sum_{n=0}^p\frac{A^n}{m(n)}\frac{B^{p-n}}{m(p-n)}= \sum_{p\geq 0}\sum_{n=0}^p\frac{B^{p-n}}{m(n)}\frac{A^n}{m(p-n)}&= \sum_{p\geq 0}\sum_{l=0}^p\frac{B^{l}}{m(l)}\frac{A^{p-l}}{m(p-l)}=\sum_{p\geq 0}\frac{B^p}{m(p)}\sum_{p\geq 0}\frac{A^p}{m(p)}=E(B)E(A).
\lim_{\varepsilon \to 0}\frac{1}{\sigma}\int_{\mathbb{R}^d \times (0,\infty)}-g^\varepsilon\nabla \varphi^\varepsilon \cdot \Phi \, dxdt = \int_{\mathbb{R}^d \times (0,\infty)}g \cdot \Phi \, d\mu
\mathrm{t r}\frac{1}{\lambda^{n}}= \mathrm{t r}\frac{1}{\eta^{n}}\pm \biggl ( - \, 2 \varphi \, \frac{N}{\left( - \xi \right)^{n}}+ 2 N \delta_{n , 1}- N \delta_{n , 2}\biggl ) ,
\left\| \psi \left(\frac{\mathbf y}{\tau}\right)\right\|_{L^8(\partial \mathcal C)}\lesssim \tau^{\frac{1}{4}}\|\psi(\mathbf x)\|_{H^{\frac{7}{8},8}(\partial \mathcal C)}\lesssim \tau^{\frac{1}{4}}\|\psi(\mathbf x)\|_{H^{1,8}( \mathcal C)}\lesssim \tau^{-\frac{3}{20}},
\frac{1}{2 \pi}\, \int_{0}^{\infty}\, d \omega^{\prime \prime}\, \tilde{\beta}_{\omega^{\prime \prime}\omega}^{R *}\, \tilde{\beta}_{\omega^{\prime \prime}\omega}^{R}= \langle \, a_{\omega}^{R \dag}a_{\omega}^{R}\, \rangle \,_{\mathrm{{i n}}}\, .
F'_{k,j}(x,y)={\alpha+j-1 \choose j}\sum_{k_2=0}^{k}\dots \sum_{k_{j}=0}^{k_{j-1}}d_{k_j}\prod_{\ell = 1}^{j-1}d_{k_{\ell}-k_{\ell+1}}(x^{2k_j+4}+y^{2k_j+4}) \prod_{\ell = 1}^{j-1}(x^{2(k_{\ell}-k_{\ell+1})+4}+y^{2(k_{\ell}-k_{\ell+1})+4}).
I(l,f,\beta,d)= \underset{\eta \in [0,1]}{}\left\{\beta\alpha\eta+2l\sqrt{2\beta m}(1-\eta)\right\}= \underset{\eta \in [0,1]}{}\left\{\eta(\beta\alpha-2l\sqrt{2\beta m})+2l\sqrt{2\beta m}\right\}.
\begin{aligned}g_{D_I}(x)= - \sum_{i=1}^N \min_{l_i \leq y_i\leq u_i}\Big \{\beta y^2_i + \Big{(}\beta \sigma_{(-i)}(x) - \alpha \Big{)}y_i + h_i(y_i)\Big\}+ \beta \sum_{i=1}^N x^2_i - a^T x + \sum_{i=1}^N h_i(x_i),\end{aligned}
L^{A}\, = \, \frac{1}{2 R}\sum_{n \geq 0}\left[ \omega_{n + \frac{1}{2}}( a_{n}\dot{b_{n}}\, - \, \dot{a_{n}}b_{n}) \, \, - \, \, \omega_{n + \frac{1}{2}}^{2}( a_{n}^{2}\, + \, b_{n}^{2}) \right] .
\frac{\partial r_t(\eta)}{\partial t}= \sum \limits_{x \in \eta^+}r_t(\eta^+ \backslash x, \eta^-)\frac{\partial \rho_t^+(x)}{\partial t}+ \sum \limits_{x \in \eta^-}r_t(\eta^+, \eta^- \backslash x)\frac{\partial \rho_t^-(x)}{\partial t}.
H^n\left(x^0, \bar{P}_n(\vec{a}), 0 \right) = \left[k_{12}^n \pm k_{21}^n\right]\left(\{x^0\}_{\tau=1}^{r+s}, \bar{P}_n(\vec{a})\right) &= \left[h_{12}^n \pm h_{21}^n\right]\left(\{x^0\}_{\tau=1}^{r+s}, \bar{P}_n(\vec{a})\right).
\begin{array}{c}(t_*,x_*) \in \Omega_\delta :=\big\{(t,x) \in [\tau, \tau + \delta] \times \mathbb R^n \colon \min\limits_{l \in L}\|x - z - l (t - \tau)\| \leq \delta\big\}, begin{align*}0.2cm] (p_0,p) \in D^-\varphi(t_*,x_*,\kappa_{t_*}(\cdot)),\kappa(\cdot) \in \Lambda_0(\tau,z,w(\cdot)),\end{array}