harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{}^{M\!L\!}{\Phi}_{p}(b;c;z) = \frac{1}{2\pi i}\int_{-i\infty}^{-i\infty}\frac{{}^{M\!L\!}\Gamma(s)B(b+s,c+s-b)}{B(b,\; c-b)}\Phi(b+s;c+2s;z)p^{-s}ds.
cG_-^m=\langle c^m,G_-\rangle aF_+^m=\langle a^m,F_+\rangle dG_+^m=\langle d^m,G_+\rangle aF_-^m=\langle a^m,F_-\rangle cF_-^m=\langle c^m,F_-\rangle bG_+^m=\langle b^m,G_+\rangle dF_+^m=\langle d^m,F_+\rangle bG_-^m=\langle b^m,G_-\rangle
\Delta E ( \vartheta ) = - 2 \cos \left( \vartheta \right) \left| \frac{\det^{\prime}M}{\det M_{0}}\right|^{- 1 / 2}\left( \frac{S_{E}\left[ \varphi_{\mathrm{i n s t}}\right]}{2 \pi}\right)^{1 / 2}e^{- S_{E}\left[ \varphi_{\mathrm{i n s t}}\right]}\, ,
1 + v^{- 4}= \sum_{l \in \bf{N}_{0}}\epsilon_{l}\frac{( ( l + 3 ) ! )^{2}}{( 2 l + 7 ) !}( v - 1 )^{l}_{2}F_{1}( l + 4 , l + 4 ; 2 l + 8 ; 1 - v )
\Delta_v ={1 \over 2}+{1 \over 2}\pi \beta_{n=1}q^2 ={1 \over 2}+{q^2 \over |{\cal D}-9|}, \;\; \Delta_m ={1 \over 2}+{1 \over 8 \pi \beta_{n=1}}e^2 ={1 \over 2}+{e^2 |{\cal D}-9| \over 16}
\lefteqn{\theta_S(\bar H(k, \alpha k) \ltimes \bar H(n, \beta n))}&= \max \left\{{\hat f(0,0)\over f(0,0)}: \hat f \ge 0, f(x,y) = 0 \forall (x,y) \in\mathcal{D}\setminus(0,0) \right\}= \min\left\{2^{k+n}{g(0,0)\over \hat g(0,0)}: \hat g \ge 0, g(x,y)\le 0, \forall (x,y) \in \mathcal{D}^c \setminus (0,0) \right\}
\delta S^{(d)}=\int^{t_{2}}_{t_{1}}dt\sum_{i=1}^{n}\left[\sum^{d}_{\alpha=0}(-D)^{\alpha}\frac{\partial L^{(d)}}{\partial (D^{\alpha}q^{i})}\right]\delta q^{i}+\left[\sum_{i=1}^{n}\sum^{d}_{\beta=\alpha\geq1}(-D)^{\beta-\alpha}\frac{\partial L^{(d)}}{\partial (D^{\beta}q^{i})}\delta(D^{\alpha-1}q^{i})\right]^{t_{2}}_{t_{1}}.
\epsilon^{- 1}( y ) \delta ( y ) \bar{\Psi}_{\mu}^{A}( \delta - \gamma_{5}\sigma_{3})_{A}^{B}\gamma_{5}= - \delta ( y ) \alpha_{0}\bar{\Psi}_{\mu}^{A}( \delta + \gamma_{5}\sigma_{3}) ( \sigma_{1})_{A}^{B}\ .
\left \lbrace \begin{array}{l}\displaystyle{d \xi^{\mu}_s = \dot{\xi}_s^{\mu}ds}, \displaystyle{d\xi^{\mu}_s= -\Gamma_{\nu \rho}^{\mu}(\xi_s)\, \xi^{\nu}_s \xi^{\rho}_s ds + d \times \frac{\sigma^2}{2}\, \xi^{\mu}_s ds+ \sigma d M^{\mu}_s}, \end{array}\right.
\frac{1}{n}\sum_{i=1}^n E\left[|V_{n,i}|^{2+\delta}\right] \leq \frac{\|a\|^{2+\delta}}{n}\sum_{i=1}^n E\left[\|X_{n,i}\|^{2+\delta}\big(|Y_{n,i}|+ \|X_{n,i}\|\|\theta_n\|+\|Z_{n,i}\|\|\gamma_n\|\big)^{2+\delta}\right].
\lim_{\lambda\to0^+}\lambda\left\|\left[\int_{\mathbb{R}^n}\mathbf{1}_{E_{\lambda,\frac{\gamma}{q}}[f]}(\cdot,y)\left|\cdot-y\right|^{\gamma-n}\,dy\right]^\frac{1}{q}\right\|_{L^r_\omega(\mathbb{R}^n)}=\left[-\frac{\kappa(q,n)}{\gamma}\right]^\frac{1}{q}\left\|\,\left|\nabla f\right|\,\right\|_{L^r_\omega(\mathbb{R}^n)}.
\widetilde W(r) \sim \begin{cases}\left(\frac{n- P}{P}\right)^P r^{-P}1< P< n \left(\frac{P-1}{P}\right)^P|r\log(r)|^{-P}P= n C_{P,f}\,\, r^{- \frac{P(n-1)}{P-1}}n< P< \infty \end{cases}r\to 0,
\begin{array}{r c l c r c l}{\displaystyle [ t_{a}, t_{b}]}&{=}&{i F_{a b c}t_{c}}&{,}&{F_{a b 0}}&{=}&{0}\\{\displaystyle \{t_{a}, t_{b}\}}&{=}&{D_{a b c}t_{c}}&{,}&{D_{a b 0}}&{=}&{\displaystyle \sqrt{\frac{2}{N}}\delta_{a b}}\\ \end{array}
\left\| \mathbf{f}\right\|_{p}:=\left( f_{1}^{p}+\mathellipsis +f_{d}^{p}\right)^{\raise0.7ex\hbox{1}\!\mathord{\left/{\vphantom{1 p}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{p}},\thinspace p>1,
\Pi_{1B}~=~ \frac{2\pi^{2\mu}}{(\mu-1)^2\Gamma^2(\mu)}\left[1 - \frac{\eta_1}{N}\left( \frac{2}{(\mu-1)}- \frac{3\mu(2\mu-3)}{2}\left( \Theta+ \frac{1}{(\mu-1)^2}\right) \right) \right]
[{L}_{u^{\alpha (i,j)}}]\Delta_\nu u^\alpha (x)^{(i,j)}+\Sigma_{\mu=1,2}\{\Delta_\mu E_\mu^{-1}{{T}_D}_\nu^{\mu(i,j)}+\frac{\partial{{L}_D}^{(i,j)}}{\partial x^{\nu (i,j)}}\}=0.
S(B|M)_{(\Phi\otimes\mathbb{I}_M)(\hat{\rho}_{AM})}= S\left((\Phi\otimes\mathbb{I}_M)(\hat{\rho}_{AM})\right) - S(\hat{\rho}_M) = S\left(\tilde{\Phi}(\hat{\rho}_A)\right) - S(\hat{\rho}_A)\;,
\boldsymbol{K}_1\left(\gamma_{1,3}^{\eta_1, \delta}, \mu_1\right) = \mathbb{E}\left[ c_1( S_1, \widetilde{S}_1 ) \right] \leq \eta_0, S_1 = (Y_1, X) \widetilde{S}_1 = (\widetilde{Y}_1, \widetilde{X}).
\int_{M_k\setminus\{p\}}|\psi|^{\frac{n-2}{n-1}}L_g(|\psi|^{\frac{n-2}{n-1}})\,dv^g = I_g(u_k)-\int_{A_k}\chi_k |\psi|^{\frac{n-2}{n-1}}L_g(\chi_k |\psi|^{\frac{n-2}{n-1}})\,dv^g.
g(u_1,u_2)= \alpha_1 |u_1^2+u_2^2|^2 + \alpha_2 |u_1^2-u_2^2|^2 -4\alpha_2 |u_1|^2|u_2|^2 +2\alpha_3 \cos \eta (|u_1|^4-|u_2|^4) & +4 \alpha_3 \sin \eta (|u_1|^2+|u_2|^2) \Re (\overline{u_1}u_2) + (2\alpha_1 + r) (|u_1|^2+|u_2|^2)^2,
\left(\begin{array}{c}\dot I \dot\varphi\end{array}\right)=\left(\begin{array}{c}A^0I \omega^0\end{array}\right)+P\left(\begin{array}{c}u_0^0(\varphi)+u_1^0(\varphi)I+H_1^0 w^0(\varphi)+H_2^0\end{array}\right)+P (G-G^1),
V_+(1, (x_0,1)) = \min_{\gamma \in \{0,1\}}\Bigg\{c_1 \gamma - (m_1^+) + \mathbb{E}_+\left[V_+\left(2, (x_0,1,\Psi(1,\gamma,\xi_2))\right)\right] \Bigg\}& = - \beta + \min \Bigg\{0, c_1 - \frac{5}{16}\beta \Bigg\}.
P_{\infty}^{({\pm}N ) ( J )}( g ) = \lim_{l{\rightarrow}{\infty}}P^{({\pm}N ) ( J )}( u ) = \left( \begin{array}{c c}{\frac{1 +{\gamma}}{2}{\cal P}^{({\pm}N )}( g )}&{0}\\{0}&{\frac{1 -{\gamma}}{2}{\cal P}^{({\pm}N )}( g )}\\ \end{array}\right) .
\Gamma_{kij}=\frac{1}{2}(\frac{\partial}{\partial x^{j}}g_{ki}+\frac{\partial}{\partial x^{i}}g_{kj}-\frac{\partial}{\partial x^{k}}g_{ij})=\frac{1}{2}(g_{ki,j}+g_{kj,i}-g_{ij,k}),
m( t) = \pi+\arctan\left(\frac{(1-a^2)\sin t}{2a-(1+a^2)\cos t}\right), m'( t) = \frac{1-a^2}{2a \cos t -1-a^2}, \textrm{and}\,\,\, m''( t) = \frac{2a(1-a^2)\sin t}{(2a \cos t -1-a^2)^2}\,.
\frac{d^k}{d\xi^k}\frac{d^2}{d\sigma^2}h(\sigma, \xi) = \frac{d^2}{d\sigma^2}\frac{d^k}{d\xi^k}h(\sigma, \xi) = i^k\frac{d^2}{d\sigma^2}\int{(x-m)^k\exp(i(x-m)\xi)\mu_{\sigma}(dx)},
D_{\vec f}\left(P_{\vec \varphi}, Q_{\vec \varphi}\right)\ = \ \frac{(2 \pi)^{n/2}}{\sqrt{\det (A)}}\ \prod_{i=1}^{n}\left[ c_i\f_i \bigg( \frac{\det (A)}{c_i^2}\bigg) \right]^{\frac{1}{n}}.
\lim_{\varepsilon\rightarrow0}I_{\varepsilon}(t_{\varepsilon}(Z_{\varepsilon,q})Z_{\varepsilon,q})=\frac{1}{2}\int_{\mathbb{R}_{+}^{n}}|\nabla V|^{2}+(a-\omega^{2})V^{2}dy-\frac{1}{p}\int_{\mathbb{R}_{+}^{n}}V^{p}dy=m_{e}^{+}
\mathrm{Tr}(A_L) = \mathrm{N}_{q^m/q}(\alpha_1/\alpha_2)(G_{m}- u^{\sigma^{-1}}G^\sigma_{m-2})= \mathrm{N}_{q^m/q}(\alpha_1/\alpha_2)(G_m + G^\sigma_m+ G^\sigma_{m-1}).
\begin{cases}\vec{r}_k=\frac{1}{2}\sum_{i,j=1}^3\epsilon_{ijk}(\vec{x}_i-\vec{x}_j), \vec{\rho}_k=\frac{3}{2}\,\vec{x}_k-\frac{1}{2}\sum_{\ell=1}^3 \vec{x}_\ell, \end{cases}k\in\{1,2,3\}
\int_1^{\infty}\int_1^{x}\int_1^{x}\frac{dzdydx}{\left(x+y\ln x+z\ln^2 x\right)^3}=\sum_{n=0}^{\infty}\int_{2^n}^{2^{n+1}}dx\int_1^{x}\int_1^{x}\frac{dzdy}{\left(x+y\ln x+z\ln^2 x\right)^3}&\leq C\sum_{n=0}^{\infty}\int_1^{2^{n+1}}\int_1^{2^{n+1}}\frac{2^n\, dzdy}{\left(2^{n}+ny+n^2z\right)^3}&\leq C\sum_{n=0}^{\infty}\frac{1}{n^2}<\infty.
\partial_{\mu}S_{0}^{\mu}= \frac{1}{3}\sum_{i}\left( \gamma_{i}^{\ast}+ \gamma_{i}\right) \Phi_{i}^{\alpha}\frac{\partial W}{\partial \Phi_{i}^{\alpha}}\Bigg | + \frac{1}{4 8 \pi^{2}}\left[ 3 T ( G ) - \sum_{i}T ( R_{i}) ( 1 - ( \gamma_{i}^{\ast}+ \gamma_{i}) ) \right] F_{\mu v}^{a}\widetilde{F}^{\mu v a}\, ,
(\theta+dd^c q)^n(t_0,x_0) \geq \nu(x_0)= e^{\partial_t u (t_0,x_0)+u(t_0,x_0)+F(t_0,x_0,u(t_0,x_0))}\mu(t_0,x_0) &\geq e^{\partial_t q(t_0,x_0))+F(t_0,x_0,q(t_0,x_0))}\mu(t_0,x_0).
-\int_{0}^{t}\frac{1}{x}\log_{\lambda}(1-x)dx =\int_{0}^{t}\sum_{n=1}^{\infty}\frac{\lambda^{n-1}}{n!}(1)_{n,\frac{1}{\lambda}}(-1)^{n-1}x^{n-1}dx &=\sum_{n=1}^{\infty}\frac{\lambda^{n-1}}{(n-1)!n^{2}}(1)_{n,\frac{1}{\lambda}}(-1)^{n-1}t^{n}=\mathrm{Li}_{2,\lambda}(t).
\leq (-1-d)\sum_{i=1}^{2^n}|I_i| - 2^n + d\left(\sum_{i=1}^{2^n}|I_i| -|I|\right) + n &= -\sum_{i=1}^{2^n}|I_i| - 2^n - d|I| + n &\leq -2^{n+1}- d|I| + n
\tilde{\mathcal{B}}_{\mathrm{rot}}\colon D(\tilde{\mathcal{A}}^{-1/2}) \to (L^{2}(\Omega))^{2}, f \mapsto \nabla \varphi + \mathrm{rot}' \psi (\varphi, \psi)' := \tilde{\mathcal{A}}^{-1}(f, 0)' \in \tilde{\mathcal{V}}
x_{\Sigma A}(t)=x_{A}(n_{A}(0)-\lfloor\frac{x_{\Sigma D}(0)\cdot t_{1}}{y_{A}}\rfloor-\lfloor\frac{\int_{t_{1}}^{t}x_{D}(n_{D}(0)-\lfloor\frac{\int_{t_{1}}^{t}x_{\Sigma A}(\tau)d\tau}{y_{A}}\rfloor)d\tau}{y_{A}}\rfloor)
J =\nabla (\phi \nabla)+m; Q(h,f) =\int_\Sigma \phi \nabla f\nabla h-m fh, S(h,f) =\int_\Sigma \phi \nabla f\nabla h-mfh-\alpha \int_{\partial \Sigma}\phi fh &=Q(h,f)-\alpha\int_{\partial \Sigma}\phi fh.
=\left.\widehat P_{a+1,b+1}(s'')\right|_{s''_1=0}+\left.\left[c_{b+1}\, s''_{b}\,\widehat P_{1,a+1}(s'')+c_{a+1}\, s''_{a}\,\widehat P_{1,b+1}(s'')+c_{a+1}\,c_{b+1}\, s''_{a}\, s''_{b}\,\widehat P_{1,1}(s'')\right]\right|_{s''_1=0},
Q_{3}= - \sum_{n = 0}^{\infty}\mathrm{{s g n}}( n + z ) b_{n}^{\dagger}b_{n}+ \sum_{n = 1}^{\infty}{\mathrm{s g n}}( n - z ) d_{n}^{\dagger}d_{n}+ ( z -{\frac{1}{2}}) \ .
\sum_{\substack{1\leq j\leq g d_i\geq 0}}(-1)^j<\lambda_j\tau_1\prod_{i=1}^n\tau_{d_i}>u^{2j}\prod_{i=1}^n\widetilde{\phi_{d_i}}(u,q) = (2g-2+n)\sum_{\substack{1\leq j\leq g d_i\geq 0}}(-1)^j<\lambda_j\prod_{i=1}^n\tau_{d_i}>u^{2j}\prod_{i=1}^n\widetilde{\phi_{d_i}}(u,q).
\frac{d z}{d\lambda}=\mathcal{B}R(\zeta)\left\{\frac{\omega(\zeta,\gamma_1^{(0)})\omega(\zeta, \gamma_2^{(0)})}{\prod_{j=1}^{M}\omega(\zeta,\gamma_{1}^{(j)}) \omega(\zeta,\gamma_{2}^{(j)})}\prod_{j=0}^{M}{\prod_{k=1}^{n_j}\left[\omega(\zeta,a_{k}^{(j)})\right]^{\beta_k^{(j)}}}\right\}.
[x^{\tilde i}, x^{\tilde j}] = i \theta^{\ \! \! \tilde i\tilde j}, \quad [\partial_{\tilde i}, x^{\tilde j}] = i \delta_{\tilde i}^{\tilde j}, \quad[\partial_{\tilde i}, \partial_{\tilde j}] = -i \Phi'_{\tilde i\tilde j}\quad (\tilde i, \tilde j = \tilde 1, \tilde 2)
p^{-1}(g) = \overline{{\mathcal{M}}_{0,r}}(Fl_J, \d) \times_{(Fl_I)^r}\left(g \cdot (Fl_J)^r \right) \simeq \pi^{-1}(g) & \simeq g \cdot \left(\overline{{\mathcal{M}}_{0,r}}(Fl_J, \d) \times_{(Fl_I)^r}(Fl_J)^r \right) \simeq \overline{{\mathcal{M}}_{0,r}}(Fl_J, \d) \times_{(Fl_I)^r}(Fl_J)^r
\displaystyle{\left. \Pi_{2 2}( p ) \right|_{\mathrm{G}}}= \displaystyle{\frac{1}{4 \pi}\left[ - \frac{p^{2}}{8}\mathrm{l n}\left( \frac{p^{2}}{\mu^{2}}\right) - \frac{5 3}{6}\frac{( p_{1})^{2}( p_{2})^{2}}{p^{2}}\right] \, .}
\{(n,m): \#(R_{n})=m\}= \{(n,m): \exists \; X \; (\forall \; x \; x\in X\leftrightarrow Rnx) \; \& \; \#X=m\}= \{(n,m): \forall \; X \; (\forall \; x \; x\in X\leftrightarrow Rnx) \; \rightarrow \; \#X=m\}
\lim\limits_{N \rightarrow\infty}\frac{K\lambda_0}{N}= (K+M)N, \lim\limits_{N \rightarrow\infty}\frac{K|\lambda_1|}{2N\cos^2 \frac{\pi}{2N-1}}= \left ( \frac{2K}{\pi^2}-\frac{M}{2}\right )N.
|\Omega| =\int_\Omega u^{\frac{1}{r}}(t,x;s,u_0) u^{-\frac{1}{r}}(t,x;s,u_0)dx &\le \left(\int_\Omega u(t,x;s,u_0)dx\right)^{\frac{p}{p+1}}\left(\int_\Omega u^{-p}(t,x;s,u_0)dx \right)^{\frac{1}{p+1}}\forall\, t>s.
F_{2n}=\sum_{k=1}^n{n+k-1\choose n-k},F_{2n-1}=\sum_{k=1}^n{n+k-2\choose n-k}, J_{2n}=\sum_{k=1}^n2^{n-k}{n+k-1\choose n-k},J_{2n-1}=\sum_{k=1}^n2^{n-k}{n+k-2\choose n-k}.
\Delta' \lambda = \left(\frac{d^4 G_E}{d \varphi_0^4}\right)_{s=1}= -\frac{3\lambda^2}{32\pi^3}\, \sqrt{a_3 a_4}\, \lim_{s\rightarrow 1}A_2^{c^2}(s,a_3,a_4) \,.
\max_{\vec v\in\mathcal{V}_{x}\setminus \vec 0}\left|\log\frac{\|\vec v\|_{2\,(x)}}{\|\vec v\|_{1\,(x)}}\right|=\max_{\vec v\in\mathcal{V}_{x}\setminus \vec 0}\left|\log\frac{\|\vec v\|_{1\,(x)}}{\|\vec v\|_{2\,(x)}}\right|
CG_{2k}(z) = \prod_{n=0}^\infty(1+zq^{n+1})^{2k}(1+z^{-1}q^n)^{2k}\equiv \prod_{n=0}^\infty(1+z^2q^{2n+2})^{k}(1+z^{-2}q^{2n})^{k}\pmod{2}.
\|R_j f_q\|_\ell = \|R_j \Delta_q \tilde{f}_q\|_\ell & \leq \|R_j \Delta_{q-1}f_q\|_\ell+\|R_j \Delta_q f_q\|_\ell+\|R_j \Delta_{q+1}f_q\|_\ell & \lesssim\|f_q\|_\ell,
\Phi_2(\varrho)=\frac{(2\,\varrho)^{-\frac{N-1}{2}}}{(1-\varrho)^{1+s\,p}}\int_{0}^{\frac{\varrho}{(1-\varrho)^2}}\frac{\left(2-\frac{(1-\varrho)^2}{2\,\varrho}\,\tau\right)^\frac{N-3}{2}\,\tau^\frac{N-3}{2}}{(1+\tau)^\frac{N+s\,p}{2}}\, d\tau.
\mathcal{A}= \frac{1}{r R^{2}}\begin{pmatrix}r(\sigma r_{1}^{2}+\overline{\sigma}r_{2}^{2}+ \sigma r_{3}^{2}) (\overline{\sigma}-\sigma)r_{3}r_{2}^{2}(\overline{\sigma}-\sigma)r_{3}r_{2}^{2}r_{3}(\sigma r_{1}^{2}+\overline{\sigma}r_{2}^{2}+ \sigma r^{2})\end{pmatrix}.
(\Lambda_{N}(t)-\mid\gamma+t\mid^{2})(\Psi_{N,t},e^{i\left\langle\gamma+t,x\right\rangle})=\sum_{\gamma_{1}}\dfrac{q_{\gamma_{1}}(q\Psi_{N,t},e^{i\left\langle \gamma-\gamma_{1}+t,x\right\rangle})}{\Lambda_{N}(t)-\mid\gamma-\gamma_{1}+t\mid^{2}}.
B_\eta \left( \left(\overline{x}_{n(p)+\i-1,\widehat{n(p)+\eta-1}}\right)_{\i=1}^{\infty},\left(\overline{x}_{m(p)+\i-1,\widehat{m(p)+\eta-1}}\right)_{\i=1}^{\infty}\right) & = d\left(\overline{x}_{n(p)+\eta-1},\overline{x}_{m(p)+\eta-1}\right).
\chi(G) \leq 1 + f(n - s, t - y s) = 1 + 2 \sqrt{n - s}+ 6^{1/3}(t - y s)^{1/3}&\leq 1 + 2 \sqrt{n}- \frac{s}{\sqrt{n}}+ 6^{1/3}(t^{1/3}- \frac{y s}{3 t^{2/3}}) = f(n,t) + 1 - \frac{s}{\sqrt{n}}- \frac{6^{1/3}y s}{3 t^{2/3}}
( \widetilde \psi , \widetilde \psi ) \ = \ - \eta \int d^{3}x^{T}4 P^{2}\ \overline{{{\widetilde \Psi}}}\ \frac{\partial}{\partial s}[ \widetilde g_{0}^{- 1}+ \widetilde V ] \ \widetilde \psi \ = \ 2 P_{L}\ ,
{|N_G(S')|=|N_G{(S_{1}')}| \geq \mu |S_{1}'| \binom{n-32}{2}\geq \mu \dfrac{|S'|}{\lambda_2}\binom{n-32}{2}}\geq \binom{n-32}{2}\dfrac{|S'|}{\dfrac{(n-2)(n-3)}{12}+\dfrac{\alpha}{\mu}}\geq |S'|,
\frac{\kappa p+1}{2(\kappa+1)^p}\|u u_h^{\kappa}\|_{1,p}^p &\leq \left[\frac{\kappa p+1}{(\kappa+1)^p}+M_9(\kappa p +1)L(\kappa, u)\right]\|u u_h^{\kappa}\|_{p}^p begin{align*}1ex] +M_{10}G(\kappa, u) \| u u_h^{\kappa}\|_{p,\partial\Omega}^p +M_{11}\kappa.
g^{*}(\tau)=(\frac{i}{2})^{k-1}\int_{-\overline{\tau}}^{i\infty}(z+\tau)^{-k}\overline{g(-\overline{z})}dz=\sum_{n}n^{k-1}\overline{b_{n}}\beta_{k}(4ny)q^{-n+1}
\left( \begin{matrix}{\alpha}&{\beta}\\{\gamma}&{\delta}\\ \end{matrix}\right) = \left( \begin{matrix}{{\frac{b_{-}}{\sqrt{1 - b_{3}}}}}&{\sqrt{1 - b_{3}}}\\{- \sqrt{1 - b_{3}}}&{{\frac{b_{+}}{\sqrt{1 - b_{3}}}}}\\ \end{matrix}\right) \left( \begin{matrix}{e^{\imath \phi}}&{0}\\{0}&{e^{- \imath \phi}}\\ \end{matrix}\right) .
\frac{\partial\tilde{\rho}}{\partial t}=\frac{1}{2}\nabla\cdot\left[D(\tilde{\rho})\nabla\tilde{\rho}\right]-\nabla\cdot\left[A(\tilde{\rho})\left(\tilde{b}-\frac{1}{\lambda}\tilde{\Gamma}\tilde{\rho}\right)\right]
\begin{aligned}\mathcal{VW}:\mathcal{C}_k(P)\to\mathcal{C}_{k-1}'(P), \mathcal{VW}(A,B,C) :=\begin{pmatrix}d_A^*B+d_AC F_A^++\frac{1}{8}[B\centerdot B]+\frac{1}{2}[B,C] \end{pmatrix}.\end{aligned}
\bar{\Psi}^{\downarrow}= \psi : e^{\frac{i}{\sqrt{2}}( \varphi_{c}+ \varphi_{s})}: \, , \qquad \Delta_{s}^{\uparrow \uparrow}= \, : e^{i \sqrt{2}\varphi_{s}}: \, , \qquad \Delta_{c}= \, : e^{i \sqrt{2}\varphi_{c}}: \ ,
\mathrm{Mant}_{\mathrm{GL}_{n_1}, (1^2, 0^{n_1-2})}(LJ (\rho_1(n_2/2)))=[\rho_1(n_2/2)][r_{(-1^2, 0^{n_1-2})}\circ LL(\rho_1(n_2/2))\otimes |\cdot |^{2-n_1}],
{\cal V}(x_0 ,\theta_0, \bar\beta, \bar\zeta,\bar\omega,\rho )=e^{iPx_0}e^{-iQ\theta_0}e^{-i\bar{S}\bar\beta}e^{-i\bar Q\bar \zeta}e^{i\bar M \bar\omega}e^{iD\ln\rho}\,.
\mathcal{R}\left( t \right) = \mathrm{cl}\left\lbrace \phi_{t}\left( f \right) \middle\vert f \in \mathcal{S}\left( \left[ 0,\infty \right) ; \mathcal{E}\right) \right\rbrace = \left\lbrace \phi_{t}\left( f \right) \middle\vert f \in \mathcal{D}\left( \left[ 0,\infty \right) ; \mathcal{E}\right) \right\rbrace.
\Delta \hat E^{(q)}_n=\frac{f^2}{6}\int_{-\infty}^\infty d\hat x_0 \psi^{(0)*}_n(\hat x_0)\Bigl\{V(\hat x_0) \Bigl[1-4\mu\hat x_0^2 \Bigl(V(\hat x_0) -E^{\rm (un)}_n\Bigr)\Bigr]-{\textstyle\frac{2}{3}}\mu E^{\rm(un)}_n \hat x_0^3 V'(\hat x_0)\Bigr\}\psi^{(0)}_n(\hat x_0).
b_M(m \otimes \{a_1 | ... | a_n \}) := d_Mm \otimes \{a_1 | ... |a_n\}- \sum_{i=1}^n m \otimes (-1)^{\epsilon_i}\{a_1| ...|d_Aa_i| ... |a_n\}&+ (-1)^{|m|}(m \cdot a_1) \otimes \{a_2 | ... | a_n \}+ \sum_{i=1}^{n-1}(-1)^{\epsilon_i}m \otimes \{a_1 | ... |a_i a_{i+1}|... | a_n \}
V_{N ; g}={\cal C}_{g}^{o p e n}\prod_{i = 1}^{N}\bigg [{}_{i}\! < \! p_{i}; O_{a}| \bigg ] \delta \left( \sum_{i = 1}^{N}p_{i}\right) \int [{\mathrm{d}}m ]_{g}^{N}{\hat{V}}_{N ; 0}
\int_{0}^{\infty}t^{- \lambda}J_{\nu}^{2}( \alpha t ) d t = \frac{\alpha^{\lambda - 1}\Gamma ( \lambda ) \Gamma \left( \displaystyle{\frac{2 \nu - \lambda + 1}{2}}\right)}{2^{\lambda}\Gamma^{2}\left( \displaystyle{\frac{\lambda + 1}{2}}\right) \Gamma \left( \displaystyle{\frac{2 \nu + \lambda + 1}{2}}\right)}\, ,
S_{1}= \frac{V_{1}}{R_{1}}= \frac{0 . 7 5 \times 1 0^{- 3}m^{3}}{0 . 0 3 m}= \frac{0 . 9 \times 1 0^{3}k g / m^{3}}{2 5 \times 1 0^{3}m^{3}}S_{2}= S_{1}\times 2 = 2 5 \times 1 0^{- 3}m^{2}= 5 \times 1 0^{- 2}m^{3}
\sum_{\substack{X^{\eta}\le p_1\le \dots\le p_\ell X^{1-\theta_2}\le\prod_{i\in\mathcal{I}}p_j\le X^{1-\theta_1}p_1\cdots p_\ell\le X/p_j}}^*S_{p_1\cdots p_\ell}(p_j)=o_{\mathcal{L},\eta}\Bigl(\frac{\#\mathcal{A}}{\log{X}}\Bigr),
\frac{1}{8\pi G}(R_{\mu\nu}-\frac{1}{2}R g_{\mu\nu}) =T_{\mu\nu}+ \langle T_{\mu\nu}\rangle^{G}_{\lambda=0}+\langle T_{\mu\nu}\rangle^{M}_{\lambda=0}+\langle T_{\mu\nu}\rangle_{a(\lambda)}+\langle T_{\mu\nu}\rangle_{b(\lambda)},
\bar{\Pi}^{g}_{ij}=\frac{\partial{\cal L}}{\partial (\partial_{0}g_{ij})}\,=\,\frac{1}{\lambda^2}\partial^{0}g^{-1}_{ji}\,-\,\frac{n}{4\pi}A_{ji}(g)\,+\,\frac{i\alpha}{4\lambda^2}\bar{\psi}_{jk}\gamma^1\psi_{km}g^{-1}_{mi}
\delta A = - \delta \bar{\theta}\Gamma_{1 1}\Gamma_{m}\theta \Pi^{m}+{\frac{1}{2}}\delta \bar{\theta}\Gamma_{1 1}\Gamma_{m}\theta \bar{\theta}\Gamma^{m}d \theta -{\frac{1}{2}}\delta \bar{\theta}\Gamma^{m}\theta \bar{\theta}\Gamma_{1 1}\Gamma_{m}d \theta .
\sum_{n=1}^{\infty}R^{E_n}_{M_D(\cdot, z)}(x_0)<\infty \textrm{if and only if}\sum_{n=1}^{\infty}2^{nd}\frac{\Phi(2^{-n})}{g(A_{2^{-n}}(z))^2}\gamma_{g}(E_n)<\infty\, .
\psi(g,s)=2\zeta(2s-1/2) \sum_{\mathfrak{a}\in \mathcal{E}_k}\sum_{\gamma\in \Gamma_{k,\vec{h}_\mathfrak{a}}\backslash \Gamma_k}\frac{1}{(\vec{h}_{\mathfrak{a}}^{\intercal}(\gamma g)^{-\intercal}P_0(\gamma g)^{-1}\vec{h}_{\mathfrak{a}})^{s-1/4}},
f^{(\pm)}=e^{\mp ikx}+e^{\mp ikx}\sum_{n=0}^{\infty}(\frac{-1}{\pi})^{n+1}\int_{1}^{\infty}d\alpha_o...\int_{1}^{\infty}d\alpha_n\frac{(\prod_{j=0}^{n}D(\nu;\alpha_j)e^{-2\alpha_jx})}{[\prod_{j=0}^{n-1}(\alpha_j+\alpha_{j+1})][\alpha_o\pm ik]}
I_1 = [D_uG_{n_0}(\ell x, G_{m-n_0}(hx,z))-D_uG_{n_0}(\ell x, G_{m-n_0}(hx,z'))] \D\ell(x), I_2 = [D_sG_{n_0}(\ell x, G_{m-n_0}(hx,z))-D_sG_{n_0}(\ell x, G_{m-n_0}(hx,z'))]\D_uG_{m-n_0}(hx,z) \Dh(x), I_3 = D_sG_{n_0}(\ell x, G_{m-n_0}(hx,z')) \ [D_uG_{m-n_0}(hx,z)-D_uG_{m-n_0}(hx,z')] \Dh(x) .
|E_2| = \int_K \Phi_{E_2}(x)dx =\int_{E_2}1~ dx ~\leq ~\dfrac{4}{{\lambda}^2}\|B f_2(x)\|^2_{L^2(K)}~\leq ~ \dfrac{4}{{\lambda}^2}\| f_2\|^2_{L^2(K)}~\leq~ \dfrac{4}{{\lambda}^2}\lambda q\| f\|_{L^1(K)}~= ~\dfrac{4q}{\lambda}\|f\|_{L^1(K)}.
J_K^{2,+}u(x) := \{(p,X)\in \mathbb{R}^n \times \mathbb{S}^n| \u(y)\le u(x) + p\cdot (y-x) + \frac{1}{2}X(y-x)\cdot (y-x) + o(|y-x|^2) \{\rm as}\y\in K\rightarrow x\},
\phi(\psi\otimes\bar{v}; z,\bar{z}) \;\phi(\chi\otimes\bar{w}; \zeta,\bar{\zeta})\; = \;\phi\left( \phi\left(\psi\otimes\bar{v};z-\zeta,\bar{z}-\bar{\zeta}\right)(\chi\otimes\bar{w}); \zeta,\bar{\zeta}\right),
\varphi(x) = c_k^{-\frac{1}{2}}\left| \frac{J}{u}\right|^{\frac{1}{2}}\int_{-\infty}^\infty \left(x_1' + \frac{\sqrt{-1}}{v}x_2' \right)^k e^{- \frac{\pi}{2}(v x_1'^2 + \frac{1}{v}x_2'^2)}e^{2 \pi \sqrt{-1}x_2'y_2'}\, dx_2' \int_{-\infty}^\infty e^{- \frac{\pi}{2}(v x_3'^2 + \frac{1}{v}x_4'^2)}e^{2 \pi \sqrt{-1}x_4'y_4'}\, dx_4'.
\begin{array}{c c}{{\left\{\, b ( \sigma , \tau ) , c ({\sigma}^{\prime}, \tau ) \right\}\, = 2 \pi \delta ( \sigma -{\sigma}^{\prime})}}&{{}}\\{{\left[ \beta ( \sigma , \tau ) , \gamma ({\sigma}^{\prime}, \tau ) \right] = 2 \pi \delta ( \sigma -{\sigma}^{\prime})}}&{{}}\\ \end{array}
\mu_{k}^{2}(\mathcal{N}) =\mu_{k}\left(\mu_{k}(\mathcal{N})\right) &=\widetilde{\mu_{k}}(\mu_{k}\left(\mathcal{N}\right))_{red}&=\widetilde{\mu_{k}}\left( \widetilde{\mu_{k}}(\mathcal{N}\right)_{red})_{red}
xu = xu_j-\sum_{i<j}y^{f_i}\rho_ixu_i = \alpha_j + y^{e_j}s_j - \sum_{i<j}y^{f_i}\rho_i(\alpha_i+y^{e_i}s_i) = \left(\alpha_j-\sum_{i<j}y^{f_i}\rho_i\alpha_i\right)+y^{e_j}\left(s_j-\sum_{i<j}\rho_is_i\right)
\Upsilon_+(z)=\Upsilon_-(z)\begin{pmatrix}1 0 -1 1\end{pmatrix}, z\in \Sigma_{A,1}, &\Upsilon_+(z)=\Upsilon_-(z)\begin{pmatrix}0 -1 1 0\end{pmatrix}, z\in \Sigma_{A,2}, &\Upsilon_+(z)=\Upsilon_-(z)\begin{pmatrix}1 0 -1 1\end{pmatrix}, z\in \Sigma_{A,3}, &\Upsilon_+(z)=\Upsilon_-(z)\begin{pmatrix}1 -1 0 1\end{pmatrix}, z\in \Sigma_{A,4}.
H^{2}= \frac{\rho_{m}^{2}}{3 6}+ \frac{U_{B}\rho_{m}}{1 2}- \frac{1}{1 6 a^{4}}\int d \tau \frac{d a^{4}}{d \tau}( \dot{\phi}^{2}- 2 U ) - \frac{1}{1 2 a^{4}}\int d \tau a^{4}\rho_{m}\frac{d U_{B}}{d \tau},
\sum_{\ell=1}^{k}(-1)^{\ell}(n+\ell)!L(k,\ell)=\sum_{\ell=0}^\infty(-1)^{\ell}(n+\ell)!L(k,\ell) =-k! (n+1)!{}_2F_1(1-k,n+2;2;1).
\frac{d^{2}\Delta z}{dt^{2}}=-g^{\prime}=-g(\rho_{0}(z_{0})-\rho_{0}(z_{0}+\Delta z))/\rho_{0}(z_{0})\simeq-g(-\frac{d\rho_{0}}{dz}\Delta z)/\rho_{0}(z_{0})
=\frac{1}{r}\frac{d\psi}{dr}+x_{i}\frac{d\psi}{dr}\frac{\partial r}{\partial x_{i}}\frac{d}{dr}\frac{1}{r}+\frac{x_{i}}{r}\frac{\partial r}{\partial x_{i}}\frac{d}{dr}\frac{d\psi}{dr}
\bar{b}(\theta) = \sum_{k=1}^{q}\bar{b}_k \theta^k \bar{b}(\theta) = \begin{bmatrix}\bar{b}_1(\theta) \vdots \bar{b}_s(\theta) \end{bmatrix}\bar{b}_{k}= \begin{bmatrix}\bar{b}_{1k}\vdots \bar{b}_{sk}\end{bmatrix}\, k=1,2,\ldots, q
\Vert \Lambda \Vert_{L_2(\Omega)}\leqslant \vert \Omega \vert^{1/2}M_1,M_1:= m^{1/2}(2r_0)^{-1}\alpha_0^{-1/2}\Vert g\Vert^{1/2}_{L_\infty}\Vert g^{-1}\Vert^{1/2}_{L_\infty}, &\Vert \mathbf{D}\Lambda \Vert_{L_2(\Omega)}\leqslant \vert \Omega \vert^{1/2}M_2,M_2:= m^{1/2}\alpha_0^{-1/2}\Vert g\Vert^{1/2}_{L_\infty}\Vert g^{-1}\Vert^{1/2}_{L_\infty}.
J^2 = \left( \sum_{j=1}^{n-1}e_{j+1}e_j^T\right)\left(\sum_{j=1}^{n-1}e_{j+1}e_j^T\right) = \sum_{j=2}^{n-1}(e_{j+1}e_j^T)(e_je_{j-1}^T) = \sum_{j=1}^{n-2}e_{j+2}e_j^T.
G ( \beta , L ) = \frac{3 \mu}{8 \pi m}\left[ ( e^{m \beta}- 1 )^{- 1}+ ( e^{m L}- 1 )^{- 1}+ 2 \sum_{n_{1}, n_{2}= 1}^{\infty}e^{- m \sqrt{\beta^{2}n_{1}^{2}+ L^{2}n_{2}^{2}}}\right] ,
\lim_{n\rightarrow\infty}a_n^{-\beta}\sum_{x,y\in\mathbb Z^{d_0}}\left|\sum_{i=1}^m\theta_i\Big(N_{\lfloor nt_i\rfloor}(x)N_{\lfloor nt_i\rfloor}(y)-N_{\lfloor nt_{i-1}\rfloor}(x)N_{\lfloor nt_{i-1}\rfloor}(y)\Big)\right|_\pm^\beta=K_\beta^2\sum_{j=1}^m|\theta_j|_\pm^\beta(t_j^2-t_{j-1}^2) .
\frac{\big(q^{\lambda+K+1}\xi/x, q^K\xi\alpha_1, q^{K-1}\xi\alpha_2, q^K\xi\alpha_3 ;q\big)_{\infty}}{\big(q^K\xi/x, q^K\xi\beta_1, q^K\xi\beta_2, q^K\xi\beta_3 ;q\big)_{\infty}}=0
\int_{\mathbb{R}^d}|x|^2d\mu_{t}(x)= (2\pi t)^{-n/2}\int_{B_R}\int_{\mathbb{R}^d}(|x|^2+|y|^2)\exp\left(-\frac{|x|^2}{2t}\right)dx\,d\mu(y) \leq (2\pi t)^{-n/2}\int_{\mathbb{R}^d}\int_{B_R}(|x|^2+R^2)\exp\left(-\frac{|x|^2}{2t}\right)d\mu(y)dx = (2\pi t)^{-n/2}\int_{\mathbb{R}^d}(|x|^2+R^2)\exp\left(-\frac{|x|^2}{2t}\right)dx = nt+R^2,
+\frac{2}{AB-1}(AX_{n}^{0}\tilde{X}_{n}^{1}-BX_{n}^{1}\tilde{X}_{n}^{0})e^{-nl_{0}}e^{-nl_{1}}]exp\sum_{n \geq 1}[X_{-n}^{0}\tilde{X}_{-n}^{1}-X_{-n}^{1}\tilde{X}_{-n}^{0}]|0>
\mathcal{L}(t_{k+1}-t_k \mid X_{t_k}^{(0)}<0) =\mathcal{L}(t_1 \mid X_0^{(0)}=x) \mathcal{L}(t_{k+1}-t_k \mid X_{t_k}^{(0)}\geq 0) =\mathcal{L}(t_1 \mid X_0^{(0)}=y)
\frac{1}{2}\frac{\varphi(z) + \varphi(w)^{*}}{1 - z \overline{w}}= \int_{{\mathbb T}}\frac{1}{(1 - z \overline{\zeta}) (1 - \overline{w}\zeta)}(I - S(\zeta)^{*}S(\zeta)) \frac{|{\tt d}\zeta|}{2 \pi}.
[st,(1,1)]_g[1,(bp_n(x),cq_n(x))]_g =\Theta_{n-1}([st,(1,1)]_f) &=\Theta_{n-1}([s,(1,1)]_f)\Theta_{n-1}([t,(1,1)]_f) &=[st,(1,1)]_g[1,(b_sp_n(x_s)b_tp_n(x_t),c_sg_n(x_s)c_tg_n(x_t))]_g
|c_{\alpha^{(1)}}(B_{K}(a))c_{\alpha^{(2)}}(B_{K}(a))|^{\frac{1}{2}}\lesssim \iint_{\mathbb{R}^{6}\times \mathbb{R}^{6}}\prod_{i=1}^{2}|E_{\alpha^{(i)}}f(x-y_{i})|^{\frac{1}{2}}\zeta_{K}(y_{i})\,dy_{1}dy_{2}.
F_- =T(\phi) F_+ = \left( \begin{array}{llll}e^{-2 i q_1 \phi}0 \cdots 0 0 e^{-2 i q_2 \phi}0 \vdots \ddots \vdots 0 0 \cdots e^{-2 i q_n \phi}\end{array}\right)F_+
-(\lambda_{n,1}^{(1)}-\lambda_{n,1}^{(2)})+\log\frac{\lambda_{n,1}^{(1)}}{\lambda_{n,1}^{(2)}}=\log\left(1+O\left(\sum_{i=1}^2\frac{1}{\lambda_{n,1}^{(i)}}\right)\right)=O\left(\sum_{i=1}^2\frac{1}{\lambda_{n,1}^{(i)}}\right).
\kappa_n := \sum_{\pi \in \mathfrak{S}(n)}\left( \mathrm{sign}(\pi) \left( w\kappa_{d_\pi}\right) \otimes \bigotimes_{\ell \vert (n/d_\pi)}\rho_\ell (P_\ell (\mathrm{Fr}^{-1}_{\pi(\ell)}) ) \right) \in \mathrm{H}^1(\mathbb{Q}, T_{\overline{f}}(1) / I_n T_{\overline{f}}(1) ) \otimes G_n