harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
P\{\tilde{T}_1 = \tilde{T}_{\min}\}= \int_0^{\infty}\cdots \int_0^{\infty}\int_0^{t_2 \wedge \cdots \wedge t_g}f_g(t_g) \cdots f_2(t_2) f_1(t_1) \, dt_1 dt_2 \cdots dt_g & = \int_0^{\infty}\cdots \int_0^{\infty}f_g(t_g) \cdots f_2(t_2) F_1(t_2 \wedge \cdots \wedge t_g) \, dt_2 \cdots dt_g,
x_{2}= 1 - 1 / ( N + 1 ) - 1 / r^{N + 1}+{\cal O}\bigl ( 1 / r^{2 ( N + 1 )}\bigr ) = 1 - 1 / N + 1 / N^{2}+{\cal O}\bigl ( 1 / N^{3}\bigr ) .
\mathcal{A}:= \left\{(\rho, \phi): -\nabla \cdot(\epsilon(x) \nabla \phi) = f(x)+\sum_{i=1}^{s}z_i\rho_i, \alpha \phi+\beta\frac{\partial \phi}{\partial \mathbf{n}}=\phi^b, x\in \partial \Omega, \rho\in [\mathcal{P}(\Omega)]^s \right\}.
Q_\ell^q\Big(\ell^{q_1}a_1,\ell^{q_2}a_2\Big)=(-1)^q \ \Big(\ell^{q_2-q_1}\big(\ell^{r-2(q-q_1)}a_1\big) a_2\Big)=0, a_1 \in P_\ell^{q-q_1}, \ \a_2 \in P_\ell^{q-q_2}.
Uniform_{\mathbb{GT}_c(N)}^{x^{(N)}}(dx^{(1)},\cdots,dx^{(N-1)})=\frac{\prod_{j=1}^{N-1}j!}{\Delta_N(x^{(N)})}\textbf{1}\left(x^{(1)}\prec x^{(2)}\prec \cdots \prec x^{(N-1)}\prec x^{(N)}\right)dx^{(1)}\cdots dx^{(N-1)},
\textsf{E}\exp(C_{0}\lambda \Vert B_{i}\Vert_{2}^{2}(\max_{j\le n}\eta_{j}-\textsf{E}\max_{j\le n}\eta_{j}))=\prod_{l=1}^{n}\textsf{E}\exp(C_{0}\lambda \Vert B_{i}\Vert_{2}^{2}\frac{T_{l}-2}{2(n-l+1)}),
ds_4^2 = - \frac{1}{\sqrt{-{1 \over 6}H^0 \, C_{ABC}H^A H^B H^C}}\,dt^2 + \sqrt{-{\textstyle{1 \over 6}}H_0 \, C_{ABC}H^A H^B H^C}\, d\vec{x}^2 \;.
\lim_{k\rightarrow \infty}H(G^\prime_k,\frac{\mathbf 1}{m_k}) =\lim_{k\rightarrow \infty}\log \chi_f(G^\prime_k) = \lim_{k\rightarrow \infty}H(G,P_k) = H(G,P) \leq \log\chi_f(G).
\sup_U\Phi_{L,C,\eta}=u(l,\bar{t})-u(\bar{y},\bar{t})-L\phi(|l-\bar{y}|)-C|l-\hat{x}|^2-\frac{2}{\eta(T-\bar{t})}&\le u(l,\bar{t})-u(\bar{y},\bar{t})-L\phi(|l-\bar{y}|) &\le (L_2-L(1-\hat{r}^\alpha))|l-\bar{y}|,
\sum_{i}\gamma_{i}^{\left( D\right)}\left( a,b,c\right) \,\alpha_{il}\left( a,b,c\right) \alpha_{il^{\prime}}\left( a,c,b\right) =\left(-1\right)^{\left( m-2\right) \left( l-1\right)}\gamma_{l}^{\left(D\right)}\left( b,a,c\right) \delta_{ll^{\prime}}.
\Biggl [ - \frac{1}{2}\frac{d^{2}}{d x_{\parallel}^{2}}- \frac{1}{2}\Biggl ( \omega + q E_{0}x_{\parallel}\Biggr )^{2}\Biggr ] \phi_{\omega ,{\bf k}_{\perp}}( x_{\parallel}) = - \frac{1}{2}( m^{2}+{\bf k}_{\perp}^{2}) \phi_{\omega ,{\bf k}_{\perp}}( x_{\parallel}) .
\P(\Omega \setminus \Omega_{\mathrm{coupling}}) = \P(R_1 \neq R_2 \ \mbox{or}\T_2^z \leq T_1^y) = \P(R_1 \neq R_2 \ \mbox{or}\T_2^z \leq T_1^z + (z-y)/2 ) \leq \P(R_1 \neq R_2) + \P(T_2^z \leq T_1^z + (z-y)/2) = c_1(y,z) + c_2(y,z).
\phi(-q^{\frac{j-i}{2}}x_{j-1}x_{j-2}\cdots x_{i}) =\displaystyle \sum_{m\geq 0}(-1)^{m}\frac{q^{\frac{m(m-1)}{2}}(-q^{\frac{j-i}{2}}x_{j-1}\cdots x_{i})^{m}}{(q)_{m}}&= \displaystyle \sum_{m\geq 0}\frac{q^{\frac{m^{2}}{2}}q^{\frac{m(j-i)}{2}-\frac{m}{2}}(x_{j-1}\cdots x_{i})^{m}}{(q)_{m}},
A_L = \mathrm{N}_{q^m/q}(\alpha_1/ \alpha_2)\begin{pmatrix}-u^{\sigma^{-1}}G_{m-2}^{\sigma}-(\alpha_0/\alpha_1)G^\sigma_{m-1}(\alpha_2/\alpha_1)^{\sigma^{-1}}G_{m-1}G_m\end{pmatrix},
\frac{s_{2}^{n_{2}}}{n_{2}}=\frac{s_{2}^{n_{2}}(\rho_{2})}{n_{2}}=\int_{0}^{\rho_{2}}t_{2}^{n_{2}-1-\frac{\gamma_{2}}{p-1}}dt=\frac{p-1}{n_{2}(p-1)-\gamma_{2}}\rho_{2}^{\frac{n_{2}(p-1)-\gamma_{2}}{p-1}}
\mathrm{d}Y_1(t) = A_1(Y_1,Y_2,t) \mathrm{d}t + b_{11}(Y_1,Y_2,t)\mathrm{d}W_1(t) + b_{12}(Y_1,Y_2,t) \mathrm{d}W_2(t) \mathrm{d}Y_2(t) = A_2(Y_1,Y_2,t) \mathrm{d}t + b_{21}(Y_1,Y_2,t)\mathrm{d}W_1(t) + b_{22}(Y_1,Y_2,t) \mathrm{d}W_2(t).
W_{\alpha}= - i \lambda_{\alpha}+ \theta \cdot \sigma^{\mu}\cdot \bar{\theta}\partial_{\mu}\lambda_{\alpha}- \frac{i}{4}\theta^{2}\bar{\theta}^{2}\partial^{2}\lambda_{\alpha}+ \theta_{\alpha}D - \frac{i}{2}\theta^{2}\sigma^{\mu}{}_{\alpha \dot{\alpha}}\bar{\theta}^{\dot{\alpha}}\partial_{\mu}D
\zeta(s) = \Re(\zeta(s))+i\Im(\zeta(s)) = \sum_{n=1}^\infty \frac{1}{n^\sigma}\left[ \cos( t\ln n) - i\sin(t\ln n)\right] \Rightarrow \Re(\zeta(s)) = \sum_{n=1}^\infty n^{-\sigma}\cos(t\ln n) \Im(\zeta(s)) = \sum_{n=1}^\infty n^{-\sigma}\sin(t \ln n)
\overline{\Psi}_{f_{1}}\left(u,T\right)=J\left(u,T\right)\int_{\left(S_{p}\times S_{q}\right)^{n}}f_{1}\left(\left\Vert g_{0}^{-1}\left(u_{1}\eta_{1},\sqrt{u_{1}^{2}-u_{2}/T^{2}}\eta_{2},\dots,u_{2n}\eta_{2n}\right)\right\Vert \right)d\eta
S_{1}=\mathrm{tr}(\mathcal{A}_{\widehat{W}})=-\sum_{i=1}^{n}\epsilon_{i}\overline{g}(\overline{\nabla}_{Z_{i}}\widehat{W},Z_{i})=-\sum_{i=1}^{n+1}\epsilon_{i}\overline{g}(\overline{\nabla}_{Z_{i}}\widehat{W},Z_{i})=-\mathrm{div}^{\overline{\nabla}}(\widehat{W}),
\mathbf{E}^{\prime}( \mathbf{x}) = \frac{1}{4 \pi}\int \rho_{e}^{\prime}( \mathbf{x}^{\prime}) \frac{\mathbf{r}}{r^{3}}\, d^{3}x^{\prime}\, - \frac{1}{4 \pi}\int \mathbf{J}_{m}^{\prime}( \mathbf{x}^{\prime}) \,{\bf \times}\, \frac{\mathbf{r}}{r^{3}}\, d^{3}x^{\prime},
J_{i,k}^{(s)}(l_1,l_2) =\frac{1}{[2\Gamma(1/s)]^{k+1}}\int_{l_1}^{l_2}y^{\frac{i+1}{s}-1}\,\exp(-y)\left[2\Gamma(1/s)-\Gamma(1/s,y)\right]^k\mathrm{d}y.
\beta = \left( \frac{\beta_{1}- \beta_{0}e^{k_{2}y_{1}}}{1 - e^{( k_{2}- k_{1}) y_{1}}}\right) e^{k_{1}( y - y_{1})}+ \left( \frac{\beta_{0}- \beta_{1}e^{- k_{1}y_{1}}}{1 - e^{( k_{2}- k_{1}) y_{1}}}\right) e^{k_{2}y},
Y_{n}^{\left( k_{1}+k_{2}\right)}\left( \lambda \right)=\sum_{v_{1}=0}^{n}\left( \begin{array}{c}n v_{1}\end{array}\right) Y_{v_{1}}^{\left( k_{1}\right)}\left( \lambda \right)Y_{n-v_{1}}^{\left( k_{2}\right)}\left( \lambda \right) .
\partial_n^{(0)}(f)(g_1,\dots, g_{n-1}) =\sum_{g_0\in G}g_0^{-1}f(g_0,g_1, \dots, g_{n-1}), \partial_n^{(i)}(f)(g_1,\dots, g_{n-1}) =\sum_{\substack{g, \bar{g}\in G g\bar{g}=g_i}}f(g_1, \dots, g_{i-1}, g, \bar{g}, g_{i+1}, g_{n-1})~ ~1\leq i\leq n-1, \partial_n^{(n)}(f)(g_1,\dots, g_{n-1}) =\sum_{g_n\in G}f(g_1, \dots, g_{n-1}, g_n).
{\rm d}\mu= e^{-8\pi^2/g^2}\,(M_{\rm PV})^{n_b-\frac{1}{2}n_f}\,(2\pi)^{-n_b/2}\,\prod_i{\rm d}\eta_i \, \left\{\mbox{Ber}\, \left\langle\frac{\partial\Phi(\eta)}{\partial \eta_j}\bigg|\frac{\partial\Phi(\eta)}{\partial \eta_k}\right\rangle\right\}^{\frac{1}{2}}\,,
s_i := \sum_{k=1}^\infty\frac{1}{k!}\sum_{\substack{x_1,\ldots,x_k\in \Lambda, m_{i-1}\le d(x_1,\ldots,x_k)< m_i}}( \rho_+(x_1, \ldots,x_k)^2 + \rho_-(x_1, \ldots,x_k)^2).
\mathrm{Blow}_{Z_n^{I;\vec{i}}}V^I_{\vec{i}}= \left\{(z^I(\vec{i}) , l^I(\vec{i}) ) \in \mathbb{C}^N \times \mathbb{CP}^{N-n-1}| z^I(\vec{i})_i l^I(\vec{i})_j = z^I(\vec{i})_j l^I(\vec{i})_i i,j \in [N] - \vec{i}\right\}
I_{5}= 2 4 C \, \, T r \left( \theta^{\left( 1 \right)}G^{- 1}\theta^{\left( 1 \right)}G^{- 1}\right) P_{9}W_{9}^{\left( 2 \right)}X_{5 6 7 8 9}^{\left( 2 \right)}. \nonumber
q^{\varepsilon}(x)=\alpha(s)+\frac{1}{1-\varepsilon M}\left( \frac{M}{2}|x-s|^{2}+\langle v(s),x-s\rangle+\frac{\varepsilon}{2}|v(s)|^{2}\right), & q_{\varepsilon}(x)= \alpha(s)+\frac{1}{1+\varepsilon M}\left( \frac{M}{2}|x-s|^{2}+\langle v(s),x-s\rangle-\frac{\varepsilon}{2}|v(s)|^{2}\right).
\begin{aligned}(KU_1^p e, U_2^q\bar e)= (U_1^p e, K^\ast U_2^q C e)=(C K^\ast U_2^q C e, CU_1^p e) = (C K^\ast C U_1^{-q}e, U_2^{-p}C e)=(C K^\ast C \, U_1^{-q}e, U_2^{-p}\bar e).\end{aligned}
\begin{gathered}\mathbb{E}(M(n+1)-M(n)|\mathcal{F}_n) =\sum_{i=1}^{\infty}\mathbb{P}(d_{n+1}=i)\left(1 - \left(1-\frac{(M(n)+\beta)L(n)}{(2+\beta)n+\beta}\right)^{i}\right).\end{gathered}
\Theta^L_{\lambda, T}= \sum_{\gamma \in L_1}\ \left( t_{\gamma}\frac{e^{\lambda}}{\prod_{\beta \in T_1}(1-e^{-\beta})}\right) \ \Phi^{[\tfrac{k}{2}| \gamma_1 |^2; (\lambda| \gamma_1)]}(\tau,- \beta_1, \beta_1 + \gamma^\vee_1 + \tau (\gamma | \gamma^\vee_1 + 2 \beta_1)).
P_{n,r,s}(q,t_c) := (-q)^{rs(n^2-1)}\epsilon_c(\gamma_{r,s}(p_{n-1}(y))\cdot 1_c) J_{n,r,s}(q,t) := (-q)^{rs(n^2-1)}\epsilon(\gamma_{r,s}(p^-_{n-1}(y))\cdot \varnothing))
[\begin{matrix}(\begin{matrix}1&0\\ 0&0\end{matrix})&(\begin{matrix}0&1\\ 0&0\end{matrix})\\ (\begin{matrix}0&0\\ 1&0\end{matrix})&(\begin{matrix}0&0\\ 0&1\end{matrix})\end{matrix}]=[\begin{matrix}1&0&0&1\\ 0&0&0&0\\ 0&0&0&0\\ 1&0&0&1\\ \end{matrix}].
v_{\tilde{u}}(x) =-\frac{1}{\gamma_n}\int_{\mathbb{R}^n}\ln \left(\frac{1+|y|}{|x|\left|\frac{x}{|x|^2}-y\right|}\right)e^{nu(y)}dy &=-\frac{1}{\gamma_n}\int_{\mathbb{R}^n}\ln \left(\frac{1+|y|}{|x|\left|\frac{x}{|x|^2}-y\right|}\right)|y|^{n\beta}e^{n\left(v(y)+p(y)+q\left(\frac{y}{|y|^2}\right)\right)}dy.
\gamma:{\P}(n)\otimes{\P}(k_1)\otimes\cdots\otimes{\P}(k_n) \longrightarrow{\P}(k_1+\cdots +k_n) (a,b_1,\ldots,b_n) \longmapsto \gamma(a;b_1,\ldots ,b_n)=\Big(...\big((a\circ_n b_n)\circ_{n-1}b_{n-1}\big)\cdots\Big)\circ_1 b_1,
\Bigl ( \frac{{\dot{a}}}{a}\Bigr )^{2}= \frac{8 \pi G_{N}}{3}\rho - \frac{K}{a^{2}}+ \frac{\Lambda_{4}}{3}+ \frac{\kappa_{5}^{4}}{3 6}\rho^{2}+ \frac{\mu}{a^{4}},
u_{N,\varepsilon_1\varepsilon_2 \ldots \varepsilon_n}(t) := \left(u_{N,\varepsilon_1}\circ u_{N,\varepsilon_2}\circ \ldots \circ u_{N,\varepsilon_n}\right)(t) = \frac{N}{\displaystyle \varepsilon_1 +\frac{N}{\displaystyle \varepsilon_2 + \ddots +\frac{N}{\varepsilon_n +t}}}.
B = \left(\begin{tabular}{ccc}$1$ $0$ $1$ $1$ $1$ $0$ $0$ $1$ $1$\end{tabular}\right)\begin{tabular}{c}\begin{tiny}{double}\end{tiny}begin{align*}-5pt]$\longrightarrow$\end{tabular}\left(\begin{tabular}{rrr}$2$ $0$ $2$ $2$ $2$ $0$ $0$ $2$ $2$\end{tabular}\right)
\tilde{Q}^{\alpha,\{k_1\}}_*(\theta(k_1))= e^{-\theta(k_1)}\tilde{Q}^{\alpha,\{k_1\}}_{-1}+\tilde{Q}^{\alpha,\{k_1\}}_0+ e^{\theta(k_1)}\tilde{Q}^{\alpha,\{k_1\}}_1.
\prod_{j=1}^{m}f_{j}(y_{j}) = \prod_{j=1}^{m}\Big( f_{j}^{0}(y_{j}) + f_{j}^{\infty}(y_{j}) \Big) = \sum_{\rho_{1},\dots,\rho_{m}\in \{0,\infty\}}f_{1}^{\rho_{1}}(y_{1}) \cdots f_{m}^{\rho_{m}}(y_{m}) = \prod_{j=1}^{m}f_{j}^{0}(y_{j}) + \sum_{(\rho_{1},\dots,\rho_{m})\in \rho}f_{1}^{\rho_{1}}(y_{1}) \cdots f_{m}^{\rho_{m}}(y_{m}),
(E^{e_1}\circ O^{o_1}\circ E^{e_2}\circ \dotsb \circ O^{o_l}\circ E^{e_{l+1}})(n) = \lceil(E^{e_1 + \dotsb + e_{l+1}}\circ O^{o_1 + \dotsb + o_{l}})(n)\rceil.
\lambda^{-1-2\alpha+\epsilon}\sum\limits_{i,j=1}^{n+1}[Y_i Y_j \tilde{v}]_{C^{0,\epsilon}(\mathcal{B}_{\lambda/4}(y))}\leq C\left(\|v\|_{L^2(\mathcal{B}_1^+)}+\|f\|_{Y_{\alpha,\epsilon}}\right).
I_{1}= \int_{0}^{+ \infty}d t_{1}d t_{2}\exp \Bigl [ - \frac{t_{1}^{2}}{2 \alpha}- \frac{t_{2}^{2}}{2 \alpha}- \frac{t_{1}t_{2}}{2 \alpha}\Bigr ] \sin ( t_{1}\frac{\alpha_{1}}{\alpha}) \sin ( t_{2}\frac{\alpha_{1}}{\alpha}) .
\mathcal J_{n}^{2}={}- 4 \beta^{2}\left( 2 \mathcal J_{n}^{1}+{\cal W}\right) - \frac{1}{2}( 4 \beta^{2})^{2}{\cal S}_{\Phi}\left({\cal S}_{\Phi}+ n \sigma_{3}\right) - \frac 1 2{\cal C}.
S ={\frac{1}{1 6 \pi \alpha^{( n - 1 ) / 2}}}\int d^{n + 1}x \sqrt{| g |}e^{- \varphi}\left[ R + ( \nabla \varphi )^{2}+{\frac{k \alpha^{\prime}}{4}}\left( R_{G B}^{2}- ( \nabla \varphi )^{4}\right) \right] .
\tilde{P}(\mu) \geq\int_{0}^{1}\xi''(s)\phi_{\mu}(s)+2\sqrt{\eta''}-\phi_{\mu}\eta''-\frac{1}{1-s}ds+h^{2}\phi_{\mu}(0) =\int_{0}^{1}(\eta''(s)-\xi''(s))(1-s-\phi_{\mu}(s))+\int_{0}^{1}2\sqrt{\eta''}-\eta''(s)\cdot(1-s)-\frac{1}{1-s}&+\int_{0}^{1}\xi''(s)\cdot(1-s)ds+h^{2}\phi_{\mu}(0).
\kappa_{00}=\sum_{n\geq 0}\prod_{k=1}^{n-1}q_{(10)^k0}(0)\prod_{k=0}^{n-1}q_{(01)^k00(1)}+\sum_{n\geq 1}\prod_{k=1}^{n-2}q_{(10)^k0}(0)\prod_{k=0}^{n-1}q_{(01)^k00(1)}\kappa_{11}=\sum_{n\geq 0}\prod_{k=0}^{n-1}q_{(10)^k11}(0)\prod_{k=1}^{n-1}q_{(01)^k1(1)}+\sum_{n\geq 1}\prod_{k=0}^{n-1}q_{(10)^k11}(0)\prod_{k=1}^{n-2}q_{(01)^k1(1)}.
(f_2\circ f_1)\star f_3+(-1)^{(m_1-1)m_3}f_2\star (f_3 \circ f_1)-(-1)^{(m_3-1)(m_1-1)}(f_2\star f_3 )\circ f_1 = (-1)^{(m_1+m_2)(m_3-1)+(m_1-1)(m_2-1)}f_2\star_0(f_1\star_0f_3)+ f_2\star_1(f_1\star_1 f_3).
\mathcal{R}_{G, b, \mu}=\{(M_{S_1}, \mu_{S_1}) \in \mathcal{C}_G: (M_{S_1}, \mu_{S_1}) \leq (M_{S_2}, \mu_{S_2}) \,\ \,\ (M_{S_2}, \mu_{S_2}) \in \mathcal{T}_{G, b, \mu}\}.
0<d_0 \leq d_4^1:= \min \Big\{\frac23 -2\varrho_\perp - \hat q_1\, ,\frac{4}{3}{\mathfrak a}-\varrho_\perp\, , \frac 13 - \frac 34 (\hat q_1 + \varrho_\perp)\Big\}\,,
b a^{\dagger}b^{\dagger}= b^{\dagger}a^{\dagger}b , \; a b^{\dagger}a^{\dagger}= a^{\dagger}b^{\dagger}a , \; b b^{\dagger}a^{\dagger}= a^{\dagger}b^{\dagger}b + 2 a^{\dagger}, \; a a^{\dagger}b^{\dagger}= b^{\dagger}a^{\dagger}a + 2 b^{\dagger}
\lim_{y\rightarrow 1}\left\{\frac{\frac{d}{dy}\left[R_I(y\sqrt{v-\epsilon})\right]}{R_I(y\sqrt{v-\epsilon})}\right\}= \lim_{y\rightarrow 1}\left\{\frac{\frac{d}{dy}\left[R_E(y\sqrt{\epsilon})\right]}{R_E(y\sqrt{\epsilon})}\right\}.
\left( \begin{array}{c}z^\prime \bar z^\prime \bf 1\end{array}\right) ={\bf M}\left( \begin{array}{c}z \bar z \bf 1\end{array}\right) \equiv\left( \begin{array}{ccc}u 0 t 0 \bar u \bar t 0 0{\bf 1}\end{array}\right)\left( \begin{array}{c}z \bar z \bf 1\end{array}\right) \ .
\Theta (gh)(x) =\mathcal{I}_{gh}(x) =(gh)x(gh)^{-1}=(gh)x(h^{-1}g^{-1}) =g(hxh^{-1})g^{-1}=\mathcal{I}_g(\mathcal{I}_h(x))=(\mathcal{I}_g\circ \mathcal{I}_h)(x) =(\Theta (g)\circ \Theta (h))(x).
{_{*}\Phi}^{(s+1)}_{\alpha}=\{{_{*}\Phi}^{(s)}_{\alpha},H_{T}\}+\frac{\partial{_{*}\Phi}^{(s)}_{\alpha}}{\partial t}:\approx0, &{_{*}\Phi}^{(m_{\alpha}+1)}_{\alpha}:={_{*}X}_{T}{_{*}\Phi}^{(m_{\alpha})}_{\alpha}={_{*}C}_{\alpha s}^{\beta}{_{*}\Phi}^{(s)}_{\beta},
P_A \left( \sigma(x^1) = y^1,\ldots,\sigma(x^p) = y^p \right) := \frac{1}{Y_{m_1,m_2}(A)}\sum_{\sigma}\prod_{k=1}^p \mathbf{1}\{\sigma(x^i) = y^i \}\prod_{x \in \mathbb{T}_{m_1,m_2}}w_A(x^i,y^i) &= \frac{\prod_{i=1}^k w_A(x^i,y^i)}{Y_{m_1,m_2}(A)}\frac{\partial}{\partial W_{x^1,y^1}}\ldots \frac{\partial}{\partial W_{x^p,y^p}}Y_{m_1,m_2}(A).
E_1 =-E_{01}+E_{27}-E_{36}+E_{45}, E_2 =-E_{02}-E_{17}+E_{35}+E_{46}, E_3 =-E_{03}+E_{16}-E_{25}+E_{47}, E_4 =-E_{04}-E_{15}-E_{26}-E_{37}, E_5 =-E_{05}+E_{14}+E_{23}-E_{67}, E_6 =-E_{06}-E_{13}+E_{24}+E_{57}, E_7 =-E_{07}+E_{12}+E_{34}-E_{56},
-4i\Omega L_\zeta = \xi\eta(\xi^2 + \eta^2 - 2\zeta^2) J(\zeta)(J(\zeta) - J(\xi) - J(\eta)) &- \xi^2\eta(\xi - \eta) J(\zeta)(J(\xi) - J(\eta) - J(\zeta)) &- \xi\eta^2(\xi - \eta) J(\zeta)(J(\xi) - J(\eta) + J(\zeta)).
\begin{array}{rll}\partial_t \bar v - \Delta_H \bar v + \nabla_H\pi = -\bar v\cdot\nabla_H\bar v - \frac1h \int_{-h}^0 (\tilde v\cdot\nabla_H\tilde v + \mathrm{div}_H\, v\,\tilde v)\,dz - \frac1h v_z|_{\Gamma_b}\; G, \mathrm{div}_H\,\bar v = 0 \; G,\end{array}
R(q_i)-cf^{-1}(q_i) \geq R(q_i')-cf^{-1}(q_i') \geq \frac{cf^2}{2+f^3}(q_i') -cf^{-1}(q_i')=\frac{-2c^2}{f(q_i') \left( f^3(q_i') +c\right)}.
(\varphi E(\lambda))(T_{n_w}) = \varphi(E(\lambda)T_{n_s}T_{n_{sw}}) & = \varphi(T_{n_s}E(n_s^{-1}(\lambda))T_{n_{sw}}) & = \varphi (T_{n_w})\sigma_\mathcal{A}(E(n_w^{-1}(\lambda))).
\zeta(z, q) = 2\pi i \left(\frac{e^{2\pi i z}}{e^{2\pi i z}-1}- \frac{1}{2}- \sum_{n=1}^\infty \frac{q^n}{1-q^n}\left[ e^{2\pi i n z}- e^{-2\pi i n z}\right] \right) \wp(z, q) = (2\pi i)^2 \left(\frac{e^{2\pi i z}}{(e^{2\pi i z}-1)^2}+ \sum_{n=1}^\infty \frac{n q^n}{1-q^n}\left[ e^{2\pi i n z}+ e^{-2\pi i n z}\right] \right),
e.(f.m)=\frac{\Phi(e,f,g)}{\Phi(e,f,h)}(ef).m, (m.e).f=\frac{\Phi(h,e,f)}{\Phi(g,e,f)}m.(ef), (e.m).f=\frac{\Phi(e,h,f)}{\Phi(e,g,f)}e.(m.f),
(\sigma^{-1}\circ \mathcal{I}_g\circ \sigma)(x) =\sigma^{-1}(\mathcal{I}_g(\sigma(x))) =\sigma^{-1}(g\sigma (x)g^{-1}) =\sigma^{-1}(g)x\sigma^{-1}(g^{-1}) =\sigma^{-1}(g)x(\sigma^{-1}(g))^{-1}=\mathcal{I}_{\sigma^{-1}(g)}(x).
J(ru_1'(r)+2, ru_2'(r)+2) \le J(ru_1'(r)+2, ru_2'(r)+2)+(A-B)r^2\bigg[a_2e^{u_1}\quad+a_1e^{u_2}-\frac{a_2(1+a_1)}{2}e^{2u_1}-\frac{a_1(1+a_2)}{2}e^{2u_2}+a_1a_2e^{u_1+u_2}\bigg] &=J(R_3u_1'(R_3)+2, R_3u_2'(R_3)+2)+o(1)
I_k^\flat(u) = \int_{\Sigma_{k}^\flat}u^{1}(s_1)\dots u^k(s_k) d\L^k , I_{n-k}(u) = \int_{\Sigma_{n-k}}u^{k+1}(s_{k+1})\dots u^{n}(s_n) d\L^{n-k}.
B = \frac{1}{e}Q = - \frac{i}{e}\int_{- \infty}^{+ \infty}d x ~ J_{2}^{E M}( x , 0 ) = \frac{1}{\sqrt{\pi}}\int_{- \infty}^{+ \infty}d x \sum_{i}\chi_{i}^{\prime}= \frac{1}{\sqrt{\pi}}\sum_{i}[ \chi_{i}( \infty , 0 ) - \chi_{i}( - \infty , 0 ) ] .
p^5=\left[\begin{array}{cc|c}1 3 1 2 2 2 3 1 3 \end{array}\right],p^6=\left[\begin{array}{cc|c}2 3 1 3 1 2 1 2 3 \end{array}\right],p^{7}=\left[\begin{array}{cc|c}2 1 1 1 3 2 3 2 3 \end{array}\right],p^{8}=\left[\begin{array}{cc|c}1 2 1 2 1 2 3 3 3 \end{array}\right],
\langle P\nabla f_2(Px+q)-P \nabla f_2(Py+q),x-y\rangle &=\langle \nabla f_2(Px+q)-\nabla f_2(Py+q),P(x-y)\rangle &=\langle \nabla f_2(Px+q)-\nabla f_2(Py+q),(Px+q)-(Py+q))\rangle
\left\Vert\mathbb{G}^{(2m)}-\mathbb{G}_{}^{(\infty)}\right\Vert_{\mathcal{F}}=\left\Vert (S_{1,}^{(m-1)}-S_{1,})^{t}\cdot\mathbb{G}[\mathcal{A}]+(S_{2,}^{(m-2)}-S_{2,})^{t}\cdot\mathbb{G}[\mathcal{B}]\right\Vert_{\mathcal{F}}& \leqslant c_{0}\max(\lambda_{1},\lambda_{2})^{m-2}Z,
\begin{aligned}-\frac{p'(x^{*})}{k_1(x^{*})^{k_1-1}}x^{k_1}+p(x)-\left(-\frac{p'(x^{*})}{k_1(x^{*})^{k_1-1}}(x^{*})^{k_1}+p(x^{*})\right)\left(\frac{x^{*}}{x}\right)^{-k_1}&=\frac{1}{l_2}\left[\frac{1-\theta}{-\left(\frac{K}{-\frac{p'(x^{*})}{k_1(x^{*})^{k_1-1}}(x^{*})^{k_1}+p(x^{*})}+\theta\right)}-l_1\right].\end{aligned}
\epsilon^{\mu \nu \rho \lambda \sigma}\Lambda \cdot x ( \tilde{H}_{\mu \nu}g_{1}+ ( \tilde{H}^{3})_{\mu \nu}g_{2}) \partial_{\rho}( \tilde{H}_{\lambda \sigma}g_{1}+ ( \tilde{H}^{3})_{\lambda \sigma}g_{2})
p^{X}_{j}(e^{tA}-I) = p^{X}_{j}\left( \sum_{n=1}^{\infty}\dfrac{(tA)^n}{n!}\right)\leqslant \sum_{n=1}^{\infty}\frac{\big(t \, p^{X}_{j}(A)\big)^{n}}{n!}= e^{t p^{X}_{j}(A)}-1,
D\theta^{a}=\tau^{a}+\overline{\tau}^{a}\mathbf{,}D\tau^{a}-\zeta^{a}-\beta_{b}^{a}\theta^{b}=0, D\zeta^{a}+\beta_{b}^{a}\overline{\tau}^{b}-\overline{\beta}_{b}^{a}\tau^{b}+\gamma_{b}^{a}\theta^{b}=0,
\Lambda_{\mathcal{R}_X}(n)=\sum_{\substack{n=p_1\cdots p_\ell p_j\in (X^{a_j},X^{a_j+\delta}]\, p_\ell\ge X^{\eta/4},X^{1-\sum_ia_i-\ell\delta}}}\prod_{i=1}^\ell \log{p_i}=\sum_{\substack{n=mp p\ge X^{\eta/4}p\ge X^{1-\sum_ia_i-\ell\delta}}}\Lambda_{\mathcal{C}}(m)\log{p},
S\;=\;\int^{\infty}_{-\infty}\Bigl[\;{1\over 2}\;(1+{x^2\over \tau^2}\;(1-\varepsilon))\left({\partial_{x}\phi}\right)^2+{\lambda\over4}{1\over{1+{x^2\over \tau^2}\;(1-\varepsilon)}}\;\phi^4 \;\Bigr]\;dx,
2\lambda_n\beta_n\langle\nabla\Psi_1(x_n),x_{n+1}-u\rangle=2\lambda_n\beta_n\langle\nabla\Psi_1(x_n),x_{n+1}-x_n\rangle+2\lambda_n\beta_n\langle\nabla\Psi_1(x_n),x_{n}-u\rangle.
\liminf_{n \to \infty}W_n(x) \geq \liminf_{n \to \infty}W_n^{\prime}(x) - \lim_{n \to \infty}\frac{\log8}{\sqrt{n}}- \lim_{n \to \infty}\sum_{m=1}^3\frac{\log a_{k_n(x)+m}(x)}{\sqrt{n}}\geq 0.
\left|\int_{B(z,R)}G_1(x)\, dx\right| \leq |B(z,R)|^{1/2}\|G_1\|_{L^2(B(z,R))}\leq \left(\frac{4\pi\,R^3}{3}\right)^{1/2}\|G_1\|_{L^2(Q_1)}=\frac{1}{\pi}\left(\frac{4\pi\,R^3}{3}\sum_{k\in\mathbb{Z}^3\setminus\{0\}}\frac{1}{|k|^4}\right)^{1/2}.
{\frac{1}{z K}}\sqrt{\frac{A_{T^{2}}^{\mathrm{( O M )}}}{A_{T^{2}}^{\mathrm{( b u l k )}}}}\sim \left({\frac{l_{p}}{l_{g}}}\right)^{3}\propto{\frac{\alpha^{\prime}}{\alpha_{( e f f )}^{\prime}}}.
\theta_{n}^{(h)}=\sum_{k=1}^{K}(x_{k}-m_{1}^{(h)})^{n}H(x_{k})\Delta x_{k}=\frac{1}{A}\sum_{k=1}^{K}(x_{k}-m_{1}^{(h)})^{n}h(x_{k})\Delta x_{k}
\begin{aligned}\frac{1}{Q(\pi(q))Q(\pi''(q))}\sim_{E(\pi(q)) E(\pi'(q))}(2\pi i)^{dn(n+1)/2}\frac{L^S(\tfrac{1}{2},\Pi\otimes \Pi')}{L^S(1,\Pi,{\rm As}^{(-1)^n})L^S(1,\Pi',{\rm As}^{(-1)^{n-1}})}.\end{aligned}
\left\vert\int_{\Gamma_t}(\mathcal B_1{\mathbf V}_{x_2}\cdot{\mathbf V}_{x_2})\vert_{x_1=0}\,dx_2ds\right\vert\le C_2\left\{\int_0^t(I_{1,\ast}+I_{1,n})(s)ds+\int_0^t\Vert\varphi(s)\Vert_{L^2(\mathbb R)}^2 ds\right\}\,.
|v|_{k,p,\Omega}:= \biggl(\sum_{|\delta|=k}|\partial^\delta v|_{0,p,\Omega}^p \biggr)^{1/p}, \|v\|_{k,p,\Omega}:= \biggl(\sum_{0 \le m \le k}|v|_{m,p,\Omega}^p \biggr)^{1/p},
\psi_1(0,\rho) = (\tfrac{T}{T_0})^{\frac{2}{p-1}}c_p + T^{\frac{2}{p-1}}f(T\rho ), \psi_2(0,\rho) = (\tfrac{T}{T_0})^{\frac{p+1}{p-1}}\tfrac{2}{p-1}c_p + T^{\frac{p+1}{p-1}}g(T \rho).
r_1 = -R_{1212}(R_{2323}S_{12131}+R_{1313}S_{12232})=0, & r_4 = R_{1313}(R_{2323}S_{12131}-R_{1212}S_{13233})=0, & r_6 = R_{2323}(R_{1313}S_{12232}+R_{1212}S_{13233})=0.
\Gamma^{0}= \mathrm{i}\epsilon_{2}\otimes \epsilon_{1}\Gamma^{1}= \mathrm{i}\epsilon_{1}\otimes{\boldmath 1}\Gamma^{2}= \mathrm{i}\epsilon_{3}\otimes{\boldmath 1}\Gamma^{3}= \mathrm{i}\epsilon_{2}\otimes \epsilon_{2}
\Psi_{n_{12}}^{\mu_{12},\epsilon_{12}}(\rho,\phi) = \displaystyle\sum\limits_{n1,n2}C^{n_1 n_2}_{n_{12}j}\psi_{n_1}^{\mu_1}(\rho \cos \phi)\psi_{n_2}^{\mu_2}(\rho \sin \phi).
\operatorname*{Res}_{\mathrm{i}\kappa_{n}}\varphi\left( k\right)=\mathrm{i}c_{n}^{2}\left( t\right) \psi\left( \mathrm{i}\kappa_{n}\right),\operatorname*{Res}_{-\mathrm{i}\kappa_{n}}\varphi\left( k\right)=-\mathrm{i}c_{n}^{2}\left( t\right) \psi\left( \mathrm{i}\kappa_{n}\right) ,
\langle Y u, u\rangle = \langle Y u_{\varepsilon}, u_{\varepsilon}\rangle+ \langle Y u - (Y u)_{\varepsilon}, u\rangle &+ \langle (Y u)_{\varepsilon}- Y u_{\varepsilon}, u\rangle+ \langle Y u_{\varepsilon}, u - u_{\varepsilon}\rangle & =: I_1 + I_2 + I_3 + I_4.
\big[M_{\mu\nu},Q_\alpha\big] = -{1\over 2}(\sigma_{\mu\nu})_\alpha{}^\beta Q_\beta, \qquad \big[M_{\mu\nu},\bar Q_{\dot\alpha}\big] ={1\over 2}(\tilde\sigma_{\mu\nu})^{\dot\beta}{}_{\dot\alpha}\bar Q_{\dot\beta}\;,
\left\Vert \left( \begin{array}{c}\int_{\mathbb{S}^{1}}\cos 2\theta d\Psi_{\mu}\int_{\mathbb{S}^{1}}\sin 2\theta d\Psi_{\mu}\end{array}\right) \right\Vert =\frac{\left\vert \alpha -1\right\vert}{\left\vert\alpha -3\right\vert}\left\Vert \Psi_{\mu}\right\Vert \ .
{\cal B}\Bigl ( \frac{v^{2}}{N_{F}}, \tilde{\alpha}, \tilde{\lambda}, \{\zeta_{i}\}\Bigr ) = \frac{v^{2}}{N_{F}}{\cal E}_{\mathrm{c l}}\bigl ( \tilde{\alpha}, \tilde{\lambda}, \{\zeta_{i}\}\bigr ) +{\cal E}^{F}( \{\zeta_{i}\}) - 1 \, .
q_n(x) := x^{2n}(x^2 - 1), t_n(x) := 2x^{\tfrac{3n+1}{2}}+2x^{n+1}+2x^{\tfrac{n+3}{2}}+x^2 +1, \xi_n(x) := \frac{q(x)}{t(x)}= \frac{x^{\tfrac{n-1}{2}}(x^2 -1)}{2 + 2x^{\tfrac{-n+1}{2}}+ 2 x^{-n +1}+ x^{\tfrac{-3n+3}{2}}+ x^{-\tfrac{3n+1}{2}}}
[det(\eta_{\mu\nu}+M_{\mu\nu})]^{\frac{1}{2}}=\frac{[1+(H_{i5})^2-\frac{1}{4!}\epsilon^{ijkl}\epsilon^{mnpq}H_{mi}H_{nj}H_{pk}H_{ql}]}{[-det(\eta_{ab}+i{}^*H_{ab})]^{\frac{1}{2}}}.
b_{i+1}(n) =\sum\limits_{m=0}^{B^k-1}b_i(B^k n+m) &=\sum\limits_{m=0}^{B^k-1}\sum\limits_{j\in S_i}a(B^{ki}(B^k n+m)+j) =\sum\limits_{j\in S_i}\sum\limits_{m=0}^{B^k-1}a(B^{ki+k}n+B^{ki}m + j).
\ |\mathcal H(k,d,2)|-|\mathcal H(k,d,l)| \geq \ (d-1)\binom{n-d-1}{k-d}-\sum_{i=3}^{d+1}\binom{n-d-i+1}{k-d}-(d-1)\binom{n-d-l+1}{k-d-l+2}= \ \sum_{i=3}^{d+1}\left(\binom{n-d-1}{k-d}-\binom{n-d-i+1}{k-d}\right)-(d-1)\binom{n-d-l+1}{k-d-l+2}.
\mbox{\boldmath${\nabla}$}\times \left[ f_{l}(r)\mathbf{Y}_{l,m}\right]=\frac{i\,l(l+1)}{r}f_{l}(r){Y}_{l,m}\,\hat{\mathbf{r}}+\frac{1}{r}\frac{d}{dr}[r f_{l}(r)]\,\hat{\mathbf{r}}\times\mathbf{Y}_{l,m},
L^{}(x;\phi) = \sum_{n\leq x}\frac{(\phi*\phi)(n)}{n}\log \frac{n}{e}+ \frac{1}{2}\sum_{n\leq x}\frac{(\phi*\Lambda)(n)}{n}&+ \log\sqrt{2\pi}\sum_{n\leq x}\frac{\phi(n)}{n}+{\Theta}\sum_{n\leq x}\frac{(\phi* \phi_{-1})(n)}{n}&:= I_{2,1}+ I_{2,2}+I_{2,3}+I_{2,4},
S_1 =\{(\gamma,Y): 0\leq Y\leq \gamma\leq (p-1)^{-1},\,Y<\gamma^{-1}\}, S_2 =\big\{(\gamma,Y):\gamma<\min\{Y,(p-1)^{-1}\}\big\}, S_3 =\{(\gamma,Y):\gamma=(p-1)^{-1},\, Y\geq (p-1)^{-1}\}.
\Psi_\epsilon (\alpha)\equiv \frac{\mu^{\epsilon}}{2R}\sum_{n=-\infty}^{\infty}\int\frac{d^{d-\epsilon}p}{(2\pi)^{d-\epsilon}}\,\left[p^2+\left(\frac{2\pi n}{R}\right)^2+m^2\right]^{\alpha}.
\left( \omega_{\alpha}^{g_{1}g_{2}, g_{3}}\right) \left( g_{3}^{*}\omega_{\alpha}^{g_{1}, g_{2}}\right) \: = \: \left( \omega_{\alpha}^{g_{1}, g_{2}g_{3}}\right) \left( \omega_{\alpha}^{g_{2}, g_{3}}\right)
Z_f(s, A_1) = q^{-(\omega+1)-ls}\int_{\mathcal{O}_K^2}\chi(ac(\pi^{k+r}u^pw^{k+r+l}+w^{r+l}v^l\mathbb{H}_r^k(w, t\pi vw-w) \times |\pi^{k+r}u^pw^{k+r+l}+w^{r+l}v^l\mathbb{H}_r^k(w, t\pi vw-w)|^s|-w^{\frac{k+r+l}{p}+1}||dudvdw| := q^{-(\omega+1)-ls}Z_{f_1}(s, \chi)\int_{\mathcal{O}_K^{\times}}\chi(ac(w^{k+r+l}))|dw|,
U:=e^{\sum_{i=1}^N\left[\int_0^tB_i(s)d\beta_i+\theta_i\beta_i-\frac{1}{2}\int_0^tB_i^2(s)ds-\frac{1}{2}\theta_i^2t-\theta_i\int_0^tB_i(s)ds\right]}y.
R_{a}^{i}= \left( \begin{matrix}{\partial T + 2 T \partial}&{- 3 / 2 \partial G - 5 / 2 G \partial}&{0}&{0}\\{\partial G + 5 / 2 G \partial}&{T^{2}}&{0}&{0}\\{\nabla^{( - 1 )}}&{- f T}&{0}&{2 T^{2}}\\{\partial f - 3 / 2 f \partial}&{\nabla^{( - 3 / 2 )}}&{9 \partial T + 4 T \partial}&{- 1 5 \partial G}\\ \end{matrix}\right) \ ,
(f \circ_i (g \circ_j h)) ([r]; a_1, \ldots, a_{m+n+p-2}) &= f ( [r]; a_1, \ldots, a_{i-1}, (g \circ_j h) ([1]+ \cdots+[n+p]; a_i, \ldots, a_{i+n+p-2}), \ldots, a_{m+n+p-2}).