harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\big[K_i E_i, [F_i ,F_j]_{q^3}\big]_q =q^3[3] F_j K_i K_i', \Big[K_i E_i ,\big[F_i , [F_i ,F_j]_{q^3}\big]_q \Big]_{q^{-1}}=q(1+[3])[B_i^\sigma,F_j]_{q^3}K_i K_i'.
H^{(1)}_{\rm spin\frac{1}{2}}[N] = - \frac{i}{2\kappa^2}\int d^3x N(x) \left(\epsilon^{ijk}\epsilon^{abc}4\frac{\{A_a^i(x),V(x,\delta)\}\{A_b^j(x),V(x,\delta)\}}{\sqrt{q}(x)}\right)[\pi^T\tau_k{\cal D}_c\xi -c.c.]
a'_1 =\sum_{k=3}^g \tbinom{k-1}{2}\tfrac{k!(k-1)!}{12}(-1)^{g-k}(g-k)!\tfrac{g!}{k!}\tbinom{g-1}{k-1}\tbinom{g}{k}&=\tfrac{g!(g-1)!}{12}(-1)^{g-1}\sum_{k=3}^{g}(-1)^{k-1}\tbinom{k-1}{2}\tbinom{g}{k}=\tfrac{g!(g-1)!}{12}(-1)^{g-1}.
I_B := \int_B\mathrm{d}z \;g(z), g(z) := \frac{\alpha}{6\pi}z^2 \, \chi_{\Lambda}^2(z)\, \left\| \, \frac{(h_{1}-e_{1})^{1/2}}{(\alpha^2(h_{1}-e_{1})+z)^{1/2}}\, x u_{1}\right\|^2 \left(\frac{i}{z y}- \frac{1}{z^2 y^2}- \frac{i}{2 z^3 y^3}\right) e^{2 i z y}.
\mu_1 (a, b) + \varphi_1 (a \cdot b) =~ \mu_1' (a, b) + \varphi_1 (a) \cdot b + a \cdot \varphi_1 (b), R_1 + \varphi_1 \circ R =~ R_1' + R \circ \varphi_1, .
d s^{2}= - 2 d \hat{x}^{+}d \hat{x}^{-}+ d \hat{x}^{2}+ ( \hat{x}^{2}+ R^{2}) d \hat{z}^{2}+ 2 ( \hat{x}^{+}d \hat{x}- \hat{x}d \hat{x}^{+}) d \hat{z}+ d x_{\perp}^{2},
A_{4 \pi}= i ( \delta_{i j}\delta_{k l}+ c y c l i c . . ) [ 4 f^{\prime \prime}-{\frac{1 6 v^{2}( f^{\prime \prime}( v^{2}) )^{2}}{2 f^{\prime}( v^{2}) + 4 v^{2}f^{\prime \prime}( v^{2})}}]
u_X(\nu t) \le \nu^{d_n}u_X(t) \le \frac{1-s \mu_X}{1-s}\cdot u_X(t) \le \frac{u_X(t) - s v_X(t)}{1-s}= \frac{u_X(t)-v_X(t)}{1-s}+ v_X(t)
G(x,t;\l) = \begin{cases}\ \ \Phi(x,\l) (C+D\Phi(1,\l))^{-1}C \Phi^{-1}(t,\l) i B, 0 \leqslant t \leqslant x, -\Phi(x,\l) (C+D\Phi(1,\l))^{-1}D \Phi(1,\l) \Phi^{-1}(t,\l) i B, x < t \leqslant 1. \end{cases}
Y_g(P_{i,n}, z^{1/m})= Y_g(\frac{T^{n}}{n!}P_{i,0}, z^{1/m}) = \frac{\partial^{n}_z}{n!}Y_g(P_{i,0},z^{1/m}) = \frac{\partial^{n}_z}{n!}P_i(x_1 (z^{1/m}), ... , x_k (z^{1/m}))
\hat{\rm P}_{\pm}= \pm \hat{\cal H}_{\rm F}^{\pm}(0) +\sum_{n>0}\hat{\rm G}_{\pm}^a(-n){A}_{1,\pm}^a(n) +\sum_{n \leq 0}{A}_{1,\pm}^a(n) \hat{\rm G}_{\pm}^a(-n),
\mathfrak{F}_{\lambda}(u)+\frac{1}{p}\,[u_n-u]^p_{W^{s,p}(\mathbb{R}^N)}-\frac{\lambda}{p}\,\int_\Omega \frac{|u_n-u|^p}{d_\Omega^{s\,p}}\,dx=m(\lambda)+o(1),\mbox{as}n\to \infty.
(S^2)_\mathfrak{U}\equiv 1;\left((ST)^3\right)_\mathfrak{U}\equiv 1; (G_k S^2 G^{-1}_k)_\mathfrak{U}\equiv 1;\left(G_k(TS)^3 G^{-1}_k\right)_\mathfrak{U}\equiv 1 &(k=0,1,\ldots,p-1).
Z = \mathrm{d e t}( \Delta_{0}\oplus \Delta_{0})^{- \frac{1}{4}}\mathrm{d e t}( \Delta_{1}\oplus \Delta_{1})^{\frac{3}{4}}= \mathrm{d e t}( \Delta_{0})^{- \frac{1}{2}}\mathrm{d e t}( \Delta_{1})^{\frac{3}{2}},
Z_{\nu}= \int d U \sum_{m_{1}= 0}^{+ \infty}\sum_{m_{2}= 0}^{+ \infty}\cdots \sum_{m_{N - 1}= 0}^{+ \infty}\sum_{n_{N}= - \infty}^{+ \infty}\det ( I_{n_{j}+ i - j}( V \Sigma m ) ) \left( \det U \right)^{\nu - n_{N}}\chi_{( \ell_{1}, \ell_{2}, \cdots , \ell_{N})}( U^{\dagger}) ,
\boldsymbol{\alpha}^{k}:= (\alpha_{i}^{k})_{1 \leq i \leq k}, \boldsymbol{\beta}^{k}:= (\beta_{i}^{k})_{1 \leq i \leq k}, \boldsymbol{\tau}^{k}:= (\tau_{i}^{k})_{1 \leq i \leq k},
\|(S_{\mu,\gamma}-S_{\mu,\gamma,m})(f)(z)\|_{H^2}^{2}\asymp\sum_{n=m+1}^{\infty}\left|\sum_{k=0}^{\infty}(n+1)^{\frac{2\gamma-\beta-1}{2}}(k+1)^{\frac{\alpha-1}{2}}\mu_{n,k}a_k\right|^2 &\preceq \epsilon^2\sum_{n=1}^{\infty}\left(\sum_{k=1}^{\infty}n^{\frac{2\gamma-\beta-1}{2}}k^{\frac{\alpha-1}{2}}\frac{|a_k|}{(n+k)^{\gamma-\frac{\beta-\alpha}{2}}}\right)^2
\frac{s(x,k)}{\frac{e^{kx}}{2k}}= 1 - e^{-2kx}= 1 + O(\lvert k\rvert^{-3}), \frac{c(x,k)}{\frac{e^{kx}}{2}}= 1 + e^{-2kx}= 1+ O(\lvert k\rvert^{-3}),
T_\sigma (\bar r + K \kappa) (x) = M_\sigma W (\bar r + K \kappa) (x) \leq M_\sigma (W \bar r + W (K \kappa)) (x) &= M_\sigma W \bar r(x) + W(K \kappa) (x, \sigma(x))\leq T_\sigma \bar r(x) + \alpha\beta K \kappa(x)
[g]^m (x_1, \dots, x_m) = [A_1 \circ f \circ A_2 + A_0]^m (x_1, \dots, x_m) = [A_1 \circ f \circ A_2]^m (x_1, \dots, x_m) &= \alpha_1 \left( [f]^{m+1}(\alpha_2 (x_1), \dots, \alpha_2 (x_m), c_2) + [f]^m (\alpha_2 (x_1), \dots, \alpha_2 (x_m))\right).
\sum_{m\in \mathbb{Z}}\mu(T_m(\phi_2)) y^m = \left(\frac{1+y^{-1}}{2}\right) \left(\frac{y^{-1}}{6}+ \frac{5}{6}\right) \left(\frac{27}{48}+\frac{21}{48}y^{-1}\right) &= \frac{7}{192}y^{-2}+ \frac{17}{64}y^{-1}+ \frac{89}{192}+ \frac{15}{64}y.
2 7 m_{d y n}^{2}\equiv \bar{\Sigma}_{0}^{2}= \Lambda^{2}e^{2 \delta}\left[ \left( \frac{1 - \omega}{1 + \omega}\right) \frac{g - \left( \frac{\omega + 1}{2}\right)^{2}}{g - \left( \frac{1 - \omega}{2}\right)^{2}}\right]^{1 / \omega}.
\beta(dx,dy) = \delta_{(x_+,z_+)}+ \frac{z_--z_+}{z_--y}\delta_{(x_-,y)}+ \frac{z_+-y}{z_--y}\delta_{(x_-,z_-)}, \gamma(dx,dy) = \delta_{(x_-,z_+)}+ \frac{z_--z_+}{z_--y}\delta_{(x_+,y)}+ \frac{z_+-y}{z_--y}\delta_{(x_+,z_-)},
\mbox{\textbf{NP1:}}\;\; \max_{\mathbf{x}}\mathbf{c}^T \mathbf{x}\mbox{s.t.}\;\; \sum_{k=1}^K x_{(j-1)K+k}\leq L_j,~j=1,2,\cdots,J \;\; \sum_{j=1}^J x_{k+(j-1)K}\leq 1,~k=1,2,\cdots,K \;\; 0 \leq x_{k}\leq 1,~k=1,2,\cdots,K \;\; \mbox{Constraints (\ref{eqn:mmhn80})}.
\begin{aligned}H(t) = H^{(0)}+tH^{(1)}+t^{2}H^{(1)}&\psi_{n}(t) = \psi^{(0)}_{n}+t\psi^{(1)}_{n}+t^{2}\psi_{n}^{(2)}+... &\lambda_{n}(t) = \lambda^{(0)}_{n}+t\lambda^{(1)}_{n}+t^{2}\lambda_{n}^{(2)}+...\end{aligned}
\varphi (g_1 g_3) \varphi (g_1 g_2) = \varphi ((g_3 g^{2}_1) g_2) \mbox{or}\varphi (g_1 g_3) \varphi (g_1 g_2) = \varphi (g_2 (g_3 g^{2}_1)) \,.
\begin{array}{ll}\theta''(X;H,W) =\displaystyle \sum_{i=1}^{s-1}{\rm Tr}(\widehat V_i(H,W))- \displaystyle \sum_{i=s+1}^r{\rm Tr}(\widehat V_i(H,W))+{\rm Tr}({Q^s_{b^s_+}}^T\widehat V_s(H,W)Q^s_{b^s_+}) begin{align*}4pt] -{\rm Tr}({Q^s_{b^s_-}}^T\widehat V_s(H,W)Q^s_{b^s_-}) + \|{Q^s_{b^s_0}}^T\widehat V_s(H,W)Q^s_{b^s_0}\|_*. \end{array}
|S(n,k)| \leq\frac{1}{k!}\sum_{l_{1}+\dots+l_{k}=n}\frac{n!}{l_{1}!\dots l_{k}!}|g^{(l_{1})}(0)|\dots|g^{(l_{k})}(0)| \leq\frac{1}{k!}\sum_{l_{1}+\dots+l_{k}=n}\frac{n!l_{1}!\dots l_{k}!}{l_{1}!\dots l_{k}!}G_{\varepsilon}^{k}\varepsilon^{-n}\leq\frac{n!}{k!}G_{\varepsilon}^{k}2^{n}\varepsilon^{-n}.
(1-\dfrac{\Delta t}{2}\{(1-r_2)\delta_{\alpha_2 y}^{V^-}+r_2\delta_{\alpha_2 y}^{V^+}\}) V_{i,j}^{n+1}+(1+\dfrac{\Delta t}{2}\{(1-r_2)\delta_{\alpha_2 y}^{V^-}+r_2\delta_{\alpha_2 y}^{V^+}\}) V_{i,j}^{n}=2 V_{i,j}^*
u_{l}\leftarrow(\sum_{n=(s',d',f)}\mid h_{d}^{s'}[f]|^{2}|v_{d'}^{s'}[f]|^{2}+\sigma_{d}^{2})^{-1}h_{d}^{s}[f]v_{d}^{s}[f] w_{l}\leftarrow (1-(h_{d}^{s}[f]v_{d}^{s}[f])^{*}u_{l})^{-1}
b^cax =b^{s_0}ab^{s_1}a\cdots b^{s_n}ab^mb^{x_{n+1}}\cdots b^{x_{k-1}}a, b^cay =b^{s_0}ab^{s_1}a\cdots b^{s_n}ab^mb^{y_{n+1}}\cdots b^{y_{k-1}}a.
\begin{matrix}\phi_{1}=p_{11}Q_{1}+\cdot\cdot\cdot+p_{1n}Q_{n}\\ \phi_{2}=p_{21}Q_{1}+\cdot\cdot\cdot+p_{2n}Q_{n}\\ \vdots\\ \phi_{n}=p_{n1}Q_{1}+\cdot\cdot\cdot+p_{nn}Q_{n}\end{matrix}.
G_1^{(2,0)}+ u_{(3,0)}= 0 ; G_1^{(1,1)}= 0 ; G_1^{(0,2)}= 0 \, , G_2^{(2,0)}+ u_{(2,1)}= 0 ; G_2^{(1,1)}= 0 ; G_2^{(0,2)}= 0 \, .
\varphi(P)^\sigma = (u^2x_0,u^3y_0)^\sigma = (\sigma(u^2x_0),\sigma(u^3y_0)) = (u^2\sigma(x_0),\pm u^3\sigma(y_0)) &=\varphi(\pm P^\sigma) = \varphi(\pm m_\sigma P) = \pm m_\sigma \varphi(P).
\Phi_{\{\{1,4\}\{2,5,6\}\{3,7\}\}}= M_{\{\{1,4\}\{2,5,6\}\{3,7\}\}}+ M_{\{\{1,2,4,5,6\}\{3,7\}\}}+ M_{\{\{1,3,4,7\}\{2,5,6\}\}}+ M_{\{\{1,4\}\{2,3,5,6,7\}\}}+ M_{\{\{1,2,3,4,5,6,7\}\}}.
\eta= \left(n-2-\left(k+\frac{-h^2+2h+11}{2}-i\right),n-2-(h-1+i_{h-4}),\ldots,n-2-(4+i_1)\right) = \left(\frac{2n-2k+h^2-2h-15}{2}+i,n-h-1-i_{h-4},\dots,n-6-i_1\right).
ds_{}^{2}={\displaystyle\sum\limits_{j=1}^{l}}\frac{1}{\sigma_{2j-1}^{2}}\left( d\mu_{2j-1}^{2}+2\rho_{2j-1}d\mu_{2j-1}d\sigma_{2j-1}+2d\sigma_{2j-1}^{2}\right)
\left (\dot \Psi_k-i\left(\sigma_k^2+2\alpha \mathrm{Re}\big[\frac{(u\ |\ 1)(1\ |\u'_k)}{(u'_k\ |\u'_k)}\big]\right)\Psi_k \right )u'_k=-i[T_{|u|^2},\Psi_k](u'_k)\ .
\int_\Omega |g_{S_1,n}(x) - g_{S_2,n}(x)|^2 \d x = \int_\Omega \Big| \sum_{j\in S_1 \vartriangle S_2}x h_{h,j}(x) \Big|^2 \d x \geq c \int \Big| \sum_{j\in S_1 \vartriangle S_2}x h_{h,j}(x) \Big|^2 \d x = c \int |g_{S_1,n}(x) - g_{S_2,n}(x)|^2 \d x
[x_{i, r+1}^{-}, x_{i-1, s}^{-}] - [x_{i, r}^{-}, x_{i-1, s+1}^{-}] = \varepsilon_2 x_{i, r}^{-}x_{i-1, s}^{-}+ \varepsilon_1 x_{i-1, s}^{-}x_{i, r}^{-},
R_2(p_2,\cdots, p_s)= \sum_{j=3}^s 2^{j-p_j+p_1}(2^{p_j-p_{j-1}}-2)\left[(3\ 2^{j-1}-1)\Delta_{2^{p_j}+\cdots+2^{p_s}}\right. &\left.+(2^{j-2}(2^{p_j-p_{j-1}}-2^2)+1)(1+2^{p_{j+1}-p_j}+\cdots+2^{p_s-p_j})\right].
\begin{array}{c}{R_{i j}= R_{i}{}^{k}{}_{j k}, R_{i a}= P_{i}{}^{k}{}_{a k}= - P_{i a},}\\{R_{a i}= P_{a}{}^{b}{}_{i b}= P_{a i}, R_{a b}= S_{a}{}^{c}{}_{b c}= S_{a b}}\\ \end{array}
\dot{r}= \sin \phi \dot{s}= \cos \phi \dot{\phi}= \left(\frac{-r}{2}+ \frac{(n-1)2r}{r^2-s^2}\right)\cos \phi + \left(\frac{s}{2}+ \frac{(n-1)2s}{r^2-s^2}\right)\sin \phi + \lambda.
A = |S^{d-1}| \int_0^{D_p}f_p^{d-1}(s) \, ds = |S^{d-1}| \int_0^{D_{p}}f_{p^*}^{d-1}(s) \, ds = |S^{d-1}| \int_0^{D_p/2}f_p^{d-1}(s) \, ds + |S^{d-1}| \int_0^{D_{p}/2}f_{p^*}^{d-1}(s) \, ds \leq \frac{|S^{d-1}|}{2^{d-1}d}\, D_p^d = \frac{|B^{d}|}{2^{d-1}}\, D_p^d \,.
T_{a, \vec{v}}^{j, l}f:=\sum_{\substack{Q\in\mathcal{D}_{\vec{v}}l(Q)=2^{\lfloor{-j\rho+l+10}\rfloor}}}T_{a}^{j, l}(f\chi_{\frac{1}{3}Q}) \:\: l > j \epsilon.
X_{h}^{r}= Y^{r}_{h}\cap H_{0}^{1}(\Omega), \mathcal{X}_{h}^{r}= X_{h}^{r}\oplus{\rm i}X_{h}^{r},\mathbf{X}_{h}= \big( Y^{2}_{h}\big)^{3}\cap \mathbf{H}^{1}_{\rm t}(\Omega).
\begin{array}{@{}l@{}}{\displaystyle \Delta t \mathrm{Re}\left[B\left(\overline{\mathbf{A}}_{h}^{k};\overline{\theta}_{\Psi}^{k},\partial \theta_{\Psi}^{k}\right)\right] =\mathrm{Re}\big(J_1^{(k)}\big)+\mathrm{Re}\big(J_2^{(k)}\big)+\mathrm{Re}\big(J_3^{(k)}\big)+\mathrm{Re}\big(J_4^{(k)}\big).}\end{array}
dF(Y,\widetilde Y) = F_1\,dY + F_2\,d\widetilde Y + \frac{1}{2}F_{11}\,d\langle Y,Y\rangle + F_{12}\,d\langle Y,\widetilde Y\rangle + \frac{1}{2}F_{22}\,d\langle \widetilde Y,\widetilde Y\rangle.
\delta I_{C S}= \frac{1}{8 \pi^{2}}\int_{T^{2}}d^{2}x \epsilon^{i j}\mathrm{T r}[ G_{m}\partial_{i}G_{m}^{- 1}({\cal U}_{s}^{- 1}\partial_{j}{\cal U}_{s}-{\cal U}_{s}\partial_{j}{\cal U}_{s}^{- 1}) ] .
\Gamma_{12}^1=\frac{e^{-2s_2}}{1+e^{-2s_2}}\;,\Gamma_{11}^2=-\frac{e^{-2s_2}\left(1-e^{-2s_2}\right)}{\left(1+e^{-2s_2}\right)^2}\;,\;\;\Gamma_{22}^2=-\frac{e^{-2s_2}}{1-e^{-2s_2}}.
H(x,y) := (J_{\eta^{-1}}\circ \eta(x,y))(J_{\eta^{-1}}\circ \eta (x,y))^T = \begin{pmatrix}\frac{1}{L^2(y)}+\frac{x^2 L'^2(y)}{L^2(y)}-\frac{x L'(y)}{L(y)}-\frac{x L'(y)}{L(y)}1\end{pmatrix}.
\int_0^\infty\exp_p(\mu|u_n|^{p'})r^{\alpha_0}\mathrm dr =\int_0^\infty\left[\exp_p(\mu|u_n|^{p'})-\dfrac{\mu^{p-1}|u_n|^p}{\Gamma(p)}\right]r^{\alpha_0}\mathrm dr+\dfrac{\mu^{p-1}}{\Gamma(p)}\int_0^\infty|u_n|^pr^{\alpha_0}\mathrm dr &\overset{n\to\infty}\longrightarrow\int_0^\infty\exp_p(\mu|u|^{p'})r^{\alpha_0}\mathrm dr,
(t,1)\cdot \tilde{x}_0 = \lim_{z\rightarrow 0}(t,1)\cdot (\lambda(z),z^m)\cdot \tilde{x}& = \lim_{z\rightarrow 0}(\lambda(z),z^m)\cdot (t,1)\cdot \tilde{x}& = \lim_{z\rightarrow 0}(\lambda(z),z^m) \tilde{x}\curvearrowleft t^{-1}H & = \tilde{x}_0 \curvearrowleft t^{-1}H.
\frac{1}{2}- \frac{\mu_{u, t}}{2}\sum_{r = 0}^{K - 1}\binom{2 r}{\!r}\left(\frac{1 - \mu_{u, t}^2}{4}\right)^{\!r}= \left(\frac{1 - \mu_{u, t}}{2}\right)^{\!K}\, \sum_{r = 0}^{K - 1}\binom{K - 1 + r}{r}\left(\frac{1 + \mu_{u, t}}{2}\right)^{\!r}.
\mathcal{P}_{\rm D}-U\mathcal{P}_{\rm D}U=\left(\begin{matrix}\mathcal{P}_{\rm D}-\mathcal{U}^\ast \mathcal{P}_{\rm D}\mathcal{U}0 \cr0 \mathcal{P}_{\rm D}-\mathcal{U}\mathcal{P}_{\rm D}\mathcal{U}^\ast \cr\end{matrix}\right).
R[E_2](M(t):\:t\in\Omega_2) = R(\{M_f :\: \|f\|_{E_2}\leq 1\}) & = R(\{N_{K'f}:\: \|f\|_{E_2}\leq 1\}) & \leq \|K'\| R(\{N_{K'f}:\: \|K'f\|_{E_1}\leq 1\}) & \leq \|K\| R(\{N_g:\: \|g\|_{E_1}\leq 1\}) & = \|K\| R[E_1](N(t):\:t\in\Omega_1).
U (d_1, d_2) = \begin{pmatrix}1 d_1 0 -d_1^2/2 0 1 0 -d_1 0 0 1 0 0 0 0 1\end{pmatrix}\begin{pmatrix}1 0 d_2 d_2^2/(2 \epsilon) 0 1 0 0 0 0 1 d_2/ \epsilon 0 0 0 1\end{pmatrix},
\ell_{\rho}(\gamma) = \sum_A \ell_{\rho}(A) + \sum_{\Gamma_i}\ell_{\rho}(\Gamma_i) &\sim N + \sum_{\Gamma_i}\{i(\Gamma_i, \alpha)(-2\log \ell_{\rho}(\alpha)) + i(\Gamma_i, \tau)\ell_{\rho}(\alpha)\}&\sim[i(\gamma, \alpha) + 1] + i(\gamma, \alpha)(-2\log \ell_{\rho}(\alpha)) + (\sum_{\Gamma_i}i(\Gamma_i, \tau))\ell_{\rho}(\alpha).
{\frac{3}{r_{1}^{( 1 +{\frac{3 b}{4}})}( r_{1}^{3}- R_{1}^{3})^{( 1 -{\frac{3 b}{4}})}}}\, \left[ \left( 1 +{\frac{( 4 + 3 b )}{3 ( 4 - 3 b )}}\right) \, r_{1}^{3}- R_{1}^{3}\right] = -{\frac{V}{4}}\, ({\frac{4}{3}}- b ) \, e^{{\frac{3 b}{2}}\, \Phi_{0}}.
A =\{\omega\in\Omega:\, \sup_{i\in I}|V_{\omega}(i)-V_{\bar\omega}(i)|<\epsilon\}&=\bigcap_{n=1}^r\{\omega\in\Omega:\, b_n |\omega_{k_n}-\bar\omega_{p_n}|<\epsilon\, \}&=\bigcap_{n=1}^r\{\omega\in\Omega:\, \omega_{k_n}\in(\bar\omega_{p_n}-\epsilon\, b_n^{-1}, \bar\omega_{p_n}+\epsilon\, b_n^{-1})\},
\widehat{T_\ell}[v_1,\cdots, v_p]_k = \sum_{\substack{k=k_1-k_2+\cdots-k_{p-1}+k_p |k_\ell|\gg k_2^* Case\, B}}\frac{\prod_{\substack{j=1 odd}}^{p}\widehat{v}_{k_i}\prod_{\substack{j=2 even}}^p\overline{\widehat{v}_{k_i}}}{\Phi(k, k_1\cdots, k_p)},
{\bar{\beta}}^{B}{\bar{\pi}}_{B}= \pm{\sqrt E}e^{-{\frac{i \varphi}{2}}}e^{- i \psi}\cos{\frac{\theta}{2}}, \qquad \qquad \beta^{A^{\prime}}\pi_{A^{\prime}}= \pm{\sqrt E}e^{{\frac{i \varphi}{2}}}e^{i \psi}\cos{\frac{\theta}{2}},
\int_{\mathbb R^n}x^\gamma Ta(x)\,dx =\lim_{j\to\infty}\int_{\mathbb R^n}\phi_j(x)x^\gamma Ta(x)\,dx=\lim_{j\to\infty}\langle\phi_jx^\gamma,Ta\rangle &=\lim_{j\to\infty}\langle T^*(\phi_j x^\gamma),a\rangle=\langle T^*(x^\gamma),a\rangle=0
\alpha(\mathfrak{a})\mathfrak{pa}^\ast =-2(a_1\!+\!a_2)a_3e_3\!-\!2(a_1\!+\!a_2)a_4e_4 - (a_1^2\!+\!a_2^2\!+\!a_3^2\!+\!a_4^2\!+\!a_5^2\!-\!a_6^2\!+\!2a_2a_1)e_1 - (a_1^2\!+\!a_2^2\!-\!a_3^2\!-\!a_4^2\!-\!a_5^2\!+\!a_6^2\!+\!2a_2a_1)e_2 - 2(a_1\!+\!a_2)a_5e_5\!-\!2(a_1\!+\!a_2)a_6e_6.
\frac{(b;q^2)_N}{(a;q^2)_N}\frac{(a;q^2)_{N-n}}{(b;q^2)_{N-n}}\left(\frac{a}{b}\right)^n=\frac{(q^{2-2N}/b;q^2)_n}{(q^{2-2N}/a;q^2)_n}
\eta_{\pm}=\lim_{t\rightarrow 0_{\pm}}\eta, \ \ \eta(\nabla_i)=\pm(*\nabla_i-\nabla_i*),\ \ \eta_+=\eta_0+\sum_{i\ge 1}\sigma_i, \ \ \eta_-=\eta_0+\sum_{i\ge 1}(-1)^i\sigma_i
I_{P\varphi}(x,t):=J_{3}-J_{4}=\int_{0}^{t}\int_{\partial\Omega}\bigg( \frac{\partial \varepsilon_{n}\big(x-\xi,b(t,\tau)\big)}{\partial\nu(\xi)}a(\tau)(P\varphi)(\xi,\tau) &-\varepsilon_{n}\big(x-\xi,b(t,\tau)\big)a(\tau)\frac{\partial (P\varphi)(\xi,\tau)}{\partial\nu(\xi)}\bigg) dS_{\xi}d\tau=0.
p(t,x) :=\pi^{-\frac{d}{2}}|x|^{-d}\textnormal{H}_{12}^{20}\left[ \frac{1}{4}t^{-\alpha}|x|^{2}\ \Big|\begin{array}{cc}(1,\alpha) (\frac{d}{2},1) (1,1)\end{array}\right] =\pi^{-\frac{d}{2}}|x|^{-d}\frac{1}{2\pi\textnormal{i}}\int_{L_{v}}\frac{\Gamma(\frac{d}{2}+ z)\Gamma(1+z)}{\Gamma(1+\alpha z)}\left(\frac{1}{4}t^{-\alpha}|x|^{2}\right)^{-z}dz.
E_{g}^{2}=V_{b}^{*}E_{a}V_{b}V_{b}^{*}E_{a}V_{b}= V_{b}^{*}E_{a}E_{b}E_{a}V_{b}=V_{b}^{*}E_{b}E_{a}^{2}V_{b}~~~\textrm{( Since $E_{a}$ and $E_{b}$ commute)}=V_{b}^{*}E_{a}V_{b}=E_{g}.
\left(J_b^\alpha g_n\right)(x)= & \sum_{k=0}^n (-\alpha)^k C_n^k (b-x)^{n-k}\left[\frac{1}{k+1}\left(\frac{b-x}{\alpha}\right)^{k+1}E_1\left(\frac{b-x}{\alpha}\right)-\frac{k!}{k+1}e_k\left(\frac{b-x}{\alpha}\right)e^{\frac{-(b-x)}{\alpha}}+1\right].
\theta=(\frac{1}{2}-\frac{b_{k}}{4}-\frac{b_{k}b_{k-1}}{8}-\frac{b_{k}b_{k-1}b_{k-2}}{16}-\cdot\cdot\cdot-\frac{b_{k}b_{k-1}\cdot\cdot\cdot b_{1}}{2^{k+1}})\pi.
\int_{B_1\times B_{4}^c}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\, dx\, dy\leq 2^{p-1}\int_{B_1\times B_{4}^c}\frac{|u(x)|^p+|u(y)|^p}{|x-y|^{N+ps}}\, dx\, dy &\leq C\left(\int_{B_1}|u|^p\, dx+\int_{B_4^c}\frac{|u|^p}{|y|^{N+ps}}\, dy\right)\leq C \int_{B_3\setminus B_2}|D^su|^p+|u|^p\, dx.
LS = \int_1^T \int B Q^3 B \left[- 2\partial^L_{XY}\Delta_L^{-1}[ Q^3 - G \partial_{YY}^L U^3 - 2\psi_z \partial_{YZ}^L U^3 - \Delta_t C \partial_Y^L U^3] \right] \,dV dt & + \int_1^T \int B Q^3 B \left(\psi_y \partial_{XY}^t U^3 \right) \, dV dt = LS_1 + LS_2 + LS_3 + LS_4 + LS_5.
N ( \lambda , \omega ) \sim{\frac{\lambda^{d / 2}~ a_{0}}{( 4 \pi )^{d / 2}\Gamma ( d / 2 + 1 )}}\sim r{\frac{\lambda^{d / 2}~ V_{d}}{( 4 \pi )^{d / 2}\Gamma ( d / 2 + 1 )}},
\gamma^{0}= \left( \begin{matrix}{0}&{i}\\{- i}&{0}\\ \end{matrix}\right) \quad \gamma^{1}= \left( \begin{matrix}{0}&{1}\\{1}&{0}\\ \end{matrix}\right) \quad \gamma^{*}= \left( \begin{matrix}{1}&{0}\\{0}&{- 1}\\ \end{matrix}\right) \quad .
\log ( p (n \, \boldsymbol{|}\, | A_1 |, \dots , | A_m| ; |s_1|, \dots, |s_m| ) ) = \log ( p ( n ) ) + & + \sum_{j=1}^m \log (p ( | A_j | \, \boldsymbol{|}\, | s_j | , n ) ) - \sum_{j=1}^m \log ( p ( | A_j | \, \boldsymbol{|}\, | s_j | ) ) .
\sum_{k=1}^{r_1+r_2+1}\sum_{i=0}^{r_2}(-1)^i\binom{r_2}{i}\binom{r_1+i}{k-1}x^k = \sum_{i=0}^{r_2}(-1)^i\binom{r_2}{i}\sum_{k=0}^{r_1+r_2}\binom{r_1+i}{k}x^{k+1}=\sum_{i=0}^{r_2}(-1)^i\binom{r_2}{i}x(1+x)^{r_1+i}= x(1+x)^{r_1}(1-(1+x))^{r_2}=\sum_{l=0}^{r_1}\binom{r_1}{l}(-1)^{r_2}x^{r_2+l+1}=\sum_{k=0}^{r_1+r_2+1}(-1)^{r_2}\binom{r_1}{k-r_2-1}x^k.
M^{(-)}(\nu,k)=M^{(-)}_o(k)+ gM_1^{(-)}(k)+g\int_{1}^{\infty}d\alpha(\sum_{\ell=1}^{5}\frac{\Phi_{\ell}^{(2)}(\alpha)}{(\alpha-ik)^{\ell}})[\alpha^{\frac{\nu}{2}}+\alpha^{\frac{-\nu}{2}}]
\mathcal{\hat{T}}_{\alpha\beta}(t) =\xi*\int_{0}^{t}\hat{\mathcal{S}}_{\alpha\beta}(\tau,t-\tau)d\tau, \mathcal{T}_{\alpha\beta}(t) =\xi*\int_{0}^{t}\mathcal{S}_{\alpha\beta}(\tau,t-\tau)d\tau,
\mathfrak{F}[u(t,\cdot)](p)=\Phi(t,-\psi(p))\mathfrak{F}[u_0](p) &=\mathfrak{F}[u_0](p)+\int_0^t k(t,s)(-\psi(p))\Phi(s,-\psi(p))\mathfrak{F}[u_0](p)ds &=\mathfrak{F}[u_0](p)+\int_0^t k(t,s) \mathfrak{F}[Lu(s,\cdot)](p) ds
\hat{Z}_{f}[ A , \epsilon , \kappa ] = \int{\cal D}\psi{\cal D}\bar{\psi}e^{\int_{\Sigma}d^{2}x \sqrt{g}\bar{\psi}( - i{D \! \! \! /}+{A \! \! /}) \psi}= d e t [ - i{D \! \! \! /}+{A \! \! /}] ,
\overline{\mathcal{D}^c}=\overline{\bigcap_{t>0}\{y:\|a+ty\|_\sharp>\|a\|_\sharp\}}\subseteq\bigcap_{t>0}\overline{\{y:\|a+ty\|_\sharp>\|a\|_\sharp\}}=\bigcap_{t>0}\{y:\|a+ty\|_\sharp\geq\|a\|_\sharp\}.
\begin{aligned}\int H(z, y, 1)\lambda(y)dy = \int_{B(x, 4R)}H(z, y, 1)\lambda(y)dy &\geq \int_{B(x, 2R)\backslash E}H(z, y, 1)\lambda(y)dy +\int_{B(x, 4R)\backslash B(x, 2R)}H(z, y, 1)\lambda(y)dy &+ \int_{E\cap B(z, 1)}H(z, y, 1)\lambda(y)dy.\end{aligned}
\pi_n^*\left(\overline{\mathcal{H}yp}_{2,n-1}\right) \cdot \rho_n^*\left(\overline{\mathcal{H}yp}_{2,1}\right) \equiv a \,\overline{\mathcal{H}yp}_{2,n}+ b \sum_{i=1}^{n-1}{\sigma_{i}}_\ast\left( \overline{\mathcal{H}yp}_{2,n-1}\right)+ c \sum_{i=1}^{n-1}\Phi_i+ d \sum_{1\leq i<j\leq n-1}\Gamma_{i,j}
{P_{\ge \frac{1}{{{\mu}}}}}u(t) ={S_{1,\alpha}}{P_{\ge \frac{1}{{{\mu}}}}}{u_0}+{S_{2,\alpha}}{P_{\ge \frac{1}{{{\mu}}}}}{u_1}+ \int_0^t{{S_{2,\alpha}}}(t - s){P_{\ge \frac{1}{{{\mu}}}}}\left({{{\left| u \right|}^{p - 1}}u}\right)(s)ds.
(x-1)(x-2) \left(f_j(x-2,y) - f_{j-1}(x-2,y) \right) + (x-1)(2x-5) f_{j-1}(x-1, y) &-(x-1)(x-y-1)f_j(x-1,y) - (x-2)^2 f_{j-1}(x,y) &= \begin{cases}(x-1)(y-1) j= 0, 0 j > 0,\end{cases}
\begin{cases}J^1V_1=\sqrt{-1}V_1,~J^1V_2=\sqrt{-1}V_2, J^2V_1=\tau\overline{V}_1+\lambda e^{-i\theta_1}\overline{V}_2,~J^2{V}_2=-\mu e^{-i\theta_1}\overline{V}_1-\overline{\tau}e^{-2i\theta_1}\overline{V}_2, J^3=J^1J^2.\end{cases}
\sum_{i=1}^{k}\sum_{p:s_{i}\rightarrow t_{i}}f_{p}^{*}\cdot\sum_{e\in p}l_{e}(f_{e})<\sum_{i=1}^{k}\sum_{p:s_{i}\rightarrow t_{i}}f_{p}\cdot\sum_{e\in p}l_{e}(f_{e})
K(\alpha_i,{\bar\alpha}_i)\approx \sum_i^n\alpha_i\bar\alpha_i+{1\over4}\sum_i^n\alpha^2_i\bar\alpha^2_i+\sum_{i<j}^n(\alpha_i\bar\alpha_i\alpha_j\bar\alpha_j-{1\over4}\alpha^2_i\bar\alpha^2_j-{1\over4}\alpha^2_j\bar\alpha^2_i),
d_X\big(\big(\Phi_T(c_{b,x}) \big)(s), \big(\Phi_T(c_{b,\pi_{R'}(x)})\big)(s)\big) = \; d_X\big(c_{b,x}(s+T),c_{b,\pi_{R'}(x)}(s+T)\big) = s + T - R' = s - (R'-T).
\left[ a\right]_{q}=\frac{1-q^{a}}{1-q},\ \ \ \left( q\neq1,\a\in\mathbb{C}\right) ;\ \ \ \ \ \left[ 0\right]_{q}! =1,\left[ n\right]_{q}!=\left[ n\right]_{q}\left[ n-1\right]_{q}!\ \ \ \ \ ,\ .
C_1=\left(\begin{tabular}{ccc}-0.8660 0.5000 0 0.4830 0.8365 0.2241 0.1294 0.2241 -0.8365\end{tabular}\right),\;C_2=\left(\begin{tabular}{ccc}0.8365 0.2241 -0.1294 0.2241 -0.8365 0.4830 0 0.4830 0.8365\end{tabular}\right),
(n+1)^d-\left(n^{d/(1-p)}+1\right)^{1-p}\ge(n+1)^d-n^d-(1-p)n^{-\frac{dp}{1-p}}&=n^d+d\xi^{d-1}-n^d-(1-p)n^{-\frac{dp}{1-p}}&\ge n^d+dn^{d-1}-n^d-(1-p)n^{-\frac{dp}{1-p}}&=dn^{d-1}-(1-p)n^{-\frac{dp}{1-p}}.
{\mathcal{M}_{2^{-(2+2/{\bf \tilde{p}})}\cdot h}(B_{\rho}(x,h)|E)}~\geq~ \left({h\over 2\cdot 2^{-(2+2/{\bf \tilde{p}})}\cdot h}\right)^{\bf{\tilde{p}}}~=~2^{{\bf \tilde{p}}+2}\qquad\forall h>0~.
c_{v_0, \omega}= \begin{cases}\displaystyle \frac{(\ell - 1) q^{- \deg v_0}}{1 + (\ell -1) q^{- \deg v_0}}\omega = , \displaystyle \frac{1}{\ell (1 + (\ell - 1) q^{- \deg v_0})}\omega = , \end{cases}
\mathcal{A}^- =\left\{\begin{aligned}\left(\tau,v_{s+k+1},\omega_{k+1}\right)\subset[0,\infty)\times\mathbb{R}^d\times\mathbb{S}^{d-1}\textnormal{such that}& \omega_{k+1}\cdot \left(v_{s+k+1}-v_{i_{k+1}}^\prime\right)\leq 0\end{aligned}\right\}
\mathrm{T r}[ F^{\mu \nu}( p ) ] ~ \mathrm{T r}[ F_{\mu \nu}( - p ) ] = \mathrm{T r}[ f^{\mu \nu}( p ) ] ~ \mathrm{T r}[ f_{\mu \nu}( - p ) ] = N ~ \mathrm{T r}[ f^{\mu \nu}( p ) ~ f_{\mu \nu}( - p ) ] ~ .
T(f_{\mathbb{Z}_2^2}) = T(f_{\mathbb{Z}_4}) = 2 \cdot 4^{n-1}W(n) + 8^{n-1}= = \frac{1}{16}\cdot 4^{n-1}\left(6^n - 3 \cdot 2^n\right) + 8^{n-1}= \frac{3}{8}\cdot 24^{n-1}+ 5 \cdot 8^{n-2}.
\begin{aligned}\int_{\mathbb{T}^\infty}\varphi(\log^{+}|F_{[r]}|)dm_\infty =\int_{E_r}\varphi(0)dm_\infty+\int_{E_r^c}\varphi(\log|F_{[r]}|)dm_\infty &\leq\varphi(0)+\int_{\mathbb{T}^\infty}\varphi(\log|F_{[r]}|)dm_\infty &\leq\varphi(0)+\int_{\mathbb{T}^\infty}\varphi(\log|F^{*}|)dm_\infty.\end{aligned}
\gamma(u) = -\frac{\pi}{2\sin(\pi\mu)}\left(1+O\left(\frac{1}{u^{2N}}\right)\right), \delta(u) = \frac{\pi}{2\sin(\pi\mu)}\left(1-2\mu\sum_{s=0}^{N-1}\frac{B_s'(0)}{u^{2s+2}}+O\left(\frac{1}{u^{2N+2}}\right)\right).
d_+:=\frac{1}{\sqrt{\Sigma^2(S)}}\left\{\log\left(\frac{P(0,T)}{KP(0,S)}\right) + \frac{1}{2}\Sigma^2(S)\right\}, &d_-:=d_+ - \sqrt{\Sigma^2(S)},\Sigma^2(S):=\int_0^S v_{S,T}^2(t)dt, &v_{S,T}(t):=\frac{\sigma}{b}\left\{e^{-b(T-t)}-e^{-b(S-t)}+ l(t)(m(T-t) - m(S-t))\right\}.
\check{\mathbf{u}}^{(t+1)}_N = e^{j\arg\left(\mathbf{T}_N^H\mathrm{Diag}(\mathbf{y})^H\mathbf{F}^{H}\mathbf{B}\mathbf{F}\mathrm{Diag}(\mathbf{y})\mathbf{T}_N\check{\mathbf{u}}^{(t)}_N\right)}.
v_p =\xi'(q_{p+1})-\xi'(q_p)\,\,\mbox{if $0\leq p\leq k$,}d_p' =\sum_{p\leq \ell \leq \tau-1}n_\ell v_\ell=\frac{1}{1+t}\int_{q_p}^{q_\tau}\xi''(s)x(s)ds\,\,\mbox{if $0\leq p\leq \tau-1$},\,\,d_{\tau}'=0, d_p = \sum_{p\leq \ell \leq k}n_\ell v_\ell=\int_{q_p}^1\xi''(s)x(s)ds\,\,\mbox{if $\tau\leq p\leq k$},\,\,d_{k+1}=0.
\int_{\Omega}\lambda |v^\pm|^r+\mu\frac{|v^\pm|^q}{|x|^\alpha}\, dx =\lim_n\int_{\Omega}\lambda |v_n^\pm|^r+\mu\frac{|v_n^\pm|^q}{|x|^\alpha}\, dx &=\lim_n\langle (-\Delta_p)^s v_n, v_n^\pm\rangle\geq \liminf_n[v_n^\pm]_{s,p}^p\geq \delta_0^p>0.
\left( \begin{array}{c c}{A}&{( m_{i}+ \partial_{i}) I}\\{( m_{i}- \partial_{i}) I}&{- A}\\ \end{array}\right)^{2}= ( A^{2}+ ( m_{i}+ \partial_{i}) ( m_{i}- \partial_{i}) I ) \left( \begin{array}{c c}{I}&{0}\\{0}&{I}\\ \end{array}\right)
\lim_{z\to 0}zK_1(2\pi\tau_2 nz)=\frac{1}{2\pi n\tau_2}, \lim_{z\to 0}z^2K_1(2\pi\tau_2 n z)=0, \lim_{z\to 0}z^2K_0(2\pi \tau_2 nz)=0\,.
\int_\Omega|\partial_zT|^4dxdydz=-\int_\Omega T\partial_z(|\partial_zT|^2\partial_zT)dxdydz = -3\int_\Omega T|\partial_zT|^2\partial_z^2Tdxdydz\leq3\|T\|_\infty\|\partial_z^2T\|_2\|\partial_zT\|_4^2 \leq C\|\partial_z^2T\|_2\|\partial_zT\|_4^2,
(\phi^{\dagger}_{p})_{ij}=\frac{ig}{4\sqrt{2}\pi^2}\frac{1}{(y^2+\rho^2)}\left\{\left(\frac{N_c-2}{2N_c}\right)\delta_{ij}\delta_{i,j\leq2}-\frac{1}{N_c}\delta_{ij}\delta_{i,j\geq3}\right\}\tilde{\eta}\eta.
\mu^{N,n,F}=(\Phi^{N,n,F}_{\varepsilon,\theta})_\# \omega^{N,n,F}_{\varepsilon,\theta}, \quad\omega^{N,n,F}_{\varepsilon,\theta}:=\frac{\gamma^{N^{F}n}|_{X^F_{N,n,\varepsilon,\theta}}}{\gamma^{N^{F}n}(X^F_{N,n,\varepsilon,\theta})}.
\partial_t u= -v(1+\psi_0^2) \partial_t\ln h+\frac{1}{4}(-\psi_0-v)\partial_t\ln k^* &+\frac{\mu}{v+\psi_0}\left\{(\partial_x v)^2(-1+\frac{2\psi_0}{v+\psi_0})+\psi_0\partial_{xx}u \right\}
B_{I,\delta}(x) = \beta_{i_1,\delta_1}(x_1) \dots \beta_{i_n,\delta_n}(x_n) \beta_{i_j,\delta_j}(x_j)=\left(\begin{array}{c}\delta_j i_j\end{array}\right) x_j^{i_j}(1-x_j)^{\delta_j-i_j}.