harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
H_{q,n}(F_{f}/2)=\begin{vmatrix}\gamma_{n}\gamma_{n+1}\cdots \gamma_{n+q-1}\gamma_{n+1}\gamma_{n+2}\cdots \gamma_{n+q}\vdots \vdots \vdots \vdots \gamma_{n+q-1}\gamma_{n+q}\cdots \gamma_{n+2(q-1)}\end{vmatrix}.
\omega_{0k_l}^\mathrm{h}= \mathop{\arg\max}\limits_{\omega \in \left[0,2\pi \right)}\mathbf{e}_{M_\mathrm{h}}^\mathrm{H}\left[\omega \right] \mathbf{R}_{k_l}^\mathrm{h}\mathbf{e}_{M_\mathrm{h}}\left[\omega \right],
R_p=\left(\begin{array}{rrrrrr}0 0 0 0 0.5 0.25 0 0 0.5 0.25 0 0 0 0.5 0 0.25 0 0 0.5 0 0 0 0 0.25 0.5 0 0 0.25 0 0 0 0.5 0 0 0 0.25 0 0 0.5 0 0 0.25 0 0 0 0.25 0.5 0 \end{array}\right)
I ( D ) = \int_{0}^{\infty}\! \! \! d \hat{T}_{1}d \hat{T}_{2}\int_{0}^{1}\! \! \! d a d b d c d d{\biggl [ ( \hat{T}_{1}+ G_{B a b}) ( \hat{T}_{2}+ G_{B c d}) -{\frac{C^{2}}{4}}\biggr ]}^{-{\frac{D}{2}}}
\langle \mu^{sp}|\Gamma_+(\{\mathbf{x}^{\pm},z\})|\lambda^{sp}\rangle=sp_{\lambda/\mu}(\mathbf{x}^{\pm};z), &\langle \mu^{o}|\Gamma_+(\{\mathbf{x}^{\pm},z\})|\lambda^{o}\rangle=o_{\lambda/\mu}(\mathbf{x}^{\pm};z),
\sigma_h(x,y) = y + h\sum_{i=1}^sb_iw(k_i) + h\sum_{i=1}^sb_iv(l_i(d_h,\ldots,d_h,y+he_h)) = (y + he_h) +h\sum_{i=1}^s b_i d_h + h\sum_{i=1}^sb_iv(l_i(d_h,\ldots,d_h,y+he_h)) + h\left(\sum_{i=1}^s b_i(w(k_i)-d_h)-e_h\right) = \chi_h(d_h(x),y+he_h(x)) + hc_h(x),
m\in \{0,1,\cdots,n-1\}, &1=s_1<s_2<\cdots <s_{m+1}\le n,1=t_1<t_2<\cdots <t_{n-m}\le n, &\{s_j\}_{j=2}^{m+1}\cup \{t_k\}_{k=2}^{n-m}=\{2,3,\cdots,n\}, \{s_j\}_{j=2}^{m+1}\cap \{t_k\}_{k=2}^{n-m}=\emptyset.
\theta_{j,k_j}^{(N)}:=e_{k_j}\big(\theta_{N_{j-1}+1}^{(N)},\ldots,\theta_{N_{j}}^{(N)}\big), 1 \le j \le r, 1 \le k_j \le n_j, N_j:=n_1+\cdots+n_j, N_0=0,
g = \kappa M^{2 - 2 \Delta_{V}}\quad , \quad \kappa = \frac{2 \Gamma ( \Delta_{V})}{\pi \Gamma ( 1 - \Delta_{V})}\left( \frac{\sqrt{\pi}}{2 \Gamma \left( \frac{1}{2 - 2 \Delta_{V}}\right) \Gamma \left( \frac{h}{4 - 4 \Delta_{V}}\right)}\right)^{2 - 2 \Delta_{V}}.
\langle J_0'(u), \varphi\rangle=\lim_n\langle J_0'(u_n), \varphi\rangle=\lim_n\langle J_0'(u_n)-J'_{\rho_n}(u_n), \varphi\rangle=\lim_n\int_\Omega\frac{|u_n|^{p^*_\alpha-1}-|u_n|^{p^*_\alpha-1-\rho_n}}{|x|^\alpha}\varphi\, dx=0,
-2\frac{a_1a_2}{1-a_3}+ 2a_3(1-a_3)\frac{a_1}{1-a_3}\frac{a_2}{1-a_3}= \frac{-2a_1a_2 + 2a_1a_2a_3}{1-a_3}= -2a_1a_2.
M_{1}= \left( \begin{array}{c c c c}{{1}}&{{2}}&{{0}}&{{2}}\\{{- 2}}&{{- 2}}&{{- 2}}&{{- 1}}\\{{1}}&{{0}}&{{1}}&{{0}}\\{{2}}&{{1}}&{{2}}&{{0}}\\ \end{array}\right) \, .
\lim_{p\downarrow p_0}\frac{(d+2)(p-1)}{2p}=\frac{d+2-6\alpha}{6}, &\lim_{\theta\uparrow 3}\frac{(d+2)[\theta-(\theta-1)p]}{2\alpha\theta p}=\frac{(d+2)(3-2p)}{6\alpha p}\in(0,1),p\in(p_0,3/2).
A(r,t)=\frac{\mu_{0}}{4\pi}(\frac{qv}{|r-r_{s}|(1-\beta_{s}\cdot(r-r_{s})/|r-r_{s}|)})_{t_{r}}=\frac{\mu_{0}c}{4\pi}(\frac{q\beta_{s}}{(1-n_{s}\cdot\beta_{s})|r-r_{s}|})_{t_{r}}
I ={\sum}_{{\sigma}{\sigma}^{\prime}={\pm}}{\int}_{\infty}d^{d}x d{\omega}( - i ) ( - 1 )^{( 1 -{\sigma}^{\prime}) / 2}| x^{\mu},{\omega},{\sigma},{\sigma^{\prime}}> < x^{\mu},{\omega},{- \sigma},{- \sigma^{\prime}}|
C=\left( \bigoplus_{i=1}^s \langle \theta_i \rangle \Box\mathfrak{C}_i \right) \oplus \left( \bigoplus_{j=1}^t \left( \langle \theta_j' \rangle \Box\mathfrak{C}_j' \right) \oplus \left( \langle \theta_j'' \rangle \Box\mathfrak{C}_j'' \right) \Bigr) \right),
\left|\frac{zf_3'(z)}{f_3(z)}\right| = \left|1+\frac{\rho_3}{1-\rho_3^2}\left(\frac{u \rho_3^2+4 \rho_3+u}{\rho_3^2+u \rho_3+1}+\frac{q \rho_3^2+4 \rho_3+q}{\rho_3^2+q \rho_3 +1}\right)\right|=1+\sin(1).
{\tilde{V}}_{e f f}^{S Q C D}\equiv{\frac{V_{e f f}^{S Q C D}}{2 / \pi^{2}L^{4}}}= - 2 F ( t ) + \bigl [ F ( t - \beta ) + F ( t + \beta ) \bigr ] \equiv - 2 F ( t ) + G ( t , \beta ) ,
[\Theta_{g_k^*},\Theta_{g_i^*}]=-\frac{1}{2}\sum\limits_{\alpha\in S(-1)}(\Theta_{[g_k^*,z_\alpha]^{\sharp}}\Theta_{[z_\alpha^*,g_i^*]^{\sharp}}+\Theta_{[g_i^*,z_\alpha]^{\sharp}}\Theta_{[z_\alpha^*,g_k^*]^{\sharp}})\in\sum_{\alpha\in\Phi^+_{e,0}}U(\ggg,e)\cdot\Theta_{e_\alpha}\in I_{\lambda,c}.
\mathcal{L}=\frac 12\left( \mathbf{E}^2-\mathbf{H}^2\right) -\frac{2s(s+1)\sigma g}{s(s+1)(g^2-\sigma^2)-1}\left( \mathcal{G}-\mathcal{G}_0\right) +\frac{\epsilon \alpha^2(2s+1)}{90m^4}
r_{00}= -3,r_{01}= \sqrt{d_1},r_{10}= -\frac{1}{\sqrt{d_1}},r_{jj}= -1,\j=1,\ldots,2n-1, r_{i\,i+1}= \frac{d_{i+1}}{d_i}\sqrt{-\frac{d_i}{d_{i+1}}}, r_{i+1\,i}= - \sqrt{-\frac{d_i}{d_{i+1}}},\i=2,\ldots,2n-2.
\sum_{k = 1}^n ((n_s(\lambda_s)^{-k}\cdot c')\theta(n_s(\lambda_s)^{-k}n_s^2\lambda_s^{k - 1}) + (n_s(\lambda_s)^{-k}\cdot c_{n_s})\theta(n_s(\lambda_s)^{-k}\lambda_s^k))\theta(\lambda).
\left(\frac{\partial}{\partial t}\right)^2 \Omega_r[u](t,x)+ \frac{\mu}{1+t}\left(\frac{\partial}{\partial t}\right)\, \Omega_r[u](t,x)+ \frac{\nu^2}{(1+t)^2}\, \Omega_r[u](t,x) = \Omega_r[\Delta u](t,x)+\Omega_r[f](t,x).
(-\Delta)^s w +cw =(-\Delta)^sv +cv +(-\Delta)^s(w-v) &\geqslant -C \bigg ( 1 + \tau d^{-n-2}+(\theta d)^{-n-2s-2}\bigg )x_1 &\geqslant -C \bigg ( (\theta d)^{-n-2s-2}+ \tau d^{-n-2}\bigg )x_1
\begin{pmatrix}1 s_2 \cdots s_d \varphi_{d+1}\cdots \varphi_n \sum_j \lambda_j \Phi_j -\sum_j \lambda_j \varphi_{j2}\cdots -\sum_j \lambda_j \varphi_{jd}\lambda_{d+1}\cdots \lambda_n\end{pmatrix}
\vec{B}=\vec{B}_{1}+\vec{B}_{2}=\mu_{0}\frac{\vec{I_{1}}\times\vec{r}_{1}}{2\pi|\vec{r}_{1}|^{2}}+\mu_{0}\frac{\vec{I_{2}}\times\vec{r}_{2}}{2\pi|\vec{r}_{2}|^{2}}
\zeta^{\prime}=\frac{x^{\prime}+iy^{\prime}}{\frac{1}{2}-z^{\prime}}=\frac{e^{i\phi}(x+iy)}{\frac{1}{2}-z}=e^{i\phi}\zeta=\frac{e^{\frac{i\phi}{2}}\zeta+0}{0\zeta+e^{-\frac{i\phi}{2}}},
\begin{aligned}{\rm Bias}(\hat D_l) = \sum_{j\in J}\gamma_j\varphi_j(l)\varGamma^{-j/2d}+\rho_{bias}(\varGamma^{-1/2}), \end{aligned}\begin{aligned}{\rm Var}(\hat D_l) = \rho_{var}(\varGamma^{-1/2}), \end{aligned}
(\mu^2 +1)(\|x -r\|^2 +\|r -s\|^2) \geq (\mu\|x -r\| +\|r -s\|)^2 &=(\mu\|x -r\| +\mu\|r -b\|)^2 &\geq (\mu\|x -b\|)^2 \geq \mu^2d_B^2(x).
\sup_{B^{(l)}(1-\theta)}|\tilde{u}_l|\leq C(n, p, \theta)\Big[J_{\tilde{u}_l}^{(l)}(1)\Big]^{\frac{1}{2}}\leq C(n, p, \theta, \alpha)\Big[J_{\tilde{u}_l}^{(l)}(\frac{1}{2})\Big]^{\frac{1}{2}}= C(n, p, \theta, \alpha) &\sup_{B^{(l)}(1-\theta)}|\nabla \tilde{u}_l|_{g_l}\leq C(n, p, \theta, \alpha)
\nabla_x F^n_t(^{x}X_t) - (\nabla^w_xF_t)(^{x}X_t) = \int_{-M}^M\rho(z)\Big((\nabla^w_xF_t) (^{x-z/n}X_t) - (\nabla^w_xF_t)(^{x}X_t)\Big)dz~a.s
\begin{aligned}-\frac{2}{t^5}+\frac{2}{t^8}< \frac{x}{y}<-\frac{1}{t^6}\ \ \ \ \ \ \ t+\frac{1}{t^6}< \frac{x}{y}<t+\frac{2}{t^5}-\frac{1}{t^6}\ \ \ \ \ \ \ t^4+3t-\frac{1}{t^8}< \frac{x}{y}<t^4+3t-\frac{1}{t^9}\ \ \ \ \ \ \end{aligned}
\int_{\mathbb B^k}(1-\|\eta\|^2)^{\alpha\cdot(\mathbf a+\mathbf 1)}|{\eta}|^{2\mathbf c}dV=\pi^k\int_{\mathbf B^k_{+}}(1-\sum_{j=1}^{k}r_j)^{\alpha\cdot(\mathbf a+\mathbf 1)}r^{\mathbf c}dV,
\tilde{V}(t) = \int_0^t \left( \frac{\mbox{d}}{\mbox{d}u}D_u Y(t) \right)^2 \, \mbox{d}u & = \int_0^t \left\{\frac{\ 1 \}{t}\left( \int_u^t f^\prime (X(s)) \, Z(s,u) \, \mbox{d}s \right) \, A_1 (u, X_u) \right\}^2 \mbox{d}u.
\frac{\delta \Gamma [ A_{k}]}{\delta \omega ( x )}= \int d y \, \frac{\delta A_{i}( y )}{\delta \omega ( x )}\frac{\delta \Gamma [ A_{k}]}{\delta A_{i}( y )}= D_{i}\, \frac{\delta \Gamma [ A_{k}]}{\delta A_{i}( x )}= 0
Q_{i i}^{[ 2 ]}= \sum_{m_{0}+ \cdots + m_{{D - 1}}= m}{\frac{D_{m_{0}\cdots m_{{D - 1}}}}{( m_{i}+ 1 ) ( m_{i}+ 2 )}}x_{0}^{m_{0}}\cdots x_{i}^{m_{i}+ 2}\cdots x_{{D - 1}}^{m_{{D - 1}}},
\|f_1\partial_zf_2\|_{L^2}=\|\widetilde{f}_1\partial_x\widetilde{f}_2\|_{L^2}\leq C\big(\|\partial_x\widetilde{f}_1\|_{L^2}+\|\widetilde{f}_1\|_{L^2}\big)\|\Delta \widetilde{f}_2\|_{L^2}&=C\big(\|\partial_zf_1\|_{L^2}+\|f_1\|_{L^2}\big)\|\Delta f_2\|_{L^2}.
\bar{u}_{j}(x):=\varepsilon_{0}^{s}u_{j}(\varepsilon_{0}(x+4e_{1})),\quad\mathrm{and}\;\,\bar{K}_{j}(x):=\varepsilon_{0}^{\sigma p-s(p-1)(q-1)}K_{j}(\varepsilon_{0}(x+4e_{1})).
2e(H) \ge \delta\cdot |A|+\Delta(n-|A|) =\Delta n-(\Delta-\delta)|A| &\ge \Delta n-(\Delta-\delta)(\delta+1) &=\Delta n+\delta^2-(\Delta -1)\delta-\Delta &=\Delta n+\left(\delta-\frac{\Delta -1}2\right)^2 -\frac{(\Delta +1)^2}4.
\mathbb{P}(H^0(\mathcal{M},\mathcal{L})^\vee) = \mathbb{P}(H^0(\mathcal{U},\mathcal{L})^\vee) \cong \mathbb{P}(H^0(\mathcal{U}',\mathcal{L}')^\vee) = \mathbb{P}(H^0(\mathcal{M}',\mathcal{L}')^\vee)
F_{2}( z ) = \sum_{n = - \infty}^{\infty}\frac{1}{1 - 4 n^{2}}\frac{\coth \left( \frac{\sqrt{4 \pi^{2}n^{2}+ z^{2}}}{2}\right)}{\sqrt{4 \pi^{2}n^{2}+ z^{2}}}
\sum_{i\in d_1(\vec{x},\vec{y})\bigcup d_2(\vec{x},\vec{y})}M_i[H(\nu,\cdot)](\mu)=H(\nu,\mu^{*})\sum_{i\in d_1(\vec{x},\vec{y})\bigcup d_2(\vec{x},\vec{y})}M_i[t^{-\chi(\cdot)}](\vec{x},\vec{y}).
\beta_{\Lambda^c;i,j}:=\lim_{\Lambda_{1}\uparrow\uparrow V}\prod_{x\in \Lambda_{1}\setminus\Lambda}\hbox{Tr}_{\mathcal{H}_{x}}\left(h_{x, i}h_{x, j}^* \right)=\lim_{\Lambda_{1}\uparrow\uparrow V}e^{\sum_{x\in \Lambda_{1}\setminus\Lambda}H_{x,i,j}};
\alpha_{\lambda+1-\frac n2}=(\lambda-p+1),\alpha_{\frac n2-\lambda}=(n-\lambda-p), \alpha_{\frac n2-\lambda-1}=(n-\lambda-p-1), \beta_{\lambda+1-\frac n2}=(n-\lambda-p-1),\beta_{\frac n2-\lambda}=(\lambda-p), \beta_{\frac n2-\lambda-1}=(\lambda-p+1),
\theta_0=\frac{1}{2}i\sum_{k=1}^KL_k\ln\left[\frac{z-z_k}{z^*-z^*_k}\right]\ + frac{1}{4}\sum_{k=1}^K\sum_{n_k=1}^{N_k}\left[\theta_{k,n_k}(z)+\theta^*_{k,n_k}(z^*)\right].
\{|\tau_c^N-\tau_c|\leq \delta\}\cap \{\frac{N}{\gamma_N}(\tau_c^N-\tau_c)>x\}&\subseteq \left\{\sum_{k=1}^3\eta^N_{\tau_c^N,k}(f_k)<-\zeta_1(\epsilon, x)\right\}\cap\{|\tau_c^N-\tau_c|\leq \delta\}
B_{m,m-n}(b_1, b_2, b_3, b_4, \ldots) = \sum_{\substack{m-2n \leq j 0 \leq j j \leq m-n-1}}\frac{m!}{n!j!}B_{n,m-n-j}(b_2/2, b_3/3, b_4/4, \ldots) & = \sum_{k=1}^n\frac{m!}{n!(m-n-k)!}B_{n,k}(b_2/2, b_3/3, b_4/4, \ldots)
\begin{cases}-\Delta v_k = \mu_k \left(b(x) (v + \epsilon_{n_k})^{q-2}v_k + a(x) v_k^{p-1}\right) \mbox{in $\Omega$}, \dfrac{\partial v_k}{\partial \mathbf{n}}= 0 \mbox{on $\partial \Omega$}.\end{cases}
\frac{\partial^{k_{1}+\cdot\cdot\cdot+k_{n}}f}{\partial{z_{1}}^{k_{1}}\cdot\cdot\cdot\partial{z_{n}}^{k_{n}}}=\sum_{v=1}^{\infty}\frac{\partial^{k_{1}+\cdot\cdot\cdot+k_{n}}f_{v}}{\partial{z_{1}}^{k_{1}}\cdot\cdot\cdot\partial{z_{n}}^{k_{n}}}
\tilde{I}_{1}^{\zeta}( t ) = I_{1}^{\zeta}( t ) - \left[ \left( a ( t ) \Lambda \vphantom{^2}\right)^{2}- M^{2}( t ) \ln \left( a ( t ) \Lambda \vphantom{^2}\right) \right] / ( 8 \pi^{2}) .
T \prod_{i = 1}^{n^{\prime}}: \! \varphi ( x_{i})^{k_{i}}\! : = T \prod_{i = 1}^{l}: \! \varphi ( x_{i})^{k_{i}}\! : T \prod_{i = l + 1}^{n^{\prime}}: \! \varphi ( x_{i})^{k_{i}}\! :
\begin{pmatrix}\frac{a_\ell}{2}0 \beta_\ell \frac{a_\ell}{2}\end{pmatrix}= \begin{pmatrix}a 0 c d \end{pmatrix}\begin{pmatrix}\frac{a_\ell}{2}0 \beta_\ell' \frac{a_\ell}{2}\end{pmatrix}\begin{pmatrix}a 0 c d \end{pmatrix}^{-1}
Veff(\phi)=V_0(\phi)+\frac{N}{2}\int \frac{d^Dp}{(2\pi)^D}\ln \left [p^2+M^2(\phi) \right ] -\frac{N}{4!}(\lambda_0+\frac{\eta_0 \phi^2}{10N}) F(\phi)^2-\frac{2N\eta F(\phi)^3}{6!}.
g=(1-\frac{2M}{r})dt_{r}^{2}+2(1-\frac{2M}{r})a^{\prime}dt_{r}dr-(\frac{1}{1-\frac{2M}{r}}-(1-\frac{2M}{r}){a^{\prime}}^{2})dr^{2}-r^{2}d\theta^{2}-r^{2}sin^{2}\theta d\phi^{2}
P_{\textrm{out,N}}^{\textrm{ISAOC}}= \textrm{Pr}^{\textrm{OFDMA}}\left\lbrace \textrm{FHN=0,NDN=0,FDF=1,FDN=1}\right\rbrace &+\textrm{Pr}^{\textrm{OFDMA}}\left\lbrace \textrm{NDN=0}\right\rbrace \times \left(1- \textrm{Pr}^{\textrm{OFDMA}}\left\lbrace \textrm{FDF=1,FDN=1}\right\rbrace\right).
\widetilde{\psi}(s) = \prod_p \frac{\widetilde{W}(p,p^{-s})}{1-p^{15-5s}+p^{30-10s}}= \prod_p \left( 1 + \frac{p^{14-5s}- p^{59-21s}+ p^{74-26s}- p^{75-26s}- p^{89 -31s}}{1-p^{15-5s}+p^{30-10s}}\right);
\Delta\tilde{Q}_\mu+\mu\Lambda\tilde{Q}_\mu-\nabla\cdot(\tilde{Q}_\mu\nabla\phi_{\tilde{Q}_\mu}) =-\tilde{\mu}{\mathcal{L}_\mu^y}(\partial_\mu Q_\mu)-\tilde{\mu}^2\nabla\cdot(\partial_\mu Q_\mu\nabla\phi_{\partial_\mu Q_\mu}) &=\tilde{\mu}\Lambda Q_\mu -\tilde{\mu}^2\nabla\cdot(\partial_\mu Q_\mu\nabla\phi_{\partial_\mu Q_\mu}),
Op_h^w(\widetilde{\Gamma}(\frac{x+p'(\xi)}{\sqrt{h}}))Op_h^w(\Sigma(\xi)\chi(h^{\beta}\xi))v = Op_h^w\left(\Sigma(\xi)\chi(h^{\beta}\xi)\widetilde{\Gamma}(\frac{x+p'(\xi)}{\sqrt{h}})\right) v + Op_h(r_0)v \, ,
\lambda \mu(\{0\}) \delta_0(\d x) + \lambda \mu'(\d x) + \lambda \mu(\{1\}) \delta_1(\d x) = \lambda \mu(\d x) = \mathcal P^*(\mu)(\d x) = (\mu(\{1\}) + \mu\{0\}) \delta_0(\d x) + \mathcal P^* (\mu').
c_{{\bf{k + q / 2}}}^{\dagger}c_{{\bf{k - q / 2}}}= O ({\bf{k}}) \mathrm{~}\delta_{{\bf{q}},{\bf{0}}}+ [ \sqrt{n_{{\bf{k}}+{\bf{q}}/ 2}}A_{{\bf{k}}}( -{\bf{q}}) + A_{{\bf{k}}}^{\dagger}({\bf{q}}) \sqrt{n_{{\bf{k}}-{\bf{q}}/ 2}}]
F (x) = (\theta-1)\big[ a_2 x^{\frac{\gamma-\theta}{1-\theta}}- a_{-1}x^{\frac{\theta+1}{\theta-1}}+ a_0 x^{\frac{\theta}{\theta-1}}+ \frac{1}{2}b^2 \theta x^{-1}\big] \leq M , x \geq 1.
\begin{array}{llll}d = d_- \tilde d = d_\mathrm{pcv}\tilde \phi = \phi_\mathrm{pcv}\mbox{for Bregman primal Condat--V\~u~\eqref{e-bcv},}d = d_+ \tilde d = d_\mathrm{dcv}\tilde \phi = \phi_\mathrm{dcv}\mbox{for Bregman dual Condat--V\~u~\eqref{e-bcv-dual}.}\end{array}
< G ( p , \tau ) > = \frac{- i}{( m^{2}- p^{2})}\sum_{n = 0}^{\infty}\frac{( - e^{2}A )^{n}}{n ! \epsilon^{n}}\Gamma ( 1 + n \epsilon ) ( \frac{2 i \mu^{2}}{m^{2}- p^{2}})^{n \epsilon}
d s^{2}= e^{4 \phi}( d t + \vec{\omega}\cdot d \vec{x})^{2}- d \vec{x}^{2}- \left( d x^{4}- e^{2 \phi}( d t + \vec{\omega}\cdot d \vec{x}) \right)^{2}- \sum_{5}^{9}d x^{i}d x^{i}\ .
\sum_{r\notin \delta}\Vert v_{tr}\Vert^2=\sum_{r\notin \delta}\Vert d_{t1}e_{1r}+\cdots + d_{tn}e_{nr}\Vert^2 \leq\sum_{r\notin \delta}(\sum_{i=1}^n \vert d_{ti}\vert^2)(\sum_{i=1}^n \vert e_{ir}\vert^2) \leq \Vert v_0\Vert^2\sum_{i=1}^n ( \sum_{r\notin \delta}\Vert e_{ir}\Vert^2 ) <\epsilon^2.
W[\vec{V}^R]=\int d^4x{\cal L}_{\rm red}=\int d^4x{1\over 2}\left(\dot{V}^R_iR_{ij}^{-1}\dot{V}^R_j-V^R_i\left(\vec{\partial}^2-M^2\right)R_{ij}^{-1}V^R_j\right)~.
\Vert u_{\lambda}(t) - u_{\mu}(t) \Vert = \Vert \int_0^t (u_{\lambda}^{\prime}- u_{\mu}^{\prime})\, dt \Vert \le C_{13}{\Vert u_{\lambda}^{\prime}- u_{\mu}^{\prime}\Vert}_{X_T}, \ 0\le t \le T.
\|\hat f- f_0\|^2_n= \left\langle \sum_{i=0}^{n-1}(\hat f_i-f_i)\psi_i, \sum_{i=0}^{n-1}(\hat f_i-f_i)\psi_i\right\rangle_n=\sum_{i=0}^{n-1}(\hat f_i-f_i)^2.
\Big(\mathbb{E}\mathbb{E}\Big\|\sum_{j,k=1}^{m}r_{k}r_{j}'\alpha_{j,k}x_{j,k}\Big\|^{2}\Big)^{1/2}\leq C\Big(\mathbb{E}\mathbb{E}\Big\|\sum_{j,k=1}^{m}r_{k}r_{j}'x_{j,k}\Big\|^{2}\Big)^{1/2}.
w_z = \frac{1}{4}e^{\frac{z}{4}}(y(-\frac{z}{4}) - y'(-\frac{z}{4})), w_{zz}= \frac{1}{16}e^{\frac{z}{4}}(y(-\frac{z}{4}) - 2y'(-\frac{z}{4}) + y''(-\frac{z}{4})).
|k|x\left|\int_{2|k|x}^2\frac{e^{i\xi}-1}{\xi}\frac{1}{\xi-kx}~d\xi\right| \le C|k|x\int_{2|k|x}^2\frac{1}{\xi-|k|x}~d\xi &=C|k|x\log\left(\frac{2}{|k|x}-1\right) &\le C|k|^{\epsilon}|x|^{\epsilon}.
D=\frac{k_{B}T}{4\pi\eta_{m}h}[ln(2/\epsilon)-\gamma+4\epsilon/\pi-(\epsilon^{2}/2)ln(2/\epsilon)][1-(\epsilon^{3}/\pi)ln(2/\epsilon)+c_{1}\epsilon^{b_{1}}/(1+c_{2}\epsilon^{b_{2}})]^{-1}
\begin{cases}\partial_5{\cal H}[q,u,\mu,p,p_{\alpha}](t)=\dot{q}(t)\, , \partial_6{\cal H}[q,u,\mu,p,p_{\alpha}](t)={_a^CD_t}^\alpha q(t) \, , \partial_2{\cal H}[q,u,\mu,p,p_{\alpha}](t)=-\dot{p}(t)+_tD_b^\alpha p_{\alpha}(t) \, ;\end{cases}
\lVert f\rVert_{M^{p,q}({\mathbb{R}^d})}:= \left( \int_{{\mathbb{R}^d}}\left( \int_{{\mathbb{R}^d}}\left| V_g f(x,\xi) \right|^p dx\right)^{q/p}d\xi \right)^{1/q}< \infty,
\begin{array}{ll}\frac{1}{2\lambda^2}\{h(F_\ast (\nabla_X Z), F_\ast Y) + h(F_\ast(\nabla_Y Z), F_\ast X)\}+ Ric(X,Y) + \frac{\mu}{\lambda^2}h(F_\ast X,F_\ast Y) = 0.\end{array}
\left\Vert I_{2}(g)\right\Vert_{L^{p}\left( L^{\infty}\right)}\lesssim_{\, p}\left( \int\left\Vert g(z,~\cdot~)\right\Vert_{L^{q}\cap L^{q^{\prime}}}^{p}~dz\right)^{1/p}& =\left\Vert g\right\Vert_{L^{p}\left( L^{q}\cap L^{q^{\prime}}\right)}.
\left(\phi\right)^+_{i+\frac{1}{2},v}= \lim_{\eth \to 0^+}\phi_h\left(x_{i+\frac{1}{2}}+\eth,v\right) \mbox{and}\left(\phi\right)^-_{i+\frac{1}{2},v}= \lim_{\eth \to 0^+}\phi_h\left(x_{i+\frac{1}{2}}-\eth,v\right).
\nu\left(\partial_{p^e}c_{\bar\gamma_{\ell +1}}\mathbf{Q}_{\ell +1}^{\bar\gamma_{\ell +1}}\right)=\nu_\ell \left(\partial_{p^e}c_{\bar\gamma_{\ell +1}}\mathbf{Q}_{\ell +1}^{\bar\gamma_{\ell +1}}\right),
\psi_1(u,v,w)= \beta uw+ \alpha(1+v)^2, \psi_2(u,v,w)=\beta uw+ \alpha(1+u)^2, \psi_3(u,v,w)= \beta vw+\alpha\left(1+v\right)^2\psi_4(v,w)=\beta vw+\alpha(1+u)^2.
\lambda_i^{pq}={}\delta_{pi}\delta_{qi}\lambda_i^{pq,rs}={}(1-\delta_{ip})\frac{\delta_{iq}\delta_{ir}\delta_{ps}}{\lambda_i - \lambda_p}+(1-\delta_{ir})\frac{\delta_{is}\delta_{ip}\delta_{rq}}{\lambda_i - \lambda_r},
(\tfrac 14 + t^2)^{\ell}\langle E(\cdot, \tfrac 12 + it), \phi \rangle = \langle E(\cdot, \tfrac 12 + it), \Delta^{\ell}\phi \rangle \ll (CM)^{2\ell}(1+|t|) \int_{\mathcal F}\frac{dx dy}{y^{3/2}}.
\begin{array}{|l|c|l|c|}\hline{\rm Genus}{\rm Curvature}{\rm Group}\Lambda \hline0 1 SO(1,2) -1 1 0 ISO(1,1) \,\,\,\,0 \geq 2 -1 SO(2,1) \,\,\,\, 1 \hline\end{array}
||r_G||_2^2=\sum_{j\in\mathbb{Z}^d\backslash\{0\}}\tilde{a}_j^2+\tilde{b}_j^2=\underbrace{\sum_{0<||j||_{\infty}\leq G}\tilde{a}_j^2+\tilde{b}_j^2}_{(*)}+\underbrace{\sum_{||j||_{\infty}>G}\tilde{a}_j^2+\tilde{b}_j^2}_{(**)}.
\Omega(A_{n},B_{kn})=(-1)^{N_{kn}}\sum\limits^{N_{kn}-M_{kn}}_{l=0}\sum\limits^{l}_{m=0}(-1)^{m}\sigma_{m}(M_{kn},N_{kn})S_{l-m}^{n}T_{N-M-l}^{kn}
\tau^*(\mu_R,\mu_T)\ge\tau_{L2}\triangleq\max\limits_{\substack{l=1,\cdots,\min\{N_T,N_R\}s_1=0,1,\ldots,l s_2=0,1,\ldots,N_R-l}}\frac{1}{l}\Big\{(s_1+s_2)-(N_T-l)s_2\mu_T &\left.-\left(\frac{2s_2+s_1+1}{2}\cdot s_1+s_2^2\right)\mu_R\right. &\left.+\left(\frac{2s_2+s_1}{2}(s_1-1)+s_2^2\right)(1-N_T\mu_T)^+\right\}
f(r) = \sqrt{1-\kappa r^2}, V(r) = V(r;l,Q) = \frac{L(L+1)}{r^2}- \frac{Q}{r}f(r), L = l + \frac{d-3}{2}, E ={\cal E}+ \frac{1}{4}\kappa (d-1)^2, \psi(r) = r^{(d-1)/2}f^{-1/2}(r) R(r),.
\begin{aligned}{\mathfrak a}_{2}= \bigl( \delta_{ab}+ \frac{1}{3}R_{acbd}x^{c}x^{d}\bigr) \xi_{a}\xi_{b}+ o({\mathbf{x^{2}}}),{\mathfrak a}_{1}= \frac{2i}{3}\mathrm{Ric}_{ab}x^{a}\xi_{b}+ o({\mathbf{x^{2}}}).\end{aligned}
\bigg| \frac{1}{2}y_1 + \frac{\sqrt{3}}{2}y_n \bigg| \leq \frac{1}{2}|y_1|+ \frac{\sqrt{3}}{2}|y_n| \leq \frac{1}{2}(|z_1|+r)+ \frac{\sqrt{3}}{2}(|z_n|+r) & \leq \frac{1+\sqrt{3}}{2}\frac{2-|z_1|}{4(1+\sqrt{3})}+ \frac{|z_1|+\frac{1}{2}(2-|z_1|)}{2}= \lambda_n.
\left((-1)^{m-1}\xi_1, (-1)^{m-2}\xi_2,\dots, -\xi_{m-1},\xi_m, \xi_{m+1},\xi_{m+2},\dots, \xi_n\right)=\left(\xi_m,\xi_{m-1},\dots, \xi_1\right)\chi_\Gamma\left(\left(X_f\right)_f\right).
[\frac{3\theta_{4}(q^{3})^{2}}{\theta_{4}(q)^{2}}-\frac{\theta_{4}(q^{1/3})^{2}}{\theta_{4}(q)^{2}}]^{2}=4+4[\frac{2\theta_{2}(q)^{2}\theta_{3}(q)^{2}}{\theta_{4}(q)^{4}}]^{2/3}
(\lambda- \mid a+n_{1}v_{k}+t\mid^{2})c(a,n_{1})=q_{a-\delta+\left(n_{1}-n_{2}\right) v_{k}}+ &{\textstyle\sum\limits_{m=1}^{n_{1}-n_{2}-1}}{\textstyle\sum\limits_{u\in\Gamma(k)}}c(a-u,n-m)q_{u+mv_{k}}+(\Psi,e^{i(a+n_{1}v_{k}}).
D_{\vec f}\left(P_{\vec \varphi}, Q_{\vec \varphi}\right) = \ \frac{( 2 n \pi)^{\frac{n}{2}}}{\big( \det( \sum_{i=1}^n A_i) \big)^{\frac{1}{2}}}\ \prod_{i=1}^{n}\left[ c_i \f_i \left( \frac{\det(A_i)}{c_i^{2}}\right) \right]^{\frac{1}{n}}.
\sum_{i = 1}^{r}\Theta ( x - l H \alpha_{i}) - \sum_{1 \le i < j \le r}\Theta ( x - l H ( \alpha_{i}+ \alpha_{j}) ) + \cdots + ( - 1 )^{r - 1}\Theta ( x - l H \sum_{i = 1}^{r}\alpha_{i}) .
\Pi_V(w) =\Pi_V(P_D(g))+\Pi_V(R^D\mu)+\Pi_V(R^{D}f(\cdot,w)) &=P_Dg-P_V(P_D((g))+R^{V}f(\cdot,u)+R^V\mu=R^{V}f(\cdot,w)+R^V\mu\mbox{q.e.}
c_{n}= 2 ( 2^{5 + 2 n}- 1 ) \Gamma ( 5 + 2 n ) \zeta ( 5 + 2 n ) + 8 \pi^{2}( 2^{3 + 2 n}- 1 ) \Gamma ( 3 + 2 n ) \zeta ( 3 + 2 n )
G = \cup_{i}K_{i}, \, \, \, \left| G \right| = \sum_{i = 0}^{r - 1}\left| K_{i}\right| , \, \, \, \, K_{e}= K_{0}, \, \, \, K_{1 ,}. . . . K_{r - 1}, \, \, \, \, r = \# c l a s s e s
g(\tau) = \begin{pmatrix}q^{-1/60}\displaystyle{\sum_{n=0}^\infty \frac{q^{n^2}}{(q;q)_n}}, q^{11/60}\displaystyle{\sum_{n=0}^\infty \frac{q^{n^2 + n}}{(q;q)_n}}\end{pmatrix}^\mathsf{T}
f_{\nu}\left(z\right)=f_{\nu}\left(1\right)+\sum_{l=1}^{k}\frac{1}{l!}f_{\nu}^{\left(l\right)}\left(1\right)\cdot\left(z-1\right)^{l}+\frac{1}{\left(k+1\right)!}f_{\nu}^{\left(k+1\right)}\left(\xi\right)\cdot\left(z-1\right)^{k+1}
c_{-}= \left( \begin{array}{c c c}{{0}}&{{0}}&{{0}}\\{{C_{-}}}&{{0}}&{{0}}\\{{0}}&{{- C_{-}^{T}}}&{{0}}\\ \end{array}\right) , \qquad c_{+}= \left( \begin{array}{c c c}{{0}}&{{C_{+}}}&{{0}}\\{{0}}&{{0}}&{{- C_{+}^{T}}}\\{{0}}&{{0}}&{{0}}\\ \end{array}\right) .
\begin{array}{ll}\mathop{\min}\limits_{\mathbf{c}}\sum_{j=1}^{i-1}u( <\mathbf{C}[:,j],\mathbf{c}> - m^{(i)})+ \sum_{j=1}^{i-1}u( -<\mathbf{C}[:,j],\mathbf{c}> - m^{(i)}) \mathbf{c}\in \{-1,+1\}^{L_{s}\times 1}\end{array}
\begin{aligned}|||u-v|||^2 = \frac{1}{2}\|y(u) - y(v)\|^2 + \frac{\lambda}{2}\|u-v\|^2 = \frac{1}{2}\|y(u) - y(v)\|^2 + \frac{\lambda}{2}\|u_d - \frac{1}{\lambda}p(u) - (u_d - \frac{1}{\lambda}p(v))\|^2 = \frac{1}{2}\|y(u) - y(v)\|^2 + \frac{1}{2\lambda}\|p(u) - p(v)\|^2. \end{aligned}
\prod_{\substack{m,n \geq 1 m \leq n ; \, (m,n)=1}}\left( \frac{1}{1-z^n}\right)^{\frac{m^3}{n^4}}= \sqrt[3]{\frac{1}{1-z}}\; \exp\left\{\frac{1}{4}Li_2(z) + \frac{1}{4}\frac{z}{1-z}\right\},
J = \frac{\partial ( \ldots , \bar{A}_{- 1}^{a}, \bar{A}_{0}^{a}, \bar{A}_{1}^{a}, \ldots )}{\partial ( \ldots , \, \bar{\varphi}_{- 1}^{a}, a_{1}^{a}\, , \bar{\varphi}_{1}^{a}\, , \ldots )}\; .
\mathbf{V}_{\alpha, p}\mu(x) \ge c_1 \, \mathbf{W^{c}}_{\alpha, p}\mu(x)=c_1 \int_0^{\infty}\left(\frac{\mu(B_r(x))}{r^{n-\alpha p}}\right)^{\frac{1}{p-1}}e^{-c r}\frac{dr}{r}, x \in \mathbb{R}^n.
\begin{alignedat}{4}a_1 =\alpha_1n+1, a_2 =\alpha_2n+1, a_3 =\alpha_3n+1, a_4 =\alpha_4n+1, b_1 =\beta_1n+1, b_2 =\beta_2n+1, b_3 =\beta_3n+1, b_4 =\beta_4n+2,\end{alignedat}
F = \int d^{4}x \{- \frac{1}{4}B_{\mu \nu}B^{\mu \nu}- \frac{1}{4}\varepsilon_{a b}C_{\mu}^{a}C^{\mu b}- \frac{1}{1 2}\varepsilon_{a b}\varepsilon_{c d}C^{a c}C^{b d}\}
\sup_{\varepsilon \in (0,\epsilon_1)}\int_0^T \int_{\Omega}\frac{1}{\varepsilon}(g^\varepsilon )^2 \, dxdt= \sup_{\varepsilon \in (0,\epsilon_1)}\int_0^T (\lambda^\varepsilon )^2 \int_{\Omega}\frac{2W(\varphi^\varepsilon)}{\varepsilon}\, dxdt \leq 2c_1D_1(1+T).
\hat{T}_{m}^{(j)}\rightarrow U(R)^{\dagger}\hat{T}_{m}^{(j)}U(R)=exp(i\frac{\theta}{\hbar}\hat{n}\cdot ad_{\vec{J}})\hat{T}_{m}^{(j)}=\sum_{m^{\prime}}D_{m^{\prime}m}^{(j)}(R^{-1})\hat{T}_{m^{\prime}}^{(j)}
2(m-n)\big(\sum_k\beta_{m+n,k}P_k+\rho_{m+n}'c_2\big) = 2(m-n)\psi_1(L_{m+n})=[\psi_1(L_m),L_n]+[L_m, \psi_1(L_n)] = -\sum_k\beta_{m,k}[L_n,P_k]+ \sum_k\beta_{n,k}[L_m,P_k] = -\sum_k\beta_{m,k}(\ell n - k)P_{n+k}-c_2\beta_{m,-n}(n^3-n)\delta_{\ell,1}&+\sum_k\beta_{n,k}(\ell m - k)P_{m+k}+ c_2\beta_{n,-m}(m^3-m)\delta_{\ell,1}.
\Phi(r_{1},\ldots,r_{s}):= \left\{\sum_{i=1}^{s+2}\alpha_{i}\tilde{A}_{i}\mid0\leq\alpha_{i}<1\right\}\tilde{A}_{i}:= (e^{-r_{1}}\sigma_{1}(a_{i}),\ldots,e^{-r_{s}}\sigma_{s}(a_{i}),\sigma_{s+1}(a_{i}))^{t}
\epsilon^{f_{p}\ldots f_{1}f}( \Gamma_{\{f_{1}\}\ldots \{f_{p}\}}C^{-{1}})_{{b a}}= - i^{{{\frac{p ( p - 1 )}{2}}}}p ! ( \Gamma^{\{f \}}C^{-{1}}\Gamma^{\prime})_{a}^{c}
|f(b)-m_{k'}| =(-1)^{k'+\alpha+1}(f(b)-m_{k'}) =(-1)^{k'+\alpha+1}\left(f(b)-(f(\tau_{k'})+(-1)^{k'+\alpha}\lambda)\mathbf{1}_{k'\ge1}-f(a)\mathbf{1}_{k'=0}\right) =(-1)^{k'+\alpha+1}[(f(b)-f(\tau_{k'}))\mathbf{1}_{k'\ge1}+(f(b)-f(a))\mathbf{1}_{k'=0}]+\lambda\mathbf{1}_{k'\ge1}
{\widehat G}= \left( \begin{array}{c c}{{G_{R R},}}&{{G_{R A}}}\\{{G_{A R},}}&{{G_{A A}}}\\ \end{array}\right) = \left( \begin{array}{c c}{{G^{\mathrm{r e t}},}}&{{0}}\\{{0 ,}}&{{G^{\mathrm{a d v}}}}\\ \end{array}\right) \, ,
F(f)(m \otimes s \lambda) = f^0(m)\otimes s \lambda + \sum_{i=1}^{n}f^1(\gamma_{i})(m) \otimes p_i s \lambda = f^0(m)s \otimes \lambda + \sum_{i=1}^{n}f^1(s\gamma_i)(m) \otimes p_i \lambda = f^0(ms) \otimes \lambda + \sum_{i=1}^{n}f^1(\gamma_i)(ms) \otimes p_i \lambda = F(f)(ms \otimes \lambda);
\int_{M}\alpha A ={\frac{1}{2}}\langle \int_{M}\alpha ({\tilde{\overline{\Psi}}}_{L}{\frac{\delta{S}^{-}}{\delta{\tilde{\overline{\Psi}}}_{L}}}+{\tilde{\Psi}}_{L}{\frac{\delta{S}^{-}}{\delta{\tilde{\Psi}}_{L}}}) \rangle .
G_{r}^{( 2 )}= x^{\frac{r - 1}{2}}\cdot \prod_{m = 1}^{\nu - r}\left( \frac{U_{m}^{2}}{U_{m}^{2}- 1}\right)^{r}\times \prod_{m = \nu - r + 1}^{\nu - 1}\left( \frac{U_{m}^{2}}{U_{m}^{2}- 1}\right)^{\nu - m}.