harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
H'(t) \leq -a_1 \int_U |u_t(x,t)|^2 \,dx - a_1 \int_U |u_t(x,t)|^m \,dx + (\frac{3}{2}+ a^2_2 B) \epsilon \int_U (u_t^2(x,t)) \,dx - \frac{1}{4}\epsilon \int_U (u_{xx}(x,t))^2 \,dx + a_2 \epsilon (\gamma \delta ||u||^2_2 + c(\delta)||u_t||^m_m) - \epsilon E(t),
H_1:=\left\langle\begin{pmatrix}1 1 0 1\end{pmatrix},\begin{pmatrix}2 0 0 1\end{pmatrix},\begin{pmatrix}2 0 0 2\end{pmatrix}\right\rangle,H_2:=\left\langle\begin{pmatrix}1 2 3 1\end{pmatrix},\begin{pmatrix}2 0 0 1\end{pmatrix},\begin{pmatrix}2 0 0 2\end{pmatrix}\right\rangle.
|v(y) - v(x)| \leq \sum_{q=1}^{q(k)}\mathrm{osc}(v,J^{(k)}(q)) + \sum_{l > k}\sum_{\epsilon = \pm}\sum_{q=1}^{q_\epsilon(l)}\mathrm{osc}(v,J_\epsilon^{(l)}(q))
S_0 = - N \int d\tilde t \,STr \left( - \frac{1}{2}\partial_{\tilde t}\tilde\Phi \partial_{\tilde t}\tilde \Phi + \frac{1}{4}[\tilde \Phi, \tilde \Phi]^2+{1\over 2}\tilde \Theta \dot{\tilde \Theta}-{1\over 2}\tilde \Theta \gamma^i [ \tilde \Phi_i, \tilde \Theta] \right)
C_0^* := \frac{16 \bigl( 1 +| a|_{W^{1, \infty}(Q)}+|b|_{L^\infty(Q)}+|\mu|_{L^\infty(0, T; H)}^2 +|\omega|_{L^\infty(Q)}^2 \bigr)}{\min \{1, \nu^2, \delta_*(a)\}}.
P\left(\mbox{there exists}\,s\in S(g): a_s> t+\Delta t, ~b_s> T-t, ~s-U\in [t,t+\Delta t]\right) &=P\left(\mbox{there exists}\, s\in S(g)\cap [0,1) \,\right. &~~~\left.\mbox{such that}\, a_s>t+\Delta t, ~b_s>T-t, ~\mbox{and}~s-U-\lfloor s-U \rfloor\in[t,t+\Delta t]\right).
e^{\mu(t,x^5)}= \frac{1}{(k+f)}\frac{[R_0R_0^{\prime}+(f^{\prime}/2)^2 (t-\hat{t_0})^2 -f\hat{t_0}^{\prime}(t-\hat{t_0})]^2}{[R^2_0 +f(t-\hat{t_0})^2]},
\Delta_{\varepsilon ,\pm 1,\overline{\varepsilon}}=\frac{\sigma^2}2\cdot \frac{m-i\varepsilon \widehat{p}}{2m}\cdot\frac{m-i\overline{\varepsilon}\overline{p}}{2m}\cdot \frac12\sigma_p\left( \sigma_p\pm 1\right) =\Psi_{\varepsilon ,\pm1,\overline{\varepsilon}}\cdot \overline{\Psi}_{\varepsilon,\pm 1,\overline{\varepsilon}},
G_{\pm}( \lambda ) \equiv \sum_{n = 0}^{\infty}P_{\pm}( n ) \lambda^{n}= 2 | \widetilde{N_{\pm}}|^{2}e^{- | \alpha |^{2}}\left( e^{| \alpha |^{2}\lambda}\pm \mathrm{R e}\, e^{\alpha^{2}\lambda}\right)
\pi_{\rho}^{o,\kappa,h}(n) = \pi_{\rho}^{o,\kappa,h}(0)\frac{(\rho h -\lfloor \rho h \rfloor)h\lambda_{\ell_{\kappa}}}{n \mu_{i_{\kappa}}}\prod_{n'=1}^{n-1}\frac{h\lambda_{\ell_{\kappa}}}{n'\mu_{i_{\kappa}}};
z_3=\Gamma h_2 c_{\check u_3 e^{-\hat T_1}+\sigma}b_{\eta}=\Gamma h_2 c_{\check \tau_3}a_{-\hat T_1}c_\sigma b_\eta =\Gamma h_3a_{-\check \tau_3}b_{-\check u_3}a_{-\hat T_1}c_\sigma b_\eta.
{\bar{G}}^{( \zeta )}( \beta ) ={\frac{( \beta + i \pi ) \sinh \left({\frac{\displaystyle \pi \beta}{\displaystyle 2 \pi + \zeta}}\right)}{\Gamma \left( 1 -{\frac{\displaystyle \pi - i \beta}{\displaystyle 2 \pi + \zeta}}\right) \Gamma \left( 1 -{\frac{\displaystyle \pi + i \beta}{\displaystyle 2 \pi + \zeta}}\right)}}\ ,
G^{(j)}_{y}(x,z):=\sum\limits_{l,m\geq 0}s^{(j)}_{l+m-1}\frac{B_{m}(y)}{m!}\frac{x^{l}}{l!}z^{m-1}\in \mathbb{C}[y,x,z,z^{-1}][[s_{0}^{(j)},s_{1}^{(j)},s_{2}^{(j)},\ldots]],
I := \int_{0}^{T}\int_{0}^{t}\left(\sum_{(t-s)2^{j\alpha}\le1}2^{j\alpha/p}\exp(-c(t-s)2^{j\alpha}) \Big(\int_{{\mathbb R}^d}|F_{j}(s,y)|^{p}dy\Big)^{1/p}\right)^{p}dsdt II := \int_{0}^{T}\int_{0}^{t}\left(\sum_{(t-s)2^{j\alpha}>1}2^{j\alpha/p}\exp(-c(t-s)2^{j\alpha}) \Big(\int_{{\mathbb R}^d}|F_{j}(s,y)|^{p}dy\Big)^{1/p}\right)^{p}dsdt.
\begin{aligned}0 \leq \sum_i F^{ii}W_{ii}- W_t \leq C_2 \beta \sum_i F^{ii}- (1 + \beta\rho) \sum_i ( 2\rho_{ii}- \phi_{u}) F^{ii}U_{ii}&+ (A + \frac{M}{2}) \sum_i F^{ii}\rho_{ii}- 2 \beta \sum_i F^{ii}U_{ii}\rho_i^2 ,\end{aligned}
G_{1}(j;k;0)={\displaystyle \sum_{n=1}^{k}}\,\mathrm{cos}({\displaystyle \frac{2\,\pi \,j\,n^{2}}{k}}) +{\displaystyle i\,\sum_{n=1}^{k}}\,\mathrm{sin}({\displaystyle \frac{2\,\pi \,j\,n^{2}}{k}})\,.
D_{a,k}:= \begin{pmatrix}D_{a}& \ddots & D_{a}& 1_{2n-2k}\end{pmatrix}= \left(\bigoplus_{1}^{k}D_{a}\right) \bigoplus 1_{2(n-k)}, D_{a}:=\begin{pmatrix}a 0 0 a^{-1}\end{pmatrix}.
\begin{array}{ll}\displaystyle\partial_{t}u_j +a_j(x,t)\partial_{x}u_j +\sum_{k=1}^{n}b_{jk}(x,t)u_k +\sum_{k=1}^{n}\int_{0}^{x}g_{jk}(y,t)u_k(y,t)dy= f_j(x,t), \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\;\;\; (x,t)\in(0,1)\times(0,\infty),\; j\le n, \end{array}
\log x ( t ) = \sum_{k = 1}^{\infty}\frac{a_{k}}{k !}( i t )^{k}- \frac{1}{2}( \sum_{k = 1}^{\infty}\frac{a_{k}}{k !}( i t )^{k})^{2}+ \frac{1}{3}( \sum_{k = 1}^{\infty}\frac{a_{k}}{k !}( i t )^{k})^{3}- \ldots
e_0 \sharp \widehat{\eta}(B)=\widehat{\eta}(e_0^{-1}(B))= \sum_{i=1}^{2}\lambda_i \eta_i(e_0^{-1}(B))=\sum_{i=1}^{2}\lambda_i e_0 \sharp \eta_i(B)=\sum_{i=1}^{2}\lambda_i m_0(B) =m_0 (B).
-{\biggl (}\gamma^2-\frac{\omega^2}{\alpha^2}{\biggr )}\alpha^2 \rho^2 dt^2+ \frac{d \rho^2}{\alpha^2 \rho^2}+{\biggl (}\gamma^2-\frac{\omega^2}{\alpha^2}{\biggr )}\rho^2 d \varphi^2 +\alpha^2 \rho^2 dz^2\:,
d\left(T \{x_n^i\},T \{x_n^j\}\right) = d\left( \{x_n^{i+11}\}, \{x_n^{j+11}\}\right) &= \frac{10}{i+j+22}+1 &< \frac{100}{i+j}+1 &= d\left(\{x_n^i\},\{x_n^j\}\right).
\frac{{{m_w}^{{m_w}}}}{{\Gamma\!\left({{m_w}}\right)}}\!\sum\limits_{j = 0}^M \!{\frac{{{K_w}^j{\alpha_{wj}}}}{{j!}}}{\left({{\varepsilon_k}\! -\!{\kappa^2}}\right)^{q +\!{m_{fw}}\!- 2}}\!\geqslant\!{\left({{\varepsilon_{th}}\! -\!{\kappa^2}}\right)^{q +{m_{fw}}- 2}}\!.
p^1=\left[\begin{array}{cc|c}1 1 1 2 2 2 3 3 3 \end{array}\right],p^2=\left[\begin{array}{cc|c}3 3 1 2 2 2 1 1 3 \end{array}\right],p^3=\left[\begin{array}{cc|c}3 3 1 1 1 2 2 2 3 \end{array}\right],p^4=\left[\begin{array}{cc|c}1 1 1 3 3 2 2 2 3 \end{array}\right],
\langle \mathrm{g e o n}| d_{n , \epsilon}d_{n^{\prime}, \epsilon^{\prime}}| \mathrm{g e o n}\rangle = \frac{{( - 1 )}^{n}\delta_{n , n^{\prime}}\delta_{\epsilon , - \epsilon^{\prime}}}{2 \sinh ( \pi n / r_{+})}\ \ ,
\sum_{n=0}^\infty \sum_{k=0}^n{n\choose k}2^k F_{jk}(\sqrt{5}F_j)^{n-k}B_{n-k}(x) \frac{z^n}{n!}= F_j z e^{\big(\frac{\sqrt{5}F_j}{2}(2x-1) + L_j\big) z}\Big (e^{\frac{\sqrt{5}F_j}{2}z}+ e^{-\frac{\sqrt{5}F_j}{2}z}\Big ) & = F_j z \Big (e^{(\sqrt{5}F_j x + L_j) z}+ e^{(\sqrt{5}F_j(x-1) + L_j)z}\Big ).
\|\Phi_{\#}^{\infty}\|^2_{\#}- \|\Phi_{\#}^y\|^2_{\#}= - \alpha \int_{\mathbb{R}^3}\mathrm{d}k \; \left\| \frac{(h_{\alpha}-e_{\alpha}) x u_{\alpha}}{(h_{\alpha}-e_{\alpha}+|k|)^{1/2}}\right\|_{L^2}^2 f_y(k) +O(\alpha^3 L^2 e^{-L}) .
(\left\{y_j=S_{x,\tau}^{-1}\theta_x^*e_j+Ve_j-V\theta_\tau S_{x,\tau}^{-1}\theta_x^*e_j=S_{x,\tau}^{-1}x_j+Ve_j-V\theta_\tau S_{x,\tau}^{-1}x_j\right\}_{j\in \mathbb{J}}, \left\{\omega_j=S_{x,\tau}^{-1}\theta_\tau^*e_j+Ue_j-U\theta_xS_{x,\tau}^{-1}\theta_\tau^*e_j=S_{x,\tau}^{-1}\tau_j+Ue_j-U\theta_xS_{x,\tau}^{-1}\tau_j\right\}_{j\in \mathbb{J}})
\begin{array}{l l}{{V_{\phi}( 1 , \cdots , N ) =}}&{{e x p \left\{\frac{1}{2}\sum_{r \neq s}\phi_{0}^{( r )}\, \tilde{N}_{0 , 0}^{r , s}\phi_{0}^{( s )}+ \right.}}\\{{}}&{{\left. + \sum_{r , s}\sum_{n}\phi_{0}^{( r )}\tilde{N}_{0 , n}^{r , s}\phi_{n}^{( s )}+ \frac{1}{2}\sum_{r , s}\sum_{n , m}\phi_{0}^{( r )}\tilde{N}_{n , m}^{r , s}\phi_{m}^{( s )}\right\}.}}\\ \end{array}
\left(\frac{\pi}{\sqrt{c}}\right)^{-s}\Gamma(s)\,\zeta_{\infty,p^{\prime}}\left(s,c\right)=\left(\frac{\pi}{\sqrt{c}}\right)^{-(1-s)}\Gamma\left(1-s\right)\,\tilde{\zeta}_{p^{\prime},\infty}\left(1-s,c\right).
{\rm SIR}_{g,n,\ell}=\frac{d_{0}^{-\alpha}|h_{0}|^2}{\sum_{b\in\Phi_b\setminus\{B_{g,n,\ell,0}\}}d_{b}^{-\alpha}|h_{b}|^2},g\in\{{\rm SVC},{\rm DASH}\}.
\sum_{l_{1}= - 1}^{n}\sum_{l_{2}= - 1}^{k}\left( \begin{array}{c}{{n + 1}}\\{{l_{1}+ 1}}\\ \end{array}\right) z^{n - l_{1}}\left( \begin{array}{c}{{k + 1}}\\{{l_{2}+ 1}}\\ \end{array}\right) z^{k - l_{2}}( l_{1}- l_{2}) = ( n - k ) ( z + 1 )^{n + k + 1}.
m_{d+1,1^k}(x_1,x_2,\ldots,x_i) = \begin{cases}m_{d,1^k}(x_1,x_2,\ldots,x_i) x_{i+1}(d\ge2), (k+1) m_{1^{k+1}}(x_1,x_2,\ldots,x_i) x_{i+1}(d=1). \end{cases}
|DG(m,t)| =\frac{2t^{-3/2}4^m}{\sqrt{\pi}\,\Gamma(m+1/2)}\int_{0}^te^{-4w}w^{m-1}w^{3/2}\big(1-\tfrac{w}{t}\big)^{m-1/2}\,dw &\le 2\frac{4^m}{\sqrt{\pi}\,\Gamma(m+1/2)(m+1)^{3/2}}\int_{0}^te^{-4w}w^{m-1}\,dw &\le C\frac{\Gamma(m)}{\Gamma(m+1/2)(m+1)^{3/2}}\sim \frac{1}{m^2},
\prod_{v\notin S}{q(\hat{\lambda}(c_v))}\prod_{v\in S}{q({\lambda}(H^1(K_v)))}=\prod_{v\notin S}{q({\lambda}(O_v))}\prod_{v\in S}\frac{q({\lambda}(H^0(K_v)))}{{[\mathrm{cok}(\hat{\lambda}(H^0(K_v)))]}}.
\left| \langle a_l\cos\theta_l, \sum_{k\neq l}a_{k}\cos\theta_{k}\rangle \right| \leq \left[ \sum_{k\neq l}\mu_{k,l}\|a_l\cos\theta_l\|_{L^2}\|a_k\cos\theta_k\|_{L^2}\right] &=\delta_1 \|a_l\cos\theta_l\|^2,
\sum_{j=-\infty}^{\infty}z^{j}q^{j(j-1)/2}A_{q}\left(zq^{j+(m-n)}\right)A_{q}\left(zq^{j-(m-n)}\right)= \frac{4\left(q;q\right)_{\infty}^{2}}{\left(-1;q^{1/2}\right)_{\!\infty}^{\!2}\left(q^{1/2};q\right)_{\!\infty}^{\!2}}\theta_{q}\left(-z\right)\delta_{m,n}
d^{4}x = \frac{d^{4}x^{\prime}}{x^{8}}\hspace{1 c m}/ \hspace{- 2 m m}x^{\prime}\gamma_{\mu}/ \hspace{- 2 m m}x^{\prime}= - x^{2}\, J_{\mu \nu}( x^{\prime}) \gamma_{\nu}\, .
\sum_{n=1}^{\infty}\intop_{0}^{\infty}\,\left(J_{0}(2\pi xy)-1\right)e^{-2\pi ny}\,dy=-\,\sum_{n=1}^{\infty}\frac{x}{n}\,\intop_{0}^{\infty}e^{-2\pi ny}\,J_{1}(2\pi xy)\,dy,
Q_{1,\vec{v}}=\left( \begin{pmatrix}1 0 0 0 1 0 0 0 1 \end{pmatrix},\begin{pmatrix}0 0 2 1 0 0 0 1 0 \end{pmatrix},\begin{pmatrix}0 2 0 0 0 2 1 0 0\end{pmatrix}\right).
\mathrm{t r}[ a_{1}^{\dagger}( \frac{P^{+}}{3}) a_{2}^{\dagger}( \frac{2 P^{+}}{3}) ] | 0 \rangle + \mathrm{t r}[ a_{2}^{\dagger}( \frac{P^{+}}{3}) a_{1}^{\dagger}( \frac{2 P^{+}}{3}) ] | 0 \rangle ,
\int_0^{\eta_{-N-1}}v_1^{-p'}(x)\biggl(\int_x^{a^{-1}(x)}|\chi_{[\eta_{-N},\eta_N]}g|\biggr)^{p'}dx=0=\int_{\eta_{N}}^\infty v_1^{-p'}(x)\biggl(\int_x^{a^{-1}(x)}|\chi_{[\eta_{-N},\eta_N]}g|\biggr)^{p'}dx.
\underline{{{\underline{{R}}}}}_{a 0}= - \frac 1{\ell^{2}}\underline{{{\Theta}}}_{a}\wedge \underline{{{\Theta}}}_{0}-{\frac{1}{r}}\left( d \underline{{{\gamma}}}_{a}+ \underline{{{\omega}}}_{a b}\wedge \underline{{{\gamma}}}^{b}\right) + O ( r^{- 3}) \qquad ,
\left( \begin{array}{c}{m^{\prime}}\\{n^{\prime}}\\ \end{array}\right) = ( ( \Lambda_{0}^{5})^{- 1})^{T}\left( \begin{array}{c}{0}\\{\bar{\Delta}_{( m^{\prime}, n^{\prime})}^{1 / 2}}\\ \end{array}\right) .
\left( 1 - \frac{wN}{6}e^{2\phi}\right) \left( \dot{\phi}' -\dot{\rho}\phi' - \rho' \dot{\phi}\right) - \dot{\phi}\phi'= \frac{N}{12}e^{2\phi}\left[ \dot{Z}Z' + 2\dot{Z}' - 2 \left( \dot{\rho}Z' + \rho' \dot{Z}\right) \right].
\frac{dx^2}{x^4}\otimes (x^2\partial_x), \,\frac{dx^2}{x^4}\otimes (x \partial_{y_j}), \,\frac{dx}{x^2}\frac{dy_i}{x}\otimes (x^2 \partial_x), & \frac{dx}{x^2}\frac{dy_i}{x}\otimes (x^2\partial_{y_j}), \,\frac{dy_kdy_i}{x^2}\otimes (x^2\partial_x), \,\frac{dy_kdy_i}{x^2}\otimes (x\partial_{y_j}),
f^{2}= 1 - \frac{1}{\kappa^{2}u_{b}^{2}}g \ \ , \ \{\cal E}\simeq - \frac{\pi \Delta u}{u_{b}}( \kappa^{2}u_{b}^{2}) \left[ 1 - \frac{1}{\kappa^{2}u_{b}^{2}}\left( L - \frac{1}{2}u_{b}^{2}b_{\mathrm{e x t}}\right)^{2}\right]^{2}.
r_2(z)-\widetilde{r}_2(z)=\frac{2 i kb}{a}-\frac{2 i k\tilde{b}}{\tilde{a}}, &z^{-2}(r_2(z)-\widetilde{r}_2(z))=\frac{2 i kz^{-2}b}{a}-\frac{2 i kz^{-2}\tilde{b}}{\tilde{a}}.
\check{R}( z_{1}- z_{2}) \phi_{a d}( z_{1}) \otimes \phi_{d c}( z_{2}) = \sum_{b}W_{z_{1}- z_{2}}\left( \begin{array}{c c}{{a}}&{{b}}\\{{d}}&{{c}}\\ \end{array}\right) \phi_{a b}( z_{2}) \otimes \phi_{b c}( z_{1}) .
a_j^2 = 6m^{\ell-1}N^{-d}(2^\ell M_0\log N)^2 A = \sum_{j=1}^m a_j^2=6m^{\ell}N^{-d}(2^\ell M_0\log N)^2 \delta =(2^\ell M_0 \log N)^{-1}t = 2\sqrt{A(d+h+1)\log N}= M_1 (m^{\ell}N^{-d})^{1/2}(\log N)^{3/2}
\Sigma_-\ :\ \tilde{x}^3 - \frac{3}{4}\tilde u_0^4 \tilde{x}- \frac{1}{4}(\tilde u_0^6 + \tilde u_1)+ \frac{1}{4}(\zeta + \frac{1}{\zeta}) +{\cal O}(\tilde \epsilon) = 0 \ ,
\zeta(\{2p\}_n)=\frac{2p\cdot(2\pi)^{2p n}}{(2p n+p)!}\left(\frac{1}{2\sin\frac{\pi}{2p}}\right)^{2p n+p}\bigg\{1+\sum_{k=2}^{N_{p}}R_{p,k}^{2p n+p}\bigg\}\,,
\dot{y}=\Re \int_{\mathbb{R}^N}|x|^2 \dot{u}\bar{u}dx = -2 \Im \int_{\mathbb{R}^N}|x|^2 (\Delta u) \bar{u}dx =4 \Im \int_{\mathbb{R}^N}x \cdot \nabla u \bar{u}dx,
f(x_1, x_2, x_3) &=\sum \limits_{j,k\in\mathbb Z}\sum_{R=I\times J\times S\in \mathcal R^N_{\frak z}(j,k)}|R| \widetilde \psi_{j,k}(x_1, x_2, x_3,x_I,x_J,x_S)\psi_{j,k}*f(x_I,x_J,x_S),
\|f\|^2_{\mathbb{H}^s(\mathbb{R}^d)}=\sum_{k\in\mathbb{N}^d}(1+k_1)^s\cdots(1+k_d)^s |\hat{f}_{k}|^2, \|f\|^2_{\mathbb{E}^{p}_q(\mathbb{R}^d)}= \sum_{k\in\mathbb{N}^d}^\infty \mathrm{e}^{-q(k_1^p+\ldots+k_d^p)}|\hat{f}_{k}|^2 \, .
\varphi_1^{(d+1)}(t k) (t k)^{d+1}- \varphi_1^{(d+1)}(t k') (t k')^{d+1}=\big(\varphi_1^{(d+2)}(x) x^{d+1}+ (d+1) \varphi_1^{(d+1)}(x) x^d\big)(k - k') t,
\begin{array}{c}{{( q^{-{\frac{\sigma^{3}}{2}}}\otimes \sigma^{+}+ \sigma^{+}\otimes q^{\frac{\sigma^{3}}{2}}) \downarrow \otimes \downarrow =}}\\{{= q^{1 / 2}\downarrow \otimes \uparrow + q^{- 1 / 2}\uparrow \otimes \downarrow}}\\ \end{array}
\begin{array}{r c l}{{1}}&{{=}}&{{\langle t , 1 \rangle = \langle t ,{\bar{t}}\cdot{\bar{t}}^{- 1}\rangle}}\\{{}}&{{=}}&{{\langle t , \bar{t}\rangle \langle t , \bar{t}^{- 1}\rangle ~ ,}}\\ \end{array}
r_\varepsilon^2\sim\varepsilon^2\sqrt{\sum_{k=1}^{D_\varepsilon}b_k^{-4}}\sim \varepsilon^2 D_\varepsilon^{(1+4t)/2}\sim D_\varepsilon^{-(1+4(s+t))/2}D_\varepsilon^{(1+4t)/2}=D_\varepsilon^{-2s}.
\frac{dV_{n}^{m}}{dt}=[-i_{ion,n}+\frac{d\Delta x}{4\rho_{i}L}\cdot(\frac{V_{n-1}^{m}-2V_{n}^{m}+V_{n+1}^{m}}{\Delta x^{2}}+\frac{V_{n-1}^{e}-2V_{n}^{e}+V_{n+1}^{e}}{\Delta x^{2}})]/c
f(\theta) = -a_3 \cos^3\theta+(3a_2+a_5) \cos^2\theta\sin\theta & \phantom{=}+(2a_3+a_4+a_6)\cos\theta\sin^2\theta-a_2 \sin^3\theta, g(\theta) = a_2 \cos^3\theta+(3a_3+a_4)\cos^2\theta\sin\theta &\phantom{=}-(3a_2+a_5)\cos\theta \sin^2\theta - a_6\sin^3\theta,
\frac{1}{(\tau+m_1)^{n_1}\cdots (\tau+m_r)^{n_r}}&= \sum_{q=1}^r\sum_{\substack{k_1+\cdots+k_r=N k_1,\ldots,k_r\ge1}}\left( \prod_{j=1}^{q-1}\frac{\binom{k_j-1}{n_j-1}}{(m_q-m_j)^{k_j}}\right) \frac{(-1)^{N+n_q}}{(\tau+m_q)^{k_q}}\left( \prod_{j=q+1}^r \frac{(-1)^{k_j}\binom{k_j-1}{n_j-1}}{(m_j-m_q)^{k_j}}\right),
\gamma^{i}= - i \left( \begin{array}{c c}{0}&{\sigma_{i}}\\{- \sigma_{i}}&{0}\\ \end{array}\right) , \quad \gamma^{4}= \left( \begin{array}{c c}{0}&{{\bf 1}_{2}}\\{{\bf 1}_{2}}&{0}\\ \end{array}\right) ,
j_{e}=-\frac{e_{c}}{\hbar^{3}}\sum_{p}u_{e}f_{e}^{\prime}=-\frac{e_{c}}{\hbar^{3}k_{B}T}\sum_{p}u_{e}\tau_{e}(-\frac{\partial f_{e}^{o}}{\partial E_{e}})(u_{e}\cdot F_{te}),
1+\alpha (T(x)) - \alpha (x) \ =\ 1 + \phi ( \pi (T(x))-\phi ( \pi (x)) = \ 1+\big( \phi ( \pi (T(x))-\phi ( T (x))\big) &+ \big(\phi ( T (x)) -\phi (x) \big)+\big( \phi(x) -\phi ( \pi (x)) \big) > \ \big(1+\phi ( T (x)) -\phi (x) \big) -2\frac{\mu}{3}\ >\ \frac{\mu}{3}
\tilde{\cal L}^{( 0 )}= \pi_{a}\dot{\sigma}^{a}+ \Psi \dot{\theta}- \frac{1}{2}{\sigma^{\prime}}_{a}^{2}- \frac{1}{2}\lambda \bigl ( \sigma_{a}^{2}- 1 \bigr ) - \frac{1}{2}\pi_{a}^{2}+ \sigma_{a}\pi^{a}\theta - \frac{1}{2}\sigma_{a}^{2}\theta^{2}.
\begin{array}{ll}\{Q^\alpha,Q^\beta\}=-2\tilde\Gamma^{m\alpha\beta}P_m, [P_m,Q^\alpha]=-i\tilde\Gamma_m{}^{\alpha\beta}\Sigma_\beta,{}[M_{mn},Q^\alpha]=\displaystyle\frac i4 Q^\beta (\Gamma_{mn})_\beta{}^\alpha, [M_{mn},\Sigma_\alpha]=-\displaystyle\frac i4(\Gamma_{mn})_\alpha{}^\beta\Sigma_\beta\end{array}
N_{Q 5}^{c a}( p ) = - \frac{i}{2}d_{Q}( R ) \int \frac{d^{4}l}{{( 2 \pi )}^{4}}\mathrm{t r}\left[ i \gamma_{5}i S^{c a}( l , m_{Q}) i \gamma_{5}i S^{a c}( l + p , m_{Q}) \right] ,
X_{0}^{\lambda ;p^{\prime}}=\left\| \bigwedge_{i=1}^{p}\nabla D_i\wedge\bigwedge_{j=1}^{k}\nabla I_j \right\|_{k+p}^{-2}\cdot\sum_{i=1}^{p^{\prime}}(-1)^{n-i}(-\lambda) (D- d_i) \Theta_i,
\langle \phi_{i_{1}}. . . \phi_{i_{s}}\rangle_{W}^{g = 0}= r e s ( \frac{\phi_{i_{1}}. . . \phi_{i_{s}}}{\partial W}) \equiv \frac{1}{2 \pi i}\oint_{\gamma}d X \frac{\phi_{i_{1}}( X ) . . . \phi_{i_{s}}( X )}{\partial W}
a_{2r}=\left( -1\right)^{r}b_{r}+\sum_{r<k\leq m}\dbinom{k}{2r-k}\left( -1\right)^{k}\partial^{2(k-r)}b_{k}& =\left( -1\right)^{r}b_{r}+\sum_{r<k\leq m}U_{r,k}\left( -1\right)^{k}b_{k}
K(x)= -\frac{2\ell}{\sqrt{2\pi}}\int_0^{+\infty}\frac{y e^{-|x|y}}{(c-y^2)^2 +\ell^2 y^2}dy +\frac{2 \sqrt{2\pi}}{\sqrt{4c-\ell^2}}e^{-\frac{\sqrt{4c-\ell^2}}{2}|x|}cos(\ell x/2).
\langle \, X \, [ \, \phi \, , \, \phi^{\ast}\, ] \, \rangle_{_J}\, \, = \, \, \int \,{\cal D}\phi \, X \, \, \, e x p{\frac{i}{\hbar}}\, \{S [ \phi \, , \, \phi^{\ast}\, ] \, + \, J^{A}\, \phi^{A}\, \, \}
\begin{array}{c}\int_{\Omega}(\nabla v_{n}\nabla \varphi_{2}+v_{n}\varphi_{2})\,\mathrm{d}x=\int_{\Omega}g(x,\mathrm{T}_{1}(z_{1,n}),\mathrm{T}_{2}(z_{2,n}))\varphi_{2}\,\mathrm{d}x\end{array}
\gamma_M a+(\ell-k)\begin{pmatrix}1 1\end{pmatrix}=a^*, \gamma_M a^*+(\ell-k-1)\begin{pmatrix}1 1\end{pmatrix}=a,\quad\gamma_M b-\begin{pmatrix}1 1\end{pmatrix}=b^*, \quad\gamma_M b^*=b,
T_s=\frac{1}{4}\left(\!\!\!\begin{array}{ccccccccc}2 0 -2 -4 2 2 3 -1 -2 -1 3 -2 2 -1 -1 0 -1 1 0 6 -6 0 0 0 3 -3 0\end{array}\right)^{\!\!\sim}
\|R_N(f)\|_{\dot{\mathcal B}^{\alpha,q}_{p,\mathcal F}}=\bigg\{\sum_{k\in \Bbb Z}\Big(2^{k\alpha}\big\|D_kR_N(f)\big\|_{L^p_\mu}\Big)^q\bigg\}^{1/q}&=\bigg\{\sum_{k\in \Bbb Z}\bigg(2^{k\alpha}\Big\|D_kR_N\Big(\sum_{k'\in \Bbb Z}T_N^{-1}D_{k'}^ND_{k'}(f)\Big)\Big\|_{L^p_\mu}\bigg)^q\bigg\}^{1/q}.
W(x,t) = P_t \ast Kf(x) = \lim_{\ell\to+\infty}P_t \ast \left(c \sum_{k\leq \ell}a_k K\tilde{H}(x-x_k) \right) & = \lim_{\ell\to+\infty}c \sum_{k\leq \ell}a_k P_t \ast K\tilde{H}(x-x_k).
\mathbb{K}_{\rm cr}\Phi_k = \frac{1}{\sqrt{k!(k+2)!}}\!\sum_{\sigma\in\mathfrak{S}_{k+2}}\!\!K_2(x_{\sigma(1)},x_{\sigma(2)})\,\Phi_k(x_{\sigma(3)},\dots,x_{\sigma(k+2)}),
\int_{S^2}(x^3)^2(-6(x^2)^8+6(x^1)^2 (x^2)^6)\,d\mu_{S^2}= 6\left(-2\int_{S^2}(x^3)^2 (x^2)^8d\mu_{S^2}+\int_{S^2}(x^3)^2 (x^2)^6d\mu_{S^2}-\int_{S^2}(x^3)^4 (x^2)^6 \,d\mu_{S^2}\right) = \frac{-16 \pi}{7\cdot 11}.
\frac{1}{2}\sum_{- N \leq k \leq N}\log\left( 1 + \left(\frac{p + k}{q}\right)^2 \right) = \sum_{-N \leq j \leq N}\log | p + j| -(2N+1) \log q + \frac{1}{2}\sum_{-N \leq j \leq N}\log \left( 1 + \left( \frac{q}{p + j}\right)^2 \right).
Q_{m}^{1}=\displaystyle\prod_{n\in\mathbb{N}^*, n\notin I(m)}\frac{|\lambda_n|^2}{|\lambda_{n-m}||\lambda_{n+m}|}&=\prod_{n=1}^{m_1-1}\frac{|\lambda_n|^2}{|\lambda_{n-m}||\lambda_{n+m}|}\prod_{m_1+1}^{m_2-1}\frac{|\lambda_n|^2}{|\lambda_{n-m}||\lambda_{n+m}|}\cdots \prod_{m_5+1}^\infty\frac{|\lambda_n|^2}{|\lambda_{n-m}||\lambda_{n+m}|}.
\nu(1+\varpi^l x+\varpi^l y\tau)\nu'(1+\varpi^l x+\varpi^l y\sigma(\tau))=\psi(\varpi^l{\tau}(x+y{\tau})+\varpi^l \sigma ({\tau}) (x+y\sigma({\tau}))), \forall x,y\in{\mathcal O}_r.
\lbrack f,g]_{\phi}(x)=(1-x)(1+x)^{\beta+3}(f^{\prime\prime\prime}(x)\overline{g}^{\prime\prime}(x)-f^{\prime\prime}(x)\overline{g}^{\prime\prime\prime}(x))\quad(f,g\in\mathcal{D}(T_{2}))
g''_1(A_n) = A_n^{-1}e^{-\left(\sqrt{A_n/2}+\sqrt{P/2}\right)^2}\cdot\left(-\frac{1}{4\pi}e^{-\left(\sqrt{A_n/2}+\sqrt{P/2}\right)^2}\right. \left.+ \frac{1}{4\sqrt{2\pi}}(1+{\rm{erf}}(\sqrt{A_n/2}+ \sqrt{P/2}))\left( A_n^{1/2}+ \sqrt{P}+ A_n^{-1/2}\right)\right).
[f,g]_n(x)=\sum_{k=1}^n\sum_{j=1}^k (-1)^{k+j}\Big\{(a_k(x) \overline{g}^{(k)}(x))^{(k-j)}f^{(j-1)}(x) &-(a_k(x)f^{(k)}(x)^{(k-j)})\overline{g}^{(j-1)}(x)\Big\},
{\bf P}_{c_L}= \sum\limits_{\nu=1}^{M_0}Z_\nu \sum\limits_{q=1}^{R}(\sum\limits_{k_1=0}^{L-1}{\cal W}_{\nu({k_1})}{\bf p}^{(1)}_{q}) \otimes (\sum\limits_{k_2=0}^{L-1}{\cal W}_{\nu({k_2})}{\bf p}^{(2)}_{q}) \otimes (\sum\limits_{k_3=0}^{L-1}{\cal W}_{\nu({k_3})}{\bf p}^{(3)}_{q}).
\begin{dcases}\partial_t \bar{u}+\bar{u}\cdot \nabla \bar{u}= -\nabla \big(h_1^\prime(\bar{\rho})+\bar{\phi}\big) \varepsilon \big( \partial_t\bar{v}+\bar{v}\cdot \nabla \bar{v}\big)= - \nabla \big(h_2^\prime(\bar{n})-\bar{\phi}\big) + \varepsilon \frac{\bar{e}}{\bar{n}}\ , \end{dcases}
\bar{S}_F(x-y) = - i\left\langle 0 \left| T^{+}\Psi(x)\bar{\Psi}(y)\right| 0 \right\rangle = \left( i\gamma^\mu\partial_\mu^{x}- M\right) \Delta_F(x-y,M^2) - \frac{\gamma^{+}}{2}\frac{1}{i\partial_{-}}* \delta(x-y).
E(\rho_s)-E(\bar{\rho})=\frac{d}{2}\ln\left[\frac{1-\frac{2}{k}c_\beta+\left(\frac{k-s}{k}+s\right)\frac{c_\beta^2}{k^2}}{(1-\frac{c_\beta}{k})^2}\right]=\frac{d}{2}\ln\left[1+\frac{\left(s-\frac{s}{k}\right)\frac{c_\beta^2}{k^2}}{\left(1-\frac{c_\beta}{k}\right)^2}\right]<\frac{s}{k}\ln k.
\hat{{Q\kern-0.1em\raise0.3ex\llap{/}\kern0.15em\relax}}_\tau \, \gamma^\mu \, \hat{{R\kern-0.1em\raise0.3ex\llap{/}\kern0.15em\relax}}_\tau \, \left( \equiv \hat{{Q\kern-0.1em\raise0.3ex\llap{/}\kern0.15em\relax}}_\tau^{(n + 1)}\, \gamma^\mu \, \hat{{R\kern-0.1em\raise0.3ex\llap{/}\kern0.15em\relax}}_\tau^{(n + 1)}\right) \, ,
S_2(z) = \frac{1}{2n}- \frac{\pi z}{2}\int_0^1 x^{n}\coth(\pi z x)dx & = \frac{1}{2n}- \frac{\pi z}{2}\int_0^1 x^{n}\Big (1 + \frac{2e^{-2\pi zx}}{1-e^{-2\pi zx}}\Big ) dx & = \frac{1}{2n}- \frac{\pi z}{2}\Big ( \frac{1}{n+1}+ 2 \sum_{m=1}^\infty \int_0^1 x^{n}e^{-2\pi z m x}dx \Big ).
J_2^2 \le\iint_{{\mathcal{R}}_R}\big(v(x)-v(y)\big)^2\,\big(\tau(x)+\tau (y)\big)^2\,\,u_2(x)\,u_2(y)\,K(x-y)\,dx\,dy &\qquad\qquad\cdot\iint_{{\mathcal{R}}_R}\big(\tau(x)-\tau (y)\big)^2\,v^2(y)\,u_2(x)\,u_2(y)\,K(x-y)\,dx\,dy.
\int_{\mathbb{R}^n}\left(\sqrt{c^2 |\xi|^2 + m^2 c^4}- mc^2 \right) |\hat{\phi}(\xi)|^2 d\xi = \int_{\mathbb{R}^n}\frac{|\xi|^2}{\sqrt{\frac{|\xi|^2}{c^2}+m^2}+ m}|\hat{\phi}(\xi)|^2 d\xi & \leq \frac{1}{2m}\int_{\mathbb{R}^n}|\xi|^2 |\hat{\phi}(\xi)|^2 d\xi.
Q_t(X) \geq \frac{1}{2}a_0 t M_0(X)+\frac{1}{4}\sum_{j=1}^{k_0}a_j t^{2j+1}M_j(X)\geq \frac{1}{4}\sum_{j=0}^{k_0}a_j t^{2j+1}M_j(X).
\begin{aligned}u(t,x)= \int_DK(t,x-y)(J+T+1)(1+u_0(y))dy+\int_DS(t,x-y)(J+T+1)v_0(y)dy & +c_1(S\ast u)(t,x)+\int_0^t\int_DS(t-s,x-y)(c_2u+f(u))\dot{W}(ds,dy),\end{aligned}
\operatorname{Re}\psi_3(i\rho,i\tau,\xi+i\eta)=\frac{\alpha\eta}{\rho}\leqslant 0,\shortintertext{and}\operatorname{Re}\psi_4(i\rho,i\tau, \xi+i\eta)=\frac{\alpha\eta}{\tau}\leqslant 0,
|\tilde v(t)|^2 = |h(0)|^2 + 2\sum_{i=1}^m \int_{(0,t]}\langle v^{\ast}_i(s),v(s)\rangle\,dA(s) + 2\int_{(0,t]}( \tilde v(s-)\,dh(s)) - \int_{(0,t]}|v^{\ast}(s)|^2 \Delta A(s) dA(s) + [h]_t,
\mathbb{E}|\mathcal{S}_{1,5}| \leq N^{-2}\sum_{p<k}\left(\mathbb{E}|R_{p}|^{2}\,\mathbb{E}\sum_{j1\neq k,j2\neq k}c_{pj1}c_{pj2}(|W_{pj1}|^{2}-1)(|W_{pj2}|^{2}-1)\right)^{1/2}&=N^{-2}\sum_{p<k}\left(\mathbb{E}|R_{p}|^{2}\sum_{j \neq k}c_{pj}^{2}\mathbb{E}(|W_{pj}|^{2}-1)^{2}\right)^{1/2}\leq c\sqrt{d_{N}^{2}/N}
g^{\mu \nu}= - ( \varepsilon_{5})^{\mu}( \varepsilon_{5})^{\nu}+ ( \varepsilon_{1})^{\mu}( \varepsilon_{1})^{\nu}+ ( \varepsilon_{2})^{\mu}( \varepsilon_{2})^{\nu}+ \frac{1}{\kappa^{2}}k^{\mu}k^{\nu}
\renewcommand{\arraystretch}{1 . 3}\begin{array}{r r c}{{T^{k}T^{k}\left| a \right> = t ( t + 1 ) \left| a \right> ,}}&{{\, \, T^{3}\left| a \right> = t^{z}\left| a \right> ,}}&{{\, \, F \left| a \right> = f \left| a \right> ,}}\\{{Q \left| a \right> = q \left| a \right> ,}}&{{}}&{{}}\\ \end{array}
\delta S ={1 \over 2}\int_{{\cal M}^2}d^2 \xi \sqrt{-g}\delta g^{mn}(\xi )\left(\partial_{m}{\hat X}^{\underline{m}}\partial_{n}{\hat X}_{\underline{m}}-{1 \over 2}g_{mn}g^{kl}\partial_{k}{\hat X}^{\underline{m}}\partial_{l}{\hat X}_{\underline{m}}\right) -
0= (\theta A-A\theta)\xi_b-2\sum^3_{a=1}\{g(A\xi_b,\phi\xi_a)\phi\xi_a-g(\phi\xi_a,\xi_b)A\phi\xi_a\}= f_3\Big\{\big(\eta_{b+2}(\xi)^2+\eta_{b+1}(\xi)^2\big)\xi_b -\eta_b(\xi)\eta_{b+1}(\xi)\xi_{b+1}-\eta_b(\xi)\eta_{b+2}(\xi)\xi_{b+2}&-\eta_{b+2}(\xi)\phi\xi_{b+1}+\eta_{b+1}(\xi)\phi\xi_{b+2}\Big\}.
\dim(\mathcal{M}(L/\gamma_2(L))) =\frac{1}{2}[(m+n-r-s)^2+n-s-m+r] &=\frac{1}{2}[(m+n-r-s)^2+2(n-s)-(m+n-r-s)] &=\frac{1}{2}[(m+n-r-s)(m+n-r-s-1)+2(n-s)].
\left\|(I-F)^{-1}\right\|_{L_x^{\infty}L_z^2\rightarrow L_x^{\infty}L_z^2}\leq \sum_{n=0}^{\infty}\frac{1}{n!}\left\|\widetilde{Q}(u)\right\|_{L^1}^n=\mathrm{e}^{\left\|\widetilde{Q}(u)\right\|_{L^1}}.
|M_i|= |\phi_{i-1}\circ\cdots\circ \phi_{1}(A_i)|=2^p\int_{A_i}(1/2)^p\det\nabla(\phi_{i-1}\circ\cdots\circ \phi_{1})~dx \leq 2^p\int_{A_i}\frac{f^p}{(\det \nabla (\phi_{i-1}\circ \ldots \circ \phi_1))^p}\det\nabla(\phi_{i-1}\circ\cdots\circ \phi_{1})~dx = 2^pI_i
{{\cal G}^{- 1}}_{\mu \nu}^{\alpha \beta}\eta_{\alpha \beta}^{\mathrm{L T T}}= \left( k^{2}- \Lambda_{l}+ \frac{1}{2 r^{2}}( N - 1 ) - \frac{N + d}{N + d - 2}\Lambda \right) \eta_{\mu \nu}^{\mathrm{L T T}}.
(B^\prime_1(s), B^\prime_2(s)) := \begin{cases}(B_1(\lambda^n_0 + s) - y^n_0, B_2(\lambda^n_0 + s) - y^n_0) s \in [0, \lambda^n_1 - \lambda^n_0] (B_1(\lambda^n_0 + s) - y^n_0,B_1(\lambda^n_0 + s) - y^n_0) s > \lambda^n_1 - \lambda^n_0.\end{cases}