harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
F_{1}(\phi,\psi)(\xi): =\beta\phi(\xi)+d_{1}\mathcal{N}_{1}[\phi](\xi)+\phi(\xi)f\left(\phi,\psi\right)(\xi), begin{align*}0.2cm] F_{2}(\phi,\psi)(\xi): =\beta\psi(\xi)+d_{2}\mathcal{N}_{2}[\psi](\xi)+ \psi(\xi)g\left(\phi,\psi\right)(\xi),
\|e^{-tH^{\beta}}g\|_{L^{\phi}}\leq \inf \left\{\lambda>0: t^{\frac p{p-1}}\, \left(\exp \left\{\left({C}t^{-\frac p{p-1}}\frac{\|g\|_{L^1}}{\lambda}\right)^{2p}\right\}-1\right) \leq 1\right\}={C}t^{-\frac p{p-1}}\left(\log (t^{-\frac p{p-1}}+1)\right)^{-\frac 1{2p}}\|g\|_{L^1}.
W_{\bf a}^{r e n}[ M^{4}] = \kappa_{0}^{r e n}A ( \Sigma ) S_{2}( e^{\frac{2 \pi}{{\bf a}_{r e n}}}) + B ( \Sigma ) ( \kappa_{1}^{r e n}+ S_{0}( e^{\frac{2 \pi}{{\bf a}_{r e n}}}) ) ~ ~ .
\eta_{\ell,m}(Z)S^{m-j}T^j =(\ell+m-j)S^{m-j}T^j, \eta_{\ell,m}(Z')S^{m-j}T^j =(\ell+j)S^{m-j}T^j, \eta_{\ell,m}(N_+)S^{m-j}T^j =jS^{m-j+1}T^{j-1}, \eta_{\ell,m}(N_-)S^{m-j}T^j =-(m-j)S^{m-j-1}T^{j+1}.
\vec{r}_{G V}={\frac{1}{2}}{\frac{1}{p^{2}+ 2}}{\frac{1}{2 p^{2}+ 1}}\left( \begin{array}{c}{{{\frac{p^{2}}{4}}\sin{4 u}+ ( p^{2}+ 1 )^{2}\sin{2 u}}}\\{{-{\frac{2 \sqrt{2}}{3}}p \cos{3 u}- 2 \sqrt{2}p ( p^{2}+ 2 ) \cos{u}}}\\{{-{\frac{p^{2}}{4}}\cos{4 u}- ( ( p^{2}+ 1 )^{2}- 2 ) \cos{2 u}}}\\ \end{array}\right)
\langle e_{i+1},e_{i+1}\rangle_{L^2_{p,\mathcal{I}}}=\frac{1}{b_0m}\int_{\mathbb{R}^3}\int_0^\infty (p^i)^2 F_E^0\left(1+\frac{\mathcal{I}}{mc^2}\right)^2\phi(\mathcal{I})\,d\mathcal{I}dp\cr =1.
f_{U U U}= 8 \pi^{2}\frac{( T -{\bar{T}})}{( U -{\bar{U}})^{2}}\int \frac{d^{2}\tau}{\tau_{2}^{2}}{\bar{F}}({\bar{\tau}}) \partial_{\bar{\tau}}\left( \tau_{2}^{2}\partial_{\tau}( \tau_{2}^{2}\sum_{P_{L}, P_{R}}P_{L}P_{R}P_{R}P_{R}e^{\pi i \tau | P_{L}|^{2}}e^{- \pi i{\bar{\tau}}| P_{R}|^{2}}) \right) .
I_\frac{p-2}{2}(-r)=(-1)^\frac{p-2}{2}\frac{1}{\Gamma(\frac{p-1}{2})\sqrt{\pi}}\big(\frac{r}{2}\big)^\frac{p-2}{2}\int_{-1}^1 e^{ru}(1-u^2)^{\frac{p-3}{2}}du \Leftrightarrow\ \int_{-1}^1 e^{ru}(1-u^2)^{\frac{p-3}{2}}du=(-1)^\frac{p-2}{2}\Gamma(\frac{p-1}{2})\sqrt{\pi}\big(\frac{2}{r}\big)^\frac{p-2}{2}I_\frac{p-2}{2}(-r)
P_h(x')= \left(\frac{\partial z_j}{\partial x_i}\right), M_{h}(x')=P_h \operatorname{diag}\big[z_1^0,z_2^0,\dots,z_n^0\big] P_h^{-1}, M_{\mathcal{V}^{(h)}}(x)=-P_h \operatorname{diag}[z_1,z_2,\dots,z_n] P_h^{-1},
\Psi^{\dag}\Psi = \left( h_1^{\dag}- \sigma_{123}h_2^{\dag}\right) \left( h_1 + \sigma_{123}h_2 \right) = |h_1 |^2 + | h_2 |^2 + \sigma_{123}\left( h_1^{\dag}h_2 - \mbox{h.c.}\right)~.
K_{4}=2(\alpha^{2}+\gamma^{2}-2\beta\delta)(\delta^{2}+\beta^{2}-2\alpha\gamma), K_{5}=2\Big((\alpha^{2}-\beta\delta)(\gamma^{2}-\beta\delta)+(\beta^{2}-\alpha\gamma)(\delta^{2}-\alpha\gamma)\Big) -(\alpha\beta-\gamma\delta)^{2}-(\beta\gamma-\alpha\delta)^{2}.
\Theta_{k k^{\prime}}( t , t^{\prime}) = \mathrm{R e}\left\{\left( \alpha_{k}( t ) \beta_{k^{\prime}}( t ) + \alpha_{k^{\prime}}( t ) \beta_{k}( t ) \right) \left( \alpha_{k}( t^{\prime}) \beta_{k^{\prime}}( t^{\prime}) + \alpha_{k^{\prime}}( t^{\prime}) \beta_{k}( t^{\prime}) \right)^{*}\right\}.
\mathbb{P}(T_{(k)}<X_{r}) =\mathbb{E}\left[\mathbb{P}(T_{(k)}< X_{r})|X_{r}\right] &=\mathbb{E}\left(\sum_{m=k}^{n}\binom{n}{m}(1-e^{-X_{r}})^{m}e^{-(n-m)X_{r}}\right) ~ (~ \eqref{order12}) &=\sum_{m=k}^{n}\binom{n}{m}\left(\sum_{j=0}^{m}(-1)^{j}\binom{m}{j}\mathbb{E}e^{-(n-m+j)X_{r}}\right)
pp'm^2+n^2+mb_\pm+\frac{b_\pm^2+pp'}{4pp'}+n =\frac{p'}{\Delta}\left(x+\frac{b_\pm+p'}{2p'}\right)^2 +\frac{p}{\Delta}\left(y+\frac{b_\pm-p}{2p}\right)^2.
I(T;x) =\!\begin{multlined}[t] x^{15}+ 65x^{14}+ 4436x^{13}+ 31558x^{12}+ 106408x^{11}+ 218649x^{10}+ 303540x^9 +299860x^8 + 216493x^7 + 115475x^6 + 45423x^5 + 12987x^4 + 2617x^3 + 351x^2 + 28x + 1, \end{multlined}
\begin{aligned}\langle \widetilde{\pi}(f)l', \widetilde{v}\rangle = \int_{G_n}f(g) \langle \widetilde{\pi}(g)l', \widetilde{v}\rangle dg &= \int_{G_n}f(g) \langle l', \widetilde{\pi}(g^{-1})(\widetilde{v}) \rangle dg.\end{aligned}
|z_0|^2 \leqslant \frac{|c-a|}{|a|}+ \frac{|c||c-a|}{|s|-1}\leqslant \frac{|c-a|}{|a|}+\frac{2|c|\cdot|c-a|}{|a|}&= \frac{|c-a|}{|a|}(2|c|+1) \leqslant \left(\Big|\frac{c}{a}\Big|+1\right)(2|c|+1) &\leqslant \left(\frac{|c|}{4}+1\right)(2|c|+1) = \frac{|c|^2}{2}+\frac 94 |c|+1 < |c|^2
k_{spec}=\frac{\rho_{s}}{\sqrt{(N\cdot L)(N\cdot V)}}\frac{N\cdot L}{4\pi\alpha_{x}\alpha_{y}}exp[-2\frac{(\frac{H\cdot X}{\alpha_{x}})^{2}+(\frac{H\cdot Y}{\alpha_{y}})^{2}}{1+(H\cdot N)}]
\begin{cases}N_{1}= p_{1}+p_{2}^{3}+p_{3}^{3}+p_{4}^{3}+p_{5}^{3}+2^{v_{1}}+2^{v_{2}}+\cdots+2^{v_{k}}N_{2}= p_{6}+p_{7}^{3}+p_{8}^{3}+p_{9}^{3}+p_{10}^{3}+2^{v_{1}}+2^{v_{2}}+\cdots+2^{v_{k}}.\end{cases}
Z_N(s,\alpha) = \sum_{n=0}^{\infty}(n+\alpha)^{-s}\phi \left(\frac{n+\alpha}{N}\right), Z_N(s,\mathbb{X}_\alpha) = \sum_{n=0}^{\infty}\frac{\mathbb{X}_\alpha(n)}{(n+\alpha)^s}\phi \left(\frac{n+\alpha}{N}\right)
l_{P}(\mathcal{S},F;t) =l_{F,0}(1+t+t^2+ \cdots+t^{n-\dim{F}}) &+l_{F,1}(t+t^2+ \cdots +t^{n-\dim{F}-1}) &+l_{F,2}(t^2+ \cdots +t^{n-\dim{F}-2}) &+ \cdots \cdots.
\sum_{n=0}^{\infty}\frac{1}{n^{2}+a^{2}}=\frac{a\pi\cdot coth(a\pi)-1}{2a^{2}}=\frac{\pi}{2a}\cdot\frac{e^{2a\pi}+1}{e^{2a\pi}-1}-\frac{1}{2a^{2}}
\det X^{1,\ldots,n-1}_{1,\ldots,n-1}\det X^{2,\ldots,n}_{2,\ldots,n}-\det X^{1,\ldots,n-1}_{2,\ldots,n}\det X^{2,\ldots,n}_{1,\ldots,n-1}=\det X^{1,2,\ldots,n}_{1,2,\ldots,n}\,\det X^{2.\ldots,n-1}_{2,\ldots,n-1}.
c_a(x_1,\dots,x_N) = \sum^N_{i=1}\sum^N_{j=i+1}\vert x_j-x_i\vert^2, c_w(x_1,\dots,x_N) = -\sum^N_{i=1}\sum^N_{j=i+1}\vert x_j-x_i\vert^2.
Z_{Q}(\lambda +i\pi )=Z_{Q}(\lambda ),\lim_{\lambda \rightarrow\pm \infty}\frac{Z_{Q}(\lambda )}{e^{\pm 4(\mathsf{N}+1)\lambda}}=\frac{\kappa_{+}\kappa_{-}\cosh (\alpha_{+}+\alpha_{-}-\beta_{+}+\beta_{-}-(\mathsf{N}+1)\eta )}{2^{(2\mathsf{N}+1)}\sinh \zeta_{+}\sinh \zeta_{-}},
\sum_{n=2}^{\infty}|T_m^{l,(n)}(\beta_1)- T_m^{l,(n)}(\beta_2)|_0\le c \beta_1^{-\frac{1}{2}}c_0^{-\frac{m}{2}}M^{-l-1}\alpha^{-4},\ (\forall m\in\{2,4\}).
B_2, K_{i_1}, \ldots, K_{i_n}, S_{p_m,q_m,r_m}, C_1 = \Psi_1, \Psi_2, \Psi_3; & \vdash \Psi_1 \to H_k^?; & \vdash \Psi_2 \to H_\ell^?; & \vdash \Psi_3 \to S_{p_i,q_i,r_i}.
\left|u_{\beta}(\phi\psi_{\beta})\right| \leq C\sup_{\stackrel{{|p| \leq m+n}}{z \in K}}\left|D^{p}(\phi\psi_{\beta})(z)\right|\,\,,\quad \phi\psi_{\beta}\in G^{\infty}_{0}(\widetilde{X}_{\beta})\,\,
L_{k j_1 \cdots j_s l_1\cdots l_{s-1}}D_{l_1 \cdots l_{s-1}}= \frac{2s-1}{s-1}\delta_{\hat{kj_1}}D_{\hat{j}_2 \cdots \hat{j}_s}- \delta_{\hat{j}_1\hat{j}_2}D_{\hat{j}_3 \cdots \hat{j}_sk}.
\begin{array}{l}{< \phi_{q}^{4}> = \frac 1{1 2 8 \pi^{3}C}\log \left[ 1 + \frac{\Lambda^{2}}{( m_{r}^{2}+ ( \xi_{r}- \frac 1 8 ) R - 1 2 \kappa_{r}\phi_{c}^{2}+ 1 5 \lambda_{r}^{2}\phi_{c}^{4}+ \frac Q C )}\right]}\\ \end{array}
\langle \mathcal{H}\rangle =0,\langle \mathcal{H}^{2}\rangle = N(1+\lambda^2+\alpha^2),\langle \mathcal{H}^{3}\rangle = -6N\alpha^2, \langle \mathcal{H}^{4}\rangle =3N^2(1+\lambda^2+\alpha^2)^2+N( 24\alpha^2-2\alpha^4-2-2\lambda^4 -8\lambda^2 -4\lambda^2\alpha^2)
(\mathbf{R}_{\mathbf{r}x}^{\mathrm{mu}})_n=h_{jn}\mathcal{E}_\mathrm{s}\left[\delta_n+\bar{\delta}_n\cdot\sqrt{\frac{4}{\pi(\|\mathbf{h}_n\|^2\mathcal{E}_\mathrm{s}+1)}}\right],
\partial_t L(t,s) = k\ \frac{1 - L(t,s)}{s}+ \partial_s \ell(t,\zeta(t,s))\ \partial_t\zeta(t,s) = k\ \frac{1 - L(t,s)}{s}- \partial_s \ell(t,\zeta(t,s))\ \frac{\partial_t\Sigma(t,\zeta(t,s))}{\partial_s\Sigma(t,\zeta(t,s))}= k\ \frac{1 - L(t,s)}{s}- \partial_s L(t,s)\ (L(t,s)-k-1)\ ,
E_{m}=\frac{RT}{F}ln(\frac{\sum_{i}^{n}P_{M_{i}^{+}}[M_{i}^{+}]_{out}+\sum_{j}^{m}P_{A_{j}^{-}}[A_{j}^{-}]_{in}}{\sum_{i}^{n}P_{M_{i}^{+}}[M_{i}^{+}]_{in}+\sum_{j}^{m}P_{A_{j}^{-}}[A_{j}^{-}]_{out}})
\mathcal{U}^1(h,k)= Q \sum_{\substack{1\leq c \leq C (c ,hk)=1}}\frac{\mu(c)}{c}\sum_{\substack{1\leq m,n<\infty (mn,c )=1 mh\neq nk}}\frac{\tau_A(m) \tau_B(n)}{\sqrt{mn}}V\left( \frac{m}{X}\right) V\left( \frac{n}{X}\right) \frac{\phi(mnhk)}{mnhk}\int_0^{\infty}W(u)\,du.
[{\mit \Psi}( u , x^{\prime}, y^{\prime}, p^{\prime}) ,{\mit \Psi}^{\dagger}( u , x^{\prime \prime}, y^{\prime \prime}, p^{\prime \prime}) ] = \delta ( x^{\prime}- x^{\prime \prime}) \, \delta ( y^{\prime}- y^{\prime \prime}) \, \delta ( p^{\prime}- p^{\prime \prime}) ,
t_{a}=\sqrt{\frac{2b^{2}+2c^{2}-a^{2}}{4}}=\frac{a\sqrt{3}}{2},t_{b}=\sqrt{\frac{2a^{2}+2c^{2}-b^{2}}{4}}=\frac{c\sqrt{3}}{2},t_{c}=\sqrt{\frac{2a^{2}+2b^{2}-c^{2}}{4}}=\frac{b\sqrt{3}}{2},
\partial_ta_{0,0}=0 \partial_tb_{3,0}= v^{3(2-t)}\cdot a_{0,0}\partial c_{0,3}= v^{3t}\cdot a_{0,0}\partial d_{3,3}= v^{3(2-t)}\cdot b_{3,0}+ v^{3t}\cdot c_{0,3}\partial e_{1,1}= v^2\cdot a_{0,0},
U_{2^{p_1}+2^{p_2}+\cdots+2^{p_s}}=\frac{1}{2}U_{2+2^{p_2-p_1}+\cdots+2^{p_s-p_1}}+2U_{2^{p_2}+\cdots+2^{p_s}}+\frac{1}{2}.
\frac{1}{2y}\bigg(-\frac{m}{\sqrt{mn}}A+B\bigg)y^{1+\sqrt{mn}-c}~~= -\frac{mx^{1+\sqrt{mn}-c}}{\sqrt{mn}}\frac{\alpha}{4t^\alpha}\mathrm{e}^{-\frac{\alpha(x^2+y^2)}{t^\alpha}}\bigg(\frac yx\bigg)^{\frac{1+\sqrt{mn}-c}{2}}I_{\frac{1+\sqrt{mn}-c}{2}}\bigg(\frac{\alpha\sqrt{xy}}{2t^\alpha}\bigg),
v_{\mu}^{( 1 )}= - \frac{1}{2 \pi \kappa}H_{\mu}^{( 2 )}+ G_{\mu}^{( 1 )}, v_{\mu}^{( 2 )}= - \frac{1}{2 \pi \kappa}H_{\mu}^{( 1 )}+ G_{\mu}^{( 2 )},
\psi \, = - 2 i \pi \, g^{0}\, \tau \, \int d p_{0}^{\prime}d p_{0}\delta ( p_{0}^{\prime}\! - \! s_{1}) \beta_{1}\beta_{2}K^{T}( p_{0}^{\prime}, p_{0}) \delta ( p_{0}\! - \! s_{1}) \, \psi .
K\begin{bmatrix}e_{(m+1)(p-1)}(x) e_{(m+1)(p-1)+1}(x) \cdots e_{(m+1)(p-1)+m}(x)\end{bmatrix}^{T}=\begin{bmatrix}e_{(m+1)p}(x) e_{(m+1)p+1}(x) \cdots e_{(m+1)p+m}(x)\end{bmatrix}^{T}.
\begin{aligned}(-\Delta)^s_pu(x) =\frac{(2x)^{sp}}{(1-x^2)^s}\lim_{\epsilon\rightarrow0}\int\limits_{\{w\in\mathbb{R}^n: |w|\geq\frac{2x\epsilon}{1-x^2}\}}\frac{\left|1-\left(1+w_1-\frac{1-x^2}{4x^2}|w|^2\right)^s_+\right|^{p-2}\left[1-\left(1+w_1-\frac{1-x^2}{4x^2}|w|^2\right)^s_+\right]}{|w|^{n+sp}}\mathrm{d}w.\end{aligned}
\mathbb{E}\left(e^{-sa_{n}W_{1}}\right) =\mathbb{E}\left[e^{-U_{1}\left(sa_{n}b+\int_{0}^{\infty}\left(1-e^{-sa_{n}y}\right)\mu\left(dy\right)\right)}\right] =\mathbb{E}\left[e^{-n^{-1}U_{1}\left(sa_{n}nb+\int_{0}^{\infty}\left(1-e^{-sy}\right)n\mu\left(a_{n}^{-1}dy\right)\right)}\right],
\left\|\Delta v\right\|_{L^q(\Omega)}\leq \sum\limits_{k = k_0}^\infty\|\Delta\psi_{2^{-k}}\|_{L^q(\Omega)}= \sum\limits_{k = k_0}^\infty2^{-k(\beta-2+2/q)}\|\Delta\psi\|_{L^q(\Omega)}\rightarrow 0
f_k =[ u^k] F(u)=\frac{z(z+1)}{(1+z^2)s_1^k}, g_k =[ u^k] G(u)=\frac{z^2(2zs_1+1)}{(1+z^2)s_1^{k+2}}, \mbox{and}h_k =[ u^k] H(u)=\frac{z((z-1)s_1+1)}{(1+z^2)s_1^{k+1}}.
(e^{\frac{1}{T}}-1)\sum_{n=T}^{N-1}\frac{e^{\frac{n}{T}}}{n+\xi}= \frac{e^{\frac{N}{T}}}{N}+ O\Bigg(\frac{1}{T}+ \sum_{n=T}^{N}\frac{e^{\frac{n}{T}}}{(n+1)^2}\Bigg), (N \geq T+1).
\int_0^t (t_0 + s)^{\lambda - \frac{p+1}{p-1}}h(s) \,ds \le C \begin{dcases}1 \left( \lambda < \frac{2}{p-1}\right), \log(t_0+t) \left( \lambda = \frac{2}{p-1}\right), (t_0+t)^{\lambda - \frac{2}{p-1}}\left( \lambda > \frac{2}{p-1}\right); \end{dcases}
g^{- 1}= \left( \begin{array}{c c c c}{0}&{0}&{0}&{1}\\{0}&{0}&{\left( 1 + e^{2}\right)^{- 1}}&{0}\\{0}&{- \left( 1 + e^{2}\right)^{- 1}}&{0}&{0}\\{- 1}&{0}&{0}&{0}\\ \end{array}\right) ,
(\hat y_i^1,\dots, \hat y_i^n):=(-\hat f_i^1,\dots, -\hat f_i^m, \varphi_{i,\delta}^{m+1},\dots, \varphi_{i,\delta}^n):B_{R}^n(0,\hat g_{i,\delta}^*)\to \mathbb R^n
\omega_Q^{1/\bar{p}_2-1/\bar{p}_1}\int_{\mathbb H^n}\frac{|y|_h^{-Q/p}}{\max(1,|y|_h^Q)}dy=\omega_Q^{1/\bar{p}_2-1/\bar{p}_1}\frac{\omega_QQ}{(Q-Q/p)Q/p},
\begin{cases}S'=G(t,S)-a(t)f(S,I,P)P-\beta(t)SI I'=\beta(t)SI-\eta(t)g(S,I,P)I-c(t)I P'=h(t,P)P+\gamma(t)a(t)f(S,I,P)P+\theta(t)\eta(t)g(S,I,P)I\end{cases},
\widetilde{\Psi}_{R}^{c}( x ; k ) = \left( \begin{array}{c}{{\frac{\Psi_{R}^{1}( x )}{\frac{\pi}{L}( k + \frac{1}{2}) - g q}}}\\{{\frac{\Psi_{R}^{2}( x )}{\frac{\pi}{L}( k + \frac{1}{2}) + g q}}}\\ \end{array}\right)^{c},
\Psi = \int d^3x \{\overline{C}\chi + \overline{P}N + \overline{C}^i\chi_i + \overline{P}^iN_i + \overline{C}_{(1)}\chi_{(1)}+ \overline{P}_{(1)}N_{(1)}+ \overline{C}^1_1\chi^1_1\},
(Au)_{d-1}g_1\cdots g_{d-1}= \theta_{d-1}\left(u_{(d-1)+}g_1\cdots g_{d-2}g_{d-1}' + \sum_{i=1}^{d-2}(Au)_i g_1 \cdots g_{i-1}g_i' g_{i+1}\cdots g_{d-1}\right).
T_{2,2}(m)=\left\lfloor\frac m3\left\lfloor\frac{m-1}2\right\rfloor\right\rfloor-[m\equiv5\bmod6] T_{2,b}(m)=\begin{cases}\left\lfloor\frac{b-2}3\binom m2\right\rfloor+T_{2,2}(m) 2\mid m\land2\mid b \left\lfloor\frac{b-1}3\binom m2\right\rfloor-[m\equiv5\bmod6\lor b\equiv5\bmod6] \end{cases}
g^{(k)}(x)-g^{(k)}(0) = \displaystyle\lim_{n\rightarrow\infty}f_n^{(k)}(x)-f_n^{(k)}(0) &= \displaystyle\lim_{n\rightarrow\infty}\int_0^x f_n^{(k+1)}&= \int_0^x a^{k+1}.
\left( ( 5 h_{( p , r ; q , s )}+ 1 ) \, h_{( p , r ; q , s )}\; Q_{- 2}- 1 2 q_{( p , r ; q , s )}\; L_{- 1}^{2}+ 6 q_{( p , r ; q , s )}\, ( h_{( p , r ; q , s )}+ 1 ) \; L_{- 2}\right) \phi_{( p , r ; q , s )}.
\omega_l^2 =(\omega_l^{(0)})^2+\varepsilon[\frac{2}{a_l}(\Delta q^{(1)})(\pm e_l,n^{(l)})](\omega,a) &\ \ +\delta[\frac{2}{a_l}(q^{(1)})_{*}^{p+1}(\pm e_l,n^{(l)})](\omega,a), \, \l= 1, 2, ..., b.
< T_{\mu \nu}( x ) \, T_{\rho \sigma}( 0 ) > ={\frac{1}{4 8 0 \pi^{4}}\prod}_{\mu \nu , \rho \sigma}^{( 2 )}\left({\frac{c ( g ( x ) )}{x^{4}}}\right) + \pi_{\mu \nu}\pi_{\rho \sigma}\left({\frac{f ( \ln x \mu , g ) )}{x^{4}}}\right) .
\begin{array}{l}D\,\varphi^{(n,m)}= \displaystyle \frac 1{h}\left (\varphi^{(n-1,m)}-\varphi^{(n-1,m)}(\cdot-h) \right )\,, D^2\,\varphi^{(n,m)}=\displaystyle \frac 1{h^2}\left (\varphi^{(n-1,m)}-2\,\varphi^{(n-1,m)}(\cdot-h)+\varphi^{(n-1,m)}(\cdot-2\,h) \right )\,,\end{array}
- \, \frac{\pi^{2 \mu}a ( 1 ) a ( 2 \mu - 2 )}{2 ( \mu - 1 )}\left[ 3 ( \mu - 1 ) \Theta \Xi + \frac{3 \Xi}{( \mu - 1 )}- ( 3 \mu + 2 ) \Theta + \frac{2 \mu \Psi}{( \mu - 1 )}\right]
\alpha_{i}^{\pm}= \frac{1}{\sqrt{2}}\left[ \alpha_{i}^{1}\mp i \alpha_{i}^{2}\right] \ \ , \ \{\alpha_{i}^{\pm}}^{\dagger}= \frac{1}{\sqrt{2}}\left[{\alpha_{i}^{1}}^{\dagger}\pm i{\alpha_{i}^{2}}^{\dagger}\right] \ \ \ ,
\sum_{j=1}^{\infty}s_j F_j(1, \varphi)\cdot \frac{1}{2\pi}\int_0^{2\pi}\int_0^1 u_0(\rho,\theta) \overline{Q_j(\rho,\theta)}\rho d\rho d\theta= -\frac{1}{4}\int_0^1 u_0(\rho,\varphi)d\rho.
d s^{2}= - \frac{1}{\kappa_{+}^{2}r^{2}}( r + r_{+})^{2}( r^{2}+ r_{-}^{2}) e^{- 2 \frac{r_{-}}{r_{+}}\operatorname{a r c t a n}\frac{r}{r_{-}}}d U d V + r^{2}d \Sigma_{q}^{2}.
p_{m(p+1)+\ell,0}^{2}(\tau)\,d|\nu_{m(p+1)+\ell,0}|(\tau)=p_{m(p+1)+\ell,0}^{2}(\tau)\,\frac{d\sigma_{m(p+1)+\ell,0}(\tau)}{|P_{m(p+1)+\ell,1}(\tau)|}.
\sum_{i \in M_C}\sum_{j=1}^{r_i - 1}a_{i r_i}x^*_{ij}+ \sum_{i \in M_C}\sum_{j=r_i}^{n_i}\max \left\{a_{ij}, b - \sum_{k \in M_C - i}a_{k r_k}\right\}x^*_{ij}> b?
\psi_i^{(1)}= (b_i c_{\alpha}^{-1}) (\alpha \otimes \alpha + e_i \otimes \alpha^2), \psi_i^{(2)}= (b_i c_{\alpha}^{-1})^2 (\alpha \otimes \alpha^2 + e_i \otimes \alpha^3), \psi_i^{(3)}= (b_i c_{\alpha}^{-1})^3 (\alpha \otimes \alpha^3).
\frac{\langle \Phi \mid T\left\{\hat{A}_I\left(x_1\right)\hat{A}_I\left(x_2\right) \hat{A}_I\left(x_3\right)...\exp \left(-\int\limits_{-\infty}^\infty H_i\left(t\right) dt\right)\right\}\mid \Phi \rangle}{\langle \Phi \mid T\left\{\exp\left(-\int\limits_{-\infty}^\infty H_i\left(t\right) dt\right)\right\}\mid \Phi \rangle},
\mathrm{Fitt}_{\Lambda_n}\left( \mathrm{Sel}(\mathbb{Q}_n, E[p^\infty])^\vee \right) = \mathrm{Fitt}_{\Lambda_n}\left( \mathcal{A}_n \right) \cdot \mathrm{Fitt}_{\Lambda_n}\left( \mathcal{S}_{\Gamma_n}\right) & \subseteq \mathrm{Fitt}_{\Lambda_n}\left( \mathcal{Y}_n \right) \cdot \mathrm{Fitt}_{\Lambda_n}\left( \mathcal{S}_{\Gamma_n}\right)
V_{\beta}( \phi ) = V_{0}( \phi ) + \frac{N}{2}L F_{\beta}( \phi )_{R}- \frac{N}{4 !}( \lambda_{0}+ \frac{\eta_{0}\phi^{2}}{1 0 N}) ( F_{\beta}( \phi )_{R})^{2}- \frac{2 N \eta ( F_{\beta}( \phi )_{R})^{3}}{6 !}.
\begin{array}{lcl}g^{ab}\left<T_{ab}\right> = \left(\eta^{ab}+h^{ab}\right)\left<T_{ab}\right>=2\eta^{z\overline z}\left<T_{z\overline z}\right>+\cal O\left(h^2\right) \&= 4\left<T_{z\overline z}\right>+\cal O\left(h^2\right)\;,\end{array}
\frac{1}{2}\frac{d}{dt}\|\tilde{u}_N\|_{\mathcal{H}}+2\nu\int_{\mathcal{F}_0}|D(\tilde{u}_N)|^2 dx+2\nu \alpha \int_{\partial \mathcal{S}_0}|\tilde{u}_N- \tilde{u}_{N,\mathcal{S}}|^2ds = 2\nu a(H,\tilde{u}_N)-b(\tilde{u}_N,\tilde{u}_N,H).
\dfrac{{\rm d}}{{\rm d}t}\left[ \int_{0}^{L}\frac{1}{2}\left( \gamma|\partial_{xx}{b}|^2 + \frac{1}{{b}}\right) - \int_{0}^{L}\rho_s\partial_{t}{b}\, \partial_{xx}{b}\right] + \int_{0}^{L}\left( \beta |\partial_{xx}{b}|^2 + \alpha |\partial_{xxx}{b}|^2 \right) = \int_{0}^{L}\rho_s|\partial_{tx}{b}|^2\,.
{\cal S}[ e , \psi ] = \int \, d^{4}x \sqrt{g}\left\{\frac{i}{2}[ \overline{{{\psi}}}\gamma^{\hat{\alpha}}D_{\hat{\alpha}}\psi - ( \overline{{{D_{\hat{\alpha}}\psi}}}) \gamma^{\hat{\alpha}}\psi ] - m \overline{{{\psi}}}\psi \right\}
{\cal D}\Psi_D =\left( \begin{array}{c}-B (\bar Z e^2 + \bar Y,_3 e^3) dB + B[h,_4 +(Z-\bar Z) h/2] e^3 dC + C[h,_4 -(Z-\bar Z) h/2] e^3 C ( Z e^1 + Y,_3 e^3)\end{array}\right).
\mathcal{X}_E*\vartheta_{\tilde{\eta}}((Q_1)_{ii})=\tilde{\eta}\int_{(Q_1)_{ii}-E_U}^{(Q_1)_{ii}-E^-}\frac{1}{y^2+\tilde{\eta}^2}dy \leq \tilde{\eta}\int_{(Q_1)_{ii}-E_U}^{(Q_1)_{ii}-E^-}\frac{1}{y^2}dy \leq \frac{7}{2\tau^2}N^{-4/3+\epsilon-9\epsilon_0},
\vec{X}_{01}= \Bigg(0 , 1 , - \frac{\lambda}{m-n}\bigg(\frac{1}{1- A^{-1}}\bigg)\Bigg), \mu_{01}=1, \vec{X}_{02}= \Bigg( 1 , ab , - \frac{\lambda ad}{m-n}\bigg(\frac{1}{1- 2A^{-1}}\bigg)\Bigg), \mu_{02}=2, \vec{X}_{03}= (0 , 0 , 1), \mu_{03}= A.
C(\omega,\omega') = \delta_{{\omega+\omega'},0}f(\omega , \omega' ,\theta) -4\sum_{\omega_1 ,\omega_2 >0}^N{\omega_1}^2{\omega_2}^2 C_{\omega_1,\omega_2}^{-\omega'}\: C_{\omega_1 ,\omega_2}^{\omega}+ \sum_{\omega_1>0}^N{\omega_1}^2\:(\omega - \omega_1)^2\:C_{\omega_1,\omega-\omega_1}^{-\omega'}
\begin{aligned}\|\nabla^k u_t\|_{L^2}^2 =\!\int \!\nabla^k(-u \cdot \nabla u+g(\varrho)[\mu\Delta u+(\mu+\nu)\nabla{\rm div}u])\nabla^k u_tdx &-\!\int \!\nabla^k[(f(\varrho)+1)\nabla\varrho\!+\!g(\varrho)\nabla n\Delta n]\nabla^k u_tdx &=IV_1+IV_2+IV_3+IV_4.\end{aligned}
\mathcal{J}^{(n)}_1\times I^{(n)}_k\times \mathcal{J}^{(n)}_2 &:= [2^{-(\eta+1)n},\infty)\times I^{(n)}_k\times [t-2^{-2 n}n^{-2m},t-2^{-2(n+1)}(n+1)^{-2m}),
g_{20}=-c_{1}\left( \sqrt{3}+i\right) ,g_{11}=\frac{c_{1}}{2}\left( \sqrt{3}-i\right) ,g_{02}=0, g_{21}=\frac{1}{4}\left( 3A_{1}+i3A_{2}-i B_{1}+ B_{2}+ C_{1}+iC_{2}+-i3D_{1}+3D_{2}\right) .
P_n(t_1,\ldots,t_{n-1};\beta)= \left(\prod_{i= 1}^{n-1}t_i^{n-i}\right)\mathfrak{G}_{\pi}^{(\beta)}\big(t_1^{-1},\dots,t_{n-1}^{-1}\big) = \sum_{{\cal{T}}}wt({\cal{T}}),
F_{_{( 1 )}}= \left( \begin{array}{c c c}{f}&{0}&{M_{_{( 1 )}}}\\{0}&{0}&{R_{_{( 1 )}}}\\{- M_{_{( 1 )}}^{^T}}&{- R_{_{( 1 )}}^{^T}}&{0}\\ \end{array}\right) .
\begin{array}{l}{\varphi_{1 E}\left( r , \theta \right) = \sum_{n = - \infty}^{\infty}f_{n}\left( r \right) e^{i \left( n + \frac{1}{2}\right) \theta}}{\chi_{1 E}\left( r , \theta \right) = \sum_{n = - \infty}^{\infty}g_{n}\left( r \right) e^{i \left( n + \frac{1}{2}\right) \theta}}\end{array}
\left\|\zeta_N(s,\mathbb{X}_\alpha)(\omega)- \zeta_N(s,\mathbb{Y}_\alpha)(\omega_0)\right\| \leq \sum_{n=0}^{N}\left|\mathbb{X}_\alpha(n)(\omega)-\mathbb{Y}_\alpha(n)(\omega_0)\right| \|(n+\alpha)^{-s}\| &\leq \sum_{n=0}^{N}2\pi \delta \sqrt{M}(n+c/2)^{-1/2}&\leq \delta B_1 \sqrt{N}
\sum_{k=0}^{n-1}\frac{(q^{d+1};q^d)_k^{d-1}(q^{1-d};q^d)_kq^{dk}}{(q^d;q^d)_k^d}&=[d]\sum_{k=0}^{n-1}\frac{(q^{d+1};q^d)_k^{d-2}(q,q^{1-d};q^d)_kq^{dk}}{(q^d;q^d)_k^d}-q[d-1]\sum_{k=0}^{n-1}\frac{(q^{d+1};q^d)_k^{d-2}(q;q^d)_k^2q^{dk}}{(q^d;q^d)_k^d},
\nu_- = \left( \begin{array}{cc}I_n 0 \nu_{-21}I_k\end{array}\right), \qquad\nu_+ = \left( \begin{array}{cc}I_n \nu_{+12}0 I_k\end{array}\right), \qquad\eta = \left( \begin{array}{cc}\eta_{11}0 0 \eta_{22}\end{array}\right),
(i) \,\,P_\lambda(x_1,x_2;q,t,T)=m_\lambda+\sum_{\mu\in P^+,\mu<\lambda}a_{\lambda\mu}(q,t,T) m_\mu, (ii) \,\,E_{\omega_1}(q,t,T)P_\lambda(x_1,x_2;q,t,T)=c_\lambda P_\lambda(x_1,x_2;q,t,T),
\begin{aligned}\left|a\nabla J_{\delta}^{0}u\right|= \left|a\nabla\int_{0}^{\infty}e^{-t}P_{\delta t}^{0}u\,dt\right|=\left|a\nabla\int_{0}^{\infty}e^{-t+2K\delta t}P_{\delta t}^{a}u\,dt\right| \le \int_{0}^{\infty}e^{-t+2K\delta t}\left|a\nabla P_{\delta t}^{a}u\right|\,dt\le\int_{0}^{\infty}e^{-t}P_{\delta t}^{a}\left|a\nabla u\right|\,dt=J_{\delta}^{a}|a\nabla u|,\end{aligned}
\widetilde z''(\omega_\beta(z))=\frac{F''_{\mu_\alpha}(\omega_\beta(z))}{F'_{\mu_\beta}(\omega_\alpha(z))}\big(1-F'_{\mu_\beta}(\omega_\alpha(z))\big)-\frac{1}{(F'_{\mu_\beta}(\omega_\alpha(z)))^3}F''_{\mu_\beta}(\omega_\alpha(z))\cdot\big( F'_{\mu_\alpha}(\omega_\beta(z))\big)^2\,.
\frac{\phi_{x,y-1,z}(a,b)}{\phi_{x,y-1,z-1}(a,b)}\frac{\phi_{x,y,z-1}(a,b)}{\phi_{x,y,z}(a,b)}+\frac{\phi_{x+1,y-1,z-1}(a,b)}{\phi_{x,y-1,z-1}(a,b)}\frac{\phi_{x-1,y,z}(a,b)}{\phi_{x,y,z}(a,b)}&=\frac{(2x+y+z+2a+2b)(2y+2z+2a+2b-1)}{(2x+2y+2z+2a+2b-1)(x+y+z+2a+2b)}&+\frac{x(2x+2a+2b+1)}{(2x+2y+2z+2a+2b-1)(x+y+z+2a+2b)}=1.
j_{\mu}^{matter}= \bar{\varphi}_2 \partial_{\mu}\bar{\varphi}_1 - (\bar{\varphi}_1 + \hbox{$\frac me$}) \partial_{\mu}\bar{\varphi}_2 + e A_{\mu}(\bar{\varphi}_2^2 + (\bar{\varphi}_1 + \hbox{$\frac me$})^2) + O(\hbar)
\gamma_{m}(t)=i\int_{t_{0}}^{t}\langle\psi_{m}(t)|\partial_{\vec{R}}|\psi_{m}(t)\rangle\dot{\vec{R}}dt^{\prime}=i\int_{C}\langle\psi_{m}(\vec{R})|\partial_{\vec{R}}|\psi_{m}(\vec{R})\rangle d\vec{R}
\boldsymbol{y}_{\mathrm{t}}= \left(\boldsymbol{G}_{\mathrm{t}}\boldsymbol{\Theta}_{\mathrm{t}}\boldsymbol{H}_{\mathrm{s,t}}+\boldsymbol{H}_{\mathrm{d,t}}\right)\boldsymbol{s}_{\mathrm{t}}+\boldsymbol{z},
E(\Phi) M(\Phi)^{\frac{1-s_c}{s_c}}={}\tfrac{2s_c}{d}\mathfrak{I}_3 ={}\tfrac{s_c}{1-s_c}(\tfrac12(-g_{\min})^{s_c-\frac{d}{2}}\|Q\|_{L^2}^2)^{\frac{1}{s_c}}
\alpha\beta\gamma \cdot s_1\ldots s_k = [\alpha^\circ (\beta\cdot s_1) (\gamma\cdot \epsilon)] \cdot s_2\ldots s_n &= \ldots = \alpha^\circ (\beta\cdot s_1\ldots s_n) (\gamma\cdot \epsilon^n) &= \alpha^\circ (\beta\cdot s_1\ldots s_n) (\gamma\cdot \epsilon),
\begin{array}{ll}\dfrac{d}{dt}\left( \dfrac{D}{t^2}\right) =\dfrac{1}{t}(M_{-1}-2Id) \left( \dfrac{D}{t^2}\right) + M' \left( t, \dfrac{D}{t^2}\right) &=\dfrac{1}{t}M_{-1}' \left( \dfrac{D}{t^2}\right) + M' \left( t, \dfrac{D}{t^2}\right),\end{array}
\mathcal{E}_1' \ll \frac{W}{(MNL_1L_3V)^{\frac{1}{2}}}\sum_{w \asymp W}\frac{1}{\varphi(w)}\sum_{v \asymp V}\left|\mathop{\sum \sum}_{(mn,v)=1}\chi(n) \overline{\chi}(m) \right| \left|\mathop{\sum \sum}_{(\ell_1\ell_3,v)=1}\overline{\chi}(\ell_1\ell_3) \right|,
-{\beta f}= \ln{\{2 \sqrt3 \exp(-{\frac{\Delta}{4}})[1 + \exp(-2{\Delta})]\}}+{\frac{1}{8{\pi}^2}}\int_{0}^{2\pi}\d{\vartheta}\int_{0}^{2\pi}\d{\varphi}\ln[c^2 + s(\cos{\vartheta}+ \cos{\varphi})],
\frac{r!}{[(r+1)!]^2}\sum^r_{i=0}\sum^\mu_{j=0}{\mu\choose j}{\nu\choose i-j}i!(r-i)!\int^\infty_0(\cosh t)^{2j-\mu-i-\beta-3}\tilde{c}_i(t)D(t)dt \|\phi\|^2.
\begin{aligned}\Vert \nabla w_2\Vert_{L^2}\lesssim 1+ \sup_Y\left|\frac{h^0_e}{u^0_e}\right|\Vert \nabla w_1\Vert_{L^2}+L\left(\sup_Y \left|\frac{h^0_e}{u^0_e}\right|+\Vert \partial_Y h^0_e\Vert_{L^\infty}\right)\Vert \partial_x w_1\Vert_{L^2}.\end{aligned}
\frac{\delta \Sigma}{\delta c}+\varepsilon^{ab}\overline{c}^a\frac{\delta\Sigma}{\delta b^b}=-\partial^2\,\overline{c}-\partial^\mu A_\mu^{*}+\varepsilon^{ab}A_\mu^{a*}A^{b\mu}-\varepsilon^{ab}c^{a*}c^b+(\xi-\alpha )\varepsilon^{ab}b^a\overline{c}^b\;.
{\cal L}^{( 0 )}= - \frac{e^{2}B^{2}}{2 4 \pi^{2}}\ln \frac{\Lambda^{2}}{m^{2}}- \frac{e^{2}B^{2}}{8 \pi^{2}}\int_{0}^{\infty}\frac{d s}{s^{2}}\; e^{- m^{2}s / ( e B )}( \coth s - \frac{1}{s}- \frac{s}{3})
x''(t)\in A(t)x(t)+F(t,x(t))+Bu(t), t\in I=[0,b], x(0)+ g(x)=x_0\in X,x'(0)+h(x)=y_0\in X, \Delta x(t_i)= I_i(x(t_i)), \Delta x'(t_i)= J_i(x(t_i)),i=1,2,\cdots,n,
\mathrm{d}\varepsilon = \frac{1}{24\pi^2}|J| \, \mathrm{d}^{(1/12,2/12)}+ \frac{1}{24\pi^2}|J| \, \mathrm{d}^{(1/12,4/12)}+ \frac{1}{384\pi^2}|J| \, \mathrm{d}_6 \times \mathrm{d}_6,
A_{N}= \frac{{\cal N}_N}{2\pi^3(2\pi^2)}\int_0^1 dz \frac{z}{\sqrt{z}\sqrt{1-z}}\int_0^{z^2}\frac{dy}{y}\left(\frac{-\ln(y)}{4\pi^2}\right)^{{\textstyle{\frac{k+1-D}{2}}}}e^{{\textstyle 16ly}}
B_{t 1 2}^{( 1 1 )}={\frac{2 m \cosh \delta_{e}\sinh \delta_{e}}{r^{D - 3}}}f_{D}T , B_{\phi_{i}1 2}^{( 1 1 )}= -{\frac{2 m l_{i}\mu_{i}^{2}\sinh \delta_{e}}{r^{D - 3}}}f_{D}T .
= \sum_{\substack{t_1,\ldots,t_{\nu}\ge 0 t_1+2t_2+\cdots + \nu t_{\nu}=\nu}}(-1)^{t_1+\cdots + t_{\nu}}\binom{t_1+\cdots +t_{\nu}}{t_1,\ldots,t_{\nu}}a_1^{t_1}\cdots a_{\nu}^{t_{\nu}}