harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left[ B_{a}^{0 i}\left( x \right) , A_{j}^{b}\left( y \right) \right]_{x^{0}= y^{0}}^{*}= \delta_{a}^{\; \; b}\delta_{\; \; j}^{i}\delta^{3}\left( \mathbf{x}- \mathbf{y}\right) ,
V_1=\vert \Omega\vert^{-1}\int_\Omega (\nabla \Psi (\mathbf{x}))^t g(\mathbf{x})\nabla \widetilde{\Lambda}_2(\mathbf{x})\,d\mathbf{x}, &V_2=-\vert \Omega\vert^{-1}\int_\Omega (\nabla \Psi (\mathbf{x}))^t g(\mathbf{x})\nabla \widetilde{\Lambda}_1(\mathbf{x})\,d\mathbf{x}.
\langle C,X\rangle-\langle b,y\rangle=\langle C,X\rangle-\sum_{i=1}^{m}y_{i}b_{i}=\langle C,X\rangle-\sum_{i=1}^{m}y_{i}\langle A_{i},X\rangle=\langle C-\sum_{i=1}^{m}y_{i}A_{i},X\rangle\ge0,
\|P(fk_z)-P(\overline{f_z}k_z)\| \leq \|P\| \|fk_z-\overline{f_z}k_z\| = \|P\| \|\overline{f}k_z -f_z k_z\| &= \|P\| \|\overline{f}k_z-P(\overline{f}k_z)\|=\|P\| \|H_{\overline{f}}k_z\|.
\int_{\Gamma_{0}}f(\gamma )\lambda_{z}(\mathrm{d}\gamma )=f(\emptyset)+\sum_{n=1}^{\infty}\frac{z^{n}}{n!}\int_{X^{n}}f_{n}(x_{1},\dots ,x_{n}) mathrm{d}x_{1}\cdots \mathrm{d}x_{n},
\tilde{c}\,q_X(\alpha) = \frac{k}{2}\int_X \alpha^2 (\sigma\bar{\sigma})^{k-1}+ (1-k)\left(\int_X\alpha\,\sigma^{n-1}\bar{\sigma}^{n}\right)\left(\int_X\alpha\,\sigma^{n}\bar{\sigma}^{n-1}\right).
O_{1}^{( k )}= \left< \sum_{i}( q_{i}^{( k )}- 6 )^{2}\right> \mathrm{a n d}O_{2}^{( k )}= \left< \sum_{< i j >}( q_{i}^{( k )}- 6 ) ( q_{j}^{( k )}- 6 ) \right> ,
\hat{U}_{1}(x,y,z) = \frac{f_{1}(x) - g_{1,[z,1]}(x)}{x}, \hat{U}_{2}(x,y,z) = \frac{f_{1}(y) - g_{1,[z,1]}(y)}{z - y}, \hat{U}_{3}(x,y,z) = \frac{f_{2}(z) - g_{2,[0,y]}(z)}{z - y},
N_{1 1}= \left( \begin{matrix}{{{\frac{{\partial^{2}}V}{\partial{T_{1}}^{2}}}}}\\ \end{matrix}\right)_{0}\; , \; N_{1 2}= \left( \begin{matrix}{{{\frac{{\partial^{2}}V}{\partial{T_{1}}\partial{T_{2}}}}}}\\ \end{matrix}\right)_{0}\; , \; N_{2 2}= \left( \begin{matrix}{{{\frac{{\partial^{2}}V}{\partial{T_{2}}^{2}}}}}\\ \end{matrix}\right)_{0}
F=\alpha\mathcal L^* ( N^{in}_0-N^{out}) (N^{in}_n - N^{out})^{-1}(N^{in}_0 -N^{in}_n)\mathcal L, \alpha=\frac{1}{4\pi}\left (\frac{k}{2\pi i}\right )^{\frac{d-3}{2}}.
\hat{C}^{l_3 l_1 l_2}_{0\,0\,0}= \rho({l_3 , l_1, l_2})\sum_k{(-1)^k\over k!}{(l_1+k)! (l_2+l_3-k)!\over (l_1-k)! (l_2-l_3+k)! (l_3-k)!},
\frac{{\rm d}Y}{{\rm d}x}=\left\{\begin{pmatrix}0 1 0 0 0 0 0 0 0\end{pmatrix}+\frac{A_0}{x}+\frac{A_1}{x-1}\right\}Y, A_* \sim \operatorname{diag}(0, \theta^*_1, \theta^*_2), *=0,1.
\frac{d\mathcal{F}_t}{dt}(x)=-t<\frac{dG_t}{dx}(\frac{dg_t(u)}{dv})\cdot \bigl(\frac{dg_1}{dv}(u)-\frac{dg_0}{dv}(u)\bigr), \frac{dg_1}{dv}(u)-\frac{dg_0}{dv}(u) >.
\Delta_{\mu \nu}^{a b}= g^{2}\delta^{a b}\frac{\delta_{\mu \nu}}{p^{2}}- g^{4}N \frac{\Gamma ( 2 - \omega ) \Gamma ( \omega ) \Gamma ( \omega - 1 )}{( 4 \pi )^{\omega}\Gamma ( 2 \omega )}\cdot 4 ( 2 \omega - 1 ) \delta^{a b}\frac{\delta_{\mu \nu}- p_{\mu}p_{\nu}/ p^{2}}{p^{6 - 2 \omega}}.
\frac{f(2,\ell+1)}{f(2,\ell)}= \frac{(2\ell+2)(2\ell+1)\ell^6}{2(\ell+1)^7}= (2\ell+1)\left(\frac{\ell}{\ell+1}\right)^{6}\geq (2\cdot 21+1)\left(\frac{21}{21+1}\right)^{6}>1,
-\frac{\partial L_{d}D_{\alpha\beta}I(p\|q)}{\partial q_{j}}=T\;\left[\left(\frac{a-1}{a-b}\right)(X.Y)^{a-2}-\left(\frac{b-1}{a-b}\right)(X.Y)^{b-2}\right]\frac{\partial (X.Y)}{\partial q_{j}}
\partial_{\boldsymbol e}\eta^{t,\bar t}= \left( \frac{\partial_N (\partial_{\boldsymbol e}u^{t,\bar t})}{(N\cdot \nu^{t,\bar t})^2 \,\Delta h^{t,\bar t}}\right) (z, \eta^{t,\bar t}(x)),
\begin{aligned}\int_0^\infty\rho(r^2u_x)^2dx+2\int_0^\infty\rho^{-1}\frac{u^2}{r^2}dx\leq C\left(\int_0^\infty(\rho^\frac{\gamma-1}{2})_t^2dx+\left(\frac{dR}{dt}\right)^2\right),\end{aligned}
M_{i,j}^{K}= ( ~\varphi_i \circ F_K, \varphi_j \circ F_K)_K, K_{i,j}^{K}= ( a_h \nabla \varphi_i \circ F_K, \nabla \varphi_j \circ F_K)_K, G_{i,j}^{\partial K}= (\partial_n \varphi_i \circ F_K, \varphi_j \circ F_K)_{\partial K},
\lambda_{k\ell}=\left[ \left(k+\frac{1}{2}\right)^2+\left(\ell +\frac{1}{2}\right)^2\right]\pi^2 &\phi_{k\ell}(x,y)=2\cos \left(\left(k+\frac{1}{2}\right)\pi x\right)\cos \left(\left(\ell +\frac{1}{2}\right)\pi y\right).
\Big| \iint \displaylimits_{\max(|x-y|,|x-z|)>\rho}\big(K(x,y,z)-K(x',y,z) \big) f(y) g(z) \, d\mu(y) \, d\mu(z)\Big| &+ \Big| \iint \displaylimits_{\max(|x-y|,|x-z|)>\rho}K(x',y,z) f(y) g(z) \, d\mu(y) \, d\mu(z)-T_{\mu, \rho}( f , g )(x') \Big|.
g_{m}(d)=e^{-G(d)}\mathsf{D}_{d}^{m}e^{G(d)}=\sum_{(\ast)}c_{(\ast)}\prod_{i=1}^{m}\left( \mathsf{D}_{d}^{i}G(d)\right)^{j_{i}}=\sum_{(\ast)}c_{(\ast)}\prod_{i=1}^{m}\left( \mathsf{D}_{d}^{i-1}g(d)\right)^{j_{i}},
f^{-}( z ) ={\frac{1}{P^{+}}}\int_{0}^{z}\left({\frac{\alpha}{2}}{f^{\prime}}_{\mathrm{t r}}^{2}+{\frac{\beta}{2}}{h^{\prime}}^{2}- 2 \beta h^{\prime \prime}\right) d z^{\prime}+ c^{-}.
|x|^\frac{2p}{1-q}= |x|^\frac{2(p-q+2)}{1-q}\eta^{-\frac{4}{1-q}}|\xi|^{-2\gamma}\cdot |x|^\frac{2(q-2)}{1-q}\eta^\frac{4}{1-q}|\xi|^{2\gamma}= \eta^\frac{2(p-q)}{1-q}|\xi|^{\frac{2(p-q+2)}{1-q}-2\gamma}\cdot |x|^\frac{2(q-2)}{1-q}\eta^\frac{4}{1-q}|\xi|^{2\gamma},
[ra_m, sa_n] = r\pi_h h^{m-1}[y^m, s\pi_h h^{n-1}] y^n - s\pi_h h^{n-1}[ y^n, r\pi_h h^{m-1}] y^m &= r\pi_h h^{m-1}\sum_{k=1}^m{m \choose k}(s\pi_hh^{n-1})^{(k)}y^{m+n-k}& - s\pi_h h^{n-1}\sum_{k=1}^n{n \choose k}(r\pi_h h^{m-1})^{(k)}y^{m+n-k}.
P_{0,\pi}^V \left(\frac{X_n^+ - \Gamma(n)}{n/(\log n)^2}< x \right)= P_{0,\pi}^V\left(\frac{\tau^X_{k_{n,x}}- k_{n,x}D(k_{n,x})}{k_{n,x}}> \frac{n - k_{n,x}D(k_{n,x})}{k_{n,x}}\right).
u_{\ell}\tilde r_{k+1}+ u_{\ell+1}= u_{\ell}+ u_{\ell+1}= 2 u_{\ell}+ u_{\ell-1}&> u_\ell \sqrt{q_{k+1}\over q_k}+ u_{\ell-1}\sqrt{q_k \over q_{k+1}}={u_\ell q_{k+1}+ u_{\ell-1}q_{k}\over \sqrt{q_k q_{k+1}}}= \frac{q_{k+\ell}}{\sqrt{q_k q_{k+1}}}.
I(\mathcal{C}(\mathbf{a})) = \left( \left\lbrace \mathfrak{n}^\prime - \sum_{\mathfrak{n}\in \mathcal{N}}F^{\mathfrak{n}^\prime}_\mathfrak{n}\mathfrak{n}\in \mathbb{F}[\mathbf{x}; \sigma,\delta] \mid \mathfrak{n}^\prime \in \mathcal{N}^c \right\rbrace \right) .
c_i^{\rm in}(x) \geq 0 \Longrightarrow c_i(t, x) \geq 0 \forall t>0, \; x\in \Omega, & \int_{\Omega}c_i(t,x)\,dx=\int_{\Omega}c_i^{\rm in}(x)\,dx \forall t>0, & \frac{d}{dt}F = -\sum_{i=1}^m \int_{\Omega}c_i^{-1}|\nabla c_i +q_ic_i \nabla \psi|^2 dx +\int_{\partial \Omega}\partial_t \sigma \psi ds,
\Box^{(q)}_{s,k}\mathcal{N}_{k}+\mathcal{S}_{k}\equiv\hat{\mathcal{I}}_k\mod O(k^{-\infty})\ \ \mbox{on $D_0$}, &\mathcal{N}^*_{k}\Box^{(q)}_{s,k}+\mathcal{S}^*_{k}\equiv\hat{\mathcal{I}}^*_k\mod O(k^{-\infty})\ \ \mbox{on $D_0$},
s(\delta(x))= s(\delta(s(x))x+s(x)\delta(x)s(x)+x\delta(s(x))) = r(\delta(s(x)x)\vee l(\delta(s(x)x)\vee r(x\delta(s(x)))\vee l(x\delta(s(x))) &\vee r(s(x)\delta(x)s(x))\vee l(s(x)\delta(x)s(x)).
\int_0^\delta(R-s)^{N-1}\{v^p s^{\beta(p-1)}+ \beta \frac{N-1}{R(R-s)}v\}\:ds=\int_0^\delta(R-s)^{N-1}v^p s^{\beta(p-1)}\:ds +\beta v(0)(N-1)R^{N-3}\delta +v(0) \frac{\beta(N-1)}{R}\int_0^\delta (R-s)^{N-2}\eta\:ds +\eta_3.
\sum_{n = - \infty}^{+ \infty}\int_{m a}^{\infty}d z \, z \left( z^{2}- m^{2}a^{2}\right)^{D / 2 - 2}= \frac{1}{2}B ( D / 2 - 1 , 1 - D / 2 ) ( 1 + 2 \zeta_{R}( 0 ) ) = 0 ,
S \, = \, \int d^{3}x \left[ - \frac{1}{4}( \frac{1}{2 \rho}) F_{\mu \nu}( A ) F^{\mu \nu}( A ) + 2 \pi A^{\mu}{\tilde{J}}_{\mu}+ \frac{1}{4 \rho}( \partial \rho )^{2}- V ( \rho ) \right]
\Vert \overline{u}^{\varepsilon_n}_{\tau_n + h_n}- \overline{u}^{\varepsilon_n}_{\tau_n}\Vert_{-r}^2 =\sum_{k=1}^{\infty}( 1 + k^2 )^{-r}\left(\langle \overline{u}^{\varepsilon_n}_{\tau_n + h_n}, \phi_{k}\rangle - \langle \overline{u}^{\varepsilon_n}_{\tau_n}, \phi_{k}\rangle\right)^2.
m=\left(\begin{array}{cc}e^b \left(a e^{-d}p+e^{-d}\right) e^{-b-d}p-e^b \left(a e^{-d}p+e^{-d}\right) q e^b \left(a \left(e^d-c e^d p\right)-c e^d\right) e^{-b}\left(e^d-c e^d p\right)-e^b \left(a \left(e^d-c e^d p\right)-c e^d\right) q \end{array}\right).
\begin{aligned}\dot{x}= - y + a_{20}x^2 +(-2 b_{20}+ (2 a_{20}- b_{11}) i) x y - a_{20}y^2 + r_{20}x^3 + r_{11}x^2 y - r_{20}x y^2, \dot{y}= x + b_{20}x^2 + b_{11}x y - b_{20}y^2 + r_{20}x^2 y + r_{11}x y^2 - r_{20}y^3.\end{aligned}
\mathbf{p}\left( t\right) :=\underset{\eta \rightarrow 0}{\lim}\ \underset{l\rightarrow \infty}{\lim}\left\{\left( \eta^{2}\left\vert \Lambda_{l}\right\vert \right)^{-1}\mathbf{P}^{(\omega ,\eta \mathbf{A}_{l})}\left( t\right) \right\}\ .
f\circ g\ =\ (f^{\uparrow 3}\circ \ell_3)\circ g\ =\f^{\uparrow 3}\circ(\ell_3\circ g)\ =\f^{\uparrow 3}\circ (\ell_\gamma+\epsilon), f\ast g\ =\ (f^{\uparrow 3}\ast \ell_3)\ast g\ =\f^{\uparrow 3}\ast(\ell_3\ast g)\ =\f^{\uparrow 3}\ast(\ell_{\gamma}+\epsilon),
N_{(1,10)}(n) = 8\sigma(n) - 32\sigma(\frac{n}{4}) + 8\sigma(\frac{n}{10}) -32\sigma(\frac{n}{40}) + 64\, W_{(1,10)}(n) + 1024\, W_{(1,10)}(\frac{n}{4}) - 256\, \biggl( W_{(2,5)}(\frac{n}{2}) + W_{(1,40)}(n) \biggr),
\theta(e_d\cdots e_1) =\theta(e_1)\cdots\theta(e_d) &=g_1\cdots g_d e_1\cdots e_d &=gg_1\cdots g_d e_{\sigma^{-1}(1)}\ldots e_{\sigma^{-1}(d)}&=g\theta(e_{\sigma^{-1}(1)})\ldots \theta(e_{\sigma^{-1}(d)}) &=g\theta(e_{\sigma^{-1}(d)}\ldots e_{\sigma^{-1}(1)}).
x^{-1}\phi_{g}(h)h^{-1}x= \; \phi_{g}(h_1^{-1}) \phi_{g}(h)h^{-1}\phi_{g}(h_1) = \; \big(\phi_{g}(h_1^{-1}h) \big) \big(h^{-1}h_1\big) \big(h^{-1}_1 \phi_{g}(h_1)\big) = \; \big(\phi_{g}(h_1^{-1}h) \; (h^{-1}_1 h)^{-1}\big)\; \big(h^{-1}_1 \phi_{g}(h_1)\big) \in H^{(2)},
[ \widehat{J}_{\mu}, \tilde{\cal R}_{\nu}] = - i \epsilon_{\mu \nu \lambda}\tilde{\cal R}{}^{\lambda}, \quad [ p_{\mu}, \tilde{\cal R}_{\nu}] = [ \Pi_{-}, p_{\mu}] = [ \Pi_{-}, \widehat{J}_{\mu}] = 0 .
\mathbf{\bar{U}_n^{*}}= \mathbf{\bar{M}_{0n}^*}\mathbf{\bar{S}_n^*}+ (\mathbf{{\bar{S}^{*'}_n}}\mathbf{\bar{M}_{1n}^*}\mathbf{\bar{S}_n^*},\ldots, \mathbf{{\bar{S}^{*'}_n}}\mathbf{\bar{M}_{pn}^*\bar{S}_n^*})'
f(2x+y)+f(x+2y)=\frac{f(x)f(y)\sum_{k=0}^{n}(2^{n-k}+2^k)\binom{n}{k}f(x)^{\frac{k}{n}}f(y)^{\frac{n-k}{n}}}{\left(2f(x)^{\frac{2}{n}}+5f(x)^{\frac{1}{n}}f(y)^{\frac{1}{n}}+2f(y)^{\frac{2}{n}}\right)^n}, .
s_1=p_1=-2\cdot 5 -5^2 t, &s_2=(s_1p_1-p_2)/2= 11\cdot 5+ 2\cdot 5^3 t+5^4t^2, &s_3=(s_2p_1-s_1p_2+p_3)/3=-(28\cdot 5 + 11\cdot 5^3 t+ 2\cdot 5^5 t^2+5^6t^3), &s_4=(s_3p_1-s_2p_2+s_1p_3-p_4)/4=7\cdot 5^2+ 28\cdot 5^3t+11\cdot 5^5 t^2+ 2\cdot 5^7t^3+5^8t^4.
u(t) =\gamma_1(t)(H_1(\alpha_1[u])+(d_{11}I_1+e_{11}N_1)(u(\tau_1)))+\delta_1(t)L_1(\beta_1[v])+ F_1(u,v)(t)+\mu e(t) &\geq \gamma_1(t)(H_1(\alpha_1[u])+(d_{11}I_1+e_{11}N_1)(u(\tau_1)))+ F_1(u,v)(t)+\mu e(t).
\mathrm{I m}F_{+ - + -}= - 1 2 \pi \alpha^{2}\frac{s - t}{u}- 1 6 \pi \alpha^{2}\left[ \frac{u}{s}\ln \Big | \frac{u}{m_{W}^{2}}\Big | + \frac{u^{2}}{s t}\ln \Big | \frac{t}{m_{W}^{2}}\Big | \right] ,
\mathcal{B}\mathrm{k}\langle AB\rangle=\mathcal{B}\sum_{i=1}^{n}V_{\alpha_{i}}=\sum_{i=1}^{n}\mathcal{A}_{\alpha_{i}}V_{\alpha_{i}}=\sum_{i=1}^{n}V_{\alpha_{i}}=\mathrm{k}\langle AB\rangle.
\langle Dv,w\rangle =\sum_{i,j}w_j\frac{\partial Z_j}{\partial m_{i}}v_i = \sum_{i,j}\sum_{S\neq S_1,\ldots,S_k}w_j \left(\frac{\partial Z_j}{\partial x_S}\right) \left(\frac{\partial x_S}{\partial m_i}\right)v_i
{\rm codim\,}((L_1 \cap C)\cap (L_2 \cap C), L_1 \cap C) ={\rm codim\,}(L_1 \cap L_2 \cap C,L_1 \cap C) \phantom{{\rm codim\,}((L_1 \cap C)\cap (L_2 \cap C), L_1 \cap C)}\leq{\rm codim\,}(L_1\cap L_2,L_1)= d(L_1,L_2).
G^{[F]}=-dx^{0}\otimes dx^{0}+dx^{1}\otimes dx^{1}+g_{i^{\prime}j^{\prime}}^{[F]}(x,y)dx^{i^{\prime}}\otimes dx^{j^{\prime}}+g_{i^{\prime}j^{\prime}}^{[F]}(x,y)\delta y^{i^{\prime}}\otimes \delta y^{j^{\prime}}
x_1^3 y_1^3 + x_2^3 y_2^3 + ax_1 y_1 x_2 y_2 (x_1 y_1 + x_2 y_2) + b y_1 y_2 (x_1^3 y_2 + x_2^3 y_1) + + c x_1 x_2 (x_1 y_2^3 + x_2 y_1^3) = 0 \, .
\left[ g_{\mu \nu}( u ) \right] = \left( \begin{array}{c | c}{1}&{u^{0}\vec{u}^{\mathrm{T}}}\\ \hline{u^{0}\vec{u}}&{- I + \vec{u}\otimes \vec{u}^{\mathrm{T}}( u^{0})^{2}}\\ \end{array}\right) ,
S^\mathrm{old}_{2k}(\Gamma_0(N)) = \bigoplus_{dM \mid N,\M \ne N}V(d)S^\mathrm{new}_{2k}(\Gamma_0(M)) = \sum_{p_i \in S}S_{2k}(\Gamma_0(N/p_i)) + V(p_i)S_{2k}(\Gamma_0(N/p_i)).
\Omega=A_1\cdots A_n, C(n)=C(n,\kappa)= \frac{16}{n! \kappa}e^n (2n +1+2 \kappa)(n+2)(4(n+1))^{n+1}\left( \frac 12 en\right)^{\kappa}, C_0= \log\left(e^{4.4n+7}n^{5.5}D^2 \log(eD)\right), W_0=\log(1.5eBD \log(eD)).
\det_{1\leq k,\ell\leq n}\big(\, m_{k,\ell}(q_\ell)\,\big) = \prod_{1\leq i<j\leq n}(b_i+a_j) \det_{1\leq k,\ell\leq n}\big(\, \tilde{m}_{k,\ell}(q_\ell)\,\big) \,,
{B}=\left(\begin{array}{ccc}M_n^{N_h^0\times N_h^0}0 M_n^{N_h^0\times N_h^\mathcal{B}}0 M^{N_h^0\times N_h^0}M^{N_h^0\times N_h^\mathcal{B}}(M_n^{N_h^0\times N_h^\mathcal{B}})^T -(M^{N_h^0\times N_h^\mathcal{B}})^T M_n^{N_h^\mathcal{B}\times N_h^\mathcal{B}}-M^{N_h^B\times N_h^\mathcal{B}}\end{array}\right).
\begin{aligned}JO_{n}^{(3)}=J_{n}^{(3)}+\sum_{s=1}^{7}J_{n+s}^{(3)}e_{s},\n\geq 0 &=J_{n}^{(3)}+J_{n+1}^{(3)}e_{1}+J_{n+2}^{(3)}e_{2}+J_{n+3}^{(3)}e_{3}&\ \ +J_{n+4}^{(3)}e_{4}+J_{n+5}^{(3)}e_{5}+J_{n+6}^{(3)}e_{6}+J_{n+7}^{(3)}e_{7}\end{aligned}
\widehat{\nabla \mathrm{H}}- \alpha \ \nabla\int_\mathrm{B}\nabla\mathbb{G}^{(0)}(\mathrm{\xi},\mathrm{\eta})\cdot\widehat{\nabla \mathrm{H}}(\mathrm{\eta}) d\mathrm{\eta}= \widehat{\nabla\mathrm{H}}^{\textbf{in}}+ \alpha \delta^2 \mathbb{T}_\delta\Big[\widehat{\nabla\mathrm{H}}\Big].
w^3 (z) = \left(z -\tfrac{1}{2}\right)^3 \sum\limits_{k=0}^\infty\frac{(-2)^{k+2}}{k+3}\left(z-\tfrac{1}{2}\right)^k,\;\;(2z)^{- 1/2}=\sum\limits_{k=0}^\infty \frac{(-1)^k}{2^k}\binom{2k}{k}\left(z-\tfrac{1}{2}\right)^k.
\begin{cases}-\Delta v_i=f(v_i)+\lambda v_i \ \ \ \ \mathbb R^N, v_i(x)>0,\ \lim_{|x|\to\infty}v_i(x)=0,i=1,2, \cdots, \ell, \sum_{i=1}^{\ell}|v_i|_2^2=\alpha. \end{cases}
r^{-m}\int_{B_{r}(0)}\left( r^{2}|D_{\hat{L}}u(z)|^{2}+ |D_{v}u(z)|^{2}\right) \, \mathrm{d}z \leq C(m,\rho) \sum_{i=0}^{k}\left( \theta(y_{i}, 4r) - \theta(y_{i},2r) \right),
\pi^{-1}(\ker \beta_{0+})=\pi^{-1}\Big(\bigcap_{i=0}^{m^+}\ker \beta_i \Big) = \bigcap_{i=0}^{m^+}\pi^{-1}(\ker \beta_i)= \bigcap_{i=0}^{m^+}(V+\ker B_i),
\hat{r}^{(D)}:=\sum_{d=1}^{\infty}w_{d}\bigg{[}\hat{r}^{(D)}_{d}-(1-2\beta)\sum_{i=1}^{D}(X_{i}1_{i\leq d}-\hat{\theta}_{\mathrm{MA},\beta,i})^2 \bigg{]}.
0 =V[F_{\hat{\nabla}^K}(X,Y)] +X[F_{\hat{\nabla}^K}(Y,V)] +Y[F_{\hat{\nabla}^K}(V,X)] =i V[\rho(X,Y)] -X\left [\frac{i}{2}\delta \tilde{G}(V)\cdot \omega\cdot Y \right] +Y\left [\frac{i}{2}\delta \tilde{G}(V)\cdot \omega\cdot X \right] &=i V[\rho(X,Y)] -\frac{i}{2}d(\delta \tilde{G}(V)\cdot \omega) (X,Y),
\hat{V}_{\mu \nu \rho}^{( 1 x , y ) A B}= - 2 \cdot 2 \mathrm{T r}T^{A}T^{B}( Y_{R}^{( d )}- Y_{L}^{( d )}) V_{\mu \nu \rho}^{( 1 )}= 4 \mathrm{T r}T^{A}T^{B}V_{\mu \nu \rho}^{( 1 )}.
S^{reg}[{\mathcal A}]\;=\;\frac{\kappa}{2}\langle A_\mu | u(\frac{\partial^2}{M^2}) R_{\mu\nu}| A_\nu \rangle \,+\,\frac{1}{4}\langle F_{\mu\nu}| v(\frac{\partial^2}{M^2})|F_{\mu\nu}\rangle \;,
(p-q)P_{q+1}^{n-2,p-q}(x)+2s\Big(P_{q+1}^{n-2,p-q}(x)\Big)'=(p+1)P_{q+1}^{n-1,p-q-1}(x) &(q+1)P_{q+1}^{n-2,p-q}(x)-2s\Big(P_{q+1}^{n-2,p-q}(x)\Big)'=-(p+1)P_{q}^{n-1,p-q}(x) &\binom{n-1+p}{p}P_q^{n-1,p-q}(x)=\sum_{j=0}^q\kappa_{p-j,q-j}P_{q-j}^{n-2,p-q}(x),
\partial_{\chi_{(A)}}\Gamma_{Y_1}(0) \; \; \Gamma_{\varphi_1 \varphi_1}(0) + \partial_{\chi_{(A)}}\Gamma_{Y_1 \varphi_1}(0) \; \; \Gamma_{\varphi_1}(0) = - \partial_{\xi_{(A)}}\Gamma_{\varphi_1}(0)
U=\begin{pmatrix}0 -\frac{1}{3}0 \frac{2}{3}\frac{2}{3}0 0 begin{align*}2.5mm]0 0 0 0 0 0 1 begin{align*}2.5mm]0 \frac{2}{3}0 \frac{2}{3}\frac{1}{3}0 0 begin{align*}2.5mm]0 \frac{2}{3}0 -\frac{1}{3}\frac{2}{3}0 0 begin{align*}2.5mm]0 0 1 0 0 0 0 begin{align*}2.5mm]1 0 0 0 0 0 0 begin{align*}2.5mm]0 0 0 0 0 1 0\end{pmatrix}.
{}{}\textrm{maximize :}\;\;\;E \left\{\sum_{i=1}^{M_R}\ln \left( 1 + \frac{P}{N_0}l_i \right) \right\}&\textrm{s.t.}\;\;\;{\mathcal U}= E\left\{\eta P \right \}\leq Q, \;\; 0 \leq P \leq P_{max}.
T'_{i,e}(v) =\sum_{\substack{a,b,c \geq 0; a- b+c=m}}(-1)^b q_i^{e(b-ac)}F_i^{(a)}E_i^{(b)}F_i^{(c)}v,v\in M_m, T''_{i,e}(v) =\sum_{\substack{a,b,c \geq 0; -a+b-c=m}}(-1)^b q_i^{e(b-ac)}E_i^{(a)}F_i^{(b)}E_i^{(c)}v,v\in M_m.
\begin{aligned}X^\ast T^\ast TX-X^\ast X =U_{(\theta)1}^\ast X^\ast X U_{(\theta)1}-X^\ast X &= (K^\ast+U_{(\theta)c}^\ast)|\varphi|^2(U_{(\theta)c})(K+U_{(\theta)c}\bigr)-|\varphi|^2(U_{(\theta)c}) &=K^\ast|\varphi|^2(U_{(\theta)c})K+(\overline\chi|\varphi|^2)(U_{(\theta)c})K+K^\ast(\chi|\varphi|^2)(U_{(\theta)c}). \end{aligned}
\frac{d^{n}}{dc^{n}}\,{}_{p}F_{q}\left[\begin{matrix}\vec{a}\vec{b}\end{matrix};c\right] =\frac{( \vec{a})_{n}}{(\vec{b})_{n}}\,{}_{p}F_{q}\left[\begin{matrix}\vec{a}+n \vec{b}+n\end{matrix};c\right] ,
\frac{1}{r_1^2}\int_{\{\Psi=-\infty\}}|f|^2=\lim_{p\rightarrow0+0}\frac{1}{r_1^2}\int_{\{p\Psi<2\log r_1\}}|f|^2\ge G(0;\Psi,I_+(2a_o^f(\Psi)\Psi)_o,f).
F\left( t\right) =\exp \left[ \left( \frac{4t}{4t+1}\right) x^{2}-\frac{1}{2}\left[\ln \left( 4t+1\right) \right] \right] =\frac{1}{\sqrt{4t+1}}\exp \left( \frac{4tx^{2}}{4t+1}\right),
A = - \frac{\kappa^{2}}{4}\frac{p_{1 1}^{( 1 )}p_{1 1}^{( 2 )}}{\left( p_{\perp}^{( 1 )}- p_{\perp}^{( 4 )}\right)^{2}}\left( \frac{p_{\perp}^{( 1 )}}{p_{1 1}^{( 1 )}}- \frac{p_{\perp}^{( 2 )}}{p_{1 1}^{( 2 )}}\right)^{4}
P(W,Z,\theta,\varphi;\alpha,\beta)=\prod_{i=1}^{K}P(\varphi_{i};\beta)\prod_{j=1}^{M}P(\theta_{j};\alpha)\prod_{t=1}^{N}P(Z_{j,t}|\theta_{j})P(W_{j,t}|\varphi_{Z_{j,t}}),
\begin{aligned}\||u\||_\mathcal H\le \|u\|_\mathcal H\le (|\Omega|+\|a\|_1)\||u\||_\mathcal H, \quad \mbox{if}|\Omega|+\|a\|_1\ge1, (|\Omega|+\|a\|_1)\||u\||_\mathcal H\le\|u\|_\mathcal H\le \||u\||_\mathcal H, \quad \mbox{if}|\Omega|+\|a\|_1<1.\end{aligned}
\mathsf{Q}(l) = \begin{cases}\left\lfloor \textsf{clip}(l)/\Delta + \frac{1}{2}\right\rfloor, l > \frac{\Delta}{2}\left\lceil \textsf{clip}(l)/\Delta - \frac{1}{2}\right\rceil, l < -\frac{\Delta}{2}0, . \end{cases}
+{\frac{1}{2 m^{2}}}\Pi_{\theta}^{2}+{\frac{m^{2}}{2}}\partial_{i}\theta \partial_{i}\theta + m^{2}( \partial =_{i}A_{i}) \theta - A_{0}( \partial_{i}\Pi_{i}+{\frac{\mu}{2}}\epsilon_{i j}\partial =_{i}A_{j}+{\frac{m^{2}}{2}}A_{0}) ,
\int\limits_{\pi/2}^{\pi-\psi_0}Cf(0,e_n,\psi)g(\psi_0,\psi)d\psi =\int\limits_{\psi_0}^{\pi/2}Cf(0,e_n,\pi-\psi)g(\psi_0,\pi-\psi)d\psi &= \int\limits_{\psi_0}^{\pi/2}\int\limits_0^{\infty}\int\limits_{S^{n-2}}f(\rho (\sin\psi) \omega, -\rho \cos \psi)(\rho \sin\psi)^{n-2}g(\psi_0,\psi)d\omega d\rho d\psi.
\frac{\Phi(\epsilon) - \Phi(0)}{\epsilon}= \frac{1}{\epsilon}\int_0^\epsilon G(x) = \frac{1}{\epsilon}\int_0^\epsilon \exp\left(-Kf(x) \right)\exp \left( i f(x) \right) & = \exp(if(0)) \cdot \frac{1}{\epsilon}\int_0^\epsilon \exp\left(-Kf(x) \right)\exp \left( i \alpha(x) \right),
\hat{H}_{tot}=\sum_{i}\frac{p_{i}^{2}}{2m}+\sum_{I}\frac{P_{I}^{2}}{2M_{I}}+\sum_{i}V_{nucl}(r_{i})+\frac{1}{2}\sum_{i\ne j}\frac{e^{2}}{|r_{i}-r_{j}|}+\frac{1}{2}\sum_{I\ne J}\frac{Z_{I}Z_{J}e^{2}}{|R_{I}-R_{J}|}
V = -{\frac{1}{2 \bar{\bf Z}}}\left[ \left( \begin{matrix}{{\bf 0}}&{{\bf I}}\\{-{\bf I}}&{{\bf 0}}\\ \end{matrix}\right) \cdot{\bf M}({\cal F}) + i \left( \begin{matrix}{{\bf I}}&{{\bf 0}}\\{{\bf 0}}&{{\bf I}}\\ \end{matrix}\right) \right] \cdot{\cal Q},
||\dot{\gamma}||_{\Sigma,L}=\sqrt{(\overline{q}\frac{\dot{\gamma}_1}{\gamma_1}-\overline{p}\dot{\gamma}_3)^2+(\frac{l_L}{l})^2L\omega(\dot{\gamma}(t))^2}\sim L^{\frac{1}{2}}|\omega(\dot{\gamma}(t))|,~~~~{\rm as}~~~~L\rightarrow +\infty.
\begin{gathered}v_\varepsilon \buildrel\ast\over\rightharpoonup v_p ~~~ ~ L^\infty(0,T; H^2(\Omega_h\times 2\mathbb T)), v_\varepsilon \rightharpoonup v_p ~~~ ~ L^2(0,T; H^3(\Omega_h\times 2\mathbb T)),\end{gathered}
\ddot{q}_{n n^{\prime}}^{l}+ \omega_{l n^{\prime}}^{2}( t ) q_{n n^{\prime}}^{l}= 2 h \sum_{p = 1}^{\infty}\dot{q}_{n p}^{l}a_{p n^{\prime}}^{l}+ \dot{h}\sum_{p = 1}^{\infty}q_{n p}^{l}a_{p n^{\prime}}^{l}+ h^{2}\sum_{p , s = 1}^{\infty}q_{n p}^{l}a_{p s}^{l}a_{n^{\prime}s}^{l}.
\begin{aligned}u^n(q) = F_\tau(n;q)(v-D(q)b) + D(q)b +\tau\sum_{j=1}^{n}E_\tau(j;q) f &= F_\tau(n;q)(v-D(q)b) + D(q)b +(I-F_\tau(n;q))A(q)^{-1}f.\end{aligned}
\begin{array}{rcl}V_{\mathrm{D}}(\vec{x}):= \max\limits_{\boldsymbol{\lambda}}-\dfrac{1}{2}\boldsymbol{\lambda}^{\top}\vec{G}\vec{H}^{-1}\vec{G}^{\top}\boldsymbol{\lambda}- \boldsymbol{\lambda}^{\top}(\vec{W}+\vec{S}\vec{x}) begin{align*}1ex] \mathrm{s.t.}\boldsymbol{\lambda}\geq \vec{0}_{m}.\end{array}
\begin{aligned}{}\big|\psi(\tau x',\tau^{2}y_{2}\cdots,\tau^{k}y_{k})-\psi(\tau\overline{x}',\tau^{2}y_{2}\cdots,\tau^{k}y_{k}) \big| \leq\tilde{\delta}\tau|x'-\overline{x}'|\omega_{\nabla\psi}(\tau) \leq C_{2}\tilde{\delta}\tau\omega_{\nabla\psi}(\tau)d(p,\overline{p}) \leq C_{2}\tilde{\delta}\tau\omega(\tau)d(p,\overline{p})\end{aligned}
l_{\langle\rangle}(\mathcal I_{\forall_x\psi,b}P) =(l_{\langle\rangle}(P)\backslash\{\forall_x\psi\})\cup\{\psi(b)\}, o_{\langle\rangle}(\mathcal I_{\forall_x\psi,b}P) =o_{\langle\rangle}(P)
= \frac{x + 2}{( x - 2 ) ( x + 3 )^{2}}- \frac{5 ( x - 2 )}{( x + 3 )^{2}( x - 2 )}= \frac{( x + 2 ) - 5 x + 1 0}{( x - 2 ) ( x + 3 )^{2}}= \frac{1 2 - 4 x}{( x - 2 ) ( x + 1 )}= \frac{4 ( 3 - x )}{( x - 2 ) ( x + 2 )}= - \frac{4}{( x - 2 ) ( x + 2 )}
\nabla \Psi_{t,\eta_i}^{q(t)}\cdot \nabla v_k-\frac{\mathrm{tr}_g(\nabla \Psi_{t,\eta_i}^{q(t)}\cdot \nabla v_k)}{n}g+\nabla v_k\cdot \nabla \Psi_{t,\eta_i}^{q(t)}-\frac{\mathrm{tr}_g(\nabla v_k\cdot \nabla \Psi_{t,\eta_i}^{q(t)})}{n}g+\nabla v_k\cdot \nabla v_k-\frac{\mathrm{tr}_g(\nabla v_k\cdot \nabla v_k)}{n}g=h.
\Lambda_S(f,g):=\sum_{j>N}\iint \left|\sum_{n\in S_j}f_{n,m,j}\left(x-\frac{t^a+\epsilon_P(t)}{2^{aj}}\right)\right| &\left|\sum_{n\in S_j}g_{n,m,j}\left(x-\frac{t^b+\epsilon_Q(t)}{2^{bj}}\right)\right||\rho(t)|\,dtdx
\sum_{k=1}^N \Delta t_k \left| P_{T-t_{k}}\bar \L f (y_{t_{k-1\over \epsilon}}^\epsilon)-P_{T-t_{k-1}}\bar \L f(y_{t_{k-1\over \epsilon}}^\epsilon)\right|\le 2\epsilon |f|_{2,\infty}( 1+\gamma_1(y_0)).
\langle s_{1},\ldots,s_{n-1}\,|\,s_{i}^2=1,\, s_{i}s_{i+1}s_{i}=s_{i+1}s_{i}s_{i+1},\, s_{i}s_{j}=s_{j}s_{i}\,{\rm if}\, |i-j| \ge 2 \rangle.
\tau(L_1,L_2,\dots,L_m)= \mu_c^{+}(Y_1,Y_2,\dots,Y_m)-\mu_c^{-}(Y_1,Y_2,\dots,Y_m)=\nu_c^{+}(Y_1,Y_2,\dots,Y_m)-\nu_c^{-}(Y_1,Y_2,\dots,Y_m),
\int^{\pi}_0 \Psi (Y(\theta )+ Z(\theta )\cos\varphi ) \d\varphi &=(\int^{\frac{\pi}{2}}_0 + \int^{\pi}_{\frac{\pi}{2}}) \ \Psi (Y(\theta ) + Z(\theta ) \cos\varphi ) \d\varphi &=\int^{\frac{\pi}{2}}_0 \{\Psi (Y(\theta )+ Z(\theta )\cos\varphi )+\Psi (Y(\theta )- Z(\theta )\cos\varphi )-2\Psi (Y(\theta )) \}\d\varphi & \qquad+\pi \Psi (Y(\theta )),
\int_{Yu(K)}f^* \geq \int_{\cap_{i=1}^\infty\cup_{k=i}^\infty Yu_k(K)}f^* = \lim_{i \rightarrow \infty}\int_{\cup_{k=i}^\infty Yu_k(K)}f^* \geq\lim_{i \rightarrow \infty}\limsup_{k \rightarrow \infty}\int_{Yu_k(K)}f^* = \limsup_{k \rightarrow \infty}\int_{Yu_k(K)}f^*.
-661608 - 1138860 s - 551250 s^2 - 47625 s^3 + 3125 s^4 - 41472 a_1 + 11160 s a_1 - 24000 s^2 a_1 &\quad- 1750 s^3 a_1 + 176904 a_2 + 18360 s a_2 + 450 s^2 a_2=0.
(m\alpha+1-Cm\epsilon^{m'})\int \eta^m\overline{u}_s^{m\alpha}\phi(|\nabla \overline{u}_s|)|\nabla \overline{u}_s|^2 \le \int Cmg_1(x,\epsilon)\phi(\overline{u})\overline{u}^2\overline{u}_s^{m\alpha}+\int |f|\eta^m\overline{u}_s^{m\alpha}\overline{u}.
\begin{aligned}\mathrm{R}_{\Phi}(h): =-\sum_{k\in S}(h, \beta_1 \partial_x e^{\mathrm{i}k x})_{L^2(\mathbb{T})}e^{\mathrm{i}k x}+\sum_{k\in S}(h, (\beta_1)_x e^{\mathrm{i}k x})_{L^2(\mathbb{T})}e^{\mathrm{i}k x}+\sum_{k\in S}(h, \beta_1 \partial_x e^{\mathrm{i}k x})_{L^2(\mathbb{T})}e^{\mathrm{i}k x}begin{align*}2mm] =\Pi_S[(\beta_1)_x \,h].\end{aligned}
\frac{\tau^{-\alpha}}{\Gamma(2-\alpha)}\sum_{j=0}^n b_{n-j}\left (U_h^j, v_h \right )+ \left(\nabla U_h^{n},\nabla v_h \right)= \left(f(U_h^{n-1}), v_h \right), \forall v_h\in V_h.
\sum_{k=1}^{M}\frac{1}{(\begin{matrix}2k\\ 2\end{matrix})}=\frac{1}{(\begin{matrix}2\\ 2\end{matrix})}+\frac{1}{(\begin{matrix}4\\ 2\end{matrix})}+\frac{1}{(\begin{matrix}6\\ 2\end{matrix})}+\frac{1}{(\begin{matrix}8\\ 2\end{matrix})}+\frac{1}{(\begin{matrix}10\\ 2\end{matrix})}+...+\frac{1}{(\begin{matrix}2M\\ 2\end{matrix})}
K(u(t)) =-\dfrac{2tu'(t)+t^2u''(t)}{(u't)^3}, &=-\dfrac{1+2t\phi'(t)+t^2\phi''(t)}{(1+t\phi'(t))^3}, &=-\dfrac{g_1(t)+tg_1'(t)}{g_1(t)^3}, &=-(g_2(t)-\frac{t}{2}g_2'(t)),
I_{TOP}={\rm Tr}\int_{X}\left( \tau R^{+}\wedge R^{+}+\overline\tau R^{-}\wedge R^{-}\right)={\rm Tr}\int_{X}\left( \tau R^{+}\wedge R^{+}+\overline\tau\overline{R^{+}}\wedge\overline{R^{+}}\right),
\begin{aligned}\boldsymbol u_{n2}=\boldsymbol u_n+ \tau\varphi_1(\tau K)(K\boldsymbol u_n+ \boldsymbol g(t_n,\boldsymbol u_n)), \boldsymbol u_{n+1}=\boldsymbol u_{n2}+ \tau\varphi_2(\tau K)(\boldsymbol g(t_{n+1},\boldsymbol u_{n2})- \boldsymbol g(t_n,\boldsymbol u_n)). \end{aligned}