\begin{equation*} \begin{aligned} \displaystyle {\mathbf{i}}\hbar \partial _t\varphi _1(t,{\mathbf{x}})&= {} \displaystyle \left[ -\frac{\hbar ^2}{2m}\nabla ^2+V({\mathbf{x}}) +E_1+c_0|\Phi |^2+ c_2(|\varphi _1|^2+|\varphi _0|^2-|\varphi _{-1}|^2) -\omega \widehat{L}_z\right] \varphi _1 + c_2{\varphi }_{-1}^*\varphi _0^2,\nonumber \\ \displaystyle {\mathbf{i}}\hbar \partial _t\varphi _0(t,{\mathbf{x}}) &= {} \displaystyle \left[ -\frac{\hbar ^2}{2m}\nabla ^2+V({\mathbf{x}}) +E_0+c_0|\Phi |^2 +c_2(|\varphi _1|^2+|\varphi _{-1}|^2)-\omega \widehat{L}_z\right] \varphi _0 +2c_2\varphi _{-1}{\varphi }_0^*\varphi _1,\nonumber\\ \displaystyle {\mathbf{i}}\hbar \partial _t\varphi _{-1}(t,{\mathbf{x}} ) &= {} \displaystyle \left[ -\frac{\hbar ^2}{2m}\nabla ^2+V({\mathbf{x}}) +E_{-1}+c_0|\Phi |^2+c_2(|\varphi _{-1}|^2+|\varphi _0|^2-|\varphi _1|^2) -\omega \widehat{L}_z\right] \varphi _{-1} + c_2\varphi _0^2{\varphi }_1^*. \end{aligned} \end{equation*}

\begin{align*} & p_{n+1}=p_n-\frac{f(p_n)}{f'(p_n)} \end{align*}

\( [t,y]=\textrm{ModifiedEuler}(a,b,y0,N) \quad \text{sloves Diff. Eqn.}\) \(\frac{dy}{dt} =y-t^{2}+1, \quad y(0)=0.5\) The universe is written in the language of mathematics (Galileo)

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