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\lhead{{\sf Math 9850, Algebraic Systems Biology}: Homework 7}
\chead{}\rhead{\thepage}
\lfoot{Due Friday, November 22, 2024}
\cfoot{}\rfoot{Written by M.~Macauley}
\pagestyle{fancy}

\begin{document}

$\;$ \vspace{-4mm}

\begin{enumerate}

%%--------------------------------------------------------------------- 

\item \textbf{Reverse-engineering a Boolean model}. Consider a Boolean
  model $f=(f_1,f_2,f_3)$ whose (synchronous) phase spaces consists of
  the following:
  \[
  (0,0,1)\stackrel{f}{\longto}(1,0,1)\stackrel{f}{\longto}(1,1,1)
  \stackrel{f}{\longto}(1,1,0)
  \stackrel{f}{\longto}(0,1,0)
  \stackrel{f}{\longto}(0,0,0).
  \]
  In this problem, you will reverse-engineer the wiring diagram and
  then the models space.
  \begin{enumerate}
  \item Do the following steps for each $k=1,2,3$.
    \begin{enumerate}
    \item Write down the corresponding set of \emph{data}
      \[
      \mathcal{D}_k:=\big\{(\s_1,t_{1k}),\dots,(\s_5,t_{5k})\big\}
      \]
      that arises from the $k^{\rm th}$ coordinate of this time-series.
    \item Find the monomial ideal $I_{\Delta_k^c}$ of non-disposable
      sets, and compute its primary decomposition to find the min-sets.     
    \item Find the pseudomonomial ideal $J_{\Delta_k^c}$, and compute
      its primary decomposition to find the signed min-sets.
    \end{enumerate}
  \item Find all min-sets and signed min-sets of the original Boolean model.
  \item Use Macaulay2 to compute the vanishing ideal $I(\D)$. How big
    is it?
  \item Find the model space
    $\Mod(\D)=\Prod_{i=1}^3\big[f+I(\D)\big]$. Use Lagrange
    interpolation to find $f=(f_1,f_2,f_3)$, and Macaulay2 to reduce
    them modulo $I(\D)$. \\
  \end{enumerate}

%%--------------------------------------------------------------------- 


\item \textbf{Reverse-engineering a ternary model}. Repeat the
  previous problem but for the following \emph{time series} of a
  $3$-node algebraic model over $\F_3$:
  \[
  (1,1,1)\stackrel{f}{\longto}(2,0,1)\stackrel{f}{\longto}(2,0,0)
  \stackrel{f}{\longto}(0,2,2)\stackrel{f}{\longto}(0,2,2).
  \]


  
\end{enumerate}

\end{document}
  
%%====================================================================
  
  

