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Exercise 22 - Chapter 4

edited June 2018 in Exercises

Consider the \( \textbf{Cost}\)-profunctors \( \Phi : \mathcal{X} \nrightarrow \mathcal{Y} \) and \( \Psi : \mathcal{Y} \nrightarrow \mathcal{Z} \) shown below:

fig

$$ \begin{array}{c | c c c} \Phi . \Psi & p & q & r & s \\ \hline A & ? & 24 & ? & ? \\ B & ? & ? & ? & ? \\ C & ? & ? & ? & ? \\ D & ? & ? & 9 & ? \end{array} $$ Previous Next

Comments

  • 1.

    $$ \begin{array}{c | c c c} \Phi . \Psi & p & q & r & s \\ \hline A & ? & 24 = 3+3+5+9+0+1+1+2 & ? & ? \\ B & ? & ? & ? & ? \\ C & ? & ? & ? & ? \\ D & ? & ? & 9 = 9+0 & ? \end{array} $$

    Comment Source:\[ \begin{array}{c | c c c} \Phi . \Psi & p & q & r & s \\\\ \hline A & ? & 24 = 3+3+5+9+0+1+1+2 & ? & ? \\\\ B & ? & ? & ? & ? \\\\ C & ? & ? & ? & ? \\\\ D & ? & ? & 9 = 9+0 & ? \end{array} \]
  • 2.

    From my answer to Exercise 4.17:

    $$ \begin{array}{c | c c c c} M_\mathcal{X}^4 * M_\Phi * M_\mathcal{Y}^3 & x & y & z \\\\ \hline A & 17 & 20 & 20 \\\\ B & 11 & 14 & 14 \\\\ C & 14 & 17 & 17 \\\\ D & 12 & 9 & 15 \end{array} $$

    The matrices associated with \(\Psi\) and \(\mathcal{Z}\) are: $$\begin{matrix}\begin{array}{c | c c c c} M_{\Psi} & p & q & r & s \\\\ \hline x & \infty & \infty & \infty & \infty \\\\ y & \infty & \infty & 0 & \infty \\\\ z & 4 & \infty & \infty & 4 \\\\ \end{array} & \begin{array}{c | c c c c} M_\mathcal{Z} & p & q & r & s \\\\ \hline p & 0 & 2 & \infty & \infty \\\\ q & \infty & 0 & 2 & \infty \\\\ r & \infty & \infty & 0 & 1 \\\\ s & 1 & \infty & \infty & 0 \end{array}\end{matrix}$$ The second matrix has powers: $$\begin{matrix} \begin{array}{c | c c c c} M_\mathcal{Z}^2 & p & q & r & s \\\\ \hline p & 0 & 2 & 4 & \infty \\\\ q & \infty & 0 & 2 & 3 \\\\ r & 2 & \infty & 0 & 1 \\\\ s & 1 & 3 & \infty & 0 \end{array} & \begin{array}{c | c c c c} M_\mathcal{Z}^4 & p & q & r & s \\\\ \hline p & 0 & 2 & 4 & 5 \\\\ q & 4 & 0 & 2 & 3 \\\\ r & 2 & 4 & 0 & 1 \\\\ s & 1 & 3 & 5 & 0 \end{array}\end{matrix}$$ So, one has that $$\begin{matrix}\begin{array}{c | c c c c} M_\Psi * M_\mathcal{Z}^4 & p & q & r & s \\\\ \hline x & \infty & \infty & \infty & \infty \\\\ y & 2 & 4 & 0 & 1 \\\\ z & 4 & 6 & 8 & 4 \\\\ \end{array} & \begin{array}{c | c c c c} M_{\Phi . \Psi} & p & q & r & s \\\\ \hline A & 22 & 24 & 20 & 21 \\\\ B & 16 & 18 & 14 & 15 \\\\ C & 19 & 21 & 17 & 18 \\\\ D & 11 & 13 & 9 & 10 \end{array}\end{matrix}$$

    Comment Source:From [my answer](https://forum.azimuthproject.org/discussion/comment/20095/#Comment_20095) to Exercise 4.17: >\[ \begin{array}{c | c c c c} M_\mathcal{X}^4 * M_\Phi * M_\mathcal{Y}^3 & x & y & z \\\\ \hline A & 17 & 20 & 20 \\\\ B & 11 & 14 & 14 \\\\ C & 14 & 17 & 17 \\\\ D & 12 & 9 & 15 \end{array} \] The matrices associated with \\(\Psi\\) and \\(\mathcal{Z}\\) are: \[\begin{matrix}\begin{array}{c | c c c c} M_{\Psi} & p & q & r & s \\\\ \hline x & \infty & \infty & \infty & \infty \\\\ y & \infty & \infty & 0 & \infty \\\\ z & 4 & \infty & \infty & 4 \\\\ \end{array} & \begin{array}{c | c c c c} M_\mathcal{Z} & p & q & r & s \\\\ \hline p & 0 & 2 & \infty & \infty \\\\ q & \infty & 0 & 2 & \infty \\\\ r & \infty & \infty & 0 & 1 \\\\ s & 1 & \infty & \infty & 0 \end{array}\end{matrix}\] The second matrix has powers: \[\begin{matrix} \begin{array}{c | c c c c} M_\mathcal{Z}^2 & p & q & r & s \\\\ \hline p & 0 & 2 & 4 & \infty \\\\ q & \infty & 0 & 2 & 3 \\\\ r & 2 & \infty & 0 & 1 \\\\ s & 1 & 3 & \infty & 0 \end{array} & \begin{array}{c | c c c c} M_\mathcal{Z}^4 & p & q & r & s \\\\ \hline p & 0 & 2 & 4 & 5 \\\\ q & 4 & 0 & 2 & 3 \\\\ r & 2 & 4 & 0 & 1 \\\\ s & 1 & 3 & 5 & 0 \end{array}\end{matrix}\] So, one has that \[\begin{matrix}\begin{array}{c | c c c c} M_\Psi * M_\mathcal{Z}^4 & p & q & r & s \\\\ \hline x & \infty & \infty & \infty & \infty \\\\ y & 2 & 4 & 0 & 1 \\\\ z & 4 & 6 & 8 & 4 \\\\ \end{array} & \begin{array}{c | c c c c} M_{\Phi . \Psi} & p & q & r & s \\\\ \hline A & 22 & 24 & 20 & 21 \\\\ B & 16 & 18 & 14 & 15 \\\\ C & 19 & 21 & 17 & 18 \\\\ D & 11 & 13 & 9 & 10 \end{array}\end{matrix}\]
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