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| -rw-r--r-- | figs/connected.pdf | bin | 0 -> 135349 bytes | |||
| -rw-r--r-- | figs/shattered.pdf | bin | 0 -> 164526 bytes | |||
| -rw-r--r-- | topology.tex | 17 |
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diff --git a/figs/connected.pdf b/figs/connected.pdf Binary files differnew file mode 100644 index 0000000..8fccdaf --- /dev/null +++ b/figs/connected.pdf diff --git a/figs/shattered.pdf b/figs/shattered.pdf Binary files differnew file mode 100644 index 0000000..d3d2957 --- /dev/null +++ b/figs/shattered.pdf diff --git a/topology.tex b/topology.tex index de4576c..032fca9 100644 --- a/topology.tex +++ b/topology.tex @@ -190,6 +190,23 @@ $\mathbb M(1)=\frac1N\phi(1)\cdot\mathbf x_0$, the result is \end{aligned} \end{equation} +\begin{figure} + \includegraphics[width=0.49\columnwidth]{figs/connected.pdf} + \hfill + \includegraphics[width=0.49\columnwidth]{figs/shattered.pdf} + + \caption{ + Cartoon of the topology of the CCSP solution manifold implied by our + calculation. The arrow shows the vector $\mathbf x_0$ defining the height + function. The region of solutions is shaded orange, and the critical points + of the height function restricted to this region are marked with a red + point. For $\alpha<1$, there are few simply connected regions with most of + the minima and maxima contributing to the Euler characteristic concentrated + at the height $m_\mathrm a^*$. For $\alpha\geq1$, there are many simply + connected regions and most of their minima and maxima are concentrated at + the equator. + } +\end{figure} \begin{equation} |