From c0f6b823fee72db898728ee655235fba0d93644c Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Thu, 1 Aug 2024 11:41:19 +0200 Subject: First try at sphere figures. --- figs/connected.pdf | Bin 0 -> 135349 bytes figs/shattered.pdf | Bin 0 -> 164526 bytes topology.tex | 17 +++++++++++++++++ 3 files changed, 17 insertions(+) create mode 100644 figs/connected.pdf create mode 100644 figs/shattered.pdf diff --git a/figs/connected.pdf b/figs/connected.pdf new file mode 100644 index 0000000..8fccdaf Binary files /dev/null and b/figs/connected.pdf differ diff --git a/figs/shattered.pdf b/figs/shattered.pdf new file mode 100644 index 0000000..d3d2957 Binary files /dev/null and b/figs/shattered.pdf differ diff --git a/topology.tex b/topology.tex index 8c21ee9..58e341a 100644 --- a/topology.tex +++ b/topology.tex @@ -139,6 +139,23 @@ $\mathbb M(1)=\frac1N\phi(1)\cdot\mathbf x_0$, 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{acknowledgements} JK-D is supported by a \textsc{DynSysMath} Specific Initiative of the INFN. -- cgit v1.2.3-70-g09d2