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Diffstat (limited to 'topology.tex')
| -rw-r--r-- | topology.tex | 5 |
1 files changed, 2 insertions, 3 deletions
diff --git a/topology.tex b/topology.tex index d3d90d4..56af0c6 100644 --- a/topology.tex +++ b/topology.tex @@ -247,9 +247,8 @@ points, one must take pains to eliminate the sign of the determinant \cite{Fyodorov_2004_Complexity}. Here it is correct to preserve it. We need to choose a function $H$ for our calculation. Because $\chi$ is -a topological invariant, any choice will work so long as it does not share some -symmetry with the underlying manifold, i.e., that $H$ satisfies the Smale condition. Because our manifold of random -constraints has no symmetries, we can take a simple height function $H(\mathbf +a topological invariant, any choice will work so long as it does not have degenerate stationary points on the manifold, i.e., that it is a Morse function, and does not share some +symmetry with the underlying manifold, i.e., that it satisfies the Smale condition. Because our manifold is random and has no symmetries, we can take a simple height function $H(\mathbf x)=\mathbf x_0\cdot\mathbf x$ for some $\mathbf x_0\in\mathbb R^N$ with $\|\mathbf x_0\|^2=N$. We call $H$ a height function because when $\mathbf x_0$ is interpreted as the polar axis of a spherical coordinate system, $H$ gives the height on the sphere relative to the equator. |