Mathematics – Differential Geometry
Scientific paper
2010-08-30
Mathematics
Differential Geometry
76 pages
Scientific paper
We prove a Chern-Lashof type formula computing the expected number of critical points of smooth function on a smooth manifold $M$ randomly chosen from a finite dimensional subspace $V\subset C^\infty(M)$ equipped with a Gaussian probability measure. We then use this formula this formula to find the asymptotics of the expected number of critical points of a random linear combination of a large number eigenfunctions of the Laplacian on the round sphere, tori, or a products of two round spheres. In the case $M=S^1$ we show that the number of critical points of a trigonometric polynomial of degree $\leq \nu$ is a random variable $Z_\nu$ with expectation $E(Z_\nu)\sim 2\sqrt{0.6}\,\nu$ and variance $var(Z_\nu)\sim c\nu$ as $\nu\ra \infty$, $c\approx 0.35$.
No associations
LandOfFree
Critical sets of random smooth functions on compact manifolds does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Critical sets of random smooth functions on compact manifolds, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Critical sets of random smooth functions on compact manifolds will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-400924