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Register a new userA Comparison of Popular Point Configurations on $mathbb{S}^2$
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There are many ways to generate a set of nodes on the sphere for use in a variety of problems in numerical analysis. We present a survey of quickly generated point sets on $mathbb{S}^2$, examine their equidistribution properties, separation, covering, and mesh ratio constants and present a new point set, equal area icosahedral points, with low mesh ratio. We analyze numerically the leading order asymptotics for the Riesz and logarithmic potential energy for these configurations with total points $N<50,000$ and present some new conjectures.
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Observations suggest that configurations of points on a sphere that are stable with respect to a Riesz potential distribute points uniformly over the sphere. Further, these stable configurations have a local structure that is largely hexagonal. Minimal configurations differ from stable configurations in the arrangement of defects within the hexagonal structure. This paper reports the asymptotic difference between the average energy of stable states and the lowest reported energies. We use this to infer the energy scale at which defects in the hexagonal structure are manifest. We report results for the Riesz potentials for s=0, s=1, s=2 and s=3. Additionally we compare existing theory for the asymptotic expansion in N of the minimal $N$-point energy with experimental results. We report a case of two distinct stable states that have the same Voronoi structure. Finally, we report the observed growth of the number of stable states as a function of N.
We provide a method for fast and exact simulation of Gaussian random fields on spheres having isotropic covariance functions. The method proposed is then extended to Gaussian random fields defined over spheres cross time and having covariance functions that depend on geodesic distance in space and on temporal separation. The crux of the method is in the use of block circulant matrices obtained working on regular grids defined over longitude $times$ latitude.
In this note, we study symmetry of solutions of the elliptic equation begin{equation*} -Delta _{mathbb{S}^{2}}u+3=e^{2u} hbox{on} mathbb{S}^{2}, end{equation*} that arises in the study of rigidity problem of Hawking mass in general relativity. We provide various conditions under which this equation has only constant solutions, and consequently imply the rigidity of Hawking mass for stable constant mean curvature (CMC) sphere.
We establish theorems on the existence and compactness of solutions to the $sigma_2$-Nirenberg problem on the standard sphere $mathbb S^2$. A first significant ingredient, a Liouville type theorem for the associated fully nonlinear Mobius invariant elliptic equations, was established in an earlier paper of ours. Our proof of the existence and compactness results requires a number of additional crucial ingredients which we prove in this paper: A Liouville type theorem for the associated fully nonlinear Mobius invariant degenerate elliptic equations, a priori estimates of first and second order derivatives of solutions to the $sigma_2$-Nirenberg problem, and a B^ocher type theorem for the associated fully nonlinear Mobius invariant elliptic equations. Given these results, we are able to complete a fine analysis of a sequence of blow-up solutions to the $sigma_2$-Nirenberg problem. In particular, we prove that there can be at most one blow-up point for such a blow-up sequence of solutions. This, together with a Kazdan-Warner type identity, allows us to prove $L^infty$ a priori estimates for solutions of the $sigma_2$-Nirenberg problem under some simple generic hypothesis. The higher derivative estimates then follow from classical estimates of Nirenberg and Schauder. In turn, the existence of solutions to the $sigma_2$-Nirenberg problem is obtained by an application of the by now standard degree theory for second order fully nonlinear elliptic operators.
We use generating functions to relate the expected values of polynomial factorization statistics over $mathbb{F}_q$ to the cohomology of ordered configurations in $mathbb{R}^3$ as a representation of the symmetric group. Our methods lead to a new proof of the twisted Grothendieck-Lefschetz formula for squarefree polynomial factorization statistics of Church, Ellenberg, and Farb.
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