We address the two novel types of point processes on the multidimensional sphere proposed by Beltrán and Etayo. First, for the point process on the odd-dimensional sphere introduced by Beltrán and Etayo (Constr. Approx., 2018), we give an explicit form of the expectation of the p-frame potential of the process and investigate its asymptotic behavior. Second, for the point processes, termed generalized spherical ensemble, on the even-dimensional sphere introduced by Beltrán and Etayo (J. Math. Anal. Appl., 2019), we provide an upper bound of the expectation of the p-frame potential of the process.
The Sinc approximation is known to yield quite accurate results for double-exponentially decaying functions. To date, this approximation has been successfully applied to derivatives, initial value problems, boundary value problems, and so forth. However, an error analysis of the approximation of the m-th derivatives has not been explicitly given, except in the case of m = 1. This study provides an error analysis for all positive integers m.
Surfaces with constant mean curvature (CMC) are critical points of the area with volume constraint. They serve as a mathematical model of surfaces of soap bubbles and tiny liquid drops. CMC surfaces are said to be stable if the second variation of the area is nonnegative for all volume-preserving variations satisfying the given boundary condition. In this paper, we examine the stability of CMC hypersurfaces in general Euclidean space possibly having boundaries on two parallel hyperplanes. We reveal the stability of equilibrium hypersurfaces without self-intersection for the first time in all dimensions. The analysis is assisted by numerical computations.