Energy minimization, periodic sets and spherical designs

We study energy minimization for pair potentials among periodic sets in Euclidean spaces. We derive some sufficient conditions under which a point lattice locally minimizes the energy associated to a large class of potential functions. This allows in particular to prove a local version of Cohn and Kumar's conjecture that $\mathsf{A}_2$, $\mathsf{D}_4$, $\mathsf{E}_8$ and the Leech lattice are globally universally optimal, regarding energy minimization, and among periodic sets of fixed point density.Comment: 16 pages; incorporated referee comment

A note on entropy estimation

We compare an entropy estimator $H_z$ recently discussed in [10] with two estimators $H_1$ and $H_2$ introduced in [6][7]. We prove the identity $H_z \equiv H_1$, which has not been taken into account in [10]. Then, we prove that the statistical bias of $H_1$ is less than the bias of the ordinary likelihood estimator of entropy. Finally, by numerical simulation we verify that for the most interesting regime of small sample estimation and large event spaces, the estimator $H_2$ has a significant smaller statistical error than $H_z$.Comment: 7 pages, including 4 figures; two references adde

On the uncertainty principle in Rindler and Friedmann spacetimes

We revise the extended uncertainty relations for the Rindler and Friedmann spacetimes recently discussed by Dabrowski and Wagner in [9]. We reveal these results to be coordinate dependent expressions of the invariant uncertainty relations recently derived for general 3-dimensional spaces of constant curvature in [10]. Moreover, we show that the non-zero minimum standard deviations of the momentum in [9] are just artifacts caused by an unfavorable choice of coordinate systems which can be removed by standard arguments of geodesic completion

A reinterpretation of the cosmological vacuum

In this paper, we make a proposal for addressing the cosmological constant problem. Our approach will be based on a reinterpretation of two non-standard de Sitter solutions given by the Einstein vacuum equations with Λ\u3e0. As a first result, we derive an uncertainty principle for both variants of the de Sitter space (Theorem). Subsequently, a decomposition of the cosmological constant in a pair of time-dependent pieces is introduced (Corollary). The time-dependence of the corresponding energy and dark energy density is discussed and especially matched at the Planck scale. Furthermore, we show that for every instant of cosmic time this approach can be revealed in terms of a Schwarzschild-Anti-de Sitter cosmology with Λ\u3e0. The corresponding field equations are provided