arXiv subjects

D. P. Hardin

Publications and source records attributed to D. P. Hardin.

25 records · Page 2Linked to original sources

arXiv2008

Riesz $s$-equilibrium measures on $d$-rectifiable sets as $s$ approaches $d$

Let $A$ be a compact set in ${\mathbb R}^p$ of Hausdorff dimension $d$. For $s\in(0,d)$, the Riesz $s$-equilibrium measure $μ^s$ is the unique Borel probability measure with support in $A$ that minimizes $$ I_s(μ):=\iint\frac{1}{|x-y|^s}dμ(y)dμ(x)$$ over all such probability measures. If $A$ is strongly $({\mathcal H}^d, d)$-rectifiable, then $μ^s$ converges in the weak-star topology to normalized $d$-dimensional Hausdorff measure restricted to $A$ as $s$ approaches $d$ from below.

math-ph↗

arXiv2008

The Riesz energy of the $N$-th roots of unity: an asymptotic expansion for large $N$

We derive the complete asymptotic expansion in terms of powers of $N$ for the Riesz $s$-energy of $N$ equally spaced points on the unit circle as $N\to \infty$. For $s\ge -2$, such points form optimal energy $N$-point configurations with respect to the Riesz potential $1/r^{s}$, $s\neq0$, where $r$ is the Euclidean distance between points. By analytic continuation we deduce the expansion for all complex values of $s$. The Riemann zeta function plays an essential role in this asymptotic expansion.

math-ph↗

arXiv2007

The support of the limit distribution of optimal Riesz energy points on sets of revolution in $\mathbb{R}^{3}$

Let A be a compact set in the right-half plane and $Γ(A)$ the set in $\mathbb{R}^{3}$ obtained by rotating A about the vertical axis. We investigate the support of the limit distribution of minimal energy point charges on $Γ(A)$ that interact according to the Riesz potential 1/r^{s}, 0<s<1, where r is the Euclidean distance between points. Potential theory yields that this limit distribution coincides with the equilibrium measure on $Γ(A)$ which is supported on the outer boundary of $Γ(A)$. We show that there are sets of revolution $Γ(A)$ such that the support of the equilibrium measure on $Γ(A)$ is {\bf not} the complete outer boundary, in contrast to the Coulomb case s=1. However, the support of the limit distribution on the set of revolution $Γ(R+A)$ as R goes to infinity, is the full outer boundary for certain sets A, in contrast to the logarithmic case (s=0).

math-ph↗

arXiv2006

The support of the logarithmic equilibrium measure on sets of revolution in $\R^3$

For surfaces of revolution $B$ in $\R^3$, we investigate the limit distribution of minimum energy point masses on $B$ that interact according to the logarithmic potential $\log (1/r)$, where $r$ is the Euclidean distance between points. We show that such limit distributions are supported only on the ``out-most'' portion of the surface (e.g., for a torus, only on that portion of the surface with positive curvature). Our analysis proceeds by reducing the problem to the complex plane where a non-singular potential kernel arises whose level lines are ellipses.

math-ph↗

arXiv2006

Asymptotics of Best-Packing on Rectifiable Sets

We investigate the asymptotic behavior, as $N$ grows, of the largest minimal pairwise distance of $N$ points restricted to an arbitrary compact rectifiable set embedded in Euclidean space, and we find the limit distribution of such optimal configurations. For this purpose, we compare best-packing configurations with minimal Riesz $s$-energy configurations and determine the $s$-th root asymptotic behavior (as $s\to \infty)$ of the minimal energy constants. We show that the upper and the lower dimension of a set defined through the Riesz energy or best-packing coincides with the upper and lower Minkowski dimension, respectively. For certain sets in ${\rm {\bf R}}^d$ of integer Hausdorff dimension, we show that the limiting behavior of the best-packing distance as well as the minimal $s$-energy for large $s$ is different for different subsequences of the cardinalities of the configurations.

math-ph↗

arXiv2006

Asymptotics for Discrete Weighted Minimal Riesz Energy Problems on Rectifiable Sets

Given a compact $d$-rectifiable set $A$ embedded in Euclidean space and a distribution $ρ(x)$ with respect to $d$-dimensional Hausdorff measure on $A$, we address the following question: how can one generate optimal configurations of $N$ points on $A$ that are "well-separated" and have asymptotic distribution $ρ(x)$ as $N\to \infty$? For this purpose we investigate minimal weighted Riesz energy points, that is, points interacting via the weighted power law potential $V=w(x,y)|x-y|^{-s}$, where $s>0$ is a fixed parameter and $w$ is suitably chosen. In the unweighted case ($w\equiv 1$) such points for $N$ fixed tend to the solution of the best-packing problem on $A$ as the parameter $s\to \infty$.

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arXiv2004

Minimal Riesz Energy Point Configurations for Rectifiable d-Dimensional Manifolds

For a compact set A in Euclidean space we consider the asymptotic behavior of optimal (and near optimal) N-point configurations that minimize the Riesz s-energy (corresponding to the potential 1/t^s) over all N-point subsets of A, where s>0. For a large class of manifolds A having finite, positive d-dimensional Hausdorff measure, we show that such minimizing configurations have asymptotic limit distribution (as N tends to infinity with s fixed) equal to d-dimensional Hausdorff measure whenever s>d or s=d. In the latter case we obtain an explicit formula for the dominant term in the minimum energy. Our results are new even for the case of the d-dimensional sphere.

math-ph↗