arXiv subjects

D. P. Hardin

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

At least 19 recordsLinked to original sources

arXiv2025

Universal polar dual pairs of spherical codes found in $E_8$ and $Λ_{24}$

We identify universal polar dual pairs of spherical codes $C$ and $D$ such that for a large class of potential functions $h$ the minima of the discrete $h$-potential of $C$ on the sphere occur at the points of $D$ and vice versa. Moreover, the minimal values of their normalized potentials are equal. These codes arise from the known sharp codes embedded in the even unimodular extremal lattices $E_8$ and $Λ_{24}$ (Leech lattice). This embedding allows us to use the lattices' properties to find new universal polar dual pairs. In the process we extensively utilize the interplay between the binary Golay codes and the Leech lattice. As a byproduct of our analysis, we identify a new universally optimal (in the sense of energy) code in the projective space $\mathbb{RP}^{21}$ with $1408$ points (lines). Furthermore, we extend the Delsarte-Goethals-Seidel definition of derived codes from their seminal $1977$ paper and generalize their Theorem 8.2 to show that if a $τ$-design is enclosed in $k\leq τ$ parallel hyperplanes, then each of the hyperplane's sub-code is a $(τ+1-k)$-design in the ambient subspace.

math.CO↗

arXiv2018

Condensers with touching plates and constrained minimum Riesz and Green energy problems

We study minimum energy problems relative to the $α$-Riesz kernel $|x-y|^{α-n}$, $α\in(0,2]$, over signed Radon measures $μ$ on $\mathbb R^n$, $n\geqslant3$, associated with a generalized condenser $(A_1,A_2)$, where $A_1$ is a relatively closed subset of a domain $D$ and $A_2=\mathbb R^n\setminus D$. We show that, though $A_2\cap\mathrm{Cl}_{\mathbb R^n}A_1$ may have nonzero capacity, this minimum energy problem is uniquely solvable (even in the presence of an external field) if we restrict ourselves to $μ$ with $μ^+\leqslantξ$, where a constraint $ξ$ is properly chosen. We establish the sharpness of the sufficient conditions on the solvability thus obtained, provide descriptions of the weighted $α$-Riesz potentials of the solutions, single out their characteristic properties, and analyze their supports. The approach developed is mainly based on the establishment of an intimate relationship between the constrained minimum $α$-Riesz energy problem over signed measures associated with $(A_1,A_2)$ and the constrained minimum $α$-Green energy problem over positive measures carried by $A_1$. The results are illustrated by examples.

math.CA↗

arXiv2018

Constrained minimum Riesz energy problems for a condenser with intersecting plates

We study the constrained minimum energy problem with an external field relative to the $α$-Riesz kernel $|x-y|^{α-n}$ of order $α\in(0,n)$ for a generalized condenser $\mathbf A=(A_i)_{i\in I}$ in $\mathbb R^n$, $n\geqslant 3$, whose oppositely charged plates intersect each other over a set of zero capacity. Conditions sufficient for the existence of minimizers are found, and their uniqueness and vague compactness are studied. Conditions obtained are shown to be sharp. We also analyze continuity of the minimizers in the vague and strong topologies when the condenser and the constraint both vary, describe the weighted equilibrium vector potentials, and single out their characteristic properties. Our arguments are based particularly on the simultaneous use of the vague topology and a suitable semimetric structure on a set of vector measures associated with $\mathbf A$, and the establishment of completeness theorems for proper semimetric spaces. The results remain valid for the logarithmic kernel on $\mathbb R^2$ and $\mathbf A$ with compact $A_i$, $i\in I$. The study is illustrated by several examples.

math.CA↗

arXiv2018

On spherical codes with inner products in a prescribed interval

We develop a framework for obtaining linear programming bounds for spherical codes whose inner products belong to a prescribed subinterval $[\ell,s]$ of $[-1,1)$. An intricate relationship between Levenshtein-type upper bounds on cardinality of codes with inner products in $[\ell,s]$ and lower bounds on the potential energy (for absolutely monotone interactions) for codes with inner products in $[\ell,1)$ (when the cardinality of the code is kept fixed) is revealed and explained. Thereby, we obtain a new extension of Levenshtein bounds for such codes. The universality of our bounds is exhibited by a unified derivation and their validity for a wide range of codes and potential functions.

math.MG↗

arXiv2017

Local properties of Riesz minimal energy configurations and equilibrium measures

We investigate separation properties of $N$-point configurations that minimize discrete Riesz $s$-energy on a compact set $A\subset \mathbb{R}^p$. When $A$ is a smooth $(p-1)$-dimensional manifold without boundary and $s\in [p-2, p-1)$, we prove that the order of separation (as $N\to \infty$) is the best possible. The same conclusions hold for the points that are a fixed positive distance from the boundary of $A$ whenever $A$ is any $p$-dimensional set. These estimates extend a result of Dahlberg for certain smooth $(p-1)$-dimensional surfaces when $s=p-2$ (the harmonic case). Furthermore, we obtain the same separation results for `greedy' $s$-energy points. We deduce our results from an upper regularity property of the $s$-equilibrium measure (i.e., the measure that solves the continuous minimal Riesz $s$-energy problem), and we show that this property holds under a local smoothness assumption on the set $A$.

math.CA↗

arXiv2017

Optimal discrete measures for Riesz potentials

For $s\geqslant d$, we obtain the leading term as $N\to \infty$ of the maximal weighted $N$-point Riesz $s$-polarization (or Chebyshev constant) for a certain class of $d$-rectifiable compact subsets of $\mathbb{R}^p$. This class includes compact subsets of $d$-dimensional $C^1$ manifolds whose boundary relative to the manifold has $\mathcal{H}_d$-measure zero, as well as finite unions of such sets when their pairwise intersections have $\mathcal{H}_d$-measure zero. We also explicitly find the weak$^*$ limit distribution of asymptotically optimal $N$-point polarization configurations as $N\to \infty$.

math.CA↗

arXiv2016

A Comparison of Popular Point Configurations on $\mathbb{S}^2$

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.

math.NA↗

arXiv2016

Generating Point Configurations via Hypersingular Riesz Energy With an External Field

For a compact $ d $-dimensional rectifiable subset of $ \mathbb{R}^{p} $ we study asymptotic properties as $ N\to\infty $ of $N$-point configurations minimizing the energy arising from a Riesz $ s $-potential $ 1/r^s $ and an external field in the hypersingular case $ s\geq d$. Formulas for the weak$ ^* $ limit of normalized counting measures of such optimal point sets and the first-order asymptotic values of minimal energy are obtained. As an application, we derive a method for generating configurations whose normalized counting measures converge to a given absolutely continuous measure supported on a rectifiable subset of $ \mathbb{R}^{p} $. Results on separation and covering properties of discrete minimizers are given. Our theorems are illustrated with several numerical examples.

math.CA↗

arXiv2015

Universal upper and lower bounds on energy of spherical designs

Linear programming (polynomial) techniques are used to obtain lower and upper bounds for the potential energy of spherical designs. This approach gives unified bounds that are valid for a large class of potential functions. Our lower bounds are optimal for absolutely monotone potentials in the sense that for the linear programming technique they cannot be improved by using polynomials of the same or lower degree. When additional information about the structure (upper and lower bounds for the inner products) of the designs is known, improvements on the bounds are obtained. Furthermore, we provide `test functions' for determining when the linear programming lower bounds for energy can be improved utilizing higher degree polynomials. We also provide some asymptotic results for these energy bounds.

math.MG↗

arXiv2015

Universal lower bounds for potential energy of spherical codes

We derive and investigate lower bounds for the potential energy of finite spherical point sets (spherical codes). Our bounds are optimal in the following sense -- they cannot be improved by employing polynomials of the same or lower degrees in the Delsarte-Yudin method. However, improvements are sometimes possible and we provide a necessary and sufficient condition for the existence of such better bounds. All our bounds can be obtained in a unified manner that does not depend on the potential function, provided the potential is given by an absolutely monotone function of the inner product between pairs of points, and this is the reason for us to call them universal. We also establish a criterion for a given code of dimension $n$ and cardinality $N$ not to be LP-universally optimal, e.g. we show that two codes conjectured by Ballinger et al to be universally optimal are not LP-universally optimal.

math.MG↗

arXiv2014

Periodic Discrete Energy for Long-Range Potentials

We consider periodic energy problems in Euclidean space with a special emphasis on long-range potentials that cannot be defined through the usual infinite sum. One of our main results builds on more recent developments of Ewald summation to define the periodic energy corresponding to a large class of long-range potentials. Two particularly interesting examples are the logarithmic potential and the Riesz potential when the Riesz parameter is smaller than the dimension of the space. For these examples, we use analytic continuation methods to provide concise formulas for the periodic kernel in terms of the Epstein Hurwitz Zeta function. We apply our energy definition to deduce several properties of the minimal energy including the asymptotic order of growth and the distribution of points in energy minimizing configurations as the number of points becomes large. We conclude with some detailed calculations in the case of one dimension, which shows the utility of this approach.

math-ph↗

arXiv2013

Polarization optimality of equally spaced points on the circle for discrete potentials

We prove a conjecture of Ambrus, Ball and Erdélyi that equally spaced points maximize the minimum of discrete potentials on the unit circle whenever the potential is of the form \sum_{k=1}^n f(d(z,z_k)), where $f:[0,π]\to [0,\infty]$ is non-increasing and strictly convex and $d(z,w)$ denotes the geodesic distance between $z$ and $w$ on the circle.

math-ph↗

arXiv2013

Low complexity methods for discretizing manifolds via Riesz energy minimization

Let $A$ be a compact $d$-rectifiable set embedded in Euclidean space $\RR^p$, $d\le p$. For a given continuous distribution $σ(x)$ with respect to $d$-dimensional Hausdorff measure on $A$, our earlier results provided a method for generating $N$-point configurations on $A$ that have asymptotic distribution $σ(x)$ as $N\to \infty$; moreover such configurations are "quasi-uniform" in the sense that the ratio of the covering radius to the separation distance is bounded independent of $N$. The method is based upon minimizing the energy of $N$ particles constrained to $A$ interacting via a weighted power law potential $w(x,y)|x-y|^{-s}$, where $s>d$ is a fixed parameter and $w(x,y)=\left(σ(x)σ(y)\right)^{-({s}/{2d})}$. Here we show that one can generate points on $A$ with the above mentioned properties keeping in the energy sums only those pairs of points that are located at a distance of at most $r_N=C_N N^{-1/d}$ from each other, with $C_N$ being a positive sequence tending to infinity arbitrarily slowly. To do this we minimize the energy with respect to a varying truncated weight $v_N(x,y)=Φ$\left|x-y\right|/r_N$w(x,y)$, where $Φ:(0,\infty)\to [0,\infty)$ is a bounded function with $Φ(t)=0$, $t\geq 1$, and $\lim_{t\to 0^+}Φ(t)=1$. This reduces, under appropriate assumptions, the complexity of generating $N$ point `low energy' discretizations to order $N C_N^d$ computations.

math-ph↗

arXiv2012

Mesh ratios for best-packing and limits of minimal energy configurations

For $N$-point best-packing configurations $ω_N$ on a compact metric space $(A,ρ)$, we obtain estimates for the mesh-separation ratio $γ(ω_N,A)$, which is the quotient of the covering radius of $ω_N$ relative to $A$ and the minimum pairwise distance between points in $ω_N$. For best-packing configurations $ω_N$ that arise as limits of minimal Riesz $s$-energy configurations as $s\to \infty$, we prove that $γ(ω_N,A)\le 1$ and this bound can be attained even for the sphere. In the particular case when N=5 on $S^2$ with $ρ$ the Euclidean metric, we prove our main result that among the infinitely many 5-point best-packing configurations there is a unique configuration, namely a square-base pyramid $ω_5^*$, that is the limit (as $s\to \infty$) of 5-point $s$-energy minimizing configurations. Moreover, $γ(ω_5^*,S^2)=1$.

math-ph↗

arXiv2012

Minimal $N$-Point Diameters and $f$-Best-Packing Constants in $R^d$

In terms of the minimal $N$-point diameter $D_d(N)$ for $R^d,$ we determine, for a class of continuous real-valued functions $f$ on $[0,+\infty],$ the $N$-point $f$-best-packing constant $\min\{f(\|x-y\|)\, :\, x,y\in \R^d\}$, where the minimum is taken over point sets of cardinality $N.$ We also show that $$ N^{1/d}Δ_d^{-1/d}-2\le D_d(N)\le N^{1/d}Δ_d^{-1/d}, \quad N\ge 2,$$ where $Δ_d$ is the maximal sphere packing density in $\R^d$. Further, we provide asymptotic estimates for the $f$-best-packing constants as $N\to\infty$.

math-ph↗

arXiv2012

The next-order term for optimal Riesz and logarithmic energy asymptotics on the sphere

We survey known results and present estimates and conjectures for the next-order term in the asymptotics of the optimal logarithmic energy and Riesz $s$-energy of $N$ points on the unit sphere in $\mathbb{R}^{d+1}$, $d\geq 1$. The conjectures are based on analytic continuation assumptions (with respect to $s$) for the coefficients in the asymptotic expansion (as $N\to \infty$) of the optimal $s$-energy.

math-ph↗

arXiv2011

Quasi-uniformity of Minimal Weighted Energy Points on Compact Metric Spaces

For a closed subset $K$ of a compact metric space $A$ possessing an $α$-regular measure $μ$ with $μ(K)>0$, we prove that whenever $s>α$, any sequence of weighted minimal Riesz $s$-energy configurations $ω_N=\{x_{i,N}^{(s)}\}_{i=1}^N$ on $K$ (for `nice' weights) is quasi-uniform in the sense that the ratios of its mesh norm to separation distance remain bounded as $N$ grows large. Furthermore, if $K$ is an $α$-rectifiable compact subset of Euclidean space ($α$ an integer) with positive and finite $α$-dimensional Hausdorff measure, it is possible to generate such a quasi-uniform sequence of configurations that also has (as $N\to \infty$) a prescribed positive continuous limit distribution with respect to $α$-dimensional Hausdorff measure. As a consequence of our energy related results for the unweighted case, we deduce that if $A$ is a compact $C^1$ manifold without boundary, then there exists a sequence of $N$-point best-packing configurations on $A$ whose mesh-separation ratios have limit superior (as $N\to \infty$) at most 2.

math-ph↗

arXiv2009

Discrete Energy Asymptotics on a Riemannian circle

We derive the complete asymptotic expansion in terms of powers of $N$ for the geodesic $f$-energy of $N$ equally spaced points on a rectifiable simple closed curve $Γ$ in ${\mathbb R}^p$, $p\geq2$, as $N \to \infty$. For $f$ decreasing and convex, such a point configuration minimizes the $f$-energy $\sum_{j\neq k}f(d(\mathbf{x}_j, \mathbf{x}_k))$, where $d$ is the geodesic distance (with respect to $Γ$) between points on $Γ$. Completely monotonic functions, analytic kernel functions, Laurent series, and weighted kernel functions $f$ are studied. % Of particular interest are the geodesic Riesz potential $1/d^s$ ($s \neq 0$) and the geodesic logarithmic potential $\log(1/d)$. By analytic continuation we deduce the expansion for all complex values of $s$.

math-ph↗