Douglas P. Hardin
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View article: Notes on the discretization of TV-norm regularized inverse potential problems
Notes on the discretization of TV-norm regularized inverse potential problems Open
We describe a method to discretize optimization problems arising in the regularization of linear inverse problem having compact forward operator defined on 3-D valed measures, compactly supported on a fixed set. The criterion is a quadrati…
View article: Universal minima of discrete potentials for sharp spherical codes
Universal minima of discrete potentials for sharp spherical codes Open
This article is devoted to the study of discrete potentials on the sphere in \mathbb{R}^{n} for sharp codes. We show that the potentials of most of the known sharp codes attain the universal lower bounds for polarization for spherical \tau…
View article: Bounds on energy and potentials of discrete measures on the sphere
Bounds on energy and potentials of discrete measures on the sphere Open
We establish upper and lower universal bounds for potentials of weighted designs on the sphere $\mathbb{S}^{n-1}$ that depend only on quadrature nodes and weights derived from the design structure. Our bounds hold for a large class of pote…
View article: Energy bounds for weighted spherical codes and designs via linear programming
Energy bounds for weighted spherical codes and designs via linear programming Open
Universal bounds for the potential energy of weighted spherical codes are obtained by linear programming. The universality is in the sense of Cohn-Kumar -- every attaining code is optimal with respect to a large class of potential function…
View article: Universally Optimal Periodic Configurations in the Plane
Universally Optimal Periodic Configurations in the Plane Open
We develop lower bounds for the energy of configurations in $\mathbb{R}^d$ periodic with respect to a lattice. In certain cases, the construction of sharp bounds can be formulated as a finite dimensional, multivariate polynomial interpolat…
View article: Estimating the Net Magnetic Moment of Geological Samples From Planar Field Maps Using Multipoles
Estimating the Net Magnetic Moment of Geological Samples From Planar Field Maps Using Multipoles Open
Recent advances in magnetic microscopy have enabled studies of geological samples whose weak and spatially nonuniform magnetizations were previously inaccessible to standard magnetometry techniques. A quantity of central importance is the …
View article: Universal minima of discrete potentials for sharp spherical codes
Universal minima of discrete potentials for sharp spherical codes Open
This article is devoted to the study of discrete potentials on the sphere in $\mathbb{R}^n$ for sharp codes. We show that the potentials of most of the known sharp codes attain the universal lower bounds for polarization for spherical $τ$-…
View article: On polarization of spherical codes and designs
On polarization of spherical codes and designs Open
In this article we investigate the $N$-point min-max and the max-min polarization problems on the sphere for a large class of potentials in $\mathbb{R}^n$. We derive universal lower and upper bounds on the polarization of spherical designs…
View article: Asymptotics of $k$-nearest neighbor Riesz energies
Asymptotics of $k$-nearest neighbor Riesz energies Open
We obtain new asymptotic results about systems of $ N $ particles governed by Riesz interactions involving $ k $-nearest neighbors of each particle as $N\to\infty$. These results include a generalization to weighted Riesz potentials with e…
View article: Eigenfunctions of the Fourier transform with specified zeros
Eigenfunctions of the Fourier transform with specified zeros Open
Eigenfunctions of the Fourier transform with prescribed zeros played a major role in the proof that the E 8 and the Leech lattice give the best sphere packings in respective dimensions 8 and 24 by Cohn, Kumar, Miller, Radchenko and Viazovs…
View article: Inverse potential problems in divergence form for measures in the plane
Inverse potential problems in divergence form for measures in the plane Open
We study inverse potential problems with source term the divergence of some unknown (ℝ 3 -valued) measure supported in a plane; e.g. , inverse magnetization problems for thin plates. We investigate methods for recovering a magnetization μ …
View article: Dynamics of particles on a curve with pairwise hyper-singular repulsion
Dynamics of particles on a curve with pairwise hyper-singular repulsion Open
We investigate the large time behavior of particles restricted to a smooth closed curve in and subject to a gradient flow with respect to Euclidean hyper-singular repulsive Riesz -energy with We show that regardless of their initial pos…
View article: Asymptotic properties of short-range interaction functionals
Asymptotic properties of short-range interaction functionals Open
We describe a framework for extending the asymptotic behavior of a short-range interaction from the unit cube to general compact subsets of $ \mathbb R^d $. This framework allows us to give a unified treatment of asymptotics of hypersingul…
View article: Hyperuniform point sets on the sphere: probabilistic aspects
Hyperuniform point sets on the sphere: probabilistic aspects Open
We give a unified description of the modular and quasi-modular functions used in Viazovska's proof of the best packing bounds in dimension 8 and the proof by Cohn, Kumar, Miller, Radchenko, and Viazovska of the best packing bound in dimens…
View article: Divergence-free measures in the plane and inverse potential problems in divergence form
Divergence-free measures in the plane and inverse potential problems in divergence form Open
We show that a divergence-free measure on the plane is a continuous sum of unit tangent vector fields on rectifiable Jordan curves. This loop decomposition is more precise than the general decomposition in elementary solenoids given by S.K…
View article: Sparse recovery for inverse potential problems in divergence form
Sparse recovery for inverse potential problems in divergence form Open
We discuss recent results from [10] on sparse recovery for inverse potential problem with source term in divergence form. The notion of sparsity which is set forth is measure- theoretic, namely pure 1-unrectifiability of the support. The t…
View article: On the Search for Tight Frames of Low Coherence
On the Search for Tight Frames of Low Coherence Open
We introduce a projective Riesz $s$-kernel for the unit sphere $\mathbb{S}^{d-1}$ and investigate properties of $N$-point energy minimizing configurations for such a kernel. We show that these configurations, for $s$ and $N$ sufficiently l…
View article: Universal Bounds for Size and Energy of Codes of Given Minimum and Maximum Distances
Universal Bounds for Size and Energy of Codes of Given Minimum and Maximum Distances Open
We employ signed measures that are positive definite up to certain degrees to establish Levenshtein-type upper bounds on the cardinality of codes with given minimum and maximum distances, and universal lower bounds on the potential energy …
View article: Upper bounds for energies of spherical codes of given cardinality and separation
Upper bounds for energies of spherical codes of given cardinality and separation Open
We introduce a linear programming framework for obtaining upper bounds for the potential energy of spherical codes of fixed cardinality and minimum distance. Using Hermite interpolation we construct polynomials to derive corresponding boun…
View article: Universal bounds for spherical codes: the Levenshtein framework lifted
Universal bounds for spherical codes: the Levenshtein framework lifted Open
Based on the Delsarte-Yudin linear programming approach, we extend Levenshtein's framework to obtain lower bounds for the minimum $h$-energy of spherical codes of prescribed dimension and cardinality, and upper bounds on the maximal cardin…
View article: Magnetic moment estimation and bounded extremal problems
Magnetic moment estimation and bounded extremal problems Open
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View article: Inverse Potential Problems for Divergence of Measures with Total\n Variation Regularization
Inverse Potential Problems for Divergence of Measures with Total\n Variation Regularization Open
We study inverse problems for the Poisson equation with source term the\ndivergence of an $\\mathbf{R}^3$-valued measure, that is, the potential $\\Phi$\nsatisfies $$\n \\Delta \\Phi= \\text{div} \\boldsymbol{\\mu},\n $$ and $\\boldsymbol{…
View article: Energy Bounds for Codes in Polynomial Metric Spaces
Energy Bounds for Codes in Polynomial Metric Spaces Open
In this article we present a unified treatment for obtaining bounds on the potential energy of codes in the general context of polynomial metric spaces (PM-spaces). The lower bounds we derive via the linear programming (LP) techniques of D…
View article: On spherical codes with inner products in a prescribed interval
On spherical codes with inner products in a prescribed interval Open
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 cardinali…