Ryan Matzke
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View article: Minimizers for an Aggregation Model with Attractive–Repulsive Interaction
Minimizers for an Aggregation Model with Attractive–Repulsive Interaction Open
We solve explicitly a certain minimization problem for probability measures involving an interaction energy that is repulsive at short distances and attractive at large distances. We complement earlier works by showing that in an optimal p…
View article: Riesz Energy with a Radial External Field: When is the Equilibrium Support a Sphere?
Riesz Energy with a Radial External Field: When is the Equilibrium Support a Sphere? Open
We consider Riesz energy problems with radial external fields. We study the question of whether or not the equilibrium measure is the uniform distribution on a sphere. We develop general necessary and general sufficient conditions on the e…
View article: Geodesic Distance Riesz Energy on Projective Spaces
Geodesic Distance Riesz Energy on Projective Spaces Open
We study probability measures that minimize the Riesz energy with respect to the geodesic distance $\vartheta (x,y)$ on projective spaces $\mathbb{FP}^d$ (such energies arise from the 1959 conjecture of Fejes Tóth about sums of non-obtuse …
View article: Babai Numbers and Babai Spectra of Paths and Cycles
Babai Numbers and Babai Spectra of Paths and Cycles Open
We study Babai numbers and Babai $k$-spectra of paths and cycles. We completely determine the Babai numbers of paths $P_n$ for $n>1$ and $1 \leq k \leq n-1$, and the Babai $k$-spectra for $P_n$ when $1 \leq k \leq n/2$. We also completely …
View article: Riesz Energy with a Radial External Field: When is the Equilibrium Support a Sphere?
Riesz Energy with a Radial External Field: When is the Equilibrium Support a Sphere? Open
We consider Riesz energy problems with radial external fields. We study the question of whether or not the equilibrium is the uniform distribution on a sphere. We develop general necessary as well as general sufficient conditions on the ex…
View article: Riesz energy, L2$L^2$ discrepancy, and optimal transport of determinantal point processes on the sphere and the flat torus
Riesz energy, L2$L^2$ discrepancy, and optimal transport of determinantal point processes on the sphere and the flat torus Open
Determinantal point processes exhibit an inherent repulsive behavior, thus providing examples of very evenly distributed point sets on manifolds. In this paper, we study the so‐called harmonic ensemble, defined in terms of Laplace eigenfun…
View article: Riesz Energy, $L^2$ Discrepancy, and Optimal Transport of Determinantal Point Processes on the Sphere and the Flat Torus
Riesz Energy, $L^2$ Discrepancy, and Optimal Transport of Determinantal Point Processes on the Sphere and the Flat Torus Open
Determinantal point processes exhibit an inherent repulsive behavior, thus providing examples of very evenly distributed point sets on manifolds. In this paper, we study the so-called harmonic ensemble, defined in terms of Laplace eigenfun…
View article: Riesz and Green energy on projective spaces
Riesz and Green energy on projective spaces Open
In this paper we study Riesz, Green and logarithmic energy on two-point homogeneous spaces. More precisely we consider the real, the complex, the quaternionic and the Cayley projective spaces. For each of these spaces we provide upper esti…
View article: Minimizers for an aggregation model with attractive-repulsive interaction
Minimizers for an aggregation model with attractive-repulsive interaction Open
We solve explicitly a certain minimization problem for probability measures involving an interaction energy that is repulsive at short distances and attractive at large distances. We complement earlier works by showing that part of the rem…
View article: A random line intersects $\mathbb{S}^2$ in two probabilistically independent locations
A random line intersects $\mathbb{S}^2$ in two probabilistically independent locations Open
We consider random lines in $\mathbb{R}^3$ (random with respect to the kinematic measure) and how they intersect $\mathbb{S}^2$. It is known that the entry point and the exit point behave like \textit{independent} uniformly distributed ran…
View article: Optimal Measures for Multivariate Geometric Potentials
Optimal Measures for Multivariate Geometric Potentials Open
We study measures and point configurations optimizing energies based on multivariate potentials. The emphasis is put on potentials defined by geometric characteristics of sets of points, which serve as multi-input generalizations of the we…
View article: Optimizers of three-point energies and nearly orthogonal sets
Optimizers of three-point energies and nearly orthogonal sets Open
This paper is devoted to spherical measures and point configurations optimizing three-point energies. Our main goal is to extend the classic optimization problems based on pairs of distances between points to the context of three-point pot…
View article: Polarization and Greedy Energy on the Sphere
Polarization and Greedy Energy on the Sphere Open
We investigate the behavior of a greedy sequence on the sphere $\mathbb{S}^d$ defined so that at each step the point that minimizes the Riesz $s$-energy is added to the existing set of points. We show that for $0
View article: Riesz and Green energy on projective spaces
Riesz and Green energy on projective spaces Open
In this paper we study Riesz, Green and logarithmic energy on two-point homogeneous spaces. More precisely we consider the real, the complex, the quaternionic and the Cayley projective spaces. For each of these spaces we provide upper esti…
View article: Optimal measures for $p$-frame energies on spheres
Optimal measures for $p$-frame energies on spheres Open
We provide new answers about the distribution of mass on spheres so as to minimize energies of pairwise interactions. We find optimal measures for the p -frame energies, i.e., energies with the kernel given by the absolute value of the inn…
View article: Optimal measures for $p$-frame energies on spheres
Optimal measures for $p$-frame energies on spheres Open
We provide new answers about the distribution of mass on spheres so as to minimize energies of pairwise interactions. We find optimal measures for the p -frame energies, i.e., energies with the kernel given by the absolute value of the inn…
View article: Positive definiteness and the Stolarsky invariance principle
Positive definiteness and the Stolarsky invariance principle Open
In this paper we elaborate on the interplay between energy optimization, positive definiteness, and discrepancy. In particular, assuming the existence of a $K$-invariant measure $μ$ with full support, we show that conditional positive defi…
View article: Problems with a lot of Potential: Energy Optimization on Compact Spaces
Problems with a lot of Potential: Energy Optimization on Compact Spaces Open
University of Minnesota Ph.D. dissertation. 2021. Major: Mathematics. Advisor: Dmitriy Bilyk. 1 computer file (PDF); 259 pages.
View article: Potential theory with multivariate kernels
Potential theory with multivariate kernels Open
In the present paper we develop the theory of minimization for energies with multivariate kernels, i.e. energies, in which pairwise interactions are replaced by interactions between triples or, more generally, $n$-tuples of particles. Such…
View article: On subgraphs with prescribed eccentricities
On subgraphs with prescribed eccentricities Open
A well-known result by Hedetniemi states that for every graph G there is a graph H whose center is G.We extend this result by showing under which conditions there exists, for a given graph G in which each vertex v has an integer label ℓ(v)…
View article: THE MAXIMUM SIZE OF -SUM-FREE SETS IN CYCLIC GROUPS
THE MAXIMUM SIZE OF -SUM-FREE SETS IN CYCLIC GROUPS Open
A subset $A$ of a finite abelian group $G$ is called $(k,l)$ -sum-free if the sum of $k$ (not necessarily distinct) elements of $A$ never equals the sum of $l$ (not necessarily distinct) elements of $A$ . We find an explicit formula for th…
View article: The Maximum Size of $(k,l)$-Sum-Free Sets in Cyclic Groups
The Maximum Size of $(k,l)$-Sum-Free Sets in Cyclic Groups Open
A subset $A$ of a finite abelian group $G$ is called $(k,l)$-sum-free if the sum of $k$ (not-necessarily-distinct) elements of $A$ never equals the sum of $l$ (not-necessarily-distinct) elements of $A$. We find an explicit formula for the …
View article: On the Fejes Tóth problem about the sum of angles between lines
On the Fejes Tóth problem about the sum of angles between lines Open
In 1959 Fejes Tóth posed a conjecture that the sum of pairwise non-obtuse angles between $N$ unit vectors in $\mathbb S^d$ is maximized by periodically repeated elements of the standard orthonormal basis. We obtain new improved upper bound…
View article: Connected minimum secure-dominating sets in grids
Connected minimum secure-dominating sets in grids Open
For any (finite simple) graph the secure domination number of satisfies . Here we find a secure-dominating set in such that in all cases when is a grid, and in the majority of cases when is a cylindrical or toroidal grid. In all such cases…
View article: Stolarsky principle and energy optimization on the sphere
Stolarsky principle and energy optimization on the sphere Open
The classical Stolarsky invariance principle connects the spherical cap $L^2$ discrepancy of a finite point set on the sphere to the pairwise sum of Euclidean distances between the points. In this paper we further explore and extend this p…