arXiv subjects

P. D. Dragnev

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

13 recordsLinked to original sources

arXiv2026

Universal optimality of $T$-avoiding spherical codes and designs

Given an open set $T\subset [-1,1)$, we introduce the concepts of $T$-avoiding spherical codes and designs, that is, spherical codes that have no inner products in the set $T$. We show that certain codes found in the minimal vectors of the Leech lattice, as well as the minimal vectors of the Barnes--Wall lattice and codes derived from strongly regular graphs, are universally optimal in the restricted class of $T$-avoiding codes. We also extend a result of Delsarte--Goethals--Seidel about codes with three inner products $α, β, γ$ (in our terminology $(α,β)$-avoiding $γ$-codes). Parallel to the notion of tight spherical designs, we also derive that these codes are minimal (tight) $T$-avoiding spherical designs of fixed dimension and strength. In some cases, we also find that codes under consideration have maximal cardinality in their $T$-avoiding class for given dimension and minimum distance.

math.CO↗

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↗

arXiv2019

Logarithmic Equilibrium on the Sphere in the Presence of Multiple Point Charges

With the sphere $\mathbb{S}^2 \subset \mathbb{R}^3$ as a conductor holding a unit charge with logarithmic interactions, we consider the problem of determining the support of the equilibrium measure in the presence of an external field consisting of finitely many point charges on the surface of the sphere. We determine that for any such configuration, the complement of the equilibrium support is the stereographic preimage from the plane of a union of classical quadrature domains, whose orders sum to the number of point charges.

math.CV↗

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

Probability, Minimax Approximation and Nash-Equilibrium. Estimating the Parameter of a Biased Coin

This paper deals with the application of Approximation Theory type techniques to study a classical problem in Probability: estimating the parameter of a biased coin. For this purpose, a Minimax Estimation problem is considered and the characterization of the optimal estimator is shown, together with the weak asymptotics of such optimal choices as the number of coin tosses approaches infinity; in addition, a number of numerical examples and graphs are displayed. At the same time, the problem is also discussed from the Game Theory viewpoint, as a non-cooperative, two-player game, and a Nash-equilibrium is established. The particular case of n=2 tosses is completely solved.

math.PR↗

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↗

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

Minimum Riesz energy problems for a condenser with "touching plates"

Minimum Riesz energy problems in the presence of an external field are analyzed for a condenser with touching plates. We obtain sufficient and/or necessary conditions for the solvability of these problems in both the unconstrained and the constrained settings, investigate the properties of minimizers, and prove their uniqueness. Furthermore, characterization theorems in terms of variational inequalities for the weighted potentials are established. The results obtained are illustrated by several examples.

math.CA↗

arXiv2015

Log-optimal configurations on the sphere

In this article we consider the distribution of $N$ points on the unit sphere $\mathbb{S}^{d-1}$ in $\mathbb{R}^d$ interacting via logarithmic potential. A characterization theorem of the stationary configurations is derived when $N=d+2$ and two new log-optimal configurations minimizing the logarithmic energy are obtained for six points on $\mathbb{S}^3$ and seven points on $\mathbb{S}^4$. A conjecture on the log-optimal configurations of $d+2$ points on $\mathbb{S}^{d-1}$ is stated and three auxiliary results supporting the conjecture are presented.

math-ph↗

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↗

arXiv2011

A Fascinating Polynomial Sequence arising from an Electrostatics Problem on the Sphere

A positive unit point charge approaching from infinity a perfectly spherical isolated conductor carrying a total charge of +1 will eventually cause a negatively charged spherical cap to appear. The determination of the smallest distance $ρ(d)$ ($d$ is the dimension of the unit sphere) from the point charge to the sphere where still all of the sphere is positively charged is known as Gonchar's problem. Using classical potential theory for the harmonic case, we show that $1+ρ(d)$ is equal to the largest positive zero of a certain sequence of monic polynomials of degree $2d-1$ with integer coefficients which we call Gonchar polynomials. Rather surprisingly, $ρ(2)$ is the Golden ratio and $ρ(4)$ the lesser known Plastic number. But Gonchar polynomials have other interesting properties. We discuss their factorizations, investigate their zeros and present some challenging conjectures.

math-ph↗

arXiv2009

Minimal Riesz energy on the sphere for axis-supported external fields

We investigate the minimal Riesz s-energy problem for positive measures on the d-dimensional unit sphere S^d in the presence of an external field induced by a point charge, and more generally by a line charge. The model interaction is that of Riesz potentials |x-y|^(-s) with d-2 <= s < d. For a given axis-supported external field, the support and the density of the corresponding extremal measure on S^d is determined. The special case s = d-2 yields interesting phenomena, which we investigate in detail. A weak* asymptotic analysis is provided as s goes to (d-2)^+.

math-ph↗