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Liudmyla Kryvonos

Publications and source records attributed to Liudmyla Kryvonos.

7 recordsLinked to original sources
arXiv2026

Energy minimization for eight points on the sphere

We study the energy minimization problem for eight points on the unit sphere. For the logarithmic and Coulomb energies, we show that the unique global minimizer up to congruence is a square antiprism with height characterized by a unique stationarity equation. The proof is computer-assisted and fully verified in Lean. After this, we consider generalizations of the result to other important energies. For the Riesz $s$-energies, we provide a Lean-verified, non-computer-assisted proof that the square antiprism with height depending on $s$ is the unique global minimizer for all sufficiently large $s$, and a computer-assisted proof that this in fact holds for all $s\geq 0$. We also give examples of energies arising from completely monotonic potentials for which the square antiprism is not a global minimizer, answering in the negative a universality question of Cohn and Woo.

math.MG↗
arXiv2026

Energy, Polarization, and Separation of Greedy Sequences for Riesz and Green Kernels

We investigate the asymptotic behavior of greedy $s$-Riesz and Green energy sequences $\{x_{n}\}_{n=1}^{\infty}$ on the unit sphere $\mathbb{S}^{d} \subset \mathbb{R}^{d+1}$, where each point $x_n$ is defined as the minimizer of the discrete potential generated by the preceding points $x_1, x_2, ..., x_{n-1}$. We show that the greedy sequence attains optimal growth behavior for the second-order term of the Green and Riesz $s$-energies when $d-2 \leq s < d$. The main idea is to establish the bounds on polarization using well-separation properties of the greedy configurations.

math.CA↗
arXiv2024

On a problem of E. Meckes for the unitary eigenvalue process on an arc

We study the problem originally communicated by E. Meckes on the asymptotics for the eigenvalues of the kernel of the unitary eigenvalue process of a random $n \times n$ matrix. The eigenvalues $p_{j}$ of the kernel are, in turn, associated with the discrete prolate spheroidal wave functions. We consider the eigenvalue counting function $|G(x,n)|:=\#\{j:p_j>Ce^{-x n}\}$, ($C>0$ here is a fixed constant) and establish the asymptotic behavior of its average over the interval $x \in (λ-\varepsilon, λ+\varepsilon)$ by relating the function $|G(x,n)|$ to the solution $J(y)$ of the following energy problem on the unit circle $S^{1}$, which is of independent interest. Namely, for given $θ$, $0<θ< 2 π$, and given $q$, $0<q<1$, we determine the function $J(q) =\inf \{I(μ): μ\in \mathcal{P}(S^{1}), μ(A_θ) = q\}$, where $I(μ):= \iint \log\frac{1}{|z - ζ|} dμ(z) dμ(ζ)$ is the logarithmic energy of a probability measure $μ$ supported on the unit circle and $A_θ$ is the arc from $e^{-i θ/2}$ to $e^{i θ/2}$.

math.PR↗
arXiv2023

Weighted Minkowski's Existence Theorem and Projection Bodies

The Brunn-Minkowski Theory has seen several generalizations over the past century. Many of the core ideas have been generalized to measures. With the goal of framing these generalizations as a weighted Brunn-Minkowski theory, we prove the Minkowski existence theorem for a large class of Borel measures with continuous density, denoted by $Λ^n$: for $ν$ a finite, even Borel measure on the unit sphere and even $μ\inΛ^n$, there exists a symmetric convex body $K$ such that $$dν(u)=c_{μ,K}dS^μ_{K}(u),$$ where $c_{μ,K}$ is a quantity that depends on $μ$ and $K$ and $dS^μ_{K}(u)$ is the surface area-measure of $K$ with respect to $μ$. Examples of measures in $Λ^n$ are homogeneous measures (with $c_{μ,K}=1$) and probability measures with radially decreasing densities (e.g. the Gaussian measure). We will also consider weighted projection bodies $Π_μK$ by classifying them and studying the isomorphic Shephard problem: if $μ$ and $ν$ are even, homogeneous measures with density and $K$ and $L$ are symmetric convex bodies such that $Π_μ K \subset Π_ν L$, then can one find an optimal quantity $\mathcal{A}>0$ such that $μ(K)\leq \mathcal{A}ν(L)$? Among other things, we show that, in the case where $μ=ν$ and $L$ is a projection body, $\mathcal{A}=1$.

math.FA↗
arXiv2021

Polynomial approximation of piecewise analytic functions on quasi-smooth arcs

For a function $f$ that is piecewise analytic on a quasi-smooth arc $\mathcal{L}$ and any $0<σ<1$ we construct a sequence of "near-best" polynomials that converge at a rate $e^{-n^σ}$ at each point of analyticity of $f$ and are close to the best polynomial approximants on the whole $\mathcal{L}$. Also we give examples of quasi-smooth arcs for which convergence of "near-best" approximants at points of analiticity of $f$ is geometric, i.e. has a rate $e^{-cn}$ with some $c>0$.

math.CV↗