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We consider the minimum Riesz $s$-energy problem on the unit disk $\mathbb D:=\{(x_1,\ldots,x_d)\in\mathbb R^d: x_1=0, x_2^2+x_3^2+\ldots+x_d^2\leq 1\}$ in the Euclidean space $\mathbb R^d$, $d\geq 3$, immersed into a smooth rotationally…
Specific data compression techniques, formalized by the concept of coresets, proved to be powerful for many optimization problems. In fact, while tightly controlling the approximation error, coresets may lead to significant speed up of the…
The nature of the interplay between fluctuations and quenched random disorder is a long-standing open problem, particularly in systems with a continuous order parameter. This lack of a full theoretical treatment has been underscored by…
We show that the $L^p$ boundedness, $p>2$, of the Riesz transform on a complete non-compact Riemannian manifold with upper and lower Gaussian heat kernel estimates is equivalent to a certain form of Sobolev inequality. We also characterize…
A closed set of a Euclidean space is said to be Chebyshev if every point in the space has one and only one closest point in the set. Although the situation is not settled in infinite-dimensional Hilbert spaces, in 1932 Bunt showed that in…
This paper presents new quadrature rules for functions in a reproducing kernel Hilbert space using nodes drawn by a sampling algorithm known as randomly pivoted Cholesky. The resulting computational procedure compares favorably to previous…
The manuscript is devoted to the boundary behavior of mappings with bounded and finite distortion. We consider mappings of domains of the Euclidean space that satisfy weighted Poletsky inequality. Assume that, the definition domain is…
In the setting of a Gaussian channel without power constraints, proposed by Poltyrev, the codewords are points in an n-dimensional Euclidean space (an infinite constellation) and the tradeoff between their density and the error probability…
We consider the problem of choosing Euclidean points to maximize the sum of their weighted pairwise distances, when each point is constrained to a ball centered at the origin. We derive a dual minimization problem and show strong duality…
In this paper, we propose criteria for unboundedness of the images of set-valued mappings having closed graphs in Euclidean spaces. We focus on mappings whose domains are non-closed or whose values are connected. These criteria allow us to…
We study the electrostatics of a thin, finite-length conducting cylindrical shell held at constant potential V0. Exploiting axial symmetry, we recast the problem as a one-dimensional singular integral equation for the axial surface-charge…
We present a dynamic programming algorithm for selecting a representative subset of size $k$ from a given set with $n$ points such that the Riesz $s$-energy is near minimized. While NP-hard in general dimensions, the one-dimensional case…
This article is concerned with an extension of univariate Chebyshev polynomials of the first kind to the multivariate setting, where one chases best approximants to specific monomials by polynomials of lower degree relative to the uniform…
This paper considers sparse polynomial optimization with unbounded sets. When the problem possesses correlative sparsity, we propose a sparse homogenized Moment-SOS hierarchy with perturbations to solve it. The new hierarchy introduces one…
The control of active colloidal particles via optical traps is a cornerstone for research of matter at the micron and nanometer scale. A central challenge in this domain is the derivation of optimal transport protocols that minimize the…
Motivated by a geometric problem, we introduce a new non-convex graph partitioning objective where the optimality criterion is given by the sum of the Dirichlet eigenvalues of the partition components. A relaxed formulation is identified…
This paper investigates fractional Riesz-Bessel equations with random initial conditions that exhibit either classical or cyclic long-range dependence. It studies zoom-in asymptotics for the corresponding solutions and establishes…
We develop a rigorous framework for global non-convex optimization by reformulating the minimization problem as a discounted infinite-horizon optimal control problem. For non-convex, continuous, and possibly non-smooth objective functions…
We extend the notion of some energy-type expressions based on two sets, developed in the abstract potential theory. We also give the discretized version of the quantities defined, similar to Chebyshev constant. This extension allows to…
In this paper, we consider a well-known sparse optimization problem that aims to find a sparse solution of a possibly noisy underdetermined system of linear equations. Mathematically, it can be modeled in a unified manner by minimizing…