Related papers: On universal sets and sumsets
We study generalized splines from the perspective of the representation theory of the category of graphs with contractions. Our main theorem proves a kind of finite generation, which in turn implies the existence of a ``universal generating…
Categorical universal algebra can be developed either using Lawvere theories (single-sorted finite product theories) or using monads, and the category of Lawvere theories is equivalent to the category of finitary monads on Set. We show how…
In this paper we consider a notion of universal sets for ideals. We show that there exist universal sets of minimal Borel complexity for classic ideals like null subsets of $2^\omega$ and meager subsets of any Polish space, and demonstrate…
We define the notion of universal lift of a projective complex based on non-commutative parameter algebras, and prove its existence and uniqueness. We investigate the properties of parameter algebras for universal lifts.
We discuss a structural approach to subset-sum problems in additive combinatorics. The core of this approach are Freiman-type structural theorems, many of which will be presented through the paper. These results have applications in various…
We consider asymptotics of ratios of random characteristic polynomials associated with orthogonal polynomial ensembles. Under some natural conditions on the measure in the definition of the orthogonal polynomial ensemble we establish a…
In the present paper, classical tools of convex analysis are used to study the solution set to a certain class of set-inclusive generalized equations. A condition for the solution existence and global error bounds is established, in the…
Consider the following variant of the set cover problem. We are given a universe $U=\{1,...,n\}$ and a collection of subsets $\mathcal{C} = \{S_1,...,S_m\}$ where $S_i \subseteq U$. For every element $u \in U$ we need to find a set $\phi(u)…
We give a conceptual explanation of universal deformation formulas for unital associative algebras and prove some results on the structure of their moduli spaces. We then generalize universal deformation formulas to other types of algebras…
We study the universal groups of inverse semigroups associated with point sets and with tilings. We focus our attention on two classes of examples. The first class consists of point sets which are obtained by a cut and projection scheme…
In the present paper, some aspects of the finite-dimensional theory of set-inclusive generalized equations are studied. Set-inclusive generalized equations are problems arising in several contexts of optimization and variational analysis,…
Models of computation operating over the real numbers and computing a larger class of functions compared to the class of general recursive functions invariably introduce a non-finite element of infinite information encoded in an arbitrary…
Several results about the union-closed sets conjecture are presented.
Systems of equations with sets of integers as unknowns are considered. It is shown that the class of sets representable by unique solutions of equations using the operations of union and addition $S+T=\makeset{m+n}{m \in S, \: n \in T}$ and…
We provide a spectrum of results for the Universal Guard Problem, in which one is to obtain a small set of points ("guards") that are "universal" in their ability to guard any of a set of possible polygonal domains in the plane. We give…
We further investigate the uniform regularity property of collections of sets via primal and dual characterizing constants. These constants play an important role in determining convergence rates of projection algorithms for solving…
The concept of a uniform set is introduced for an ergodic, measure-preserving transformation on a non-atomic, infinite Lebesgue space. The uniform sets exist as much as they generate the underlying $\sigma$-algebra. This leads to the result…
We develop universal algebra over an enriched category $\mathcal K$ and relate it to finitary enriched monads over $\mathcal K$. Using it, we deduce recent results about ordered universal algebra where inequations are used instead of…
In this paper we introduce the concept of completeness of sets. We study this property on the set of integers. We examine how this property is preserved as we carry out various operations compatible with sets. We also introduce the problem…
In contrast to finite arithmetic configurations, relatively little is known about which infinite patterns can be found in every set of natural numbers with positive density. Building on recent advances showing infinite sumsets can be found,…