Related papers: Minimal Constructible Sets
We establish an upper bound on the cardinality of a minimal generating set for the fundamental group of a large family of connected, balanced simplicial complexes and, more generally, simplicial posets.
We study the parametrizations of simple modules provided by the theory of basic sets for all finite Weyl groups. In the case of type B, we show the existence of basic sets for the matrices of constructible representations. Then we study…
We study the smallest possible number of points in a topological space having k open sets. Equivalently, this is the smallest possible number of elements in a poset having k order ideals. Using efficient algorithms for constructing a…
We prove in Theorem $2.2$ that the multiplicatively closed subset generated by at most two elements in the set of natural numbers $\mathbb{N}$ has arbitrarily large gaps by explicitly constructing large integer intervals with known prime…
We investigate the problem of constructing unextendible product bases in the qubit case - that is, when each local dimension equals 2. The cardinality of the smallest unextendible product basis is known in all qubit cases except when the…
We give an alternative construction for a family of partition generating functions due to Kanade and Russell. In our alternative construction, we use ordinary partitions instead of jagged partitions. We also present new generating functions…
The mapping class group of an orientable surface, which records its symmetries up to isotopy, plays a central role in low-dimensional topology. This chapter explores the foundational problem of determining minimal generating sets for these…
We determine the groups of minimal order in which all groups of order n can embedded for 1 < n < 16. We further determine the order of a minimal group in which all groups or order n or less can be embedded, also for 1 < n < 16.
This paper presents a novel approach to constructing finite generating sets for infinitely generated ideals. By integrating algebraic and computational techniques, we provide a method to identify finite generators, demonstrated through…
Motivated by a problem in graph theory, this article introduces an algebra called the balanced algebra. This algebra is defined by generators and relations, and the main goal is to find a minimal set of relations for it.
We determine whether each known generating set of arbitrary oriented Reidemeister moves is minimal. We then provide a complete classification of minimal generating sets that include a coherent Reidemeister move of type II. We also classify…
We display a new family of prime ideals with unbounded minimal number of generators in a three-dimensional power series ring over a field of characteristic zero. These primes are obtained as the kernel of a quasi-monomial algebra…
Our focus is on the set of lower-triangular, infinite matrices that have natural operations like addition, multiplication by a number, and matrix multiplication. With respect to addition this set forms and abelian group while with respect…
Cylindrical Algebraic Decomposition (CAD) algorithms typically produce a decomposition adapted to a finite family of semi-algebraic sets $\mathcal{F}$ (i.e. every member of $\mathcal{F}$ is a union of cells). Different algorithms may…
We describe a procedure for generating families of cyclic cubic fields with explicit fundamental units. This method generates all known families and gives new ones.
First we solve the problem of finding minimal degree families on toric surfaces by reducing it to lattice geometry. Then we describe how to find minimal degree families on, more generally, rational complex projective surfaces.
The set splittability problem is the following: given a finite collection of finite sets, does there exits a single set that contains exactly half the elements from each set in the collection? (If a set has odd size, we allow the floor or…
We study a recursively defined sequence which is constructed using the least common multiple. It has been conjectured that every term of that sequence is $1$ or a prime. In this paper we show that this claim is connected to a strong version…
In this paper, we study the structure of the set of tilings produced by any given tile-set. For better understanding this structure, we address the set of finite patterns that each tiling contains. This set of patterns can be analyzed in…
In this work, we carry out structural and algorithmic studies of a problem of barrier forming: selecting theminimum number of straight line segments (barriers) that separate several sets of mutually disjoint objects in the plane. The…