Related papers: Trimming and Building Freezing Sets
Freezing sets and cold sets have been introduced as part of the theory of fixed points in digital topology. In this paper, we introduce a generalization of these notions, the limiting set, and examine properties of limiting sets.
We continue the work of [10], studying properties of digital images determined by fixed point invariants. We introduce pointed versions of invariants that were introduced in [10]. We introduce freezing sets and cold sets to show how the…
Several recent papers in digital topology have sought to obtain fixed point results by mimicking the use of tools from classical topology, such as complete metric spaces. We show that in many cases, researchers using these tools have…
Cone and suspension constructions have been introduced in digital topology, modeled on those of classical topology. For digital cones and suspensions, and for some related digital images, we find (m, n)-limiting sets; especially (0,…
Several recent papers in digital topology have sought to obtain fixed point results by mimicking the use of tools from classical topology, such as complete metric spaces and homotopy invariant fixed point theory. We show that in many cases,…
We continue the study of freezing sets in digital topology, introduced in [2]. We show how to find a minimal freezing set for a "thick" convex disk X in the digital plane Z^2. We give examples showing the significance of the assumption that…
In this paper, we examine some properties of the fixed point set of a digitally continuous function. The digital setting requires new methods that are not analogous to those of classical topological fixed point theory, and we obtain results…
In this work we consider triangulations of point sets in the Euclidean plane, i.e., maximal straight-line crossing-free graphs on a finite set of points. Given a triangulation of a point set, an edge flip is the operation of removing one…
The topic of fixed points in digital metric spaces has drawn yet more publications with assertions that are incorrect, incorrectly proven, trivial, or incoherently stated. We discuss publications with bad assertions concerning fixed points…
Exceptional points are universal level degeneracies induced by non-Hermiticity. Whereas past decades witnessed their new physics, the unified understanding has yet to be obtained. Here we present the complete classification of generic…
An articulation point in a network is a node whose removal disconnects the network. Those nodes play key roles in ensuring connectivity of many real-world networks, from infrastructure networks to protein interaction networks and terrorist…
We discuss published assertions concerning fixed points in digital metric spaces that are incorrect or incorrectly proven, or reduce to triviality.
A point of a digital space is called simple if it can be deleted from the space without altering topology. This paper introduces the notion simple set of points of a digital space. The definition is based on contractible spaces and…
The topic of fixed points in digital metric spaces continues to draw publications with assertions that are incorrect, incorrectly proven, trivial, or incoherently stated. We continue the work of our earlier papers that discuss publications…
Nonexpansive mappings play a central role in modern optimization and monotone operator theory because their fixed points can describe solutions to optimization or critical point problems. It is known that when the mappings are sufficiently…
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…
Methods were developed in Ref. [1] for constructing reference metrics (and from them differentiable structures) on three-dimensional manifolds with topologies specified by suitable triangulations. This note generalizes those methods by…
We define a point-free construction of real exponentiation and logarithms, i.e.\ we construct the maps $\exp\colon (0, \infty)\times \mathbb{R} \rightarrow \!(0,\infty),\, (x, \zeta) \mapsto x^\zeta$ and $\log\colon (1,\infty)\times (0,…
Model sets (or cut and project sets) provide a familiar and commonly used method of constructing and studying nonperiodic point sets. Here we extend this method to situations where the internal spaces are no longer Euclidean, but instead…
The fixed point construction is a method for designing tile sets and cellular automata with highly nontrivial dynamical and computational properties. It produces an infinite hierarchy of systems where each layer simulates the next one. The…