arXiv subjects
Publications and source records attributed to Laurence Boxer.
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 of self-functions on digital images, as in some of our previous papers
S.E. Hans paper, Remarks on Pseudocovering Spaces in a Digital Topological Setting: A Corrigendum, is meant to address errors in previous papers. However, this paper is also marked by errors in its mathematics, as well as improprieties in its citations. We address these flaws in the current work.
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 of self-functions on digital images, as in some of our previous papers.
Most of the literature of computational geometry concerns geometric properties of sets of static points. M.J. Atallah introduced dynamic computational geometry, concerned with both momentary and long-term geometric properties of sets of moving point-objects. This area of research seems to have been dormant recently. The current paper examines new problems in dynamic computational geometry.
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 with bad assertions concerning fixed points of self-functions on digital images.
We continue a discussion of published assertions that are incorrect, incorrectly proven, or trivial, in the theory of fixed points in digital topology.
We develop new tools for the construction of fixed point sets in digital topology. We define excludable points and show that these may be excluded from all freezing sets. We show that articulation points are excludable. We also present results concerning points that must belong to a freezing set and often are easily recognized. These include points of degree~1 and some local extrema.
This paper continues a series in which we study deficiencies in previously published works concerning fixed point assertions for digital images.
We study how the properties of irreducibility and rigidity in digital images interact with Cartesian products, wedges, and cold and 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.
This paper continues a series discussing flaws in published assertions concerning fixed points in digital images.
SE Han's paper [11] discusses several variants of digital covering maps. We show several equivalences among these variants and discuss shortcomings in Han's paper.
This paper continues a series discussing flaws in published assertions concerning fixed points in digital metric spaces.
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, 0)-limiting (freezing) sets.
We adapt the study of hyperspaces and function spaces from classical topology to digital topology. We define digital hyperspaces and digital function graphs, and study some of their relationships and graphical properties.
We show how to obtain freezing sets for digital images in $\mathbb{Z}^n$ for arbitrary $n$, using the $c_1$ and $c_n$ adjacencies.
We study the consequences of some restrictions on digitally continuous functions. One of our results modifies easily to yield an analogous result for topological spaces.
We discuss published assertions concerning fixed points in digital metric spaces that are incorrect or incorrectly proven, or reduce to triviality.