arXiv subjects
Publications and source records attributed to Laurence Boxer.
Cold sets and freezing sets belong to the theory of (approximate) fixed points for continuous self-maps on digital images. We study some properties of cold sets for digital images in the digital plane, and we examine some relationships between cold sets and freezing sets.
Two objects may be close in the Hausdorff metric, yet have very different geometric and topological properties. We examine other methods of comparing digital images such that objects close in each of these measures have some similar geometric or topological property. Such measures may be combined with the Hausdorff metric to yield a metric in which close images are similar with respect to multiple properties.
We use results of [6] to enlarge our knowledge of the approximate fixed point property (AFPP) for digital images in $\mathbb{Z}^2$. In particular, we study conditions under which the union of two convex digital disks has the AFPP.
We examine the relationship between convexity and the approximate fixed point property (AFPP) for digital images in Z^2.
We analyze the running time of the Saukas-Song algorithm for selection on a coarse grained multicomputer without expressing the running time in terms of communication rounds. This shows that while in the best case the Saukas-Song algorithm runs in asymptotically optimal time, in general it does not. We propose other algorithms for coarse grained selection that have optimal expected running time.
We continue the study of freezing sets for digital images introduced in [4, 2, 3]. We prove methods for obtaining freezing sets for digital images (X, c_i) for X \subset Z^2 and i \in {1, 2}. We give examples to show how these methods can lead to the determination of minimal freezing sets.
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 X is convex.
We add to our knowledge of the approximate fixed point property (AFPP) in digital topology. We show that a digital image that is a tree has the AFPP. Given two digital images (X, κ) and (Y, λ) that have the approximate fixed point property, does their Cartesian product have the AFPP? We explore conditions that yield an affirmative answer. A general answer to this question is not known at the current writing.
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 existence of a fixed point set for a continuous self-map restricts the map on the complement of the fixed point set.
We continue the work of [4, 2, 3], in which we discuss published assertions that are incorrect or incorrectly proven; that are severely limited or reduce to triviality; or that we improve upon.
We study the approximate fixed point property (AFPP) for continuous single-valued functions and for continuous multivalued functions in digital topology. We extend what is known about these notions and discuss errors that have appeared in the literature.
The paper [10] incorrectly asserts that the digital image MSS_18, a digital model of the Euclidean 2-sphere S^2, is not 18-contractible. We show this assertion is false.
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 that often differ greatly from standard results in classical topology. We introduce several measures related to fixed points for continuous self-maps on digital images, and study their properties. Perhaps the most important of these is the fixed point spectrum $F(X)$ of a digital image: that is, the set of all numbers that can appear as the number of fixed points for some continuous self-map. We give a complete computation of $F(C_n)$ where $C_n$ is the digital cycle of $n$ points. For other digital images, we show that, if $X$ has at least 4 points, then $F(X)$ always contains the numbers 0, 1, 2, 3, and the cardinality of $X$. We give several examples, including $C_n$, in which $F(X)$ does not equal $\{0,1,\dots,\#X\}$. We examine how fixed point sets are affected by rigidity, retraction, deformation retraction, and the formation of wedges and Cartesian products. We also study how fixed point sets in digital images can be arranged; e.g., in some cases the fixed point set is always connected.
We continue the work of [5] and [3], in which are considered papers in the literature that discuss fixed point assertions in digital topology. We discuss published assertions that are incorrect or incorrectly proven; that are severely limited or reduce to triviality under "usual" conditions; or that we improve upon.
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 derived conclusions that are incorrect, trivial, or limited.
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, researchers using these tools have derived conclusions that are incorrect or trivial.
There is a concept in digital topology of a shy map. We define an analogous concept for topological spaces: We say a function is shy if it is continuous and the inverse image of every path-connected subset of its image is path-connected. Some basic properties of such maps are presented. For example, every shy map onto a semilocally simply connected space induces a surjection of fundamental groups (but a shy map onto a space that is not semilocally simply connected need not do so).
We study properties of Cartesian products of digital images for which adjacencies based on the normal product adjacency are used. We show that the use of such adjacencies lets us obtain many "product properties" for which the analogous statement is either unknown or invalid if, instead, we were to use $c_u$-adjacencies.