arXiv research

Study of digital topology concepts like hyperspaces and function graphs.

Critiques incorrect fixed point assertions in digital topology.

Corrects incorrect assertions about fixed points in digital topology.

Find limiting sets for digital cones and suspensions.

The paper highlights issues in fixed point claims in digital topology.

Study minimal freezing sets in convex digital disks.

Incorrect fixed point assertions in digital topology are discussed.

Incorrect fixed point assertions in digital topology are discussed.

The paper highlights issues with fixed point claims in digital images.

The paper addresses flaws in fixed point assertions for digital images.

Fixed point assertions in digital topology are often incorrect or poorly stated.

Study restrictions on digitally continuous functions and their effects.

Digital trees have approximate fixed point property, and conditions for products are explored.

We study properties of shy maps in digital topology.

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.

New tools for constructing fixed point sets in digital topology.

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 tr…

In this paper, we show how to construct graph theoretical models of n-dimensional continuous objects and manifolds. These models retain topological properties of their continuous counterparts. An LCL collection of n-cells in Euclidean space is introduced and investigated. If an LCL collection of n-cells is a cover of a…

We study properties of Cartesian products of digital images, using a variety of adjacencies that have appeared in the literature.

We give an answer to the question given by T.Y.Kong in his article "Can 3-D Digital Topology be Based on Axiomatically Defined Digital Spaces?" In this article he asks the question, if so called "good pairs" of neighborhood relations can be found on the set Z^n such that the existence of digital manifolds of dimension …

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.

Corrects errors in Hans' pseudocovering spaces paper.

Recently, in the paper "Weight Agnostic Neural Networks" Gaier & Ha utilized architecture search to find networks where the topology completely encodes the knowledge. However, architecture search in topology space is expensive. We use the existing framework of binarized networks to find performant topologies by constra…

The paper presents a new set of axioms of digital topology, which are easily understandable for application developers. They define a class of locally finite (LF) topological spaces. An important property of LF spaces satisfying the axioms is that the neighborhood relation is antisymmetric and transitive. Therefore any…

We present a way to use Topological Data Analysis (TDA) for machine learning tasks on grayscale images. We apply persistent homology to generate a wide range of topological features using a point cloud obtained from an image, its natural grayscale filtration, and different filtrations defined on the binarized image. We…

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 c…

In this paper, we study two classes of planar self-similar fractals $T_\varepsilon$ with a shifting parameter $\varepsilon$. The first one is a class of self-similar tiles by shifting $x$-coordinates of some digits. We give a detailed discussion on the disk-likeness ({\it i.e., the property of being a topological disk}…

This study redefines probability for finite outcomes using axioms and examples.

We construct an elementary, combinatorial kind of topological quantum field theory, based on curves, surfaces, and orientations. The construction derives from contact invariants in sutured Floer homology and is essentially an elaboration of a TQFT defined by Honda--Kazez--Matic. This topological field theory stores inf…

Let $M$ be a $3\times 3$ integer matrix each of whose eigenvalues is greater than $1$ in modulus and let $\mathcal{D}\subset\mathbb{Z}^3$ be a set with $|\mathcal{D}|=|\det M|$, called digit set. The set equation $MT = T+\mathcal{D}$ uniquely defines a nonempty compact set $T\subset \mathbb{R}^3$. If $T$ has positive L…

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.

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 …

Given an integer $n\geq 2$ and a digit set ${\mathcal D}\subsetneq {0,1,...,n-1}^2$, there is a self-similar set $F \subset {\Bbb R}^2$ satisfying the set equation: $F=(F+{\mathcal D})/n$. We call such $F$ a fractal square. By studying a periodic extension $H= F+ {\mathbb Z}^2$, we classify $F$ into three types accordi…

Self-affine tiles homeomorphic to a ball proven for a specific digit set.

The paper uses topological concepts to analyze neural networks, revealing complex structure and dynamics.

We perform topological data analysis on the internal states of convolutional deep neural networks to develop an understanding of the computations that they perform. We apply this understanding to modify the computations so as to (a) speed up computations and (b) improve generalization from one data set of digits to ano…

ISOMORPH creates a digital twin for supply chain logistics, advancing time-series forecasting benchmarks.

Digital money could reduce germ spread during coronavirus.

Study AFPP of unions of convex digital disks in 2D.

This paper optimizes cybersecurity resource allocation in networks with heterogeneous attacker and defender valuations.

Study on cold and freezing sets in digital images.

We analyze the time series of four major cryptocurrencies (Bitcoin, Ethereum, Litecoin, and Ripple) before the digital market crash at the end of 2017 - beginning 2018. We introduce a methodology that combines topological data analysis with a machine learning technique -- $k$-means clustering -- in order to automatical…

Study convexity and AFPP in digital images.

Han discusses variants of digital covering maps and their equivalences.

Examines how irreducibility and rigidity affect digital images.

A new method streamlines digital payment programming using smart contracts.

New framework explains leading digit patterns without probabilistic assumptions.

Study freezing sets for digital images in a 2D grid.