Electronic Research Archive

Special issue on computational topology: Foundations & applications

Hemen Dutta, Department of Mathematics, Gauhati University, Guwahati-781014, India duttah@gauhati.ac.in

James F. Peters, Computational Intelligence Laboratory, University of Manitoba, WPG, MB, R3T5V6, Canada and Department of Mathematics, Faculty of Arts and Sciences,Adiyaman University, 02040 Adiyaman, Turkey james.peters3@umanitoba.ca

The digital smash productSpecial Issues
Ismet Cinar, Ozgur Ege and Ismet Karaca
2020, 28(1): 459-469 doi: 10.3934/era.2020026 +[Abstract](12)+[HTML](15) +[PDF](371.39KB)

In this paper, we construct the smash product from the digital viewpoint and prove some its properties such as associativity, distributivity, and commutativity. Moreover, we present the digital suspension and the digital cone for an arbitrary digital image and give some examples of these new concepts.

Generalizations of some ordinary and extreme connectedness properties of topological spaces to relator spacesSpecial Issues
Muwafaq Salih and Árpád Száz
2020, 28(1): 471-548 doi: 10.3934/era.2020027 +[Abstract](10)+[HTML](4) +[PDF](771.11KB)

Motivated by some ordinary and extreme connectedness properties of topologies, we introduce several reasonable connectedness properties of relators (families of relations). Moreover, we establish some intimate connections among these properties.

More concretely, we investigate relationships among various minimalness (well-chainedness), connectedness, hyper- and ultra-connectedness, door, superset, submaximality and resolvability properties of relators.

Since most generalized topologies and all proper stacks (ascending systems) can be derived from preorder relators, the results obtained greatly extends some standard results on topologies. Moreover, they are also closely related to some former results on well-chained and connected uniformities.

Existence of best proximity points satisfying two constraint inequalitiesSpecial Issues
Duraisamy Balraj, Muthaiah Marudai, Zoran D. Mitrovic, Ozgur Ege and Veeraraghavan Piramanantham
2020, 28(1): 549-557 doi: 10.3934/era.2020028 +[Abstract](11)+[HTML](12) +[PDF](251.01KB)

In this paper, we prove the existence of best proximity point and coupled best proximity point on metric spaces with partial order for weak proximal contraction mappings such that these critical points satisfy some constraint inequalities.

On the mod p Steenrod algebra and the Leibniz-Hopf algebraSpecial Issues
Neşet Deniz Turgay
2020, 28(2): 951-959 doi: 10.3934/era.2020050 +[Abstract](20)+[HTML](8) +[PDF](310.83KB)

Let \begin{document}$ p $\end{document} be a fixed odd prime. The Bockstein free part of the mod \begin{document}$ p $\end{document} Steenrod algebra, \begin{document}$ \mathcal{A}_p $\end{document}, can be defined as the quotient of the mod \begin{document}$ p $\end{document} reduction of the Leibniz Hopf algebra, \begin{document}$ \mathcal{F}_p $\end{document}. We study the Hopf algebra epimorphism \begin{document}$ \pi\colon \mathcal{F}_p\to \mathcal{A}_p $\end{document} to investigate the canonical Hopf algebra conjugation in \begin{document}$ \mathcal{A}_p $\end{document} together with the conjugation operation in \begin{document}$ \mathcal{F}_p $\end{document}. We also give a result about conjugation invariants in the mod 2 dual Leibniz Hopf algebra using its multiplicative algebra structure.

Rough semi-uniform spaces and its image proximitiesSpecial Issues
Surabhi Tiwari and Pankaj Kumar Singh
2020, 28(2): 1095-1106 doi: 10.3934/era.2020060 +[Abstract](25)+[HTML](19) +[PDF](3941.13KB)

In this paper, we introduce the concept of rough semi-uniform spaces as a supercategory of rough pseudometric spaces and approximation spaces. A completion of approximation spaces has been constructed using rough semi-uniform spaces. Applications of rough semi-uniform spaces in the construction of proximities of digital images is also discussed.

Colimits of crossed modules in modified categories of interestSpecial Issues
Ali Aytekin and Kadir Emir
2020, 28(3): 1227-1238 doi: 10.3934/era.2020067 +[Abstract](8)+[HTML](8) +[PDF](324.17KB)

In this paper, we give the constructions of the coequalizer and coproduct objects for the category of crossed modules, in a modified category of interest (MCI). In other words, we prove that the corresponding category is finitely cocomplete.

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