scholarly journals Fixed Point Sets of k-Continuous Self-Maps of m-Iterated Digital Wedges

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1617 ◽  
Author(s):  
Sang-Eon Han
Keyword(s):  
Fixed Point ◽  
Natural Number ◽  
Digital Image ◽  
Digital Images ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Point Sets ◽  
Digital Curve ◽  
Curve Theory ◽  
Fixed Point Sets

Let Ckn,l be a simple closed k-curves with l elements in Zn and W:=Ckn,l∨⋯∨Ckn,l︷m-times be an m-iterated digital wedges of Ckn,l, and F(Conk(W)) be an alignment of fixed point sets of W. Then, the aim of the paper is devoted to investigating various properties of F(Conk(W)). Furthermore, when proceeding with this work, this paper addresses several unsolved problems. To be specific, we firstly formulate an alignment of fixed point sets of Ckn,l, denoted by F(Conk(Ckn,l)), where l(≥7) is an odd natural number and k≠2n. Secondly, given a digital image (X,k) with X♯=n, we find a certain condition that supports n−1,n−2∈F(Conk(X)). Thirdly, after finding some features of F(Conk(W)), we develop a method of making F(Conk(W)) perfect according to the (even or odd) number l of Ckn,l. Finally, we prove that the perfectness of F(Conk(W)) is equivalent to that of F(Conk(Ckn,l)). This can play an important role in studying fixed point theory and digital curve theory. This paper only deals with k-connected digital images (X,k) such that X♯≥2.

scholarly journals Fixed poin sets in digital topology, 1

2020 ◽  
Vol 21 (1) ◽  
pp. 87 ◽  
Author(s):  
Laurence Boxer ◽  
P. Christopher Staecker
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Digital Images ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Point Sets ◽  
Fixed Point Set ◽  
Complete Computation ◽  
Point Set ◽  
Fixed Point Sets

<p>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.</p><p>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<sub>n</sub>) where C<sub>n</sub> 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<sub>n</sub>, in which F(X) does not equal {0, 1, . . . , #X}.</p><p>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., for some digital images the fixed point set is always connected.</p>

scholarly journals Digital Topological Properties of an Alignment of Fixed Point Sets

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 921 ◽  
Author(s):  
Sang-Eon Han
Keyword(s):  
Fixed Point ◽  
Unsolved Problem ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Digital Topology ◽  
Fundamental Groups ◽  
Topological Properties ◽  
Point Sets ◽  
Fixed Point Sets ◽  
Connected Spaces

The present paper investigates digital topological properties of an alignment of fixed point sets which can play an important role in fixed point theory from the viewpoints of computational or digital topology. In digital topology-based fixed point theory, for a digital image ( X , k ) , let F ( X ) be the set of cardinalities of the fixed point sets of all k-continuous self-maps of ( X , k ) (see Definition 4). In this paper we call it an alignment of fixed point sets of ( X , k ) . Then we have the following unsolved problem. How many components are there in F ( X ) up to 2-connectedness? In particular, let C k n , l be a simple closed k-curve with l elements in Z n and X : = C k n , l 1 ∨ C k n , l 2 be a digital wedge of C k n , l 1 and C k n , l 2 in Z n . Then we need to explore both the number of components of F ( X ) up to digital 2-connectivity (see Definition 4) and perfectness of F ( X ) (see Definition 5). The present paper addresses these issues and, furthermore, solves several problems related to the main issues. Indeed, it turns out that the three models C 2 n n , 4 , C 3 n − 1 n , 4 , and C k n , 6 play important roles in studying these topics because the digital fundamental groups of them have strong relationships with alignments of fixed point sets of them. Moreover, we correct some errors stated by Boxer et al. in their recent work and improve them (see Remark 3). This approach can facilitate the studies of pure and applied topologies, digital geometry, mathematical morphology, and image processing and image classification in computer science. The present paper only deals with k-connected spaces in DTC. Moreover, we will mainly deal with a set X such that X ♯ ≥ 2 .

scholarly journals Fixed Point Sets of Digital Curves and Digital Surfaces

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1896
Author(s):  
Sang-Eon Han
Keyword(s):  
Fixed Point ◽  
Digital Image ◽  
Digital Object ◽  
Point Sets ◽  
Surface Theory ◽  
Digital Curve ◽  
Fixed Point Sets

Given a digital image (or digital object) (X,k), we address some unsolved problems related to the study of fixed point sets of k-continuous self-maps of (X,k) from the viewpoints of digital curve and digital surface theory. Consider two simple closed k-curves with li elements in Zn, i∈{1,2},l1⪈l2≥4. After initially formulating an alignment of fixed point sets of a digital wedge of these curves, we prove that perfectness of it depends on the numbers li,i∈{1,2}, instead of the k-adjacency. Furthermore, given digital k-surfaces, we also study an alignment of fixed point sets of digital k-surfaces and digital wedges of them. Finally, given a digital image which is not perfect, we explore a certain condition that makes it perfect. In this paper, each digital image (X,k) is assumed to be k-connected and X♯≥2 unless stated otherwise.

scholarly journals Discrete Group Actions on Digital Objects and Fixed Point Sets by Isok(·)-Actions

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 290
Author(s):  
Sang-Eon Han
Keyword(s):  
Fixed Point ◽  
Digital Image ◽  
Discrete Group ◽  
Dihedral Group ◽  
Group Structure ◽  
Topological Invariant ◽  
Digital Object ◽  
Point Sets ◽  
Digital Objects ◽  
Fixed Point Sets

Given a digital image (or digital object) (X,k),X⊂Zn, this paper initially establishes a group structure of the set of self-k-isomorphisms of (X,k) with the function composition, denoted by Isok(X) or Autk(X). In particular, let Ckn,l be a simple closed k-curve with l elements in Zn. Then, the group Isok(Ckn,l) is proved to be isomorphic to the standard dihedral group Dl with order l. The calculation of this quantity Isok(Ckn,l) is a key step for obtaining many new results. Indeed, it is essential for exploring many features of Isok(X). Furthermore, this quantity is proved to be a digital topological invariant. After proceeding with an Isok(X)-action on (X,k), we investigate some properties of fixed point sets by this action. Finally, we explore various features of fixed point sets by this action from the viewpoint of digital k-curve theory. This paper only deals with k-connected digital images (X,k) whose cardinality is equal to or greater than 2. Besides, we discuss some errors that have appeared in the lilterature.

scholarly journals The Most Refined Axiom for a Digital Covering Space and Its Utilities

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1868
Author(s):  
Sang-Eon Han
Keyword(s):  
Fixed Point ◽  
Digital Images ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Covering Space ◽  
Fundamental Groups ◽  
Crucial Step ◽  
Covering Map ◽  
Five Elements ◽  
Digital Covering

This paper is devoted to establishing the most refined axiom for a digital covering space which remains open. The crucial step in making our approach is to simplify the notions of several types of earlier versions of local (k0,k1)-isomorphisms and use the most simplified local (k0,k1)-isomorphism. This approach is indeed a key step to make the axioms for a digital covering space very refined. In this paper, the most refined local (k0,k1)-isomorphism is proved to be a (k0,k1)-covering map, which implies that the earlier axioms for a digital covering space are significantly simplified with one axiom. This finding facilitates the calculations of digital fundamental groups of digital images using the unique lifting property and the homotopy lifting theorem. In addition, consider a simple closed k:=k(t,n)-curve with five elements in Zn, denoted by SCkn,5. After introducing the notion of digital topological imbedding, we investigate some properties of SCkn,5, where k:=k(t,n),3≤t≤n. Since SCkn,5 is the minimal and simple closed k-curve with odd elements in Zn which is not k-contractible, we strongly study some properties of it associated with generalized digital wedges from the viewpoint of fixed point theory. Finally, after introducing the notion of generalized digital wedge, we further address some issues which remain open. The present paper only deals with k-connected digital images.

scholarly journals Fixed point sets in digital topology, 2

2020 ◽  
Vol 21 (1) ◽  
pp. 111
Author(s):  
Laurence Boxer
Keyword(s):  
Fixed Point ◽  
Digital Images ◽  
Digital Topology ◽  
Point Sets ◽  
Fixed Point Set ◽  
Point Set ◽  
Fixed Point 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 existence of a fixed point set for a continuous self-map restricts the map on the complement of the fixed point set.

Modeling the dynamics of hepatitis E via the Caputo–Fabrizio derivative

2019 ◽  
Vol 14 (3) ◽  
pp. 311 ◽  
Author(s):  
Muhammad Altaf Khan ◽  
Zakia Hammouch ◽  
Dumitru Baleanu
Keyword(s):  
Fixed Point ◽  
Existence And Uniqueness ◽  
Numerical Scheme ◽  
Arbitrary Order ◽  
Fixed Point Theory ◽  
Hepatitis E ◽  
Point Theory ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Essential Properties

A virus that causes hepatitis E is known as (HEV) and regarded on of the reason for lever inflammation. In mathematical aspects a very low attention has been paid to HEV dynamics. Therefore, the present work explores the HEV dynamics in fractional derivative. The Caputo–Fabriizo derivative is used to study the dynamics of HEV. First, the essential properties of the model will be presented and then describe the HEV model with CF derivative. Application of fixed point theory is used to obtain the existence and uniqueness results associated to the model. By using Adams–Bashfirth numerical scheme the solution is obtained. Some numerical results and tables for arbitrary order derivative are presented.

scholarly journals A new method in fixed point theory

1960 ◽  
Vol 34 (1) ◽  
pp. 1-16 ◽  
Author(s):  
Richard G. Swan
Keyword(s):  
Fixed Point ◽  
Fixed Point Theory ◽  
Point Theory ◽  
New Method
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