scholarly journals Coalgebras on Digital Images

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2082
Author(s):  
Sunyoung Lee ◽  
Dae-Woong Lee
Keyword(s):  
Commutative Ring ◽  
Digital Images ◽  
Continuous Functions ◽  
Contravariant Functor ◽  
Fundamental Properties

In this article, we investigate the fundamental properties of coalgebras with coalgebra comultiplications, counits, and coalgebra homomorphisms of coalgebras over a commutative ring R with identity 1R based on digital images with adjacency relations. We also investigate a contravariant functor from the category of digital images and digital continuous functions to the category of coalgebras and coalgebra homomorphisms based on digital images via the category of unitary R-modules and R-module homomorphisms.

scholarly journals On the Digital Cohomology Modules

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1451 ◽  
Author(s):  
Dae-Woong Lee
Keyword(s):  
Commutative Ring ◽  
Digital Image ◽  
Finite Subset ◽  
Digital Images ◽  
Continuous Functions ◽  
Contravariant Functor ◽  
Adjacency Relation ◽  
The Relationship ◽  
Cohomology Modules

In this paper, we consider the digital cohomology modules of a digital image consisting of a bounded and finite subset of Zn and an adjacency relation. We construct a contravariant functor from the category of digital images and digital continuous functions to the category of unitary R-modules and R-module homomorphisms via the category of cochain complexes of R-modules and cochain maps, where R is a commutative ring with identity 1R. We also examine the digital primitive cohomology classes based on digital images and find the relationship between R-module homomorphisms of digital cohomology modules induced by the digital convolutions and digital continuous functions.

On Finite Part Integrals and Hadamard-Type Fractional Derivatives

2018 ◽  
Vol 13 (9) ◽  
Author(s):  
Li Ma ◽  
Changpin Li
Keyword(s):  
Fractional Derivative ◽  
Singular Integral ◽  
Fractional Derivatives ◽  
Continuous Functions ◽  
Finite Part ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous ◽  
Fundamental Properties ◽  
Logarithmic Series ◽  
The Right

This paper is devoted to investigating the relation between Hadamard-type fractional derivatives and finite part integrals in Hadamard sense; that is to say, the Hadamard-type fractional derivative of a given function can be expressed by the finite part integral of a strongly singular integral, which actually does not exist. Besides, our results also cover some fundamental properties on absolutely continuous functions, and the logarithmic series expansion formulas at the right end point of interval for functions in certain absolutely continuous spaces.

scholarly journals Digital fixed points, approximate fixed points, and universal functions

2016 ◽  
Vol 17 (2) ◽  
pp. 159 ◽  
Author(s):  
Laurence Boxer ◽  
Ozgur Ege ◽  
Ismet Karaca ◽  
Jonathan Lopez ◽  
Joel Louwsma
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Digital Images ◽  
Continuous Functions ◽  
Universal Functions ◽  
Approximate Fixed Point ◽  
Fixed Point Properties ◽  
Digitally Continuous ◽  
Approximate Fixed Point Property ◽  
The Relationship

A. Rosenfeld [23] introduced the notion of a digitally continuous function between digital images, and showed that although digital images need not have fixed point properties analogous to those of the Euclidean spaces modeled by the images, there often are approximate fixed point properties of such images. In the current paper, we obtain additional results concerning fixed points and approximate fixed points of digitally continuous functions. Among these are several results concerning the relationship between universal functions and the approximate fixed point property (AFPP).

scholarly journals The higher topological complexity in digital images

2020 ◽  
Vol 21 (2) ◽  
pp. 305
Author(s):  
Melih İs ◽  
İsmet Karaca
Keyword(s):  
Lower Bounds ◽  
Algebraic Topology ◽  
Digital Images ◽  
Topological Complexity ◽  
Fundamental Properties ◽  
The Future

Y. Rudyak develops the concept of the topological complexity TC(X) defined by M. Farber. We study this notion in digital images by using the fundamental properties of the digital homotopy. These properties can also be useful for the future works in some applications of algebraic topology besides topological robotics. Moreover, we show that the cohomological lower bounds for the digital topological complexity TC(X,κ) do not hold.

scholarly journals Compactness Property of Fuzzy Soft Metric Space and Fuzzy Soft Continuous Function

2021 ◽  
pp. 3031-3038
Author(s):  
Raghad I. Sabri
Keyword(s):  
Functional Analysis ◽  
Continuous Function ◽  
Metric Space ◽  
Metric Spaces ◽  
Continuous Functions ◽  
Compactness Property ◽  
Locally Compact ◽  
Fuzzy Metric Spaces ◽  
Fundamental Properties ◽  
Definition Of

      The theories of metric spaces and fuzzy metric spaces are crucial topics in mathematics.    Compactness is one of the most important and fundamental properties that have been widely used in Functional Analysis. In this paper, the definition of compact fuzzy soft metric space is introduced and some of its important theorems are investigated. Also, sequentially compact fuzzy soft metric space and locally compact fuzzy soft metric space are defined and the relationships between them are studied. Moreover, the relationships between each of the previous two concepts and several other known concepts are investigated separately. Besides, the compact fuzzy soft continuous functions are studied and some essential theorems are proved.

scholarly journals On the Digital Pontryagin Algebras

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 875 ◽  
Author(s):  
Sunyoung Lee ◽  
Yeonjeong Kim ◽  
Jeong-Eun Lim ◽  
Dae-Woong Lee
Keyword(s):  
Computer Science ◽  
Digital Images ◽  
Base Point ◽  
Fundamental Properties

In the current study, we explore digital homology modules, and investigate their fundamental properties on (pointed) digital images as one of the developments of symmetries. We also examine pointed digital Hopf spaces and base point preserving digital Hopf functions between pointed digital Hopf spaces with suitable digital multiplications, and explore the digital primitive homology classes, digital Pontryagin algebras on digital Hopf spaces as a symmetric phenomenon in mathematics and computer science.

scholarly journals L-Hollow modules

2019 ◽  
Vol 24 (7) ◽  
pp. 104
Author(s):  
Thaer Z. Khlaif ◽  
Nada K. Abdullah
Keyword(s):  
Commutative Ring ◽  
Strong Form ◽  
Fundamental Properties ◽  
Maximal Submodule

To consider R is a commutative ring with unity,  be a nonzero unitary left   R-module,  is known hollow module if each proper submodule of  is small.  L-hollow module is a strong form of hollow module, where an R-module  is known L-hollow module if  has a unique maximal submodule which contains each small submodules. The current study deals with this class of modules and give several fundamental properties  related with this concept.   http://dx.doi.org/10.25130/tjps.24.2019.136

scholarly journals Continuous functions between sets with operations

2020 ◽  
Vol 24 (2) ◽  
pp. 225-239
Author(s):  
Fumie Nakaoka ◽  
Nobuyuki Oda
Keyword(s):  
Topological Space ◽  
Continuous Functions ◽  
Compact Spaces ◽  
Fundamental Properties ◽  
Connected Spaces

A set with an operation is a generalization of a topological space. Two types of continuous functions are dened between sets with operations. They are characterized making use of two types of closures and interiors. Homeomorphisms between sets with operations are also characterized. Variants of subspaces, connected spaces and compact spaces are introduced in a set with an operation and some fundamental properties of them are proved.

scholarly journals Digital homotopy relations and digital homology theories

2021 ◽  
Vol 22 (2) ◽  
pp. 223
Author(s):  
P. Christopher Staecker
Keyword(s):  
Digital Images ◽  
Continuous Functions ◽  
Homology Theory ◽  
Simplicial Homology ◽  
Digitally Continuous ◽  
New Type ◽  
The Difference ◽  
Cubical Homology

In this paper we prove results relating to two homotopy relations and four homology theories developed in the topology of digital images.<br /><br />We introduce a new type of homotopy relation for digitally continuous functions which we call ``strong homotopy.'' Both digital homotopy and strong homotopy are natural digitizations of classical topological homotopy: the difference between them is analogous to the difference between digital 4-adjacency and 8-adjacency in the plane.<br /><br />We also consider four different digital homology theories: a simplicial homology theory by Arslan et al which is the homology of the clique complex, a singular simplicial homology theory by D. W. Lee, a cubical homology theory by Jamil and Ali, and a new kind of cubical homology for digital images with $c_1$-adjacency which is easily computed, and generalizes a construction by Karaca \&amp; Ege. We show that the two simplicial homology theories are isomorphic to each other, but distinct from the two cubical theories.<br /><br />We also show that homotopic maps have the same induced homomorphisms in the cubical homology theory, and strong homotopic maps additionally have the same induced homomorphisms in the simplicial theory.

Sign in / Sign up

Export Citation Format

Share Document