Approximation Theory in Computer and Computational Sciences

Topic Information

Dear Colleagues,

In recent years, we have observed increasing interest of Mathematicians in Approximation theory and computational methods, due to its large applications in Engineering sciences and related areas. Approximating functions, some data, or a member of a given set are some of the examples of the approximation calculations. From constructive proof of the Weierstrass approximation theorem via Bernstein polynomials to the theory of positive linear operators, role of Bernstein type polynomials to mimic shapes of curves and surfaces in Computer-Aided Geometric Design and its applications to finite element analysis, approximation by non-linear operators, Neural network approximation and the theory of sampling operators to overcome complexity of Mathematical models are some of the key areas. Approximation theory links both theoretical and applied Mathematics from a need to represent functions in computer calculations to Numerical analysis and development of mathematical software etc. Any development can be used in many industrial and commercial fields and thus requires advances in the subject.

In this Topic Issue, we will cover the field of approximations in Neural Networks, Linear and Non-linear approximation operators, Numerical analysis, Special function classes, Computer Aided Geometric design and applications of approximation theory. Our goal is to gather articles reflecting new trends in approximation theory and related topics.

Dr. Faruk Özger
Dr. Asif Khan
Dr. Syed Abdul Mohiuddine
Dr. Zeynep Ödemiş Özger
Topic Editors

Keywords

Participating Journals

Journal Name	Impact Factor	CiteScore	Launched Year	First Decision (median)	APC
Algorithms algorithms	1.8	4.1	2008	18.9 Days	CHF 1600
Axioms axioms	1.9	-	2012	22.8 Days	CHF 2400
Fractal and Fractional fractalfract	3.6	4.6	2017	23.7 Days	CHF 2700
Mathematics mathematics	2.3	4.0	2013	18.3 Days	CHF 2600
Symmetry symmetry	2.2	5.4	2009	17.3 Days	CHF 2400

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Published Papers (3 papers)

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All Algorithms Axioms Fractal Fract Mathematics Symmetry

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16 pages, 341 KiB  

Open AccessArticle

Finiteness of One-Valued Function Classes in Many-Valued Logic

by Elmira Yu. Kalimulina

Fractal Fract. 2024, 8(1), 29; https://doi.org/10.3390/fractalfract8010029 - 29 Dec 2023

Cited by 2 | Viewed by 1486

Abstract

This paper addresses the theoretical issues in k-valued logic, which are crucial for developing solutions in various fields of science and technology. One of the fundamental issues is a complete description of the closed classes of functions of three-valued logic. The explicit description [...] Read more.

This paper addresses the theoretical issues in k-valued logic, which are crucial for developing solutions in various fields of science and technology. One of the fundamental issues is a complete description of the closed classes of functions of three-valued logic. The explicit description of closed classes in multivalued logic is an open problem. In this study, we consider a special case of the finite generation of all closed classes of three-valued logic through the operation of superposition. Previously, we considered the issue of the finite generation of classes containing a subset of single-variable functions. We have also provided a description of superlattices (lattices of lattices) containing a precomplete class of unary functions. The finite generation of these superlattices is proved. On the basis of these results, in this paper, we have proven that any class containing any of the precomplete classes from the set of single-valued functions is also finitely generated. The main result of this paper consists of three theorems on the finite generation of classes containing precomplete classes of single-valued functions and classes including all monotone unary functions. Thus, the obtained theoretical result provides easily verifiable criteria for the finiteness of classes of multivalued logic functions. It allows you to use simple procedures instead of cumbersome explicit constructs. The finite generation of overlattices allows the development of digital computing circuits that are crucial for practical applications. The proofs are based on an explicit description of these classes by an induction in the number of variables and essentially use the properties of functionally closed (Burle) classes of functions. Full article

(This article belongs to the Topic Approximation Theory in Computer and Computational Sciences)

20 pages, 3754 KiB  

Open AccessArticle

Influence of Spatial Dispersal among Species in a Prey–Predator Model with Miniature Predator Groups

by Shivam, Turki Aljrees, Teekam Singh, Neeraj Varshney, Mukesh Kumar, Kamred Udham Singh and Vrince Vimal

Symmetry 2023, 15(5), 986; https://doi.org/10.3390/sym15050986 - 26 Apr 2023

Cited by 1 | Viewed by 2176

Abstract

Dispersal among species is an important factor that can govern the prey–predator model’s dynamics and cause a variety of spatial structures on a geographical scale. These structures form when passive diffusion interacts with the reaction part of the reaction–diffusion system in such a [...] Read more.

Dispersal among species is an important factor that can govern the prey–predator model’s dynamics and cause a variety of spatial structures on a geographical scale. These structures form when passive diffusion interacts with the reaction part of the reaction–diffusion system in such a way that even if the reaction lacks symmetry-breaking capabilities, diffusion can destabilize the symmetry and allow the system to have them. In this article, we look at how dispersal affects the prey–predator model with a Hassell–Varley-type functional response when predators do not form tight groups. By considering linear stability, the temporal stability of the model and the conditions for Hopf bifurcation at feasible equilibrium are derived. We explored spatial stability in the presence of diffusion and developed the criterion for diffusion-driven instability. Using amplitude equations, we then investigated the selection of Turing patterns around the Turing bifurcation threshold. The examination of the stability of these amplitude equations led to the discovery of numerous Turing patterns. Finally, numerical simulations were performed to validate the outcomes of the analysis. The outcomes of the theoretical study and numerical simulation were accorded. Our findings demonstrate that spatial patterns are sensitive to dispersal and predator death rates. Full article

(This article belongs to the Topic Approximation Theory in Computer and Computational Sciences)

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21 pages, 330 KiB  

Open AccessArticle

On the Approximation by Bivariate Szász–Jakimovski–Leviatan-Type Operators of Unbounded Sequences of Positive Numbers

by Abdullah Alotaibi

Mathematics 2023, 11(4), 1009; https://doi.org/10.3390/math11041009 - 16 Feb 2023

Cited by 6 | Viewed by 1416

Abstract

In this paper, we construct the bivariate Szász–Jakimovski–Leviatan-type operators in Dunkl form using the unbounded sequences ${\alpha }_n$, ${\beta }_m$ and ${\xi }_m$ of positive numbers. Then, we obtain the rate of convergence in terms of the weighted modulus of continuity of [...] Read more.

In this paper, we construct the bivariate Szász–Jakimovski–Leviatan-type operators in Dunkl form using the unbounded sequences ${\alpha }_n$, ${\beta }_m$ and ${\xi }_m$ of positive numbers. Then, we obtain the rate of convergence in terms of the weighted modulus of continuity of two variables and weighted approximation theorems for our operators. Moreover, we provide the degree of convergence with the help of bivariate Lipschitz-maximal functions and obtain the direct theorem. Full article

(This article belongs to the Topic Approximation Theory in Computer and Computational Sciences)

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