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Symmetry
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Complex Variable in Approximation Theory
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Complex Variable in Approximation Theory

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (31 January 2022) | Viewed by 6416

Special Issue Editors

Prof. Dr. Paolo Emilio Ricci

E-Mail Website
Guest Editor
Section of Mathematics Uninettuno University, Roma, Italy
Interests: special functions; matrix functions; eigenvalues; differential and integral equations; number theory
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Clemente Cesarano

E-Mail Website
Guest Editor
Section of Mathematics, International Telematic University Uninettuno, 00186 Roma, Italy
Interests: special functions, orthogonal polynomials, differential equations, operator theory, multivariate approximation theory, Lie algebra
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear colleagues,

The close connection between the real and the complex variable is well known, not only for the closure of the complex field with respect to the roots of algebraic equations with real coefficients, which is proven in the so-called fundamental theorem of algebra, but also in many problems of mathematical analysis.

In fact, phenomena such as the length of the convergence radius of the McLaurin series expansion of the arctangent function or even the Runge phenomenon in the Lagrange interpolation over a set of equispaced points would be incomprehensible without the knowledge of the behavior of the considered functions in the complex plane.

The latter are only a few examples of the influence of the complex variable in the approximation problems of real functions.  Recently, the calculation of the roots of a non singular matrix with real or complex entires has been obtained using the Dunford–Taylor integral, a classic tool of functional analysis that extends Cauchy's integral formula for complex functions to the case of operators.

In the opinion of the Guest Editors, there are many other possibilities for the application of the use of complex analysis tools to solve problems of approximation of the real variable.

This Special Issue is intended to encourage scholars to submit their research in this interesting field of study.

Prof. Dr. Paolo Emilio Ricci
Prof. Dr. Clemente Cesarano
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • special functions
  • matrix functions
  • eigenvalues
  • differential and integral equations
  • number theory

Published Papers (3 papers)

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Research

9 pages, 229 KiB  
Article
Inversion of Tridiagonal Matrices Using the Dunford-Taylor’s Integral
by Diego Caratelli and Paolo Emilio Ricci
Symmetry 2021, 13(5), 870; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13050870 - 13 May 2021
Cited by 4 | Viewed by 1901
Abstract
We show that using Dunford-Taylor’s integral, a classical tool of functional analysis, it is possible to derive an expression for the inverse of a general non-singular complex-valued tridiagonal matrix. The special cases of Jacobi’s symmetric and Toeplitz (in particular symmetric Toeplitz) matrices are [...] Read more.
We show that using Dunford-Taylor’s integral, a classical tool of functional analysis, it is possible to derive an expression for the inverse of a general non-singular complex-valued tridiagonal matrix. The special cases of Jacobi’s symmetric and Toeplitz (in particular symmetric Toeplitz) matrices are included. The proposed method does not require the knowledge of the matrix eigenvalues and relies only on the relevant invariants which are determined, in a computationally effective way, by means of a dedicated recursive procedure. The considered technique has been validated through several test cases with the aid of the computer algebra program Mathematica©. Full article
(This article belongs to the Special Issue Complex Variable in Approximation Theory)
16 pages, 334 KiB  
Article
Fractional Reverse Coposn’s Inequalities via Conformable Calculus on Time Scales
by Mohammed Zakarya, Mohamed Altanji, Ghada AlNemer, Hoda A. Abd El-Hamid, Clemente Cesarano and Haytham M. Rezk
Symmetry 2021, 13(4), 542; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13040542 - 25 Mar 2021
Cited by 19 | Viewed by 1397
Abstract
This paper provides novel generalizations by considering the generalized conformable fractional integrals for reverse Copson’s type inequalities on time scales. The main results will be proved using a general algebraic inequality, chain rule, Hölder’s inequality, and integration by parts on fractional time scales. [...] Read more.
This paper provides novel generalizations by considering the generalized conformable fractional integrals for reverse Copson’s type inequalities on time scales. The main results will be proved using a general algebraic inequality, chain rule, Hölder’s inequality, and integration by parts on fractional time scales. Our investigations unify and extend some continuous inequalities and their corresponding discrete analogues. In addition, when α = 1, we obtain some well-known time scale inequalities due to Hardy, Copson, Bennett, and Leindler inequalities. Full article
(This article belongs to the Special Issue Complex Variable in Approximation Theory)
21 pages, 871 KiB  
Article
Bifurcation Analysis of Time-Delay Model of Consumer with the Advertising Effect
by Mahmoud A. Abd-Rabo, Mohammed Zakarya, Clemente Cesarano and Shaban Aly
Symmetry 2021, 13(3), 417; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13030417 - 04 Mar 2021
Cited by 5 | Viewed by 2026
Abstract
Given the economic importance of advertising and product promotions, we have developed a diffusion model to describe the impact of advertising on sales. The main message of this study is to show the effect of advertising diffusion to convert potential buyers into actual [...] Read more.
Given the economic importance of advertising and product promotions, we have developed a diffusion model to describe the impact of advertising on sales. The main message of this study is to show the effect of advertising diffusion to convert potential buyers into actual customers which may result in persistent alteration in marketing over time. This work is devoted to studying the dynamic behavior of a reaction-diffusion model and its delayed version with the advertising effect. For the non-delay model, it is proven the existence of Hopf bifurcation. Moreover, the stability and direction of bifurcation of periodic solutions are detected. On the other hand, we consider there is a lag for responding of potential buyers to the advertising. Therefore, the time delay τ is deemed as an additional factor in the diffusion model. We have determined the critical values for the delay parameter that yield periodic solutions. Furthermore, the direction and the stability of bifurcating periodic solutions is studied. For supporting the theoretical analysis and demonstrate complex dynamic behaviors, numerical simulations including families of periodic curves are given. Full article
(This article belongs to the Special Issue Complex Variable in Approximation Theory)
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