Advances in Mathematics and Its Applications

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: closed (31 March 2023) | Viewed by 18329

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Guest Editor
Department of Mathematics, Ankara Hacı Bayram Veli University, 06900 Ankara, Turkey
Interests: applications of matrix theory; graph theory; number theory and mathematical education based on projects and competencies
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Mathematical modeling is an active area of applied mathematics.  At its beginning, engineers were the main practitioners of this area of mathematics, developing mathematical models for solving engineering problems in natural sciences.

However, analysis methods and models in social sciences are similar to those of nature sciences including engineering, with the only difference that instead of using principles of the nature, one uses principles or theories from experts of such social sciences.

Models based on ordinary or partial differential equations describe a wide variety of phenomena such as sound, heat, diffusion, electrostatics, electrodynamics, fluid dynamics, elasticity, or quantum mechanics, for example.  Additionally, stochastic models have recently attracted increased attention. Obviously, some of these types of complex problems also require a deep analysis of the tools utilized to solve these situations.

In this Special Issue, we try to integrate models, methods, and also applications, not only in the scope of traditional natural sciences, but also opening the scope to education and other social sciences. Theory and data-driven models, even in a synergy that gives rise to producing fertile, multidisciplinary, and hybrid models, can be considered. Potential topics include, but are not limited to:

  • Numerical and modelling analysis;
  • Optimization and evolutionary algorithms models and methods;
  • Deterministic differential equations: methods and models;
  • Random differential equations: methods and models;
  • Numerical solution of large systems of linear and nonlinear equations;
  • Educational models and methods;
  • Social networks models and methods;
  • Engineering models and simulation;
  • Analysis modelling in economics and finance;
  • Algebraic models and methods with applications;
  • Intelligent data analysis models and methods.

Before submission, authors should carefully read over the journal’s Author Guidelines, which are located at:

https://www.mdpi.com/journal/axioms/instructions

Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at:

https://susy.mdpi.com/

according to the timetable.

Dr. Jesús Martín Vaquero
Prof. Dr. Deolinda M. L. Dias Rasteiro
Prof. Dr. Araceli Queiruga-Dios
Dr. Fatih Yilmaz
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. Axioms 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.

Published Papers (10 papers)

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Research

21 pages, 376 KiB  
Article
Periodic and Almost Periodic Solutions of Stochastic Inertial Bidirectional Associative Memory Neural Networks on Time Scales
by Mingshuo Liu, Huanhe Dong, Yong Zhang and Yong Fang
Axioms 2023, 12(6), 574; https://doi.org/10.3390/axioms12060574 - 09 Jun 2023
Viewed by 1315
Abstract
The stochastic inertial bidirectional associative memory neural networks (SIBAMNNs) on time scales are considered in this paper, which can unify and generalize both continuous and discrete systems. It is of primary importance to derive the criteria for the existence and uniqueness of both [...] Read more.
The stochastic inertial bidirectional associative memory neural networks (SIBAMNNs) on time scales are considered in this paper, which can unify and generalize both continuous and discrete systems. It is of primary importance to derive the criteria for the existence and uniqueness of both periodic and almost periodic solutions of SIBAMNNs on time scales. Based on that, the criteria for their exponential stability on time scales are studied. Meanwhile, the effectiveness of all proposed criteria is demonstrated by numerical simulation. The above study proposes a new way to unify and generalize both continuous and discrete SIBAMNNs systems, and is applicable to some other practical neural network systems on time scales. Full article
(This article belongs to the Special Issue Advances in Mathematics and Its Applications)
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18 pages, 322 KiB  
Article
Discrete Analogue of Fishburn’s Fractional-Order Stochastic Dominance
by Hoover H. F. Yin, Xishi Wang, Hugo Wai Leung Mak, Chun Sang Au Yong and Ian Y. Y. Chan
Axioms 2023, 12(6), 564; https://doi.org/10.3390/axioms12060564 - 07 Jun 2023
Viewed by 685
Abstract
A stochastic dominance (SD) relation can be defined by two different perspectives: One from the view of distributions, and the other one from the view of expected utilities. In the early days, Fishburn investigated SD from the view of distributions, and we refer [...] Read more.
A stochastic dominance (SD) relation can be defined by two different perspectives: One from the view of distributions, and the other one from the view of expected utilities. In the early days, Fishburn investigated SD from the view of distributions, and we refer this perspective as Fishburn’s SD. One of his many results was the development of fractional-order SD for continuous distributions. However, discrete fractional-order SD cannot be directly generalized, because some properties of fractional calculus may not possess a discrete counterpart. In this paper, we develop a discrete analogue of fractional-order SD for discrete utilities from the view of distributions. We generalize the order of SD by Lizama’s fractional delta operator, show the preservation of SD hierarchy, and formulate the utility classes that are congruent with our SD relations. This work brings a message that some results of discrete SD cannot be directly generalized from continuous SD. We characterize the difference between discrete and continuous fractional-order SD, as well as the way to handle it for further applications in mathematics and computer science. Full article
(This article belongs to the Special Issue Advances in Mathematics and Its Applications)
14 pages, 313 KiB  
Article
Second-Order Multiparameter Problems Containing Complex Potentials
by Ibrahim Erdal and Ekin Uğurlu
Axioms 2022, 11(12), 706; https://doi.org/10.3390/axioms11120706 - 08 Dec 2022
Viewed by 805
Abstract
In this work, we provide some lower bounds for the number of squarly integrable solutions of some second-order multiparameter differential equations. To obtain the results, we use both Sims and Sleeman’s ideas and the results are some generalization of the known results. To [...] Read more.
In this work, we provide some lower bounds for the number of squarly integrable solutions of some second-order multiparameter differential equations. To obtain the results, we use both Sims and Sleeman’s ideas and the results are some generalization of the known results. To be more precise, we firstly construct the Weyl–Sims theory for the singular second-order differential equation with several spectral parameters. Then, we obtain some results for the several singular second-order differential equations with several spectral parameters. Full article
(This article belongs to the Special Issue Advances in Mathematics and Its Applications)
29 pages, 13638 KiB  
Article
Interval Type-2 Fuzzy Approach for Dynamic Parameter Adaptation in the Bird Swarm Algorithm for the Optimization of Fuzzy Medical Classifier
by Ivette Miramontes and Patricia Melin
Axioms 2022, 11(9), 485; https://doi.org/10.3390/axioms11090485 - 19 Sep 2022
Cited by 3 | Viewed by 1520
Abstract
Optimization is essential for applications since it can improve the results provided in different areas; for this task, it is beneficial to use soft computing techniques, such as bio-inspired algorithms. In addition, it has been shown that if dynamic parameter adaptation is applied [...] Read more.
Optimization is essential for applications since it can improve the results provided in different areas; for this task, it is beneficial to use soft computing techniques, such as bio-inspired algorithms. In addition, it has been shown that if dynamic parameter adaptation is applied to these algorithms, they can provide a better result. For this work, the main contribution is to carry out the dynamic parameter adaptation to the bird swarm algorithm using interval type-2 fuzzy systems to realize a new fuzzy bio-inspired algorithm. The design of the proposed fuzzy system consists of two inputs corresponding to the iterations and diversity. As outputs, it takes the values of C and S, which are parameters to be adjusted by the algorithm. Once the design and the experimentation are realized, they are divided into two study cases. The first consists of a set of complex functions of the Congress of Evolutionary Competition 2017. The second case study consists of optimizing the membership functions in a fuzzy system designed to provide the nocturnal blood pressure profile, which corresponds to a neuro-fuzzy hybrid model to obtain the risk of hypertension. Analyzing the 30 experiments performed in both case studies, we can observe that the results obtained are improved when compared with the original method and other proposed methodologies, achieving good results in the complex functions. In addition, the optimized fuzzy system will reach an average of 97% correct classification. Statistically, it can be concluded that there is significant evidence to affirm that the proposed method provides good results. Full article
(This article belongs to the Special Issue Advances in Mathematics and Its Applications)
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8 pages, 263 KiB  
Article
On the Generalized Gaussian Fibonacci Numbers and Horadam Hybrid Numbers: A Unified Approach
by Fatih Yılmaz and Mustafa Özkan
Axioms 2022, 11(6), 255; https://doi.org/10.3390/axioms11060255 - 26 May 2022
Cited by 4 | Viewed by 2029
Abstract
In this paper, we consider an approach based on the elementary matrix theory. In other words, we take into account the generalized Gaussian Fibonacci numbers. In this context, we consider a general tridiagonal matrix family. Then, we obtain determinants of the matrix family [...] Read more.
In this paper, we consider an approach based on the elementary matrix theory. In other words, we take into account the generalized Gaussian Fibonacci numbers. In this context, we consider a general tridiagonal matrix family. Then, we obtain determinants of the matrix family via the Chebyshev polynomials. Moreover, we consider one type of tridiagonal matrix, whose determinants are Horadam hybrid polynomials, i.e., the most general form of hybrid numbers. Then, we obtain its determinants by means of the Chebyshev polynomials of the second kind. We provided several illustrative examples, as well. Full article
(This article belongs to the Special Issue Advances in Mathematics and Its Applications)
7 pages, 258 KiB  
Article
Statistically Convergent Sequences in Intuitionistic Fuzzy Metric Spaces
by Banu Pazar Varol
Axioms 2022, 11(4), 159; https://doi.org/10.3390/axioms11040159 - 01 Apr 2022
Cited by 3 | Viewed by 1748
Abstract
In this paper, we introduce the concepts of statistical convergence and statistical Cauchy sequences with respect to the intuitionistic fuzzy metric spaces inspired by the idea of statistical convergence in fuzzy metric spaces. Then, we give useful characterizations for statistically convergent sequences and [...] Read more.
In this paper, we introduce the concepts of statistical convergence and statistical Cauchy sequences with respect to the intuitionistic fuzzy metric spaces inspired by the idea of statistical convergence in fuzzy metric spaces. Then, we give useful characterizations for statistically convergent sequences and statistically Cauchy sequences. Full article
(This article belongs to the Special Issue Advances in Mathematics and Its Applications)
11 pages, 285 KiB  
Article
On Solutions and Stability of Stochastic Functional Equations Emerging in Psychological Theory of Learning
by Ali Turab, Janusz Brzdęk and Wajahat Ali
Axioms 2022, 11(3), 143; https://doi.org/10.3390/axioms11030143 - 21 Mar 2022
Cited by 3 | Viewed by 1743
Abstract
We show how to apply the well-known fixed-point approach in the study of the existence, uniqueness, and stability of solutions to some particular types of functional equations. Moreover, we also obtain the Ulam stability result for them. The functional equations that we consider [...] Read more.
We show how to apply the well-known fixed-point approach in the study of the existence, uniqueness, and stability of solutions to some particular types of functional equations. Moreover, we also obtain the Ulam stability result for them. The functional equations that we consider can be used to explain various experiments in mathematical psychology and arise in a natural way in the stochastic approach to the processes of perception, learning, reasoning, and cognition. Full article
(This article belongs to the Special Issue Advances in Mathematics and Its Applications)
19 pages, 280 KiB  
Article
De Moivre’s and Euler Formulas for Matrices of Hybrid Numbers
by Mücahit Akbıyık, Seda Yamaç Akbıyık, Emel Karaca and Fatih Yılmaz
Axioms 2021, 10(3), 213; https://doi.org/10.3390/axioms10030213 - 06 Sep 2021
Cited by 6 | Viewed by 2760
Abstract
It is known that the hybrid numbers are generalizations of complex, hyperbolic and dual numbers. Recently, they have attracted the attention of many scientists. At this paper, we provide the Euler’s and De Moivre’s formulas for the 4×4 matrices associated with [...] Read more.
It is known that the hybrid numbers are generalizations of complex, hyperbolic and dual numbers. Recently, they have attracted the attention of many scientists. At this paper, we provide the Euler’s and De Moivre’s formulas for the 4×4 matrices associated with hybrid numbers by using trigonometric identities. Also, we give the roots of the matrices of hybrid numbers. Moreover, we give some illustrative examples to support the main formulas. Full article
(This article belongs to the Special Issue Advances in Mathematics and Its Applications)
15 pages, 266 KiB  
Article
A Parametric Type of Cauchy Polynomials with Higher Level
by Takao Komatsu
Axioms 2021, 10(3), 207; https://doi.org/10.3390/axioms10030207 - 30 Aug 2021
Viewed by 1363
Abstract
There are many kinds of generalizations of Cauchy numbers and polynomials. Recently, a parametric type of the Bernoulli numbers with level 3 was introduced and studied as a kind of generalization of Bernoulli polynomials. A parametric type of Cauchy numbers with level 3 [...] Read more.
There are many kinds of generalizations of Cauchy numbers and polynomials. Recently, a parametric type of the Bernoulli numbers with level 3 was introduced and studied as a kind of generalization of Bernoulli polynomials. A parametric type of Cauchy numbers with level 3 is its analogue. In this paper, as an analogue of a parametric type of Bernoulli polynomials with level 3 and its extension, we introduce a parametric type of Cauchy polynomials with a higher level. We present their characteristic and combinatorial properties. By using recursions, we show some determinant expressions. Full article
(This article belongs to the Special Issue Advances in Mathematics and Its Applications)
15 pages, 846 KiB  
Article
The Soliton Solutions for Some Nonlinear Fractional Differential Equations with Beta-Derivative
by Erdoğan Mehmet Özkan and Ayten Özkan
Axioms 2021, 10(3), 203; https://doi.org/10.3390/axioms10030203 - 26 Aug 2021
Cited by 15 | Viewed by 1850
Abstract
Nonlinear fractional differential equations have gained a significant place in mathematical physics. Finding the solutions to these equations has emerged as a field of study that has attracted a lot of attention lately. In this work, He’s semi-inverse variation method and the ansatz [...] Read more.
Nonlinear fractional differential equations have gained a significant place in mathematical physics. Finding the solutions to these equations has emerged as a field of study that has attracted a lot of attention lately. In this work, He’s semi-inverse variation method and the ansatz method have been applied to find the soliton solutions for fractional Korteweg–de Vries equation, fractional equal width equation, and fractional modified equal width equation defined by Atangana’s conformable derivative (beta-derivative). These two methods are effective methods employed to get the soliton solutions of these nonlinear equations. All of the calculations in this work have been obtained using the Maple program and the solutions have been replaced in the equations and their accuracy has been confirmed. In addition, graphics of some of the solutions are also included. The found solutions in this study have the potential to be useful in mathematical physics and engineering. Full article
(This article belongs to the Special Issue Advances in Mathematics and Its Applications)
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