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Axioms
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Advances in Mathematical Modeling and Related Topics
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Advances in Mathematical Modeling and Related Topics

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

Deadline for manuscript submissions: 31 May 2025 | Viewed by 7154

Special Issue Editor

Prof. Dr. En-Bing Lin

E-Mail Website
Guest Editor
School of Computing and Data Science, Wentworth Institute of Technology, Boston, MA 02132, USA
Interests: mathematical physics; complex geometry; applied and computational mathematics; wavelet analysis and applications
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Mathematical modeling is a powerful tool for understanding systems across engineering, biology, economics, environmental science, and more. We are excited to announce this Special Issue, entitled "Advances in Mathematical Modeling and Related Topics", showcasing the latest developments in this field. This Special Issue will provide a platform for researchers to present innovative models addressing complex real-world problems. It also offers opportunities to review themes, explore unaddressed aspects, propose new approaches, exchange perspectives, and inspire new research directions.

Topics of interest include, but are not limited to, the following: numerical methods for PDEs; dynamical systems; mathematical models of tumor growth; symmetry in nonlinear dynamics; AI and machine learning models; biomechanical systems modeling; stochastic processes in finance; optimization in industry applications; population dynamics in mathematical biology; topological data analysis; simulation in computational fluid dynamics; epidemiological models for public health; climate change mathematical predictions; quantum mechanics models; big data analysis techniques; rough set theory; bioinformatics for proteomics; formal concept analysis; fuzzy set theory and applications; granular computing in data analysis; wavelet-based image compression and denoising; rough-fuzzy hybrid models; wavelet analysis in time-series forecasting; supply chain optimization; game theory in economics; wave propagation in media; heat transfer models; quantum computing algorithms; signal processing techniques; image processing algorithms; elasticity and plasticity in materials; population genetics models; Bayesian inference applications; complex network dynamics; discrete mathematics applications; inverse problems in engineering; geophysical phenomena modeling; neural network optimization; materials science mathematical models; mathematical ecology; health care system modeling; and genomic data analysis.

We invite researchers specializing in these fields to submit their work for consideration. Contributions may be submitted on a rolling basis until the deadline and will undergo a peer-review process to ensure selection based on quality and relevance. 

Prof. Dr. En-Bing Lin
Guest Editor

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.

Keywords

  • mathematical modeling
  • numerical methods for PDEs
  • AI and machine learning models
  • wavelet analysis in time-series forecasting
  • mathematical models of tumor growth

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Published Papers (7 papers)

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Research

13 pages, 347 KiB  
Article
On the Dynamics of a Modified van der Pol–Duffing Oscillator
by Oscar A. R. Cespedes and Jaume Llibre
Axioms 2025, 14(4), 321; https://doi.org/10.3390/axioms14040321 - 21 Apr 2025
Viewed by 172
Abstract
The 3-dimensional modified van der Pol–Duffing oscillator has been studied by several authors. We complete its study, first characterizing its zero-Hopf equilibria and then its zero-Hopf bifurcations—i.e., we provide sufficient conditions for the existence of three, two or one periodic solutions, bifurcating from [...] Read more.
The 3-dimensional modified van der Pol–Duffing oscillator has been studied by several authors. We complete its study, first characterizing its zero-Hopf equilibria and then its zero-Hopf bifurcations—i.e., we provide sufficient conditions for the existence of three, two or one periodic solutions, bifurcating from the zero-Hopf equilibrium localized at the origin of coordinates. Recall that an equilibrium point of a 3-dimensional differential system whose eigenvalues are zero and a pair of purely imaginary eigenvalues is a zero-Hopf equilibrium. Finally, we determine the dynamics of this system near infinity, i.e., we control the orbits that escape to or come from the infinity. Full article
(This article belongs to the Special Issue Advances in Mathematical Modeling and Related Topics)
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24 pages, 8243 KiB  
Article
Emergence of Self-Identity in Artificial Intelligence: A Mathematical Framework and Empirical Study with Generative Large Language Models
by Minhyeok Lee
Axioms 2025, 14(1), 44; https://doi.org/10.3390/axioms14010044 - 7 Jan 2025
Viewed by 2957
Abstract
This paper introduces a mathematical framework for defining and quantifying self-identity in artificial intelligence (AI) systems, addressing a critical gap in the theoretical foundations of artificial consciousness. While existing approaches to artificial self-awareness often rely on heuristic implementations or philosophical abstractions, we present [...] Read more.
This paper introduces a mathematical framework for defining and quantifying self-identity in artificial intelligence (AI) systems, addressing a critical gap in the theoretical foundations of artificial consciousness. While existing approaches to artificial self-awareness often rely on heuristic implementations or philosophical abstractions, we present a formal framework grounded in metric space theory, measure theory, and functional analysis. Our framework posits that self-identity emerges from two mathematically quantifiable conditions: the existence of a connected continuum of memories C⊆M in a metric space (M,dM), and a continuous mapping I:M→S that maintains consistent self-recognition across this continuum, where (S,dS) represents the metric space of possible self-identities. To validate this theoretical framework, we conducted empirical experiments using the Llama 3.2 1B model, employing low-rank adaptation (LoRA) for efficient fine-tuning. The model was trained on a synthetic dataset containing temporally structured memories, designed to capture the complexity of coherent self-identity formation. Our evaluation metrics included quantitative measures of self-awareness, response consistency, and linguistic precision. The experimental results demonstrate substantial improvements in measurable self-awareness metrics, with the primary self-awareness score increasing from 0.276 to 0.801 (190.2% improvement) after fine-tuning. In contrast to earlier methods that view self-identity as an emergent trait, our framework introduces tangible metrics to assess and measure artificial self-awareness. This enables the structured creation of AI systems with validated self-identity features. The implications of our study are immediately relevant to the fields of humanoid robotics and autonomous systems. Additionally, it opens up new prospects for controlled adjustments of self-identity in contexts that demand different levels of personal involvement. Moreover, the mathematical underpinning of our framework serves as the basis for forthcoming investigations into AI, linking theoretical models to real-world applications in current AI technologies. Full article
(This article belongs to the Special Issue Advances in Mathematical Modeling and Related Topics)
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15 pages, 1807 KiB  
Article
A First Application of the Backward Technique in Social Sciences: Exploring Demographic Noise in a Model with Three Personality Types
by Roberto Macrelli, Margherita Carletti and Vincenzo Fano
Axioms 2025, 14(1), 9; https://doi.org/10.3390/axioms14010009 - 27 Dec 2024
Viewed by 541
Abstract
In the realm of dynamical systems described by deterministic differential equations used in biomathematical modeling, two types of random events influence the populations involved in the model: the first one is called environmental noise, due to factors external to the system; the second [...] Read more.
In the realm of dynamical systems described by deterministic differential equations used in biomathematical modeling, two types of random events influence the populations involved in the model: the first one is called environmental noise, due to factors external to the system; the second one is called demographic noise, deriving from the inherent randomness of the modeled phenomenon. When the populations are small, only space-discrete stochastic models are capable of describing demographic noise; when the populations are large, these discrete models converge to continuous models described by stochastic ordinary differential systems, maintaining the essence of intrinsic noise. Moving forward again from a continuous stochastic framework, we get to the continuous deterministic setting described by ordinary differential equations if we assume that noise can be neglected. The inverse process has recently been explored in the literature by means of the so-called “backward technique” in a biological context, starting from a system of continuous ordinary differential equations and going “backward” to the reconstruction and numerical simulation of the underlying discrete stochastic process, that models the demographic noise intrinsic to the biological phenomenon. In this study, starting from a predictable, deterministic system, we move beyond biology and explore the effects of demographic noise in a novel model arising from the social sciences. Our field will be psychosocial, that is, the connections and processes that support social relationships between individuals. We consider a group of individuals having three personality types: altruistic, selfish, and susceptible (neutral). Applying the backward technique to this model built on ordinary differential equations, we demonstrate how demographic noise can act as a switching factor, i.e., moving backward from the deterministic continuous model to the discrete stochastic process using the same parameter values, a given equilibrium switches to a different one. This highlights the importance of addressing demographic noise when studying complex social interactions. To our knowledge, this is also the first time that the backward technique has been applied in social contexts. Full article
(This article belongs to the Special Issue Advances in Mathematical Modeling and Related Topics)
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17 pages, 352 KiB  
Article
Parametric Inference in Biological Systems in a Random Environment
by Manuel Molina-Fernández and Manuel Mota-Medina
Axioms 2024, 13(12), 883; https://doi.org/10.3390/axioms13120883 - 20 Dec 2024
Viewed by 619
Abstract
This research focuses on biological systems with sexual reproduction in which female and male individuals coexist together, forming female–male couples with the purpose of procreation. The couples can originate new females and males according to a certain probability law. Consequently, in this type [...] Read more.
This research focuses on biological systems with sexual reproduction in which female and male individuals coexist together, forming female–male couples with the purpose of procreation. The couples can originate new females and males according to a certain probability law. Consequently, in this type of biological systems, two biological phases are involved: a mating phase in which the couples are formed, and a reproduction phase in which the couples, independently of the others, originate new offspring of both sexes. Due to several environmental factors of a random nature, these phases usually develop over time in a non-predictable (random) environment, frequently influenced by the numbers of females and males in the population and by the number of couples participating in the reproduction phase. In order to investigate the probabilistic evolution of these biological systems, in previous papers, by using a methodology based on branching processes, we had introduced a new class of two-sex mathematical models. Some probabilistic properties and limiting results were then established. Additionally, under a non-parametric statistical framework, namely, not assuming to have known the functional form of the offspring law, estimates for the main parameters affecting the reproduction phase were determined. We now continue this research line focusing the attention on the estimation of such reproductive parameters under a parametric statistical setting. In fact, we consider offspring probability laws belonging to the family of bivariate power series distributions. This general family includes the main probability distributions used to describe the offspring dynamic in biological populations with sexual reproduction. Under this parametric context, we propose accurate estimates for the parameters involved in the reproduction phase. With the aim of assessing the quality of the proposed estimates, we also determined optimal credibility intervals. For these purposes, we apply the Bayesian estimation methodology. As an illustration of the methodology developed, we present a simulated study about the demographic dynamics of Labord’s chameleon populations, where a sensitivity analysis on the prior density is included. Full article
(This article belongs to the Special Issue Advances in Mathematical Modeling and Related Topics)
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17 pages, 1633 KiB  
Article
Stochastic Models for Ontogenetic Growth
by Chau Hoang, Tuan Anh Phan and Jianjun Paul Tian
Axioms 2024, 13(12), 861; https://doi.org/10.3390/axioms13120861 - 9 Dec 2024
Viewed by 632
Abstract
Based on allometric theory and scaling laws, numerous mathematical models have been proposed to study ontogenetic growth patterns of animals. Although deterministic models have provided valuable insight into growth dynamics, animal growth often deviates from strict deterministic patterns due to stochastic factors such [...] Read more.
Based on allometric theory and scaling laws, numerous mathematical models have been proposed to study ontogenetic growth patterns of animals. Although deterministic models have provided valuable insight into growth dynamics, animal growth often deviates from strict deterministic patterns due to stochastic factors such as genetic variation and environmental fluctuations. In this study, we extend a general model for ontogenetic growth proposed by West et al. to stochastic models for ontogenetic growth by incorporating stochasticity using white noise. According to data variance fitting for stochasticity, we propose two stochastic models for ontogenetic growth, one is for determinate growth and one is for indeterminate growth. To develop a universal stochastic process for ontogenetic growth across diverse species, we approximate stochastic trajectories of two stochastic models, apply random time change, and obtain a geometric Brownian motion with a multiplier of an exponential time factor. We conduct detailed mathematical analysis and numerical analysis for our stochastic models. Our stochastic models not only predict average growth well but also variations in growth within species. This stochastic framework may be extended to studies of other growth phenomena. Full article
(This article belongs to the Special Issue Advances in Mathematical Modeling and Related Topics)
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27 pages, 383 KiB  
Article
Qualitative Analysis of Stochastic Caputo–Katugampola Fractional Differential Equations
by Zareen A. Khan, Muhammad Imran Liaqat, Ali Akgül and J. Alberto Conejero
Axioms 2024, 13(11), 808; https://doi.org/10.3390/axioms13110808 - 20 Nov 2024
Cited by 3 | Viewed by 849
Abstract
Stochastic pantograph fractional differential equations (SPFDEs) combine three intricate components: stochastic processes, fractional calculus, and pantograph terms. These equations are important because they allow us to model and analyze systems with complex behaviors that traditional differential equations cannot capture. In this study, we [...] Read more.
Stochastic pantograph fractional differential equations (SPFDEs) combine three intricate components: stochastic processes, fractional calculus, and pantograph terms. These equations are important because they allow us to model and analyze systems with complex behaviors that traditional differential equations cannot capture. In this study, we achieve significant results for these equations within the context of Caputo–Katugampola derivatives. First, we establish the existence and uniqueness of solutions by employing the contraction mapping principle with a suitably weighted norm and demonstrate that the solutions continuously depend on both the initial values and the fractional exponent. The second part examines the regularity concerning time. Third, we illustrate the results of the averaging principle using techniques involving inequalities and interval translations. We generalize these results in two ways: first, by establishing them in the sense of the Caputo–Katugampola derivative. Applying condition β=1, we derive the results within the framework of the Caputo derivative, while condition β→0+ yields them in the context of the Caputo–Hadamard derivative. Second, we establish them in Lp space, thereby generalizing the case for p=2. Full article
(This article belongs to the Special Issue Advances in Mathematical Modeling and Related Topics)
16 pages, 495 KiB  
Article
A Kinematic Approach to the Classical SIR Model
by Fernando Córdova-Lepe, Juan Pablo Gutiérrez-Jara and Katia Vogt-Geisse
Axioms 2024, 13(10), 718; https://doi.org/10.3390/axioms13100718 - 16 Oct 2024
Viewed by 758
Abstract
Given the risk and impact of infectious-contagious X diseases, which are expected to increase in frequency and unpredictability due to climate change and anthropogenic penetration of the wilderness, it is crucial to advance descriptions and explanations that improve the understanding and applicability of [...] Read more.
Given the risk and impact of infectious-contagious X diseases, which are expected to increase in frequency and unpredictability due to climate change and anthropogenic penetration of the wilderness, it is crucial to advance descriptions and explanations that improve the understanding and applicability of current theories. An inferential approach is to find analogies with better-studied contexts from which new questions and hypotheses can be raised through their concepts, propositions, and methods. Kinematics emerges as a promising analog field in physics by interpreting states’ changes in a contagion process as a movement. Consequently, this work explores, for a contagion process, the representations and conceptual equivalents for position, displacement, velocity, momentum, and acceleration, introducing some metrics. It also discusses some epistemological aspects and proposes future perspectives. Full article
(This article belongs to the Special Issue Advances in Mathematical Modeling and Related Topics)
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