AppliedMath, Volume 5, Issue 3 (September 2025) – 47 articles

This work breaks a 180-year-old framework created by Hamilton both with regard to the use of imaginary quantities and the definition of a quaternion product. The general quaternionic algebraic structure we are considering was provided by the author in a previous work with a commutative product and will be provided here with a non-commutative product. We replace the imaginary units usually used in the theory of quaternions by linearly independent vectors and the usual Hamilton product rule by a Hamiltonian-adapted vector-valued vector product and prove both a new geometric property of this product and a vectorial adopted Euler type formula. Full article

Parameterized quantum circuits play a key role in quantum computing. Measuring the suitability of such a circuit for solving a class of problems is needed. One such promising measure is the expressivity of a circuit, which is defined in two main variants. The variant in focus of this contribution is the so-called dimensional expressivity, which measures the dimension of the submanifold of states produced by the circuit. Understanding this measure needs a lot of background from differential topology, which makes it hard to comprehend. In this article, we provide this background in a vivid as well as pedagogical manner. Especially, it strives towards being self-contained for understanding expressivity, e.g., the required mathematical foundations are provided, and examples are given. Also, the literature makes several statements about expressivity, the proofs of which are omitted or only indicated. In this article, we give proof for key statements from dimensional expressivity, sometimes revealing limits for generalizing them, and sketching how to proceed in practice to determine this measure. Full article

We study equations with real polynomials of arbitrary degree, such that each coefficient has a small, individual error; this may originate, for example, from imperfect measuring. In particular, we study the influence of the errors on the roots of the polynomials. The errors are modeled by imprecisions of Sorites type: they are supposed to be stable to small shifts. We argue that such imprecisions are appropriately reflected by (scalar) neutrices, which are convex subgroups of the nonstandard real line; examples are the set of infinitesimals, or the set of numbers of order $\epsilon$, where $\epsilon$ is a fixed infinitesimal. The Main Theorem states that the imprecisions of the roots are neutrices, and determines their shape. Full article

This paper investigates quiescent solitons in optical fibers and crystals, modeled by the complicated Ginzburg–Landau equation incorporating nonlinear chromatic dispersion and six self-phase modulation structures introduced by Kudryashov. The model is formulated with linear temporal evolution and analyzed using Lie symmetry methods. The study also identified parameter constraints under which solutions exist. Full article

This paper presents a non-intrusive method for estimating the distance between a camera and a human subject using a monocular vision system and statistically derived interpupillary distance (IPD) values. The proposed approach eliminates the need for individual calibration by utilizing average IPD values based on biological sex, enabling accurate, scalable distance estimation for diverse users. The algorithm, implemented in Python 3.12.11 using the MediaPipe Face Mesh framework, extracts pupil coordinates from facial images and calculates IPD in pixels. A sixth-degree polynomial calibration function, derived from controlled experiments using a uniaxial displacement system, maps pixel-based IPD to real-world distances across three intervals (20–80 cm, 80–160 cm, and 160–240 cm). Additionally, a geometric correction is applied to compensate for in-plane facial rotation. Experimental validation with 26 participants (15 males, 11 females) demonstrates the method’s robustness and accuracy, as confirmed by relative error analysis against ground truth measurements obtained with a Bosch GLM120C laser distance meter. Males exhibited lower relative errors across the intervals (3.87%, 4.75%, and 5.53%), while females recorded higher mean relative errors (6.0%, 6.7%, and 7.27%). The results confirm the feasibility of the proposed method for real-time applications in human–computer interaction, augmented reality, and camera-based proximity sensing. Full article

This paper presents the convergence analysis of a newly proposed algorithm for approximating solutions to split equality variational inequality and fixed point problems in real Hilbert spaces. We establish that, under reasonably mild conditions, specifically when the involved mappings are quasimonotone, uniformly continuous, and quasi-nonexpansive, the sequences generated by the algorithm converge strongly to a solution of the problem. Furthermore, we provide several numerical experiments to demonstrate the practical effectiveness of the proposed method and compare its performance with that of existing algorithms. Full article

In this paper, we present Lie symmetry analysis of a generalized (1+1)-dimensional porous medium equation characterized by parameters m and d. Through group classification, we examine how these parameters influence the Lie symmetry structure of the equation. Our analysis establishes conditions under which the equation admits either a three-dimensional or a five-dimensional Lie algebra. Using the obtained symmetry algebras, we construct optimal systems of one-dimensional subalgebras. Subsequently, we derive invariant solutions corresponding to each subalgebra, providing explicit formulas in relevant parameter regimes. These solutions deepen our understanding of the nonlinear diffusion processes modeled by porous medium equations and offer valuable benchmarks for analytical and numerical studies. Full article

This study deals with P-type iterative learning control (ILC) techniques for switched impulsive systems governed by composite fractional derivatives. The systems considered incorporate non-instantaneous impulses and an initial state offset, with the objective of accurately tracking time-varying reference trajectories over a finite time interval using a finite number of iterations. By implementing a P-type learning law integrated with an initial iteration mechanism, we derive sufficient conditions that guarantee the convergence of the tracking error. The effectiveness and robustness of the proposed control concepts are validated through a comprehensive illustrative example. Full article

Our research goal is to provide the mathematical guidance to enable any combination of “intelligent” machines, artificial intelligence (AI) and humans to be able to interact with each other in roles that form the structure of a team interdependently performing a team’s tasks. Our quantum-like model, representing one of the few, if only, mathematical models of interdependence, captures the tradeoffs in energy expenditures a team chooses as it consumes its available energy on its structure versus its performance, measured by the uncertainty (entropy) relationship generated. Here, we outline the support for our quantum-like model of uncertainty relations, our goals in this study, and our future plans: (i) Redundancy reduces interdependence. This first finding confirms the existence of interdependence in systems, both large and small. (ii) Teams with orthogonal roles perform best. This second finding is the root cause of humans, including scientists, being unable to appreciate the role of interdependence in “squeezing” states of teams. (iii) Cognitive reports may not equal behavior. The last finding allows us to tie our research together and to account for the absence of social scientists from leading the mathematical science of teams. In this article, we review the need for a mathematics for the future of team operations, the literature, the mathematics in our model of agents with full agency (viz., intelligent and interdependent), our hypothesis that freely organized teams enjoy significant advantages over command decision-making (CDM) systems, and results from the field. We close with future plans and a generalization about squeezing states to control interdependent systems. Full article

Our study tracks the development of 105 Roma children between 3 and 5 (median age: 51 months), enrolled in an NGO-aided developmental program. Each child undergoes pre- and post-assessment based on the Developmental Assessment of Young Children (DAYC), a standard tool used to track the progress in early childhood development and detect delays. Data are gathered from three sources, teacher, parent/caregiver and specialist, covering four developmental domains and adaptive behavior scale. There are subjective biases; however, in the post-assessment, the teachers’ and parents’ evaluations converge. The test results confirm significant improvement in all areas ($p<0.0001$), with the highest being in cognitive skills $32.2\%$ and the lowest being in physical development $14.4\%$. We also apply machine learning methods to impute missing data and predict the likely future progress for a given student in the program based on the initial input, while also evaluating the influence of environmental factors. Our weighted ensemble regression models are coupled with principal component analysis (PCA) and yield average coefficients of determination $R^2\approx 0.7$ for the features of interest. Also, we perform k-means clustering in the plane cognitive vs. social–emotional progress and consider the classification problem of predicting the group in which a given student would eventually be assigned to, with a weighted $F_1$-score of $0.83$ and a macro-averaged area under the curve (AUC) of $0.94$. This could be useful in practice for the optimized formation of study groups. We explore classification as a means of imputing missing categorical data too, e.g., education, employment or marital status of the parents. Our algorithms provide solutions with the $F_1$-score ranging from $0.92$ to $0.97$ and, respectively, an AUC between $0.99$ and 1. Full article

The 4-body equations of motion are derived in our previously published paper. Here we prove the existence–uniqueness of a periodic solution by applying the fixed-point method for a suitable introduced operator. To apply the fixed-point theorem, we need to derive appropriate analytical inequalities for the right-hand sides of the equations that ensure that the operator for periodic solutions maps the set of periodic functions into itself. In this way, we prove the existence of the Bohr–Sommerfeld orbits for the 4-body problem in the relativistic case. That allows us to estimate the minimal distances between the electrons on the first and second Bohr–Sommerfeld stationary states. A natural example of such a problem is the Lithium atom, which has three electrons orbiting the nucleus. Full article

The present study proposes a novel connection between the susceptible–infected–recovered $\left(SIR\right)$ model and genetic algebras through the construction of quadratic stochastic processes (QSPs). To determine the underlying dynamics, explicit formulations of the QSP are generated and their long-term behavior is examined under different cases. Moreover, we evaluate the attributes of the related limiting genetic algebras, focusing on fundamental characteristics like Rota–Baxter operators, automorphisms, and derivations. Full article

This paper proposes a mathematical model based on modified geometric progressions for supplementary retirement planning. Unlike traditional annuity models that assume fixed contributions and withdrawals, the proposed method incorporates inflation-indexed contributions and withdrawals. This allows for accurate simulations aligned with real-world financial behavior. The model has practical applicability in pension fund policy, personal financial planning tools, and governmental simulations. A case study is developed, demonstrating that with a 3% annual geometric annuity and a 0.5% monthly interest rate, an initial deposit of R$ 767.67 over 25 years results in a monthly retirement income of R$ 3049.19 for 30 years, with preserved purchasing power. The model offers a practical and realistic tool for individual retirement planning and paves the way for future applications in both public and private pension systems. Full article

This paper introduces the Intensional Conceptualization Model for Open Environments (ICMOE), a formal framework designed to enable semantic integration in dynamic and distributed systems. Grounded in intensional logic and formalized via a domain-specific language (ICMOE-L) built on Description Logic (DL), the model distinguishes between intensional and extensional semantics, allowing structured representation and evolution of concepts, relations, and domain rules under the open world assumption. ICMOE supports advanced semantic reasoning through an interpretation function that bridges relational data and ontological structures. A formal complexity analysis shows that reasoning with ICMOE-L has a worst-case complexity of O(n) ), where n is the total number of TBox and ABox axioms. To validate its effectiveness, ICMOE is evaluated using both qualitative and quantitative metrics. The model achieves a Concept Coverage score of 0.94, Semantic Depth of 0.89, Dynamic Adaptability Index of 0.91, Semantic Rule Density of 0.85, and Ontology Alignment Efficiency of 0.88. These results demonstrate ICMOE’s superior scalability, semantic richness, and adaptability when compared to foundational models such as those by Guarino and Bealer—making it a robust solution for open distributed environments. Full article

This research paper delves into the application of optimality results in orthogonal fuzzy metric spaces to demonstrate the existence and uniqueness of solutions of nonlinear differential equations with boundary conditions and nonlinear integral equations, emphasizing the importance of orthogonal fuzzy metric spaces in extending fixed-point theory. Through introducing this innovative concept, the study provides a theoretical framework for analyzing mappings in diverse scenarios. In this study, we introduce the concept of best proximity point (BPP) within the framework of orthogonal fuzzy metric spaces by employing orthogonal fuzzy proximal contractive mappings. Moreover, this research explores the implications of the established results, considering both self-mappings and non-self mappings that share the same parameter set. Additionally, some examples are provided to illustrate the practical relevance of the proven results and consequences in various mathematical contexts. The findings of this study can open up avenues for further exploration and application in solving real-world problems. Full article

The application of contagion risk spread modeling within the banking sector is a relatively recent development, emerging as a response to the persistent threat of liquidity risk that has affected financial institutions globally. Liquidity risk is recognized as one of the most destructive financial threats to banks, capable of causing severe and irreparable damage if overlooked or underestimated. This study aims to identify the most effective control strategy for managing financial contagion using a Susceptible–Infected–Recovered (SIR) epidemic model, incorporating time delays in both state and control variables. The proposed strategy seeks to maximize the number of resilient (vulnerable) banks while minimizing the number of infected institutions at risk of bankruptcy. Our goal is to formulate intervention policies that can curtail the propagation of financial contagion and mitigate associated systemic risks. Our model remains a simplification of reality. It does not account for strategic interactions between banks (e.g., panic reactions, network coordination), nor for adaptive regulatory mechanisms. The integration of these aspects will be the subject of future work. We establish the existence of an optimal control strategy and apply Pontryagin’s Maximum Principle to characterize and analyze the control dynamics. To numerically solve the control system, we employ a discretization approach based on forward and backward finite difference approximations. Despite the model’s simplifications, it captures key dynamics relevant to major European banks. Simulations performed using Python 3.12 yield significant results across three distinct scenarios. Notably, in the most severe case (${\alpha }_3=1.0$), the optimal control strategy reduces bankruptcies from 25% to nearly 0% in Spain, and from 12.5% to 0% in France and Germany, demonstrating the effectiveness of timely intervention in containing financial contagion. Full article

This paper introduces a new unit-Burr distortion function constructed via a transformation of the Burr random variable. The distortion can be applied to existing base copulas to create new copula families. The relationships of tail dependence coefficients and tail orders between the base bivariate copula and the unit-Burr distorted copula are derived. The unit-Burr distortion-induced family of copulas includes well-known copula classes, such as the BB1, BB2, and BB4 copulas, as special cases. The unit-Burr distortion of existing bivariate copulas may result in a family of copulas with both lower and upper tail coefficients ranging from 0 to 1. An empirical application to the rates of return for Microsoft and Google stocks is presented. Full article

There is an increasing demand for robust methodologies to rigorously evaluate the psychometric properties of measurement scales used in quantitative research across various scientific disciplines. This article proposes an integrative method that combines structural equation modelling (SEM) with machine learning (ML) to jointly assess model fit and predictive accuracy, limitations often addressed separately in traditional approaches. Using a measurement scale for voluntary employee turnover intention, the method demonstrates clear improvements: RMSEA decreased from 0.073 to 0.065, and classifier accuracy slightly increased from 0.862 to 0.863 after removing three redundant items. Compared to standalone SEM or ML, the integrated framework yields a shorter, better-fitting scale without compromising predictive power. For practitioners, this method enables the creation of more efficient, theoretically grounded, and predictive tools, facilitating faster and more accurate assessments in organisational settings. To this end, this study employs Covariance-Based SEM (CB-SEM) in conjunction with classifiers such as naive Bayes, linear and nonlinear support vector machines, decision trees, k-nearest neighbours, and logistic regression. Full article

Constitutive modeling in plasticity is a critical topic in solid mechanics. However, modeling nonlinear plasticity remains a challenge due to the theoretical complexity in representing realistic material behavior. This work aims to develop a general material model based on a rational polynomial function for plasticity and to use Bayesian optimization to identify its parameters. As a data-driven approach, Bayesian optimization effectively estimates model parameters for a high-computational-cost model. In this work, automotive structural steel is selected as a representative example to benchmark the proposed approach. Our results demonstrate that the rational polynomial function effectively models the plasticity behavior for metallic alloy before the failure point, and Bayesian optimization successfully estimates the parameters. This novel framework also has the potential to be applied to other materials, whose constitutive models can be defined by stress–strain curves. Full article

This work proposes a novel multi-objective genetic algorithm to solve the Periodic Vehicle Routing Problem with Time Windows (PVRPTWs) tailored for sales teams with diverse geographic scales and visit frequency requirements. Unlike existing models, our approach incorporates workload balancing and applies a clustering-based preprocessing step for long-distance routes using multidimensional scaling and fuzzy clustering, improving initial route grouping. When tested on three salesperson profiles (short-, mid-, and long-distance), the model achieved up to a 69% reduction in total travel time compared to a nearest neighbor baseline. These results demonstrate substantial improvements over existing methods and underscore the model’s flexibility and potential for extension to dynamic or real-time sales routing applications. Full article

This paper proposes a new index, the “distribution non-uniformity index (DNUI)”, for quantitatively measuring the non-uniformity or unevenness of a probability distribution relative to a baseline uniform distribution. The proposed DNUI is a normalized, distance-based metric ranging between 0 and 1, with 0 indicating perfect uniformity and 1 indicating extreme non-uniformity. It satisfies our axioms for an effective non-uniformity index and is applicable to both discrete and continuous probability distributions. Several examples are presented to demonstrate its application and to compare it with two distance measures, namely, the Hellinger distance (HD) and the total variation distance (TVD), and two classical evenness measures, namely, Simpson’s evenness and Buzas and Gibson’s evenness. Full article

The rising computational and energy demands of artificial intelligence systems urge the exploration of alternative software and hardware solutions that exploit physical effects for computation. According to machine learning theory, a neural network-based computational system must exhibit nonlinearity to effectively model complex patterns and relationships. This requirement has driven extensive research into various nonlinear physical systems to enhance the performance of neural networks. In this paper, we propose and theoretically validate a reservoir-computing system based on a single bubble trapped within a bulk of liquid. By applying an external acoustic pressure wave to both encode input information and excite the complex nonlinear dynamics, we showcase the ability of this single-bubble reservoir-computing system to forecast a Hénon benchmarking time series and undertake classification tasks with high accuracy. Specifically, we demonstrate that a chaotic physical regime of bubble oscillation—where tiny differences in initial conditions lead to wildly different outcomes, making the system unpredictable despite following clear rules, yet still suitable for accurate computations—proves to be the most effective for such tasks. Full article

The Statistics Anxiety Rating Scale (STARS) is a 51-item scale commonly used to measure college students’ anxiety regarding statistics. To date, however, limited empirical research exists that examines statistics anxiety among ethnically diverse or first-generation graduate students. We examined the factor structure and reliability of STARS scores in a diverse sample of students enrolled in graduate courses at a Minority-Serving Institution (n = 194). To provide guidance on assessing dimensionality in small college samples, we compared the performance of best-practice factor analysis techniques: confirmatory factor analysis (CFA), exploratory structural equation modeling (ESEM), and Bayesian structural equation modeling (BSEM). We found modest support for the original six-factor structure using CFA, but ESEM and BSEM analyses suggested that a four-factor model best captures the dimensions of the STARS instrument within the context of graduate-level statistics courses. To enhance scale efficiency and reduce respondent fatigue, we also tested and found support for a reduced 25-item version of the four-factor STARS scale. The four-factor STARS scale produced constructs representing task and process anxiety, social support avoidance, perceived lack of utility, and mathematical self-efficacy. These findings extend the validity and reliability evidence of the STARS inventory to include diverse graduate student populations. Accordingly, our findings contribute to the advancement of data science education and provide recommendations for measuring statistics anxiety at the graduate level and for assessing construct validity of psychometric instruments in small or hard-to-survey populations. Full article

This study explores the role of artificial intelligence (AI) in human resource management (HRM), with a focus on recruitment, employee retention, and performance optimization. Through a PRISMA-based systematic literature review, the paper examines many machine learning algorithms including XGBoost, SVM, random forest, and linear regression in decision-making related to employee-attrition prediction and talent management. The findings suggest that these technologies can automate HR processes, reduce bias, and personalize employee experiences. However, the implementation of AI in HRM also presents challenges, including data privacy concerns, algorithmic bias, and organizational resistance. To address these obstacles, the study highlights the importance of adopting ethical AI frameworks, ensuring transparency in decision-making, and developing effective integration strategies. Future research should focus on improving explainability, minimizing algorithmic bias, and promoting fairness in AI-driven HR practices. Full article

The objective of this investigation is to obtain numerical solutions for a variety of mathematical models in a wide range of disciplines, such as chemical kinetics, neurosciences, nonlinear optics, metallurgical separation/alloying processes, and asset dynamics in mathematical finance. This research features numerical simulations conducted with a remarkably low error measure, providing a visual representation of the examined models in these areas. The proposed method is the double Laplace–Adomian decomposition method, which facilitates the numerical acquisition and analysis of solutions. This paper presents the first report of numerical simulations employing this innovative methodology to address these problems. The findings are expected to benefit the natural sciences, mathematical modeling, and their practical applications, representing the innovative aspect of this article. Additionally, this method can analyze many classes of partial differential equations, whether linear or nonlinear, without the need for linearization or discretization. Full article

This paper develops a unified framework for mathematical finance under general semimartingale models that allow for dividend payments, negative asset prices, and unbounded jumps. We present a rigorous approach to the mathematical modeling of financial markets with dividend-paying assets by defining appropriate concepts of numéraires, discounted processes, and self-financing trading strategies. While most of the mathematical results are not new, this unified framework has been missing in the literature. We carefully examine the transition between nominal and discounted price processes and define appropriate notions of admissible strategies that work naturally in both settings. By establishing the equivalence between these models and providing clear conditions for their applicability, we create a mathematical foundation that encompasses a wide range of realistic market scenarios and can serve as a basis for future work on mathematical finance and derivative pricing. We demonstrate the practical relevance of our framework through a comprehensive application to dividend-paying equity markets where the framework naturally handles discrete dividend payments. This application shows that our theoretical framework is not merely abstract but provides the rigorous foundation for pricing derivatives in real-world markets where classical assumptions need extension. Full article

The Korteweg–de Vries (KdV) equation is a nonlinear evolution equation that reflects a wide variety of dispersive wave occurrences with limited amplitude. It has also been used to describe a range of major physical phenomena, such as shallow water waves that interact weakly and nonlinearly, acoustic waves on a crystal lattice, lengthy internal waves in density-graded oceans, and ion acoustic waves in plasma. The KdV equation is one of the most well-known soliton models, and it provides a good platform for further research into other equations. The KdV equation has several forms. The aim of this study is to introduce and investigate a (2+1)-dimensional generalized fifth-order KdV equation with power law nonlinearity (gFKdVp). The research methodology employed is the Lie group analysis. Using the point symmetries of the gFKdVp equation, we transform this equation into several nonlinear ordinary differential equations (ODEs), which we solve by employing different strategies that include Kudryashov’s method, the $(G^{\prime }/G)$ expansion method, and the power series expansion method. To demonstrate the physical behavior of the equation, 3D, density, and 2D graphs of the obtained solutions are presented. Finally, utilizing the multiplier technique and Ibragimov’s method, we derive conserved vectors of the gFKdVp equation. These include the conservation of energy and momentum. Thus, the major conclusion of the study is that analytic solutions and conservation laws of the gFKdVp equation are determined. Full article

This paper addresses the numerical evaluation of highly oscillatory integrals by developing a spectral Galerkin–Levin approach that efficiently solves Levin’s differential equation formulation for such integrals. The method employs Legendre polynomials as basis functions to approximate the solution, leveraging their orthogonality and favorable numerical properties. A key finding is that the Galerkin–Levin formulation is invariant with respect to the choice of polynomial basis—be it monomials or classical orthogonal polynomials—although the use of Legendre polynomials leads to a more straightforward derivation of practical quadrature rules. Building on this, this paper derives a simple and efficient numerical quadrature for both scalar and matrix-valued highly oscillatory integrals. The proposed approach is computationally stable and well-conditioned, overcoming the limitations of collocation-based methods. Several numerical examples validate the method’s high accuracy, stability, and computational efficiency. Full article

Hydropressor systems are of paramount importance in keeping water supplies running properly. A typical such device consists of two (or more) identical electropumps operating alternately, so as to avoid downtime as much as possible. We consider a dual pump configuration to identify the ideal usage proportion of each pump (from 0%-100%, meaning interchange only upon failure, to 50%-50%, where each pump works half the time) in order to improve availability, accounting solely for corrective maintenance. We also address the possibility of improving the availability of a single pump under the hazard of failure in three different ways (with their own occurrence frequencies), while also accounting for preventive maintenance. Both settings are tackled through Monte Carlo simulation and the models are implemented with the Python 3.12 programming language. The results indicate that significant improvements to standard industry practices can be made. Full article

We present a formal framework for modeling neural network dynamics using Category Theory, specifically through Markov categories. In this setting, neural states are represented as objects and state transitions as Markov kernels, i.e., morphisms in the category. This categorical perspective offers an algebraic alternative to traditional approaches based on stochastic differential equations, enabling a rigorous and structured approach to studying neural dynamics as a stochastic process with topological insights. By abstracting neural states as submeasurable spaces and transitions as kernels, our framework bridges biological complexity with formal mathematical structure, providing a foundation for analyzing emergent behavior. As part of this approach, we incorporate concepts from Interacting Particle Systems and employ mean-field approximations to construct Markov kernels, which are then used to simulate neural dynamics via the Ising model. Our simulations reveal a shift from unimodal to multimodal transition distributions near critical temperatures, reinforcing the connection between emergent behavior and abrupt changes in system dynamics. Full article