AppliedMath doi: 10.3390/appliedmath6070118

Authors: Ronobir Chandra Sarker Shelly Arora Atiqur Rahman Mahede- Ul-Hassan Sharandeep Singh Pandher

The generalized Burgers–Huxley (gBH) equation arises as a canonical model in nerve-pulse propagation (generalizing the Hodgkin–Huxley/FitzHugh–Nagumo excitable-media framework), in population dynamics with Allee-threshold reaction kinetics, and in nonlinear wave propagation in dispersive media; accurate benchmark solutions are essential for quantitative predictions in these domains. We couple the fourth-order exponential time differencing scheme ETDRK4 with a Chebyshev collocation spatial discretization and a linear boundary-lifting procedure to solve the gBH equation on a bounded interval with non-homogeneous Dirichlet data. On the canonical Ismail–Raslan–Rabboh travelling-wave benchmark the scheme attains L∞ errors at the level of floating-point round-off (∼10−19 absolute, ∼10−15 relative) with as few as N=2 collocation points and a single time step of size Δt=1.0—that is, three total nodes and one ETDRK4 advance. In strongly nonlinear regimes (γ=0.1, 0.3, 0.5, 0.9) the scheme exhibits approximately O(Δt2.45) temporal convergence across all four parameter values, consistent with the classical Hochbruck–Ostermann order reduction for exponential integrators on parabolic PDEs with non-homogeneous Dirichlet data. Used as a high-accuracy probe, the scheme provides a diagnostic of independent interest: the wave-speed formula of Wang, Zhu and Lu, still appearing as the exact-solution benchmark in numerical studies as recently as 2020, does not satisfy the partial differential equation. The corrected formula stated by Deng and verified symbolically by Appadu and Tijani is the unique value that makes the travelling-wave ansatz a genuine solution. We derive the residual associated with Wang’s formula in closed form, R=γA12(A2−A2W)(1−v2), and show both analytically and numerically that reported errors for schemes benchmarked against Wang’s formula coincide with the analytical wave-profile gap γA12|A2−A2W| rather than with true scheme accuracy. At the Ismail benchmark this gap equals 3.748×10−7, which matches the N- and Δt-independent plateau observed when the scheme is measured against Wang’s profile. In the nerve-pulse and excitable-media interpretation, the two formulas correspond to action-potential propagation speeds of opposite sign at the Ismail benchmark, underscoring that the correction is not a mere algebraic curiosity but changes the qualitative physical prediction of the model.

AppliedMath doi: 10.3390/appliedmath6070117

Authors: Yumu Lu Su Hoe Yeak

Many applications in finance and biology involve multiplicative noise geometric Brownian motion (GBM)-type stochastic differential equations (SDEs) whose drift carries a single dominant periodic component. Such structures arise in seasonal Black–Scholes option pricing, commodity derivatives with annual price cycles, and stochastic biological oscillators driven by a known frequency; the primary contribution of this paper is a high-precision numerical scheme validated on GBM-type test problems with periodic drift. This paper proposes a Strang operator splitting scheme within the Logarithmic Drift-Diffusion Splitting (LDDS) framework, which splits the SDE in y-space into a deterministic drift ODE sub-step (Step A) and an exactly solvable multiplicative diffusion sub-step (Step B). Step A employs the Periodically Fitted Adams–Bashforth–Moulton fourth-order predictor–corrector method (PABM4), which achieves zero local truncation error for trigonometric forcing terms by introducing additional shift terms and simultaneously imposing polynomial exactness conditions and trigonometric fitting conditions. When the Step A forcing belongs to the PABM4 exact function class Fω=span{1,t,t2,sinωt,cosωt}, the Strang+PABM4 scheme achieves floating-point precision saturation. We investigate three test problems: the cosine-drift GBM (Test Problem 1), the polynomial–trigonometric mixed drift GBM (Test Problem 2), and a dual-frequency drift applicability test (Test Problem 3). Monte Carlo strong error experiments (M=1000 paths) validate that Strang+PABM4 achieves saturation at machine precision (≈10−15) on Test Problems 1 and 2, improving precision by ≈102× over the best algebraically convergent reference. Test Problem 3 identifies the method’s applicability boundary: when the drift contains a second frequency outside Fω, Strang+PABM4 degrades gracefully to order ≈ 4 without catastrophic failure. The floating-point saturation of Strang+PABM4 is contingent on the drift belonging to F^ω; when this condition is violated, the method degrades gracefully to algebraic order ≈ 4, as demonstrated in Test Problem 3.

AppliedMath doi: 10.3390/appliedmath6070116

Authors: Darija Semenoja Minna Sinkkonen Patrik Eklund

The focus of this paper is the semi-quantitative approach to risk analysis in HACCP (Hazard Analysis and Critical Control Points), where we show how alternative computations in the risk matrix, combining probability and impact, can be provided based on algebraic and many-valued logical techniques. Doing so, we further show how applying such computations requires being formal concerning underlying information structures, which in turn enables being formal concerning functional representation of mappings between information structures appearing within risk analysis in risk management. Our mathematical algebraic framework also enables a more formal treatment of the duality between threat and opportunity.

AppliedMath doi: 10.3390/appliedmath6070115

Authors: Guojin Wang Wei Yao

Cruise ships, characterized by high density, enclosed environments and shared facilities, have become amplifiers for infectious disease outbreaks; however, predicting transmission in such settings remains challenging due to small initial case numbers. In this study, a mathematical modeling framework—including compartmental, small-world (SW), and scale-free (SF) network models—was developed to analyze the dynamics of hantavirus and norovirus transmission on recent cruise ship outbreaks. The results show that when the initial number of exposed or infected individuals is extremely small (e.g., 1–2 persons), disease spread is dominated by stochasticity, causing substantial variation between individual simulations and mean model predictions. Notably, SF networks exhibited lower transmission risk under these conditions, contradicting the theoretical expectation that when the population is large, SF networks facilitate the spread of disease. The compartmental model, while consistent with average simulation outcomes, failed to reliably predict any single outbreak event. In conclusion, for closed environments such as cruise ships with very few initial cases, stochastic variability is essential, and single-outbreak outcomes should be understood as highly contingent rather than deterministically predictable.

AppliedMath doi: 10.3390/appliedmath6070114

Authors: Bojana Stojčetović

This paper presents a procedure for constructing a matrix A⊗A* of a size 3×3 in max-plus algebra. The max-algebraic product of a matrix A and its conjugate (negative transpose) matrix A* always yields a pseudo skew-symmetric matrix. We address the inverse problem: given a matrix A˜ of this type, can one determine a matrix A such that A⊗A*=A˜? A particular case involving 3×3 matrices is examined, and one procedure is explained in detail. The method illustrates how such a matrix A can be constructed column by column while also indicating directions for further generalization and computational exploration.

AppliedMath doi: 10.3390/appliedmath6070113

Authors: Ioannis Kosmas Theofanis Papadopoulos Christos Michalakelis

The operationalization of machine learning (ML) introduces distinct engineering and lifecycle management challenges—such as extreme data dependence, silent model degradation (concept drift), and inherent non-determinism—which traditional software engineering workflows fail to adequately address. This systematic literature review provides a rigorous, comprehensive mapping of the ML lifecycle domain between 2015 and 2025 using the PRISMA protocol. Out of an initial pool of 12,450 articles, a highly specialized cohort of 22 primary studies was extracted, classified, and synthesized to map out contemporary Machine Learning Operations (MLOps) patterns, technical debt structures, governance models, and security vulnerabilities. To address the documented “production gap,” this paper formalizes the findings into a synthesized operational mapping and introduces a preliminary conceptual layout for an Adaptive Lifecycle Framework (ALF), juxtaposing it with legacy paradigms like CRISP-DM. Furthermore, we expand the scope to investigate domain-specific lifecycle complexities in healthcare systems and Large Language Model (LLM) pipelines, providing an essential evolutionary baseline for sustainable MLOps.

AppliedMath doi: 10.3390/appliedmath6070112

Authors: Ndivhuwo Ndou

This study presents a comparative numerical investigation of second-order and fourth-order Runge–Kutta methods for solving chaotic dynamical systems. The Lorenz, Genesio–Tesi, and Rössler systems are considered because of their nonlinear behavior and high sensitivity to initial conditions. The numerical schemes investigated include the Midpoint, Improved Euler, Ralston, and fourth-order Runge–Kutta (RK4) methods. The performance of the methods is evaluated in terms of convergence behavior, numerical accuracy, stability characteristics, and computational cost. A stability analysis of each chaotic system is carried out through equilibrium point determination and Jacobian eigenvalue analysis. Numerical simulations are implemented in MATLAB 2023 version, and comparisons are performed using different step sizes. The results indicate that all numerical methods converge as the step size decreases; however, the RK4 method consistently provides significantly smaller errors and improved stability properties compared with the second-order schemes. The findings further demonstrate that higher-order numerical integration methods provide superior performance for highly sensitive chaotic systems where accuracy and reliability are essential.

AppliedMath doi: 10.3390/appliedmath6070111

Authors: Yung-Ning Cheng Ching-Wen Yeh Kuo-Chen Hung

This paper solves the open problem mentioned by Lin that was published in 2026 of Algorithms to prove the uniqueness property for the solution procedure proposed by Lin. The developed mathematical verification fulfills the research gap left by Lin. Moreover, for the local maximum point near the starting point (denoted as 3×10−7) of the amelioration inventory model, this study presents an alternative explanation to help researchers execute their numerical methods with confidence such that, without the analytical procedure developed by Lin, only a rough numerical method can help researchers locate two local maximum points.

AppliedMath doi: 10.3390/appliedmath6070110

Authors: Jiazheng Du Bing Lin Hongling Ye Zhichao Guo

To address jagged boundaries in conventional fixed-mesh topology optimization and the discontinuous transfer of topology information after remeshing, this study proposes an improved Movable Morphable Smooth-Boundary (MMSB) method for displacement-constrained continuum topology optimization. A topology optimization model is established using the Independent Continuous Mapping (ICM) method, with structural weight minimization as the objective and displacement as the constraint. In the proposed framework, a triangular mesh is adopted as the current analysis mesh, threshold boundary points are identified using a holographic scanning strategy, and a fixed background mesh is introduced as an intermediate carrier for topology-variable transfer before and after remeshing. Three numerical examples are used to validate the proposed method and to compare it with the original MMSB method. The results show that the proposed method produces clearer boundary representations and more distinct load-transfer paths. In Examples 1–3, the number of result analyses is reduced from 60 to 24, from 144 to 36, and from 112 to 24, respectively. The number of boundary movements is also reduced from 5 to 4, from 8 to 6, and from 7 to 4, respectively. Meanwhile, the final structural weights are 22.0 kg, 101.4 kg, and 1.64 kg, which are close to or slightly lower than those obtained by the original MMSB method. These results indicate that the proposed method improves topology-information continuity and boundary representation while maintaining structural performance.

AppliedMath doi: 10.3390/appliedmath6070109

Authors: Ahmed Nuino Omar Bahou Senhaji Yassine Mustapha Ez-zaiym Karim El Moutaouakil Savin Treanta

Complex-valued knowledge graph embedding (KGE) models like ComplEx effectively capture asymmetric relations but are fundamentally constrained by integer-order transformations. This restriction limits their ability to model multi-scale interactions, hierarchical correlations, and non-local semantic dependencies inherent in heterogeneous graphs. To address these limitations, this paper introduces FracComplEx, a novel fractional-order extension that embeds fractional calculus into the complex latent space. By leveraging fractional operators, the framework introduces non-local dynamics and memory-aware mechanisms to continuously generalize standard linear transformations. The core architecture employs a fractional-order parameter α as a controllable scaling mechanism that balances local relational details with global topology, optimizing representation smoothness and flexibility. We provide rigorous theoretical findings demonstrating that fractional transformations enhance the embedding’s expressive capacity, spectral characteristics, and perturbation robustness beyond conventional integer-order benchmarks. Extensive experiments on FB15k-237, WN18RR, and CoDEx-M establish the empirical superiority of FracComplEx, yielding significant improvements in Mean Reciprocal Rank (MRR) and Hits@K metrics over classical baselines, particularly under severe structural data sparsity.

AppliedMath doi: 10.3390/appliedmath6070108

Authors: Ahmadjan Muhammadhaji Hui Zhang

This study constructs a novel fractional-order eco-epidemiological predator–prey model, in which disease spreads among predators through environmental transmission, and both cooperative hunting behavior and disease latency delay are incorporated simultaneously. Different from classical integer-order predator–prey models, fractional derivative is adopted to describe the memory-dependent mechanism of ecological populations, and the infection can alter the hunting strategy of diseased predators. The existence, non-negativity, and boundedness of system solutions are proved theoretically. The local stability of all equilibrium points is analyzed, and the conditions for the occurrence of Hopf bifurcation induced by latency delay are derived. Numerical simulations further verify the theoretical results, and quantitatively reveal the separate and combined effects of the fractional order, cooperative hunting coefficient, and latency delay on the dynamical evolution of the population system.

AppliedMath doi: 10.3390/appliedmath6070107

Authors: Aayush Samant Alexander Y. Klimenko Yuanshen Lu Mayank Kumar

This study presents an ideal-cycle thermodynamic analysis of an advanced compressed air energy storage (A-CAES) system with single thermal energy storage (TES) and an external heat boost. The additional heat is represented by a solar heat source, although the analysis is equally applicable to other forms of externally supplied thermal energy. Following the classical thermodynamic approach used for ideal cycles such as the Brayton, Otto and Diesel cycles, the objective is to establish analytical relationships and performance bounds for the integrated system rather than to model a specific engineering configuration. Three principal performance measures are examined: the electrical round-trip coefficient of performance (CoP), the marginal thermal coefficient of performance associated with external heat addition, and the overall second-law efficiency. Closed-form analytical expressions are derived for these quantities under idealised but still practically relevant assumptions. The analysis identifies distinct operating regimes governed by the level of external heat input and establishes analytical transition conditions between them. It is shown that external heat addition can substantially increase the round-trip coefficient of performance and lead to high marginal heat-utilisation effectiveness. A rigorous upper bound on the second-law efficiency is also obtained from a complete-cycle exergy analysis, demonstrating consistency with the laws of thermodynamics. The results provide analytical insight into the fundamental thermodynamic structure of solar-assisted A-CAES systems and establish performance bounds that are independent of any particular engineering implementation.

AppliedMath doi: 10.3390/appliedmath6070106

Authors: Ibrahim Sadok Hennia Douini Saqer Abdullah Faqih Ramy A. Aldallal

The rapid growth of high-dimensional predictive models in science and industry has intensified the need for statistically rigorous interpretability tools. Although model-agnostic feature importance methods are widely used to explain black-box models, they lack formal uncertainty quantification, leading to unreliable conclusions in high-dimensional settings where spurious correlations are common. We propose a Bootstrap-of-Bootstrap (BoB) inference framework that enables valid uncertainty quantification and hypothesis testing for any model-agnostic feature importance measure. To overcome the high computational cost of nested resampling, we develop an efficient analytical approximation based on influence function theory. The proposed approach provides calibrated confidence intervals and a stability score for each feature, strengthening the statistical foundations of explainable AI. Simulation studies and real-world applications in cancer genomics and credit risk modeling demonstrate its effectiveness, providing reliable, auditable explanations for high-stakes decision-making.

AppliedMath doi: 10.3390/appliedmath6070105

Authors: Teng Long

Viscous effects play an important role in the mechanical characterization of soft biological tissues under high-strain-rate loading. Accurate modeling of these behaviors is important for impact biomechanics, injury prediction, and crash safety analysis, in which biological tissues may experience high-strain-rate deformation. To describe the dynamic mechanical responses of soft tissues, a reliable constitutive framework is therefore needed to represent the dynamic response of soft tissues under high-strain-rate loading. The objective of this study is to develop and evaluate a viscohyperelastic constitutive framework for describing the dynamic compressive responses of multiple soft tissues. The proposed formulation is constructed within a continuum mechanics framework, in which the viscous contribution is expressed using objective invariant functions, namely J2, J6, and J7. The developed analytical formulations are calibrated against high-strain-rate experimental data from different soft biological tissues, namely porcine meniscus, bovine liver, and ovine brain tissues. To find the material model parameters, genetic algorithm optimization is used to identify the material parameters and assess the robustness of the fitting procedure. In order to assess the robustness of the proposed constitutive framework across different loading rates, a multi-objective optimization strategy is used to calibrate the model parameters by fitting multiple strain-rate-dependent responses at the same time. This approach enables the model predictive capability to be evaluated over a range of high-strain-rate conditions. These results show that the proposed framework can reasonably describe the nonlinear and rate-dependent mechanical responses of different soft tissues under dynamic compression.

AppliedMath doi: 10.3390/appliedmath6070104

Authors: Sangsung Park Sunghae Jun

This study proposes a transformed two-part bootstrap confidence interval (TTB-CI) for zero-inflated count data. The method combines a standard zero-inflated mixture formulation, parametric bootstrap, and monotone transformations to improve inference for practically meaningful estimands, including the marginal mean, zero probability, and positive-part mean. Simulation studies under zero-inflated Poisson (ZIP) and zero-inflated negative binomial (ZINB) data-generating processes show that the proposed method maintains nominal or near-nominal coverage while reducing interval width, particularly for the positive-part mean. Compared with conventional Poisson- and negative binomial-based confidence intervals, the proposed TTB-CI provides a more favorable coverage and width tradeoff and yields more informative intervals for positive count inference. These results indicate that the proposed method offers a practical and efficient confidence interval framework for zero-inflated count data.

AppliedMath doi: 10.3390/appliedmath6070103

Authors: Janet Kez Mohamed Shenify Fokrul Alom Mazarbhuiya

Bipolar fuzzy sets provide an effective framework for representing both positive and negative aspects of information. The necessity of a mathematically rigorous and valid distance measure in bipolar fuzzy environments motivates us to introduce a new real-valued function on the set of bipolar fuzzy sets defined over both discrete and continuous universes of discourse. The proposed function is shown to define a valid metric on the set of bipolar fuzzy sets, as it satisfies all the metric axioms. The metric induced by the real-valued function is inspired by the Canberra distance, and it can effectively quantify the dissimilarity between bipolar fuzzy sets in a normalized and interpretable manner. The practical utility of the proposed metric is demonstrated in a pattern recognition problem, where it successfully recognizes an unknown pattern using known bipolar fuzzy patterns. Using the proposed metric, a bipolar fuzzy C-means clustering algorithm is developed for sentiment analysis. The time complexity of the aforementioned algorithm is also analysed. Experiments conducted on the IMDb Movie Review Dataset demonstrate that the proposed algorithm outperforms k-means, fuzzy C-means, and intuitionistic fuzzy C-means algorithms. The proposed bipolar fuzzy C-means algorithm achieves an accuracy of 90.04%, a precision of 90.51%, a recall of 89.01%, an F1-score of 89.75%, a Root mean square error of 0.1191, and a Silhouette score of 0.75. The findings establish that the proposed metric and the associated bipolar fuzzy clustering approach provide a robust and effective framework of handling sentiment data associated with simultaneous positive and negative opinions.

AppliedMath doi: 10.3390/appliedmath6070102

Authors: Danail Brezov

Reference Replacement [...]

AppliedMath doi: 10.3390/appliedmath6070101

Authors: Yasser A. Davizon Alexander Mendoza-Acosta Rafael García-Martinez Aureliano Quiñonez-Ruiz Jaime Sanchez-Leal Eric D. Smith Neale R. Smith

This technical note aims to present a theoretical analysis for cybertronics engineering (CE) for a class of dynamic supply chains for artificial intelligence (AI)-based products, services or hybrid solutions. The cybertronics-based analysis encompasses three classes of supply chains: (1) energy-based dynamic supply chains (DSC); (2) semiconductor manufacturing and quantum supply chains; and (3) retailing of the AI-based DSC solutions generated. Considering the nonlinear nature of DSC, to provide solutions for products and services based on AI-chained supply chains, novel robust control is addressed via sliding mode control (SMC) with proper stability analysis for the DSC.

AppliedMath doi: 10.3390/appliedmath6060100

Authors: Gabriela Vasileva Dilyana Karova Mariyan Milev Penko Mitev

We compare classical and modern regression models for next-day cryptocurrency forecasting on 14 USD-denominated coins across three liquidity tiers from 2018 through 2025, and we use the resulting panel to formally test three pre-specified hypotheses. The features are a strictly past-only 28-element set; the evaluation uses expanding-window walk-forward cross-validation with nested hyperparameter tuning, stationary block-bootstrap 95% confidence intervals, and pairwise Diebold–Mariano tests. Methodologically, we derive a bias-variance bound that turns the ‘no model beats the mean’ observation from a null finding into a predicted outcome under weak-form market efficiency. Empirically, (H1) the threshold–effect interaction is not supported (slope −1.7 × 10−4, 95% CI [−4.8 × 10−4, +1.4 × 10−4], p = 0.25). (H2) Statistical loss minimisation is decoupled from risk-adjusted economic outcome: the cluster-bootstrapped 95% CI for the Spearman rank correlation between the within-ticker MAE rank and within-ticker post-cost Sharpe rank is [−0.39, +0.10] overall, lies *strictly below zero* on the mid-cap (CI [−0.71, −0.04]) and long-tail (CI [−0.26, −0.09]) tiers, and decisively rejects perfect alignment (ρ = +1) on every tier. None of the seven (ticker, model) pairs with annualised Sharpe ≥ 0.5 has a hit rate significantly different from 0.5; high-Sharpe outcomes reflect return skew, not directional skill—formally predicted by a closed-form Sharpe–MSE decoupling proposition we derive in Section 3.6 under non-zero return skewness. (H3) Lo–MacKinlay variance ratio tests show top-tier coins are indistinguishable from a random walk (|z| ≤ 1.5 at q ∈ {2, 5, 10}), while mid- and long-tail tiers reject the random-walk null at q = 2 (z = −2.36, z = −2.60). The findings extend across two robustness layers. An AR(1)-GARCH(1,1) baseline produces R2 ≈ −0.005 on every tier and is indistinguishable from Lasso, supporting the bias-variance bound; Giacomini–White conditional predictive ability tests reject equal predictive ability between Lasso and tree-based models on every coin in every tier, complicating naive DM interpretations; and a forward-walking 2026-Q1 holdout—83 daily observations per coin entirely outside the training window—confirms that H1 is even more decisively null on unseen data and that the H3 efficiency conclusion holds. Together, these results give a formally tested EMH-style picture for daily crypto: no model meaningfully forecasts log-returns; statistical accuracy and trading P&L are decoupled by an analytically derived mechanism; and weak-form efficiency is approximately satisfied in most liquid coins and in the convergence across the cross-section.

AppliedMath doi: 10.3390/appliedmath6060099

Authors: Darko Tomaš Bojan Baškot Andrej Ševa Dalibor Tomaš

This paper presents a methodological framework for anomaly detection in child benefit administration based on Long Short-Term Memory (LSTM) neural networks. The content of this analysis, in general, is situated within the social (S) pillar of the environmental, social, and governance (ESG) accountability framework. We construct a framework applied to 305,338 child allowance claim records from the Fund for Child Protection of Republika Srpska, Bosnia and Herzegovina (February 2017 to December 2025), construct behavioural and demographic features at the applicant and household level, encode sequential claim histories as three-dimensional tensors, and conduct a systematic architecture sweep across six LSTM configurations. The target variable, the guardianship anomaly flag, identifies 172 anomalous records (0.056%) among 305,338 claims, and yields a class weighting ration of approximately 1515:1. Across all six configurations, ROC-AUC values range from 0.706 to 0.870 and PR-AUC from 0.002 to 0.071. The reference configuration (L1_U10_T20_he_normal, ROC-AUC = 0.870) flags 170 applications (0.37% of the test set) for priority manual review at the operational audit threshold of τ=0.05. The highest-risk application identified (anomaly probability 0.935) is characterised by a four-child household with below-poverty declared income, elevated benefit-to-income ratios, home delivery payment method, and a persistent high-risk sequential claim pattern not previously flagged by the Fund’s rule-based administrative system. The results confirm that LSTM-based sequential anomaly detection is a viable and principled complement to rule-based eligibility screening in public social transfer administration.

AppliedMath doi: 10.3390/appliedmath6060098

Authors: Fatma Al-Musalhi Nasser Al-Salti Erkinjon Karimov

A non-homogeneous fractional differential equation with a variable coefficient involving a Caputo fractional derivative is considered. The equation is first transformed into an integral equation and then solved using the method of successive approximations. The obtained general solution involves a generalized Mittag–Leffler-type function and Meijer G-functions. Example solutions corresponding to particular choices of the non-homogeneous term are presented. As an application of the considered non-homogeneous equation, direct and inverse source problems are studied. The solutions are expressed in the form of series expansions using an orthogonal basis obtained through separation of variables. Illustrative examples for the direct and inverse problems are also presented for specific choices of the initial and final time data and the source function.

AppliedMath doi: 10.3390/appliedmath6060097

Authors: Qingli Luo Zhe Wang Yufeng Zhang

This paper investigates recursion operators and nonlocal symmetry structures for the modified Veronese web equation. The novelty of the work lies in the explicit construction of a direct recursion operator and its inverse in the tangent-covering framework. Starting from a compatible linear covering with a spectral parameter, we derive both operators and interpret them as auto-Bäcklund transformations for the corresponding linearized equation. We also determine the contact symmetry algebra and compute the action of the two recursion operators on its infinitesimal generators. In particular, the inverse recursion operator produces shadows of nonlocal symmetries associated with conservation-law coverings. These results provide a concrete recursive mechanism for the symmetry space of the modified Veronese web equation and clarify its covering-based nonlocal geometric structure.

AppliedMath doi: 10.3390/appliedmath6060096

Authors: Tarikul Islam Marco Martins Afonso Sílvio Gama

This study investigates the thermal optimization of unsteady nanofluid natural convection within a quarter-circular domain with an inner wavy boundary under inclined periodic magnetic forcing. A combined finite-element method (FEM) and central composite design-based response surface methodology (RSM) is employed for optimizing both the geometric configuration and the parametric setting. For the geometric optimization, we find that the wavy-wall amplitude is the key parameter to determine the optimal configuration, followed by the inner radius and undulation number. The parametric analysis shows that strong magnetic effects suppress convection, while increasing the Rayleigh number and the nanoparticle volume fraction significantly enhances heat transport. Additionally, rising magnetic field wavelength and/or inclination angle result in a reduction in heat transmission under strong magnetic intensity. A statistical quadratic correlation equation with the help of the RSM method between Rayleigh number, Hartmann number, and nanoparticle volume fraction, and the mean Nusselt number is formulated, which gives a good match with numerical FEM analysis results in which about 99% of the variation in the response variable is predicted (R2 = 0.9975). The results obtained in this study offer valuable information along with computational efficiency in predicting the behavior of such advanced thermal systems.

AppliedMath doi: 10.3390/appliedmath6060095

Authors: C. Jeeva Jose Aneesh P. Baiju Riya Roy A. Harikrishnan Rahul Sanju P. B. Vinod Kumar K. K. Sherly Rinku Jacob G. Sreekumar

In this study, a layer-wise point cloud representation of CT scan images is proposed, from which persistence diagrams are constructed and persistent entropy is computed as a compact topological feature for three-class lung tumor classification. Two parallel approaches are investigated: the direct computation of persistence diagrams from CT images, and computation from subsampled point clouds derived from image intensity layers. The proposed method is evaluated on the publicly available IQ-OTH/NCCD lung cancer dataset, comprising 1097 CT scan images from 110 individuals, annotated by expert oncologists and radiologists. Classification is performed using K-Nearest Neighbors (KNN), Random Forest, Support Vector Machine, and eXtreme Gradient Boosting, and compared against Convolutional Neural Network (CNN) and traditional image feature-based methods. The persistent entropy approach applied to layer-wise subsampled point clouds achieves 97.67% accuracy, a Precision–Recall AUC of 96.63%, and a ROC-AUC of 99.46% using KNN, outperforming direct image-based analysis (95.91%) and achieving comparable accuracy to the CNN method (97.21%) with a computational speedup of approximately 478×. These results demonstrate that persistent homology applied to subsampled point clouds provides an accurate, mathematically interpretable, and computationally efficient alternative to deep learning for lung tumor classification.

AppliedMath doi: 10.3390/appliedmath6060094

Authors: Adam Dudáš Peter Babic

Graphical methods of correlation analysis, such as correlation n-ptychs or hotspots, focus on the identification of the strength and direction of functional relationships between sets of attributes in multidimensional datasets. Since these correlation structures only take into account values of the attributes, situations arise when the relationship is coincidental, meaning that there is no real-world causality between the values of the observed attributes but these values still exhibit significant correlation. This problem of correlation analysis as a whole motivates the need for semantic evaluation of significant relationships identified using its methods—a task that could potentially be time- and resource-intensive when conducted manually. However, modern results in the large language model area provide tools for the automatization of such tasks. Hence, this work focuses on the design and implementation of a novel large language model-based method for semantic evaluation of correlation structures embedded in a correlation graph, specifically correlation n-ptychs for n∈{3, 4, 5} and correlation hotspots. In the method, the large language model is automatically prompted to assess the semantic nature of relationships in the set of correlation substructures of the dataset, identify their real-world relevance, and visualize the result in the form of a Semantic evaluation card. The proposed approach is evaluated using two benchmarking datasets focusing on the visualization method used in the model, large language model interaction with the correlation substructures, and comparative analysis with previously used tools in the area.

AppliedMath doi: 10.3390/appliedmath6060093

Authors: Sonja Orlić José Luis Palacios Aleksandar Petojević

Thehitting time index HT(G) is a recently introduced topological descriptor based on expected hitting times of a random walk on a graph. In this paper, we derive a closed-form formula for HT(G) for broom graphs Bn,d that holds for all parameters 2≤d≤n−1, HT(Bn,d)=S1+S2+S3+(n−d)∑i=1d−1max{A(i),B(i)}, where S1,S2,S3,A(i),B(i) are explicitly defined. For d≥2 and n≥4d−8 we derive a simpler cubic polynomial formula in n, HT(Bn,d)=n3+adn2+bdn+cd, with explicitly given coefficients ad,bd,cd depending only on d. We also consider quartic polynomial formulas for special cases.

AppliedMath doi: 10.3390/appliedmath6060092

Authors: Khadijetou Maaloum Din Rabii El Maani Ahmed Tchvagha Zeine Rachid Ellaia

Multi-objective optimization plays a fundamental role in solving complex engineering design problems characterized by conflicting objectives and nonlinear constraints. In this study, a novel hybrid optimization algorithm, named Multi-objective Lemurs Optimizer-Backtracking Search Algorithm (MOLOBSA), is proposed to improve the exploration and exploitation capabilities of existing metaheuristic methods. The proposed approach integrates the global exploration ability of the Lemurs Optimizer (LO) with the efficient mutation and crossover mechanisms of the Backtracking Search Algorithm (BSA) within a multi-objective optimization framework. The effectiveness of the proposed algorithm is evaluated using the CEC2020 multimodal multi-objective benchmark suite, where its performance is assessed using the PSP and IGDX performance indicators. In addition, the proposed method was successfully applied to the multi-objective design optimization of an I-beam structure, where the objectives were to minimize the structural weight and the maximum displacement under mechanical constraints. The obtained Pareto solutions exhibit better diversity and improved trade-off characteristics compared with those produced by the baseline algorithm.

AppliedMath doi: 10.3390/appliedmath6060091

Authors: Wafa Louafi Houda Tadjer Yacine Lafifi

The adaptive regulation of student productivity remains a challenging problem in technology-enhanced learning environments due to the continuous and uncertain nature of cognitive effort, attention, and behavioral fluctuations. While existing educational intervention models are predominantly based on discrete-time decision frameworks, they often provide limited support for the representation of stochastic productivity dynamics and continuous effort adaptation. This paper proposes a continuous-time stochastic optimal control framework for adaptive effort allocation in student productivity regulation. The learner productivity level is modeled as a bounded stochastic diffusion process evolving on the interval ([0, 1]), where the drift and diffusion coefficients depend on both effort allocation and learner-specific psychological characteristics. The control objective is formulated as the maximization of an expected cumulative productivity reward penalized by excessive cognitive effort over a finite study horizon. Using the Hamilton–Jacobi–Bellman (HJB) framework, we derive an optimal state-dependent feedback policy that dynamically adjusts effort allocation according to the current productivity level, the remaining study horizon, and the learner profile. We establish the well-posedness of the controlled stochastic dynamics and show that the productivity state remains invariant within the admissible interval. The resulting HJB equation is solved numerically using a semi-implicit finite-difference approximation combined with iterative feedback updates. Simulation experiments conducted on synthetic learner profiles illustrate the qualitative behavior of the proposed controller under heterogeneous psychological configurations. Compared with constant-effort and threshold-based heuristic strategies, the adaptive feedback policy produces smoother productivity trajectories and more stable effort allocation patterns under stochastic perturbations. The proposed framework provides a mathematically grounded approach for studying adaptive productivity regulation under uncertainty and establishes a foundation for future data-driven calibration and personalized intervention systems.

AppliedMath doi: 10.3390/appliedmath6060090

Authors: Kaloyan N. Vitanov Nikolay K. Vitanov

We present a review on the application of the mathematical models for research on social processes, social structures, and actors in social systems. The scope of the review is not restricted to the classical applications of mathematics such as theory of probability, statistics, stochastic processes, differential equations, and game theory. We also discuss applications of the theory of networks for social network analysis and the numerical research on dynamics of social systems. The number of these applications has increased very fast in recent years. Special attention is given to the results from the area of sociophysics, where mathematical methodology is used to analyze social systems in cooperation with the models and concepts of physics. Another special topic in his review is connected to the results from econophysics, where the mathematical methodology and theories and methods of physics are used in the studies on the dynamics of economic systems. In addition, we give several examples for the application of mathematical methods to social systems: (a) application of difference equations to model the flow of substances in channels of networks; (b) analytical solution of nonlinear equations connected to the model of waves of popularity; (c) numerical results of the waves of popularity in a model that accounts for the change in the opinion of the supporters of the ideas for positive or negative popularity of a person, material item, or a piece of information (idea, theory, ideology, etc.) In the last case, we illustrate the effectiveness of the numerical analysis to discover new effects on the studied social system. The review ends with a large list of references. These references can be used as a guide of the way of new researchers to the large field of mathematical social dynamics.

AppliedMath doi: 10.3390/appliedmath6060089

Authors: Adjété Lionel Wilson Toyo Koffi Edarh Bossou

We introduce a mathematical model describing a dynamical system made up of interdependent compartments subject to intrinsic fragility and cross-support. The dynamics are driven by the competition between hierarchical sublinear dissipation and cyclic coupling, enabling a natural interpretation of transitions between distinct regimes (e.g., resilience versus degradation, vulnerability and crisis/extinction in finance or biology). We establish the existence, uniqueness, positivity, and global continuation of solutions. We also investigate the qualitative behavior of the system by studying the stability of equilibrium points and deriving threshold conditions characterizing increasing, decreasing, and stationary regimes. Numerical simulations are provided to illustrate the theoretical results and the transition phenomena between extinction and self-sustained dynamics.

AppliedMath doi: 10.3390/appliedmath6060088

Authors: Stathis Hadjidemetriou

The classical Regiomontanus problem asks for the position that maximizes the angle subtended by a fixed line segment, a problem originating in Euclidean geometry and observational astronomy. This work extends the problem to observers constrained to move along a circular path. The solution uses the geometric fact that the optimal viewpoint occurs precisely where the trajectory is tangent to an isoangle circle. This framework unifies configurations both exterior and interior to the circular path. The analytical solution is validated against direct numerical optimization, with agreement exceeding ten decimal places, and the subtended angle is shown to attain a unique maximum along each half of the circular trajectory. In the limit as the radius becomes large, the analytical result reduces to the classical linear case. Beyond its historical interest, a potential extension illustrates how this simple geometric condition can govern optimal viewing and highlights connections to modern imaging and vision systems.

AppliedMath doi: 10.3390/appliedmath6060087

Authors: Wiam Zeid Haissam Chehade Issam Kaddoura Yahia Awad

Let k be a positive integer. A polynomial A∈F2[x] is called k-unitary perfectif the sum of the k-th powers of its distinct unitary divisors equals Ak. In this paper, we study the case k=2n and prove that every 2n-unitary perfect polynomial over F2 is even. Moreover, we completely classify all even 2n-unitary perfect polynomials having at most three distinct irreducible factors. In particular, we characterize all such polynomials of the form A=xa(x+1)bPh, where P is a Mersenne prime over F2 and a,b,h∈N. As a consequence, several explicit infinite families of k-unitary perfect polynomials over F2 are obtained.

AppliedMath doi: 10.3390/appliedmath6060086

Authors: Rekha Panicker Rahul Shukla

This paper investigates common fixed point theorems for (β,α)-generalized enriched contractions in Banach spaces. We provide a corrected proof of an existing theorem on enriched contractions, thereby strengthening the reliability of results in this area. Our analysis further shows that a previously published illustrative example does not satisfy the proposed enriched contraction condition, since the condition fails for x=0, y=18, and b=45. We then introduce a generalized pair of mappings and prove two common fixed point theorems for single-valued (β,α)-generalized enriched contractions under weak commutativity and compatibility conditions. Strong convergence to the unique common fixed point is established through a Mann-type iteration process, and an example is provided to validate the proposed generalizations.

AppliedMath doi: 10.3390/appliedmath6060085

Authors: Mathias Wagner Thomas Harweg Roland Linder Frank Weichert

In this paper, we introduce an agent-based pedestrian simulation with aerosol modeling, which we use for analyzing the risk of infection with airborne diseases, with special attention to indoor scenarios and the corresponding geometry. For our analysis, we simulate a realistic supermarket scenario, and analyze the influence of geometric factors for the risk of infection regarding aerosol concentration. Using such a defined set of geometry allows for a targeted analysis of risk factors. Specifically, we examine if angular structures bear higher viral loads than flat structures, which is confirmed by our experiments. An artificial neural network (ANN) specifically trained on simulation data is able to identify adjacent geometric structures based on aerosol concentration with up to 94% accuracy.

AppliedMath doi: 10.3390/appliedmath6060084

Authors: Olga Kozlovska Felix Sadyrbaev

The system of ordinary differential equations that arises in models of genetic networks is considered. This system has both nonlinear and linear parts. As a nonlinear part, we use a piecewise linear function. This allows us to treat systems with multiple (up to ten) equations. Special attention is paid to the four-dimensional systems that have an attractor in the form of a periodic solution. In the final part, the 10th-dimensional system is considered, which describes a subnetwork of a larger network that arises in a practical biomedical problem.

AppliedMath doi: 10.3390/appliedmath6060083

Authors: Sapna Amit Sharma Naveen Mani Rahul Shukla

This study determines the time period of vibrational modes for a non-uniform orthotropic parallelogram plate, featuring a one-dimensional circular thickness variation and subjected to clamped (CCCC) edges. The authors make assumptions about the material; they assume a one-dimensional circular density variation and incorporate Poisson’s ratio to address material non-uniformity. The motivation to choose circular variation in the plate parameter was due to its numerous applications in engineering and real-life applications like engine covers, rotor blades, etc. The authors also account for a parabolic temperature gradient across the plate. Utilizing the Rayleigh–Ritz technique, they derive the frequency equation and solve it to determine the vibration mode time periods. This study includes a convergence analysis of the plate across (CCCC) edge conditions. The primary aim is to demonstrate the advantage of selecting a variable (circular) density and Poisson’s ratio simultaneously over solely varying the density parameter. The secondary aim is to show that the variable Poisson’s ratio is a much better choice in comparison to variable density as a non-homogeneity parameter.

AppliedMath doi: 10.3390/appliedmath6060082

Authors: César Catalán José M. Calabuig Luis M. García-Raffi Enrique A. Sánchez-Pérez

This study introduces a novel physics-inspired framework to quantify defensive pressure in football from tracking data. We model defender–attacker interactions as a variable-mass dynamical system, translating Newtonian mechanics into operational metrics that combine spatial configuration and motion. From this formulation we derive interpretable quantities at dyad, player, and team level, including a Center of Pressure (CP), Defensive Momentum, Defensive Force, and Defensive Work. We illustrate the framework in a single-match proof-of-concept using professional optical tracking data, analysing both full-match behaviour and football-specific phases such as counter-pressing, set-pieces, and throw-ins. Results show how the proposed metrics separate persistent spatial constraint (pressure) from energetically demanding defensive actions (work), enable identification of high-cost match-ups and workload concentration, and support time-resolved descriptions of coordinated pressing sequences. The framework provides a transferable, mechanically grounded toolkit for applied defensive performance analysis and motivates future validation on larger datasets.

AppliedMath doi: 10.3390/appliedmath6050081

Authors: Daniel B. Wright Sage N. Stafford

Count outcomes are commonly analyzed using Poisson regression, but empirical data often exhibit overdispersion, excess ties, heaping, or other departures from the Poisson distribution. This paper evaluates a rank-Poisson transformation, denoted poisrank, designed to map observed counts onto Poisson quantiles before fitting a Poisson regression model. Our goal is to test whether a rank-Poisson transformation offers a useful general-purpose strategy when count data do not satisfy Poisson assumptions. Using an empirical example and a Monte Carlo simulation study with Poisson, overdispersed, rounded, and gapped count distributions, we compared Poisson regression on raw counts, Poisson regression after the poisrank transformation, quasi-Poisson regression, and additional comparison approaches. Although the transformation made the marginal distribution more similar to a Poisson distribution, it generally did not outperform standard alternatives for inference. In particular, quasi-Poisson regression more consistently maintained appropriate rejection rates with overdispersion whereas poisrank tended to be conservative and often reduced power. These findings suggest that the rank-Poisson transformation is better understood as an exploratory robustness device than as a preferred replacement for established count-data methods.

AppliedMath doi: 10.3390/appliedmath6050080

Authors: Danail Brezov

This paper builds upon a previous study suggesting an optimization procedure for rotation sequences by introducing a fourth factor in Euler-type decompositions, thus allowing for an additional degree of freedom used both as a variational parameter and a means to avoid the gimbal lock singularity. Here, an analogous result is derived for generic rigid motions, which is of potential interest in 3D robot manipulators, aircraft, and spacecraft using gimbals to navigate in space. The idea is based on Kotelnikov’s principle of transference, which extends the properties of pure rotations to arbitrary Galilean transformations, interpreted as screw motions. To do that in practice, it is convenient to use dual quaternions or their projective version, referred to as dual Rodrigues’ vectors. With this approach, the explicit solutions are easy to extend and therefore optimization is rather straightforward: we show, both analytically and with numerical examples, that factorizing motion into sequences of four consecutive screws is, in general, significantly more energy-efficient compared to using three.

AppliedMath doi: 10.3390/appliedmath6050079

Authors: Daniel O. Aikhuele Shahryar Sorooshian Harold U. Nwosu

This study develops a stochastic degradation-based reliability framework for mechanical systems subject to interacting operational stresses and imperfect maintenance. The degradation dynamics are formulated in cumulative damage space and modeled using a geometric Itô diffusion process, in which the drift term incorporates a multiplicative degradation kernel representing the combined influence of load, speed, misalignment, and environmental exposure. Imperfect maintenance is represented through a continuous attenuation functional embedded within the drift structure, allowing maintenance actions to reduce degradation growth without restoring the system to an as-good-as-new condition. Using a logarithmic transformation, the multiplicative stochastic differential equation is converted into an additive diffusion process, enabling analytical treatment via Itô’s lemma. A closed-form reliability expression is then obtained through first-passage analysis, yielding a lognormal survival function governed directly by the degradation dynamics. Numerical evaluation demonstrates physically consistent wear-out behavior and confirms the stability of the derived reliability formulation. The model further enables reliability-based maintenance optimization through preventive replacement analysis. Sensitivity results indicate that system reliability is strongly influenced by the degradation growth parameter governing the stochastic drift. The proposed framework provides a mathematically tractable connection between stochastic degradation modeling, reliability theory, and maintenance optimization. Beyond its application to conveyor belt systems, the formulation offers a general analytical structure for reliability assessment of degrading engineering systems governed by multiplicative stochastic dynamics.

AppliedMath doi: 10.3390/appliedmath6050078

Authors: Mario Alberto Sandoval-Hernandez Arturo Sarmiento-Reyes Fernando Ivan Molina-Herrera Hugo Jimenez-Islas Uriel Antonio Filobello-Nino Gerardo Ulises Diaz-Arango Francisco Marroquin-Gutierrez Rogelio Alejandro Callejas-Molina Sandra Ysabel Campos-Dominguez Cristian Dumay Hernandez-Garcia Hector Vazquez-Leal

This paper presents analytical approximations for the inverse error function, its complementary inverse, and the cumulative distribution function using the Power Series Extender Method (PSEM). The proposed expressions exhibit high accuracy over a wide portion of the domain, particularly in the central region, while maintaining a compact structure based on elementary functions. This formulation ensures practical implementation and computational efficiency without the need for specialized numerical algorithms. The use of strategically selected cancellation points further enhances the accuracy of the approximations, especially in regions of interest. As expected for this class of elementary approximations, a gradual loss of accuracy is observed near the boundaries of the domain due to the asymptotic behavior of the inverse functions. To demonstrate the effectiveness and practical relevance of the proposed expressions, two case studies are presented, involving applications in statistical analysis and engineering contexts.

AppliedMath doi: 10.3390/appliedmath6050077

Authors: Dinis Pestana Maria Luísa Rocha

Rising temperatures in industrial processes are a serious alert that the system can be shifting from an In Control (InC) to an Out of Control (OutC) state, causing waste, financial losses and, eventually, disaster. Consultation in a case study analyzing the Statistical Quality Control (SQC) routines in a potato chip factory revealed that laymen dealing with data may naively spoil and misuse traditional SQC tools, downgrading the interval-scale temperature data to a simple nominal classification, true or false Negative (N) or Positive (P) symptoms that the production line is InC or OutC. Appropriate scores, negative for true N and false P, and positive for false N and true P, were designed so that their moving averages upcrossing 0 detect clusters of suspicious temperature deregulation, in order to effectively salvage the InC/OutC prognostic value of data.

AppliedMath doi: 10.3390/appliedmath6050076

Authors: Remo Pacella Angelo Di Egidio

A reduced analytical formulation for the linear elastic behavior of axisymmetric shells subjected to axisymmetric load distributions is presented. The mechanical response of the shell is interpreted through the interaction between two families of one–dimensional structural elements, namely meridian fibers and circumferential fibers, whose kinematic coupling emerges naturally from the compatibility relations of the classical Reissner–Mindlin shell theory. By exploiting a geometric reinterpretation of the shell kinematics in terms of auxiliary curvature radii, a simplified mechanical model is derived by neglecting the kinematic contributions associated with one of these radii, which become negligible for shells sufficiently far from the degenerative planar membrane/plate configuration. The resulting formulation leads to a reduced set of compatibility, equilibrium, and constitutive equations that preserve the essential mechanical features of the shell response while significantly simplifying the mathematical structure of the problem. Two internally constrained variants of the reduced model are introduced, corresponding, respectively, to shear–indeformable and inextensible meridian fibers. Within this framework, the governing equations reduce to ordinary differential equations that, for specific shell geometries such as spherical and conical shells, admit closed-form analytical solutions. Based on these reduced models, two approximate solution strategies are developed. The first relies directly on the reduced shear–indeformable shell formulation to describe the overall structural behavior, whereas the second combines membrane solutions with the more internally constrained shell model to capture boundary effects through a superposition procedure. The effectiveness of the proposed approaches is assessed through comparison with numerical solutions obtained from the classical Reissner–Mindlin axisymmetric shell model. The results show that the proposed formulations provide an accurate approximation of both displacements and stress resultants for a sufficiently large range of spherical and conical shell configurations under distributed loads.

AppliedMath doi: 10.3390/appliedmath6050075

Authors: Rajai S. Alassar Mohammed Abushoshah Husain Al-Attas Said Algarni

An analytical solution is presented for transient heat conduction in a two-layer composite cylinder subjected to outer-surface convection with a general time-dependent ambient temperature. Using Duhamel’s principle, closed-form series expressions are derived and then specialized to harmonic ambient fluctuations, recovering the classical constant-ambient solution in the zero-frequency limit. A parametric study shows that the ratio of the inner layer conductivity to the conductivity of the outer layer strongly shapes interfacial gradients and mean-temperature evolution, with sensitivity concentrated at small ratios and diminishing when the ratio is larger than 0.1. Increasing Biot number accelerates the heat transfer and approaches the isothermal-surface limit as it becomes extremely large. The geometric aspect ratio is most influential when the inner layer is resistive, and becomes weak for large conductivity ratio, supporting thin-coating approximations. Under harmonic ambient fluctuations, the response rapidly reaches a periodic steady state; higher frequency decreases amplitude and increases phase lag, while larger Biot numbers amplify oscillations and reduce delay. The coupled effects of the aspect ratio and the conductivity ratio govern penetration and phase behavior.

AppliedMath doi: 10.3390/appliedmath6050074

Authors: Julian Allagan Vladimir Deriglazov Kevin Pereyra Matthew Hill

Adversarial perturbations measured by ℓp norms do not reflect key structural constraints in tabular security data, including anisotropic geometry, feature dependence, and distributional plausibility. We introduce a composite attacker cost functional Catk(x,x′)=τmax{0,m(x′)}+λ1δ⊤G⊥(γ)(x)δ+λ∑jωj|δj|+λ2δ⊤LHδ+λ3logf^1(ϵ)(x)−logf^1(ϵ)(x′)+∑j∈supp(δ)cj+β|M(supp(δ))|ν, which integrates a spectrally truncated geometric term, a graph-based coherence penalty, a smooth copula density barrier, and a superlinear module-spread term. Under spectral degeneracy of the legitimate-class covariance, we establish nonnegativity under density dominance, exact zero self-cost, lower semicontinuity, and λ3κK-weak convexity of the continuous component on compact convex sets, for both affine and ρm-weakly convex scoring functions. These properties yield existence of constrained minimizers. The continuous component is locally Lipschitz, whereas the full functional is not due to the support-counting term. A component feasibility result shows that each term eliminates a distinct class of degenerate perturbations. Limiting regimes and refined evasion cost bounds are derived. An empirical instantiation on PHIUSIIL indicates that perturbations with identical ℓ2 norm can incur costs differing by an order of magnitude.

AppliedMath doi: 10.3390/appliedmath6050073

Authors: Saba N. Al-Khafaji Emad Kadhim Mouajeeb

Bazilevič mappings are considered very important in the theory of geometric mappings because they provide a way to generalize and study the properties of important classes of univalent mappings. Their importance is not only in the deepening of the theory, but also in the practical means of modeling phenomena in applied science and engineering, physics, and differential equations. This paper, in this sense, provides a new subclass of bi-Bazilevič mappings with the use of advanced analytical methods, Chebyshev polynomials on one side, and a Miller–Ross-type Poisson distribution on the other side. The Poisson distribution is considered one of the most important models of probability distributions with a large scope of application in the various sciences. The main components of this study are the definition and the study of this new class of functions, in which the initial Taylor–Maclaurin coefficients, in particular, q2 and q3, are determined and estimated for mappings in this subclass. Also, the classical Fekete–Szegö problem is solved and the first-order limits of this important functional are obtained with respect to the newly introduced bi-Bazilevič mappings. The outcomes contribute to expanding both the theoretical and practical aspects of this type of mapping.

AppliedMath doi: 10.3390/appliedmath6050072

Authors: Noppanon Teangthae Dawud Thongtha

A warrant is a financial derivative that grants the holder the right to purchase company shares at a predetermined price within a specified period. Generally, upon exercise, the total number of outstanding shares increases because of the issuance of new shares, reducing the stock price. In this study, an analytical formula for warrant valuation is developed without relying on the restrictive assumptions of log-normal asset return distributions or constant volatility. The model incorporates key financial variables, including the current asset price, the strike price, the risk-free interest rate, the time to maturity, and the dilution factor. To capture dynamic market conditions, asset return volatility is estimated using GARCH-type models. The performance of this analytical approach is evaluated by comparing its numerical results with those obtained using alternative methods, such as Monte Carlo simulations and the conventional warrant valuation framework. An empirical analysis based on data from the Stock Exchange of Thailand indicates that the proposed method yields improved pricing accuracy with lower estimation errors than existing benchmarks.

AppliedMath doi: 10.3390/appliedmath6050071

Authors: Van Thanh Phan Duc Trien Nguyen Ngoc Xuan Quynh Nguyen Xuan Hau Huynh

Energy consumption demand forecasting plays a critical role in the planning and development of national energy security, which underpins the Vietnam’s Eighth National Power Development Plan (PDP VIII) and Vietnam’s ambitious Net-Zero 2050 commitment. However, this task becomes more difficult because the big data environment is filled with a lot of noise and highly fluctuating data. In order to deal with the problem, this paper evaluates five models: Linear Regression, Holt’s Exponential Smoothing, PSO-GM (1,1), Support Vector Regression (SVR), and Random Forest as a benchmark to conduct a rigorous comparative analysis to identify the most accurate forecasting model. The performance was evaluated by MAE, RMSE, and MAPE indexes based on Vietnam’s total primary energy demand data from 1986 to 2024. To check the accuracy of the forecasting model, this study split the data into two periods: first time for the training data (1986–2016), and second for the testing data set (2017–2024). Furthermore, a five-fold rolling-window time-series cross-validation method and Diebold–Mariano tests were employed to ensure the statistical robustness of the findings on the small-sample datasets (n = 39). The results decisively identified that the Holt’s model as the superior framework, maintaining high stability and achieving a testing MAPE of 7.19% (training MAPE of 5.52%), while the complex machine learning benchmark shows severe over-fitting. Applying this model, Vietnam’s energy demand will reach 1528.08 TWh and 1882.55 TWh in 2025 and 2030, respectively. Furthermore, this study provides empirical evidence that simpler, well-chosen statistical models can surpass complex alternatives in small-sample scenarios, offering a reliable quantitative baseline for policymakers to navigate infrastructure development and decarbonization challenges.

AppliedMath doi: 10.3390/appliedmath6050070

Authors: Rakhal Das Satyendra Narayan

In this paper, we introduce and systematically study the class of interpolative Geraghty-type contractive mappings within the framework of complete bicomplex-valued metric spaces (bi-CVMS). We prove seven new results: (i) a fixed point theorem for a single interpolative Geraghty contraction; (ii) a common fixed point theorem for a pair of such mappings; (iii) a fixed point theorem for interpolative Reich–Rus–Ćirić type contractions in bi-CVMS; (iv) a coincidence point and common fixed point theorem for weakly compatible maps; (v) a fixed point theorem for Jaggi-type hybrid contractions in bi-CVMS; (vi) a stability result for the Picard iteration associated with the main contraction; and (vii) an application theorem establishing the existence and uniqueness of solutions to a boundary value problem governed by a Caputo fractional differential equation. All results are furnished with complete proofs and non-trivial illustrative examples. Several well-known theorems—including those of Banach, Kannan, Reich, Geraghty, and their complex-valued analogues—follow as special cases. The paper significantly advances the fixed point theory in bicomplex-valued metric spaces.

AppliedMath doi: 10.3390/appliedmath6050069

Authors: Alexandru-Dragoș Adam Alina Nirvana Popescu Jair Gonzalez

This paper presents a proposed architecture for an artificial intelligence-driven unmanned aerial vehicle (UAV) system intended for tactical intelligence, surveillance, and reconnaissance (ISR) missions. The architecture brings together electro-optical imaging, long-wave infrared sensing, two-dimensional light detection and ranging (LiDAR), inertial navigation support, onboard edge computing, and resilient communication links within a unified system-level framework. Unlike many existing approaches that treat perception, autonomy, communication, and safety as loosely coupled functions, the proposed architecture combines multi-modal sensing, operator-supervised autonomy, and a safety-oriented decision validation layer intended for future integration with Ansys SCADE. The system is structured around operational and sensor-performance requirements used to justify the selection and interaction of the main onboard subsystems. At the architectural level, the proposed framework is intended to support target detection, tracking, environment awareness, and mission-level decision support under degraded visibility, constrained communication, and contested operating conditions. The paper therefore contributes a requirement-driven and safety-aware ISR UAV architecture that provides a scalable basis for future implementation, validation, and multi-UAV extension.

AppliedMath doi: 10.3390/appliedmath6050068

Authors: Mudassir Shams Bruno Carpentieri

This paper presents a hybrid two-stage implicit scheme for the numerical solution of fractional initial value problems involving Caputo derivatives. The proposed formulation incorporates the nonlinear source term directly into the time-stepping procedure, leading to improved stability and accuracy compared with classical fractional implicit schemes. The resulting nonlinear systems are solved using a parallel iterative strategy based on the Weierstrass-type method, combined with OpenMP-style parallelization to ensure efficient workload distribution and accelerated convergence. In addition, a data-driven module is introduced to generate high-quality initial guesses, thereby enhancing the robustness and efficiency of the nonlinear solver. The main contributions include the development of a unified fractional-parallel-data-driven framework, improved stability properties with enlarged real-axis stability regions, and reduced computational cost through parallel implementation and informed initialization. A theoretical analysis establishes consistency, boundedness, and convergence under standard Lipschitz assumptions. Numerical experiments on representative fractional models demonstrate that the proposed schemes achieve higher accuracy and improved efficiency compared with classical implicit methods, with significant reductions in error and iteration counts. The ANN-enhanced variant further attains near machine-precision accuracy for a range of fractional orders. Overall, the proposed approach provides a robust and scalable computational framework for the efficient solution of nonlinear fractional dynamical systems.

AppliedMath doi: 10.3390/appliedmath6050067

Authors: B. Divya K. Ganesan A. Selvam

This research presents a fractional-order formulation and mathematical analysis of the Rössler chaotic attractor. By utilizing the Nabla discrete Atangana–Baleanu fractional difference derivative in the Caputo sense, the classical integer-order attractor is extended into the fractional domain. The existence and uniqueness of solutions for the resulting fractional system are established via the fixed-point theorem, thereby ensuring that the recommended attractor is well-posed. Furthermore, the Ulam–Hyers stability is investigated within the Nabla discrete Atangana–Baleanu fractional difference derivative in the Caputo sense framework. For numerical investigations, an Euler numerical scheme adapted to the fractional difference derivative is developed and implemented, yielding high-quality phase portraits of a chaotic attractor. The results highlight the effectiveness of fractional-order modeling and numerical methods in capturing the dynamics and stability of the Rössler chaotic system.

AppliedMath doi: 10.3390/appliedmath6050066

Authors: Ali Shojaei-Fard

The article applies the working platform of topological Hopf algebra of renormalization to address a new construction program for the fabric of space-time from the perspective of Feynman graphon models.

AppliedMath doi: 10.3390/appliedmath6050065

Authors: V. L. Stass

The problem that this study is concerned with is the ontogenetic growth of humans and animals. The aim of this research is to analyze a model of the ontogenetic growth of animals. The target of the analyses is to show a link between growth and longevity. This model has implications for modelling the growth of humans as well. In this study, pigs were considered model animals for humans. Humans and pigs have a number of comparable physiological features as well as a few analogous aspects of growth. The resemblance of the biological functions leads us to think that by modelling the growth of pigs, one can gain a better look into the growth of humans. In this research, we model growth, which is operationalized as weight gain; weight loss was not considered. In this study, a discussion of the translation of the results to the growth and longevity of humans was provided. The lifespan or longevity of animals was not modelled explicitly; predictions were made in accordance with the results of the growth model. The main result of the model is that growth promotes, if not causes, longevity.

AppliedMath doi: 10.3390/appliedmath6050064

Authors: Elvira Rolón José G. Méndez Roberto Pichardo

Crime forecasting in heterogeneous urban contexts remains challenging due to the combined effects of territorial heterogeneity and complex temporal dynamics. However, a large portion of the existing literature tends to address territorial segmentation and predictive modeling separately, or to combine them within unified workflows that may obscure their distinct analytical roles. This study presents a modular spatial–temporal analytical approach that treats territorial segmentation and short-term crime prediction as complementary but methodologically independent components. Unsupervised segmentation captures territorial heterogeneity, while a supervised ensemble model estimates short-term crime occurrence. A chronological expanding-window validation scheme is implemented, reserving the most recent period as a blind test set to prevent temporal leakage. Across municipalities, recall values in 2022 range from 0.36 to 0.77, with corresponding F1-scores ranging from 0.174 to 0.696, while blind-test recall ranges from 0.184 to 0.856, with F1-scores ranging from 0.000 to 0.784, and AUC values up to 0.88, indicating that predictive performance is context-dependent rather than uniform. The proposed approach provides a replicable and context-aware analytical approach for spatially differentiated crime risk estimation under strict forward-looking evaluation.

AppliedMath doi: 10.3390/appliedmath6040063

Authors: Amna M. Grgar Lucy J. Campbell

Asymptotic solutions are derived to model the development of atmospheric internal gravity waves generated by latent heating in a two-dimensional configuration involving a vertically-sheared background flow. The mathematical model comprises nonlinear partial differential equations derived from the conservation laws of fluid dynamics under the anelastic approximation where the background density and temperature vary with altitude. The latent heating is represented by a horizontally-periodic but vertically-localized nonhomogeneous forcing term in the energy conservation equation. This generates gravity waves that are considered as perturbations to the background flow and are expressed as perturbation series, with the leading-order contributions being the solutions of linearized equations. Taking into account the nonlinear terms at the next order gives expressions for the effects of the waves on the background mean flow. Due to the vertical shear, there is a critical level where momentum and energy are transferred from the wave modes to the mean flow. The asymptotic solutions show that the wave–mean-flow interaction is nonlocal and occurs over the range of altitudes from the thermal forcing level up the critical level. This is in contrast to what occurs in the case of waves forced by an oscillatory lower boundary, where the interaction is typically localized around the critical level. It is found that the wave drag is negative above the thermal forcing level, making the mean flow velocity more negative, but it becomes positive as the waves approach the critical level, indicating wave absorption in this region. There is wave transmission through the critical level, as well as absorption, and the extent of transmission depends on the depth of the latent heating profile. The mean potential temperature is reduced above the thermal forcing level and enhanced at the critical level, a situation that could ultimately lead to the development of convective instabilities.

AppliedMath doi: 10.3390/appliedmath6040062

Authors: Mahmoud B. Mahmoud Amira M. Salama Mustafa AL-Tawfiq Khaled H. Ibrahim Eslam M. Abd Elaziz

Harmonic distortion in power systems, primarily caused by nonlinear loads, leads to significant power quality issues such as increased losses, reduced power factor, and equipment malfunctions. To mitigate these effects, passive power filters (PPFs) are widely employed due to their cost-effectiveness and simplicity. This paper presents an optimized design of a single-tuned passive filter (STPF) using the Firefly Algorithm (FFA) and its multi-objective extension, the Multi-Objective Firefly Algorithm (MOFA). The optimization aims to minimize both voltage total harmonic distortion (VTHD) and power loss and to maximize the power factor (PF) while complying with IEEE 519-2014 standards. The study evaluates the proposed method under two different industrial case studies with varying system parameters and harmonic profiles. Simulation results demonstrate that the proposed FFA-based optimization outperforms the Mixed Integer Distributed Ant Colony Optimization (MIDACO) method, achieving superior VTHD reduction, power loss minimization, and power factor enhancement. The MOFA approach provides a Pareto-optimal front, offering trade-offs among competing objectives. Comparative analysis confirms the efficiency, robustness, and faster convergence of FFA-based optimization, making it a promising approach for optimal filter design in power systems.

AppliedMath doi: 10.3390/appliedmath6040061

Authors: Stelios Arvanitis Pantelis Argyropoulos Spyros Vassilakis

We develop a categorical framework for modeling recursive uncertainty over preferences in decision theory. Classical models of ambiguity allow for uncertainty over outcomes or beliefs but usually rely on finite or exogenously truncated representations when agents face uncertainty about their own evaluative criteria. Given that such recursive preference formation generates an infinite hierarchy that may not stabilize at any finite level, we introduce a contractive von Neumann–Morgenstern utility functor on a category of compact metric spaces enriched over complete metric spaces, and establish the existence and uniqueness of its canonical fixed point. This fixed point is interpreted as a universal preference space that contains all levels of recursive ambiguity in a consistent and metrically stable form. We further extend the construction to multi-utility representations and discuss its relation to existing models of ambiguity and universal choice spaces. This framework offers a minimal unified representation of recursive preference structures.

AppliedMath doi: 10.3390/appliedmath6040060

Authors: Meriem Keddali Hamida Talhi Mohammed Amine Meraou Ali Slimani

A novel flexible single-parameter polynomial distribution is presented in this study. The forms of hazard rate and density functions are examined. Additionally, exact formulas for a number of numerical characteristics of distributions are obtained. Stochastic ordering, the moment technique, the maximum likelihood, and a Bayesian analysis of this novel distribution based on type II censored data are used to derive the extreme order statistics. We construct Bayes estimators and the associated posterior risks using a variety of loss functions, such as the generalized quadratic, entropy, and Linex functions. Since tractable analytical formulations of these estimators are unattainable, we suggest using a simulation technique based on Markov chain Monte-Carlo (MCMC) to examine their performance. Furthermore, we construct maximum likelihood estimators given initial values for the model’s parameters. Additionally, we use integrated mean square error and Pitman’s proximity criteria to compare their performance with that of the Bayesian estimators. Lastly, we apply the new family to many real-world datasets to show its versatility, and we model cancer survival data using this new distribution to explain our methodology.

AppliedMath doi: 10.3390/appliedmath6040059

Authors: Obiora Cornelius Collins Oludolapo Akanni Olanrewaju

Substance abuse addictions and hepatitis B infections are two major public health problems facing humanity globally, especially in areas where the two problems co-exist. A mathematical model was used in this work to study the co-dynamics of substance abuse addictions and hepatitis B infections and investigate their possible control strategies. The mathematical features of the model, such as the disease-free equilibrium, endemic equilibrium, and basic reproduction number, were computed. The stability analysis of the disease-free equilibrium and endemic equilibrium was conducted analytically. The impact of multiple control measures, including public enlightenment, rehabilitation of individuals with substance abuse disorders, treatment of persons infected with hepatitis B, and vaccination of susceptible individuals, was examined numerically. The study reveals how co-existence fundamentally alters system behavior and control effectiveness and offers new insights for designing effective control management strategies.

AppliedMath doi: 10.3390/appliedmath6040058

Authors: Douglas Kwasi Boah Suleman Abudu Fiele Christian John Etwire

This study introduces a deterministic fractional-penalty refinement of Vogel’s Approximation Method (VAM) for generating high-quality initial basic feasible solutions (IBFS) in classical transportation problems. Unlike the traditional additive regret measure employed in VAM, the proposed method uses a multiplicative contrast ratio between the two smallest admissible costs in each row and column. This modification preserves the allocation structure of VAM while introducing scale-invariant prioritization that improves sensitivity to relative cost differences.The method was evaluated on thirty-four benchmark transportation problems drawn from the literature and self-constructed large-scale instances (up to 10×20). Performance was assessed using percentage optimality gaps relative to optimal solutions obtained via the Stepping–Stone and MODI procedures. Across all instances, the proposed approach achieved a mean optimality gap of 2.78%, compared to 5.22% for classical VAM, 14.97% for the Least Cost Method (LCM), and 45.78% for the Northwest Corner Method (NWCM). Dispersion of deviations was also reduced, indicating improved robustness across heterogeneous cost structures Statistical validation confirms the improvement over VAM: the paired t-test yielded t=−3.17 (p=0.00163, one-sided), and the Wilcoxon signed-rank test produced p=6.10×10−5. Computational experiments further show that the refinement does not increase runtime relative to classical IBFS procedures.The proposed method therefore constitutes a structured enhancement of VAM that improves initial solution quality while maintaining computational simplicity.

AppliedMath doi: 10.3390/appliedmath6040057

Authors: Senne Aerts Eleonora Iachini Urszula Kochanska Eleni Koutrouli Polychronis Manousopoulos

Crypto-asset markets have been rapidly evolving during the past years, being under the spotlight of a diverse set of actors in the financial ecosystem, including investors, financial institutions, regulators and academics. Their potential interconnections with the traditional financial markets are important, and identifying them can provide useful insight in a diversity of areas such as risk contagion and mitigation, price formation, portfolio management and regulatory framework design. In order to identify such interconnections, various lines of research are followed. Specifically, the correlation between prominent stock market indices and crypto-assets from 2018 to 2025 is examined, while their volatility is also evaluated. Furthermore, the relevant effect of news, events and announcements is explored. The results are based on both daily and high-frequency datasets, with the use of the latter focusing on intra-day variation. The analysis of the results identifies existing interconnections between 2020 and 2025, as well as the important respective impact of news and announcements. An additional generic outcome is the usefulness of high-frequency datasets in the crypto-asset context. The conclusions are useful for all actors in the financial ecosystem. Future work can focus on the extension of the research to additional markets or crypto-assets.

AppliedMath doi: 10.3390/appliedmath6040056

Authors: Salvador E. Ayala-Raggi Manuel Rendón-Marín

The exact perimeter of an ellipse involves the complete elliptic integral of the second kind, which lacks a closed-form expression in elementary functions. As a result, analytical approximations have been proposed for applications requiring fast and accurate evaluation of elliptical geometries. In this study, we present a new ultra-accurate and compact closed-form approximation for the ellipse perimeter based on an exponential correction applied to Ramanujan’s second formula. The proposed expression preserves simplicity—using only three exponential functions and six constants—while achieving a maximum relative error of approximately 0.57 ppm observed over the tested grids covering the full eccentricity range. This represents a significant accuracy improvement over classical and modern approximations while maintaining a single-line analytical form with low computational cost. Due to its robustness, quasi-exact behavior at both circular and highly eccentric limits, and its suitability for numerical algorithms and embedded implementations, the proposed approximation is particularly useful in engineering computations involving elliptical boundaries.

AppliedMath doi: 10.3390/appliedmath6040055

Authors: Daniel López-Rodríguez Jorge Jordán-Núñez Bàrbara Micó-Vicent Antonio Belda

Sectional warping requires selecting a final operating length when only a small sample of residual cone masses can be measured. This paper proposes a Bayesian chance-constrained planning rule that combines a conjugate log-space model with fast posterior predictive simulation of the population minimum to recommend a risk-limited band length. The method provides a transparent risk parameter, efficient computation, and direct comparison with heuristic, bootstrap, distribution-free, and tail-model baselines. In an industrial-like synthetic study, the Bayesian policy reduced the mean remainder relative to a tuned sample-minimum rule while maintaining controlled shortage risk, and the results clarify why fully distribution-free guarantees are impractical under typical sampling budgets.

AppliedMath doi: 10.3390/appliedmath6040054

Authors: Kadrzhan Shiyapov Zhanars Abdiramanov Zhuldyz Issa Aruzhan Zhumaseyitova

We develop a fast numerical method for solving nonlinear diffusion equations with memory phenomena, a class of problems arising within viscoelastic materials, anomalous transport, and hereditary systems. The primary computational problem is the nonlocal temporal dependence captured by Volterra-type memory operators, which makes direct evaluation scale quadratically with the number of time steps (O(Nt2)), rendering prolonged simulations prohibitively expensive. To address this bottleneck, we develop a novel synthesis that combines a high-order spectral method for spatial discretization with a fast memory algorithm based on a sum-of-exponentials approximation. The spectral method obtains exponential spatial convergence for smooth solutions. At the same time, the fast memory algorithm reduces memory usage and computational complexity to O(Nt), yielding computational speedups exceeding 414x for prolonged simulations. We rigorously prove that the proposed scheme preserves the discrete energy dissipation law of the continuous system under mild assumptions on the memory kernel, thereby ensuring unconditional stability. Error analysis verifies spectral accuracy in space and first-order temporal convergence. Extensive numerical experiments using exponentially decaying and weakly singular kernels validate the theoretical results and illustrate the method’s effectiveness for modeling viscoelastic transport phenomena and irregular diffusion in complex systems.

AppliedMath doi: 10.3390/appliedmath6040053

Authors: Gorenc Mateja

This paper studies subsidy allocation in interconnected industrial systems using the spectral theory of positive matrices. The allocation is characterized by the Perron eigenvector of a cost matrix describing inter-factory interactions. We show that convergence to equilibrium is exponential and governed by the spectral ratio. A systemic resilience index based on spectral separation is introduced to quantify both stability and robustness under perturbations. The results demonstrate that stability and fairness arise from the spectral structure of the system.

AppliedMath doi: 10.3390/appliedmath6040052

Authors: Rajendra Prasad Najwa Mohammed Al-Asmari Abdul Haseeb Sushmita Sen

The main object of this work is to study the generalized B-curvature tensor in an n-dimensional Lorentzian para-Kenmotsu (briefly, (LPK)n) manifold along a semi-symmetric metric connection ∇¯. First, in an (LPK)n-manifold, we explore certain flatness conditions, namely, B¯(Y,Z)X=0, B¯(Y,Z)ζ=0, g(B¯(φY,φZ)φX,φW)=0, and B¯(Y,Z)·φ=0 conditions, which all result in an η-Einstein manifold. Furthermore, in an (LPK)n-manifold, we study the curvature conditions B¯.Q=0 and B¯.Q¯ = 0, which provide the scalar curvature. The generalized B-curvature tensor blends the features of different curvature tensors, allowing researchers to study conditions like semi-symmetry, pseudo-symmetry in a unified framework. Conditions like B-semi-symmetry correspond to conservation laws or stability properties in physical systems.

AppliedMath doi: 10.3390/appliedmath6030051

Authors: Stefan Brock

Time delays are inherent in modern motion-control and electric-drive loops due to sensing, filtering, sampling and computation, communication, and actuation scheduling. When such delays are only partially known, they can markedly reduce stability margins and narrow the admissible range of state-feedback gains, especially in high-bandwidth servo applications. This paper develops a design-oriented state-feedback framework for delay-affected plants based on the Manabe polynomial concept and the Coefficient Diagram Method (CDM). The plant is represented as a chain of integrators of order two to four with an effective input gain, and the feedback gain is synthesized for the nominal delay-free model by matching a standard Manabe/CDM characteristic polynomial using the classical CDM stability-index pattern. When an unmodeled input delay is present, the closed loop is governed by a delay-dependent characteristic equation. By introducing a normalized representation, the analysis yields explicit delay-stability limits that directly translate into a lower bound on the equivalent time constant used for tuning. The degradation of the phase margin and gain margin with increasing normalized delay is quantified as design charts, and a simple phase-margin-based inequality is proposed for selecting the tuning time constant, with gain-margin checks recommended as a verification step.

AppliedMath doi: 10.3390/appliedmath6030050

Authors: Sunil Kumar Yadav Najwa Mohammed Al-Asmari Abdul Haseeb

The objective of this work is to characterize certain geometric aspects of LP-Sasakian (LPS) manifolds admitting a generalized almost Schouten soliton (GASS) and to prove that a such manifold with GASS is of constant scalar curvature. Initially, we examine the solitonic behavior of ϕ-recurrent LPS manifolds with GASS in view of certain curvature conditions. Moreover, we also deliberate the geometric properties of a perfect fluid LPS spacetime with a unit torse-forming vector field (UTVF) in connection with a GASS. Also, the behavior of a GASS is studied in the broader framework of special types of perfect fluid LPS spacetime such as dust fluid, dark fluid, and radiation era. Overall, the main novelty of this work is its study of the geometrical phenomena and characteristics of a GASS on LPS manifolds and their application in a perfect fluid LPS spacetime.

AppliedMath doi: 10.3390/appliedmath6030049

Authors: Abdul-Mumin Khalid Musah Sulemana Wahab Abdul Iddrisu

Multimodal epileptic seizure detection using physiological biosignals remains challenging due to signal noise, inter-subject variability, weak cross-modal alignment, and the limited interpretability of many machine learning models. To address these challenges, this study proposes a parameterized multimodal feature-fusion framework that unifies normalization, modality weighting, and nonlinear cross-modal interaction within a single mathematical representation. Four fusion parameters, the fusion exponent ρ, interaction weight (δ), normalization factor (λ), and the cross-modal interaction term (η), are introduced at the feature-fusion level, while all classifiers retain their original learning mechanisms. The framework is evaluated using synchronized EEG, ECG, EMG, and accelerometer signals from 120 subjects, segmented into 2 s windows at 512 Hz and analyzed using twelve classical and deep learning classifiers. Principal Component Analysis (PCA) applied to the fused feature space reveals improved class separability compared to unimodal representations, with EEG exhibiting the strongest intrinsic discrimination and peripheral modalities contributing complementary structure when fused. SHapley Additive exPlanations (SHAP) further identify entropy as the most influential feature across all modalities, followed by RMS and energy, yielding physiologically coherent attributions. Quantitative performance evaluation and ablation analysis confirm that the observed improvements arise from the proposed representation design rather than classifier-specific modifications. Unlike existing architecture-dependent fusion strategies, the proposed method introduces a mathematically parameterized feature-space formulation that enhances separability and interpretability without modifying classifier architectures, thereby establishing a representation-driven paradigm for explainable multimodal seizure detection. These results demonstrate that mathematically principled feature-space modeling can simultaneously enhance predictive performance and interpretability, providing a transparent and robust foundation for explainable multimodal seizure detection.

AppliedMath doi: 10.3390/appliedmath6030048

Authors: Rinaldi (Unciuleanu) Oana Costin-Gabriel Chiru

This paper develops a rigorous mathematical and computational framework for four-dimensional chess defined on the discrete hypercubic lattice {1,…, 8}4. We formalize piece movement using displacement sets in Z4, define adjacency via the Chebyshev metric, and analyze the resulting move graphs for rooks, bishops, knights, queens, and kings. We establish exact mobility formulas, parity invariants, and connectivity properties, consolidating known product-graph results for rooks and kings while introducing a boundary-sensitive analysis of the four-dimensional knight verified by exhaustive enumeration. The mathematical framework is complemented by a fully implemented 4D chess engine and interactive visualization environment rendering all 64 (z,w)-slices of the hypercube simultaneously. The system supports full move legality, generalized special rules, multi-king checkmate detection, and reproducible state enumeration. Performance measurements and exploratory branching-factor estimates are obtained through reproducible random playouts using the publicly available implementation. We contextualize this ruleset within existing work on move graphs on Znm, higher-dimensional leapers, spectral properties of grid graphs, toroidal analogs, and multidimensional visualization. Exploratory qualitative feedback (N = 18) is included to examine whether the visualization design is interpretable and navigable in practice, providing feasibility-oriented observations on how slice-based 4D projection and layered board rendering are perceived by non-expert users in an exploratory context. Together, the mathematical results, implemented engine, and visualization form a coherent foundation for the study of strategy, complexity, and human interaction in four-dimensional game systems. The framework provides a basis for future investigations into spectral analysis of move graphs, symmetry-aware search, hierarchical planning, and educational applications in high-dimensional geometry.

AppliedMath doi: 10.3390/appliedmath6030047

Authors: Tatsuya Sekiguchi Hiroyuki Hamada Masahiro Okamoto

Parameter estimation for biochemical reaction networks is computationally demanding, especially for systems with oscillatory nonlinear dynamics, where standard iterative optimization strategies, including genetic algorithms, often struggle with prohibitive computational costs. We introduce an efficient parameter estimation framework that combines a real-coded genetic algorithm with a novel adaptive simulation termination strategy. This strategy defines a time-dependent termination boundary based on population quantiles, which is permissive during early transients and becomes progressively stricter as simulations advance, explicitly accounting for the temporal structure of oscillatory behavior. Crucially, this mechanism facilitates the efficient identification and early simulation termination of poor parameter candidates, thus avoiding the computational expense of full-horizon simulations. The framework further integrates global exploration with the modified Powell method for rapid local refinement. Numerical experiments on two benchmark oscillatory models—the Lotka–Volterra and Goodwin oscillators—demonstrate that the framework reduces computational cost by approximately 30–50% compared to a baseline GA without this strategy. For the parameter-sensitive Goodwin model, the framework efficiently identifies candidates evolving toward damped oscillations caused by subtle parameter variations. Sensitivity analysis also confirms robustness across diverse hyperparameter settings, indicating that adaptive simulation termination provides a practical acceleration mechanism for inverse problems in systems biology where iterative objective function evaluation dominates runtime.

AppliedMath doi: 10.3390/appliedmath6030046

Authors: Seweryn Lipiński Łukasz Dziubiński Paweł Chwietczuk

The paper presents a mathematical validation and a local sensitivity analysis of a reduced-order thermal balance model designed to predict nighttime heat losses from an above-ground outdoor pool. The model expresses the total heat flux as a linear function of the water–air temperature difference through an effective overall heat-transfer coefficient aggregating convective, evaporative, and radiative mechanisms, as well as cover-related effects. The analysis is explicitly restricted to quasi-steady nighttime conditions. Field data were segmented into 13 independent nighttime realizations (∆T ≈ 5.5–26.9 °C, wind ≈ 0.00–1.32 m∙s−1). Across the entire dataset, the model achieved a mean relative error of 0.39% and a maximum absolute deviation of 3.72%, with no monotonic error growth versus ∆T or wind speed. Normalized local sensitivities reveal that the convective (hc) and evaporative (he) components dominate the response, whereas the radiative contribution is smaller under typical nighttime boundaries; the cover-permeability factor gains influence as wind speed increases. The additive structure limits independent identifiability of individual mechanisms, supporting an interpretation in terms of effective parameters. The results delineate the domain where the reduced-order formulation is predictive without refitting and provide a compact, interpretable reference for analyzing energy-balance models of open-water systems under nighttime operation.

AppliedMath doi: 10.3390/appliedmath6030045

Authors: Tristan Guillaume

This paper investigates the finite-horizon survival probability for a system of correlated arithmetic Brownian motions with heterogeneous drifts and volatilities, focusing on the event in which one component remains strictly below all others. Using a whitening transformation of the covariance structure, we reduce the problem to the survival of a standard Brownian motion in a simplicial cone, characterized by its spherical cross-section. While explicit solutions are available in low dimensions, we address the computationally challenging tetrahedral angular case. We derive a semi-analytic formula for the survival probability via an eigenfunction expansion of the Dirichlet Laplace–Beltrami operator on this curved domain. For efficient implementation, we construct a diffeomorphism from the spherical tetrahedron to a fixed Euclidean tetrahedron, enabling the computation of angular eigenpairs through a stable finite-element scheme. For higher-dimensional regimes, we also introduce a covariance-based difficulty index and geometric bounds based on an inscribed spherical cap to assess spectral convergence and estimate long-time decay rates. Numerical experiments show that this offline–online approach achieves high accuracy and substantial speedups relative to Monte Carlo benchmarks.

AppliedMath doi: 10.3390/appliedmath6030044

Authors: Vicente Peña Caballero Abraham Efraim Rodríguez-Mata Pablo Antonio López-Pérez Dulce J. Hernández-Melchor Víctor Alejandro González-Huitrón

This work proposes an integral-enhanced nonlinear PI2 state observer for the robust estimation of unmeasured states in nonlinear dynamic systems, with experimental validation on a flat-panel photobioreactor. The observer is designed as a virtual sensor to reconstruct key biological variables using a reduced set of online measurements and known operating conditions. Compared with a conventional extended Luenberger observer, the proposed structure improves estimation accuracy and robustness against constant disturbances and model mismatch, which are common in bioprocess applications. The experimental results show a clear performance advantage during transient growth phases while highlighting that the method relies on a locally valid model structure and appropriate gain tuning. Overall, the proposed observer provides a practical and scalable monitoring tool for nonlinear systems where the direct measurement of critical state is not feasible.

AppliedMath doi: 10.3390/appliedmath6030043

Authors: Mudassir Shams Bruno Carpentieri

Nonlinear systems with multiple roots arise frequently in biomedical and engineering models, yet their reliable numerical solution remains a challenging task. Many classical methods suffer from sensitivity to initial guesses, reduced convergence rates, and loss of accuracy in the presence of multiple or clustered solutions. In addition, the exploitation of parallelism to improve robustness and computational efficiency has received limited attention. In this work, we propose a high-accuracy parallel numerical framework of fourth-order convergence for the simultaneous approximation of all solutions of nonlinear systems with multiple roots. The proposed scheme is derivative-free and structurally decoupled, enabling efficient parallel implementation and robust convergence even when reliable initial approximations are unavailable. The effectiveness of the method is demonstrated on representative biomedical engineering models, including a glucose–insulin–glucagon regulatory network and a multi-compartment pharmacokinetic system, both exhibiting strong nonlinearity and multistability. Numerical experiments confirm stable convergence toward distinct solution clusters, machine-level accuracy, reduced residual norms, and improved computational performance when compared with existing approaches. These results indicate that the proposed framework provides a reliable and efficient alternative for solving nonlinear systems with multiple roots in complex applied settings.

AppliedMath doi: 10.3390/appliedmath6030042

Authors: Chih-Chiang Hong

An advanced frequency study in thick-walled functionally graded material (FGM) spherical shells is investigated with advanced shear correction. The values of advanced shear correction can be greater than one, be a negative value, and be affected by a nonlinear term of third-order shear deformation theory (TSDT) of displacements, FGM power law index, and temperature. It is novel and interesting to consider using TSDT and advanced shear correction to derive a simple homogeneous equation with reasonable simplifications into a symmetrical sparse matrix subjected to free vibration. The zero determinant of the symmetrical sparse matrix can be expressed to calculate the natural frequency by Newton’s method. The parameter effects of advanced shear correction, a nonlinear TSDT term, temperature, and the FGM power-law index on the natural frequencies of thick-walled FGM spherical shells are presented. The natural-frequency data for the axial and circumferential mode shapes are obtained. This is a new finding, as the assumed simplification in a sparse matrix causes a numerical truncation error; the natural-frequency values of the presented sparse matrix are much greater than those in a full matrix for thick-walled FGM spherical shells.

AppliedMath doi: 10.3390/appliedmath6030041

Authors: Felipe J. Torres Israel Martínez Antonio J. Balvantín Edgar H. Robles

Professors, students, and researchers from universities around the world use software distributed under licenses for numerical simulation purposes, which requires a computer with considerable hardware capabilities. This implies a high cost of simulations in engineering applications that require dynamic modeling using numerical methods, particularly in robotics and nonlinear control. This article compares and analyzes the performance of a frugal simulation scheme based on the use of low-cost, free, and open-source technology, specifically a low-power, single-board minicomputer (Raspberry Pi) in conjunction with GNU-Octave software. The benchmark is a numerical simulation of trajectory tracking control in the joint space of a Selective Conformal Assembly Robot Arm (SCARA). To perform this task, a system of coupled nonlinear differential equations is solved in matrix form using a numerical method known as an ODE solver. This solution includes the control law and the dynamic system model derived from Euler–Lagrange formalism. The time complexity and accuracy are analyzed to compare the performance of the frugal simulation tool with that of a conventional simulation setup consisting of a personal computer and MATLABTM running the same simulation code. The analysis shows minimal deviations in the numerical solutions and reasonable time complexity. Moreover, the frugality score of this approach and the low acquisition cost of the simulation tool enable the creation of simulation laboratories at universities with limited budgets for education and research.

AppliedMath doi: 10.3390/appliedmath6030040

Authors: Yanhui Ding Qiong Tang Sijia Tang

The paper utilizes the continuous finite element method to solve stiff ordinary differential equations and proves that the linear finite element method and the quadratic finite element method have A-stability in solving autonomous ordinary differential equations, and exponential dichotomy in solving non-autonomous ordinary differential equations. In the numerical experiments of nonlinear autonomous and non-autonomous strongly and moderately stiff ordinary differential equations, a relatively large step size of h=0.1 was adopted over a longer period of time, with the numerical solution accuracy reaching 10−4. The superconvergence order maintained the theoretical order. A new approach is provided for solving stiff ordinary differential equations.

AppliedMath doi: 10.3390/appliedmath6030039

Authors: Fotini Sereti Dimitrios Georgiou Theodoros Karakasidis

Genetic sequences play a central role in biological and medical research, and mathematics provides powerful means for their representation and analysis. Conventional approaches, such as the fuzzy polynucleotide space [0, 1]12, model codons as 12-dimensional vectors, but this comes at the cost of high dimensionality. In this study, we introduce two new models, Vector-Fuzzy-I and Vector-Fuzzy-II, that map codons and genetic sequences into the 4-dimensional Euclidean space ℝ4 using vector algebra and fuzzy set theory. In the first model, sequence structure is represented by successive vector addition, while in the second, it is represented by positional frequencies normalized by nucleotide locations. These low-dimensional representations are unique, preserve sequence order, and allow effective measurement of similarity and difference via Euclidean metrics. Compared with the fuzzy polynucleotide space, the proposed models achieve dimensionality reduction while enhancing the resolution of sequence differentiation. Our approach offers new mathematical perspectives for sequence analysis in theoretical biology.

AppliedMath doi: 10.3390/appliedmath6030038

Authors: Tahmineh Azizi

The complex and heterogeneous nature of cancer necessitates advanced modeling techniques to better understand tumor dynamics and inform treatment strategies. This paper explores the application of stochastic modeling in cancer research, focusing on five key areas: tumor growth kinetics, evolutionary dynamics of cancer, treatment response and resistance, spatial modeling of tumor progression, and clinical applications of stochastic models. We first examine how stochastic models capture the randomness in tumor growth and proliferation, providing insights into cellular behaviors that deterministic models may overlook. Next, we investigate the evolutionary dynamics that govern tumor heterogeneity and the emergence of resistance, highlighting the role of genetic mutations and environmental pressures. The paper also discusses how stochastic modeling can improve predictions of treatment responses, elucidating mechanisms behind therapy resistance in various tumor subpopulations. Furthermore, we address the significance of spatial modeling in understanding tumor interactions within their microenvironment, shedding light on processes such as metastasis. Finally, we emphasize the translational potential of these mathematical frameworks, demonstrating how they can enhance personalized medicine approaches in oncology. By integrating stochastic modeling into cancer research, this work contributes to a deeper understanding of cancer biology and paves the way for improved patient outcomes.

AppliedMath doi: 10.3390/appliedmath6030037

Authors: Chengxi Zhang

This paper addresses the consensus problem in multi-agent systems via a self-learning control scheme that directly reuses prior control information to accelerate transient coordination while maintaining robustness. I study agents with linear dynamics and external disturbances, and design a lightweight self-learning consensus control law for the distributed consensus domain, formulated as ui(t)=k1ui(t−τ)+k2si(t) with learning intensity k1 and learning interval τ. I provide a Lyapunov-based stability proof showing uniform ultimate boundedness of the consensus error under bounded disturbances. Compared to non-learning consensus laws, the proposed strategy achieves faster agreement with reduced long-term effort and retains simplicity suitable for resource-constrained multi-agent platforms, while also achieving decent performance against external disturbances. Simulations validate the improved transient speed and steady accuracy. The full-version-source code is open-sourced.

AppliedMath doi: 10.3390/appliedmath6030036

Authors: Calisto B. Marime Justin B. Munyakazi

Standard numerical methods such as Runge–Kutta and Euler methods have been widely used to approximate solutions to nonlinear systems. These methods converge to the solution only for small step sizes; for larger time steps, they generally generate spurious or chaotic solutions. In this paper, we consider a malaria propagation model with control for which we construct a second-order nonstandard finite difference scheme that preserves the important mathematical properties of the continuous model, which are positivity, boundedness, and stability of solutions irrespective of the step size. Moreover, we show that the equilibrium points of the discrete model are the same as those of the continuous model. By applying the double mesh principle, we provide evidence that the second-order NSFD scheme approximates the true solution with small errors. Theoretical assertions and numerical results show the advantages of the developed second-order nonstandard finite difference method.

AppliedMath doi: 10.3390/appliedmath6030035

Authors: Shi-Chu Li Hong-Yu Ma Bo-Wen Zhang Li-Yong Shen

Triangular mesh surface representation is widely adopted in geometric design and reverse engineering applications. However, in high-precision Computer Numerical Control (CNC) machining, significant limitations persist in automated Computer-Aided Manufacturing (CAM) tool path generation for such representations. Conventional CAM workflows heavily rely on manual engineering interventions, such as creating drive surfaces or tuning extensive parameters—a dependency that becomes particularly acute for generic free-form models. To address this critical challenge, this paper proposes a novel end-to-end single-step end-milling tool path generation methodology for triangular mesh surfaces in high-precision five-axis CNC machining. The framework includes clustering analysis for optimal workpiece orientation, normal vector distribution analysis to identify shallow and steep regions, Graphics Processing Unit (GPU)-accelerated collision detection for feasible tool orientation domains, and iso-planar tool path generation with Traveling Salesman Problem (TSP) optimization for efficient tool lifting and movement. Experimental validation confirms the framework ensures machining quality and algorithmic robustness.

AppliedMath doi: 10.3390/appliedmath6020034

Authors: Olugbade Ezekiel Faniyi Mark Ifeanyi Modebei Matthew Olanrewaju Oluwayemi Ikechukwu Jackson Otaide

This paper addresses the numerical integration of first-order ordinary differential equations by developing a continuous linear multistep block method. The method is constructed through the approximation of the exact solution using a linear combination of shifted Vieta–Lucas polynomials defined on the interval [0, 4]. The use of this polynomial basis extends traditional approximation approaches and provides improved stability while maintaining high-order accuracy. Theoretical analysis shows that the proposed method attains sixth-order convergence and possesses an extended stability interval of [−19.5,0], ensuring reliable performance for moderately stiff problems. Numerical experiments confirm that the method achieves lower errors and higher computational efficiency than conventional methods. These results demonstrate the suitability of the proposed approach for scientific computing applications, including engineering simulations and mathematical modeling, where accurate numerical integration of first-order differential equation is required.

AppliedMath doi: 10.3390/appliedmath6020033

Authors: Elsayed M. E. Zayed Mona El-Shater Ahmed H. Arnous Lina S. Calucag Anjan Biswas

The paper retrieves quiescent dispersive solitons in dispersion-flattened optical fibers having nonlinear chromatic dispersion and the Kerr law of self-phase modulation. The platform model is the Schrödinger–Hirota equation. The enhanced direct algebraic method has made this retrieval possible. The intermediary functions are Jacobi’s elliptic function and Weierstrass’ elliptic function. The final results appear with parameter constraints for the existence of such solitons.

AppliedMath doi: 10.3390/appliedmath6020032

Authors: Wullianallur Raghupathi

Modeling legal reasoning with artificial intelligence and machine learning presents formidable challenges. Legal decisions emerge from a complex interplay of factual circumstances, statutory interpretation, case precedent, jurisdictional variation, and human judgment—including the behavioral characteristics of judges and juries. This paper takes an exploratory approach to investigating how contemporary ML techniques might capture aspects of this complexity. Using pharmaceutical patent litigation as an illustrative domain, we develop a multi-layer analytical pipeline integrating text mining, clustering, topic modeling, and classification to analyze 698 U.S. federal district court decisions spanning January 2016 through December 2018, comprising substantive validity and infringement rulings under the Hatch-Waxman regulatory framework. Results demonstrate that the pipeline achieves 85–89% prediction accuracy—substantially exceeding the 42% baseline majority-class rate and comparing favorably with prior legal prediction studies—while producing interpretable intermediate outputs: clusters that correspond to recognized doctrinal categories (Abbreviated New Drug Application—ANDA litigation, obviousness, written description, claim construction) and topics that capture recurring legal themes. We discuss what these findings reveal about both the possibilities and limitations of computational approaches to legal reasoning, acknowledging the significant gap between statistical prediction and genuine legal understanding.

AppliedMath doi: 10.3390/appliedmath6020031

Authors: Edoardo Ballico

In Algebraic Statistics, M.A. Cueto, J. Morton and B. Sturmfels introduced a statistical model, the Restricted Boltzmann Machine, which introduced the Hadamard product of two or more vectors of an affine or projective space, i.e., the componentwise product of their entries, forcing Algebraic Geometry to enter. The Hadamard product X⋆Y of two subvarieties X,Y⊂Pn is defined as the Zariski closure of the Hadamard product of its elements. Recently, D. Antolini and A. Oneto introduced and studied the definition of Hadamard rank, and we prove some results on it. Moreover, we prove some theorems on the dimension and shape of the Hadamard powers of X. The aim is to describe the images of the Hadamard products without taking the Zariski closure. We also discuss several scenarios describing the case in which some of the data, i.e., the variety X, is wrong or it is not possible to recover it.

AppliedMath doi: 10.3390/appliedmath6020030

Authors: Moritz Sohns Ali Zakaria Idriss

This paper presents a topology-based approach to the general vector-valued stochastic integral for predictable integrands and semimartingale integrators. The integral is defined as a unique mapping that achieves closure under the semimartingale topology. While the topology and the closedness of the integral operator are well known, the method of defining the integral via this mapping is new and offers a significantly more efficient path to understanding the general stochastic integral compared to existing techniques. Instead of defining a basic integral and then extending it through a sequence of case distinctions, our construction performs a single topological closure: we define the vector stochastic integral as the unique continuous extension of the simple-predictable integral under the Émery topology, within the predictable σ-algebra. This single step yields the general predictable, vector-valued integral without invoking semimartingale decompositions, Doob–Meyer, or detours through H2/quasimartingale frameworks and without re-engineering from the componentwise to the vector case.

AppliedMath doi: 10.3390/appliedmath6020029

Authors: Preet Mishra Shyam Kumar Sorokhaibam Cha Captain Vyom R. K. Brojen Singh

Evolutionary changes can significantly impact interactions among populations and disrupt ecosystems by driving extinctions or collapsing population oscillations, posing substantial challenges to biodiversity conservation. This study addresses the ecological rescue of a predator population threatened by a mutant prey population using the optimal control method. To study this, we study a model that incorporates a genotypically structured prey population comprising wild-type, heterozygous, and mutant prey types, as well as the predator population. We prove that this model has both local and global existence and uniqueness of solutions, ensuring the model’s robustness. Then, we applied the optimal control method, incorporating Pontryagin’s Maximum Principle, to introduce a control input into the model and minimize the mutant population, thereby stabilizing the ecosystem. We utilize a reproduction number and a control efficacy measure to numerically demonstrate that the undesired dynamics of the model can be controlled, leading to the suppression of the mutant and the restoration of the oscillatory dynamics of the system. These findings demonstrate the applicability of optimal control strategies and provide a mathematical framework for managing such ecological disruptions.

AppliedMath doi: 10.3390/appliedmath6020028

Authors: Len Meas Chhaunny Chhum Phichhang Ou Mara Mong

This paper explores the intersection of three foundational areas—partial differential equations, financial mathematics, and probability—by providing a rigorous framework for the classical Black-Scholes–Merton option pricing model and its generalized extensions. For the classical model, a change in variables is employed to transform the Black-Scholes partial differential equation into the linear heat equation. The resulting formulation enables the use of Fourier transform techniques and the fundamental solution (heat kernel) to derive the closed-form Black-Scholes–Merton formula. To extend the classical setting, the interest rate in the discount factor and the stock’s rate of return are modeled using a multifactor Vasicek process, leading to a more sophisticated and realistic option pricing framework. In addition, a complementary derivation based on probabilistic methods, using a change in measure, yields an alternative explicit pricing formula. Numerical experiments based on Monte Carlo simulation show excellent agreement with the closed-form solutions and illustrate notable gains in computational efficiency. The comparative analysis further demonstrates that stochastic interest rates systematically produce lower option prices than the classical constant-rate model, underscoring the importance of accurate interest-rate modeling in practical valuation.

AppliedMath doi: 10.3390/appliedmath6020027

Authors: Jené Mercia van Schalkwyk Peter Joseph Witbooi Sibaliwe Maku Vyambwera Mozart Umba Nsuami

This article introduces and thoroughly examines a novel deterministic compartmental model of COVID-19 dynamics. The model uniquely incorporates compartments for symptomatic and asymptomatic individuals alongside an environmental reservoir for the pathogen. It also accounts for a steady inflow of infected visitors and a steady outflow from the removed class. The mathematical soundness of the model is established by identifying the invariant region and ensuring positivity of solutions. Notably, during surges of infected visitors, certain classes maintain positive minimum values. We analytically determine endemic equilibrium points and prove the global stability of the disease-free equilibrium. Sensitivity analysis highlights the significant roles of transmission rates and asymptomatic individuals in disease spread. Simulation results corroborate the theoretical findings and provide additional insights into the model’s predictive capabilities.

AppliedMath doi: 10.3390/appliedmath6020026

Authors: Muhammad Azam Imran Shabir Chuhan Muhammad Shafiq Ahmed Kaleem Arshid

Recent advances in physics-informed neural networks (PINN) have highlighted the need for systematic criteria for selecting appropriate algorithms to solve differential equations. This paper presents a numerical comparison between standard PINNs and gradient-enhanced PINNs (gPINNs) used to solve a high-order partial differential equations (PDE). To verify the accuracy and convergence behavior of all the methods, we solve a fourth-order PDE whose analytical solution is known. gPINN is recommended for problems requiring high accuracy in gradient fields or operating with sparse data, whereas standard PINN is advised for strongly nonlinear or computationally constrained scenarios. We synthesize our findings into a practical selection guide; gPINN is recommended for problems requiring high accuracy in gradient fields or operating with sparse data, whereas standard PINN is advised for strongly nonlinear or computationally constrained scenarios. This framework provides a clear, evidence-based policy for algorithm choice in SciML. Beyond numerical comparison, we provide an analytical interpretation linking solver performance to the spectral and stiffness properties of each PDE class, offering a principled basis for algorithm selection.

AppliedMath doi: 10.3390/appliedmath6020025

Authors: Sroor M. Elnady Mohamed A. El-Beltagy Mohammed E. Fouda Ahmed G. Radwan

The chain rule is a foundational concept in calculus, critical for differentiating composite functions, especially those appearing in modern AI techniques. Its extension to fractional calculus presents challenges due to the integral-based nature and intrinsic memory effects of these fractional operators. This survey provides a review of chain-rule formulations across major known FDs, including Riemann-Liouville (RL), Caputo, Caputo-Fabrizio (CF), Atangana-Baleanu-Riemann (ABR), Atangana-Baleanu-Caputo (ABC), and Caputo-Fabrizio with Gaussian kernel (CFG). The main contribution here is the introduction of a unified criterion, denoted as C, which synthesizes and extends previous classification frameworks for systematically formulating the chain rule across different operators. Each chain rule is examined in terms of its derivation, operator structure, and scope of applicability. In addition, the survey analyzes series-based approximations that appear in computing these derivatives, highlighting the minimum number of terms required to achieve acceptable mean absolute error (MAE). By consolidating theoretical developments, derivation methods, and numerical strategies, this paper provides a comprehensive resource for researchers and practitioners working with fractional-order models.

AppliedMath doi: 10.3390/appliedmath6020024

Authors: Pablo Ramos-Ruiz Antonio Miguel Fuentes-Jiménez José E. Ramos-Ruiz Inmaculada Jiménez-Manchado

In recent years, several quantum algorithms have been proposed for addressing combinatorial optimization problems. Among them, the Quantum Approximate Optimization Algorithm (QAOA) has become a widely used approach. However, reported limitations of QAOA have motivated the development of multiple algorithmic variants, including recursive hybrid methods such as the Recursive Quantum Approximate Optimization Algorithm (RQAOA), as well as the Quantum-Informed Recursive Optimization (QIRO) framework. In this work, we integrate the Quantum Alternating Operator Ansatz within the QIRO framework in order to improve its quantum inference stage. Both the original and the enhanced versions of QIRO are applied to the Minimum Vertex Cover problem, an NP-complete problem of practical relevance. Performance is evaluated on a benchmark of Erdös-Rényi graph instances with varying sizes, densities, and random seeds. The results show that the proposed modification leads to a higher number of successfully solved instances across the considered benchmark, indicating that refinements of the variational layer can improve the effectiveness of the QIRO framework.

AppliedMath doi: 10.3390/appliedmath6020023

Authors: Ioannis G. Tsoulos Vasileios Charilogis Dimitrios Tsalikakis

Artificial neural networks are reliable machine learning models that have been applied to a multitude of practical and scientific applications in recent decades. Among these applications, there are examples from the areas of physics, chemistry, medicine, etc. To effectively apply them to these problems, it is necessary to adapt their parameters using optimization techniques. However, in order to be effective, optimization techniques must know the range of values for the parameters of the artificial neural network, so that they can adequately train the artificial neural network. In most cases, this is not possible, as these ranges are also significantly affected by the inputs to the artificial neural network from the objective problem it is called upon to solve. This situation usually results in artificial neural networks becoming trapped in local minima of the error function or, even worse, in the phenomenon of overfitting, where although the training error achieves low values, the artificial neural network exhibits low performance in the corresponding test set. To address this limitation, this work proposes a novel two-stage training approach in which a simulated annealing (SA)-based preprocessing stage is employed to automatically identify optimal parameter value intervals before the application of any optimization method to train the neural network. Unlike similar approaches that rely on fixed or heuristically selected parameter bounds, the proposed preprocessing technique explores the parameter space probabilistically, guided by a temperature-controlled acceptance mechanism that balances global exploration and local refinement. The proposed method has been successfully applied to a wide range of classification and regression problems and comparative results are presented in detail in the present work.

AppliedMath doi: 10.3390/appliedmath6020022

Authors: Luisa Carracciuolo Ugo D’Amora Raffaele Dubbioso Ines Fasolino

Amyotrophic Lateral Sclerosis (ALS) is a progressive neurodegenerative disorder for which despite its severity, no validated biomarker currently exists to support early diagnosis, limiting therapeutic effectiveness and patient survival. In this context, mathematical modeling therefore becomes essential: it allows us to maximize the information obtainable from a limited number of samples, identify patterns that may not be directly observable, and estimate the relative contribution of different molecular markers to ALS progression. In this work, we propose methods for qualitatively and quantitatively evaluating the relevance of selected biomarkers in ALS classification and disease-state identification and laying the foundations for the definition of a protocol useful for constructing “digital twins” of the entire process of study, diagnosis, and treatment of the disease from the perspective of innovative precision medicine.

AppliedMath doi: 10.3390/appliedmath6020021

Authors: Ibrahim Mohammed Dibal Yeak Su Hoe

This study introduces a novel single-step hybrid block method with three intra-step points that attains fifth-order accuracy, offering an accurate and computationally economical tool for solving first-order differential equations. The method is specifically designed to handle first-order differential equations with efficiency and precision while employing a constant step size throughout the computation. To further enhance accuracy, interpolation techniques are incorporated to approximate function values at specific positions, addressing the fundamental properties of the method and verifying its mathematical soundness. These analyses confirm that the scheme satisfies the essential requirements of stability, consistency, and convergence, ensuring reliability in practical applications. In addition, the method demonstrates strong adaptability, making it suitable for a broad spectrum of problem settings that involve both stiff and non-stiff systems. Numerical experiments are carried out, and the results consistently demonstrate that the proposed method is robust and effective under various test cases. The outcomes further reveal that it frequently outperforms several existing numerical approaches in terms of both accuracy and computational efficiency.

AppliedMath doi: 10.3390/appliedmath6020020

Authors: Vasiliki Pantoula Vasileios Mandikas Tryfon Daras

The analysis of multivariate data is a central issue in biomedical research, where the accurate classification of patients and the extraction of reliable conclusions are of critical importance. Linear Discriminant Analysis (LDA) remains one of the most established methods for both dimensionality reduction and classification of data. In this paper, we examine in detail the theoretical foundations, assumptions, and statistical properties of LDA, and apply the method step by step to real data from the Breast Cancer Wisconsin (Diagnostic) database, which includes cellular features from breast biopsy samples with the aim of distinguishing benign from malignant tumors. Emphasis is placed on the importance of the method’s assumptions, such as multivariate normality, equality of covariance matrices, and absence of multicollinearity, demonstrating that their fulfillment leads to significant improvements in model performance. Specifically, careful preprocessing and strict adherence to these assumptions increase classification accuracy from 95.6% (94.7% cross-validated) to 97.8% (97.4% cross-validated). To our knowledge, this study is the first to demonstrate the dual use of LDA as both a dimensionality-reduction tool and a predictive classification model for this medical database within the same biomedical analysis framework. Moreover, we provide, for the first time, a systematic comparison between our assumption-aware LDA model and related studies employing the most accurate machine-learning classifiers reported in the literature for this dataset, showing that classical LDA achieves accuracy comparable to these more complex methods. The resulting discriminant model, which uses 13 variables out of the original 30, can be applied easily by clinical researchers to classify new cases as benign or malignant, while simultaneously providing interpretable coefficients that reveal the underlying relationships among variables. The implementation is carried out in the SPSS environment, following the theoretical steps described in the paper, thus offering a user-friendly and reproducible framework for reliable application. In addition, the study establishes a structured and transparent workflow for the proper application of LDA in biomedical research by explicitly linking assumption verification, preprocessing, dimensionality reduction, and classification.

AppliedMath doi: 10.3390/appliedmath6020019

Authors: Hening Huang

In many scientific and engineering fields (e.g., measurement science), a probability density function often models a system comprising a signal embedded in noise. Conventional measures, such as the mean, variance, entropy, and informity, characterize signal strength and uncertainty (or noise level) separately. However, the true performance of a system depends on the interaction between signal and noise. In this paper, we propose a novel measure, called “inforpower”, for quantifying the system’s informational power that explicitly captures the interaction between signal and noise. We also propose a new measure of central tendency, called “information-energy center”. Closed-form expressions for inforpower and information-energy center are provided for ten well known continuous distributions. Moreover, we propose a maximum inforpower criterion, which can complement the Akaike information criterion (AIC), the minimum entropy criterion, and the maximum informity criterion for selecting the best distribution from a set of candidate distributions. Two examples (synthetic Weibull distribution data and Tana River annual maximum streamflow) are presented to demonstrate the effectiveness of the proposed maximum inforpower criterion and compare it with existing goodness-of-fit criteria.