AppliedMath, Volume 6, Issue 5 (May 2026) – 18 articles

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. Full article

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. Full article

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. Full article

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. Full article

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. Full article

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. Full article

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. Full article

Adversarial perturbations measured by ${\mathsf{\ell }}_p$ 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 $C_{\mathrm{atk}}(\mathit{x},{\mathit{x}}^{\prime })=\tau max\{0,m\left({\mathit{x}}^{\prime }\right)\}+{\lambda }_1{\mathit{\delta }}^{\top }{\mathit{G}}_{\perp }^{\left(\gamma \right)}\left(\mathit{x}\right)\mathit{\delta }+\lambda {\displaystyle \sum _j}{\omega }_j|{\delta }_j|+{\lambda }_2{\mathit{\delta }}^{\top }{\mathit{L}}_H\mathit{\delta }+{\lambda }_3\left[log{\widehat{f}}_1^{\left(ϵ\right)}\left(\mathit{x}\right)-log{\widehat{f}}_1^{\left(ϵ\right)}\left({\mathit{x}}^{\prime }\right)\right]+{\displaystyle \sum _{j\in supp\left(\mathit{\delta }\right)}}{c}_j+\beta {\left|\mathcal{M}(supp\left(\mathit{\delta }\right))\right|}^{\nu },$ 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 ${\lambda }_3{\kappa }_K$-weak convexity of the continuous component on compact convex sets, for both affine and ${\rho }_m$-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 ${\mathsf{\ell }}_2$ norm can incur costs differing by an order of magnitude. Full article

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, $q_2$ and $q_3$, 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. Full article

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. Full article

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. Full article

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. Full article

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. Full article

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. Full article

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. Full article

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. Full article

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. Full article