Abstract: Customer segmentation is a core application of data mining in the retail industry. Traditional K-means clustering is widely adopted here for its simple principle and high computational efficiency, yet it has notable drawbacks: random initial clustering centers easily lead to local optimal solutions, it is highly sensitive to abnormal data, and the cluster number K relies on manual experience, resulting in unstable clustering performance. This paper designs an improved K-means algorithm, which filters outliers through a two-layer mechanism combining Local Outlier Factor and distance threshold. It also constructs a multi-index system with Silhouette Coefficient, Calinski-Harabasz and Davies-Bouldin indices to automatically determine the optimal K-value, optimizes initial centers via density peak clustering, and introduces weighted Euclidean distance to enhance clustering compactness. Experiments on the Mall Customer Segmentation dataset compare the proposed algorithm with traditional K-means, K-medoids and DBSCAN. Results show it achieves a Silhouette Coefficient of 0.5821, a CH index of 1025.36 and a DB index of 0.5107, outperforming all comparison algorithms in all indicators with more reasonable and stable clustering results. Applied to mall customer segmentation, this algorithm divides customers into 5 groups with distinct characteristics, providing solid data support for malls to formulate scientific and differentiated marketing strategies.

Abstract: In this paper, the general moment form of Green’s Theorem curve integral, for the calculation of the area and moments values of the planar region enclosed by arbitrary curve, is derived from the numerical version of its belonging moment integrals that are obtained from discrete vector product and differential vector product properties. Then, all six area and moments integrals are derived in a new, uniform integral form, for calculation of: area itself, its static moments, area centroids and moments of inertia of observed enclosed area region below arbitrary curve, based on the initial Green’s theorem curve integral for the calculation of the area enclosed by general curve, and the properties of discrete and differential vector products.

Abstract: This paper is a comparative analysis of classical and fractional derivatives models using Mittag-Leffler function. The Caputo fractional derivative is used to generalize the classical exponential decay model in order to include memory effects. Transform methods are used to obtain solutions of the forms of the Mittag-Leffler function. Numerical simulations are performed to analyze the behavior of the system for different fractional orders. The findings reveal that the fractional model generalizes the classical solution and it features slower decay because of memory effects. The analysis is further generalized to second order system, in which the system results in a damped oscillatory behavior due to the presence of a fractional dynamics. The results indicate the significance of the fractional calculus in the modeling of complicated physical systems.

Abstract: This paper builds upon a previous study of rotation sequences with four factors, in which the additional parameter is used for optimization, extending the result to generic rigid motions in three-dimensional Euclidean space. To do that in practice, one uses dual projective quaternions (dual Rodrigues’ vectors) describing screw motions and applies the well-known principle of transference in a rather straightforward manner. There are, however, some technicalities worth discussing, like the famous gimbal lock problem emerging in Euler-type decompositions. Also, there is ambiguity in the cost function which in this case includes both spherical and Euclidean distance—for pure rotations and translations, respectively. Since the relative cost of each counterpart depends on engineering details, we consider them separately. Explicit closed-form solutions are derived, based only on geometry, but numerical examples are also provided for illustration.

Abstract: This paper deals with some expressions of the fractional generalized Gronwall inequality when associated with both non-negative and non-positive singular kernels and establishes sharp Mittag- Leffler bounds containing different ingredients. The long-term behavior of non-autonomous fractional order systems by means of modified fractional Lyapunov theorems is analyzed. As an application, we give a few examples that use quadratic Lyapunov functions for typical fractional order systems to predict trajectories that ultimately aim to reach vector 0 as t → ∞.

Abstract: Nonlinear equations arise extensively in engineering and applied sciences. This study introduces a family of Caputo and Atangana–Baleanu–Caputo (ABC) fractional order iterative methods for solving nonlinear problems. The proposed schemes are designed to enhance convergence behavior and improve robustness compared to existing fractional Newton-type methods. Local convergence is analyzed using fractional Taylor expansions, establishing the order of convergence and associated error equations. In addition, a dynamical systems perspective is adopted to investigate global convergence properties through basin of attraction analysis, including fractal structures and the Wada measure. Numerical experiments on application-inspired nonlinear models demonstrate that the proposed methods achieve faster error reduction, lower residuals, and improved computational efficiency compared to existing schemes. These results indicate that the proposed framework provides an effective and flexible approach for solving nonlinear equations, combining accuracy, stability, and dynamical insight.

Abstract: Traditional time-series analysis methods, such as Fourier and wavelet transforms, excel at identifying frequency components and their temporal localization. While powerful for spectral analysis, these methods do not explicitly capture the global geometric structure of state transitions or the emergence of cyclic (non-conservative) dynamics within the signal. In this paper, we propose a novel geometric framework that encodes the local complexity dynamics of a time series as a simplicial complex. Using a sliding Hann window, we map the signal into a sequence of local power spectral density (PSD) distributions. We construct a Vietoris-Rips complex using the Wasserstein distance to preserve the physical metric of frequency shifts, and define a directed edge flow based on the asymmetry of Kullback-Leibler (KL) divergence. Applying discrete Hodge decomposition to this flow separates the dynamics into gradient, curl, and harmonic components.Baseline experiments with synthetic signals demonstrate that our method robustly discriminates commensurable signals (gradient-dominant), incommensurable quasi-periodic signals (emergence of curl flow), and stochastic noise (curl-dominant decomposition). An exploratory application to empirical photoplethysmography (PPG) data demonstrates the framework's capability to characterize real-world biological fluctuations, showing that PPG trajectory patterns are structurally similar to those of incommensurable quasi-periodic signals. In a pilot study with 53 PPG recordings, the harmonic component showed a statistically significant correlation with heart rate that is not explained by standard heart rate variability (HRV) features, suggesting the framework extracts genuinely novel information from physiological signals. This framework offers a potential new mathematical lens for quantifying and classifying the hidden topological structures of time-series data, laying a foundation for future empirical applications and explorations across diverse scientific domains.

Abstract: We study the center problem for polynomial maps y=f(x)=−∑n=0∞anxn+1, arising from homogeneous algebraic curves x+y+∑k=0nαn−k,kxn−kyk=0. While explicit conditions were previously known only for low even degrees n = 2,4,6,8,10, their general structure remained conjectural. In this paper we resolve the case n = 12 and prove that the observed algebraic patterns completely characterize the center for all even degrees n. More precisely, we show that the center condition is equivalent to one of two explicit families of algebraic relations. This provides a complete classification of the center problem in the homogeneous case.

Abstract: We study the problem of constructing group-invariant embeddings that faithfully represent data modulo group symmetries, a task that arises naturally in signal processing, physics, and machine learning. A central challenge is to design embeddings that are simultaneously orbit-separating, stable, and computationally tractable. To address this, we develop a general lifting framework for constructing such embeddings. The key idea is to start from a group-invariant embedding defined on a low-dimensional reduced space, and lift it to the ambient space by composing it with a finite family of parameterized linear maps, followed by an aggregation step that produces a global embedding. This framework provides a unified perspective that connects classical problems such as phase retrieval and permutation-invariant embeddings. We demonstrate the effectiveness of this framework in the finite group setting. In this setting, we establish general sufficient conditions for orbit separation and prove that any orbit-separating lifting embedding is automatically bi-Lipschitz. We further extend the bi-Lipschitz result to sparse regimes, and show that, when applied to phase retrieval, it yields an equivalence between uniqueness and stability for real sparse phase retrieval.

Abstract: Servo motors typically utilize Field-Oriented Control (FOC). However, the conventional cascaded PI control framework is inherently constrained by its fixed-parameter design, making it highly susceptible to parameter variations and unmodeled disturbances. While intelligent control strategies—such as model predictive control (MPC)—provide a robust, multi-objective alternative, their intensive stepwise computational demand often degrades transient response. Motivated by the stochastic dynamics of motor operation, we propose a novel physics-informed control paradigm. Specifically, we formulate the FOC-based motor control as an online stochastic optimization problem, wherein the objective function is updated iteratively using stochastic gradient estimates, and the resulting time-varying subproblems are solved efficiently by the MSALM algorithm. Our approach significantly outperforms conventional PI controllers in environmental adaptability and disturbance rejection. Experimental results demonstrate that the proposed method achieves comparable high-precision tracking performance while significantly reducing computational time per iteration, ensuring rapid dynamic response and strict enforcement of physical constraints.

Abstract: China served as the primary source of novel materials and innovations that significantly contributed to the development of medieval Europe. In this study, I employ an unconventional approach grounded in the mathematics of ornamental arts to trace the trajectory of Chinese goods to theWest. Utilizing the concept of the wallpaper group, this research analyzes Chinese ornaments to discern similarities with the artwork of the Arabs and Turkish Seljuks during the 8th to 12th centuries. Furthermore, it elucidates the mechanisms through which Chinese art reached theWest, thereby providing insights into the migration of technology.

Abstract: Spring flood modeling is a major tool used to understand the risks of severe hydrological events in the context of climate change. In Kazakhstan spring of 2024 was a rapid shift from cold to mild temperatures in just a few weeks. While traditional flood forecasting methods have been limited in their ability to consider the interactions of the natural processes that create runoff, this research sets out to address these limitations by modelling the formation of spring floods using a numerical approach. The study discusses the effects of snowmelt and freeze-thaw processes on surface runoff and the flooding that results from it in the northern and western regions of Kazakhstan. A comprehensive model has been developed that considers the heat transfer in soil, infiltration of meltwater, and propagation of runoff. Numerical modelling indicates that in 2024, relative to 2021, there was earlier soil thawing and shallower depths of soil freezing. However, an increase in the intensity of snowmelt leads to the fact that the infiltration capacity of the soil is insufficient, despite the formation of a thawed layer. As a result, a higher surface runoff is formed. Using Saint-Venant’s equations to perform calculations indicates that higher values of current depth and velocity of runoff were observed in 2024 than in 2021, indicating a greater likelihood of flooding. Therefore, it can be concluded that increases in winter temperatures have the potential to create an increased flooding impact due to changes in the proportions of surface runoff and infiltration.

Abstract: The paper proposes a new model of Tuberculosis (TB) dynamics taking into account multi-drug resistant forms, which takes into account the detection of infected people with and without bacterial excretion. The model is described by a system of nine nonlinear ordinary differential equations united by the law of mass action and controlled by 10 epidemiological parameters. The conditions for the stability of the system’s equilibrium states are obtained, and the sensitivity-based identifiability analysis of the model is conducted using the Sobol method. Based on Bayesian optimization, the boundaries of sensitive parameters are specified and posterior distributions of the model parameters are obtained for five regions of the Russian Federation based on statistics from 2009 to 2020. It is shown the heterogeneity of epidemic situation by wide credible intervals of correlated parameters of virus contagiousness, the proportion of infected TB converting to the bacterial excretion form and the rate of detection of TB infected with bacterial excretion. Probabilistic forecasts of the expected number of TB infections to 2025 are constructed and validated to the 2021-2023 data.

Abstract: Fractional quantum calculus provides a flexible mathematical framework for incorporating memory and scaling effects into numerical models. However, classical iterative methods for nonlinear equations often suffer from limited stability, strong dependence on initial guesses, and restricted convergence domains, particularly for highly nonlinear problems. In this work, we introduce a new Caputo fractional--quantum iterative scheme, denoted by MSB$_{\mathfrak{q}:\alpha}$, formulated as a parameterized two-step method based on a Caputo-type fractional quantum derivative. The proposed framework incorporates additional structural parameters that regulate the iterative dynamics and provide enhanced control over convergence behavior and stability properties. To investigate the performance of the proposed scheme, we employ tools from complex dynamical systems, including stability analysis and fractal basin investigations in the complex plane. These analyses illustrate how the fractional and quantum parameters influence the geometry of attraction domains and the global convergence behavior of the method. Numerical experiments on representative nonlinear test problems motivated by engineering and biomedical applications demonstrate improved robustness with respect to initial guesses, reduced residual errors, and competitive computational efficiency compared with classical iterative solvers. Overall, the results indicate that the proposed fractional--quantum framework provides an effective and flexible approach for the numerical solution of challenging nonlinear equations.

Abstract: This work presents a distributed optimal control problem for steady-state heat conduction in a system made up of two solids in thermal contact, with heat flux continuity and a temperature jump at the interface. The control affects the system’s energy source. The existence and uniqueness of optimal control are established, and the corresponding optimality conditions are derived. Additionally, it is shown that optimal control can be considered a fixed point of a well-defined operator. Moreover, an iterative algorithm is introduced to approximate the solution to the optimal control problem, which converges regardless of the initial data. Finally, an explicit solution related to one-dimensional case in Cartesian coordinates is given.

Abstract: Energy consumption demand forecasting plays a critical role in the planning and development of the nation’s energy security, which underpins the 8-year Power Development Plan (PDP8) and Vietnam’s ambitious Net-Zero 2050 commitment. However, this task becomes more difficult while challenges the big data environment is filled with a lot of noise and high fluctuation data. In order to deal with the problem, this paper using four distinct models which are Linear Regression, Holt’s (Additive), PSO-GM (1,1), and Support Vector Regression (SVR) to conduct a rigorous comparative analysis to identify the most accurate forecasting model. The performance evaluated by MAE, RMSE, MAPE indexes based on the Vietnam’s total primary energy demand data from 1986 to 2024. To check the accuracy of forecasting model, this study slits the length of data was into two period time, first time for the training data (1986- 2016) and next time for the testing data set (2017-2024). The results decisively identified that the Holt’s model achieving significantly outperforming all counterparts with the lowest error metrics (MAE = 89.33, RMSE = 99.50, and a MAPE of 7.19%). This model is strongly suggested to forecast the Vietnam’s energy demand in the period time 2025 to 2030. Based on this model, the 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.

Abstract: In this comprehensive study, we meticulously investigated multidimensional data analysis techniques, particularly focusing on Tucker decomposition methods, spanning the period from 2000 to 2025. Our primary objective was to discern trends, advancements, and applications of these techniques across various domains of knowledge and how they have evolved over time. An extensive corpus of 288 scientific articles related to tensor decompositions, Tucker models and applications was previously reviewed. Multivariate methods such as text mining using IraMuteq software and MANOVA-Biplot were employed to visualize identified data patterns, and the analytical capability of ChatGPT artificial intelligence was assessed to provide contextual insights and add another layer of information to the research. Our conclusions underscore the importance of blending traditional statistical approaches with natural language processing prowess to achieve a profound understanding of the data. This analysis offers a comprehensive perspective on the evolution and application of multidimensional data analysis techniques, with a special emphasis on the enduring relevance of Tucker techniques in this new millennium.

Abstract: Quantitative analytics has assumed a growing role in Philippine policy research as government and sectoral databases have become increasingly central to planning, monitoring, and resource allocation. This review synthesizes recent work in education, energy, agriculture, health, and finance to examine how forecasting, statistical forensics, and predictive modeling have been applied to Philippine policy problems. Across these sectors, the literature shows a clear methodological progression from descriptive diagnostics and classical time-series models toward comparative machine learning, deep learning, explainable artificial intelligence, nonlinear embedding, and Benford-based anomaly detection. Several recurring strengths emerge, particularly the consistent use of official Philippine datasets, transparent model benchmarking, and close alignment with practical policy concerns such as dropout reduction, electricity and crop planning, disease surveillance, and financial forecasting. At the same time, important limitations remain, including limited multivariate and spatial modeling, uneven validation practices, and relatively little attention to uncertainty quantification and operational deployment. In comparison with the broader international literature, the strongest contributions are those that position analytics as a support tool for planning and monitoring, while the main gaps lie in external validation, richer explanatory structures, and decision-oriented system integration. The evidence suggests that the next phase of Philippine policy analytics should move beyond isolated single-series applications toward integrated frameworks that combine forecasting, data-quality assessment, explainable modeling, and sector-specific decision thresholds for routine governance.

Abstract: Let f (z) = ∑_n≥0 a_nzⁿ be analytic at the origin, and assume that no Taylor coefficient vanishes. We study the normalized Taylor tails T^f_n (w) := ∑_k≥0 a_n+k/a_nw^k, not as isolated remainders but as a discrete renormalization orbit on the space of normalized analytic germs. The governing map is the nonlinear operator S(F)(w) := F(w)−1 wF′ (0) , which acts as a coefficient shift followed by canonical normalization. The exact identity T^f_n+1 = S(T^f_n) turns the Taylor coefficients of f into a dynamical system. We develop a self-contained theory of this dynamics. First, we prove that the nonlinearity of S is exactly linearized in ratio coordinates: the map F→ (Sⁿ F)′(0)
_n≥0 conjugates S to the one-sided shift on an explicit space of admissible ratio sequences. This yields complete reconstruction of the orbit and a realization theorem for all admissible orbits. Second, we classify the rigid orbit types: fixed points are exactly geometric series, periodic points are exactly rational functions with denominator 1 − Λw^m, and eventual periodicity is equivalent to a polynomial plus a rational tail. Third, we exhibit genuinely rich internal dynamics by constructing compact invariant subsystems on which S is conjugate to full shifts on finite alphabets. Fourth, on the asymptotic side, ratio limits force universal geometric profiles, while first- and second-order corrections to the coefficient ratios produce universal corrections to the tail orbit. In particular, dominant algebraic singularities leave a precise first asymptotic fingerprint on the renormalized tails. We also prove exact transport laws under differentiation and Hadamard products. The basic normalized tail object overlaps, up to an index shift, with the normalized remainders recently studied in the special-functions literature. The present contribution is different in focus: it isolates the renormalization operator itself, proves exact shift linearization and orbit realization, identifies symbolic invariant subsystems, and develops rigidity and asymptotic classification results for the resulting dynamical flow.

Abstract: Distributed heterogeneous hybrid flow shop scheduling with job deadlines and priorities (DHHFSP-JDP) is a combination of scheduling problem and distributionary environment. Addressing complex work sequences and energy consumption in distributed manufacturing with heterogeneous plants is a major challenge. It is necessary for optimizing total weighted delay (TWD) and total energy consumption (TEC) in distributed heterogeneous green hybrid flowshops. A model using mixed integer linear programming is applied to describe DHGHFSP-JDP and a decomposition-based coevolutionary algorithm (DBCEA) is considered to be the solution in this article. In this approach, (1) a decomposition-based heuristic initialization is proposed, in which an initialization strategy with a randomly sized population is adopted to establish effective initial schedules. (2) elite selection strategy based on the integration of an external archive and an elite archive. (3) four problem-based operator selection strategies embedded in a cooperative local search framework, and an Upper Confidence Bound (UCB) mechanism to design a strategy for selecting local search operators. In the end, The superiority of DBCEA is validated through comparative experiments against several advanced algorithms across 20 benchmarks, with results showing it often provides the best Pareto solution set.