Mathematical Applications and Statistical Rigor (MASR)
Mathematical Applications and Statistical Rigor (MASR) is an international, peer-reviewed scholarly journal devoted to advancing research at the intersection of deep mathematical theory, rigorous statistical methodology, and impactful real-world applications. MASR is founded on the conviction that mathematics and statistics achieve their highest value when they are not only theoretically sound but also decisively relevant to contemporary scientific, technological, and societal challenges. MASR is more than a publication venue; it is a global research community committed to building a durable bridge between abstraction and application. The journal serves as a meeting point for theorists, applied scientists, and interdisciplinary researchers who seek to translate mathematical and statistical innovation into robust, actionable solutions. Researchers are welcome to submit papers to MASR after preparing them according to the SIRG template and following the SIRG Publisher Ethics and Malpractice Statement. (MASR) follows the SIRG Publisher peer-review process in handling all submitted papers. By submitting their paper to MASR, authors confirm their agreement to SIRG Publisher terms, conditions, and publication ethics and Copyrights page. ISSN (Online): 3071-3811 Frequency: 6 Volumes Annually Editor-in-Chiefs: Prof. Haroon Barakat Department of Mathematics, Faculty of Science, Zagazig University, Zagazig- Egypt. hbarakat@zu.edu.eg, hbarakat2@hotmail.com, hbarakat2@yahoo.com Prof. Haroon Mohamed Abdel Fattah Barakat, a distinguished Professor of Mathematical Statistics at the Faculty of Science, Zagazig University, Egypt. His academic background includes a Ph.D. in Mathematical Statistics from Moscow State University (1986), an M.Sc. from Alexandria University (1980), and a B.Sc. in Statistics (1976). His research profile is strongly oriented toward probability theory, statistical inference, reliability theory, extreme value theory, and order statistics, with a particular emphasis on asymptotic theory and stochastic modeling. A significant part of his work contributes to the development of order statistics, generalized order statistics, record values, and their applications in statistical reliability and environmental modeling. Prof. Barakat has made substantial contributions to advanced distribution theory, including multivariate and bivariate models, copula-based distributions, and generalized families of probability distributions. His research also extends to stress–strength reliability models, Bayesian and non-Bayesian inference, censoring schemes, and bootstrap methodologies, which are widely applied in survival analysis and engineering statistics. He has an extensive publication record with more than two hundred peer-reviewed papers in international journals, covering topics such as: Limit theorems and asymptotic behavior of order statistics Extreme value theory and tail modeling Statistical dependence structures and copula models Information measures including entropy and extropy Reliability analysis and stochastic processes Bayesian estimation under complex sampling and censoring schemes In addition to theoretical contributions, his research includes applied statistical modeling in environmental studies, air pollution analysis, engineering systems, and biomedical data. Prof. Barakat has also supervised a large number of Ph.D. and M.Sc. theses, many of which focus on modern topics in statistical inference, stochastic modeling, and applied probability. His academic service includes editorial roles in international journals, refereeing for major statistical journals, and participation in national and international scientific committees. He has been an active participant in international conferences and workshops across Europe, Asia, and the Middle East, frequently serving as invited speaker, session chair, and scientific committee member. His contributions have been recognized through multiple national and international awards in mathematics and statistical sciences. Overall, his scientific work reflects a sustained and influential contribution to modern mathematical statistics, particularly in the theory of ordered data, reliability modeling, and stochastic processes, with strong integration between theoretical development and real-world applications. ================================================= For more details, please get in touch with MASR: Journal Contact Email: Email: ( editorial.masr@scientificirg.com )
Scientific Innovation Research Group (SIRG)
Mathematical Applications and Statistical Rigor (MASR) content is published under a Creative Commons Attribution License (CCBY). This means that content is freely available to all readers upon publication, and content is published as soon as production is complete. Mathematical Applications and Statistical Rigor (MASR) seeks to publish the most influential papers that will significantly advance scientific understanding. Selected articles must present new and widely significant data, syntheses, or concepts. They should merit recognition by the wider scientific community and the general public through publication in a reputable scientific journal.
Dynamical Characteristics and Approximate Technique to Solve the Model of Nonlinear Biological Reactions
In this paper, we present a rigorous mathematical analysis of the nonlinear Michaelis-Menten biochemical reaction model, formulated as a system of non-dimensional coupled nonlinear ordinary differential equations (ODEs). Using the classical theory of ordinary differential equations, comparison principles, and Lyapunov's direct method, fundamental qualitative properties of the model, such as existence, uniqueness, non-negativity, boundedness, and local and global asymptotic stability of solutions, are established. The Elzaki Transform Homotopy Perturbation Method (ETHPM) is used to obtain an accurate approximate analytical solution, and its accuracy is studied by direct comparison with the fourth-order Runge-Kutta (RK4) method and error analysis. Numerical simulations confirm the temporal dynamics of biochemical systems and validate the efficiency and accuracy of the proposed semi-analytical approach. The results indicate that the Elzaki transform decomposition framework provides a computationally efficient, linearization-free, and highly accurate tool for analyzing nonlinear biochemical reaction models with wider applicability to a broad class of nonlinear problems arising in mathematical biology and applied science.
Mathematical and Statistical Methodology
Computational and Data-Driven Mathematics
Copyright (c) 2026 Mathematical Applications and Statistical Rigor (MASR) https://creativecommons.org/licenses/by/4.0
Hybrid Entropy–Decimal Approaches for Isomorphism Detection of Factorial Designs
In experimental design, selecting an optimal design from a vast space of candidates is a fundamental challenge. Many candidate designs are isomorphic-structurally equivalent under relabeling factors, reordering runs, or switching levels and thus statistically indistinguishable. Retaining such redundant designs wastes computational resources and obscures truly distinct experimental strategies. Existing criteria for detecting non-isomorphic designs fail to distinguish all non‑isomorphic classes (e.g., the 68 classes of 27-run three-level designs). More powerful projection‑based methods achieve perfect discrimination but are computationally expensive. This paper bridges the gap by introducing simple yet powerful hybrid methods that combine entropy with a decimal‑system transformation. The approach converts the structural information of a design into a compact decimal vector and then computes an enhanced entropy that incorporates the magnitude of the values, producing a scalar signature invariant under isomorphism. The proposed algorithms work seamlessly for both regular and irregular factorial designs. Extensive experiments on three challenging families-18-run three‑level designs (3 non‑isomorphic classes), 27‑run three-level designs (68 non‑isomorphic classes), and 32‑run four‑level designs (20 non‑isomorphic classes) show that the method achieves perfect discrimination at low computational cost. The framework is simple, universal, and efficient, making it a valuable tool for design theorists and practitioners.
Stochastic, Dynamical, and Complex Systems (Mathematical Perspective)
Bernoulli Polynomials for solving three-dimensional Volterra-Fredholm integral equations of the second kind
In this work. our approach for solving three-dimensional linear Volterra-Fredholm integral equations (3D-VFIEs) based on Bernoulli polynomials. This approach was previously applied to solve two-dimensional Volterra-Fredholm integral equations. This method transforms the 3D-VFIEs into a system of linear algebraic equations. The Banach fixed-point theorem is employed to demonstrate the existence and uniqueness of the 3D-VFIE. It has demonstrated its efficiency and effectiveness in achieving accurate numerical results, outperforming other methods. The numerical method that we relied on in solving (3D-VFIEs) as explained in this paper, and the approximate solution that we reached it by comparing it to other solutions that were deduced by other methods showed us very close results for some methods such as the Lucas and Shifted Chebyshev polynomials methods and more efficient than other ones such as Haar Wavelet’s technique, Block-pulse functions and Modified block-pulse functions. Compared to other error rates, the error coefficient was the best. This is supported by examples of numerical solutions to linear integral equations, statistical tables, and figures, which provide the strongest evidence of the convergence of the exact and approximate solutions. The high accuracy of this method is verified through some numerical examples. The Maple 18 program outputs all of the results.
Spectral Analysis of Long Memory in Financial Volatility Dynamics: Evidence from the CAC 40 Stock Market Index
Long-memory processes are widely used in time series analysis to characterize persistent dependence structures over time. In financial markets, this phenomenon is particularly relevant for modeling volatility dynamics, which measure the variability and risk of asset returns. This study investigates the presence of long-memory behavior in financial volatility using a spectral analysis approach. While the persistence of volatility is a well-documented stylized fact in financial econometrics, traditional time-domain models such as ARCH, GARCH, and their extensions primarily capture short- and medium-term dependence in conditional variance, often failing to fully account for long-range dependence. This paper addresses this gap by employing a frequency-domain approach based on spectral analysis to detect long memory in volatility dynamics. The empirical study is conducted using daily data from the CAC 40 stock market index. The price series is transformed into logarithmic returns, with volatility proxied by squared returns. A comparative analysis between returns and squared returns reveals that returns exhibit weak autocorrelation, whereas squared returns display strong persistence—consistent with established stylized facts. To further examine this persistence, we perform spectral density analysis and a simulation study. The results reveal a significant concentration of spectral power at low frequencies, providing strong evidence of long-memory dynamics in volatility. These findings have important implications for volatility forecasting, risk management, and the specification of econometric models in financial applications.
Benchmarking Low-Dimensional Projection Uniformity in Mixed Two- and Three-Level Designs
The uniformity of low-dimensional projections is a fundamental consideration in the construction of experimental designs, particularly in screening experiments where the effect hierarchy principle dictates that lower-order effects dominate. This paper provides comprehensive benchmarks for assessing this uniformity in mixed two- and three-level balanced designs—a class of designs ubiquitous in industrial, pharmaceutical, and computer experiments. We investigate the projection-weighted symmetric discrepancy (PWSDisc) as a uniformity criterion that explicitly prioritizes low-dimensional projections. A closed-form analytical expression for PWSDisc is derived in terms of elementary coincidence counts, revealing its intrinsic combinatorial structure. Building on this representation, we establish a sharp, easily computable lower bound that serves as a gold standard for design optimality. A catalog of lower bounds for practical parameter combinations is provided, offering immediate benchmarks for practitioners. Furthermore, an updated threshold-accepting algorithm incorporating these bounds as a stopping rule is presented, enabling efficient construction of uniform mixed two- and three-level designs. The results fill a critical gap in the theory of projection-based uniformity and have direct applications in computer experiments, robust parameter design, and beyond.
Copyright (c) 2026 Mathematical Applications and Statistical Rigor (MASR) https://creativecommons.org/licenses/by/4.0/
Lower Bounds of Unanchored Discrepancy for Mixed-Level U-Type Designs
Uniform $U$-type designs are fundamental to modern experimental methodology, particularly in computer experiments and robust parameter design, owing to their ability to uniformly distribute points across the design space. This space-filling property confers resilience against model misspecification and provides an ideal foundation for nonparametric estimation. Despite their practical significance, constructing uniform designs that minimize a given discrepancy poses an inherently intractable combinatorial optimization problem. This challenge has motivated the pursuit of theoretical lower bounds, which serve as optimality benchmarks and termination criteria for heuristic search algorithms. Mixed two- and three-level designs are ubiquitous in practical experimental settings, including industrial experiments, clinical trials, and computer simulations. In this paper, we derive sharp, analytically tractable lower bounds for the unanchored discrepancy in mixed two- and three-level U-type designs. Leveraging Jensen's inequality and combinatorial optimization, we obtain closed-form expressions that are both computationally expedient and demonstrably tight. We further delineate necessary conditions under which these bounds are attainable and propose constructive methodologies for generating designs that achieve or approach these theoretical limits. The resulting benchmarks fill a critical gap in the theory of uniform designs and offer tangible utility in constructing optimal experimental plans for computer simulations and robust parameter studies.
Bi-OSW-MF-DFA: An Enhanced Multifractal Method for Gold and Silver Market Risk Assessment
Amid global economic uncertainty and inflationary pressures, precious metals have gained prominence as safe-haven assets, yet accurately quantifying their risk remains challenging due to the inherent limitations of existing fractal analysis methods. Conventional Multifractal Detrended Fluctuation Analysis (MF-DFA) suffers from boundary-induced artifacts and computational inefficiencies, which can distort risk measurements and obscure true market dynamics. To address this gap, we introduce an enhanced method—Binary Overlapped Sliding Window MF-DFA (Bi-OSW-MF-DFA)—that mitigates segmentation discontinuities and improves computational robustness through overlapping windows and binary series partitioning. Applied to spot gold and silver markets, our approach reveals that silver exhibits stronger multifractality and higher inherent risk than gold, with long-range temporal correlations identified as the primary driver of multifractal behavior. This study offers a refined analytical framework for market risk assessment and provides actionable insights for investors and portfolio managers navigating precious metals volatility.
Advances in Uniform Experimental Designs: A Decade Selective Review of Algorithmic Search and Deterministic Construction Methods
This paper presents a comprehensive and selective review of the last decade's progress in the construction of uniform experimental designs. Confronting the growing complexity of modern experiments—characterized by high-dimensional factor spaces and constrained resources—recent research has produced promising methods tools for constructing efficient, cost-effective designs. The review is organized around three pivotal themes: (1) enhanced stochastic search algorithms, including adjusted threshold accepting and permutation-projection methods, for constructing (nearly) uniform minimum aberration designs, supported by benchmarks that reduce computational search; (2) frameworks for constructing uniform fold-over designs in two-stage sequential experimentations, enabling the breaking of aliasing structures across symmetric and asymmetric designs; and (3) deterministic construction algorithms—such as multiple doubling, tripling, and quadrupling, and their integration—for efficiently generating large-scale uniform designs with low, high, or mixed levels without exhaustive search. Collectively, these advances offer researchers a robust, computationally efficient, and theoretically coherent toolkit for designing experiments across scientific and industrial domains, representing a substantial leap beyond conventional methodologies. A critical discussion and comparative analysis of the reviewed methods are also provided, along with practical recommendations for implementation.