Mathematics

18 pages, 1187 KB

Open AccessFeature PaperArticle

A Bi-Population Co-Evolutionary Multi-Objective Optimization Algorithm for Production Scheduling Problems in a Metal Heat Treatment Process with Time Window Constraints

by Jiahui Gu, Boheng Liu and Ziyan Zhao

Mathematics 2025, 13(16), 2696; https://doi.org/10.3390/math13162696 - 21 Aug 2025

Viewed by 235

Abstract

Heat treatment is a critical intermediate process in copper strip manufacturing, where strips go through an air-cushion annealing furnace. The production scheduling for the air-cushion annealing furnace can contribute to cost reduction and efficiency enhancement throughout the overall copper strip production process. The [...] Read more.

Heat treatment is a critical intermediate process in copper strip manufacturing, where strips go through an air-cushion annealing furnace. The production scheduling for the air-cushion annealing furnace can contribute to cost reduction and efficiency enhancement throughout the overall copper strip production process. The production scheduling problem must account for time window constraints and gas atmosphere transition requirements among jobs, resulting in a complex combinatorial optimization problem that necessitates dual-objective optimization of the total atmosphere transition cost of annealing and the total penalties for time window violations. Most multi-objective optimization algorithms rely on the evolution of a single population, which makes them prone to premature convergence, leading to local optimal solutions and insufficient exploration of the solution space. To address the challenges above effectively, we propose a Bi-population Co-evolutionary Multi-objective Optimization Algorithm (BCMOA). Specifically, the BCMOA initially constructs two independent populations that evolve separately. When the iterative process meets predefined conditions, elite solution sets are extracted from each population for interaction, thereby generating new offspring individuals. Subsequently, these new offspring participate in elite solution selection alongside the parent populations via a non-dominated selection mechanism. The performance of the BCMOA has undergone extensive validation on benchmark datasets. The results show that the BCMOA outperforms its competitive peers in solving the relevant problem, thereby demonstrating significant application potential in industrial scenarios. Full article

(This article belongs to the Special Issue Intelligent Systems and Dynamic Scheduling: Optimization and Management)

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28 pages, 1982 KB

Open AccessFeature PaperArticle

Mathematical Modeling and Finite Element Analysis of Torsional Divergence of Carbon Plates with an AIREX Foam Core

by Mirko Dinulović, Mato Perić, Dragi Stamenković, Marta Trninić and Jovan Bengin

Mathematics 2025, 13(16), 2695; https://doi.org/10.3390/math13162695 - 21 Aug 2025

Viewed by 269

Abstract

This study presents a novel analytical–numerical framework for investigating the torsional divergence of composite sandwich structures composed of carbon fiber-reinforced skins and an AIREX foam core. A divergence differential equation is derived and modified to accommodate the anisotropic behavior of composite materials through [...] Read more.

This study presents a novel analytical–numerical framework for investigating the torsional divergence of composite sandwich structures composed of carbon fiber-reinforced skins and an AIREX foam core. A divergence differential equation is derived and modified to accommodate the anisotropic behavior of composite materials through an equivalent shear modulus, extending classical formulations originally developed for isotropic structures. The resulting equation is solved using the Galerkin method, yielding structural section rotations as a continuous function along the wing span. These torsional modes are then applied as boundary inputs in a high-fidelity finite element model of the composite fin to determine stress distributions across the structure. The method enables evaluation of not only in-plane (membrane) stresses, but also out-of-plane responses such as interlaminar stresses and local core-skin interactions critical for assessing failure modes in sandwich composites. This integrated workflow links analytical aeroelastic modeling with detailed structural analysis, offering valuable insights into the interplay between global torsional stability and local stress behavior in laminated composite systems. Full article

(This article belongs to the Special Issue Numerical Analysis and Finite Element Method with Applications)

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21 pages, 1434 KB

Open AccessFeature PaperArticle

Estimating Skewness and Kurtosis for Asymmetric Heavy-Tailed Data: A Regression Approach

by Joseph H. T. Kim and Heejin Kim

Mathematics 2025, 13(16), 2694; https://doi.org/10.3390/math13162694 - 21 Aug 2025

Viewed by 277

Abstract

Estimating skewness and kurtosis from real-world data remains a long-standing challenge in actuarial science and financial risk management, where these higher-order moments are critical for capturing asymmetry and tail risk. Traditional moment-based estimators are known to be highly sensitive to outliers and often [...] Read more.

Estimating skewness and kurtosis from real-world data remains a long-standing challenge in actuarial science and financial risk management, where these higher-order moments are critical for capturing asymmetry and tail risk. Traditional moment-based estimators are known to be highly sensitive to outliers and often fail when the assumption of normality is violated. Despite numerous extensions—from robust moment-based methods to quantile-based measures—being proposed over the decades, no universally satisfactory solution has been reported, and many existing methods exhibit limited effectiveness, particularly under challenging distributional shapes. In this paper we propose a novel method that jointly estimates skewness and kurtosis based on a regression adaptation of the Cornish–Fisher expansion. By modeling the empirical quantiles as a cubic polynomial of the standard normal variable, the proposed approach produces a reliable and efficient estimator that better captures distributional shape without strong parametric assumptions. Our comprehensive simulation studies show that the proposed method performs much better than existing estimators across a wide range of distributions, especially when the data are skewed or heavy-tailed, as is typical in actuarial and financial applications. Full article

(This article belongs to the Special Issue Actuarial Statistical Modeling and Applications)

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27 pages, 521 KB

Open AccessArticle

RMVC: A Validated Algorithmic Framework for Decision-Making Under Uncertainty

by Abdurrahman Dayioglu, Fatma Ozen Erdogan and Basri Celik

Mathematics 2025, 13(16), 2693; https://doi.org/10.3390/math13162693 - 21 Aug 2025

Viewed by 267

Abstract

The reliability of decision-making algorithms within soft set theory is fundamentally constrained by their underlying membership functions. Traditional binary approaches overlook the implicit connections between the attributes a candidate possesses and those it lacks—connections that can be inferred from the wider candidate pool. [...] Read more.

The reliability of decision-making algorithms within soft set theory is fundamentally constrained by their underlying membership functions. Traditional binary approaches overlook the implicit connections between the attributes a candidate possesses and those it lacks—connections that can be inferred from the wider candidate pool. To address this core challenge, this paper puts forward the Relational Membership Value Calculation (RMVC), an algorithmic framework whose core is a fine-grained relational membership function. Our approach moves beyond binary logic to capture these nuanced interrelationships. We provide a rigorous theoretical analysis of the proposed algorithm, including its computational complexity and robustness, which is validated through a comprehensive sensitivity analysis. Crucially, a comparative analysis using the Gini Index quantitatively demonstrates that our method provides significantly higher granularity and discriminatory power on a representative case study. The RMVC is implemented as an open-source Python program, providing a foundational tool to enhance the reasoning capabilities of AI-driven decision support and expert systems. Full article

(This article belongs to the Section E1: Mathematics and Computer Science)

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16 pages, 3305 KB

Open AccessArticle

A Continuous-Time Distributed Optimization Algorithm for Multi-Agent Systems with Parametric Uncertainties over Unbalanced Digraphs

by Qing Yang and Caiqi Jiang

Mathematics 2025, 13(16), 2692; https://doi.org/10.3390/math13162692 - 21 Aug 2025

Viewed by 276

Abstract

This paper investigates distributed optimization problems for multi-agent systems with parametric uncertainties over unbalanced directed communication networks. To settle this class of optimization problems, a continuous-time algorithm is proposed by integrating adaptive control techniques with an output feedback tracking protocol. By systematically employing [...] Read more.

This paper investigates distributed optimization problems for multi-agent systems with parametric uncertainties over unbalanced directed communication networks. To settle this class of optimization problems, a continuous-time algorithm is proposed by integrating adaptive control techniques with an output feedback tracking protocol. By systematically employing Lyapunov stability theory, perturbed system analysis, and input-to-state stability theory, we rigorously establish the asymptotic convergence property of the proposed algorithm. A numerical simulation further demonstrates the effectiveness of the algorithm in computing the global optimal solution. Full article

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26 pages, 9294 KB

Open AccessArticle

Bayesian Analysis of Bitcoin Volatility Using Minute-by-Minute Data and Flexible Stochastic Volatility Models

by Makoto Nakakita, Tomoki Toyabe and Teruo Nakatsuma

Mathematics 2025, 13(16), 2691; https://doi.org/10.3390/math13162691 - 21 Aug 2025

Viewed by 697

Abstract

This study analyzes the volatility of Bitcoin using stochastic volatility models fitted to one-minute transaction data for the BTC/USDT pair between 1 April 2023, and 31 March 2024. Bernstein polynomial terms were introduced to accommodate intraday and intraweek seasonality, and flexible return distributions [...] Read more.

This study analyzes the volatility of Bitcoin using stochastic volatility models fitted to one-minute transaction data for the BTC/USDT pair between 1 April 2023, and 31 March 2024. Bernstein polynomial terms were introduced to accommodate intraday and intraweek seasonality, and flexible return distributions were used to capture distributional characteristics. Seven return distributions—normal, Student-t, skew-t, Laplace, asymmetric Laplace (AL), variance gamma, and skew variance gamma—were considered. We further incorporated explanatory variables derived from the trading volume and price changes to assess the effects of order flow. Our results reveal structural market changes, including a clear regime shift around October 2023, when the asymmetric Laplace distribution became the dominant model. Regression coefficients suggest a weakening of the volume–volatility relationship after September and the presence of non-persistent leverage effects. These findings highlight the need for flexible, distribution-aware modeling in 24/7 digital asset markets, with implications for market monitoring, volatility forecasting, and crypto risk management. Full article

(This article belongs to the Special Issue Quantitative Finance and Stochastic Volatility with Their Numerical Models)

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22 pages, 2782 KB

Open AccessArticle

A Novel Optimization Method and Its Application for Hazardous Materials Vehicle Routing Problem Under Different Road Conditions

by Fangwei Zhang, Lu Ding, Jun Jiang, Fanyi Kong and Xiaoyu Liu

Mathematics 2025, 13(16), 2690; https://doi.org/10.3390/math13162690 - 21 Aug 2025

Viewed by 290

Abstract

With the increasing demand for hazardous materials (hazmat) from enterprises, port chemical industrial parks face growing risks in hazardous material transportation. By using internal road network information of parks, this study investigates the hazmat vehicle routing problem (HVRP) under different road conditions, with [...] Read more.

With the increasing demand for hazardous materials (hazmat) from enterprises, port chemical industrial parks face growing risks in hazardous material transportation. By using internal road network information of parks, this study investigates the hazmat vehicle routing problem (HVRP) under different road conditions, with a bi-objective of minimizing total transportation risk and cost. The two main innovations are as follows. First, according to the grid-like road conditions in parks, the research scope of transportation segments of hazmat vehicles is divided into straight segments and curved segments. Second, the potential affected area of an accident is defined as a type of geometric shape associated with a series of factors refined from transportation situations. Finally, the effectiveness of the proposed two-stage ant colony optimization (TSACO) algorithm is verified through one instance using field data from a real port chemical industry park, and twelve instances from the classical capacitated vehicle routing problem (CVRP) resource. Full article

(This article belongs to the Special Issue Multi-Criteria Decision-Making and Operations Research)

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19 pages, 319 KB

Open AccessArticle

Eigenvalue Characterizations for the Signless Laplacian Spectrum of Weakly Zero-Divisor Graphs on

Z_{n}

by Nazim, Alaa Altassan and Nof T. Alharbi

Mathematics 2025, 13(16), 2689; https://doi.org/10.3390/math13162689 - 21 Aug 2025

Viewed by 244

Abstract

Let

R

be a commutative ring with identity

1 \neq 0

. The weakly zero-divisor graph of

R

, denoted

W_{Γ} (R)

, is the simple undirected graph whose vertex set consists of the nonzero zero-divisors of

R

, where [...] Read more.

Let

R

be a commutative ring with identity

1 \neq 0

. The weakly zero-divisor graph of

R

, denoted

W_{Γ} (R)

, is the simple undirected graph whose vertex set consists of the nonzero zero-divisors of

R

, where two distinct vertices

a

and

b

are adjacent if and only if there exist

r \in ann (a)

and

s \in ann (b)

such that

r s = 0

. In this paper, we study the signless Laplacian spectrum of

W_{Γ} (Z_{n})

for several composite forms of n, including

n = p^{2} q^{2}

,

n = p^{2} q r

,

n = p^{m} q^{m}

and

n = p^{m} q r

, where

p, q, r

are distinct primes and

m \geq 2

. By using generalized join decomposition and quotient matrix methods, we obtain explicit eigenvalue formulas for each case, along with structural bounds, spectral integrality conditions and Nordhaus–Gaddum-type inequalities. Illustrative examples with computed spectra are provided to validate the theoretical results, demonstrating the interplay between the algebraic structure of

Z_{n}

and the spectral properties of its weakly zero-divisor graph. Full article

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19 pages, 2631 KB

Open AccessArticle

Porosity-Dependent Frequency Analysis of Bidirectional Porous Functionally Graded Plates via Nonlocal Elasticity Theory

by Hela D. El-Shahrany

Mathematics 2025, 13(16), 2688; https://doi.org/10.3390/math13162688 - 21 Aug 2025

Viewed by 233

Abstract

Elastic solutions of a differential system of vibrational responses of a bidirectional porous functionally graded plate (BPFG) are described by employing high-order normal and shear deformation theory, in the present study. Natural frequency values are computed for the plates with simply supported boundary [...] Read more.

Elastic solutions of a differential system of vibrational responses of a bidirectional porous functionally graded plate (BPFG) are described by employing high-order normal and shear deformation theory, in the present study. Natural frequency values are computed for the plates with simply supported boundary conditions and taking into consideration the thickness stretching effect. Grading of the effective material property for the BPFG plate is defined according to a power-law distribution. Navier’s approach is applied to determine the governing differential equations solution of the studied model derived by Hamilton’s principle. To confirm the reliability of the solution and the model accuracy, a comparison study with various studies that are presented in the literature is carried out. Numerical illustrations are presented to discuss the influences of the plate geometry, the porosity, the volume fraction distribution, and the nonlocality on the vibration behaviors of the model. The dynamic responses of unidirectional and bidirectional porous functionally graded nanoplates are analyzed in detail, employing two parametric numerical examples. Numerical results show the sensitivity of frequencies to the studied parametric factors and their dependence on porosity and nonlocality coefficients. Frequencies of BPFG with uneven/even distribution porosity decrease when increasing the transverse and axial power-law indexes (

P \neq 0

), and the same effect appears when increasing the nonlocal parameter. Full article

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18 pages, 1993 KB

Open AccessArticle

Fault Line Selection in Distribution Networks Based on Dual-Channel Time-Frequency Fusion Network

by Yuyi Ma, Wei Guo, Yuntao Shi, Jianing Guan, Yushuai Qi, Xiang Yin and Gang Liu

Mathematics 2025, 13(16), 2687; https://doi.org/10.3390/math13162687 - 21 Aug 2025

Viewed by 308

Abstract

In distribution networks, single-phase ground faults often lead to abnormal changes in voltage and current signals. Traditional single-modal fault diagnosis methods usually struggle to accurately identify the fault line under such conditions. To address this issue, this paper proposes a fault line identification [...] Read more.

In distribution networks, single-phase ground faults often lead to abnormal changes in voltage and current signals. Traditional single-modal fault diagnosis methods usually struggle to accurately identify the fault line under such conditions. To address this issue, this paper proposes a fault line identification method based on a multimodal feature fusion model. The approach combines time-frequency images—generated using a Short-Time Fourier Transform (STFT) and Wigner–Ville Distribution (WVD) fusion algorithm with one-dimensional time-series signals for classification. The time-frequency images visualize both temporal and spectral features of the signal and are processed using the RepLKNet model for deep feature extraction. Meanwhile, the raw one-dimensional time-series signals preserve the original temporal dependencies and are analyzed using a BiGRU network enhanced with a global attention mechanism to improve feature representation. Finally, features from both modalities are extracted in parallel and fused to achieve accurate fault line identification. Experimental results demonstrate that the proposed method effectively leverages the complementary nature of multimodal data and shows strong robustness in the presence of noise interference. Full article

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13 pages, 2088 KB

Open AccessArticle

Shock Waves of the Gerdjikov–Ivanov Equation Using the Adomian Decomposition Schemes

by Fadwa Althrwi, Aisha S. H. Farhat, A. A. AlQarni, H. O. Bakodah and A. A. Alshaery

Mathematics 2025, 13(16), 2686; https://doi.org/10.3390/math13162686 - 20 Aug 2025

Viewed by 260

Abstract

Analytical solutions for the complex-valued nonlinear Gerdjikov–Ivanov (GI) equation have been studied extensively using integrability-based methods. In contrast, numerical and semi-analytical exploration remains relatively underdeveloped. Thus, the present study deploys both the traditional Adomian decomposition method (ADM) and its improved version (IADM) to [...] Read more.

Analytical solutions for the complex-valued nonlinear Gerdjikov–Ivanov (GI) equation have been studied extensively using integrability-based methods. In contrast, numerical and semi-analytical exploration remains relatively underdeveloped. Thus, the present study deploys both the traditional Adomian decomposition method (ADM) and its improved version (IADM) to explore the computational relevance of the GI equation to shock waves against a benchmark exact soliton solution. The findings indicate that both methods are effective in addressing the GI equation, with the improved method demonstrating an enhancement in the stability of the convergence under specific conditions. This work offers the first systematic semi-analytic and numerical evaluation of the GI equation, introducing practical implementation guidelines. Full article

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40 pages, 725 KB

Open AccessFeature PaperArticle

Upper and Lower Bounds of Performance Metrics in Hybrid Systems with Setup Time

by Ken’ichi Kawanishi and Yuki Ino

Mathematics 2025, 13(16), 2685; https://doi.org/10.3390/math13162685 - 20 Aug 2025

Viewed by 235

Abstract

To address the increasing demand for computational and communication resources, modern networked systems often rely on heterogeneous servers, including those requiring setup times, such as virtual machines or servers, and others that are always active. In this paper, we model and analyze the [...] Read more.

To address the increasing demand for computational and communication resources, modern networked systems often rely on heterogeneous servers, including those requiring setup times, such as virtual machines or servers, and others that are always active. In this paper, we model and analyze the performance of such hybrid systems using a level-dependent quasi-birth-and-death (LDQBD) process. Building upon an existing queueing model, we extend the analysis by considering scalable approximation methods. Since matrix analytic methods become computationally expensive in large-scale settings, we propose a stochastic bounding approach that derives upper and lower bounds for the stationary distribution, thereby significantly reducing computational cost. This approach further provides bounds on the performance metrics of the hybrid system. Full article

(This article belongs to the Special Issue Recent Research in Queuing Theory and Stochastic Models, 2nd Edition)

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15 pages, 290 KB

Open AccessFeature PaperArticle

General Decay for a Viscoelastic Equation with Acoustic Boundary Conditions and a Logarithmic Nonlinearity

by Jum-Ran Kang and Hye-Jin Kim

Mathematics 2025, 13(16), 2684; https://doi.org/10.3390/math13162684 - 20 Aug 2025

Viewed by 272

Abstract

In this work, we investigate the stability of solutions in a situation where the logarithmic source term competes with the viscoelastic dissipation under acoustic boundary conditions. We assume minimal conditions on the relaxation function g, namely, [...] Read more.

In this work, we investigate the stability of solutions in a situation where the logarithmic source term competes with the viscoelastic dissipation under acoustic boundary conditions. We assume minimal conditions on the relaxation function g, namely,

g^{'} (t) \leq - ξ (t) H (g (t))

, where H is a strictly increasing and strictly convex function near the origin, and

ξ (t)

is a non-increasing function. Under these general assumptions, we establish a general decay estimate for the solution. This result extends and improves some previous results. Full article

(This article belongs to the Special Issue Stability and Stabilization of Partial Differential Equations)

25 pages, 2958 KB

Open AccessArticle

An Improved Pareto Local Search-Based Evolutionary Algorithm for Multi-Objective Shortest-Path Network Counter-Interdiction Problem

by Chenghui Mao, Ronghuan Gao, Qizhang Luo and Guohua Wu

Mathematics 2025, 13(16), 2683; https://doi.org/10.3390/math13162683 - 20 Aug 2025

Viewed by 287

Abstract

Most existing studies on the Shortest-Path Network Interdiction Problem (SPIP) adopt the attacker’s perspective, often overlooking the critical role of defender-oriented strategies. To support proactive defense, this paper introduces a novel problem named the Multi-Objective Shortest-Path Counter-Interdiction Problem (MO-SPCIP). The problem incorporates a [...] Read more.

Most existing studies on the Shortest-Path Network Interdiction Problem (SPIP) adopt the attacker’s perspective, often overlooking the critical role of defender-oriented strategies. To support proactive defense, this paper introduces a novel problem named the Multi-Objective Shortest-Path Counter-Interdiction Problem (MO-SPCIP). The problem incorporates a backup-based defense strategy from the defender’s viewpoint and addresses the inherent trade-offs among minimizing the shortest path length, minimizing backup resource consumption, and maximizing the attacker’s resource usage. To solve this complex problem, we propose an Improved Pareto Local Search-based Evolutionary Algorithm (IPLSEA). The algorithm integrates several problem-specific components, including a tailored initial solution generation method, a customized solution representation, and specialized genetic operators. In addition, an improved Pareto Local Search (IPLS) is incorporated into the algorithm framework, allowing an adaptive and selective search. To further enhance local refinement, three problem-specific neighborhood search operations are designed and embedded within the Pareto Local Search. The experimental results demonstrate that IPLSEA significantly outperforms state-of-the-art algorithms in terms of its convergence quality and solution diversity, enabling a more robust performance in network counter-interdiction scenarios. Full article

(This article belongs to the Special Issue Evolutionary Multi-Criteria Optimization: Methods and Applications)

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38 pages, 1267 KB

Open AccessArticle

Aggregation Operator-Based Trapezoidal-Valued Intuitionistic Fuzzy WASPAS Algorithm and Its Applications in Selecting the Location for a Wind Power Plant Project

by Bibhuti Bhusana Meher, Jeevaraj Selvaraj and Melfi Alrasheedi

Mathematics 2025, 13(16), 2682; https://doi.org/10.3390/math13162682 - 20 Aug 2025

Viewed by 216

Abstract

Trapezoidal-valued intuitionistic fuzzy numbers (TrVIFNs) are the real generalizations of intuitionistic fuzzy numbers, interval-valued intuitionistic fuzzy numbers, and triangular intuitionistic fuzzy numbers, which effectively model real-life problems that consist of imprecise and incomplete data. This study incorporates the Aczel-Alsina aggregation operators (which consist [...] Read more.

Trapezoidal-valued intuitionistic fuzzy numbers (TrVIFNs) are the real generalizations of intuitionistic fuzzy numbers, interval-valued intuitionistic fuzzy numbers, and triangular intuitionistic fuzzy numbers, which effectively model real-life problems that consist of imprecise and incomplete data. This study incorporates the Aczel-Alsina aggregation operators (which consist of parameter-based flexibility) for solving any group of decision-making problems modeled in a trapezoidal-valued intuitionistic fuzzy (TrVIF) environment. In this study, we first define new operations on TrVIFNs based on the Aczel-Alsina operations. Secondly, we introduce new trapezoidal-valued intuitionistic fuzzy aggregation operators, such as the TrVIF Aczel-Alsina weighted averaging operator, the TrVIF Aczel-Alsina ordered weighted averaging operator, and the TrVIF Aczel-Alsina hybrid averaging operator, and we discuss their fundamental mathematical properties by examining various theorems. This study also includes a new algorithm named ‘three-stage multi-criteria group decision-making’, where we obtain the criteria weights using the newly proposed TrVIF-MEREC method. Additionally, we introduce a new modified algorithm called TrVIF-WASPAS to solve the multi-criteria decision-making (MCDM) problem in the trapezoidal-valued intuitionistic fuzzy environment. Then, we apply this proposed method to solve a model case study problem involving location selection for a wind power plant project. Then, we discuss the proposed algorithm’s sensitivity analysis by changing the criteria weights concerning different parameter values. Finally, we compare our proposed methods with various existing methods, like some subclasses of TrVIFNs such as IVIFWA, IVIFWG, IVIFEWA, and IVIFEWG, and also with some MCGDM methods of TrVIFNs, such as the Dombi aggregation operator-based method in TrVIFNs and the TrVIF-Topsis method-based MCGDM, to show the efficacy of our proposed algorithm. This study has many advantages, as it consists of a total ordering principle in ranking alternatives in the newly proposed TrVIF-MCGDM techniques and TrVIF-WASPAS MCDM techniques for the first time in the literature. Full article

(This article belongs to the Special Issue Applications of Fuzzy Set Theory and Decision-Making in Business, Finance, and Engineering Using Empirical Mathematical Models)

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16 pages, 294 KB

Open AccessArticle

On Some Mean Value Results for the Zeta-Function and a Rankin–Selberg Problem

by Jing Huang, Yukun Liu and Deyu Zhang

Mathematics 2025, 13(16), 2681; https://doi.org/10.3390/math13162681 - 20 Aug 2025

Viewed by 281

Abstract

Denote by

Δ_{1} (x; φ)

the error term in the classical Rankin–Selberg problem. Denote by

ζ (s)

the Riemann zeta-function. We establish an upper bound for this integral [...] Read more.

Denote by

Δ_{1} (x; φ)

the error term in the classical Rankin–Selberg problem. Denote by

ζ (s)

the Riemann zeta-function. We establish an upper bound for this integral

\int_{0}^{T} Δ_{1} (t; φ) {|ζ (\frac{1}{2} + i t)|}^{2} d t .

In addition, when

2 \leq k \leq 4

is a fixed integer, we will derive an asymptotic formula for the integral

\int_{1}^{T} Δ_{1}^{k} (t; φ) {|ζ (\frac{1}{2} + i t)|}^{2} d t .

The results rely on the power moments of

Δ_{1} (t; φ)

and

E (t)

, the classical error term in the asymptotic formula for the mean square of

|ζ (\frac{1}{2} + i t)|

. Full article

(This article belongs to the Special Issue Recent Studies in Number Theory and Algebraic Geometry)

30 pages, 5491 KB

Open AccessFeature PaperArticle

ε-Algorithm Accelerated Fixed-Point Iteration for the Three-Way GIPSCAL Problem in Asymmetric MDS

by Yuefeng Qin, Chen Mao and Jiaofen Li

Mathematics 2025, 13(16), 2680; https://doi.org/10.3390/math13162680 - 20 Aug 2025

Viewed by 252

Abstract

The Generalized Inner Product SCALing (GIPSCAL) model is a specialized tool for analyzing square asymmetric tables within asymmetric multidimensional scaling (MDS), with applications in sociology (e.g., social mobility tables) and marketing (e.g., brand switching data). This paper presents the development of an efficient [...] Read more.

The Generalized Inner Product SCALing (GIPSCAL) model is a specialized tool for analyzing square asymmetric tables within asymmetric multidimensional scaling (MDS), with applications in sociology (e.g., social mobility tables) and marketing (e.g., brand switching data). This paper presents the development of an efficient numerical algorithm for solving the three-way GIPSCAL problem. We focus on vector

ε

-algorithm-accelerated fixed-point iterations, detailing the underlying acceleration principles. Extensive numerical experiments show that the proposed method achieves acceleration performance comparable to polynomial extrapolation and Anderson acceleration. Furthermore, compared to continuous-time projected gradient flow methods and first- and second-order Riemannian optimization algorithms from the Manopt toolbox, our approach demonstrates superior computational efficiency and scalability. Full article

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30 pages, 1031 KB

Open AccessFeature PaperArticle

Test-Path Scheduling for Interposer-Based 2.5D Integrated Circuits Using an Orthogonal Learning-Based Differential Evolution Algorithm

by Chunlei Li, Libao Deng, Guanyu Yuan, Liyan Qiao, Lili Zhang and Chu Chen

Mathematics 2025, 13(16), 2679; https://doi.org/10.3390/math13162679 - 20 Aug 2025

Viewed by 314

Abstract

2.5D integrated circuits (ICs), which utilize an interposer to stack multiple dies side by side, represent a promising architecture for improving system performance, integration density, and design flexibility. However, the complex interconnect structures present significant challenges for post-fabrication testing, especially when scheduling test [...] Read more.

2.5D integrated circuits (ICs), which utilize an interposer to stack multiple dies side by side, represent a promising architecture for improving system performance, integration density, and design flexibility. However, the complex interconnect structures present significant challenges for post-fabrication testing, especially when scheduling test paths under constrained test access mechanisms. This paper addresses the test-path scheduling problem in interposer-based 2.5D ICs, aiming to minimize both total test time and cumulative inter-die interconnect length. We propose an efficient orthogonal learning-based differential evolution algorithm, named OLELS-DE. The algorithm combines the global optimization capability of differential evolution with an orthogonal learning-based search strategy and an elites local search strategy to enhance the convergence and solution quality. Comprehensive experiments are conducted on a set of benchmark instances with varying die counts, and the proposed method is compared against five state-of-the-art metaheuristic algorithms and CPLEX. Experimental results demonstrate that OLELS-DE consistently outperforms the competitors in terms of test cost reduction and convergence reliability, confirming its robustness and effectiveness for complex test scheduling in 2.5D ICs. Full article

(This article belongs to the Special Issue Intelligence Optimization Algorithms and Applications)

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20 pages, 1619 KB

Open AccessArticle

Hybrid Shifted Gegenbauer Integral–Pseudospectral Method for Solving Time-Fractional Benjamin–Bona–Mahony–Burgers Equation

by Kareem T. Elgindy

Mathematics 2025, 13(16), 2678; https://doi.org/10.3390/math13162678 - 20 Aug 2025

Viewed by 316

Abstract

This paper introduces a novel hybrid shifted Gegenbauer integral–pseudospectral (HSG-IPS) method to solve the time-fractional Benjamin–Bona–Mahony–Burgers (FBBMB) equation with high accuracy. The approach transforms the equation into a form with only a first-order derivative, which is approximated using a stable shifted Gegenbauer differentiation [...] Read more.

This paper introduces a novel hybrid shifted Gegenbauer integral–pseudospectral (HSG-IPS) method to solve the time-fractional Benjamin–Bona–Mahony–Burgers (FBBMB) equation with high accuracy. The approach transforms the equation into a form with only a first-order derivative, which is approximated using a stable shifted Gegenbauer differentiation matrix (SGDM), while other terms are computed with precise quadrature rules. By integrating advanced techniques such as the shifted Gegenbauer pseudospectral method (SGPS), fractional derivative and integral approximations, and barycentric integration matrices, the HSG-IPS method achieves spectral accuracy. Numerical results show it reduces average absolute errors (AAEs) by up to 99.99% compared to methods like Crank–Nicolson linearized difference scheme (CNLDS) and finite integration method using Chebyshev polynomial (FIM-CBS), with computational times as low as 0.04–0.05 s. The method’s stability is improved by avoiding ill-conditioned high-order derivative approximations, and its efficiency is boosted by precomputed matrices and Kronecker product structures. Robust across various fractional orders, the HSG-IPS method offers a powerful tool for modeling wave propagation and nonlinear phenomena in fractional calculus applications. Full article

(This article belongs to the Topic Numerical Methods for Partial Differential Equations)

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22 pages, 311 KB

Open AccessArticle

A Dempster–Shafer, Fusion-Based Approach for Malware Detection

by Patricio Galdames, Simon Yusuf Enoch, Claudio Gutiérrez-Soto and Marco A. Palomino

Mathematics 2025, 13(16), 2677; https://doi.org/10.3390/math13162677 - 20 Aug 2025

Viewed by 373

Abstract

Dempster–Shafer theory (DST), a generalization of probability theory, is well suited for managing uncertainty and integrating information from diverse sources. In recent years, DST has gained attention in cybersecurity research. However, despite the growing interest, there is still a lack of systematic comparisons [...] Read more.

Dempster–Shafer theory (DST), a generalization of probability theory, is well suited for managing uncertainty and integrating information from diverse sources. In recent years, DST has gained attention in cybersecurity research. However, despite the growing interest, there is still a lack of systematic comparisons of DST implementation strategies for malware detection. In this paper, we present a comprehensive evaluation of DST-based ensemble mechanisms for malware detection, addressing critical methodological questions regarding optimal mass function construction and combination rules. Our systematic analysis was tested with 630,504 benign and malicious samples collected from four public datasets (BODMAS, DREBIN, AndroZoo, and BMPD) to train malware detection models. We explored three approaches for converting classifier outputs into probability mass functions: global confidence using fixed values derived from performance metrics, class-specific confidence with separate values for each class, and computationally optimized confidence values. The results establish that all approaches yield comparable performance, although fixed values offer significant computational and interpretability advantages. Additionally, we introduced a novel linear combination rule for evidence fusion, which delivers results on par with conventional DST rules while enhancing interpretability. Our experiments show consistently low false positive rates—ranging from 0.16% to 3.19%. This comprehensive study provides the first systematic methodology comparison for DST-based malware detection, establishing evidence-based guidelines for practitioners on optimal implementation strategies. Full article

(This article belongs to the Special Issue Analytical Frameworks and Methods for Cybersecurity, 2nd Edition)

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29 pages, 3101 KB

Open AccessArticle

Optimizing Efficiency for Logistics Training Using Virtual Reality Movies

by Qiaoling Zou, Xinyan Jiang, Xiangling Hu, Wanyu Zheng and Dongning Li

Mathematics 2025, 13(16), 2676; https://doi.org/10.3390/math13162676 - 20 Aug 2025

Viewed by 250

Abstract

(1) Background: Traditional logistics training faces challenges like high costs, limited scalability, and safety risks. Virtual Reality Movie Training (VRMT) enhances operational accuracy, safety, and accessibility through immersive simulation. However, adoption faces barriers including high equipment costs, immature technology, and coordination challenges among [...] Read more.

(1) Background: Traditional logistics training faces challenges like high costs, limited scalability, and safety risks. Virtual Reality Movie Training (VRMT) enhances operational accuracy, safety, and accessibility through immersive simulation. However, adoption faces barriers including high equipment costs, immature technology, and coordination challenges among logistics enterprises, design companies, and government entities. This study explores strategic interactions to optimize VRMT adoption. (2) Methods: A tripartite evolutionary game model was used to analyze strategic interactions between logistics enterprises, design companies, and government. (3) Results: System stability occurs when logistics enterprises adopt VRMT, design companies deliver high-quality solutions, and government provides active support. Simulations reveal stronger adoption coefficients through increased employee acceptance and enhanced training quality. Government incentives and brand premiums significantly influence quality design provision, though excessive subsidies may reduce governmental willingness to support initiatives. (4) Conclusions: Cost minimization and accessibility improvement require batch hardware purchasing, optimized training cycles, and shared platforms at logistics enterprises. Design companies should optimize content development for cost-effectiveness while maintaining quality standards to leverage brand benefits. Governments should establish VRMT quality certification, invest in public VR platforms for SMEs, and convert accident savings into fiscal supplements. This tripartite collaboration enables efficient, safe, and sustainable logistics training transformation. Full article

(This article belongs to the Special Issue Optimization and Mathematical Modelling in Transport and Logistics Network)

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28 pages, 1356 KB

Open AccessArticle

Short Video Marketing or Live Streaming Marketing: Choice of Marketing Strategies for Retailers

by Shuai Feng, Rui Yuan and Jiqiong Liu

Mathematics 2025, 13(16), 2675; https://doi.org/10.3390/math13162675 - 20 Aug 2025

Viewed by 390

Abstract

This study investigates retailers’ strategic choices between short video marketing (SVM) and live streaming marketing (LSM) in the social media era, with a focus on the synergistic effects and decision-making mechanisms of these two digital marketing models. Using game theory, we construct a [...] Read more.

This study investigates retailers’ strategic choices between short video marketing (SVM) and live streaming marketing (LSM) in the social media era, with a focus on the synergistic effects and decision-making mechanisms of these two digital marketing models. Using game theory, we construct a game analysis model to analyze retailers’ optimal selection among three marketing strategies: S (sole implementation of SVM), L (sole implementation of LSM), and H (integration of both SVM and LSM). The findings reveal that retailers should make different strategic choices based on the different stages of development. In the early market entry phase, characterized by both a low mixed commission rate and a low slotting fee, the H strategy emerges as the optimal choice. As the market enters its growth phase, retailers should shift to the L strategy, driven by “influencer LSM”. When the market enters a mature stage, retailers should be more inclined to adopt the S strategy or the L strategy dominated by “merchant self-LSM”. These findings provide new theoretical insights into the dynamic selection mechanisms of digital marketing strategies while offering practical decision-making guidance for retailers in allocating marketing resources across different development stages. The conclusions have direct implications for optimizing corporate marketing mix strategies. Full article

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12 pages, 225 KB

Open AccessFeature PaperArticle

On Solving the Knapsack Problem with Conflicts

by Roberto Montemanni and Derek H. Smith

Mathematics 2025, 13(16), 2674; https://doi.org/10.3390/math13162674 - 20 Aug 2025

Viewed by 255

Abstract

A variant of the well-known Knapsack Problem is studied in this paper. In the classic problem, a set of items is given, with each item characterized by a weight and a profit. A knapsack of a given capacity is provided, and the problem [...] Read more.

A variant of the well-known Knapsack Problem is studied in this paper. In the classic problem, a set of items is given, with each item characterized by a weight and a profit. A knapsack of a given capacity is provided, and the problem consists of selecting a subset of items such that the total weight does not exceed the capacity of the knapsack, while the total profit is maximized. In the variation considered in the present work, pairs of items are conflicting, and cannot be selected at the same time. The resulting problem, which can be used to model several real applications, is considerably harder to approach than the classic one. In this paper, we consider a mixed-integer linear program representing the problem and we solve it with a state-of-the-art black-box software. A vast experimental procedure on the instances available from the literature, and adopted in the last decade by the community, indicates that the approach we propose achieves results comparable with, and in many cases better than, those of state-of-the-art methods, notwithstanding that the latter are typically based on more complex and problem-specific ideas and algorithms than the idea we propose. Full article

(This article belongs to the Special Issue Computational Optimization and Operations Research: Theory and Applications)

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9 pages, 267 KB

Open AccessArticle

Laplacian Conditions and Sphericity of Hypersurfaces in the Nearly Kähler 6-Sphere

by Ibrahim Al-Dayel

Mathematics 2025, 13(16), 2673; https://doi.org/10.3390/math13162673 - 20 Aug 2025

Viewed by 270

Abstract

In this paper, we investigate hypersurfaces in the nearly Kähler 6-sphere

S^{6}

and establish several foundational results. In particular, under certain conditions of the function

ξ (f) = g (\nabla f, ξ)

, we demonstrate that a [...] Read more.

In this paper, we investigate hypersurfaces in the nearly Kähler 6-sphere

S^{6}

and establish several foundational results. In particular, under certain conditions of the function

ξ (f) = g (\nabla f, ξ)

, we demonstrate that a hypersurface M of

S^{6}

must be a sphere. Here,

f \in C^{\infty} (M)

is a smooth vector field,

ξ = - J N

denotes the characteristic vector field, J is the almost complex structure on

S^{6}

, and N is the unit vector field normal to the hypersurface. We also support our results with illustrative examples. Full article

(This article belongs to the Special Issue Differential Geometry, Geometric Analysis and Their Related Applications, 2nd Edition)

23 pages, 994 KB

Open AccessArticle

A Random Forest-Enhanced Genetic Algorithm for Order Acceptance Scheduling with Past-Sequence-Dependent Setup Times

by Yu-Yan Zhang, Shih-Hsin Chen, Yen-Wen Wang, Chia-Hsuan Liao and Chen-Hsiang Yu

Mathematics 2025, 13(16), 2672; https://doi.org/10.3390/math13162672 - 19 Aug 2025

Viewed by 304

Abstract

This study developed a simple genetic algorithm (SGA) enhanced by a random forest (RF) surrogate model, namely

{S G A}_{R F}

, to solve the permutation flow-shop scheduling problem with order acceptance under the conditions of limited capacity, weighted-tardiness, and past-sequence-dependent (PSD) [...] Read more.

This study developed a simple genetic algorithm (SGA) enhanced by a random forest (RF) surrogate model, namely

{S G A}_{R F}

, to solve the permutation flow-shop scheduling problem with order acceptance under the conditions of limited capacity, weighted-tardiness, and past-sequence-dependent (PSD) setup times (PFSS-OAWT with PSD). To the best of our knowledge, this is the first study to investigate this problem. Our proposed algorithm increases the setup time for each successive job by a constant proportion of the cumulative processing time of preceding jobs to capture the progressive slowdown that often occurs on real production lines. In the developed algorithm with maximum

10^{5}

fitness evaluations, the RF surrogate model predicts objective function values and guides crossover and mutation. On the PFSS-OAWT with PSD benchmark (up to 500 orders and 20 machines, 160 instances),

{S G A}_{R F}

represents improvements of 0.9% over SGA, 0.8% over

{S G A}_{L S}

, and 5.6% over SABPO. Although the surrogate incurs additional runtime, the gains in both profit and order-acceptance rates justify its use for high-margin, offline planning. Overall, the results of this study suggest that integrating evolutionary search into data-driven prediction is an effective strategy for solving complex capacity-constrained scheduling problems. Full article

(This article belongs to the Special Issue Advances in Intelligent Computing, Machine Learning and Pattern Recognition)

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12 pages, 244 KB

Open AccessFeature PaperArticle

New Sufficient Conditions for Moment Determinacy via Probability Density Tails

by Gwo Dong Lin and Jordan M. Stoyanov

Mathematics 2025, 13(16), 2671; https://doi.org/10.3390/math13162671 - 19 Aug 2025

Viewed by 263

Abstract

One of the ways to characterize a probability distribution is to show that it is moment-determinate, uniquely determined by knowing all its moments. The uniqueness, in the absolutely continuous case, depends entirely on the behaviour of the tails of the probability density function [...] Read more.

One of the ways to characterize a probability distribution is to show that it is moment-determinate, uniquely determined by knowing all its moments. The uniqueness, in the absolutely continuous case, depends entirely on the behaviour of the tails of the probability density function f. We find and exploit a condition, (D), in terms only of f which is of a ‘general’ form and easy to check. Condition (D), showing the ‘speed’ for f to tend to zero, is sufficient to conclude the moment determinacy. We establish a series of theorems and corollaries in both Stieltjes and Hamburger cases and provide an interesting illustrative example. The results in this paper are either new or extend some recently published results. Full article

16 pages, 274 KB

Open AccessArticle

Revisiting Black–Scholes: A Smooth Wiener Approach to Derivation and a Self-Contained Solution

by Alessandro Saccal and Andrey Artemenkov

Mathematics 2025, 13(16), 2670; https://doi.org/10.3390/math13162670 - 19 Aug 2025

Viewed by 347

Abstract

This study presents a self-contained derivation and solution of the Black and Scholes partial differential equation (PDE), replacing the standard Wiener process with a smoothed Wiener process, which is a differentiable stochastic process constructed via normal kernel smoothing. By presenting a self-contained, Itô-free [...] Read more.

This study presents a self-contained derivation and solution of the Black and Scholes partial differential equation (PDE), replacing the standard Wiener process with a smoothed Wiener process, which is a differentiable stochastic process constructed via normal kernel smoothing. By presenting a self-contained, Itô-free derivation, this study bridges the gap between heuristic financial reasoning and rigorous mathematics, bringing forth fresh insights into one of the most influential models in quantitative finance. The smoothed Wiener process does not merely simplify the technical machinery but further reaffirms the robustness of the Black and Scholes framework under alternative mathematical formulations. This approach is particularly valuable for instructors, apprentices, and practitioners who may seek a deeper understanding of derivative pricing without relying on the full machinery of stochastic calculus. The derivation underscores the universality of the Black and Scholes PDE, irrespective of the specific stochastic process adopted, under the condition that the essential properties of stochasticity, volatility, and of no arbitrage may be preserved. Full article

12 pages, 808 KB

Open AccessArticle

Robust Angular Frequency Control of Incommensurate Fractional-Order Permanent Magnet Synchronous Motors via State-Sequential Sliding Mode Control

by Guo-Hsin Hu, Chia-Wei Ho and Jun-Juh Yan

Mathematics 2025, 13(16), 2669; https://doi.org/10.3390/math13162669 - 19 Aug 2025

Viewed by 315

Abstract

This paper proposes an innovative state-sequential sliding mode control (SS-SMC) to suppress chaotic behavior and achieve angular frequency control of incommensurate fractional-order permanent magnet synchronous motor (IFOPMSM) systems. The method is designed to handle both input perturbations and mismatched external disturbances. Conventional sliding [...] Read more.

This paper proposes an innovative state-sequential sliding mode control (SS-SMC) to suppress chaotic behavior and achieve angular frequency control of incommensurate fractional-order permanent magnet synchronous motor (IFOPMSM) systems. The method is designed to handle both input perturbations and mismatched external disturbances. Conventional sliding mode control (SMC) is robust to matched uncertainties. However, the use of discontinuous sign functions causes chattering. This reduces control accuracy and overall performance. Many methods have been proposed to reduce chattering. Yet, for IFOPMSMs, achieving both robust stabilization and chattering suppression under mismatched disturbances and input uncertainties remains challenging. To address these issues, this study introduces an SS-SMC strategy that combines a fractional-order integral-type sliding surface with a continuous control law. Unlike conventional SMC methods that rely on discontinuous sign functions, the proposed approach uses a continuous control function. This preserves the robustness of traditional SMC while effectively eliminating chattering. The SS-SMC utilizes state-sequential control, allowing a single input to stabilize all system states sequentially and achieve the control objectives while reducing system complexity. Simulation results and comparative analyses confirm the effectiveness of the proposed method. The findings show that the SS-SMC ensures robust angular frequency regulation of the IFOPMSM and suppresses chattering effectively. Full article

(This article belongs to the Special Issue Dynamic Modeling and Simulation for Control Systems, 3rd Edition)

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23 pages, 7175 KB

Open AccessArticle

Prunability of Multi-Layer Perceptrons Trained with the Forward-Forward Algorithm

by Mitko Nikov, Damjan Strnad and David Podgorelec

Mathematics 2025, 13(16), 2668; https://doi.org/10.3390/math13162668 - 19 Aug 2025

Viewed by 329

Abstract

We explore the sparsity and prunability of multi-layer perceptrons (MLPs) trained using the Forward-Forward (FF) algorithm, an alternative to backpropagation (BP) that replaces the backward pass with local, contrastive updates at each layer. We analyze the sparsity of the weight matrices during training [...] Read more.

We explore the sparsity and prunability of multi-layer perceptrons (MLPs) trained using the Forward-Forward (FF) algorithm, an alternative to backpropagation (BP) that replaces the backward pass with local, contrastive updates at each layer. We analyze the sparsity of the weight matrices during training using multiple metrics, and test the prunability of FF networks on the MNIST, FashionMNIST and CIFAR-10 datasets. We also propose FFLib—a novel, modular PyTorch-based library for developing, training and analyzing FF models along with a suite of FF-based architectures, including FFNN, FFNN+C and FFRNN. In addition to structural sparsity, we describe and apply a new method for visualizing the functional sparsity of neural activations across different architectures using the HSV color space. Moreover, we conduct a sensitivity analysis to assess the impact of hyperparameters on model performance and sparsity. Finally, we perform pruning experiments, showing that simple FF-based MLPs exhibit significantly greater robustness to one-shot neuron pruning than traditional BP-trained networks, and a possible 8-fold increase in compression ratios while maintaining comparable accuracy on the MNIST dataset. Full article

(This article belongs to the Section E1: Mathematics and Computer Science)

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32 pages, 9357 KB

Open AccessArticle

On the Dynamics of a Synchronous Binary Asteroid System with Non-Uniform Mass Distribution

by Leandro Forne Brejão, Antonio F. Bertachini de Almeida Prado, Diogo Merguizo Sanchez and Jean P. dos Santos Carvalho

Mathematics 2025, 13(16), 2667; https://doi.org/10.3390/math13162667 - 19 Aug 2025

Viewed by 260

Abstract

In this work, particle dynamics in a binary asteroid system is analyzed within the Circular Restricted Three-Body Problem (CRTBP) framework, assuming the largest body is treated as a mass point. The secondary body is modeled as a mass dipole in synchronous rotation with [...] Read more.

In this work, particle dynamics in a binary asteroid system is analyzed within the Circular Restricted Three-Body Problem (CRTBP) framework, assuming the largest body is treated as a mass point. The secondary body is modeled as a mass dipole in synchronous rotation with its orbital motion, which leads to the spin–orbit resonance. The third body is a point of negligible mass whose motion is restricted to the orbital plane of the primary bodies. We considered asymmetrical and symmetrical dipole cases. The number and positions of the equilibrium points are determined for the dynamical analysis, and the zero-velocity curves are studied. This model preserves the number and geometric arrangement of the equilibrium points compared to the CRTBP. The equilibrium points adjacent to the dipole are the most sensitive in position to the variations in physical parameters. Considering the solar radiation pressure on the third body, different initial conditions for its motion in the vicinity of the dipole are analyzed. As a result, the particle survival time in orbital motion is estimated before colliding or suffering gravitational ejection from the system. Full article

(This article belongs to the Section E: Applied Mathematics)

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Mathematics, Volume 13, Issue 16 (August-2 2025) – 155 articles

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