Mathematics

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Open AccessFeature PaperArticle

A Diophantine Inequality Involving Mixed Powers of Primes with a Specific Type

by Tatiana L. Todorova and Atanaska Georgieva

Mathematics 2025, 13(19), 3065; https://doi.org/10.3390/math13193065 (registering DOI) - 23 Sep 2025

Let

λ_{1}, λ_{2}, λ_{3}

be nonzero real numbers, not all of the same sign; let

λ_{1} / λ_{2}

be irrational; and let

η

be any real number. We investigate the solvability of the inequality [...] Read more.

Let

λ_{1}, λ_{2}, λ_{3}

be nonzero real numbers, not all of the same sign; let

λ_{1} / λ_{2}

be irrational; and let

η

be any real number. We investigate the solvability of the inequality

| λ_{1} p_{1} + λ_{2} p_{2} + λ_{3} p_{3}^{2} + η | < {(max p_{j})}^{- 1 / 12 + θ}

,

θ > 0

in the prime variables

p_{1}

,

p_{2}

, and

p_{3}

. We require that

p_{1} + 2

and

p_{2} + 2

have no more than 20 prime factors, while

p_{3} + 2

has no more than 42 prime factors. Full article

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

418 KB

Open AccessFeature PaperArticle

Exploiting Generalized Cyclic Symmetry to Find Fast Rectangular Matrix Multiplication Algorithms Easier

by Charlotte Vermeylen, Nico Vervliet, Lieven De Lathauwer and Marc Van Barel

Mathematics 2025, 13(19), 3064; https://doi.org/10.3390/math13193064 (registering DOI) - 23 Sep 2025

Abstract

The quest to multiply two large matrices as fast as possible is one that has already intrigued researchers for several decades. However, the `optimal’ algorithm for a certain problem size is still not known. The fast matrix multiplication (FMM) problem can be formulated [...] Read more.

The quest to multiply two large matrices as fast as possible is one that has already intrigued researchers for several decades. However, the `optimal’ algorithm for a certain problem size is still not known. The fast matrix multiplication (FMM) problem can be formulated as a non-convex optimization problem—more specifically, as a challenging tensor decomposition problem. In this work, we build upon a state-of-the-art augmented Lagrangian algorithm, which formulates the FMM problem as a constrained least squares problem, by incorporating a new, generalized cyclic symmetric (CS) structure in the decomposition. This structure decreases the number of variables, thereby reducing the large search space and the computational cost per iteration. The constraints are used to find practical solutions, i.e., decompositions with simple coefficients, which yield fast algorithms when implemented in hardware. For the FMM problem, usually a very large number of starting points are necessary to converge to a solution. Extensive numerical experiments for different problem sizes demonstrate that including this structure yields more ‘unique’ practical decompositions for a fixed number of starting points. Uniqueness is defined relative to the known scale and trace invariance transformations that hold for all FMM decompositions. Making it easier to find practical decompositions may lead to the discovery of faster FMM algorithms when used in combination with sufficient computational power. Lastly, we show that the CS structure reduces the cost of multiplying a matrix by itself. Full article

(This article belongs to the Special Issue Spectral Theory of Tensors, Tensor (Rank) Decompositions, and Their Applications)

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Open AccessArticle

Mathematical Aspects of ANM/FEM Numerical Model, Applied to Nonlinear Elastic, and Thermo Elastic Analysis of Wrinkles in Film/Substrate Systems, and a New Implementation in the FreeFEM++ Language

by Pascal Ventura, Frédéric Hecht, Michel Potier-Ferry, Hamid Zahrouni, Fan Xu, Hamza Azzayani, Michael Brun and Anh-Khoa Chau

Mathematics 2025, 13(19), 3063; https://doi.org/10.3390/math13193063 - 23 Sep 2025

Abstract

The main purposes of the present paper are to present the mathematical and algorithmic aspects of the ANM/FEM numerical model and to show how it is applied to analyze elastic and thermo-elastic nonlinear solid mechanical problems. ANM is a robust continuation method based [...] Read more.

The main purposes of the present paper are to present the mathematical and algorithmic aspects of the ANM/FEM numerical model and to show how it is applied to analyze elastic and thermo-elastic nonlinear solid mechanical problems. ANM is a robust continuation method based on a perturbation technique for solving nonlinear problems dependent on a loading parameter. Historically, this technique has been successfully applied to problems in various fields of solid and fluid mechanics. This paper shows how ANM is used to solve nonlinear elastic and nonlinear thermo-elastic problems involving elastic behavior and geometrical nonlinearities. The implementation of ANM for FEM in the FreeFEM++ language is then presented. The FEM software development platform, called FreeFEM++, is structured to work with variational formulations and, therefore, is well adapted to implement ANM for instability problems in solid mechanics. In order to illustrate the great efficiency of FreeFEM++, scripts will be presented for computing the different steps of ANM continuation for solid elastic structures, considering simple geometries subjected to conservative loading. For the purpose of validation, the problem of a cantilever subjected to an applied force is presented. Next, the new numerical model is applied to study wrinkles appearing in a planar film/substrate system that is subjected to compressive surface forces at the lateral faces of the film. Finally, the model is applied to a spherical film/substrate system subjected to thermo-elastic shrinkage. In both cases, the ANM/FEM prediction method, together with a Newton–Riks correction (if needed), identifies the equilibrium paths efficiently, especially after the post-buckling regime. Full article

(This article belongs to the Special Issue Recent Developments in Numerical Methods for Partial Differential Equations)

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Open AccessReview

From Mobile Media to Generative AI: The Evolutionary Logic of Computational Social Science Across Data, Methods, and Theory

by Hua Li, Qifang Wang and Ye Wu

Mathematics 2025, 13(19), 3062; https://doi.org/10.3390/math13193062 - 23 Sep 2025

Abstract

Since its articulation in 2009, Computational Social Science (CSS) has grown into a mature interdisciplinary paradigm, shaped first by mobile media-generated digital traces and more recently by generative AI. With over a decade of development, CSS has expanded its scope across data, methods, [...] Read more.

Since its articulation in 2009, Computational Social Science (CSS) has grown into a mature interdisciplinary paradigm, shaped first by mobile media-generated digital traces and more recently by generative AI. With over a decade of development, CSS has expanded its scope across data, methods, and theory: data sources have evolved from mobile traces to multimodal records; methods have diversified from surveys and experiments to agent-based modeling, network analysis, and computer vision; and theory has advanced by revisiting classical questions and modeling emergent digital phenomena. Generative AI further enhances CSS through scalable annotation, experimental design, and simulation, while raising challenges of validity, reproducibility, and ethics. The evolutionary logic of CSS lies in coupling theory, models, and data, balancing innovation with normative safeguards to build cumulative knowledge and support responsible digital governance. Full article

(This article belongs to the Special Issue Mathematical Models and Methods in Computational Social Science)

243 KB

Open AccessArticle

Maximum General Sum-Connectivity Index of Trees and Unicyclic Graphs with Given Order and Number of Pendant Vertices

by Elize Swartz and Tomáš Vetrík

Mathematics 2025, 13(19), 3061; https://doi.org/10.3390/math13193061 - 23 Sep 2025

Abstract

For

a \in R

, the general sum-connectivity index of a graph G is defined as [...] Read more.

For

a \in R

, the general sum-connectivity index of a graph G is defined as

χ_{a} (G) = \sum_{u v \in E (G)} {[d_{G} (u) + d_{G} (v)]}^{a}

, where

E (G)

is the set of edges of G and

d_{G} (u)

and

d_{G} (v)

are the degrees of vertices u and v, respectively. For trees and unicyclic graphs with given order and number of pendant vertices, we present upper bounds on the general sum-connectivity index

χ_{a}

, where

0 < a < 1

. We also present the trees and unicyclic graphs that attain the maximum general sum-connectivity index for

0 < a < 1

. Full article

(This article belongs to the Special Issue Topological Analysis and Computation of Chemical Graphs and Physical Networks)

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Open AccessFeature PaperArticle

Marshall’s Quotient and the Arason–Pfister Hauptsatz for Reduced Special Groups

by Kaique Matias de Andrade Roberto and Hugo Luiz Mariano

Mathematics 2025, 13(19), 3060; https://doi.org/10.3390/math13193060 - 23 Sep 2025

Abstract

We provide a new proof of the Arason–Pfister Hauptsatz (APH) in the setting of reduced special groups, as developed by Dickmann and Miraglia. Our approach avoids the use of Boolean invariants and instead relies on a construction inspired by Marshall’s quotient, suitably adapted [...] Read more.

We provide a new proof of the Arason–Pfister Hauptsatz (APH) in the setting of reduced special groups, as developed by Dickmann and Miraglia. Our approach avoids the use of Boolean invariants and instead relies on a construction inspired by Marshall’s quotient, suitably adapted to the context of special groups. We establish structural properties of this quotient and show that it generalizes the Pfister quotient by a Pfister subgroup. Using this framework, we define iterated quadratic extensions of special groups and develop a theory of Arason–Pfister sequences. These tools allow us to prove that any anisotropic form

φ \in I^{n} (G)

over a reduced special group G satisfies the inequality

dim (φ) \geq 2^{n}

, where

I^{n} (G)

denotes the n-th power of the fundamental ideal of the Witt ring of G. Our methods are purely algebraic and internal to the theory of special groups, contributing with novel tools to the categorical study of abstract theories of quadratic forms. Full article

(This article belongs to the Section A: Algebra and Logic)

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Open AccessFeature PaperArticle

A SEIQRS Model for Interbank Financial Risk Contagion and Rescue Strategies in Complex Networks

by Bo Sun and Yujia Liu

Mathematics 2025, 13(19), 3059; https://doi.org/10.3390/math13193059 - 23 Sep 2025

Abstract

Our paper employs complex network theory and the SEIQRS epidemic model based on the dynamics of differential equations to investigate the contagion mechanisms of financial risk within banking systems and to evaluate rescue strategies. A scale-free interbank network of 36 listed Chinese banks [...] Read more.

Our paper employs complex network theory and the SEIQRS epidemic model based on the dynamics of differential equations to investigate the contagion mechanisms of financial risk within banking systems and to evaluate rescue strategies. A scale-free interbank network of 36 listed Chinese banks is constructed using the minimum-density method. Under the SEIQRS epidemic model, we simulate risk propagation pathways and analyze how key parameters affect systemic risk. Simulation of various rescue interventions demonstrates that, building on the existing support framework, coordinated adjustment of the quarantine rate, exposed-to-infectious transition rate, and quarantine-recovery rate can substantially curb the spread of risk. Among the strategies tested, the high-degree-first rescue strategy yields the best outcomes but requires precise timing, specifically, implementation at the first non-worsening time point. Finally, we offer some policy recommendations, which provide theoretical support and practical enlightenment for preventing cross-system financial risk contagion. Full article

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Open AccessFeature PaperArticle

An Approach for Sustainable Supplier Segmentation Using Adaptive Network-Based Fuzzy Inference Systems

by Ricardo Antonio Saugo, Francisco Rodrigues Lima Junior, Luiz Cesar Ribeiro Carpinetti, Ana Paula Duarte and Jurandir Peinado

Mathematics 2025, 13(19), 3058; https://doi.org/10.3390/math13193058 - 23 Sep 2025

Abstract

Due to the globalization of supply chains and the resulting increase in the quantity and diversity of suppliers, the segmentation of suppliers has become fundamental to improving relationship management and supplier performance. Moreover, given the need to incorporate sustainability into supply chain management, [...] Read more.

Due to the globalization of supply chains and the resulting increase in the quantity and diversity of suppliers, the segmentation of suppliers has become fundamental to improving relationship management and supplier performance. Moreover, given the need to incorporate sustainability into supply chain management, criteria based on economic, environmental, and social performance have been adopted for evaluating suppliers. However, few studies present sustainable supplier segmentation models in the literature, and none of them make it possible to predict individual supplier performance for each TBL dimension in a non-compensatory manner. Moreover, none of them permits the use of historical performance data to adapt the model to the usage environment. Given this, this study aims to propose a decision-making model for sustainable supplier segmentation using an adaptive network-based fuzzy inference system (ANFIS). Our approach combines three ANFIS computational models in a tridimensional quadratic matrix, based on diverse criteria associated with the triple bottom line (TBL) dimensions. A pilot application of this model in a sugarcane mill was performed. We implemented 108 candidate topologies using the Neuro-Fuzzy Designer of the MATLAB^® software (R2014a). The cross-validation method was applied to select the best topologies. The accuracy of the selected topologies was confirmed using statistical tests. The proposed model can be adopted for supplier segmentation processes in companies that wish to monitor and/or improve the sustainability performance of their suppliers. This study may also be helpful to researchers in developing computational solutions for segmenting suppliers, mainly regarding ANFIS modeling and providing appropriate topological parameters to obtain accurate results. Full article

(This article belongs to the Special Issue Advances in Fuzzy Logic and Artificial Neural Networks, 2nd Edition)

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5817 KB

Open AccessArticle

Unsupervised Segmentation and Alignment of Multi-Demonstration Trajectories via Multi-Feature Saliency and Duration-Explicit HSMMs

by Tianci Gao, Konstantin A. Neusypin, Dmitry D. Dmitriev, Bo Yang and Shengren Rao

Mathematics 2025, 13(19), 3057; https://doi.org/10.3390/math13193057 (registering DOI) - 23 Sep 2025

Abstract

Learning from demonstration with multiple executions must contend with time warping, sensor noise, and alternating quasi-stationary and transition phases. We propose a label-free pipeline that couples unsupervised segmentation, duration-explicit alignment, and probabilistic encoding. A dimensionless multi-feature saliency (velocity, acceleration, curvature, direction-change rate) yields [...] Read more.

Learning from demonstration with multiple executions must contend with time warping, sensor noise, and alternating quasi-stationary and transition phases. We propose a label-free pipeline that couples unsupervised segmentation, duration-explicit alignment, and probabilistic encoding. A dimensionless multi-feature saliency (velocity, acceleration, curvature, direction-change rate) yields scale-robust keyframes via persistent peak–valley pairs and non-maximum suppression. A hidden semi-Markov model (HSMM) with explicit duration distributions is jointly trained across demonstrations to align trajectories on a shared semantic time base. Segment-level probabilistic motion models (GMM/GMR or ProMP, optionally combined with DMP) produce mean trajectories with calibrated covariances, directly interfacing with constrained planners. Feature weights are tuned without labels by minimizing cross-demonstration structural dispersion on the simplex via CMA-ES. Across UAV flight, autonomous driving, and robotic manipulation, the method reduces phase-boundary dispersion by 31% on UAV-Sim and by 30–36% under monotone time warps, noise, and missing data (vs. HMM); improves the sparsity–fidelity trade-off (higher time compression at comparable reconstruction error) with lower jerk; and attains nominal 2σ coverage (94–96%), indicating well-calibrated uncertainty. Ablations attribute the gains to persistence plus NMS, weight self-calibration, and duration-explicit alignment. The framework is scale-aware and computationally practical, and its uncertainty outputs feed directly into MPC/OMPL for risk-aware execution. Full article

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

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Open AccessArticle

The Prediction of Tea Production Using Dynamic Rolling Update Grey Model: A Case Study of China

by Suwen Xie, Wai Kuan Wong, Hui Shan Lee and Kee Seng Kuang

Mathematics 2025, 13(19), 3056; https://doi.org/10.3390/math13193056 - 23 Sep 2025

Abstract

China is one of the world’s largest tea-producing countries, and its fluctuations in production affect the international market and domestic economic stability. Existing research often uses limited predictive models at the local scale and lacks systematic national analysis. This study evaluated five models—autoregressive [...] Read more.

China is one of the world’s largest tea-producing countries, and its fluctuations in production affect the international market and domestic economic stability. Existing research often uses limited predictive models at the local scale and lacks systematic national analysis. This study evaluated five models—autoregressive integrated moving average model (ARIMA), grey model (GM (1,1)), Markov chain grey model (Markov-GM (1,1)), particle swarm optimization Markov chain grey model (PSO-Markov-GM), and dynamic rolling update grey model (DRUGM (1,1))—using three stages of annual tea production data from China (2004–2023). The results indicate that DRUGM (1,1) has the lowest prediction error, demonstrating superior ability to capture production trends. The dynamic update mechanism of this model enhances its adaptability, providing an efficient and scalable framework for predicting the production level of tea and other crops. Accurate predictions are crucial for improving agricultural planning, optimizing resource allocation, and providing information for trade policy design. This study provides practical tools for sustainable agricultural decision-making, helping to strengthen rural economic stability and resilient food systems. Full article

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23 pages, 2537 KB

Open AccessArticle

Dynamic Scheduling for Security Protection Re-2 Sources in Cloud–Edge Collaboration Scenarios Using Deep Reinforcement Learning

by Lin Guan, Hongmei Shi, Haoran Chen and Yi Wang

Mathematics 2025, 13(19), 3055; https://doi.org/10.3390/math13193055 - 23 Sep 2025

Abstract

Current cloud–edge collaboration collaboration architectures face challenges in security resource scheduling due to their mostly static nature, which cannot keep up with real-time attack patterns and dynamic security needs. To address this, this paper proposes a dynamic scheduling method using Deep Reinforcement Learning [...] Read more.

Current cloud–edge collaboration collaboration architectures face challenges in security resource scheduling due to their mostly static nature, which cannot keep up with real-time attack patterns and dynamic security needs. To address this, this paper proposes a dynamic scheduling method using Deep Reinforcement Learning (DQN) and SRv6 technology. The method establishes a multi-dimensional feature space by collecting network threat indicators and security resource states; constructs a dynamic decision-making model with DQN to optimize scheduling strategies online by encoding security requirements, resource constraints, and network topology into a Markov Decision Process; and enables flexible security service chaining through SRv6 for precise policy implementation. Experimental results demonstrate that this approach significantly reduces security service deployment delays (by up to 56.8%), enhances resource utilization, and effectively balances the security load between edge and cloud. Full article

(This article belongs to the Special Issue Research and Application of Network and System Security)

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Mathematics, Volume 13, Issue 19 (October-1 2025) – 11 articles

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