Mathematics

10 pages, 275 KB

Open AccessArticle

On p-Hardy–Rogers and p-Zamfirescu Contractions in Complete Metric Spaces: Existence and Uniqueness Results

by Zouaoui Bekri, Nicola Fabiano, Mohammed Ahmed Alomair and Abdulaziz Khalid Alsharidi

Mathematics 2025, 13(24), 4011; https://doi.org/10.3390/math13244011 - 16 Dec 2025

In this paper, we introduce and investigate two generalized forms of classical contraction mappings, namely the p-Hardy–Rogers and p-Zamfirescu contractions. By incorporating the integer parameter

p \geq 1

, these new definitions extend the traditional Hardy–Rogers and Zamfirescu conditions to iterated [...] Read more.

In this paper, we introduce and investigate two generalized forms of classical contraction mappings, namely the p-Hardy–Rogers and p-Zamfirescu contractions. By incorporating the integer parameter

p \geq 1

, these new definitions extend the traditional Hardy–Rogers and Zamfirescu conditions to iterated mappings

ℏ^{p}

. We establish fixed-point theorems, ensuring both existence and uniqueness of fixed points for continuous self-maps on complete metric spaces that satisfy these p-contractive conditions. The proofs are constructed via geometric estimates on the iterates and by transferring the fixed point from the p-th iterate

ℏ^{p}

to the original mapping ℏ. Our results unify and broaden several well-known fixed-point theorems reported in previous studies, including those of Banach, Hardy–Rogers, and Zamfirescu as special cases. Full article

(This article belongs to the Section C: Mathematical Analysis)

46 pages, 5390 KB

Open AccessArticle

A Simulated Weather-Driven Bio-Economic Optimization Model for Agricultural Planning

by Bunnel Bernard, David Riegert, Kenzu Abdella and Suresh Narine

Mathematics 2025, 13(24), 4010; https://doi.org/10.3390/math13244010 - 16 Dec 2025

Abstract

This study develops a weather-driven bio-economic optimization framework for agricultural planning in Guyana by integrating weather simulation, crop modeling, and multi-objective optimization. Precipitation was modeled using a first-order Markov chain with fitted distribution, while temperature and relative humidity were simulated using stochastic differential [...] Read more.

This study develops a weather-driven bio-economic optimization framework for agricultural planning in Guyana by integrating weather simulation, crop modeling, and multi-objective optimization. Precipitation was modeled using a first-order Markov chain with fitted distribution, while temperature and relative humidity were simulated using stochastic differential equations. Reference evapotranspiration was estimated using an artificial neural network. These simulated weather variables were then used as inputs to AquaCrop to estimate rice, maize, and soybean yields across multiple planting intervals. A multi-objective optimization model was then applied to optimize gross profit, economic water productivity, and land use efficiency. Validation at the Rose Hall Estate showed strong accuracy for rice and maize (MAPE < 10%) and moderate accuracy for soybeans. Scenario analyses for the 2024–2025 season, assuming 25% and 50% export targets, revealed that rice–maize double cropping produced the highest profitability, while soybean–maize combinations were less favorable. The framework replaces static yield assumptions with dynamic, simulation-driven models that incorporate price forecasts and allow substitution of alternative forecasting or crop simulators to enhance precision. The scenario-based design provides a flexible decision-support platform for optimizing crop selection, planting intervals, and resource allocation under climate variability and market uncertainty. Moreover, the framework is scalable and well-suited for evidence-based agricultural planning. Full article

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

► Show Figures

Figure 1

21 pages, 1087 KB

Open AccessArticle

Dynamic Event-Triggered, Fixed-Time Control for Heterogeneous Multi-Agent Systems with Hybrid DoS Attacks

by Ji Han and He Jiang

Mathematics 2025, 13(24), 4009; https://doi.org/10.3390/math13244009 - 16 Dec 2025

Abstract

In this article, the fixed-time, quasi-consensus control problem for heterogeneous multi-agent systems (HMASs) under denial-of-service (DoS) attacks is investigated. Unlike most previous studies in this area, which focus on periodic (or single-type) DoS attacks with static event-triggered control, this paper ensures that HMASs [...] Read more.

In this article, the fixed-time, quasi-consensus control problem for heterogeneous multi-agent systems (HMASs) under denial-of-service (DoS) attacks is investigated. Unlike most previous studies in this area, which focus on periodic (or single-type) DoS attacks with static event-triggered control, this paper ensures that HMASs achieve fixed-time quasi-consensus under aperiodic hybrid DoS attacks via dynamic event-triggered control. According to whether DoS attacks are known, two control protocols based on dynamic event-triggered conditions are given, which both ensure that HMASs achieve output quasi-consensus within a fixed time and exhibit less conservative triggering conditions than static event-triggered protocols. Moreover, the proof that the given dynamic event-triggered conditions can avoid Zeno-behavior is provided. Lastly, simulation examples are presented to support the obtained points. Full article

(This article belongs to the Section E2: Control Theory and Mechanics)

17 pages, 12789 KB

Open AccessArticle

EGAN: Encrypting GAN Models Based on Self-Adversarial

by Yujie Zhu, Wei Li, Yuhang Jiang, Yanrong Huang and Faming Fang

Mathematics 2025, 13(24), 4008; https://doi.org/10.3390/math13244008 - 16 Dec 2025

Abstract

The increasing prevalence of deep learning models in industry has highlighted the critical need to protect the intellectual property (IP) of these models, especially generative adversarial networks (GANs) capable of synthesizing realistic data. Traditional IP protection methods, such as watermarking model parameters (white-box) [...] Read more.

The increasing prevalence of deep learning models in industry has highlighted the critical need to protect the intellectual property (IP) of these models, especially generative adversarial networks (GANs) capable of synthesizing realistic data. Traditional IP protection methods, such as watermarking model parameters (white-box) or verifying outputs (black-box), are insufficient against non-public misappropriation. To address these limitations, we introduce EGAN (Encrypted GANs), which secures GAN models by embedding a novel self-adversarial mechanism. This mechanism is trained to actively maximize the feature divergence between authorized and unauthorized inputs, thereby intentionally corrupting the outputs from non-key inputs and preventing unauthorized operation. Our methodology utilizes key-based transformations applied to GAN inputs and incorporates a generator loss regularization term to enforce model protection without compromising performance. This technique is compatible with existing watermark-based verification methods. Extensive experimental evaluations reveal that EGAN maintains the generative capabilities of original GAN architectures, including DCGAN, SRGAN, and CycleGAN, while exhibiting robust resistance to common attack strategies such as fine-tuning. Compared with prior work, EGAN provides comprehensive IP protection by ensuring unauthorized users cannot achieve desired outcomes, thus safeguarding both the models and their generated data. Full article

(This article belongs to the Special Issue Information Security and Image Processing)

21 pages, 843 KB

Open AccessArticle

Multi-Condition Degradation Sequence Analysis in Computers Using Adversarial Learning and Soft Dynamic Time Warping

by Yuanhong Mao, Xi Liu, Pengchao He, Bo Chai, Ling Li, Yilin Zhang, Xin Hu and Yunan Li

Mathematics 2025, 13(24), 4007; https://doi.org/10.3390/math13244007 - 16 Dec 2025

Abstract

Predicting the degradation and lifespan of embedded computers relies critically on the accurate evaluation of key parameter degradation within computing systems. Accelerated high-temperature tests are frequently employed as an alternative to ambient-temperature degradation tests, primarily due to the excessive duration and cost of [...] Read more.

Predicting the degradation and lifespan of embedded computers relies critically on the accurate evaluation of key parameter degradation within computing systems. Accelerated high-temperature tests are frequently employed as an alternative to ambient-temperature degradation tests, primarily due to the excessive duration and cost of ambient-temperature testing. However, the scarcity of effective methodologies for correlating degradation trends across distinct temperature conditions persists as a prominent challenge. This study addresses this gap by leveraging adversarial learning to generate low-temperature degradation sequences from high-temperature datasets. The adversarial learning framework enables feature transfer across diverse operating conditions and facilitates domain adaptation learning. This empowers the model to extract features invariant to degradation trends across multiple temperature conditions. Furthermore, soft dynamic time warping (SDTW) is utilized to precisely align the generated low-temperature sequences with their real-world counterparts. This alignment methodology enables elastic matching of time series data exhibiting nonlinear temporal variations, thereby ensuring accurate comparison and synchronization of degradation sequences. Compared with prior methodologies, our proposed approach delivers superior performance on computer degradation data. It offers a more accurate and reliable solution for the degradation analysis and lifespan prediction of embedded computers, thereby advancing the reliability of computational systems. Full article

(This article belongs to the Special Issue Recent Advances in Time Series Analysis: Methods, Theory, and Applications)

22 pages, 509 KB

Open AccessArticle

Mathematical Properties of the Inverted Topp–Leone Family of Distributions

by Daya K. Nagar, Edwin Zarrazola and Santiago Echeverri-Valencia

Mathematics 2025, 13(24), 4006; https://doi.org/10.3390/math13244006 - 16 Dec 2025

Abstract

This article defines an inverted Topp–Leone distribution. Several mathematical properties and maximum likelihood estimation of parameters of this distribution are considered. The shape of the distribution for different sets of parameters is discussed. Several mathematical properties such as the cumulative distribution function, mode, [...] Read more.

This article defines an inverted Topp–Leone distribution. Several mathematical properties and maximum likelihood estimation of parameters of this distribution are considered. The shape of the distribution for different sets of parameters is discussed. Several mathematical properties such as the cumulative distribution function, mode, moment-generating function, survival function, hazard rate function, stress-strength reliability R, moments, Rényi entropy, Shannon entropy, Fisher information matrix, and partial ordering associated with this distribution, have been derived. Distributions of the sum and quotient of two independent inverted Topp–Leone variables have also been obtained. Full article

► Show Figures

Figure 1

27 pages, 402 KB

Open AccessArticle

Completeness and Hereditary Transfer of Exactness Properties for Internal Group Objects in D-Modules

by Jian-Gang Tang, Miao Liu, Huangrui Lei, Nueraminaimu Maihemuti, Quan-Guo Chen and Jia-Yin Peng

Mathematics 2025, 13(24), 4005; https://doi.org/10.3390/math13244005 - 16 Dec 2025

Abstract

This paper establishes a comprehensive framework for the hereditary transfer of categorical completeness and cocompleteness to categories of internal group objects in D-modules. We prove that while completeness of

Grp (D - Mod)

follows unconditionally from the completeness of the base [...] Read more.

This paper establishes a comprehensive framework for the hereditary transfer of categorical completeness and cocompleteness to categories of internal group objects in D-modules. We prove that while completeness of

Grp (D - Mod)

follows unconditionally from the completeness of the base category

D - Mod

, cocompleteness requires

D - Mod

to be regular, cocomplete, and admit a free group functor left adjoint to the forgetful functor. Explicit constructions are provided for limits via componentwise operations and for colimits through coequalizers of relations induced by group axioms over free group objects. The theory reveals fundamental geometric obstructions: differentially constrained subcategories such as holonomic D-modules fail to be cocomplete due to characteristic variety constraints that prevent free group constructions. Applications demonstrate cocompleteness in topological D-module groups and D-module sheaves, while counterexamples in differential geometric groups exhibit necessary analytic constraints. Additional results include regularity inheritance under product-preserving free group functors, internal hom-object constructions in locally Cartesian closed settings yielding Tannaka-type dualities, and monadicity criteria for locally presentable base categories. This work unifies categorical algebra with differential geometric obstruction theory, resolving fundamental questions on exactness transfer while enabling new constructions in homotopical algebra and internal representation theory for D-modules. Full article

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

27 pages, 755 KB

Open AccessArticle

OO-IB-MPRE: A Post-Quantum Secure Online/Offline Identity-Based Matchmaking Proxy Re-Encryption Scheme for Exercise Physiology Data

by You Zhao, Ye Song, Weiyi Song and Juyan Li

Mathematics 2025, 13(24), 4004; https://doi.org/10.3390/math13244004 - 16 Dec 2025

Abstract

As smart education evolves, there is an increasing need for the cloud-centric management and sharing of student exercise physiological data gathered through wearable devices in the physical education domain. However, challenges arise in achieving authentication for data sources, ensuring the security of sensitive [...] Read more.

As smart education evolves, there is an increasing need for the cloud-centric management and sharing of student exercise physiological data gathered through wearable devices in the physical education domain. However, challenges arise in achieving authentication for data sources, ensuring the security of sensitive data, and implementing efficient dynamic access control. Traditional cryptographic schemes face limitations in resisting quantum attacks, authenticating data sources, protecting identity privacy, handling dynamic permission changes, and computational efficiency. To tackle these challenges, we put forward a lattice-based Online/Offline Identity-Based Matchmaking Proxy Re-Encryption (OO-IB-MPRE) scheme. The scheme offers post-quantum security assurances grounded in lattice cryptography (under the LWE/ISIS assumptions); incorporates Identity-Based matchmaking encryption (IB-ME) to realize bidirectional identity matching, which not only enables identity authentication for data sources but also safeguards the sender’s identity privacy from exposure to other entities; leverages Proxy Re-Encryption (PRE) to support dynamic management of access control; and combines online/offline encryption to adapt to resource constrained sensors. The security of the OO-IB-MPRE scheme is verified under standard lattice assumptions to meet the security requirements of semi-selective privacy and authenticity. Performance analysis and experimental validation demonstrate that in comparison to existing lattice-based PRE schemes, the devised scheme shows notable advantages in both space and computational overhead. Therefore, the proposed OO-IB-MPRE offers a secure, efficient, and scalable solution for the sensitive health data in smart physical education. Full article

(This article belongs to the Special Issue Applications of Cryptography Theory in Network Security)

20 pages, 1030 KB

Open AccessArticle

TFGCRN: Temporal–Frequency Graph Convolutional Recurrent Network for Incomplete Traffic Forecasting

by Jiazhan Hu and Tao Feng

Mathematics 2025, 13(24), 4003; https://doi.org/10.3390/math13244003 - 16 Dec 2025

Abstract

Traffic forecasting is a crucial component that underpins an intelligent transportation system. Among the current mainstream forecasting algorithms, spatial–temporal graph neural networks (STGNNs), as the mainstream solution, have been used in traffic forecasting due to their ability to model spatial–temporal dependencies effectively. However, [...] Read more.

Traffic forecasting is a crucial component that underpins an intelligent transportation system. Among the current mainstream forecasting algorithms, spatial–temporal graph neural networks (STGNNs), as the mainstream solution, have been used in traffic forecasting due to their ability to model spatial–temporal dependencies effectively. However, sensor failures caused by factors such as bad weather often lead to incomplete traffic data, which severely prevents STGNNs from modeling spatial–temporal dependencies and consequently degrades forecasting performance. To achieve accurate forecasting under incomplete traffic conditions, this paper proposes a Temporal–Frequency Graph Convolutional Recurrent Network (TFGCRN) model that embeds a Temporal–Frequency Graph Convolutional Network into gated recurrent units. During the recursive modeling process, TFGCRN fully leverages both global and local information to resist the adverse effects of missing values while generating more accurate spatial relationships, thereby achieving precise incomplete traffic forecasting. The experiments on four real-world datasets show that TFGCRN can achieve satisfactory results superior to multiple baselines and effectively adapt to different missing rates. Compared with the state-of-the-art baseline, TFGCRN can reduce forecasting error by 2–6%. Full article

(This article belongs to the Special Issue Optimization and Modeling in Spatio-Temporal Data Mining Using Graph Neural Networks)

► Show Figures

Figure 1

22 pages, 2204 KB

Open AccessArticle

A Lightweight YOLOv8-Based Network for Efficient Corn Disease Detection

by Deao Song, Yiran Peng, Xinyuan Gu and KinTak U

Mathematics 2025, 13(24), 4002; https://doi.org/10.3390/math13244002 - 16 Dec 2025

Abstract

To address the pressing need for accurate and efficient detection of corn diseases, we propose a novel, lightweight object detection framework, CBS-YOLOv8 (C2f-BiFPN-SCConv YOLOv8), which builds upon the YOLOv8 architecture to enhance performance for corn disease detection. The model incorporates two key components, [...] Read more.

To address the pressing need for accurate and efficient detection of corn diseases, we propose a novel, lightweight object detection framework, CBS-YOLOv8 (C2f-BiFPN-SCConv YOLOv8), which builds upon the YOLOv8 architecture to enhance performance for corn disease detection. The model incorporates two key components, the GhostNetV2 block and SCConv (Selective Convolution). The GhostNetV2 block improves feature representation by reducing computational complexity, while SCConv optimizes convolution operations dynamically, adjusting based on the input to ensure minimal computational overhead. Together, these features maintain high detection accuracy while keeping the network lightweight. Additionally, the model integrates the C2f-GhostNetV2 module to eliminate redundancy, and the SimAM attention mechanism improves lesion-background separation, enabling more accurate disease detection. The Bi-directional Feature Pyramid Network (BiFPN) enhances feature representation across multiple scales, strengthening detection across varying object sizes. Evaluated on a custom dataset of over 6000 corn leaf images across six categories, CBS-YOLOv8 achieves improved accuracy and reliability in object detection. With a lightweight architecture of just 8.1M parameters and 21 GFLOPs, it enables real-time deployment on edge devices in agricultural settings. CBS-YOLOv8 offers high detection performance while maintaining computational efficiency, making it ideal for precision agriculture. Full article

(This article belongs to the Special Issue Intelligent Mathematics and Applications)

► Show Figures

Figure 1

31 pages, 1923 KB

Open AccessArticle

Handwriting-Based Mathematical Assistant Software System Using Computer Vision Methods

by Ahmet Alkan and Gozde Yolcu Oztel

Mathematics 2025, 13(24), 4001; https://doi.org/10.3390/math13244001 - 15 Dec 2025

Abstract

Mathematics is a discipline that forms the foundation of many fields and should be learned gradually, starting from early childhood. However, some subjects can be difficult to learn due to their abstract nature, the need for attention and planning, and math anxiety. Therefore, [...] Read more.

Mathematics is a discipline that forms the foundation of many fields and should be learned gradually, starting from early childhood. However, some subjects can be difficult to learn due to their abstract nature, the need for attention and planning, and math anxiety. Therefore, in this study, a system that contributes to mathematics teaching using computer vision approaches has been developed. In the proposed system, users can write operations directly in their own handwriting on the system interface, learn their results, or test the accuracy of their answers. They can also test themselves with random questions generated by the system. In addition, visual graph generation has been added to the system, ensuring that education is supported with visuals and made enjoyable. Besides the character recognition test, which is applied on public datasets, the system was also tested with images obtained from 22 different users, and successful results were observed. The study utilizes CNN networks for handwritten character detection and self-created image processing algorithms to organize the obtained characters into equations. The system can work with equations that include single and multiple unknowns, trigonometric functions, derivatives, integrals, etc. Operations can be performed, and successful results can be achieved even for users who write in italicized handwriting. Furthermore, equations written within each closed figure on the same page are evaluated locally. This allows multiple problems to be solved on the same page, providing a user-friendly approach. The system can be an assistant for improving performance in mathematics education. Full article

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

19 pages, 929 KB

Open AccessArticle

Some Modified Mann-Type Inertial Forward–Backward Iterative Methods for Monotone Inclusion Problems

by Mohammad Dilshad, Ibrahim Al-Dayel, Esmail Alshaban and Md. Nasiruzzaman

Mathematics 2025, 13(24), 4000; https://doi.org/10.3390/math13244000 - 15 Dec 2025

Abstract

In this paper, we propose three variants of Mann-type inertial forward–backward iterative methods for approximating the minimum-norm solution of the monotone inclusion problem and the fixed points of nonexpansive mappings. In the first two methods, we compute the Mann-type iteration together with the [...] Read more.

In this paper, we propose three variants of Mann-type inertial forward–backward iterative methods for approximating the minimum-norm solution of the monotone inclusion problem and the fixed points of nonexpansive mappings. In the first two methods, we compute the Mann-type iteration together with the inertial extrapolation and fixed-point iteration in the initiation of the process, while the last method computes only the Mann-type iteration with inertial extrapolation at the start of the process. We establish the strong convergence results for each method with appropriate assumptions and discuss some applications of the presented methods. Finally, we present numerical examples in both finite- and infinite-dimensional Hilbert spaces to demonstrate their efficiency. A comparative analysis with existing methods is also provided. Full article

(This article belongs to the Special Issue Fixed Point, Optimization, and Applications: 3rd Edition)

13 pages, 277 KB

Open AccessArticle

Existence of Mild Solutions to Impulsive Fractional Equations with Almost-Sectorial Operators

by Mostefa Seghier, Kadda Maazouz and Rosana Rodríguez-López

Mathematics 2025, 13(24), 3999; https://doi.org/10.3390/math13243999 - 15 Dec 2025

Abstract

This study investigates the existence of mild solutions to impulsive fractional differential equations involving almost-sectorial operators. Through the application of solution operator techniques, fixed-point theory, and Laplace transform, we demonstrate the existence of a unique mild solution to the considered system. Full article

(This article belongs to the Special Issue Advances in Fractional Calculus for Modeling and Applications)

24 pages, 347 KB

Open AccessArticle

Fuzzy and Gradual Prime Ideals

by Pascual Jara and Salwa Mohamed

Mathematics 2025, 13(24), 3998; https://doi.org/10.3390/math13243998 - 15 Dec 2025

Abstract

There is a correspondence between equivalence classes of fuzzy ideals, on a commutative ring, and decreasing gradual ideals. In this paper, we explore how to construct a fuzzy ideal starting from any decreasing gradual ideal

σ

. To achieve this, we consider an [...] Read more.

There is a correspondence between equivalence classes of fuzzy ideals, on a commutative ring, and decreasing gradual ideals. In this paper, we explore how to construct a fuzzy ideal starting from any decreasing gradual ideal

σ

. To achieve this, we consider an interior operator,

σ^{d}

, and a closure operator,

σ^{e}

, and show that the pair

(σ^{d}, σ^{e})

is always an F-pair, which defines a fuzzy ideal. Furthermore, this correspondence, and its inverse, preserves sums, intersections and products. This therefore provides an algebraic framework for studying fuzzy ideals. In particular, prime fuzzy ideals and weakly prime fuzzy ideals have their counterparts in the theory of decreasing gradual ideals, offering us a new perspective on these particular objects. One of the main objectives is to characterize fuzzy prime ideals using single fuzzy elements and gradual ideals. Full article

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

32 pages, 1453 KB

Open AccessArticle

Robust Explosion Point Location Detection via Multi–UAV Data Fusion: An Improved D–S Evidence Theory Framework

by Xuebin Liu and Hanshan Li

Mathematics 2025, 13(24), 3997; https://doi.org/10.3390/math13243997 - 15 Dec 2025

Abstract

The Dempster–Shafer (D–S) evidence theory, while powerful for uncertainty reasoning, suffers from mathematical limitations in high–conflict scenarios where its combination rule produces counterintuitive results. This paper introduces a reformulated D–S framework grounded in optimization theory and information geometry. We rigorously construct a dynamic [...] Read more.

The Dempster–Shafer (D–S) evidence theory, while powerful for uncertainty reasoning, suffers from mathematical limitations in high–conflict scenarios where its combination rule produces counterintuitive results. This paper introduces a reformulated D–S framework grounded in optimization theory and information geometry. We rigorously construct a dynamic weight allocation mechanism derived from minimizing systemic Jensen–Shannon divergence and propose a conflict–adaptive fusion rule with theoretical guarantees. We formally prove that our framework possesses the Conflict Attenuation Property and Robustness to Outlier Evidence. Extensive Monte Carlo simulations in multi–UAV explosion point localization demonstrate the framework’s superiority, reducing localization error by 75.6% in high–conflict scenarios compared to classical D–S. This work provides not only a robust application solution but also a theoretically sound and generalizable mathematical framework for multi–source data fusion under uncertainty. Full article

24 pages, 12737 KB

Open AccessArticle

Curriculum-Enhanced Adaptive Sampling for Physics-Informed Neural Networks: A Robust Framework for Stiff PDEs

by Hasan Cetinkaya, Fahrettin Ay, Mehmet Tunçel, Hazem Nounou, Mohamed Numan Nounou, Hasan Kurban and Erchin Serpedin

Mathematics 2025, 13(24), 3996; https://doi.org/10.3390/math13243996 - 15 Dec 2025

Abstract

Physics-Informed Neural Networks (PINNs) often struggle with stiff partial differential equations (PDEs) exhibiting sharp gradients and extreme nonlinearities. We propose a Curriculum-Enhanced (CE) Adaptive Sampling framework that integrates curriculum learning with adaptive refinement to improve PINN training. Our framework introduces four methods: CE-RARG [...] Read more.

Physics-Informed Neural Networks (PINNs) often struggle with stiff partial differential equations (PDEs) exhibiting sharp gradients and extreme nonlinearities. We propose a Curriculum-Enhanced (CE) Adaptive Sampling framework that integrates curriculum learning with adaptive refinement to improve PINN training. Our framework introduces four methods: CE-RARG (greedy sampling), CE-RARD (probabilistic sampling), and their novel difficulty-aware dynamic counterparts, CED-RARG and CED-RARD, which adjust refinement effort based on task difficulty. We test these methods on five challenging stiff PDEs: the Allen–Cahn, Burgers’ (I and II), Korteweg–de Vries (KdV), and Reaction equations. Our methods consistently outperform both Vanilla PINNs and curriculum-only baselines. In the most difficult regimes, CED-RARD achieves errors up to 100 times lower for the Burgers’ and KdV equations. For the Allen–Cahn and Reaction equations, CED-RARG proves most effective, reducing errors by over 40% compared to its non-dynamic counterpart and by over two orders of magnitude relative to Vanilla PINN. Visualizations confirm that our methods effectively allocate collocation points to high-gradient regions. By demonstrating success across a wide range of stiffness parameters, we provide a robust and reproducible framework for solving stiff PDEs, with all code and datasets publicly available. Full article

(This article belongs to the Special Issue Physics-Informed Machine Learning: Methodologies and Applications)

► Show Figures

Figure 1

27 pages, 2307 KB

Open AccessArticle

An Energy-Aware AIoT Framework for Intelligent Remote Device Control

by Daniel Stefani, Iosif Viktoratos, Albin Uruqi, Alexander Astaras and Chris Christodolou

Mathematics 2025, 13(24), 3995; https://doi.org/10.3390/math13243995 - 15 Dec 2025

Abstract

This paper presents an energy-aware Artificial Intelligence of Things framework designed for intelligent remote device control in residential settings. The system architecture is grounded in the Power Administration Device (PAD), a cost-effective and non-intrusive smart plug prototype that measures real-time electricity consumption and [...] Read more.

This paper presents an energy-aware Artificial Intelligence of Things framework designed for intelligent remote device control in residential settings. The system architecture is grounded in the Power Administration Device (PAD), a cost-effective and non-intrusive smart plug prototype that measures real-time electricity consumption and actuates appliance power states. The PAD transmits data to a scalable, cross-platform cloud infrastructure, which powers a web-based interface for monitoring, configuration, and multi-device control. Central to this framework is Cross-Feature Time-MoE, a novel neural forecasting model that processes the ingested data to predict consumption patterns. Integrating a Transformer Decoder with a Top-K Mixture-of-Experts (MoE) layer for temporal reasoning and a Bilinear Interaction Layer for capturing complex cross-time and cross-feature dependencies, the model generates accurate multi-horizon energy forecasts. These predictions drive actionable recommendations for device shut-off times, facilitating automated energy efficiency. Simulation results indicate that this system yields substantial reductions in energy consumption, particularly for high-wattage appliances, providing a user-friendly, scalable solution for household cost savings and environmental sustainability. Full article

(This article belongs to the Special Issue Application of Neural Networks and Deep Learning, 2nd Edition)

► Show Figures

Figure 1

29 pages, 901 KB

Open AccessArticle

Stability Study for an Age-Structured Epidemic Model with Latent Phase, Relapse and Nonlinear Infection Rate

by Abdeljabbar Ghanmi and Abderrazak Nabti

Mathematics 2025, 13(24), 3994; https://doi.org/10.3390/math13243994 - 15 Dec 2025

Abstract

This paper is dedicated to the analytical investigation of the global dynamics of an SEIR epidemiological model that incorporates latency age (the time spent by an individual in the exposed class before becoming infectious) and a general nonlinear incidence rate. In this model, [...] Read more.

This paper is dedicated to the analytical investigation of the global dynamics of an SEIR epidemiological model that incorporates latency age (the time spent by an individual in the exposed class before becoming infectious) and a general nonlinear incidence rate. In this model, to reflect the dependence of disease progress on the latency age, the exposed class is structured by the latency age, and the rate at which the latent individual becomes infected, and the removal rate are assumed to depend on the latency age. By analyzing the characteristic equations associated with each equilibrium, we study the local stability of both the disease-free and endemic steady states of the model. Moreover, it is proven that the semiflow generated by this system is asymptotically smooth, and if the basic reproduction number is greater than unity, the system is uniformly persistent. Furthermore, based on Lyapunov functional and LaSalle’s invariance principle, the global dynamics of the model are established. It is obtained that if the basic reproduction number is less than unity, the disease-free steady state is globally asymptotically stable and hence the disease dies out; however, if the basic reproduction number is greater than unity, the endemic steady state is globally asymptotically stable, and the disease persists. Numerical simulations are carried out to illustrate the main analytic results. Full article

(This article belongs to the Section E3: Mathematical Biology)

27 pages, 391 KB

Open AccessArticle

Analysis of λ-Hölder Stability of Economic Equilibria and Dynamical Systems with Nonsmooth Structures

by Anna V. Aleshina, Andrey L. Bulgakov, Yanliang Xin and Igor Y. Panarin

Mathematics 2025, 13(24), 3993; https://doi.org/10.3390/math13243993 - 15 Dec 2025

Abstract

This paper develops a mathematical approach to the analysis of the stability of economic equilibria in nonsmooth models. The

λ

-Hölder apparatus of subdifferentials is used, which extends the class of systems under study beyond traditional smooth optimization and linear approximations. Stability conditions [...] Read more.

This paper develops a mathematical approach to the analysis of the stability of economic equilibria in nonsmooth models. The

λ

-Hölder apparatus of subdifferentials is used, which extends the class of systems under study beyond traditional smooth optimization and linear approximations. Stability conditions are obtained for solutions to intertemporal choice problems and capital accumulation models in the presence of nonsmooth dependencies, threshold effects, and discontinuities in elasticities. For

λ

-Hölder production and utility functions, estimates of the sensitivity of equilibria to parameters are obtained, and indicators of the convergence rate of trajectories to the stationary state are derived for

λ > 1

. The methodology is tested on a multisectoral model of economic growth with technological shocks and stochastic disturbances in capital dynamics. Numerical experiments confirm the theoretical results: a power-law dependence of equilibrium sensitivity on the magnitude of parametric disturbances is revealed, as well as consistency between the analytical

λ

-Hölder convergence rate and the results of numerical integration. Stochastic disturbances of small variance do not violate stability. The results obtained provide a rigorous mathematical foundation for the analysis of complex economic systems with nonsmooth structures, which are increasingly used in macroeconomics, decision theory, and regulation models. Full article

(This article belongs to the Section E5: Financial Mathematics)

Journal Description

Mathematics

Latest Articles

Journal Menu

Journal Browser

Highly Accessed Articles

Latest Books

E-Mail Alert

News

Topics

Conferences

Special Issues

Topical Collections

Further Information

Guidelines

MDPI Initiatives

Follow MDPI