Mathematics

9 pages, 200 KiB

Open AccessArticle

The Fourth Hybrid Power Mean Involving the Character Sums and Exponential Sums

by Guohui Chen and Tingting Du

Mathematics 2025, 13(10), 1680; https://doi.org/10.3390/math13101680 - 20 May 2025

In this paper, we consider the fourth hybrid power mean involving two-term exponential sums and third-order character sum modulo p, a topic of significant importance in analytic number theory. These results generalize prior research, and provide new insights for studying the relationship [...] Read more.

In this paper, we consider the fourth hybrid power mean involving two-term exponential sums and third-order character sum modulo p, a topic of significant importance in analytic number theory. These results generalize prior research, and provide new insights for studying the relationship between character sums and exponential sums. Full article

(This article belongs to the Special Issue Analytic Methods in Number Theory and Allied Fields)

15 pages, 666 KiB

Open AccessArticle

A Simultaneous Decomposition for a Quaternion Tensor Quaternity with Applications

by Jia-Wei Huo, Yun-Ze Xu and Zhuo-Heng He

Mathematics 2025, 13(10), 1679; https://doi.org/10.3390/math13101679 - 20 May 2025

Abstract

Quaternion tensor decompositions have recently been the center of focus due to their wide potential applications in color data processing. In this paper, we establish a simultaneous decomposition for a quaternion tensor quaternity under Einstein product. The decomposition brings the quaternity of four [...] Read more.

Quaternion tensor decompositions have recently been the center of focus due to their wide potential applications in color data processing. In this paper, we establish a simultaneous decomposition for a quaternion tensor quaternity under Einstein product. The decomposition brings the quaternity of four quaternion tensors into a canonical form, which only has 0 and 1 entries. The structure of the canonical form is discussed in detail. Moreover, the proposed decomposition is applied to a new framework of color video encryption and decryption based on discrete wavelet transform. This new approach can realize simultaneous encryption and compression with high security. Full article

(This article belongs to the Special Issue Advanced Numerical Linear Algebra)

22 pages, 344 KiB

Open AccessArticle

Inverse Source Problem for a Singular Parabolic Equation with Variable Coefficients

by Xue Qin and Shumin Li

Mathematics 2025, 13(10), 1678; https://doi.org/10.3390/math13101678 - 20 May 2025

Abstract

We consider a parabolic equation with a singular potential in a bounded domain

Ω \subset R^{n} .

The main result is a Lipschitz stability estimate for an inverse source problem of determining a spatial varying factor

f (x)

of the [...] Read more.

We consider a parabolic equation with a singular potential in a bounded domain

Ω \subset R^{n} .

The main result is a Lipschitz stability estimate for an inverse source problem of determining a spatial varying factor

f (x)

of the source term

R (x, t) f (x) .

We obtain a consistent stability result for any

μ \leq p_{1} μ^{*},

where

p_{1} > 0

is the lower bound of

p (x)

and

μ^{*} = {(n - 2)}^{2} / 4,

and this condition for

μ

is also almost a consistently optimal condition for the existence of solutions. The main method we used is the Carleman estimate, and the proof for the inverse source problem relies on the Bukhgeim–Klibanov method. Full article

45 pages, 3641 KiB

Open AccessReview

Certified Neural Network Control Architectures: Methodological Advances in Stability, Robustness, and Cross-Domain Applications

by Rui Liu, Jianhua Huang, Biao Lu and Weili Ding

Mathematics 2025, 13(10), 1677; https://doi.org/10.3390/math13101677 - 20 May 2025

Abstract

Neural network (NN)-based controllers have emerged as a paradigm-shifting approach in modern control systems, demonstrating unparalleled capabilities in governing nonlinear dynamical systems with inherent uncertainties. This comprehensive review systematically investigates the theoretical foundations and practical implementations of NN controllers through the prism of [...] Read more.

Neural network (NN)-based controllers have emerged as a paradigm-shifting approach in modern control systems, demonstrating unparalleled capabilities in governing nonlinear dynamical systems with inherent uncertainties. This comprehensive review systematically investigates the theoretical foundations and practical implementations of NN controllers through the prism of Lyapunov stability theory, NN controller frameworks, and robustness analysis. The review establishes that recurrent neural architectures inherently address time-delayed state compensation and disturbance rejection, achieving superior trajectory tracking performance compared to classical control strategies. By integrating imitation learning with barrier certificate constraints, the proposed methodology ensures provable closed-loop stability while maintaining safety-critical operation bounds. Experimental evaluations using chaotic system benchmarks confirm the exceptional modeling capacity of NN controllers in capturing complex dynamical behaviors, complemented by formal verification advances through reachability analysis techniques. Practical demonstrations in aerial robotics and intelligent transportation systems highlight the efficacy of controllers in real-world scenarios involving environmental uncertainties and multi-agent interactions. The theoretical framework synergizes data-driven learning with nonlinear control principles, introducing hybrid automata formulations for transient response analysis and adjoint sensitivity methods for network optimization. These innovations position NN controllers as a transformative technology in control engineering, offering fundamental advances in stability-guaranteed learning and topology optimization. Future research directions will emphasize the integration of physics-informed neural operators for distributed control systems and event-triggered implementations for resource-constrained applications, paving the way for next-generation intelligent control architectures. Full article

(This article belongs to the Special Issue Artificial Intelligence and Mathematical Models in Robotics and Automation)

22 pages, 5231 KiB

Open AccessArticle

Parametric Analysis of Auxetic Honeycombs

by Stefan Tabacu, Ana-Gabriela Badea, Alina-Ionela Aparaschivei and Daniel-Constantin Anghel

Mathematics 2025, 13(10), 1676; https://doi.org/10.3390/math13101676 - 20 May 2025

Abstract

The auxetic honeycomb can change mechanical behavior from positive to negative Poisson’s ratio by slightly adjusting the geometrical feature. As a first step, the paper investigates existing analytical solutions to the problem to provide the theoretical background. The present study discusses the methodology [...] Read more.

The auxetic honeycomb can change mechanical behavior from positive to negative Poisson’s ratio by slightly adjusting the geometrical feature. As a first step, the paper investigates existing analytical solutions to the problem to provide the theoretical background. The present study discusses the methodology used to examine these structures using the finite element method and how to adapt simple numerical models to capture structural behavior. Subsequently, the numerical model is used to run parametric analyses to determine the performance and provide the background for discussing the influence of the dimensional set on the response. Following this set of observations, an empirical solution connecting the unit cell to the global structure was formulated. A large dataset is generated to train and check machine-learning applications to develop a helpful tool for predicting the response without numerical and theoretical evaluation models and methods. The machine-learning section investigates the influence of several parameters on the response and the effect of complete versus incomplete datasets. Full article

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

17 pages, 2483 KiB

Open AccessFeature PaperArticle

Robust Closed–Open Loop Iterative Learning Control for MIMO Discrete-Time Linear Systems with Dual-Varying Dynamics and Nonrepetitive Uncertainties

by Yawen Zhang, Yunshan Wei, Zuxin Ye, Shilin Liu, Hao Chen, Yuangao Yan and Junhong Chen

Mathematics 2025, 13(10), 1675; https://doi.org/10.3390/math13101675 - 20 May 2025

Abstract

Iterative learning control (ILC) typically requires strict repeatability in initial states, trajectory length, external disturbances, and system dynamics. However, these assumptions are often difficult to fully satisfy in practical applications. While most existing studies have achieved limited progress in relaxing either one or [...] Read more.

Iterative learning control (ILC) typically requires strict repeatability in initial states, trajectory length, external disturbances, and system dynamics. However, these assumptions are often difficult to fully satisfy in practical applications. While most existing studies have achieved limited progress in relaxing either one or two of these constraints simultaneously, this work aims to eliminate the restrictions imposed by all four strict repeatability conditions in ILC. For general finite-duration multi-input multi-output (MIMO) linear discrete-time systems subject to multiple non-repetitive uncertainties—including variations in initial states, external disturbances, trajectory lengths, and system dynamics—an innovative open- closed loop robust iterative learning control law is proposed. The feedforward component is used to make sure the tracking error converges as expected mathematically, while the feedback control part compensates for missing tracking data from previous iterations by utilizing real-time tracking information from the current iteration. The convergence analysis employs an input-to-state stability (ISS) theory for discrete parameterized systems. Detailed explanations are provided on adjusting key parameters to satisfy the derived convergence conditions, thereby ensuring that the anticipated tracking error will eventually settle into a compact neighborhood that meets the required standards for robustness and convergence speed. To thoroughly assess the viability of the proposed ILC framework, computer simulations effectively illustrate the strategy’s effectiveness. Further simulation on a real system, a piezoelectric motor system, verifies that the ILC tracking error converges to a small neighborhood in the sense of mathematical expectation. Extending the ILC to complex real-world applications provides new insights and approaches. Full article

(This article belongs to the Special Issue Analysis and Applications of Control Systems Theory)

19 pages, 2170 KiB

Open AccessArticle

Economic Model Predictive Control for Wastewater Treatment Processes Based on Global Maximum Error POD-TPWL

by Zhiyu Wang, Jing Zeng and Jinfeng Liu

Mathematics 2025, 13(10), 1674; https://doi.org/10.3390/math13101674 - 20 May 2025

Abstract

To address the challenge of low computational efficiency in nonlinear Economic Model Predictive Control (EMPC) for large-scale systems such as wastewater treatment plants (WWTPs), this paper proposes a Trajectory Piecewise Linearization (TPWL)-based EMPC framework integrated with global maximum error control (GMEC) and Proper [...] Read more.

To address the challenge of low computational efficiency in nonlinear Economic Model Predictive Control (EMPC) for large-scale systems such as wastewater treatment plants (WWTPs), this paper proposes a Trajectory Piecewise Linearization (TPWL)-based EMPC framework integrated with global maximum error control (GMEC) and Proper Orthogonal Decomposition (POD). The TPWL method constructs a reduced-order model framework, while GMEC iteratively refines the linearization point selection process. A two-stage strategy is employed: first, coarse selection of candidate linearization points along the original nonlinear model’s state trajectory based on Euclidean distance, followed by refinement to determine optimal points that minimize global approximation errors. Simulation results demonstrate that the proposed method reduces computational time by at least 65% under identical weather conditions while maintaining effluent quality and total cost indices within acceptable thresholds. Compared with conventional TPWL-POD approaches, this framework achieves higher model accuracy and superior EMPC control performance. These advancements underscore the method’s potential for real-time implementation in complex industrial systems, balancing computational efficiency with control precision. Additionally, the framework’s modular design enables integration with existing optimization techniques to further reduce computational complexity without compromising effluent quality compliance. Full article

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28 pages, 5434 KiB

Open AccessArticle

A Deep Learning Framework of Super Resolution for License Plate Recognition in Surveillance System

by Pei-Fen Tsai, Jia-Yin Shiu and Shyan-Ming Yuan

Mathematics 2025, 13(10), 1673; https://doi.org/10.3390/math13101673 - 20 May 2025

Abstract

Recognizing low-resolution license plates from real-world scenes remains a challenging task. While deep learning-based super-resolution methods have been widely applied, most existing datasets rely on artificially degraded images, and common quality metrics poorly correlate with OCR accuracy. We construct a new paired low- [...] Read more.

Recognizing low-resolution license plates from real-world scenes remains a challenging task. While deep learning-based super-resolution methods have been widely applied, most existing datasets rely on artificially degraded images, and common quality metrics poorly correlate with OCR accuracy. We construct a new paired low- and high-resolution license plate dataset from dashcam videos and propose a specialized super-resolution framework for license plate recognition. Only low-resolution images with OCR accuracy ≥5 are used to ensure sufficient feature information for effective perceptual learning. We analyze existing loss functions and introduce two novel perceptual losses—one CNN-based and one Transformer-based. Our approach improves recognition performance, achieving an average OCR accuracy of 85.14%. Full article

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

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15 pages, 280 KiB

Open AccessArticle

Integral Formulae and Applications for Compact Riemannian Hypersurfaces in Riemannian and Lorentzian Manifolds Admitting Concircular Vector Fields

by Mona Bin-Asfour, Kholoud Saad Albalawi and Mohammed Guediri

Mathematics 2025, 13(10), 1672; https://doi.org/10.3390/math13101672 - 20 May 2025

Abstract

This paper investigates compact Riemannian hypersurfaces immersed in

(n + 1)

-dimensional Riemannian or Lorentzian manifolds that admit concircular vector fields, also known as closed conformal vector fields (CCVFs). We focus on the support function of the hypersurface, which is defined [...] Read more.

This paper investigates compact Riemannian hypersurfaces immersed in

(n + 1)

-dimensional Riemannian or Lorentzian manifolds that admit concircular vector fields, also known as closed conformal vector fields (CCVFs). We focus on the support function of the hypersurface, which is defined as the component of the conformal vector field along the unit-normal vector field, and derive an expression for its Laplacian. Using this, we establish integral formulae for hypersurfaces admitting CCVFs. These results are then extended to compact Riemannian hypersurfaces isometrically immersed in Riemannian or Lorentzian manifolds with constant sectional curvatures, highlighting the crucial role of CCVFs in the study of hypersurfaces. We apply these results to provide characterizations of compact Riemannian hypersurfaces in Euclidean space

R^{n + 1}

, Euclidean sphere

S^{n + 1}

, and de Sitter space

S_{1}^{n + 1}

. Full article

(This article belongs to the Special Issue Analysis on Differentiable Manifolds)

43 pages, 14479 KiB

Open AccessArticle

Finite Volume Incompressible Lattice Boltzmann Framework for Non-Newtonian Flow Simulations in Complex Geometries

by Akshay Dongre, John Ryan Murdock and Song-Lin Yang

Mathematics 2025, 13(10), 1671; https://doi.org/10.3390/math13101671 - 20 May 2025

Abstract

Arterial diseases are a leading cause of morbidity worldwide, necessitating the development of robust simulation tools to understand their progression mechanisms. In this study, we present a finite volume solver based on the incompressible lattice Boltzmann method (iLBM) to model complex cardiovascular flows. [...] Read more.

Arterial diseases are a leading cause of morbidity worldwide, necessitating the development of robust simulation tools to understand their progression mechanisms. In this study, we present a finite volume solver based on the incompressible lattice Boltzmann method (iLBM) to model complex cardiovascular flows. Standard LBM suffers from compressibility errors and is constrained to uniform Cartesian meshes, limiting its applicability to realistic vascular geometries. To address these issues, we developed an incompressible LBM scheme that recovers the incompressible Navier–Stokes equations (NSEs) and integrated it into a finite volume (FV) framework to handle unstructured meshes while retaining the simplicity of the LBM algorithm. The FV-iLBM model with linear reconstruction (LR) scheme was then validated against benchmark cases, including Taylor–Green vortex flow, shear wave attenuation, Womersley flow, and lid-driven cavity flow, demonstrating improved accuracy in reducing compressibility errors. In simulating flow over National Advisory Committee for Aeronautics (NACA) 0012 airfoil, the FV-iLBM model accurately captured vortex shedding and aerodynamic forces. After validating the FV-iLBM solver for simulating non-Newtonian flows, pulsatile blood flow through an artery afflicted with multiple stenoses was simulated, accurately predicting wall shear stress and flow separation. The results establish FV-iLBM as an efficient and accurate method for modeling cardiovascular flows. Full article

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

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27 pages, 4374 KiB

Open AccessArticle

Quantifying Temporal Dynamics in Global Cyber Threats: A GPT-Driven Framework for Risk Forecasting and Strategic Intelligence

by Fahim Sufi and Musleh Alsulami

Mathematics 2025, 13(10), 1670; https://doi.org/10.3390/math13101670 - 20 May 2025

Abstract

Despite the exponential rise in cybersecurity incidents worldwide, existing analytical approaches often fail to detect subtle temporal dynamics in cyber threats, particularly on a quarterly scale. This paper addresses a critical research gap in the domain of temporal cyber risk analysis by introducing [...] Read more.

Despite the exponential rise in cybersecurity incidents worldwide, existing analytical approaches often fail to detect subtle temporal dynamics in cyber threats, particularly on a quarterly scale. This paper addresses a critical research gap in the domain of temporal cyber risk analysis by introducing a mathematically rigorous and AI-augmented framework capable of identifying, validating, and forecasting quarterly shifts in global cyber-attack patterns. The methodology integrates a hybrid data acquisition pipeline with GPT-based AI classification to construct a structured, high-dimensional dataset comprising 11,497 cybersecurity incidents spanning from October 2023 to March 2025. These incidents cover 106 attack types, 29 industries, and 257 countries. The framework decomposes the dataset into quarterly intervals and applies mathematical formulations to compute frequency shifts across categorical variables (attack types, industries, countries) and numerical variables (attack significance), followed by robust statistical validations (Chi-square and ANOVA tests), time-series forecasting via ARIMA, and the computation of a Quarterly Composite Index (QCI). Key results reveal dominant attack types—Social Engineering (ing1733) and Zero-Day Exploits (1657)—and highlight sectoral vulnerabilities in IT (5959) and Government (2508). Statistically significant quarterly variations were confirmed (

χ^{2} = 2319.13

,

F = 3.78

,

p < 0.001

). ARIMA forecasts predict 1782–2080 incidents per quarter for 2025–2026, while QCI trends average around 0.75, signifying sustained volatility. The research delivers both theoretical and practical advancements by combining generative AI, temporal segmentation, and statistical modeling to create an operationalizable intelligence system. This contribution enhances strategic cybersecurity preparedness and policymaking in a complex, evolving threat landscape. Full article

(This article belongs to the Topic Soft Computing and Machine Learning)

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22 pages, 1130 KiB

Open AccessArticle

Two-Mode Hereditary Model of Solar Dynamo

by Evgeny Kazakov, Gleb Vodinchar and Dmitrii Tverdyi

Mathematics 2025, 13(10), 1669; https://doi.org/10.3390/math13101669 - 20 May 2025

Abstract

The magnetic field of the Sun is formed by the mechanism of hydromagnetic dynamo. In this mechanism, the flow of the conducting medium (plasma) of the convective zone generates a magnetic field, and this field corrects the flow using the Lorentz force, creating [...] Read more.

The magnetic field of the Sun is formed by the mechanism of hydromagnetic dynamo. In this mechanism, the flow of the conducting medium (plasma) of the convective zone generates a magnetic field, and this field corrects the flow using the Lorentz force, creating feedback. An important role in dynamo is played by memory (hereditary), when a change in the current state of a physical system depends on its states in the past. Taking these effects into account may provide a more accurate description of the generation of the Sun’s magnetic field. This paper generalizes classical dynamo models by including hereditary feedback effects. The feedback parameters such as the presence or absence of delay, delay duration, and memory duration are additional degrees of freedom. This can provide more diverse dynamic modes compared to classical memoryless models. The proposed model is based on the kinematic dynamo problem, where the large-scale velocity field is predetermined. The field in the model is represented as a linear combination of two stationary predetermined modes with time-dependent amplitudes. For these amplitudes, equations are obtained based on the kinematic dynamo equations. The model includes two generators of a large-scale magnetic field. In the first, the field is generated due to large-scale flow of the medium. The second generator has a turbulent nature; in it, generation occurs due to the nonlinear interaction of small-scale pulsations of the magnetic field and velocity. Memory in the system under study is implemented in the form of feedback distributed over all past states of the system. The feedback is represented by an integral term of the type of convolution of a quadratic form of phase variables with a kernel of a fairly general form. The quadratic form models the influence of the Lorentz force. This integral term describes the turbulent generator quenching. Mathematically, this model is written with a system of integro-differential equations for amplitudes of modes. The model was applied to a real space object, namely, the solar dynamo. The model representation of the Sun’s velocity field was constructed based on helioseismological data. Free field decay modes were chosen as components of the magnetic field. The work considered cases when hereditary feedback with the system arose instantly or with a delay. The simulation results showed that the model under study reproduces dynamic modes characteristic of the solar dynamo, if there is a delay in the feedback. Full article

(This article belongs to the Special Issue Advances in Nonlinear Dynamical Systems of Mathematical Physics)

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22 pages, 793 KiB

Open AccessFeature PaperArticle

Decision Support System to Solve Single-Container Loading Problem Considering Practical Constraints

by Natalia Romero-Olarte , Santiago Amézquita-Ortiz, John Willmer Escobar and David Álvarez-Martínez

Mathematics 2025, 13(10), 1668; https://doi.org/10.3390/math13101668 - 19 May 2025

Abstract

The container loading problem (CLP) has a broad spectrum of applications in industry and has been studied for over 60 years due to its high complexity. This paper addresses a realistic single-container loading scenario with practical constraints, including orientation limitations, maximum stacking weight, [...] Read more.

The container loading problem (CLP) has a broad spectrum of applications in industry and has been studied for over 60 years due to its high complexity. This paper addresses a realistic single-container loading scenario with practical constraints, including orientation limitations, maximum stacking weight, static stability, overall container weight limit, and fractional loading for multiple drop-off points (multidrop). We propose an open-source decision support system (DSS) implemented on a widely used platform (MS Excel®), which employs a heuristic algorithm to find efficient loading solutions under these constraints. The DSS uses a multi-start randomized constructive algorithm based on a maximal residual space representation. The constructive phase builds the loading pattern in vertical layers (columns or walls), while respecting all practical constraints. The performance of the proposed heuristic is validated through extensive computational experiments on classical benchmark instances, comparing its results against the recent state-of-the-art methods. We also analyze the impact of multi-drop constraints on utilization metrics. The DSS features an interactive interface for creating/loading instances, visualizing step-by-step packing patterns, and displaying key statistics, thus providing a user-friendly decision tool for practitioners. Full article

(This article belongs to the Section D2: Operations Research and Fuzzy Decision Making)

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28 pages, 1450 KiB

Open AccessFeature PaperReview

N00N State Generation by Floquet Engineering

by Yusef Maleki

Mathematics 2025, 13(10), 1667; https://doi.org/10.3390/math13101667 - 19 May 2025

Abstract

We review quantum architectures for engineering the N00N state, a bipartite maximally entangled state essential in quantum metrology. These schemes transform the initial state

| N ⟩ \otimes | 0 ⟩

into the N00N state, [...] Read more.

We review quantum architectures for engineering the N00N state, a bipartite maximally entangled state essential in quantum metrology. These schemes transform the initial state

| N ⟩ \otimes | 0 ⟩

into the N00N state,

\frac{1}{\sqrt{2}} (| N ⟩ \otimes | 0 ⟩ + | 0 ⟩ \otimes | N ⟩),

where

| N ⟩

and

| 0 ⟩

are Fock states with N and 0 excitations, respectively. We demonstrate that this state can be generated through superpositions of quantum light modes, hybrid light–matter interactions, or spin ensembles. Our approach also enables the creation of mesoscopic and macroscopic entangled states, including entangled coherent and squeezed states. Furthermore, we show that a broad class of maximally entangled states can be realized within this framework. Extensions to multi-mode state engineering are also explored. Full article

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

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20 pages, 766 KiB

Open AccessArticle

Representation and Abstraction

by Antal Jakovác and András Telcs

Mathematics 2025, 13(10), 1666; https://doi.org/10.3390/math13101666 - 19 May 2025

Abstract

In this paper we propose a mathematical model for cognitive phenomena such as abstraction, generalization and extension. The main concept is the coordination of the observable world with relevant features, a concept defined mathematically in the paper. These features represent necessary conditions for [...] Read more.

In this paper we propose a mathematical model for cognitive phenomena such as abstraction, generalization and extension. The main concept is the coordination of the observable world with relevant features, a concept defined mathematically in the paper. These features represent necessary conditions for performing a task. There are different strategies to choose a coordination, and some of them can be associated with the System 1 and System 2 ways of thinking. System 2 uses multiple, context dependent relevant features for coordination, which makes it possible to speak about abstraction and the generalization of concepts. We also discuss the way extension works in these systems. Full article

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

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36 pages, 1587 KiB

Open AccessFeature PaperArticle

Analysis of MCP-Distributed Jammers and 3D Beam-Width Variations for UAV-Assisted C-V2X Millimeter-Wave Communications

by Mohammad Arif, Wooseong Kim, Adeel Iqbal and Sung Won Kim

Mathematics 2025, 13(10), 1665; https://doi.org/10.3390/math13101665 - 19 May 2025

Abstract

Jamming devices introduce unwanted signals into the network to disrupt primary communications. The effectiveness of these jamming signals mainly depends on the number and distribution of the jammers. The impact of clustered jamming has not been investigated previously for an unmanned aerial vehicle [...] Read more.

Jamming devices introduce unwanted signals into the network to disrupt primary communications. The effectiveness of these jamming signals mainly depends on the number and distribution of the jammers. The impact of clustered jamming has not been investigated previously for an unmanned aerial vehicle (UAV)-assisted cellular-vehicle-to-everything (C-V2X) communications by considering multiple roads in the given region. Also, exploiting three-dimensional (3D) beam-width variations for a millimeter waveband antenna in the presence of jamming for vehicular node (V-N) links has not been evaluated, which influences the UAV-assisted C-V2X system’s performance. The novelty of this paper resides in analyzing the impact of clustered jamming for UAV-assisted C-V2X networks and quantifying the effects of fluctuating antenna 3D beam width on the V-N performance by exploiting millimeter waves. To this end, we derive the analytical expressions for coverage of a typical V-N linked with a line-of-sight (LOS) UAV, non-LOS UAV, macro base station (MBS), and recipient V-N for UAV-assisted C-V2X networks by exploiting beam-width variations in the presence of jammers. The results show network performance in terms of coverage and spectral efficiencies by setting V-Ns equal to 3 km⁻², MBSs equal to 3 km⁻², and UAVs equal to 6 km⁻². The findings indicate that the performance of millimeter waveband UAV-assisted C-V2X communications is decreased by introducing clustered jamming in the given region. Specifically, the coverage performance of the network decreases by 25.5% at −10 dB SIR threshold in the presence of clustered jammers. The performance further declines by increasing variations in the antenna 3D beam width. Therefore, network designers must focus on considering advanced counter-jamming techniques when jamming signals, along with the beam-width fluctuations, are anticipated in vehicular networks. Full article

(This article belongs to the Section D1: Probability and Statistics)

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23 pages, 2863 KiB

Open AccessArticle

Using Physics-Informed Neural Networks for Modeling Biological and Epidemiological Dynamical Systems

by Amer Farea, Olli Yli-Harja and Frank Emmert-Streib

Mathematics 2025, 13(10), 1664; https://doi.org/10.3390/math13101664 - 19 May 2025

Abstract

Physics-Informed Neural Networks (PINNs) have emerged as a powerful approach for integrating physical laws into a deep learning framework, offering enhanced capabilities for solving complex problems. Despite their potential, the practical implementation of PINNs presents significant challenges. This paper explores the application of [...] Read more.

Physics-Informed Neural Networks (PINNs) have emerged as a powerful approach for integrating physical laws into a deep learning framework, offering enhanced capabilities for solving complex problems. Despite their potential, the practical implementation of PINNs presents significant challenges. This paper explores the application of PINNs to systems of ordinary differential equations (ODEs), focusing on two key challenges: the forward problem of solution approximation and the inverse problem of parameter estimation. We present three detailed case studies involving dynamical systems for tumor growth, gene expression, and the SIR (Susceptible, Infected, Recovered) model for disease spread. This paper outlines the core principles of PINNs and their integration with physical laws during neural network training. It details the steps involved in implementing PINNs, emphasizing the critical role of network architecture and hyperparameter tuning in achieving optimal performance. Additionally, we provide a Python package, ODE-PINN, to reproduce results for ODE-based systems. Our findings demonstrate that PINNs can yield accurate and consistent solutions, but their performance is highly sensitive to network architecture and hyperparameters tuning. These results underscore the need for customized configurations and robust optimization strategies. Overall, this study confirms the significant potential of PINNs to advance the understanding of dynamical systems in biology and epidemiology. Full article

(This article belongs to the Special Issue Machine Learning Methods and Mathematical Modeling with Applications)

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30 pages, 1422 KiB

Open AccessArticle

A Dynamic Threat Assessment Method for Multi-Target Unmanned Aerial Vehicles at Multiple Time Points Based on Fuzzy Multi-Attribute Decision Making and Fuse Intention

by Qianru Niu, Shuangyin Ren, Wei Gao and Chunjiang Wang

Mathematics 2025, 13(10), 1663; https://doi.org/10.3390/math13101663 - 19 May 2025

Abstract

In response to the threat assessment challenge posed by unmanned aerial vehicles (UAVs) in air defense operations, this paper proposes a dynamic assessment model grounded in fuzzy multi-attribute decision making. First, a three-dimensional evaluation index system is established, encompassing capability, opportunity, and intention. [...] Read more.

In response to the threat assessment challenge posed by unmanned aerial vehicles (UAVs) in air defense operations, this paper proposes a dynamic assessment model grounded in fuzzy multi-attribute decision making. First, a three-dimensional evaluation index system is established, encompassing capability, opportunity, and intention. Quantification functions for assessing the threat level of each attribute are then designed. To account for the temporal dynamics of the battlefield, an innovative fusion approach is developed, integrating inverse Poisson distribution time weights with subjective–objective comprehensive weighting, thereby establishing a dynamic variable weight fusion mechanism. Among these, the subjective weights are determined by integrating the intention probability matrix, effectively incorporating the intentions into the threat assessment process to reflect their dynamic changes and enhancing the overall evaluation accuracy. Leveraging the improved technique for order preference by similarity to ideal solution (TOPSIS), the model achieves threat prioritization. Experimental results demonstrate that this method significantly enhances the reliability of threat assessments in uncertain and dynamic battlefield environments, offering valuable support for air defense command and control systems. Full article

(This article belongs to the Topic Fuzzy Sets Theory and Its Applications)

25 pages, 2882 KiB

Open AccessArticle

Exact Solutions for Strong Nonlinear Oscillators with Linear Damping

by Livija Cveticanin

Mathematics 2025, 13(10), 1662; https://doi.org/10.3390/math13101662 - 19 May 2025

Abstract

This paper presents the derivation of an exact solution for a damped nonlinear oscillator of arbitrary order (both integer and non-integer). A coefficient relationship was defined under which such a solution exists. The analytical procedure was developed based on the application of the [...] Read more.

This paper presents the derivation of an exact solution for a damped nonlinear oscillator of arbitrary order (both integer and non-integer). A coefficient relationship was defined under which such a solution exists. The analytical procedure was developed based on the application of the Ateb (inverse beta) function. It has been shown that an exact solution exists for a specific relationship between the damping coefficient and the coefficient of the linear elastic term, and that this relationship depends on the order of nonlinearity. The exact amplitude of vibration was found to be a time-decreasing function, depending on the initial amplitude, damping coefficient, and the order of nonlinearity. The period of vibration was also shown to depend not only on the amplitude but also on both the nonlinearity coefficient and its order. For cases where the damping coefficient of the exact oscillator is slightly perturbed, an approximate solution based on the exact one was proposed. Three illustrative examples of oscillators with different orders of nonlinearity were considered: a nearly linear oscillator, a Duffing oscillator, and one with strong nonlinearity. For all cases, the high accuracy of the asymptotic solution was confirmed. Since no exact analytic solution exists for a purely nonlinear damped oscillator, an approximate solution was constructed using the solution of the corresponding undamped oscillator with a time-varying amplitude and phase. In the case of a purely cubic damped oscillator, the approximate solution was compared with numerical results, and good agreement was demonstrated. Full article

(This article belongs to the Special Issue Mathematical Modeling and Simulation of Oscillatory Phenomena, 2nd Edition)

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