Symmetry

26 pages, 6532 KiB

Open AccessArticle

Analysis of the Impact of Different Road Conditions on Accident Severity at Highway-Rail Grade Crossings Based on Explainable Machine Learning

by Zhen Yang, Chen Zhang, Gen Li and Hongyi Xu

Symmetry 2025, 17(1), 147; https://doi.org/10.3390/sym17010147 - 20 Jan 2025

Viewed by 613

Abstract

Previous studies on highway_rail grade crossing collisions have primarily focused on identifying factors contributing to the frequency and severity of driver injuries. In recent years, increasing attention has been given to modeling driver injury severity at these crossings. Recognizing the variations in injury [...] Read more.

Previous studies on highway_rail grade crossing collisions have primarily focused on identifying factors contributing to the frequency and severity of driver injuries. In recent years, increasing attention has been given to modeling driver injury severity at these crossings. Recognizing the variations in injury severity under different road surface conditions, this study investigates the impact of road surface conditions on driver injury severity at highway_rail grade crossings. Using nearly a decade of accident data (2012–2021), thi study employs a LightGBM model to predict factors influencing injury severity and utilizes SHAP values for result interpretation. The symmetry principle of SHAP esures that factors with identical influence receive equal values, enhancing the reliability of predictive outcomes. The findings reveal that driver injury severity at highway_rail grade crossings varies significantly under different road surface conditions. Key factors identified include train speed, driver age, vehicle speed, annual average daily traffic (AADT), driver presence inside the vehicle, weather conditions, and location. The results indicate that collisions are more frequent when either the vehicle or train travels at high speed. Implementing speed limits for both vehicles and trains under varying road conditions could effectively reduce accident severity. Additionally, older drivers are more prone to severe accidents, highlighting the importance of installing control devices, such as warning signs or signals, to enhance driver alertness and mitigate injury risks. Furthermore, adverse weather conditions, such as rain, snow, and fog, exacerbate accident severity on road surfaces like sand, mud, dirt, oil, or gravel. Timely removal of surface obstacles may help reduce the severity of such accidents. Full article

(This article belongs to the Special Issue Advances in Machine Learning with Symmetry/Asymmetry in Transportation)

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25 pages, 382 KiB

Open AccessArticle

New Numerical Quadrature Functional Inequalities on Hilbert Spaces in the Framework of Different Forms of Generalized Convex Mappings

by Waqar Afzal and Luminita-Ioana Cotîrlă

Symmetry 2025, 17(1), 146; https://doi.org/10.3390/sym17010146 - 20 Jan 2025

Viewed by 514

Abstract

The purpose of this article is to investigate some tensorial norm inequalities for continuous functions of self-adjoint operators in Hilbert spaces. Our first approach is to develop a gradient descent inequality and some relational properties for continuous functions involving Huber convex functions, as [...] Read more.

The purpose of this article is to investigate some tensorial norm inequalities for continuous functions of self-adjoint operators in Hilbert spaces. Our first approach is to develop a gradient descent inequality and some relational properties for continuous functions involving Huber convex functions, as well as several new bounds for Simpson type inequality that is twice differentiable using different types of generalized convex mappings. It is believed that this study will provide a valuable contribution towards developing a new perspective on functional inequalities by utilizing some other types of generalized mappings. Full article

(This article belongs to the Special Issue Advance in Functional Equations, Second Edition)

4 pages, 195 KiB

Open AccessCorrection

Correction: Buchbinder, I.L.; Reshetnyak, A.A. Covariant Cubic Interacting Vertices for Massless and Massive Integer Higher Spin Fields. Symmetry 2023, 15, 2124

by I. L. Buchbinder and A. A. Reshetnyak

Symmetry 2025, 17(1), 145; https://doi.org/10.3390/sym17010145 - 20 Jan 2025

Viewed by 275

Abstract

The authors wish to make the following corrections in their paper [...] Full article

(This article belongs to the Section Physics)

23 pages, 3677 KiB

Open AccessArticle

A Robust Large-Scale Multi-Criteria Decision Algorithm for Financial Risk Management with Interval-Valued Picture Fuzzy Information

by Na Shang, Hongfei Wang and Jie Fan

Symmetry 2025, 17(1), 144; https://doi.org/10.3390/sym17010144 - 19 Jan 2025

Viewed by 454

Abstract

Financial Risk Management (FRM) is crucial for organizations navigating complex and volatile economic conditions, as it aids in identifying and mitigating potential losses. Conventional FRM approaches are inadequate because they do not incorporate vagueness and variability in financial data. To overcome these challenges, [...] Read more.

Financial Risk Management (FRM) is crucial for organizations navigating complex and volatile economic conditions, as it aids in identifying and mitigating potential losses. Conventional FRM approaches are inadequate because they do not incorporate vagueness and variability in financial data. To overcome these challenges, this research presents interval-valued picture fuzzy measurement alternatives and rankings according to the Compromise Solution (IVPF-MARCOS) method. The IVPF-MARCOS method ranks investment strategies under uncertainty by assessing ten distinct investment options across seven key factors, including market risk and return on investment. It evidences its usefulness in enhancing decision-making, increasing accuracy in FRM, and developing Multi-Criteria Group Decision-Making (MCGDM) methodologies involving aggregation operators that are symmetric in nature. Consequently, this research establishes a compelling need to adopt improved fuzzy techniques in formulating the FRM to achieve more robust financial strategies. Full article

(This article belongs to the Section Computer)

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19 pages, 4102 KiB

Open AccessArticle

On Fractional Discrete Memristive Model with Incommensurate Orders: Symmetry, Asymmetry, Hidden Chaos and Control Approaches

by Hussein Al-Taani, Ma’mon Abu Hammad, Mohammad Abudayah, Louiza Diabi and Adel Ouannas

Symmetry 2025, 17(1), 143; https://doi.org/10.3390/sym17010143 - 18 Jan 2025

Viewed by 578

Abstract

Memristives provide a high degree of non-linearity to the model. This property has led to many studies focusing on developing memristive models to provide more non-linearity. This article studies a novel fractional discrete memristive system with incommensurate orders using

ϑ_{i}

-th Caputo-like [...] Read more.

Memristives provide a high degree of non-linearity to the model. This property has led to many studies focusing on developing memristive models to provide more non-linearity. This article studies a novel fractional discrete memristive system with incommensurate orders using

ϑ_{i}

-th Caputo-like operator. Bifurcation, phase portraits and the computation of the maximum Lyapunov Exponent

(L E_{m a x})

are used to demonstrate their impact on the system’s dynamics. Furthermore, we employ the sample entropy approach (SampEn),

C_{0}

complexity and the 0-1 test to quantify complexity and validate chaos in the incommensurate system. Studies indicate that the discrete memristive system with incommensurate fractional orders manifests diverse dynamical behaviors, including hidden chaos, symmetry, and asymmetry attractors, which are influenced by the incommensurate derivative values. Moreover, a 2D non-linear controller is presented to stabilize and synchronize the novel system. The work results are provided by numerical simulation obtained using MATLAB R2024a codes. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry in Chaos Theory and Application)

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22 pages, 5681 KiB

Open AccessArticle

The Research on Strategic Choices of Food Supply Chain Considering Information Symmetry and Cost Sharing

by Jianhua Wang and Ji Xu

Symmetry 2025, 17(1), 142; https://doi.org/10.3390/sym17010142 - 18 Jan 2025

Viewed by 458

Abstract

In the digital economy era, information symmetry, transparency, and traceability in food supply chains have increasingly garnered consumer attention. To motivate supply chain members to engage in product traceability, this paper examines the competitive and cooperative dynamics among participants in the food supply [...] Read more.

In the digital economy era, information symmetry, transparency, and traceability in food supply chains have increasingly garnered consumer attention. To motivate supply chain members to engage in product traceability, this paper examines the competitive and cooperative dynamics among participants in the food supply chain over continuous time. By developing a differential game model involving manufacturers and retailers with three decision-making modes, we solve the model using the Hamilton–Jacobi–Bellman (HJB) equation and perform a simulation analysis to assess the impact of different modes on overall supply chain profits. Additionally, we analyze how various parameters affect the profits of manufacturers and retailers. The key findings of this study indicate that centralized decision-making enhances the overall benefits of the food supply chain. Among the three decision-making models, the cost-sharing model proves to be the optimal approach, as it leads to a Pareto improvement in the profits of both manufacturers and retailers. These conclusions provide valuable insights for supply chain members seeking to optimize product traceability and enhance supply chain efficiency, as well as for government authorities involved in traceable supply chain governance. Full article

(This article belongs to the Section Mathematics)

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19 pages, 456 KiB

Open AccessArticle

Mathematical Models for Coverage with Star Tree Backbone Topology for 5G Millimeter Waves Networks

by Sergio Cordero, Pablo Adasme, Ali Dehghan Firoozabadi, Renata Lopes Rosa and Demóstenes Zegarra Rodríguez

Symmetry 2025, 17(1), 141; https://doi.org/10.3390/sym17010141 - 18 Jan 2025

Viewed by 492

Abstract

This paper proposes mathematical optimization models for solving the network planning problem using millimeter wave technology for 5G wireless communications networks. To this end, it is assumed that a set of users,

M = {1, \dots, m}

, and [...] Read more.

This paper proposes mathematical optimization models for solving the network planning problem using millimeter wave technology for 5G wireless communications networks. To this end, it is assumed that a set of users,

M = {1, \dots, m}

, and a set of base stations,

N = {1, \dots, n}

, are deployed randomly in a square area. In particular, the base stations should be connected, forming a star backbone so that users can connect to their nearest active base stations forming the backbone where the connections are symmetric. In particular, the first two models maximize the number of users connected to the backbone and minimize the distance costs of connecting users to the base stations, and distances of connecting the base stations themselves. Similarly, the last two models maximize and minimize the same objectives and the number of base stations to be activated to form the star backbone. Each user is allowed to connect to a unique active base station. In general, the millimeter wave technology presents a high path loss. Consequently, the transmission distances should be no larger than 300 m at most for different radial transmissions. Thus, a direct line of sight between users and base stations is assumed. Finally, we propose local search-based algorithms that allow finding near-optimal solutions for all our tested instances. Our numerical results indicate that we can solve network instances optimally with up to

k = 100

,

n = 200

, and

m = 5000

users. Full article

(This article belongs to the Section Engineering and Materials)

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15 pages, 2686 KiB

Open AccessArticle

Quantum-Chemical Investigations on the Structure and Stability of Mixed Trimers Containing HC₃N in Combination with H₂C₂ and/or HCN Analyzed by QTAIM, NBO and SAPT Methods

by Andrea Pietropolli Charmet, Paolo Stoppa, Alessandra De Lorenzi and Patrizia Canton

Symmetry 2025, 17(1), 140; https://doi.org/10.3390/sym17010140 - 18 Jan 2025

Viewed by 455

Abstract

The present work deals with the computational study of HC₃N

\cdot \cdot

HCN

\cdot \cdot

H₂C₂-, (HC₃N)₂

\cdot \cdot

H₂C₂-, and HC₃N

\cdot \cdot

(H₂C₂ [...] Read more.

The present work deals with the computational study of HC₃N

\cdot \cdot

HCN

\cdot \cdot

H₂C₂-, (HC₃N)₂

\cdot \cdot

H₂C₂-, and HC₃N

\cdot \cdot

(H₂C₂)₂-mixed trimers. The different equilibrium structures of the different low-lying minima on the corresponding potential energy surface (PES) were accurately determined, and the relative stabilities were computed by extrapolation procedures to the complete basis set limit. For each mixed trimer, the non-covalent interactions ruling the structure of the most stable isomer were analyzed using the QTAIM (Quantum Theory of Atoms in Molecules) approach. Additional insights into these interactions were provided by the Natural Bond Orbital (NBO) and Symmetry-Adapted Perturbation Theory (SAPT) methods. These results can be used to assist further theoretical investigations and experimental studies on the formation of larger molecules potentially relevant in astrochemistry. Full article

(This article belongs to the Special Issue Chemistry: Symmetry/Asymmetry—Feature Papers and Reviews)

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18 pages, 7744 KiB

Open AccessArticle

High-Performance Carrier Phase Recovery for Local Local Oscillator Continuous-Variable Quantum Key Distribution

by Jiayu Ma, Chao Zhou, Dengke Qi, Ziyang Chen, Yongmei Sun, Song Yu and Xiangyu Wang

Symmetry 2025, 17(1), 139; https://doi.org/10.3390/sym17010139 - 18 Jan 2025

Viewed by 548

Abstract

Continuous-variable quantum key distribution (CV-QKD) has been increasingly studied, which offers the advantage of compatibility with modern coherent optical communication systems. In contrast to CV-QKD with a transmitting local oscillator, the local local oscillator CV-QKD avoids the security vulnerabilities of a local oscillator [...] Read more.

Continuous-variable quantum key distribution (CV-QKD) has been increasingly studied, which offers the advantage of compatibility with modern coherent optical communication systems. In contrast to CV-QKD with a transmitting local oscillator, the local local oscillator CV-QKD avoids the security vulnerabilities of a local oscillator by generating a local oscillator at the receiver. In practice, the frequency offset of the two lasers introduces extra phase noise, which is generally suppressed by various carrier phase recovery algorithms. However, the accuracy of carrier phase recovery can be influenced by the power of the pilot tone, particularly as the transmission distance increases. To further improve accuracy, we propose a method based on the unscented particle filter algorithm, to increase the accuracy of phase estimation, effectively restore the quantum signal and reduce excess noise. In our work, we demonstrated a local local oscillator CV-QKD experiment with a finite-size block of

1 \times 10^{8}

under a transmission distance of 50 km. Through our method, we achieved a secret key rate of 525 kbps, which represents a 28% improvement. These results confirm that our proposed method not only improves the accuracy of carrier phase recovery, but also provides a new approach for future research on algorithms for long-distance CV-QKD. Furthermore, our study improves the phase compensation performance, enabling the orthogonal components of the quantum signal to exhibit enhanced symmetry in phase space. Full article

(This article belongs to the Section Physics)

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33 pages, 4104 KiB

Open AccessArticle

Prediction of Marine Shaft Centerline Trajectories Using Transformer-Based Models

by Jialin Han, Qingbo Zhu, Sheng Yang, Wan Xia and Yongjun Yao

Symmetry 2025, 17(1), 137; https://doi.org/10.3390/sym17010137 - 18 Jan 2025

Viewed by 389

Abstract

The accurate prediction of marine shaft centerline trajectories is essential for ensuring the operational performance and safety of ships. In this study, we propose a novel Transformer-based model to forecast the lateral and longitudinal displacements of ship main shafts. A key challenge in [...] Read more.

The accurate prediction of marine shaft centerline trajectories is essential for ensuring the operational performance and safety of ships. In this study, we propose a novel Transformer-based model to forecast the lateral and longitudinal displacements of ship main shafts. A key challenge in this prediction task is capturing both short-term fluctuations and long-term dependencies in shaft displacement data, which traditional models struggle to address. Our Transformer-based model integrates Bidirectional Splitting–Agg Attention and Sequence Progressive Split–Aggregation mechanisms to efficiently process bidirectional temporal dependencies, decompose seasonal and trend components, and handle the inherent symmetry of the shafting system. The symmetrical nature of the shafting system, with left and right shafts experiencing similar dynamic conditions, aligns with the bidirectional attention mechanism, enabling the model to better capture the symmetric relationships in displacement data. Experimental results demonstrate that the proposed model significantly outperforms traditional methods, such as Autoformer and Informer, in terms of prediction accuracy. Specifically, for 96 steps ahead, the mean absolute error (MAE) of our model is 0.232, compared to 0.235 for Autoformer and 0.264 for Informer, while the mean squared error (MSE) of our model is 0.209, compared to 0.242 for Autoformer and 0.286 for Informer. These results underscore the effectiveness of Transformer-based models in accurately predicting long-term marine shaft centerline trajectories, leveraging both temporal dependencies and structural symmetry, thus contributing to maritime monitoring and performance optimization. Full article

(This article belongs to the Section Engineering and Materials)

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18 pages, 293 KiB

Open AccessArticle

On Symmetrically Stochastic System of Fractional Differential Equations and Variational Inequalities

by Yue Zhang, Lu-Chuan Ceng, Jen-Chih Yao, Yue Zeng, Yun-Yi Huang and Si-Ying Li

Symmetry 2025, 17(1), 138; https://doi.org/10.3390/sym17010138 - 17 Jan 2025

Viewed by 476

Abstract

In this work, we are devoted to discussing a system of fractional stochastic differential variational inequalities with Lévy jumps (SFSDVI with Lévy jumps), that comprises both parts, that is, a system of stochastic variational inequalities (SSVI) and a system of fractional stochastic differential [...] Read more.

In this work, we are devoted to discussing a system of fractional stochastic differential variational inequalities with Lévy jumps (SFSDVI with Lévy jumps), that comprises both parts, that is, a system of stochastic variational inequalities (SSVI) and a system of fractional stochastic differential equations(SFSDE) with Lévy jumps. Here it is noteworthy that the SFSDVI with Lévy jumps consists of both sections that possess a mutual symmetry structure. Invoking Picard’s successive iteration process and projection technique, we obtain the existence of only a solution to the SFSDVI with Lévy jumps via some appropriate restrictions. In addition, the major outcomes are invoked to deduce that there is only a solution to the spatial-price equilibria system in stochastic circumstances. The main contributions of the article are listed as follows: (a) putting forward the SFSDVI with Lévy jumps that could be applied for handling different real matters arising from varied domains; (b) deriving the unique existence of solutions to the SFSDVI with Lévy jumps under a few mild assumptions; (c) providing an applicable instance for spatial-price equilibria system in stochastic circumstances affected with Lévy jumps and memory. Full article

(This article belongs to the Special Issue Functional Analysis, Fractional Operators and Symmetry/Asymmetry: Second Edition)

14 pages, 1143 KiB

Open AccessArticle

On the Horizontal Divergence Asymmetry in the Gulf of Mexico

by Tianshu Zhou, Jin-Han Xie and Dhruv Balwada

Symmetry 2025, 17(1), 136; https://doi.org/10.3390/sym17010136 - 17 Jan 2025

Viewed by 314

Abstract

Due to the geostrophic balance, horizontal divergence-free is often assumed when analyzing large-scale oceanic flows. However, the geostrophic balance is a leading-order approximation. We investigate the statistical feature of weak horizontal compressibility in the Gulf of Mexico by analyzing drifter data (the Grand [...] Read more.

Due to the geostrophic balance, horizontal divergence-free is often assumed when analyzing large-scale oceanic flows. However, the geostrophic balance is a leading-order approximation. We investigate the statistical feature of weak horizontal compressibility in the Gulf of Mexico by analyzing drifter data (the Grand LAgrangian Deployment (GLAD) experiment and the LAgrangian Submesoscale ExpeRiment (LASER)) based on the asymptotic probability density function of the angle between velocity and acceleration difference vectors in a strain-dominant model. The results reveal a notable divergence at scales between 10 km and 300 km, which is stronger in winter (LASER) than in summer (GLAD). We conjecture that the divergence is induced by wind stress with its curl parallel to the Earth’s rotation. Full article

(This article belongs to the Special Issue Applications Based on Symmetry/Asymmetry in Fluid Mechanics)

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25 pages, 1773 KiB

Open AccessArticle

A Robust Kalman Filter Based on the Pearson Type VII-Inverse Wishart Distribution: Symmetrical Treatment of Time-Varying Measurement Bias and Heavy-Tailed Noise

by Shen Liang and Xun Zhang

Symmetry 2025, 17(1), 135; https://doi.org/10.3390/sym17010135 - 17 Jan 2025

Viewed by 391

Abstract

This paper introduces a novel robust Kalman filter designed to leverage symmetrical properties within the Pearson Type VII-Inverse Wishart (PVIW) distribution, enhancing state estimation accuracy in the presence of time-varying biases and non-stationary heavy-tailed (NSHT) noise. The filter includes a shape parameter from [...] Read more.

This paper introduces a novel robust Kalman filter designed to leverage symmetrical properties within the Pearson Type VII-Inverse Wishart (PVIW) distribution, enhancing state estimation accuracy in the presence of time-varying biases and non-stationary heavy-tailed (NSHT) noise. The filter includes a shape parameter from the normal distribution and an extra variable from the Gamma distribution, which are used to symmetrically adjust the average and variation measures of the data to fit better under difficult noise conditions. To deal with unknown noise that changes over time, the filter uses the Inverse Wishart distribution to model and estimate the scale matrix deviations, making it easier to adapt to changes. The filter also uses a technique called Variational Bayesian to estimate both the state and the parameters at the same time. The results from simulations show that this new filter greatly improves the accuracy and strength of the estimation compared to the usual Kalman filters that assume a normal distribution, especially when there is non-stationary heavy-tailed noise. The main objective is to improve estimation in signal processing and control systems where heavy-tailed noise is prevalent. Full article

(This article belongs to the Section Engineering and Materials)

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18 pages, 4976 KiB

Open AccessArticle

Second-Order Kinematic Invariants for the Design of Compliant Auxetic Symmetrical Structures

by Marco Cirelli, Matteo Autiero, Luca D’Angelo and Pier Paolo Valentini

Symmetry 2025, 17(1), 134; https://doi.org/10.3390/sym17010134 - 17 Jan 2025

Viewed by 430

Abstract

Auxetic structures have great potential in modern engineering, and their design represents an emerging field in industrial applications. The accurate synthesis of the elements of such structures requires multidisciplinary approaches that combine kinematics and structural mechanics. Design methodologies are often based on complex [...] Read more.

Auxetic structures have great potential in modern engineering, and their design represents an emerging field in industrial applications. The accurate synthesis of the elements of such structures requires multidisciplinary approaches that combine kinematics and structural mechanics. Design methodologies are often based on complex time-consuming numerical methods and with considerable computational burden for exploring a large set of alternatives. The aim of the present work is to propose a novel method for designing symmetrical auxetic structures based on the use of pseudo-rigid mechanisms that can reproduce their nonlinear elasto-kinematic behavior with a limited set of parameters. For the definition of these pseudo-rigid mechanisms, the theory of kinematic invariants is proposed. It allows for the deduction of surrogate rigid-link mechanisms with a simpler structure but remarkable accuracy. This approach is an emerging method employed in the generic synthesis and analysis of compliant mechanisms, and, in this study, it is extended for the first time to support the design of auxetic structures. This paper describes the analytical process to deduce the design equations and discusses the example of an application to a symmetrical re-entrant structure, comparing the results with those of a numerical flexible multibody model, a finite element model, and with experimental tests. All the comparisons demonstrate the considerable potential of the proposed methodology, which can also be adapted to other types of auxetic structures. Full article

(This article belongs to the Special Issue Advances in Mechanics of Rigid and Flexible Systems: Mathematical Models, Numerical Modelling, Experiments, Symmetry and Applications)

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33 pages, 3753 KiB

Open AccessArticle

Matching Polynomials of Symmetric, Semisymmetric, Double Group Graphs, Polyacenes, Wheels, Fans, and Symmetric Solids in Third and Higher Dimensions

by Krishnan Balasubramanian

Symmetry 2025, 17(1), 133; https://doi.org/10.3390/sym17010133 - 17 Jan 2025

Viewed by 759

Abstract

The primary objective of this study is the computation of the matching polynomials of a number of symmetric, semisymmetric, double group graphs, and solids in third and higher dimensions. Such computations of matching polynomials are extremely challenging problems due to the computational and [...] Read more.

The primary objective of this study is the computation of the matching polynomials of a number of symmetric, semisymmetric, double group graphs, and solids in third and higher dimensions. Such computations of matching polynomials are extremely challenging problems due to the computational and combinatorial complexity of the problem. We also consider a series of recursive graphs possessing symmetries such as D_2h-polyacenes, wheels, and fans. The double group graphs of the Möbius types, which find applications in chemically interesting topologies and stereochemistry, are considered for the matching polynomials. Hence, the present study features a number of vertex- or edge-transitive regular graphs, Archimedean solids, truncated polyhedra, prisms, and 4D and 5D polyhedra. Such polyhedral and Möbius graphs present stereochemically and topologically interesting applications, including in chirality, isomerization reactions, and dynamic stereochemistry. The matching polynomials of these systems are shown to contain interesting combinatorics, including Stirling numbers of both kinds, Lucas polynomials, toroidal tree-rooted map sequences, and Hermite, Laguerre, Chebychev, and other orthogonal polynomials. Full article

(This article belongs to the Collection Feature Papers in Chemistry)

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16 pages, 3343 KiB

Open AccessArticle

Behavior Safety Decision-Making Based on Deep Deterministic Policy Gradient and Its Verification Method

by Yi Zhu, Zexin Li, Jinyong Wang, Ying Zhao and Miaoer Li

Symmetry 2025, 17(1), 132; https://doi.org/10.3390/sym17010132 - 17 Jan 2025

Viewed by 449

Abstract

As an emerging mode of transportation, autonomous vehicles are increasingly attracting widespread attention. To address the issues of the traditional reinforcement learning algorithm, which only considers discrete actions within the system and cannot ensure the safety of decision-making, this paper proposes a behavior [...] Read more.

As an emerging mode of transportation, autonomous vehicles are increasingly attracting widespread attention. To address the issues of the traditional reinforcement learning algorithm, which only considers discrete actions within the system and cannot ensure the safety of decision-making, this paper proposes a behavior decision-making method based on the deep deterministic policy gradient. Firstly, to enable autonomous vehicles to drive as close to the center of the road as possible while sensitively avoiding surrounding obstacles, the reward function for reinforcement learning is constructed by comprehensively considering road boundaries and nearby vehicles. We account for the symmetry of the road by calculating the distances between the vehicle and both the left and right road boundaries, ensuring that the vehicle remains centered within the road. Secondly, to ensure the safety of decision-making, the safety constraints in autonomous driving scenarios are described using probabilistic computation tree logic, and the scenario is modeled as a stochastic hybrid automaton. Finally, the model is verified by the statistical model checker UPPAAL. The above method enables autonomous vehicles not only to independently acquire driving skills across diverse driving environments but also significantly enhances their obstacle avoidance capabilities, thereby ensuring driving safety. Full article

(This article belongs to the Section Mathematics)

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25 pages, 1811 KiB

Open AccessArticle

Symmetric and Asymmetric Expansion of the Weibull Distribution: Features and Applications to Complete, Upper Record, and Type-II Right-Censored Data

by Mahmoud El-Morshedy, M. El-Dawoody and Adel A. El-Faheem

Symmetry 2025, 17(1), 131; https://doi.org/10.3390/sym17010131 - 17 Jan 2025

Viewed by 516

Abstract

This paper introduces a new continuous lifetime model called the Odd Flexible Weibull-Weibull (OFW-W) distribution, which features three parameters. The new model is capable of modeling both symmetric and asymmetric datasets, regardless of whether they are positively or negatively skewed. Its hazard rate [...] Read more.

This paper introduces a new continuous lifetime model called the Odd Flexible Weibull-Weibull (OFW-W) distribution, which features three parameters. The new model is capable of modeling both symmetric and asymmetric datasets, regardless of whether they are positively or negatively skewed. Its hazard rate functions can exhibit various behaviors, including increasing, decreasing, unimodal, or bathtub-shaped. The key characteristics of the OFW-W model are discussed, including the quantile function, median, reliability and hazard rate functions, kurtosis and skewness, mean waiting (residual) lifetimes, moments, and entropies. The unknown parameters of the model are estimated using eight different techniques. A comprehensive simulation study evaluates the performance of these estimators based on bias, mean squared error (MSE), and mean relative error (MRE). The practical usefulness of the OFW-W distribution is demonstrated through four real datasets from the fields of engineering and medicine, including complete data, upper record data, and type-II right-censored data. Comparisons with five other lifetime distributions reveal that the OFW-W model exhibits superior flexibility and capability in fitting various data types, highlighting its advantages and improvements. In conclusion, we anticipate that the OFW-W model will prove valuable in various applications, including human health, environmental studies, reliability theory, actuarial science, and medical sciences, among others. Full article

(This article belongs to the Special Issue Symmetrical and Asymmetrical Distributions in Statistics and Data Science II)

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34 pages, 13140 KiB

Open AccessArticle

Characterization of Spatial Cognitive EEG Signals Using Normalized Adjusted Permutation Conditional Mutual Information

by Xianglong Wan, Yue Sun, Zhenzhen Wu and Dong Wen

Symmetry 2025, 17(1), 130; https://doi.org/10.3390/sym17010130 - 17 Jan 2025

Viewed by 472

Abstract

Spatial cognitive ability, a fundamental domain within the human cognitive system, involves the perception, interpretation, and manipulation of spatial environments. This study introduces a new EEG feature extraction algorithm, Normalized Adjusted Permutation Conditional Mutual Information (NAPCMI), to improve the accuracy of spatial cognition [...] Read more.

Spatial cognitive ability, a fundamental domain within the human cognitive system, involves the perception, interpretation, and manipulation of spatial environments. This study introduces a new EEG feature extraction algorithm, Normalized Adjusted Permutation Conditional Mutual Information (NAPCMI), to improve the accuracy of spatial cognition assessments. By capturing the symmetry and temporal dependencies within EEG signals during spatial cognition tasks, NAPCMI enhances the ability to extract relevant features. The study validates NAPCMI using a BCI-VR spatial cognition assessment system, incorporating gesture recognition. Results demonstrate that NAPCMI outperforms traditional methods in feature extraction, highlighting its potential for advancing the understanding and assessment of spatial cognitive abilities. The findings also emphasize the significance of specific EEG frequency bands, such as Delta and Beta1, in spatial cognition tasks, further validating NAPCMI’s effectiveness. Full article

(This article belongs to the Special Issue Advances in Symmetry/Asymmetry and Biomedical Engineering)

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19 pages, 2185 KiB

Open AccessArticle

State Compensation Model in Adaptive Event-Triggered Predictive Control: A Novel Approach to Mitigating Moving Bottlenecks

by Jingwen Yang and Ping Wang

Symmetry 2025, 17(1), 129; https://doi.org/10.3390/sym17010129 - 17 Jan 2025

Viewed by 396

Abstract

Moving bottlenecks, characterized by their high frequency and unpredictability, pose significant challenges to timely response and management, often resulting in road congestion and increased risk of traffic accidents. To address these issues, this paper proposes an adaptive event-triggered variable speed limit (AET-VSL) method [...] Read more.

Moving bottlenecks, characterized by their high frequency and unpredictability, pose significant challenges to timely response and management, often resulting in road congestion and increased risk of traffic accidents. To address these issues, this paper proposes an adaptive event-triggered variable speed limit (AET-VSL) method based on a state compensation model, which emphasizes the concept of symmetry in the optimization of multi-segment speed limits. This symmetry approach facilitates a balanced and efficient control strategy that adjusts speed limits in a way that harmonizes traffic flow across multiple road segments, reducing congestion and improving overall traffic stability. The state compensation model builds on the classical METANET traffic flow model, incorporating coordination between road segments to reduce congestion while minimizing disruptions to traffic flow stability. By dynamically adjusting speed limits using real-time traffic data, the AET-VSL method addresses fluctuations in traffic conditions and ensures adaptive control to manage bottlenecks efficiently. A simulation framework was employed to evaluate the proposed strategy across varying traffic scenarios. Results demonstrate that AET-VSL outperforms traditional methods, providing consistent improvements in traffic performance. For instance, under low-traffic-flow conditions, AET-VSL reduced waiting time (WT) by 41.36%, potential collisions (PCs) by 51.92%, and fuel consumptionfuel consumption (FC) by 34.07%. This study highlights the novelty and effectiveness of AET-VSL, offering a scalable and reliable solution for dynamic traffic management and showcasing its potential to enhance traffic safety and efficiency. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry of Applications in Automation and Control Systems)

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19 pages, 4712 KiB

Open AccessArticle

An Independent Suspension and Trafficability Analysis for an Unmanned Ground Platform

by Jianying Li, Yinghong Xie and Yongwang Huo

Symmetry 2025, 17(1), 128; https://doi.org/10.3390/sym17010128 - 17 Jan 2025

Viewed by 450

Abstract

The objective of this paper was to investigate and design a novel vertical- and horizontal-arm independent suspension system aimed at enhancing the autonomous obstacle-crossing capabilities of unmanned ground platforms in complex, unstructured environments such as mountainous regions, hills, and mining areas. By thoroughly [...] Read more.

The objective of this paper was to investigate and design a novel vertical- and horizontal-arm independent suspension system aimed at enhancing the autonomous obstacle-crossing capabilities of unmanned ground platforms in complex, unstructured environments such as mountainous regions, hills, and mining areas. By thoroughly considering factors such as the suspension structure design, changes in the centroid position, distribution of driving forces, and dynamic stability analysis, we proposed an innovative suspension structure. An unmanned ground platform model equipped with this suspension system was developed using ADAMS and MATLAB/Simulink. Subsequently, a joint simulation was conducted to validate the performance of the suspension system. The results indicated that the unmanned ground platform could successfully traverse vertical steps up to 370 mm high and trenches measuring up to 600 mm wide. Furthermore, when confronted with intricate obstacles including vertical barriers, trenches, and side slopes, the platform demonstrated exceptional traversing capabilities. In conclusion, the proposed suspension system significantly enhances both the obstacle-surmounting ability and the terrain adaptability of unmanned ground platforms while providing crucial technical support for their deployment in complex unstructured environments. Full article

(This article belongs to the Section Computer)

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20 pages, 312 KiB

Open AccessArticle

Existence and Uniqueness of a Solution of a Boundary Value Problem Used in Chemical Sciences via a Fixed Point Approach

by Umar Ishtiaq, Fahad Jahangeer, Mubariz Garayev and Ioan-Lucian Popa

Symmetry 2025, 17(1), 127; https://doi.org/10.3390/sym17010127 - 16 Jan 2025

Viewed by 407

Abstract

In this paper, we present Proinov-type fixed point theorems in the setting of bi-polar metric spaces and fuzzy bi-polar metric spaces. Fuzzy bi-polar metric spaces with symmetric property extend classical metric spaces to address dual structures and uncertainty, ensuring consistency and balance. We [...] Read more.

In this paper, we present Proinov-type fixed point theorems in the setting of bi-polar metric spaces and fuzzy bi-polar metric spaces. Fuzzy bi-polar metric spaces with symmetric property extend classical metric spaces to address dual structures and uncertainty, ensuring consistency and balance. We provide different concrete conditions on the real-valued functions

Ω, Π : (0, \infty) \to R

for the existence of fixed points via the

(Ω, Π)

-contraction in bi-polar metric spaces. Further, we define real-valued functions

Ω, Π : (0, 1] \to R

to obtain fixed point theorems in fuzzy bi-polar metric spaces. We apply

(Ω, Π)

fuzzy bi-polar version of a Banach fixed point theorem to show the existence of solutions. Furthermore, we provide some non-trivial examples to show the validity of our results. In the end, we find the existence and uniqueness of a solution of integral equations and boundary value problem used in chemical sciences by applying main results. Full article

(This article belongs to the Section Mathematics)

13 pages, 2067 KiB

Open AccessArticle

Study on the Influence of Two-Step Filling Mechanic Characteristics on the Stability of Single-Side Exposed Cemented Backfill

by Yao Li, Dan Mei, Xingyu Hu and Binyu Luo

Symmetry 2025, 17(1), 126; https://doi.org/10.3390/sym17010126 - 16 Jan 2025

Viewed by 416

Abstract

This study analyzed the influence of the mechanical properties of two-step backfill on the stability of mining sites. The study focused on the one-step adhesive backfill of segmented backfill mining in a mine in Shandong Province, where the front wall was exposed and [...] Read more.

This study analyzed the influence of the mechanical properties of two-step backfill on the stability of mining sites. The study focused on the one-step adhesive backfill of segmented backfill mining in a mine in Shandong Province, where the front wall was exposed and the back wall was compressed. A three-dimensional mechanical model of the front wall exposed, back wall compressed cemented filling material considering the mechanical properties of the two-step weakly cemented filling material was established through theoretical analysis. On this basis, considering the influence of different mechanical properties (elastic modulus, internal friction angle, cohesion, and Poisson’s ratio) of two-step weakly cemented filling on one-step cemented filling, FLAC 3D 6.00.60 numerical simulation software was used to study the influence of various factors on the horizontal displacement distribution of cemented filling under single-sided exposure conditions using numerical simulation methods. The results show that the adhesive filling material exposed on one side is subjected to lateral pressure from adjacent weak adhesive filling materials, and its stability is affected by the contact area and mechanical properties of the weak adhesive filling material. Increasing the elastic modulus of the two-step weak adhesive filling material from 100 MPa to 500 MPa can reduce the maximum horizontal displacement of the one-step adhesive filling material from 116 mm to 32 mm, a decrease of about 72%. Similarly, increasing the cohesive force from 0.09 MPa to 0.21 MPa can reduce displacement from 96 mm to 33 mm, a decrease of 66%. Improving the mechanical properties of the two-step weakly cemented filling material can reduce the tendency of tailings to slide and collapse, and can reduce the lateral pressure applied by the cemented filling material. The horizontal displacement law of the two-step cemented filling material with front wall exposure and rear wall compression is basically similar under different mechanical properties of the one-step weakly cemented filling material. In the vertical direction, as the height of the filling material increases, the horizontal displacement first slowly increases to the maximum value and then slowly decreases. As the mechanical properties of the two-step weakly cemented filling increase, the horizontal displacement of the one-step cemented filling decreases. Full article

(This article belongs to the Section Engineering and Materials)

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16 pages, 313 KiB

Open AccessArticle

On Superization of Nonlinear Integrable Dynamical Systems

by Anatolij K. Prykarpatski, Radosław A. Kycia and Volodymyr M. Dilnyi

Symmetry 2025, 17(1), 125; https://doi.org/10.3390/sym17010125 - 15 Jan 2025

Viewed by 344

Abstract

We study an interesting superization problem of integrable nonlinear dynamical systems on functional manifolds. As an example, we considered a quantum many-particle Schrödinger–Davydov model on the axis, whose quasi-classical reduction proved to be a completely integrable Hamiltonian system on a smooth functional manifold. [...] Read more.

We study an interesting superization problem of integrable nonlinear dynamical systems on functional manifolds. As an example, we considered a quantum many-particle Schrödinger–Davydov model on the axis, whose quasi-classical reduction proved to be a completely integrable Hamiltonian system on a smooth functional manifold. We checked that the so-called “naive” approach, based on the superization of the related phase space variables via extending the corresponding Poisson brackets upon the related functional supermanifold, fails to retain the dynamical system super-integrability. Moreover, we demonstrated that there exists a wide class of classical Lax-type integrable nonlinear dynamical systems on axes in relation to which a superization scheme consists in a reasonable superization of the related Lax-type representation by means of passing from the basic algebra of pseudo-differential operators on the axis to the corresponding superalgebra of super-pseudodifferential operators on the superaxis. Full article

(This article belongs to the Special Issue Symmetry in Nonlinear Dynamics and Chaos II)

45 pages, 4574 KiB

Open AccessReview

Chiral Effective Model of Cold and Dense Two-Color QCD: The Linear Sigma Model Approach

by Daiki Suenaga

Symmetry 2025, 17(1), 124; https://doi.org/10.3390/sym17010124 - 15 Jan 2025

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Abstract

This review is devoted to summarizing recent developments of the linear sigma model (LSM) in cold and dense two-color QCD (QC₂D), in which lattice simulations are straightforwardly applicable thanks to the disappearance of the sign problem. In QC₂D, both [...] Read more.

This review is devoted to summarizing recent developments of the linear sigma model (LSM) in cold and dense two-color QCD (QC₂D), in which lattice simulations are straightforwardly applicable thanks to the disappearance of the sign problem. In QC₂D, both theoretical and numerical studies derive the presence of the so-called baryon superfluid phase at a sufficiently large chemical potential (

μ_{q}

), where diquark condensates govern the ground state. The hadron mass spectrum simulated in this phase shows that the mass of an iso-singlet (

I = 0

) and

0^{-}

state is remarkably reduced, but such a mode cannot be described by the chiral perturbation theory. Motivated by this fact, I have invented a LSM constructed upon the linear representation of chiral symmetry, more precisely Pauli–Gürsey symmetry. It is shown that my LSM successfully reproduces the low-lying hadron mass spectrum in a broad range of

μ_{q}

simulated on the lattice. As applications of the LSM, topological susceptibility and sound velocity in cold and dense QC₂D are evaluated to compare with the lattice results. Additionally, the generalized Gell–Mann–Oakes–Renner relation and hardon mass spectrum in the presence of a diquark source are analyzed. I also introduce an extended version of the LSM incorporating spin-1 hadrons. Full article

(This article belongs to the Special Issue Chiral Symmetry, and Restoration in Nuclear Dense Matter)

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25 pages, 2721 KiB

Open AccessArticle

Spatial Kinetic Modeling of Crowd Evacuation: Coupling Social Behavior and Infectious Disease Contagion

by Juan Pablo Agnelli, Claudio Armas and Damián A. Knopoff

Symmetry 2025, 17(1), 123; https://doi.org/10.3390/sym17010123 - 15 Jan 2025

Viewed by 605

Abstract

This paper introduces a kinetic model of crowd evacuation from a bounded domain, integrating social behavior and contagion dynamics. The model describes the spatial movement of individuals in a crowd, taking into account interactions with other people and the geometry of the environment. [...] Read more.

This paper introduces a kinetic model of crowd evacuation from a bounded domain, integrating social behavior and contagion dynamics. The model describes the spatial movement of individuals in a crowd, taking into account interactions with other people and the geometry of the environment. Interactions between healthy and infectious individuals can lead to disease transmission and are considered. The approach is grounded in the kinetic theory of active particles, where the activity variable represents both the infectious disease status of individuals (e.g., susceptible, infected) and the psychological state of pedestrians, including contagion awareness. Varying awareness levels influence individual behavior, leading to more cautious movement patterns, potentially reducing the overall infection rate. The performance of the model is evaluated through a series of numerical simulations. Different scenarios are examined to investigate the impact of awareness levels on pedestrian behavior, infectious disease spread, and evacuation times. Additionally, the effects of population immunization and individual contagion awareness are assessed to determine the most effective strategy for reducing infections. The results provide valuable insights into targeted strategies to mitigate contagion. Full article

(This article belongs to the Special Issue Mathematical Modeling of Symmetry in Collective Biological Dynamics)

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35 pages, 3375 KiB

Open AccessArticle

Optimization in Symmetric Trees, Unicyclic Graphs, and Bicyclic Graphs with Help of Mappings Using Second Form of Generalized Power-Sum Connectivity Index

by Muhammad Yasin Khan, Gohar Ali and Ioan-Lucian Popa

Symmetry 2025, 17(1), 122; https://doi.org/10.3390/sym17010122 - 15 Jan 2025

Viewed by 450

Abstract

The topological index (TI), sometimes referred to as the connectivity index, is a molecular descriptor calculated based on the molecular graph of a chemical compound. Topological indices (TIs) are numeric parameters of a graph used to characterize its topology and are usually graph-invariant. [...] Read more.

The topological index (TI), sometimes referred to as the connectivity index, is a molecular descriptor calculated based on the molecular graph of a chemical compound. Topological indices (TIs) are numeric parameters of a graph used to characterize its topology and are usually graph-invariant. The generalized power-sum connectivity index (GPSCI) for the graph is

Ω

Y_{α} (Ω) = \sum_{ζ ϱ \in E (Ω)} {(d_{Ω} {(ζ)}^{d_{Ω} (ζ)} + d_{Ω} {(ϱ)}^{d_{Ω} (ϱ)})}^{α}

, while the second form of the GPSCI is defined as

Y_{β} (Ω) = \sum_{ζ ϱ \in E (Ω)} {(d_{Ω} {(ζ)}^{d_{Ω} (ζ)} \times d_{Ω} {(ϱ)}^{d_{Ω} (ϱ)})}^{β}

. In this paper, we investigate

Y_{β}

in the family of trees, unicyclic graphs, and bicyclic graphs. We determine optimal graphs in the desired families for

Y_{β}

using certain mappings. For graphs with maximal values, two mappings are used, namely A and B, while for graphs with minimal values, mapping C and mapping D are considered. Full article

(This article belongs to the Special Issue Symmetry and Graph Theory, 2nd Edition)

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22 pages, 2930 KiB

Open AccessArticle

Type-2 Backstepping T-S Fuzzy Bionic Control Based on Niche Symmetry Function

by Yunli Hao, Maohua Wang, Jian Tang, Ziyue Zhang and Jiangling Xiong

Symmetry 2025, 17(1), 121; https://doi.org/10.3390/sym17010121 - 14 Jan 2025

Viewed by 523

Abstract

Niche can reflect the changes in the quality of the ecological environment and the balance of ecological state. The more advanced the ecosystem, the more complex and higher-order nonlinearities and uncertainties that are presented. For such an uncertain parameter system with complex nonlinearity, [...] Read more.

Niche can reflect the changes in the quality of the ecological environment and the balance of ecological state. The more advanced the ecosystem, the more complex and higher-order nonlinearities and uncertainties that are presented. For such an uncertain parameter system with complex nonlinearity, backstepping fuzzy control is a good control method. When the backstepping control method is introduced into the Type-2 fuzzy T-S control principle, the equality index symmetry function composed of ecological factors is used as the backstepping control consequence, and the Lyapunov function is constructed to analyze the stability and find out the adaptive law of the ecological factors in the equality index symmetry function of the control consequence. This reflects that the individual organisms always develop in their own favorable direction, highlighting the bionic intelligent control of the method. Through simulation analysis, the Type-2 Backstepping control method is effective in stability and parameter tracking, which reflects the self-development ability and self-coordination ability of individual organisms, highlighting the physical background and symmetry of the bionic intelligent control of this method. Full article

(This article belongs to the Special Issue Symmetry in Mathematical Optimization Algorithm and Its Applications)

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19 pages, 3272 KiB

Open AccessArticle

A Systematic Method Combining Rotated Convolution and State Space Augmented Transformer for Digitizing and Classifying Paper ECGs

by Xiang Wang and Jie Yang

Symmetry 2025, 17(1), 120; https://doi.org/10.3390/sym17010120 - 14 Jan 2025

Viewed by 562

Abstract

Billions of paper Electrocardiograms (ECGs) are recorded annually worldwide, particularly in the Global South. Manual review of this massive dataset is time-consuming and inefficient. Accurate digital reconstruction of these records is essential for efficient cardiac disease diagnosis. This paper proposes a systematic framework [...] Read more.

Billions of paper Electrocardiograms (ECGs) are recorded annually worldwide, particularly in the Global South. Manual review of this massive dataset is time-consuming and inefficient. Accurate digital reconstruction of these records is essential for efficient cardiac disease diagnosis. This paper proposes a systematic framework for digitizing paper ECGs with 12 symmetrically distributed leads and identifying abnormal samples. This method consists of three main components. First, we introduce an adaptive rotated convolution network to detect the positions of lead waveforms. By exploiting the symmetric distribution of 12 leads, a novel loss is proposed to improve the detection model’s performance. Second, image processing techniques, including denoising and connected component analysis, are employed to digitize ECG waveforms. Finally, we propose a transformer-based classification method combined with a state space model. Our process is evaluated on a large synthetic dataset, including ECG images characterized by rotations, noise, and creases. The results demonstrate that the proposed detection method can effectively reconstruct paper ECGs, achieving an 11% improvement in SNR compared to the baseline. Moreover, our classification model exhibits slightly higher performance than other counterparts. The proposed approach offers a promising solution for the automated analysis of paper ECGs, supporting clinical decision-making. Full article

(This article belongs to the Special Issue Multidimensional Signal Processing and Deep Learning—Symmetry Approach)

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13 pages, 491 KiB

Open AccessArticle

Low-Frequency Longitudinal Vibrations of a Fluid-Loaded Elastic Layer

by Sheeru Shamsi and Ludmila Prikazchikova

Symmetry 2025, 17(1), 119; https://doi.org/10.3390/sym17010119 - 14 Jan 2025

Viewed by 310

Abstract

This paper concerns the low-frequency symmetric (extensional) motions of a thin elastic layer submerged in a fluid. This problem is less investigated than that for antisymmetric motion corresponding to bending vibrations, partly because the classical theory for thin-plate extension is not oriented to [...] Read more.

This paper concerns the low-frequency symmetric (extensional) motions of a thin elastic layer submerged in a fluid. This problem is less investigated than that for antisymmetric motion corresponding to bending vibrations, partly because the classical theory for thin-plate extension is not oriented to model the transverse compression of the plate caused by the pressure of the fluid. It is also worth noting that, in contrast to a fluid-borne bending wave, the extensional wave radiates into the fluid, resulting in complex-valued terms in the associated dispersion relation. In this paper, we derive a refined asymptotic formulation for symmetric motion starting from the 2D plane strain problem regarding fluid–structure interaction. The obtained results have the potential to be implemented for interpreting numerical and experimental data for a variety of modern engineering setups. Full article

(This article belongs to the Section Engineering and Materials)

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25 pages, 8892 KiB

Open AccessArticle

A Symmetry-Inspired Hierarchical Control Strategy for Preventing Rollover in Articulated Rollers

by Quanzhi Xu, Wei Qiang and Hui Xie

Symmetry 2025, 17(1), 118; https://doi.org/10.3390/sym17010118 - 14 Jan 2025

Viewed by 345

Abstract

In off-road environments, the lateral rollover stability of articulated unmanned rollers (URs) is critical to ensure operational safety and efficiency. This paper introduces the concept of a rollover energy barrier (REB), a symmetry-based metric that quantifies the energy margin between the current state [...] Read more.

In off-road environments, the lateral rollover stability of articulated unmanned rollers (URs) is critical to ensure operational safety and efficiency. This paper introduces the concept of a rollover energy barrier (REB), a symmetry-based metric that quantifies the energy margin between the current state and the critical rollover threshold of articulated rollers. URs exhibit dynamic asymmetry due to their hydraulic steering systems, which differ significantly from traditional passenger vehicles. To address these challenges, we propose a hierarchical control framework inspired by the principles of dynamic symmetry. This framework integrates Nonlinear Model Predictive Control (NMPC) and Active Disturbance Rejection Control (ADRC): NMPC is used for trajectory planning by incorporating the REB into the cost function, ensuring rollover stability, while ADRC compensates for dynamic asymmetries, model uncertainties, and external disturbances during trajectory tracking. Simulation and experimental results validate the effectiveness of the proposed control strategy in enhancing the rollover stability and tracking performance of the URs under off-road conditions. Full article

(This article belongs to the Section Engineering and Materials)

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