Journal Description
Symmetry
Symmetry
is an international, peer-reviewed, open access journal covering research on symmetry/asymmetry phenomena wherever they occur in all aspects of natural sciences. Symmetry is published monthly online by MDPI.
- Open Access— free for readers, with article processing charges (APC) paid by authors or their institutions.
- High Visibility: indexed within SCIE (Web of Science), Scopus, CAPlus / SciFinder, Inspec, Astrophysics Data System, and other databases.
- Journal Rank: JCR - Q2 (Multidisciplinary Sciences) / CiteScore - Q1 (General Mathematics )
- Rapid Publication: manuscripts are peer-reviewed and a first decision is provided to authors approximately 17.1 days after submission; acceptance to publication is undertaken in 2.8 days (median values for papers published in this journal in the first half of 2025).
- Recognition of Reviewers: reviewers who provide timely, thorough peer-review reports receive vouchers entitling them to a discount on the APC of their next publication in any MDPI journal, in appreciation of the work done.
- Testimonials: See what our editors and authors say about Symmetry.
Impact Factor:
2.2 (2024);
5-Year Impact Factor:
2.1 (2024)
Latest Articles
A Symmetric Multi-Scale Convolutional Transformer Network for Plant Disease Image Classification
Symmetry 2025, 17(8), 1232; https://doi.org/10.3390/sym17081232 (registering DOI) - 4 Aug 2025
Abstract
Plant disease classification is critical for effective crop management. Recent advances in deep learning, especially Vision Transformers (ViTs), have shown promise due to their strong global feature modeling capabilities. However, ViTs often overlook local features and suffer from feature extraction degradation during patch
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Plant disease classification is critical for effective crop management. Recent advances in deep learning, especially Vision Transformers (ViTs), have shown promise due to their strong global feature modeling capabilities. However, ViTs often overlook local features and suffer from feature extraction degradation during patch merging as channels increase. To address these issues, we propose PLTransformer, a hybrid model designed to symmetrically capture both global and local features. We design a symmetric multi-scale convolutional module that combines two different-scale receptive fields to simultaneously extract global and local features so that the model can better perceive multi-scale disease morphologies. Additionally, we propose an overlap-attentive channel downsampler that utilizes inter-channel attention mechanisms during spatial downsampling, effectively preserving local structural information and mitigating semantic loss caused by feature compression. On the PlantVillage dataset, PLTransformer achieves 99.95% accuracy, outperforming DeiT (96.33%), Twins (98.92%), and DilateFormer (98.84%). These results demonstrate its superiority in handling multi-scale disease features.
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(This article belongs to the Section Computer)
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Open AccessArticle
Dual Chaotic Diffusion Framework for Multimodal Biometric Security Using Qi Hyperchaotic System
by
Tresor Lisungu Oteko and Kingsley A. Ogudo
Symmetry 2025, 17(8), 1231; https://doi.org/10.3390/sym17081231 (registering DOI) - 4 Aug 2025
Abstract
The proliferation of biometric technology across various domains including user identification, financial services, healthcare, security, law enforcement, and border control introduces convenience in user identity verification while necessitating robust protection mechanisms for sensitive biometric data. While chaos-based encryption systems offer promising solutions, many
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The proliferation of biometric technology across various domains including user identification, financial services, healthcare, security, law enforcement, and border control introduces convenience in user identity verification while necessitating robust protection mechanisms for sensitive biometric data. While chaos-based encryption systems offer promising solutions, many existing chaos-based encryption schemes exhibit inherent shortcomings including deterministic randomness and constrained key spaces, often failing to balance security robustness with computational efficiency. To address this, we propose a novel dual-layer cryptographic framework leveraging a four-dimensional (4D) Qi hyperchaotic system for protecting biometric templates and facilitating secure feature matching operations. The framework implements a two-tier encryption mechanism where each layer independently utilizes a Qi hyperchaotic system to generate unique encryption parameters, ensuring template-specific encryption patterns that enhance resistance against chosen-plaintext attacks. The framework performs dimensional normalization of input biometric templates, followed by image pixel shuffling to permutate pixel positions before applying dual-key encryption using the Qi hyperchaotic system and XOR diffusion operations. Templates remain encrypted in storage, with decryption occurring only during authentication processes, ensuring continuous security while enabling biometric verification. The proposed system’s framework demonstrates exceptional randomness properties, validated through comprehensive NIST Statistical Test Suite analysis, achieving statistical significance across all 15 tests with p-values consistently above 0.01 threshold. Comprehensive security analysis reveals outstanding metrics: entropy values exceeding 7.99 bits, a key space of , negligible correlation coefficients (< ), and robust differential attack resistance with an NPCR of 99.60% and a UACI of 33.45%. Empirical evaluation, on standard CASIA Face and Iris databases, demonstrates practical computational efficiency, achieving average encryption times of 0.50913s per user template for 256 × 256 images. Comparative analysis against other state-of-the-art encryption schemes verifies the effectiveness and reliability of the proposed scheme and demonstrates our framework’s superior performance in both security metrics and computational efficiency. Our findings contribute to the advancement of biometric template protection methodologies, offering a balanced performance between security robustness and operational efficiency required in real-world deployment scenarios.
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(This article belongs to the Special Issue New Advances in Symmetric Cryptography)
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An Efficient Spectral Method for a Class of Asymmetric Functional-Order Diffusion–Wave Equations Using Generalized Chelyshkov Wavelets
by
Quan H. Do and Hoa T. B. Ngo
Symmetry 2025, 17(8), 1230; https://doi.org/10.3390/sym17081230 - 4 Aug 2025
Abstract
Asymmetric functional-order (variable-order) fractional diffusion–wave equations (FO-FDWEs) introduce considerable computational challenges, as the fractional order of the derivatives can vary spatially or temporally. To overcome these challenges, a novel spectral method employing generalized fractional-order Chelyshkov wavelets (FO-CWs) is developed to efficiently solve such
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Asymmetric functional-order (variable-order) fractional diffusion–wave equations (FO-FDWEs) introduce considerable computational challenges, as the fractional order of the derivatives can vary spatially or temporally. To overcome these challenges, a novel spectral method employing generalized fractional-order Chelyshkov wavelets (FO-CWs) is developed to efficiently solve such equations. In this approach, the Riemann–Liouville fractional integral operator of variable order is evaluated in closed form via a regularized incomplete Beta function, enabling the transformation of the governing equation into a system of algebraic equations. This wavelet-based spectral scheme attains extremely high accuracy, yielding significantly lower errors than existing numerical techniques. In particular, numerical results show that the proposed method achieves notably improved accuracy compared to existing methods under the same number of basis functions. Its strong convergence properties allow high precision to be achieved with relatively few wavelet basis functions, leading to efficient computations. The method’s accuracy and efficiency are demonstrated on several practical diffusion–wave examples, indicating its suitability for real-world applications. Furthermore, it readily applies to a wide class of fractional partial differential equations (FPDEs) with spatially or temporally varying order, demonstrating versatility for diverse applications.
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(This article belongs to the Topic Numerical Methods for Partial Differential Equations)
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Examples for BPS Solitons Destabilized by Quantum Effects
by
Willem J. Meyer and Herbert Weigel
Symmetry 2025, 17(8), 1229; https://doi.org/10.3390/sym17081229 - 4 Aug 2025
Abstract
We investigate serval models for two scalar fields in one space dimension with topologically stable solitons that are constructed from BPS equations. The asymptotic behavior of these solitons fully determines their classical energies. A particular feature of the considered models is that there
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We investigate serval models for two scalar fields in one space dimension with topologically stable solitons that are constructed from BPS equations. The asymptotic behavior of these solitons fully determines their classical energies. A particular feature of the considered models is that there are several translationally invariant ground states that we call primary and secondary vacua. The former are those that are asymptotically assumed by the solitons. Solitons that occupy a secondary vacuum in finite but eventually large portions of space are classically degenerate. Thus the quantum contributions to the energies are decisive for the energetically favored soliton. While some of these solitons were constructed previously, we, for the first time, compute the leading (one-loop) quantum contribution their energies. In all cases considered we find that this contribution is not bounded from below and that it is the more negative the larger the region is in which the soliton approaches a secondary vacuum. This corroborates the conjecture, earlier inferred from the Shifman-Voloshin soliton, that the availability of secondary vacua destabilizes these solitons on the quantum level.
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(This article belongs to the Section Physics)
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A Priori Sample Size Determination for Estimating a Location Parameter Under a Unified Skew-Normal Distribution
by
Cong Wang, Weizhong Tian and Jingjing Yang
Symmetry 2025, 17(8), 1228; https://doi.org/10.3390/sym17081228 - 4 Aug 2025
Abstract
The a priori procedure (APP) is concerned with determining appropriate sample sizes to ensure that sample statistics to be obtained are likely to be good estimators of corresponding population parameters. Previous researchers have shown how to compute a priori confidence interval means or
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The a priori procedure (APP) is concerned with determining appropriate sample sizes to ensure that sample statistics to be obtained are likely to be good estimators of corresponding population parameters. Previous researchers have shown how to compute a priori confidence interval means or locations for normal and skew-normal distributions. However, two critical limitations persist in the literature: (1) While numerous skewed models have been proposed, the APP equations for location parameters have only been formally established for the basic skew-normal distributions. (2) Even within this fundamental framework, the APPs for sample size determinations in estimating locations are constructed on samples of specifically dependent observations having multivariate skew-normal distributions jointly. Our work addresses these limitations by extending a priori reasoning to the more comprehensive unified skew-normal (SUN) distribution. The SUN family not only encompasses multiple existing skew-normal models as special cases but also enables broader practical applications through its capacity to model mixed skewness patterns and diverse tail behaviors. In this paper, we establish APP equations for determining the required sample sizes and set up confidence intervals for the location parameter in the one-sample case, as well as for the difference in locations in matched pairs and two independent samples, assuming independent observations from the SUN family. This extension addresses a critical gap in the literature and offers a valuable contribution to the field. Simulation studies support the equations presented, and two applications involve real data sets for illustrations of our main results.
Full article
(This article belongs to the Special Issue Skewed (Asymmetrical) Probability Distributions and Applications Across Disciplines, Fourth Edition)
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Fingerprint-Based Secure Query Scheme for Databases over Symmetric Mirror Servers
by
Yu Zhang, Rui Zhu, Yin Li and Wenjv Hu
Symmetry 2025, 17(8), 1227; https://doi.org/10.3390/sym17081227 - 4 Aug 2025
Abstract
The Karp and Rabin (KR) fingerprint is a special hash-like function widely utilized for efficient string matching. Recently, Sharma et al.leveraged its linear and symmetric properties to facilitate private database queries. However, their approach mainly protects encrypted or secret-shared databases rather than public
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The Karp and Rabin (KR) fingerprint is a special hash-like function widely utilized for efficient string matching. Recently, Sharma et al.leveraged its linear and symmetric properties to facilitate private database queries. However, their approach mainly protects encrypted or secret-shared databases rather than public databases, where only the query privacy is required. In this paper, we focus explicitly on privacy-preserving queries over public read-only databases. We propose a novel fingerprint-based keyword query scheme using the distributed point function (DPF), which effectively hides users’ data access patterns across two symmetric mirror servers. Moreover, we provide a rigorous analysis of the false positive probability inherent in fingerprinting and discuss strategies for its minimization. Our scheme achieves efficiency close to plaintext methods, significantly reducing deployment complexity.
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(This article belongs to the Section Computer)
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Construction of Typical Scenarios for Multiple Renewable Energy Plant Outputs Considering Spatiotemporal Correlations
by
Yuyue Zhang, Yan Wen, Nan Wang, Zhenhua Yuan, Lina Zhang and Runjia Sun
Symmetry 2025, 17(8), 1226; https://doi.org/10.3390/sym17081226 - 3 Aug 2025
Abstract
A high-quality set of typical scenarios is significant for power grid planning. Existing construction methods for typical scenarios do not account for the spatiotemporal correlations among renewable energy plant outputs, failing to adequately reflect the distribution characteristics of original scenarios. To address the
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A high-quality set of typical scenarios is significant for power grid planning. Existing construction methods for typical scenarios do not account for the spatiotemporal correlations among renewable energy plant outputs, failing to adequately reflect the distribution characteristics of original scenarios. To address the issues mentioned above, this paper proposes a construction method for typical scenarios considering spatiotemporal correlations, providing high-quality typical scenarios for power grid planning. Firstly, a symmetric spatial correlation matrix and a temporal autocorrelation matrix are defined, achieving quantitative representation of spatiotemporal correlations. Then, distributional differences between typical and original scenarios are quantified from multiple dimensions, and a scenario reduction model considering spatiotemporal correlations is established. Finally, the genetic algorithm is improved by incorporating adaptive parameter adjustment and an elitism strategy, which can efficiently obtain high-quality typical scenarios. A case study involving five wind farms and fourteen photovoltaic plants in Belgium is presented. The rate of error between spatial correlation matrices of original and typical scenario sets is lower than 2.6%, and the rate of error between temporal autocorrelations is lower than 2.8%. Simulation results demonstrate that typical scenarios can capture the spatiotemporal correlations of original scenarios.
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(This article belongs to the Special Issue New Power System and Symmetry)
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GAPSO: Cloud-Edge-End Collaborative Task Offloading Based on Genetic Particle Swarm Optimization
by
Wu Wen, Yibin Huang, Zhong Xiao, Lizhuang Tan and Peiying Zhang
Symmetry 2025, 17(8), 1225; https://doi.org/10.3390/sym17081225 - 3 Aug 2025
Abstract
In the 6G era, the proliferation of smart devices has led to explosive growth in data volume. The traditional cloud computing can no longer meet the demand for efficient processing of large amounts of data. Edge computing can solve the energy loss problems
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In the 6G era, the proliferation of smart devices has led to explosive growth in data volume. The traditional cloud computing can no longer meet the demand for efficient processing of large amounts of data. Edge computing can solve the energy loss problems caused by transmission delay and multi-level forwarding in cloud computing by processing data close to the data source. In this paper, we propose a cloud–edge–end collaborative task offloading strategy with task response time and execution energy consumption as the optimization targets under a limited resource environment. The tasks generated by smart devices can be processed using three kinds of computing nodes, including user devices, edge servers, and cloud servers. The computing nodes are constrained by bandwidth and computing resources. For the target optimization problem, a genetic particle swarm optimization algorithm considering three layers of computing nodes is designed. The task offloading optimization is performed by introducing (1) opposition-based learning algorithm, (2) adaptive inertia weights, and (3) adjustive acceleration coefficients. All metaheuristic algorithms adopt a symmetric training method to ensure fairness and consistency in evaluation. Through experimental simulation, compared with the classic evolutionary algorithm, our algorithm reduces the objective function value by about 6–12% and has higher algorithm convergence speed, accuracy, and stability.
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(This article belongs to the Special Issue Evolutionary Computation, Metaheuristics, Nature-Inspired Algorithms, and Symmetry)
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Enhancing Complex Decision-Making Under Uncertainty: Theory and Applications of q-Rung Neutrosophic Fuzzy Sets
by
Omniyyah Saad Alqurashi and Kholood Mohammad Alsager
Symmetry 2025, 17(8), 1224; https://doi.org/10.3390/sym17081224 - 3 Aug 2025
Abstract
This thesis pioneers the development of q-Rung Neutrosophic Fuzzy Rough Sets (q-RNFRSs), establishing the first theoretical framework that integrates q-Rung Neutrosophic Sets with rough approximations to break through the conventional constraint of existing
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This thesis pioneers the development of q-Rung Neutrosophic Fuzzy Rough Sets (q-RNFRSs), establishing the first theoretical framework that integrates q-Rung Neutrosophic Sets with rough approximations to break through the conventional constraint of existing fuzzy–rough hybrids, achieving unprecedented capability in extreme uncertainty representation through our generalized model ( ). The work makes three fundamental contributions: (1) theoretical innovation through complete algebraic characterization of q-RNFRSs, including two distinct union/intersection operations and four novel classes of complement operators (with Theorem 1 verifying their involution properties via De Morgan’s Laws); (2) clinical breakthrough via a domain-independent medical decision algorithm featuring dynamic q-adaptation (q = 2–4) for criterion-specific uncertainty handling, demonstrating 90% diagnostic accuracy in validation trials—a 22% improvement over static models ( ); and (3) practical impact through multi-dimensional uncertainty modeling (truth–indeterminacy–falsity), robust therapy prioritization under data incompleteness, and computationally efficient approximations for real-world clinical deployment.
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(This article belongs to the Special Issue The Fusion of Fuzzy Sets and Optimization Using Symmetry)
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Data-Driven Symmetry and Asymmetry Investigation of Vehicle Emissions Using Machine Learning: A Case Study in Spain
by
Fei Wu, Jinfu Zhu, Hufang Yang, Xiang He and Qiao Peng
Symmetry 2025, 17(8), 1223; https://doi.org/10.3390/sym17081223 - 2 Aug 2025
Abstract
Understanding vehicle emissions is essential for developing effective carbon reduction strategies in the transport sector. Conventional emission models often assume homogeneity and linearity, overlooking real-world asymmetries that arise from variations in vehicle design and powertrain configurations. This study explores how machine learning and
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Understanding vehicle emissions is essential for developing effective carbon reduction strategies in the transport sector. Conventional emission models often assume homogeneity and linearity, overlooking real-world asymmetries that arise from variations in vehicle design and powertrain configurations. This study explores how machine learning and explainable AI techniques can effectively capture both symmetric and asymmetric emission patterns across different vehicle types, thereby contributing to more sustainable transport planning. Addressing a key gap in the existing literature, the study poses the following question: how do structural and behavioral factors contribute to asymmetric emission responses in internal combustion engine vehicles compared to new energy vehicles? Utilizing a large-scale Spanish vehicle registration dataset, the analysis classifies vehicles by powertrain type and applies five supervised learning algorithms to predict CO2 emissions. SHapley Additive exPlanations (SHAPs) are employed to identify nonlinear and threshold-based relationships between emissions and vehicle characteristics such as fuel consumption, weight, and height. Among the models tested, the Random Forest algorithm achieves the highest predictive accuracy. The findings reveal critical asymmetries in emission behavior, particularly among hybrid vehicles, which challenge the assumption of uniform policy applicability. This study provides both methodological innovation and practical insights for symmetry-aware emission modeling, offering support for more targeted eco-design and policy decisions that align with long-term sustainability goals.
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(This article belongs to the Section Engineering and Materials)
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Variable Step-Size FxLMS Algorithm Based on Cooperative Coupling of Double Nonlinear Functions
by
Jialong Wang, Jian Liao, Lin He, Xiaopeng Tan and Zongbin Chen
Symmetry 2025, 17(8), 1222; https://doi.org/10.3390/sym17081222 - 2 Aug 2025
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Based on the principle of symmetry, we propose a variable step-size FxLMS algorithm with double nonlinear functions cooperative coupling (DNVSS-FxLMS), aiming to optimize the contradiction between convergence rate and steady-state error in the active pressure pulsation control system of hydraulic systems. The algorithm
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Based on the principle of symmetry, we propose a variable step-size FxLMS algorithm with double nonlinear functions cooperative coupling (DNVSS-FxLMS), aiming to optimize the contradiction between convergence rate and steady-state error in the active pressure pulsation control system of hydraulic systems. The algorithm innovatively couples two types of nonlinear mechanisms (rational-fractional and exponential-function-based), constructing a refined error-step mapping relationship to achieve a balance between rapid convergence and low steady-state error. Simulation experiments were conducted considering the complex time-varying operating environment of a simulation-based hydraulic system. The results demonstrate that, when the system undergoes unstable random changes, the DNVSS-FxLMS algorithm converges at least twice as fast as traditional and existing variable step size algorithms, while reducing steady-state error by 2–5 dB. The proposed DNVSS-FxLMS algorithm exhibits significant advantages in convergence rate, steady-state error reduction, and tracking capability, providing a highly efficient and robust solution for real-time active control of hydraulic system pressure pulsation under complex operating conditions.
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An Image Encryption Method Based on a Two-Dimensional Cross-Coupled Chaotic System
by
Caiwen Chen, Tianxiu Lu and Boxu Yan
Symmetry 2025, 17(8), 1221; https://doi.org/10.3390/sym17081221 - 2 Aug 2025
Abstract
Chaotic systems have demonstrated significant potential in the field of image encryption due to their extreme sensitivity to initial conditions, inherent unpredictability, and pseudo-random behavior. However, existing chaos-based encryption schemes still face several limitations, including narrow chaotic regions, discontinuous chaotic ranges, uneven trajectory
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Chaotic systems have demonstrated significant potential in the field of image encryption due to their extreme sensitivity to initial conditions, inherent unpredictability, and pseudo-random behavior. However, existing chaos-based encryption schemes still face several limitations, including narrow chaotic regions, discontinuous chaotic ranges, uneven trajectory distributions, and fixed pixel processing sequences. These issues substantially hinder the security and efficiency of such algorithms. To address these challenges, this paper proposes a novel hyperchaotic map, termed the two-dimensional cross-coupled chaotic map (2D-CFCM), derived from a newly designed 2D cross-coupled chaotic system. The proposed 2D-CFCM exhibits enhanced randomness, greater sensitivity to initial values, a broader chaotic region, and a more uniform trajectory distribution, thereby offering stronger security guarantees for image encryption applications. Based on the 2D-CFCM, an innovative image encryption method was further developed, incorporating efficient scrambling and forward and reverse random multidirectional diffusion operations with symmetrical properties. Through simulation tests on images of varying sizes and resolutions, including color images, the results demonstrate the strong security performance of the proposed method. This method has several remarkable features, including an extremely large key space (greater than ), extremely high key sensitivity, nearly ideal entropy value (greater than 7.997), extremely low pixel correlation (less than 0.04), and excellent resistance to differential attacks (with the average values of NPCR and UACI being and , respectively). Compared to existing encryption algorithms, the proposed method provides significantly enhanced security.
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(This article belongs to the Special Issue Symmetry in Chaos Theory and Applications)
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Model-Agnostic Meta-Learning in Predicting Tunneling-Induced Surface Ground Deformation
by
Wei He, Guan-Bin Chen, Wenlian Qian, Wen-Li Chen, Liang Tang and Xiangxun Kong
Symmetry 2025, 17(8), 1220; https://doi.org/10.3390/sym17081220 - 2 Aug 2025
Abstract
The present investigation presents the field measurement and prediction of tunneling-induced surface ground settlement in Tianjin Metro Line 7, China. The cross-section of a metro tunnel exhibits circular symmetry, thereby making it suitable for tunneling with a circular shield machine. The ground surface
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The present investigation presents the field measurement and prediction of tunneling-induced surface ground settlement in Tianjin Metro Line 7, China. The cross-section of a metro tunnel exhibits circular symmetry, thereby making it suitable for tunneling with a circular shield machine. The ground surface may deform during the tunneling stage. In the early stage of tunneling, few measurement data can be collected. To obtain a better usable prediction model, two kinds of neural networks according to the model-agnostic meta-learning (MAML) scheme are presented. One kind of deep learning strategy is a combination of the Back-Propagation Neural Network (BPNN) and the MAML model, named MAML-BPNN. The other prediction model is a mixture of the MAML model and the Long Short-Term Memory (LSTM) model, named MAML-LSTM. Founded on several measurement datasets, the prediction models of the MAML-BPNN and MAML-LSTM are successfully trained. The results show the present models possess good prediction ability for tunneling-induced surface ground settlement. Based on the coefficient of determination, the prediction result using MAML-LSTM is superior to that of MAML-BPNN by 0.1.
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(This article belongs to the Special Issue Symmetry, Asymmetry and Other Patterns in Structural Engineering with Its Application)
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The Analysis of Three-Dimensional Time-Fractional Helmholtz Model Using a New İterative Method
by
Yasin Şahin, Mehmet Merdan and Pınar Açıkgöz
Symmetry 2025, 17(8), 1219; https://doi.org/10.3390/sym17081219 - 1 Aug 2025
Abstract
This paper proposes a novel analytical method to address the Helmholtz fractional differential equation by combining the Aboodh transform with the Adomian Decomposition Method, resulting in the Aboodh–Adomian Decomposition Method (A-ADM). Fractional differential equations offer a comprehensive framework for describing intricate physical processes,
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This paper proposes a novel analytical method to address the Helmholtz fractional differential equation by combining the Aboodh transform with the Adomian Decomposition Method, resulting in the Aboodh–Adomian Decomposition Method (A-ADM). Fractional differential equations offer a comprehensive framework for describing intricate physical processes, including memory effects and anomalous diffusion. This work employs the Caputo–Fabrizio fractional derivative, defined by a non-singular exponential kernel, to more precisely capture these non-local effects. The classical Helmholtz equation, pivotal in acoustics, electromagnetics, and quantum physics, is extended to the fractional domain. Following the exposition of fundamental concepts and characteristics of fractional calculus and the Aboodh transform, the suggested A-ADM is employed to derive the analytical solution of the fractional Helmholtz equation. The method’s validity and efficiency are evidenced by comparisons of analytical and approximation solutions. The findings validate that A-ADM is a proficient and methodical approach for addressing fractional differential equations that incorporate Caputo–Fabrizio derivatives.
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(This article belongs to the Special Issue Symmetries and Fractional Differential Equations: Theory and Applications)
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Comparative Analysis of the Gardner Equation in Plasma Physics Using Analytical and Neural Network Methods
by
Zain Majeed, Adil Jhangeer, F. M. Mahomed, Hassan Almusawa and F. D. Zaman
Symmetry 2025, 17(8), 1218; https://doi.org/10.3390/sym17081218 - 1 Aug 2025
Abstract
In the present paper, a mathematical analysis of the Gardner equation with varying coefficients has been performed to give a more realistic model of physical phenomena, especially in regards to plasma physics. First, a Lie symmetry analysis was carried out, as a result
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In the present paper, a mathematical analysis of the Gardner equation with varying coefficients has been performed to give a more realistic model of physical phenomena, especially in regards to plasma physics. First, a Lie symmetry analysis was carried out, as a result of which a symmetry classification following the different representations of the variable coefficients was systematically derived. The reduced ordinary differential equation obtained is solved using the power-series method and solutions to the equation are represented graphically to give an idea of their dynamical behavior. Moreover, a fully connected neural network has been included as an efficient computation method to deal with the complexity of the reduced equation, by using traveling-wave transformation. The validity and correctness of the solutions provided by the neural networks have been rigorously tested and the solutions provided by the neural networks have been thoroughly compared with those generated by the Runge–Kutta method, which is a conventional and well-recognized numerical method. The impact of a variation in the loss function of different coefficients has also been discussed, and it has also been found that the dispersive coefficient affects the convergence rate of the loss contribution considerably compared to the other coefficients. The results of the current work can be used to improve knowledge on the nonlinear dynamics of waves in plasma physics. They also show how efficient it is to combine the approaches, which consists in the use of analytical and semi-analytical methods and methods based on neural networks, to solve nonlinear differential equations with variable coefficients of a complex nature.
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(This article belongs to the Section Physics)
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Multi-Granulation Covering Rough Intuitionistic Fuzzy Sets Based on Maximal Description
by
Xiao-Meng Si and Zhan-Ao Xue
Symmetry 2025, 17(8), 1217; https://doi.org/10.3390/sym17081217 - 1 Aug 2025
Abstract
Rough sets and fuzzy sets are two complementary approaches for modeling uncertainty and imprecision. Their integration enables a more comprehensive representation of complex, uncertain systems. However, existing rough fuzzy sets models lack the expressive power to fully capture the interactions among structural uncertainty,
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Rough sets and fuzzy sets are two complementary approaches for modeling uncertainty and imprecision. Their integration enables a more comprehensive representation of complex, uncertain systems. However, existing rough fuzzy sets models lack the expressive power to fully capture the interactions among structural uncertainty, cognitive hesitation, and multi-level granular information. To address these limitations, we achieve the following: (1) We propose intuitionistic fuzzy covering rough membership and non-membership degrees based on maximal description and construct a new single-granulation model that more effectively captures both the structural relationships among elements and the semantics of fuzzy information. (2) We further extend the model to a multi-granulation framework by defining optimistic and pessimistic approximation operators and analyzing their properties. Additionally, we propose a neutral multi-granulation covering rough intuitionistic fuzzy sets based on aggregated membership and non-membership degrees. Compared with single-granulation models, the multi-granulation models integrate multiple levels of information, allowing for more fine-grained and robust representations of uncertainty. Finally, a case study on real estate investment was conducted to validate the effectiveness of the proposed models. The results show that our models can more precisely represent uncertainty and granularity in complex data, providing a flexible tool for knowledge representation in decision-making scenarios.
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(This article belongs to the Section Mathematics)
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A Qualitative Analysis and Discussion of a New Model for Optimizing Obesity and Associated Comorbidities
by
Mohamed I. Youssef, Robert M. Maina, Duncan K. Gathungu and Amr Radwan
Symmetry 2025, 17(8), 1216; https://doi.org/10.3390/sym17081216 - 1 Aug 2025
Abstract
This paper addresses the problem of optimizing obesity, which has been a challenging issue in the last decade based on recent data revealed in 2024 by the World Health Organization (WHO). The current work introduces a new mathematical model of the dynamics of
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This paper addresses the problem of optimizing obesity, which has been a challenging issue in the last decade based on recent data revealed in 2024 by the World Health Organization (WHO). The current work introduces a new mathematical model of the dynamics of weight over time with embedded control parameters to optimize the number of obese, overweight, and comorbidity populations. The mathematical formulation of the model is developed under certain sufficient conditions that guarantee the positivity and boundedness of solutions over time. The model structure exhibits inherent symmetry in population group transitions, particularly around the equilibrium state, which allows the application of analytical tools such as the Routh–Hurwitz and Metzler criteria. Then, the analysis of local and global stability of the obesity-free equilibrium state is discussed based on these criteria. Based on the Pontryagin maximum principle (PMP), the deviation from the obesity-free equilibrium state is controlled. The model’s effectiveness is demonstrated through simulation using the Forward–Backward Sweeping algorithm with parameters derived from recent research in human health. Incorporating symmetry considerations in the model enhances the understanding of system behavior and supports balanced intervention strategies. Results suggest that the model can effectively inform strategies to mitigate obesity prevalence and associated health risks.
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(This article belongs to the Special Issue Mathematical Modeling of the Infectious Diseases and Their Controls)
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Special Curves and Tubes in the BCV-Sasakian Manifold
by
Tuba Ağırman Aydın and Ensar Ağırman
Symmetry 2025, 17(8), 1215; https://doi.org/10.3390/sym17081215 - 1 Aug 2025
Abstract
In this study, theorems and proofs related to spherical and focal curves are presented in the BCV-Sasakian space. An approximate solution to the differential equation characterizing spherical curves in the BCV-Sasakian manifold is obtained using the Taylor matrix collocation method. The
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In this study, theorems and proofs related to spherical and focal curves are presented in the BCV-Sasakian space. An approximate solution to the differential equation characterizing spherical curves in the BCV-Sasakian manifold is obtained using the Taylor matrix collocation method. The general equations of canal and tubular surfaces are provided within this geometric framework. Additionally, the curvature properties of the tubular surface constructed around a non-vertex focal curve are computed and analyzed. All of these results are presented for the first time in the literature within the context of the BCV-Sasakian geometry. Thus, this study makes a substantial contribution to the differential geometry of contact metric manifolds by extending classical concepts into a more generalized and complex geometric structure.
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(This article belongs to the Section Mathematics)
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FAMNet: A Lightweight Stereo Matching Network for Real-Time Depth Estimation in Autonomous Driving
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Jingyuan Zhang, Qiang Tong, Na Yan and Xiulei Liu
Symmetry 2025, 17(8), 1214; https://doi.org/10.3390/sym17081214 - 1 Aug 2025
Abstract
Accurate and efficient stereo matching is fundamental to real-time depth estimation from symmetric stereo cameras in autonomous driving systems. However, existing high-accuracy stereo matching networks typically rely on computationally expensive 3D convolutions, which limit their practicality in real-world environments. In contrast, real-time methods
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Accurate and efficient stereo matching is fundamental to real-time depth estimation from symmetric stereo cameras in autonomous driving systems. However, existing high-accuracy stereo matching networks typically rely on computationally expensive 3D convolutions, which limit their practicality in real-world environments. In contrast, real-time methods often sacrifice accuracy or generalization capability. To address these challenges, we propose FAMNet (Fusion Attention Multi-Scale Network), a lightweight and generalizable stereo matching framework tailored for real-time depth estimation in autonomous driving applications. FAMNet consists of two novel modules: Fusion Attention-based Cost Volume (FACV) and Multi-scale Attention Aggregation (MAA). FACV constructs a compact yet expressive cost volume by integrating multi-scale correlation, attention-guided feature fusion, and channel reweighting, thereby reducing reliance on heavy 3D convolutions. MAA further enhances disparity estimation by fusing multi-scale contextual cues through pyramid-based aggregation and dual-path attention mechanisms. Extensive experiments on the KITTI 2012 and KITTI 2015 benchmarks demonstrate that FAMNet achieves a favorable trade-off between accuracy, efficiency, and generalization. On KITTI 2015, with the incorporation of FACV and MAA, the prediction accuracy of the baseline model is improved by 37% and 38%, respectively, and a total improvement of 42% is achieved by our final model. These results highlight FAMNet’s potential for practical deployment in resource-constrained autonomous driving systems requiring real-time and reliable depth perception.
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(This article belongs to the Special Issue Computer Vision, Pattern Recognition, Machine Learning, and Symmetry, 2nd Edition)
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Adaptive Switching Surrogate Model for Evolutionary Multi-Objective Community Detection Algorithm
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Nan Sun, Siying Lv, Xiaoying Xiang, Shuwei Zhu, Hengyang Lu and Wei Fang
Symmetry 2025, 17(8), 1213; https://doi.org/10.3390/sym17081213 - 31 Jul 2025
Abstract
Community detection is widely recognized as a crucial area of research in network science. In recent years, multi-objective evolutionary algorithms (MOEAs) have been extensively employed in community detection tasks. Continuous coding is able to transform the discrete problem into a continuous one. However,
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Community detection is widely recognized as a crucial area of research in network science. In recent years, multi-objective evolutionary algorithms (MOEAs) have been extensively employed in community detection tasks. Continuous coding is able to transform the discrete problem into a continuous one. However, conventional continuous coding methodologies frequently disregard the relationships between node structures, resulting in low-quality encoded populations that subsequently diminish community detection performance. Furthermore, continuous coding needs to be decoded into to label-based coding during the optimization process to compute objective functions. To alleviate this, we design the surrogate model adaptive switching strategy that selects the optimal surrogate model for the task. Subsequently, the surrogate-assisted evolutionary multi-objective community detection algorithm with core node learning is proposed. The core node learning method is employed to enhance the connection between nodes in augmented sequential coding, which helps initialize the population using the node similarity matrix. The core nodes of the network are subsequently identified based on node weights, which can be utilized to construct a surrogate model between the continuous coding and the objective function. The surrogate model is updated during the optimization process, which effectively improves both the accuracy and efficiency of community detection tasks. Experimental results obtained from synthetic and real-world networks demonstrate that the proposed algorithm exhibits superior performance compared to seven community detection algorithms.
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(This article belongs to the Special Issue Symmetry/Asymmetry in Evolutionary Computation and Machine Learning)
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