Symmetry

16 pages, 34025 KB

Open AccessArticle

On Some Incommensurate Fractional-Order Reaction–Diffusion Systems: The Degn–Harrison and Its Stability

by Omar Kahouli, Amel Hioual, Adel Ouannas, Waleed Mohammed Abdelfattah, Younès Bahou, Ilyes Abidi, Sameir Hamed, Mohamed Chaabane and Sarra Elgharbi

Symmetry 2026, 18(5), 862; https://doi.org/10.3390/sym18050862 (registering DOI) - 19 May 2026

Abstract

In this paper, we consider a reaction–diffusion system governed by incommensurate fractional time derivatives based on the Degn–Harrison model. Its formulation incorporates various memory effects on axial position through Caputo derivatives of variable orders, producing a more realistic modeling of the temporal dynamics. [...] Read more.

In this paper, we consider a reaction–diffusion system governed by incommensurate fractional time derivatives based on the Degn–Harrison model. Its formulation incorporates various memory effects on axial position through Caputo derivatives of variable orders, producing a more realistic modeling of the temporal dynamics. This paper starts with a study of the spatially homogeneous system and establishes conditions for local stability by using the Matignon criterion. The spectral decomposition method under Neumann boundary condition is then applied to study the complete reaction–diffusion system and describe diffusion-induced instabilities. Our results indicate that the noninteger fractional orders lead to significant changes in stability regions, as well as the initiation of pattern formation. Specifically, the orders of fractions induced as a control variable are regarded to be effective in controlling the stability of the system, thus they are global (or positive) control variables when their values achieved at some levels apply to the entire saturation, etc. Our numerical simulations are in excellent agreement with the theoretical predictions and show that memory asymmetry induces complex spatiotemporal dynamics not seen for classical integer-order systems. Full article

(This article belongs to the Special Issue Symmetry in Mathematical and Statistical Sciences: Theory, Methods, and Applications)

18 pages, 1341 KB

Open AccessArticle

Approximation of RCI Set Under p-Norm Ball Constraints for Linear Discrete-Time Systems

by Hongli Yang, Longfei Yang and Ivan Ganchev Ivanov

Symmetry 2026, 18(5), 861; https://doi.org/10.3390/sym18050861 (registering DOI) - 19 May 2026

Abstract

This paper investigates the approximation of robust control invariant (RCI) sets for linear discrete-time systems subject to p-norm ball constraints (

p \geq 1

). Unlike classical results focusing on specific cases like polytope or ellipsoid constraints, we propose a unified framework [...] Read more.

This paper investigates the approximation of robust control invariant (RCI) sets for linear discrete-time systems subject to p-norm ball constraints (

p \geq 1

). Unlike classical results focusing on specific cases like polytope or ellipsoid constraints, we propose a unified framework for arbitrary p-norm ball constraints. Sufficient conditions for a non-empty set to be contained within a p-norm ball are established, revealing the geometric insight that the set is essentially contained within the inscribed 2-norm ball. Utilizing these results, the approximation problem of the RCI set is formulated as a standard linear programming problem that requires verifying constraints at only n standard basis vectors, significantly reducing computational complexity. Specific optimization models are derived, and numerical experiments demonstrate the effectiveness and competitive accuracy of the proposed method. Full article

38 pages, 472 KB

Open AccessArticle

On the Cohomological Understanding of Interactions Between Weyl Graviton and Photon

by Eugen-Mihaita Cioroianu and Stefan-Sabin Manolescu

Symmetry 2026, 18(5), 860; https://doi.org/10.3390/sym18050860 (registering DOI) - 19 May 2026

Abstract

The problem of constructing consistent interactions between a Weyl graviton—with its free limit expressed by the linearized Weyl action—and a photon—with its free dynamics generated from the standard Maxwell action—is analyzed as a deformation problem for the antifield-BRST generator associated with the non-interacting [...] Read more.

The problem of constructing consistent interactions between a Weyl graviton—with its free limit expressed by the linearized Weyl action—and a photon—with its free dynamics generated from the standard Maxwell action—is analyzed as a deformation problem for the antifield-BRST generator associated with the non-interacting free model. By relaxing the standard working hypotheses to allow at most four spacetime derivatives in the interaction vertices, while not restricting the number of derivatives on the photon potentials, the most general cross-couplings are derived. This proves the uniqueness of the previously geometrically prescribed, overall fourth-order Lagrangian dynamics of the electromagnetic field in the presence of dynamical-type full Weyl gravity. Full article

(This article belongs to the Special Issue Gravitational Physics and Symmetry)

26 pages, 2603 KB

Open AccessArticle

Perceiving Symmetry and Variability: A Probabilistic Vision–Language Framework for Medical Image Segmentation

by Jiu Jiang, Qi Zhou and Chu He

Symmetry 2026, 18(5), 859; https://doi.org/10.3390/sym18050859 (registering DOI) - 19 May 2026

Abstract

Medical image segmentation is challenging due to subtle pathological patterns and the inherent ambiguity of clinical descriptions. Although vision–language models have shown promise, they frequently lack fine-grained perception of structural variability. To address these limitations, we propose the Symmetry- and Variability-Perceiving Conditional Variational [...] Read more.

Medical image segmentation is challenging due to subtle pathological patterns and the inherent ambiguity of clinical descriptions. Although vision–language models have shown promise, they frequently lack fine-grained perception of structural variability. To address these limitations, we propose the Symmetry- and Variability-Perceiving Conditional Variational Autoencoder (SVP-CVAE). The proposed method integrates a clinical attribute encoder with a morphology-aware enhancement module that incorporates a cross-bilateral symmetry mechanism to explicitly capture symmetry-related variations. By reformulating the segmentation task as a probabilistic prior-to-posterior inference process, SVP-CVAE models the one-to-many mapping between textual attributes and visual realizations. Furthermore, we introduce an attribute-latent contrastive objective to ensure that the latent space encodes discriminative morphological information. Extensive experiments demonstrate that the proposed framework achieves superior segmentation accuracy compared to state-of-the-art methods. Results indicate that SVP-CVAE effectively captures diverse yet anatomically plausible structural variations while maintaining high sensitivity to bilateral symmetry. Comprehensive ablation studies confirm that the performance gains are synergistically driven by the proposed symmetry-perceiving module and the contrastive semantic alignment objective, rather than relying solely on the probabilistic formulation. In conclusion, integrating explicit symmetry perception with probabilistic modeling significantly enhances the reliability and interpretability of multimodal medical image segmentation in complex clinical scenarios. Full article

(This article belongs to the Special Issue Hybrid Deep Learning and Explainable AI for Symmetry-Aware and Multiscale Medical Image Analysis)

29 pages, 1875 KB

Open AccessArticle

UAV Flight Path Planning Based on HPSOCAOA Optimization Algorithm

by Kaijun Xu, Hongda Luo, Yilin Hong, Yong Yang and Weiqi Feng

Symmetry 2026, 18(5), 858; https://doi.org/10.3390/sym18050858 (registering DOI) - 18 May 2026

Abstract

To address the issues with the Crocodile Ambush Optimization Algorithm (CAOA) in UAV trajectory planning—such as its tendency to get stuck in local optima, the difficulty in balancing global search and local exploration, and low convergence accuracy—this study proposes a three-dimensional trajectory planning [...] Read more.

To address the issues with the Crocodile Ambush Optimization Algorithm (CAOA) in UAV trajectory planning—such as its tendency to get stuck in local optima, the difficulty in balancing global search and local exploration, and low convergence accuracy—this study proposes a three-dimensional trajectory planning method based on the Hybrid Particle Swarm and Crocodile Ambush Optimization Algorithm (HPSOCAOA). First, a collaborative search structure combining the Particle Swarm Optimization (PSO) algorithm and the Crocodile Ambush Optimization Algorithm (CAOA) is established; second, an adaptive energy consumption coefficient is designed to address the issues of premature individual elimination in the early stages and insufficient convergence momentum in the later stages, thereby further balancing global exploration and local exploitation; finally, crossover learning is introduced. Using a cross-group replacement mechanism for superior individuals, PSO’s fine-tuning identifies high-quality individuals, which are then substituted for lower-quality individuals in CAOA. This resolves the problems of redundant low-quality individuals within the population and low search efficiency, and enhances overall optimization performance. Standard test functions demonstrate that HPSOCAOA outperforms the comparison algorithms in terms of optimization accuracy and stability. In simulation experiments for path planning in complex 3D mountainous environments, HPSOCAOA was compared with classical intelligent algorithms, verifying its superiority and practicality in complex 3D scenarios. Full article

(This article belongs to the Section Mathematics)

29 pages, 5903 KB

Open AccessArticle

A Symmetric Fault Diagnosis Method for Power Batteries Based on Digital Battery Passport and Knowledge Graph-Fuzzy Bayesian Network

by Tongzhou Ji and Jie Li

Symmetry 2026, 18(5), 857; https://doi.org/10.3390/sym18050857 (registering DOI) - 18 May 2026

Abstract

The safe operation of power battery systems relies on the dynamic symmetric equilibrium of electrochemical distribution and thermal management states, whereas fault occurrence is often accompanied by symmetry breaking. To achieve accurate fault diagnosis and symmetry restoration, this study proposes a symmetrical closed-loop [...] Read more.

The safe operation of power battery systems relies on the dynamic symmetric equilibrium of electrochemical distribution and thermal management states, whereas fault occurrence is often accompanied by symmetry breaking. To achieve accurate fault diagnosis and symmetry restoration, this study proposes a symmetrical closed-loop framework (DBP-KG-FBN) that integrates digital battery passport (DBP) text mining, knowledge graph (KG), and fuzzy Bayesian network (FBN). Power battery fault diagnosis is critical to new energy vehicle (NEV) safety; however, conventional methods face two key limitations: (1) they inadequately exploit multi-source heterogeneous textual data in DBPs; and (2) they fail to handle uncertainty in fault propagation. The methodology proceeds as follows. First, a BERT-BiLSTM-CRF model extracts fault-related entities and relations from unstructured DBP text, which are structured into a Neo4j-based knowledge graph. Second, via rule-based topological mapping, the KG topology is transformed into a Bayesian network through structurally symmetric transformation between the semantic and probabilistic layers, with cyclic dependencies resolved by introducing latent variables. Third, network parameters are determined by integrating fuzzy set theory with game theory-based weighting to quantify uncertainty and subjectivity in expert evaluations, thereby achieving symmetric utilization of subjective and objective information. This enables bidirectional symmetric reasoning for forward fault prediction and backward fault traceability. Experimental results demonstrate that while maintaining symmetric stability of the diagnostic knowledge topology, the proposed DBP-KG-FBN method achieves a diagnostic accuracy of 0.92 (Top-3). This symmetrical closed-loop framework significantly outperforms fault tree analysis (FTA) and event tree analysis (ETA) in diagnostic accuracy and reasoning efficiency. It transforms unstructured DBP data into computable knowledge for intelligent battery diagnosis. Future work will expand the corpus via transfer learning and optimize adaptive weighting algorithms for expert evaluations. Full article

(This article belongs to the Section Engineering and Materials)

25 pages, 1396 KB

Open AccessArticle

Key Technologies of Near-Bit Multi-Parameter MWD for Directional Drilling in Underground Engineering

by Zhiwei Chu, Shijun Hao, Quanxin Li, Long Chen, Yunhong Wang, Jun Fang, Dongdong Yang, Jiguan Zhang, Fei Liu and Guo Chen

Symmetry 2026, 18(5), 856; https://doi.org/10.3390/sym18050856 (registering DOI) - 18 May 2026

Abstract

Near-bit multi-parameter MWD (measurement while drilling) is a key technology for achieving precise and efficient directional drilling in underground and tunnel engineering. The near-bit multi-parameter MWD method was studied, and a “center + side wall” distributed measurement scheme was proposed, based on an [...] Read more.

Near-bit multi-parameter MWD (measurement while drilling) is a key technology for achieving precise and efficient directional drilling in underground and tunnel engineering. The near-bit multi-parameter MWD method was studied, and a “center + side wall” distributed measurement scheme was proposed, based on an analysis of special application scenarios in underground and tunnel engineering. The transmission characteristics of Bluetooth wireless signals in water were investigated. An analysis of the underwater Bluetooth signal link was conducted. When the transmission distance is 100 mm, the received signal strength is −17.5 dBm, and the link margin is 69.5 dB. Wireless Bluetooth was used to transmit the near-bit data. A Bluetooth wireless communication simulation model was established using ANSYS software, and the influence of transmission power, transmission medium, and transmission distance on the Bluetooth signal strength was analyzed. The results indicate that: (1) the received signal strength increases with transmission power, and appropriately increasing the transmission power can improve the effect of Bluetooth wireless communication and extend the communication distance. (2) When the transmission medium is water, the received signal is unstable, and the echo loss curve shows a high and low oscillation form, presenting a frequency shift feature; when the transmission medium is air, the received signal is relatively stable, and the echo loss curve shows a parabolic form. The echo loss of Bluetooth wireless signal in water transmission is significantly higher than that in air transmission, indicating that the Bluetooth signal attenuates more rapidly when transmitted in water. (3) When the transmission distance increases near the optimal transmission frequency of 2.4 GHz, the echo loss increases accordingly, and the received signal strength of the wireless receiving module gradually decreases. The theoretical analysis, simulation, and indoor test results are in good agreement. The reasonable Bluetooth transmission power is 1 mW, and the transmission distance is 100 mm. After completing the overall scheme design and simulation analysis optimization, the structure, circuit, and program development were carried out, and the near-bit multi-parameter MWD device was developed. A laboratory water supply test was conducted, and the power supply, collection, and wireless transmission were all normal. A drilling test was carried out at an underground engineering of a coal mine in Wuhai City, achieving a drilling depth of 2328 m. A continuous and stable collection of various parameters such as WOB (weight on bit), torque, rotation speed, vibration, and gamma was carried out. A wireless transmission channel for near-bit data was established across the screw drilling tool. It can provide key technical support for the research and development of near-bit MWD in underground and tunnel engineering. Full article

(This article belongs to the Section Engineering and Materials)

37 pages, 1171 KB

Open AccessArticle

IID-DAKD: An Incremental Intrusion Detection Method for Encrypted Traffic Based on Dual Augmentation and Fusion Knowledge Distillation

by Liangchen Chen, Deyin Fu, Shu Gao, Shuo Zhang and Baoxu Liu

Symmetry 2026, 18(5), 855; https://doi.org/10.3390/sym18050855 (registering DOI) - 18 May 2026

Abstract

To address the pronounced degradation in detection accuracy of encrypted traffic intrusion models after incremental updates, which is primarily caused by catastrophic forgetting and task-level overfitting during incremental learning, this paper proposes a novel incremental intrusion detection method for encrypted traffic based on [...] Read more.

To address the pronounced degradation in detection accuracy of encrypted traffic intrusion models after incremental updates, which is primarily caused by catastrophic forgetting and task-level overfitting during incremental learning, this paper proposes a novel incremental intrusion detection method for encrypted traffic based on dual augmentation and fusion knowledge distillation, termed IID-DAKD. First, both known and previously unseen attacks identified during detection are leveraged to update a representative sample set. An encrypted traffic representative sample augmentation strategy based on Gaussian noise is then devised to reduce storage requirements and classifier bias, thereby effectively mitigating catastrophic forgetting. Second, a self-supervised learning framework driven by encrypted traffic class augmentation is constructed to alleviate representation bias and suppress task-level overfitting. Finally, three complementary knowledge distillation strategies are jointly employed to extract and transfer attack classification knowledge from the old model to the updated model, further improving detection accuracy and robustness while enhancing training efficiency. Extensive experimental results demonstrate that the proposed IID-DAKD approach alleviates catastrophic forgetting and task-level overfitting while maintaining symmetrical knowledge transfer during incremental learning, enabling efficient model updates and high detection accuracy for encrypted traffic intrusion detection. Full article

(This article belongs to the Section Computer)

17 pages, 25463 KB

Open AccessArticle

Wave-DIP: Unsupervised Image Decomposition Fusing Wavelet Multi-Scale Representation and Deep Image Prior

by Zirui Mao, Liwen Feng, Quanyou Xu and Yihang Liu

Symmetry 2026, 18(5), 854; https://doi.org/10.3390/sym18050854 (registering DOI) - 18 May 2026

Abstract

To address the issues of texture residuals and structural detail loss often encountered in traditional image decomposition methods, this paper proposes an unsupervised decomposition model that integrates the Wavelet Transform with Deep Image Prior (DIP). Leveraging the multi-scale and multi-directional characteristics of the [...] Read more.

To address the issues of texture residuals and structural detail loss often encountered in traditional image decomposition methods, this paper proposes an unsupervised decomposition model that integrates the Wavelet Transform with Deep Image Prior (DIP). Leveraging the multi-scale and multi-directional characteristics of the Wavelet Transform, the model carefully models the structural information of the cartoon component. Meanwhile, capitalizing on the unsupervised learning advantages of Deep Image Prior and incorporating low-rank constraints, it accurately extracts texture details. The model is solved via the Alternating Direction Method of Multipliers (ADMM). Experimental results on multiple test images demonstrate that, compared with existing methods, the proposed model achieves a more thorough separation of image structure and texture, yielding high-quality visual decomposition performance. Full article

(This article belongs to the Section Computer)

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23 pages, 2959 KB

Open AccessArticle

Block Cipher Generation Model: A Step Towards Generative Ciphers

by Muhammad Fahad Khan, Khalid Saleem, Ali Alshehri, Abdullah Aljuhni, Sarah Abu Ghazalah and Tehreem Sabir

Symmetry 2026, 18(5), 853; https://doi.org/10.3390/sym18050853 (registering DOI) - 18 May 2026

Abstract

The protection of confidential information is a worldwide challenge, and block ciphers are the most reliable option by which data security is accomplished. To the best of our knowledge, this type of research is being performed for the first time, unlocking new avenues [...] Read more.

The protection of confidential information is a worldwide challenge, and block ciphers are the most reliable option by which data security is accomplished. To the best of our knowledge, this type of research is being performed for the first time, unlocking new avenues for block cipher design research, shifting the paradigm from cryptographer-designed ciphers to computationally generated. We propose a computational model named Block Cipher Design Generation Model (BCDGM) to generate a complete design of novel block ciphers and their primitives, such as cipher structures, S-boxes, inverse S-boxes, P-boxes, round functions, half-round functions, and derived keys, in an automated manner without the participation of cryptographers. To accomplish this goal, BCDGM only requires high-quality quantum random bits as input to generate a myriad of new block ciphers. A quantum circuit is designed over the International Business Machines Corporation (IBM) Quantum Santiago computer to generate high-quality random bits. BCDGM itself and all its generated cipher structures and primitives are invariably transparent, unreproducible, and nondeterministic due to their sole reliance on quantum random bits. Furthermore, every decision made by BCDGM is randomized. As a result, potential vulnerabilities and attacks that exist in other ciphers are bypassed in BCDGM-generated ciphers. Extensive experimentation was conducted, generating more than fifty thousand new block cipher designs tested over 107 terabytes of data. Generated ciphers are compared with twelve reputable standard block ciphers, including five AES competition finalists. The results show that the proposed block ciphers occasionally surpass standard ciphers and achieve equivalent security strength in many cases. The implementation of the proposed model is publicly available. Full article

(This article belongs to the Special Issue New Advances in Symmetric Cryptography)

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12 pages, 2519 KB

Open AccessArticle

On Generalized Criterion for the Non-Isothermic Spacelike Surfaces

by Filiz Kanbay and Burcu Yüksekdağ

Symmetry 2026, 18(5), 852; https://doi.org/10.3390/sym18050852 (registering DOI) - 17 May 2026

Abstract

In this work, we examine the generalized criterion to determine a non-isothermic spacelike Bonnet surface in three-dimensional Lorentzian space

L^{3}

. For this aim, we generalized the criterion presented by Soyuçok in the Euclidean space to

L^{3}

. By applying the [...] Read more.

In this work, we examine the generalized criterion to determine a non-isothermic spacelike Bonnet surface in three-dimensional Lorentzian space

L^{3}

. For this aim, we generalized the criterion presented by Soyuçok in the Euclidean space to

L^{3}

. By applying the generalized criterion, we identify the non-isothermic spacelike Bonnet surfaces among the helicoid surfaces classified by Beneki et al. Full article

(This article belongs to the Section Mathematics)

33 pages, 6642 KB

Open AccessArticle

An Adaptive MPA-RUN Framework for Multilevel Thresholding of Multispectral Satellite Images

by Ataberk Köşger, Arda Güneş, Enes Altındirek, İsmail Buğra Kuru and Muhammed Faruk Şahin

Symmetry 2026, 18(5), 851; https://doi.org/10.3390/sym18050851 (registering DOI) - 17 May 2026

Abstract

Multispectral satellite image segmentation constitutes a challenging optimization problem due to high dimensionality and complex inter-band correlation structures. As the number of thresholds increases, the search space grows exponentially, causing metaheuristic methods to suffer from convergence instability by getting trapped in local optima [...] Read more.

Multispectral satellite image segmentation constitutes a challenging optimization problem due to high dimensionality and complex inter-band correlation structures. As the number of thresholds increases, the search space grows exponentially, causing metaheuristic methods to suffer from convergence instability by getting trapped in local optima on highly multimodal landscapes. In this study, a hybrid optimization method is proposed by integrating the Marine Predators Algorithm (MPA) with the Runge–Kutta (RUN) approach. The proposed framework enhances global exploration through Cauchy-based perturbation, while improving exploitation capability via a mutation-based local refinement mechanism, and reduces spectral redundancy using Principal Component Analysis (PCA). The MPA-RUN hybrid structure, combined with a Cauchy-driven exploration strategy and an adaptive local search mechanism, significantly improves the exploration–exploitation balance in multispectral image thresholding problems. Experiments are conducted on Sentinel-2 multispectral images, and the proposed method is evaluated against conventional metaheuristic algorithms over a wide threshold range (2–26), encompassing both low- and high-dimensional configurations. At high threshold levels, the proposed method achieves Peak Signal-to-Noise Ratio (PSNR) = 23.66, Structural Similarity Index Measure (SSIM) = 0.863, and Feature Similarity Index Measure (FSIM) = 0.797, while providing approximately 35% lower computational time at moderate levels, demonstrating superior efficiency. These results demonstrate that a balanced trade-off between accuracy and computational cost is achieved. The proposed approach offers a fast and reliable solution for processing high-dimensional data by effectively balancing segmentation quality and computational complexity. Full article

(This article belongs to the Special Issue Symmetry Applied in Remote Sensing Technology)

29 pages, 1315 KB

Open AccessArticle

An Improved Hydro-Mechanical Coupling Shear Creep Model for Fully Persistent Rock Joints

by Hantao Xu, Yuhang Chen, Jiapeng Li, Haojie Wang and Qun Sui

Symmetry 2026, 18(5), 850; https://doi.org/10.3390/sym18050850 (registering DOI) - 17 May 2026

Abstract

The model is based on the periodic translational symmetry of regular saw-toothed joint surfaces and reveals the time-dependent breaking of this symmetry under hydro-mechanical coupling through the introduction of damage evolution. Traditional creep models typically rely on static constants, which fail to capture [...] Read more.

The model is based on the periodic translational symmetry of regular saw-toothed joint surfaces and reveals the time-dependent breaking of this symmetry under hydro-mechanical coupling through the introduction of damage evolution. Traditional creep models typically rely on static constants, which fail to capture the nonlinear, time-dependent degradation of rock under complex conditions. To address this, this paper proposes a novel nonlinear shear creep model for regular saw-toothed joint surfaces under hydro-mechanical coupling. First, a calculation method for effective shear stress is established, accounting for normal stress, asperity height, and water pressure. Next, traditional static parameters are transformed into dynamic variables to accurately model the primary and steady-state creep stages. Finally, a plastic damage element is introduced to simulate the accelerated creep stage, revealing that damage accumulates with time and is exacerbated by higher seepage pressure. By integrating early-stage viscoelastic and late-stage viscoplastic characteristics, this model captures the complete nonlinear shear creep process, providing a robust theoretical basis for long-term stability evaluations. Full article

(This article belongs to the Section Engineering and Materials)

40 pages, 1667 KB

Open AccessArticle

Fast Feature Selection in Interval Data Using Rough Sets with Fuzzy Tolerance Relation-Based Hierarchical Approximations

by Nan Zhang, Jinming Gu, Yuanzhao Gong and Heqing Kong

Symmetry 2026, 18(5), 849; https://doi.org/10.3390/sym18050849 (registering DOI) - 16 May 2026

Abstract

As a highly active research field, feature selection plays a critical role in multi-criteria decision making, hierarchical data analysis, and artificial intelligence. Rough set (RS) theory, presented by Pawlak, is a useful and important methodology for feature selection. Discernibility matrix-based and heuristic methods [...] Read more.

As a highly active research field, feature selection plays a critical role in multi-criteria decision making, hierarchical data analysis, and artificial intelligence. Rough set (RS) theory, presented by Pawlak, is a useful and important methodology for feature selection. Discernibility matrix-based and heuristic methods are two important rough set approaches for feature selection in interval-valued data. However, they still face significant computational challenges when dealing with high-dimensional and large-scale interval-valued data. To improve the efficiency of feature selection in an interval-valued decision system (IvDS), we construct a hierarchical approximation model and establish the corresponding theoretical results based on the fuzzy tolerance relation, which is symmetric and reflexive. Based on this model, we explore the order-preservation property of attribute significance in an IvDS and achieve attribute reduction on a gradually decreasing universe, which improves the efficiency of feature selection. Motivated by this idea, we develop two fast algorithms, named FFSDF and FFSCE. The proposed algorithms can achieve higher efficiency while preserving the same reduction results as the original feature selection algorithms. Finally, experiments are conducted on fifteen datasets to demonstrate that the proposed fast algorithms are more effective and efficient. Full article

(This article belongs to the Section Mathematics)

46 pages, 4599 KB

Open AccessArticle

Multi-Strategy Enhanced Beaver Behavior Optimizer for Global Optimization and Enterprise Bankruptcy Prediction

by Haoyuan He and Mingyang Yu

Symmetry 2026, 18(5), 848; https://doi.org/10.3390/sym18050848 (registering DOI) - 15 May 2026

Abstract

Enterprise bankruptcy prediction is a critical research issue in financial risk early warning, credit evaluation, and investment decision-making. To address the limitations of traditional methods in handling high-dimensional, nonlinear, and complex financial data, including parameter sensitivity, susceptibility to local optima, and insufficient prediction [...] Read more.

Enterprise bankruptcy prediction is a critical research issue in financial risk early warning, credit evaluation, and investment decision-making. To address the limitations of traditional methods in handling high-dimensional, nonlinear, and complex financial data, including parameter sensitivity, susceptibility to local optima, and insufficient prediction stability, this study proposes a multi-strategy enhanced Beaver Behavior Optimizer and applies it to optimize kernel extreme learning machines, constructing the MEBBO KELM prediction model. Three improvement mechanisms are introduced, including an elite pool enhanced exploration strategy, a stochastic centroid reverse learning strategy, and a leader guided boundary control strategy, which improve population diversity, global search capability, boundary handling capacity, and convergence accuracy. The proposed algorithm is evaluated on CEC2017 and CEC2022 benchmark datasets and compared with EWOA, HPHHO, MELGWO, TACPSO, CFOA, ALA, AOO, RIME, and BBO. Statistical analyses are conducted using the Wilcoxon rank sum test and the Friedman test. The results demonstrate that MEBBO achieves superior solution accuracy and stability, indicating strong global optimization capability and robustness. Further experiments on the Wieslaw Corporate Bankruptcy Dataset show that MEBBO-KELM achieves strong and robust performance across multiple evaluation metrics, including ACC, MCC, Sensitivity, Specificity, Precision, Recall, and F1 score. Specifically, ACC reaches 79.7578, MCC reaches 0.6050, and F1 score reaches 78.8504, confirming its effectiveness. Full article

(This article belongs to the Special Issue Symmetry and Metaheuristic Algorithms)

17 pages, 998 KB

Open AccessArticle

Symmetry-Aware Vehicle State Estimation Using a Chaotic-Gradient-Optimized Extended Kalman Filter

by Qianyu Cheng, Wenguang Liu, Xi Liu, Huajun Che and Bei Ding

Symmetry 2026, 18(5), 847; https://doi.org/10.3390/sym18050847 (registering DOI) - 15 May 2026

Abstract

To address the uncertainty of the measurement noise covariance matrix in vehicle state estimation, this paper proposes a symmetry-aware extended Kalman filter optimized by a chaotic-gradient strategy. The symmetry-aware concept is introduced from the approximate mirror symmetry of vehicle lateral dynamics under left [...] Read more.

To address the uncertainty of the measurement noise covariance matrix in vehicle state estimation, this paper proposes a symmetry-aware extended Kalman filter optimized by a chaotic-gradient strategy. The symmetry-aware concept is introduced from the approximate mirror symmetry of vehicle lateral dynamics under left and right steering excitations. Under identical road adhesion and vehicle operating conditions, the yaw-rate and sideslip-angle responses should exhibit balanced statistical characteristics for positive and negative lateral motions. However, a fixed measurement noise covariance matrix may break this balance and lead to direction-dependent estimation bias or delayed convergence. To improve the statistical consistency of the estimation process, the proposed method adaptively tunes the measurement noise covariance matrix according to the innovation covariance mismatch. A chaotic search mechanism is first used to enhance global exploration, and a variable-step gradient method is then applied to refine the local optimal solution. Through the iterative combination of chaotic traversal and gradient-based refinement, the proposed observer improves the balance between model prediction and measurement correction under stochastic disturbances. The effectiveness of the proposed method is verified through CarSim and MATLAB/Simulink co-simulation. The results show that, compared with EKF, UKF, and AEKF benchmark observers, the proposed CG_EKF provides more accurate estimation of vehicle yaw rate and sideslip angle. Full article

(This article belongs to the Section Engineering and Materials)

33 pages, 2272 KB

Open AccessArticle

Statistical Inference of Stress–Strength Reliability for Multi-State System Based on Exponentiated Pareto Distribution Using Generalized Survival Signature

by Jiaojiao Guo, Jialin Su, Jianhui Li and Tian Guo

Symmetry 2026, 18(5), 846; https://doi.org/10.3390/sym18050846 (registering DOI) - 15 May 2026

Abstract

The stress–strength reliability model is widely applied in various fields such as mechanical engineering, materials science, and aerospace engineering to identify weak links in systems and thereby improve system reliability. This paper analyzes the stress–strength reliability for multi-state systems composed of multi-state components. [...] Read more.

The stress–strength reliability model is widely applied in various fields such as mechanical engineering, materials science, and aerospace engineering to identify weak links in systems and thereby improve system reliability. This paper analyzes the stress–strength reliability for multi-state systems composed of multi-state components. One of the main contributions is the derivation of a multi-state stress–strength reliability model under combined stresses based on the generalized survival signature theory. In the model analysis, it is assumed that each component of the system is subjected to two different stresses corresponding to two different strengths, and that the stress variables and strength variables are mutually independent and all follow the exponentiated Pareto distribution with the common second shape parameter. Another contribution is the use of maximum likelihood estimation, empirical Bayesian estimation, and weakly informative Bayesian estimation to estimate the variable parameters and the stress–strength reliability under the progressive first-failure censoring scheme. In addition, the asymptotic confidence intervals for the stress–strength reliability model are derived, and the Bayesian credible intervals are constructed based on MCMC sampling. Finally, through MCMC simulation of a three-state consecutive 3-out-of-5: G system, the accuracy of the variable parameters and the stress–strength reliability under the aforementioned point estimation and interval estimation methods is analyzed, and the performance of these estimation methods is compared under different sample sizes. In addition, sensitivity analyses were conducted on the common shape parameter and the hyperparameters of the weakly informative prior distributions. Furthermore, a real data set is applied to illustrate the proposed procedures. Full article

(This article belongs to the Section Mathematics)

20 pages, 1815 KB

Open AccessArticle

Directional Anisotropy of Admissible Tilt in Rectangular Sinking Wells: A Closed-Form Analytical Model

by Dawid Karasiewicz, Tomasz Garbowski and Anna Szymczak-Graczyk

Symmetry 2026, 18(5), 845; https://doi.org/10.3390/sym18050845 (registering DOI) - 15 May 2026

Abstract

Rectangular sinking wells are widely used in underground and hydraulic engineering, where maintaining vertical alignment during construction is essential for safe serviceability and long-term performance. Even moderate inclination may cause eccentric transfer of the vertical load to the concrete plug, leading to non-uniform [...] Read more.

Rectangular sinking wells are widely used in underground and hydraulic engineering, where maintaining vertical alignment during construction is essential for safe serviceability and long-term performance. Even moderate inclination may cause eccentric transfer of the vertical load to the concrete plug, leading to non-uniform contact stresses beneath the base. This study presents a closed-form analytical framework for assessing the admissible tilt of rectangular sinking wells based on stress redistribution under one-axis inclination. The well is modeled as a rigid body, and the resulting load eccentricity is related to the shaft height and tilt angle. Classical eccentric compression theory is then used to derive explicit expressions for the maximum and minimum base stresses, from which a serviceability-based admissibility criterion is formulated. The obtained solution provides the allowable inclination directly as a function of shaft height, plan dimensions, and an adopted stress amplification factor. Parametric analyses show that the admissible tilt decreases with increasing shaft height and increases with larger base dimensions. A distinct directional anisotropy is observed for rectangular plans, whereas square wells recover symmetric behavior with identical limits in both orthogonal directions. This identifies square geometry as the symmetry-preserving limit case of the proposed model. Sensitivity analyses further demonstrate the influence of admissible stress amplification on permissible inclination levels. The proposed formulation offers a transparent screening tool for construction monitoring and post-construction assessment, while also illustrating how geometric symmetry reduction governs the mechanical response of inclined foundation systems. Full article

(This article belongs to the Special Issue Symmetry and Asymmetry in Structural Engineering)

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42 pages, 2468 KB

Open AccessArticle

ARQ2: Toward Stability-Aware Hybrid Optimization on Complex and Noisy Search Problems

by Vasileios Charilogis, Ioannis G. Tsoulos and Anna Maria Gianni

Symmetry 2026, 18(5), 844; https://doi.org/10.3390/sym18050844 (registering DOI) - 15 May 2026

Abstract

ARQ2 is introduced as a next-generation extension of the ARQ (Archive-guided Roulette-based with Quarantine) optimization framework, which was originally established as a cohesive strategy for achieving stable performance on difficult and noisy search landscapes. While rooted in the core principles of the original [...] Read more.

ARQ2 is introduced as a next-generation extension of the ARQ (Archive-guided Roulette-based with Quarantine) optimization framework, which was originally established as a cohesive strategy for achieving stable performance on difficult and noisy search landscapes. While rooted in the core principles of the original design, ARQ2 advances this foundation through a more refined integration of adaptive search, archive-guided exploration, controlled replacement, and robustness-aware population management. In doing so, it moves beyond the level of a simple algorithmic modification and emerges as a distinct methodological development within hybrid continuous optimization. Its significance lies in shaping a more mature and experimentally substantiated variant that promotes dependable behavior, consistent search quality, and balanced exploration–exploitation dynamics across diverse optimization environments. Empirical evaluation against nine established optimizers on 36 continuous optimization problems yields an overall average rank of 1.958, with 25 rank-1 placements and a mean-value improvement over ARQ on 21 out of 36 problems, confirming the stronger robustness and repeated-run reliability of the proposed design. As such, ARQ2 contributes to the ongoing development of stability-oriented optimization methodologies and reinforces the scientific relevance of this design line in the contemporary literature. Full article

(This article belongs to the Special Issue Symmetry in Stochastic Models for Machine Learning Applications: Theoretical Insights and Applications)

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