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Foldable/Deployable Spherical Mechanisms Based on Regular Polygons
Bi-Symmetric Polyhedral Cages with Three, Four, Five or Six Connected Faces and Small Holes
Emergence and Localization of Exceptional Points in an Exactly Solvable Toy Model
Machine Learning-Driven Conservative-to-Primitive Conversion in Hybrid Piecewise Polytropic and Tabulated Equations of State
Transition of Natural Convection in Liquid Metal Within an Annular Enclosure with Various Angular Partitions

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, and 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.
Journal Cluster of Mathematics and Its Applications: AppliedMath, Axioms, Computation, Fractal and Fractional, Geometry, International Journal of Topology, Logics, Mathematics and Symmetry.

Impact Factor: 2.2 (2024); 5-Year Impact Factor: 2.1 (2024)

Imprint Information Journal Flyer Open Access ISSN: 2073-8994

Latest Articles

30 pages, 1592 KB

Open AccessArticle

Hybrid Mathematical Modeling and Optimization Framework for Branch Flow Estimation at Y-Intersections: A Constraint- Aware Approach with Minimal Sensing Requirements

by Mindong Liu, Jiahao Hu, Chenhao Wu, Qiuquan Sun and Xiaojie Zhao

Symmetry 2025, 17(12), 2052; https://doi.org/10.3390/sym17122052 (registering DOI) - 1 Dec 2025

Accurate estimation of branch-level traffic flows at urban Y-intersections from limited mainline measurements remains a critical challenge in intelligent transportation systems. Y-intersections, with their symmetric geometric configuration where multiple branches converge, pose unique challenges from flow coupling, signal-induced periodicity, and merging delays. This [...] Read more.

Accurate estimation of branch-level traffic flows at urban Y-intersections from limited mainline measurements remains a critical challenge in intelligent transportation systems. Y-intersections, with their symmetric geometric configuration where multiple branches converge, pose unique challenges from flow coupling, signal-induced periodicity, and merging delays. This study develops a hybrid mathematical modeling framework that integrates piecewise linear segments with periodic components for each branch flow. The model enforces physical constraints including flow conservation, non-negativity, and segment continuity, while incorporating operational features such as signal timing and merging delays. Parameter estimation employs a two-stage optimization approach combining least-squares fitting with constrained nonlinear programming, utilizing sparse mainline detector data and minimal historical priors. Experimental validation across five progressive problem formulations demonstrates robust performance, achieving RMSE values of 3.3432 and 5.4467 for complex multi-branch scenarios while accurately capturing 10-min green/8-min red signal cycles and 2-min merging delays. The method successfully reconstructs branch flow profiles at required time points (07:30 and 08:30), reducing observation requirements by 60–80% while maintaining estimation accuracy. The proposed framework provides a practical and interpretable solution for branch flow estimation under sparse sensing conditions, bridging physics-based modeling with data-driven techniques and offering transportation agencies a deployable tool for intersection monitoring without extensive instrumentation. Full article

(This article belongs to the Section Engineering and Materials)

28 pages, 345 KB

Open AccessArticle

m-Polar Picture Fuzzy Bi-Ideals and Their Applications in Semigroups

by Warud Nakkhasen, Atthchai Chada and Teerapan Jodnok

Symmetry 2025, 17(12), 2051; https://doi.org/10.3390/sym17122051 (registering DOI) - 1 Dec 2025

The concept of symmetry is fundamental to the study of algebra; it serves as the basis for a branch of group theory that is essential to abstract algebra. A semigroup is a structure that builds upon the concept of a group, similarly extending [...] Read more.

The concept of symmetry is fundamental to the study of algebra; it serves as the basis for a branch of group theory that is essential to abstract algebra. A semigroup is a structure that builds upon the concept of a group, similarly extending the idea of symmetry found within groups. In this study, we specifically focus on semigroups. The main objective of this research is to apply the notion of m-polar picture fuzzy sets (m-PPFSs), with m being a natural number, in investigations into semigroups, as this concept generalizes m-polar fuzzy sets (m-PFSs) and picture fuzzy sets (PFSs). This research introduces the concepts of m-polar picture fuzzy left ideals (m-PPFLs), m-polar picture fuzzy right ideals (m-PPFRs), m-polar picture fuzzy ideals (m-PPFIs), m-polar picture fuzzy bi-ideals (m-PPFBs), and m-polar picture fuzzy generalized bi-ideals (m-PPFGBs) in semigroups. This study examines the relationships between these concepts, showing that every m-PPFL (m-PPFR) in the semigroups is also an m-PPFB, and that every m-PPFB in the semigroups is an m-PPFGB. However, the opposite is not true. Additionally, we provide the characteristics of the m-PPFLs, m-PPFRs, m-PPFIs, m-PPFBs, and m-PPFGBs in semigroups. We further discuss the connections between the m-PPFLs (m-PPFIs) and the m-PPFBs within the framework of regular semigroups, and most importantly, we show that, if the semigroup is regular, then the m-PPFBs and m-PPFGBs are equal. Finally, we utilize the properties of the m-PPFLs, m-PPFRs, m-PPFIs, m-PPFBs, and m-PPFGBs within semigroups to explore the classifications of regular semigroups. Full article

(This article belongs to the Special Issue Research on Fuzzy Logic and Mathematics with Applications, 3rd Edition)

14 pages, 335 KB

Open AccessArticle

Degenerate Appell Polynomials Connecting Beta Function as a Family of Operators and Their Approximations

by Mohammad Farid, Harun Çiçek, Mohammad Ayman-Mursaleen and Nadeem Rao

Symmetry 2025, 17(12), 2050; https://doi.org/10.3390/sym17122050 (registering DOI) - 1 Dec 2025

This paper introduces a novel family of positive linear operators constructed by blending degenerate Appell polynomials with a classical Beta kernel in the Durrmeyer setting. The operators are defined as [...] Read more.

This paper introduces a novel family of positive linear operators constructed by blending degenerate Appell polynomials with a classical Beta kernel in the Durrmeyer setting. The operators are defined as

H_{n} (g; u) = \sum_{ƿ = 0}^{\infty} h_{ƿ} (n + u; ƛ) \int_{0}^{1} K_{n} (ƿ, ʈ) g (ʈ) d ʈ

, where

h_{ƿ} (n u; ƛ)

is derived from degenerate Appell polynomials (

n u

denotes the product of n and u) and

K_{n} (ƿ, ʈ)

is a Beta-type kernel. We establish the linearity and positivity of these operators and derive crucial moment estimates. Approximation properties are examined via Korovkin-type theorems, and the asymptotic behavior is investigated through a Voronovskaja-type theorem. The results extend and unify earlier work on Appell-based approximation operators and offer new tools for approximating functions in weighted spaces. Numerical examples and error estimates are provided to illustrate the efficacy of the proposed operators. In addition, the inherent symmetry in the structure of the proposed operators-arising from the symmetric nature of the Beta kernel and the generating functions of degenerate Appell polynomials is discussed. Such symmetry plays a key role in ensuring balanced approximation and convergence characteristics. Full article

(This article belongs to the Special Issue Symmetries and Fractional Differential Equations: Theory and Applications)

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25 pages, 1934 KB

Open AccessArticle

A Tripartite Analytical Framework for Nonlinear (1+1)-Dimensional Field Equations: Painlevé Analysis, Classical Symmetry Reduction, and Exact Soliton Solutions

by Muhammad Uzair, Aljethi Reem Abdullah and Irfan Mahmood

Symmetry 2025, 17(12), 2049; https://doi.org/10.3390/sym17122049 (registering DOI) - 1 Dec 2025

This study presents a tripartite analytical framework for the (1+1)-dimensional nonlinear Klein–Fock–Gordon equation, a key model for spinless particles in relativistic quantum mechanics. The investigation begins with a Painlevé analysis showing that the equation is completely integrable via the Painlevé test by using [...] Read more.

This study presents a tripartite analytical framework for the (1+1)-dimensional nonlinear Klein–Fock–Gordon equation, a key model for spinless particles in relativistic quantum mechanics. The investigation begins with a Painlevé analysis showing that the equation is completely integrable via the Painlevé test by using Maple. Subsequently, classical Lie symmetry analysis is employed to derive the infinitesimal generators of the equation. A Lagrangian formulation is constructed for these generators, from which similarity variables are systematically obtained. This framework enables a complete similarity reduction, transforming the complex nonlinear partial differential equation into a more tractable ordinary differential equation. To solve this reduced ordinary differential equation and to obtain a spectrum of soliton solutions, we implement the new generalized exponential differential rational function method. This advanced technique utilizes a rational trial function based on the

i t h

derivatives of exponentials, generating a diverse spectrum of closed-form soliton solutions through strategic choices of arbitrary constants. The novelty of this approach provides a unified framework for handling higher-order nonlinearities, yielding solutions such as multi-peakons and lump solitons, which are vividly characterized using Mathematica-generated 3D, 2D, and contour plots. These findings provide significant insights into nonlinear wave dynamics with potential applications in quantum field theory, nonlinear optics, plasma physics, etc. Full article

(This article belongs to the Special Issue Symmetry in Integrable Systems and Soliton Theories)

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40 pages, 4276 KB

Open AccessArticle

UAV-Based Hybrid Fuzzy Inference Framework for Symmetry and Asymmetry in Real-Time Air Quality Monitoring

by Svetlana Beryozkina and Inga Zicmane

Symmetry 2025, 17(12), 2048; https://doi.org/10.3390/sym17122048 (registering DOI) - 1 Dec 2025

This study presents a UAV-based fuzzy inference framework for real-time air quality monitoring that integrates symmetric and asymmetric fuzzy rules. Symmetric rules capture baseline pollutant dynamics, ensuring computational stability, while asymmetric rules account for local anomalies, turbulence, and environmental disturbances, effectively regularizing the [...] Read more.

This study presents a UAV-based fuzzy inference framework for real-time air quality monitoring that integrates symmetric and asymmetric fuzzy rules. Symmetric rules capture baseline pollutant dynamics, ensuring computational stability, while asymmetric rules account for local anomalies, turbulence, and environmental disturbances, effectively regularizing the inherently ill-posed backward problem of reconstructing pollutant concentrations from noisy UAV measurements. Simulation and field experiments demonstrate that this hybrid fuzzy approach provides both mathematical robustness and practical reliability, outperforming purely symmetric models in dynamic, asymmetric environments. The proposed framework offers a generalizable methodology for environmental monitoring, emphasizing the critical role of symmetry and its breaking in modeling real-world ecological processes. Future developments will focus on atmospheric dispersion integration, fuzzy rule optimization, and large-scale UAV deployment. The results indicate that the hybrid fuzzy inference system can enhance the accuracy and reliability of UAV-based air quality monitoring under real-world disturbances, providing a robust framework applicable for urban planning, environmental policy, and large-scale deployment scenarios. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry in Fuzzy Control)

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24 pages, 2143 KB

Open AccessArticle

Symmetry-Aided Active RIS for Physical Layer Security in WSN-Integrated Cognitive Radio Networks: Green Interference Regulation and Joint Beamforming Optimization

by Yixuan Wu

Symmetry 2025, 17(12), 2047; https://doi.org/10.3390/sym17122047 (registering DOI) - 1 Dec 2025

Driven by 5G/6G and the Internet of Things (IoT), wireless sensor networks (WSNs) are confronted with core challenges such as limited energy constraints, unbalanced resource allocation, and security vulnerabilities. To address these, WSNs are integrated with cognitive radio networks (CRNs) to alleviate spectrum [...] Read more.

Driven by 5G/6G and the Internet of Things (IoT), wireless sensor networks (WSNs) are confronted with core challenges such as limited energy constraints, unbalanced resource allocation, and security vulnerabilities. To address these, WSNs are integrated with cognitive radio networks (CRNs) to alleviate spectrum scarcity, and reconfigurable intelligent surfaces (RIS) are adopted to enhance performance, but traditional passive RIS suffers from “double fading” (signal path loss from transmitter to RIS and RIS to receiver), which undermines WSNs’ energy efficiency and the physical layer security (PLS) (e.g., secrecy rate, SR) of primary users (PUs) in CRNs. This study leverages symmetry to develop an active RIS framework for WSN-integrated CRNs, constructing a tripartite collaborative model where symmetric beamforming and resource allocation improve WSN connectivity, reduce energy consumption, and strengthen PLS. Specifically, three symmetry types—resource allocation symmetry, beamforming structure symmetry, and RIS reflection matrix symmetry—are formalized mathematically. These symmetries reduce the degrees of freedom in optimization (e.g., cutting precoding complexity by ~50%) and enhance the directionality of green interference, while ensuring balanced resource use for WSN nodes. The core objective is to minimize total transmit power while satisfying constraints of PU SR, secondary user (SU) quality-of-service (QoS), and PU interference temperature, achieved by converting non-convex SR constraints into solvable second-order cone (SOC) forms and using an alternating optimization algorithm to iteratively refine CBS/PBS precoding matrices and active RIS reflection matrices, with active RIS generating directional “green interference” to suppress eavesdroppers without artificial noise, avoiding redundant energy use. Simulations validate its adaptability to WSN scenarios: 50% lower transmit power than RIS-free schemes (with four CBS antennas), 37.5–40% power savings as active RIS elements increase to 60, and a 40% lower power growth slope in multi-user WSN scenarios, providing a symmetry-aided, low-power solution for secure and efficient WSN-integrated CRNs to advance intelligent WSNs. Full article

(This article belongs to the Special Issue Symmetry and Asymmetry in Wireless Sensor Networks)

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17 pages, 1697 KB

Open AccessArticle

Football-YOLO: A Lightweight and Symmetry-Aware Football Detection Model with an Enlarged Receptive Field

by Jingjing Zhou, Hongyang Liu, Gang Zhao and Ying Gao

Symmetry 2025, 17(12), 2046; https://doi.org/10.3390/sym17122046 (registering DOI) - 1 Dec 2025

In modern elite football, accurate ball localization is increasingly vital for smooth match flow and reliable officiating. Yet mainstream detectors still struggle with small objects like footballs in cluttered scenes due to limited receptive fields, weak feature representations, and non-trivial computational cost. To [...] Read more.

In modern elite football, accurate ball localization is increasingly vital for smooth match flow and reliable officiating. Yet mainstream detectors still struggle with small objects like footballs in cluttered scenes due to limited receptive fields, weak feature representations, and non-trivial computational cost. To address these issues and introduce structural symmetry, we propose a lightweight framework that balances model complexity and representational completeness. Concretely, we design a Dynamic clustering C3k2 module (DcC3k2) to enlarge the effective receptive field and preserve local–global symmetry and a SegNeXt-based noise-attentive C3k2 module (SNAC3k2) to perform multi-scale suppression of background interference. For efficient feature extraction, we adopt GhostNetV2—a lightweight convolutional backbone—thereby maintaining computational symmetry and speed. Experiments on a Football dataset show that our approach improves mAP by 3.4% over strong baselines while reducing computation by 2.2%. These results validate symmetry-aware lightweight design as a promising direction for high-precision small-object detection in football analytics. Full article

(This article belongs to the Special Issue Computer Vision, Pattern Recognition, Machine Learning, and Symmetry, 2nd Edition)

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9 pages, 347 KB

Open AccessArticle

Effects of Sports Flooring on Peak Ground Reaction Forces During Bilateral Drop Landings

by Nicholas A. Busuttil, Marcus Dunn, Alexandra H. Roberts, Anthony J. Viglione and Kane J. Middleton

Symmetry 2025, 17(12), 2045; https://doi.org/10.3390/sym17122045 (registering DOI) - 1 Dec 2025

With continued advancement in flooring technology, modular sports flooring tiles have emerged as a potential alternative flooring solution for sports performance. However, there is limited empirical evidence regarding their effects on ground reaction forces in landing tasks (GRFs). Therefore, the aim of this [...] Read more.

With continued advancement in flooring technology, modular sports flooring tiles have emerged as a potential alternative flooring solution for sports performance. However, there is limited empirical evidence regarding their effects on ground reaction forces in landing tasks (GRFs). Therefore, the aim of this study was to assess the effects of flooring surface on peak GRFs during bilateral drop landings. Eighteen physically active adults (10 males, 8 females) completed three bilateral drop landings from a 50 cm height across each of three flooring types: modular sport tiles, athletic track, and bare force plates, measuring contacts from both the left and right limb. GRFs were captured using two in-ground force platforms, normalised to body mass, and then analysed using a linear mixed-effects model with post-hoc comparisons where significant interactions were recorded. Peak anterior GRFs were significantly lower in the modular tiles compared with athletic sports track and bare metal surfaces (p < 0.001, η²_p ≥ 0.430). Additionally, anterior (p = 0.048, η²_p = 0.040), lateral (p < 0.001, η²_p = 0.280), and vertical (p = 0.001, η²_p = 0.100) GRFs were significantly greater in the right limb compared with the left limb within each flooring surface condition. Future research should investigate sport-specific movement patterns and long-term adaptations associated with training on modular surfaces to assess their potential role in enhancing performance and mitigating injuries. Full article

(This article belongs to the Special Issue Symmetry Application in Motor Control in Sports and Rehabilitation)

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16 pages, 295 KB

Open AccessArticle

The Power Spectrum of Compressible Turbulence: A Story in Symmetries

by Olivier Coquand

Symmetry 2025, 17(12), 2044; https://doi.org/10.3390/sym17122044 (registering DOI) - 1 Dec 2025

This article presents a theoretical study of the scaling properties of the kinetic energy spectrum in compressible turbulence. From the fundamental symmetries and linear transformations of the microscopic action, we derive exact relations between the correlation functions and their generators. These relations put [...] Read more.

This article presents a theoretical study of the scaling properties of the kinetic energy spectrum in compressible turbulence. From the fundamental symmetries and linear transformations of the microscopic action, we derive exact relations between the correlation functions and their generators. These relations put strong constraints on the possible scaling relations in the system as a function of scale. One of the main results of this study is that the action can be split between an incompressible part, which is the same as the usual stochastic Navier–Stokes theory in whatever the value of the Mach number is, and a longitudinal part, whose behavior is to be compared to the three-dimensional Burgers equation, which presents a much richer phase diagram as its usually discussed one-dimensional counterpart. Full article

(This article belongs to the Special Issue Symmetries in Applied and Theoretical Physics Selected Papers from the “12th BPU Congress”)

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20 pages, 359 KB

Open AccessArticle

The Spacetime Geodesy of Perfect Fluid Spheres

by Christopher Simmonds and Matt Visser

Symmetry 2025, 17(12), 2043; https://doi.org/10.3390/sym17122043 (registering DOI) - 1 Dec 2025

Herein we shall argue for the utility of “spacetime geodesy”, a point of view where one delays as long as possible worrying about dynamical equations, in favour of the maximal utilization of both symmetries and geometrical features. This closely parallels Weinberg’s distinction between [...] Read more.

Herein we shall argue for the utility of “spacetime geodesy”, a point of view where one delays as long as possible worrying about dynamical equations, in favour of the maximal utilization of both symmetries and geometrical features. This closely parallels Weinberg’s distinction between “cosmography” and “cosmology”, wherein maximal utilization of both the symmetries and geometrical features of Friedmann–Lemaître–Robertson–Walker (FLRW) spacetimes is emphasized. This “spacetime geodesy” point of view is particularly useful in those situations where, for one reason or another, the dynamical equations of motion are either uncertain or completely unknown. Several specific examples are discussed—we shall illustrate what can be done by considering the physics implications of demanding spatially isotropic Ricci tensors as a way of automatically implementing the (isotropic) perfect fluid condition, without committing to a specific equation of state. We also consider the structure of the Weyl tensor in spherical symmetry, with and without the (isotropic) perfect fluid condition, and relate this to the notion of “complexity”. In closing, we indicate some ways in which these considerations might be further generalized to more physically complicated (and technically very much more complicated) situations such as axisymmetric spacetimes. Full article

(This article belongs to the Section Physics)

17 pages, 1482 KB

Open AccessArticle

Integrated Adaptive Robust Control for Pipe Robots Facing Compound Uncertainties

by Ke Han, Dong Zhang, Xiangli Tao, Yilong Li, Haonan Li, Haiwei Wang and Xuefei Liu

Symmetry 2025, 17(12), 2042; https://doi.org/10.3390/sym17122042 (registering DOI) - 1 Dec 2025

To address the challenges of low control accuracy and poor robustness for pipe robots operating in complex environments with compound uncertainties, including parameter variations, nonlinear friction, and external disturbances, this paper proposes an integrated adaptive robust control strategy. The framework begins by establishing [...] Read more.

To address the challenges of low control accuracy and poor robustness for pipe robots operating in complex environments with compound uncertainties, including parameter variations, nonlinear friction, and external disturbances, this paper proposes an integrated adaptive robust control strategy. The framework begins by establishing a high-fidelity dynamic model that considers arbitrary pipe inclination. Subsequently, a Finite-Time Disturbance Observer (FTDO) is designed to rapidly and accurately estimate and compensate for the lumped uncertainties, thereby enhancing the system’s disturbance rejection capabilities. Based on this compensation, an Adaptive Non-singular Terminal Sliding Mode Control (ANTSMC) law is synthesized. By adjusting its gain online, the controller handles unknown uncertainty bounds, ensures the finite-time convergence of tracking errors, and effectively suppresses chattering. Simulation results validate that the proposed method achieves high-precision trajectory tracking with smooth control inputs and strong robustness, even in the presence of significant uncertainties. This study offers an effective solution for the high-precision and robust motion control of pipe robots. Full article

(This article belongs to the Section Engineering and Materials)

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23 pages, 48303 KB

Open AccessArticle

Symmetric UAV Cooperative Lifting Motion Planning in Confined Space

by Jingwen Huang, Tianyi Jia and Xiulan Wei

Symmetry 2025, 17(12), 2041; https://doi.org/10.3390/sym17122041 (registering DOI) - 1 Dec 2025

This paper investigates the motion planning problem for symmetric UAV cooperative lifting in confined spaces. A dynamic model of the symmetric UAV cooperative lifting system is established, and differential flatness analysis is employed to transform nonlinear dynamics into constraints on flat outputs, thereby [...] Read more.

This paper investigates the motion planning problem for symmetric UAV cooperative lifting in confined spaces. A dynamic model of the symmetric UAV cooperative lifting system is established, and differential flatness analysis is employed to transform nonlinear dynamics into constraints on flat outputs, thereby simplifying the motion planning process. The planning framework consists of two levels: path planning and trajectory planning. For path planning, a reinforcement learning-based bidirectional RRT (RLDB-BiRRT) method is proposed, which integrates the random tree expansion mechanism with the DDPG algorithm to achieve adaptive directional bias. This approach effectively mitigates the issues of low search efficiency and excessive redundant nodes inherent in traditional RRT algorithms. For trajectory planning, an adaptive safe flight corridor (SFC) construction method is introduced, combining symmetric ellipsoids and convex polyhedra to generate high-quality linear constraints. Building upon the proposed motion planning method and leveraging differential flatness analysis, a unified planning framework is developed that seamlessly integrates the reinforcement learning-enhanced path planning with adaptive safe corridor construction and differential-flatness-based trajectory optimization, specifically designed for symmetric UAV cooperative lifting tasks in confined spaces. This integrated approach enhances corridor space utilization and ensures trajectory continuity. Simulation experiments validate the effectiveness of the proposed methods, demonstrating their capability to generate dynamically feasible, smooth, and safe transportation trajectories in confined environments, while effectively constraining load swing and UAV attitude angles. This study provides theoretical foundations and practical references for the application of symmetric UAV cooperative lifting in low-altitude logistics and emergency transportation scenarios. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry Studies in Data Mining & Machine Learning)

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17 pages, 1519 KB

Open AccessArticle

PHASE: Progressive Hierarchical Adaptation for Sample-Efficient Rebalancing in Long-Tail Classification

by Jiale Li, Jicong Duan, Changbin Shao and Hualong Yu

Symmetry 2025, 17(12), 2040; https://doi.org/10.3390/sym17122040 (registering DOI) - 30 Nov 2025

When trained on long-tailed distributions, deep neural networks often suffer performance degradation and model bias due to the dominance of head classes. Existing reweighting and sampling strategies have significant limitations, such as reliance on fixed heuristics and inability to adapt to dynamic sample [...] Read more.

When trained on long-tailed distributions, deep neural networks often suffer performance degradation and model bias due to the dominance of head classes. Existing reweighting and sampling strategies have significant limitations, such as reliance on fixed heuristics and inability to adapt to dynamic sample difficulty and class imbalance. Additionally, they fail to integrate sample-level granularity with class-level balance, further inadequately addressing the global imbalance issue. Motivated by these challenges, we introduce the Progressive Hierarchical Adaptation for Sample-Efficient rebalancing (PHASE) training framework, which employs a double-layer tuning paradigm to optimize performance under long-tailed distributions. Specifically, the double-layer tuning paradigm adopted by PHASE runs as follows: (1) an early-stage difficulty-aware mechanism targets those difficult-to-classify samples to guide representation learning; and (2) a later-stage multi-scale reweighting strategy integrates class distribution statistics with sample characteristics. This method ensures fine-grained adaptability and global balance, and thus outperforming those static or localized techniques. Extensive experiments on CIFAR10-LT, CIFAR100-LT, and ImageNet-LT datasets demonstrate that PHASE can significantly improve the accuracy of tail classes without degenerating head class performance, and acquire state-of-the-art classification results. PHASE provides a novel paradigm for long-tailed image recognition. Full article

(This article belongs to the Topic Applications of Machine Learning in Large-Scale Optimization and High-Dimensional Learning)

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28 pages, 17592 KB

Open AccessArticle

Symmetry-Aware Bayesian-Optimized Gaussian Process Regression for Remaining Useful Life Prediction of Lithium-Ion Batteries Under Real-World Conditions

by Vikraman Karkuzhali, Nesamony Jothi Swaroopan, Nagalingam Rajendiran Shanker and Sarangapani Senthilraj

Symmetry 2025, 17(12), 2039; https://doi.org/10.3390/sym17122039 (registering DOI) - 30 Nov 2025

Lithium-ion batteries are widely used in electric vehicles (EVs) due to their high energy and power density. The accurate prediction of Remaining Useful Life (RUL) is critical for ensuring safety, reliability, and optimal battery utilization. However, RUL estimation remains challenging because battery degradation [...] Read more.

Lithium-ion batteries are widely used in electric vehicles (EVs) due to their high energy and power density. The accurate prediction of Remaining Useful Life (RUL) is critical for ensuring safety, reliability, and optimal battery utilization. However, RUL estimation remains challenging because battery degradation is influenced not only by electrochemical factors but also by real-world operating conditions, which often exhibit complex symmetric and asymmetric patterns. Existing RUL prediction models neglect the impact of micro-climatic conditions and road-induced vehicle vibrations, which leads to reduced prediction accuracy and limited application in practical driving environments. This paper proposes a Bayesian-optimized Gaussian process regression model (BO_GPR) for RUL prediction by integrating internal resistance data, battery degradation characteristics, micro-climatic parameters (temperature, humidity, wind speed), and vehicle vibration data under diverse driving scenarios. Vibration signals are preprocessed using the Discrete Wavelet Transform (DWT) and band-specific features are extracted using Tunable Q-factor Wavelet Transform (TQWT) to enhance feature sensitivity. The proposed BO-GPR model achieves an accuracy of 98.1%, outperforming conventional machine learning approaches. Experimental analysis shows that Z-axis vibrations, aggressive driving patterns, and urban terrain roads, in combination with micro-climatic variability, play a crucial role in accelerating RUL degradation. By explicitly modeling these factors, the proposed method provides a more realistic, data-driven framework for the health monitoring of electric vehicle batteries. These findings highlight the importance of incorporating environmental influences, vehicle dynamics, and degradation symmetry considerations in RUL prediction, supporting predictive maintenance, fleet management, and battery warranty optimization, improving the reliability and lifecycle cost-effectiveness of electric mobility systems. Full article

(This article belongs to the Section Engineering and Materials)

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39 pages, 44721 KB

Open AccessArticle

Numerical Investigation of Wake Interference in Tandem Square Cylinders at Low Reynolds Numbers

by Sarath R S, R Ajith Kumar and K Suresh Kumar

Symmetry 2025, 17(12), 2038; https://doi.org/10.3390/sym17122038 (registering DOI) - 29 Nov 2025

This study numerically investigates laminar flow around two prismatic bodies, specifically square cylinders, arranged in tandem. The analysis covered gap ratios (

L / D = 2

–7) and Reynolds numbers (

R e

= 100–200), focusing on quantifying the aerodynamic characteristics and [...] Read more.

This study numerically investigates laminar flow around two prismatic bodies, specifically square cylinders, arranged in tandem. The analysis covered gap ratios (

L / D = 2

–7) and Reynolds numbers (

R e

= 100–200), focusing on quantifying the aerodynamic characteristics and examining the wake flow structures within the established interference regimes. The time-averaged and unsteady parameters, including the drag and lift coefficients, RMS lift, vortex formation length, Strouhal number, recirculation length, wake width, and pressure distribution, were evaluated for both cylinders. A consistent critical spacing of

L / D \approx 4.5

was observed across all Reynolds numbers, coinciding with the minimum Strouhal number, a sharp increase in unsteady lift, and divergence in wake width between cylinders. Notably, in the range

4.5 ≲ L / D ≲ 6.5

at higher

R e

, the DC exhibited a mean drag exceeding that of an isolated cylinder, attributed to base-pressure reduction and accelerated inflow from the upstream wake. A critical spacing in the co-shedding regime produced strong drag amplification on the DC, attaining an overall maximum value of 50.41% at

R e = 200

and

L / D = 6.0

. To note, unlike mean drag, mean lift is found to be zero in all interference cases for both cylinders, irrespective of spacing ratio and Re, owing to the symmetry of the time-averaged pressure distribution on either side of the cylinders. Spectral and phase analyses reveal a transition from broadband, desynchronised oscillations to a frequency-locked state, with the phase angle between the cylinders reducing sharply to

Δ ϕ \approx 0^{\circ}

at the critical spacing. This indicates complete in-phase synchronisation or symmetry of the vortex-shedding process between the cylinders at the critical spacing. This confirmed the hydrodynamic transition between the coupled and independent shedding modes of the cylinders. The recirculation lengths for the DC reduce to as low as

0.6 D

in the co-shedding regime, highlighting rapid wake recovery. The research presented here offers new insights into force modulation, the evolution of wake structures, and the sensitivity to the Re that occurs when laminar flow occurs between two tandem square cylinders. These findings can be utilised to develop methods for controlling VIV and designing thermal-fluid systems. Full article

(This article belongs to the Special Issue Symmetry in Fluid Mechanics: New Challenges in Fluid–Structure Interaction)

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19 pages, 3230 KB

Open AccessArticle

Traveling Wave Solutions and Symmetries of Reverse Space-Time Nonlocal Nonlinear Schrödinger Equations

by Yu-Xiang Li, Yu-Shan Bai and Yan-Ting Ren

Symmetry 2025, 17(12), 2037; https://doi.org/10.3390/sym17122037 (registering DOI) - 29 Nov 2025

This paper investigates the reverse space-time nonlocal nonlinear Schrödinger (NNLS) equation, which arises in nonlinear optics, Bose–Einstein condensation, integrable systems, and plasma physics. Several classes of exact solutions are constructed using multiple analytical techniques. First, traveling wave solutions of Jacobi elliptic, hyperbolic, and [...] Read more.

This paper investigates the reverse space-time nonlocal nonlinear Schrödinger (NNLS) equation, which arises in nonlinear optics, Bose–Einstein condensation, integrable systems, and plasma physics. Several classes of exact solutions are constructed using multiple analytical techniques. First, traveling wave solutions of Jacobi elliptic, hyperbolic, and trigonometric function types are ultimately obtained by employing a traveling wave transformation combined with a Weierstrass-type Riccati equation expansion method. Second, Lie symmetry analysis is applied to the NNLS equation, and the corresponding infinitesimal generators are determined. Using these generators, the original equation is reduced to local and nonlocal ordinary differential equations (ODEs), whose invariant solutions are subsequently obtained through integration. Finally, the NNLS equation is generalized to a multi-component system, for which the general form of the infinitesimal symmetries is derived. Symmetry reductions of the extended system yield further classes of reduced ODEs. In particular, the general form of the multi-component solutions is derived. Full article

(This article belongs to the Section Mathematics)

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24 pages, 8378 KB

Open AccessArticle

Study on Cross-Section Transition Form and Stability of Super-Large Variable-Span Tunnel

by Can Xie, Wenbo Gong, Shuguang Song and Rui Wang

Symmetry 2025, 17(12), 2036; https://doi.org/10.3390/sym17122036 - 28 Nov 2025

In order to clarify the influence of cross-section change mode of a large-span variable cross-section tunnel on the stability of the surrounding rock of a tunnel, four three-dimensional finite element models were established for four typical cross-section conversion forms: Sudden changes in the [...] Read more.

In order to clarify the influence of cross-section change mode of a large-span variable cross-section tunnel on the stability of the surrounding rock of a tunnel, four three-dimensional finite element models were established for four typical cross-section conversion forms: Sudden changes in the cross-section of the symmetric tunnel, Gradual changes in the cross-section of the symmetric tunnel, Sudden changes in the cross-section of the asymmetric tunnel, and Gradual changes in the cross-section of the asymmetric tunnel. The Stress characteristics and deformation laws of the surrounding rock under different cross-sectional changes were systematically analyzed. The simulation results were compared with the field monitoring results, and the construction scheme was optimized based on the results of the numerical analysis and field monitoring. The results demonstrate that the final settlement values for sudden changes in the cross-section of the symmetric tunnel and sudden changes in the cross-section of the asymmetric tunnel were 21.23 mm and 21.98 mm, respectively, representing reductions of 14.6% and 15.7% compared to gradual changes in the cross-section of the symmetric tunnel and gradual changes in the cross-section of the asymmetric tunnel. The final support stress values of sudden changes in the cross-section of the symmetric tunnel and sudden changes in the cross-section of the asymmetric tunnel were 8.59 and 7.88 MPa, respectively, representing reductions of 21.5% and 26.9% compared with gradual changes in the cross-section of the symmetric tunnel and gradual changes in the cross-section of the asymmetric tunnel. Compared with asymmetric construction, symmetrical construction is more likely to lead to a stress concentration effect on both sides of the tunnel. Considering the overall construction feasibility and economy, the asymmetric sudden-change construction scheme provides better comprehensive benefits. The research results can provide a theoretical basis and technical reference for solving engineering problems such as complex stress in a large section transition zone and difficult control of tunnel construction. Full article

(This article belongs to the Special Issue Asymmetric Problems in Computational Mechanics and Their Engineering Applications)

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13 pages, 2812 KB

Open AccessArticle

Research on a Class of Optimization Problems of Higher Education Cost with Cobb–Douglas Constraint Condition

by Mingxia Lv, Ping Ji and Yirong Ying

Symmetry 2025, 17(12), 2035; https://doi.org/10.3390/sym17122035 - 28 Nov 2025

This study focuses on a class of optimization problems about higher education costs with industrial production characteristics, in which the objective function is a quadratic cost function and the nonlinear constraints follow the Cobb–Douglas form. Through variable substitution, this class of nonlinear-constrained optimization [...] Read more.

This study focuses on a class of optimization problems about higher education costs with industrial production characteristics, in which the objective function is a quadratic cost function and the nonlinear constraints follow the Cobb–Douglas form. Through variable substitution, this class of nonlinear-constrained optimization problems is converted into optimization problems with linear constraints. Further, this study derives the conditions for the existence of optimal solutions to this optimization problem and conducts numerical simulations for the two-variable scenario to demonstrate its applicability in the field of higher education. Full article

(This article belongs to the Special Issue Symmetry in Data Analysis and Optimization)

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16 pages, 277 KB

Open AccessArticle

Identities Involving the Higher-Order Degenerate Type 2 ω-Daehee Polynomials

by Pengfei Zhang, Yonglin Yang and Huihui Wang

Symmetry 2025, 17(12), 2034; https://doi.org/10.3390/sym17122034 - 28 Nov 2025

In this paper, based on previous study of type 2

ω

-Daehee polynomials and some of their properties, we further introduce the generating function definition for the higher-order degenerate type 2

ω

-Daehee polynomials. By employing the methods of generating functions and Riordan [...] Read more.

In this paper, based on previous study of type 2

ω

-Daehee polynomials and some of their properties, we further introduce the generating function definition for the higher-order degenerate type 2

ω

-Daehee polynomials. By employing the methods of generating functions and Riordan arrays, we investigate the properties of these higher-order degenerate polynomials in depth and establish identities that relate them to certain special combinatorial sequences. Full article

(This article belongs to the Section Mathematics)

11 pages, 1728 KB

Open AccessArticle

A Symmetric Deep Learning Approach for Dynamic Reserve Evaluation of Tight Sandstone Gas Wells

by Yi Zhang, Bin Zhang, Banghua Liu, Haikun Zeng, Junhui Bai, Xijun Tian, Peng Liu, Jiahui Wu and Chaoqiang Feng

Symmetry 2025, 17(12), 2033; https://doi.org/10.3390/sym17122033 - 28 Nov 2025

Traditional dynamic storage calculation methods face challenges such as difficult data acquisition and prolonged testing periods. To address the industry’s need for rapid yet accurate estimation of single-well dynamic reserves in tight sandstone gas formations, a deep learning architecture combining convolutional neural network [...] Read more.

Traditional dynamic storage calculation methods face challenges such as difficult data acquisition and prolonged testing periods. To address the industry’s need for rapid yet accurate estimation of single-well dynamic reserves in tight sandstone gas formations, a deep learning architecture combining convolutional neural network (CNN) and long short-term memory (LSTM) network is proposed. This model enables fast and accurate reserve evaluation, outperforming other machine learning methods in overall capability while achieving a symmetric improvement in both training efficiency and prediction accuracy—reaching up to 95.9%. Based on this model, dynamic reserves of gas wells in the Sulige Gas Field were predicted. The single-well dynamic reserve test showed a relative error of less than 10%, and the method demonstrated strong stability and high precision in localized multi-well group tests, with errors distributed symmetrically within a narrow margin. All results satisfy engineering standards. The feasibility of the method has been verified, proving it can deliver fast and accurate gas well dynamic reserve predictions, greatly reduce evaluation costs, and enhance work efficiency. Full article

(This article belongs to the Section Computer)

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