Symmetry

12 pages, 272 KiB

Open AccessArticle

GUP, Lorentz Invariance (Non)-Violation, and Non-Commutative Geometry

by Michael Bishop, Daniel Hooker, Peter Martin and Douglas Singleton

Symmetry 2025, 17(6), 923; https://doi.org/10.3390/sym17060923 - 10 Jun 2025

In this work, we formulate a generalized uncertainty principle with both position and momentum operators modified from their canonical forms. We study whether Lorentz symmetry is violated and whether it can be saved with these modifications. The requirement that Lorentz invariance is not [...] Read more.

In this work, we formulate a generalized uncertainty principle with both position and momentum operators modified from their canonical forms. We study whether Lorentz symmetry is violated and whether it can be saved with these modifications. The requirement that Lorentz invariance is not violated places restrictions on the way the position and momentum operators can be modified. We also investigate the connection between general uncertainty principle and non-commutative geometry models, e.g., laying out the connection between area/area operators and angular momentum in both models. Full article

(This article belongs to the Special Issue Nature and Origin of Dark Matter and Dark Energy, 2nd Edition)

24 pages, 1157 KiB

Open AccessArticle

New Perspectives on the Irregular Singular Point of the Wave Equation for a Massive Scalar Field in Schwarzschild Space-Time

by Giampiero Esposito and Marco Refuto

Symmetry 2025, 17(6), 922; https://doi.org/10.3390/sym17060922 - 10 Jun 2025

Abstract

For a massive scalar field in a fixed Schwarzschild background, the radial wave equation obeyed by Fourier modes is first studied. After reducing such a radial wave equation to its normal form, we first study approximate solutions in the neighborhood of the origin, [...] Read more.

For a massive scalar field in a fixed Schwarzschild background, the radial wave equation obeyed by Fourier modes is first studied. After reducing such a radial wave equation to its normal form, we first study approximate solutions in the neighborhood of the origin, horizon and point at infinity, and then we relate the radial with the Heun equation, obtaining local solutions at the regular singular points. Moreover, we obtain the full asymptotic expansion of the local solution in the neighborhood of the irregular singular point at infinity. We also obtain and study the associated integral representation of the massive scalar field. Eventually, the technique developed for the irregular singular point is applied to the homogeneous equation associated with the inhomogeneous Zerilli equation for gravitational perturbations in a Schwarzschild background. Full article

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32 pages, 833 KiB

Open AccessReview

Boltzmann Equation and Its Cosmological Applications

by Seishi Enomoto, Yu-Hang Su, Man-Zhu Zheng and Hong-Hao Zhang

Symmetry 2025, 17(6), 921; https://doi.org/10.3390/sym17060921 - 10 Jun 2025

Abstract

We review the derivation of the Boltzmann equation and its cosmological applications in this paper. A novel derivation of the Boltzmann equation, especially the collision term, is discussed in detail in the language of quantum field theory without any assumption of a finite [...] Read more.

We review the derivation of the Boltzmann equation and its cosmological applications in this paper. A novel derivation of the Boltzmann equation, especially the collision term, is discussed in detail in the language of quantum field theory without any assumption of a finite temperature system. We also discuss the integrated Boltzmann equation, incorporating the temperature parameter as an extension of the standard equation. Among a number of its cosmological applications, we mainly target two familiar examples, the dynamics of the dark matter abundance through the freeze-out/in process and a baryogenesis scenario. The formulations in those systems are briefly discussed with techniques in their calculations. Full article

(This article belongs to the Special Issue Quantum Gravity and Cosmology: Exploring the Astroparticle Interface)

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28 pages, 1262 KiB

Open AccessArticle

Mathematical Modeling of Impurity Diffusion Processes in a Multiphase Randomly Inhomogeneous Body Using Feynman Diagrams

by Petro Pukach, Yurii Chernukha, Olha Chernukha, Yurii Bilushchak and Myroslava Vovk

Symmetry 2025, 17(6), 920; https://doi.org/10.3390/sym17060920 - 10 Jun 2025

Abstract

Modeling of impurity diffusion processes in a multiphase randomly inhomogeneous body is performed using the Feynman diagram technique. The impurity diffusion equations are formulated for each of the phases separately. Their random boundaries are subject to non-ideal contact conditions for concentration. The contact [...] Read more.

Modeling of impurity diffusion processes in a multiphase randomly inhomogeneous body is performed using the Feynman diagram technique. The impurity diffusion equations are formulated for each of the phases separately. Their random boundaries are subject to non-ideal contact conditions for concentration. The contact mass transfer problem is reduced to a partial differential equation describing diffusion in the body as a whole, which accounts for jump discontinuities in the searched function as well as in its derivative at the stochastic interfaces. The obtained problem is transformed into an integro-differential equation involving a random kernel, whose solution is constructed as a Neumann series. Averaging over the ensemble of phase configurations is performed. The Feynman diagram technique is developed to investigate the processes described by parabolic partial differential equations. The mass operator kernel is constructed as a sum of strongly connected diagrams. An integro-differential Dyson equation is obtained for the concentration field. In the Bourret approximation, the Dyson equation is specified for a multiphase randomly inhomogeneous medium with uniform phase distribution. The problem solution, obtained using Feynman diagrams, is compared with the solutions of diffusion problems for a homogeneous layer, one having the coefficients of the base phase and the other having the characteristics averaged over the body volume. Full article

(This article belongs to the Section Mathematics)

20 pages, 312 KiB

Open AccessArticle

An Analysis of Existing Hash-Based Post-Quantum Signature Schemes

by Cristina Maria Pacurar, Razvan Bocu and Maksim Iavich

Symmetry 2025, 17(6), 919; https://doi.org/10.3390/sym17060919 - 10 Jun 2025

Abstract

The rapid development of quantum computing poses challenges to the foundations of traditional cryptography. The threats are significant in terms of both asymmetric cryptography (which exposes schemes like RSA and ECC to efficient attacks) and symmetric cryptography, where key sizes must be increased [...] Read more.

The rapid development of quantum computing poses challenges to the foundations of traditional cryptography. The threats are significant in terms of both asymmetric cryptography (which exposes schemes like RSA and ECC to efficient attacks) and symmetric cryptography, where key sizes must be increased to mitigate these threats. In this paper, we review the evolution of hash-based digital signatures, from early one-time signatures to modern stateless schemes, with an emphasis on their security properties, efficiency, and practical constraints. Moreover, we propose a simple comparative metric that reflects structural symmetry across key parameters such as key size, signature size, and computational cost, enabling a visual clustering of the schemes. We give particular attention to recent developments such as Verkle trees, which preserve symmetric design principles while improving scalability and proof compactness. The study highlights ongoing tradeoffs between stateful and stateless designs and argues for the continued relevance of symmetric cryptographic constructions in building secure, efficient post-quantum systems. Full article

(This article belongs to the Section Computer)

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12 pages, 283 KiB

Open AccessArticle

The Reliability of Cayley Graphs Generated by Transposition Trees Based on Edge Failures

by Xiang-Jun Li, Lin-Fei Dong, Ling-Xing Qin, Chai Shu and Mei-Jie Ma

Symmetry 2025, 17(6), 918; https://doi.org/10.3390/sym17060918 - 10 Jun 2025

Abstract

Extra edge connectivity is an important parameter for measuring the reliability of interconnection networks. Given a graph G and a non-negative integer h, the h-extra edge connectivity of G, denoted by

λ_{h} (G)

, is the minimum cardinality of a [...] Read more.

Extra edge connectivity is an important parameter for measuring the reliability of interconnection networks. Given a graph G and a non-negative integer h, the h-extra edge connectivity of G, denoted by

λ_{h} (G)

, is the minimum cardinality of a set of edges in G (if it exists) whose deletion disconnects G such that each remaining component contains at least

h + 1

vertices. In this paper, we obtain the h-extra edge connectivity of Cayley graphs generated by transposition trees for

h \leq 5

. As byproducts, we derive the h-extra edge connectivity of the star graph

S_{n}

and the bubble-sort graph

B_{n}

for

h \leq 5

. Full article

(This article belongs to the Section Mathematics)

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22 pages, 3432 KiB

Open AccessArticle

The Formation Mechanism of Residual Stress in Friction Stir Welding Based on Thermo-Mechanical Coupled Simulation

by Tianlei Yang, Xiao Wei, Jiangfan Zhou, Hao Jiang, Xinyu Liu and Zongzhe Man

Symmetry 2025, 17(6), 917; https://doi.org/10.3390/sym17060917 - 10 Jun 2025

Abstract

Friction Stir Welding (FSW) is widely used for high-strength aluminum alloys due to its solid-state bonding, which ensures superior weld quality and service stability. However, thermo-mechanical interactions during welding can induce complex residual stress distributions, compromising joint integrity. Previous studies have primarily focused [...] Read more.

Friction Stir Welding (FSW) is widely used for high-strength aluminum alloys due to its solid-state bonding, which ensures superior weld quality and service stability. However, thermo-mechanical interactions during welding can induce complex residual stress distributions, compromising joint integrity. Previous studies have primarily focused on thermal load-driven stress evolution, often neglecting mechanical factors such as the shear force generated by the stirring pin. This study develops a three-dimensional thermo-mechanical coupled finite element model based on a moving heat source. The model incorporates axial pressure from the tool shoulder and torque-derived shear force from the stirring pin. A hybrid surface–volumetric heat source is applied to represent frictional heating, and realistic mechanical boundary conditions are introduced to reflect actual welding conditions. Simulations on AA6061-T6 aluminum alloy show that under stable welding, the peak temperature in the weld zone reaches approximately 453 °C. Residual stress analysis indicates a longitudinal tensile peak of ~170 MPa under thermal loading alone, which reduces to ~150 MPa when mechanical loads are included, forming a characteristic M-shaped distribution. Further comparison with a Coupled Eulerian–Lagrangian (CEL) model reveals stress asymmetry, with higher tensile stress on the advancing side. This is primarily attributed to the directional shear force, which promotes greater plastic deformation on the advancing side than on the retreating side. The consistency between the proposed model and CEL results confirms its validity. This study provides a reliable framework for residual stress prediction in FSW and supports process parameter optimization. Full article

(This article belongs to the Special Issue Symmetry in Impact Mechanics of Materials and Structures)

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19 pages, 2866 KiB

Open AccessArticle

Optimization of Dismantling Line for Used Electric Vehicle Battery Packs

by Jia Mao, Shenggang Li, Ziang Zhao, Yanzhi Zhou, Jinyuan Cheng and Weiwen Li

Symmetry 2025, 17(6), 916; https://doi.org/10.3390/sym17060916 - 10 Jun 2025

Abstract

Recycling used battery packs for electric vehicles has important economic and ecological benefits. However, existing studies lack a systematic comparison of the different models. There is also a lack of exploration of symmetrical bilateral disassembly lines in comparison to other disassembly lines. Therefore, [...] Read more.

Recycling used battery packs for electric vehicles has important economic and ecological benefits. However, existing studies lack a systematic comparison of the different models. There is also a lack of exploration of symmetrical bilateral disassembly lines in comparison to other disassembly lines. Therefore, this paper takes the battery pack dismantling line of a company as the object of study. The disassembly line model is established by considering the number of workstations, disassembly time, disassembly costs, and disassembly risks. This study quantifies the risk of disassembly at each process into specific indicators for modeling. The solving algorithm adopts the improved Northern Goshawk Optimization Algorithm and the Non-dominated Sorting Genetic Algorithm. A simulation demonstration was conducted using plant simulation to compare the advantages and disadvantages of the unilateral disassembly and symmetric bilateral disassembly line programs. The results show that the optimized unilateral and symmetric bilateral disassembly line schemes increase the efficiency by 23.08% and 38.46%, respectively, compared to the original program. The symmetric bilateral disassembly line scheme is the most efficient among the programs. The optimized schemes significantly improve the overall operational efficiency of the disassembly line. Programs to promote increased efficiency in battery recycling will contribute to environmental sustainability. Full article

(This article belongs to the Section Engineering and Materials)

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26 pages, 2613 KiB

Open AccessArticle

A Cognitive Load Assessment Method for Airtight Cabin Operators Based on a One-Dimensional Convolutional Neural Network

by Lei Wang, Jingluan Wang, Dengkai Chen and Jie Song

Symmetry 2025, 17(6), 915; https://doi.org/10.3390/sym17060915 - 10 Jun 2025

Abstract

Airtight cabins with highly complex human–machine systems impose an excessive cognitive load on operators. However, the traditional cognitive load assessment methods often cannot fully extract physiological features such as electroencephalogram and electrocardiogram signals, relying heavily on artificial feature extraction. Therefore, this study proposes [...] Read more.

Airtight cabins with highly complex human–machine systems impose an excessive cognitive load on operators. However, the traditional cognitive load assessment methods often cannot fully extract physiological features such as electroencephalogram and electrocardiogram signals, relying heavily on artificial feature extraction. Therefore, this study proposes an evaluation method based on a one-dimensional convolutional neural network to evaluate the cognitive load of airtight cabin workers. This evaluation method preprocesses and intercepts raw physiological signals such as electroencephalogram and electrocardiogram signals and then inputs them into the model for evaluation. The experimental results demonstrate that the training accuracy rate of the one-dimensional convolutional neural network is 97.6%, and the test classification accuracy rate is 86.5%. Despite sample size limitations, the proposed method demonstrates valid effectiveness in this study. Finally, taking a manned submersible as an example, cognitive load in different difficult tasks is identified, evaluated, and classified. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry in Computer-Aided Industrial Design)

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23 pages, 1601 KiB

Open AccessArticle

Level-Wise Feature-Guided Cascading Ensembles for Credit Scoring

by Yao Zou and Guanghua Cheng

Symmetry 2025, 17(6), 914; https://doi.org/10.3390/sym17060914 - 10 Jun 2025

Abstract

Accurate credit scoring models are essential for financial risk management, yet conventional approaches often fail to address the complexities of high-dimensional, heterogeneous credit data, particularly in capturing nonlinear relationships and hierarchical dependencies, ultimately compromising predictive performance. To overcome these limitations, this paper introduces [...] Read more.

Accurate credit scoring models are essential for financial risk management, yet conventional approaches often fail to address the complexities of high-dimensional, heterogeneous credit data, particularly in capturing nonlinear relationships and hierarchical dependencies, ultimately compromising predictive performance. To overcome these limitations, this paper introduces the level-wise feature-guided cascading ensemble (LFGCE) model, a novel framework that integrates hierarchical feature selection with cascading ensemble learning to systematically uncover latent feature hierarchies. The LFGCE framework leverages symmetry principles in its cascading architecture, where each ensemble layer maintains structural symmetry in processing its assigned feature subset while asymmetrically contributing to the final prediction through hierarchical information fusion. The LFGCE model operates through two synergistic mechanisms: (1) a hierarchical feature selection strategy that quantifies feature importance and partitions the feature space into progressively predictive subsets, thereby reducing dimensionality while preserving discriminative information, and (2) a cascading ensemble architecture where each layer specializes in learning risk patterns from its assigned feature subset, while iteratively incorporating outputs from preceding layers to enable cross-level information fusion. This dual process of hierarchical feature refinement and layered ensemble learning allows the LFGCE to extract deep, robust representations of credit risk. Empirical validation on four public credit datasets (Australian Credit, German Credit, Japan Credit, and Taiwan Credit) demonstrates that the LFGCE achieves an average AUC improvement of 0.23% over XGBoost (Python 3.13) and 0.63% over deep neural networks, confirming its superior predictive accuracy. Full article

(This article belongs to the Special Issue Symmetric Studies of Distributions in Statistical Models)

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18 pages, 15092 KiB

Open AccessArticle

Ultra-Low Bitrate Predictive Portrait Video Compression with Diffusion Models

by Xinyi Chen, Weimin Lei, Wei Zhang, Yanwen Wang and Mingxin Liu

Symmetry 2025, 17(6), 913; https://doi.org/10.3390/sym17060913 - 10 Jun 2025

Abstract

Deep neural video compression codecs have shown great promise in recent years. However, there are still considerable challenges for ultra-low bitrate video coding. Inspired by recent diffusion models for image and video compression attempts, we attempt to leverage diffusion models for ultra-low bitrate [...] Read more.

Deep neural video compression codecs have shown great promise in recent years. However, there are still considerable challenges for ultra-low bitrate video coding. Inspired by recent diffusion models for image and video compression attempts, we attempt to leverage diffusion models for ultra-low bitrate portrait video compression. In this paper, we propose a predictive portrait video compression method that leverages the temporal prediction capabilities of diffusion models. Specifically, we develop a temporal diffusion predictor based on a conditional latent diffusion model, with the predicted results serving as decoded frames. We symmetrically integrate a temporal diffusion predictor at the encoding and decoding side, respectively. When the perceptual quality of the predicted results in encoding end falls below a predefined threshold, a new frame sequence is employed for prediction. While the predictor at the decoding side directly generates predicted frames as reconstruction based on the evaluation results. This symmetry ensures that the prediction frames generated at the decoding end are consistent with those at the encoding end. We also design an adaptive coding strategy that incorporates frame quality assessment and adaptive keyframe control. To ensure consistent quality of subsequent predicted frames and achieve high perceptual reconstruction, this strategy dynamically evaluates the visual quality of the predicted results during encoding, retains the predicted frames that meet the quality threshold, and adaptively adjusts the length of the keyframe sequence based on motion complexity. The experimental results demonstrate that, compared with the traditional video codecs and other popular methods, the proposed scheme provides superior compression performance at ultra-low bitrates while maintaining competitiveness in visual effects, achieving more than 24% bitrate savings compared with VVC in terms of perceptual distortion. Full article

(This article belongs to the Special Issue 2025 9th International Conference on Electronic Information Technology and Computer Engineering)

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18 pages, 835 KiB

Open AccessArticle

Reliability Evaluation of Two-Stage Uncertain Multi-State Weighted k-Out-of-n Systems

by Chun Wei, Haiyan Shi and Zhiqiang Zhang

Symmetry 2025, 17(6), 912; https://doi.org/10.3390/sym17060912 - 9 Jun 2025

Abstract

A class of complex systems can be structurally decomposed into several typical multi-state weighted k/n systems. In view of this, this paper proposes a reliability model to evaluate the reliability of such systems. The system reliability evaluation process is divided into two critical [...] Read more.

A class of complex systems can be structurally decomposed into several typical multi-state weighted k/n systems. In view of this, this paper proposes a reliability model to evaluate the reliability of such systems. The system reliability evaluation process is divided into two critical stages: (1) Primary stage: Structural reliability analysis (system-level) and (2) Secondary stage: Subsystem reliability verification (component-level). In practical applications, when there is a lack of operational data, expert experience is needed to evaluate the states and availability weights of components, which is a typical application scenario for uncertainty theory. Under the framework of uncertainty theory, two-stage reliability measures and importance measures are defined, and corresponding calculation formulas are derived. To improve computational efficiency, a binary search algorithm is proposed. Finally, an application case of SDN was proposed to demonstrate the effectiveness of the theory and method. This study provides an idea for the reliability modeling of complex systems (multiple pipeline transmissions, power systems, distributed computing, etc.) in the absence of operational data. Full article

(This article belongs to the Section Mathematics)

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19 pages, 640 KiB

Open AccessArticle

A Hypergraph-Based Approach to Attribute Reduction in an Incomplete Decision System

by Lirun Su and Chunmao Jiang

Symmetry 2025, 17(6), 911; https://doi.org/10.3390/sym17060911 - 9 Jun 2025

Abstract

Attribute reduction has been demonstrated to be an effective approach for addressing fuzziness and uncertainty in data analysis, especially for data dimension reduction. As an extension of graphs, hypergraphs have been established by prior research as a potent mathematical framework for attribute reduction [...] Read more.

Attribute reduction has been demonstrated to be an effective approach for addressing fuzziness and uncertainty in data analysis, especially for data dimension reduction. As an extension of graphs, hypergraphs have been established by prior research as a potent mathematical framework for attribute reduction in decision systems. However, current studies rarely explore the integration of hypergraphs and rough set theories for attribute reduction in incomplete decision systems. To bridge this theoretical gap, this paper proposes a novel hypergraph-based attribute reduction method for incomplete decision systems through a matrix. Firstly, we introduce two types of construction methods for the characteristic matrices of a hypergraph, and the characteristic matrix decomposition relationship between them is examined. Moreover, some features in hypergraphs including transversal are systematically investigated via these characteristic matrices. Secondly, using the characteristic matrices of the hypergraphs derived from an incomplete information system, a hypergraph-based method is developed for the process of attribute reduction in incomplete information systems via a discernibility matrix. Finally, we discuss the attribute reduction method of incomplete decision systems, and establish a new judgment method for the attribute reduction in incomplete decision systems through the constructed characteristic matrices of hypergraphs. Full article

(This article belongs to the Section Mathematics)

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19 pages, 3412 KiB

Open AccessArticle

Neutron Stars in the Theory of Gravity with Non-Minimal Derivative Coupling and Realistic Equations of State

by Pavel E. Kashargin, Alexander A. Lebedev and Sergey V. Sushkov

Symmetry 2025, 17(6), 910; https://doi.org/10.3390/sym17060910 - 9 Jun 2025

Abstract

We numerically construct compact stars in the scalar–tensor theory of gravity with non-minimal derivative coupling of a scalar field to the curvature and nonzero cosmological constant. There are two free parameters in this model of gravity: the non-minimal derivative coupling parameter ℓ and [...] Read more.

We numerically construct compact stars in the scalar–tensor theory of gravity with non-minimal derivative coupling of a scalar field to the curvature and nonzero cosmological constant. There are two free parameters in this model of gravity: the non-minimal derivative coupling parameter ℓ and the cosmological constant parameter

ξ

. We study the relationship between the model parameters and characteristic of the neutron star, which allowed us to limit the permissible range of

ξ

and ℓ. In particular, in the case

ξ = - 1

, the external geometry of the neutron star coincides with the Schwarzschild–anti-de Sitter geometry, while the internal geometry of the star differs from the case of the standard gravity theory. Many realistic equations of the state of neutron star matter were considered. In general, the neutron star model in the theory of gravity with a non-minimal derivative coupling does not contradict astronomical data and is viable. Full article

(This article belongs to the Special Issue Feature Papers in 'Physics' Section 2025)

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1 pages, 121 KiB

Open AccessCorrection

Correction: El-hady et al. On Approximate Multi-Cubic Mappings in 2-Banach Spaces. Symmetry 2025, 17, 475

by El-sayed El-hady, Ghazyiah Alsahli, Abasalt Bodaghi and Mehdi Dehghanian

Symmetry 2025, 17(6), 909; https://doi.org/10.3390/sym17060909 - 9 Jun 2025

Abstract

In the published publication [...] Full article

17 pages, 359 KiB

Open AccessArticle

Importance Measures Analysis of Two-Stage Uncertain Weighted k-out-of-n Systems

by Chun Wei, Haiyan Shi and Zhiqiang Zhang

Symmetry 2025, 17(6), 908; https://doi.org/10.3390/sym17060908 - 8 Jun 2025

Abstract

This paper proposes a two-stage weighted k-out-of-n (W-k/n) system reliability model based on uncertainty theory to address the problem of traditional W-k/n system theory being difficult to apply to the reliability evaluation of complex systems with two-layer structures. This model effectively solves the [...] Read more.

This paper proposes a two-stage weighted k-out-of-n (W-k/n) system reliability model based on uncertainty theory to address the problem of traditional W-k/n system theory being difficult to apply to the reliability evaluation of complex systems with two-layer structures. This model effectively solves the evaluation problem in the case of insufficient data by decomposing the reliability assessment of complex systems into two stages: system structure reliability and subsystem reliability. In the model, the system state and weights are described by uncertain variables that follow a certain distribution. Under the framework of uncertainty theory, two-stage reliability measures and importance measures are defined, and corresponding calculation formulas are derived. In order to reduce computational complexity, a corresponding numerical solution method including traverse search algorithm and binary search algorithm is proposed. The effectiveness of the method is also demonstrated through a project management case of electronic consumer goods. This study provides new theoretical ideas and methodological support for reliability modeling of complex systems under data loss conditions. Full article

(This article belongs to the Section Mathematics)

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19 pages, 1053 KiB

Open AccessArticle

Symmetry-Aware Dynamic Scheduling Optimization in Hybrid Manufacturing Flexible Job Shops Using a Time Petri Nets Improved Genetic Algorithm

by Xuanye Lin, Zhenxiong Xu, Shujun Xie, Fan Yang, Juntao Wu and Deping Li

Symmetry 2025, 17(6), 907; https://doi.org/10.3390/sym17060907 - 8 Jun 2025

Abstract

Dynamic scheduling in hybrid flexible job shops (HFJSs) presents a critical challenge in modern manufacturing systems, particularly under dynamic and uncertain conditions. These systems often exhibit inherent structural and behavioral symmetry, such as uniform machine–job relationships and repeatable event response patterns. To leverage [...] Read more.

Dynamic scheduling in hybrid flexible job shops (HFJSs) presents a critical challenge in modern manufacturing systems, particularly under dynamic and uncertain conditions. These systems often exhibit inherent structural and behavioral symmetry, such as uniform machine–job relationships and repeatable event response patterns. To leverage this, we propose a time Petri nets (TPNs) model that integrates time and logic constraints, capturing symmetric processing and setup behaviors across machines as well as dynamic job and machine events. A transition select coding mechanism is introduced, where each transition node is assigned a normalized priority value in the range [0, 1], preserving scheduling consistency and symmetry during decision-making. Furthermore, we develop a symmetry-aware time Petri nets-based improved genetic algorithm (TPGA) to solve both static and dynamic scheduling problems in HFJSs. Experimental evaluations show that TPGA significantly outperforms classical dispatching rules such as Shortest Job First (SJF) and Highest Response Ratio Next (HRN), achieving makespan reductions of 23%, 10%, and 13% in process, discrete, and hybrid manufacturing scenarios, respectively. These results highlight the potential of exploiting symmetry in system modeling and optimization for enhanced scheduling performance. Full article

(This article belongs to the Special Issue Symmetry and Asymmetry in Intelligent Control and Computing)

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24 pages, 9307 KiB

Open AccessArticle

DASS-YOLO: Improved YOLOv7-Tiny with Attention-Guided Shape Awareness and DySnakeConv for Spray Code Defect Detection

by Yixuan Shi, Shiling Zheng, Meiyue Bian, Xia Zhang and Lishan Yang

Symmetry 2025, 17(6), 906; https://doi.org/10.3390/sym17060906 - 8 Jun 2025

Viewed by 40

Abstract

To address the challenges of detecting spray code defects caused by complex morphological variations and the discrete characterization of dot-matrix spray codes, an improved YOLOv7-tiny algorithm named DASS-YOLO is proposed. Firstly, the DySnakeConv module is employed in Backbone–Neck cross-layer connections. With a dynamic [...] Read more.

To address the challenges of detecting spray code defects caused by complex morphological variations and the discrete characterization of dot-matrix spray codes, an improved YOLOv7-tiny algorithm named DASS-YOLO is proposed. Firstly, the DySnakeConv module is employed in Backbone–Neck cross-layer connections. With a dynamic structure and adaptive learning, it can capture the complex morphological features of spray codes. Secondly, we proposed an Attention-guided Shape Enhancement Module with CAA (ASEM-CAA), which adopts a symmetrical dual-branch structure to facilitate bidirectional interaction between local and global features, enabling precise prediction of the overall spray code shape. It also reduces feature discontinuity in dot-matrix codes, ensuring a more coherent representation. Furthermore, Slim-neck, which is famous for its more lightweight structure, is adopted in the Neck to reduce model complexity while maintaining accuracy. Finally, Shape-IoU is applied to improve the accuracy of the bounding box regression. Experiments show that DASS-YOLO improves the detection accuracy by 1.9%. Additionally, for small defects such as incomplete code and code spot, the method achieves better accuracy improvements of 8.7% and 2.1%, respectively. Full article

(This article belongs to the Special Issue Symmetry and Its Applications in Computer Vision)

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20 pages, 758 KiB

Open AccessArticle

Dynamics of Bone Remodeling by Using Mathematical Model Under ABC Time-Fractional Derivative

by Kamonchat Trachoo, Inthira Chaiya, Sirawit Phakmee and Din Prathumwan

Symmetry 2025, 17(6), 905; https://doi.org/10.3390/sym17060905 - 8 Jun 2025

Viewed by 43

Abstract

Bone remodeling is a dynamic biological process that preserves bone strength and structure through the coordinated actions of osteoblasts, osteoclasts, osteocytes, and bone mass density. Traditional models based on ordinary differential equations often fail to capture the memory-dependent nature of these interactions. In [...] Read more.

Bone remodeling is a dynamic biological process that preserves bone strength and structure through the coordinated actions of osteoblasts, osteoclasts, osteocytes, and bone mass density. Traditional models based on ordinary differential equations often fail to capture the memory-dependent nature of these interactions. In this study, we propose a novel mathematical model of bone remodeling using the Atangana–Baleanu–Caputo fractional derivative, which accounts for the non-local and hereditary characteristics of biological systems. The model introduces fractional-order dynamics into a previously established ODE framework while maintaining the intrinsic symmetry between bone-forming and bone-resorbing mechanisms, as well as the balance mediated by porosity-related feedback. We establish the existence, uniqueness, and positivity of solutions, and analyze the equilibrium points and their global stability using a Lyapunov function. Numerical simulations under various fractional orders demonstrate symmetric convergence toward equilibrium across all biological variables. The results confirm that fractional-order modeling provides a more accurate and balanced representation of bone remodeling and reveal the underlying symmetry in the regulation of bone tissue. This work contributes to the growing use of fractional calculus in modeling physiological processes and highlights the importance of symmetry in both mathematical structure and biological behavior. Full article

(This article belongs to the Section Mathematics)

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23 pages, 2266 KiB

Open AccessArticle

Macro-Financial Condition Index Construction and Forecasting Based on Machine Learning Techniques: Empirical Evidence from China

by Xinlong Li, Liqing Xue and Jiayuan Liang

Symmetry 2025, 17(6), 904; https://doi.org/10.3390/sym17060904 - 7 Jun 2025

Viewed by 174

Abstract

Identifying and forecasting macro-financial conditions is critical to stabilizing the economy. This study aims to develop a novel methodology for constructing China’s Financial Conditions Index, utilizing monthly data from six major Chinese financial markets (comprising 33 key financial indicators) along with 25 external [...] Read more.

Identifying and forecasting macro-financial conditions is critical to stabilizing the economy. This study aims to develop a novel methodology for constructing China’s Financial Conditions Index, utilizing monthly data from six major Chinese financial markets (comprising 33 key financial indicators) along with 25 external macroeconomic variables from both China and the United States, spanning January 2002 to June 2024. Although the traditional TVP-FAVAR model can capture the linear relationship in the financial market, it cannot adequately characterize the nonlinear or asymmetric nature of the macro-financial conditions exhibited when major risk events occur at home and abroad. In this paper, we propose an innovative kernel factor-augmented time-varying parameter vector autoregressive model (TVP-KFAVAR), which can better capture the nonlinear nature of the macro-financial situation. It is shown that the TVP-KFAVAR model successfully reflects the impact of major domestic and international risk events on China’s Financial Conditions Index. Meanwhile, the ARIMA model and five machine learning techniques (GRU, LSTM, BiLSTM, TCN and Transformer) are used in this study to predict the Macro-Financial Conditions Index, and it is found that the vast majority of the machine learning techniques outperform the traditional time-series models in terms of forecasting performance. TCN has the outstanding prediction performance under different input configurations. This study can provide policymakers with a powerful tool for macro-financial regulation and risk early warning, and help improve macro-financial management in emerging markets. Full article

(This article belongs to the Section Computer)

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25 pages, 2838 KiB

Open AccessArticle

BHE+ALBERT-Mixplus: A Distributed Symmetric Approximate Homomorphic Encryption Model for Secure Short-Text Sentiment Classification in Teaching Evaluations

by Jingren Zhang, Siti Sarah Maidin and Deshinta Arrova Dewi

Symmetry 2025, 17(6), 903; https://doi.org/10.3390/sym17060903 - 7 Jun 2025

Viewed by 152

Abstract

This study addresses the sentiment classification of short texts in teaching evaluations. To mitigate concerns regarding data security in cloud-based sentiment analysis and to overcome the limited feature extraction capacity of traditional deep-learning methods, we propose a distributed symmetric approximate homomorphic hybrid sentiment [...] Read more.

This study addresses the sentiment classification of short texts in teaching evaluations. To mitigate concerns regarding data security in cloud-based sentiment analysis and to overcome the limited feature extraction capacity of traditional deep-learning methods, we propose a distributed symmetric approximate homomorphic hybrid sentiment classification model, denoted BHE+ALBERT-Mixplus. To enable homomorphic encryption of non-polynomial functions within the ALBERT-Mixplus architecture—a mixing-and-enhancement variant of ALBERT—we introduce the BHE (BERT-based Homomorphic Encryption) algorithm. The BHE establishes a distributed symmetric approximation workflow, constructing a cloud–user symmetric encryption framework. Within this framework, simplified computations and mathematical approximations are applied to handle non-polynomial operations (e.g., GELU, Softmax, and LayerNorm) under the CKKS homomorphic-encryption scheme. Consequently, the ALBERT-Mixplus model can securely perform classification on encrypted data without compromising utility. To improve feature extraction and enhance prediction accuracy in sentiment classification, ALBERT-Mixplus incorporates two core components: 1. A meta-information extraction layer, employing a lightweight pre-trained ALBERT model to capture extensive general semantic knowledge and thereby bolster robustness to noise. 2. A hybrid feature-extraction layer, which fuses a bidirectional gated recurrent unit (BiGRU) with a multi-scale convolutional neural network (MCNN) to capture both global contextual dependencies and fine-grained local semantic features across multiple scales. Together, these layers enrich the model’s deep feature representations. Experimental results on the TAD-2023 and SST-2 datasets demonstrate that BHE+ALBERT-Mixplus achieves competitive improvements in key evaluation metrics compared to mainstream models, despite a slight increase in computational overhead. The proposed framework enables secure analysis of diverse student feedback while preserving data privacy. This allows marginalized student groups to benefit equally from AI-driven insights, thereby embodying the principles of educational equity and inclusive education. Moreover, through its innovative distributed encryption workflow, the model enhances computational efficiency while promoting environmental sustainability by reducing energy consumption and optimizing resource allocation. Full article

(This article belongs to the Section Computer)

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19 pages, 602 KiB

Open AccessArticle

FGeo-Eval: Evaluation System for Plane Geometry Problem Solving

by Qike Huang, Xiaokai Zhang, Na Zhu, Fangzhen Zhu and Tuo Leng

Symmetry 2025, 17(6), 902; https://doi.org/10.3390/sym17060902 - 7 Jun 2025

Viewed by 76

Abstract

Plane geometry problem solving has been a long-term challenge in mathematical reasoning and symbolic artificial intelligence. With the continued advancement of automated methods, the need for large-scale datasets and rigorous evaluation frameworks has become increasingly critical for benchmarking and guiding system development. However, [...] Read more.

Plane geometry problem solving has been a long-term challenge in mathematical reasoning and symbolic artificial intelligence. With the continued advancement of automated methods, the need for large-scale datasets and rigorous evaluation frameworks has become increasingly critical for benchmarking and guiding system development. However, existing resources often lack sufficient scale, systematic difficulty modeling, and quantifiable, process-based evaluation metrics. To address these limitations, we propose FGeo-Eval, a comprehensive evaluation system for plane geometry problem solving, and introduce the FormalGeo30K dataset, an extended dataset derived from FormalGeo7K. The evaluation system includes a problem completion rate metric PCR to assess partial progress, theorem weight computation to quantify knowledge importance, and a difficulty coefficient based on reasoning complexity. By analyzing problem structures and solution dependencies, this system enables fine-grained difficulty stratification and objective performance measurement. Concurrently, FormalGeo30K expands the dataset to 30,540 formally annotated problems, supporting more robust model training and evaluation. Experimental results demonstrate that the proposed metrics effectively evaluate problem difficulty and assess solver capabilities. With the augmented dataset, the average success rate across all difficulty levels for the FGeo-HyperGNet model increases from 77.43% to 85.01%, while the average PCR increases from 88.57% to 91.79%. These contributions provide essential infrastructure for advancing plane geometry reasoning systems, offering standardized benchmarks for model development and guiding optimization of geometry-solving models. Full article

(This article belongs to the Special Issue Symmetry and Asymmetry in Machine Learning)

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18 pages, 5190 KiB

Open AccessArticle

Flow Field Evaluation Method of High Water-Cut Reservoirs Based on K-Means Clustering Algorithm

by Chen Liu, Qihong Feng, Wensheng Zhou, Chi Zhang and Xianmin Zhang

Symmetry 2025, 17(6), 901; https://doi.org/10.3390/sym17060901 - 6 Jun 2025

Viewed by 178

Abstract

In this paper, the concept of symmetry is utilized to evaluate the distribution characteristics of flow fields—that is, flow fields with balanced displacement generally exhibit good spatial symmetry. In the late stage of water-flooding reservoir development, identifying flow field distribution and implementing targeted [...] Read more.

In this paper, the concept of symmetry is utilized to evaluate the distribution characteristics of flow fields—that is, flow fields with balanced displacement generally exhibit good spatial symmetry. In the late stage of water-flooding reservoir development, identifying flow field distribution and implementing targeted adjustments are crucial for improving development efficiency and enhancing oil recovery. This study establishes a quantitative evaluation index system integrating both static geological and dynamic production factors to comprehensively characterize flow field distribution in ultra-high water-cut reservoirs. The system incorporates residual oil potential abundance, water-flooding ratio, and water influx intensity as key indicators. A flow field classification method based on the K-Means clustering algorithm was proposed, with the Davies–Bouldin index applied to evaluate clustering validity. The approach was validated using the Egg model, where the flow field was effectively classified into four types: inefficient retention field, effective displacement field, dominant displacement field, and extreme displacement field. Adjustment measures were then applied based on classification results. The findings demonstrate that the proposed method weakens dominant displacement areas while expanding effective and inefficient displacement zones, leading to a 1.1 percentage point increase in recovery factor. This research provides a practical and quantitative tool for flow field diagnosis and adjustment, offering valuable technical guidance for managing ultra-high water-cut reservoirs. Full article

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

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25 pages, 1603 KiB

Open AccessArticle

Decoding the Symmetry of Influence: A Machine Learning Study of Reading Exposure and Social Attitudes Across Social Groups

by Yuanqing Wang, Hao Chen, Wei Zhao and Qixia Zhang

Symmetry 2025, 17(6), 900; https://doi.org/10.3390/sym17060900 - 6 Jun 2025

Viewed by 124

Abstract

The relationship between reading exposure and social attitudes across demographic groups remains a pivotal yet underexplored topic in computational social science. This study adopts a machine learning framework to examine the symmetry of reading’s influence on social attitude formation. Models including Random Forest, [...] Read more.

The relationship between reading exposure and social attitudes across demographic groups remains a pivotal yet underexplored topic in computational social science. This study adopts a machine learning framework to examine the symmetry of reading’s influence on social attitude formation. Models including Random Forest, XGBoost, LightGBM, and linear regression were employed on data from the 2021 Chinese General Social Survey (CGSS). The results show that reading volume is a key predictor of social attitudes. Moreover, a SHAP-based subgroup analysis revealed that the impact of reading exposure remained stable across gender groups, indicating a symmetric pattern of cognitive influence. This study proposes a methodological pipeline for assessing the symmetry of feature importance in social data, offering actionable insights for researchers and policymakers into the equitable role of media consumption in shaping social cognition. Full article

(This article belongs to the Special Issue Applications Based on Symmetry in Machine Learning and Data Mining)

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13 pages, 2360 KiB

Open AccessArticle

New Bayesian Estimation Method Based on Symmetric Projection Space and Particle Flow Velocity

by Juan Tan, Zijun Wu and Lijuan Chen

Symmetry 2025, 17(6), 899; https://doi.org/10.3390/sym17060899 - 6 Jun 2025

Viewed by 149

Abstract

Aiming at the state estimation problem of nonlinear systems (NLSs), the traditional typical nonlinear filtering methods (e.g., Particle Filter, PF) have large errors in system state, resulting in low accuracy and high computational speed. To perfect the imperfections, a new Bayesian estimation method [...] Read more.

Aiming at the state estimation problem of nonlinear systems (NLSs), the traditional typical nonlinear filtering methods (e.g., Particle Filter, PF) have large errors in system state, resulting in low accuracy and high computational speed. To perfect the imperfections, a new Bayesian estimation method based on particle flow velocity (PFV-BEM) is proposed in this paper. Firstly, a symmetrical projection space based on the state information is selected, the basis function is determined by a set of Fourier series with symmetric properties, the state update is carried out according to the projection principle to calculate the prior information of the state, and select its particle points. Secondly, the particle flow velocity is defined, which describes the evolution process of random samples from the prior distribution to the posterior distribution. The posterior information of the state is calculated by solving the parameters related to the particle flow velocity. Finally, the estimated mean and standard deviation of the state are solved. Simulation experiments are carried out based on two instances of one-dimensional general nonlinear examples and multi-target motion tracking, The newly proposed algorithm is compared with the Particle Filter (PF), and the simulation results clearly indicate the feasibility of this novel Bayesian estimation algorithm. Full article

(This article belongs to the Section Mathematics)

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19 pages, 964 KiB

Open AccessArticle

SGMNet: A Supervised Seeded Graph-Matching Method for Cyber Threat Hunting

by Chenghong Zhang and Lingyin Su

Symmetry 2025, 17(6), 898; https://doi.org/10.3390/sym17060898 - 6 Jun 2025

Viewed by 173

Abstract

Proactively hunting known attack behaviors within system logs, termed threat hunting, is gaining traction in cybersecurity. Existing methods typically rely on constructing a query graph representing known attack patterns and identifying it as a subgraph within a system-wide provenance graph. However, the large [...] Read more.

Proactively hunting known attack behaviors within system logs, termed threat hunting, is gaining traction in cybersecurity. Existing methods typically rely on constructing a query graph representing known attack patterns and identifying it as a subgraph within a system-wide provenance graph. However, the large scale and redundancy of provenance data lead to poor matching efficiency and high false-positive rates. To address these issues, this paper introduces SGMNet, a supervised seeded graph-matching network designed for efficient and accurate threat hunting. By selecting indicators of compromise (IOCs) as initial seed nodes, SGMNet extracts compact subgraphs from large-scale provenance graphs, significantly reducing graph size and complexity. It then learns adaptive node-expansion strategies to capture relevant context while suppressing irrelevant noise. Experiments on four real-world system log datasets demonstrate that SGMNet achieves a runtime reduction of over 60% compared to baseline methods, while reducing false positives by 35.2% on average. These results validate that SGMNet not only improves computational efficiency but also enhances detection precision, making it well suited for real-time threat hunting in large-scale environments. Full article

(This article belongs to the Section Computer)

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22 pages, 7744 KiB

Open AccessArticle

Optimization and Design of Built-In U-Shaped Permanent Magnet and Salient-Pole Electromagnetic Hybrid Excitation Generator for Vehicles

by Keqi Chen, Shilun Ma, Changwei Li, Yongyi Wu and Jianwei Ma

Symmetry 2025, 17(6), 897; https://doi.org/10.3390/sym17060897 - 6 Jun 2025

Viewed by 115

Abstract

In this paper, the concept of symmetry is utilized to optimize the structural parameters and output characteristics of the generator design—that is, the construction and solution of the equivalent magnetic circuit method for the hybrid excitation generator are symmetrical. To address the issues [...] Read more.

In this paper, the concept of symmetry is utilized to optimize the structural parameters and output characteristics of the generator design—that is, the construction and solution of the equivalent magnetic circuit method for the hybrid excitation generator are symmetrical. To address the issues of high excitation loss and low power density in purely electrically excited generators, as well as the difficulty in adjusting the magnetic field in purely permanent magnet generators, a new topology for a built-in permanent magnet and salient-pole electromagnetic hybrid excitation generator is proposed. Firstly, an equivalent magnetic circuit model of the generator is established. Secondly, expressions are derived to describe the relationships between the dimensions of the salient-pole rotor and the permanent magnets and the generator’s no-load induced electromotive force, cogging torque, and air gap flux density. These expressions are then used to analyze the structural parameters that influence the generator’s performance. Thirdly, optimization targets are selected through sensitivity analysis, with the no-load induced electromotive force, cogging torque, and air gap flux density serving as the optimization objectives. A multi-objective genetic algorithm is employed to optimize these parameters and determine the optimal structural matching parameters for the generator. As a result, the optimized no-load induced electromotive force increased from 18.96 V to 20.14 V, representing a 6.22% improvement; the cogging torque decreased from 177.08 mN·m to 90.52 mN·m, a 48.88% reduction; the air gap flux density increased from 0.789 T to 0.829 T, a 5.07% improvement; and the air gap flux density waveform distortion rate decreased from 6.22% to 2.38%, a 39.3% reduction. Finally, a prototype is fabricated and experimentally tested, validating the accuracy of the simulation analysis, the feasibility of the optimization method, and the rationality of the generator design. Therefore, the proposed topology and optimization method can effectively enhance the output performance of the generator, providing a valuable theoretical reference for the design of hybrid excitation generators for vehicles. Full article

(This article belongs to the Section Engineering and Materials)

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24 pages, 576 KiB

Open AccessArticle

Asymmetry in the Mean Free Path of Neutrinos in Hot Neutron Matter Under Strong Magnetic Fields

by Eduardo Bauer and Vanesa D. Olivera

Symmetry 2025, 17(6), 896; https://doi.org/10.3390/sym17060896 - 6 Jun 2025

Viewed by 93

Abstract

We investigate the asymmetry in the mean free path of massive neutrinos propagating through hot neutron matter under strong magnetic fields. The system is studied at temperatures up to 30 MeV and baryon densities up to ρ/ρ₀ = 2.5, where [...] Read more.

We investigate the asymmetry in the mean free path of massive neutrinos propagating through hot neutron matter under strong magnetic fields. The system is studied at temperatures up to 30 MeV and baryon densities up to ρ/ρ₀ = 2.5, where ρ₀ is the nuclear saturation density. Magnetic field strengths up to B = 10¹⁸ G are considered. We analyze three different equations of state: one corresponding to a non-interacting Fermi gas and two derived from Skyrme-type interactions. The impact of a finite neutrino mass is assessed and found to be negligible within the energy range considered. The neutrino mean free path is computed for various angles of incidence with respect to the magnetic field direction, revealing a clear angular asymmetry. We show that quantum interference terms contribute significantly to this asymmetry, enhancing neutrino emission in directions perpendicular to the magnetic field at high densities. This result contrasts with previous expectations and suggests a revised interpretation of neutrino transport in magnetized nuclear matter. Full article

(This article belongs to the Special Issue Neutrino Physics and Symmetries)

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8 pages, 1018 KiB

Open AccessCommunication

Construction of a Symmetrical Bi-Hydroxamate Metal–Organic Framework with Chemical Robustness

by Yue Dong, Chaozhi Xiong, Zhen-Wu Shao and Chong Liu

Symmetry 2025, 17(6), 895; https://doi.org/10.3390/sym17060895 - 6 Jun 2025

Viewed by 153

Abstract

Recently, the emerging class of hydroxamate-based metal–organic frameworks (MOFs) has demonstrated significant structural diversity and chemical robustness, both essential for potential applications. Combining the favorable hard–hard Bi-O interactions and chelating chemistry of hydroxamate groups, a rigid and symmetrical three-dimensional bismuth-hydroxamate metal–organic framework was [...] Read more.

Recently, the emerging class of hydroxamate-based metal–organic frameworks (MOFs) has demonstrated significant structural diversity and chemical robustness, both essential for potential applications. Combining the favorable hard–hard Bi-O interactions and chelating chemistry of hydroxamate groups, a rigid and symmetrical three-dimensional bismuth-hydroxamate metal–organic framework was successfully prepared via solvothermal synthesis and structurally elucidated via X-ray crystallography. The MOF, namely SUM-91 (SUM = Sichuan University Materials), features one-dimensional Bi-oxo secondary building blocks (SBUs), which are bridged by chelating 1,4-benzenedihydroxamate linkers. With the demonstrated permanent porosity and molecular sieving effect (CO₂ vs. N₂), SUM-91 was also found to be stable under harsh chemical conditions (aqueous solutions with pH = 2–12 and various organic solvents). As the structural robustness of SUM-91 could be attributed to the finetuning of the coordinative sphere of Bi centers, this work shed light on the further development of (ultra-)microporous materials with high stability and selective adsorption properties. Full article

(This article belongs to the Section Chemistry: Symmetry/Asymmetry)

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12 pages, 360 KiB

Open AccessArticle

Identification of Source Term from Part of the Boundary Conditions

by Yunjie Ma

Symmetry 2025, 17(6), 894; https://doi.org/10.3390/sym17060894 - 6 Jun 2025

Viewed by 129

Abstract

This paper identifies a source term depending on spatial variable in a heat equation from just part of the boundary conditions. The measurement data are specified at an internal moment of time. The ill-posedness of the problem is higher than most of the [...] Read more.

This paper identifies a source term depending on spatial variable in a heat equation from just part of the boundary conditions. The measurement data are specified at an internal moment of time. The ill-posedness of the problem is higher than most of the previous source identification problems. This is because the problem becomes a noncharacteristic Cauchy problem for the heat equation if the source term is given, which is known as severely ill-posed. The method of fundamental solutions (MFS) in conjunction with the classical Tikhonov regularization method is proposed to reconstruct a stable approximation. The fundamental solutions for the heat equation are spherically symmetric in spatial variable and satisfy the equation automatically, and thus only the boundary conditions need to be satisfied. This characteristic allows the discretization to be performed only on boundary-like geometry and improve the computational efficiency. In this paper, several numerical examples are listed to show the feasibility and effectiveness of the suggested method. Full article

(This article belongs to the Section Mathematics)

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Symmetry, Volume 17, Issue 6 (June 2025) – 121 articles

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