Symmetry

16 pages, 3804 KiB

Open AccessArticle

Study of Numerical Modeling Method for Precooling of Spherical Horticultural Produce Stacked Symmetrically in Vented Package

by Xifang Wang, Zhongyang Fan, Chuanhui Zhu and Hongbin Liu

Symmetry 2025, 17(6), 810; https://doi.org/10.3390/sym17060810 - 22 May 2025

Abstract

Numerical simulation has become a pivotal tool for analyzing airflow dynamics and temperature patterns during the precooling of postharvest horticultural products stacked in vented package. In this study, a three-dimensional mathematical model for iceberg lettuces stacked symmetrically in plastic crate was developed. The [...] Read more.

Numerical simulation has become a pivotal tool for analyzing airflow dynamics and temperature patterns during the precooling of postharvest horticultural products stacked in vented package. In this study, a three-dimensional mathematical model for iceberg lettuces stacked symmetrically in plastic crate was developed. The influence of the physical model at different gap sizes on the simulation accuracy was studied by assessing wall drag coefficient, airflow distribution, and heat transfer efficiency. The results show a reasonable decrease in the drag coefficient with an increasing gap size to 6 mm in terms of airflow distribution inside the plastic crate; any further increase in gap size and the average airflow velocity in both windward and leeward sides decreases rapidly. Interestingly, the gap size exhibited a limited impact on heat transfer characteristics during the cooling process. Thus, 6 mm was found to be the optimal distance to ensure good accuracy in simulation results and reduce the complexity of grid division. The numerical model was verified by experimental data. Moreover, the validation confirms good consistency between the simulated predictions and experimental measurements. This study provides a theoretical basis for establishing a reliable numerical model for the precooling of agricultural produce stacked symmetrically in a vented package. Full article

(This article belongs to the Section Engineering and Materials)

11 pages, 277 KiB

Open AccessArticle

On the Theory of Nonlinear Landau Damping

by Leon Kos, Ivona Vasileska and Davy D. Tskhakaya

Symmetry 2025, 17(6), 809; https://doi.org/10.3390/sym17060809 - 22 May 2025

Abstract

An exact solution of the collisionless time-dependent Vlasov equation is found. For the first time in a century, an analytical solution to the one-dimensional time-dependent Vlasov–Boltzmann equation has been found. It has been found that instead of the widely discussed damping, waves are [...] Read more.

An exact solution of the collisionless time-dependent Vlasov equation is found. For the first time in a century, an analytical solution to the one-dimensional time-dependent Vlasov–Boltzmann equation has been found. It has been found that instead of the widely discussed damping, waves are subject to instability. By means of this solution, the behavior of the Langmuir waves in the nonlinear stage is considered. A symmetry method is found that allows us to establish the dependence on time of the desired quantity based on the dependence on the previous time. The analysis is restricted by the consideration of the first nonlinear approximation—keeping the second power of the electric strength. It is shown that in general the waves with finite amplitudes are not subjected to the damping. Conditions have been found under which waves can be unstable. Full article

(This article belongs to the Section Mathematics)

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17 pages, 5610 KiB

Open AccessArticle

The Detection of Maize Leaf Disease Based on an Improved Real-Time Detection Transformer Model

by Jianbin Yao, Zhenghao Zhu, Mengqi Yuan, Linyuan Li and Meijia Wang

Symmetry 2025, 17(6), 808; https://doi.org/10.3390/sym17060808 - 22 May 2025

Abstract

Maize is one of the most important global crops. It is highly susceptible to diseases during its growth process, meaning that the timely detection and prevention of maize diseases is critically important. However, simple deep learning classification tasks do not allow for the [...] Read more.

Maize is one of the most important global crops. It is highly susceptible to diseases during its growth process, meaning that the timely detection and prevention of maize diseases is critically important. However, simple deep learning classification tasks do not allow for the accurate identification of multiple diseases present in a single leaf, and the existing RT-DETR (Real-Time Detection Transformer) detection methods suffer from issues such as excessive model parameters and inaccurate recognition of multi-scale features on maize leaves. The aim of this paper is to address these challenges by proposing an improved RT-DETR model. The model enhances the feature extraction capability by introducing a DAttention (Deformable Attention) module and optimizes the feature fusion process through the symmetry structure of spatial and channel in the SCConv (Spatial and Channel Reconstruction Convolution) module. In addition, the backbone network of the model is reconfigured, which effectively reduces the parameter size of the model and achieves a balanced symmetry between the model precision and the parameter count. Experimental results demonstrate that the proposed improved model achieves an mAP@0.5 of 92.0% and a detection precision of 89.2%, representing improvements of 7.3% and 8.4%, respectively, compared to the original RT-DETR model. Additionally, the model’s parameter size has been reduced by 18.9 M, leading to a substantial decrease in resource consumption during deployment and underscoring its extensive application potential. Full article

(This article belongs to the Section Life Sciences)

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40 pages, 40212 KiB

Open AccessReview

Fundamentals of Symmetry and Topology: Applications to Materials Science and Condensed Matter Physics

by Mengdi Yin, Jing Zhang and Dimitri D. Vvedensky

Symmetry 2025, 17(6), 807; https://doi.org/10.3390/sym17060807 - 22 May 2025

Abstract

We review the connections between condensed matter physics, symmetry, and topology. Physics goes back to at least the time of Galileo, but condensed matter physics, or solid-state physics, is a much newer, emerging only as a separate subject in the 1940s. The subject [...] Read more.

We review the connections between condensed matter physics, symmetry, and topology. Physics goes back to at least the time of Galileo, but condensed matter physics, or solid-state physics, is a much newer, emerging only as a separate subject in the 1940s. The subject of symmetry, which is the mathematics of groups and representations, only came to the fore with the advent of quantum mechanics. Early applications to crystalline solids include Bloch’s theorem, the symmetry of electronic and phononic energy bands, and selection rules. Topology, on the other hand, did not exist as a mathematical subject before the twentieth century, but has had a profound influence on physics in general, and on condensed matter physics in particular. The quantum Hall effect is recognized as the first solid-state topological phenomenon and, along with the Berry phase, led to the development of topological materials. This, in turn, led to the topological description of energy bands and to the development of topological quantum chemistry and the energy band representation. Topology has also led to the description of martensitic transformations and the shape memory effect in terms of topological transformations. Apart from a concise statement of martensitic transformations, topology provides a fast-screening method for the discovery of new shape-memory materials. We review these phenomena, providing background material in topology and differential geometry to enable the reader to understand applications to topological materials and to materials physics. Full article

(This article belongs to the Special Issue The Benefits That Physics Derives from the Concept of Symmetry)

16 pages, 250 KiB

Open AccessArticle

J˜-Tangent Affine Hypersurfaces with an Induced Almost Paracontact Structure

by Zuzanna Szancer

Symmetry 2025, 17(6), 806; https://doi.org/10.3390/sym17060806 - 22 May 2025

Abstract

The subjects of our study are affine hypersurfaces

f : M \to R^{2 n + 2}

considered with a transversal vector field C, which is

\tilde{J}

-tangent. By

\tilde{J}

we understand the canonical paracomplex structure on [...] Read more.

The subjects of our study are affine hypersurfaces

f : M \to R^{2 n + 2}

considered with a transversal vector field C, which is

\tilde{J}

-tangent. By

\tilde{J}

we understand the canonical paracomplex structure on

R^{2 n + 2}

. The vector field C induces on the hypersurface f an almost paracontact structure

(φ, ξ, η)

. We obtain a complete classification of hypersurfaces admitting a metric induced almost paracontact structure with respect to the second fundamental form. We show that, in this case, the

\tilde{J}

-tangent transversal vector field is restricted to centroaffine and the hypersurface must be a piece of hyperquadric. It is demonstrated that these hyperquadrics have a very specific form. A three-dimensional example is also given. Moreover, we establish an equivalence relation between almost paracontact metric structures, para

α

-contact metric structures, and para

α

-Sasakian structures. Methods of affine differential geometry, as well as paracomplex/paracontact geometry, are used. Full article

(This article belongs to the Section Mathematics)

22 pages, 9548 KiB

Open AccessArticle

A BiGRUSA-ResSE-KAN Hybrid Deep Learning Model for Day-Ahead Electricity Price Prediction

by Nan Yang, Guihong Bi, Yuhong Li, Xiaoling Wang, Zhao Luo and Xin Shen

Symmetry 2025, 17(6), 805; https://doi.org/10.3390/sym17060805 - 22 May 2025

Abstract

In the context of the clean and low-carbon transformation of power systems, addressing the challenge of day-ahead electricity market price prediction issues triggered by the strong stochastic volatility of power supply output due to high-penetration renewable energy integration, as well as problems such [...] Read more.

In the context of the clean and low-carbon transformation of power systems, addressing the challenge of day-ahead electricity market price prediction issues triggered by the strong stochastic volatility of power supply output due to high-penetration renewable energy integration, as well as problems such as limited dataset scales and short market cycles in test sets associated with existing electricity price prediction methods, this paper introduced an innovative prediction approach based on a multi-modal feature fusion and BiGRUSA-ResSE-KAN deep learning model. In the data preprocessing stage, maximum–minimum normalization techniques are employed to process raw electricity price data and exogenous variable data; the complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) and variational mode decomposition (VMD) methods are utilized for multi-modal decomposition of electricity price data to construct a multi-scale electricity price component matrix; and a sliding window mechanism is applied to segment time-series data, forming a three-dimensional input structure for the model. In the feature extraction and prediction stage, the BiGRUSA-ResSE-KAN multi-branch integrated network leverages the synergistic effects of gated recurrent units combined with residual structures and attention mechanisms to achieve deep feature fusion of multi-source heterogeneous data and model complex nonlinear relationships, while further exploring complex coupling patterns in electricity price fluctuations through the knowledge-adaptive network (KAN) module, ultimately outputting 24 h day-ahead electricity price predictions. Finally, verification experiments conducted using test sets spanning two years from five major electricity markets demonstrate that the introduced method effectively enhances the accuracy of day-ahead electricity price prediction, exhibits good applicability across different national electricity markets, and provides robust support for electricity market decision making. Full article

(This article belongs to the Section Computer)

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22 pages, 2168 KiB

Open AccessArticle

Research on Ship Equipment Health State Assessment Method Based on BP Neural Network-Random Forest (BP-RF) and Combined Weighting

by Yuanwei Zeng, Jing Li, Hao Chen, Zhigang Hu and Yanzhou Wu

Symmetry 2025, 17(6), 804; https://doi.org/10.3390/sym17060804 - 22 May 2025

Abstract

In view of the diversity and varying complexity of ship equipment, and the difficulty of existing state assessment methods in effectively handling the differences in the influence of different characteristic parameters on the equipment’s health state, leading to poor evaluation results, this paper [...] Read more.

In view of the diversity and varying complexity of ship equipment, and the difficulty of existing state assessment methods in effectively handling the differences in the influence of different characteristic parameters on the equipment’s health state, leading to poor evaluation results, this paper proposes a ship equipment health state assessment method based on BP-RF and combined weighting. This method utilizes the BP-RF model to mine the implicit relationship between ship equipment feature parameters and state patterns, converting monitoring data into state pattern probability information. A trapezoidal membership function is used to determine the membership degree of each state pattern probability to different health state levels. The combined weighting method, which reflects a symmetric concept, balances expert experience and data information by integrating the subjective and objective weights of each state pattern probability, thus determining the equipment’s health state level. Through a case study of a specific type of ship’s gas turbine, the BP-RF model achieves a diagnostic accuracy of 98.3%, with

F_{1}

scores improved by 5.1%, 5.0%, 8.6%, 9.9%, 6.7%, 3.4%, and 5.3% compared to the BP neural network, RF, Support Vector Machine (SVM), Convolutional Neural Network (CNN), BP-SVM, SVM-RF, and CNN-SVM models, respectively. Additionally, the evaluation results of this method exhibit clear boundaries for each state membership degree, effectively addressing the problem of unbalanced contributions from characteristic parameters and comprehensively reflecting the relative importance and correlation of each parameter. Overall, this method provides a more accurate and comprehensive assessment of ship equipment health compared to other methods, offering reliable support for ship equipment maintenance and assurance. Full article

(This article belongs to the Section Engineering and Materials)

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22 pages, 597 KiB

Open AccessArticle

Dynamics of a Symmetric Seasonal Influenza Model with Variable Recovery, Treatment, and Fear Effects

by Rubayyi T. Alqahtani, Abdelhamid Ajbar and Manal Alqhtani

Symmetry 2025, 17(6), 803; https://doi.org/10.3390/sym17060803 - 22 May 2025

Abstract

This study proposes and examines the dynamics of a susceptible–exposed–infectious–recovered (SEIR) model for the spread of seasonal influenza. The population is categorized into four distinct groups: susceptible (S), exposed (E), infectious (I), and recovered (R) individuals. The symmetric model integrates a bilinear incidence [...] Read more.

This study proposes and examines the dynamics of a susceptible–exposed–infectious–recovered (SEIR) model for the spread of seasonal influenza. The population is categorized into four distinct groups: susceptible (S), exposed (E), infectious (I), and recovered (R) individuals. The symmetric model integrates a bilinear incidence rate alongside a nonlinear recovery rate that depends on the quality of healthcare services. Additionally, it accounts for the impact of fear related to the disease and includes a constant vaccination rate as well as a nonlinear treatment function. The model advances current epidemiological frameworks by simultaneously accounting for these interrelated mechanisms, which are typically studied in isolation. We derive the expression for the basic reproduction number and analyze the essential stability properties of the model. Key analytical results demonstrate that the system exhibits rich dynamic behavior, including backward bifurcation (where stable endemic equilibria persist even when the basic reproduction number is less than one) and Hopf bifurcation. These phenomena emerge from the interplay between fear-induced suppression of transmission, treatment saturation, and healthcare quality. Numerical simulations using Saudi Arabian demographic and epidemiological data quantify how increased fear perception shrinks the bistability region, facilitating eradication. Healthcare capacity improvements, on the other hand, reduce the critical reproduction number threshold while treatment accessibility suppresses infection loads. The model’s practical significance lies in its ability to identify intervention points where small parameter changes yield disproportionate control benefits and evaluate trade-offs between pharmaceutical (vaccination/treatment) and non-pharmaceutical (fear-driven distancing) strategies. This work establishes a versatile framework for public health decision making and the integrated approach offers policymakers a tool to simulate combined intervention scenarios and anticipate nonlinear system responses that simpler models cannot capture. Full article

(This article belongs to the Special Issue Three-Dimensional Dynamical Systems and Symmetry)

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24 pages, 5378 KiB

Open AccessArticle

Assessment of the Measured Mixing Time in a Water Model of Asymmetrical Gas-Stirred Ladle with a Low Gas Flowrate Part II: Effect of the Salt Solution Tracer Volume and Concentration

by Yansong Zhao, Xin Tao, Linbo Li, Zhijie Guo, Hongyu Qi, Jia Wang, Kun Yang, Wanming Lin, Jinping Fan and Chao Chen

Symmetry 2025, 17(5), 802; https://doi.org/10.3390/sym17050802 - 21 May 2025

Abstract

Mixing time, as a key parameter for evaluating ladle refining efficiency, has long attracted extensive attention from researchers. In typical experimental studies, salt solution tracers are introduced into ladle water models to assess the degree of mixing within the ladle. Previous studies have [...] Read more.

Mixing time, as a key parameter for evaluating ladle refining efficiency, has long attracted extensive attention from researchers. In typical experimental studies, salt solution tracers are introduced into ladle water models to assess the degree of mixing within the ladle. Previous studies have demonstrated that the volume of tracer can significantly influence the measured mixing time. However, the gas flow rates employed in these studies are generally relatively high, whereas, in industrial operations, especially during final composition adjustments, lower gas flow rates are often applied. To systematically investigate the effect of the salt solution tracer volume on the mixing efficiency in a ladle water model under asymmetrical gas stirring with a low gas flow rate, a 1:3-scaled water model was developed based on a 130-ton industrial ladle. The mixing behaviors corresponding to different tracer volumes were comprehensively analyzed. The results indicate that the relationship between tracer volume and mixing time is non-monotonic. As the tracer volume increases, the mixing time first decreases and then increases, reaching a minimum at 185 mL. When the tracer volume was small, the dimensionless concentration curves at Monitoring Point 4 exhibited two distinct patterns: A parabolic profile, which was when the tracer initially moved through the left and central regions and then slowly crossed the gas plume to reach the monitoring point. A sinusoidal profile, which was when the tracer predominantly circulated along the right side of the ladle. When the tracer volume exceeded 277 mL, the concentration curves at Monitoring Point 4 consistently exhibited a sinusoidal pattern. Compared with moderate gas flow conditions (8.3 L/min), the peak concentration at Monitoring Point 3 was significantly lower under a low gas flow (2.3 L/min), and the overall mixing time was longer, indicating reduced mixing efficiency. Based on the findings, a recommended tracer volume range of 185–277 mL is proposed for low gas flow conditions (2.3 L/min) to achieve accurate and efficient mixing time measurements with minimal disturbance to the flow field. It was also observed that when the tracer concentration was relatively low, the mixing behavior throughout the ladle became more uniform. Full article

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

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19 pages, 38532 KiB

Open AccessArticle

A Novel 2D Hyperchaotic Map with Homogeneous Multistability and Its Application in Image Encryption

by Xin Huang, Wenhao Yan, Wenjie Dong and Qun Ding

Symmetry 2025, 17(5), 801; https://doi.org/10.3390/sym17050801 - 21 May 2025

Abstract

This study proposes a novel two-dimensional hyperchaotic map model based on an orthogonal feedback mechanism, exhibiting dynamic behaviors with multistable characteristics and high complexity. By analyzing the homogeneous multistability of the system, it is revealed that the initial states determine the positions of [...] Read more.

This study proposes a novel two-dimensional hyperchaotic map model based on an orthogonal feedback mechanism, exhibiting dynamic behaviors with multistable characteristics and high complexity. By analyzing the homogeneous multistability of the system, it is revealed that the initial states determine the positions of attractors. An image encryption scheme for color images is developed by integrating confusion and diffusion strategies with this hyperchaotic map. The effectiveness of the proposed scheme in enabling secure image transmission is validated through comprehensive numerical simulations and rigorous security assessments. Full article

(This article belongs to the Section Computer)

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17 pages, 1222 KiB

Open AccessArticle

The Mittag-Leffler–Caputo–Fabrizio Fractional Derivative and Its Numerical Approach

by Manal Alqhtani, Lakhlifa Sadek and Khaled Mohammed Saad

Symmetry 2025, 17(5), 800; https://doi.org/10.3390/sym17050800 - 21 May 2025

Abstract

This study introduces a novel fractional-order derivative, termed the Mittag-Leffler–Caputo–Fabrizio (MLCF) fractional derivative, which is characterized by a singular kernel. Symmetry plays a key role in the structure and behavior of fractional operators, and our formulation reflects this by incorporating symmetric properties of [...] Read more.

This study introduces a novel fractional-order derivative, termed the Mittag-Leffler–Caputo–Fabrizio (MLCF) fractional derivative, which is characterized by a singular kernel. Symmetry plays a key role in the structure and behavior of fractional operators, and our formulation reflects this by incorporating symmetric properties of the Mittag-Leffler function and its integral representation. To numerically approximate the MLCF derivative, we apply a two-point finite forward difference scheme to estimate the first-order derivative of the function

u (λ)

within the integral component of the definition. This leads to the construction of a new numerical differentiation scheme. Our analysis demonstrates that the proposed approximation exhibits first-order convergence, with absolute errors decreasing as the time step size h diminishes. These errors are quantified by comparing our numerical results with exact analytical solutions, reinforcing the accuracy of the method. Full article

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

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18 pages, 312 KiB

Open AccessArticle

Lipschitz and Second-Order Regularities for Non-Homogeneous Degenerate Nonlinear Parabolic Equations in the Heisenberg Group

by Huiying Wang, Chengwei Yu, Zhiqiang Zhang and Yue Zeng

Symmetry 2025, 17(5), 799; https://doi.org/10.3390/sym17050799 - 21 May 2025

Abstract

In the Heisenberg group

H^{n}

, we establish the local regularity theory for weak solutions to non-homogeneous degenerate nonlinear parabolic equations of the form [...] Read more.

In the Heisenberg group

H^{n}

, we establish the local regularity theory for weak solutions to non-homogeneous degenerate nonlinear parabolic equations of the form

\partial_{t} u - \sum_{i = 1}^{2 n} X_{i} A_{i} (X u) = K (x, t, u, X u)

, where the nonlinear structure is modeled on non-homogeneous parabolic p-Laplacian-type operators. Specifically, we prove two main local regularities: (i) For

2 \leq p \leq 4

, we establish the local Lipschitz regularity (

u \in C_{loc}^{0, 1}

), with the horizontal gradient satisfying

X u \in L_{loc}^{\infty}

; (ii) For

2 \leq p < 3

, we establish the local second-order horizontal Sobolev regularity (

u \in H W_{loc}^{2, 2}

), with the second-order horizontal derivative satisfying

X X u \in L_{loc}^{2}

. These results solve an open problem proposed by Capogna et al. Full article

(This article belongs to the Special Issue Advances in Mathematical Models and Partial Differential Equations: 2nd Edition)

27 pages, 10202 KiB

Open AccessArticle

WIGformer: Wavelet-Based Illumination-Guided Transformer

by Wensheng Cao, Tianyu Yan, Zhile Li and Jiongyao Ye

Symmetry 2025, 17(5), 798; https://doi.org/10.3390/sym17050798 - 20 May 2025

Abstract

Low-light image enhancement remains a challenging task in computer vision due to the complex interplay of noise, asymmetrical artifacts, illumination non-uniformity, and detail preservation. Existing methods such as traditional histogram equalization, gamma correction, and Retinex-based approaches often struggle to balance contrast improvement and [...] Read more.

Low-light image enhancement remains a challenging task in computer vision due to the complex interplay of noise, asymmetrical artifacts, illumination non-uniformity, and detail preservation. Existing methods such as traditional histogram equalization, gamma correction, and Retinex-based approaches often struggle to balance contrast improvement and naturalness preservation. Deep learning methods such as CNNs and transformers have shown promise, but face limitations in modeling multi-scale illumination and long-range dependencies. To address these issues, we propose WIGformer, a novel wavelet-based illumination-guided transformer framework for low-light image enhancement. The proposed method extends the single-stage Retinex theory to explicitly model noise in both reflectance and illumination components. It introduces a wavelet illumination estimator with a Wavelet Feature Enhancement Convolution (WFEConv) module to capture multi-scale illumination features and an illumination feature-guided corruption restorer with an Illumination-Guided Enhanced Multihead Self-Attention (IGEMSA) mechanism. WIGformer leverages the symmetry properties of wavelet transforms to achieve multi-scale illumination estimation, ensuring balanced feature extraction across different frequency bands. The IGEMSA mechanism integrates adaptive feature refinement and illumination guidance to suppress noise and artifacts while preserving fine details. The same mechanism allows us to further exploit symmetrical dependencies between illumination and reflectance components, enabling robust and natural enhancement of low-light images. Extensive experiments on the LOL-V1, LOL-V2-Real, and LOL-V2-Synthetic datasets demonstrate that WIGformer achieves state-of-the-art performance and outperforms existing methods, with PSNR improvements of up to 26.12 dB and an SSIM score of 0.935. The qualitative results demonstrate WIGformer’s superior capability to not only restore natural illumination but also maintain structural symmetry in challenging conditions, preserving balanced luminance distributions and geometric regularities that are characteristic of properly exposed natural scenes. Full article

(This article belongs to the Section Computer)

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19 pages, 758 KiB

Open AccessArticle

Closed Forms and Structural Properties of Lucas Matrices Derived from Tridiagonal Toeplitz Matrices

by Fikri Koken and Muaz Aksoy

Symmetry 2025, 17(5), 797; https://doi.org/10.3390/sym17050797 - 20 May 2025

Abstract

This study investigates the closed forms of Lucas matrices, with a particular emphasis on the

n^{t h}

powers of the tridiagonal symmetric Toeplitz matrix

S_{4} (x, y)

, whose entries are associated with Lucas numbers

L_{n}

. [...] Read more.

This study investigates the closed forms of Lucas matrices, with a particular emphasis on the

n^{t h}

powers of the tridiagonal symmetric Toeplitz matrix

S_{4} (x, y)

, whose entries are associated with Lucas numbers

L_{n}

. The analysis extends Filipponi’s foundational work by examining distinct cases of ordered pairs

(x, y)

, thereby determining the precise conditions under which

S_{4} (x, y)

qualifies as a Fibonacci–Lucas matrix. Furthermore, it identifies specific conditions under which

S_{4} (x, y)

can be classified as any Fibonacci–Lucas matrix. These findings contribute to the theoretical framework of Fibonacci–Lucas matrices and provide novel insights into their structural properties. Full article

(This article belongs to the Section Mathematics)

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24 pages, 1736 KiB

Open AccessArticle

ProFusion: Multimodal Prototypical Networks for Few-Shot Learning with Feature Fusion

by Jia Zhao, Ziyang Cao, Huiling Wang, Xu Wang and Yingzhou Chen

Symmetry 2025, 17(5), 796; https://doi.org/10.3390/sym17050796 - 20 May 2025

Abstract

Existing few-shot learning models leverage vision-language pre-trained models to alleviate the data scarcity problem. However, such models usually process visual and text information separately, which causes still inherent disparities between cross-modal features. Therefore, we propose the ProFusion model, which leverages multimodal pre-trained models [...] Read more.

Existing few-shot learning models leverage vision-language pre-trained models to alleviate the data scarcity problem. However, such models usually process visual and text information separately, which causes still inherent disparities between cross-modal features. Therefore, we propose the ProFusion model, which leverages multimodal pre-trained models and prototypical networks to construct multiple prototypes. Specifically, ProFusion generates image and text prototypes symmetrically using the visual encoder and text encoder, while integrating visual and text information through the fusion module to create more expressive multimodal feature fusion prototypes. Additionally, we introduce the alignment module to ensure consistency between image and text prototypes. During inference, ProFusion calculates the similarity of test images to the three types of prototypes separately and applies a weighted sum to generate the final prediction. Experiments demonstrate that ProFusion performs outstanding classification tasks on 15 benchmark datasets. Full article

(This article belongs to the Special Issue Symmetry and Asymmetry in Computer Vision and Graphics)

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34 pages, 356 KiB

Open AccessArticle

Anisotropic Four-Dimensional Spaces of Real Numbers

by Maksut M. Abenov, Mars B. Gabbassov, Tolybay Z. Kuanov and Berik I. Tuleuov

Symmetry 2025, 17(5), 795; https://doi.org/10.3390/sym17050795 - 20 May 2025

Abstract

This article constructs all the anisotropic spaces of four-dimensional numbers in which the commutative and associative operations of addition and multiplication are defined. In this case, so-called “zero divisors” appear in these spaces. The structures of zero divisors in each space are described [...] Read more.

This article constructs all the anisotropic spaces of four-dimensional numbers in which the commutative and associative operations of addition and multiplication are defined. In this case, so-called “zero divisors” appear in these spaces. The structures of zero divisors in each space are described and their properties are investigated. It is shown that there are two types of zero divisors and they form a two-dimensional subspace of the four-dimensional space. A space of 4 × 4 matrices is constructed that is isomorphic to the space of four-dimensional numbers. The concept of the spectrum of a four-dimensional number is introduced and a bijective mapping between four-dimensional numbers and their spectra is constructed. Thanks to this, methods for solving linear and quadratic equations in four-dimensional spaces are developed. It is proven that a quadratic equation in a four-dimensional space generally has four roots. The concept of the spectral norm is introduced in the space of four-dimensional numbers and the equivalence of the spectral norm to the Euclidean norm is proved. Full article

(This article belongs to the Section Mathematics)

16 pages, 1509 KiB

Open AccessArticle

A Reliable Deep Learning Model for ECG Interpretation: Mitigating Overconfidence and Direct Uncertainty Quantification

by Xuedong Li, Qingxiao Zheng, Shibin Zhang, Shipeng Fu, Yingke Chen and Ke Ye

Symmetry 2025, 17(5), 794; https://doi.org/10.3390/sym17050794 - 20 May 2025

Abstract

Electrocardiogram (ECG) interpretation using deep learning models holds immense potential for improving cardiac diagnosis. However, existing models often suffer from overconfident predictions and lack the capability to directly quantify uncertainty, leading to unreliable clinical guidance. To address this challenge, we propose a model [...] Read more.

Electrocardiogram (ECG) interpretation using deep learning models holds immense potential for improving cardiac diagnosis. However, existing models often suffer from overconfident predictions and lack the capability to directly quantify uncertainty, leading to unreliable clinical guidance. To address this challenge, we propose a model for uncertainty-aware ECG interpretation. The model employs a deep convolutional architecture with max-pooling residual modules to capture both local and global spatiotemporal features from raw ECG signals. The architectural design respects the symmetry inherent in ECG waveforms—such as periodicity and morphological consistency across cardiac cycles—enabling the network to extract clinically relevant features more effectively. Then, unlike conventional models that rely on softmax-based probability outputs, our approach parameterizes class distributions using the Dirichlet distribution, while Subjective Logic translates these parameters into interpretable belief masses and uncertainty scores. We evaluate the model on the PhysioNet Challenge 2017 dataset, our model achieves an accuracy of 86.12%, an F1 score of 83.14%, a Precision-Recall Area Under the Curve (PR-AUC) of 85.25%, and a Receiver Operating Characteristic Area Under the Curve (ROC-AUC) of 92.87%—outperforming baseline models in three out of four metrics. Critically, the model reduces overconfidence to 0.59% (compared to 12–22% in softmax-based baselines), aligning prediction confidence with true accuracy. By progressively increasing the uncertainty threshold u, the model dynamically filters low-confidence predictions, leading to consistently improved performance—reaching up to 93.59% accuracy, 93.22% F1 score, 89.17% PR-AUC, and 95.10% ROC-AUC at u = 0.1. These results validate the model’s capacity for reliable ECG interpretation while leveraging physiological signal symmetry for enhanced feature extraction. Full article

(This article belongs to the Section Computer)

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19 pages, 6004 KiB

Open AccessArticle

Remote Sensing Image Change Detection Based on Dynamic Adaptive Context Attention

by Yong Xie, Yixuan Wang, Xin Wang, Yin Tan and Qin Qin

Symmetry 2025, 17(5), 793; https://doi.org/10.3390/sym17050793 - 20 May 2025

Abstract

Although some progress has been made in deep learning-based remote sensing image change detection, the complexity of scenes and the diversity of changes in remote sensing images lead to challenges related to background interference. For instance, remote sensing images typically contain numerous background [...] Read more.

Although some progress has been made in deep learning-based remote sensing image change detection, the complexity of scenes and the diversity of changes in remote sensing images lead to challenges related to background interference. For instance, remote sensing images typically contain numerous background regions, while the actual change regions constitute only a small proportion of the overall image. To address these challenges in remote sensing image change detection, this paper proposes a Dynamic Adaptive Context Attention Network (DACA-Net) based on an exchanging dual encoder–decoder (EDED) architecture. The core innovation of DACA-Net is the development of a novel Dynamic Adaptive Context Attention Module (DACAM), which learns attention weights and automatically adjusts the appropriate scale according to the features present in remote sensing images. By fusing multi-scale contextual features, DACAM effectively captures information regarding changes within these images. In addition, DACA-Net adopts an EDED architectural design, where the conventional convolutional modules in the EDED framework are replaced by DACAM modules. Unlike the original EDED architecture, DACAM modules are embedded after each encoder unit, enabling dynamic recalibration of T1/T2 features and cross-temporal information interaction. This design facilitates the capture of fine-grained change features at multiple scales. This architecture not only facilitates the extraction of discriminative features but also promotes a form of structural symmetry in the processing pipeline, contributing to more balanced and consistent feature representations. To validate the applicability of our proposed method in real-world scenarios, we constructed an Unmanned Aerial Vehicle (UAV) remote sensing dataset named the Guangxi Beihai Coast Nature Reserves (GBCNR). Extensive experiments conducted on three public datasets and our GBCNR dataset demonstrate that the proposed DACA-Net achieves strong performance across various evaluation metrics. For example, it attains an F1 score (F1) of 72.04% and a precision(P) of 66.59% on the GBCNR dataset, representing improvements of 3.94% and 4.72% over state-of-the-art methods such as semantic guidance and spatial localization network (SGSLN) and bi-temporal image Transformer (BIT), respectively. These results verify that the proposed network significantly enhances the ability to detect critical change regions and improves generalization performance. Full article

(This article belongs to the Section Computer)

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18 pages, 1890 KiB

Open AccessArticle

Symmetry-Entropy-Constrained Matrix Fusion for Dynamic Dam-Break Emergency Planning

by Shuai Liu, Dewei Yang, Hao Hu and Junping Wang

Symmetry 2025, 17(5), 792; https://doi.org/10.3390/sym17050792 - 20 May 2025

Abstract

Existing studies on ontology evolution lack automated mechanisms to balance semantic coherence and adaptability under real-time uncertainties, particularly in resolving spatiotemporal asymmetry and multidimensional coupling imbalances in dam-break scenarios. Traditional methods such as WordNet’s tree symmetry and FrameNet’s frame symmetry fail to formalize [...] Read more.

Existing studies on ontology evolution lack automated mechanisms to balance semantic coherence and adaptability under real-time uncertainties, particularly in resolving spatiotemporal asymmetry and multidimensional coupling imbalances in dam-break scenarios. Traditional methods such as WordNet’s tree symmetry and FrameNet’s frame symmetry fail to formalize dynamic adjustments through quantitative metrics, leading to path dependency and delayed responses. This study addresses this gap by introducing a novel symmetry-entropy-constrained matrix fusion algorithm, which integrates algebraic direct sum operations and Hadamard product with entropy-driven adaptive weighting. The original contribution lies in the symmetry entropy metric, which quantifies structural deviations during fusion to systematically balance semantic stability and adaptability. This work formalizes ontology evolution as a symmetry-driven optimization process. Experimental results demonstrate that shared concepts between ontologies (s = 3) reduce structural asymmetry by 25% compared to ontologies (s = 1), while case studies validate the algorithm’s ability to reconcile discrepancies between theoretical models and practical challenges in evacuation efficiency and crowd dynamics. This advancement promotes the evolution of traditional emergency management systems towards an adaptive intelligent form. Full article

(This article belongs to the Section Mathematics)

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Symmetry

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