Symmetry

19 pages, 1620 KB

Open AccessArticle

Secure Quantum Teleportation of Squeezed Thermal States

by Alexei Zubarev, Marina Cuzminschi and Aurelian Isar

Symmetry 2025, 17(11), 1804; https://doi.org/10.3390/sym17111804 (registering DOI) - 26 Oct 2025

Quantum teleportation is a fundamental protocol in quantum information science. It represents a critical resource for quantum communication and distributed quantum computing. We derive an analytical expression of the fidelity of teleportation of an input squeezed thermal state using for teleportation a bipartite [...] Read more.

Quantum teleportation is a fundamental protocol in quantum information science. It represents a critical resource for quantum communication and distributed quantum computing. We derive an analytical expression of the fidelity of teleportation of an input squeezed thermal state using for teleportation a bipartite Gaussian resource state shared between Alice and Bob. Each mode of the resource state is susceptible to the influence of the environment. We employ the characteristic function approach in conjunction with the covariance matrix formalism. The fidelity of teleportation is expressed in terms of input and resource state covariance matrices. We investigate, as an example, the feasibility of secure quantum teleportation of a squeezed thermal state using a two-mode resource state whose modes are placed in separate thermal baths. A successful quantum teleportation requires meeting two criteria: the presence of two-way quantum steering and a teleportation fidelity exceeding the classical threshold. The quantum steering is by nature asymmetric and has found applications in quantum cryptography and secure quantum teleportation. Weak squeezing and a high number of average thermal photons in the input states lead to an increase in the fidelity of teleportation. Generally, steering disappears much faster than the fidelity of teleportation decreases below its classical limit. Full article

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

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27 pages, 2176 KB

Open AccessArticle

Intelligent Fault Diagnosis of Rolling Bearings Based on Digital Twin and Multi-Scale CNN-AT-BiGRU Model

by Jiayu Shi, Liang Qi, Shuxia Ye, Changjiang Li, Chunhui Jiang, Zhengshun Ni, Zheng Zhao, Zhe Tong, Siyu Fei, Runkang Tang, Danfeng Zuo and Jiajun Gong

Symmetry 2025, 17(11), 1803; https://doi.org/10.3390/sym17111803 (registering DOI) - 26 Oct 2025

Abstract

Rolling bearings constitute critical rotating components within rolling mill equipment. Production efficiency and the operational safety of the whole mechanical system are directly governed by their operational health state. To address the dual challenges of the over-reliance of conventional diagnostic methods on expert [...] Read more.

Rolling bearings constitute critical rotating components within rolling mill equipment. Production efficiency and the operational safety of the whole mechanical system are directly governed by their operational health state. To address the dual challenges of the over-reliance of conventional diagnostic methods on expert experience and the scarcity of fault samples in industrial scenarios, we propose a virtual–physical data fusion-optimized intelligent fault diagnosis framework. Initially, a dynamics-based digital twin model for rolling bearings is developed by leveraging their geometric symmetry. It is capable of generating comprehensive fault datasets through parametric adjustments of bearing dimensions and operational environments in virtual space. Subsequently, a symmetry-informed architecture is constructed, which integrates multi-scale convolutional neural networks with attention mechanisms and bidirectional gated recurrent units (MCNN-AT-BiGRU). This architecture enables spatiotemporal feature extraction and enhances critical fault characteristics. The experimental results demonstrate 99.5% fault identification accuracy under single operating conditions. It maintains stable performance under low SNR conditions. Furthermore, the framework exhibits superior generalization capability and transferability across the different bearing types. Full article

(This article belongs to the Section Computer)

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21 pages, 2252 KB

Open AccessArticle

A Physics-Constrained Heterogeneous GNN Guided by Physical Symmetry for Heavy-Duty Vehicle Load Estimation

by Lizhuo Luo, Leqi Zhang, Hongli Wang, Yunjing Wang and Hang Yin

Symmetry 2025, 17(11), 1802; https://doi.org/10.3390/sym17111802 (registering DOI) - 26 Oct 2025

Abstract

Accurate heavy-duty vehicle load estimation is crucial for transportation and environmental regulation, yet current methods lack precision in data accuracy and practicality for field implementation. We propose a Self-Supervised Reconstruction Heterogeneous Graph Convolutional Network (SSR-HGCN) for load estimation using On-Board Diagnostics (OBD) data. [...] Read more.

Accurate heavy-duty vehicle load estimation is crucial for transportation and environmental regulation, yet current methods lack precision in data accuracy and practicality for field implementation. We propose a Self-Supervised Reconstruction Heterogeneous Graph Convolutional Network (SSR-HGCN) for load estimation using On-Board Diagnostics (OBD) data. The method integrates physics-constrained heterogeneous graph construction based on vehicle speed, acceleration, and engine parameters, leveraging graph neural networks’ information propagation mechanisms and self-supervised learning’s adaptability to low-quality data. The method comprises three modules: (1) a physics-constrained heterogeneous graph structure that, guided by the symmetry (invariance) of physical laws, introduces a structural asymmetry by treating kinematic and dynamic features as distinct node types to enhance model interpretability; (2) a self-supervised reconstruction module that learns robust representations from noisy OBD streams without extensive labeling, improving adaptability to data quality variations; and (3) a multi-layer feature extraction architecture combining graph convolutional networks (GCNs) and graph attention networks (GATs) for hierarchical feature aggregation. On a test set of 800 heavy-duty vehicle trips, SSR-HGCN demonstrated superior performance over key baseline models. Compared with the classical time-series model LSTM, it achieved average improvements of 20.76% in RMSE and 41.23% in MAPE. It also outperformed the standard graph model GraphSAGE, reducing RMSE by 21.98% and MAPE by 7.15%, ultimately achieving < 15% error for over 90% of test samples. This method provides an effective technical solution for heavy-duty vehicle load monitoring, with immediate applications in fleet supervision, overloading detection, and regulatory enforcement for environmental compliance. Full article

(This article belongs to the Section Computer)

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18 pages, 324 KB

Open AccessArticle

A Comparative Study of Trimmed L-Moments, Direct L-Moments, and Maximum Likelihood Estimation for the Inverse Weibull Distribution Under Type-I Right Censoring

by Hager Ahmad Ibrahim and Ahmed R. El-Saeed

Symmetry 2025, 17(11), 1801; https://doi.org/10.3390/sym17111801 (registering DOI) - 25 Oct 2025

Abstract

This paper presents a comparative evaluation of three distinct methodologies for estimating the parameters of the Inverse Weibull distribution under Type-I right censoring: trimmed linear moments (using Type-AT and Type-BT variants), direct linear moments (using Type-AD and Type-BD variants), and maximum likelihood estimation. [...] Read more.

This paper presents a comparative evaluation of three distinct methodologies for estimating the parameters of the Inverse Weibull distribution under Type-I right censoring: trimmed linear moments (using Type-AT and Type-BT variants), direct linear moments (using Type-AD and Type-BD variants), and maximum likelihood estimation. The performance of these methods is assessed through Monte Carlo simulations, focusing on estimation accuracy, relative absolute bias, and root mean square error to identify the most appropriate approach. The practical applicability of these techniques is demonstrated through a real-world dataset analysis. Full article

(This article belongs to the Section Mathematics)

33 pages, 1433 KB

Open AccessArticle

Hybrid Time Series Transformer–Deep Belief Network for Robust Anomaly Detection in Mobile Communication Networks

by Anita Ershadi Oskouei, Mehrdad Kaveh, Francisco Hernando-Gallego and Diego Martín

Symmetry 2025, 17(11), 1800; https://doi.org/10.3390/sym17111800 (registering DOI) - 25 Oct 2025

Abstract

The rapid evolution of 5G and emerging 6G networks has increased system complexity, data volume, and security risks, making anomaly detection vital for ensuring reliability and resilience. However, existing machine learning (ML)-based approaches still face challenges related to poor generalization, weak temporal modeling, [...] Read more.

The rapid evolution of 5G and emerging 6G networks has increased system complexity, data volume, and security risks, making anomaly detection vital for ensuring reliability and resilience. However, existing machine learning (ML)-based approaches still face challenges related to poor generalization, weak temporal modeling, and degraded accuracy under heterogeneous and imbalanced real-world conditions. To overcome these limitations, a hybrid time series transformer–deep belief network (HTST-DBN) is introduced, integrating the sequential modeling strength of TST with the hierarchical feature representation of DBN, while an improved orchard algorithm (IOA) performs adaptive hyper-parameter optimization. The framework also embodies the concept of symmetry and asymmetry. The IOA introduces controlled symmetry-breaking between exploration and exploitation, while the TST captures symmetric temporal patterns in network traffic whose asymmetric deviations often indicate anomalies. The proposed method is evaluated across four benchmark datasets (ToN-IoT, 5G-NIDD, CICDDoS2019, and Edge-IoTset) that capture diverse network environments, including 5G core traffic, IoT telemetry, mobile edge computing, and DDoS attacks. Experimental evaluation is conducted by benchmarking HTST-DBN against several state-of-the-art models, including TST, bidirectional encoder representations from transformers (BERT), DBN, deep reinforcement learning (DRL), convolutional neural network (CNN), and random forest (RF) classifiers. The proposed HTST-DBN achieves outstanding performance, with the highest accuracy reaching 99.61%, alongside strong recall and area under the curve (AUC) scores. The HTST-DBN framework presents a scalable and reliable solution for anomaly detection in next-generation mobile networks. Its hybrid architecture, reinforced by hyper-parameter optimization, enables effective learning in complex, dynamic, and heterogeneous environments, making it suitable for real-world deployment in future 5G/6G infrastructures. Full article

(This article belongs to the Special Issue AI-Driven Optimization for EDA: Balancing Symmetry and Asymmetry)

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19 pages, 1856 KB

Open AccessArticle

Multiscale Texture Fractal Analysis of Thermo-Mechanical Coupling in Micro-Asperity Contact Interfaces

by Jiafu Ruan, Xigui Wang, Yongmei Wang and Weiqiang Zou

Symmetry 2025, 17(11), 1799; https://doi.org/10.3390/sym17111799 (registering DOI) - 25 Oct 2025

Abstract

The line contact behavior of multiscale meshing interfaces necessitates synergistic investigation spanning nano-to centimeter-scale ranges. When nominally smooth gear teeth surfaces come into contact, the mechanical–thermal coupling effect at the meshing interface actually occurs over a collection of microscale asperities (roughness peaks) exhibiting [...] Read more.

The line contact behavior of multiscale meshing interfaces necessitates synergistic investigation spanning nano-to centimeter-scale ranges. When nominally smooth gear teeth surfaces come into contact, the mechanical–thermal coupling effect at the meshing interface actually occurs over a collection of microscale asperities (roughness peaks) exhibiting hierarchical distribution characteristics. The emergent deformation phenomena across multiple asperity scales govern the self-organized evolution of interface conformity, thereby regulating both the load transfer efficiency and thermal transport properties within the contact zone. The fractal nature of the roughness topography on actual meshing interfaces calls for the development of a cross-scale theoretical framework that integrates micro-texture optimization with multi-physics coupling contact behavior. Conventional roughness characterization methods based on statistical parameters suffer from inherent limitations: their parameter values are highly dependent on measurement scale, lacking uniqueness under varying sampling intervals and instrument resolutions, and failing to capture the scale-invariant nature of meshing interface topography. A scale-independent parameter system grounded in fractal geometry theory enables essential feature extraction and quantitative characterization of three-dimensional interface morphology. This study establishes a progressive deformation theory for gear line contact interfaces with fractal geometric characteristics, encompassing elastic, elastoplastic transition, and perfectly plastic stages. By systematically investigating the force–thermal coupling mechanisms in textured meshing interfaces under multiscale conditions, the research provides a theoretical foundation and numerical implementation pathways for high-precision multiscale thermo-mechanical analysis of meshing interfaces. Full article

(This article belongs to the Section Engineering and Materials)

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15 pages, 340 KB

Open AccessArticle

Nonlinear Almost Relational Contractions via a Triplet of Test Functions and Applications to Second-Order Ordinary Differential Equations

by Doaa Filali and Faizan Ahmad Khan

Symmetry 2025, 17(11), 1798; https://doi.org/10.3390/sym17111798 (registering DOI) - 24 Oct 2025

Abstract

After the introduction of the relation-theoretic contraction principle, the branch of metric fixed-point theory has attracted much attention in this direction, and various fixed-point results have been proven in the framework of relational metric space via different approaches. The aim of this article [...] Read more.

After the introduction of the relation-theoretic contraction principle, the branch of metric fixed-point theory has attracted much attention in this direction, and various fixed-point results have been proven in the framework of relational metric space via different approaches. The aim of this article is to establish some fixed-point outcomes in the framework of relational metric space verifying a generalized nonlinear contraction utilizing three test functions

Φ

,

Ψ

and

Θ

satisfying the appropriate characteristics. The findings obtained herein expand, sharpen, improve, modify and unify a few well-known findings. To demonstrate the utility of our outcomes, several examples are furnished. We utilized our outcomes to investigate a unique solution of second-order ordinary differential equations prescribed with specific boundary conditions. Full article

(This article belongs to the Special Issue Nonlinear Analysis and Applications, Geometry of Banach Spaces and Symmetry II)

14 pages, 271 KB

Open AccessArticle

On Finitely Generated Neutrosophic Modules with Finite Value Distribution

by Amr Elrawy and Ali Yahya Hummdi

Symmetry 2025, 17(11), 1797; https://doi.org/10.3390/sym17111797 (registering DOI) - 24 Oct 2025

Abstract

This paper presents the novel framework of neutrosophic modules, an algebraic structure that arises by superimposing neutrosophic sets on classical module theory. The core of this study lies in the investigation of the structural symmetry between the axioms of a module and the [...] Read more.

This paper presents the novel framework of neutrosophic modules, an algebraic structure that arises by superimposing neutrosophic sets on classical module theory. The core of this study lies in the investigation of the structural symmetry between the axioms of a module and the trivalent nature of the neutrosophic set. We define a new class of modules based on the neutrosophic set. In addition, this study establishes and examines the categorical structure corresponding to neutrosophic

R

-modules. Furthermore, it presents the procedures by which finitely generated variants of these modules can be formulated and studied. Partial characterizations are given for a particular type in which the distribution of neutrosophic values remains within a finite set. Full article

(This article belongs to the Section Mathematics)

20 pages, 1982 KB

Open AccessArticle

Bias Term for Outlier Detection and Robust Regression

by Felix Ndudim and Thanasak Mouktonglang

Symmetry 2025, 17(11), 1796; https://doi.org/10.3390/sym17111796 (registering DOI) - 24 Oct 2025

Abstract

Noisy data and outliers are among the main challenges in machine learning, as their presence in training data can significantly reduce model performance and generalization. Detecting and handling these anomalous samples is particularly difficult because it is hard to distinguish them from normal [...] Read more.

Noisy data and outliers are among the main challenges in machine learning, as their presence in training data can significantly reduce model performance and generalization. Detecting and handling these anomalous samples is particularly difficult because it is hard to distinguish them from normal data. In this study, we propose a novel bias-based method (BT-SVR) to detect outliers and noisy inputs. The method uses a bias term derived from pairwise relationships among data points, which captures structural information about input distances. Outliers and noisy samples typically produce near-zero bias responses, allowing them to be identified effectively. A root-mean-square (RMS) scoring mechanism is then applied to quantify the anomaly strength of each sample, enabling the impact of outliers to be underweighted before training. Experiments demonstrate that BT-SVR improves the performance of Support Vector Regression (SVR) and enhances its robustness against noisy and anomalous data. Full article

(This article belongs to the Special Issue Machine Learning and Data Analysis III)

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23 pages, 882 KB

Open AccessArticle

A Gauss Hypergeometric-Type Model for Heavy-Tailed Survival Times in Biomedical Research

by Jiju Gillariose, Mahmoud M. Abdelwahab, Joshin Joseph and Mustafa M. Hasaballah

Symmetry 2025, 17(11), 1795; https://doi.org/10.3390/sym17111795 - 24 Oct 2025

Abstract

In this study, we introduced and analyzed the Slash–Log–Logistic (SlaLL) distribution, a novel statistical model developed by applying the slash methodology to log–logistic and beta distributions. The SlaLL distribution is particularly suited for modeling datasets characterized by heavy tails and extreme [...] Read more.

In this study, we introduced and analyzed the Slash–Log–Logistic (SlaLL) distribution, a novel statistical model developed by applying the slash methodology to log–logistic and beta distributions. The SlaLL distribution is particularly suited for modeling datasets characterized by heavy tails and extreme values, frequently encountered in survival time analyses. We derived the mathematical representation of the distribution involving Gauss hypergeometric and beta functions, explicitly established the probability density function, cumulative distribution function, hazard rate function, and reliability function, and provided clear definitions of its moments. Through comprehensive simulation studies, the accuracy and robustness of maximum likelihood and Bayesian methods for parameter estimation were validated. Comparative empirical analyses demonstrated the SlaLL distribution’s superior fitting performance over well-known slash-based models, emphasizing its practical utility in accurately capturing the complexities of real-world survival time data. Full article

(This article belongs to the Section Mathematics)

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24 pages, 384 KB

Open AccessArticle

h-Almost Conformal η-Ricci–Bourguignon Solitons and Spacetime Symmetry in Barotropic Fluids Within f(

R, T

) Gravity

by Sunil Kumar Yadav, Sameh Shenawy, Hanan Alohali and Carlo Mantica

Symmetry 2025, 17(11), 1794; https://doi.org/10.3390/sym17111794 - 23 Oct 2025

Abstract

We investigate the geometric and physical properties of the h-almost conformal

η

-Ricci–Bourguignon soliton and its gradient form by employing a barotropic equation of state within the framework of

f (R, T)

gravity. We derive this barotropic equation of [...] Read more.

We investigate the geometric and physical properties of the h-almost conformal

η

-Ricci–Bourguignon soliton and its gradient form by employing a barotropic equation of state within the framework of

f (R, T)

gravity. We derive this barotropic equation of state under the assumption that the matter content of

f (R, T)

gravity is modeled by a barotropic perfect fluid. We also examine the way in which these soliton structures both reveal and limit the underlying symmetries of the spacetime geometry. Furthermore, we obtain modified Poisson and Liouville equations associated with these solitons in such a gravitational setting. Additionally, we explore certain harmonic aspects of the h-almost conformal

η

-Ricci–Bourguignon soliton on a spacetime filled with a barotropic perfect fluid, considering a harmonic potential function

Ψ

. Finally, we present physical interpretations of the conformal pressure

\tilde{p}

in the context of the h-almost conformal

η

-Ricci–Bourguignon soliton within

f (R, T)

gravity. Full article

(This article belongs to the Section Physics)

14 pages, 1036 KB

Open AccessArticle

Biomedical Knowledge Graph Embedding with Hierarchical Capsule Network and Rotational Symmetry for Drug-Drug Interaction Prediction

by Sensen Zhang, Xia Li, Yang Liu, Peng Bi and Tiangui Hu

Symmetry 2025, 17(11), 1793; https://doi.org/10.3390/sym17111793 - 23 Oct 2025

Abstract

The forecasting of Drug-Drug Interactions (DDIs) is essential in pharmacology and clinical practice to prevent adverse drug reactions. Existing approaches, often based on neural networks and knowledge graph embedding, face limitations in modeling correlations among drug features and in handling complex BioKG relations, [...] Read more.

The forecasting of Drug-Drug Interactions (DDIs) is essential in pharmacology and clinical practice to prevent adverse drug reactions. Existing approaches, often based on neural networks and knowledge graph embedding, face limitations in modeling correlations among drug features and in handling complex BioKG relations, such as one-to-many, hierarchical, and composite interactions. To address these issues, we propose Rot4Cap, a novel framework that embeds drug entity pairs and BioKG relationships into a four-dimensional vector space, enabling effective modeling of diverse mapping properties and hierarchical structures. In addition, our method integrates molecular structures and drug descriptions with BioKG entities, and it employs capsule network–based attention routing to capture feature correlations. Experiments on three benchmark BioKG datasets demonstrate that Rot4Cap outperforms state-of-the-art baselines, highlighting its effectiveness and robustness. Full article

(This article belongs to the Section Computer)

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12 pages, 500 KB

Open AccessArticle

A New Mathematical Model for Eye Vision Disability Dynamics: Investigation of Visual Acuity, Lens Power, Pupil Diameter, and Environmental Factors

by Ibrahim Al-Dayel

Symmetry 2025, 17(11), 1792; https://doi.org/10.3390/sym17111792 - 23 Oct 2025

Abstract

This study presents a mathematical model to understand the dynamics of eye vision disability, incorporating key physiological and environmental variables such as visual acuity, lens power, pupil diameter, and environmental influence. The novel model developed in this work consists of four state variables [...] Read more.

This study presents a mathematical model to understand the dynamics of eye vision disability, incorporating key physiological and environmental variables such as visual acuity, lens power, pupil diameter, and environmental influence. The novel model developed in this work consists of four state variables that describe their interactions, aiming to simulate changes in visual acuity and the progression of vision disability under different circumstances. Various mathematical aspects including positivity and equilibrium points along with local and global stability are demonstrated. In addition, the sensitivity analysis is also presented to examine the impact of variations in parameters on the model’s outcomes, highlighting the factors that significantly influence visual acuity and pupil adjustments. Furthermore, the phase portraits explore the dynamic interactions between variables, revealing different insights into the stability of the system and adaptive responses. These results provide a thorough understanding of factors influencing eye vision, offering new insights into vision disability dynamics through an integrated, non-autonomous model and practical applications in developing corrective strategies. Full article

(This article belongs to the Special Issue Symmetry in Nonlinear Equations: Mathematical Models, Methods and Applications Ⅱ)

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33 pages, 2812 KB

Open AccessArticle

A Symmetry-Aware Predictive Framework for Olympic Cold-Start Problems and Rare Events Based on Multi-Granularity Transfer Learning and Extreme Value Analysis

by Yanan Wang, Yi Fei and Qiuyan Zhang

Symmetry 2025, 17(11), 1791; https://doi.org/10.3390/sym17111791 - 23 Oct 2025

Abstract

This paper addresses the cold-start problem and rare event prediction challenges in Olympic medal forecasting by proposing a predictive framework that integrates multi-granularity transfer learning with extreme value theory. The framework comprises two main components, a Multi-Granularity Transfer Learning Core (MG-TLC) and a [...] Read more.

This paper addresses the cold-start problem and rare event prediction challenges in Olympic medal forecasting by proposing a predictive framework that integrates multi-granularity transfer learning with extreme value theory. The framework comprises two main components, a Multi-Granularity Transfer Learning Core (MG-TLC) and a Rare Event Analysis Module (RE-AM), which address multi-level prediction for data-scarce countries and first medal prediction tasks. The MG-TLC incorporates two key components: Dynamic Feature Space Reconstruction (DFSR) and the Hierarchical Adaptive Transfer Strategy (HATS). The RE-AM combines a Bayesian hierarchical extreme value model (BHEV) with piecewise survival analysis (PSA). Experiments based on comprehensive, licensed Olympic data from 1896–2024, where the framework was trained on data up to 2016, validated on the 2020 Games, and tested by forecasting the 2024 Games, demonstrate that the proposed framework significantly outperforms existing methods, reducing MAE by 25.7% for data-scarce countries and achieving an AUC of 0.833 for first medal prediction, 14.3% higher than baseline methods. This research establishes a foundation for predicting the 2028 Los Angeles Olympics and provides new approaches for cold-start and rare event prediction, with potential applicability to similar challenges in other data-scarce domains such as economics or public health. From a symmetry viewpoint, our framework is designed to preserve task-relevant invariances—permutation invariance in set-based country aggregation and scale robustness to macro-covariate units—via distributional alignment between data-rich and data-scarce domains and Olympic-cycle indexing. We treat departures from these symmetries (e.g., host advantage or event-program changes) as structured asymmetries and capture them with a rare event module that combines extreme value and survival modeling. Full article

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

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25 pages, 5642 KB

Open AccessArticle

A Trusted Transaction Method for Remote Sensing Image Data Based on a Homomorphic Encryption Watermark and Blockchain

by Minxuan Wang, Lei Zhang, Na Ren and Changqing Zhu

Symmetry 2025, 17(11), 1790; https://doi.org/10.3390/sym17111790 - 23 Oct 2025

Abstract

Existing methods for the secure transaction and circulation of remote sensing image data primarily focus on post-event investigation, lacking a reliable mechanism for secure distribution and fair trading of data. To address this issue, this study proposes a trusted transaction method that integrates [...] Read more.

Existing methods for the secure transaction and circulation of remote sensing image data primarily focus on post-event investigation, lacking a reliable mechanism for secure distribution and fair trading of data. To address this issue, this study proposes a trusted transaction method that integrates a watermark based on Paillier homomorphic encryption, blockchain, and smart contract. This method leverages the homomorphic property of the Paillier cryptosystem to imperceptibly embed the ciphertext of the watermark generated from transaction information into encrypted remote sensing image data. The data buyer at the receiving end decrypts the key pair using the private key, thereby decrypting the data to obtain the watermarked plaintext. Simultaneously, transaction records are immutably stored on trusted blockchain nodes via smart contracts. Throughout the entire transaction process, data encryption/decryption and watermark embedding/extraction are symmetric. The experimental results demonstrate that the watermark can be effectively extracted after encryption, thereby supporting transaction verification and traceability. Furthermore, the three smart contracts designed in this study all exhibit strong execution performance. In particular, the smart contract employed for verification demonstrated an average execution latency of only 0.19 s per instance. Through enforcing the retrieval of parameters and storage credentials from the blockchain, the proposed method effectively constrains malicious behavior from both parties, offering a novel technical approach to facilitate consensus and mutual trust. Full article

(This article belongs to the Section Computer)

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27 pages, 341 KB

Open AccessArticle

Four-Dimensional Spaces of Complex Numbers and Unitary States of Two-Qubit Quantum Systems

by Mars B. Gabbassov, Tolybay Z. Kuanov, Turganbay K. Yermagambetov and Berik I. Tuleuov

Symmetry 2025, 17(11), 1789; https://doi.org/10.3390/sym17111789 - 22 Oct 2025

Abstract

The pure states of two-qubit quantum systems are described by a four-dimensional vector of complex numbers, and unitary operators transferring a two-qubit quantum system from one state to another have the form of a

4 \times 4

matrix with complex elements. This fact [...] Read more.

The pure states of two-qubit quantum systems are described by a four-dimensional vector of complex numbers, and unitary operators transferring a two-qubit quantum system from one state to another have the form of a

4 \times 4

matrix with complex elements. This fact brings to mind the idea of studying the spaces of four-dimensional numbers with complex components. Moreover, the results obtained by the authors for four-dimensional numbers with real components inspire some optimism. In this paper we construct four-dimensional spaces of complex numbers by analogy with four-dimensional spaces of real numbers. Each four-dimensional number is mapped to a matrix formed from its components and it is proved that the constructed mapping is a bijection and a homomorphism. In the space of four-dimensional numbers of the eight basis elements, half are real and half are imaginary. The presence of such symmetry distinguishes these spaces from the space of quaternions, in which one basis element is real and the rest are imaginary. The symmetry of the basis numbers makes these spaces a natural generalization of one-dimensional and two-dimensional (complex) algebra. The conditions under which the corresponding matrices are gates for two-qubit quantum systems are defined. The notion of a unitary state of a two-qubit quantum system is introduced, to which various gates from commutative groups of gates correspond. It is shown that any gate of a unitary state transforms a unitary state into a unitary state and a non-unitary state into a non-unitary state. Almost all gates used in the construction of quantum circuits, in particular H, SWAP, CX, CY, and CZ, have the same properties. The problem of searching for a gate that transfers a quantum system from one unitary state to another unitary state has been solved. Thus, with the help of four-dimensional spaces of complex numbers it was possible to construct whole classes of two-qubit gates, which opens new possibilities for the construction of quantum algorithms. The results obtained have important theoretical and practical implications for quantum computing. Full article

(This article belongs to the Section Physics)

25 pages, 1741 KB

Open AccessArticle

Event-Aware Multimodal Time-Series Forecasting via Symmetry-Preserving Graph-Based Cross-Regional Transfer Learning

by Shu Cao and Can Zhou

Symmetry 2025, 17(11), 1788; https://doi.org/10.3390/sym17111788 - 22 Oct 2025

Abstract

Forecasting real-world time series in domains with strong event sensitivity and regional variability poses unique challenges, as predictive models must account for sudden disruptions, heterogeneous contextual factors, and structural differences across locations. In tackling these challenges, we draw on the concept of symmetry [...] Read more.

Forecasting real-world time series in domains with strong event sensitivity and regional variability poses unique challenges, as predictive models must account for sudden disruptions, heterogeneous contextual factors, and structural differences across locations. In tackling these challenges, we draw on the concept of symmetry that refers to the balance and invariance patterns across temporal, multimodal, and structural dimensions, which help reveal consistent relationships and recurring patterns within complex systems. This study is based on two multimodal datasets covering 12 tourist regions and more than 3 years of records, ensuring robustness and practical relevance of the results. In many applications, such as monitoring economic indicators, assessing operational performance, or predicting demand patterns, short-term fluctuations are often triggered by discrete events, policy changes, or external incidents, which conventional statistical and deep learning approaches struggle to model effectively. To address these limitations, we propose an event-aware multimodal time-series forecasting framework with graph-based regional transfer built upon an enhanced PatchTST backbone. The framework unifies multimodal feature extraction, event-sensitive temporal reasoning, and graph-based structural adaptation. Unlike Informer, Autoformer, FEDformer, or PatchTST, our model explicitly addresses naive multimodal fusion, event-agnostic modeling, and weak cross-regional transfer by introducing an event-aware Multimodal Encoder, a Temporal Event Reasoner, and a Multiscale Graph Module. Experiments on diverse multi-region multimodal datasets demonstrate that our method achieves substantial improvements over eight state-of-the-art baselines in forecasting accuracy, event response modeling, and transfer efficiency. Specifically, our model achieves a 15.06% improvement in the event recovery index, a 15.1% reduction in MAE, and a 19.7% decrease in event response error compared to PatchTST, highlighting its empirical impact on tourism event economics forecasting. Full article

(This article belongs to the Special Issue Symmetry in Data Sciences and Machine Learning for Multidisciplinary Research)

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24 pages, 38382 KB

Open AccessArticle

Skeleton Information-Driven Reinforcement Learning Framework for Robust and Natural Motion of Quadruped Robots

by Huiyang Cao, Hongfa Lei, Yangjun Liu, Zheng Chen, Shuai Shi, Bingquan Li, Weichao Xu and Zhi-Xin Yang

Symmetry 2025, 17(11), 1787; https://doi.org/10.3390/sym17111787 - 22 Oct 2025

Abstract

Legged robots have great potential in complex environments, but achieving robust and natural locomotion remains difficult due to challenges in generating smooth gaits and resisting disturbances. This article presents a novel reinforcement learning framework that integrates a skeleton-aware graph neural network (GNN), a [...] Read more.

Legged robots have great potential in complex environments, but achieving robust and natural locomotion remains difficult due to challenges in generating smooth gaits and resisting disturbances. This article presents a novel reinforcement learning framework that integrates a skeleton-aware graph neural network (GNN), a single-stage teacher–student architecture, a system-response model, and a Wasserstein Adversarial Motion Priors (wAMP) module. The skeleton-aware GNN enriches observations by encoding key node information and link properties, providing structured body information and better spatial awareness on irregular terrains. Unlike conventional two-stage approaches, this method jointly trains teacher and student policies to accelerate learning and improve sim-to-real transfer using hybrid advantage estimation (HAE). The system-response model further enhances robustness by predicting future observations from historical states via contrastive learning, enabling the policy to anticipate terrain variations and external disturbances. Finally, wAMP provides a more stable adversarial imitation method for fitting expert datasets of both flat ground and stair locomotion. Experiments on quadruped robots demonstrate that the proposed approach achieves more natural gaits and stronger robustness than existing baselines. Full article

(This article belongs to the Special Issue Latest Advancements in Symmetry and Asymmetry in Robotics and Mechatronics: Highlighting Emerging Trends and Their Intersections with Artificial Intelligence (AI), Machine Learning, Nonlinear Systems, and Control)

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16 pages, 18210 KB

Open AccessArticle

Basic Study of Blood Coagulation by Microplasma

by Marius Gabriel Blajan, Anca Daniela Stoica, Cristian Sevcencu, Septimiu Cassian Tripon, Vasile Surducan and Kazuo Shimizu

Symmetry 2025, 17(11), 1786; https://doi.org/10.3390/sym17111786 - 22 Oct 2025

Abstract

Plasma medicine is a field of research that focuses on the sterilization of bacteria, wounds and cancer treatment, tissue regeneration and other biomedical applications using plasma. Dielectric barrier discharge microplasma was used for biomedical applications such as sterilization of bacteria and skin treatment [...] Read more.

Plasma medicine is a field of research that focuses on the sterilization of bacteria, wounds and cancer treatment, tissue regeneration and other biomedical applications using plasma. Dielectric barrier discharge microplasma was used for biomedical applications such as sterilization of bacteria and skin treatment for transdermal drug delivery. In this study, we investigated the feasibility of using microplasma for improving blood coagulation parameters. Blood samples collected from one dog and one cat were treated with microplasma, and the blood coagulation effect of this treatment was compared with the effect achieved by treating the blood with air flow only. The microplasma electrodes were energized using a negative pulse voltage power supply and environmental air was used as discharge gas. The microplasma treatment produced clear coagulation effects that increased proportionally with treatment time, discharge voltage and frequency. In contrast, the treatment with air flow only had no coagulation effects after the same treatment time as for the microplasma treatment. The observed blood coagulation effects induced by microplasma treatment could be attributed to the reactive oxygen and nitrogen species generated by microplasma. The blood sample subjected to microplasma treatment had a slight temperature increase (≈4 °C) confirming the nonthermal operation. In conclusion, this study shows promising results that suggest the potential of using microplasma treatment as a tool for improving blood coagulation parameters. Furthermore, microplasma’s suitability for portability and integration indicates the potential for developing a compact microplasma device tailored for use by first responders in cases of bleeding. Full article

(This article belongs to the Special Issue Advances in Plasma Physics with Symmetry/Asymmetry)

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