Symmetry

18 pages, 638 KB

Open AccessArticle

Continuous-Mode Analysis of Improved Two-Way CV-QKD

by Yanhao Sun, Jiayu Ma, Xiangyu Wang, Song Yu, Ziyang Chen and Hong Guo

Symmetry 2026, 18(2), 382; https://doi.org/10.3390/sym18020382 - 20 Feb 2026

Continuous-variable quantum key distribution (CV-QKD) enables information-theoretically secure key generation between legitimate parties. To further enhance system performance, an improved two-way CV-QKD protocol has been proposed, which is accessible in practice and exhibits increased robustness against excess noise. However, in practical implementations, device [...] Read more.

Continuous-variable quantum key distribution (CV-QKD) enables information-theoretically secure key generation between legitimate parties. To further enhance system performance, an improved two-way CV-QKD protocol has been proposed, which is accessible in practice and exhibits increased robustness against excess noise. However, in practical implementations, device nonidealities inevitably drive the optical field from the single-mode regime into the continuous-mode regime. In this work, we introduce temporal modes to characterize the evolution of optical fields in the improved two-way protocol and establish a security analysis framework for the continuous-mode scenario based on adaptive normalization with calibrated shot-noise unit. In addition, finite-size effects are taken into account in the analysis. Our results demonstrate that the improved two-way protocol retains a performance advantage over its one-way counterpart. The analysis provides useful guidance for the practical implementation and performance optimization of improved two-way CV-QKD systems. Full article

(This article belongs to the Section Physics)

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27 pages, 4096 KB

Open AccessArticle

Autonomous Driving Optimization for Autonomous Robot Vehicles Based on FAST-LIO2 Algorithm Improvement

by Xuyan Ge, Gu Gong and Xiaolin Wang

Symmetry 2026, 18(2), 381; https://doi.org/10.3390/sym18020381 - 20 Feb 2026

Abstract

In urban environments, autonomous vehicles face critical challenges in localization and perception under extreme lighting conditions, including rapid illumination changes, high contrast, and nighttime low-light scenarios. To address the performance degradation of traditional LiDAR-inertial odometry systems under such conditions, this study proposes a [...] Read more.

In urban environments, autonomous vehicles face critical challenges in localization and perception under extreme lighting conditions, including rapid illumination changes, high contrast, and nighttime low-light scenarios. To address the performance degradation of traditional LiDAR-inertial odometry systems under such conditions, this study proposes a high-precision FAST-LIO2-EC algorithm that fuses event cameras into the FAST-LIO2 framework. Event cameras, with their microsecond temporal resolution and 140 dB dynamic range, provide asynchronous edge information that complements LiDAR point clouds and IMU measurements. We validate the proposed system through real-world road tests conducted on public roads and closed test tracks, covering three typical extreme lighting scenarios: tunnel entrance/exit transitions, high-contrast shadow boundaries, and nighttime sparse-lighting conditions. The experimental platform is equipped with a 32-beam LiDAR, a 6-axis IMU, a DVS event camera, and an RTK-GNSS system for ground truth trajectory acquisition. Real-world results demonstrate that the FAST-LIO2-EC system achieves significant improvements in localization accuracy and robustness. In illumination change scenarios, the Absolute Trajectory Error (ATE) is reduced by 32.5% compared to the baseline FAST-LIO2 system, with zero tracking loss events. The point cloud quality is substantially enhanced, with more uniform distribution and clearer obstacle boundaries. In high-contrast scenarios, both systems maintain comparable performance with ATE below 0.15 m. However, in nighttime scenarios, the fusion system shows moderate improvement (15.3% ATE reduction) but reveals sensitivity to event camera noise, indicating the need for adaptive thresholding strategies. Supplementary simulation experiments validate the system’s robustness under varying speeds and sensor noise levels. This work provides a practical solution for autonomous vehicle deployment in complex urban lighting environments, with a comprehensive analysis of real-world performance boundaries and deployment considerations. Full article

(This article belongs to the Section Computer)

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23 pages, 520 KB

Open AccessArticle

Time-Domain Oversampling-Enabled Multi-NS Reception for MoCDMA

by Weidong Gao, Yuanhui Wang and Jun Li

Symmetry 2026, 18(2), 380; https://doi.org/10.3390/sym18020380 - 20 Feb 2026

Abstract

In molecular communication via diffusion (MCvD) uplinks where multiple nano-sensors report concurrently to a fusion center (FC), the long channel memory and the near–far imbalance jointly create strong multiple access interference (MAI) coupled with residual inter-symbol/inter-chip effects. This paper studies an oversampling-enabled time-domain [...] Read more.

In molecular communication via diffusion (MCvD) uplinks where multiple nano-sensors report concurrently to a fusion center (FC), the long channel memory and the near–far imbalance jointly create strong multiple access interference (MAI) coupled with residual inter-symbol/inter-chip effects. This paper studies an oversampling-enabled time-domain reception for an uplink molecular code-division multiple-access (MoCDMA) system employing bipolar molecular signalling. By exploiting intra-chip oversampling at the FC, three linear detectors following the principles of maximum ratio combining (MRC), zero-forcing (ZF), and minimum mean-square error (MMSE) are developed and further enhanced through a feedback-assisted interference subtraction (FAIS) scheme that combines single-tap ISI feedback equalization with near-to-far successive MAI subtraction. Owing to the complementary structure of bipolar molecular emissions, the signal-dependent counting noise corresponding to the two molecule types can be jointly modeled in a symmetric and information-independent manner to support unified linear detection and FAIS processing. Numerical results demonstrate that oversampling effectively improves detection reliability, while increasing the molecular emission budget alone is insufficient to mitigate near–far effects. Moreover, FAIS provides significant performance gains, particularly for far NSs. Full article

(This article belongs to the Section Computer)

20 pages, 461 KB

Open AccessArticle

Mathematical Analysis of Higher-Order m-Coupled System Differential Equations with Caputo–Fabrizio Derivatives

by Neama Haron, Ali H. Tedjani, Arshad Ali, Khaled Aldwoah, Faez A. Alqarni, Ria Egami and Mohammed Messaoudi

Symmetry 2026, 18(2), 379; https://doi.org/10.3390/sym18020379 - 20 Feb 2026

Abstract

This paper examines the existence and stability of an m-cyclic coupled system of higher-order fractional differential equations with non-singular kernels. Sufficient conditions for the existence and stability of solutions are obtained using fixed-point techniques. Two numerical examples involving coupled and triply coupled [...] Read more.

This paper examines the existence and stability of an m-cyclic coupled system of higher-order fractional differential equations with non-singular kernels. Sufficient conditions for the existence and stability of solutions are obtained using fixed-point techniques. Two numerical examples involving coupled and triply coupled systems are presented to validate the theoretical results, and simulations of the triply coupled case illustrate the influence of different fractional orders on the system dynamics. Full article

(This article belongs to the Section Mathematics)

25 pages, 3538 KB

Open AccessArticle

Nonlinear Receding Contact Mechanics of Functionally Graded Layers for Aerospace Structures: A Symmetry-Based Analytical and FEM Study

by Ahmad Abushattal, Merve Terzi, Ayşegül Eyüboğlu, Murat Yaylacı, Dursun Murat Sekban, Safa Nayır, Ecren Uzun Yaylacı, Deshinta Arrova Dewi and Ahmet Birinci

Symmetry 2026, 18(2), 378; https://doi.org/10.3390/sym18020378 - 19 Feb 2026

Abstract

Functionally graded materials (FGMs) are widely applied in spacecraft structural design, thermal protection systems, and planetary landing mechanisms, benefiting from their ability to resist large thermal, pressure, and force gradients. To assess structural response behaviors for lander missions, docking maneuvers, and force transfer [...] Read more.

Functionally graded materials (FGMs) are widely applied in spacecraft structural design, thermal protection systems, and planetary landing mechanisms, benefiting from their ability to resist large thermal, pressure, and force gradients. To assess structural response behaviors for lander missions, docking maneuvers, and force transfer in layered aerospace structures, analyzing the contacts subjected to heavily stressed areas becomes very important. This article investigates the receding contact between a functionally graded top layer and a uniform substrate lying on a Winkler elastic foundation using the elasticity theory. An analytical approach has been validated using a finite element method (FEM) implemented in ANSYS. Comparison between the analytical solution and the FEM solution has been conducted for different stamp radii, elastic foundation stiffnesses, and ratios of shearing modulus for various realistic materials in the aerospace field. The data indicate very good convergence between the two solutions for both the length of contacts and the normal stress distribution, where differences are always below 3%. An increase in stamp radius leads to an extension of the contacts as well as a reduction in normal stresses and elevated stiffness and shearing modulus ratio contribute to smaller contacts and higher stresses. The validated methodological approach offers a realistic means for predicting force transfer mechanisms in spacecraft landing pads, multi-layer insulation panels, adaptive space structures, and functionally graded parts subjected to localized loads. This work offers predictive capabilities for space material interface design and optimization for harsh mechanical environments. Full article

(This article belongs to the Special Issue Aerospace Engineering and Symmetry/Asymmetry)

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28 pages, 4633 KB

Open AccessArticle

Analysis of Reaction-Diffusion Systems: Painlevé Properties and Conservation Laws of the Convective Fisher–Kolmogorov Equations

by Ali Raza, Alhussein Mohamed Alhussein Ahmed and Abdul Hamid Kara

Symmetry 2026, 18(2), 377; https://doi.org/10.3390/sym18020377 - 18 Feb 2026

Abstract

This article investigates the widespread occurrence of wave-like behavior in biological systems and reflects on the importance of reaction–diffusion frameworks to explain fast information transfer. Focusing on the classical Fisher–Kolmogorov equation, it demonstrates how the combination of nonlinear reaction dynamics and diffusion gives [...] Read more.

This article investigates the widespread occurrence of wave-like behavior in biological systems and reflects on the importance of reaction–diffusion frameworks to explain fast information transfer. Focusing on the classical Fisher–Kolmogorov equation, it demonstrates how the combination of nonlinear reaction dynamics and diffusion gives rise to stable traveling wavefronts that propagate at a constant speed while preserving their shape. We adopt a number of approaches (some novel) in dealing with the analyses, namely the symmetry method, soliton approach (algebraic) and Painlevé analysis, among others. Full article

(This article belongs to the Special Issue Symmetry in Lie Groups and Lie Algebras)

22 pages, 4286 KB

Open AccessArticle

Symmetry-Enhanced Indoor Occupant Locating and Motionless Alarm System: Fusion of BP Neural Network and DS-TWR Technology

by Li Wang, Zhe Wang, Xinhe Meng, Wentao Chen and Aijun Sun

Symmetry 2026, 18(2), 376; https://doi.org/10.3390/sym18020376 - 18 Feb 2026

Abstract

To address the critical demand for real-time dynamic tracking of personnel in complex buildings during emergency rescue, a novel system was proposed integrating Back Propagation (BP) neural networks with Double-Sided Two-Way Ranging (DS-TWR) technology to achieve precise indoor localization and motionless detection. Comprising [...] Read more.

To address the critical demand for real-time dynamic tracking of personnel in complex buildings during emergency rescue, a novel system was proposed integrating Back Propagation (BP) neural networks with Double-Sided Two-Way Ranging (DS-TWR) technology to achieve precise indoor localization and motionless detection. Comprising hardware (positioning base stations, tags, POE switches, routers, and a computer) and software (developed on LabVIEW), the system leverages the symmetric signal transmission of DS-TWR and the adaptive learning capability of BP neural networks to effectively mitigate multipath interference, enhancing positioning consistency and accuracy. Thresholds of time period and movement distance were set to determine whether the occupant was trapped. When tested in several common building structures, it demonstrated good stability and high accuracy—the average RMSE of the positioning system was within 0.012–0.018 m (static state) and 0.048–0.065 m (dynamic state). Furthermore, the system could real-time monitor and display the movement trajectory of each person, and automatically alarm when anyone was trapped in a fire scene. Hence, rescue measures can be taken timely according to the alarm information provided by the system, effectively ensuring the safety of personnel and improving the efficiency of fire rescue work. The proposed approach provides a symmetry-driven framework for intelligent building safety. Full article

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26 pages, 51756 KB

Open AccessArticle

Soliton Genesis in a Novel Gross–Pitaevskii System: Analytical Construction and Dynamical Control

by Khaled Aldwoah, L. M. Abdalgadir, Shafqat Ur Rehman, Muhammad Bilal, Faez A. Alqarni, Ria Egami and M. M. Rashed

Symmetry 2026, 18(2), 375; https://doi.org/10.3390/sym18020375 - 18 Feb 2026

Abstract

The purpose of this study is to construct diverse forms of exact soliton solutions and conduct a comprehensive qualitative analysis. For this aim, we use the Gross–Pitaevskii system, which belongs to the family of nonlinear Schrödinger equations. This model is considered to be [...] Read more.

The purpose of this study is to construct diverse forms of exact soliton solutions and conduct a comprehensive qualitative analysis. For this aim, we use the Gross–Pitaevskii system, which belongs to the family of nonlinear Schrödinger equations. This model is considered to be iconic and significant because it has potential applications in applied sciences, such as in physics, where it is used to exemplify quantum systems like Bose–Einstein condensates and illustrate the propagation of waves in optical fibers. Employing analytical techniques, the modified sine–cosine/sinh–cosh and extended rational sinh–Gordon expansion methods, we extract several waves from solutions in the shape of trigonometric, hyperbolic, and rational forms. To further deepen our insights related to the system’s behavior, we execute a detailed dynamical analysis, including sensitivity, bifurcation, and chaos, using the corresponding Hamiltonian structure. We also derive the instability modulation using linear stability theory. Using Mathematica, we systematically simulate and verify all constructed results and present some solutions for appropriate parameter values using 2D, 3D, and contour plots. The outcomes provide fruitful insights relevant to multiple scientific domains, including optical fiber technology, plasma, and condensed matter physics. This work contributes to the ongoing study of nonlinear models by applying novel solution techniques and offering a broader perspective on the complex behavior of such systems. The novelty of this study lies in the fact that the proposed model has not been previously explored using the aforementioned advanced methods and comprehensive dynamical analyses. Full article

(This article belongs to the Section Physics)

19 pages, 2302 KB

Open AccessArticle

A Gradient-Norm-Aware Optimizer for Symmetry-Preserving and Stable Deep Learning

by Hong Chen, Xuefeng Zhang, Haijing Sun and Yichuan Shao

Symmetry 2026, 18(2), 374; https://doi.org/10.3390/sym18020374 - 18 Feb 2026

Abstract

The stability of deep neural networks during training is related to the structural symmetry between gradient dynamics and parameter updates. Existing Adam-type optimizers are prone to breaking this dynamic symmetry when the gradient scale changes drastically, leading to training instability. In addition, quantitative [...] Read more.

The stability of deep neural networks during training is related to the structural symmetry between gradient dynamics and parameter updates. Existing Adam-type optimizers are prone to breaking this dynamic symmetry when the gradient scale changes drastically, leading to training instability. In addition, quantitative metrics of stability are serious lacking. A novel Optimizer, Gradient-Norm-Aware Adam (GNAdam), is proposed in this paper. It introduces a gradient norm-based smooth learning rate mechanism to adjust the step size, effectively maintaining the structural symmetry and exhibiting strong training stability. A new stability evaluation metric, Stability Score Plus (SS+), combining gradient norm and parameter update norm, is designed in this paper to quantify algorithm stability. Experimental results on the CIFAR-10 dataset demonstrate that GNAdam reduces gradient ripple and parameter update instability. The stability quantification results are highly consistent with the visualization results, with the SS+ value being only 24% of Adam’s and 8% of RAdam’s. Experimental results on glaucoma datasets demonstrate that the proposed method shows significant advantages in classification metrics such as accuracy and AUC, as well as medical metrics such as sensitivity and specificity. This indicates that improved training stability can effectively translate into more reliable clinical diagnosis in complex medical image classification tasks. Full article

(This article belongs to the Section Computer)

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29 pages, 4557 KB

Open AccessArticle

Post-Experimental Analysis of the Light-Harvesting Protein–Pigment Complex Present in Green Sulfur Bacteria: An Approach in Quantum Biology

by Francisco Delgado and Estela Delgado-Ceballos

Symmetry 2026, 18(2), 373; https://doi.org/10.3390/sym18020373 - 17 Feb 2026

Abstract

Quantum biology is a multidiscipline which analyses possible critical aspects of life that could be based on the macroscopic expression of quantum phenomena. The high efficiency of light energy harvesting in green sulfur bacteria during photosynthesis is associated with entanglement and tunneling effects [...] Read more.

Quantum biology is a multidiscipline which analyses possible critical aspects of life that could be based on the macroscopic expression of quantum phenomena. The high efficiency of light energy harvesting in green sulfur bacteria during photosynthesis is associated with entanglement and tunneling effects in the Fenna–Mathew–Olson complex. This has been studied to assess itscontribution, when conducting the light energy captured by the chlorosome, to the reaction center, where it is transformed into chemical energy. This work analyses, in the quantum domain, the coherence and entanglement between those two components associated with a general non-localized absorption spectrum in the pigments serving as input antennas. This study first imposes a more symmetric structure on the absorption spectrum, revealing certain relations which, when it is partially broken and parametrized on the most feasible pigments, displays a characteristic spectrum associated with the nature of the bacteria studied, in terms of their habitat and evolutionary survival. Finally, a brief insight analysis of similarities and differences in the protein sequence of the complex is conducted to trace possible traits relating them to some of the previous quantum features and suggesting some responsible positions within the FMO protein sequence. Full article

(This article belongs to the Special Issue Symmetry and Asymmetry in Quantum Models)

25 pages, 4998 KB

Open AccessArticle

Pareto-Aware Dual-Preference Optimization for Task-Oriented Dialogue

by Shenghui Bao and Mideth Abisado

Symmetry 2026, 18(2), 372; https://doi.org/10.3390/sym18020372 - 17 Feb 2026

Abstract

Task-oriented dialogue systems face a tension between comprehensive constraint elicitation (task adequacy) and conversational efficiency (minimizing turns). Current preference learning frameworks treat preferences as static, unable to capture the dynamic evolution of interaction states that evolve across dialogue progression. We present Dual-DPO, a [...] Read more.

Task-oriented dialogue systems face a tension between comprehensive constraint elicitation (task adequacy) and conversational efficiency (minimizing turns). Current preference learning frameworks treat preferences as static, unable to capture the dynamic evolution of interaction states that evolve across dialogue progression. We present Dual-DPO, a framework that embeds multi-objective preferences into data construction via turn-aware scoring. Our approach decouples objective balancing from policy updates through offline preference scalarization, addressing the optimization instability challenges in online multi-objective reinforcement learning. Experiments on MultiWOZ 2.4 demonstrate 28–35% dialogue turn reduction while maintaining Joint Goal Accuracy > 89% (

p < 0.001

). Pareto frontier analysis shows 94% coverage with hypervolume

H V = 0.847

. Independent expert evaluation by 10 PhD-level researchers (

n = 300

assessments, inter-rater agreement

α = 0.78

) confirms 32% user satisfaction improvement (

p < 0.001

). Theoretical analysis demonstrates that offline scalarization, which correlates with improved optimization stability, achieves

3.2 \times

lower gradient variance than online multi-reward optimization by eliminating sampling stochasticity. Our approach enables balanced treatment of competing objectives through Pareto-optimal trade-offs. These results highlight a symmetric and balanced treatment of competing objectives within a Pareto-optimal optimization framework. Full article

(This article belongs to the Section Computer)

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24 pages, 7887 KB

Open AccessArticle

A Novel Multi-Cooperative Neural Radiance Field Reconstruction Method Based on Optical Properties for 3D Reconstruction of Scenes Containing Transparent Objects

by Xiaopeng Sha, Wenbo Sun, Kai Sun, Xinqi Sang and Shuyu Wang

Symmetry 2026, 18(2), 371; https://doi.org/10.3390/sym18020371 - 17 Feb 2026

Abstract

Due to phenomena, such as refraction, reflection, and light scattering, the three-dimensional (3D) reconstruction of transparent objects with complex geometric symmetry or contours is confronted with the challenges of insufficient extraction of feature points and recognition of contour detail. To solve this challenge, [...] Read more.

Due to phenomena, such as refraction, reflection, and light scattering, the three-dimensional (3D) reconstruction of transparent objects with complex geometric symmetry or contours is confronted with the challenges of insufficient extraction of feature points and recognition of contour detail. To solve this challenge, a novel reconstruction method based on multi-cooperative Neural Radiance Fields (NeRF) is proposed in the paper. This method incorporates angular offset fields and local reconstruction fields, explicitly modeling the effects of refraction and reflection during light propagation. The angular offset field simulates the internal refractive deflection within transparent materials, while the localized reconstruction field performs secondary reconstruction in regions affected by specular reflection. This approach effectively captures surface contours of transparent objects and accurately reconstructs scene details. Experimental results demonstrate that our method achieves approximately 10% improvement in reconstruction accuracy compared to traditional neural radiance field techniques, with a PSNR of 25, an increased SSIM of 0.87, and a reduced LPIPS value of 0.365. The proposed method offers a new perspective for reconstructing transparent objects and scenes containing such materials, holding significant theoretical and practical value. Full article

(This article belongs to the Section Computer)

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17 pages, 2181 KB

Open AccessArticle

Numerical Investigation into the Effects of Geometric Symmetry Breaking on Low-Frequency Noise in Urban Rail Transit Viaducts

by Xinting Dong, Bing Zhong and Bin Wang

Symmetry 2026, 18(2), 370; https://doi.org/10.3390/sym18020370 - 17 Feb 2026

Abstract

The expansion of urban rail transit has exacerbated environmental issues related to low-frequency noise (LFN), yet the impact of geometric symmetry breaking on structure-borne noise remains underexplored. This study aims to quantify the mechanism by which cross-sectional asymmetry influences the vibro-acoustic coupling of [...] Read more.

The expansion of urban rail transit has exacerbated environmental issues related to low-frequency noise (LFN), yet the impact of geometric symmetry breaking on structure-borne noise remains underexplored. This study aims to quantify the mechanism by which cross-sectional asymmetry influences the vibro-acoustic coupling of viaducts. A 2.5D Hybrid Finite Element-Boundary Element Method (FEM-BEM) was employed to model a parametric box girder under eccentric track loading, and the numerical framework was validated against analytical benchmarks. The “Modal Symmetry Index” (MSI) and “Acoustic Asymmetry Indicator” (AAI) were defined to evaluate the effects of the asymmetry parameter (

α

) on sound field distribution. Numerical results reveal a nonlinear “V-shaped” relationship between geometric asymmetry and acoustic directivity. While severe asymmetry (

α > 0.15

) exacerbates noise deflection via flexural–torsional coupling, a critical “self-balance zone” exists. Specifically, moderate asymmetry (

α \approx 0.07

) effectively neutralizes load eccentricity, reducing the AAI from 1.5 dB (in strictly symmetric designs) to nearly 0 dB. Robustness analysis under right-side loading conditions further confirms a “reverse deflection” phenomenon, verifying that the proposed self-balance design minimizes directional sensitivity. These findings challenge the traditional assumption that geometric symmetry is acoustically optimal. A “competition–compensation” mechanism is identified, suggesting that deliberate, slight geometric asymmetry can serve as an effective passive noise control strategy for viaducts. Full article

(This article belongs to the Section Mathematics)

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27 pages, 2102 KB

Open AccessArticle

Hub Location and Truck Platoon Routing Optimization for Courier Line-Haul Networks with Carbon Benefits Under Undirected Symmetry

by Yinan Zhao and Hanwen Jiang

Symmetry 2026, 18(2), 369; https://doi.org/10.3390/sym18020369 - 16 Feb 2026

Abstract

Truck platooning enabled by V2X and cooperative driving can reduce aerodynamic drag and consequently decrease fuel consumption and CO₂ emissions. Meanwhile, hub-and-spoke courier networks require strategic decisions on hub locations, allocation, and line-haul routing. This paper introduces an integrated Hub Location-Platoon Routing [...] Read more.

Truck platooning enabled by V2X and cooperative driving can reduce aerodynamic drag and consequently decrease fuel consumption and CO₂ emissions. Meanwhile, hub-and-spoke courier networks require strategic decisions on hub locations, allocation, and line-haul routing. This paper introduces an integrated Hub Location-Platoon Routing Problem (HLPRP) that jointly optimizes (i) hub selection and single allocation of spokes; (ii) the departure hubs where platoons are formed; (iii) line-haul (inter-hub) service design and route selection; and (iv) demand routing, while internalizing monetized carbon benefits from platooning. A variable neighborhood search-based simulated annealing solution framework is developed to eliminate duplicated hub pair representations induced by network symmetry. Computational experiments on benchmark and large-scale North China instances demonstrate that the proposed approach consistently produces high-quality solutions within practical runtimes. The results indicate that the optimal network structure is primarily driven by transportation cost trade-offs and is further shaped by platoon-enabling investment and the associated carbon benefit, which concentrates on a subset of high-volume inter-hub corridors. Overall, the proposed framework provides a decision support approach for designing low-carbon courier line-haul networks. Full article

(This article belongs to the Section Engineering and Materials)

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36 pages, 141661 KB

Open AccessArticle

Design Optimization of Eccentric Pole PM Motors Using the Bilinear Mapping Method

by Ali Jabbari and Frédéric Dubas

Symmetry 2026, 18(2), 368; https://doi.org/10.3390/sym18020368 - 16 Feb 2026

Abstract

The eccentric permanent-magnet (PM) pole technique is widely recognized as an effective technique for reducing cogging torque in surface-mounted PM motors (SMPMMs). This paper proposes a novel analytical approach based on bilinear mapping to determine the optimal PM reduction parameters. In this method, [...] Read more.

The eccentric permanent-magnet (PM) pole technique is widely recognized as an effective technique for reducing cogging torque in surface-mounted PM motors (SMPMMs). This paper proposes a novel analytical approach based on bilinear mapping to determine the optimal PM reduction parameters. In this method, the outer surface of the PM and the stator inner bore are modeled as eccentric circles. Bilinear mapping is then used to transform a slotted stator bore into an equivalent slotless configuration with small slot-openings, allowing the optimal PM reduction to be identified. The key electromagnetic performance characteristics of SMPMMs—including torque, efficiency, mean air-gap flux density, and related parameters—are formulated as explicit mathematical functions of the PM reduction factor. The influence of the optimal PM reduction on both static and dynamic rotor eccentricity is also investigated. The results reveal that the bilinear mapping equations yield two distinct roots for the optimal PM reduction. Once the optimal values are known for a reference motor, those of other motors with different dimensions can be readily derived by scaling according to the ratio of the outer PM radii, without repeating the full calculation process. The proposed method is applicable to various SMPMM geometries, including radial, parallel, and bread-loaf configurations. Full article

(This article belongs to the Special Issue Computational Mathematics and Its Applications in Numerical Analysis)

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25 pages, 1014 KB

Open AccessArticle

Stabilization of Nonlinear Coupled Parametric Oscillators of Mathieu’s Type in Fractal Space

by Ji-Huan He, Yusry O. El-Dib and Haifa A. Alyousef

Symmetry 2026, 18(2), 367; https://doi.org/10.3390/sym18020367 - 16 Feb 2026

Abstract

In this work, the Renormalization Method (RM) is used to analyze the dynamics of a nonlinear two-degree-of-freedom (2DOF) system under parametric excitation, with a focus on fractal vibration behavior. This procedure comprises transforming the system into a comparable form. An equivalent linearized model [...] Read more.

In this work, the Renormalization Method (RM) is used to analyze the dynamics of a nonlinear two-degree-of-freedom (2DOF) system under parametric excitation, with a focus on fractal vibration behavior. This procedure comprises transforming the system into a comparable form. An equivalent linearized model is produced by isolating the system’s nonlinear interactions using a two-scale formulation and mean-square analysis. The non-autonomous fractal equations are transformed into an autonomous representation using the RM, and then the system is described in traditional derivative form using El-Dib’s fractal transformation. The fractal-coupled Mathieu system’s stability behavior can be effectively identified using this framework. An agreement with the analytical solutions is shown by numerical results. All things considered, the integrated RM-based approach provides a reliable tool for forecasting and managing intricate nonlinear fractal systems. Full article

(This article belongs to the Section Mathematics)

28 pages, 17682 KB

Open AccessArticle

Causal-Enhanced Spatio-Temporal Markov Graph Convolutional Network for Traffic Flow Prediction

by Jing Hu and Shuhua Mao

Symmetry 2026, 18(2), 366; https://doi.org/10.3390/sym18020366 - 15 Feb 2026

Abstract

Traffic flow prediction is a pivotal task in intelligent transportation systems. The primary challenge lies in accurately modeling the dynamically evolving and directional spatio-temporal dependencies inherent in road networks. Existing graph neural network-based methods suffer from three main limitations: (1) symmetric adjacency matrices [...] Read more.

Traffic flow prediction is a pivotal task in intelligent transportation systems. The primary challenge lies in accurately modeling the dynamically evolving and directional spatio-temporal dependencies inherent in road networks. Existing graph neural network-based methods suffer from three main limitations: (1) symmetric adjacency matrices fail to capture the causal propagation of traffic flow from upstream to downstream; (2) the serial combination of graph and temporal convolutions lacks an explicit modeling of joint spatio-temporal state transition probabilities; (3) the inherent low-pass filtering property of temporal convolutional networks tends to smooth high-frequency abrupt signals, thereby weakening responsiveness to sudden events. To address these issues, this paper proposes a causal-enhanced spatio-temporal Markov graph convolutional network (CSHGCN). At the spatial modeling level, we construct an asymmetric causal adjacency matrix by decoupling source and target node embeddings to learn directional traffic flow influences. At the spatio-temporal joint modeling level, we design a spatio-temporal Markov transition module (STMTM) based on spatio-temporal Markov chain theory, which explicitly learns conditional transition patterns through temporal dependency encoders, spatial dependency encoders, and a joint transition network. At the temporal modeling level, we introduce differential feature enhancement and high-frequency residual compensation mechanisms to preserve key abrupt change information through frequency-domain complementarity. Experiments on four datasets—PEMS03, PEMS04, PEMS07, and PEMS08—demonstrate that CSHGCN outperforms existing baselines in terms of MAE, RMSE, and MAPE, with ablation studies validating the effectiveness of each module. Full article

(This article belongs to the Section Computer)

22 pages, 6091 KB

Open AccessArticle

Coupled Impact Dynamics of Double Droplets on Curved Cylindrical Surfaces

by Haibin Zhang, Zhenqiang Ma and Min Wei

Symmetry 2026, 18(2), 365; https://doi.org/10.3390/sym18020365 - 15 Feb 2026

Abstract

Droplet impact dynamics is a critical subject in interfacial fluid mechanics, with applications in aerospace, energy transport, microfluidics, and chemical engineering. This study investigates the impact behavior of double droplets on cylindrical surfaces, focusing on the interaction dynamics and the effects of parameters [...] Read more.

Droplet impact dynamics is a critical subject in interfacial fluid mechanics, with applications in aerospace, energy transport, microfluidics, and chemical engineering. This study investigates the impact behavior of double droplets on cylindrical surfaces, focusing on the interaction dynamics and the effects of parameters such as Weber number, droplet spacing, and surface curvature. Using numerical simulations, the study identifies three distinct rebound modes—twin-wing rebound, vertical rebound, and arc-shaped rebound—regulated by the Weber number and droplet spacing. Results show that increasing the Weber number enhances spreading and reduces contact time, with the arc-shaped rebound mode resulting in the shortest contact times. Droplet spacing further influences the dynamics, with wider spacing increasing contact time due to additional retraction phases. The findings provide valuable insights into the complex multi-field interactions governing droplet behavior on curved surfaces, offering new perspectives for the design of anti-icing coatings and curved microfluidic devices. Full article

(This article belongs to the Section Physics)

27 pages, 1250 KB

Open AccessArticle

High-Dimensional Semiparametric Analysis of Differential Network Analysis for Matrix-Variate

by Tianying Li and Lu Niu

Symmetry 2026, 18(2), 364; https://doi.org/10.3390/sym18020364 - 15 Feb 2026

Abstract

Differential network analysis provides a powerful framework for characterizing how biological networks change across different conditions or external stimuli. With recent advances in high-throughput technologies, matrix-valued data have become increasingly common in biostatistics and medical research. However, most existing differential network methods are [...] Read more.

Differential network analysis provides a powerful framework for characterizing how biological networks change across different conditions or external stimuli. With recent advances in high-throughput technologies, matrix-valued data have become increasingly common in biostatistics and medical research. However, most existing differential network methods are designed for vector-valued observations. As a result, they do not fully exploit the structural information inherent in matrix-valued data. In addition, many of these approaches rely on multivariate normality, an assumption that is often violated in practice, motivating the need for robust inference procedures. In this paper, we develop a semiparametric matrix-variate differential network model that accommodates heavy-tailed or non-Gaussian data distributions. We introduce a rank-based D-trace loss with an

ℓ_{1}

penalty to directly estimate the spatial differential partial correlation matrix. The resulting optimization problem is solved using an efficient ADMM algorithm. We establish the theoretical properties of the proposed estimator, including its consistency under high-dimensional scaling. Extensive simulation studies and real-data analyses further demonstrate that our robust procedure substantially outperforms existing non-robust alternatives. Full article

(This article belongs to the Special Issue Mathematics: Feature Papers 2025)

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