Symmetry

29 pages, 7532 KB

Open AccessArticle

Methodology to Quantify the Water Content of Axisymmetric Cylindrical Cement-Based Material Samples Using Neutron Radiography

by Luiz Antonio de Siqueira Neto, Osman Burkan Isgor, Steven Richard Reese and William Jason Weiss

Symmetry 2026, 18(1), 114; https://doi.org/10.3390/sym18010114 - 7 Jan 2026

Abstract

Neutron radiography has been used to assess water content and transport in porous media like cement paste, mortar and concrete. This paper presents a methodology to evaluate water content profiles from neutron radiographs of axisymmetric cylindrical samples along the radial direction. Three examples [...] Read more.

Neutron radiography has been used to assess water content and transport in porous media like cement paste, mortar and concrete. This paper presents a methodology to evaluate water content profiles from neutron radiographs of axisymmetric cylindrical samples along the radial direction. Three examples are proposed as potential applications of this methodology: a drying mortar specimen, a sample of fluid cement paste during high-shear flow, and a steel-reinforced mortar specimen with a gap at the steel–mortar interface. In each case, a simulated neutron image is generated to represent experimental data. The proposed methodology is used to back-calculate the water content distribution of the original sample. The proposed approach can accurately quantify the distribution of water in all three theoretical cylindrical samples. For neutron radiographs created using a random distribution of neutron cross-section values for each constituent, emulating the experimental variability of the imaging process, the proposed method was able to quantify the distribution of water along the radial direction with an average error less than 1.5% for the drying mortar specimen, 3% for the cement paste sample during high-shear flow, and 4% for the reinforced sample with a gap at the steel–mortar interface. Full article

(This article belongs to the Special Issue Applications Based on Symmetry in Additive Manufacturing)

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25 pages, 351 KB

Open AccessArticle

Symmetry-Aware LLM-Driven Generation and Repair of Interactive Fiction Graphs in Twine/Twee

by Marcin Puchalski and Bożena Woźna-Szcześniak

Symmetry 2026, 18(1), 113; https://doi.org/10.3390/sym18010113 - 7 Jan 2026

Abstract

We present a hybrid system that combines large language models (LLMs) with formal graph-analytic methods to generate and automatically repair interactive fiction (IF) stories written in the Twine/Twee format. We chronologically describe the practical challenges encountered when attempting to produce fully playable branching [...] Read more.

We present a hybrid system that combines large language models (LLMs) with formal graph-analytic methods to generate and automatically repair interactive fiction (IF) stories written in the Twine/Twee format. We chronologically describe the practical challenges encountered when attempting to produce fully playable branching narratives using contemporary state-of-the-art LLMs, including missing passages, trap-like cycles without exits, dead-end passages, narrative discontinuities, incorrect use of Twine macro commands, and inconsistent handling of story variables. To address these limitations, we deliberately abandon all macro- and variable-based logic and instead encode story state directly within passage names through structured, token-based naming. We formalize symmetry and asymmetry in the resulting narrative graphs: symmetrical convergence occurs when multiple branches with compatible states merge into a common passage, whereas asymmetry reveals incorrect or logically inconsistent merging of divergent states (for example, entering a scene in which an item or companion is present via paths where they were never acquired or met). We propose algorithms to detect naming-based asymmetries, cycles, unreachable endings, and structurally defective branches, and we integrate these diagnostics into a repair loop that prompts the LLM to rewrite missing or inconsistent parts of the story. Experiments with several LLM backends indicate that this approach can yield structurally robust and locally coherent interactive stories by reducing state inconsistencies and structural defects. Beyond the specific case of Twine, we argue that symmetry/asymmetry analysis offers a powerful lens for evaluating and correcting AI-generated narrative graphs in general. Full article

(This article belongs to the Special Issue Applications of Symmetry/Asymmetry in Artificial Intelligence and Deep Metaheuristics)

31 pages, 13997 KB

Open AccessArticle

Deep Reinforcement Learning for Financial Trading: Enhanced by Cluster Embedding and Zero-Shot Prediction

by Haoran Zhang, Xiaofei Li, Tianjiao Wan and Junjie Du

Symmetry 2026, 18(1), 112; https://doi.org/10.3390/sym18010112 - 7 Jan 2026

Abstract

Deep reinforcement learning (DRL) plays a pivotal role in decision-making within financial markets. However, DRL models are highly reliant on raw market data and often overlook the impact of future trends on model performance. To address these challenges, we propose a novel framework [...] Read more.

Deep reinforcement learning (DRL) plays a pivotal role in decision-making within financial markets. However, DRL models are highly reliant on raw market data and often overlook the impact of future trends on model performance. To address these challenges, we propose a novel framework named Cluster Embedding-Proximal Policy Optimization (CE-PPO) for trading decision-making in financial markets. Specifically, the framework groups feature channels with intrinsic similarities and enhances the original model by leveraging clustering information instead of features from individual channels. Meanwhile, zero-shot prediction for unseen samples is achieved by assigning them to appropriate clusters. Future Open, High, Low, Close, and Volume (OHLCV) data predicted from observed values are integrated with actually observed OHLCV data, forming the state space inherent to reinforcement learning. Experiments conducted on five real-world financial datasets demonstrate that the time series model integrated with Cluster Embedding (CE) achieves significant improvements in predictive performance: in short-term prediction, the Mean Absolute Error (MAE) is reduced by an average of 20.09% and the Mean Squared Error (MSE) by 30.12%; for zero-shot prediction, the MAE and MSE decrease by an average of 21.56% and 31.71%, respectively. Through data augmentation using real and predicted data, the framework substantially enhances trading performance, achieving a cumulative return rate of 137.94% on the S&P 500 Index. Beyond its empirical contributions, this study also highlights the conceptual relevance of symmetry in the domain of algorithmic trading. The constructed deep reinforcement learning framework is capable of capturing the inherent balanced relationships and nonlinear interaction characteristics embedded in financial market behaviors. Full article

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

19 pages, 1362 KB

Open AccessArticle

Parallel Darboux Equidistant Ruled Surfaces in E³

by Ceyda Cevahir Yıldız, Süleyman Şenyurt and Luca Grilli

Symmetry 2026, 18(1), 111; https://doi.org/10.3390/sym18010111 - 7 Jan 2026

Abstract

In this study, equidistant ruled surfaces generated by the Darboux vector, which has significant kinematic importance and characterizes the instantaneous rotation of a moving frame, are investigated specifically for the Frenet frame. By establishing a structural relationship between a surface and its equidistant [...] Read more.

In this study, equidistant ruled surfaces generated by the Darboux vector, which has significant kinematic importance and characterizes the instantaneous rotation of a moving frame, are investigated specifically for the Frenet frame. By establishing a structural relationship between a surface and its equidistant ruled surface, transition formulas are provided for shape operators, Gaussian and mean curvatures, and fundamental forms, revealing that the equidistant surface is a scaled transformation of the original one. The obtained results demonstrate that both surfaces are developable and that the geometric properties of the equidistant ruled surfaces can be expressed dependently on each other. Furthermore, it is shown that the geometric character of the equidistant surface, including the invariance of asymptotic lines and the preservation of umbilical points under constant angle conditions, is determined by the rotational dynamics of the base curve. These findings constitute a theoretical foundation for cases involving the use of Darboux axes of different frames in higher dimensions or the investigation of similar structures in different geometric spaces. The geometric interpretation of this theoretical framework is elucidated through the fundamental properties of the surfaces. Finally, a concrete example is presented, where the symmetry of the central planes of the equidistant ruled surfaces at appropriate points is visualized using Maple 2017 software. Full article

(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology, 4th Edition)

18 pages, 598 KB

Open AccessArticle

A Biased-Randomized Algorithm for the Bi-Objective Capacitated Dispersion Problem with Symmetries

by Juan F. Gomez, Wenwen Chen, Laura Calvet, Majsa Ammouriova and Angel A. Juan

Symmetry 2026, 18(1), 110; https://doi.org/10.3390/sym18010110 - 7 Jan 2026

Abstract

Given a network of nodes and a certain demand that needs to be satisfied, the capacitated dispersion problem (CDP) involves selecting a subset of nodes to maximize dispersion between them. In many practical instances, symmetry in the structure of the selected nodes (e.g., [...] Read more.

Given a network of nodes and a certain demand that needs to be satisfied, the capacitated dispersion problem (CDP) involves selecting a subset of nodes to maximize dispersion between them. In many practical instances, symmetry in the structure of the selected nodes (e.g., using nodes of the same type) can lead to synergies. Hence, this paper studies a bi-objective variant of the CDP to account for these symmetries. The first goal seeks to maximize the minimum distance between opened nodes, while the second goal accounts for symmetry by penalizing the use of nodes of different types (in our case, represented by different colors). We formalize the problem as a bi-objective mathematical program and address it through a classical multiobjective strategy, the

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-constraint method. Exact methods are used when the problem size allows, even though the problem is NP-hard. To tackle larger instances, we design a biased-randomized algorithm based on a constructive heuristic. Computational experiments show that our biased-randomized algorithm provides high-quality approximations of the Pareto frontier. Full article

(This article belongs to the Special Issue Applications of Symmetry/Asymmetry in Artificial Intelligence and Deep Metaheuristics)

46 pages, 1021 KB

Open AccessArticle

Confidence Intervals for the Difference and Ratio Means of Zero-Inflated Two-Parameter Rayleigh Distribution

by Sasipong Kijsason, Sa-Aat Niwitpong and Suparat Niwitpong

Symmetry 2026, 18(1), 109; https://doi.org/10.3390/sym18010109 - 7 Jan 2026

Abstract

The analysis of road traffic accidents often reveals asymmetric patterns, providing insights that support the development of preventive measures, reduce fatalities, and improve road safety interventions. The Rayleigh distribution, a continuous distribution with inherent asymmetry, is well suited for modeling right-skewed data and [...] Read more.

The analysis of road traffic accidents often reveals asymmetric patterns, providing insights that support the development of preventive measures, reduce fatalities, and improve road safety interventions. The Rayleigh distribution, a continuous distribution with inherent asymmetry, is well suited for modeling right-skewed data and is widely used in scientific and engineering fields. It also shares structural characteristics with other skewed distributions, such as the Weibull and exponential distributions, and is particularly effective for analyzing right-skewed accident data. This study considers several approaches for constructing confidence intervals, including the percentile bootstrap, bootstrap with standard error, generalized confidence interval, method of variance estimates recovery, normal approximation, Bayesian Markov Chain Monte Carlo, and Bayesian highest posterior density methods. Their performance was evaluated through Monte Carlo simulation based on coverage probabilities and expected lengths. The results show that the HPD method achieved coverage probabilities at or above the nominal confidence level while providing the shortest expected lengths. Finally, all proposed confidence intervals were applied to fatalities recorded during the seven hazardous days of Thailand’s Songkran festival in 2024 and 2025. Full article

(This article belongs to the Special Issue Skewed (Asymmetrical) Probability Distributions and Applications Across Disciplines, Fourth Edition)

31 pages, 3447 KB

Open AccessArticle

Improved Chimpanzee Optimization Algorithm Based on Multi-Strategy Fusion and Its Application in Multiphysics Parameter Optimization

by Bin Zhou, Chaoyun Shi, Ning Yan and Yangyang Chu

Symmetry 2026, 18(1), 108; https://doi.org/10.3390/sym18010108 - 7 Jan 2026

Abstract

To address the challenges of high computational costs, susceptibility to local optima, and heavy reliance on manual intervention in multi-physics parameter optimization for symmetric acoustic metamaterials, an enhanced Chimp Optimization Algorithm (DADCOA) is proposed in this paper. This algorithm integrates the double chaotic [...] Read more.

To address the challenges of high computational costs, susceptibility to local optima, and heavy reliance on manual intervention in multi-physics parameter optimization for symmetric acoustic metamaterials, an enhanced Chimp Optimization Algorithm (DADCOA) is proposed in this paper. This algorithm integrates the double chaotic initialization strategy (DCS), adaptive multimodal convergence mechanism (AMC), and dual-weight pinhole imaging update operator (DWPI). It employs a Logistic–Tent composite chaotic mapping strategy for population initialization, significantly enhancing distribution uniformity within high-dimensional parameter spaces. An AMC factor is then introduced to dynamically balance global exploration and local exploitation based on the real-time evolutionary state of the population. A dual-weight population update mechanism, incorporating distance and historical contributions, is integrated with a pinhole imaging opposition-based learning strategy to improve population diversity. Additionally, a composite single objective error feedback local differential mutation operation is introduced to improve optimization accuracy for coupled multi-physics objectives. Experimental validation based on the CEC 2022 test function suite and an acoustic metamaterial parameter optimization model demonstrates that compared to the standard COA algorithm and existing improved algorithms, the DADCOA algorithm reduces simulation time by 28.46% to 60.76% while maintaining high accuracy. This approach effectively addresses the challenges of high computational cost, stringent accuracy requirements, and composite single objective coupling in COMSOL physical parameter optimization, providing an effective solution for the design of acoustic metamaterials based on symmetric structures. Full article

(This article belongs to the Section Engineering and Materials)

20 pages, 5903 KB

Open AccessArticle

Bound Optimization by Quadratic Approximation for Heat-Dissipation-Oriented Design of an Air-Cooled Lithium Battery Energy Storage Cabinet

by Liqun Wang, Yunqing Tang, Jianbin Yu, Wei Qin, Yangyang Zhang, Guoyan Wang, Dongjing Liu, Yukui Cai and Zhanqiang Liu

Symmetry 2026, 18(1), 107; https://doi.org/10.3390/sym18010107 - 7 Jan 2026

Abstract

With the increasing energy density of lithium-ion batteries, the heat dissipation performance of air-cooled battery energy storage cabinets has become a critical determinant of both system performance and service life. This performance depends strongly on the geometry of the airflow channels and their [...] Read more.

With the increasing energy density of lithium-ion batteries, the heat dissipation performance of air-cooled battery energy storage cabinets has become a critical determinant of both system performance and service life. This performance depends strongly on the geometry of the airflow channels and their influence on the internal flow distribution. In this study, the internal flow field of a battery energy storage cabinet was analyzed, and the airflow-channel geometry was optimized using the BOBYQA algorithm. The results indicate that the risk of thermal runaway is largely associated with inadequate airflow design, which leads to localized heat accumulation. Geometric optimization of the airflow channels reduced the maximum hotspot temperature from 72.9 °C to 57.6 °C. The hotspots were concentrated at the tops of the battery modules. Modifications to the channel geometry increased the airflow velocity and improved its directionality in these regions, thereby reducing both the hotspot temperature and the extent of the affected area. Moreover, slightly increasing the inlet pressure while reducing the outlet pressure produced a more uniform temperature distribution across the tops of the battery modules. Full article

(This article belongs to the Special Issue Symmetry in Mechanical Engineering: Properties and Applications)

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21 pages, 7915 KB

Open AccessArticle

Analysis of Wind Erosion Resistance Enhancement of Aeolian Sand by Microbially Induced Carbonate Precipitation Technology

by Fangcan Ji, Junhui Zhang, Weiming Guan, Hui Chen, Xin Wang, Meng Xie, Haosen Wang and Defeng Hou

Symmetry 2026, 18(1), 106; https://doi.org/10.3390/sym18010106 - 7 Jan 2026

Abstract

Aeolian sand in arid mining regions is highly susceptible to wind erosion, posing serious threats to ecological stability and surface engineering safety. To enhance its resistance, this study applied the microbially induced carbonate precipitation (MICP) technique and conducted wind tunnel experiments combined with [...] Read more.

Aeolian sand in arid mining regions is highly susceptible to wind erosion, posing serious threats to ecological stability and surface engineering safety. To enhance its resistance, this study applied the microbially induced carbonate precipitation (MICP) technique and conducted wind tunnel experiments combined with SEM and XRD analyses to examine the effects of cementing solution type and concentration, bacteria-to-cementation-solution ratio (B/C ratio), and spraying volume on the wind erosion behavior of MICP-treated aeolian sand. Results show that the cementing solution type and concentration jointly control erosion resistance. The MgO-based system exhibited the best performance at a B/C ratio of 1:2, reducing erosion loss by 47.2% compared with the CaCl₂ system, while a 1.0 mol/L concentration further decreased loss by 97.4% relative to 0.5 mol/L. Increasing the spraying volume from 0.6 to 1.2 L/m² reduced erosion loss by 70–99%, and a moderate B/C ratio (1:2) ensured balanced microbial activity and uniform CaCO₃ deposition. Microstructural observations confirmed that MICP strengthened the sand through CaCO₃ crystal attachment, pore filling, and interparticle bridging, forming a dense surface crust with enhanced integrity. From a symmetry perspective, the microbially induced mineralization process promotes a more symmetric and spatially uniform distribution of carbonate precipitates at particle contacts and within pore networks. This symmetry-enhanced microstructural organization plays a key role in improving the macroscopic stability and wind erosion resistance of aeolian sand. Overall, MICP improved wind erosion resistance through a coupled biological induction–chemical precipitation–structural reconstruction mechanism, providing a sustainable approach for eco-friendly sand stabilization and wind erosion control in arid mining regions. Full article

(This article belongs to the Section Engineering and Materials)

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22 pages, 19863 KB

Open AccessArticle

A Generalized Mixing Length Model with Adverse-Pressure-Gradient Effects

by Mingze Ma, Yuxuan Shi, Yalin Zhu, Anxiao Han and Xi Chen

Symmetry 2026, 18(1), 105; https://doi.org/10.3390/sym18010105 - 7 Jan 2026

Abstract

The adverse-pressure-gradient (APG) severely disrupts the inherent symmetry of the turbulent boundary layer (TBL). The universal physical behavior and scaling laws observed under zero-pressure-gradient (ZPG) conditions are significantly altered under APG, thereby challenging the predictive accuracy of turbulence models. To address this breakdown, [...] Read more.

The adverse-pressure-gradient (APG) severely disrupts the inherent symmetry of the turbulent boundary layer (TBL). The universal physical behavior and scaling laws observed under zero-pressure-gradient (ZPG) conditions are significantly altered under APG, thereby challenging the predictive accuracy of turbulence models. To address this breakdown, we develop a generalized mixing length model by incorporating APG effects into a baseline ZPG framework. The model is directly constructed using APG-extended wall laws—superposing the log law and the half-power law for the mean velocity—and the logarithmic decay law for the total shear stress. Validated against ten databases spanning the entire APG region, various Reynolds numbers, and both equilibrium and non-equilibrium flows, the model demonstrates accurate predictions for both mixing length and mean velocity profiles. The key achievements are as follows: (i) a significant simplification of model parameters, wherein the inner model can be reduced to a parameterless scaling law, while outer parameters were reduced from five to two; and (ii) the unification of predictions across the entire APG range. This work demonstrates that consistent modeling of APG effects bridges slight and strong APGs, enabling a unified description of the APG region before separation and providing critical insights for turbulence modeling. Full article

(This article belongs to the Section Engineering and Materials)

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19 pages, 5679 KB

Open AccessArticle

SDDNet: Two-Stage Network for Forgings Surface Defect Detection

by Shentao Wang, Depeng Gao, Byung-Won Min, Yue Hong, Tingting Xu and Zhongyue Xiong

Symmetry 2026, 18(1), 104; https://doi.org/10.3390/sym18010104 - 6 Jan 2026

Abstract

Detecting surface defects in forgings is crucial for ensuring the reliability of automotive components such as steering knuckles. In fluorescent magnetic particle inspection (FDMPI) images, normal forging surfaces generally exhibit locally symmetric texture patterns, whereas cracks and other flaws appear as locally asymmetric [...] Read more.

Detecting surface defects in forgings is crucial for ensuring the reliability of automotive components such as steering knuckles. In fluorescent magnetic particle inspection (FDMPI) images, normal forging surfaces generally exhibit locally symmetric texture patterns, whereas cracks and other flaws appear as locally asymmetric regions. Traditional FDMPI inspection relies on manual visual judgement, which is inefficient and error-prone. This paper introduces SDDNet, a symmetry-aware deep learning model for surface defect detection in FDMPI images. A dedicated FDMPI dataset is constructed and further expanded using a denoising diffusion probabilistic model (DDPM) to improve training robustness. To better separate symmetric background textures from asymmetric defect cues, SDDNet integrates a UPerNet-based segmentation layer for background suppression and a Scale-Variant Inception Module (SVIM) within an RTMDet framework for multi-scale feature extraction. Experiments show that SDDNet effectively suppresses background noise and significantly improves detection accuracy, achieving a mean average precision (mAP) of 45.5% on the FDMPI dataset, 19% higher than the baseline, and 71.5% mAP on the NEU-DET dataset, outperforming existing methods by up to 8.1%. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry in Image Processing and Computer Vision)

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20 pages, 3754 KB

Open AccessArticle

Scheduling Intrees with Unavailability Constraints on Two Parallel Machines

by Khaoula Ben Abdellafou, Kamel Zidi and Wad Ghaban

Symmetry 2026, 18(1), 103; https://doi.org/10.3390/sym18010103 - 6 Jan 2026

Abstract

This paper considers the two parallel-machine scheduling problem with intree-precedence constraints where machines are subject to non-availability constraints. In the literature, this problem is considered to be an open problem of unknown complexity. The proposed solution proves that the problem under consideration has [...] Read more.

This paper considers the two parallel-machine scheduling problem with intree-precedence constraints where machines are subject to non-availability constraints. In the literature, this problem is considered to be an open problem of unknown complexity. The proposed solution proves that the problem under consideration has polynomial complexity. Periods of machine unavailability are predetermined, and both task execution and inter-task communication are modeled as requiring one unit of time. The optimization criterion central to this study is the minimization of the makespan. Such a scheduling challenge is directly applicable to manufacturing environments, where production equipment can be intermittently offline for reasons such as unscheduled repairs or planned preventative maintenance. Adopting a unit-time task model offers a valuable framework for subsequently scheduling larger, preemptable jobs.This work presents a new method, called Scheduling Intrees with Unavailability Constraints (SIwUC), which operates by aggregating tasks into distinct groups. The analysis establishes that the SIwUC algorithm produces optimal schedules and reveals how the underlying problem architecture and its solutions demonstrate a symmetrical property in the distribution of tasks across the two parallel machines. This paper demonstrates that the proposed SIwUC algorithm builds optimal schedules and highlight how the problem structure and its solutions exhibit a form of symmetry in balancing task allocation between the two parallel machines. Full article

(This article belongs to the Special Issue Symmetry in Process Optimization)

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39 pages, 15362 KB

Open AccessArticle

IAVOA–EATCN: An Adaptive Deep Framework for Accurate Power Load Forecasting

by Ziang Peng, Haotong Han and Jun Ma

Symmetry 2026, 18(1), 102; https://doi.org/10.3390/sym18010102 - 6 Jan 2026

Abstract

With the large-scale integration of renewable energy, the operational complexity of power systems has increased, placing higher demands on the accuracy of load forecasting. To address the nonlinear characteristics of load variations and improve feature utilization, this paper proposes an IAVOA–EATCN load forecasting [...] Read more.

With the large-scale integration of renewable energy, the operational complexity of power systems has increased, placing higher demands on the accuracy of load forecasting. To address the nonlinear characteristics of load variations and improve feature utilization, this paper proposes an IAVOA–EATCN load forecasting model. In the feature engineering stage, an expand–reduce transformation is employed to reconstruct the original multi-feature inputs, and variational mode decomposition (VMD) is further applied to extract low- and high-frequency components, thereby compressing redundant features while preserving essential information structures. In terms of model architecture, the nonlinear representation capability of the temporal convolutional network (TCN) is enhanced by introducing the FlexSwish activation function, and an Efficient Channel Attention (ECA) mechanism is integrated to strengthen the perception of critical features. For parameter optimization, an improved African Vulture Optimization Algorithm (IAVOA) is proposed, which initializes the population using perturbation-enhanced dynamic Tent mapping, balances global exploration and local exploitation through adaptive parameter control, and incorporates elite retention and migration mechanisms to avoid premature convergence. Experimental results on real-world load data demonstrate that the proposed model achieves RMSE, R², and MAE values of 26.5544, 0.9804, and 18.5589, respectively, significantly outperforming benchmark methods and exhibiting strong generalization capability and practical potential for intelligent load forecasting. Full article

(This article belongs to the Section Engineering and Materials)

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23 pages, 6574 KB

Open AccessArticle

Waiting Time in an MSP Queue with Active Management

by Andrzej Chydzinski

Symmetry 2026, 18(1), 101; https://doi.org/10.3390/sym18010101 - 6 Jan 2026

Abstract

We study waiting times in a queue with active management and correlated job/packet sizes, which induce correlated service times. In the transient case, formulae for the distribution tail, the probability density, and the expected virtual waiting time at any time t, are [...] Read more.

We study waiting times in a queue with active management and correlated job/packet sizes, which induce correlated service times. In the transient case, formulae for the distribution tail, the probability density, and the expected virtual waiting time at any time t, are found. Then, grounded on these results, stationary versions of the probability density and the expected value are obtained. The correlation of service times resulting from correlated job sizes is modeled through an MSP (Markovian service process). Theoretical results are reinforced by numerical examples, in which we examine the impact of symmetric positive and negative correlation of service times, and the impact of symmetric weak and strong active management, on transient and stationary waiting times. We also compare the effects of these factors on the waiting time with their effects on the queue length. In these examples, we can see a surprisingly large expected virtual waiting time, much greater than the product of the expected service time and the queue length. This effect is observed for both weak and strong management functions when the correlation is positive, but it vanishes when a symmetric negative correlation is applied. We also observe a weaker effect of active management on virtual waiting times than that of service time correlation, as well as a weaker impact of active management on virtual waiting time densities than on queue length distributions. Full article

(This article belongs to the Section Mathematics)

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25 pages, 908 KB

Open AccessArticle

Statistical Estimation of Common Percentile in Birnbaum–Saunders Distributions: Insights from PM_2.5 Data in Thailand

by Warisa Thangjai, Sa-Aat Niwitpong, Suparat Niwitpong and Rattana Prommai

Symmetry 2026, 18(1), 100; https://doi.org/10.3390/sym18010100 - 6 Jan 2026

Abstract

This study develops approaches for estimating the common percentile of Birnbaum–Saunders (BS) distributions and applies them to daily PM_2.5 concentration data from six monitoring stations in Chiang Mai Province, Thailand. Percentiles provide a robust representation of typical pollutant exposure, being less sensitive [...] Read more.

This study develops approaches for estimating the common percentile of Birnbaum–Saunders (BS) distributions and applies them to daily PM_2.5 concentration data from six monitoring stations in Chiang Mai Province, Thailand. Percentiles provide a robust representation of typical pollutant exposure, being less sensitive to outliers and suitable for skewed environmental data. Estimating the same percentile across multiple monitoring sites offers a standardized metric for regional air quality assessment, enabling meaningful comparisons and informing evidence-based environmental policy. Four statistical approaches—Generalized Confidence Interval (GCI), bootstrap, Bayesian, and Highest Posterior Density (HPD)—were employed to construct confidence intervals (CIs) for the common percentile. Simulation studies evaluated the methods in terms of average length (AL) and coverage probability (CP), showing that the GCI approach offers the best balance between precision and reliability. Application to real PM_2.5 data confirmed that the BS distribution appropriately models pollutant concentrations and that the common percentile provides a meaningful measure for environmental assessment. These findings highlight the GCI method as a robust tool for constructing CIs in environmental data analysis. Full article

(This article belongs to the Special Issue Skewed (Asymmetrical) Probability Distributions and Applications Across Disciplines, Fourth Edition)

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16 pages, 2844 KB

Open AccessArticle

Dynamic Analysis of a Symmetrical Frustum-Shaped Metal Rubber Isolator Under Random Vibration

by Yun Xiao, Jin Gao, Jinfa Lin, Hanbin Wang and Xin Xue

Symmetry 2026, 18(1), 99; https://doi.org/10.3390/sym18010099 - 6 Jan 2026

Abstract

During orbital service, precision aerospace equipment is frequently subjected to harsh vibration environments that can significantly affect reliability and service life. Consequently, the development of effective vibration isolation technologies has become a crucial aspect of aerospace structural design. In this study, random vibration [...] Read more.

During orbital service, precision aerospace equipment is frequently subjected to harsh vibration environments that can significantly affect reliability and service life. Consequently, the development of effective vibration isolation technologies has become a crucial aspect of aerospace structural design. In this study, random vibration theory and frequency-domain analysis methods were employed to investigate the dynamic response characteristics of a symmetrical frustum-shaped metal rubber (FSMR) isolation device under complex operating conditions. The influence of metal rubber density, spring stiffness, and input vibration level on its isolation performance was systematically examined. This work presents the first systematic experimental investigation into the nonlinear dependencies of the performance of a symmetrical frustum-shaped metal rubber isolator on multiple parameters (density, stiffness, excitation level) under random vibration. The test results show that under identical excitation conditions, the device achieves optimal damping ratio and isolation efficiency (59.71%) when the metal rubber density is 2.0 g/cm³. A moderate increase in spring stiffness reduces the resonance peak and improves stability, with a stiffness of 100 kN/m exhibiting the best overall performance. In addition, higher input vibration levels markedly elevate the acceleration response and the resonant peak amplification factor of the isolator, demonstrating that high-intensity excitation magnifies the vibration response and degrades the isolation efficiency. Full article

(This article belongs to the Special Issue Advances in Dynamics, Planning and Control of Spacecraft and Related Systems)

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13 pages, 15829 KB

Open AccessArticle

A Dual-Band Full-Polarized Annular-Ring Slot Antenna with High Polarization Isolation

by Xuenan Wang, Hongcheng Zhou, Yong Bao, Yan Hu, Xiaoqing Xing, Mian Zhong and Chao Zhou

Symmetry 2026, 18(1), 98; https://doi.org/10.3390/sym18010098 - 6 Jan 2026

Abstract

This paper proposes a dual-band, full-polarized (dual-sense circular polarization and arbitrary linear polarization) annular-ring slot antenna centered at 2.4 GHz and 5.8 GHz, which effectively overcomes the limitations of narrow bandwidth and limited polarization diversity in conventional designs. By employing an eccentric annular-ring [...] Read more.

This paper proposes a dual-band, full-polarized (dual-sense circular polarization and arbitrary linear polarization) annular-ring slot antenna centered at 2.4 GHz and 5.8 GHz, which effectively overcomes the limitations of narrow bandwidth and limited polarization diversity in conventional designs. By employing an eccentric annular-ring slot and two orthogonal 50-ohm patches, the antenna achieves dual-band circular polarization (CP) radiation with single-port feeding. Based on the theory of orthogonal dual-circular polarization synthesis, arbitrary linear polarization (LP) can be generated by adjusting the phase difference when both ports are fed. The measured results show that the 10 dB return loss bandwidth of LP spans 2–2.8 GHz (relative bandwidth of 33.3%) and 4.5–7.5 GHz (relative bandwidth of 50%), with polarization isolation exceeding 50 dB. For CP mode, the measured bandwidth (for 10 dB return loss and 3 dB axial ratio) ranges from 2.24 to 2.58 GHz (relative bandwidth of 14.1%) and from 5.1 to 6.6 GHz (relative bandwidth of 25.64%), with polarization isolation greater than 15 dB. The proposed antenna simultaneously features a high frequency ratio (2.42), full polarization, high polarization isolation, a low profile (0.008 λ₀), and bidirectional radiation, which can meet the urgent demand of modern information systems for dual-band, full-polarized antennas. Full article

(This article belongs to the Section Engineering and Materials)

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18 pages, 307 KB

Open AccessArticle

Structural Properties of Pascal Pyramids and Pascal Simplexes: Classical Results and Some Extensions

by Hui Li

Symmetry 2026, 18(1), 97; https://doi.org/10.3390/sym18010097 - 5 Jan 2026

Abstract

Pascal’s Triangle, renowned for its geometric elegance and profound applications across combinatorics, algebra, and probability, has fascinated mathematicians for centuries. While its origins can be traced to Chinese, Persian, and European mathematical traditions, the study of its higher-dimensional analogues remains notably underexplored. This [...] Read more.

Pascal’s Triangle, renowned for its geometric elegance and profound applications across combinatorics, algebra, and probability, has fascinated mathematicians for centuries. While its origins can be traced to Chinese, Persian, and European mathematical traditions, the study of its higher-dimensional analogues remains notably underexplored. This paper offers a systematic and self-contained study of Pascal Pyramids and Pascal Simplexes with their proofs. It encompasses both classical results (such as multinomial identities) and novel contributions (including boundary and scaling properties), as well as fresh perspectives (such as graph-theoretic interpretations) that are rarely documented in the existing literature. Full article

(This article belongs to the Section Computer)

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20 pages, 566 KB

Open AccessArticle

Bayesian and Classical Inferences of Two-Weighted Exponential Distribution and Its Applications to HIV Survival Data

by Asmaa S. Al-Moisheer, Khalaf S. Sultan and Mahmoud M. M. Mansour

Symmetry 2026, 18(1), 96; https://doi.org/10.3390/sym18010096 - 5 Jan 2026

Abstract

The paper presents a statistical model based on the two-weighted exponential distribution (TWED) to examine censored Human Immunodeficiency Virus (HIV) survival information. Identifying HIV as a disability, the study endorses an inclusive and sustainable health policy framework through some predictive findings. The TWED [...] Read more.

The paper presents a statistical model based on the two-weighted exponential distribution (TWED) to examine censored Human Immunodeficiency Virus (HIV) survival information. Identifying HIV as a disability, the study endorses an inclusive and sustainable health policy framework through some predictive findings. The TWED provides an accurate representation of the inherent hazard patterns and also improves the modelling of survival data. The parameter estimation is achieved in both a classical maximum likelihood estimation (MLE) and a Bayesian approach. Bayesian inference can be carried out under general entropy loss conditions and the symmetric squared error loss function through the Markov Chain Monte Carlo (MCMC) method. Based on the symmetric properties of the inverse of the Fisher information matrix, the asymptotic confidence intervals (ACLs) for the MLEs are constructed. Moreover, two-sided symmetric credible intervals (CRIs) of Bayesian estimates are also constructed based on the MCMC results that are based on symmetric normal proposals. The simulation studies are very important for indicating the correctness and probability of a statistical estimator. Implementing the model on actual HIV data illustrates its usefulness. Altogether, the paper supports the idea that statistics play an essential role in promoting disability-friendly and sustainable research in the field of public health in general. Full article

(This article belongs to the Section Mathematics)

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