Symmetry

40 pages, 586 KiB

Open AccessArticle

Advanced Lifetime Modeling Through APSR-X Family with Symmetry Considerations: Applications to Economic, Engineering and Medical Data

by Badr S. Alnssyan, A. A. Bhat, Abdelaziz Alsubie, S. P. Ahmad, Abdulrahman M. A. Aldawsari and Ahlam H. Tolba

Symmetry 2025, 17(7), 1118; https://doi.org/10.3390/sym17071118 - 11 Jul 2025

Abstract

This paper introduces a novel and flexible class of continuous probability distributions, termed the Alpha Power Survival Ratio-X (APSR-X) family. Unlike many existing transformation-based families, the APSR-X class integrates an alpha power transformation with a survival ratio structure, offering a new mechanism for [...] Read more.

This paper introduces a novel and flexible class of continuous probability distributions, termed the Alpha Power Survival Ratio-X (APSR-X) family. Unlike many existing transformation-based families, the APSR-X class integrates an alpha power transformation with a survival ratio structure, offering a new mechanism for enhancing shape flexibility while maintaining mathematical tractability. This construction enables fine control over both the tail behavior and the symmetry properties, distinguishing it from traditional alpha power or survival-based extensions. We focus on a key member of this family, the two-parameter Alpha Power Survival Ratio Exponential (APSR-Exp) distribution, deriving essential mathematical properties including moments, quantile functions and hazard rate structures. We estimate the model parameters using eight frequentist methods: the maximum likelihood (MLE), maximum product of spacings (MPSE), least squares (LSE), weighted least squares (WLSE), Anderson–Darling (ADE), right-tailed Anderson–Darling (RADE), Cramér–von Mises (CVME) and percentile (PCE) estimation. Through comprehensive Monte Carlo simulations, we evaluate the estimator performance using bias, mean squared error and mean relative error metrics. The proposed APSR-X framework uniquely enables preservation or controlled modification of the symmetry in probability density and hazard rate functions via its shape parameter. This capability is particularly valuable in reliability and survival analyses, where symmetric patterns represent balanced risk profiles while asymmetric shapes capture skewed failure behaviors. We demonstrate the practical utility of the APSR-Exp model through three real-world applications: economic (tax revenue durations), engineering (mechanical repair times) and medical (infection durations) datasets. In all cases, the proposed model achieves a superior fit over that of the conventional alternatives, supported by goodness-of-fit statistics and visual diagnostics. These findings establish the APSR-X family as a unique, symmetry-aware modeling framework for complex lifetime data. Full article

(This article belongs to the Section Computer)

37 pages, 1848 KiB

Open AccessArticle

Chaos, Local Dynamics, Codimension-One and Codimension-Two Bifurcation Analysis of a Discrete Predator–Prey Model with Holling Type I Functional Response

by Muhammad Rameez Raja, Abdul Qadeer Khan and Jawharah G. AL-Juaid

Symmetry 2025, 17(7), 1117; https://doi.org/10.3390/sym17071117 - 11 Jul 2025

Abstract

We explore chaos, local dynamics, codimension-one, and codimension-two bifurcations of an asymmetric discrete predator–prey model. More precisely, for all the model’s parameters, it is proved that the model has two boundary fixed points and a trivial fixed point, and also under parametric conditions, [...] Read more.

We explore chaos, local dynamics, codimension-one, and codimension-two bifurcations of an asymmetric discrete predator–prey model. More precisely, for all the model’s parameters, it is proved that the model has two boundary fixed points and a trivial fixed point, and also under parametric conditions, it has an interior fixed point. We then constructed the linearized system at these fixed points. We explored the local behavior at equilibria by the linear stability theory. By the series of affine transformations, the center manifold theorem, and bifurcation theory, we investigated the detailed codimensions-one and two bifurcations at equilibria and examined that at boundary fixed points, no flip bifurcation exists. Furthermore, at the interior fixed point, it is proved that the discrete model exhibits codimension-one bifurcations like Neimark–Sacker and flip bifurcations, but fold bifurcation does not exist at this point. Next, for deeper understanding of the complex dynamics of the model, we also studied the codimension-two bifurcation at an interior fixed point and proved that the model exhibits the codimension-two 1:2, 1:3, and 1:4 strong resonances bifurcations. We then investigated the existence of chaos due to the appearance of codimension-one bifurcations like Neimark–Sacker and flip bifurcations by OGY and hybrid control strategies, respectively. The theoretical results are also interpreted biologically. Finally, theoretical findings are confirmed numerically. Full article

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

25 pages, 7696 KiB

Open AccessArticle

YOLO-StarLS: A Ship Detection Algorithm Based on Wavelet Transform and Multi-Scale Feature Extraction for Complex Environments

by Yihan Wang, Shuang Zhang, Jianhao Xu, Zhenwen Cheng and Gang Du

Symmetry 2025, 17(7), 1116; https://doi.org/10.3390/sym17071116 - 11 Jul 2025

Abstract

Ship detection in complex environments presents challenges such as sea surface reflections, wave interference, variations in illumination, and a range of target scales. The interaction between symmetric ship structures and wave patterns challenges conventional algorithms, particularly in maritime wireless networks. This study presents [...] Read more.

Ship detection in complex environments presents challenges such as sea surface reflections, wave interference, variations in illumination, and a range of target scales. The interaction between symmetric ship structures and wave patterns challenges conventional algorithms, particularly in maritime wireless networks. This study presents YOLO-StarLS (You Only Look Once with Star-topology Lightweight Ship detection), a detection framework leveraging wavelet transforms and multi-scale feature extraction through three core modules. We developed a Wavelet Multi-scale Feature Extraction Network (WMFEN) utilizing adaptive Haar wavelet decomposition with star-topology extraction to preserve multi-frequency information while minimizing detail loss. We introduced a Cross-axis Spatial Attention Refinement module (CSAR), which integrates star structures with cross-axis attention mechanisms to enhance spatial perception. We constructed an Efficient Detail-Preserving Detection head (EDPD) combining differential and shared convolutions to enhance edge detection while reducing computational complexity. Evaluation on the SeaShips dataset demonstrated YOLO-StarLS achieved superior performance for both mAP50 and mAP50–95 metrics, improving by 2.21% and 2.42% over the baseline YOLO11. The approach achieved significant efficiency, with a 36% reduction in the number of parameters to 1.67 M, a 34% decrease in complexity to 4.3 GFLOPs, and an inference speed of 162.0 FPS. Comparative analysis against eight algorithms confirmed the superiority in symmetric target detection. This work enhances real-time ship detection and provides foundations for maritime wireless surveillance networks. Full article

(This article belongs to the Section Computer)

30 pages, 804 KiB

Open AccessArticle

A Novel Heterogeneous Federated Edge Learning Framework Empowered with SWIPT

by Yinyin Fang, Sheng Shu, Yujun Zhu, Heju Li and Kunkun Rui

Symmetry 2025, 17(7), 1115; https://doi.org/10.3390/sym17071115 - 11 Jul 2025

Abstract

Federated edge learning (FEEL) is an innovative approach that facilitates collaborative training among numerous distributed edge devices while eliminating the need to transfer sensitive information. However, the practical deployment of FEEL faces significant constraints, owing to the limited and asymmetric computational and communication [...] Read more.

Federated edge learning (FEEL) is an innovative approach that facilitates collaborative training among numerous distributed edge devices while eliminating the need to transfer sensitive information. However, the practical deployment of FEEL faces significant constraints, owing to the limited and asymmetric computational and communication resources of these devices, along with their energy availability. To this end, we propose a novel asymmetry-tolerant training approach for FEEL, enabled via simultaneous wireless information and power transfer (SWIPT). This framework leverages SWIPT to offer sustainable energy support for devices while enabling them to train models with varying intensities. Given a limited energy budget, we highlight the critical trade-off between heterogeneous local training intensities and the quality of wireless transmission, suggesting that the design of local training and wireless transmission should be closely integrated, rather than treated as separate entities. To elucidate this perspective, we rigorously derive a new explicit upper bound that captures the combined impact of local training accuracy and the mean square error of wireless aggregation on the convergence performance of FEEL. To maximize overall system performance, we formulate two key optimization problems: the first aims to maximize the energy harvesting capability among all devices, while the second addresses the joint learning–communication optimization under the optimal energy harvesting solution. Comprehensive experiments demonstrate that our proposed framework achieves significant performance improvements compared to existing baselines. Full article

(This article belongs to the Section Computer)

► Show Figures

Figure 1

28 pages, 418 KiB

Open AccessArticle

Geometric Accumulation Operators of Dombi Weighted Trapezoidal-Valued Fermatean Fuzzy Numbers with Multi-Attribute Group Decision Making

by M. Kaviyarasu, J. Angel and Mohammed Alqahtani

Symmetry 2025, 17(7), 1114; https://doi.org/10.3390/sym17071114 - 10 Jul 2025

Abstract

Trapezoidal-valued fermatean fuzzy numbers (TpVFFNs) are essential for handling daily decision-making issues in the engineering and management fields. Accumulation processes on the set of TpVFFN are used to address decision-making problems described in this environment as necessary. The primary goal of this paper [...] Read more.

Trapezoidal-valued fermatean fuzzy numbers (TpVFFNs) are essential for handling daily decision-making issues in the engineering and management fields. Accumulation processes on the set of TpVFFN are used to address decision-making problems described in this environment as necessary. The primary goal of this paper is to provide the concept of Dombi t-norm

(D_{t n})

- and Dombi t-conorm

(D_{t c n})

-based accumulation operators on the class of TpVFFN, emphasizing how they behave symmetrically in aggregation processes to maintain consistency and fairness. To use s to illustrate mathematical circumstances, we first create a trapezoidal-valued fermatean fuzzy Dombi’s weighted geometric operator, hexagonal hybird geometric operator, fermatean fuzzy order weighted geometric operator. Second, we use a multi-attribute group decision-making (MAGDM) approach to compute the recommended accumulation operators. Finally, we demonstrate the potential practical application of the proposed decision-making problem related to the pink cab. Full article

(This article belongs to the Topic Fuzzy Sets Theory and Its Applications)

► Show Figures

Figure 1

35 pages, 5154 KiB

Open AccessArticle

A Monte Carlo Simulation Framework for Evaluating the Robustness and Applicability of Settlement Prediction Models in High-Speed Railway Soft Foundations

by Zhenyu Liu, Liyang Wang, Taifeng Li, Huiqin Guo, Feng Chen, Youming Zhao, Qianli Zhang and Tengfei Wang

Symmetry 2025, 17(7), 1113; https://doi.org/10.3390/sym17071113 - 10 Jul 2025

Abstract

Accurate settlement prediction for high-speed railway (HSR) soft foundations remains challenging due to the irregular and dynamic nature of real-world monitoring data, often represented as non-equidistant and non-stationary time series (NENSTS). Existing empirical models lack clear applicability criteria under such conditions, resulting in [...] Read more.

Accurate settlement prediction for high-speed railway (HSR) soft foundations remains challenging due to the irregular and dynamic nature of real-world monitoring data, often represented as non-equidistant and non-stationary time series (NENSTS). Existing empirical models lack clear applicability criteria under such conditions, resulting in subjective model selection. This study introduces a Monte Carlo-based evaluation framework that integrates data-driven simulation with geotechnical principles, embedding the concept of symmetry across both modeling and assessment stages. Equivalent permeability coefficients (EPCs) are used to normalize soil consolidation behavior, enabling the generation of a large, statistically robust dataset. Four empirical settlement prediction models—Hyperbolic, Exponential, Asaoka, and Hoshino—are systematically analyzed for sensitivity to temporal features and resistance to stochastic noise. A symmetry-aware comprehensive evaluation index (CEI), constructed via a robust entropy weight method (REWM), balances multiple performance metrics to ensure objective comparison. Results reveal that while settlement behavior evolves asymmetrically with respect to EPCs over time, a symmetrical structure emerges in model suitability across distinct EPC intervals: the Asaoka method performs best under low-permeability conditions (EPC ≤ 0.03 m/d), Hoshino excels in intermediate ranges (0.03 < EPC ≤ 0.7 m/d), and the Exponential model dominates in highly permeable soils (EPC > 0.7 m/d). This framework not only quantifies model robustness under complex data conditions but also formalizes the notion of symmetrical applicability, offering a structured path toward intelligent, adaptive settlement prediction in HSR subgrade engineering. Full article

(This article belongs to the Section Engineering and Materials)

22 pages, 323 KiB

Open AccessArticle

A System of Parabolic Laplacian Equations That Are Interrelated and Radial Symmetry of Solutions

by Xingyu Liu

Symmetry 2025, 17(7), 1112; https://doi.org/10.3390/sym17071112 - 10 Jul 2025

Abstract

We utilize the moving planes technique to prove the radial symmetry along with the monotonic characteristics of solutions for a system of parabolic Laplacian equations. In this system, the solutions of the two equations are interdependent, with the solution of one equation depending [...] Read more.

We utilize the moving planes technique to prove the radial symmetry along with the monotonic characteristics of solutions for a system of parabolic Laplacian equations. In this system, the solutions of the two equations are interdependent, with the solution of one equation depending on the function of the other. By use of the maximal regularity theory that has been established for fractional parabolic equations, we ensure the solvability of these systems. Our initial step is to formulate a narrow region principle within a parabolic cylinder. This principle serves as a theoretical basis for implementing the moving planes method. Following this, we focus our attention on parabolic systems with fractional Laplacian equations and deduce that the solutions are radial symmetric and monotonic when restricted to the unit ball. Full article

(This article belongs to the Special Issue Advance in Functional Equations, Second Edition)

► Show Figures

Figure 1

23 pages, 4119 KiB

Open AccessArticle

Cross-Scenario Interpretable Prediction of Coal Mine Water Inrush Probability: An Integrated Approach Driven by Gaussian Mixture Modeling with Manifold Learning and Metaheuristic Optimization

by Qiushuang Zheng and Changfeng Wang

Symmetry 2025, 17(7), 1111; https://doi.org/10.3390/sym17071111 - 10 Jul 2025

Abstract

Predicting water inrush in coal mines faces significant challenges due to limited data, model generalization, and a lack of interpretability. Current approaches often neglect the inherent geometrical symmetries and structured patterns within the complex hydrological parameter space, rely on local parameter optimization, and [...] Read more.

Predicting water inrush in coal mines faces significant challenges due to limited data, model generalization, and a lack of interpretability. Current approaches often neglect the inherent geometrical symmetries and structured patterns within the complex hydrological parameter space, rely on local parameter optimization, and struggle with interpretability, leading to insufficient predictive accuracy and engineering applicability under complex geological conditions. This study addresses these limitations by integrating Gaussian mixture modeling (GMM), manifold learning, and data augmentation to effectively capture multimodal hydrological data distributions and reveal their intrinsic symmetrical configurations and manifold structures, thereby reducing feature dimensionality. We then apply a whale optimization algorithm (WOA)-enhanced XGBoost model to forecast water inrush probabilities. Our model achieved an R² of 0.92, demonstrating a greater than 60% error reduction across various metrics. Validation at the Yangcheng Coal Mine confirmed that this balanced approach significantly enhances predictive accuracy, interpretability, and cross-scenario applicability. The synergy between high accuracy and transparency provides decision makers with reliable risk insights, enabling bidirectional validation with geological mechanisms and supporting the implementation of targeted, proactive safety measures. Full article

(This article belongs to the Section Mathematics)

► Show Figures

Figure 1

18 pages, 3719 KiB

Open AccessArticle

Energy-Efficient Bipedal Locomotion Through Parallel Actuation of Hip and Ankle Joints

by Prabhu Manoharan and Karthikeyan Palanisamy

Symmetry 2025, 17(7), 1110; https://doi.org/10.3390/sym17071110 - 10 Jul 2025

Abstract

Achieving energy-efficient, human-like gait remains a major challenge in bipedal humanoid robotics, as traditional serial actuation architectures often lead to high instantaneous power peaks and uneven load distribution. This study addresses the lack of research on how mechanical symmetry, achieved through parallel actuation, [...] Read more.

Achieving energy-efficient, human-like gait remains a major challenge in bipedal humanoid robotics, as traditional serial actuation architectures often lead to high instantaneous power peaks and uneven load distribution. This study addresses the lack of research on how mechanical symmetry, achieved through parallel actuation, can improve power management in lower-limb joints. We developed a 14-degree-of-freedom (DOF) hip-sized bipedal robot model and conducted simulations comparing a conventional serial configuration—using single-DOF rotary actuators—with a novel parallel configuration that employs paired linear actuators at the hip pitch, hip roll, ankle pitch, and ankle roll joints. Simulation results over a standardized walking cycle show that the parallel configuration reduces peak hip-pitch power by 80.4% and peak ankle-pitch power by 53.5%. These findings demonstrate that incorporating actuator symmetry through parallel joint design significantly reduces actuator stress, improves load sharing, and enhances overall energy efficiency in bipedal locomotion. Full article

(This article belongs to the Section Engineering and Materials)

► Show Figures

Figure 1

32 pages, 2917 KiB

Open AccessArticle

Self-Adapting CPU Scheduling for Mixed Database Workloads via Hierarchical Deep Reinforcement Learning

by Suchuan Xing, Yihan Wang and Wenhe Liu

Symmetry 2025, 17(7), 1109; https://doi.org/10.3390/sym17071109 - 10 Jul 2025

Abstract

Modern database systems require autonomous CPU scheduling frameworks that dynamically optimize resource allocation across heterogeneous workloads while maintaining strict performance guarantees. We present a novel hierarchical deep reinforcement learning framework augmented with graph neural networks to address CPU scheduling challenges in mixed database [...] Read more.

Modern database systems require autonomous CPU scheduling frameworks that dynamically optimize resource allocation across heterogeneous workloads while maintaining strict performance guarantees. We present a novel hierarchical deep reinforcement learning framework augmented with graph neural networks to address CPU scheduling challenges in mixed database environments comprising Online Transaction Processing (OLTP), Online Analytical Processing (OLAP), vector processing, and background maintenance workloads. Our approach introduces three key innovations: first, a symmetric two-tier control architecture where a meta-controller allocates CPU budgets across workload categories using policy gradient methods while specialized sub-controllers optimize process-level resource allocation through continuous action spaces; second, graph neural network-based dependency modeling that captures complex inter-process relationships and communication patterns while preserving inherent symmetries in database architectures; and third, meta-learning integration with curiosity-driven exploration enabling rapid adaptation to previously unseen workload patterns without extensive retraining. The framework incorporates a multi-objective reward function balancing Service Level Objective (SLO) adherence, resource efficiency, symmetric fairness metrics, and system stability. Experimental evaluation through high-fidelity digital twin simulation and production deployment demonstrates substantial performance improvements: 43.5% reduction in p99 latency violations for OLTP workloads and 27.6% improvement in overall CPU utilization, with successful scaling to 10,000 concurrent processes maintaining sub-3% scheduling overhead. This work represents a significant advancement toward truly autonomous database resource management, establishing a foundation for next-generation self-optimizing database systems with implications extending to broader orchestration challenges in cloud-native architectures. Full article

(This article belongs to the Section Computer)

► Show Figures

Figure 1

30 pages, 34072 KiB

Open AccessArticle

ARE-PaLED: Augmented Reality-Enhanced Patch-Level Explainable Deep Learning System for Alzheimer’s Disease Diagnosis from 3D Brain sMRI

by Chitrakala S and Bharathi U

Symmetry 2025, 17(7), 1108; https://doi.org/10.3390/sym17071108 - 10 Jul 2025

Abstract

Structural magnetic resonance imaging (sMRI) is a vital tool for diagnosing neurological brain diseases. However, sMRI scans often show significant structural changes only in limited brain regions due to localised atrophy, making the identification of discriminative features a key challenge. Importantly, the human [...] Read more.

Structural magnetic resonance imaging (sMRI) is a vital tool for diagnosing neurological brain diseases. However, sMRI scans often show significant structural changes only in limited brain regions due to localised atrophy, making the identification of discriminative features a key challenge. Importantly, the human brain exhibits inherent bilateral symmetry, and deviations from this symmetry—such as asymmetric atrophy—are strong indicators of early Alzheimer’s disease (AD). Patch-based methods help capture local brain changes for early AD diagnosis, but they often struggle with fixed-size limitations, potentially missing subtle asymmetries or broader contextual cues. To address these limitations, we propose a novel augmented reality (AR)-enhanced patch-level explainable deep learning (ARE-PaLED) system. It includes an adaptive multi-scale patch extraction network (AMPEN) to adjust patch sizes based on anatomical characteristics and spatial context, as well as an informative patch selection algorithm (IPSA) to identify discriminative patches, including those reflecting asymmetry patterns associated with AD; additionally, an AR module is proposed for future immersive explainability, complementing the patch-level interpretation framework. Evaluated on 1862 subjects from the ADNI and AIBL datasets, the framework achieved an accuracy of 92.5% (AD vs. NC) and 85.9% (AD vs. MCI). The proposed ARE-PaLED demonstrates potential as an interpretable and immersive diagnostic aid for sMRI-based AD diagnosis, supporting the interpretation of model predictions for AD diagnosis. Full article

► Show Figures

Figure 1

23 pages, 2267 KiB

Open AccessArticle

Special Basis for Efficient Numerical Solutions of Differential Equations: Application to the Energy Transfer Equation

by Fahir Talay Akyildiz and Fehaid Salem Alshammari

Symmetry 2025, 17(7), 1107; https://doi.org/10.3390/sym17071107 - 9 Jul 2025

Abstract

We introduce a novel family of compactly supported basis functions, termed Legendre Delta-Shaped Functions (LDSFs), constructed using the eigenfunctions of the Legendre differential equation. We begin by proving that LDSFs form a basis for a Haar space. We then demonstrate that interpolation using [...] Read more.

We introduce a novel family of compactly supported basis functions, termed Legendre Delta-Shaped Functions (LDSFs), constructed using the eigenfunctions of the Legendre differential equation. We begin by proving that LDSFs form a basis for a Haar space. We then demonstrate that interpolation using classical Legendre polynomials is a special case of interpolation with the proposed Legendre Delta-Shaped Basis Functions (LDSBFs). To illustrate the potential of LDSBFs, we apply the corresponding series to approximate a rectangular pulse. The results reveal that Gibbs oscillations decay rapidly, resulting in significantly improved accuracy across smooth regions. This example underscores the effectiveness and novelty of our approach. Furthermore, LDSBFs are employed within the collocation framework to solve Poisson-type equations and systems of nonlinear differential equations arising in energy transfer problems. We also derive new error bounds for interpolation polynomials in a special case, expressed in both the discrete

{(L}_{2})

norm and the Sobolev

(H_{p})

norm. To validate the proposed method, we compare our results with those obtained using the Legendre pseudospectral method. Numerical experiments confirm that our approach is accurate, efficient, and highly competitive with existing techniques. Full article

► Show Figures

Figure 1

39 pages, 5065 KiB

Open AccessFeature PaperReview

Electroexcitation of Nucleon Resonances and Emergence of Hadron Mass

by Patrick Achenbach, Daniel S. Carman, Ralf W. Gothe, Kyungseon Joo, Victor I. Mokeev and Craig D. Roberts

Symmetry 2025, 17(7), 1106; https://doi.org/10.3390/sym17071106 - 9 Jul 2025

Abstract

Developing an understanding of phenomena driven by the emergence of hadron mass (EHM) is one of the most challenging problems in the Standard Model. This discussion focuses on the impact of results on nucleon resonance (

N^{*}

) electroexcitation amplitudes (or [...] Read more.

Developing an understanding of phenomena driven by the emergence of hadron mass (EHM) is one of the most challenging problems in the Standard Model. This discussion focuses on the impact of results on nucleon resonance (

N^{*}

) electroexcitation amplitudes (or

γ_{v} p N^{*}

electrocouplings) obtained from experiments during the 6 GeV era in Hall B at Jefferson Lab on understanding EHM. Analyzed using continuum Schwinger function methods (CSMs), these results have revealed new pathways for the elucidation of EHM. A good description of the

Δ (1232) 3 / 2^{+}

,

N (1440) 1 / 2^{+}

, and

Δ (1600) 3 / 2^{+}

electrocouplings, achieved by CSM analyses that express a realistic dressed quark mass function, sheds light on the strong interaction dynamics underlying EHM. Extensions to

N^{*}

studies for higher-mass states are outlined, as well as experimental results anticipated in the 12 GeV era at Jefferson Lab and those that would be enabled by a further increase in the beam energy to 22 GeV. Full article

(This article belongs to the Special Issue The Symmetry of QCD Matter and Functional QCD Approaches)

► Show Figures

Figure 1

22 pages, 2171 KiB

Open AccessArticle

A Multi-Objective Method for Enhancing the Seismic Resilience of Urban Water Distribution Networks

by Li Long, Ziang Pan, Huaping Yang, Yong Yang and Feiyu Liu

Symmetry 2025, 17(7), 1105; https://doi.org/10.3390/sym17071105 - 9 Jul 2025

Abstract

Enhancing the seismic resilience of urban water distribution networks (WDNs) requires the improvement of both earthquake resistance and rapid recovery capabilities within the system. This paper proposes a multi-objective method to enhance the seismic resilience of the WDNs, focusing on system restoration capabilities [...] Read more.

Enhancing the seismic resilience of urban water distribution networks (WDNs) requires the improvement of both earthquake resistance and rapid recovery capabilities within the system. This paper proposes a multi-objective method to enhance the seismic resilience of the WDNs, focusing on system restoration capabilities while comprehensively considering the hydraulic recovery index, maintenance time, and maintenance cost. The method utilizes a random simulation approach to generate various damage scenarios for the WDN, considering pipe leakage, pipe bursts, and variations in node flow resulting from changes in water pressure. It characterizes the functions of the WDN through hydraulic service satisfaction and quantifies system resilience using a performance response function. Additionally, it determines the optimal dispatch strategy for emergency repair teams and the optimal emergency repair sequence for earthquake-damaged networks using a genetic algorithm. Furthermore, a comprehensive computational platform has been developed to systematically analyze and optimize seismic resilience strategies for WDNs. The feasibility of the proposed method is demonstrated through an example involving the WDN in Xi’an City. The results indicate that the single-objective seismic resilience improvement method based on the hydraulic recovery index is the most effective for enhancing the seismic resilience of the WDN. In contrast, the multi-objective method proposed in this article reduces repair time by 17.9% and repair costs by 3.4%, while only resulting in a 0.2% decrease in the seismic resilience of the WDN. This method demonstrates the most favorable comprehensive restoration effect, and the success of our method in achieving a symmetrically balanced restoration outcome demonstrates its value. The proposed methodology and software can provide both theoretical frameworks and technical support for urban WDN administrators. Full article

(This article belongs to the Section Engineering and Materials)

► Show Figures

Figure 1

20 pages, 2516 KiB

Open AccessArticle

Visual Attention Fusion Network (VAFNet): Bridging Bottom-Up and Top-Down Features in Infrared and Visible Image Fusion

by Yaochen Liu, Yunke Wang and Zixuan Jing

Symmetry 2025, 17(7), 1104; https://doi.org/10.3390/sym17071104 - 9 Jul 2025

Abstract

Infrared and visible image fusion aims to integrate useful information from the source image to obtain a fused image that not only has excellent visual perception but also promotes the performance of the subsequent object detection task. However, due to the asymmetry between [...] Read more.

Infrared and visible image fusion aims to integrate useful information from the source image to obtain a fused image that not only has excellent visual perception but also promotes the performance of the subsequent object detection task. However, due to the asymmetry between image fusion and object detection tasks, obtaining superior visual effects while facilitating object detection tasks remains challenging in real-world applications. Addressing this issue, we propose a novel visual attention fusion network for infrared and visible image fusion (VAFNet), which can bridge bottom-up and top-down features to achieve high-quality visual perception while improving the performance of object detection tasks. The core idea is that bottom-up visual attention is utilized to extract multi-layer bottom-up features for ensuring superior visual perception, while top-down visual attention determines object attention signals related to object detection tasks. Then, a bidirectional attention integration mechanism is designed to naturally integrate two forms of attention into the fused image. Experiments on public and collection datasets demonstrate that VAFNet not only outperforms seven state-of-the-art (SOTA) fusion methods in qualitative and quantitative evaluation but also has advantages in facilitating object detection tasks. Full article

(This article belongs to the Special Issue Symmetry in Next-Generation Intelligent Information Technologies)

► Show Figures

Figure 1

27 pages, 392 KiB

Open AccessArticle

Non-Autonomous Soliton Hierarchies

by Maciej Błaszak, Krzysztof Marciniak and Błażej M. Szablikowski

Symmetry 2025, 17(7), 1103; https://doi.org/10.3390/sym17071103 - 9 Jul 2025

Abstract

A formalism for the systematic construction of integrable non-autonomous deformations of soliton hierarchies is presented. The theory is formulated as an initial value problem (IVP) for an associated Frobenius integrability condition on a Lie algebra. It is shown that this IVP has a [...] Read more.

A formalism for the systematic construction of integrable non-autonomous deformations of soliton hierarchies is presented. The theory is formulated as an initial value problem (IVP) for an associated Frobenius integrability condition on a Lie algebra. It is shown that this IVP has a formal solution, and within the framework of two particular subalgebras of the hereditary Lie algebra, the explicit forms of this formal solution are derived. Finally, this formalism is applied to the Korteveg-de Vries, dispersive water waves and Ablowitz–Kaup–Newell–Segur soliton hierarchies. The zero-curvature representations and Hamiltonian structures of the considered non-autonomous soliton hierarchies are also provided. Full article

(This article belongs to the Special Issue Symmetry in Integrable Systems and Soliton Theories)

23 pages, 5304 KiB

Open AccessArticle

Improvement and Optimization of Underwater Image Target Detection Accuracy Based on YOLOv8

by Yisong Sun, Wei Chen, Qixin Wang, Tianzhong Fang and Xinyi Liu

Symmetry 2025, 17(7), 1102; https://doi.org/10.3390/sym17071102 - 9 Jul 2025

Abstract

The ocean encompasses the majority of the Earth’s surface and harbors substantial energy resources. Nevertheless, the intricate and asymmetrically distributed underwater environment renders existing target detection performance inadequate. This paper presents an enhanced YOLOv8s approach for underwater robot object detection to address issues [...] Read more.

The ocean encompasses the majority of the Earth’s surface and harbors substantial energy resources. Nevertheless, the intricate and asymmetrically distributed underwater environment renders existing target detection performance inadequate. This paper presents an enhanced YOLOv8s approach for underwater robot object detection to address issues of subpar image quality and low recognition accuracy. The precise measures are enumerated as follows: initially, to address the issue of model parameters, we optimized the ninth convolutional layer by substituting certain conventional convolutions with adaptive deformable convolution DCN v4. This modification aims to more effectively capture the deformation and intricate features of underwater targets, while simultaneously decreasing the parameter count and enhancing the model’s ability to manage the deformation challenges presented by underwater images. Furthermore, the Triplet Attention module is implemented to augment the model’s capacity for detecting multi-scale targets. The integration of low-level superficial features with high-level semantic features enhances the feature expression capability. The original CIoU loss function was ultimately substituted with Shape IoU, enhancing the model’s performance. In the underwater robot grasping experiment, the system shows particular robustness in handling radial symmetry in marine organisms and reflection symmetry in artificial structures. The enhanced algorithm attained a mean Average Precision (mAP) of 87.6%, surpassing the original YOLOv8s model by 3.4%, resulting in a marked enhancement of the object detection model’s performance and fulfilling the real-time detection criteria for underwater robots. Full article

(This article belongs to the Special Issue Computer Vision, Pattern Recognition, Machine Learning, and Symmetry, 2nd Edition)

► Show Figures

Figure 1

20 pages, 922 KiB

Open AccessArticle

Distributed Time Delay Models: An Alternative to Fractional Calculus-Based Models for Fractional Behavior Modeling

by Jocelyn Sabatier

Symmetry 2025, 17(7), 1101; https://doi.org/10.3390/sym17071101 - 9 Jul 2025

Abstract

This paper illustrates that distributed time delay models can exhibit fractional behaviors, addressing the limitations of fractional calculus-based models outlined in the introduction. Given the extensive results generated by these models, they present a compelling alternative to fractional models. The demonstration is done [...] Read more.

This paper illustrates that distributed time delay models can exhibit fractional behaviors, addressing the limitations of fractional calculus-based models outlined in the introduction. Given the extensive results generated by these models, they present a compelling alternative to fractional models. The demonstration is done both in discrete time and in continuous time. The two cases yield fractional behavior within a defined time/frequency range. To conclude and using two examples, the article highlights that modeling fractional behaviors using distributed delay systems allows for coherent physical interpretations, which a fractional model representation struggles to achieve. Full article

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

► Show Figures

Figure 1

18 pages, 301 KiB

Open AccessArticle

Casorati-Type Inequalities for Submanifolds in S-Space Forms with Semi-Symmetric Connection

by Md Aquib

Symmetry 2025, 17(7), 1100; https://doi.org/10.3390/sym17071100 - 9 Jul 2025

Abstract

The primary aim of this paper is to establish two sharp geometric inequalities concerning submanifolds of S-space forms equipped with semi-symmetric metric connections (SSMCs). Specifically, we derive new inequalities involving the generalized normalized

δ

-Casorati curvatures [...] Read more.

The primary aim of this paper is to establish two sharp geometric inequalities concerning submanifolds of S-space forms equipped with semi-symmetric metric connections (SSMCs). Specifically, we derive new inequalities involving the generalized normalized

δ

-Casorati curvatures

δ_{c} (t; q_{1} + q_{2} - 1)

and

{\hat{δ}}_{c} (t; q_{1} + q_{2} - 1)

for bi-slant submanifolds. The cases in which equality holds are thoroughly examined, offering a deeper understanding of the geometric structure underlying such submanifolds. In addition, we present several immediate applications that highlight the relevance of our findings, and we support the article with illustrative examples. Full article

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

21 pages, 2471 KiB

Open AccessArticle

Attention-Based Mask R-CNN Enhancement for Infrared Image Target Segmentation

by Liang Wang and Kan Ren

Symmetry 2025, 17(7), 1099; https://doi.org/10.3390/sym17071099 - 9 Jul 2025

Abstract

Image segmentation is an important method in the field of image processing, while infrared (IR) image segmentation is one of the challenges in this field due to the unique characteristics of IR data. Infrared imaging utilizes the infrared radiation emitted by objects to [...] Read more.

Image segmentation is an important method in the field of image processing, while infrared (IR) image segmentation is one of the challenges in this field due to the unique characteristics of IR data. Infrared imaging utilizes the infrared radiation emitted by objects to produce images, which can supplement the performance of visible-light images under adverse lighting conditions to some extent. However, the low spatial resolution and limited texture details in IR images hinder the achievement of high-precision segmentation. To address these issues, an attention mechanism based on symmetrical cross-channel interaction—motivated by symmetry principles in computer vision—was integrated into a Mask Region-Based Convolutional Neural Network (Mask R-CNN) framework. A Bottleneck-enhanced Squeeze-and-Attention (BNSA) module was incorporated into the backbone network, and novel loss functions were designed for both the bounding box (Bbox) regression and mask prediction branches to enhance segmentation performance. Furthermore, a dedicated infrared image dataset was constructed to validate the proposed method. The experimental results demonstrate that the optimized model achieves higher segmentation accuracy and better segmentation performance compared to the original network and other mainstream segmentation models on our dataset, demonstrating how symmetrical design principles can effectively improve complex vision tasks. Full article

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

► Show Figures

Figure 1

21 pages, 7528 KiB

Open AccessArticle

A Fine-Tuning Method via Adaptive Symmetric Fusion and Multi-Graph Aggregation for Human Pose Estimation

by Yinliang Shi, Zhaonian Liu, Bin Jiang, Tianqi Dai and Yuanfeng Lian

Symmetry 2025, 17(7), 1098; https://doi.org/10.3390/sym17071098 - 9 Jul 2025

Abstract

Human Pose Estimation (HPE) aims to accurately locate the positions of human key points in images or videos. However, the performance of HPE is often significantly reduced in practical application scenarios due to environmental interference. To address this challenge, we propose a ladder [...] Read more.

Human Pose Estimation (HPE) aims to accurately locate the positions of human key points in images or videos. However, the performance of HPE is often significantly reduced in practical application scenarios due to environmental interference. To address this challenge, we propose a ladder side-tuning method for the Vision Transformer (ViT) pre-trained model based on multi-path feature fusion to improve the accuracy of HPE in highly interfering environments. First, we extract the global features, frequency features and multi-scale spatial features through the ViT pre-trained model, the discrete wavelet convolutional network and the atrous spatial pyramid pooling network (ASPP). By comprehensively capturing the information of the human body and the environment, the ability of the model to analyze local details, textures, and spatial information is enhanced. In order to efficiently fuse these features, we devise an adaptive symmetric feature fusion strategy, which dynamically adjusts the intensity of feature fusion according to the similarity among features to achieve the optimal fusion effect. Finally, a multi-graph feature aggregation method is developed. We construct graph structures of different features and deeply explore the subtle differences among the features based on the dual fusion mechanism of points and edges to ensure the information integrity. The experimental results demonstrate that our method achieves 4.3% and 4.2% improvements in the AP metric on the MS COCO dataset and a custom high-interference dataset, respectively, compared with the HRNet. This highlights its superiority for human pose estimation tasks in both general and interfering environments. Full article

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

► Show Figures

Figure 1

22 pages, 2009 KiB

Open AccessArticle

Transient Analysis of a Continuous-Service Markovian Queueing Model with Offline and Online Customers

by Ramupillai Sudhesh, Paulsamy Balakrishnan and Ratchaga Dass Sebasthi Priya

Symmetry 2025, 17(7), 1097; https://doi.org/10.3390/sym17071097 - 9 Jul 2025

Abstract

This study examines a single-server Markovian queueing system featuring continuous service and an infinite number of customers at both ends—namely, offline and online clients. Offline customers are conventional clients who arrive at the system following a Poisson process, while online customers are assumed [...] Read more.

This study examines a single-server Markovian queueing system featuring continuous service and an infinite number of customers at both ends—namely, offline and online clients. Offline customers are conventional clients who arrive at the system following a Poisson process, while online customers are assumed to be endlessly present in the system. All service times are exponentially and identically distributed and independent. Utilizing generating functions and Laplace transform techniques, this study derives exact analytical expressions for the system size probabilities in both transient and steady states. Furthermore, it evaluates key performance measures for each state and provides graphical representations to illustrate the system’s dynamics, thereby enriching the understanding of its operational behavior. This work contributes to the advancement of priority-based queueing models and proposes a novel framework applicable to hybrid service architectures in contemporary digital ecosystems. Full article

(This article belongs to the Section Mathematics)

► Show Figures

Figure 1

15 pages, 295 KiB

Open AccessArticle

Neutrosophic Quadruple Metric Spaces

by Memet Şahin and Arif Sarıoğlan

Symmetry 2025, 17(7), 1096; https://doi.org/10.3390/sym17071096 - 8 Jul 2025

Abstract

Instead of measuring the distance between two points with a positive real number, determining the degree to which the distance between these two points is close, not close, or uncertain allows for more detailed measurement. Recently, researchers have overcome this grading problem by [...] Read more.

Instead of measuring the distance between two points with a positive real number, determining the degree to which the distance between these two points is close, not close, or uncertain allows for more detailed measurement. Recently, researchers have overcome this grading problem by using probability distribution functions, along with fuzzy, intuitionistic fuzzy, and neutrosophic sets. This study pioneers neutrosophic quadruple metric spaces as a powerful new tool for quantifying distances under complex, multi-dimensional uncertainty. It provides a comprehensive mathematical structure, including topology, convergence theory, and completeness, and handles both symmetric and asymmetric cases, generalising previous neutrosophic metric results. For this purpose, neutrosophic quadruple metric spaces were derived from neutrosophic metric spaces in order to better model situations involving uncertainty. Also, we generalised the findings obtained with the neutrosophic metric to the quadruple neutrosophic metric. Full article

(This article belongs to the Special Issue Theory and Applications on the Recent Extensions of Ordinary Fuzzy Sets)

20 pages, 49600 KiB

Open AccessArticle

An Improved Symmetric Network with Feature Difference and Receptive Field for Change Detection

by Botao Zhang, Yixuan Wang, Jia Lu and Qin Wang

Symmetry 2025, 17(7), 1095; https://doi.org/10.3390/sym17071095 - 8 Jul 2025

Viewed by 19

Abstract

Change detection (CD) is essential for Earth observation tasks, as it identifies alterations in specific geographic areas over time. The advancement of deep learning has significantly improved the accuracy of CD. However, encoder–decoder architectures often struggle to effectively capture temporal differences. Encoders may [...] Read more.

Change detection (CD) is essential for Earth observation tasks, as it identifies alterations in specific geographic areas over time. The advancement of deep learning has significantly improved the accuracy of CD. However, encoder–decoder architectures often struggle to effectively capture temporal differences. Encoders may lose critical spatial details, while decoders can introduce bias due to inconsistent receptive fields across layers. To address these limitations, this paper proposes an enhanced symmetric network, termed FDRF (feature difference and receptive field), which incorporates two novel components: the multibranch feature difference extraction (MFDE) module and the adaptive ensemble decision (AED) module. MFDE independently extracts differential features from bitemporal images at each encoder layer, using multiscale fusion to retain image content and improve the quality of feature difference modeling. AED assigns confidence weights to predictions from different decoder layers based on their receptive field sizes and then combines them adaptively to reduce scale-related bias. To validate the effectiveness and robustness of FDRF, experiments were conducted on five public datasets (SYSU, LEVIR-CD, WHU, NJDS, and CLCD), as well as a UAV-based dataset collected from two national coastal nature reserves in Guangxi Beihai, China. The results demonstrate that FDRF consistently outperforms existing methods in accuracy and robustness across diverse scenarios. Full article

(This article belongs to the Section Computer)

► Show Figures

Figure 1

27 pages, 13752 KiB

Open AccessArticle

Robust Watermarking of Tiny Neural Networks by Fine-Tuning and Post-Training Approaches

by Riccardo Adorante, Alessandro Carra, Marco Lattuada and Danilo Pietro Pau

Symmetry 2025, 17(7), 1094; https://doi.org/10.3390/sym17071094 - 8 Jul 2025

Viewed by 52

Abstract

Because neural networks pervade many industrial domains and are increasingly complex and accurate, the trained models themselves have become valuable intellectual properties. Developing highly accurate models demands increasingly higher investments of time, capital, and expertise. Many of these models are commonly deployed in [...] Read more.

Because neural networks pervade many industrial domains and are increasingly complex and accurate, the trained models themselves have become valuable intellectual properties. Developing highly accurate models demands increasingly higher investments of time, capital, and expertise. Many of these models are commonly deployed in cloud services and on resource-constrained edge devices. Consequently, safeguarding them is critically important. Neural network watermarking offers a practical solution to address this need by embedding a unique signature, either as a hidden bit-string or as a distinctive response to specially crafted “trigger” inputs. This allows owners to subsequently prove model ownership even if an adversary attempts to remove the watermark through attacks. In this manuscript, we adapt three state-of-the-art watermarking methods to “tiny” neural networks deployed on edge platforms by exploiting symmetry-related properties that ensure robustness and efficiency. In the context of machine learning, “tiny” is broadly used as a term referring to artificial intelligence techniques deployed in low-energy systems in the mW range and below, e.g., sensors and microcontrollers. We evaluate the robustness of the selected techniques by simulating attacks aimed at erasing the watermark while preserving the model’s original performances. The results before and after attacks demonstrate the effectiveness of these watermarking schemes in protecting neural network intellectual property without degrading the original accuracy. Full article

(This article belongs to the Section Computer)

► Show Figures

Graphical abstract

25 pages, 906 KiB

Open AccessArticle

Query-Efficient Two-Phase Reinforcement Learning Framework for Black-Box Adversarial Attacks

by Zerou Ma and Tao Feng

Symmetry 2025, 17(7), 1093; https://doi.org/10.3390/sym17071093 - 8 Jul 2025

Viewed by 47

Abstract

Generating adversarial examples under black-box settings poses significant challenges due to the inaccessibility of internal model information. This complexity is further exacerbated when attempting to achieve a balance between the attack success rate and perceptual quality. In this paper, we propose QTRL, a [...] Read more.

Generating adversarial examples under black-box settings poses significant challenges due to the inaccessibility of internal model information. This complexity is further exacerbated when attempting to achieve a balance between the attack success rate and perceptual quality. In this paper, we propose QTRL, a query-efficient two-phase reinforcement learning framework for generating high-quality black-box adversarial examples. Unlike existing approaches that treat adversarial generation as a single-step optimization problem, QTRL introduces a progressive two-phase learning strategy. The initial phase focuses on training the agent to develop effective adversarial strategies, while the second phase refines the perturbations to improve visual quality without sacrificing attack performance. To compensate for the unavailability of gradient information inherent in black-box settings, QTRL designs distinct reward functions for the two phases: the first prioritizes attack success, whereas the second incorporates perceptual similarity metrics to guide refinement. Furthermore, a hard sample mining mechanism is introduced to revisit previously failed attacks, significantly enhancing the robustness and generalization capabilities of the learned policy. Experimental results on the MNIST and CIFAR-10 datasets demonstrate that QTRL achieves attack success rates comparable to those of state-of-the-art methods while substantially reducing query overhead, offering a practical and extensible solution for adversarial research in black-box scenarios. Full article

(This article belongs to the Section Computer)

► Show Figures

Figure 1

19 pages, 9926 KiB

Open AccessArticle

Deep Learning-Based Optimal Condition Monitoring System for Plant Growth in an Indoor Smart Hydroponic Greenhouse

by Oybek Eraliev Maripjon Ugli and Chul-Hee Lee

Symmetry 2025, 17(7), 1092; https://doi.org/10.3390/sym17071092 - 8 Jul 2025

Viewed by 23

Abstract

This study introduces a deep learning (DL)-based optimal condition monitoring and control system tailored to indoor smart greenhouses, with a novel focus on maintaining symmetry—defined as a dynamic equilibrium among temperature, humidity, and CO₂ levels—critical in plant growth. A hydroponic greenhouse prototype [...] Read more.

This study introduces a deep learning (DL)-based optimal condition monitoring and control system tailored to indoor smart greenhouses, with a novel focus on maintaining symmetry—defined as a dynamic equilibrium among temperature, humidity, and CO₂ levels—critical in plant growth. A hydroponic greenhouse prototype was developed to capture real-time climate data at high temporal resolution. A custom 1D convolutional neural network (1D-CNN) optimized via a genetic algorithm (GA) was employed to predict environmental fluctuations, achieving R² scores up to 0.99 and a standard error of prediction (SEP) as low as 0.35%. The system then actuated climate control mechanisms to restore and maintain symmetry. Experimental validation revealed that plants grown under the symmetry-aware control system exhibited significantly improved growth metrics. The results underscore the potential of integrating symmetry-aware DL strategies into precision agriculture in achieving sustainable and resilient plant production systems. Full article

(This article belongs to the Section Computer)

► Show Figures

Figure 1

12 pages, 1601 KiB

Open AccessArticle

Effects of Residual Stresses on the Structures and Mechanical Behavior of ZrO_xN_y/V₂O₃ Nano-Multilayers

by Wenjie Cheng, Lingran Wang and Zhiming Li

Symmetry 2025, 17(7), 1091; https://doi.org/10.3390/sym17071091 - 8 Jul 2025

Viewed by 12

Abstract

Residual stress plays a crucial role in determining the structural reliability and mechanical performance of nano-multilayers. In the present study, nano-multilayers composed of ZrO_xN_y and V₂O₃ were deposited via magnetron sputtering, with the N:Ar flow ratio systematically [...] Read more.

Residual stress plays a crucial role in determining the structural reliability and mechanical performance of nano-multilayers. In the present study, nano-multilayers composed of ZrO_xN_y and V₂O₃ were deposited via magnetron sputtering, with the N:Ar flow ratio systematically varied during the process. Through the precise control of the deposition conditions, the compressive residual stress within the films was effectively reduced to approximately 0 GPa, thereby improving their mechanical robustness. It was observed that the optimization of the stress distribution was strongly influenced by the structural symmetry of the multilayer configuration. This symmetrical design not only mitigated stress accumulation but also ensured uniform mechanical response throughout the multilayer structure. The results from nanoindentation testing revealed a steady hardness value near 10.6 GPa. Furthermore, the maximum H³/E² and H/E ratios recorded were 0.054 GPa and 0.073, respectively, suggesting enhanced resistance to both plastic deformation and cracking. Full article

(This article belongs to the Section Engineering and Materials)

► Show Figures

Figure 1

19 pages, 1256 KiB

Open AccessArticle

More Details on Two Solutions with Ordered Sequences for Binomial Confidence Intervals

by Lorentz Jäntschi

Symmetry 2025, 17(7), 1090; https://doi.org/10.3390/sym17071090 - 8 Jul 2025

Viewed by 14

Abstract

While many continuous distributions are known, the list of discrete ones (usually derived from counting) often reported is relatively short. This list becomes even shorter when dealing with dichotomous observables: binomial, hypergeometric, negative binomial, and uniform. Binomial distribution is important for medical studies, [...] Read more.

While many continuous distributions are known, the list of discrete ones (usually derived from counting) often reported is relatively short. This list becomes even shorter when dealing with dichotomous observables: binomial, hypergeometric, negative binomial, and uniform. Binomial distribution is important for medical studies, since a finite sample from a population included in a medical study with yes/no outcome resembles a series of independent Bernoulli trials. The problem of calculating the confidence interval (CI, with conventional risk of 5% or otherwise) is revealed to be a problem of combinatorics. Several algorithms dispute the exact calculation, each according to a formal definition of its exactness. For two algorithms, four previously proposed case studies are provided, for sample sizes of 30, 50, 100, 150, and 300. In these cases, at 1% significance level, ordered sequences defining the confidence bounds were generated for two formal definitions. Images of the error’s alternation are provided and discussed. Both algorithms propose symmetric solutions in terms of both CIs and actual coverage probabilities. The CIs are not symmetric relative to the observed variable, but are mirrored symmetric relative to the middle of the observed variable domain. When comparing the solutions proposed by the algorithms, with the increase in the sample size, the ratio of identical confidence levels is increased and the difference between actual and imposed coverage is shrunk. Full article

(This article belongs to the Section Mathematics)

► Show Figures

Figure 1

26 pages, 1884 KiB

Open AccessArticle

A Symmetry-Based Spherical Fuzzy MCDM Approach for the Strategic Assessment of Alternative Fuels Toward Sustainable Energy Policies

by Adnan Abdulvahitoğlu

Symmetry 2025, 17(7), 1089; https://doi.org/10.3390/sym17071089 - 8 Jul 2025

Viewed by 26

Abstract

Alternative fuels obtained from renewable sources, providing low greenhouse gas emissions and high energy efficiency, offer significant advantages in terms of sustainability. In addition, the wide applicability of these fuel types in sectors such as housing, transportation, and industry creates significant opportunities in [...] Read more.

Alternative fuels obtained from renewable sources, providing low greenhouse gas emissions and high energy efficiency, offer significant advantages in terms of sustainability. In addition, the wide applicability of these fuel types in sectors such as housing, transportation, and industry creates significant opportunities in terms of reducing dependence on fossil fuels. Alternative fuels should be evaluated not only according to their environmental contributions but also based on multi-dimensional criteria such as economic cost, technical suitability, sustainability level, fuel properties, infrastructure requirements, and social acceptance. In this context, a comparative analysis of alternative fuel types in terms of various basic parameters is no longer optional, but a necessity. These parameters generally include symmetrical relationships such as balanced trade-offs between economic and environmental dimensions or mutual effects between technical and social criteria. However, they also show variability and uncertainty depending on the fuel type. Therefore, Spherical Fuzzy Multi-Criteria Decision Making (SF-MCDM) methods, which can effectively represent symmetry in membership and hesitation degrees, have been used to achieve more realistic and reliable results in uncertain decision environments. The proposed model provides a systematic and flexible evaluation structure that helps decision makers determine the most appropriate alternative fuel options and contributes to the formation of sustainable energy policies. Full article

(This article belongs to the Section Mathematics)

► Show Figures

Figure 1

Journal Menu

Journal Browser

Symmetry, Volume 17, Issue 7 (July 2025) – 143 articles

Further Information

Guidelines

MDPI Initiatives

Follow MDPI