Symmetry

14 pages, 1105 KB

Open AccessArticle

Exact Soliton Structures and Modulation Instability in Extended Kadomtsev–Petviashvili–Boussinesq Equation

by Nadiyah Hussain Alharthi, Rubayyi T. Alqahtani and Melike Kaplan

Symmetry 2026, 18(4), 626; https://doi.org/10.3390/sym18040626 - 8 Apr 2026

In this study, we consider an extended form of the Kadomtsev–Petviashvili–Boussinesq equation motivated by wave propagation phenomena in dissipative media. The primary aim of this work is to construct exact analytical solutions and clarify the types of nonlinear wave structure admitted by the [...] Read more.

In this study, we consider an extended form of the Kadomtsev–Petviashvili–Boussinesq equation motivated by wave propagation phenomena in dissipative media. The primary aim of this work is to construct exact analytical solutions and clarify the types of nonlinear wave structure admitted by the considered model. For this purpose, the Riccati equation expansion method is applied for the first time within this framework. This method allows us to obtain several distinct families of solitary wave solutions whose qualitative behaviors and physical characteristics are illustrated through graphical representations. In addition, modulation instability analysis is carried out to assess the stability of continuous wave solutions and further elucidate the underlying nonlinear dynamics of the system. Full article

(This article belongs to the Special Issue New Trends on the Mathematical Models and Solitons Arising in Real-World Problems, 3rd Edition)

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27 pages, 1999 KB

Open AccessArticle

Uncertainty-Driven Risk Evaluation for Safety-Critical Software Under Conflicting Evidence Judgments: A Dual-Dimensional Evidence Fusion Approach

by Wenguang Xie, Wuhan Yang and Kenian Wang

Symmetry 2026, 18(4), 625; https://doi.org/10.3390/sym18040625 - 8 Apr 2026

Abstract

Risk assessment of safety-critical software relies heavily on expert reviews prone to high epistemic uncertainty and conflicting judgments. While evidence theory is widely used for information fusion, classical rules often yield counter-intuitive results in high-conflict scenarios. To address this, we propose an uncertainty-driven [...] Read more.

Risk assessment of safety-critical software relies heavily on expert reviews prone to high epistemic uncertainty and conflicting judgments. While evidence theory is widely used for information fusion, classical rules often yield counter-intuitive results in high-conflict scenarios. To address this, we propose an uncertainty-driven risk evaluation model based on a dual-dimensional evidence fusion approach. The framework integrates an improved Belief Entropy (BE) and an Evidence Conflict Coefficient (ECC) to quantify reliability from two perspectives: (1) Internal Dimension, using BE to measure inherent uncertainty within individual judgments; and (2) External Dimension, using ECC to measure divergence among multiple sources. By adaptively modifying Basic Probability Assignments (BPAs) with these dual-dimensional weights, the model effectively harmonizes data prior to fusion. Validated through an avionics software airworthiness case study, the methodology significantly enhances fusion stability and accuracy. Results confirm it effectively suppresses extreme deviations and raises the performance floor, providing a robust decision-support tool for safety-critical engineering. Full article

(This article belongs to the Section Computer)

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19 pages, 3805 KB

Open AccessArticle

Dynamics of Rotor–Bearing Systems Under Time-Varying Stiffness Excitation of Helical Gears

by Yuanxing Huang, Yutong Fu, Wanying Huang, Yuanxin Fang and Xuezhong Fu

Symmetry 2026, 18(4), 624; https://doi.org/10.3390/sym18040624 (registering DOI) - 8 Apr 2026

Abstract

The time-varying mesh stiffness excitation of helical gears impacts the vibration state of the rotor–bearing systems, while the existence of mechanical dynamic eccentricity makes the rotor–bearing dynamics equation a system of parametric excitation. To address this situation, the time-varying mesh stiffness of the [...] Read more.

The time-varying mesh stiffness excitation of helical gears impacts the vibration state of the rotor–bearing systems, while the existence of mechanical dynamic eccentricity makes the rotor–bearing dynamics equation a system of parametric excitation. To address this situation, the time-varying mesh stiffness of the helical gear is substituted into the coupled bending–torsion–axial dynamic equation of the rotor–bearing system. By considering dynamic eccentricity, the rotor’s vibration displacement response is calculated. The unified strength theory is introduced to compute the complex stress state. The study’s results indicate that time-varying stiffness significantly influences the system’s vibration characteristics, with the equivalent stress values exceeding those under twin-shear stress. This finding demonstrates the advantage of using the unified strength theory under high-load conditions, providing an essential reference for optimizing the dynamic performance of high-speed helical gear transmission systems. Full article

(This article belongs to the Section Engineering and Materials)

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21 pages, 2743 KB

Open AccessArticle

SOC and SOH Joint Estimation of Lithium-Ion Batteries Under Dynamic Current Rates Based on Machine Learning

by Mingyu Zhang, Xiaoqiang Dai, Qingjun Zeng, Ye Tian and Xiaohui Xu

Symmetry 2026, 18(4), 623; https://doi.org/10.3390/sym18040623 - 8 Apr 2026

Abstract

It is critical to accurately estimate the state of charge (SOC) and state of health (SOH) of lithium-ion batteries to ensure the safety and reliability of marine power systems, where the inherent symmetry of lithium-ion battery charge–discharge dynamics is often disrupted. However, the [...] Read more.

It is critical to accurately estimate the state of charge (SOC) and state of health (SOH) of lithium-ion batteries to ensure the safety and reliability of marine power systems, where the inherent symmetry of lithium-ion battery charge–discharge dynamics is often disrupted. However, the accuracy of conventional methods significantly deteriorates under dynamic current rates induced by fluctuating electrical loads, leading to unreliable SOC and SOH estimates. This article proposes a novel SOC and SOH joint estimation method based on a long short-term memory network with a rate awareness attention mechanism (RAAM-LSTM) and support vector regression optimized by greylag goose algorithm (GGO-SVR). RAAM-LSTM improves SOC estimation accuracy by adaptively weighting enhanced rate-related features. For SOH estimation, the GGO-SVR model incorporates the SOC as a coupling feature and applies physical constraints to ensure consistency with irreversible battery degradation. The comparative experimental results show that the error of the SOC is less than 1.6%, and that of the SOH is less than 0.5%, which are much smaller compared with those of conventional methods. Full article

(This article belongs to the Section Computer)

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24 pages, 5938 KB

Open AccessArticle

Fault Diagnosis of 2RRU-RRS Parallel Robots Based on Multi-Scale Efficient Channel Attention Residual Network

by Shuxiang He, Wei Ye, Ying Zhang, Shanyi Liu, Zhen Wu and Lingmin Xu

Symmetry 2026, 18(4), 622; https://doi.org/10.3390/sym18040622 - 8 Apr 2026

Abstract

Parallel robots are widely applied in many fields because of their unique advantages. To ensure their operational safety and reduce maintenance costs, designing an accurate and reliable fault diagnosis method is essential. Focusing on the 2RRU-RRS parallel robot, this paper proposes an intelligent [...] Read more.

Parallel robots are widely applied in many fields because of their unique advantages. To ensure their operational safety and reduce maintenance costs, designing an accurate and reliable fault diagnosis method is essential. Focusing on the 2RRU-RRS parallel robot, this paper proposes an intelligent fault diagnosis method based on a multi-scale convolutional residual network integrated with an Efficient Channel Attention mechanism (MS-ECA-ResNet). Firstly, to fully retain the time-frequency features of the signals, the one-dimensional vibration signals are converted into two-dimensional images using the Continuous Wavelet Transform (CWT). Secondly, a multi-scale convolutional feature extraction structure is designed to enhance the model’s feature extraction ability at different time scales. Furthermore, the ECA mechanism is introduced into the residual network to reinforce important feature channels and suppress noise interference. Comparative experiments, noise environment experiments, and ablation experiments were conducted on a 2RRU-RRS parallel robot experimental platform with a vibration signal dataset. The results demonstrate that the proposed method achieves superior diagnostic accuracy and robustness compared to typical deep learning models, particularly in maintaining high performance under simulated noise conditions. This provides a preliminary validation of the method’s effectiveness in capturing fault-related impacts, offering a potential technical reference for the health monitoring of parallel robots in real-world scenarios. Full article

(This article belongs to the Special Issue Symmetry in Intelligent Spindle Modelling and Vibration Analysis)

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20 pages, 2638 KB

Open AccessArticle

Design and Implementation of Underwater Robotic Systems for Visual–Inertial Trajectory Estimation and Robust Motion Control

by Yangyang Wang, Tianzhu Gao, Yongqiang Zhao, Ziyu Liu, Hang Yu and Xijun Du

Symmetry 2026, 18(4), 621; https://doi.org/10.3390/sym18040621 - 6 Apr 2026

Abstract

Reliable trajectory estimation and precise motion control are the prerequisites for underwater robotic systems to perform complex autonomous tasks, which are essential for enhancing the operational efficiency of intelligent underwater facilities. However, the inherent asymmetry of underwater hydrodynamics, featureless images caused by complex [...] Read more.

Reliable trajectory estimation and precise motion control are the prerequisites for underwater robotic systems to perform complex autonomous tasks, which are essential for enhancing the operational efficiency of intelligent underwater facilities. However, the inherent asymmetry of underwater hydrodynamics, featureless images caused by complex environments, and the lack of high-frequency state feedback significantly hinder stable trajectory tracking and robust autonomous navigation. To address these challenges, this paper proposes an integrated autonomous navigation and robust control scheme for underwater robotic systems. Specifically, we first propose a visual–inertial trajectory estimation method for underwater robotic systems, which effectively overcomes the challenges of featureless images and provides consistent, real-time pose feedback for motion execution. Furthermore, we develop a hierarchical robust motion control strategy for autonomous underwater robots, which integrates model predictive control with incremental nonlinear dynamic inversion to achieve precise positioning performance and reliable operation under environmental disturbances. Finally, we design and implement a customized, highly integrated underwater robotic platform that integrates the proposed trajectory estimation and robust control modules, with its performance validated through extensive field experiments in underwater scenarios. The experimental results demonstrate that the proposed system can effectively achieve high-precision trajectory tracking and maintain operational stability, providing a comprehensive engineering solution for the autonomous navigation of underwater robots in complex environments. Full article

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

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28 pages, 736 KB

Open AccessArticle

Frequency-Based Prediction Study of Burr X Distribution Under Type II Censoring

by Wenyu Tong and Wenhao Gui

Symmetry 2026, 18(4), 620; https://doi.org/10.3390/sym18040620 - 6 Apr 2026

Abstract

This paper investigates frequentist prediction methods for the Burr X distribution under Type II censored data. To address the challenges of small sample sizes and high censoring rates commonly encountered in survival analysis, we derive four point prediction methods: best unbiased prediction (BUP), [...] Read more.

This paper investigates frequentist prediction methods for the Burr X distribution under Type II censored data. To address the challenges of small sample sizes and high censoring rates commonly encountered in survival analysis, we derive four point prediction methods: best unbiased prediction (BUP), maximum likelihood prediction (MLP), conditional median prediction (CMP), and median unbiased prediction (MUP). For interval prediction, we examine four approaches—the pivotal method, the Wald method, the highest conditional density (HCD) method, and the shortest-length method to construct prediction intervals. The finite-sample performance of these methods is evaluated through Monte Carlo simulations and illustrated using three real-world datasets. The results demonstrate that CMP provides the most stable point predictions, with its advantage being particularly pronounced in small samples due to the conditional median’s robustness to extreme values. For interval prediction, the pivotal method yields the most consistently reliable coverage. The shortest-length method exhibits high accuracy and efficiency. Analyses of three real datasets further validate the applicability of these methods to both complete and right-censored data. Full article

(This article belongs to the Section Mathematics)

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15 pages, 297 KB

Open AccessArticle

A Pair of Hermitian and Anti-Hermitian Solutions of a Generalized Sylvester Matrix Equation over Commutative Quaternion Algebra

by Haixia Chang, Lvming Xie and Longsheng Liu

Symmetry 2026, 18(4), 619; https://doi.org/10.3390/sym18040619 - 6 Apr 2026

Abstract

In this paper, we investigate a pair of Hermitian and anti-Hermitian solutions of a generalized Sylvester matrix equation

A X B + C \bar{X} D + E Y F = G

over commutative quaternion algebra by using a complex representation of commutative [...] Read more.

In this paper, we investigate a pair of Hermitian and anti-Hermitian solutions of a generalized Sylvester matrix equation

A X B + C \bar{X} D + E Y F = G

over commutative quaternion algebra by using a complex representation of commutative quaternion matrices, the Kronecker product, vec-operation, and Moore–Penrose-generalized inverse. We establish the necessary and sufficient conditions for the existence of solutions. Moreover, we derive explicit expressions when they are solvable. We also provide two numerical examples to illustrate the main results. Full article

(This article belongs to the Section Mathematics)

23 pages, 6176 KB

Open AccessArticle

A New Image Denoising Model Based on Low-Rank and Deep Image Prior

by Liwen Feng, Yan Hao, Zirui Mao, Jiaojiao Xu and Jianlou Xu

Symmetry 2026, 18(4), 618; https://doi.org/10.3390/sym18040618 - 5 Apr 2026

Abstract

Low-rank recovery has emerged as a powerful methodology for the restoration of degraded images. Conventional low-rank recovery techniques, however, predominantly rely on nuclear norm or weighted nuclear norm minimization to separate sparse noise. A significant limitation of this approach is its dependence on [...] Read more.

Low-rank recovery has emerged as a powerful methodology for the restoration of degraded images. Conventional low-rank recovery techniques, however, predominantly rely on nuclear norm or weighted nuclear norm minimization to separate sparse noise. A significant limitation of this approach is its dependence on full singular value decomposition, which imposes a substantial computational burden, thereby hindering its practical applicability. This paper presents a novel image denoising model integrating the weighted nuclear norm and deep image prior. The weighted nuclear norm is introduced to accurately characterize the global structural properties of images, ensuring the consistency of the overall image structure after denoising. Meanwhile, the deep image prior is employed to effectively capture local details, which helps avoid the blurring of textures and edges often caused by excessive noise removal. The complementary advantages of the two components enable the proposed model to achieve superior performance compared with existing denoising methods. To efficiently compute the proposed model, we design the bilinear factorization method and the alternating direction method of multipliers. Experiments show that the proposed method outperforms mainstream approaches in both restoration accuracy and computational efficiency, exhibiting rapid convergence and robust algorithm stability, thereby demonstrating excellent comprehensive performance. Full article

(This article belongs to the Section Computer)

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7 pages, 656 KB

Open AccessArticle

On the

CF

-Connectedness of the k-Power of P_n

by Michal Staš

Symmetry 2026, 18(4), 617; https://doi.org/10.3390/sym18040617 - 5 Apr 2026

Abstract

The study of graph connectivity is a central topic in graph theory, with

CF

-connectedness being a specialized property of interest. A connected graph is

CF

-connected if, in every optimal drawing, there exists a path between every pair of vertices such that [...] Read more.

The study of graph connectivity is a central topic in graph theory, with

CF

-connectedness being a specialized property of interest. A connected graph is

CF

-connected if, in every optimal drawing, there exists a path between every pair of vertices such that no edges cross. This paper explores the

CF

-connectedness of the k-power of

P_{n}

, denoted

P_{n}^{k}

, where

P_{n}

is a path on n vertices,

P_{n}^{k}

is a graph on the same vertex set as

P_{n}

, and an edge

{u, v}

exists in

P_{n}^{k}

if the distance between u and v on

P_{n}

is at most k. The paper concludes with a controversial drawing of the graph

P_{10}^{5}

with only 16 crossings, which refutes the truth of Zheng et al.’s conjecture that the upper bound

cr (P_{n}^{5}) \leq 4 n - 23

holds with equality for all

n \geq 8

. Full article

(This article belongs to the Special Issue Symmetry in Combinatorics and Discrete Mathematics, 2nd Edition)

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15 pages, 6079 KB

Open AccessArticle

Research on the Influence of Welding Heat Source and Welding Speed on Welding Residual Stress and Temperature Field of H-Shaped Steel: A Numerical Simulation Study

by Wei Cao, Bocheng Guo and Xinye Wu

Symmetry 2026, 18(4), 616; https://doi.org/10.3390/sym18040616 - 5 Apr 2026

Abstract

To explore the influence mechanism of welding process parameters on the residual stress and temperature field of complex welded components, this paper takes H-shaped steel, which is widely used in engineering, as the research object. Based on the thermal-force coupling finite element method, [...] Read more.

To explore the influence mechanism of welding process parameters on the residual stress and temperature field of complex welded components, this paper takes H-shaped steel, which is widely used in engineering, as the research object. Based on the thermal-force coupling finite element method, a three-dimensional numerical model of its welding process is established using the ANSYS Workbench platform. Based on the heat conduction equation and structural constraint theory, in accordance with the classification criteria for thin plates and medium-thick plates in the standards of the International Institute of Welding, and in combination with the typical structural size characteristics, six sets of comparative working conditions were designed. The influence of two key parameters, namely, the welding heat source parameters and welding speed, on the welding residual stress and temperature field was analyzed in detail. The research results show that increasing the welding heat input will raise peak welding temperature and expand the range of the high-temperature zone, resulting in a significant increase in residual tensile stress in the weld zone after cooling. Increasing the welding speed can effectively reduce heat accumulation and decrease the temperature gradient, thereby lowering the peak residual stress by approximately 10% to 15%. Research reveals that, under the premise of ensuring thorough penetration, adopting a process combination of “lower heat input and higher welding speed” can effectively suppress the generation of welding residual stress in H-beams. The research results can provide a theoretical basis for the optimization of welding processes in actual production. Full article

(This article belongs to the Section Engineering and Materials)

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13 pages, 313 KB

Open AccessArticle

Almost Extraspecial Structures and Pseudofermionic Operators

by Daniele Ettore Otera and Francesco G. Russo

Symmetry 2026, 18(4), 615; https://doi.org/10.3390/sym18040615 - 5 Apr 2026

Abstract

We survey some recent combinatorial properties, which have been found in the context of the algebras of ladder operators in quantum mechanics. More specifically, we review dynamical systems which have nonselfadjoint Hamiltonians and are subject to a formalization in terms of pseudofermionic operators. [...] Read more.

We survey some recent combinatorial properties, which have been found in the context of the algebras of ladder operators in quantum mechanics. More specifically, we review dynamical systems which have nonselfadjoint Hamiltonians and are subject to a formalization in terms of pseudofermionic operators. For these systems, we detect structural analogies between algebras of pseudofermionic operators and the abstract notion of central product, which was originally studied for finite groups. Full article

(This article belongs to the Special Issue Advances in Topology and Algebraic Geometry)

12 pages, 834 KB

Open AccessArticle

Thermal Layer Polarization in Asymmetric Doped Bilayer Graphene

by Juan A. Lazzús and L. Palma-Chilla

Symmetry 2026, 18(4), 614; https://doi.org/10.3390/sym18040614 - 5 Apr 2026

Abstract

We present a minimal model that isolates the effect of hopping asymmetry on the thermal redistribution of particles in a bilayer graphene system composed of one pristine and one doped layer. The behavior of the system is governed by a single control parameter [...] Read more.

We present a minimal model that isolates the effect of hopping asymmetry on the thermal redistribution of particles in a bilayer graphene system composed of one pristine and one doped layer. The behavior of the system is governed by a single control parameter

α

, which uniformly reduces the intralayer hopping in the doped sheet and defines the only source of spectral asymmetry. Particle conservation fixes the total density at all temperatures, so thermal effects appear exclusively as a redistribution relative to the zero-temperature reference state. This redistribution is quantified by the temperature-induced change of the layer populations, which measures how thermal occupation modifies the particle density in each layer without creating or destroying carriers. With this, the difference between the thermal population corrections of the doped and pristine layers defines the polarization. Thus, thermal excitations probe the asymmetric spectrum, exposing the intrinsic layer imbalance. Results show that for

0 < α < 1

, band deformation produces unequal occupations between doped and pristine layers, and temperature amplifies this imbalance. The resulting polarization increases monotonically with temperature and systematically with hopping reduction, establishing a direct quantitative link between microscopic spectral asymmetry and macroscopic thermally induced layer imbalance. Full article

(This article belongs to the Section Physics)

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48 pages, 578 KB

Open AccessArticle

Invariant Threshold Symmetry in Bipolar Fuzzy Quasi-Subalgebras of Sheffer–Nelson Algebras

by Amal S. Alali, Tahsin Oner, Ravi Kumar Bandaru, Rajesh Neelamegarajan, Ibrahim Senturk and Ebrar Gunel

Symmetry 2026, 18(4), 613; https://doi.org/10.3390/sym18040613 - 5 Apr 2026

Abstract

This paper develops a rigorous algebraic framework for quasi-substructures in Sheffer-based Nelson algebras, extending the landscape of fuzzy algebraic theory. By systematically introducing

(\in, \in \lor q)

-bipolar fuzzy quasi-subalgebras and ideals, we analyze their structural properties through generalized belongingness [...] Read more.

This paper develops a rigorous algebraic framework for quasi-substructures in Sheffer-based Nelson algebras, extending the landscape of fuzzy algebraic theory. By systematically introducing

(\in, \in \lor q)

-bipolar fuzzy quasi-subalgebras and ideals, we analyze their structural properties through generalized belongingness and quasi-coincidence relations. We formalise invariant threshold symmetry as the condition

g^{+} (χ) + | g^{-} (χ) | = c

for a constant

c \in [0, 2]

and every

χ \in Ω

(Definition ) and prove its structural preservation within

(\in, \in \lor q)

-bipolar fuzzy quasi-subalgebras (Theorem , supported by Theorems , and ). This enables a balanced dual evaluation of positive and negative information. Characterization theorems are established via level subsets, revealing how quasi-substructure properties are governed by bounds at critical membership values. Equivalence results unify classical and bipolar fuzzy perspectives, demonstrating that algebraic constraints preserve structural coherence across crisp and fuzzy environments. Algorithmic verification procedures are provided for practical validation in finite systems, and illustrative examples highlight applications in uncertainty modeling and decision support. Overall, the proposed theory formalizes bipolar fuzzy structures in Sheffer-based Nelson algebras, utilizing invariant threshold symmetry, level-set decomposition, and crisp equivalence to evaluate dual information. Full article

(This article belongs to the Special Issue Algebras and Symmetry in Fuzzy Set Theory)

22 pages, 1280 KB

Open AccessArticle

Enhancing Early Skin Cancer Detection: A Deep Learning Approach with Multi-Scale Feature Refinement and Fusion

by Siyuan Wu, Pengfei Zhao, Huafu Xu and Zimin Wang

Symmetry 2026, 18(4), 612; https://doi.org/10.3390/sym18040612 - 5 Apr 2026

Abstract

The global incidence of skin cancer is rising, making it an increasingly critical public health issue. Malignant skin tumors such as melanoma originate from pathological alterations in skin cells, and their accurate early-stage segmentation is crucial for quantitative analysis, early diagnosis, and effective [...] Read more.

The global incidence of skin cancer is rising, making it an increasingly critical public health issue. Malignant skin tumors such as melanoma originate from pathological alterations in skin cells, and their accurate early-stage segmentation is crucial for quantitative analysis, early diagnosis, and effective treatment. However, achieving precise and efficient segmentation remains a major challenge, as existing methods often struggle to capture complex lesion characteristics. To address this challenge, we propose a novel deep learning framework that integrates the PVT v2 backbone with two key modules: the Spatial-Aware Feature Enhancement (SAFE) module and the Multiscale Dual Cross-attention Fusion (MDCF) module. The SAFE module enhances multi-scale encoder features through a dual-branch architecture, which adaptively extracts offset information to integrate fine-grained shallow details with deep semantic information, thereby bridging the feature gap across network depths. The MDCF module establishes bidirectional cross-attention between decoder and encoder features, followed by multi-scale deformable convolutions that capture lesion boundaries and small fragments across heterogeneous receptive fields, thereby enriching semantic details while suppressing background interference. The proposed model was evaluated on two public benchmark datasets (ISIC 2016 and ISIC 2018), achieving Intersection over Union (IoU) scores of 87.33% and 83.67%, respectively. These results demonstrate superior performance compared to current state-of-the-art methods and indicate that our framework significantly enhances skin lesion image analysis, offering a promising tool for improving early detection of skin cancer. Full article

(This article belongs to the Special Issue Symmetric/Asymmetric Study in Medical Imaging)

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24 pages, 2066 KB

Open AccessArticle

Advances in Near Soft Sets and Their Applications in Similarity-Based Decision Making

by Alkan Özkan, James Peters, Faruk Özger, Metin Duman and Merve Ersoy

Symmetry 2026, 18(4), 611; https://doi.org/10.3390/sym18040611 - 4 Apr 2026

Abstract

In this study, a generalized and advanced form of the near soft set theory (NST) framework is proposed for information aggregation (IA) processes. The primary motivation of the study is to address the lack of similarity-based uncertainty modeling in the literature by integrating [...] Read more.

In this study, a generalized and advanced form of the near soft set theory (NST) framework is proposed for information aggregation (IA) processes. The primary motivation of the study is to address the lack of similarity-based uncertainty modeling in the literature by integrating the parametric structure of soft sets with the similarity-oriented structure of nearness approximation spaces. Within this framework, the AND-product and OR-product operations are introduced as the main methodological tools, and their algebraic structures are analyzed in detail. It is mathematically demonstrated that these operations satisfy fundamental properties such as idempotency, absorption, distributivity, and De Morgan identities. The principal original contribution of the study is the development of a novel Uni–Int-based decision-making mechanism that enables the systematic distinction between strong and acceptable alternatives. In addition, the boundary frequency indicator (

b_{r}

), which quantitatively evaluates the reliability of objects under perceptual uncertainty and is introduced for the first time in the literature, is proposed. The applicability of the proposed model is demonstrated through a real-estate selection problem, and a sensitivity analysis is conducted to reveal the determining effect of the nearness parameter r on decision granularity. The obtained findings indicate that the proposed NST framework provides a more flexible, more discriminative, and structurally robust decision-support model than classical approaches, particularly for similarity-based IA problems. Full article

(This article belongs to the Section Mathematics)

24 pages, 2227 KB

Open AccessArticle

Prime-Enforced Symmetry Constraints in Thermodynamic Recoils: Unifying Phase Behaviors and Transport Phenomena via a Covariant Fugacity Hessian

by Muhamad Fouad

Symmetry 2026, 18(4), 610; https://doi.org/10.3390/sym18040610 - 4 Apr 2026

Abstract

The Zeta-Minimizer Theorem establishes that the Riemann zeta function

ζ (s)

and the primes arise variationally as unique minimizers of a phase functional defined on a symmetric measure space

(X, μ, G)

equipped with helical operators. Three fundamental axioms—strict concave entropy [...] Read more.

The Zeta-Minimizer Theorem establishes that the Riemann zeta function

ζ (s)

and the primes arise variationally as unique minimizers of a phase functional defined on a symmetric measure space

(X, μ, G)

equipped with helical operators. Three fundamental axioms—strict concave entropy maximization (Axiom 1), spectral Gibbs minima with non-vanishing ground states (Axiom 2), and irreducible bounded oscillations with flux conservation (Axiom 3)—allow for the selection of the non-proper Archimedean conical helix as the sole topology satisfying all constraints. Primes emerge as indivisible minimal cycles in the associated representation graph

Γ

(via Hilbert irreducibility and Maschke’s theorem), while the Euler product is recovered through the spectral Dirichlet mapping of the helical eigenvalues. The partial zeta product,

Z (s) = \prod_{j} \frac{1}{1 - p_{j}^{- s}}, s \in R \{0\},

constitutes the exact grand partition function of any finite subsystem. Numerical inversion of this product directly recovers the mixture frequency

s

from any experimental compressibility factor

Z_{m i x}

. Mole fractions

x_{i} (s)

, interaction parameters

Δ (x_{i})

, and the Lyapunov spectrum

λ (x_{i})

then follow deductively via the helical transfer matrix and the closed-form linear ODE for

Δ

. Occupation numbers

N (x_{i})

attain sharp maxima precisely at Fibonacci ratios

F_{r} / F_{r + 1}

, leading to the molecular prime-ID rule. For twelve representative purely binary (irreducible) systems spanning atomic noble gases, simple diatomics, polar molecules, and an aromatic ring, the residuals satisfy

| Z (s) - Z_{m i x} | < 1.5 \times 10^{- 8}

. The resulting

λ (x_{i})

curves accurately reproduce critical points, liquid ranges, and thermodynamic anomalies with zero adjustable parameters. The Riemann Hypothesis follows rigorously as a theorem: the unique fixed point of the duality functor

s \mapsto 1 - s

that preserves the orthogonality condition

\sum {c o s}^{2} θ^{k} = 1

is

R e (s) = 1 / 2

, enforced by Axiom 1 concavity and Axiom 3 irreducibility. The framework is fully deductive and parameter-free and extends naturally to arbitrary mixtures and multiplicities through the helical representation graph. It provides a variational unification of analytic number theory, spectral geometry, thermodynamic phase behavior, and the Riemann Hypothesis from first principles. Full article

(This article belongs to the Section Physics)

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29 pages, 4326 KB

Open AccessArticle

Efficient Attribute Reduction Algorithms Using Discernibility Attributes for Hierarchical Classification

by Ruilin Wei, Nan Zhang and Yuxuan He

Symmetry 2026, 18(4), 609; https://doi.org/10.3390/sym18040609 - 3 Apr 2026

Abstract

Attribute reduction, also referred to as feature selection, focuses on seeking a minimal attribute subset and is an important topic in rough set theory (RST). Classical rough set-based attribute reduction is restricted to decision systems with a single label level and fails to [...] Read more.

Attribute reduction, also referred to as feature selection, focuses on seeking a minimal attribute subset and is an important topic in rough set theory (RST). Classical rough set-based attribute reduction is restricted to decision systems with a single label level and fails to obtain attribute reducts across hierarchical label levels, leading to low computational efficiency. To tackle this issue, this paper proposes a novel method to derive bidirectional reducts across label levels. We first establish the reduction relationship from the coarse label level to the fine label level by analyzing the relationships among decision classes at different label levels. Correspondingly, we construct the reduction relationship from the fine label level to the coarse label level by investigating the connections between positive regions across different label levels. Based on these relationships, two efficient attribute reduction algorithms are developed, which can rapidly compute a reduct at one label level from the reduct at another level. Experimental results on twelve UCI datasets demonstrate that the proposed algorithms require less running time than the other four methods while still achieving high classification accuracy. Full article

(This article belongs to the Section Mathematics)

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29 pages, 5479 KB

Open AccessArticle

Hybrid Machine Learning for Optimal Design of Piezoelectric Diaphragm Energy Harvesters Using Modified Grey Wolf Optimization

by Nitin Yadav, Govind Vashishtha, Sumika Chauhan and Rajesh Kumar

Symmetry 2026, 18(4), 608; https://doi.org/10.3390/sym18040608 - 3 Apr 2026

Abstract

This study addresses the critical need for sustainable energy by optimizing diaphragm-type piezoelectric elements for efficient waste vibration energy harvesting. Traditional experimental optimization of complex, non-linear design parameters including applied load, tapper diameter, and support structures is often resource-intensive and time-consuming. To overcome [...] Read more.

This study addresses the critical need for sustainable energy by optimizing diaphragm-type piezoelectric elements for efficient waste vibration energy harvesting. Traditional experimental optimization of complex, non-linear design parameters including applied load, tapper diameter, and support structures is often resource-intensive and time-consuming. To overcome these limitations, we developed a novel hybrid machine learning framework that seamlessly integrates an Artificial Neural Network (ANN) with a Modified Grey Wolf Optimization (mGWO) algorithm. The ANN was rigorously trained on experimental data using Bayesian Regularization, establishing itself as a robust and high-fidelity surrogate model capable of accurately predicting voltage output based on diverse input parameters, evidenced by an R-value close to 1. This predictive model subsequently served as the fitness function for the mGWO algorithm, which incorporated a non-linear control parameter to efficiently explore the multi-dimensional design space and effectively balance exploration with exploitation. The framework successfully identified the optimal configuration for maximizing voltage output, predicting a theoretical maximum of approximately 70.67 V. This optimal setup notably involved a high applied load of 100 N, the 6CA multi-pointed tapper configuration, and the three-support boundary condition, which is consistent with the experimentally validated results. The computational findings demonstrated excellent agreement with empirical results while providing significantly higher resolution for design insights. This validated, predictive tool offers a substantial advancement for the future scaling and design optimization of piezoelectric energy harvesters, minimizing the need for extensive physical prototyping and ensuring efficient stress transfer without mechanical failure. Full article

(This article belongs to the Special Issue Symmetries in Machine Learning and Artificial Intelligence)

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