Symmetry

21 pages, 2108 KiB

Open AccessArticle

A Police Booth Planning Method Based on Wolf Pack Optimization Algorithm Using AAF and DGSS

by Dongxing Wang, Zhishu Sun and Fangbo Wu

Symmetry 2025, 17(5), 692; https://doi.org/10.3390/sym17050692 (registering DOI) - 30 Apr 2025

Efficient police booth deployment is vital for optimizing law enforcement and maintaining public safety. This paper tackles two key challenges in current solutions: insufficient coverage and redundant overlaps. First, this article introduces the Simultaneous Optimization for Max Coverage and Min Overlap model (SOOM-MCMO), [...] Read more.

Efficient police booth deployment is vital for optimizing law enforcement and maintaining public safety. This paper tackles two key challenges in current solutions: insufficient coverage and redundant overlaps. First, this article introduces the Simultaneous Optimization for Max Coverage and Min Overlap model (SOOM-MCMO), which formalizes the dual objectives into a unified optimization framework. Second, to address limitations of traditional wolf pack algorithm, this article presents the Adaptive-Approaching Framework with Dynamic-Grid-Siege Wolf Pack Algorithm (AAF-DGS-WPOA). This technique dynamically adjusts searcher populations using spatial symmetry and hybrid optimization strategies, thereby enhancing coverage precision and reducing computational costs. Our proposed method (AAF-DGS-WPOA-based Police Booth Planning Method, PBPM-AAFDGS-WPOA) was evaluated on 20 public datasets as well as SOOM-MCMO. Results showed 15–30% coverage improvement and a 16.65% runtime reduction versus popular benchmarks like PSO, GA, and WDX_WPOA. The improved wolf pack algorithm also outperformed traditional approaches in coverage sufficiency, resource efficiency, and system responsiveness. This work advances practical methods for police booth planning, achieving higher social security outputs with lower resource investment. Full article

(This article belongs to the Section Computer)

17 pages, 13196 KiB

Open AccessArticle

Dual Graph Laplacian RPCA Method for Face Recognition Based on Anchor Points

by Shu-Ting Zhuang, Qing-Wen Wang and Jiang-Feng Chen

Symmetry 2025, 17(5), 691; https://doi.org/10.3390/sym17050691 (registering DOI) - 30 Apr 2025

Abstract

High-dimensional data often contain noise and undancy, which can significantly undermine the performance of machine learning. To address this challenge, we propose an advanced robust principal component analysis (RPCA) model that integrates bidirectional graph Laplacian constraints along with the anchor point technique. This [...] Read more.

High-dimensional data often contain noise and undancy, which can significantly undermine the performance of machine learning. To address this challenge, we propose an advanced robust principal component analysis (RPCA) model that integrates bidirectional graph Laplacian constraints along with the anchor point technique. This approach constructs two graphs from both the sample and feature perspectives for a more comprehensive capture of the underlying data structure. Moreover, the anchor point technique serves to substantially reduce computational complexity, making the model more efficient and scalable. Comprehensive evaluations on both GTdatabase and VGG Face2 dataset confirm that anchor-based methods maintain competitive accuracy with standard graph Laplacian approaches (within 0.5–2.0% difference) while achieving significant computational speedups of 5.7–27.1% and 12.9–14.6% respectively. The consistent performance across datasets, from controlled laboratory conditions to challenging real-world scenarios, demonstrates the robustness and scalability of the proposed anchor technique. Full article

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

14 pages, 251 KiB

Open AccessArticle

On the Duality of Codes over Non-Unital Commutative Ring of Order p²

by Tamador Alihia

Symmetry 2025, 17(5), 690; https://doi.org/10.3390/sym17050690 (registering DOI) - 30 Apr 2025

Abstract

This paper establishes an extended theoretical framework centered on the duality of codes constructed over a special class of non-unital, commutative, local rings of order

p^{2}

, where p is a prime satisfying

p \equiv 1 mod 4

or [...] Read more.

This paper establishes an extended theoretical framework centered on the duality of codes constructed over a special class of non-unital, commutative, local rings of order

p^{2}

, where p is a prime satisfying

p \equiv 1 mod 4

or

p \equiv 3 mod 4

. The work expands the traditional scope of coding theory by developing and adapting a generalized recursive approach to produce quasi-self-dual and self-dual codes within this algebraic setting. While the method for code generation is rooted in the classical build-up technique, the primary focus is on the duality properties of the resulting codes—especially how these properties manifest under different congruence conditions on p. Computational examples are provided to illustrate the effectiveness of the proposed methods. Full article

(This article belongs to the Section Mathematics)

21 pages, 6959 KiB

Open AccessArticle

Multi-Domain Digital Twin and Real-Time Performance Optimization for Marine Steam Turbines

by Yuhui Liu, Duansen Shangguan, Liping Chen, Xiaoyan Liu, Guihao Yin and Gang Li

Symmetry 2025, 17(5), 689; https://doi.org/10.3390/sym17050689 (registering DOI) - 30 Apr 2025

Abstract

The digital twin model, which serves as a virtual counterpart symmetric to the physical entity, enables high-fidelity simulation and real-time monitoring. However, digital twin implementation for marine steam turbines (MSTs) faces dual multi-domain simulation fidelity and computational efficiency challenges. This study establishes a [...] Read more.

The digital twin model, which serves as a virtual counterpart symmetric to the physical entity, enables high-fidelity simulation and real-time monitoring. However, digital twin implementation for marine steam turbines (MSTs) faces dual multi-domain simulation fidelity and computational efficiency challenges. This study establishes a MST digital twin modeling methodology through two interconnected innovations: (1) a Modelica-based modular architecture enabling cross-domain coupling across mechanical, thermodynamic, and hydrodynamic systems via hierarchical decomposition, ensuring bidirectional symmetry between physical components and their virtual representations; and (2) a hybrid support vector regression-bidirectional long short-term memory (SVR-BiLSTM) surrogate model combining Gaussian radial basis function-supported SVR for steady-state mapping with Bi-LSTM networks for dynamic error compensation. Experimental validation demonstrates: (a) the SVR component achieves <1.57% absolute error under step-load conditions with 85% computational time reduction versus physics-based models; and (b) Bi-LSTM integration improves transient prediction accuracy by 14.85% in maximum absolute error compared to standalone SVR, effectively resolving static–dynamic discrepancies in telemetry simulation. This dual-approach innovation successfully bridges the critical trade-off between real-time computation and predictive accuracy while maintaining symmetric consistency between the physical turbine and its digital counterpart, providing a validated technical foundation for the intelligent operation and maintenance of MSTs. Full article

(This article belongs to the Section Engineering and Materials)

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20 pages, 29506 KiB

Open AccessArticle

On the Robustness of Individual Tree Segmentation to Data Adversarial Attacks from Remote Sensing Point Clouds

by Renhao Shen, Yongwei Miao and Haijian Liu

Symmetry 2025, 17(5), 688; https://doi.org/10.3390/sym17050688 (registering DOI) - 30 Apr 2025

Abstract

Forests play a vital role in maintaining ecological balance, making accurate forest monitoring technologies essential. Remote sensing point cloud data always capture distinctive geometric features of forests, including the cylindrical symmetry of tree trunks and the radial symmetry of canopies. However, the inherent [...] Read more.

Forests play a vital role in maintaining ecological balance, making accurate forest monitoring technologies essential. Remote sensing point cloud data always capture distinctive geometric features of forests, including the cylindrical symmetry of tree trunks and the radial symmetry of canopies. However, the inherent complexity of point cloud data, combined with their vulnerabilities to adversarial attacks, often disrupts these symmetrical patterns, significantly limiting the practical application of deep learning models in forest monitoring. This research presents a novel approach to enhance the robustness of individual tree segmentation networks by combining data augmentation and adversarial training techniques. Our method employs the FGSM algorithm and Gaussian noise attack to generate adversarial samples while utilizing data denoising and controlled noise injection for data augmentation. A dynamic adversarial training framework can adaptively adjust the proportion of adversarial samples during the network training stage to optimize the model. Using remote sensing point cloud datasets from Wisconsin, the experimental results demonstrate the effectiveness of the individual tree segmentation networks, PointNet++ and DBSCAN, in reducing attack success rates whilst improving the stability and accuracy of segmentation results under various adversarial conditions. This study highlights the potential for more robust forest monitoring systems capable of maintaining accuracy even when faced with data perturbations or intentional interference. Full article

(This article belongs to the Section Computer)

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13 pages, 856 KiB

Open AccessArticle

Shape Transition and Coexistence in ⁶⁶Se Studied with Phenomenological and Microscopic Models

by Petricǎ Buganu, Sara Chafik, Alaaeddine Lahbas and Mustapha Oulne

Symmetry 2025, 17(5), 687; https://doi.org/10.3390/sym17050687 (registering DOI) - 30 Apr 2025

Abstract

A comprehensive theoretical investigation of shape coexistence and transition phenomena in the neutron-deficient nucleus

{}^{66}{Se}

, using complementary microscopic and phenomenological approaches, is presented. The analysis employs the Covariant Density Functional Theory with the Density-Dependent Meson Exchange Model interaction to map the [...] Read more.

A comprehensive theoretical investigation of shape coexistence and transition phenomena in the neutron-deficient nucleus

{}^{66}{Se}

, using complementary microscopic and phenomenological approaches, is presented. The analysis employs the Covariant Density Functional Theory with the Density-Dependent Meson Exchange Model interaction to map the potential energy surface. This microscopic foundation is complemented by calculations using the Bohr–Mottelson Hamiltonian with a sextic oscillator potential, specifically adapted to explore shape coexistence between spherical and

γ

-unstable configurations. The latter model successfully reproduces the experimental energy spectrum, including the critical low-lying

0_{2}^{+}

state at 1226 keV—a key signature of shape coexistence. An analysis of probability density distributions indicates a distinctive manifestation of shape coexistence wherein different shapes exist without significant mixing in the states. These findings provide crucial insights into the structural dynamics of

{}^{66}{Se}

and establish it as an important case study for understanding shape evolution in neutron-deficient nuclei beyond the

N = Z

line. Full article

(This article belongs to the Special Issue Advances in Nuclear Physics and Symmetry)

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20 pages, 2822 KiB

Open AccessArticle

Multivariate Time-Series Missing Data Imputation with Convolutional Transformer Model

by Yanxia Wang, He Ding and Hongdun Li

Symmetry 2025, 17(5), 686; https://doi.org/10.3390/sym17050686 (registering DOI) - 30 Apr 2025

Abstract

The rapid progress in artificial intelligence technologies has significantly impacted the global economy, driving transformative changes in manufacturing and giving rise to intelligent manufacturing. In this context, multivariable time-series data have become an essential resource for modern industries. This paper introduces the Point [...] Read more.

The rapid progress in artificial intelligence technologies has significantly impacted the global economy, driving transformative changes in manufacturing and giving rise to intelligent manufacturing. In this context, multivariable time-series data have become an essential resource for modern industries. This paper introduces the Point Energy Technology, an advanced system for energy monitoring and data acquisition developed by our team. The system has been successfully deployed with several industrial partners, including a combined heat and power system in a local industrial park. Despite its capabilities, data loss remains a persistent issue, which is often caused by measurement or transmission errors during the data collection and transfer stages. These errors result in the loss of vital data samples for effective process monitoring and control. To tackle this issue, we present a convolutional transformer imputation model that is based on self-attention to generate missing data samples. This model effectively captures both historical and future sequence information through an enhanced masking mechanism while also incorporating local dependency information through the symmetrically balanced use of convolution and self-attention. To evaluate the performance of the proposed model against classical models, the energy-related data from a local industrial park were used in this experiment. Considering the real-world conditions, the missing data were categorized into two types: continuous missing and random missing. The experimental results demonstrate that our model produced high-quality data samples, effectively compensating for gaps in the multivariable time-series data. Full article

(This article belongs to the Section Engineering and Materials)

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18 pages, 25119 KiB

Open AccessArticle

Dynamics, Circuit Simulation and Fixed-Time Projection Synchronization in a Memristor-Based Hyperchaotic System

by Yan Zhou, Ruimei Li and Zhuang Cui

Symmetry 2025, 17(5), 685; https://doi.org/10.3390/sym17050685 (registering DOI) - 29 Apr 2025

Abstract

Introducing a memristor into chaotic systems facilitates the construction of high-dimensional hyperchaotic systems. The hyperchaotic signals generated by these systems have important applications in the field of information security. A five-dimensional hyperchaotic system is constructed by introducing a memristor. Its hyperchaotic nature is [...] Read more.

Introducing a memristor into chaotic systems facilitates the construction of high-dimensional hyperchaotic systems. The hyperchaotic signals generated by these systems have important applications in the field of information security. A five-dimensional hyperchaotic system is constructed by introducing a memristor. Its hyperchaotic nature is discussed using phase diagrams and Lyapunov exponential diagrams. The effects of the parameters on the dynamical behavior are examined by bifurcation diagrams and Lyapunov exponential diagrams. To validate the theoretical model, an electronic circuit was designed for circuit simulation. The electronic simulation of the circuit was carried out using the Multisim simulation platform. Finally, the fixed-time projection synchronization of the system was taken into consideration. Three sets of synchronization schemes were considered and simulated. The synchronization scheme has the features of fast synchronization speed and robustness. It is potentially valuable for applications in the fields of chaotic communication and chaotic encryption. Full article

(This article belongs to the Special Issue Symmetry in Complex System and Network Science)

13 pages, 309 KiB

Open AccessArticle

Factorizations and Accurate Computations with Min and Max Matrices

by Yasmina Khiar, Esmeralda Mainar and Eduardo Royo-Amondarain

Symmetry 2025, 17(5), 684; https://doi.org/10.3390/sym17050684 (registering DOI) - 29 Apr 2025

Abstract

Min and max matrices are structured matrices that appear in diverse mathematical and computational applications. Their inherent structures facilitate highly accurate numerical solutions to algebraic problems. In this research, the total positivity of generalized Min and Max matrices is characterized, and their bidiagonal [...] Read more.

Min and max matrices are structured matrices that appear in diverse mathematical and computational applications. Their inherent structures facilitate highly accurate numerical solutions to algebraic problems. In this research, the total positivity of generalized Min and Max matrices is characterized, and their bidiagonal factorizations are derived. It is also demonstrated that these decompositions can be computed with high relative accuracy (HRA), enabling the precise computations of eigenvalues and singular values and the solution of linear systems. Notably, the discussed approach achieves relative errors on the order of the unit roundoff, even for large and ill-conditioned matrices. To illustrate the exceptional accuracy of this method, numerical experiments on quantum extensions of Min and L-Hilbert matrices are presented, showcasing their superior precisions compared to those of standard computational techniques. Full article

(This article belongs to the Section Mathematics)

19 pages, 4317 KiB

Open AccessArticle

Stochastic Programming-Based Annual Peak-Regulation Potential Assessing Method for Virtual Power Plants

by Yayun Qu, Chang Liu, Xiangrui Tong and Yiheng Xie

Symmetry 2025, 17(5), 683; https://doi.org/10.3390/sym17050683 (registering DOI) - 29 Apr 2025

Abstract

The intervention of distributed loads, propelled by the swift advancement of distributed energy sources and the escalating demand for diverse load types encompassing electricity and cooling within virtual power plants (VPPs), has exerted an influence on the symmetry of the grid. Consequently, a [...] Read more.

The intervention of distributed loads, propelled by the swift advancement of distributed energy sources and the escalating demand for diverse load types encompassing electricity and cooling within virtual power plants (VPPs), has exerted an influence on the symmetry of the grid. Consequently, a quantitative assessment of the annual peak-shaving capability of a VPP is instrumental in mitigating the peak-to-valley difference in the grid, enhancing the operational safety of the grid, and reducing grid asymmetry. This paper presents a peak-shaving optimization method for VPPs, which takes into account renewable energy uncertainty and flexible load demand response. Firstly, wind power (WP), photovoltaic (PV) generation, and demand-side response (DR) are integrated into the VPP framework. Uncertainties related to WP and PV generation are incorporated through the scenario method within deterministic constraints. Secondly, a stochastic programming (SP) model is established for the VPP, with the objective of maximizing the peak-regulation effect and minimizing electricity loss for demand-side users. The case study results indicate that the proposed model effectively tackles peak-regulation optimization across diverse new energy output scenarios and accurately assesses the peak-regulation potential of the power system. Specifically, the proportion of load decrease during peak hours is 18.61%, while the proportion of load increase during off-peak hours is 17.92%. The electricity loss degrees for users are merely 0.209 in summer and 0.167 in winter, respectively. Full article

(This article belongs to the Special Issue Symmetry in Digitalisation of Distribution Power System)

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21 pages, 3055 KiB

Open AccessArticle

Integrated Scheduling Algorithm Based on the Improved Floyd Algorithm

by Yingxin Wei, Wei Zhou, Zhiqiang Xie, Ming Sun, Zhenjiang Tan and Wangcheng Cao

Symmetry 2025, 17(5), 682; https://doi.org/10.3390/sym17050682 (registering DOI) - 29 Apr 2025

Abstract

In the research and practice of integrated scheduling problems, the tree structure of complex products usually presents an asymmetric and complex form. This asymmetry is mainly reflected in the hierarchical relationship between the various components of the product, the degree of dependence, and [...] Read more.

In the research and practice of integrated scheduling problems, the tree structure of complex products usually presents an asymmetric and complex form. This asymmetry is mainly reflected in the hierarchical relationship between the various components of the product, the degree of dependence, and the sequence of production processes. Existing studies often neglect that leaf nodes with the lowest layer priority can be scheduled at any moment, leading to underutilization of parallelism potential under symmetric structures and exacerbation of critical path delays under asymmetric structures. Aiming at solving this kind of problem, an integrated scheduling algorithm based on the improved Floyd algorithm (ISA-IFA) is proposed. According to the improved Floyd algorithm, the algorithm proposed a path-weighted strategy, which constructs the vertical path value according to the processing time of the process itself. Combined with the proposed process scheduling advantage strategy, the leaf node process is especially emphasized as the priority scheduling object, which makes the connection between the processes more closely, and then significantly reduces the idle time of the equipment. The empirical results show that the ISA-IFA algorithm shortens the completion time of complex products and simultaneously improves the equipment utilization rate to 55.9%, verifying its effectiveness in dynamic scheduling and resource co-optimization. Full article

(This article belongs to the Section Mathematics)

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46 pages, 1618 KiB

Open AccessReview

Electroweak Form Factors of Baryons in Dense Nuclear Matter

by G. Ramalho, K. Tsushima and Myung-Ki Cheoun

Symmetry 2025, 17(5), 681; https://doi.org/10.3390/sym17050681 (registering DOI) - 29 Apr 2025

Abstract

There is evidence that the properties of hadrons are modified in a nuclear medium. Information about the medium modifications of the internal structure of hadrons is fundamental for the study of dense nuclear matter and high-energy processes, including heavy-ion and nucleus–nucleus collisions. At [...] Read more.

There is evidence that the properties of hadrons are modified in a nuclear medium. Information about the medium modifications of the internal structure of hadrons is fundamental for the study of dense nuclear matter and high-energy processes, including heavy-ion and nucleus–nucleus collisions. At the moment, however, empirical information about medium modifications of hadrons is limited; therefore, theoretical studies are essential for progress in the field. In the present work, we review theoretical studies of the electromagnetic and axial form factors of octet baryons in symmetric nuclear matter. The calculations are based on a model that takes into account the degrees of freedom revealed in experimental studies of low and intermediate square transfer momentum

q^{2} = - Q^{2}

: valence quarks and meson cloud excitations of baryon cores. The formalism combines a covariant constituent quark model, developed for a free space (vacuum) with the quark–meson coupling model for extension to the nuclear medium. We conclude that the nuclear medium modifies the baryon properties differently according to the flavor content of the baryons and the medium density. The effects of the medium increase with density and are stronger (quenched or enhanced) for light baryons than for heavy baryons. In particular, the in-medium neutrino–nucleon and antineutrino–nucleon cross-sections are reduced compared to the values in free space. The proposed formalism can be extended to densities above the normal nuclear density and applied to neutrino–hyperon and antineutrino–hyperon scattering in dense nuclear matter. Full article

(This article belongs to the Special Issue Chiral Symmetry, and Restoration in Nuclear Dense Matter)

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15 pages, 320 KiB

Open AccessArticle

From Axion—Neutrino Couplings to Axion Thermodynamics: Testing the Axion Mass Hierarchy

by Osvaldo Civitarese, Milva G. Orsaria and Ana V. Penacchioni

Symmetry 2025, 17(5), 680; https://doi.org/10.3390/sym17050680 (registering DOI) - 29 Apr 2025

Abstract

The composition and physical state of dark matter remain among the most pressing unresolved questions in modern physics. Addressing these questions is crucial to our understanding of the Universe’s structure. In this work, we explore the hypothesis that massive scalar bosons, such as [...] Read more.

The composition and physical state of dark matter remain among the most pressing unresolved questions in modern physics. Addressing these questions is crucial to our understanding of the Universe’s structure. In this work, we explore the hypothesis that massive scalar bosons, such as axions, constitute the majority of dark matter. We focus on two key aspects of axion physics: (i) the role of axion–neutrino coupling in generating neutrino mass and (ii) the thermodynamic properties of axion dark matter. We propose that the interaction between neutrinos and axions in the early Universe, prior to hadronic formation, could provide a mechanism for finite neutrino masses. Furthermore, to account for the observed large-scale distribution of dark matter, we extend the Bose–Einstein condensation framework and derive the critical temperature

T_{c}

that defines the onset of the condensate phase. Our calculations suggest that this temperature ranges from a few

10^{- 3}

degrees Kelvin to approximately one Kelvin, depending on the axion scale factor

f_{a}

. These findings support the plausibility of axions as viable dark matter candidates and emphasize the importance of future experimental searches for axion–neutrino interactions. Additional astrophysical and laboratory investigations could further refine axion mass constraints and shed light on the role of axion condensates in the evolution of the early Universe. Full article

(This article belongs to the Special Issue Neutrino Physics and Symmetries)

19 pages, 917 KiB

Open AccessArticle

SSRL: A Clustering-Based Reinforcement Learning Approach for Efficient Ship Scheduling in Inland Waterways

by Shaojun Gan, Xin Wang and Hongdun Li

Symmetry 2025, 17(5), 679; https://doi.org/10.3390/sym17050679 (registering DOI) - 29 Apr 2025

Abstract

Efficient ship scheduling in inland waterways is critical for maritime transportation safety and economic viability. However, traditional scheduling methods, primarily based on First Come First Served (FCFS) principles, often produce suboptimal results due to their inability to account for complex spatial–temporal dependencies, directional [...] Read more.

Efficient ship scheduling in inland waterways is critical for maritime transportation safety and economic viability. However, traditional scheduling methods, primarily based on First Come First Served (FCFS) principles, often produce suboptimal results due to their inability to account for complex spatial–temporal dependencies, directional asymmetries, and varying ship characteristics. This paper introduces SSRL (Ship Scheduling through Reinforcement Learning), a novel framework that addresses these limitations by integrating three complementary components: (1) a Q-learning framework that discovers optimal scheduling policies through environmental interaction rather than predefined rules; (2) a clustering mechanism that reduces the high-dimensional state space by grouping similar ship states; and (3) a sliding window approach that decomposes the scheduling problem into manageable subproblems, enabling real-time decision-making. We evaluated SSRL through extensive experiments using both simulated scenarios and real-world data from the Xiaziliang Restricted Waterway in China. Results demonstrate that SSRL reduces total ship waiting time by 90.6% compared with TSRS, 48.4% compared with FAHP-ES, and 32.6% compared with OSS-SW, with an average reduction of 57.2% across these baseline methods. SSRL maintains superior performance across varying traffic densities and uncertainty conditions, with the optimal information window length of 13–14 ships providing the best balance between solution quality and computational efficiency. Beyond performance improvements, SSRL offers significant practical advantages: it requires minimal computation for online implementation, adapts to dynamic maritime environments without manual reconfiguration, and can potentially be extended to other complex transportation scheduling domains. Full article

(This article belongs to the Section Engineering and Materials)

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19 pages, 308 KiB

Open AccessArticle

Some Characterizations of k-Fuzzy

γ

-Open Sets and Fuzzy

γ

-Continuity with Further Selected Topics

by Fahad Alsharari, Hind Y. Saleh and Islam M. Taha

Symmetry 2025, 17(5), 678; https://doi.org/10.3390/sym17050678 (registering DOI) - 29 Apr 2025

Abstract

In the present paper, we first introduced the notion of k-fuzzy

γ

-open (k-

F

-

γ

-open) sets as a generalized novel class of fuzzy open (

F

-open) sets on fuzzy topological spaces (

FTS s

) in [...] Read more.

In the present paper, we first introduced the notion of k-fuzzy

γ

-open (k-

F

-

γ

-open) sets as a generalized novel class of fuzzy open (

F

-open) sets on fuzzy topological spaces (

FTS s

) in the sense of Šostak. The class of k-

F

-

γ

-open sets is contained in the class of k-

F

-

β

-open sets and contains all k-

F

-semi-open and k-

F

-pre-open sets. Also, we introduced the closure and interior operators with respect to the classes of k-

F

-

γ

-closed and k-

F

-

γ

-open sets and discussed some of their properties. After that, we defined and studied the notions of

F

-

γ

-continuous (resp.

F

-

γ

-irresolute) functions between

FTS s

(M, ℑ)

and

(N, Ϝ)

. However, we displayed and investigated the notions of

F

-almost (resp.

F

-weakly)

γ

-continuous functions, which are weaker forms of

F

-

γ

-continuous functions. Next, we presented and characterized some new

F

-functions via k-

F

-

γ

-open and k-

F

-

γ

-closed sets, called

F

-

γ

-open (resp.

F

-

γ

-irresolute open,

F

-

γ

-closed,

F

-

γ

-irresolute closed, and

F

-

γ

-irresolute homeomorphism) functions. The relationships between these classes of functions were investigated with the help of some examples. We also introduced some new types of

F

-separation axioms called k-

F

-

γ

-regular (resp. k-

F

-

γ

-normal) spaces via k-

F

-

γ

-closed sets and discussed some properties of them. Lastly, we explored and studied some new types of

F

-compactness called k-

F

-almost (resp. k-

F

-nearly)

γ

-compact sets. Full article

(This article belongs to the Special Issue Research on Fuzzy Logic and Mathematics with Applications, 3rd Edition)

40 pages, 1068 KiB

Open AccessArticle

Tableau with Holes: Clarifying NP-Completeness

by Edgar Graham Daylight

Symmetry 2025, 17(5), 677; https://doi.org/10.3390/sym17050677 (registering DOI) - 29 Apr 2025

Abstract

In the context of defining

NP

-completeness, a tableau represents a hypothetical accepting computation path p of a nondeterministic polynomial time Turing machine N on an input w. The tableau is encoded by the propositional logic formula

ψ

, defined as [...] Read more.

In the context of defining

NP

-completeness, a tableau represents a hypothetical accepting computation path p of a nondeterministic polynomial time Turing machine N on an input w. The tableau is encoded by the propositional logic formula

ψ

, defined as

ψ = ψ_{c e l l} \land ψ_{r e s t}

. The component

ψ_{c e l l}

enforces the constraint that each cell in the tableau contains exactly one symbol, while

ψ_{r e s t}

incorporates constraints governing the step-by-step behavior of N on w. Intuitively,

ψ_{r e s t}

appears to pose a much greater challenge for satisfiability. This raises the question of whether the distinction between

ψ_{c e l l}

being a 3cnf formula, rather than a cheap 2cnf formula, actually matters. We show that if, hypothetically,

ψ_{r e s t}

can be succinctly represented as a Horn formula, then satisfying

ψ

can be achieved efficiently in

K^{f (n, k)}

steps, where N operates within

O (n^{k})

steps and both k and K are constants. Asymptotically,

f (n, k) \approx n^{\frac{2}{3} k}

. Our method has the potential for iterative application. Technically, we trim

ψ_{c e l l}

down to a 2cnf–Horn formula, whose satisfiability allows for empty cells, or “holes”, in the tableau. This modified tableau represents exponentially many paths of N on w, rather than a single accepting path p. While a tableau with holes conceptualizes the satisfiability of

ψ_{t r i m}

—a trimmed-down version of

ψ

—it does not directly address the satisfiability of

ψ

. Therefore, we introduce an external user who efficiently employs backtracking to fill in specific holes, ultimately verifying the satisfiability of the original

ψ

. Full article

(This article belongs to the Special Issue Symmetry in Solving NP-Hard Problems)

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19 pages, 4454 KiB

Open AccessArticle

Continuous Maximum Coverage Location Problem with Arbitrary Shape of Service Areas and Regional Demand

by Sergiy Yakovlev, Sergiy Shekhovtsov, Lyudmyla Kirichenko, Olha Matsyi, Dmytro Podzeha and Dmytro Chumachenko

Symmetry 2025, 17(5), 676; https://doi.org/10.3390/sym17050676 (registering DOI) - 29 Apr 2025

Abstract

This paper addresses the maximum coverage location problem in a generalized setting, where both facilities (service areas) and regional demand are modeled as continuous entities. Unlike traditional formulations, our approach allows for arbitrary shapes for both service areas and demand regions, with additional [...] Read more.

This paper addresses the maximum coverage location problem in a generalized setting, where both facilities (service areas) and regional demand are modeled as continuous entities. Unlike traditional formulations, our approach allows for arbitrary shapes for both service areas and demand regions, with additional constraints on facility placement. The key novelty of this work is its ability to handle complex, irregularly shaped service areas, including approximating them as unions of centrally symmetric shapes. This enables the use of an analytical approach based on spatial symmetry, which allows for efficient estimation of the covered area. The problem is formulated as a nonlinear optimization task. We analyze the properties of the objective function and leverage the Shapely library in Python 3.13.3 for efficient geometric computations. To improve computational efficiency, we develop an extended elastic model that significantly reduces processing time. This model generalizes the well-known quasi-physical, quasi-human algorithm for circle packing, extending its applicability to more complex spatial configurations. The effectiveness of the proposed approach is validated through test cases in which service areas take the form of circles, ellipses, and irregular polygons. Our method provides a robust and adaptable solution for various settings of practically interesting continuous maximum coverage location problems involving irregular regional demand and service areas. Full article

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

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13 pages, 1400 KiB

Open AccessArticle

Propagation of Tensor Perturbation in Horndeski-like Gravity

by Fabiano F. Santos and Jackson Levi Said

Symmetry 2025, 17(5), 675; https://doi.org/10.3390/sym17050675 (registering DOI) - 28 Apr 2025

Abstract

Scalar–tensor theories have shown promise in many sectors of cosmology. However, recent constraints from the speed of gravitational waves have put severe limits on the breadth of models such classes of theories can realize. In this work, we explore the possibility of a [...] Read more.

Scalar–tensor theories have shown promise in many sectors of cosmology. However, recent constraints from the speed of gravitational waves have put severe limits on the breadth of models such classes of theories can realize. In this work, we explore the possibility of a Horndeski Lagrangian that is equipped with two dilaton fields. The evolution of a two-dilaton coupled cosmology is not well known in the literature. We explore the tensor perturbations in order to assess the behavior of the model against the speed of the gravitational wave constraint. Our main result is that this model exhibits a class of cosmological theories that is consistent with this observational constraint. Full article

(This article belongs to the Special Issue Gravitational Physics and Symmetry)

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29 pages, 35856 KiB

Open AccessArticle

Symbol Recognition Method for Railway Catenary Layout Drawings Based on Deep Learning

by Qi Sun, Mengxin Zhu, Minzhi Li, Gaoju Li and Weizhi Deng

Symmetry 2025, 17(5), 674; https://doi.org/10.3390/sym17050674 (registering DOI) - 28 Apr 2025

Abstract

Railway catenary layout drawings (RCLDs) have the characteristics of upper and lower symmetry, a large drawing size, a small size, high similarity among target symbols, and an uneven distribution of symbol categories. These factors make the symbol detection task more complex and challenging. [...] Read more.

Railway catenary layout drawings (RCLDs) have the characteristics of upper and lower symmetry, a large drawing size, a small size, high similarity among target symbols, and an uneven distribution of symbol categories. These factors make the symbol detection task more complex and challenging. To address the aforementioned challenges, this paper proposes three enhancements to YOLOv8n to improve symbol detection performance and integrates an improved denoising diffusion probabilistic model (IDDPM) to mitigate the imbalance in symbol category distribution. First, the multi-scale dilated attention (MSDA) is introduced in the Neck part to enhance the model’s perception of the global context in complex RCLD scenes, so that it can more effectively capture the symbol information distributed in different scales and backgrounds. Secondly, the receptive field attention convolution (RFAConv) is used in the detection head to replace the standard convolution, to improve the ability to focus on the target symbols in RCLDs and effectively alleviate the occlusion interference between symbols. Finally, the dynamic upsampler (DySample) is used to enhance the clarity and positioning accuracy of the edge area of small target symbols in RCLDs and enhance the detection of small targets. The above design made targeted optimizations to resolve the problems of symbol and background interference, character overlap, and symbol category imbalances in complex scenes in RCLDs, effectively improving the overall detection performance of the model. Compared with the baseline YOLOv8n model, the improved YOLOv8n achieves increases of 2.9% in F1, 1.9% in mAP@0.5, and 1.7% in mAP@0.5:0.95. With the introduction of synthetic data, the recognition of minority-class symbols is further enhanced, leading to additional gains of 4%, 3.8%, and 14% in F1, mAP@0.5, and mAP@0.5:0.95, respectively. These results demonstrate the effectiveness and superiority of the proposed method in improving detection performance. Full article

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Symmetry

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