Symmetry doi: 10.3390/sym18050790

Authors: Zoulikha Kaid Fatimah A. Almulhim Mohammed B. Alamari

In this paper, we propose a new nonparametric method for estimating the regression operator of a scalar response given a functional covariate taking values in a semi-metric space. The estimator is obtained by minimizing the Least Absolute Relative Error (LARE) criterion, which provides a scale-invariant and equilibrated measure of prediction accuracy compared with classical regression loss functions. The antisymmetry property of the LARE rule ensures that overestimation and underestimation are penalized in a symmetric relative manner, which improves the robustness when the response variable varies in different scales. Next, the estimator is constructed using k-nearest neighbors (kNN). The combination of the two algorithms allows the procedure to benefit from both the robustness and scale-invariant nature of the LARE criterion and the flexibility and local adaptivity of the kNN smoothing approach, which is particularly suitable for functional or high-dimensional data. As an asymptotic result, we establish the uniform convergence with respect to the number of neighbors (UNN) of the proposed estimator under mild regularity conditions and derive its rate of convergence. We also discuss the selection of the optimal number of neighbors and their impact on performance. The practical effectiveness of the proposed kNN–FLARE regression estimator is illustrated through simulation experiments and an application to near-infrared (NIR) spectrometry data.

Symmetry doi: 10.3390/sym18050789

Authors: Jean-Renaud Pycke

The purpose of our paper is to provide a family of bilinear orthogonal expansions all based upon the same general pattern that is valid for a wide class of special functions. Our first family involves Jacobi, Laguerre, and Hermite polynomials. We give a discrete analogue of these bilinear expansions, the three families of classical orthogonal polynomials being replaced by zonal spherical functions associated with regular distance graphs. Such expansions playing a key role in the field of mathematical statistics, we show how our results apply to this field. We provide generalizations of the well-known Cramér–von Mises and Watson’s statistics, based upon an interpretation of their kernel in terms of the circular Laplacian. The product formula, well-known for zonal functions on Lie groups, is stated for distance-regular graphs, providing an elegant tool for proofs. Examples involving Hahn, q-Hahn, and Krawtchouk polynomials are given.

Symmetry doi: 10.3390/sym18050788

Authors: Yihe Feng Wuhui Chen Gengwu Zhang

This paper addresses issues in islanded DC microgrids, including voltage deviation, inaccurate current sharing, and high communication burden, by proposing a distributed secondary control strategy that integrates a dynamic consensus algorithm with an adaptive event-triggered mechanism. Within a hierarchical control framework, the secondary layer employs an improved dynamic consensus algorithm to estimate the average voltage and proportional current through information exchange among neighboring nodes. Corresponding voltage and current compensations are designed to mitigate voltage droop and ensure accurate proportional sharing of load currents. In this study, a 100 V power supply is stepped down to 47.4 V following primary control. Then, by employing the secondary controller with the proposed algorithm, the voltage is precisely restored to the desired value of 48 V. To further reduce the communication burden, a dynamic event-triggered condition is intended for the output current of each power source, enabling communication and control updates only when the state changes significantly. This approach substantially reduces redundant data transmission and the frequency of controller actions. The positions of the triggering points under the action of the event trigger are also illustrated in the corresponding figures in the following sections. The positions of the triggering points under the action of the event trigger are illustrated in the corresponding figures in the following sections. While communication is accomplished, the voltage remains stable at 48 V. Furthermore, the currents of each distributed unit are stabilized around 6.4 A, satisfying the 1:1:1 current-sharing setting. The asymptotic stability of the closed-loop system is proven based on Lyapunov theory, and Zeno behavior is effectively avoided. Simulation results demonstrate that the proposed strategy achieves rapid voltage restoration and high-precision current sharing under scenarios such as load transients and plug-and-play operations while significantly reducing communication frequency and enhancing system economy and reliability.

Symmetry doi: 10.3390/sym18050787

Authors: Ahmed A. Gaber Dalal Alhwikem

The (2+1)-D generalized B-type Kadomtsev–Petviashvili (BKP) equation is studied in this work utilizing Painlevé property, Lie-symmetry method and generalized Kudryashov method (GKM). This study aims to pass the Painlevé test and obtain variant exact solutions for the (2+1)-D BKP equation that occurs in physical dynamics. First, we demonstrated that the governing equation exceeds the Painlevé test by using the Painlevé property. Symmetry analysis is utilized to obtain infinitesimals and vector fields of the BKP equation. The governing equation was converted to several ordinary differential equations (ODEs) using linear combinations of these vectors. GKM is used to generate a novel class of closed-form solutions for the BKP equation. Many random constants and functions were included in the derived solutions to improve their dynamic characteristics. The emergence of solutions was facilitated by the optimal selection of estimates for these elective constants. There are several types of solution behavior, such as a kink wave, solitary wave, anti-kink wave, and single wave.

Symmetry doi: 10.3390/sym18050786

Authors: Xiao Lin Yu Fang

Wireless sensor network (WSN) coverage optimization is a challenging high-dimensional and nonlinear problem that directly affects network performance, including sensing quality, energy efficiency, and system reliability. Although metaheuristic algorithms have been widely applied to this problem, many existing methods still suffer from premature convergence, insufficient population diversity, and an imbalance between exploration and exploitation. To address these issues, this paper proposes a multi-strategy enhanced enterprise development optimization algorithm (MEEDOA) inspired by business management mechanisms. The proposed method integrates a hybrid population initialization strategy, an adaptive activity switching mechanism based on performance feedback, a multi-elite collaborative learning strategy, and a Lévy flight-based stagnation escape mechanism. These strategies are tightly coupled within a unified adaptive framework to improve global search capability, convergence speed, and robustness. Furthermore, a mathematical model for WSN deployment is constructed based on a binary sensing model and discrete coverage evaluation. From the perspective of symmetry, the sensing regions of sensor nodes exhibit significant geometric symmetry in both two-dimensional and three-dimensional deployment spaces. In the two-dimensional case, the sensing and communication regions are modeled as concentric circular structures, while in the three-dimensional case, the sensing regions are represented by isotropic spheres with symmetric spatial distributions. Such symmetry properties provide an effective basis for describing coverage behavior, reducing redundant overlap, and improving the uniformity of node deployment. Meanwhile, the proposed MEEDOA preserves population diversity and enhances search balance, enabling the algorithm to better capture symmetric coverage patterns and more effectively explore complex spatial deployment configurations. Extensive experiments on CEC2014, CEC2017, CEC2020, and CEC2022 benchmark functions demonstrate that MEEDOA achieves superior convergence accuracy, faster convergence speed, and stronger robustness compared with several state-of-the-art algorithms. Additional simulation results in WSN deployment applications verify its effectiveness in improving coverage performance under both symmetric and irregular spatial deployment scenarios. The results indicate that the proposed MEEDOA provides a reliable and efficient solution for complex global optimization problems and practical engineering applications.

Symmetry doi: 10.3390/sym18050785

Authors: Paulraj Gnanachandra Saeid Jafari Ananda Priya Baskar Cheirmaraj Arunthathi

The rhotrix, introduced as a rhomboidal extension of traditional matrix theory, offers a unique algebraic framework for the organization and manipulation of numerical arrays. The incorporation of refined neutrosophic concepts into rhotrix theory has enabled the modeling of uncertainty, indeterminacy, and inconsistency within algebraic frameworks. Motivated by these developments, this paper investigates the structure of invertible refined neutrosophic rhotrices, thereby extending the classical theory of rhotrices into the domain of refined neutrosophic. This study is carried out using heart-based multiplication, which acts as the fundamental operation governing interactions among rhotrix elements. This setting facilitates a rigorous analysis of rhotrix algebra, specifically identifying the conditions required for invertibility and exploring the structural response of refined neutrosophic elements to basic operations defined for rhotrices. In addition to the invertibility analysis, this study analyzes the index structures of rhotrices of small orders, establishes the general characterization of the rhotrix index set, and reveals an inherent symmetry of the rhotrix structure under the interchange of indices. It also proposes a rhotrix-based representation of binary relations, illustrating the broader applicability of refined neutrosophic rhotrices in modeling relational and algebraic structures. To demonstrate the practical relevance of the developed framework, a refined neutrosophic decision-making model is applied to the evaluation of selected women empowerment schemes. By incorporating real-valued performance measures together with indeterminacy components reflecting expert uncertainty, the proposed model provides a systematic mechanism for multi-criteria assessment and ranking of policy interventions. The results contribute to the advancement of rhotrix algebra in indeterminate environments and highlight its potential applications in mathematical modeling and decision-making.

Symmetry doi: 10.3390/sym18050784

Authors: Jianxin Wang Huiwen Xu Chaoen Xiao Lei Zhang Ding Ding

In response to the issues associated with implementing cryptographic algorithms using the traditional hardware description language Verilog Hardware Description Language (Verilog HDL)—namely, high code complexity, long development cycles, and difficult debugging—this paper proposes a hardware implementation scheme for the lightweight block cipher algorithm Midori-128 based on the Chisel language. This scheme adopts an iterative structure and optimizes the algorithm for throughput. The algorithm has been successfully deployed on both Xilinx platforms and the domestically produced PGL100H platform, and a power consumption analysis has been completed. Compared to traditional Verilog implementations, this design delivers a throughput improvement of approximately 42.5% on the Virtex-5 platform and approximately 124.63% on the Spartan-6 platform. It maintains lightweight logic resource utilization and excellent cross-platform adaptability, making it suitable for resource-constrained embedded security applications and providing efficient, independently controllable cryptographic hardware support. Furthermore, the design fully preserves the symmetric cryptographic properties of Midori-128.

Symmetry doi: 10.3390/sym18050783

Authors: Nobuchika Yamaki Tenna Churiki

Hemispheric specialization is a widespread feature of vertebrate nervous systems, but the minimal conditions under which bilateral systems differentiate, acquire polarity, and retain asymmetric states remain unclear. Here, we examined these issues using a minimal evolutionary model with two initially equivalent processing channels. Each channel evolved a spatial integration width while receiving the same input, and fitness rewarded the magnitude of a bilateral mismatch-separation signal rather than explicit anomaly localization. Under exact developmental symmetry, 40 lineages evolved robust left–right differences in integration width without significant directional fixation (median |Δa| = 2.511; 22 right-wider, 18 left-wider). Weak developmental gain asymmetry biased polarity selection in a graded manner, shifting outcomes toward right-wider or left-wider solutions depending on bias direction. Forced-symmetry, shared-parameter, and single-channel controls showed that high performance depended on allowing differentiated bilateral processing. After biased solutions were reseeded under restored symmetry, differentiation was retained and amplified (median |Δa| > 6.6), consistent with history-dependent persistence within the sampled fitness landscape. Structured backgrounds increased differentiation magnitude but imposed greater decision-time costs. These results distinguish differentiation, polarity bias, and persistence as separable components of minimal hemispheric specialization.

Symmetry doi: 10.3390/sym18050782

Authors: Purit Thammasiri Vasile Berinde Somyot Plubtieng Kasamsuk Ungchittrakool Rabian Wangkeeree

This paper introduces a novel inertial forward–backward–forward algorithm driven by a newly conceptualized moving point projection technique for solving monotone inclusion problems in real Hilbert spaces. By leveraging the properties of a Lipschitz continuous, monotone operator and a maximally monotone operator alongside this innovative projection strategy, we dynamically construct a sequence of nonempty, closed, and convex sets that contain the zeros of the sum of the two operators. This geometric construction ensures that the resulting sequence is well defined and guarantees its weak convergence to a solution. Furthermore, to validate the practical efficacy of the proposed theoretical framework, we evaluate our method on image restoration problems. Numerical experiments measuring the improvement in signal-to-noise ratio (ISNR) and the structural similarity index measure (SSIM) confirm that the proposed algorithm is highly efficient and significantly outperforms existing state-of-the-art methods.

Symmetry doi: 10.3390/sym18050781

Authors: Ozcan Dimez Alper Camci Tankut Acarman

Responding to a natural disaster is a short-lived and complex task requiring fast coordination and analytical skills. Responses to natural disasters must be fast, and resources need to be used in an efficient and effective manner. An automated system can significantly reduce human decision errors, increase speed, and lower operational costs. This study presents an automated system that leverages real-time mobile network data to optimize team deployment and coordination in order to repair base station alarms and re-deploy existing cellular communication networks whose core nodes have been damaged or congested. Algorithms presenting solutions from optimization to fast heuristics are adopted for automated assignment of technical teams to alarms as a response to a natural disaster. Algorithms are evaluated and tested for their multi-objective assignment performance, subject to constraints. A real alarm dataset logged from base stations is used. The average number of alarms assigned, assignment rate, average latency of the automated assignment system, and travel distance of the technical teams to the base station are used as the performance metrics. Cellular communication has to be re-deployed to sustain coordination and resilience as an immediate response to a natural disaster. The presented approach and comparative results show that technical teams located in neighboring cities can be assigned an immediate response, and automatic assignment of new arrival alarms with high priority can be assured by minimizing the travel distance of technical teams to the level of a few kilometers. This study fills specific gaps in comparison to prior studies by using the real alarm data logged after a destructive earthquake event, adopting a multi-objective optimization along with the performance metrics used by telecom operators.

Symmetry doi: 10.3390/sym18050780

Authors: Ehab T. Alnfrawy Hazem M. Bahig Hatem M. Bahig Reda Elbarougy

Integer factorization plays a foundational role in asymmetric cryptography systems, notably the Rivest, Shamir, and Adleman (RSA) cryptosystem. This paper presents an improvement of the integer factorization from both sequential and parallel computational perspectives. The algorithm is based on polynomial evaluation and the greatest common divisor. The objectives of these improvements are to decrease the execution time and memory consumption associated with the process of finding prime factors. Experimentally, we use different values of parameters (1) the number of bits n, (2) the difference between two factors, and (3) the number of processors in the parallel model. The experimental results indicate that both proposed methods, sequential and parallel, yield significant improvements regarding running time and memory usage when the difference between the two factors is n1/3 and n1/4. The average improvement observed is 99% in running time, with memory consumption reduced to a constant. This characteristic is important for limited hardware devices. Furthermore, the proposed parallel method demonstrates scalability and achieves sublinear speedup.

Symmetry doi: 10.3390/sym18050778

Authors: Qiongqi Fan Deshun Zheng Fengbo Sun Yi Li Ting Li

The Paleoproterozoic represents a pivotal but poorly constrained interval in the tidal evolution of the Earth–Moon system. Quantifying Earth–Moon orbital parameters over this interval is fundamental for predicting the long-term dynamical evolution of the Earth–Moon system, yet direct geological evidence remains scarce. In this study, we conducted cyclostratigraphic analyses of the middle and upper members of the Dagushi Formation on the southern margin of the North China Craton, using high-resolution magnetic susceptibility (MS) and phosphorus (P) data as paleoclimate proxies. By employing two independent astrochronologic approaches—the main obliquity estimation method (k+s3) and Bayesian inversion (TimeOptMCMC)—we reconstructed key parameters of the Earth–Moon system, including the precession constant k, Earth–Moon distance, and length of day (LOD). The k+s3 approach yields k=98.12±1.07 arcsec/yr, from which the Earth–Moon distance is derived as 329,732 (+888/−877) km, with a LOD of 17.69±0.08 h. In contrast, the TimeOptMCMC method produces k=95.20±1.68 arcsec/yr, implying an Earth–Moon distance of 331,292 (+1446/−1418) km, with a LOD of 17.91±0.13 h. The MS and P indicators exhibit remarkable symmetry and phase synchronicity between their curves in both depth and time domains, serving as a robust indicator of stable sedimentation and primary depositional signals. These results provide direct geological constraints on Earth–Moon system parameters at ∼1787 Ma, contributing to a refined understanding of its tidal evolution during the Paleoproterozoic.

Symmetry doi: 10.3390/sym18050779

Authors: Bowen Yang Wenying He

Disease staging is a critical component of clinical diagnosis, treatment, and prognosis assessment. However, structured clinical data typically exhibit high-dimensional, nonlinear feature interactions; stage-specific dominant features; and threshold-based discontinuities. These characteristics make it challenging for a single model to achieve both global feature modeling capability and local discriminative power, thereby limiting further improvements in prediction accuracy. To address this limitation, we propose a novel deep ensemble learning framework, ETGB-SEF (Entmax-TabNet Gradient Boosting Stacked Ensemble Framework), for multiclass disease staging. First, at the base model level, Entmax-1.5 replaces Sparsemax in TabNet, thereby enabling an adjustable sparse feature selection mechanism that enhances the ability to model weakly correlated clinical features while preserving interpretability. Second, at the model-fusion level, a stacked ensemble architecture in the probability space is developed. This architecture integrates the modified TabNet with Gradient Boosting Decision Trees (GBDT) in a complementary way, enabling the former to capture global nonlinear semantic dependencies while the latter captures threshold-based discriminative boundaries among clinical features. Extensive experiments on real-world datasets demonstrate that the proposed method consistently outperforms existing state-of-the-art approaches.

Symmetry doi: 10.3390/sym18050776

Authors: Dezhi Deng Wenyi Cao Ziyou Wang

In the context of intensified trade frictions and frequent financial market fluctuations, assessing the risk resilience of strategic emerging industries holds significant strategic value. Based on quarterly data from 2010 to 2025, this study empirically examines the time-varying and asymmetric shock effects of trade policy uncertainty and financial stress on the profitability of China’s strategic emerging industries using the TVP-VAR-SV model. The study finds that China’s strategic emerging industries exhibit significant asymmetric resilience differences when facing different external shocks, specifically demonstrating stronger trade resilience and weaker financial resilience. The shocks brought by trade uncertainty typically show short-term pain followed by rapid recovery, with the negative impact being largely eliminated within two quarters and subsequently turning into positive growth, reflecting outstanding recovery capability. In contrast, the impact of financial stress on corporate profitability has a profound long-tail effect, with negative disruptions often persisting for more than two years before gradually dissipating. This contrast indicates that trade policy uncertainty and financial stress affect industrial resilience through asymmetric response patterns in terms of impact intensity and persistence. Over time, as autonomy and controllability have improved, the industry’s defensive ability to cope with trade frictions has significantly strengthened, yet credit tightening and liquidity pressure in the financial sector remain the core threats to its profitability recovery. This study not only reveals the asymmetric resilience paths of strategic emerging industries under different external shocks but also provides empirical evidence and policy recommendations for the future improvement of the technology–finance system and the construction of a more resilient domestic industrial chain.

Symmetry doi: 10.3390/sym18050777

Authors: Nebras Sobahi Salih Taha Alperen Özçelik Muhammed Halil Akpınar Abdulkadir Sengur

Background/Objectives: Surface Electromyography (sEMG) presents tremendous potential as a non-invasive interface for the detection of motor intent, yet the low signal-to-noise ratio, subject variability, and the need to capture patterns at both long and short timescales make the recognition of hand gestures challenging. Methods: In this paper, the HAMSNet model is presented, which is designed for the recognition of ten different hand gestures using the sEMG signal. Sliding window segmentation is employed to segment the signal into fixed-length time windows, and channel-wise z-score normalization is applied to reduce amplitude variations. To capture the signal at different timescales, the model utilizes the Hydra 1D convolutional neural network (1D CNN), which extracts both short-range and long-range features. Furthermore, the learned features are refined using the multi-head self-attention technique, which highlights the more discriminative time regions. Finally, the Squeeze-and-Excitation (SE) technique is employed to refine the obtained features by channel-wise recalibration. Results: The model is trained in end-to-end fashion, and the results are validated using the 80/20 split method, where the model achieves 0.9894 accuracy, Macro F1 of 0.9894, and an ROC-AUC score of 0.99977. Additionally, the model achieves an MSE score of 0.001969. Furthermore, the model also achieves high accuracy under the leave-one-subject-out cross-validation (LOSO-CV) protocol, providing encouraging evidence of subject-independent performance within the evaluated dataset. Conclusions: The obtained HAMSNet model’s results are compared with the existing results from the literature on the same dataset. The comparisons show that the HAMSNet outperforms the existing methods. An ablation study is conducted to validate the contribution of each component to the proposed model and an explainability analysis is conducted to indicate the interpretability of the model’s decisions.

Symmetry doi: 10.3390/sym18050775

Authors: Qiaohong Yao Mengmeng Wang Dayu Chen Dan Liu Yubin Li

Micro-expressions, subtle and often asymmetric facial movements, play a pivotal role in nonverbal emotional communication. Addressing the core challenges of temporal misalignment, fragmented feature extraction, and slow real-time detection in micro-expression recognition (MER), we propose a novel dual-branch spatiotemporal model for dynamic sequence MER. Leveraging MediaPipe for 3D facial feature extraction and Dynamic Time Warping (DTW) for sequence alignment, our method nonlinearly maps variable-length sequences to a fixed length. A hybrid data augmentation technique enhances model robustness, while the dual-branch network simultaneously captures local spatial features and global temporal dynamics. Experimental results on the CASMEII dataset demonstrate state-of-the-art performance with 99.22% accuracy, along with a significant improvement in real-time detection speed. This approach holds substantial practical value for applications in deception detection, mental health assessment, and human–computer interaction.

Symmetry doi: 10.3390/sym18050774

Authors: Xiaoyu Jin Guiying Yan Weihua Yang

A proper vertex coloring is equitable if the sizes of any two color classes differ by at most one. Any graph G with maximum degree Δ(G) admits an equitable Δ(G)+1-coloring, computable in O(Δ(G)n2) time for n vertices. A circulant graph G(n;D) is the graph with vertex set Zn and two vertices x,y are adjacent if |x−y|∈±Dmodn. The partitioning problem in parallel decoding of multi-edge QC-LDPC codes can be interpreted as an equitable coloring problem. We prove some upper bounds for χ=(G(n;D)) and develop equitable coloring algorithms, including pattern-based periodic coloring and step-based coloring. The proposed methods typically use fewer than Δ(G)+1 colors and have computational complexity lower than O(Δ(G)n2) for circulant graphs G(n;D) with small |D|.

Symmetry doi: 10.3390/sym18050773

Authors: Hongqing Zhu Yan Wu Qi Yuan

This study employs molecular dynamics simulations to investigate the condensation behavior of water molecules on hydrophilic/hydrophobic substrates under varying electric field strengths. It reveals the synergistic regulation effect between electric field strength and surface wettability from the perspectives of condensation rate and morphological evolution. The results indicate that the condensation rate on hydrophilic surfaces first increases and then decreases with increasing electric field strength; the condensation efficiency reaches its maximum at an electric field strength of 1.6 V/nm. Conversely, the condensation efficiency on hydrophobic surfaces shows a monotonically decreasing trend with increasing electric field strength; the presence of an electric field does not facilitate condensation on hydrophobic surfaces. The orientation of water molecule dipole moments is synergistically regulated by external electric fields, intermolecular interactions, and substrate–water interactions. The weaker the wettability, the more readily the electric field assumes a dominant role. Furthermore, the electric field induces parallel alignment of dipole moments along its direction, enhancing intermolecular attractions along the electric field axis (Z-axis). This also drives the reconfiguration of hydrogen-bond networks, ultimately leading to the aggregation of water molecules into clusters aligned with the electric field, thereby transforming the condensation morphology.

Symmetry doi: 10.3390/sym18050772

Authors: Kai Qiu Junxi Liu Qinghua Shi Le Pu Mingyuan Liu

Refined decision-making of the design intent is a key factor affecting the iterative design of complex equipment products. While current research on design intent decision-making generally emphasizes methodological innovation, it often neglects the individualized and fuzzy expressive characteristics of cognitive agents, as well as the actual status of the research object. This oversight leads to uncertainty in both design intent and design outcomes. To address these issues, in this paper, a refined decision-making method for the multimodal fuzzy design intent of complex products based on noncooperative–cooperative game serialization is proposed. First, through scenario analysis, the fuzzy design intent evaluation process of different cognitive agents is transformed into a cooperative game model based on a fuzzy network, achieving a preliminary assessment of design intent. On this basis, a noncooperative game-based refined matching and decision-making model for design intent across different dimensions is constructed, thereby completing the final design intent decision-making for a specific product model. Finally, the proposed method is applied to the design intent decision-making process of a CKA6180 CNC machine tool, yielding the conclusion that the two design intents of “good protective performance” and “grand appearance” should be prioritized, thereby verifying the practicality and effectiveness of the method. The analysis of the results reveals the following: ① The application of scenario analysis theory enables a more comprehensive and precise characterization of the design intents of different cognitive agents; ② The construction of a model combining a fuzzy network with a cooperative game facilitates a more complete representation and evaluation of multimodal fuzzy design intent data; ③ The integration of a refined design concept with a noncooperative game model leads to more definitive design intent decision outcomes, thereby reducing the “disturbance” of experience dependence in the early design phase and consequently enhancing subsequent design satisfaction.

Symmetry doi: 10.3390/sym18050770

Authors: Jingyi Li Xin Sheng Xiaohui Luo

In the highly competitive e-commerce landscape, platforms must strategically balance complex operational and marketing parameters. These real-world systems inherently involve high-dimensional nonlinear interactions and strongly coupled variables, leading to complex consumer response behaviors and highly non-convex optimization landscapes. Traditional optimization approaches usually suffer from high computational costs in business environments, while conventional surrogate models are prone to premature convergence during hyperparameter estimation. To address these management and operational challenges, this study proposes a Chaos-initialized Quantum-behaved Particle Swarm Optimization Kriging (CQPSO–Kriging) framework. Chaotic mapping is introduced to enhance population diversity, while quantum-behaved particle dynamics improve global exploration capability. Utilizing large-scale real-world transaction data from the Brazilian e-commerce industry, high-fidelity surrogate response surfaces are constructed for three core business indicators: profitability, customer loyalty, and value density. Experimental results show that the proposed CQPSO–Kriging model significantly outperforms conventional approaches, such as support vector regression and radial basis function networks, achieving an exceptional coefficient of determination of R2 = 0.9586 in profit prediction. Furthermore, Sobol variance-based global sensitivity analysis is employed to extract critical managerial insights, revealing that financial variables act as interaction-driven utility multipliers in consumer decision-making. Multi-objective Pareto analysis further demonstrates that profit maximization naturally converges toward a balanced operational configuration, providing a robust quantitative tool for e-commerce precision marketing.

Symmetry doi: 10.3390/sym18050771

Authors: Chunhua Zhang Ziyue Chen

Supercritical CO2 gas explosion is an important technique for enhancing permeability in low-permeability coal seams, as it can improve gas drainage efficiency while avoiding the open-flame hazards of conventional explosion and the high water consumption associated with hydraulic fracturing. This study aims to reveal the crack propagation patterns and stress-wave dynamics under different hole configurations. Using LS-DYNA, fracture models were established for three configurations under supercritical CO2 explosions: single-hole, symmetrical double-hole, and symmetrical double-hole with a control hole. The fracture processes were analyzed to investigate the effective fracture radius of single-hole explosions, the optimal spacing for symmetrical double-hole explosions, and the influence of control holes on crack development and connectivity. The simulation results indicate that the effective fracture radius of a single-hole explosion reaches up to 2.6 m under the modeled conditions. Compared with the single-hole gas explosion case, the symmetrical double-hole configuration with a spacing of 7 m significantly enhances fracture interaction and connectivity, resulting in an approximately 98% increase in the effective damaged area. Permeability enhancement was further quantified by introducing a damage–permeability mapping (k/k0) based on the simulated damage factor, and the permeability-enhanced zone was evaluated using the criterion of k/k0 ≥ 2.

Symmetry doi: 10.3390/sym18050769

Authors: Jianjun Yao Zhikun Zhou Ruisheng Li Jiaming Zhang Zhiwei Qi

Multimodal Named Entity Recognition (MNER) aims to integrate textual and visual information to identify entities with specific semantic categories. However, existing methods often suffer from insufficient intra-modal semantic modeling, coarse cross-modal alignment, and vulnerability to noisy or ambiguous expressions in social media. To address these challenges, we propose a Visual–Semantic Guided Interaction Network (VSGN), which improves multimodal representation learning from both semantic and structural perspectives. Specifically, we first design an adaptive visual–semantic fusion module that incorporates visual descriptions as semantic guidance, enabling more informative cross-modal interactions. To further enhance feature quality, we introduce a deviation-aware channel-wise inhibitory routing (CIR) mechanism, which jointly models channel importance and distributional deviation to suppress noisy or redundant visual signals. In addition, we propose a visual–semantic guided graph structure learning module (VSG), which explicitly captures structural dependencies across modalities. By enforcing distribution-level alignment between textual and visual graph representations, the model achieves structure-aware cross-modal interaction and reduces modality inconsistency. Extensive experiments on the Twitter-2015 and Twitter-2017 datasets demonstrate the effectiveness of the proposed method, achieving F1 scores of 76.72% and 87.86%, respectively. The results show that jointly modeling semantic enhancement and structural alignment leads to more robust and discriminative multimodal representations.

Symmetry doi: 10.3390/sym18050768

Authors: Shaoshuai Xu Yan Wang Zhixun Su

Reference-driven humanoid motion tracking aims to reproduce a source motion on a target humanoid while preserving physical executability under actuation limits and changing contact conditions. The problem becomes particularly challenging for dynamic motions involving rapid support transitions, landing impacts, mixed hand–foot contacts, and moderate topology-preserving morphology variation. Existing pipelines often rely heavily on morphology-specific world-frame targets or treat balance and contact quality only indirectly during execution, which limits their reliability under dynamic contact variation. This paper presents a task and cost formulation for reference-driven humanoid tracking within the residual-based MuJoCo model predictive control (MPC) framework. The source motion is decomposed into a pelvis-centered canonical local reference, pelvis height and tilt references, and a pelvis-derived horizontal center-of-mass (CoM) velocity intent, and is tracked online with a zero moment point (ZMP)-aware contact-conditioned residual design including slip, penetration, posture, and control regularization. The formulation is compatible with standard MuJoCo MPC planners, and the evaluation is conducted under a shared iterative linear quadratic Gaussian (iLQG) setting on nominal and morphology-varied humanoids against tracking-only and two-stage inverse-kinematics (IK)-based baselines. The proposed formulation improves success rate, support quality, slip reduction, and progression accuracy, with the clearest gains on contact-sensitive motions; for example, success rate increases from 56.7% to 76.7% on Jump–Turn and from 46.7% to 70.0% on Cartwheel relative to the tracking-only MPC baseline. These results support the use of execution-oriented reference representation and contact-conditioned residual design for physically reliable reference-driven humanoid tracking.

Symmetry doi: 10.3390/sym18050767

Authors: Manuel De la Sen

This paper investigates some properties of contractive p (≥ 2)-cyclic mappings in S-metric spaces on a cyclic disposal of p, in general, non-intersecting nonempty closed subsets of the considered S-metric spaces. The convergence of the distances of points of the sequences between nonempty closed adjacent subsets of the cyclic disposal to the distance in-between such sets is proved. Also, the characterization of the best proximity points is given and it is proved that such best proximity points are also fixed points of the composed self-mappings p-times on themselves. The convergence of the sequences generated by the contractive cyclic self-mappings to best proximity points, one per subset of the cyclic disposal, is proved in the event that the S-metric space is complete. In the most general case, it is not considered that the constants defining the cyclic contraction are less than unity for each pair of adjacent subsets but that the product of all of them is less than unity.

Symmetry doi: 10.3390/sym18050766

Authors: Niks Kordjukovs Danilo Pietro Pau

Edge generative artificial intelligence (AI) increasingly combines language, perception, reasoning, audio, and action on resource-constrained devices. This paper profiles public GPT-Generated Unified Format (GGUF) checkpoints from the Hugging Face Hub (HFH) across conversational, instruct, thinking, audio, vision-language (VL), and vision-language-action (VLA) categories using a shared parser-based deployment-envelope workflow. The main category-specific run retained 21,039 profiled entries and estimated the minimum memory bandwidth, compute throughput, and unified-memory architecture (UMA) footprint needed to satisfy category-specific target throughput values. The resulting measurement protocol was symmetric, but the deployment envelopes were asymmetric: VL and thinking workloads were the heaviest on the compute–bandwidth axis, VLA formed a smaller elevated multimodal branch, and audio, instruct, and conversational workloads were lighter on average. A unified 10-tokens-per-second (TPS) sensitivity run compressed the compute–bandwidth gaps, showing that service-rate assumptions contributed strongly to cross-category separation. Welch/Games–Howell and Kruskal/Dunn analyses confirmed large category effects for bandwidth and compute in the category-specific regime, but only small memory effects. The results show that edge-model feasibility cannot be inferred from parameter count alone; throughput target, backbone family, modality, and memory budgeting must be considered jointly.

Symmetry doi: 10.3390/sym18050765

Authors: Sasipong Kijsason Sa-Aat Niwitpong Suparat Niwitpong

This study develops confidence and credible intervals for the variance of the zero-inflated two-parameter Rayleigh distribution, a flexible model for non-negative data with excess zeros. Seven approaches are proposed: Bayesian Markov chain Monte Carlo (MCMC), Bayesian highest posterior density (HPD), the standard confidence interval, the approximation normal, the percentile bootstrap, the bootstrap method with standard error, and the generalized confidence interval (GCI). Their performance is assessed through Monte Carlo simulation using coverage probability (CP) and expected length (EL). The results show that the Bayesian HPD interval performs best overall, attaining coverage close to the nominal level while yielding shorter intervals than the alternatives, especially for small samples. The methods are illustrated with road traffic fatality data from Chiang Mai Province, Thailand, recorded in March 2024. These findings support the practical usefulness of the HPD approach for variance interval estimation in zero-inflated continuous models.

Symmetry doi: 10.3390/sym18050763

Authors: Xiaoyong Liu Jing Liu Junjie Tao Haochen Hu Pan Li Liuzhi Chen

In unmanned aerial vehicle (UAV)-based small object detection, enhancing object perception under complex backgrounds remains a critical challenge for current detection models. Owing to the small scale, low pixel occupancy, and cluttered backgrounds of UAV small objects, their discriminative features are prone to attenuation in deep networks, which limits multi-scale feature fusion and consequently increases detection difficulty. To resolve these issues, this paper presents a Detail-Aware Multi-scale Fusion Network (DMF-Net) for UAV small object detection, consisting of two core modules: the Multi-Branch Detail-Enhanced Module (MBDE) and the Dual-Attention Fusion (DAF) Module. First, during feature extraction, a multi-branch detail enhancement module with serial convolutions in each branch and residual connections is introduced to strengthen high-frequency details and local textures while preserving semantic consistency. Second, at the feature fusion stage, a dual-attention-based feature fusion module with a symmetric interaction structure is designed to dynamically evaluate the significance of features at different scales via adaptive attention mechanisms, enabling symmetric cross-scale interaction and fine-grained feature complementarity. The experimental results obtained on the challenging VisDrone and TinyPerson datasets confirm that DMF-Net outperforms existing state-of-the-art detection methods in terms of the accuracy of small object detection, while maintaining high inference efficiency. On the TinyPerson dataset, DMF-Net improves AP by 1.4%, AP_50 by 4.1%, and AP_75 by 0.6% compared with YOLO11n, while maintaining 97.2 FPS. Furthermore, it shows promising performance in complex backgrounds and densely populated scenarios.

Symmetry doi: 10.3390/sym18050764

Authors: Pavel Straňák

Generative AI systems trained on synthetic data exhibit progressive degradation known as model collapse. This paper provides a theoretical explanation of this phenomenon using Shannon’s Data Processing Inequality (DPI), modeling iterative synthetic-data training as a Markov chain of lossy transformations. We show that mutual information with respect to the original data distribution must decrease monotonically, yielding qualitative predictions for exponential decay tendencies and indicating that information loss arises from general finite-precision and capacity constraints rather than from any specific architectural mechanism. Building on this analysis, we introduce the AI conceptual theorem, a generalized stability limit for computable systems. The theorem states that any purely computational system that generates outputs iteratively under finite precision, bounded capacity, and without external low-entropy input must experience cumulative information degradation after a finite number of steps. DPI-based collapse emerges as a special case of this broader principle. The framework is intended as a conceptual information-theoretic perspective rather than a fully formalized theory, with several assumptions intentionally simplified to highlight the underlying entropic mechanism. The results should therefore be interpreted as principled limits that motivate further empirical and mathematical investigation rather than as definitive closed-form predictions. Together, DPI and the AI Theorem provide a unified information-theoretic framework for understanding degradation in synthetic training, long-horizon inference, and other iterative computational processes. The resulting predictions are quantitatively falsifiable and offer guidance for designing more stable and information-preserving AI systems.

Symmetry doi: 10.3390/sym18050762

Authors: Yiming Xian Mingchao Xia Jichen Wang Qifang Chen Hang Deng

With the increasing integration of transportation and energy systems, highway energy replenishment facilities are gradually evolving into hybrid refueling stations that integrate photovoltaic generation, energy storage, battery charging, and hydrogen refueling. However, due to differences in resource conditions across stations, independently operated hybrid refueling stations find it difficult to simultaneously improve overall economic performance and renewable energy utilization. To address this issue, this paper investigates the coordinated operation and distributed optimization of highway hybrid refueling stations. First, an inter-station hydrogen–carbon–green certificate trading framework is established, and a trading model for a cluster of hybrid refueling stations is then developed on this basis. Then, the inter-station trading problem is decomposed into two subproblems: symmetric trading volume determination and asymmetric Nash bargaining-based price determination. These two subproblems are solved in a distributed manner using the alternating direction method of multipliers. In addition, a hydrogen transportation model is developed to translate trading decisions into feasible transportation arrangements under highway network and hydrogen tube trailer scheduling constraints. Finally, the case study demonstrates that the proposed model enables multi-resource sharing among hybrid refueling stations, reduces the overall system cost by 21.30%, and achieves a fairer distribution of benefits among stations.

Symmetry doi: 10.3390/sym18050761

Authors: Yixuan Liu Jing Zhang Ruipeng Luan Xuewen Yu

Relation Extraction is crucial for knowledge graph construction, but extracting complex relations in specialized domains like Satellite Navigation Countermeasures (SNCM) remains challenging due to long semantic spans and high relational density. While Large Language Models (LLMs) possess strong semantic understanding, they often suffer from severe recall deficiency and hallucinations in high-density multi-entity contexts. Conversely, traditional small models generate excessive redundant noise. To address these limitations, this paper proposes an evaluation-model-guided relation extraction method (E-guidedRE). This framework employs a two-stage collaborative mechanism. First, a lightweight evaluation model utilizing a GlobalPointer network with Rotary Position Embedding (RoPE) and a sparse multi-label loss function acts as a structural filter to generate high-coverage candidate entity pairs. Second, these candidates guide the frozen LLM to perform deep semantic discrimination and retrospective denoising. Furthermore, we construct a dedicated SNCM dataset to fill the vertical domain data void. Extensive experiments across five heterogeneous datasets, including general, biomedical, financial, and our self-built SNCM corpus, demonstrate that E-guidedRE exhibits remarkable robustness. In ablation studies on the SNCM dataset, our method improved the F1-score from 36.54% to 54.93% compared to standalone LLM extraction, boosting recall from 27.81% to 47.13%. The proposed paradigm effectively mitigates the LLM’s attention divergence in complex contexts, dynamically balancing precision and recall, and offers a highly reliable technical pathway for knowledge extraction in specialized vertical domains.

Symmetry doi: 10.3390/sym18050760

Authors: Ghazala Yasmin Aditi Sharma Georgia Irina Oros Shahid Ahmad Wani

In this paper, a new class of special polynomials, called the Sheffer-type general-λ-matrix polynomials, is introduced within the framework of the monomiality principle. This family is obtained by combining the structure of Sheffer sequences with the theory of general-λ matrix polynomials, which leads to a unified formulation encompassing several polynomial families. Fundamental properties of the proposed polynomials are established, including their generating function, explicit series representation, summation formulas, quasi-monomial structure, differential relations, and determinant representation. The proposed framework addresses an important problem in the theory of special functions: the systematic construction of matrix-valued polynomial families that simultaneously generalize both classical scalar polynomials and existing matrix polynomial hierarchies. Such a unified structure is of broad significance, with applications in quantum mechanics (wave function expansions), mathematical physics (matrix differential equations and spectral problems), approximation theory, and the study of special functions in the matrix domain. Several hybrid forms of the proposed family are derived through appropriate choices of the defining functions, which yield polynomial subclasses related to classical families such as Hermite, Laguerre, Bessel, and Poisson–Charlier polynomials. These subclasses illustrate how the proposed framework provides a systematic approach for constructing and studying generalized polynomial structures. In each case, the matrix parameter L introduces a new layer of structural richness not present in the scalar setting, enabling the modelling of phenomena governed by matrix-valued spectral data. Furthermore, a numerical and graphical investigation of selected hybrid forms is carried out using Mathematica(version 14.3, 2025; Wolfram Research, Inc.). Surface plots, distributions of complex zeros, and real-zero patterns are presented for different parameter values, highlighting the influence of the parameters on the behavior and structural characteristics of the polynomials.

Symmetry doi: 10.3390/sym18050759

Authors: Lixue Gao Shouyuan Wu Mu Li Futao Yang

New power systems with penetration of inverter-based resources (IBRs) exhibit symmetry breaking in post-disturbance frequency, as nodal trajectories depend on disturbance location, network coupling, and heterogeneous frequency channels across synchronous generators (SGs), grid-forming (GFM) converters, and grid-following (GFL) converters with phase-locked loops (PLLs). As a consequence, relying only on aggregated center-of-inertia/center-of-frequency (COI) metrics can underestimate asymmetric local risks, including worst-node rate of change of frequency (RoCoF), worst-node nadir, and nodal frequency split. This paper proposes a disturbance location-aware coordination framework that explicitly models and balances heterogeneous active-power frequency support across the network using an electromechanical-scale state-space formulation. First, a heterogeneous nodal frequency response (HNFR) model yields an explicit state-space input–output mapping from location-specific active power disturbances to nodal frequency outputs for both electromechanical and PLL-estimated channels. Second, a reproducible signal processing protocol computes nodal RoCoF/nadir/split indices and enables large-scale location sweeping via atlas-ready matrices that are naturally parallelizable for high-performance computing. Third, a constrained allocation layer schedules heterogeneous fast frequency response subject to converter limits and finite energy constraints, supporting an atlas-based gain scheduling implementation. Case studies demonstrate that the proposed symmetry-aware design improves worst-node security and suppresses frequency split while maintaining comparable COI behavior. Under budget-matched conditions on the modified IEEE 39-bus system, the proposed allocation reduces worst-node RoCoF by 32.2% and maximum nodal frequency split by 17.8% relative to the COI-based benchmark.

Symmetry doi: 10.3390/sym18050758

Authors: Heng Chi Hengdong Wang Yufeng Jia Degao Zou Wenquan Feng Zhuyin Wen Wei Wang

The stress and deformation sensitivity analysis of high earth-rock dams requires knowledge of the statistical mean and standard deviation of deformation parameters of dam materials. However, these parameters are typically determined through grouped tests and sorting. Given the small sample size in each group and the consequently large parameter errors, the inaccuracy of the resulting statistical parameters is evident. The least squares method fits all test points of each group in the same coordinate system for regression calculation, which not only helps to better address the issue of a small sample size, but also eliminates the errors caused by the grouping of test parameters. However, it is found that when the least squares method is applied to the elastic modulus and bulk modulus parameters of the Duncan–Chang E-B model, the residual errors have heteroscedasticity and correlation, which violates the use condition of the least squares method. In order to eliminate the heteroscedasticity and correlation of the fitting residuals of the elastic modulus and bulk modulus parameters of the Duncan–Chang E-B model, this paper decomposes the covariance matrix of the regression residuals to obtain its square root matrix, multiplies the explanatory variables, dependent variables and residual vectors of the regression equation by the square root matrix of the covariance, respectively, and performs variable substitution. The new regression equation has the homogeneity of variance and the irrelevance of the residual. The mean and variance of the model parameters are obtained directly by calculating all the experimental data. The variance of the new parameters is smaller than that of the classical least squares method. The results demonstrate that this generalized least squares method improves the estimation accuracy of elastic modulus and bulk modulus parameters of the Duncan–Chang E-B model.

Symmetry doi: 10.3390/sym18050757

Authors: Jorge Linares Catherine Cazelles Pierre Richard Dahoo Kamel Boukheddaden

Molecular spin-crossover (SCO) compounds constitute prototypical systems exhibiting first-order phase transitions. These transitions involve an abrupt switch between two well-defined states with distinctly different magnetic, optical, and vibrational properties. One state is diamagnetic (low-spin), while the other is paramagnetic (high-spin). Upon heating, the transition occurs at a characteristic temperature, Tup. Upon cooling, it takes place at a lower temperature, Tdown < Tup, thereby giving rise to thermal hysteresis. Accordingly, each SCO compound is defined by a distinct pair of transition temperatures, Tup and Tdown. The investigation of these molecular solids is of great importance, both for elucidating first-order phase transitions—including the potential emergence of re-entrant phases—and for their broad range of prospective applications. The critical temperatures Tup and Tdown are pivotal in defining their practical utility. We present a strategy to modify and tune the transition temperatures of spin-crossover (SCO) compounds to suit different applications. The approach combines a given SCO material with layers of a second SCO system, enabling precise control of the characteristic temperatures of the resulting heterostructure. We illustrate this method with three case studies that span the 100 K–400 K temperature range. All simulations were performed using Monte Carlo methods within the Metropolis algorithm framework.

Symmetry doi: 10.3390/sym18050756

Authors: Henry Michel Zelada Romero Cristina Vázquez Alexei Eduardo Zelada Romero Jesús Rascón Lily Juarez-Contreras Juan Carlos Altamirano-Oporto

Textile wastewater contains recalcitrant dyes and organic matter, requiring efficient, scalable treatment technologies. This study optimized an aluminum-based electrocoagulation (EC) process to maximize color removal (Y1) and chemical oxygen demand (COD) reduction (Y2) using synthetic textile wastewater (SWW), and evaluated the practical transferability of the optimized conditions using real textile wastewater (RTW). A rotatable central composite design (CCD) coupled with response surface methodology (RSM) was used to assess the effects of treatment time, NaCl concentration, and applied voltage on both responses. From a modeling perspective, the results reveal the coexistence of symmetric and asymmetric response behaviors; quadratic effects define locally symmetric regions around the optimum, while interaction terms introduce asymmetry due to coupled electrochemical phenomena. Under the optimized conditions (16.5 min, 2.9 g·L−1 NaCl, 18 V), removal efficiencies reached 99% for color and 97% for COD reduction, with a specific energy consumption of 6.6 kWh·m−3 and sludge moisture content of 92–94%. To assess applicability beyond bench scale, the optimized voltage, current, and electrolyte concentration were applied to a 50 L batch of RTW collected from the final rinsing stage of a denim dyeing process. Treatment time was extended to 84 min to compensate for the lower current density at the larger scale; under these conditions, 95% color removal and 80% COD reduction were achieved.

Symmetry doi: 10.3390/sym18050755

Authors: Mutaz Al-Sabbagh

This paper focuses on the study of quadric surfaces in three-dimensional Euclidean space that satisfy the finite II-type Gauss map condition, a concept introduced firstly by B.-Y. Chen in its first fundamental form; later it was applied by other researchers in the second and third fundamental forms. This study’s primary finding is that spheres are the only quadric surfaces with this characteristic. This suggests a particular and significant grouping in the more general class of quadric surfaces according to their finite-type properties with respect to the second fundamental form.

Symmetry doi: 10.3390/sym18050754

Authors: Yu Zhu Keyuan Ding Dejun Gao

Return-type cable suspension bridges transfer the main-cable force to the anchorage predominantly in the horizontal direction, which may induce coupled sliding–overturning instability of the anchorage–foundation system. This study examines the stability of return-type cable gravity anchorage using the composite anchorage of the Jixin Expressway Yellow River Three Gorges Bridge as the prototype. A 1:100 laboratory specimen was designed based on similarity theory and tested under incremental loading until failure. Four configurations were considered by combining two embedment ratios (1/4 and 1/2) with two base types (flat-base and shear-keyed). Horizontal displacement, overturning angle, interface contact stress, and foundation strain were monitored throughout loading. Because the return-type cable transmits a predominantly horizontal force, the anchorage–foundation contact stress exhibits pronounced asymmetry between the toe and heel regions, and this stress asymmetry governs the coupled sliding–overturning instability mode. The shallow flat-base case exhibited a distinct displacement and contact stress jump at high load levels, followed by rapid rotation, indicating slip–tilt coupled instability. Increasing embedment improved confinement and delayed the onset of nonlinear deformation, but the flat-base configuration still showed pronounced toe stress concentration. By contrast, the shear-keyed base mobilized cooperative bearing of the surrounding foundation, producing smoother stress–strain evolution and higher ultimate capacity. Moreover, the shear-keyed base mitigates the stress asymmetry at the anchorage–foundation interface, leading to a more symmetric distribution of contact pressure and improved overall stability. Three-dimensional finite-element simulations reproduced the measured trends in displacement, stress concentration near the toe, and strain development, providing independent verification. The results clarify the dominant instability mechanism of return-type cable gravity anchors and offer design implications for embedment depth and shear-keyed base detailing.

Symmetry doi: 10.3390/sym18050753

Authors: Xinyue Zhang Keying Wang Liangliang Li

Hydropower stations, as critical infrastructure for basic energy supply, play a pivotal role in ensuring the reliability of power systems through their safe and stable operation. Grounding grids operating long-term in complex soil environments are prone to corrosion and degradation, disrupting current distribution balance and causing spatial asymmetry in the voltage field, thereby compromising system safety. Corrosion branch resistance increment identification based on the electrical network method is typically modeled as a parameter inversion optimization problem. However, this problem exhibits underdetermination and other characteristics, making it difficult for traditional analytical methods to obtain stable solutions. To address this, this paper proposes a quantum perturbation scheduling candidate pool-guided sine–cosine algorithm (QSPSCA). Building upon the classical sine–cosine algorithm framework, it incorporates a dynamic candidate pool with multi-source attractor points and a quantum-inspired long-tail scheduling local refinement operator. This achieves an enhanced and smooth transition between global exploration and local refinement. Comparative experiments based on the CEC2017 benchmark and a hydropower station grounding grid corrosion diagnosis case demonstrate that QSPSCA outperforms multiple comparison algorithms in terms of average optimality and result stability. Furthermore, QSPSCA is applied to three typical engineering-constrained optimization problems. Results demonstrate that, whilst satisfying engineering constraints, this method consistently yields higher-quality feasible solutions with superior convergence accuracy and stability compared to alternative algorithms. Therefore, QSPSCA is not only applicable to underdetermined inversion diagnostics but also provides a solution framework with broad applicability for complex engineering optimization problems under structural symmetry perturbations.

Symmetry doi: 10.3390/sym18050752

Authors: Halit Nazli Osman Yildirim Yasser Guediri

Meaningful partitioning of 2D binary shapes remains a challenging problem in shape analysis because many existing methods rely mainly on local geometric rules or skeleton simplification, which often struggle to separate the main body of a shape from its protruding parts in a perceptually meaningful way. This limitation becomes more evident in shapes with thin limbs, branching structures, or irregular extensions, where preserving topology while achieving human-consistent decomposition is difficult. We present a fully automatic framework for the hierarchical partitioning of 2D binary shapes into semantically meaningful core bodies and protruding limbs (tails). The pipeline begins by generating candidate structural lines through multi-directional centroid tracking along horizontal, vertical, and diagonal (±45°) bands. Three direction-specific Sugeno fuzzy controllers first evaluate these lines based on normalized length, angular alignment, and minimum distance to the boundary. A second pair of fuzzy systems then classifies segments as either tails or core parts using thickness statistics derived from the distance transform. For ambiguous merged tail groups, iterative midpoint splitting is applied until stable labeling is achieved. High-curvature boundary corners are then detected via signed turning-angle analysis, and candidate cutting rays are assessed through exact region splitting, tail area measurement, and label purity analysis. An adaptive third-stage fuzzy controller ranks these candidates according to cut length, purity, and area. The highest-scoring non-overlapping cuts are executed iteratively, progressively peeling peripheral parts while preserving the overall topology and symmetry of the shape. The proposed framework is evaluated on a targeted subset of 32 categories from the 2D Shape Structure Dataset Results on this evaluated subset indicate that the method produces coherent and topologically consistent partitions, with competitive agreement with the available human-annotated references. This training-free framework provides an interpretable tool for 2D shape analysis, with potential applications in object recognition, computer animation, and symmetry studies.

Symmetry doi: 10.3390/sym18050751

Authors: Yosef Akhtman

The current paper studies the global shell layer of the Finite Ring Continuum framework in the symmetry-complete regime realized here by framed finite fields, Fp(t;0,1,et), with p=4t+1. We show that a single symmetry-complete shell carries a unified finite Euclidean datum for which its continuum comparison interpretation reproduces the familiar structural roles of e, π, and i of a one-phase step with an exponential kernel, a half-period, and a quarter-turn, respectively. In the same shell, the orbital geometry is generated by additive meridian action and multiplicative phase action from that same frame datum. The resulting orbital shell has a canonical spherical completion, combinatorially equivalent to the two-sphere, with labels depending on the chosen frame, but the shell type fixed up to isomorphism. Arbitrary finite-precision approximation on this external spherical comparison object is then obtained within every fixed symmetry-complete shell by the scale-periodic framed-rational refinement generated by the same frame datum. The Fourier formalism is developed strictly as a discrete Fourier transform over the shell ring, with conventional continuum Fourier language becoming a continuum large-p comparison case of that shell formalism.

Symmetry doi: 10.3390/sym18050749

Authors: Rui Liu Ziyin Xu Haiyang Hu Zhihao Zheng

Multi-UAV path planning in dynamic and complex environments is a challenging constrained optimization problem because it must simultaneously consider path efficiency, obstacle avoidance, altitude feasibility, flight smoothness, and inter-UAV path diversity. Existing methods often struggle to maintain search diversity, balance exploration and exploitation, and avoid premature convergence in high-dimensional search spaces. To address this issue, this paper proposes a Q-learning-guided Harris Hawk Optimization-Genetic Algorithm (QHHO_GA), which integrates Genetic Algorithm (GA), Harris Hawk Optimization (HHO), Q-learning, prioritized experience replay, entropy-based state partitioning, and a Rapidly exploring Random Tree (RRT)-based stagnation adjustment mechanism. In the proposed framework, GA enhances population quality and diversity, HHO performs the core search, Q-learning adaptively guides HHO behaviors, and stagnation monitoring with RRT-based stagnation adjustment improves the ability to escape locally trapped regions. Experimental results on the CEC2017 benchmark suite and a multi-UAV path planning task demonstrate the effectiveness of the proposed method. On the CEC2017 benchmark, QHHO_GA ranks among the top two on 18 out of 30 test functions and achieves the best overall ranking among the compared algorithms. In the UAV path planning experiments, it achieves an average ranking of 3.44 and also achieves the best overall rank among all compared methods. These results indicate that QHHO_GA is a robust and competitive method for high-dimensional constrained optimization, and is particularly effective for complex multi-UAV path planning tasks.

Symmetry doi: 10.3390/sym18050750

Authors: Mustafa M. Hasaballah Arvind Pandey Pragya Gupta Oluwafemi Samson Balogun Farouq Mohammad A. Alam Mahmoud E. Bakr

Control charts for monitoring time between events (T) and amplitude (X) have been developed in recent years. Many TBEA charts depend on limited models such as exponential, normal, and gamma distributions and mainly rely on the ratio statistic (XT). This representation ignores the symmetric relationship between event occurrence and event magnitude. This paper proposes Shewhart-type TBEA charts constructed from three statistics (Z1), (Z2), and (Z3) based on (X) and (T). The approach models symmetry between frequency and amplitude using generalized Weibull and generalized log-logistic distributions. The statistics maintain proportional invariance when both variables shift together, which enables balanced monitoring of the process. Several scenarios are examined for detecting upward shifts. Performance is assessed using numerical measures of detection efficiency and average run length. The results show improved detection compared with classical ratio-based TBEA charts. A real data example from a French forest fire database illustrates the ability of the proposed charts to detect simultaneous changes in occurrence rate and burn intensity.

Symmetry doi: 10.3390/sym18050748

Authors: Waseem Ahmad Khan Oğuz Yağcı Khidir Shaib Mohamed Alawia Adam Naglaa Mohammed

Polylogarithm-weighted sequences and h(x)-Fibonacci/Lucas polynomials have each been studied extensively, but a common formulation that incorporates generalized hyperharmonic weights into both these kernels and related Legendre-type kernels has not been formulated in a unified way. In this paper, the classical generating functions are deformed by the factor Lip(t)/(1−t)q, and the resulting coefficients are derived by Cauchy product arguments. This construction yields the h(x)-Fibonacci–polylogarithm and h(x)-Lucas–polylogarithm polynomials, explicit coefficient formulas, convolution identities, recurrence relations, and parity properties, together with a unified two-parameter family of generalized h(x)-Fibonacci–Lucas–polylogarithm polynomials Ph,na,b,p,q(x). The same deformation principle also gives rise to Legendre–polylogarithm polynomials and to a (q,λ)-extension obtained from a weighted Legendre generating kernel. These families provide a natural generating-function setting for models in which cumulative harmonic or hyperharmonic effects are intrinsic, while also making explicit the main analytic restrictions of the deformation, including convergence constraints and the loss of classical orthogonality in the Legendre setting.

Symmetry doi: 10.3390/sym18050747

Authors: Mujahid Naseem Samad Ali Taj Muhammad Shakeel Afzal Muhammad Shoaib Naseem Rajnish Kaur Calay

Biogas is a renewable energy resource that is not only economical but also fulfills the criteria of net-zero carbon emissions. This is highly favorable for agriculture-based developing countries with an abundance of animal and agricultural waste that can be effectively utilized for biogas production. A dual-stage reactor was designed and built to investigate the optimal conditions during the different seasons of winter and summer for mesophilic biogas production utilizing cow manure from local dairy farms. During the experiments, the pH was continuously monitored and automatically controlled between 6.8 and 7.2 over a period of fifteen days for each experiment using an Arduino Mega controller. The weight ratio (rw) of cow manure slurry was varied from 50% to 80%, and the optimal condition was found to be 70%, irrespective of the seasonal variations. However, the statistical analysis suggests that the optimal weight ratio is 66% for both seasons. A maximum reaction yield of 87% was achieved at a rw value of 60% during the summer, with an expected yield of over 95% at a rw value of 70% if similar extreme environmental conditions occur. Employing this apparatus for biogas production requires significant electrical energy to drive the stirrer and pumps, suggesting the use of a conventional underground setup for biogas production, integrated with an automatic pH control module.

Symmetry doi: 10.3390/sym18050746

Authors: Shaoqing Li Wei Li Meng Qin

Self-supervised learning (SSL) has become a powerful paradigm for graph representation learning. However, most existing methods operate in the graph topology domain and rely on complex data augmentations or contrastive objectives, which constrain them to local structural information and hinder the effective modeling of global or long-range dependencies. In this work, we propose the inverse-filter graph neural network (IF-GNN), an alternative spectral-based method for SSL. IF-GNN employs a fixed spectral operator (I+α(L^−I))−1 as a low-pass inverse filter, enabling efficient and label-free node representation learning in the spectral domain. The inverse filter can be interpreted as an infinite-order polynomial filter, allowing the model to aggregate high-order neighborhood information without layer stacking or explicit message passing. This design helps alleviate over-smoothing by avoiding repeated feature averaging and improves the interpretability of spectral SSL models. Extensive experiments on multiple benchmark datasets demonstrate that IF-GNN achieves competitive or superior performance compared to both spectral and self-supervised baselines, establishing inverse filtering as a new and effective spectral SSL paradigm.

Symmetry doi: 10.3390/sym18050745

Authors: Bilal Ben Zahra Mohammed Barrouch Charchaoui Wiam Abdellah Ahourag Karim El Moutaouakil Nuino Ahmed Vasile Palade

This paper introduces the Quantum-Inspired Impulsive Continuous Hopfield Network (Q-ICHN), a novel hybrid control framework designed to handle non-smooth, high-energy perturbations in nonlinear dynamical systems. Standard Continuous Hopfield Networks (CHNs) rely on sigmoidal activation functions that are prone to gradient saturation, which leads to an insufficient corrective response when the system undergoes large deviations from equilibrium. To overcome this shortcoming, the proposed Q-ICHN adopts a wave-packet-based activation function grounded in the stationary Schrödinger equation, yielding a non-monotonic and oscillatory activation profile that sustains effective compensatory dynamics across a broad range of states. Furthermore, the proposed framework incorporates Madelung’s quantum potential into the control architecture, thereby enabling a fundamental reshaping of the system’s energy landscape. Specifically, this induces a tunneling-like mechanism that allows the system to circumvent local minima and rapidly recover from impulsive disturbances, manifested as a sharpened attractor structure in the phase-space domain. Together, these properties yield enhanced convergence behavior and improved robustness over traditional neural control approaches. To rigorously assess its merits, the performance of the Q-ICHN is evaluated through a large-scale benchmark involving 20 established control methods, including Sliding Mode Control (SMC), Model Predictive Control (MPC), and Backstepping. The experimental results obtained across 20 heterogeneous scenarios demonstrate that the proposed model achieves a 48% reduction in Mean Squared Error (MSE) relative to the classical ICHN. In addition, the Q-ICHN exhibits improved smoothness, reflected in a 30% reduction in jerk with respect to high-gain robust controllers, and enhanced reliability, validated by superior spectral purity and a 34% reduction in integrated variance under stochastic perturbations. Collectively, these results underscore the potential of quantum-inspired activation mechanisms to favorably balance control responsiveness and harmonic stability, providing a robust framework for handling both continuous dynamics and impulsive effects.

Symmetry doi: 10.3390/sym18050744

Authors: Agustín Moreno Cañadas Andrés Sarrazola Alzate

We construct and analyze a family of support-defined Brauer-type configurations canonically associated with the Boolean geometry underlying the Grassmann algebra. The construction is governed by an x-support map on monomial labels, which identifies the vertex set with the Boolean lattice P([n]). This identification yields a Boolean support quiver isomorphic to the directed Hasse diagram of P([n]), equivalently, to an oriented hypercube. We then equip the family with a canonical cyclic ordering at each vertex and obtain a genuine connected reduced Brauer configuration in the standard sense, together with its associated Brauer configuration algebra and its standard Brauer quiver. A ghost-variable mechanism is introduced to obtain a connected realization without altering any support-controlled invariants. We prove that polygon membership, valencies, multiplicities, Boolean stratification, and the support quiver are invariant under support-preserving ghost relabelings. We also give an explicit description of the standard Brauer quiver and show that it is different from the Boolean support quiver. On the algebraic side, we derive closed formulas for the center dimension, the algebra dimension, and the normalization constant of the induced weighted distribution. On the probabilistic side, we distinguish the vertex entropy from the layer entropy, establish an exact decomposition of the former by Hamming layers, and show that the layer distribution is asymptotically concentrated on the middle layers, while extremal vertices and any fixed maximal path contribute a negligible fraction of the total weight. As a consequence, the layer entropy satisfies a logarithmic asymptotic law. We also investigate geometric consequences of the Boolean model transported through the support identification. Coordinate projections produce a rigidity phenomenon for antipodal pairs, providing a combinatorial analogue of Greenberger–Horne–Zeilinger (GHZ)-type fragility, whereas the first Boolean layer exhibits a persistence property analogous to W-type robustness. Together, these results exhibit a concrete bridge between Grassmann combinatorics, Brauer configuration theory, hypercube geometry, and entropy asymptotics.

Symmetry doi: 10.3390/sym18050743

Authors: Junfeng Wang Yongkun Xu Xinhang Ding Qing Chang Mengwei Wu Yan Wang

Offshore wind energy is an important source of clean energy. Single-post platforms, due to their simple structure and strong stability, can adapt to deep water environments through buoyancy and ballast systems, have small motion responses, and have low construction and maintenance costs. They are suitable for offshore wind energy development in deep-sea areas and help expand the application of offshore wind power. This paper conducts a coupled response analysis of offshore wind turbine foundations and mooring systems, as well as an optimization study on the form and number of mooring lines. Under the premise of considering the safety and economy of floating wind turbines, the mooring lines have been optimally arranged. The study calculates frequency-domain responses, time-domain responses, and mooring line forces under the constraints of the original three-line mooring system. Based on this benchmark, the study further optimizes the mooring forms and numbers for the same platform, analyzing four, six, and eight single mooring lines, as well as three groups of single-line, double-line, and triple-line mooring configurations. Finally, using AQWA software (2024 R1), the responses and mooring line forces of different mooring configurations were calculated, and the preferred mooring arrangement for this stepped single-post platform was determined to be a three-group, three-line system (a total of nine mooring lines). The mooring line tension decreased substantially from the original 3.2 × 106 N to 1.8 × 106 N, while the dynamic response was reduced to one-sixth of its original level. Meanwhile, this study provides strong support for the utilization of offshore wind energy and the construction of offshore wind turbine platforms and mooring systems.

Symmetry doi: 10.3390/sym18050741

Authors: Waseem Ahmad Khan Oğuz Yağcı Khidir Shaib Mohamed Mona A. Mohamed Naglaa Mohammed

We study a modified-degenerate version of the Wα,β,ν-factorial and the associated exponential, trigonometric, and hyperbolic families obtained by replacing the Euler gamma function with the modified-degenerate gamma function Γλ*, where λ∈(0,1). A main conclusion of this paper is that this construction does not generate a genuinely new transcendental family. Indeed, since Γλ*(s)=bλsΓ(s),bλ=λlog(1+λ), all modified-degenerate W-functions reduce to exact rescalings of their non-degenerate counterparts. The novelty of the present work is therefore operational rather than structural. We formulate this transport principle explicitly, derive the corresponding modified-degenerate Laplace-transform identities directly in the spectral variable s, establish the induced convolution rule, and obtain first-order asymptotic expansions as λ→0+. We further show that the associated W-derivative is a formal coefficient-shift operator, and conjugate it to the non-degenerate one under the scaling map. As an application, we present a complete Volterra integral-equation example with polynomial memory, including an explicit resolvent representation for the case m=1, together with convergence and residual-error checks supporting the numerical illustrations.

Symmetry doi: 10.3390/sym18050742

Authors: Abla Boudraa Soumia Kharfouchi Khudhayr A. Rashedi Abdullah H. Alenezy Tariq S. Alshammari

Recursive constructions derived from finite projective geometries generate scalable families of resolvable block designs exhibiting strong algebraic regularity and intrinsic symmetry. In this work, we investigate the structural optimization of recursion depth in such constructions and demonstrate that the combinatorial growth of recursive chains is governed by a quadratic scaling law originating from the asymptotic expansion of Gaussian binomial coefficients. We show that the resulting exponent is strictly concave, which guarantees the existence and uniqueness of an optimal recursion depth. This optimum occurs at the midpoint of the projective dimension and reflects the dual symmetry of the lattice of projective subspaces. To analyze this behavior, we introduce a normalized objective function that compares recursion depths and reveals a unique maximum corresponding to the midpoint of the projective dimension. Theoretical analysis is supported by exact enumeration and asymptotic validation, confirming that the optimal depth is robust to lower-order perturbations and remains invariant under the natural duality of projective geometry. The proposed framework establishes a direct connection between symmetry properties of discrete geometric structures and optimality in nonlinear discrete systems. These results provide a unified perspective on recursive design constructions, revealing that symmetry not only governs combinatorial structure but also induces a mathematically inevitable optimal configuration. The approach opens new directions for studying symmetry-induced optimality in combinatorial geometry, discrete optimization, and related nonlinear mathematical models.

Symmetry doi: 10.3390/sym18050740

Authors: Alessandra A. C. Alves Dênis E. C. Vargas Álvaro E. Eiras José L. Acebal

The Aedes aegypti mosquito exhibits a critical behavioral adaptation through its oviposition strategy, laying eggs in dry and wet environments just above the water level, allowing eggs to resist desiccation and hatch only when submerged by rain. To investigate this mechanism, we developed a nonlinear dynamic model incorporating climate-driven parameters affecting egg hatching and adult emergence. Theoretical analysis revealed an imperfect pitchfork bifurcation giving rise to a phenomenon we term basin-mediated hysteresis. Unlike classical hysteresis, which relies on coexisting stable states, this mechanism results from the progressive collapse of the extinction basin boundary. As the control parameter approaches its critical value, the basin of attraction of the trivial equilibrium shrinks. Once the population establishes itself above the threshold, returning the parameter below unity does not restore extinction, leading to an irreversible transition governing population persistence. The model was validated using field data from mosquito traps in a Brazilian city, showing strong agreement with observed seasonal patterns of female captures. Parameters were optimized using the Differential Evolution algorithm, yielding high correlation between model and field data. The results demonstrate that the dual oviposition strategy underlies population persistence and seasonal peaks, providing information for planning interventions amid global arbovirus expansion.

Symmetry doi: 10.3390/sym18050739

Authors: Madhusudhan R. Manohar Muthucumaraswamy Rajamanickam

This study investigates unsteady magnetohydrodynamic free convection flow past a rotating vertical plate embedded in a Darcy–Forchheimer porous medium. The formulation incorporates Hall current, thermal radiation, viscous dissipation, Joule heating, and an Arrhenius-type chemical reaction with activation energy to represent thermo-reactive transport in an electrically conducting fluid. The coupled nonlinear equations governing momentum, thermal energy, and species concentration are transformed into dimensionless form and solved numerically using the Crank–Nicolson scheme. Grid independence and validation tests confirm the accuracy and stability of the numerical procedure. The results show that electromagnetic forces, rotation, porous resistance, and thermo-reactive effects significantly influence wall shear stress, heat transfer, and mass transport. In particular, the interaction between magnetic field strength and Hall current alters near-wall transport behavior, highlighting the role of electromagnetic coupling in rotating porous systems. The study provides physical insight relevant to the design and analysis of transport processes in high-temperature energy systems, rotating reactors, and porous thermal management devices.

Symmetry doi: 10.3390/sym18050738

Authors: Shuang Wang Fangzheng Liu Zhiping Li Lei Ding Zhaojun Gu

To address the challenges of complex, heterogeneous, and blurred sensitivity boundaries in the sensitive data sources of civil aviation passengers, this paper proposes a hierarchical measurement method. This model integrates information entropy and random forest, achieving measurable sensitivity. Firstly, the correlation between data sensitivity level and business characteristics is established. Then, a Random Forest-based Hierarchical Measurement with Sensitivity Information Content Analysis (RF-HM-SICA) model integrating information entropy and random forest is proposed to construct a sensitivity measurable hierarchical measurement method for passenger sensitive data. The experimental results show that the RF-HM-SICA model exhibits high stability, generalization capability, and boundary sample protection ability under different data sizes and sensitivity levels, making it suitable for solving the multidimensional heterogeneity measurement problem of sensitive data of civil aviation passengers and providing support for data security sharing protection. In particular, the recognition accuracy and precision for high-sensitivity data approach 1.0 across datasets of different scales, while RF-HM-SICA exhibits the lowest misclassification rate among all compared models.

Symmetry doi: 10.3390/sym18050737

Authors: Ping Feng Siqi Xu Xinping Du Yan Chen Yuyuan Dong

Knowledge graphs play a vital role in tasks such as recommendation systems, question-answering systems, and information retrieval. However, during practical construction, they commonly suffer from structural incompleteness and sparse relationships, which limit reasoning performance and downstream applications. Existing methods typically focus solely on either structural modeling or semantic modeling: embedding models relying solely on graph structures struggle to leverage textual information about entities and relationships. In contrast, semantic approaches relying solely on pre-trained language models struggle to accurately capture complex relationship patterns. To address this challenge, this paper proposes SeSKGC, a semantic–structural fusion knowledge graph completion model. At the semantic level, the model employs the DeBERTa pre-trained language model to encode entity and relation text. It incorporates a neighbor text augmentation mechanism to introduce local semantic context and enhance representation quality. At the structural level, it adopts complex-space rotation to model relationships using a RotatE-like approach, and aggregates local topological information through relative position attention to capture complex relationship patterns. At the scoring stage, the model employs a weighted fusion strategy to combine semantic and structural scores and utilizes InfoNCE contrastive loss for joint optimization. Experiments conducted on WN18RR and FB15k-237 datasets demonstrate that SeSKGC achieves overall superior performance on metrics including MRR and Hits@N compared to multiple representative baseline methods. Ablation studies and parameter sensitivity analysis of fusion weight λ further reveal that the semantic encoding and structural modeling modules exhibit distinct complementary roles, while the weighted fusion design in the scoring layer plays a crucial role in enhancing model performance and stability.

Symmetry doi: 10.3390/sym18050736

Authors: Zetao Zhao Suduo Xue Xiongyan Li Jiuqi Luo

To investigate the influence of ambient temperature variations on the natural vibration characteristics and seismic responses of suspen-dome structures, a 1:20 geometric similarity dynamic scale model was designed using the symmetric suspen-dome roof of the Lanzhou Olympic Sports Center Gymnasium as the prototype. First, white noise excitation tests and seismic simulation tests were performed on the model, and the indoor ambient temperature was measured simultaneously. Subsequently, a corresponding numerical scaled model was developed using the ABAQUS 2024 finite element software, and its temperature was set according to the shaking table test measurements. Modal analysis and seismic time–history analysis were then performed, and the model’s natural frequencies and seismic responses (such as acceleration, displacement, and internal force) were compared with the shaking table test results, thereby validating the accuracy of the numerical model and confirming that the modeling approach reliably reproduces the natural frequencies and seismic responses measured in the tests. Finally, the ambient temperature of the numerical model was set according to the historical temperature data for Lanzhou. A comparative analysis was performed to examine the variations in the natural vibration characteristics and seismic responses of the suspen-dome structure under different temperature conditions. The result shows that, as the ambient temperature increases from −30 °C to 60 °C, the natural frequencies of the suspen-dome structure decrease by up to 21.8% (e.g., the third-order frequency drops from 9.423 Hz to 7.734 Hz), with low-order natural frequencies being the most significantly affected. Furthermore, under both unidirectional and three-dimensional earthquake excitations, the peak seismic responses increase markedly: acceleration increases by up to 35.5%, displacement increases by up to 88.3%, and internal force in critical members increases by up to 68.9%. Notably, structural members experiencing higher internal force responses demonstrate greater sensitivity to ambient temperature changes. These findings indicate that ambient temperature variation significantly reduces structural stiffness and amplifies seismic responses, providing a valuable reference for the seismic performance evaluation and safety design of suspen-dome structures in regions with large annual temperature fluctuations.

Symmetry doi: 10.3390/sym18050735

Authors: Ting Wang Reziyan Wakasi

Industrial chains in agriculture are currently fragmented and do not support developing resource-based competitive advantages. This is true under the Big Food Framework’s strategic orientation. This research seeks to develop a new analytical framework for evaluating pathways to the integration of agricultural industrial chains and their impact on the performance of companies engaged in food processing in Xinjiang. A mixed-method approach, employing both an exploratory and sequential design, will be used to do this. The primary method of data collection for this study is the case study method, along with the questionnaire method involving 145 agricultural enterprises. From these data, structural equation modeling (SEM) will be used to test the paths of causation among cognitive managers of firms who have implemented the BFF. Evidence will be presented to demonstrate the relationship among three types of integration (vertical, horizontal, and lateral) in the agricultural industrial chain, dynamic capabilities, and company performance. Additionally, network topology and optimization simulations will be conducted to determine how effectively structures are organized in training the respective companies. Important findings revealed in this research include the following: The managerial cognition constructs offered by BFFs play a key role in enhancing the depth and structural balance of industry chain integration. There were complementary performance effects found, and they are related to vertical integration achieving operational efficiency and financial efficiency; horizontal integration improving market competitiveness and brand competitiveness; and lateral integration facilitating innovative growth. Dynamic capabilities are a significant mediating mechanism linking institutional support and digital capability with the depth of integration across different modes of integration. The findings from network optimization suggest that there is a positive effect of balanced connectivity across the different dimensions of integration on overall system efficiency and reduced structural inefficiencies. Based on these findings, the authors recommend that organizations establish governance mechanisms that facilitate coordinated connectivity; strengthen adaptive capabilities within the firm; and promote balanced integration across industrial networks. Future researchers should consider applying these findings to conducting longitudinal studies on network evolution; integrating sustainability measures as part of their analysis; and conducting comparative validation studies across regions or industry systems.

Symmetry doi: 10.3390/sym18050734

Authors: M. Faruk Şahin Can Eyüpoğlu

Background/Objectives: Privacy preservation in big data environments is an NP-hard optimization task that requires the satisfaction of k-anonymity and l-diversity constraints to ensure data utility. Methods: This study proposes a novel hybrid optimization approach, adaptive hybrid FOX–RUN Intensive Mutation (PA-FRIM), to address the privacy–utility trade-off in anonymization process. The proposed approach integrates FOX-based global exploration with RUN-based local search using a hybrid adaptive control strategy and intensive mutation search to improve solution diversity in highly constrained solution spaces. Results: The experimental study on the Adult and Bank Marketing datasets shows that PA-FRIM exhibits stable convergence behavior compared to competing methods. The results indicate that full privacy is achieved on the Adult dataset with a violation value of 0.00, and information loss is minimized with an NIL measure of 0.5686. From the analytical utility perspective, PA-FRIM ensures data usability, even in the constrained region, achieving classification accuracies of 89.61% on the Bank Marketing dataset and 84.90% on the Adult dataset. Conclusions: By using a multi-metric evaluation strategy, PA-FRIM provides a robust optimization framework that eliminates privacy violations while maintaining high analytical performance.

Symmetry doi: 10.3390/sym18050733

Authors: Alexander B. Balakin Gleb B. Kiselev

We study the SU(N) symmetric model, which describes interaction of gravity with three field multiplets: first, the multiplet of pseudoscalar fields, which is at present associated with the multi-component cosmic dark matter; second, the multiplet of vector fields, which represents the so-called color aether, now known as dynamic aether; third, the multiplet of Yang–Mills fields, which provides the SU(N) invariance of the model as a whole. It was previously known that the decay of the color aether in the early Universe could have given rise to emergence of an axionic singlet according to the Peccei–Quinn mechanism; we proposed an extended scheme, according to which the color aether activates an additional internal tool for generating not only a simple axionic singlet, but an entire SU(N) symmetric multiplet of pseudoscalar fields. Late-time evolution of the considered field configuration is analyzed in the framework of the Bianchi-I cosmological model, and a hypothesis is proposed that the aforementioned pseudoscalar multiplet can be associated with the multi-component cosmic dark matter.

Symmetry doi: 10.3390/sym18050732

Authors: Antonio F. Miguel

Traditional analyses of transport phenomena rely on prescribed geometric boundaries, yet natural flow systems dynamically evolve their architecture to maximize access to currents. To address this disparity, we propose a field-theoretic framework for the constructal law that treats physical geometry as a dynamic state variable, represented by a time-dependent conductivity tensor. Using a variational approach grounded in non-equilibrium thermodynamics, we derive a general tensor evolution equation. Within this framework, macroscopic flow architecture emerges deterministically from the continuous competition between non-linear flux-induced accretion, linear entropic relaxation, and spatial smoothing. Scaling analysis reduces this dynamic to a tri-parameter dimensionless phase space: a morphogenic number driving structural growth, a structural diffusion number governing spatial coherence, and a stochastic intensity number providing the microscopic seeds for symmetry breaking. Our principal result is the analytical prediction of a critical bifurcation. When the local morphogenic number strictly exceeds unity, the system escapes its stable, isotropic configuration and branches into highly conductive, anisotropic architectures. We demonstrate the predictive validity and trans-scalar applicability of this continuum theory by mapping it to highly diverse phase transitions, successfully capturing phenomena ranging from microscopic aerosol agglomeration and microbial resistance, to macroscopic coral plasticity and crystal growth instabilities, and finally to the astrophysical launching of relativistic jets from black holes.

Symmetry doi: 10.3390/sym18050731

Authors: Chenghao Wei Pukai Wang Chen Li Zhiwei Ye

Bayesian network-based causal structure learning provides an effective framework for uncovering causal relationships among continuous variables. However, many existing methods for continuous data still rely on strong parametric distribution assumptions, which may introduce information loss and reduce Bayesian network modeling accuracy. Kernel density estimation (KDE), a non-parametric statistical method that is more flexible in density estimation form, offers a versatile framework for conducting conditional independence (CI) tests. This approach enables the estimation of mutual information and conditional mutual information, thereby facilitating the identification of underlying structural relationships. Nevertheless, the high computational cost of KDE-based CI testing restricts its practical application in continuous-variable causal learning. To address this issue, this study introduces a radial symmetric kernel-based acceleration scheme within a Fast Fourier Transform (FFT) framework to improve the efficiency of density estimation. On this basis, an enhanced Bayesian network structure learning method is developed for continuous variables, enabling more efficient estimation of mutual information and conditional mutual information while improving the computational efficiency and empirical stability of variable dependency discovery. With proper bandwidth and grid resolution, the proposed MMHC-FFTKDE framework achieves a reduction in computational runtime and improves efficiency compared to MMHC-KDE in the ablation setting, while maintaining competitive F1-scores and SHD for causal structure discovery.

Symmetry doi: 10.3390/sym18050730

Authors: Chen Yu Chuang Liu

Numerical simulation of sequential fracturing in horizontal wells for shale gas and oil extraction requires careful consideration of mechanical interactions between proppant and fracture surfaces—a challenge that remains largely unresolved. This study proposes a novel phase-field model featuring a strain-based formulation and a width-dependent proppant reaction force. Unlike previous studies, we integrate an empirical propped force solution, adapted from established work to account for rock properties and proppant support, to capture nonlinear fracture closure. Results show that reaction stress models significantly dictate propped geometry. The model’s fracture length, width, and closure predictions are validated against theoretical solutions. We conducted a sensitivity analysis to evaluate how fracture deflection angles and widths vary with dimensionless fracture spacing, in situ stress contrast, and proppant strength. Numerical results show that proppants induce pronounced morphological asymmetry and distinct geometric discrepancies. Specifically, the heterogeneous support provided by proppants and the resulting stress redistribution alter fracture propagation paths, leading to an 8% reduction in fracture length and a marked difference in fracture orientation of approximately 80° between supported and unsupported fractures, highlighting the important role of proppants in governing fracture geometry. Both dimensionless fracture spacing and in situ stress contrast strongly influence fracture deflection, with proppant strength also contributing. The propped-force formulation is further extended to nonplanar fractures, enabling application to sequential fracturing with multiple fractures. These results highlight fracture propagation mechanisms and demonstrate the robustness of the proposed phase-field model.

Symmetry doi: 10.3390/sym18050729

Authors: Danfeng Zuo Liang Qi Hao Ni Song Song Haifeng Li Xinwen Wang

Ship target detection is a prerequisite for achieving automated monitoring in ship detection systems. To address the challenge of accurately detecting ship targets in complex water environments, this study proposes a ship target detection method based on an improved YOLOv11 framework. To enhance the model’s ability to perceive and fuse features across multiple scales and in complex backgrounds, an Iterative Attention Feature Fusion (iAFF) module and a Biformer module are integrated at the end of the backbone network. The iAFF module iteratively optimizes multi-scale features through a two-stage attention mechanism, effectively focusing on key target regions, thereby improving the model’s detection capability for small, medium-sized, and occluded ships. The Biformer module leverages its innovative Bi-level Routing Attention (BRA) mechanism to enhance the modeling of global semantic information while reducing computational complexity, mitigating false detections caused by occlusions among ship targets, and consequently improving detection precision. This study employs the Minimum Point Distance Intersection over Union (MPDIoU) loss function, which more comprehensively measures the similarity between predicted and ground-truth bounding boxes by optimizing the distances of their key geometric points, effectively enhancing the accuracy of bounding box regression. Experimental results show that the proposed model achieved 93.96% mAP, 92.93% recall, and 94.97% precision on a self-built ship dataset, surpassing mainstream detection algorithms including YOLOv11 in multiple metrics. The model has only 2.90 M parameters, achieving a good balance between accuracy and efficiency. This provides an accurate and efficient solution for intelligent ship supervision.

Symmetry doi: 10.3390/sym18050727

Authors: Zonghui Wu Jiahao Li Wei Ye

The tunable coefficient of thermal expansion(CTE) and Poisson’s ratio(PR) properties of metamaterials help address issues caused by drastic temperature variations and external loads. In this work, we propose a novel bimaterial thermal expansion and PR integrated tunable 2D metamaterial structure. Under certain parameter constraints, the structure based on an Al alloy/low carbon steel (LCS) combination demonstrates a wide tunability, with the CTE ranging from −47 to 28 ppm/°C and the PR varying from −14.8 to 7.3. A general thermoelastic equation is adopted to establish the relationship between temperature, external force, and displacement, which is then assembled into a theoretical model. Through theoretical analysis and numerical simulations, the underlying mechanisms of the proposed 2D metamaterial structure’s CTE, PR, and their relationship with geometric parameters and elastic modulus ratios are revealed. CTE and PR experiments are conducted to validate the theoretical modeling. Finally, the coupling relationship between CTE and PR is revealed.

Symmetry doi: 10.3390/sym18050728

Authors: Yuxuan He Nan Zhang Ruilin Wei

Feature selection is a core step in data analysis and is referred to as attribute reduction in rough set theory. Granular ball computing has emerged as a novel data analysis paradigm characterized by high computational efficiency, robustness, and scalability. However, in previous attribute reduction methods for interval numbers, the construction of tolerance classes and the reduction iteration process suffer from inefficiency. To address these limitations, this paper proposes an efficient attribute reduction method based on fuzzy interval-valued granular balls. This method integrates fuzzy interval-valued granular balls with an acceleration strategy based on the positive region. Specifically, we first construct tolerance classes efficiently using fuzzy interval-valued granular balls, thereby enabling a reasonable partition of the universe. We then remove redundant objects in the positive region during the reduction iteration to avoid unnecessary computations. On this basis, we further propose a conditional entropy-based algorithm for attribute reduction. Experimental results show that this algorithm substantially improves computational efficiency while maintaining high classification accuracy.

Symmetry doi: 10.3390/sym18050726

Authors: Jiansong Liu Chunbo Xiu Yanxin Yuan Yue Zhou Baoquan Li

A model predictive control (MPC) strategy is proposed based on state observation and updating for image-based visual servoing (IBVS) tasks of micro aerial vehicles (MAVs). This control strategy enables precise pose adjustment of MAVs without relying on the global positioning system (GPS). Specifically, image features are first defined on a virtual image plane to decouple the translational motion of the MAV. Subsequently, a linear velocity observer is developed to provide high-quality real-time velocity information for the MAV during IBVS execution. Furthermore, the image dynamics on the virtual image plane are linearized using a first-order Taylor expansion, and a linear MPC controller is formulated to efficiently compute the optimal control inputs. Moreover, the state inputs to the MPC controller are updated at each control cycle to eliminate errors accumulated during the rolling optimization based on the linearized dynamics, thereby ensuring the precision of IBVS. Simulation and experimental results demonstrate the performance of the proposed observer and control strategy.

Symmetry doi: 10.3390/sym18050725

Authors: Steven D. P. Moore

This article introduces novel methodologies, coordinate systems, and procedures in computational geometry that further develop a Euclidean-based relationalistic framework. The objective is to describe tools using object-oriented relational elements with symmetry, anchored to a fixed point in a relational model, that generate structured point sets serving as blueprints for geometric figures and physical structures representing their source objects. Geometric operations and transformations construct ratio figures and ordered proportional structures. Using discrete N-Euclidean geometry, two relational coordinate systems are introduced—polar-vertex coordinates and radial coordinates—both formed through discrete geometric operations. A relational unit circle of fixed magnitude is defined by a 4::1 proportional equivalence between radius and angular ratios, independent of real-number or arc-length geometry. Euclid’s theory of proportion is extended from static abstract magnitudes to symmetry-driven geometric construction, and a square-pyramid geometric blueprint is produced from an Earth ratio figure with accurate dimensional magnitudes. The findings reveal a novel commensurability between the radius of a circle and the side length of a square using a shared fixed point coupled via a 3:4:5 Pythagorean-triple triangle, introducing the concept of ordered proportions.

Symmetry doi: 10.3390/sym18050724

Authors: Xiyuan Chen Xiaoyan Zhou Haibin Wang

As a novel tool for predicting stock trends, hypergraphs are used to effectively represent high-order relationships among stocks, capturing symmetric dependencies inherent in market interactions. However, the instability of hyperedges limits their ability to capture dynamic stock changes, and existing methods neglect the influence of time decay on feature importance. To address these challenges, a hybrid hypergraph–dynamic graph attention network based on temporal decay attention and conditional aggregation for stock trend prediction, namely HDGAN, is developed. Specifically, we utilize dynamic graphs to capture the dynamic relationships among stocks, which mitigates the instability of the hyperedge structure in dynamic markets. A temporal decay attention mechanism is designed to identify important feature points in the evolution of stock prices, and then a conditional aggregation method is proposed to aggregate information from different pathways. Extensive experiments on A-share, NASDAQ, and NYSE datasets demonstrate HDGAN outperforms other state-of-the-art methods in stock trend prediction and investment return.

Symmetry doi: 10.3390/sym18050723

Authors: Chengkai Tang Aomi Chen Zesheng Dan Yangyang Liu Jun Yang

Low-Earth orbit satellites are gradually becoming the core infrastructure of integrated aerospace communication networks, with their significant advantages of high communication rates, small transmission delay, and wide coverage. Interference with military communications in response to their security and protection needs is a current research challenge. Consequently, this paper introduces an interference technique optimized for low-Earth orbit satellite signals using a multimodal learning transformer model (OI-MLT). The proposed method incorporates symmetry-aware design by exploiting the inherent time–frequency structural characteristics of LEO satellite signals and the spatially distributed topology of interference sources. An optimized model for distributed interference sources is developed, and multimodal information of spectra and numerical values is processed in parallel through the self-attention mechanism. This approach effectively addresses the problem of dynamic matching between the interference signal and target signal in high-speed LEO scenarios, as well as high-precision interference synchronization under time-varying channels. Experimental results demonstrate that this technique enhances the precision of frequency tracking, reduces the time required for synchronization establishment, and improves the interference success rate by 27.52% on average compared with existing methods.

Symmetry doi: 10.3390/sym18050722

Authors: Wang Zhang Fuquan Zhao Zongwei Liu

Under the trend of AI-defined vehicles, multi-agent collaboration has become the core feature for intelligent vehicles to deliver superior user experience (UX). Traditional linear and independent evaluation methods can no longer adapt to the new technical characteristics and logic. Taking the agents of four functional domains—intelligent driving, intelligent cockpit, intelligent vehicle control, and intelligent connectivity—and their cross-domain collaborative relationships as research objects, this study constructs a UX evaluation index system consisting of five primary indicators and 14 secondary indicators. Innovatively, the analytic network process is adopted for indicator weight allocation, which effectively characterizes the interdependencies among indicators caused by multi-agent collaboration. Meanwhile, the coupling coordination theory is introduced to construct a comprehensive UX index, enabling quantitative evaluation of the balanced development level across the five dimensions. The results show that in intelligent vehicle UX, excellence in a single dimension does not equal excellent overall UX. Only through the collaborative upgrading of multiple agents and balanced development of the five dimensions can the comprehensive UX be maximized. This study further reveals the UX mechanism of multi-agent collaboration in intelligent vehicles and determines the optimal collaborative evolution path based on the dynamic programming algorithm, providing theoretical support and practical guidance for automakers in rational product development planning.

Symmetry doi: 10.3390/sym18050721

Authors: Zhichao Lu Shiyou Lin Tingting Hu

In this paper, we investigate the partial approximate controllability of a class of fractional diffusion control systems with three-parameter damping and nonlinear perturbations. First, based on the theory of (μ,ν,ξ,e,k)-resolvent families developed in our previous work, we define the mild solution of the system. Then, by constructing a proper objective functional and using the strict convexity, we prove the existence and uniqueness of the minimal norm control. Furthermore, employing the Arzelà–Ascoli theorem and variational inequalities, we establish the precompactness of the solution family and derive the key controllability estimate. Finally, we provide an example to illustrate the effectiveness of our theoretical results.

Symmetry doi: 10.3390/sym18050720

Authors: Bing Jiang Kaiyu Qin Yu Wu

The cooperative airdrop of UAV swarms by multiple transport aircraft creates a large-scale multi-agent planning problem. The mission involves heterogeneous aircraft, multi-visit airdrop areas, strict time windows, and threat-aware flight paths. To address these challenges, this work develops an integrated framework for both global task allocation and real-time replanning in complex three-dimensional operational environments. First, for the combinatorial optimization of task execution sequences across multiple aircraft, a static task assignment method is proposed. This method employs a Hybrid-encoding Constrained Black-winged Kite Algorithm (HCBKA), which incorporates optimization metrics such as mission execution time, completion rate, and load-balancing symmetry among aircraft. The HCBKA aims to find a task assignment scheme that achieves a comprehensive optimum across multiple objectives through efficient model solving. Second, to handle potential real-time dynamic changes during mission execution, a rapid-response and generalizable replanning mechanism is developed. This mechanism utilizes an event-triggered strategy based on a Time-window aware Dynamic Auction Algorithm (TDAA). It ensures that the system can promptly initiate and execute online task reallocation in response to contingencies such as changing mission requirements or losses within its own drone swarm, thus maintaining the adaptability and robustness of the overall plan. Simulation results show that the proposed framework produces high-quality global solutions and maintains strong robustness under dynamic changes. The approach provides an effective and scalable solution for coordinated multi-aircraft swarm airdrop missions.

Symmetry doi: 10.3390/sym18050719

Authors: Jun Liu Xinwei Li Lingyun Zhou

To mitigate local congestion and address the adaptability limitations of traditional static routing under dynamic traffic, this paper proposes an end-to-end routing method based on a Graph Attention Network (GAT), termed Congestion-Aware Graph Attention Routing (CA-GAR). To alleviate the issue of local optima in traditional heuristic iterative optimization, we design a dynamic link cost optimization algorithm with multi-start parallel exploration. This algorithm employs a ”penalty–reselection–reward” closed-loop feedback mechanism, performing global searches from multiple random initial states to generate a high-quality, empirically near-optimal cost matrix as supervised labels. Building on this, CA-GAR leverages a multi-head attention mechanism to adaptively aggregate high-order topological features of nodes and edges, and incorporates a staged hierarchical hyperparameter optimization strategy to map real-time network states to link costs. Simulation results demonstrate that CA-GAR outperforms traditional static routing under light, medium, and heavy loads. Under high-load burst conditions, the method exhibits effective congestion avoidance capability, reducing end-to-end delay by approximately 50% and lowering the packet loss rate to as low as 2%. Compared with QLRA, CA-GAR shows promising performance in multi-path traffic splitting and possesses robust fast rerouting capabilities during node failures, thereby achieving intelligent traffic distribution and global load balancing.

Symmetry doi: 10.3390/sym18050718

Authors: Abdelnaby Al-Husainy Alhawarat Rohaim Assar Elshafey

Home automation is undergoing a paradigm shift from connected IoT environments with rule based control to intelligent homes exhibiting ambient intelligence and proactive adaptation. Artificial intelligence, privacy-preserving sensing, and converging connectivity standards are the primary forces driving this transition. This systematic literature review synthesizes the technological foundations, architectural developments, emerging paradigms, and socio-technical challenges characterizing the next generation of smart homes, evaluated against the original Ambient Intelligence (AmI) vision. Following PRISMA 2020 guidelines, searches were conducted across four databases—IEEE Xplore, ACM Digital Library, Scopus, and Web of Science—covering studies published between January 2020 and June 2025. From 3450 records, 113 studies were selected through a two-reviewer screening procedure with inter-rater reliability assessments. Quality was assessed using a modified JBI Critical Appraisal Checklist, and findings were synthesized through thematic analysis. Three converging technological pillars were identified: multi-modal privacy-preserving sensing including mmWave radar; a hierarchical cloud-edge TinyML intelligence engine; and unified connectivity through the Matter/Thread standard. Emerging paradigms include LLM-based cognitive orchestration, hyper-personalization, Digital Twin simulation, and grid-interactive prosumer energy management. Realizing that the intelligent home vision requires addressing the privacy–security–trust trilemma, algorithmic bias, system reliability, and human–agent collaboration, a research roadmap encompassing explainable AI, privacy-by-design, lifelong learning, and standardized ethical auditing is proposed.

Symmetry doi: 10.3390/sym18050717

Authors: Jiajian Li Mingrui Li Hanshen Li

Proximal Policy Optimization (PPO) is widely adopted for robotic continuous control, yet it can suffer from insufficient exploration and unstable policy updates in high-dimensional action spaces. This paper proposes Adaptive Exploration Proximal Policy Optimization (AE-PPO), an enhanced PPO framework that integrates (i) adaptive clipping, which adjusts the clipping range according to the observed magnitude of policy updates to better balance stability and learning progress, (ii) adaptive entropy regularization, which schedules the entropy weight across training to maintain effective exploration while avoiding excessive randomness. AE-PPO is evaluated on standard MuJoCo continuous control benchmarks (e.g., Walker2d, HalfCheetah, and Humanoid) and compared with PPO and representative baselines such as Trust Region Policy Optimization (TRPO) and Soft Actor Critic (SAC). The results show that AE-PPO achieves faster convergence and an improved final performance with reduced training variance, demonstrating more stable and efficient learning in challenging high-dimensional tasks.

Symmetry doi: 10.3390/sym18050716

Authors: Iryna Chernega Roman Dmytryshyn Zoriana Novosad Serhii Sharyn Andriy Zagorodnyuk

We consider Banach spaces ℓp(C),1≤p<∞, where the index set C is the classical Cantor set and study various groups of symmetries of ℓp(C), associated with the binary representation of C. The main purpose of the paper is the investigation of polynomials on ℓp(C) that are symmetric (i.e., invariant) with respect to the constructed groups G. We are interested in finding systems of generators of algebras of G-symmetric polynomials for different groups G and we discuss possible applications of G-symmetric polynomials to highly composite physical systems. The generators are useful for descriptions of spectra of algebras of G-symmetric analytic functions on ℓp(C), and for the construction of some nontrivial complex homomorphisms of these algebras. Finally, we establish the topological transitivity and hypercyclicity of some shift-like operators on ℓp(C) and its subspaces, and translation operators on algebras of symmetric analytic functions on ℓp(C).

Symmetry doi: 10.3390/sym18050715

Authors: Sung Bum Park Ji Eun Kim

Let U,V⊂H be bounded C1 domains, and let f be quaternion-valued on U×V. We study the mixed Cauchy–Fueter system DxLf=0 and fDyR=0 on product domains by iterating the classical one-variable Borel–Pompeiu formulas in an order consistent with quaternionic multiplication. Under closure regularity on U¯×V¯, we prove an iterated representation formula and show that, in the biregular case, the boundary contribution reduces to the distinguished boundary ∂U×∂V. This leads to a distinguished boundary transform, TU,V, on continuous boundary data. We prove that TU,V maps C(∂U×∂V;H) into C∞(U×V;H), establish compact subset estimates for mixed real derivatives, and derive a local approximation theorem within the transform range by finite sums of separated one-variable Cauchy transforms. The analysis is restricted to this representation framework. In particular, the paper does not address a general solvability theory for the mixed inhomogeneous system and does not characterize the full range of TU,V.

Symmetry doi: 10.3390/sym18050714

Authors: Giovanni Montani Andrea Valletta

We analyze the Taylor expansion of the metric f(R) gravity in the Jordan frame around the General Relativity limit, expanding in the small deviation (ϕ−ϕ0) with ϕ0=1. By relating the scalar–tensor representation to the original f(R) formulation, we derive constraints on the expansion parameters from the observed value of the present-day ΛCDM (Λ Cold Dark Matter) deceleration parameter and from cosmological bounds on the variation of Newton’s constant. We show that these requirements imply that the scalar degree of freedom must have a mass exceeding the Hubble scale by several orders of magnitude. This result challenges the common assumption that the scalar mode can drive cosmological dynamics with a mass of order of the Hubble constant H0. We provide a dynamical interpretation of this hierarchy by emphasizing that a proper definition of the scalar mass, in a field-theoretical sense, requires an adiabatic separation between background evolution and perturbations, which naturally leads to a super-Hubble mass scale.

Symmetry doi: 10.3390/sym18050712

Authors: Jichao Li Luyao Chen Chengpeng Li

To enhance the performance of the Social Network Search (SNS) algorithm in solving complex numerical optimization problems, this paper proposes a Multi-strategy Enhanced Social Network Search (MESNS) algorithm. The original SNS simulates human social behaviors through four decision-making emotions—imitation, conversation, disputation, and innovation—to perform population-based search. However, its uniform emotion selection mechanism and purely random interaction strategy may reduce convergence efficiency and weaken exploitation capability, particularly in the later stages of optimization. To overcome these limitations, MESNS incorporates three improvement strategies. First, an adaptive decision-making emotion selection mechanism is developed to dynamically adjust the probabilities of exploration and exploitation behaviors according to the iteration progress, thereby promoting a more symmetric and coordinated search transition over time. Second, an elite-guided communication strategy is introduced to enhance information propagation by integrating high-quality individuals into the interaction process, which improves convergence while maintaining population diversity. Third, a dynamic interaction radius adjustment mechanism is designed to adaptively regulate the search step size, achieving a better balance and dynamic symmetry between global exploration and local refinement. Extensive experiments are conducted on the IEEE CEC2014, CEC2017, and CEC2022 benchmark suites under multiple dimensional settings. The results demonstrate that MESNS achieves superior optimization accuracy, faster convergence speed, and improved solution stability compared with several state-of-the-art metaheuristic algorithms. Furthermore, the proposed algorithm is successfully applied to the three-dimensional wireless sensor network deployment optimization problem, where it produces a more uniformly distributed and spatially balanced sensor layout, reduces coverage holes and redundant overlaps, and thus exhibits desirable symmetry in deployment structure and sensing coverage. These findings indicate that MESNS is an effective and competitive optimization framework for complex global optimization tasks with both theoretical significance and practical value from the perspective of symmetry.

Symmetry doi: 10.3390/sym18050713

Authors: Xiaolei Zhang Qun Liu Qi Li Dehui Wang Hongguang Jia

Time-series clustering is a prominent research area with extensive practical applications. Given the complexity and diversity of modern time-series data, this study proposes a novel time-series clustering method based on inter-network heterogeneity. First, each time-series is converted into a network by using two types of time-series segmentation techniques. Second, an inter-network clustering approach based on graph spectral theory is introduced: we calculate the total variation (TV) distance between the empirical spectral distributions of each network and identify distinct clusters using a hierarchical clustering algorithm. From the perspective of symmetry, networks constructed from similar time-series tend to exhibit comparable spectral structures, which reflect the underlying structural symmetries of their dynamics. Differences in spectral distributions correspond to symmetry breaking among networks, providing an effective mechanism for distinguishing heterogeneous time-series patterns. Our method effectively preserves more distinctive features inherent in the original time-series. To evaluate the performance of the proposed method, simulation studies are conducted, including the recognition of both stationary and non-stationary sequences. The method also performs well on real-world datasets, such as stock closing prices. These results demonstrate that our approach can handle non-stationary sequences and identify the intrinsic correlations in time-series.

Symmetry doi: 10.3390/sym18050711

Authors: Yijun Ling Wenjin Zhao Mengjia Zhao Jie Yang

Physics-guided computational imaging typically aggregates data fidelity, geometric reconstruction, and sensor consistency into a single scalar loss. In low signal-to-noise ratio (low-SNR) non-line-of-sight imaging, this centralized approach creates asymmetric gradient conflicts where the dominant constraints suppress physically meaningful updates. We propose treating multi-constraint training as a gradient coordination problem rather than scalar balancing. Our framework coordinates heterogeneous objectives through branch-wise gradient routing: soft conflict projection (PCGrad), hard physical constraint enforcement (PhysGuard), learnable sensor calibration, and a staged training protocol that decouples representation learning from nuisance parameter estimation. On held-out test scenes, the fully staged model improved the peak signal-to-noise ratio (PSNR) from 19.09 dB to 20.49 dB and the structural similarity index (SSIM) from 0.67 to 0.71 over the baseline, with consistent gains across the 48, 28, and 25 dB SNR levels. Qualitative evaluation on seven real-world scenes indicates sharper structure recovery and fewer artifacts. In this NLOS setting, gradient-level coordination is more reliable than scalar aggregation under heterogeneous constraints.

Symmetry doi: 10.3390/sym18050710

Authors: Jianbo Huang Gaoming He Zhicheng Liao Mengdi Hou

Accurate prediction of ultra-high-performance concrete (UHPC) compressive strength is essential for optimizing mixture design and reducing experimental iterations. Existing machine learning approaches suffer from limited algorithm diversity, insufficient statistical validation, and inadequate uncertainty quantification. This study presents a comprehensive framework through systematic evaluation of 20 algorithms across seven categories on 863 experimental observations. Six physically meaningful composite features (such as water-cement ratio, total binder content, and fiber aspect ratio) are engineered to capture intrinsic material relationships, with the Boruta algorithm employed for feature selection. Statistical robustness is ensured through 30 repeated experiments analyzed using both frequentist (p-value, effect size, 95% CI) and Bayesian approaches. CatBoost achieves optimal performance (R2 = 0.8979 ± 0.0239, RMSE = 10.58 ± 1.45 MPa), with curing age, sand content, and steel fiber volume identified as dominant predictors through multi-perspective interpretability analysis integrating SHAP, ALE, permutation importance, and LIME. External validation on 810 independent samples yields R2 = 0.5923 (RMSE = 25.68 MPa) under significant cross-dataset conditions, with performance reduction attributed to feature availability differences and distribution shift. Comprehensive uncertainty quantification yields prediction uncertainty of 3.48%, substantially below previously reported thresholds. The proposed framework offers practitioners a reliable tool for UHPC mixture screening while maintaining prediction confidence for structural engineering applications.

Symmetry doi: 10.3390/sym18050708

Authors: Xiaoyan Zhang Bin Wang Yu Shao Jianfeng Wang

To address the shortcomings of the information acquisition optimizer (IAO)—specifically its susceptibility to premature convergence, insufficient exploitation capability during later stages, and population diversity decay when applied to complex optimization problems—this paper proposes a multi-strategy improved information acquisition optimizer (MIIAO). Centered on balancing exploration and exploitation capabilities during the search process, this method incorporates several key strategies: an adaptive differential perturbation factor is designed to dynamically adjust the search step size; an elite-guided information acquisition mechanism is introduced to enhance convergence efficiency within high-quality regions; a diversity-based restart perturbation strategy is integrated to mitigate the risk of entrapment in local optima; and a mirror boundary handling technique is adopted to bolster the resilience of solutions near boundaries and improve the effectiveness of searching within the feasible domain. To validate the efficacy of the proposed method, MIIAO was applied to the CEC2014, CEC2017, and CEC2022 benchmark test suites and systematically compared against various representative intelligent optimization algorithms. Furthermore, the method was applied to multi-threshold image segmentation tasks based on Otsu’s criterion. Experimental results demonstrate that MIIAO consistently exhibits superior solution accuracy, convergence speed, stability, and statistical ranking across various dimensions and a diverse range of complex test functions; the results of the Wilcoxon rank-sum test and Friedman mean ranking further substantiate its comprehensive performance advantages. In the image segmentation experiments, MIIAO achieved superior Otsu objective function values across multiple test images and under various threshold settings, while also demonstrating higher segmentation quality and greater robustness across evaluation metrics such as PSNR, SSIM, and FSIM. In summary, the proposed MIIAO effectively enhances the original IAO’s global search capability, local exploitation capability, and ability to maintain population diversity, thereby demonstrating significant potential for practical application in both numerical optimization and multi-threshold image segmentation tasks.

Symmetry doi: 10.3390/sym18050709

Authors: Wang Zhang

Multi-threshold image segmentation is an important research topic in the fields of computer vision and image processing. Its core objective is to efficiently determine the optimal threshold combination within a high-dimensional and complex search space. However, as the number of thresholds and image complexity increase, the computational cost of traditional exhaustive search methods grows exponentially. Meanwhile, conventional swarm intelligence algorithms often suffer from unstable convergence, premature stagnation, and parameter sensitivity when dealing with high-dimensional composite functions. To address these issues, this paper proposes an enhanced optimization algorithm termed the Parrot Optimizer with Artistic Design Strategy (PO-ADS). The proposed method constructs a multi-strategy cooperative optimization framework that integrates an Evolution Feedback–Based Adaptive Control Strategy (EFACS), a Multi-Operator Cooperative Evolution Strategy (MOCES), and an Artistic Design Strategy (ADS). These strategies enable dynamic parameter adjustment, adaptive balance between global exploration and local exploitation, and structured perturbation enhancement mechanisms. Experimental results on the CEC2020 and CEC2022 benchmark suites demonstrate that PO-ADS significantly outperforms seven state-of-the-art optimization algorithms across different dimensional settings in terms of optimization accuracy, convergence speed, and stability. The Friedman test results show that, on the CEC2020 benchmark suite, PO-ADS achieves average ranks of 1.72 (30-dimensional) and 1.85 (50-dimensional), both statistically superior to the comparative algorithms. Furthermore, PO-ADS is applied to multi-threshold image segmentation based on the Otsu criterion. The results indicate that the proposed method achieves optimal or near-optimal performance in terms of SSIM, PSNR, FSIM, and objective function values. Overall, the experimental findings confirm that PO-ADS not only possesses strong numerical optimization capability but also demonstrates robust and practical applicability in real-world image segmentation tasks.

Symmetry doi: 10.3390/sym18050707

Authors: Rui Zhang Kai Zhou Shoufeng Shen Jiafu Pang Zhiyi Cao

This paper proposes a hybrid PINN + LSTM framework for the high-precision solution of malware propagation dynamics in wireless sensor networks. A seven-compartment SVEHLQR model is developed to capture this complex transmission process. To overcome the limitations of standard physics-informed neural networks (PINNs) in long-term prediction, including gradient vanishing and error accumulation, we integrate LSTM’s temporal memory capability into the PINN architecture. Comprehensive comparisons are conducted among the proposed PINN + LSTM, standard PINN, and Fourier PINN, using the fourth-order Runge–Kutta method as the benchmark. Experimental results demonstrate that PINN + LSTM significantly outperforms both baseline methods, achieving an average relative error of 3.88×10−3 compared to 7.20×10−2 for PINN and 2.81×10−1 for Fourier PINN, representing a 94.6% accuracy improvement over PINN. These results validate that incorporating LSTM’s recursive memory mechanism enables the accurate and efficient solution of complex time-dependent dynamical systems. Additionally, the model’s robustness is verified under 1%, 5%, and 10% Gaussian noise. PINN + LSTM maintains extremely low relative errors, not exceeding 0.0049, and outperforms PINN and Fourier PINN significantly, confirming its strong noise immunity and stable dynamics learning ability in realistic environments.

Symmetry doi: 10.3390/sym18050706

Authors: Zhiqiang Liu Guodong Li Pingtao Gao Honglin Liu Hongzhi Wang Haotian Fu Kangfei Zhang Guodong Zeng

To address the technical challenges of difficult deduction, limited field measurement, and ambiguous instability determination of roof separation critical values in mining roadways within the weakly cemented coal-bearing strata of Xinjiang, this paper proposes a discrete element method that integrates the fracture of anchor bolt and anchor cable support materials with the damage degree of the surrounding rock. Taking a specific mine in the Hosh Tolgay coalfield as the research object, a systematic study was conducted. The research process was as follows. (1) Model parameter calibration was performed. Intact rock parameters were obtained through laboratory basic mechanical tests, and rock mass parameters were corrected based on reduction empirical formulas and the Hoek–Brown criterion. Numerical model verification showed that the errors between the simulated and theoretical values of the elastic modulus, compressive strength, and tensile strength of the rock mass were all less than 10%, indicating that the corrected parameters are reasonable. (2) The critical damage values of the rock mass considering a non-constant confining pressure environment were proposed. Through triaxial compression simulations, the differential evolution patterns of rapid damage increase in sandy mudstone under low confining pressure and stable damage accumulation in coal were revealed, thereby clarifying the damage thresholds for rock mass instability under different confining pressures. (3) A large-scale model was established to analyze the evolution laws of the fracture field, support field, and displacement field of the roadway surrounding rock. A comprehensive determination method for the instability of the roof anchored bearing structure was proposed. By comparing the damage thresholds of the scaled rock mass and the roadway surrounding rock and analyzing the fracture conditions of the roadway support system, a dual-criterion consisting of surrounding rock damage and support material fracture was constructed. Based on this criterion theory, the critical values for deep and shallow separation were obtained. The research results indicate that the evolution patterns of damage in coal and sandy mudstone differ with confining pressure. The sandy mudstone layers in the shallow part of the roof are more sensitive to mining-induced unloading disturbances. Consequently, the surrounding rock damage and support fracture of the mine roof exhibit distinct distribution characteristics: the dominant failure of the roadway is shear failure, with wide-range coalescence of shallow fractures and gradual development of deep fractures, alongside the concentrated failure of shallow anchor bolts and partial failure of deep anchor cables. Based on the instability state of the roof monitoring zones, the critical value for shallow separation was determined to be 90.7 mm, and the critical value for deep separation was 129.03 mm. These results are very close to the field measured values, verifying the engineering applicability of the method. This paper reveals the damage characteristics of the rock mass and surrounding rock in weakly cemented strata, as well as the mechanism of roof separation initiation and evolution. The proposed method for determining critical values provides a scientific and feasible practical reference for the support optimization and monitoring and early warning of roadway roofs in weakly cemented strata, possessing significant engineering value for ensuring safe and efficient mine production.

Symmetry doi: 10.3390/sym18050705

Authors: Fake Jiang Shucheng Huang Mingxing Li

To address the bottlenecks of accuracy in head pose estimation caused by occlusion and rotational representation ambiguities, we propose Deep6DHead, a 6-degree-of-freedom (6DoF) head pose estimation method based on deep feature enhancement. This method innovatively integrates RGB and depth information to construct a four-channel input and achieves feature fusion of RGB-D through a dual-branch network. First, a Squeeze-and-Excitation (SE) module adaptively weights the depth geometric features of key anatomical regions to achieve channel recalibration. Second, based on the 6DoF rotation representation framework, we introduce an anatomical constraint loss using the nasal bridge normal. This constraint corrects rotation deviations caused by noise by enforcing consistency in local geometric orientation. Finally, the model outputs the rotation matrix end-to-end for final pose estimation. Experiments on the 300W-LP, BIWI, and AFLW2000 datasets demonstrate that our method significantly improves robustness and accuracy, particularly under extreme head poses. Notably, it achieves state-of-the-art performance on the roll axis (lowest error: 2.05) and a competitive overall MAE of 3.45, providing an effective solution for head pose estimation in complex real-world scenarios including extreme viewing angles.

Symmetry doi: 10.3390/sym18050704

Authors: Jiexi Liu Yi-Jian Du

In this note, we study the Hopf-algebra-based (HAB) formula of Yang–Mills-Scalar (YMS) amplitudes, which expands a YMS amplitude with massive scalars as a combination of propagator matrices that mix massless scalars corresponding to gluons with the original massive scalars. We propose a recursive formula that conveniently expresses the HAB formula. In this formula, gluons are converted into massless scalars. Thus it expresses a YMS amplitude with massive scalars by amplitudes with fewer gluons, massive scalars and massless scalars. We verify this formula by using the soft behavior of amplitudes. We further show the equivalence between the massless limit of the HAB formula and an earlier proposed recursive expansion formula through explicit calculations on amplitudes with one and two gluons.

Symmetry doi: 10.3390/sym18050703

Authors: Dat Anh Huong Anh Ngoc Trung Minh

The chemical composition of basil essential oil is influenced by plant developmental stage, which alters the relative distribution of volatile constituents and their functional properties. In this study, we investigated developmental-stage-dependent changes in the essential oil composition of Ocimum basilicum and evaluated their relationship with nitric oxide (NO) inhibitory activity and estragole exposure. Essential oils were obtained by hydrodistillation and analyzed by gas chromatography–tandem mass spectrometry (GC–MS/MS), resulting in the identification of 54 volatile compounds representing 98.13–98.97% of the total composition. Estragole remained the dominant constituent, ranging from 70.58% to 85.55%, with the lowest proportion at the flowering stage (Day 85). In contrast, minor constituents, including eucalyptol (2.41–3.77%), β-ocimene (0.52–1.98%), and methyleugenol (~2.00%), increased during flowering. NO inhibitory activity in lipopolysaccharide (LPS)-stimulated RAW 264.7 macrophages was strongest at Day 85. Estimated Daily Intake (EDI) ranged from 0.4215 to 1.1005 µg kg−1 bw day−1, and Margin of Exposure (MOE) ranged from 2999 to 7830. These findings indicate that developmental stage influences composition, activity, and exposure. From a compositional perspective, the observed redistribution among major chemical groups reflects a structured balance that can be interpreted within a symmetry-related framework in multicomponent systems.

Symmetry doi: 10.3390/sym18050702

Authors: Yusra Taj Sarfraz Nawaz Malik Alina Alb Lupaş

We introduce and study a new subclass of Janowski-type harmonic close-to-convex functions in the open unit disk defined via the Jackson q-derivative operator. The structure of the operator naturally reflects certain symmetric properties in the analytic representation of the considered harmonic mappings. By applying subordination techniques, we establish sufficient conditions for sense-preserving close-to-convexity and distortion estimates. The extreme points of the class are determined, and its topological properties are examined, showing that the class is convex and compact. We further obtain the radius of starlikeness and prove that the class is closed under convolution. Moreover, as q→1−, the operator reduces to the classical derivative, and our results recover several known results in the existing literature, demonstrating that the present work extends and generalizes earlier findings.

Symmetry doi: 10.3390/sym18050701

Authors: Dandan Wang Hao Li Jun Ma

The realization of the value of health big data relies on the coordinated cooperation among patients, the government, and data users. Enhancing the symmetry and balance between patient participation and the compliant use of data by data users is a critical link. This paper constructs a tripartite evolutionary game model and employs MATLAB R2023a simulation to analyze the impact of factors such as initial willingness, compliance costs, and penalties for violations on the strategic choices of the game players and the evolution of the system. The findings reveal that: (1) Patient participation is a key condition for achieving an ideal equilibrium in the system. (2) The data service income from participating in data provision and the costs associated with privacy breaches are critical factors influencing patients’ strategic choices. (3) Penalties for violations are a crucial factor in ensuring that data users choose compliant utilization; however, when compliance costs are high, their constraining effect may be somewhat diminished. (4) Enhancing regulatory efficiency is the future direction for government departments. Based on these findings, countermeasures and suggestions are proposed, including trust building, technological innovation and differentiated supervision, and constructing trusted data spaces, to provide references for health big data governance.

Symmetry doi: 10.3390/sym18040700

Authors: Simon Gluzman Zhanat Zhunussova Akylkerey Sarvarov Vladimir Mityushev

Classical homogenization theory treats critical exponents as universal quantities depending only on spatial dimension, but recent evidence shows that this assumption fails for continuum composites once the mechanism of randomness generation is taken into account. We synthesize three complementary frameworks—structural approximation, structural sums, and self-similar renormalization—to develop a unified geometric theory of criticality in random composites. Dilute-regime expansions for the effective conductivity and shear modulus are expressed in terms of structural sums whose ensemble statistics depend sensitively on the randomness protocol. To bridge the dilute and critical regimes, we employ self-similar factor approximants, iterated-root approximants, additive approximants, and renormalization schemes based on minimal-difference and minimal-sensitivity conditions, combined with Borel summation. For maximally disordered protocols P(τ), the conductivity index s and the elasticity index S fall within comparable numerical ranges, indicating a shared geometric origin and spectral response to the continuous breaking of translational symmetry. A regular periodic arrangement of inclusions (τ=0) possesses full discrete translational symmetry; as a stochastic protocol P(τ) is applied (τ increases), this symmetry is gradually degraded until statistical chaos is reached. For instance, the parameter τ can be considered as a time of stirring. During this evolution, the system traverses a continuous spectrum of critical indices, s=s[P(τ)], which encodes the geometric and topological memory of the initial ordered state. It is established that the classical “universality” of percolation corresponds to a fixed point τ within a broader manifold of protocol-dependent critical behaviors. The framework developed here provides a coherent basis for inverse design, diagnostics, and classification of random composites by their disorder history, offering a geometric alternative to the universality paradigm.

Symmetry doi: 10.3390/sym18040699

Authors: Tomomi Imai Shumpei Terada Osamu Kitagawa Magdalena Kwiatkowska Alicja Wzorek Vadim A. Soloshonok

In this study, the applicability of achiral column chromatography—including both medium-pressure liquid chromatography (MPLC) and classical gravity-driven techniques—was evaluated as a laboratory method for enantiomeric enrichment of scalemic (non-racemic) samples of axially chiral compounds. As model substrates, 3-(ortho-substituted-phenyl)quinazolin-4-one derivatives were employed. The results confirmed that self-disproportionation of enantiomers (SDE), occurring during column chromatography (SDEvCC), enabled the efficient isolation of enantiomerically pure fractions, with MPLC demonstrating particularly high effectiveness. Additionally, the parameters governing gravity-driven column chromatography were systematically optimized, with particular attention to variables such as eluent type and concentration, stationary phase composition, sample preparation protocol, and solvent purity. Furthermore, leveraging known crystallographic data and quantum chemical calculations based on Density Functional Theory (DFT), a molecular association mechanism was proposed to elucidate the physicochemical basis of the SDE phenomenon.

Symmetry doi: 10.3390/sym18040698

Authors: Wenhao Wang Kaiwen Chen Wenjun Luo Nan Zhou Yanyi Cao

Symmetric matrix factorization (SMF) plays an important role in clustering and representation learning. Nevertheless, most existing SMF-based approaches are formulated as non-convex optimization problems, which often leads to unstable convergence and high computational costs. In this paper, we develop a fast unconstrained convex symmetric matrix factorization framework, termed FUCSMF, for semi-supervised learning. By incorporating label information into the symmetric factorization formulation, the proposed model is transformed into a convex objective, which guarantees global optimality and enables efficient optimization using standard unconstrained solvers. To further improve scalability, a bipartite graph structure is introduced into SMF from a hypergraph-inspired perspective, significantly reducing the computational burden. The resulting computational complexity is reduced to O(nmd), which is substantially lower than the O(nmd+m2n+m3) complexity required by existing bipartite graph-based methods, where n, m, and d denote the numbers of samples, anchor points, and feature dimensions, respectively. In addition, we propose a correntropy-based graph construction strategy to alleviate the sensitivity of conventional adaptive neighbor bipartite graph methods. Extensive experiments on six benchmark datasets, involving comparisons with eleven state-of-the-art methods, demonstrate that FUCSMF achieves superior clustering performance while requiring significantly less computational time. Empirical results further show that the proposed method converges rapidly, typically within ten iterations.

Symmetry doi: 10.3390/sym18040697

Authors: Yuanhao Wu Renbao Li Chunyan Gao

Transcriptional and translational inhibition are fundamental regulatory mechanisms in gene networks, governing diverse processes from viral replication to neuroplasticity. Two-component genetic oscillators based on the “activator–repressor” motif serve as ideal models for studying biological rhythms due to their simplicity and rich dynamics. However, systematic theoretical comparisons of distinct inhibitory mechanisms—particularly using inhibition strength as a control variable—remain lacking. Addressing this gap, we present a comprehensive bifurcation analysis of the post-translational repression model, proving the existence and uniqueness of its positive equilibrium, deriving Hopf bifurcation conditions, and identifying critical parameter ranges for sustained oscillations. Using inhibition strength as a key comparator, we systematically contrast transcriptional and post-translational repression, elucidating how different inhibitory mechanisms modulate oscillation initiation and amplitude. We further reveal distinct symmetry–asymmetry patterns in their bifurcation dynamics: transcriptional repression exhibits asymmetric bistable regimes, while post-translational repression manifests narrow, nearly symmetric oscillatory intervals. This unified analytical framework not only advances the theoretical understanding of two-component genetic oscillators but also provides a generalizable paradigm for dissecting complex gene regulatory dynamics.

Symmetry doi: 10.3390/sym18040694

Authors: Yan Li Yanfei Bai

Digital electricity meters display readings via digits, but accurate image-based recognition faces a key challenge: the frequent omission of decimal points creates a critical asymmetry between the visual image and its true semantic meaning. To address this visual-semantic asymmetry, we propose an improved YOLOv10n approach incorporating cascaded Visual-Semantic processing. We introduce a Reparameterized Convolution Single-Shot Aggregation (RCSOSA) module and a SimAM attention mechanism to enhance feature extraction, and employ Normalized Wasserstein Distance (NWD) Loss to boost small-target detection. To rectify the visual-semantic asymmetry, we introduce domain-specific format rules based on power industry standards (taking GB/T 17215-2018 as an example) to provide structural constraints for digit recognition. Experimental results show superior performance with 0.870 precision, 0.932 mAP50, and 116 FPS inference speed, outperforming reference models in both precision and efficiency for real-time meter inspection.

Symmetry doi: 10.3390/sym18040695

Authors: Chunxiao Yan Shuanhong Wang

Firstly, we define and study the notions of a smash product for actions of multiplier left Hopf algebras on algebras and of an integral on such smash products. Then we construct an analogue of Radford’s biproduct in the framework of multiplier left Hopf algebras under assumption of a multiplier left Hopf algebra having an anti-bialgebra homomorphic left antipode. Finally, we study a duality theorem for smash products of a left Hopf algebra of dimension n which is a special multiplier left Hopf algebra.

Symmetry doi: 10.3390/sym18040696

Authors: Krassimir Atanassov Veselina Bureva Tania Pencheva

In the current research, we introduce two operations of a “symmetric difference” type over three-dimensional extended index matrices, and investigate some of their basic properties. An example of the implementation of symmetric difference-type operations is presented in the field of relational databases, aiming to demonstrate the operations’ efficiency by comparing sets with dissimilar attributes.

Symmetry doi: 10.3390/sym18040693

Authors: Tan Lu Junlin Wu Hao Qiu

Matter–antimatter asymmetry is a fundamental question in both astronomy and particle physics. Investigating antimatter is of great interest for testing the potential explanations of matter–antimatter asymmetry in our Universe. In relativistic heavy-ion collisions, the extremely high energy density and temperature are similar to the early Universe shortly after the Big Bang. In this paper, we review the recent progress in antimatter search and study heavy-ion collisions, with a focus on the RHIC-STAR and LHC-ALICE experiments, particularly the newly observed antimatter hypernuclei H¯Λ¯4 and He¯Λ¯4. The statistical thermal model and the coalescence production model can quantitatively describe the production yields and yield ratios, and the yield measurements of H¯Λ¯4, He¯Λ¯4 and their matter counterparts indicate the existence of spin-excited states of these (anti)hypernuclei. Furthermore, new measurements of the lifetimes of H¯Λ¯3, H¯Λ¯4 and their matter counterparts reveal no difference between a particle and its corresponding antiparticle, which validates the CPT theorem.

Symmetry doi: 10.3390/sym18040691

Authors: Yi Mo Xinsuo Li Kunyu Xiang Dengguo Xu

This paper presents an adaptive optimal consensus tracking control scheme for canonical nonlinear multi-agent systems (MASs) with unknown dynamics, employing an actor–critic reinforcement learning (RL) framework. The scheme integrates a sliding mode mechanism to suppress tracking errors and ensure consensus tracking between the followers and the leader. Additionally, optimal control is designed to find a Nash equilibrium in a graphical game. To address the intractability of obtaining an analytical solution for the coupled Hamilton–Jacobi–Bellman (HJB) equation, a policy iteration algorithm is utilized. Within this algorithm, a critic neural network (NN) approximates the gradient of the optimal value function, while an actor NN approximates the optimal control policy. Together, these networks form a compact actor–critic (AC) architecture that achieves optimal consensus tracking. Furthermore, the proposed method guarantees the boundedness of all closed-loop signals while ensuring consensus tracking. Finally, two simulations are conducted to verify the effectiveness and advantages of the proposed method.

Symmetry doi: 10.3390/sym18040692

Authors: Xingyu Lai Minjie Dai Yuhang Luo Xin Song

Optimized dispatch of microgrids is crucial for improving the economic performance and long-term sustainability of modern low-carbon power systems. In particular, accurate economic dispatch modeling for battery energy storage systems (BESSs) is essential for properly evaluating the operational benefits and lifetime costs of microgrids. However, when both battery cycle aging and calendar aging are considered, the resulting scheduling model becomes highly nonlinear, high-dimensional, non-convex, and multimodal, which poses substantial challenges to conventional optimization methods. To alleviate the above problem, a symmetry-guided multi-strategy differential hybrid slime mold algorithm (MDHSMA) is introduced for the day-ahead economic dispatch of microgrids under a refined battery degradation framework. First, a chaotic bimodal mirrored Latin hypercube sampling strategy is designed to exploit symmetry during population initialization, thereby enhancing diversity and improving structured coverage of the search space. Second, a history-driven adaptive differential evolution mechanism is integrated to balance global exploration and local exploitation more effectively during the iterative search process. Third, a state-aware stagnation handling framework is incorporated to maintain population vitality and further improve convergence accuracy and robustness. MDHSMA is evaluated against 12 state-of-the-art optimizers on the CEC2017 and CEC2022 benchmark suites and two representative engineering optimization problems to verify its overall performance. In addition, it is applied to a microgrid case study with refined BESS degradation modeling. The results show that MDHSMA achieves the lowest comprehensive operating cost by effectively coordinating electricity arbitrage and battery life consumption. Moreover, it guides the energy storage system toward shallow charge–-discharge patterns, thereby mitigating accelerated degradation caused by excessive cycling. These results confirm the effectiveness and practical value of the proposed method for sustainable microgrid dispatch in complex nonconvex optimization scenarios.