Symmetry doi: 10.3390/sym18091567

Authors: Shicheng Xie Dandan Wang

In the circulation of health data, the mutual trust dilemma leads to the symmetric problems of “inadequate supply” on the supply side and “impeded flow” on the demand side, which hinder the realization of data value. This paper analyzes the formation mechanism of this dilemma and its resolution pathways from a symmetry perspective. First, based on a literature review, we extract the manifestations of the mutual trust dilemma and construct a two-stage decision-making “vicious circle” model of mutual trust. Unlike conventional single-stage game models, this framework captures the sequential nature of trust decisions—entry in Stage 1 and compliance in Stage 2. Second, we establish a tripartite evolutionary game model involving data suppliers, demanders, and regulators, and employ MATLAB simulations to examine the impact of key parameters on strategic evolution. The findings reveal that: (1) beneficial data utilization scenarios are the prerequisite for participation; (2) strong regulation is the fundamental guarantee for mutual trust; and (3) while increasing penalties, reducing compliance costs, and enhancing trust-related gains can promote compliance, regulatory measures must be carefully balanced to avoid suppressing participation willingness. Accordingly, we propose a four-step progressive mutual trust mechanism comprising motivation activation, cooperation facilitation, performance guarantee, and trust reinforcement. Policy recommendations are offered regarding high-benefit scenarios, trusted data spaces, incentive policies, and data standardization, with the aim of building a trusted health data circulation ecosystem.

Symmetry doi: 10.3390/sym18091566

Authors: Minju Shin Youngjin Moon Sang Ki Lee Hwi-yeol Yun Juwon Song Jiahao Xu Zheng Dong Koon Soon Kim Seung Hwan Han

Bioelectrical impedance analysis (BIA) provides body composition estimates that may complement conventional assessments of dynamic movement asymmetry. This cross-sectional study examined the association between the two in 50 adults aged 20–65 years. Dynamic variables were obtained from gait analysis, electromyography, and isokinetic strength testing, and left–right asymmetry was expressed as an asymmetry index (AI). Spearman correlations were evaluated with Benjamini–Hochberg correction across a pre-specified family of 42 comparisons. Eight associations survived correction, all positive, linking the AIs of leg total body water (TBWleg), intracellular water (ICWleg), and soft lean mass (SLMleg) to stance time and gait speed during dual-task walking and to ankle and knee joint torque, with correlations ranging from 0.376 to 0.476; whole-body indices showed the opposite pattern. Against a composite score derived from these variables, TBWleg AI showed the strongest association (ρ = 0.508) and, in an exploratory analysis, separated participants in the highest tertile from the remainder with an area under the curve of 0.786. These findings support further evaluation of BIA-derived leg water and lean-tissue asymmetry indices as candidate markers of dynamic movement asymmetry; the design does not establish predictive or diagnostic validity.

Symmetry doi: 10.3390/sym18091565

Authors: Faisal Alhwikem Amir Raza Khan Fawwad Hassan Jaskani

Agentic large language model (LLM) networks are increasingly used in safety-critical settings where autonomous agents invoke tools, exchange context, and coordinate decisions. Prompt-injection attacks remain a significant threat to these multi-agent pipelines because they can compromise data flows between agents, bypass instruction hierarchies, and corrupt output integrity. Although defenses against injected prompts and mechanisms for cryptographically verifying model-related computations have been studied independently, no common framework unifies these complementary security perspectives in a protocol suitable for real-time agentic deployments. From the perspective of symmetry, secure inter-agent communication requires the preservation of an invariant integrity relationship between a message at its source and the corresponding message accepted at its destination. A benign communication path therefore exhibits a form of integrity symmetry, whereas prompt injection or message manipulation creates an asymmetric state in which the received payload, its semantic effect, or the receiving model’s internal activation pattern deviates from the trusted reference state. In this paper, we propose SecurePrompt-IntegrityNet (SPI-Net), a prompt-injection-resilient data integrity verification protocol that combines cryptographic attestation with anomaly-aware activation monitoring. SPI-Net provides three closely related mechanisms: a Merkle-tree-based commitment system that verifies the provenance and integrity of data payloads exchanged between agents; a layer-wise Mahalanobis-scoring Activation Anomaly Detector (AAD) that identifies distributional shifts in the intermediate representations of LLMs; and a Trust Propagation Consensus (TPC) mechanism that combines cryptographic and behavioral evidence into per-payload integrity verdicts. In this formulation, the Cryptographic Attestation Module (CAM) tests whether message-level structural symmetry is preserved between the sender and receiver, whereas the AAD detects behavioral symmetry breaking in activation space. Experiments on three multi-agent benchmarks under five adaptive attack strategies show that SPI-Net achieves a 96.8% detection rate with a 1.7% false positive rate, reduces the attack success rate by 94.3% relative to undefended baselines, verifies data integrity with 99.2% accuracy, and introduces only 38 ms of median per-message latency. These results demonstrate that jointly preserving cryptographic integrity symmetry and identifying activation-level asymmetry provides substantially stronger prompt-injection resilience than either verification mechanism alone.

Symmetry doi: 10.3390/sym18091564

Authors: Mieczysław Cichoń Kinga Cichoń

The paper has two main objectives. Firstly, it aims to identify appropriate fractional derivatives for epidemiological models and to investigate their advantages over ordinary derivatives, particularly in terms of the accuracy of epidemic simulations. The choice of a fractional derivative is motivated by its ability to represent the memory effects inherent in epidemiological processes. Different fractional derivatives may induce substantially different memory mechanisms; therefore, the relation between the choice of the derivative and the resulting memory effect is discussed in detail. The results are complemented by stability studies. The obtained uniqueness results provide a theoretical basis for the application of numerical methods, while numerical simulations demonstrate the impact of the fractional-order parameters. Secondly, the paper presents an SIQR epidemiological model incorporating quarantine, fractional derivatives, and multiple delays. The simultaneous use of fractional derivatives and delays makes it possible to account for both distributed memory effects and specific temporal lags within a unified model framework.

Symmetry doi: 10.3390/sym18091563

Authors: Fatimah A. Almulhim Mohammed B. Alamari Ali Laksaci

This paper introduces a novel estimator for the functional regression model with a scalar response variable D and a functional predictor C taking values in a semi-metric space. The proposed estimator is obtained by minimizing the Least Absolute Relative Error (LARE) loss, an asymmetric criterion that measures prediction errors relative to the magnitude of the response variable. The theoretical properties of the estimator are established in the functional time series case by deriving its asymptotic distribution. This result provides a fundamental basis for some statistical inferences, including the construction of confidence intervals and the development of hypothesis tests for the regression operator. Unlike the conventional least absolute deviation or least squares criteria, the absolute relative error criterion offers a scale-invariant measure of prediction accuracy, making it particularly suitable when the response variable exhibits substantial variability. By the relative absolute deviations, the proposed approach reduces the impact of extreme observations, improves robustness to heteroscedasticity and outliers. Thus, approach enhances prediction performance in functional time series. The practical relevance of the proposed methodology is highlighted through extensive simulation experiments and an application to a real-world dataset, proving its computational simplicity, stability and accuracy.

Symmetry doi: 10.3390/sym18091562

Authors: Zhihao Wu Lei Zhang Changqing Zhu Na Ren

Existing watermarking methods for copyright protection of OpenDRIVE high-definition maps have difficulty balancing robustness and geometric accuracy. This study proposes a geometry-constrained robust watermarking algorithm that exploits the parametric representation of OpenDRIVE. From a symmetry perspective, the signed curvature–length feature used by the geometric carriers is invariant under translation, rotation, and ideal positive uniform scaling. This transformation invariance contributes to stable watermark extraction when the map’s global position, orientation, or scale is changed. A multidimensional carrier pool comprising geometric shapes and road attributes is constructed, and absolute quantization index modulation enables orthogonally decoupled embedding across carrier types. Levenberg–Marquardt optimization and topology-aware chain correction are further used to preserve geometric continuity and topological consistency. Experiments show that the algorithm maintains visual imperceptibility and file-size stability while remaining robust to geometric transformations, cropping, and format sanitization. Scenario simulations conducted in esmini revealed no evident degradation of road smoothness in the test scenarios defined in this study, indicating that the algorithm satisfied the map usability criteria under the experimental conditions.

Symmetry doi: 10.3390/sym18091560

Authors: Yilin Li Xin Li Qixian Zhang Zhenyu Liang Ke Sun Aibing Liu Weibing Sun Jintian Bian Xudong Li Shuangquan Li

At detection ranges on the order of kilometers, cat-eye echoes degrade into pixel-level spots and exhibit responses similar to those of compact, high-echo false targets, making it difficult for single-band systems to balance weak-target detection rates with false alarm control. To address this, this paper pairs spatially registered and radiometrically calibrated 808 nm and 905 nm images to form a dual-band detection pair, utilizing the cross-band response asymmetry caused by chromatic defocus for discrimination. Experiments show that cat-eye targets exhibit coupled asymmetry in intensity, scale, and energy distribution, whereas the false targets under test generally maintain approximate symmetry; furthermore, the dominant discriminative information shifts from scale differences to intensity differences as distance increases. Based on this pattern, this paper proposes a range-conditioned “detection–reclassification” framework that dynamically matches appropriate models based on target distance. In this framework, a structurally symmetric, two-branch, multi-scale attention-fused YOLO network is used for candidate target detection, while an explicit asymmetric feature classifier further processes challenging samples to retain weak targets and suppress false alarms. Field experiments conducted at distances ranging from 100 to 2100 m achieved an overall mAP50 of 0.907 and an overall mAP50–95 of 0.475, with the mAP50 improving by 17.7 and 10.8 percentage points, respectively, compared to the 808 nm and 905 nm single-band baselines. The results demonstrate that this framework can effectively balance the detection of weak targets at long ranges with the suppression of false targets under the evaluated conditions.

Symmetry doi: 10.3390/sym18091561

Authors: Nikolay Hinov

Inductor–inductor–capacitor (LLC) resonant converters can produce nearly symmetric waveforms without achieving low-loss switching. This paper separates these properties, using a common framework for a resonant circuit containing two inductive elements and one capacitor. An analytical gain model is combined with a time-domain model that distinguishes positive secondary conduction, an open secondary circuit, and negative secondary conduction. The framework compares the two switching half-cycles, their conduction sequences, the gain at reciprocal normalized frequencies, and the current available for zero-voltage switching (ZVS). Finite magnetizing inductance breaks reciprocal gain symmetry, while a balanced converter can retain close half-cycle correspondence. Controlled bridge, timing, and rectifier asymmetries increase the waveform mismatch. Independent LTspice checks reproduce three representative mode families and agree with the low-order gain within 0.45–3.55%. They also confirm that a high-gain operating point can fail ZVS despite good waveform symmetry. Highlighted current paths, a datasheet-informed semiconductor-loss budget, and input/output ripple spectra connect the descriptors to practical design questions. The loss estimates are conditional analytical scenarios, and the filtering study uses a separate coupled reduced-order model. The framework supports reproducible design screening; it does not replace detailed device simulation or hardware validation.

Symmetry doi: 10.3390/sym18091559

Authors: Yi Zheng Qiuming Luo

Mathematical biology spans multiple computational regimes—from ordinary differential equation models of gene regulatory networks to stochastic simulation of chemical kinetics and reaction-diffusion models of spatial pattern formation—each imposing distinct computational demands on the software infrastructure that executes them. The choice of programming language for implementation of these mathematical models carries quantitative performance consequences that have not been systematically measured across the range of models employed in contemporary mathematical biology. This study provides a computational performance evaluation across ten languages (Python, Julia, Rust, C, C++, C#, F#, Go, Java, and R) and six mathematical biology modeling regimes: deterministic ODE integration, stochastic chemical kinetics, parameter-space exploration, reaction-diffusion spatial modeling, agent-based discrete simulation, and Bayesian parameter inference. Execution time, peak memory footprint, cyclomatic complexity, type-conversion density, and the semantic alignment between data structures and biological state representation were recorded under a uniform experimental protocol. Under a unified hand-coded Dormand–Prince 5(4) integrator, native implementations outperform managed-runtime counterparts by nearly two orders of magnitude for ODE integration, a differential that shrinks sharply once library-delegated solvers are removed from the comparison; the gap contracts below 55× when stochastic kinetics shift the bottleneck from arithmetic throughput to branch resolution. In memory-bandwidth-limited reaction-diffusion modeling, native and just-in-time implementations converge to within a few percent. Agent-based models expose a distinct regime-dependent overhead: immutable-by-default collection semantics impose disproportionate cost during mutation-intensive computation. Peak memory varies by roughly two orders of magnitude across languages, directly affecting deployment density for large-scale simulation. Code-structural measurements confirm that algorithmic form enforces a floor on cyclomatic complexity, irrespective of language, while type strictness and mutation semantics generate substantial differences in per-line cognitive load. These results provide a quantitative foundation for computational tool selection in mathematical biology, challenging universal language recommendations and supporting choices grounded in the algorithmic character of each modeling regime.

Symmetry doi: 10.3390/sym18091558

Authors: João P. Oliveira Daniel A. Marinho Tatiana Sampaio Tiago M. Barbosa Jorge E. Morais

This study examined inter-limb hand force asymmetry in competitive swimmers during front crawl at two paces (200 m and 50 m). Eight national-level swimmers (18.3 ± 2.5 years) performed 25 m trials at both intensities using wearable force-sensing paddles. Asymmetry was assessed with a traditional percentage-based index (BAI-1) and distribution-based estimates, namely the Minimal Detectable Difference (MDD) and an exploratory Minimal Important Difference (MID) estimate. Mean hand force was significantly higher at sprint pace (50 m) compared to 200 m pace (p < 0.001), confirming adherence to the prescribed intensities. Using BAI-1, swimmers showed a wide range of asymmetry values (<2% to >10%), highlighting substantial inter-individual variability. Absolute BAI-1 was descriptively higher during the 50 m pace than during the 200 m pace, although this difference was not statistically significant. Distribution-based estimates refined the interpretation: MDD corresponded to 4.48% and 4.83% for the 50 m and 200 m paces, respectively, whereas the exploratory MID estimate corresponded to 2.68% and 2.89%. These findings illustrate how BAI-1, MDD, and MID may lead to complementary interpretations of asymmetry. The combined use of traditional indices and distribution-based estimates may support more individualized monitoring of inter-limb hand force asymmetry in swimming and potentially other bilateral sports.

Symmetry doi: 10.3390/sym18091555

Authors: Shiang-Yi Han Ciann-Dong Yang

Discrete symmetries govern the motion and real axis crossings of interference zeros in the meromorphic continuation of a wave field, thereby determining the singular structure of its logarithmic derivatives. We considered two counter-propagating second-order rational pulses that satisfy the one-dimensional massless wave equation exactly. With q = rexp(iφ) denoting the relative complex amplitude, the field admits two antilinear reflection symmetries: real q  preserves collision-centered spacetime inversion followed by complex conjugation, whereas |q| = 1  preserves fixed time spatial reflection followed by conjugation up to an overall phase. The balanced in-phase state, q = 1 , is the non-degenerate intersection of these symmetry manifolds;  q = −1 produces global cancellation on the collision slice. For q = 1, the two interference zero branches lie on the imaginary axis and cross the real axis at ct = Xo ± l , on opposite sides of the pulse center collision. Away from the symmetry manifolds, the zero trajectories deform continuously, and the associated pairing constraints are lost, while the crossing conditions remain available in closed form. A logarithmic complex action representation yields local momentum, energy, transport ratio, and a derived second-order complex action descriptor without altering the underlying wave dynamics. Near an isolated non-characteristic moving zero, the leading simple pole factors cancel in the transport ratio, whereas the second-order term develops a double pole. The leading real axis response therefore scales as dmin−1. A reference finite-window fit yields an exponent of  −1.885 (ρ = −0.995), and the fitted exponent approaches −1.998 as the fitting interval is narrowed toward the isolated-zero regime. These results provide an exact benchmark linking antilinear symmetry, complex zero-pole geometry, and real axis differential amplification. The second-order quantity Qc is used only as a descriptor generated by the logarithmic representation; it is neither an externally imposed potential nor an additional dynamical term. The loss of reflection symmetry away from the two symmetry manifolds is explicit and parameter-induced, not spontaneous.

Symmetry doi: 10.3390/sym18091557

Authors: Songul Karakus Mehmet Burukanli Davut Ari

Leaf disease classifiers are ranked by held-out accuracy on their training corpus, where that number now exceeds 98% for many modern backbones. We ask what such a measurement can still resolve. On a four-crop, 21-class corpus of 7179 de-duplicated images, we train 26 architectures under one protocol with three seeds and shared partitions, and we execute the entire benchmark twice. The 26 architectures span 1.68 percentage points (pp) of internal macro-F1. The same configuration differs by 0.34 pp on average and by up to 1.06 pp between the two executions, and the internal ordering only partly replicates (Spearman’s ρ=0.69, with VGG-19 moving from 17th to 1st). Applied zero-shot to an independently collected mango corpus, the same checkpoints spread over 47 pp of accuracy and their ordering does replicate (ρ=0.92). We propose HMLA-Net, which adds multi-scale fusion, gated multi-head lesion-aware pooling, an auxiliary crop head and MixStyle to a CAFormer-S18 backbone and averages the posterior over the Klein four-group of flips at inference. In the reference execution, it attains the highest internal macro-F1 (0.9845) and the highest zero-shot accuracy (0.699) of the 26 architectures, with the smallest degradation (28.9 pp) and the lowest rate of predictions leaking into labels absent from the target corpus. The second execution reproduces its zero-shot rank but not its internal one. Five backbones retrained with its complete recipe show that its margin over its own backbone lies within replicate variation, and in a 16-removal ablation measured on both partitions, only the large effect of ImageNet pretraining remains consistent across the two executions. A cross-corpus duplicate audit finds no exact duplicates and no evidence of substantial overlap, and one class, mango sooty mould, collapses to near-zero recall in 19 of 26 architectures. The contribution is the protocol: replicate the benchmark, evaluate out of domain, and report results per class.

Symmetry doi: 10.3390/sym18091556

Authors: Yang Fan Lixia Xiao Wei Tao Jie Chen

We theoretically investigate the plasmonic analog of the relativistic Klein tunneling processes in systems of two coupled graphene waveguide arrays with spatially varying chemical potentials. By introducing an offset in the chemical potentials of the graphene sheets, a potential step is generated in this platform. Numerical simulations indicate that in the graphene array with λ = 10 μm, d = 70 nm, and μc∼ 0.15 eV, the surface plasmon polaritons exhibit non-unity transmission (up to 50%) across the potential step at the interface that depends on both angle and potential, which is analogous to the Klein tunneling effect. Analytical calculations of the transmission rate of Klein tunneling in these structures have been derived and are consistent with the simulations. This work provides a robust platform for controlling light propagation at deep-subwavelength scales and emulating relativistic quantum phenomena in photonic systems.

Symmetry doi: 10.3390/sym18091554

Authors: Zhikang Li Nobuo Funabiki Kadek Suarjuna Batubulan I Nyoman Darma Kotama Putu Sugiartawan Anak Agung Surya Pradhana

Nowadays, speakers are essential components for message delivery in indoor environments, including lectures, public addresses, and emergency announcements. Their physical placement should ensure adequate direct-sound audibility across occupant service areas while maintaining installation feasibility. A digital twin is a technology that allows an infrastructure layout to be designed and evaluated virtually on a computer before physical installation by reconstructing an indoor environment as a 3D model. In previous studies, we have proposed a method to reconstruct a 3D indoor model of an indoor environment from its 360∘ panoramic images using 3D Gaussian Splatting (3DGS) and a 3D point cloud, and applied it to surveillance camera placement. In this paper, we propose a digital twin-based speaker placement planning tool for indoor environments by generalizing the previous method to direct-sound acoustic simulation. This tool consists of four stages: (1) reconstructing a 3D indoor model from 360∘ panoramic images and extracting floor and desk receiver surfaces, (2) assigning the initial speaker budget based on the reconstructed floor area, (3) determining speaker mounting positions on valid ceiling regions using K-means spatial clustering under obstacle and boundary constraints, and (4) simulating broadband direct-sound sound pressure level (SPL) across floor and desk receiver surfaces. For evaluation, the proposed tool was deployed across three real-world indoor scenarios: a basketball hall (32.15m×21.99m), a furnished office (7.17m×6.17m), and a non-convex L-shaped office (23.00m2). The experimental results showed that in each scenario, the generated layout achieved complete direct-sound audibility compliance across all sampled physical test locations (≥60 dB floor/≥65 dB desk) with zero detected hotspots exceeding 85dB, maintaining SPL values between 66.91dB and 77.88dB. An on-site physical measurement campaign confirmed that the generated layouts satisfy target audibility thresholds under real room conditions, with mean absolute errors between 1.78dB and 2.41dB. These results confirm the practical utility of our approach as an initial geometry-driven planning tool for indoor audio infrastructure deployment.

Symmetry doi: 10.3390/sym18091553

Authors: Yichen Ren Xiangzhi Kong

To address the impact of uncertain information reliability in decision-making environments and decision-makers’ psychology on decision results, this paper proposes a multi-criteria decision-making (MCDM) method for q-rung orthopair fuzzy Z-number (qROFZN) based on prospect–regret theory. First, a novel score function for qROFZN is developed, accompanied by a comprehensive discussion of its fundamental properties. Second, the Jaccard similarity distance measure is innovatively proposed for qROFZN, along with an analysis of its essential characteristics. Subsequently, on this basis, the prospect–regret theory is extended to the qROFZN environment, and a novel MCDM method is established. In this method, the criteria importance through inter-criteria correlation (CRITIC) weighting method and prospect theory’s weighting function are employed to determine comprehensive weights, which are then integrated with prospect–regret theory to identify the optimal solution, thereby considering both subjective and objective factors and enhancing the practical relevance of the decision results. Finally, a case study on medical diagnosis is conducted, along with comparative analyses. The experimental results are consistent with the outcomes of the referenced case, which validates the practicality and effectiveness of the proposed method, demonstrating that the qROFZN-based MCDM method can effectively capture fuzziness and reliability, thus enabling accurate decision analysis under uncertainty.

Symmetry doi: 10.3390/sym18091552

Authors: Ibrahim Mosly

The Gaussian settlement-trough model assumes that ground settlement occurs because tunneling is symmetrical about the tunnel centerline. However, the shield causing the settlement operates in an asymmetric posture during earth pressure balance (EPB) tunneling. This study characterizes the postural asymmetry of the shield in EPB tunneling. Tail void (TV) clearance data for 1580 rings from published data on Guangzhou Metro Line 9, China, were analyzed. Two types of asymmetry indices—the horizontal asymmetry index (HAI) and the vertical asymmetry index (VAI)—were proposed. The two indices indicated systematic deviations from symmetry, with high statistical significance (all p-values < 0.001). The systematic horizontal deviation was to the left, with a pooled mean value of −0.055. In contrast, the stronger systematic vertical deviation was reduced crown clearance, with a pooled mean value of −0.097. Validation against independent shield axis records showed HAI tracked horizontal deviation (p < 0.001), but VAI did not track vertical attitude. The results indicated that postural asymmetry in EPB tunneling was a systematic, parameter-associated phenomenon, with implications for predicting ground settlement, annular grouting strategy, and risk-based monitoring design. However, the absolute values of the two indices were specific to this project and were not transferable without recalibration.

Symmetry doi: 10.3390/sym18091551

Authors: Bader Aldughayfiq Rehana Bibi Hisham Allahem Azzah Allahim Mohammed Alnusayri Hanan Aljuaid Ahmad Jalal

The integration of AI vision-based sensing and human motion analysis has grown to be a key element of intelligent perception systems, which now allow for automated interpretation of human movement, activity patterns, and complex visual behaviors. However, accurate functional movement assessment from monocular aerial and ground-view videos remains challenging due to low spatial resolution, background clutter, occlusions, and large variations in body posture, limiting the reliability of AI-assisted future healthcare applications. This study presents a multi-level framework that integrates asymmetric deep feature representation with transformer-based architecture and graph-driven optimization for robust vision-based human movement analysis. First, a Heavy Attention Transformer is employed to enhance image quality and emphasize clinically relevant anatomical and motion patterns by suppressing background interference. Panoptic segmentation and Real-Time Detection Transformer V2 are then used for subject localization, followed by skeletal keypoint extraction using YOLOv8. The proposed framework adopts an asymmetric dual-stream feature extraction strategy, where global contextual information is captured through Bag of Visual Words, Video Swin Transformer, Video Masked Autoencoder, and TimeSformer, while local biomechanical motion dynamics are modeled using DiffPose, PoseFormer, and Spatial–Temporal Graph Convolutional Networks. The key contribution lies in the asymmetric feature design that preserves the distinct information structures of visual context and skeletal dynamics. To reduce feature redundancy and select discriminative clinical representations, the Slime Mould Algorithm is utilized as a metaheuristic optimizer. The optimized features are subsequently classified using a Graph Attention Network for automated functional movement assessment. Experimental evaluation on the UAV-Human and UCF-ARG benchmark datasets achieved an accuracy of 82.50% and 78.20%, respectively. The proposed framework illustrates the potential of asymmetry-aware AI-enabled vision sensing to perform strong human movement analysis in complex viewpoints and lays the groundwork for future healthcare-related applications such as remote human movement evaluation and rehabilitation monitoring.

Symmetry doi: 10.3390/sym18091550

Authors: Yang Han Weiwei Yu Liqun Zhang Yongsong Li

Power-line work-at-height monitoring requires the simultaneous detection of visually heterogeneous evidence: supervisory markers that may occupy only a few pixels after resizing, and worker-state cues that depend on body posture, equipment, and surrounding scene geometry. In a compact single-stage detector, this scale gap affects feature preservation, training assignment, and prediction stability. Shallow downsampling can weaken the high-frequency traces needed by tiny targets, standard matching may allocate too few positives to small categories, and different detection heads may produce inconsistent predictions for the same physical instance. We view these effects through the lens of asymmetric treatment at three stages of the detector, and introduce three targeted modifications to YOLO11n: a shallow wavelet detail preservation module that enhances low- and high-frequency sub-bands before resolution is lost; a class- and head-aware TinyAssign strategy that adjusts positive-sample allocation by category scale; and a ground-truth-aligned multi-scale consistency regularizer (GT-MSCR) that anchors cross-head agreement to ground-truth indices during training without inference overhead. On a self-collected four-class power-line dataset, and averaged over five independent runs, the model raises mAP@0.5 from 68.10% to 69.45% and recall from 60.84% to 65.32%. The largest per-class gain is obtained by the category most prone to scale-induced detection failure. External validation on the Pictor-v3 PPE and SH17 benchmarks further shows consistent gains on public data, with a parameter increase of only 0.005 M and an additional 2.66 ms of latency over the baseline.

Symmetry doi: 10.3390/sym18091549

Authors: Marcin Michalak

The contemporary development of information technology leads to a constant increase in the amount of data collected in almost all areas of human activity [...]

Symmetry doi: 10.3390/sym18091548

Authors: Ling Wen Anmin Gong Caihong Ma Shengzhou Ma Yanzhe Zhang

As UAV platforms are increasingly employed in military reconnaissance, disaster monitoring, precision agriculture, and other remote-sensing applications, small-object detection in UAV imagery is a critical yet challenging task. Existing reviews primarily address either general small-object detection or general object detection in UAV imagery, whereas reviews specifically devoted to small-object detection in UAV imagery remain scarce. This review examines existing deep learning-based research on small-object detection in UAV imagery from a problem-driven and deployment-oriented perspective. A symmetry-informed perspective was also adopted to analyze how variations in object-scale, viewpoint, imaging quality, and scene conditions affected feature representation and detection stability. Building on five major challenges in UAV small-object detection, a three-tier framework encompassing detection paradigms, key technical strategies, and scenario-specific constraints was established. Representative UAV datasets were further examined to characterize object-scale distributions, while existing methods were reviewed in terms of multiscale representation, feature enhancement, training optimization, and lightweight deployment. The applicability of different technical approaches was then discussed across five typical UAV scenarios. In addition, under relatively consistent experimental conditions, representative detection models based on two-stage, one-stage, and Transformer-based paradigms were analyzed in terms of detection accuracy, computational efficiency, and deployment feasibility. Finally, current limitations and future directions were summarized for scale-robust, scenario-adaptive, and resource-aware small-object detection in UAV imagery.

Symmetry doi: 10.3390/sym18091547

Authors: Mohamed Elbadri Nidal E. Taha Walid Hdidi Hamdy M. Barakat Mohamed A. Abdoon

The main aim of this paper is to investigate the dynamics of the fractional-order Lorenz system within a unified computational framework using both semi-analytical and numerical approaches. The Laplace Residual Power Series Method (LRPSM) is employed to obtain computationally efficient semi-analytical solutions, while a numerical scheme based on the Caputo fractional derivative (SCFD) is developed for the considered system. The resulting dynamics are systematically analyzed through Lyapunov exponents, bifurcation diagrams, phase-space portraits, and time series to characterize the transition to fractional-order chaos and the geometric structures of the resulting attractors. For validation, the results obtained by LRPSM and SCFD are compared with those of the Adams–Bashforth–Moulton predictor–corrector method. The critical fractional order associated with the onset of chaotic behavior is also determined, and the corresponding fractional chaotic attractors are successfully obtained. The results demonstrate that LRPSM and SCFD provide consistent and computationally efficient approaches for studying nonlinear fractional-order dynamics.

Symmetry doi: 10.3390/sym18091544

Authors: Walla Al-Eidarous Aeshah Alsiyami

Pedestrian localisation in emergency environments is challenged by degraded Global Positioning System (GPS) availability, mobile-anchor uncertainty, radio-frequency interference, non-line-of-sight distortion, and inertial drift. Reporting only valid-estimate error may conceal localisation outages and recovery burden. This study proposes Hybrid S9, an adaptive, device-centric localisation framework with a structurally invariant hierarchical controller that selects among GPS, Bluetooth Low Energy (BLE)-assisted Extended Kalman Filter (EKF) correction, zero-velocity update (ZUPT), kinematic coasting, and Particle Filter (PF) recovery according to measurement quality, uncertainty, and mobility state. The framework is evaluated in Normal, Urban Canyon, and Search and Rescue regimes using valid-estimate mean absolute error (MAE), availability, penalised mean error (PME), and operation cost. In the degraded regimes, Hybrid S9 maintains near-complete localisation availability while reducing operation cost. Dedicated severe-degradation experiments verify PF activation, posterior state and covariance transfer, and controlled de-escalation.

Symmetry doi: 10.3390/sym18091546

Authors: Farouq Alshormani Hocine Guediri

In this paper, we represent Bloch functions over the upper half-plane as integrals involving modified Bergman reproducing kernels. This enables us to obtain a precise characterization of the dual space of the Bergman space A1(Π+) in terms of the normalized Bloch space Bi(Π+). Furthermore, we show conversely that the predual of the Bergman space A1(Π+) can be characterized by the little normalized Bloch space Bi,0(Π+).

Symmetry doi: 10.3390/sym18091545

Authors: Hisham Abdulrahman Alhulayyil

The rapidly changing digitalization of the energy sector, fueled by smart grids, the Industrial Internet of Things (IIoT), Advanced Metering Infrastructure (AMI), and Supervisory Control and Data Acquisition (SCADA) systems, is leading to a great and symmetrical improvement in workflow and the ability to have real-time insights. In parallel, the massive adoption of low-power and resource-constrained devices that are still connected makes cybersecurity threats more serious and thus necessitates lightweight authentication mechanisms to have safe, secure, and symmetrical communicative. Selecting the right authentication method involves a challenging multi-criteria decision-making (MCDM) process where various factors such as security aspects, computation power needed, communication capabilities that can be delivered, and deployment-related aspects are considered together, along with inherent uncertainties in expert evaluations. This paper proposes a Hesitant Fuzzy (HF)-based hybrid method that combines the Analytic Network Process (ANP) with the Technique for Order Preference by Similarity to the Ideal Solution (TOPSIS) for the evaluation of lightweight authentication systems in the energy domain. This symmetrical HF-ANP is used for the modeling of the interrelations between major evaluation criteria such as security strength, computation efficiency, communication effectiveness, and deployment scalability. It can also handle the experts’ hesitant and uncertain preferences. The calculated weighting factors are then input to the HF-TOPSIS method to rank five different lightweight authentication protocols: Hash-Based Authentication, Elliptic Curve Cryptography (ECC)-Based Lightweight Authentication, Physical Unclonable Function (PUF)-Based Authentication, Blockchain-Assisted Lightweight Authentication, and Certificate-less Lightweight Authentication. Furthermore, sensitivity analysis and comparison analysis are conducted to verify the strength, symmetry, consistency, and reliability of the proposed framework. Our results indicate that the integrated HF-ANP and TOPSIS procedure provides a symmetrical, comprehensive, and systematic decision-making tool to appraise lightweight authentication techniques amid uncertainties. The proposed method will be very beneficial for different types of energy industrial players, system designers, and security experts to pick secure, efficient, and scalable methods of authentication to protect the critical energy facilities against the new generation of cyber threats.

Symmetry doi: 10.3390/sym18091543

Authors: Junyuan Hu Lingfei Xiao Zhichao Ming Zhijie Zhou Chenyu Luo

Symmetry and asymmetry appear together in aero-engine prognostics and health management. A monitoring system should preserve a common decision structure across operating scenarios, while sensor evidence becomes asymmetric under changing operating conditions and fault modes. This study formulates the National Aeronautics and Space Administration (NASA) Commercial Modular Aero-Propulsion System Simulation (C-MAPSS) turbofan benchmark as a four-level health-state assessment problem using remaining useful life (RUL) thresholds of 100, 50 and 15 cycles, and proposes a neural evidential reasoning (ER) framework with decision-structure symmetry. FD001–FD004 share the same health-state space, threshold map, reliability-discounted ER operator and final argmax rule, whereas feature selection, evidence transformation and reliability parameters are estimated independently for each subset. Sliding statistical descriptors and training-fold-only minimum-redundancy maximum-relevance (mRMR) selection retain six compact sensor-derived features. FD001 and FD003 therefore use six inputs; FD002 and FD004 additionally retain the same three operating-setting variables and use nine inputs. Boundary soft labels and auxiliary ordinal/RUL supervision exploit ordered degradation information near adjacent-state boundaries. Evaluation uses engine-disjoint validation, three random seeds, identical subset-specific inputs for every comparator, ensemble/neural/ordinal baselines, component ablation and calibration/error-detection analyses. Across the four subsets, the proposed model achieves the highest mean three-metric average (Avg3), 0.664 (standard deviation 0.007), with Accuracy 0.816 (0.006), Macro-F1 0.582 (0.010) and Balanced Accuracy 0.594 (0.005). It is statistically comparable to histogram gradient boosting, random forest and a plain multilayer perceptron, and significantly exceeds the tested compact-input cumulative ordinal, Feature Transformer and correlation-graph attention controls after Holm correction. Fixed-probe diagnostics identify the five-cycle window as the best overall short-history setting. The mRMR top-6 interface reduces the sensor-feature dimension by 97.1% while retaining 98.2% and 97.0% of full-pool Avg3 on FD002 and FD004, respectively. Unknown mass ranks erroneous predictions above correct predictions on every subset (area under the receiver operating characteristic curve (ROC-AUC) 0.749–0.810), including the highest value on the most heterogeneous FD004 subset. These results support a compact and auditable evidence interface that combines competitive classification with source-wise reliability and uncertainty information.

Symmetry doi: 10.3390/sym18091542

Authors: Aref Afsharfard Kyung Chun Kim

Portable Ocean Wave Energy Harvesters (POWEH) offer a potential approach for supplying autonomous low-power marine sensing systems; however, their long-term reliability, durability, and economic viability remain insufficiently demonstrated. But conventional wave energy converters are often big, expensive, and fixed to seabed structures, which makes it hard to use them in different places and makes it hard to sell them. People have been paying more attention to portable ocean wave energy harvesters in recent years. These are small, light systems that can be set up anywhere and do not need to be moored or anchored. These devices are meant to power autonomous ocean sensors, emergency buoys, and maritime applications that do not need to be connected to the grid. This review looks closely at the development trends, structural designs, and power take-off mechanisms of portable wave energy systems. We compare different types of harvesters, such as mechanical, electromagnetic, piezoelectric, and hybrid ones, based on their design principles, conversion efficiency, and scalability. Focus is directed towards the importance of symmetry in structural design, dynamic response, and energy conversion. The structure, whether symmetric or asymmetric, has a crucial role in determining mass distribution, stiffness traits, vibrational dynamics, hydrodynamic loading, and the capture of multi-directional wave energy, thereby influencing the overall performance, resilience, and stability of portable harvesters. The review goes on to talk about the main problems with dynamic stability, frequency tuning, environmental adaptability, and energy management when things are portable. Finally, new strategies like nonlinear dynamic designs, self-tuning mooring systems, and hybrid energy integration are suggested as ways to improve performance and reliability. The insights from this review are meant to help future innovations that will make it possible to use portable ocean wave energy technologies in a practical and long-lasting way.

Symmetry doi: 10.3390/sym18091541

Authors: Changlong Wang Fan Zhang Jitao Zou

In power transmission systems, insulator strings typically exhibit a distinct periodic geometric symmetry. However, defects such as breakages and flashovers disrupt this symmetry, producing subtle, orientation-sensitive asymmetric patterns that traditional isotropic convolution kernels often struggle to capture. To tackle the challenge of identifying these symmetry-breaking defects, we propose the Direction-aware You Only Look Once version 11 (DA-YOLOv11), a lightweight detection framework designed for asymmetric, direction-aware feature learning. By integrating the PaddlePaddle Lightweight Convolutional Network (PP-LCNet) into the backbone, the model preserves essential structural features while minimizing computational redundancy. We also introduce the Insulator-oriented Simple Parameter-Free Attention Module (INS-SimAM), a direction-aware attention mechanism that employs an asymmetric spatial statistical strategy to adaptively enhance defect-related signals within non-uniform regions. Additionally, the Adaptive Numerically Stable Normalized Wasserstein Distance (ANS-NWD) loss models tiny defects as 2D Gaussian distributions, leveraging their mathematical symmetry to ensure stable regression. Evaluations on a custom dataset and three public benchmarks (IDID, CPLID, UPID) show that the final pruned DA-YOLOv11 achieves an 88.95% mean Average Precision (mAP50) on the core dataset, outperforming the baseline while reducing the parameter count by 34.36%. This work provides a robust solution for detecting geometric symmetry-breaking in complex power inspection scenarios.

Symmetry doi: 10.3390/sym18091540

Authors: Jorge Ortiz Ceballos Itzel María Abundez Barrera Eréndira Rendón-Lara

Surface electromyography (sEMG) signals acquired with low-cost sensors tend to exhibit variable degradation that can compromise the reliability of myoelectric control systems outside controlled conditions. This work presents an adaptive processing pipeline for sEMG signals composed of three chained stages: a motor-intent classifier based on dilated temporal convolutional networks, whose function is to determine whether a signal window contains muscle activity associated with a voluntary gesture; a dual-output quality assessor that estimates a continuous score and a binary acceptability label, aimed at deciding whether the signal can be used directly or requires intervention; and a convolutional autoencoder that recovers the morphology of degraded windows before they are used in prosthetic control. The decision policy for reconstruction operates on two independent thresholds and classifies each window into one of three states: signal discard, direct acceptance, or active restoration. The models within the proposed architecture are trained on the public NinaPro DB1, DB3, and DB10 datasets and validated without recalibration on signals recorded from two participants with transradial amputation over three to four weekly sessions. The results show that the pipeline correctly handles signal profiles with opposing characteristics. Inference latency remained below 5 ms at the 50th percentile across all scenarios, consistent with real-time operation. All inference was executed on a host computer; a prototype using an Arduino UNO R4 WiFi as a peripheral interface was built to display the pipeline’s decisions on physical hardware and to confirm that the serial-communication link does not introduce additional latency, not to perform on-board inference. A downstream evaluation further showed that, while restoration improved signal-level fidelity metrics, it did not translate into improved motor-intent classification accuracy relative to using the degraded signal directly, a limitation discussed explicitly in this work.

Symmetry doi: 10.3390/sym18091539

Authors: Shibo Tang Chuanyang Ruan Lin Yan

This paper introduces the concept of interval-valued T-spherical hesitant fuzzy set (IVTSHFS). Derived from the T-spherical fuzzy set, the IVTSHFS allows the membership, abstinence, and non-membership degrees of an element to be expressed as both interval and hesitant information. We present the basic concept, operational laws, and Hamming distance of IVTSHFSs and examine their fundamental properties. A novel score function governed by two adjustable coefficients is then developed, where λ serves as the preference coefficient and θ quantifies the degree of penalty for abstinence; an accompanying accuracy function is also constructed. Based on the operations of IVTSHFSs, two fundamental aggregation operators, namely the interval-valued T-spherical hesitant fuzzy weighted averaging (IVTSHF-WA) and weighted geometric (IVTSHF-WG) operators, are proposed. Furthermore, the power Heronian mean is extended to the IVTSHF environment, yielding the IVTSHF-WPHM and IVTSHF-WGPHM operators. To demonstrate the practicality of the proposed operators, a multi-attribute decision-making (MADM) method is developed within the IVTSHFS framework. A supply chain finance plan evaluation problem is then employed as a case study to validate the proposed method. Moreover, a systematic sensitivity analysis and comparative evaluation are conducted to rigorously assess the robustness and stability of the proposed method.

Symmetry doi: 10.3390/sym18091538

Authors: Siyang Li Qile Zeng Qingqi Peng Yuyan (Annie) Pan Yan Wang Rui Gao Na Zhang Chao Xia Randong Xiao Jian Huang Haitao Yu

Mixed-autonomy traffic with autonomous vehicles (AVs) and human-driven vehicles (HDVs) presents forecasting challenges because agent interactions and event-affected spatial dependencies vary over time. We propose Heterogeneous Adaptive Dynamic Spatiotemporal forecasting (HADS), a graph-attention framework that combines event-driven local topology reconfiguration, penetration-aware dynamic time warping (DTW) attention, and task-level temporal fusion for multi-horizon traffic-flow prediction. The forecasting target is traffic flow. The evaluation uses a semi-synthetic, penetration-controlled benchmark built from field-observed traffic-flow targets and 912 labeled anomalous events on a Beijing pilot-zone network (534 nodes and 3180 directed edges; April–July 2023), paired with SUMO-generated AV features at 20%, 40%, and 60% penetration. Under the reported single-seed runs, HADS obtains lower 15 min MAPE than AGCRN at 60% penetration, decreasing the point estimate from 2.92% to 2.61% under regular conditions and from 3.32% to 2.98% under anomalous conditions. Results across the reported 15 and 30 min settings indicate that event-conditioned topology and lag-aware heterogeneous attention can improve traffic-flow forecasting on this hybrid real–simulation benchmark. Results are from single-seed runs and should be read as preliminary point-estimate evidence rather than statistically demonstrated improvements; the semi-synthetic evaluation shows potential for pilot-zone applications rather than confirming real-world deployment performance.

Symmetry doi: 10.3390/sym18091537

Authors: Ulas Cinar

Spherical fuzzy sets are the extension of ordinary fuzzy sets in which membership, non-membership, and hesitancy all vary freely. This independence matters for expert-driven risk assessment: it lets a judgement carry support, opposition, and hesitancy as three quantities that vary independently, which intuitionistic and Pythagorean representations cannot do because their hesitancy is a residual of the other two degrees. This paper uses spherical fuzzy AHP to weight risk criteria and shows that the recovered hesitancy varies systematically across criteria: across an illustrative assessment of ten underground mining operations, hesitancy is largest on likelihood and smallest on severity. That ordering is reproduced by the corresponding residual-hesitancy calculation, and it is therefore reported as a description of the pooled linguistic profile rather than as evidence of a representational advantage: under the scale adopted here, a single linguistic term fixes all three coordinates at once, so the hesitancy is scale-assigned rather than separately elicited. The weighted criteria are combined with the MARCOS method. Building on the recently established equivalence of MARCOS and the weighted-sum method, the single channel through which the alternative set can affect a ranking is identified and closed, yielding a fixed-anchor modification of MARCOS that is provably invariant to the addition or deletion of alternatives. The channel is localized exactly: a MARCOS ranking depends on the alternative set through the ideal row alone, and can reverse only when the alternative withdrawn is the sole holder of a column maximum. In a deletion experiment over 360 pairwise comparisons, the modified specification produces no reversals, against counts ranging from one to fifteen for the seven benchmark methods, with five for classical MARCOS. Spherical fuzzy weighting and a rank-reversal-resistant aggregation thus address the representation and the stability of expert risk judgements, respectively.

Symmetry doi: 10.3390/sym18091536

Authors: Mingzhu Zhang Zhen Yang Ruiping Yin

Multi-task recommendation improves predictive performance by sharing knowledge across related tasks, but existing dense architectures and predefined expert-routing mechanisms do not explicitly adapt fine-grained parameter connectivity to individual tasks, potentially leading to parameter competition, negative transfer, and deployment overhead. We propose TAST, a Task-Aware Sparse Topology Learning framework that constructs task-specific subnetworks within a shared network. A task-conditioned topology generator jointly learns neuron- and connection-level masks, while explicit density budgets discourage trivial dense or excessively sparse solutions and enable controllable capacity allocation. TAST first learns shared parameters and task-specific topologies, hardens the learned masks into binary structures, and then optionally applies a lightweight sample-conditioned refinement module for local adaptation. Validation-guided branch selection retains the refined branch only for tasks with sufficient validation gains, allowing unused refinement branches to be removed during deployment. Experiments on four AliExpress subsets and the UserBehavior dataset show that TAST achieves the highest observed mean AUC across all five datasets, with statistically significant improvements over the fixed MoME comparator on AliExpress-US and UserBehavior after multiple-comparison correction. On AliExpress-NL, TAST achieves a slightly higher mean AUC than MoME while reducing deployment parameters and measured batch-inference latency by 87.7% and 77.0%, respectively. The current implementation still evaluates the hardened masked weights using dense GPU operators; therefore, the learned sparsity represents structural adaptation rather than physical sparse execution. These results demonstrate a favorable balance among predictive accuracy, structural adaptation, and deployment efficiency.

Symmetry doi: 10.3390/sym18091535

Authors: Hongfei Chang

Digital murals face degradation issues like fading and cracking, requiring objective and interpretable evaluation methods. To overcome the subjectivity and low efficiency of manual assessment, the study proposes an improved deep neural network that maps extracted features into four criteria: color quality, texture details, structural integrity, and degradation degree. Using the fuzzy analytic hierarchy process (FAHP) with triangular fuzzy numbers, the study computes global weights and obtains a comprehensive score via weighted linear aggregation. Experimental results show the network reduces root mean square error to 0.102 and mean absolute error to 0.089. Inter-class distance increases by 33.8%, and t-SNE visualization shows four quality clusters in four-corner separation. The comprehensive score monotonically decreases from 0.924 to 0.376 as degradation worsens, with texture details showing the most significant drop. The model effectively captures mural quality deterioration, aligning with real-world conservation cognition. This work provides an objective, quantitative, and automated tool to assist experts in quality screening and restoration prioritization for large-scale murals.

Symmetry doi: 10.3390/sym18091534

Authors: Alaa Ata Engy E. Elbasuny Ayman El-Zohairy Marwa Nabil

Problematic soils pose a risk to structures due to excessive or uneven settlement, swelling, and limited bearing capacity. To address some of these issues, mechanical reinforcement using geosynthetic products such as geocells can be employed, given their ease and speed of implementation. The purpose of this study was to assess the impact of geocells on the behavior of soft clay soil in Port Said, Egypt, using a three-dimensional finite element model, namely, the PLAXIS 3D software. The results show that replacing the top 1.5 m of the soft clay with sand significantly reduced settlement. This improvement was further enhanced by the addition of geocells available in the local market, where settlement was additionally reduced by approximately 51% to 63%, along with an appreciable increase in the bearing capacity. The effect of geocell material parameters was also investigated, including cell height, pocket dimensions, embedment depth, and reinforcement width. The results indicate that the best performance among the tested cases was achieved with a cell height of 300 mm, pocket dimensions of 210 × 245 mm, an embedment depth of 0.1 B, and a reinforcement width of 2 B, where B represents the foundation width. Furthermore, using gravel as both the replacement material and the geocell infill further improved performance, particularly in the presence of groundwater, by reducing settlement and increasing bearing capacity. The results indicate that combining gravel replacement with geocell reinforcement represents the most effective solution for improving the stability of soft clay soils in the study area.

Symmetry doi: 10.3390/sym18091533

Authors: Xiaochuan Liu Zhiren Xiao Miao Zhang Guangyao Li

The cavity expansion solution for loess is absent because the existing elastoplastic models fail to model the sophisticated behaviors of unsaturated loess as a classic axisymmetric boundary-value problem. This paper develops a novel solution for cavity expansion in unsaturated structural loess, taking the coupled hydro-mechanical and structural evolution into consideration. To properly model the behavior of the loess, the Barcelona Basic Model for unsaturated clay is modified by incorporating structural parameters of loess and a volumetric strain-dependent soil–water characteristic curve. The structural parameters are evolved with the degree of saturation and the plastic strain. Leveraging the hydro-mechanical coupling elastoplastic constitutive matrix of the modified constitutive model, the governing differential equations for cylindrical cavity expansion are derived by introducing an auxiliary variable to the stress equilibrium equation and the geometric equation which preserves the intrinsic radial symmetry during spatial coordinate mapping. By considering the constant water content condition, the problem considered is expressed as seven first-order differential equations with suction, three stress components, specific volume, and two structural parameters as unknowns. The governing equations are solved as an initial value problem by the Runge–Kutta scheme. After verifying the solution with well-established solutions and the numerical results from ABAQUS, parametric studies are conducted to investigate the effects of structure degradation and saturation on the expansion response. The results show that the proposed solution can capture the hydraulic yielding behavior well during cavity expansion and the effects of structure on the expansion responses. The proposed solution provides a more reasonable theoretical basis for modelling the static cone penetration test, pile installation effects and pressuremeter tests in loess.

Symmetry doi: 10.3390/sym18091532

Authors: Edwin Sánchez-León Eduardo Espinoza-Solís Huber Echeverría Antonio Blázquez-Zaballos Purificación Galindo-Villardón

Sparse Tucker decomposition (STD) is a framework for tensor decomposition that extends the classical Tucker model with sparsity constraints, with the aim of achieving better representation and analysis of multidimensional data. STD has attracted growing interest from various scientific fields, such as neuroimaging, machine learning, telecommunications, statistical computing, and brain connectivity analysis, due to its flexibility and computational efficiency. The purpose of this study is to present a systematic and structured review of the scientific literature on STD, describing its main trends, applications, and the evolution of research over time. Methods: A systematic review was conducted in accordance with the PRISMA 2020 guidelines using the Scopus and Web of Science databases. The search retrieved 21 records, of which 13 were selected for detailed analysis according to the established inclusion criteria. To improve the evaluation process, the selected studies were assessed using the plithogenic neutrosophic hypersoft set framework, which allowed for the evaluation of scientific contributions in terms of degrees of truth, indeterminacy, and falsity. The evaluation covered aspects such as methodological rigor, citation impact, thematic relevance, journal quality, and international collaboration networks. Results: Research on STD spans 2013 to 2025—the year of the first indexed record through the search date—and shows a marked increase after 2021. Five main application domains were identified: computational optimization, neuroimaging and neuropsychology, telecommunications, statistical analysis, and studies on brain connectivity. The literature on STD is still relatively limited and geographically concentrated, especially in the United States and China, despite the strong interdisciplinary potential and the growing scientific relevance of STD. Conclusions: STD is symmetric by construction, treating every tensor mode and component equally and applying the same sparsity penalty throughout, yet the literature surrounding it is markedly uneven, being concentrated in a few countries and a small number of technical fields. The PNHS framework allowed us to characterize this mismatch while accounting for the uncertainty of each judgement, scoring productivity, impact, collaboration, and thematic relevance through separate degrees of truth, indeterminacy, and falsity. This study identifies STD as a promising interdisciplinary analytical framework and demonstrates the value of plithogenic neutrosophic hypersoft methodologies for evaluating the scientific literature under uncertainty and multidimensionality. The review emphasizes the need for greater methodological standardization, increased international collaboration, and more robust interdisciplinary dissemination to establish STD as a robust framework for the analysis of multidimensional data.

Symmetry doi: 10.3390/sym18091531

Authors: Rafael Bayareh-Mancilla Texar Javier Ramírez-Guzmán Rosario Munguía-Fuentes Álvaro Anzueto-Ríos Arturo Vera-Hernández Citlalli Jessica Trujillo-Romero

Computer models and protocols commonly used for superficial thermotherapy assume anatomically symmetric tissue geometries. However, anatomical variability, particularly the variability of the adipose tissue thickness, may have a significant impact on heat transfer and cause non-uniform thermal doses in the patient population. This paper studies the influence of anatomical asymmetry due to adipose tissue thickness on heat propagation during superficial thermotherapy based on combined computational and experimental analyses. A Finite Element Model of the lower limb was designed with anatomically representative layers and tissue thermal properties. Parametric simulations were conducted for medium (M) and extra-large (XL) models at hot-pack temperatures (38–44 °C) for 15 min. The simulation results were compared with experimental measurements on ex vivo porcine tissue. The results showed that the thickness of adipose tissue significantly affected deep-tissue heating. When the same heating condition was applied, the boundary temperature of the muscle was increased by ~1.38 °C for the 44 °C hot-pack in the M-size model and was restricted to 0.21 °C for the XL-size model. The experimental muscle temperature increased from 22.26 °C to approximately 23.3 °C after 15 min. Under the same initial temperature and nominal 38 °C heating condition, the perfused and zero-perfusion M-size simulations reached approximately 26.8 °C and 30.5 °C, respectively; these endpoint differences support a qualitative comparison rather than quantitative validation. These results show that anatomical asymmetry has a strong effect on heat transfer during superficial thermotherapy and challenge the precept that standard heating protocols provide similar thermal doses in different body morphologies.

Symmetry doi: 10.3390/sym18091530

Authors: Yueyi Yang Jiacheng Li Haiquan Wang Xiaobo Nie Guolong Li Chaojie Wei Kangwei Liu

Remaining useful life (RUL) prediction of power transformers is essential for ensuring reliable operation and supporting condition-based maintenance. However, the complex nonlinear degradation characteristics and long-term temporal dependencies of transformer monitoring data pose significant challenges for accurate RUL prediction. Moreover, most existing deep learning-based prediction methods provide deterministic predictions without explicitly quantifying predictive uncertainty, while the consequences of RUL overestimation and underestimation in practical maintenance are inherently asymmetric. Therefore, this paper proposes an uncertainty-aware RUL prediction method for power transformers by integrating the Patch Time Series Transformer (PatchTST) with deep evidential regression (ER-PatchTST). PatchTST is used to transform long time-series inputs into patch-level representations, enabling the model to capture local temporal patterns and long-range dependencies through patch-wise tokenization and channel-independent modeling. In addition, the deep evidential regression module is designed by placing a Normal–Inverse-Gamma (NIG) prior over the parameters of the Gaussian likelihood, which can simultaneously predict RUL and quantify aleatoric and epistemic uncertainties in a single forward pass. Furthermore, a safety-oriented RUL indicator and hierarchical warning strategy are developed, and different maintenance actions are initiated when alarms at different levels are triggered. Experiments on the ETT dataset demonstrate that ER-PatchTST achieves competitive forecasting performance compared with state-of-the-art time-series forecasting methods while simultaneously providing predictive uncertainty estimates. On the DGA dataset, ER-PatchTST achieves the best RUL prediction performance among the compared methods and provides informative uncertainty quantification for condition-based maintenance decision support.

Symmetry doi: 10.3390/sym18091529

Authors: Xin Zhang Xiaopeng Hua

The Least Square Projection Twin Support Vector Machine (LSPTSVM) is an effective machine learning tool for solving classification problems. However, LSPTSVM does not account for the contribution of each sample during training, making it susceptible to outliers and noise. This susceptibility diminishes its generalization capability. To remedy this shortcoming, this paper introduces an Intuitionistic Fuzzy LSPTSVM (IFLSPTSVM). This model combines the LSPTSVM with the concept of Intuitionistic Fuzzy Numbers (IFN). In the training process of IFLSPTSVM, the importance of each training sample is gauged using an IFN-based score function that considers its geometric position and surrounding environment. Moreover, the weighted class mean, as opposed to the standard mean used in LSPTSVM, is employed in the calculation of intra-class scatter based on the intuitionistic fuzzy score of the sample. This approach effectively mitigates the impact of noise and outliers and more accurately captures the global information of the class samples. Experimental results on several real-world UCI benchmark datasets and the Case Western Reserve University rolling bearing datasets exhibit the efficacy of the proposed method.

Symmetry doi: 10.3390/sym18091528

Authors: Gianluigi Del Prete Salvatore Strano

The increasing adoption of autonomous sensing technologies for Structural Health Monitoring (SHM) and intelligent transportation systems has generated growing interest in self-powered monitoring platforms capable of operating without conventional power supplies. Among the available energy harvesting technologies, piezoelectric transducers represent an effective solution for converting ambient mechanical vibrations into electrical energy suitable for supplying low-power electronic devices. However, the design of such systems requires the simultaneous analysis of multiple interacting physical domains, including structural dynamics, electromechanical transduction, electrical conditioning and power management. This paper presents a comprehensive multi-domain modelling framework for piezoelectric energy harvesting systems developed within the MATLAB Simscape environment. The proposed methodology integrates the coupled electromechanical behaviour of a piezoelectric cantilever with the complete electrical conditioning circuit, including rectification, energy storage and DC/DC power management, allowing all subsystems to be analysed simultaneously within a unified simulation platform. Numerical simulations demonstrate the capability of the proposed framework to reproduce the complete energy conversion process from mechanical excitation to electrical power delivery while evaluating the interaction among the different physical domains. Beyond the specific case study investigated, the proposed approach provides a flexible virtual prototyping tool for the design and optimisation of future self-powered monitoring systems. Owing to its modular architecture, the framework can be extended to different transducer technologies, sensing configurations and energy management strategies, supporting the development of autonomous monitoring platforms for civil infrastructures, railway systems and intelligent transportation applications.

Symmetry doi: 10.3390/sym18091527

Authors: Feng Guo Xiaozhong Zhang Qing Qiao Pengfei Hao Guolong Zhang

Deep tight-reservoir fracability depends on temperature, fluid transport, in situ stress, fracture development, and injection energy, so no single response can provide a defensible ranking of fracturing conditions. This study develops a sparse second-order Choquet–Monte Carlo fracability framework. Additionally, the CRITIC method is used independently to quantify the information contrast of the five evaluation indicators. This was done using true triaxial sequential dual-well experiments conducted at 25–200 °C, horizontal stress differences of 3–9 MPa, and with either supercritical CO2 (SC-CO2) or water. CT scanning, U-Net segmentation, and three-dimensional reconstruction supplied fracture volume and fractal dimension; two-stage breakdown pressures and hydraulic injection work described initiation accessibility and energy demand. The CRITIC information weights were 0.261 for stress compatibility, 0.235 for injection-work demand, 0.182 for fluid mobility, 0.172 for specimen temperature, and 0.150 for pressure accessibility. The fitted second-order model identified positive temperature–fluid (0.122) and temperature–stress (0.071) non-additive coefficients, while the pressure–work coefficient was close to zero (−0.008). Sequential injection produced a consistent stage-dependent difference in breakdown pressure, with the second-stage value exceeding the first-stage value in every test. This observation is interpreted descriptively within the fixed sequential injection protocol. Condition No. 5 (SC-CO2, 200 °C, 3 MPa) ranked first with a fracability index of 0.935 and a 95% Monte Carlo interval of 0.908–0.945. Leave-one-out cross-validation gave a Spearman coefficient of 0.830 and R2 = 0.860. Within the tested domain, temperature was associated with improved fracability, particularly under the low stress-difference condition where enhanced SC-CO2 mobility and thermal effects acted together. The framework therefore distinguishes coupled benefits, redundancies, and uncertainty that an additive score would conceal.

Symmetry doi: 10.3390/sym18091526

Authors: Wei Chen Yunche Su Fang Liu Chuan Yuan Yuhong Wang Kuangyu Chen Jianquan Liao

As renewable generation and power-electronic interfaces account for a growing share of power systems, reduced system inertia and weakened voltage support place greater demands on the stable operation of weak and passive networks. This paper proposes a virtual synchronous generator (VSG)-based active support strategy for grid-forming voltage source converter-based high-voltage direct-current (VSC-HVDC) systems. First, the power-balance relationship between the converter station and the synchronous generator is analyzed to establish the basis for VSG control. For active-power–frequency regulation, a virtual zero-error frequency regulation (VZFR) strategy is proposed by introducing an additional power compensation term into the conventional VSG control, improving frequency recovery while retaining virtual inertia and damping. For voltage–reactive-power regulation, the transient voltage support mechanism is analyzed in two stages: the controlled voltage-source characteristic and virtual impedance provide initial voltage support, while the reactive-power–voltage loop subsequently regulates the internal voltage and reactive power output. PSCAD/EMTDC simulations under active load disturbances and different grid voltage sags show that the proposed strategy enhances both frequency regulation and voltage support provided by the VSC-HVDC system.

Symmetry doi: 10.3390/sym18091525

Authors: Mohamed Elbadri Nidal E. Taha Walid Hdidi Hamdy M. Barakat Mohamed A. Abdoon

In this paper, a novel four-dimensional variable-order fractional chaotic system is presented through the use of the Liouville–Caputo variable-order fractional derivative to represent the time-varying memory properties of nonlinear dynamical behaviors. Special attention will be paid to the chaotic behavior of the proposed system and their chaotic attractors depending on different memory effects. For calculation of the solutions of such systems, a numerical approach for the variable-order fractional systems is used, and three memory functions of the variable order are considered to investigate their effect on the behavior of the system. The obtained behavior will be analyzed by means of time responses, phase portraits, bifurcation diagrams, the spectrum of the Lyapunov exponents, and the Kaplan–Yorke fractal dimension. From the numerical simulations, it is shown that different fractional orders lead to completely different dynamical regimes and the appearance of symmetric chaotic attractors with different geometrical and topological structures while the physical parameters of the system are kept the same. These results highlight the important role of time-varying memory in controlling and generating chaotic structures in fractional-order nonlinear systems.

Symmetry doi: 10.3390/sym18091523

Authors: Yanwei Sang Yan Xu Yuxuan Zhang Zhipeng Huang Ledeng Huang Guojun Wang Zhehui Peng

Axial blood pumps serve as vital auxiliary therapeutic devices for patients with end-stage heart failure, and parameter optimization of the controller is critical to improve system performance. Existing optimization methods cannot satisfy the parameter optimization requirements of variable-speed axial blood pump controllers in terms of optimization accuracy. Therefore, this paper investigates a many-objective optimization method adapted to the operating characteristics of blood pumps. Firstly, a many-objective optimization model is established for the controller. To efficiently solve the proposed model, an optimization algorithm integrating deep reinforcement learning, named DQN-NSGA-CT, is developed. On the basis of the population evolution state, the optimal strategy learned by DQN dynamically adjusts the crossover probability and mutation probability, which adaptively balances population diversity in the early iteration stage and convergence speed in the later iteration stage. Meanwhile, a novel environmental selection strategy is used to reconcile population convergence and diversity. To select the best compromise solution, an entropy weight–Copula–TOPSIS comprehensive evaluation method is proposed, which realizes objective weight assignment, objective correlation correction and multi-attribute ranking. Experimental results verify the efficiency of the DQN-NSGA-CT algorithm in solving the many-objective optimization model of the controller. The research addresses the many-objective optimization problem of variable-speed axial blood pump control.

Symmetry doi: 10.3390/sym18091524

Authors: Lin Hu Song Jiang Xiu Liu Pucha Song Yue Yu Nan Zhou

Multi-view clustering has emerged as a significant research direction in the information age, as multiple feature representations become increasingly available. However, traditional multi-view clustering methods are often unsupervised and fail to exploit available label information. In practice, fully labeled data are scarce, while partially labeled data are more common and can significantly improve clustering performance. Moreover, real-world data are frequently corrupted by noise or outliers. To address these challenges, this paper proposes a Correntropy-guided Matrix Factorization and Tensor Graph Learning (CMTGL) framework for robust semi-supervised multi-view clustering. Specifically, CMTGL incorporates partial label information into a shared low-dimensional representation through constrained low-rank matrix factorization and employs the maximum correntropy criterion (MCC) to reduce the influence of noisy samples and outliers. In addition, view-specific graphs are stacked into a third-order tensor and regularized by the tensor Schatten p-norm to exploit high-order correlations across multiple views. A consensus graph is jointly learned to capture the common structural information shared among different views. The resulting optimization problem is efficiently solved by a block coordinate descent algorithm with an extrapolation accelerated block coordinate update (BCU) scheme. Extensive experiments on six public benchmark datasets demonstrate that the proposed method achieves superior clustering performance compared to state-of-the-art approaches.

Symmetry doi: 10.3390/sym18091522

Authors: Iqra Kanwal Jianghao Hao Ahmed Bchatnia Muhammad Fahim Aslam Muhammad Afnan

This paper studies a coupled system of viscoelastic wave equations with frictional damping, cross-tempered fractional damping, viscoelastic memory, and logarithmic source nonlinearities. The fractional damping acts across the two components, so that the fractional feedback in each equation is generated by the velocity of the other component. To handle the memory and fractional terms, we introduce suitable history and diffusive variables and reformulate the problem as an evolution system in an extended energy space. Under appropriate assumptions on the relaxation kernels, fractional parameters, and nonlinear exponent, we establish the local well-posedness of mild and strong solutions using semigroup theory. We then use a potential-well argument to prove global existence for initial data in the stable set. Under an additional decay condition on the relaxation kernels, an appropriate Lyapunov functional is constructed to establish the exponential decay of the energy. Finally, numerical simulations based on a finite-difference scheme and a physics-informed neural network (PINN) are used to illustrate the predicted decay behavior.

Symmetry doi: 10.3390/sym18091521

Authors: Mu Li Shouyuan Wu Yuman Song

The growing penetration of renewable energy sources introduces significant uncertainties into integrated energy systems (IESs). Conventional single-timescale management strategies, typically designed for static power balance, fail to address the symmetry of source-load uncertainties arising from both supply and demand sides. To address this challenge, this paper proposes a multi-timescale optimal scheduling framework that integrates demand response (DR) and multi-energy flow coupling. The framework adopts a hierarchical progressive strategy across day-ahead, intra-day, and real-time stages. The day-ahead stage optimizes the economic baseline with an hourly resolution. The intra-day stage conducts rolling correction at 15 min intervals to activate slow-response equipment flexibility, boosting combined heat and power (CHP) generation by 40.70% and increasing waste-heat cooling consumption by 41.12%. The real-time stage employs energy storage at 5 min resolution to suppress fluctuations, maintaining electricity, heat, and cooling load deviations, respectively, at remarkably low levels of 0.17%, 0.10%, and 0.06%. Comparative results show that with power-to-gas (P2G) integration, the system purchases off-peak electricity for synthetic natural gas production, cutting gas procurement costs by 12.70% and reducing net carbon emissions from 5.14 t to 4.91 t. DR mechanisms enable a gas–electricity substitution strategy that lowers electricity purchase costs by 9.97%, reduces evening peak electric vehicle (EV) charging load by 8.32%, and decreases charging expenses by 15%.

Symmetry doi: 10.3390/sym18091520

Authors: Jinhui Xu Runqi Li Mark Reiser

When many binary items are analyzed, most cells in the complete response table are empty. Full-table statistics may then be unreliable, and an omnibus result does not locate the misfit. This paper develops a local goodness-of-fit testing method for binary item response models by extending the weighted sequential sums-of-squares construction to pairwise concordance margins. The construction gives ordered one-degree-of-freedom components for individual item pairs. They are compared with Cholesky components and four local diagnostics under zero and wide symmetric intercepts. Omnibus statistics are examined separately under symmetric slopes. The sequential components have an empirical Type I error close to the nominal levels. With zero intercepts, all methods show substantial power for the target pairs, although pair order moves some shared discrepancy into non-target components. Cholesky gives higher target-pair power and broader moderate elevations across non-target pairs, whereas the sequential components show more concentrated, order-dependent peaks. With wide symmetric intercepts, power differs greatly among target pairs, and the Lagrange multiplier statistic is particularly sensitive to intercept distance. The main qualitative patterns persist across three nominal levels. Lower-order omnibus statistics generally maintain an empirical Type I error close to the nominal levels and detect the slope alternatives, whereas Pearson–Fisher is liberal in the sparse full table. A symmetric parameter vector therefore need not give uniform local diagnostic results.

Symmetry doi: 10.3390/sym18091519

Authors: Marko Živanović Vanja Luković Olga Ristić Hana Stefanović Sanja Antić Ana Savić

Real-time automatic detection of fire and smoke is a critical component of modern safety systems in surveillance, industrial, and environmental-protection applications, where conventional sensor-based systems show fundamental limitations. In this paper, a comparative analysis of four deep-learning architectures—DETR, Faster R-CNN (ResNet-50-FPN), Faster R-CNN (MobileNetV2), and RetinaNet—was conducted on a heterogeneous corpus of 67,765 annotated images originating from four different datasets. All models shared the same data-preparation, augmentation, and evaluation pipeline, while each was trained with the default configuration of its reference implementation; the comparison therefore reflects each architecture as it is typically deployed rather than a comparison under a single unified training budget. Faster R-CNN with the ResNet-50-FPN backbone achieved the highest accuracy (mAP@0.50 = 78.7% on the Indoor set; 73.8% on the SmokeAndFire set) and the highest mean mAP@0.50 across all datasets (48.7%), obtained with an inference speed of 76.9 FPS and a latency of 13.5 ms on NVIDIA RTX 4090 hardware, which makes it a promising candidate for real-time fire-detection systems on comparable hardware. The main contribution of this work is a unified evaluation of four representative object-detection architectures across four heterogeneous fire-and-smoke datasets. The study further quantifies the performance asymmetry between fire and smoke detection through a normalized morphological asymmetry index, reflecting the consistently lower detection accuracy achieved for smoke owing to its diffuse and semi-transparent appearance, and provides practical guidelines for selecting an architecture according to real-time deployment requirements. Because each configuration was trained once with a fixed random seed and all speed measurements were obtained on a single desktop GPU, the reported differences are interpreted descriptively; their statistical validation, cross-dataset evaluation, and measurement on embedded hardware are identified as future work.

Symmetry doi: 10.3390/sym18091518

Authors: Artur Piękosz

We prove that a direct sum of locally small spaces, each of which is either paracompact or has its admissible open sets closed under arbitrary unions, is mild (all its subspaces are strict) and that mildness is hereditary. We show how to produce examples of non-mild spaces (answering an earlier problem) and consider finitary locally small spaces. We apply the results to paracompact locally definable spaces over o-minimal structures.

Symmetry doi: 10.3390/sym18091516

Authors: Jinxin Liu Chuanyan Wang Chengpen Li

As a meta-heuristic inspired by human student learning behaviors, the original Student Psychology-Based Optimization (SPBO) suffers from insufficient exploitation of historical population records, simplistic individual interaction patterns and high risk of falling into local optima. This work develops an enhanced SPBO (ESPBO) embedding three dedicated learning mechanisms. The adaptive knowledge-accumulation learning component imports personal historical best, global elite and population-mean information into position update formulas to sustain coherent search trajectories and enhance convergence precision. The multi-level peer collaborative learning module categorizes agents into excellent, intermediate and under-performing groups based on fitness values. Customized learning rules are configured for each group to enable diverse information sharing: elite individuals expand promising search regions, medium-level agents learn from counterparts, and inferior individuals move toward high-quality candidates. The progressive examination feedback component dynamically modulates search intensity by measuring the fitness improvement of each individual, so as to better balance global exploration and local exploitation. Comparative numerical experiments are carried out on CEC2017 and CEC2022 benchmark test suites against multiple advanced meta-heuristic algorithms. Results indicate that ESPBO exhibits outstanding accuracy and robustness on unimodal, multimodal, hybrid and composite test functions. To explore its real-world applicability, ESPBO is adopted for mobile-robot path-planning simulations under multi-scale grid maps. Simulation results from 20 × 20, 40 × 40 and 60 × 60 environments illustrate that ESPBO stably produces collision-free trajectories, outperforming comparative algorithms in path length, smoothness and safety performance. It is demonstrated that the three embedded learning mechanisms substantially strengthen the optimization capacity of vanilla SPBO, and ESPBO possesses considerable application potential for numerical optimization as well as mobile robot path-planning scenarios.

Symmetry doi: 10.3390/sym18091517

Authors: Saeid Jafari Nodirbek Kamoldinovich Mamadaliev Said Isaev Rustam Mekhriddinovich Juraev

We study a support-based notion of fuzzy density together with fuzzy weight in Lowen fuzzy topological spaces. Two equivalent descriptions of the density invariant are considered: one based on supports of fuzzy sets meeting every nonzero fuzzy open set and the other on ordinary subsets meeting every nonzero fuzzy open set in a positive degree. The invariant is characterized by the ordinary density of the associated support topology. This representation is used to distinguish results transferred from ordinary topology from phenomena that retain membership-level information. We show that the supports of a fuzzy base form an ordinary base for the support topology, yielding a structural comparison between ordinary weight and fuzzy weight. For surjective fuzzy continuous and fuzzy open mappings, the fiber-cardinality estimate is shown to be sharp, while a counterexample demonstrates that fuzzy openness cannot in general be omitted from the reverse estimate. We also construct two Lowen fuzzy topologies with the same support topology and the same support-based fuzzy density but different fuzzy weights, with genuinely non-crisp membership functions playing an essential role in one of them. Applications to finite products, group quotients and fuzzy symmetric powers are obtained, and the density and weight of lower-semicontinuous Lowen fuzzy topologies are computed.

Symmetry doi: 10.3390/sym18091515

Authors: Mohamed Adel Ahmed Gamal Atta Omar Mazen Alqubori Naher Mohammed A. Alsafri Mohamed Abbas El-Naggar Waleed Mohamed Abd-Elhameed

In this paper, for some second-order singular functional differential equations (SOSFDEs), a tau-type spectral technique is built using non-origin-symmetric polynomials called shifted convolved Fibonacci polynomials as the approximation basis. Derivative formulae, integral identities required for the tau projection, power-form and inverse representations, and several operational relations are derived for this basis. The singular term and the shifted arguments can be handled directly within an algebraic framework thanks to these formulas, eliminating the need to introduce a grid on the computing interval. For the unknown coefficients, the resultant approximation yields a finite system. The uniform convergence of the truncated shifted convolved Fibonacci expansion is established, together with error estimates and a stability result for the proposed approximation. To demonstrate the procedure’s accuracy, three cases are utilized as benchmarks. As the truncation order increases, the reported errors gradually diminish, and the computed solutions correspond closely with the exact answers. Further evidence of the reliability and efficiency of the suggested shifted convolved Fibonacci tau procedure is provided by comparisons with existing numerical data.

Symmetry doi: 10.3390/sym18091514

Authors: Michiya Fujiki Takashi Mori Julian R. Koe Mohamed Mehawed Abdellatif

To probe a possibility of symmetry breaking within chainlike σ-conjugated polymers, we investigated the preservation of P-odd/T-even and P-odd/T-odd symmetries in semiflexible poly(n-butyl-isobutylsilane) (C4-iBS), which lacks point chirality. Using circular dichroism (CD) and circularly polarized luminescence (CPL) spectroscopy in various achiral solvents, we recorded three sets of temperature-dependent CD generation and inversion at the Siσ–Siσ* transitions, which were assigned to 114-, 83-, and 73-helices, respectively. In n-hexadecane-h34, C4-iBS exhibited a CD inversion due to an 83-helix reversal from gabs = −1.3 × 10−3 (40 °C) to +0.7 × 10−3 (20 °C). The observed H/D isotope effects in n-hexadecane-h34 and -d34 suggest that 1H and 2H nuclei contribute differently to the temperature-dependent 83-helicity. Remarkably, photoexcited C4-iBS at 20 °C displayed dynamic CPL oscillations at 330 nm, alternating between glum = −7 × 10−2 and +9 × 10−2 at ~20 s intervals. These CD and CPL results suggest a mixing of P-odd and P-even states in both S0- and S1-states. Furthermore, magnetic circular dichroism (MCD) and magnetochiral dichroism (MChD) under a reversing H0 = ±0.45 T at 23 °C produced non-mirror-image MCD/MChD spectra and their non-identical unpolarized MUV/MChUV spectra. These observations and analysis imply the simultaneous mixing of P-odd/P-even and T-odd/T-even terms in the S0- and S1-states of C4-iBS.

Symmetry doi: 10.3390/sym18091513

Authors: Peng-Yeng Yin

“Symmetry” is a salient feature widely observed in nature such as animal skeletons, plant leaves, starfish, and beehives, among others [...]

Symmetry doi: 10.3390/sym18091512

Authors: Zenab Z. Rashed Sameh E. Ahmed

Normalization of entropy generation provides a dimensionless and physically consistent framework for comparing irreversibility mechanisms under different operating conditions, while log-space third-order polynomial regression enables accurate representation of nonlinear coupled transport behavior over wide parameter ranges. Accordingly, this study investigates entropy generation associated with double-diffusive mixed convection in an oblique vented chamber filled with a nano-encapsulated phase change material suspension under local thermal non-equilibrium (LTNE) conditions in a porous medium. The governing equations are formulated by incorporating the effects of inlet and outlet sizes through the Reynolds number to provide a more realistic representation of the physical configuration. The dimensionless governing equations are discretized using the finite volume method based on the control-volume approach. The resulting entropy generation, heat transfer, and mass transfer characteristics, including the average fluid Nusselt and Sherwood numbers, are normalized and correlated using log-space third-order polynomial regression models. An effective RSM approach is employed to perform a sensitivity analysis of the key operating parameters. The results reveal that, for all inlet and outlet configurations, the normalized average fluid Nusselt number increases with increasing nanoparticle volume fraction. Moreover, both the normalized average Sherwood number and normalized fluid entropy generation increase with the Soret coefficient, demonstrating its pronounced influence on double-diffusive transport and irreversibility within the LTNE porous chamber.

Symmetry doi: 10.3390/sym18091511

Authors: Chih-Wen Chang

Backward problems, also referred to as inverse problems, aim to recover an unknown cause—an initial state, a boundary condition, a source term, a material parameter, or a hidden internal structure—from indirect observations of its effects [...]

Symmetry doi: 10.3390/sym18091510

Authors: Chunbiao Li Yi Yao Haiyan Fu Zuohua Liu Tomasz Kapitaniak

The number of coexisting attractors in a dynamical system needs to be increased in a simpler and more orderly manner. Conditional School of Artificial Intelligence symmetric chaotic systems achieve polarity balancing through slope changes of absolute-value functions induced by variable offset boosting, thereby generating two coexisting attractors. The attractor doubling method meets polarity balances by introducing pairs of absolute-value functions and signum functions, and therefore it can double coexisting attractors. Since the operation unit of attractor doubling can be repeated, the number of coexisting attractors in the system continuously doubles with each iteration, resulting in 2n coexisting attractors. In this work, the principle of polarity balancing is further developed. By introducing three-segment linear functions with complementary polarity balancing functions, coexisting attractors are generated through a tripling operation. This tripling operation can also be repeated, enabling the system to generate coexisting attractors in quantities of 3n. This operation, along with other approaches, makes the number of coexisting attractors more controllable and desirable and lays a foundation for the design and application of coexisting chaotic attractors and for the construction of multi-scroll/multi-wing attractors.

Symmetry doi: 10.3390/sym18091509

Authors: Linan Guan Nuodi Liu Jianwei Shen

Bursting is fundamental to neural information encoding and plays critical roles in rhythm generation, sensory processing, and motor control. However, the dynamical mechanisms by which multiple parameters coordinately regulate bursting properties in mesencephalic trigeminal neurons remain unclear. Here, using fast–slow decomposition we reveal three distinct asymmetries in the regulation of bursting dynamics by four key parameters, gNaR, gNaP, Iapp, and kb. First, asymmetric shifts in the SNLC and subH bifurcation curves in the two-parameter plane, rather than symmetric displacements, underlie the expansion or compression of the subH–SNLC window, directly modulating bursting duration (BD) and inter-burst interval (IBI). Second, gNaP produces a paradoxical asymmetry: although it narrows the subH–SNLC window, it simultaneously reduces the per-spike consumption of the slow variable hp via sustained depolarization, thereby prolonging BD while shortening IBI, a dissociation that cannot be predicted from the bifurcation window alone. Third, Iapp induces an opposing asymmetry: it widens the subH–SNLC window while shortening IBI, revealing that bifurcation expansion and the silent phase duration are not monotonically coupled. All four parameters increase spike count per burst (SCPB) and BD, yet their effects on IBI diverge, with gNaR and kb extending it whereas gNaP and Iapp shorten it, establishing that active and silent phases are asymmetrically regulated. Our findings demonstrate that bursting characteristics emerge from the cooperative interplay of bifurcation positions, membrane potential, slow-variable dynamics, and hysteresis. These results provide dynamical insights into the asymmetric regulation of bursting rhythms by ion channel parameters, and may offer theoretical guidance for neuromodulation strategies targeting channel-related disorders.

Symmetry doi: 10.3390/sym18091508

Authors: Erdem Gündoğdu Aydın Büyüksaraç Ercan Işık Fatih Avcil Marijana Hadzima-Nyarko

The city center of Kahramanmaraş suffered considerable damage as a result of the Mw 7.7 Pazarcık and Mw 7.6 Elbistan earthquakes, which took place on 6 February 2023. The present study examines the impact of urban development, groundwater conditions, and paleo-drainage systems on the pattern of earthquake damage in the Dulkadiroğlu and Onikişubat districts, which make up the urban center of Kahramanmaraş. Geological, hydrogeological, geomorphological, urban development, and building damage data were combined and analyzed using a GIS environment. Kernel Density Estimation (KDE) and spatial overlay analyses were carried out in order to show the spatial concentration of the damage. The results show that severely damaged or collapsed buildings were not distributed at random; rather, they were found in regions featuring largely young geological units, shallow groundwater levels, and former stream channels. In several areas where rapid urban expansion has occurred over the past thirty years, there is a clear spatial relationship between high damage densities and paleo-drainage systems. These findings reveal a spatial relationship between severe building damage and areas characterized by alluvial deposits, reconstructed paleo-drainage corridors, and potentially shallow groundwater conditions. Although the available data do not allow the damage to be directly attributed to site amplification or other geotechnical mechanisms, the observed spatial patterns identify areas where local ground conditions warrant further geotechnical and seismological investigation.

Symmetry doi: 10.3390/sym18091507

Authors: Chumsak Sibunruang Jantima Polpinij Manasawee Kaenampornpan Thananchai Khamket Jaturong Som-ard Anirut Chottanom Jatuphum Juanchaiyaphum Vuttichai Vichianchai Bancha Luaphol

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Symmetry doi: 10.3390/sym18091506

Authors: Mehmet Baysal İbrahim Yel Ahmet Sarucan

Accurate story point estimation is crucial for effective agile planning in software development. Both primary methods have inherent limitations. Expert judgment is susceptible to subjectivity, whereas machine learning models require extensive, high-quality, project-specific datasets that are often unavailable. This study proposes a hybrid framework that emphasizes expert reasoning while incorporating data-driven reinforcement. Evaluators are weighted using the Analytic Hierarchy Process (AHP), and their item-level assessments are expressed as Neutrosophic Z-numbers (NZN) and aggregated using a Bonferroni Mean operator. The resulting estimate is then combined with a prediction from an Extreme Gradient Boosting (XGBoost) model. The framework was tested on 1500 backlog items from an agile team developing a Governance, Risk, and Compliance (GRC) product. “The classification performance, as indicated by an accuracy of 41.18%, a precision of 0.39, and a recall of 0.41, was moderate. In regression analysis, the Mean Absolute Error (MAE) was 1.05, the Mean Squared Error (MSE) was 3.15, and the Root Mean Squared Error (RMSE) was 1.77. Furthermore, the proximity to expert estimates was notable, with 78.88% of predictions falling within ±1 Fibonacci position and 94.65% within ±3 levels.” In terms of classification (accuracy 41.18%, precision 0.39, recall 0.41), regression (Mean Absolute Error (MAE) 1.05, Mean Squared Error (MSE) 3.15, Root Mean Squared Error (RMSE) 1.77, and proximity to expert estimates (78.88% within ±1 Fibonacci position and 94.65% within ±3 levels), the exact-class accuracy was moderate. Nonetheless, most predictions closely matched expert judgment. Significantly, the NZN–Bonferroni configuration achieved an average sprint completion rate of 97.41%, exceeding the fuzzy AHP benchmark of 93.94%. This framework provides agile teams with a transparent and repeatable method for improving the consistency of sprint planning.

Symmetry doi: 10.3390/sym18091505

Authors: Qingdong Wu Longfan Shi Zhile Yang

Aiming at the multi-objective optimization problem of dynamic economic emission dispatch (DEED) considering plug-in electric vehicles (PEVs), this paper proposes an improved multi-objective grey wolf optimizer (IMGWO) to simultaneously minimize power generation costs and pollutant emissions. Traditional economic dispatch requires a trade-off between economic benefits and environmental sustainability. However, these two objectives are inherently conflicting, rendering single-objective optimization strategies ineffective for practical DEED scenarios. In this study, PEVs are integrated into the DEED framework to implement systematic charging and discharging scheduling, which realizes peak shaving and valley filling and further improves the operational quality of the power grid. The incorporation of PEVs also renders the DEED problem more nonlinear, non-convex, and non-smooth. To address these complex characteristics, the proposed IMGWO adopts modified logistic chaotic mapping for population initialization, combined with a nonlinear convergence factor and dynamic weight strategy. These improvements effectively enhance the global search ability and convergence accuracy while preventing premature convergence. Furthermore, the Pareto optimal solution set is utilized to resolve conflicts among multiple optimization objectives. Comparative experiments on standard benchmark test cases verify that the proposed IMGWO outperforms four existing advanced algorithms in comprehensive evaluation indicators, demonstrating its excellent effectiveness and robustness. Finally, the proposed algorithm is applied to three classic PEV-integrated DEED cases. The results confirm that the IMGWO can achieve superior scheduling performance and verify the positive effect of PEV participation in improving grid dispatching quality and unit operation efficiency.

Symmetry doi: 10.3390/sym18091504

Authors: Mengxuan Dai Xuan Chen Ling Yang Yunni Xia Jiale Zhao Xifeng Xu Xin Hao Houli Xie

Resource-intensive mobile edge computing (MEC) services are often provisioned on a per-request basis, resulting in repeated activation of equivalent service instances and redundant transmission of the same category-level state over overlapping inter-station links. Existing approaches rarely integrate demand aggregation, shared-instance provisioning, and reusable multi-target state distribution into a unified orchestration workflow. This paper proposes Coalition-based Shared Instance Provisioning and Routing (CoSIPR), a shared-service orchestration framework built around dynamic interest coalitions. CoSIPR predicts user requests and mobility, projects predicted locations onto the road network, filters unreliable or infeasible requests, and groups nearby users requesting the same service category. For each coalition, a marginal-gain-based candidate-reduction method and variable neighborhood search determine the serving stations, user assignments, and shared-instance counts. The selected stations then form the target set for a load-aware routing procedure that selects an existing state source and uses path-fusion reinforcement learning (PF-RL) to construct routes that reuse path segments across multiple targets. Experiments using real-world mobility and road-network data show that CoSIPR improves service-category matching, request satisfaction, and the average number of accepted requests per instance. It also reduces aggregate state-transfer cost and limits hotspot exposure while maintaining a controlled trade-off between end-to-end delay and state-transfer cost. These results demonstrate that dynamic interest coalitions and reusable multi-target paths can improve the efficiency of shared-service orchestration in MEC.

Symmetry doi: 10.3390/sym18091503

Authors: Linqing Zha Wenkang Chen Xuejun Zhang

In distributed optimization, communication bottlenecks and computational heterogeneity are two critical challenges that undermine system scalability and efficiency. To address these issues, we propose a synchronous algorithm, CC-GT (Conjugate Compressed Gradient Tracking), which integrates conjugate gradient acceleration with dual error-compensated compression to reduce communication overhead and accelerate convergence. Theoretically, we establish its linear convergence under strongly convex and smooth assumptions, and extensive experiments on regression, classification, and neural-network tasks validate the fast convergence and communication efficiency of the synchronous CC-GT. Building on this foundation, we develop GFLCC, an asynchronous extension that eliminates global synchronization barriers while preserving the core mechanisms of CC-GT, thereby inherently mitigating the straggler effect and enabling dynamic load balancing. For the asynchronous GFLCC, extensive comparisons against state-of-the-art asynchronous baselines under harsh network interference demonstrate its strong empirical convergence speed, communication efficiency, and robustness. These results confirm that the proposed framework offers a practical and theoretically grounded solution for communication-constrained and heterogeneous distributed optimization systems.

Symmetry doi: 10.3390/sym18091502

Authors: Yu Ke Yuchao Zhang Linchen Zhang Wuyang Yuan Gehui Liu

With the continuous improvement in the high-speed railway (HSR) network, passengers’ holiday travel demand and the scale of tourism passenger flow have grown rapidly, placing higher requirements on HSR transportation services. Current train line plans are formulated according to regular daily passenger flow and fail to adapt to the travel demand generated by various holiday tourism products, resulting in poor adaptation to holiday passenger flow characteristics. Different from existing line-planning studies that consider only regular daily passenger flow, this paper is among the first to embed hierarchical holiday tourism products as a structural input of the HSR line planning problem and to explicitly handle asymmetric passenger demand in holiday periods. To address this gap, this paper optimizes HSR line plans based on the travel characteristics of tourism products. An integer programming model is established that allocates asymmetric passenger flows to train flows while minimizing the total operating cost. The model incorporates both conventional HSR passenger demand and the differentiated travel demand corresponding to different tourism products. A real-world experiment based on the HSR network in Jiangxi Province, China demonstrates that the proposed model effectively improves the tourism transportation efficiency and passenger flow distribution of HSR with reliable practicability. Sensitivity analyses further reveal how the tourism demand scale, the tourism sub-product diversity, and the benchmark line requirements affect the feasibility and cost of the line plan. This study provides a valid reference for the optimization of holiday train line plans and the coordinated development of tourism and transportation.

Symmetry doi: 10.3390/sym18091501

Authors: Arockia Xavier Annie Rayan Ajai Ram

The rapid adoption of Android applications in mobile commerce has increased exposure to sophisticated malware capable of bypassing traditional security mechanisms through code obfuscation, dynamic code loading, and runtime-triggered malicious behaviors. Although static analysis offers efficient large-scale detection, it often fails to identify concealed runtime activities, while dynamic analysis provides richer behavioral evidence but suffers from limited execution coverage and high computational overhead. To address these complementary limitations, this paper proposes FusionDroid, a lightweight multimodal Android malware detection framework that integrates permission-based static features, permission co-occurrence graph representations, and engineered runtime behavioral features through probability-level ensemble fusion. The framework was evaluated using Android applications collected from the AndroZoo repository, comprising 24,055 valid applications for static analysis and a balanced paired benchmark of 1716 applications for multimodal evaluation. The experimental results show that complementary static and dynamic representations can improve Android malware detection, although the magnitude and nature of the improvement depend on class distribution and evaluation metric. The best-performing model, StackedFusion-LightGBM, achieved 93.31% accuracy, 96.86% precision, 89.53% recall, a 93.05% F1 score, 97.69% ROC–AUC, and 98.14% PR–AUC. A controlled five-fold evaluation on the common paired benchmark further showed an F1 score of 0.9175±0.0176 for the full Static+Graph+Dynamic configuration compared with 0.8781±0.0081 for the Static-only baseline. Paired statistical analysis further supported the improvement. These findings show that multimodal fusion can improve Android malware detection while preserving low-complexity manifest-derived representations. The results support FusionDroid as a staged, sandbox-assisted framework in which lightweight static analysis is complemented by runtime behavioral evidence when deeper inspection is required.

Symmetry doi: 10.3390/sym18091500

Authors: Zengwen Wu Changrong Liao Xiaoling Xu Jixiang Yang Tao Lin

This work presents a finite-time adaptive fuzzy control approach for a category of multi-input multi-output nonlinear systems in the presence of input saturation and external disturbances. A hyperbolic function is adopted to convert the unconstrained control command into a bounded signal so that the actuator constraints are strictly respected. Unknown nonlinearities are approximated by fuzzy logic systems, whereas external disturbances are attenuated by adaptive mechanisms together with tanh-based robust terms. In addition, dynamic surface control is incorporated into the backstepping framework, where first-order filters are employed to avoid the computational burden associated with the repeated differentiation of virtual control laws. On this basis, a Lyapunov-based design is carried out to derive the finite-time adaptive control protocol. It is shown that every signal in the resulting closed-loop system is bounded, the closed-loop system is semi-globally practically finite-time stable, and the state variables converge to a small neighborhood of the desired states within a finite settling time, and the control inputs never exceed the prescribed bounds. Simulation results obtained using a representative rigid spacecraft as an example confirm the feasibility and disturbance-rejection capability of the developed method.

Symmetry doi: 10.3390/sym18091499

Authors: Kaveesha V. Senarathna Sujeeva Wijesiri Shamon Almeida

Graph labeling is an important area of graph theory that studies the assignment of integers to the vertices or edges of a graph according to specific rules. Among these labeling methods, distance magic labeling has attracted considerable attention due to its interesting combinatorial structure and applications in network design and communication systems. A distance magic labeling of a graph is a bijection from the vertex set to the set {1,2,…,n} such that the sum of the labels of the neighbors of each vertex is equal to a constant called the magic constant. This paper investigates the existence and construction of distance magic labelings for certain families of complete bipartite graphs. Two principal cases are studied, namely graphs of the form K2m,2m and K2m−1,2m. For graphs of the form K2m,2m, an explicit construction is developed showing that these graphs admit a distance magic labeling for every integer m≥1. The corresponding magic constant is derived as k=m(4m+1) and the validity of the construction is verified by proving bijectivity of the labeling function and equality of vertex weights. A similar constructive approach is applied to graphs of the form K2m−1,2m, where distance magic labelings are obtained using structured arithmetic label distributions. These constructions are further extended by applying vertex swapping techniques and block-based arguments to generate complete bipartite graphs Kp,q that preserve the same magic constant for certain values of p and q. These findings contribute to the understanding of how arithmetic structure and partition properties influence the existence of distance magic labelings and suggest several directions for further research in graph labeling theory.

Symmetry doi: 10.3390/sym18091498

Authors: Lanju Tao Zhengji Li Yingrui Ji Chih-Ting Liao Xi Xiao Xin Cao

Financial question answering is often treated as document question answering, although financial evidence is both multimodal and time-dependent. Semantically equivalent facts expressed in narrative text, tables, page images, or Extensible Business Reporting Language (XBRL) should support consistent answers, whereas a disclosure may support a query only after becoming public. We formalise this combination as crossmodal evidence symmetry under a causal temporal boundary and introduce TempFinRAG, a multimodal temporal retrieval-augmented generation (RAG) framework for point-in-time financial question answering. Given a question, company, and as-of date, the framework enforces the information boundary defined by U.S. Securities and Exchange Commission (SEC) filing availability; aligns page layout, text, table structure, and XBRL facts; retrieves time-valid evidence; executes auditable financial calculations; and generates a cited answer. A verifier checks temporal validity, claim support, numerical consistency, and the need to abstain. We further introduce TempFinQA, a point-in-time evaluation protocol built from public filings and XBRL facts, and evaluate the framework on complementary evidence-grounded, numerical, conversational, and multi-table benchmarks. On TempFinQA, TempFinRAG improves answer accuracy from 66.7% to 78.9% over hybrid RAG while reducing temporal evidence leakage from 10.8% to 1.7% and hallucination from 17.3% to 7.9%. Reliable financial question answering therefore requires consistent treatment across evidence representations and deliberately asymmetric access across time.

Symmetry doi: 10.3390/sym18091497

Authors: Ramesh Radhakrishnan William Julius Gerald Cleaver

We investigate scalar perturbations of a rotating asymptotically anti-de Sitter (AdS)-Teo traversable wormhole with a controlled non-separable angular deformation. The geometry retains a regular wormhole throat and the required AdS asymptotics, while an explicit quadrupolar deformation generates angular-channel coupling in the intermediate region. A sufficient condition is derived for an ergoregion-free parameter regime in which the spectral analysis is performed. Projecting the geometry-derived Klein–Gordon equation onto spherical harmonics yields a matrix-valued Sturm–Liouville system and an associated quadratic operator pencil. Throat regularity together with normalizable AdS boundary conditions leads to a determinant quantization condition for the discrete normal-mode spectrum. Two-, four-, and six-channel calculations demonstrate systematic numerical convergence of the retained low-lying normal-mode frequencies under enlargement of the angular basis. The complete finite generalized eigenspectrum is also examined without imposing a near-reality selection criterion, and no growing scalar mode is found within the ergoregion-free parameter range and numerical resolutions studied. The six-channel rotation continuation exhibits a finite interior level-repulsion feature accompanied by collective redistribution of the multichannel eigenvectors. The same coupled spectral framework formally defines a matrix-valued two-boundary response whose poles are selected by the normal-mode matching condition; the response plots presented here illustrate this structure using the reduced two-channel effective model rather than a numerical reconstruction of the full six-channel response. We further derive the geometry-dependent short-distance Hadamard subtraction and identify the finite inter-channel structure entering a truncated local quantum mode sum, without claiming a complete numerical evaluation of ⟨Φ2⟩ren. Together, these results establish a geometry-to-spectrum framework in which the spacetime geometry determines the coupled operator, the global boundary conditions determine its normal-mode spectrum, and the same coupled spectral framework organizes the associated asymptotic response and local quantum mode-sum structure.

Symmetry doi: 10.3390/sym18091496

Authors: James M. Hill

Within the framework of special relativity, the author has developed an extended in vacuo Lorentz invariant particle-wave theory of Newton’s mechanics that takes account of the particle energy as a major driver of the motion that may be of comparable magnitude to any applied external potential. Lorentz invariance ensures that the proposed theory provides a more fundamental representation of applied force than the conventional notion of applied force as the rate of change of momentum. The purpose of this paper is to generalise an existing exact solution of this theory, originally derived for Einstein’s particle energy expression, to a family of Lorentz invariant power-law particle energies characterised by an arbitrary constant κ. By Lorentz invariance, we refer to invariance under the combined special relativistic space-time and energy–momentum transformations. The power-law energy expressions are important because they underpin and bear the same relationship with an extension of the Planck–de Broglie energy–momentum relations in exactly the same manner as Einstein’s energy expression does for the conventional Planck–de Broglie relations. The de Broglie wave energy as a function of particle velocity is determined as an integral expression. This integral is evaluated explicitly for a number of illustrative values of κ, and the known solution corresponding to the Einstein energy is included through the special case κ=0.

Symmetry doi: 10.3390/sym18091495

Authors: Haowen Yuan Shibin Li Yusheng Yang

Deep shale gas reservoirs are subjected to anisotropic in situ stresses that break the directional symmetry of hydraulic-fracture growth, promote propagation along the maximum horizontal principal stress, and suppress transverse spreading. This study investigates geomechanical and injection controls on fracture-network evolution in the Yi214 block of the Changning shale gas field using a field-calibrated numerical workflow. The model was calibrated against pre-design microseismic-derived metrics from Well Yi202, with relative differences of 1.8% for fracture length and 3.9% for effective fracture volume; this comparison is treated as calibration rather than independent multi-well validation. A dimensionless directionalization index, Id = (L/L0)/(V/V0), is defined relative to the equal-horizontal-stress case (Δσh = 0) as a global proxy for the concentration of longitudinal extension relative to volumetric spreading. One-factor-at-a-time screening evaluated Young’s modulus, Poisson’s ratio, horizontal stress difference, injection rate, fluid-volume intensity, and proppant loading, while final parameter combinations were treated as well-specific engineering design selections rather than mathematical global optima. As Δσh increased from 0 to 15 MPa, fracture volume decreased by approximately 44% and average fracture length increased by approximately 12%, raising Id from 1.00 to 1.99; the increase in Id was driven predominantly by reduced volume retention rather than length growth alone. The final combined design cases increased simulated fracture volume by 18–27%, while the reported model-based EUR forecasts increased by 36–40%. The results provide a symmetry-based interpretation of stress-controlled fracture directionalization and a field-calibrated basis for well-specific stimulation design in deep shale reservoirs.

Symmetry doi: 10.3390/sym18091493

Authors: Linxi Duan Huaping Yang Qiming Qi Qihong Wu Changjiang Shao Wanting Gong Yunfan Yang

Rapid post-earthquake assessment of urban rail viaducts requires nonlinear pier models that capture cyclic deterioration at manageable computational costs. This study calibrates and evaluates the established Ibarra–Medina–Krawinkler (IMK) formulation for reinforced-concrete solid piers rather than proposing a new hysteretic law or experimental database. Seven previously reported quarter-scale tests are reused for direction-specific parameter identification and calibration-set assessment. The new contribution is a multilevel accuracy–efficiency evaluation covering cyclic metrics, a scaled 12 m Fiber benchmark, nonlinear dynamics, incremental dynamic analysis (IDA), computational cost, and fragility. For the specimens, final secant stiffness achieved R2 = 0.891 and a 9.93% mean absolute percentage error; cumulative energy achieved R2 = 0.893 but a 27.79% error. For the prototype, the intermediate-backbone force difference was 3.83%. Under eight representative records, mean differences were 2.64% for peak base shear and 4.15% for peak displacement. Across 60 records scaled from PGA = 0.1 to 1.5 g, peak–displacement and peak–base–shear comparisons yielded R2 values of 0.854 and 0.931. The corresponding mean absolute percentage errors were 15.95% and 11.32%. The IMK model reduced wall-clock time by 96.37%, a 27.55-fold speed-up. Paired fragility curves differed by 3.14 percentage points on average, and median PGA capacities differed by 11.23%; Sa(T1) and capacity-dispersion checks preserved the main comparison. Because the record groups were not matched in spectral amplitude or velocity, their ranking is interpreted as an association rather than a causal pulse-period effect. The calibrated IMK model is therefore suitable for rapid global demand and fragility assessment within its calibration domain, whereas Fiber models remain preferable for local damage, residual deformation, and detailed energy dissipation.

Symmetry doi: 10.3390/sym18091494

Authors: Abdulrahman B. Albidah Jagadish Singh Muqrin A. Almuqrin Mohammed Alghazi Abdulaziz H. Alharbi Abdullah A. Ansari

This paper investigates the dynamical behaviour of an infinitesimal mass body moving under the gravitational influence of a heterogeneous triaxial primary body and a secondary body that produces a modified Newtonian gravitational force in the elliptic restricted three-body problem. In this model, the two primary bodies move around their common center of mass in elliptical orbits, while the mass of the third body is assumed to be sufficiently small so that its effect on the motion of the primary bodies can be neglected. The equations of motion of the infinitesimal body are formulated, and the corresponding mean motion of the system is determined by taking into account the triaxial structure and mass heterogeneity of the primary body as well as the modified gravitational field of the secondary. The equilibrium points of the system are then investigated. Both collinear equilibrium points, which lie along the line joining the two primary bodies, and non-collinear equilibrium points, which are located away from this line, are determined analytically. Their linear stability is examined by studying the characteristic equations associated with small perturbations around these points. In addition, the dynamical characteristics of the system are illustrated numerically through potential surfaces, the locations of equilibrium points, permissible regions of motion, periodic orbits, and basins of attraction. These numerical results provide a clear understanding of how the eccentricity of the motion, triaxiality of the primary body, and the modified gravitational force of the secondary influence the motion of the infinitesimal body. The results may be useful in understanding the dynamics of celestial bodies.

Symmetry doi: 10.3390/sym18091492

Authors: Mukaddes Ökten Turacı

Graph theory has become a fundamental mathematical framework with broad applications across various scientific fields, such as biology, network security, computer science, and chemistry. A specialized branch known as chemical graph theory employs graph-based mathematical principles to model and analyze molecular architectures. Typically, molecular graphs are generated from 2D representations of chemical structures, where physicochemical properties play a decisive role in interpreting a molecule’s physical and chemical behavior. Structural symmetry within these molecular graphs significantly aids in deriving fundamental graph parameters. Recently, a novel theoretical method has been introduced to perform a comparative analysis of seven significant domination parameters alongside the p-electronic energy of benzenoid hydrocarbons. In the field of graph theory, the independence, covering, and domination numbers stand as fundamental parameters. Inspired by this methodology and related research, this study presents an exploratory investigation into the physicochemical properties of 10 selected drug and bioactive molecules from diverse therapeutic classes to evaluate bulk size-dependent properties using these fundamental graph parameters. Various regression models were evaluated alongside leave one out cross validation (LOOCV) diagnostics for six specific properties of these compounds: boiling point (BP), enthalpy of vaporization (EV), flash point (FP), molar refractivity (MR), polarizability (P), and molar volume (MV). The results highlight the potential of these fundamental graph parameters in QSPR modeling, establishing an exploratory proof of concept that warrants future support from more comprehensive independent datasets.

Symmetry doi: 10.3390/sym18091491

Authors: Zongmin Liang Xiugang Li Libo Lan Xiugen Jiang Jing Xu Jiaheng Miao Tianyang Liu

Conventional structural analysis of multi-span glasshouses often idealises aluminium gutter–steel column connections as pinned or rigid, although their actual rotational restraint remains insufficiently quantified. In this study, 21 full-scale specimens were tested to investigate two representative configurations: STP joints between gutters and gable columns and STZ joints between gutters and intermediate columns. The measured secant rotational stiffness ranged from 108.5 to 185.5 kN·m/rad for STP joints and from 5.63 to 19.03 kN·m/rad for STZ joints. STP joints exhibited higher in-plane flexural resistance and rotational stiffness with smaller rotations, whereas STZ joints showed lower stiffness and greater nonlinear rotational deformation. The experimental results demonstrate that neither connection can be adequately represented as perfectly pinned or fully rigid. When the measured stiffnesses were introduced into an idealised elastic column model, the theoretical buckling capacity increased by 4.73–7.89% for STP end columns and 3.73–12.08% for STZ intermediate columns. The measured stiffnesses provide experimentally based rotational-spring parameters for refined glasshouse structural analysis.

Symmetry doi: 10.3390/sym18091489

Authors: Ravil Mussabayev Rustam Mussabayev

This paper presents a comparative analysis of optimization techniques for the minimum sum-of-squares clustering (MSSC) problem—widely known in applied research as the K-means clustering problem—in the context of big data. K-means is the most widely used algorithmic framework for solving this problem, but MSSC methods can suffer from scalability issues when dealing with large datasets. The paper reviews approaches for overcoming these issues, including decomposition, sampling, initialization, preprocessing, parallel and distributed computation, data summarization, acceleration of distance computations, and hybridization with various metaheuristic frameworks. The experimental evaluation compares selected big data MSSC algorithms under a common benchmark protocol and assesses them according to the dominance criterion provided by the “less is more” approach (LIMA), i.e., simultaneously along the dimensions of clustering quality, speed, and simplicity. The results reveal distinct accuracy–time regimes and a multi-algorithm Pareto front under LIMA dominance, indicating that no method is universally preferable and that greater algorithmic complexity does not by itself ensure a better practical trade-off. Lightweight simplest methods generally favor speed but may sacrifice accuracy, while more complex hybrid methods reach stricter accuracy levels at substantially greater computational cost; competitive stochastic-sampling methods occupy intermediate accuracy–time–simplicity trade-offs.

Symmetry doi: 10.3390/sym18091490

Authors: Zihan Liu Haoqian Liu Yan Wang Yanfeng Chen

High-precision urban 3D localization is a critical foundation for vehicular Internet of Things (IoT) applications, yet conventional Global Navigation Satellite System (GNSS)-based localization is vulnerable to signal obstruction, multipath effects, and non-line-of-sight propagation in complex environments. To improve localization reliability, this paper proposes a consistency-guided quantitative–qualitative fusion (CG–QQF) framework with vision-based terrain constraints. The framework integrates heterogeneous quantitative sensors, including an absolute positioning source, inertial measurement unit (IMU), wheel encoders, and a steering angle sensor, for continuous metric state estimation, while a monocular camera provides qualitative terrain-slope information. Rather than treating visual perception as a direct metric observation, the proposed method introduces it as a conditional structural constraint that is activated only when it is consistent with the quantitative estimate, thereby regularizing the localization solution and suppressing vertical drift. Although ultra-wideband (UWB) positioning is adopted as the absolute positioning source in the experimental platform, it serves as a generic positioning module and can be replaced by GNSS-based techniques such as real-time kinematic (RTK) and precise point positioning (PPP). In an indoor scaled proof-of-concept experiment over a controlled four-lap dataset, CG–QQF achieves a 3D RMSE of 0.0575 m and a vertical MAE of 0.0042 m. Its 3D RMSE is approximately 4.0% lower than quantitative sensor fusion (QSF), 37.9% lower than absolute-positioning/inertial fusion (ABS–INS), and 49.9% lower than vision-assisted quantitative fusion (VA–QF). These results demonstrate that consistency-triggered qualitative constraints can complement metric sensor fusion without directly introducing uncertain visual measurements, providing a practical mechanism for improving the robustness and vertical stability of heterogeneous localization systems.

Symmetry doi: 10.3390/sym18091488

Authors: Abdulrahman B. M. Alzahrani

This study discusses the (2+1)-dimensional Kadomtsev–Petviashvili–Benjamin–Bona– Mahony equation, which emerges in weakly nonlinear dispersive plasma waves and shallow water dynamics in ocean engineering. Logarithmic dependent-variable transformations are applied to reconstruct a one-exponential tau function as a common one-soliton profile and derive its dispersion relation. This is a standard transformed solution listed as three normalized logarithmic maps, but not a new family of solutions. Only a one-exponential soliton is claimed, no two-soliton family and no arbitrary-N soliton family. The explicit rational rogue-wave families of the first, second, and third orders are derived using a modified version of a well-known center-shifted polynomial tau-function method that is applied to the KPBBM bilinear form, with both center parameters β and γ independent. The novelty is thus limited to the specific model and is not based on a new KPBBM equation or a fundamentally novel symbolic algorithm. The rogue-wave center translates in the longitudinal and transverse directions through β and γ, respectively, for a fixed order N and fixed model parameters. They leave the pattern, localization width, background, and the arrangement of inner patterns unchanged. Lump solutions and lump–soliton interaction structures are also obtained and investigated. The auxiliary Hirota bilinear constraint and its reduced bilinear representation are explicitly given. The higher-degree equations found in the directional logarithmic maps are not new multilinear equations, but rather the denominator-cleared differential polynomial residuals. The validity of each solution family retained is guaranteed by means of analytical substitution or vanishing of symbolically identical-to-zero residual in the original KP–BBM equation. The two- and three-dimensional plots are used only to demonstrate the amplitude profile, localization, and propagation of the solutions, as verified by the analysis. In the weakly nonlinear, long-wave and weakly transverse regime where the KPBBM reduction is valid, these solutions give idealized mathematical representations of localization and interaction mechanisms. They are not predictive of coastal instability or offshore hydrodynamic loading, for which dimensional calibration and experimental/field validation would be necessary.

Symmetry doi: 10.3390/sym18091487

Authors: Jun Ma Jishen Peng Haotong Han Liye Song Hao Liu

High penetrations of wind and photovoltaic generation create simultaneous challenges for battery energy storage system (BESS) planning in distribution networks, including differences in nodal regulation value, power-energy configuration, and day-ahead operation under forecast uncertainty. This study develops a sequential planning-to-operation workflow comprising candidate-bus generation, siting and sizing within the candidate set, and finite-scenario day-ahead scheduling for a fixed configuration. First, nodal net-injection sensitivities, Jacobian-assisted pre-screening, and deterministic topology/support safeguards are used to generate the main candidate set, and alternating-current (AC) finite-difference refinement is performed only for the sensitivity-led fast set; in the IEEE-33 system, this refinement reduces the number of AC power-flow calls from 65 for full-node analysis to 17. Next, Sensitivity-Guided Envelope-Based Nonanticipative Adjustable Recourse Optimal Power Flow (SG-ENAR-OPF) is solved separately for each bus in the main candidate set; the BESS location and power/energy capacities are jointly determined subject to the P-Q LinDistFlow model, BESS duration constraints, a shared affine response, and finite-scenario constraints. The full-node audit serves only as an independent paper-level validation benchmark and is not part of the deployable workflow. After the configuration is fixed, interval forecasts for load, photovoltaic (PV) output, and wind-turbine (WT) output at q05/q50/q95 are used to construct 25 static load-renewable disturbance points and seven temporal stress paths, over which a shared finite-scenario day-ahead policy is optimized. The IEEE-33 MAIN case selects Bus 30, with BESS capacities of approximately 10.66 MW/10.66 MWh. Using scaled public time-series data, the final policy is replayed over 46 consecutive 24 h execution windows, comprising 1104 h actual trajectories; under the 0.002 MW/MWh storage-engineering criterion, all 46/46 windows pass storage engineering validation, and energy continuity is maintained across all 45/45 interday boundaries. Further nonlinear AC post-validation converges at all 1104/1104 operating points, of which 1058/1104 satisfy the complete voltage and branch-capacity constraints. Supplementary results for IEEE-69 show that the workflow can be executed on a second radial test feeder. However, the conclusions are strictly limited to the tested feeders, finite scenarios, scaled public-data settings, and stated engineering tolerances and do not constitute a formal guarantee over a continuous uncertainty domain or of general cross-system applicability.

Symmetry doi: 10.3390/sym18091486

Authors: Mansour Mahmoud Hanan Almuashi

In this study, a new analytical approximation for the Bessel function of the first kind of order two J2(t), t>0, is produced using its power series and its asymptotic series as t tends to infinity. The absolute error between the new analytical approximation and the function J2(t) is analyzed for t∈[0,104], and the results indicate that it decreases significantly as t increases for large values of t. Additionally, our new approximation formula presents estimations for the positive zeros of the function J2(t) with extremely minor relative errors; the largest relative error for the first positive zero is 3.83594×10−5 and, thereafter, the relative errors gradually decrease until they reach 9.91627×10−11 for the twelfth zero. We also showed that our findings outperform some recently published ones.

Symmetry doi: 10.3390/sym18091485

Authors: Xueting Jin Jinfeng Xu Qianxin Xie Yuxuan Zhang

Short-term traffic-flow forecasting predicts road-network states, yet spatio-temporal predictors can leave structured, context-dependent residuals. They are seldom reused at inference, while similarity-based transfer can introduce corrections that increase error. We propose Memory-Based Utility-Aligned Residual Evidence with Calibrated Activation for Selective Correction of Traffic-Flow Forecasts (MURECAST), which treats historical residuals as candidate interventions. After freezing a context-aware base forecaster, MURECAST builds a static out-of-sample residual memory from an independent period. At inference, same-node and time-valid constraints delimit records, a forecast-visible utility estimator re-ranks them, and utility-weighted top-K aggregation forms a multi-horizon proposal. A chronological calibration split provides an empirical one-sided lower score for applying the proposal or retaining the base forecast. Across PeMS03, PeMS04, PeMS07, and PeMS08, MURECAST ranked first in 11 of 12 reported dataset–metric comparisons, attaining the lowest mean absolute error (MAE) and root mean squared error (RMSE) on all four datasets and the lowest mean absolute percentage error (MAPE) on three; relative error reductions over the strongest published results were 1.69–9.76%. Non-beneficial corrections represented 17–26% of accepted proposals versus 41–48% of all valid proposals. These results show that MURECAST reuses observed errors while concentrating intervention on corrections with lower observed non-beneficial risk under the evaluated chronological protocol.

Symmetry doi: 10.3390/sym18091484

Authors: Rye Kyeong Kim Hajun Lee Nyeonju Kang

This review examined cumulative findings from recent meta-analyses to identify current challenges and possible suggestions for improving the effects of BCI systems on stroke motor recovery. Consistent with the Preferred Reporting Items for Systematic Reviews and Meta-Analyses guidelines, a systematic literature search was conducted using PubMed, Web of Science, and the Cochrane Library on 30 June 2026. A total of 17 systematic reviews and meta-analyses were included. Among three motor intent-induced (i.e., motor attempt, motor observation, and motor imagery) modalities, motor attempt was the modality most consistently associated with significant therapeutic effects across meta-analyses. Electrical stimulation was a consistently effective feedback modality, whereas robot-assisted and visual feedback showed heterogeneous effects. Higher weekly session frequencies and moderate session durations (approximately 20–60 min) showed consistent motor recovery. Stroke type, age, intervention period, total sessions, total training time, and long-term effect durability were inconsistent across the included evidence. These findings suggest that applying BCI-based training is an effective rehabilitation program for the functional recovery of upper extremities in patients with stroke who have moderate to severe motor impairments, potentially achieving greater therapeutic efficacy when combining motor attempts with electrical stimulation.

Symmetry doi: 10.3390/sym18091483

Authors: Xuye Jiang Weiguang Ma Zihang Yu Jiayuan Zhang Feifei Fang Chenyi Shao

Lattice strain engineering, rooted in symmetry-breaking lattice distortion, is an effective strategy for modulating the electronic structure and catalytic performance of electrocatalysts. Herein, non-noble metal CuCoNiCrMox high-entropy carbides with tunable Mo content (HECMo-x) were rapidly synthesized within seconds via a high-temperature shock method. By leveraging composition-dependent lattice distortion engineering to deliberately break local translational symmetry, these catalysts were developed to optimize the alkaline hydrogen evolution reaction (HER). Density functional theory calculations reveal that lattice distortion optimizes the d-band center and regulates the electronic configuration. Concurrently, kinetic isotope effect tests and variable-potential electrochemical impedance spectroscopy measurements verify that this modulation balances the reaction kinetics of water dissociation and hydrogen adsorption, thereby accelerating the alkaline HER process. Consequently, the optimized HECMo-15% electrocatalyst exhibits outstanding activity, requiring an overpotential of only 34 mV at 10 mA cm−2. Furthermore, it exposes abundant active sites and maintains long-term operational stability with negligible attenuation over 23 h. This work provides a feasible design strategy and a practical paradigm for developing non-noble metal high-entropy carbides as highly efficient electrocatalysts for energy conversion applications.

Symmetry doi: 10.3390/sym18091482

Authors: Vladica S. Stojanović Hassan S. Bakouch Snežana Stojičić Shuhrah Alghamdi

The manuscript proposes a novel bounded proportional autoregressive (PAR) process based on the quasi-Lindley half-logistic unit (QHU) distribution, termed the QHU-PAR(1) process. Fundamental probabilistic properties of the process are established, including its Markov structure, moments, and stationarity conditions. A pseudo-innovation-based parameter estimation procedure is performed, demonstrating strong consistency and asymptotic normality of the resulting estimators. A Monte Carlo study is also conducted, showing satisfactory performance of the proposed estimators on finite samples, while the practical utility of the model is illustrated through the analysis of normalized crime-related time series. Comparative results show that the proposed QHU-PAR(1) process outperforms some competing specifications, highlighting its flexibility and efficiency for modeling bounded and asymmetric stochastic phenomena.

Symmetry doi: 10.3390/sym18091481

Authors: Michele Di Cristo

Inverse inclusion problems arise in several applications such as electrical impedance tomography, nondestructive testing, and geophysical imaging. The purpose of this review is to present some recent advances concerning stability estimates for the determination of inclusions from boundary measurements. We focus on elliptic conductivity equations with discontinuous coefficients and discuss different settings including isotropic and anisotropic conductivities, variable coefficients, local boundary measurements, and layered media. Particular attention is devoted to the analytical tools underlying logarithmic stability estimates, namely singular solutions, asymptotic analysis of fundamental solutions, quantitative unique continuation, and propagation of smallness arguments. We also compare different stability regimes, emphasizing how finite-dimensional geometric priors, such as polygonal or polyhedral inclusions, may lead to Hölder or Lipschitz stability estimates in contrast with the logarithmic stability typical of broader infinite-dimensional classes. We highlight how a common methodological framework can be adapted to increasingly complex geometrical and physical configurations. Finally, we briefly discuss some open problems and possible future developments in the study of inverse inclusion problems.

Symmetry doi: 10.3390/sym18091480

Authors: Kenny Sau Kang Chu Kuew Wai Chew Yap Hoon Yoong Choon Chang Stella Morris Chen Chen

Accurate discrimination among static eccentricity fault (SEFs), dynamic eccentricity fault (DEFs), and mixed eccentricity fault (MEFs) in permanent-magnet-synchronous motors remains difficult because these conditions produce similar stator-current patterns. This paper proposes an STFT–CNN–LSTM-based Eccentricity Fault Diagnosis System (STFT-CL-EFDS) for classifying Normal, SEF, DEF, and MEF conditions using only three-phase stator currents. Six signal transformations were initially compared, after which the three leading representations—STFT, DWT, and CWT—were evaluated with seven neural-network architectures. Sensitivity and ablation analyses selected a 500-sample observation window and a compact log-magnitude STFT representation over the nominal 0–1 kHz band. Following full-schedule retraining, the proposed model achieved 99.57% accuracy and a 99.57% weighted F1-score, with recalls of 100.00%, 98.66%, 100.00%, and 99.53% for Normal, SEF, DEF, and MEF, respectively. It exceeded the strongest machine-learning benchmark, STFT–MLP, by 8.42 percentage points in accuracy and 8.52 percentage points in weighted F1-score. On an independent unseen test bench, the proposed model provided the most balanced response across the three fault types and ultimately converged to the correct class in every case, although temporary DEF–MEF confusion remained. These results demonstrate the effectiveness of STFT-CL-EFDS for current-only multiclass PMSM eccentricity diagnosis. The main contributions of this study are as follows: (1) a systematic comparison of six signal-transformation methods, namely Fast Fourier Transform (FFT), STFT, Discrete Wavelet Transform (DWT), Continuous Wavelet Transform (CWT), Hilbert–Huang Transform (HHT), and Variational Mode Decomposition (VMD), to determine their suitability for eccentricity fault diagnosis; (2) a comparative evaluation of seven neural-network architectures, including CNN, LSTM, CNN–LSTM, DNN, TCN, ModernTCN, and TimesNet, using the three best-performing transformation methods, namely STFT, CWT, and DWT; and (3) the development of a unified current-only STFT-CL-EFDS that combines STFT-based time–frequency representation with convolutional feature extraction and temporal-sequence learning for the classification of Normal, SEF, DEF, and MEF conditions.

Symmetry doi: 10.3390/sym18091479

Authors: Nevriye Takımsu Guldem Yıldız

This study examines the Gardner equation, a fundamental nonlinear partial differential equation (PDE) used in modeling various physical phenomena, and wave motions. Hyperbolic and trigonometric analytical solutions are derived using the Sardar sub-equation method. Additionally, an artificial neural network (ANN) in combination with the Sardar sub-equation method, an analytical solution approach for the Gardner equation, is proposed. The results demonstrate that the ANN-based hybrid model predicted the Gardner equation solutions with accuracy. This study can be useful in the integration of classical analytical methods and the ANN-based approaches to solutions of nonlinear partial equations.

Symmetry doi: 10.3390/sym18091478

Authors: Abdulrahman B. M. Alzahrani

In this paper, the nonlinear Nurshuak–Tolkunay–Myrkakulov (NTM-III) equation is analyzed using the methods of dynamical systems theory and exact solution techniques. The NTM-III model is reduced to a second-order nonlinear ordinary differential equation by using appropriate reductions, then written as a planar Hamiltonian dynamical system. Analytical study of the equilibrium points, bifurcation structures, and stability properties is conducted using Jacobian matrices and eigenvalue theory. Different parameter combinations are shown in various phase portraits, with the saddle and center equilibrium states present and their stability indicated. An external periodic perturbation is added to the system to study complex nonlinear dynamics. Using phase portraits, time-series analysis, return maps, Lyapunov exponents, sensitivity analysis, and multistability diagnostics, the resulting forced dynamical model is investigated. Chaotic behavior, strong dependence on initial conditions, and multiple coexisting attractors for the same set of parameters are illustrated through numerical simulation. In addition, a set of exact analytical solutions, including trigonometric, hyperbolic, and exponential wave structures, is obtained using the Multivariate Generalized Exponential Rational Integral Function (MGERIF) method. The solutions obtained display interesting nonlinear wave interactions, multi-peakon formations, and localized propagation patterns. The findings show the complex relationship among bifurcation, chaos, multistability, and nonlinear wave propagation in the NTM-III equation and provide new insight into the equation’s mathematical and physical properties.

Symmetry doi: 10.3390/sym18091477

Authors: Alize Yaprak Gül Cengiz Kahraman

Linear Diophantine fuzzy sets (LDFSs) enlarge the admissible representation space for uncertain information through membership and non-membership degrees together with reference parameters. Existing LDFS aggregation studies have mainly considered weighted, ordered, power-based, and parametric operators, while partitioned aggregation operators that model possible interrelationships within criterion groups remain limited. Maclaurin symmetric mean (MSM) operators capture interactions among criteria, and reducible weighted MSM operators have been developed to preserve idempotency and reducibility properties but have not been extended to a partitioned MSM operator. This study develops an MSM-based aggregation framework for LDFSs, including the linear Diophantine fuzzy MSM (LDFMSM), the linear Diophantine fuzzy partitioned MSM (LDFPMSM), a reducible weighted partitioned MSM (RWPMSM) and its extension to LDFSs, and the linear Diophantine fuzzy reducible weighted partitioned MSM (LDFRWPMSM) operators. LDFMSM performs symmetric k-subset aggregation over the complete criterion set, LDFPMSM restricts joint aggregation terms to criteria within the same predefined partition, and LDFRWPMSM incorporates criterion weights while preserving idempotency and reducibility properties. Theoretical properties and special cases are established. The framework is demonstrated through numerical examples and an illustrative multi-attribute decision-making (MADM) application for an AI-driven precision agriculture platform selection. Sensitivity analyses examine the effects of the interaction parameter, criterion weights and partition structure and reveal systematic changes in the ranking results. Comparative analysis shows that the LDFPMSM can distinguish alternatives that remain tied under globally symmetric aggregation, while the LDFRWPMSM identifies a different best alternative from the benchmark operators, reflecting its joint treatment of criterion importance and within-partition interactions.

Symmetry doi: 10.3390/sym18091476

Authors: Lizhi Hu Tao Ming Yunfeng Hu

Normalized Difference Vegetation Index (NDVI) is an important remote sensing indicator for characterizing vegetation growth status and ecosystem changes. Improving NDVI prediction accuracy is of great significance for regional ecological monitoring and conservation. However, existing prediction methods often rely on single meteorological drivers or numerous explanatory variables that are difficult to obtain for future periods, limiting their applicability to pixel-level NDVI forecasting over spatially distributed areas. In this study, a Dual-Factor-Driven Temporal Network (DfT-Net) was proposed for pixel-level NDVI prediction in the Hulunbuir grassland based on MODIS NDVI data and ERA5-Land temperature and precipitation data from 2013 to 2024. The model was independently applied to valid 1000 m grassland pixels, with the same model parameters shared across pixels. This pixel-wise prediction strategy enables the model to learn common meteorological–vegetation response patterns across different pixels while generating spatially distributed NDVI predictions. For temporal feature modeling, Time Series Decomposition (TSD) was first applied to the temperature and precipitation sequences to decompose them into trend, seasonal, and residual components, thereby characterizing their multi-scale temporal variations and recurrent seasonal patterns. Subsequently, one-dimensional convolutional neural networks (CNNs) and long short-term memory networks (LSTMs) were employed to extract local temporal patterns and long-term temporal dependencies, respectively. Experimental results showed that DfT-Net achieved an RMSE of 0.100, an MAE of 0.074, and an R2 of 0.828 on the independent temporal test set (2022–2024). Under the same experimental setting, DfT-Net achieved the lowest RMSE and MAE and the highest R2 among the evaluated models, including LSTM, CNN, TSD-CNN, TSD-LSTM, and CNN-LSTM. These results indicate that DfT-Net effectively integrates dual-factor meteorological driving, multi-scale temporal feature representation, and pixel-level prediction, providing a useful framework for spatially distributed grassland NDVI forecasting and ecological monitoring.

Symmetry doi: 10.3390/sym18091475

Authors: Yasemin Sağıroğlu Ziya Yapar

In this paper, a dual Lorentzian closed strip {X˜(s),N(s)} is considered within the context of Lorentzian 3-space, with a real parameter s, a timelike moving frame {A(s),G(s),N(s)} is defined, which moves along the curve of this dual Lorentzian closed strip. A timelike vector D, which is fixed in this frame, is investigated alongside the dual integral invariants (dual angles of pitch, dual pitch) and dralls of the dual Lorentzian closed strip corresponding to the dual Lorentzian closed curve traced by vectors A,G,N and D. Some fundamental relations among these integral invariants of the dual Lorentzian closed strip are studied. Furthermore, these dual results are mapped to the line-space R13 and several theorems are presented regarding the dual Lorentzian geodesic curvature Kg, dual Lorentzian normal curvature Kn and dual Lorentzian geodesic torsion Tg of the strip, accounting for the causal characters (timelike) inherent in Lorentzian geometry.

Symmetry doi: 10.3390/sym18091474

Authors: Mamoona Siddiq Rana Safdar Ali Artion Kashuri Davron Aslonqulovich Juraev Ebrahim E. Elsayed

The gradual developments in mathematical analysis have increased the demand for improving the efficiency of constraints and their validation, which have a significant contribution to resolving many real-world problems. There are many techniques that are used to modify the fractional inequalities, but all approaches are analytical. The adoption of Machine learning (ML) models is one practical approach to optimize the bounds of inequalities and their numerical validations. In this paper, we implement ML models to modify the bounds of Hermite–Hadamard-type inequalities, which consider the h-Godunova–Levin as a weight function. The classical development of inequalities by generalized fractional operators was never systematically studied to determine which value of the weight gives the best bound possible. We prove a sharp lower bound for the Hermite–Hadamard gap uniform over all admissible weights, show that it is achieved exactly by the classical convex weight, and characterize the optimal weights. To complement this analytical result, a parameterization of the weight function by a feedforward neural network is introduced, which is assumed to be admissible in the h-Godunova–Levin framework, and the validity of the resulting inequalities is proved. A Lipschitz-type error analysis is used to relate the approximation accuracy of the network to the tightness of the bound, and it is shown in numerical experiments that the network learns the optimal weight without any knowledge of the analytical weight’s form. The framework thus ensures the rigor of fractional convexity theory while offering a data-driven method of determining the optimal weight functions in cases where they are not explicitly known. The inherent symmetry properties of the fractional operators and the symmetric structure of the Hermite-Hadamard inequalities are preserved, while the neural network framework introduces a symmetry-breaking mechanism that enables the discovery of optimal weights. This dual perspective on symmetry—both preserving and breaking—provides a comprehensive understanding of the underlying mathematical structures and aligns perfectly with the scope of the journal Symmetry. This practical approach opens a new horizon for researchers and yields better results in the field of analysis.

Symmetry doi: 10.3390/sym18091473

Authors: Koray Yılmaz Elis Soylu Yılmaz Hatice Taşbozan

In this work, we study the relationship among Lie algebraic homotopy 3-types. We investigate the categorical connection between crossed squares, internal crossed modules, 2-crossed modules, quadratic modules of Lie algebras, and simplicial and bisimplicial Lie algebras.

Symmetry doi: 10.3390/sym18091472

Authors: Salim Bouzebda Sultana Didi

We establish an asymptotic theory for the Jones inverse-weighted kernel density estimator when length-biased observations form a strictly stationary short-range dependent sequence. The statistical difficulty is intrinsically composite: reciprocal weighting is singular at the origin, the normalizing mean is estimated from the same dependent sample, kernel localization shrinks with the bandwidth, and the centered summands form a row-wise stationary triangular array whose envelope diverges at rate hn−1. Under a non-negative compactly supported Lipschitz kernel, an inverse-moment condition, geometric α-mixing, local regularity of the target density, and uniform local bounds on lagged bivariate densities, we prove strong uniform consistency on compact subsets of (0,∞) and, separately, the uniform stochastic bound OP{hn2+(logn/(nhn))1/2}. A covariance-localization argument shows that the scaled serial-covariance contribution is O{hnlog(1/hn)}=o(1), so the first-order pointwise variance coincides with that of the corresponding independent length-biased estimator. Pointwise and finite-dimensional Gaussian limits are obtained by an explicit big-block/small-block argument with off-diagonal covariance control. The ratio normalization is treated directly: its variance contribution, its product with the localized fluctuation, and its cross-covariance with that fluctuation are all negligible at the nhn scale. We further derive first-order AMSE and AMISE criteria, their oracle bandwidths, and feasible pointwise studentization under undersmoothing. The numerical study separates oracle from data-driven bandwidth selection, evaluates full-ratio HAC and moving-block corrections, examines a Frank-copula Markov robustness design, and benchmarks the Jones estimator against an alternative length-biased estimator. The simulations support the first-order theory while demonstrating that persistent short-range dependence can remain consequential for finite-sample uncertainty.

Symmetry doi: 10.3390/sym18091471

Authors: Jinhui Dai Tingke Ye Peisheng Yan Zhibin Jiang Zekang Bian Jie Zhou

Epilepsy EEG signal recognition is an important method for epilepsy detection. However, the traditional machine learning methods used for epilepsy EEG signal recognition face many challenges. For example, traditional intelligent recognition methods often ignore the imbalance of epilepsy EEG signals, which misclassifies positive samples and may cause serious consequences; the existing imbalanced classification methods ignore class-guided global inter-class correlations among samples, resulting in poor classification results. In order to overcome these challenges, a cost-sensitive-based TSK method (CITSK) is proposed for imbalanced epilepsy EEG signal recognition. This proposed method constructs a Class-guided Sample Collaborative Relationship (SCR) matrix to assign higher weight to positive minority samples, and supplements global inter-class collaborative penalty information on the basis of the traditional static cost-sensitive matrix. Then, a model combining the SCR matrix and TSK is proposed, which inherits the good generalization ability and interpretability of TSK while ensuring classification performance. Experiments on real EEG datasets demonstrate the effectiveness of the proposed method and its applicability to the imbalanced classification of epileptic EEG signals. Quantitatively, the CITSK method achieves an average Accuracy of 93.96%, an average F_measure of 88.33%, and an average G_means of 91.34% across various imbalanced datasets. Furthermore, comparative evaluations demonstrate that the proposed CITSK achieves superior overall performance across comprehensive evaluation metrics compared to baseline models, offering a highly robust and interpretable solution for imbalanced data scenarios via reasonable global inter-class penalty compensation.

Symmetry doi: 10.3390/sym18091469

Authors: Li Ding Hua Shi Kuan Yang

When a lithium-ion cell’s series resistance is underestimated, the pre-update terminal-voltage innovation contains the first-order term ek≈−IkR0,kδb. This term breaks conditional sign symmetry and creates an odd response under current reversal. We test that mechanism before using it as an activation rule. The operational null is a near-zero conditional innovation centre with weak innovation–current coupling; declared falsifiers are comparable coupling under the nominal model, the wrong correlation sign under positive resistance error, failure of charge/discharge polarity reversal, or negative-control activation approaching ohmic-mismatch activation. A persistence-confirmed gate combines normalised-innovation-squared exceedances, innovation–current compatibility, a positive local resistance correction, and five consecutive qualifying windows. Sixty settings were ranked on 10 calibration seeds and frozen before disjoint holdouts. From 1.0× to 2.0× R0, the sign-imbalance index increased from 0.0040 to 0.0786, and |corre,I| increased from 0.0658 to 0.7777. The 30-seed static holdout produced 0/30 nominal activations, 27/30 detections at 1.5×, and 30/30 detections at 2.0–3.0×. A disjoint linear-drift audit yielded 0/30 pre-ramp activations and 30/30 detections at a median 1.71× multiplier, reducing late-drift SOC RMSE from 3.129% to 0.895%. A signed-current audit confirmed the predicted polarity reversal. An estimator-unseen audit gave 30/30 ohmic detections but retained 2/30 current-linked non-ohmic and 1/30 current-bias triggers, so the rule is not a unique fault classifier. NASA and LG records remain diagnostics of fixed versus always-on adaptation; they do not validate the gate or independent absolute SOC. The contribution is a falsifiable residual-symmetry mechanism with an explicit evidence boundary, rather than a post hoc symmetry label or hardware claim.

Symmetry doi: 10.3390/sym18091470

Authors: Qianxia Ma Qian Pu

The failure mode and effect analysis (FMEA), as an effective proactive reliability analysis technique, plays an important role in system fault detection and reliability maintenance due to its pre-analysis capability. Although traditional FMEA is widely used in practice, it has some limitations in the expression of fuzzy information, considering the interrelationships between risk factors, and considering relative importance. Therefore, a new FMEA method that combines spherical fuzzy set (SFS) and an improved spherical fuzzy weighted Hamy mean (SFWHM) operator is proposed. This method mainly includes four stages. Firstly, the SFS is applied to obtain risk assessment information, as it can handle more ambiguous information. Secondly, to ensure that the quality of evaluation information meets specific standards, consensus calculation and feedback mechanisms are introduced. Next, risk factor weights are obtained based on the logarithmic percentage change-driven objective weighting (LOPCOW) method and an improved spherical fuzzy score function. Finally, based on the proposed SFWHM operator, the risk factor information with interrelationships is integrated, and an improved score function is used to obtain the final ranking results. The proposed method is applied to the load-bearing guidance device of the subway door system to verify its applicability and compare its advantages with existing advanced FMEA methods. The results indicate that the proposed method provides a reliable tool for practical FMEA problems. The proposed method, as a reliable risk assessment technique, can be flexibly applied to various practical scenarios.

Symmetry doi: 10.3390/sym18091467

Authors: Ismail Ifakir El Habib Nfaoui Abderrahim Zannou

Multimodal aspect-based sentiment analysis (MABSA) predicts the sentiment expressed toward a target aspect by jointly using textual and visual information, supporting fine-grained opinion analysis in product reviews, brand monitoring, and customer feedback. However, existing approaches remain sensitive to irrelevant visual regions, weak text–image alignment, and limited use of external knowledge. Motivated by these challenges, this study systematically evaluated two text-only large language models and four open-weight large vision-language models for aspect-level sentiment classification. The open-weight models were adapted using 4-bit quantized low-rank adaptation, while GPT-4o was assessed under zero-shot, one-shot, and five-shot in-context learning without parameter updates. Experiments were conducted on Twitter-2015, Twitter-2017, and the seven-domain MASAD dataset and evaluated using accuracy and macro-F1. Among the evaluated multimodal models, Qwen3-VL-8B-Instruct achieves the strongest performance, reaching 83.22% accuracy and 81.72% macro-F1 on Twitter-2015, 79.50% and 78.93% on Twitter-2017, and up to 99.84% and 99.83% in the Plant domain of MASAD. From a symmetry perspective, semantically aligned text–image–aspect inputs provide consistent cross-modal evidence, whereas shuffled images introduce asymmetric, symmetry-breaking information. The resulting performance degradation under shuffled-image ablation indicates that reliable aspect-level sentiment prediction depends on preserving cross-modal semantic correspondence. These findings demonstrate the effectiveness of parameter-efficient LVLM adaptation for MABSA.