Symmetry

29 pages, 1188 KB

Open AccessArticle

Master-Refined MAPPO for Long-Term Joint Resource Scheduling in NOMA-MEC Systems

by Jianfei Zhang and Shangyu Wu

Symmetry 2026, 18(7), 1243; https://doi.org/10.3390/sym18071243 - 22 Jul 2026

Viewed by 273

Mobile edge computing (MEC) enables resource-constrained user devices (UDs) to obtain low-latency computing services by offloading computational tasks to the network edge. Non-orthogonal multiple access-enabled mobile edge computing (NOMA-MEC) systems feature asymmetric states across UDs, dynamic task arrivals, and competition for wireless and [...] Read more.

Mobile edge computing (MEC) enables resource-constrained user devices (UDs) to obtain low-latency computing services by offloading computational tasks to the network edge. Non-orthogonal multiple access-enabled mobile edge computing (NOMA-MEC) systems feature asymmetric states across UDs, dynamic task arrivals, and competition for wireless and edge computing resources. Under these conditions, offloading decisions affect device energy consumption, task delay, and edge computing resource allocation, making long-term system optimization difficult. This study jointly optimizes task offloading and system resource scheduling to minimize the long-term delay–energy cost. The problem is formulated as a partially observable Markov decision process (POMDP) and addressed using a master-refined multi-agent proximal policy optimization (MR-MAPPO) algorithm. MR-MAPPO combines continuous action relaxation, master action refinement, and a behavior cloning auxiliary term to learn policies in a hybrid discrete–continuous action space. A marginal congestion delay term is also introduced to capture the impact of newly admitted tasks on existing edge workloads. Simulation results show that MR-MAPPO outperforms the considered baselines, while ablation studies verify the effects of its key components. Under the main experimental setting, MR-MAPPO reduces the system cost by 17.9% and 22.9% relative to standard MAPPO and particle swarm optimization (PSO), respectively. Full article

(This article belongs to the Section A: Computer Science)

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26 pages, 2543 KB

Open AccessArticle

Enhanced Computational Efficiency in Solving Delay Fractional Partial Differential Equations Through the Yang Decomposition Method

by Mustafa Ahmed Ali and Mehmet Merdan

Symmetry 2026, 18(7), 1242; https://doi.org/10.3390/sym18071242 - 22 Jul 2026

Viewed by 858

Abstract

This study presents the Yang Transform Adomian Decomposition Method (YTADM), a semi-analytical framework for solving one-dimensional linear and nonlinear delay fractional partial differential equations involving the Caputo fractional derivative. The proposed method combines the Yang transform with the Adomian decomposition method to construct [...] Read more.

This study presents the Yang Transform Adomian Decomposition Method (YTADM), a semi-analytical framework for solving one-dimensional linear and nonlinear delay fractional partial differential equations involving the Caputo fractional derivative. The proposed method combines the Yang transform with the Adomian decomposition method to construct recursive solution series while efficiently handling delayed nonlinear terms. The applicability of the proposed framework is demonstrated through several examples, including proportional-delay Burgers-type equations, and its convergence properties are analyzed. The obtained results show that YTADM yields rapidly convergent semi-analytical approximations and provides an effective framework for solving one-dimensional delay fractional partial differential equations. Full article

(This article belongs to the Section B: Mathematics)

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27 pages, 8669 KB

Open AccessArticle

Heterogeneous Feature Integration for Class-Imbalanced Intrusion Detection in Grid Systems

by Kai Cheng, Dongkun Li, Weidong Tang, Lin Liu and Xueyu Zhang

Symmetry 2026, 18(7), 1241; https://doi.org/10.3390/sym18071241 - 22 Jul 2026

Viewed by 391

Abstract

Modern grid digitalization connects communication networks, monitoring terminals, service platforms, security devices, and operational data sources. Intrusion detection in this setting requires correlating heterogeneous security data with grid-side contextual evidence. To address class imbalance and cross-domain heterogeneity, this study proposes a heterogeneous feature [...] Read more.

Modern grid digitalization connects communication networks, monitoring terminals, service platforms, security devices, and operational data sources. Intrusion detection in this setting requires correlating heterogeneous security data with grid-side contextual evidence. To address class imbalance and cross-domain heterogeneity, this study proposes a heterogeneous feature group integration framework for intrusion detection with grid cybersecurity data. Four semantic feature subspaces are constructed symmetrically: network behaviour, power operation context, zone-derived communication/event topology, and system operation state, ensuring equal structural footing for subsequent modality-specific encoding. Transformer-based encoders model temporal dependencies in network, physical, and system state modalities, while a graph neural network encodes topology-related structural information. The resulting embeddings are integrated by a late fusion classifier for multiclass attack identification; the fusion process treats each feature group symmetrically at the decision level, without imposing a priori dominance among modalities. In the main run, the full model achieves an accuracy of 0.944, a macro F1 score of 0.891, a weighted F1 score of 0.937, a macro precision of 0.929, and a macro recall of 0.878. The corresponding balanced accuracy is 0.878, and the multiclass MCC is 0.924. Class-wise results show reliable performance on Benign, Scan, WebAtk, DDoS, DoS, and Backdoor classes, while Ransomware remains difficult and is frequently confused with WebAtk. Specifically, the Ransomware recall is 0.27, with most errors assigned to WebAtk. Modality analysis further indicates that modality contribution is class dependent: some feature groups have limited standalone discriminative power but provide complementary evidence after fusion. This finding highlights an inherent asymmetry in class-wise utility, which we counterbalance by employing both macro and weighted metrics, offering a symmetric evaluation lens that accounts for both minority and majority classes. These results show that grid-oriented intrusion detection benefits from decision-level integration of heterogeneous feature groups and imbalance-aware evaluation, where symmetric treatment of feature subspaces and evaluation perspectives jointly enhances robustness. Full article

(This article belongs to the Section A: Computer Science)

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40 pages, 6361 KB

Open AccessArticle

Adaptive Bitterling Fish Optimization with Evolutionary Game Theory: For Cross-Regional Emergency Repair Path Planning

by Shuangqing Chen, Chao Chen, Junfei Liu, Xingwang Wang, Zhe Xu, Yongbin Liu, Haibin Liang, Lulu Zhang and Yaqian Liu

Symmetry 2026, 18(7), 1240; https://doi.org/10.3390/sym18071240 - 22 Jul 2026

Viewed by 424

Abstract

Modern energy internets and large-scale industrial systems are becoming increasingly complex. Consequently, the rapid response capability of energy infrastructure during sudden failures has become a core element to ensure the stable operation of the social economy. Emergency repair path planning (ERPP) is a [...] Read more.

Modern energy internets and large-scale industrial systems are becoming increasingly complex. Consequently, the rapid response capability of energy infrastructure during sudden failures has become a core element to ensure the stable operation of the social economy. Emergency repair path planning (ERPP) is a complex nonlinear combinatorial optimization problem. It is characterized by dynamic uncertainties, such as fluctuating task durations and variable traffic accessibility. This paper proposes a cross-regional emergency repair path planning (CR-ERPP) optimization model considering dynamic path conditions. The model takes into account jurisdiction ownership, cross-regional dispatch costs, path weights (congestion coefficient, grade coefficient, quality coefficient) and accident risk levels. The primary objective of this model is to minimize the total repair cost. Furthermore, an Adaptive Bitterling Fish Optimization with Evolutionary Game Theory (ABFO-EGT) is developed. It introduces adaptive mechanisms, evolutionary game theory, and a symmetric mutation strategy. These enhancements are designed to overcome the inherent limitations of traditional swarm intelligence algorithms, namely unbalanced search behavior and premature convergence to local optima. Performance analysis demonstrates that the ABFO-EGT algorithm exhibits superior convergence stability and global search capability. Case study results show that the proposed method significantly reduces the total repair cost. Specifically, the cost is reduced by 33.2% compared to manual decision-making, 27.6% compared to the GWO algorithm, and 9.1% compared to both the ACO and PSO algorithms. This study provides an efficient and reliable decision support tool for emergency management of large-scale energy systems. Full article

(This article belongs to the Section A: Computer Science)

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23 pages, 1521 KB

Open AccessArticle

An Efficient LBlock Lightweight Block Cipher Coprocessor on RISC-V: Combinational Key Schedule Fusion and S-Box-to-LUT Mapping

by Jianxin Wang, Runze Zhou, Zixuan Wang, Lei Zhang, Chaoen Xiao, Zhao Wang, Maosheng He, Qian Cheng and Kaibo Sun

Symmetry 2026, 18(7), 1239; https://doi.org/10.3390/sym18071239 - 22 Jul 2026

Viewed by 417

Abstract

Resource-constrained Internet-of-Things (IoT) terminals require encryption engines that combine low silicon cost with adequate throughput, a balance that is hard to reach with general-purpose software alone. This paper presents an LBlock lightweight block cipher coprocessor tightly coupled to an open-source RISC-V (Hummingbird E203) [...] Read more.

Resource-constrained Internet-of-Things (IoT) terminals require encryption engines that combine low silicon cost with adequate throughput, a balance that is hard to reach with general-purpose software alone. This paper presents an LBlock lightweight block cipher coprocessor tightly coupled to an open-source RISC-V (Hummingbird E203) core through the NICE custom-instruction interface. LBlock serves here as a compact Feistel-cipher benchmark targeting legacy and low-volume IoT deployments rather than as a substitute for newer standards such as ASCON. We exploit two structural properties of LBlock: first, its 4-bit S-boxes map naturally onto the six-input look-up tables (LUTs) of modern FPGAs, so the entire substitution layer is realized as eight parallel single-LUT-depth tables instead of multi-cycle table lookups. Second, the LBlock key schedule is a one-way feedback-free recurrence, which lets us refactor key expansion from an independent multi-cycle sequential module into a pure combinational function that is fused with the round function and executed in the same clock cycle. The resulting encryption core performs one round per cycle, reducing core-only single-block latency from 226 cycles in the baseline implementation to 34 cycles, while the complete NICE coprocessor operation requires 85 cycles including data movement, instruction issue, computation, and write-back. The design is described in Chisel and integrated as a coprocessor with three custom instructions. On an FPGA-based SoC, the coprocessor produces outputs identical to the LBlock test vectors and accelerates encryption by 121.45× over a software baseline on the same core, while the encryption core reaches 321.7 MHz (643 Mbps) on Artix-7 and up to 472.2 MHz (944 Mbps) on Virtex-7 while occupying only 187 LUTs, as validated across three FPGA families. These results show that matching the algorithmic symmetry of LBlock to the underlying hardware fabric yields a lightweight and low-overhead cryptographic accelerator suitable for RISC-V IoT endpoints. Full article

(This article belongs to the Section A: Computer Science)

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25 pages, 1544 KB

Open AccessArticle

Thermal Analysis of the Downstream Spreading of a Planar Power-Law Liquid Jet with Convective Free-Surface Cooling

by Avnish Bhowan Magan

Symmetry 2026, 18(7), 1238; https://doi.org/10.3390/sym18071238 - 22 Jul 2026

Viewed by 231

Abstract

The two-dimensional thermal liquid jet of a non-Newtonian power-law fluid is investigated under shear-rate-dependent thermal diffusivity, resulting in a one-way coupled nonlinear system governing momentum and thermal transport. Two physically distinct free-surface thermal boundary conditions are examined: adiabatic insulation and convective heat loss. [...] Read more.

The two-dimensional thermal liquid jet of a non-Newtonian power-law fluid is investigated under shear-rate-dependent thermal diffusivity, resulting in a one-way coupled nonlinear system governing momentum and thermal transport. Two physically distinct free-surface thermal boundary conditions are examined: adiabatic insulation and convective heat loss. Conservation laws and conserved quantities for the governing system are derived systematically using the multiplier method. By coupling an appropriate conserved vector with an admitted Lie point symmetry, the governing partial differential equations are reduced to a coupled system of ordinary differential equations. Closed-form parametric families of solutions are then obtained for the thermal field. The analysis reveals fundamentally different thermal transport mechanisms across rheological regimes: shear-thinning fluids enhance thermal redistribution and become increasingly sensitive to convective cooling as the Biot number increases, whereas shear-thickening fluids suppress internal thermal transport, promoting greater thermal retention within the jet core and reducing the influence of free-surface cooling. These findings clarify the interplay between rheology, nonlinear thermal diffusion and free-surface cooling and provide new analytical insight into downstream thermal transport in non-Newtonian liquid jets. Full article

(This article belongs to the Section F: Engineering and Materials)

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23 pages, 341 KB

Open AccessArticle

On the Sequential (p, δ, τ)-Numerical Radius Function of Operator Sequence

by Zameddin I. Ismailov, Pembe Ipek Al and Mohammad Sababheh

Symmetry 2026, 18(7), 1237; https://doi.org/10.3390/sym18071237 - 22 Jul 2026

Viewed by 287

Abstract

In this article, firstly, some basic properties of the sequential

(p, δ, τ)

-numerical radius function are investigated. The relationships between the sequential

(p, δ, τ)

-numerical radius of an operator sequence and the sequential [...] Read more.

In this article, firstly, some basic properties of the sequential

(p, δ, τ)

-numerical radius function are investigated. The relationships between the sequential

(p, δ, τ)

-numerical radius of an operator sequence and the sequential

(p, δ, τ)

-numerical radii of its coordinate operators are analyzed. Then, the relationships between the sequential

(p, δ, τ)

-numerical radius of an operator sequence and the sequential

(p, δ, τ)

-numerical radii of its real and imaginary parts are presented. Finally, this analysis is extended to the case in which the coordinate operators are sectorial, providing additional insight into the structural behavior of the sequential

(p, δ, τ)

-numerical radius function. The obtained results are generalized to some well-known famous results about the numerical radius function from the recent literature. Also, an important contribution is made to the existing literature via different and useful results. Full article

(This article belongs to the Special Issue Symmetry in Complex Analysis Operators Theory)

19 pages, 5164 KB

Open AccessReview

Restoring Symmetry After Sport-Related Concussion: A Viewpoint on Biofeedback-Guided Rehabilitation

by James Stavitz

Symmetry 2026, 18(7), 1236; https://doi.org/10.3390/sym18071236 - 22 Jul 2026

Viewed by 305

Abstract

Sport-related concussion (SRC) rehabilitation has advanced toward active, multidomain management, yet recovery may still be judged largely through symptom resolution and broad clinical indicators that may not fully capture persistent functional deficits. Emerging evidence suggests subtle disturbances in postural control, gait, and sensorimotor [...] Read more.

Sport-related concussion (SRC) rehabilitation has advanced toward active, multidomain management, yet recovery may still be judged largely through symptom resolution and broad clinical indicators that may not fully capture persistent functional deficits. Emerging evidence suggests subtle disturbances in postural control, gait, and sensorimotor coordination may persist beyond apparent clinical recovery, raising the possibility that unresolved asymmetries represent an underrecognized dimension of dysfunction. This Viewpoint proposes symmetry restoration as a potential rehabilitative construct in SRC management and explores how biofeedback-guided approaches may provide a conceptual framework for identifying, monitoring, and retraining symmetry-related deficits. Drawing from concussion research, motor control theory, rehabilitation science, and biofeedback applications, this article discusses postural and movement asymmetries as possible markers of incomplete recovery, examines visual, wearable, neuromuscular, and auditory biofeedback strategies as potential mechanisms for symmetry-informed rehabilitation, and outlines clinical implications and future research priorities. Rather than proposing symmetry as a stand-alone determinant of recovery, this Viewpoint advances the conceptual proposition that symmetry-oriented approach may complement existing multidomain models by serving as an additional layer of functional assessment alongside symptom reporting, neurocognitive evaluation, vestibular and oculomotor examination, exertional testing, and routine clinical assessment. Within this framework, symmetry-related measures are envisioned not as independent clearance criteria, but as potentially informative indicators of residual sensorimotor function that may help guide rehabilitation progression and contribute to more functionally informed return-to-sport decision making through adjunctive measures such as center-of-pressure behavior, center-of-mass displacement, gait symmetry, stance and swing time asymmetry, limb-loading patterns, interlimb coordination, and muscle activation symmetry. Full article

(This article belongs to the Special Issue Biofeedback Applications and Symmetry in Rehabilitation, Sports and Ergonomics)

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34 pages, 6619 KB

Open AccessArticle

Symmetry-Driven Enhanced Auxiliary Classifier GAN for Data-Efficient Breast Tumor Classification

by Tea Marasović and Vladan Papić

Symmetry 2026, 18(7), 1235; https://doi.org/10.3390/sym18071235 - 21 Jul 2026

Viewed by 362

Abstract

The intricate nature of multi-class histopathological images, combined with pronounced class imbalances, complicates automated breast cancer diagnosis and demands AI models capable of generalizing well beyond often limited training data. To address these challenges, this paper explores the generative modeling capability of a [...] Read more.

The intricate nature of multi-class histopathological images, combined with pronounced class imbalances, complicates automated breast cancer diagnosis and demands AI models capable of generalizing well beyond often limited training data. To address these challenges, this paper explores the generative modeling capability of a symmetry-driven enhanced auxiliary classifier GAN (LSWACGAN) as an all-in-one, data-efficient framework for breast cancer histopathological image classification. LSWACGAN incorporates the Wasserstein loss with gradient penalty to promote greater training stability by mitigating overfitting and preventing vanishing gradients. Assigning smooth category labels to generated samples further helps alleviate the mode collapse problem. The proposed framework brings together three types of symmetry to improve its reliability: the inherent metric symmetry of the Wasserstein distance, the structural symmetry within the auxiliary classifier GAN, and the architectural symmetry between the generator and discriminator networks. Extensive experiments conducted on the well-known BreakHis dataset, supplemented by a thorough ablation study, demonstrate the framework’s competitive edge in a lower-data regime. For binary classification, LSWACGAN closely matches or slightly outperforms leading benchmarks on most selected evaluation metrics. Conversely, in the multi-class scenario, it emerges as a clear forerunner, consistently producing superior results and maintaining robust performance across varying magnification levels. Full article

(This article belongs to the Special Issue Symmetry and Asymmetry in Image Classification)

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22 pages, 12417 KB

Open AccessArticle

Coupling Effects of Dynamic Loads and Friction on the Gear Systems of Radial 3D Braiding Machines

by Lingling Yao, Zhilin Yang, Dongsheng Liang and Chenglong Wei

Symmetry 2026, 18(7), 1234; https://doi.org/10.3390/sym18071234 - 21 Jul 2026

Viewed by 245

Abstract

During the radial braiding process, spindle motion induces periodic load excitations as they move with the turntable. Based on the kinematics analysis of the spindles, this study derives a tension-load torque mapping model and establishes a multi-degree-of-freedom (MDOF) nonlinear dynamic model that incorporates [...] Read more.

During the radial braiding process, spindle motion induces periodic load excitations as they move with the turntable. Based on the kinematics analysis of the spindles, this study derives a tension-load torque mapping model and establishes a multi-degree-of-freedom (MDOF) nonlinear dynamic model that incorporates dynamic torque and gear tooth friction. The system’s governing differential equations are solved numerically using the fourth-order Runge–Kutta method to obtain steady-state responses under various combinations of tension and rotational speed. Results indicate that increasing yarn tension reduces the stability margin of the system’s phase trajectories, and the basin of attraction area for periodic motion decreases approximately linearly as the tension increases. Furthermore, friction exhibits dual characteristics across different frequency regimes: at operating frequencies below 1.05, friction acts as a damping mechanism to maintain system stability; however, beyond this threshold, the friction reversal mechanism triggers chaotic behavior. Full article

(This article belongs to the Section F: Engineering and Materials)

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42 pages, 3859 KB

Open AccessHypothesis

Gravity-Referenced Informational Symmetry Breaking as a Sensorimotor Scaffold for Brain Lateralization

by Dong-Gyun Han

Symmetry 2026, 18(7), 1233; https://doi.org/10.3390/sym18071233 - 21 Jul 2026

Viewed by 259

Abstract

Brain lateralization is a biological asymmetry in which a bilaterally organized nervous system develops direction-specific functional organization. This hypothesis distinguishes gravity-driven physical symmetry reduction from informational symmetry breaking. Gravity provides a stable vertical reference, yet matched leftward and rightward tilts become biologically relevant [...] Read more.

Brain lateralization is a biological asymmetry in which a bilaterally organized nervous system develops direction-specific functional organization. This hypothesis distinguishes gravity-driven physical symmetry reduction from informational symmetry breaking. Gravity provides a stable vertical reference, yet matched leftward and rightward tilts become biologically relevant only when noisy vestibular population responses carry decodable tilt-sign information. At fixed unsigned tilt magnitude, the criterion is nonzero conditional mutual information between binary tilt sign and vestibular population response; for equal sign priors, this is equivalent to Jensen–Shannon divergence between sign-conditioned response distributions. Shannon entropy describes within-condition response spread, Fisher information describes local continuous-angle precision, and noise-aware representational distance describes PIVC-centered state separation. The otolith-to-perceptual pathway is formulated as a constrained effective state-space transformation from vestibular population responses through an intermediate brainstem–cerebellar state to distributed parieto-insular vestibular cortex (PIVC)-centered cortical states and perceived self-orientation. The framework predicts sign-specific vestibular and PIVC information for matched tilts, reduced or reorganized sign information in bilateral vestibulopathy, and covariance among cortical geometry, orientation-estimation reliability, and orientation-dependent behavior. Auditory and visual spatial transformations provide computational precedents rather than anatomical homology. The model offers a testable sensorimotor scaffold without determining a fixed hemispheric sign. Full article

(This article belongs to the Section E: Life Sciences)

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56 pages, 515 KB

Open AccessArticle

A Structural Origin of the Charged-Lepton Hierarchy

by Bin Li

Symmetry 2026, 18(7), 1232; https://doi.org/10.3390/sym18071232 - 21 Jul 2026

Viewed by 264

Abstract

The charged-lepton masses are free Yukawa-sector parameters in the Standard Model, whereas their measured pole-mass ratios display a highly structured hierarchy and satisfy the Koide relation to notable accuracy. This paper develops a conditional mathematical-physics proposal in which these dimensionless regularities arise from [...] Read more.

The charged-lepton masses are free Yukawa-sector parameters in the Standard Model, whereas their measured pole-mass ratios display a highly structured hierarchy and satisfy the Koide relation to notable accuracy. This paper develops a conditional mathematical-physics proposal in which these dimensionless regularities arise from a charge-neutral parent carrier-defect architecture before effective Higgs–Yukawa read-out. The assumptions of the construction are stated explicitly as structural postulates and are separated from their derived consequences. The central rule assigns equal primitive weight to admissible internal sectors that are indistinguishable at the level where they first become exposed; protected sectors are removed before counting, and later refinements are conditional on previously selected sectors. Under this rule, the Koide relation follows as an equal-power theorem between the democratic parent component and the orthogonal branch-splitting component of the charged-lepton root-amplitude state. A minimal endpoint construction then yields a rapidly stabilizing charged tower for the electron–muon ratio. Because deeper charged terms are too small to remove the remaining residual, the framework assigns that residual to the continuation-dual neutral branch. The resulting neutral overlap gives a leading solar-angle target of 33.21 degrees and closes the electron–muon ratio at the present experimental precision; the Koide relation then fixes the corresponding tau ratios. The construction does not replace the Standard Model but is proposed as a selection rule for the boundary values of effective charged-lepton Yukawa parameters, with pole masses used because the claimed invariant is attached to completed asymptotic particle read-out. Running parameters, the absolute mass scale, and the full Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix remain outside the present derivation. The proposal has explicit failure conditions: improved measurements can exclude the predicted tau ratios or solar-angle target, and the claimed conditional uniqueness fails if a different counting scheme satisfies the same postulates while producing different endpoint weights. Full article

(This article belongs to the Section C: Physics)

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21 pages, 7220 KB

Open AccessArticle

Spatial Asymmetry in Topographic Controls on Flood Intensity: A Machine Learning Investigation of the Chi River Floodplain, Thailand

by Nutchanat Buasri, Patiwat Littidej, Benjamabhorn Pumhirunroj and Donald Slack

Symmetry 2026, 18(7), 1231; https://doi.org/10.3390/sym18071231 - 21 Jul 2026

Viewed by 912

Abstract

Understanding how landscape form influences inundation severity remains central to flood hazard assessment, yet many assumed relationships lack empirical scrutiny. We investigated whether five topographic attributes—elevation, slope, topographic wetness index, latitude, and longitude—could predict cumulative flood intensity across 541 hexagonal cells in Thailand’s [...] Read more.

Understanding how landscape form influences inundation severity remains central to flood hazard assessment, yet many assumed relationships lack empirical scrutiny. We investigated whether five topographic attributes—elevation, slope, topographic wetness index, latitude, and longitude—could predict cumulative flood intensity across 541 hexagonal cells in Thailand’s Chi River floodplain. Using Random Forest regression and SHAP analysis, we identified three distinct asymmetries that challenge conventional assumptions. Elevation dominated predictions (58.5% importance) but operated through a sharp threshold near 150 m rather than a smooth gradient. Below 145 m, flood intensity was consistently high regardless of other factors; above 155 m, it was uniformly low. The flood-amplifying effect of low-lying terrain (+200 SHAP units) far outweighed the protective benefit of high ground (−100 SHAP units). More strikingly, the Topographic Wetness Index—a widely used theoretical measure of wetness potential—showed negligible correlation with observed flooding (r = 0.109) and contributed only 5.6% to predictive performance. Linear regression models captured barely 30% of the variance (R² ≈ 0.305), whereas Random Forest explained 77.6% (R² = 0.7765), a performance gap that quantifies the degree of non-linearity in the system. Spatial cross-validation confirmed generalizability (R² = 0.583). The elevation threshold offers a straightforward zoning framework: high-risk areas below 145 m, transitional zones from 145 to 155 m, and low-risk areas above 155 m. We conclude that theoretical indices require empirical validation and that combining machine learning with symmetry-based reasoning can expose hidden structures in environmental systems that linear approaches miss. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry in Remote Sensing and Applications)

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77 pages, 715 KB

Open AccessArticle

Inverse Evolution and Dimensional Collapse: Operator-Theoretic Dynamics in Financial Manifolds

by Simon Gluzman

Symmetry 2026, 18(7), 1230; https://doi.org/10.3390/sym18071230 - 20 Jul 2026

Viewed by 281

Abstract

We develop an operator-theoretic framework for extreme events in reflexive financial systems, identifying inverse evolution—the deterministic contraction of the manifold of admissible futures—as the structural mechanism underlying crashes and melt-ups. The interpolation constraint, which forces all analytical continuations to match the terminal empirical [...] Read more.

We develop an operator-theoretic framework for extreme events in reflexive financial systems, identifying inverse evolution—the deterministic contraction of the manifold of admissible futures—as the structural mechanism underlying crashes and melt-ups. The interpolation constraint, which forces all analytical continuations to match the terminal empirical price, resolves the apparent continuum of stochastic paths into a discrete, countable spectrum of metastable futures. This countable manifold is stabilized by a spectral regularizer that preserves dimensionality through a “wait-and-adjust” re-categorization logic. Within this unified structure, we distinguish three pathways to collapse: (i) the Black Swan, a crisis of spectral weight; (ii) the projection operator, a rank-reducing projection that restores symmetry by exclusion; and (iii) the reactivation operator, a breakdown of spectral truncation that reactivates suppressed behaviour with large emergent return (Heavy) modes and forces the system into a regime of manifold resumption. Central to all modalities is the emergent return, an effective mass parameter whose sign determines whether collapse manifests as reflexive contraction (crash) or reflexive amplification (melt-up). The resulting dynamics exhibit cross-domain universality. The same operator grammar governs geopolitical choke-points, institutional purges, technological monopolies, retail-driven short squeezes, and other macrosystems in which dimensionality is either forcibly reduced or abruptly restored. Full article

(This article belongs to the Special Issue Symmetry and Approximation Methods, 3rd Edition)

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23 pages, 6130 KB

Open AccessArticle

Symmetry-Aware Collaborative Attention Network for Robust Weak Seismic Phase Picking

by Yunpeng Wang, Qing Li, Chao Zhang, Yatong Bai, Xiaofei Du, Jianfeng Wang and Yuda He

Symmetry 2026, 18(7), 1229; https://doi.org/10.3390/sym18071229 - 20 Jul 2026

Viewed by 273

Abstract

Reliable seismic phase picking is essential to earthquake monitoring, as it fundamentally affects earthquake location and source inversion. In challenging field conditions, nonstationary waveforms, diverse morphological features and intense background noise all hinder the detection of weak phases. Seismic time series also exhibit [...] Read more.

Reliable seismic phase picking is essential to earthquake monitoring, as it fundamentally affects earthquake location and source inversion. In challenging field conditions, nonstationary waveforms, diverse morphological features and intense background noise all hinder the detection of weak phases. Seismic time series also exhibit inherent spatiotemporal asymmetry. Nevertheless, mainstream networks rely on symmetric and uniform feature extraction strategies. They overlook asymmetric properties of waveforms and introduce additional picking errors. We therefore present SymPhase, a symmetry-aware collaborative attention network, to achieve precise and robust P- and S-phase picking. Using a 1D encoder–decoder backbone, the model combines global enhancement and local refinement. It captures both long-range dependencies and local features, reducing missed weak-phase detections and minimizing arrival-time bias. Extensive tests are conducted on the CEED and DiTing datasets. The results demonstrate that SymPhase outperforms both TCN and PhaseNet. On the CEED dataset, the F1 scores for P and S phases are 0.9797 and 0.9006, with mean absolute errors of 0.0761 s and 0.1003 s. On the difficult DiTing dataset, the S-phase F1 score reaches 0.4724 with a corresponding error of 0.7386 s. These results validate its superior performance for weak signal recognition. With strong accuracy and noise robustness, SymPhase provides a viable solution for automated earthquake monitoring systems. Full article

(This article belongs to the Special Issue Symmetry in Artificial Intelligence and Machine Learning: Current Advances)

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46 pages, 17142 KB

Open AccessArticle

Topological Continuity-Enforced Retinal Vessel Segmentation via Frequency-Aware Decomposition and Prototype Refinement

by Feng Li and Yaoyao Feng

Symmetry 2026, 18(7), 1228; https://doi.org/10.3390/sym18071228 - 20 Jul 2026

Viewed by 240

Abstract

Automated and accurate segmentation of retinal vessels in fundus images provides pivotal evidence for ophthalmologists to effectively and non-invasively diagnose prevalent ocular and systemic diseases. However, existing methods often struggle to maintain the topological continuity of fine-diameter capillaries, leading to severe vascular discontinuity [...] Read more.

Automated and accurate segmentation of retinal vessels in fundus images provides pivotal evidence for ophthalmologists to effectively and non-invasively diagnose prevalent ocular and systemic diseases. However, existing methods often struggle to maintain the topological continuity of fine-diameter capillaries, leading to severe vascular discontinuity and fragmented segmentation results in challenging scenarios such as complex, irregular microvascular branches, pathological lesions, and high-noise conditions. To address these limitations, we developed a novel symmetric dual-branch network with frequency-aware decomposition and prototype refinement (FDPR-DBNet). Specifically, the network initially utilizes the discrete wavelet transform (DWT) to decompose input retinal images into high-frequency and low-frequency components, which are then processed by a structurally symmetric dual-branch encoder. In the high-frequency branch, the parallel atrous convolution activation (PACA) module is designed to explore fine-grained contour and edge patterns related to vessel terminals and microvessels. Concurrently, within the low-frequency branch, the spatial-frequency characteristic activation (SFCA) unit is constructed by introducing the selective state-space model (

S 6

) and Fourier transform to extract salient structural backbones. Moreover, the spatial attention residual fusion (SARF) module and cross-frequency fusion (CFF) block are designed to establish a symmetric guidance mechanism, effectively reinforcing bidirectional feature interaction and alignment across different frequency spectra to eliminate vascular fragmentation. Furthermore, by embedding global and local window self-attention into the Transformer, we formulated the cross-scale enhancement (CSE) module, comprising global semantic enhancement (GSE) and local detail enhancement (LDE), to model multi-scale contextual semantic correlations and enhance the adaptive recognition of vessel structures. Ultimately, we embedded the multi-wise prototype characteristic refinement (MPCR) component into the decoder to correct cross-scale semantic features through a dynamic calibration mechanism, while introducing a new connectivity loss to strictly enforce topological continuity. Experimental results on four publicly available retinal image datasets (DRIVE, CHASE_DB1, STARE, and IOSTAR) demonstrate that the proposed model achieves competitive performance and effectively preserves vascular integrity even in the presence of fundus lesions and noise. Full article

(This article belongs to the Section A: Computer Science)

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23 pages, 666 KB

Open AccessArticle

Some Applications of Fractional Integral for Mittag-Leffler Function on Strong Differential Sandwich Results

by Shaymaa Y. Alkufi, Abbas Kareem Wanas and Alina Alb Lupas

Symmetry 2026, 18(7), 1227; https://doi.org/10.3390/sym18071227 - 20 Jul 2026

Viewed by 329

Abstract

In this paper, we introduce new geometric properties of analytic functions by utilizing the fractional integral operator associated with the Mittag-Leffler function. Specifically, we establish several framework criteria under which strong differential subordination as well as superordination hold across the product domain [...] Read more.

In this paper, we introduce new geometric properties of analytic functions by utilizing the fractional integral operator associated with the Mittag-Leffler function. Specifically, we establish several framework criteria under which strong differential subordination as well as superordination hold across the product domain

U \times \bar{U}

, wherein the coefficients are holomorphic functions in

U

. For each investigated relation, the corresponding best dominant and best subordinant are explicitly determined. Utilizing these foundational outcomes, we subsequently derive novel strong sandwich-type theorems that bridge these dual concepts. Full article

(This article belongs to the Special Issue Symmetry and Its Applications in Complex Analysis by the Means of Special Functions)

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29 pages, 629 KB

Open AccessArticle

A Symmetry-Theoretic Framework for AI-Guided Symbolic Execution in Embedded Systems

by Maksim Iavich, Tamari Kuchukhidze and Audrius Lopata

Symmetry 2026, 18(7), 1226; https://doi.org/10.3390/sym18071226 - 20 Jul 2026

Viewed by 1333

Abstract

Symbolic execution of embedded systems faces path explosion, Satisfiability Modulo Theories (SMT) solver bottlenecks, interrupt nondeterminism, and environment modeling complexity. Recent artificial intelligence (AI)-guided approaches using reinforcement learning, graph neural networks, and large language models improve exploration efficiency, yet all reason over raw [...] Read more.

Symbolic execution of embedded systems faces path explosion, Satisfiability Modulo Theories (SMT) solver bottlenecks, interrupt nondeterminism, and environment modeling complexity. Recent artificial intelligence (AI)-guided approaches using reinforcement learning, graph neural networks, and large language models improve exploration efficiency, yet all reason over raw symbolic states and ignore structural equivalences that arise from symmetry in embedded software. This paper presents S³E, a formal framework that organizes symbolic execution around equivalence classes of states under symmetry transformations. Symmetry groups partition the state space into orbits, and exploration proceeds over canonical representatives within quotient transition systems. Symmetry-aware AI components operate on orbit representatives rather than raw states. Four theoretical results support the framework: orbit preservation, quotient soundness, canonicalization correctness, and constraint reuse correctness. An illustrative case study based on a FreeRTOS-like scheduling environment shows how symmetry reduction collapses equivalent states into orbits, with the potential for reductions that scale factorially with symmetric components. S³E is a theoretical framework; a toy-model prototype validates the core quotient-exploration and constraint-caching mechanis, while empirical evaluation on production firmware remains future work. Full article

(This article belongs to the Section A: Computer Science)

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26 pages, 658 KB

Open AccessArticle

A Sign-Symmetric Reformulation of the Hassanat Distance for Data with Negative Feature Values

by Mohammad Saad Alaydaa, Gaseb N. Alotibi, Ahmad S. Tarawneh and Ahmad B. Hassanat

Symmetry 2026, 18(7), 1225; https://doi.org/10.3390/sym18071225 - 20 Jul 2026

Viewed by 284

Abstract

The Hassanat Distance (HasD) is a bounded, non-convex metric widely used in k-nearest-neighbor (KNN) classification for its robustness to noise, outliers, and heterogeneous feature scales. Its definition, however, breaks a natural symmetry: through a sign-dependent shift it assigns different distances to mirror-image [...] Read more.

The Hassanat Distance (HasD) is a bounded, non-convex metric widely used in k-nearest-neighbor (KNN) classification for its robustness to noise, outliers, and heterogeneous feature scales. Its definition, however, breaks a natural symmetry: through a sign-dependent shift it assigns different distances to mirror-image pairs such as

(1, 2)

and

(- 1, - 2)

, distorting neighborhoods exactly in the value ranges that modern preprocessing (z-scoring, principal component analysis (PCA), learned embeddings) produces. We introduce the Sign-Symmetric Hassanat Distance (SHasD), a single branch-free formula

D (a, b) = | a - b | / (1 + \max (| a |, | b |))

that is invariant under the reflection

x \mapsto - x

, coincides exactly with HasD on non-negative data, and removes the conditional shift entirely. We prove SHasD is a metric, and we derive a range-normalized companion, SHasD-R, that additionally restores ray monotonicity and the

[0, 1)

per-dimension bound. On 23 datasets across three normalization regimes and ten distance measures, SHasD improves significantly on HasD on data containing negative values (mean gain

+ 1.1

percentage points, up to

+ 7.4

; Wilcoxon

p = 0.0026

, Holm-corrected) and attains the best mean rank of the compared measures on signed, heavy-tailed, outlier-rich data, while preserving HasD’s robustness. An additive per-dimension decomposition yields a built-in interpretation of every prediction. Full article

(This article belongs to the Section B: Mathematics)

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17 pages, 333 KB

Open AccessArticle

Line Graphs and Embedding Properties Associated with Extended Zero-Divisor Graph of Commutative Rings

by Mohd Arif Raza and Majed Albaity

Symmetry 2026, 18(7), 1224; https://doi.org/10.3390/sym18071224 - 20 Jul 2026

Viewed by 320

Abstract

Let

P

be a finite commutative ring with identity, and let

Z (P)

denote the set of its zero-divisors. The extended zero-divisor graph of

P

, denoted by

\tilde{Γ} (P)

, is the undirected simple graph with vertex [...] Read more.

Let

P

be a finite commutative ring with identity, and let

Z (P)

denote the set of its zero-divisors. The extended zero-divisor graph of

P

, denoted by

\tilde{Γ} (P)

, is the undirected simple graph with vertex set

Z {(P)}^{*} = Z (P) ∖ {0}

, where two distinct vertices

α

and

β

are adjacent if and only if

α β = 0

or

α + β \in Z (P)

. For a graph G, let

L (G)

denote its line graph. In this paper, we first characterize all finite commutative rings

P

for which

\tilde{Γ} (P)

is a line graph of some graph. We then classify the finite commutative rings

P

such that

L (\tilde{Γ} (P))

is planar, outerplanar, or 2-outerplanar. Finally, we obtain a complete classification of finite commutative rings

P

for which

L (\tilde{Γ} (P))

is toroidal. Full article

(This article belongs to the Section B: Mathematics)

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30 pages, 13263 KB

Open AccessArticle

Single-Variable Multi-Criteria Optimization of Solid-Phase Volume Fraction in Solid–Liquid Mixing System of High-Viscosity Polyurethane Adhesive Based on CFD Coupled with Machine Learning

by Bin He, Xurong Teng, Long Fan and Renlong Liu

Symmetry 2026, 18(7), 1223; https://doi.org/10.3390/sym18071223 - 20 Jul 2026

Viewed by 312

Abstract

The solid-phase volume fraction is a core process parameter that determines the mixing quality of high-viscosity polyurethane adhesives. It exhibits complex nonlinear couplings with mixing homogeneity, rheological properties, and energy consumption. Traditional trial-and-error experiments and standalone CFD simulations are hindered by high costs, [...] Read more.

The solid-phase volume fraction is a core process parameter that determines the mixing quality of high-viscosity polyurethane adhesives. It exhibits complex nonlinear couplings with mixing homogeneity, rheological properties, and energy consumption. Traditional trial-and-error experiments and standalone CFD simulations are hindered by high costs, long computational cycles, and inefficient parameter optimization processes. Consequently, these limitations prevent them from satisfying industrial-scale process optimization demands. To address these challenges, this study proposes a single-variable, multi-criteria intelligent optimization method integrating CFD with machine learning. This study uses a 5000 L industrial stirred reactor as the research object. Twelve typical operating conditions were designed within the 8–32% solid-phase volume fraction range. Numerical simulations were conducted based on the laminar Mixture multiphase flow model. These simulations established a small-sample, high-fidelity operating condition database. To mitigate underfitting caused by limited CFD data, an adaptive cubic spline interpolation algorithm was employed for dataset augmentation. Subsequently, a comparative analysis was conducted between Gaussian process regression (GPR) and support vector regression (SVR) as small-sample surrogate models. Test results demonstrate that the GPR model achieves a coefficient of determination (R²) of 0.968, with significantly superior prediction accuracy and generalization performance compared to the SVR model. This study uses the solid-phase volume fraction as the sole decision variable. A five-criteria optimization model was constructed based on the optimal GPR surrogate, combined with the NSGA-II algorithm and ideal point decision criterion. Three optimal operating modes were identified—balanced, low-energy, and high-dispersion—tailored for different production scenarios. The proposed integrated collaborative optimization approach effectively reduces process development costs. It also provides comprehensive theoretical foundations and technical references for intelligent control and energy-efficient production of high-viscosity non-Newtonian solid–liquid mixing systems. Full article

(This article belongs to the Section F: Engineering and Materials)

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Graphical abstract

6 pages, 143 KB

Open AccessEditorial

Special Issue: New Power System and Symmetry

by Yongji Cao, Runjia Sun, Rui Wang and Jin Tan

Symmetry 2026, 18(7), 1222; https://doi.org/10.3390/sym18071222 - 20 Jul 2026

Viewed by 266

Abstract

Global challenges such as fossil energy depletion, environmental pollution, and climate warming are driving the rapid development of renewable energy sources (RESs), including wind power and photovoltaics [...] Full article

(This article belongs to the Special Issue New Power System and Symmetry)

27 pages, 10988 KB

Open AccessArticle

Text Image Super-Resolution via Fusion of OCR Priors and Cross-Scale Attention

by Xinyu Qiu, Jingchao Liu and Chen Fang

Symmetry 2026, 18(7), 1221; https://doi.org/10.3390/sym18071221 - 20 Jul 2026

Viewed by 414

Abstract

Text image super-resolution aims to improve the readability of low-quality text images while preserving character structures, stroke details, and semantic consistency. Compared with natural image super-resolution, this task is more sensitive to structural distortion because small changes in stroke topology may lead to [...] Read more.

Text image super-resolution aims to improve the readability of low-quality text images while preserving character structures, stroke details, and semantic consistency. Compared with natural image super-resolution, this task is more sensitive to structural distortion because small changes in stroke topology may lead to incorrect text recognition. To address this problem, this paper proposes an OCR prior-guided cross-scale framework for text image super-resolution. Specifically, character-level semantic priors extracted from a pretrained OCR model are introduced to provide structural guidance for degraded text reconstruction. A gated feature modulation mechanism is designed to adaptively regulate the contribution of OCR priors, reducing the influence of unreliable semantic predictions. A cross-scale dynamic attention module is also developed to aggregate multi-granularity visual features, enabling the model to jointly recover fine stroke boundaries and global character structures. In addition, a sequence-aware calibration module is introduced to improve structural consistency along the logical reading order of text. Experiments on mixed text image benchmarks and the TextZoom dataset show that the proposed method achieves competitive or better performance among the compared methods in terms of PSNR, SSIM, and recognition-oriented metrics. Additional ablation, OCR prior robustness, and computational complexity analyses further indicate that the proposed framework improves text readability while maintaining a reasonable accuracy–complexity trade-off. The results also suggest that OCR priors are useful for text image reconstruction, but should be used as soft constraints when external recognition predictions are uncertain. Full article

(This article belongs to the Section A: Computer Science)

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29 pages, 5135 KB

Open AccessArticle

Edge-Intelligent Industrial Inspection: A GPU-Accelerated Multiscale CNN Framework for Real-Time Visual Quality Assessment

by Gürkan Kavuran and Barış Baykant Alagöz

Symmetry 2026, 18(7), 1220; https://doi.org/10.3390/sym18071220 - 20 Jul 2026

Viewed by 433

Abstract

The transition toward Industry 4.0 requires the integration of technologically viable hardware–software–intelligence solutions into existing industrial infrastructures to enable smart and autonomous production systems. Thus, Industry 4.0 enables techno-symmetry, which refers to the balanced and interactive distribution of technological capacity, information processing ability, [...] Read more.

The transition toward Industry 4.0 requires the integration of technologically viable hardware–software–intelligence solutions into existing industrial infrastructures to enable smart and autonomous production systems. Thus, Industry 4.0 enables techno-symmetry, which refers to the balanced and interactive distribution of technological capacity, information processing ability, and decision-making capability across production networks. This study proposes a comprehensive hardware–software–intelligence framework for a real-time visual quality inspection of transformer cases during the manufacturing process by using an embedded deep learning architecture. First, a real-world dataset consisting of 232 defective and 264 non-defective printed transformer case images was collected from the production line of a transformer manufacturing facility and preprocessed to improve data quality and model generalization. Second, to enhance feature extraction capability, the classical AlexNet architecture was modified to develop a Multiscale AlexNet (MS-AN) model capable of simultaneously capturing both global and local spatial features. The proposed architecture incorporates parallel convolutional branches with 3 × 3 and 5 × 5 receptive fields, which are fused at the feature level to increase representation diversity and improve robustness against noise and degradation in printed images. Third, an experimental system was implemented using practical industrial automation technologies (e.g., CUDA-accelerated C++ programming, the NVIDIA Jetson Orin Nano edge computing platform, ROS-based communication infrastructure, IoT protocols, and programmable logic controller (PLC) integration). Experimental results demonstrate that the proposed system achieves real-time inspection performance of approximately 2 s per inspection with 99% classification accuracy on the constructed dataset. The developed framework enables efficient deployment of deep learning models on GPU-based edge devices; thus, it reduces reliance on workstation-class computers, lowers energy consumption, and supports scalable intelligent inspection architectures aligned with Industry 4.0 transformation objectives. Full article

(This article belongs to the Topic Artificial Intelligence in Intelligent Control, Robotics, and Manufacturing)

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25 pages, 6775 KB

Open AccessArticle

Research on a Fault-Diagnosis Method for Heavy-Duty Bearings of Thin Coal-Seam Shearers

by Minghao Li, Shuting Wang, Xiao-Guang Zhang, Xiaoxu Yi and Dongsheng Wu

Symmetry 2026, 18(7), 1219; https://doi.org/10.3390/sym18071219 - 19 Jul 2026

Viewed by 271

Abstract

Aiming at the problems of fault samples being difficult to obtain in fault diagnosis research on heavy-duty bearings of thin-seam shearers and the insufficient diagnostic performance of existing methods under small-sample conditions, this study focuses on dataset construction, sample augmentation and fault diagnosis. [...] Read more.

Aiming at the problems of fault samples being difficult to obtain in fault diagnosis research on heavy-duty bearings of thin-seam shearers and the insufficient diagnostic performance of existing methods under small-sample conditions, this study focuses on dataset construction, sample augmentation and fault diagnosis. First, based on the virtual prototype model of the cutting-unit transmission system of a thin-seam shearer, three-dimensional models of healthy bearings and four typical fault types (inner ring fault, outer ring fault, rolling-element fault and cage fault) of heavy-duty bearings were built using SolidWorks 2024. Vibration signals were collected through ADAMS dynamic simulation and converted into time-frequency images via Continuous Wavelet Transform (CWT), thereby constructing an original fault dataset with five states. Furthermore, a conditional generative adversarial network incorporating VGG perceptual loss (VGG-CGAN) was proposed to achieve targeted sample augmentation for the five bearing states, effectively alleviating the class-imbalance problem. On this basis, an improved ResNet50 fault-diagnosis model was constructed, and Bayesian optimization was used to automatically tune key hyperparameters. Experimental results show that the improved ResNet50 model achieved an accuracy of 87.14% on the self-built thin-seam shearer heavy-duty bearing dataset and 98.28% on the public CWRU dataset. The proposed method exhibits strong diagnostic performance and generalization ability under small-sample and imbalanced data conditions. This study can provide new ideas and useful references for fault diagnosis of heavy-duty bearings in thin-seam shearers. Full article

(This article belongs to the Section F: Engineering and Materials)

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12 pages, 10229 KB

Open AccessArticle

Morphological Variation and Multi-Level Fluctuating Asymmetry of the Caudal Peduncle Spines in the Picasso Triggerfish Rhinecanthus aculeatus from the Xisha Islands

by Haishan Wang, Yule Deng, Le Ye, Youming Li and Zhi Chen

Symmetry 2026, 18(7), 1218; https://doi.org/10.3390/sym18071218 - 19 Jul 2026

Viewed by 330

Abstract

The Picasso triggerfish Rhinecanthus aculeatus is common in Indo-Pacific coral reefs, yet quantitative data on intraspecific morphological variation and developmental stability of its caudal peduncle spines remain limited, especially in the South China Sea, where reef degradation has been extensively documented in the [...] Read more.

The Picasso triggerfish Rhinecanthus aculeatus is common in Indo-Pacific coral reefs, yet quantitative data on intraspecific morphological variation and developmental stability of its caudal peduncle spines remain limited, especially in the South China Sea, where reef degradation has been extensively documented in the literature. To address this, we examined 31 individuals from the Yongle Islands (Xisha Islands) using traditional morphometrics, truss distances, and angular cosine parameters for body shape, and developed a multi-level fluctuating asymmetry (FA) framework to dissect asymmetry in the caudal spine complex. Eye diameter showed strong negative allometry and low FA (FA5_pct = 1.56% ± 1.57%), supporting its suitability as a reference trait. Angular cosine values captured body shape variation more sensitively (mean CV = 14.78%) than traditional ratios (7.10%) or truss ratios (6.66%). Principal component analysis revealed continuous morphological dispersion without discrete clustering. Total spine FA5_abs was 2.13 ± 1.75, per-row difference FA5_abs was 3.10 ± 3.30, and within-row standard deviation FA5_abs was 0.59 ± 0.44. Per-row absolute differences declined numerically from anterior to posterior rows, although this trend was not statistically significant (Jonckheere–Terpstra test, p = 0.0955). A significant negative partial correlation between eye FA and total spine FA (partial ρ = −0.498, p = 0.0048; FDR-adjusted p = 0.0397) emerged after controlling for standard length, which we tentatively interpret as possible evidence of inter-organ developmental compensation. These findings document substantial developmental instability in the caudal spine complex of R. aculeatus from the Xisha Islands and illustrate the utility of multi-level FA analysis for functionally partitioned structures. Full article

(This article belongs to the Section E: Life Sciences)

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31 pages, 2880 KB

Open AccessArticle

Symmetric Complementarity of Co-Located Rail Systems on Urban Carbon Productivity

by Haokun He, Congzhe Liu and Xinyang Pang

Symmetry 2026, 18(7), 1217; https://doi.org/10.3390/sym18071217 - 19 Jul 2026

Viewed by 307

Abstract

Environmental protection has become a key priority for the Chinese government. This study examined the symmetric complementarity of spatial layouts between inter-city high-speed rail and urban subway systems. Their co-location formed a structurally symmetric low-carbon travel chain that bridged long-distance and short-distance trips. [...] Read more.

Environmental protection has become a key priority for the Chinese government. This study examined the symmetric complementarity of spatial layouts between inter-city high-speed rail and urban subway systems. Their co-location formed a structurally symmetric low-carbon travel chain that bridged long-distance and short-distance trips. The study evaluated their joint effect on urban carbon productivity within a difference-in-differences framework. The baseline two-way fixed-effects DID estimate showed that the simultaneous operation of HSR and subway systems significantly improved urban carbon productivity, with an average increase of approximately 10.51% in treated cities relative to non-treated ones. This core finding remained qualitatively robust across a series of tests, including conditional coarsened exact matching (CEM), placebo simulations, sample exclusions, and decomposition of joint effects relative to individual rail impacts. Complementary nonparametric causal forest estimates yielded smaller but still significant effect sizes, providing supportive evidence for causal validity under weaker functional form assumptions. Heterogeneity analysis based on dynamic GDP grouping revealed that high-GDP cities experienced stronger emission reductions, while low-GDP cities benefited more from the economic growth dimension of carbon productivity. Empirical patterns aligned with the theoretical prediction of the transportation substitution mechanism. The positive effect of dual rail availability on carbon productivity was stronger in cities with higher private car stock, as shown by double machine learning interaction models. This finding only provided indirect evidence and did not constitute direct proof of actual modal shift behavior. The results exhibited magnitude asymmetry between parametric and nonparametric methods, yet all estimates consistently pointed to a positive qualitative direction. This consistency revealed directional symmetry in the causal conclusion. The study contributed by (1) examining the combined carbon productivity effect of co-located HSR and subway systems, (2) identifying the moderating role of private car stock in the transportation substitution mechanism, (3) revealing development-stage-based heterogeneity, (4) and combining machine learning methods with parametric causal inference as complementary robustness evidence. Policy implications suggested that high-GDP cities should prioritize expanding dual rail coverage and optimizing connections to amplify emission reductions, while low-GDP cities should focus on improving HSR connectivity with existing public transit systems and fostering low-carbon industrial development in line with local conditions. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry in Complex Systems and Smart Cities)

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23 pages, 609 KB

Open AccessArticle

A Semiparametric Probit Mixture Cure Model for Interval Censored Data with a Cure Fraction

by Nan Zhang, Shishun Zhao and Dongmei Lu

Symmetry 2026, 18(7), 1216; https://doi.org/10.3390/sym18071216 - 19 Jul 2026

Viewed by 224

Abstract

Accurately estimating the incubation period of emerging infectious diseases, defined as the time interval from infection to symptom onset, is essential for informing quarantine strategies and allocating public health resources. To address the challenges of interval-censored data commonly encountered in incubation period studies, [...] Read more.

Accurately estimating the incubation period of emerging infectious diseases, defined as the time interval from infection to symptom onset, is essential for informing quarantine strategies and allocating public health resources. To address the challenges of interval-censored data commonly encountered in incubation period studies, as well as the possibility that some infected individuals may never develop symptoms, we propose a semiparametric probit mixture cure model. By introducing two latent variables to simplify the likelihood function, we develop a corresponding EM algorithm for parameter estimation. Under mild regularity conditions, we establish the consistency and asymptotic normality of the proposed estimators. Numerical simulation studies demonstrate that the proposed method performs well even with limited sample sizes. Furthermore, an empirical analysis of real-world COVID-19 data validates the practical applicability of the model. The proposed semiparametric probit mixed cure model offers methodological support for early prevention and control decision-making in response to sudden infectious disease outbreaks. Full article

(This article belongs to the Section B: Mathematics)

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23 pages, 861 KB

Open AccessArticle

Biased Nonlinear Ship Roll as a

Z_{2}

-Equivariant Oscillator: Symmetry Breaking, Closed-Form Capsize Boundaries, and a Second-Generation Intact-Stability Perspective

by Jiahao Hu, Weipeng Zhou, Changchun Liu and Jinyuan Zhu

Symmetry 2026, 18(7), 1215; https://doi.org/10.3390/sym18071215 - 19 Jul 2026

Viewed by 246

Abstract

The roll equation of a port–starboard symmetric ship is a clean physical realization of an order-two reflection (

Z_{2}

)-equivariant nonlinear oscillator: odd restoring and damping make the upright state a symmetric equilibrium whose two angles of vanishing stability form a heteroclinic-connected [...] Read more.

The roll equation of a port–starboard symmetric ship is a clean physical realization of an order-two reflection (

Z_{2}

)-equivariant nonlinear oscillator: odd restoring and damping make the upright state a symmetric equilibrium whose two angles of vanishing stability form a heteroclinic-connected pair. A steady heeling action—beam wind, off-center load or list—enters as a single symmetry-breaking parameter c. Although the qualitative effect of such a bias is known, we show that this one parameter organizes the whole capsize problem in closed form. Equivariant singularity theory identifies c as the imperfection that unfolds the symmetric pitchfork of equilibria into a cusp, turning the heteroclinic pair into a homoclinic loop. The biased Melnikov boundary then yields two directional capsize thresholds and a damping-independent asymmetry index

Δ f_{c r} = 2 c I_{c} / A (Ω)

, exactly linear in the bias: for a lightly damped hull a heel below

0.1 °

already halves the port/starboard split. For parametric roll, Floquet analysis gives the bias-corrected stability boundary, recovering

Δ G M / G M > 4 ζ

in the symmetric limit and translating the principal tongue; for the dead-ship condition, the stationary Fokker–Planck solution—exact for the adopted one-degree-of-freedom (1-DOF) model—shows that, in the light-damping energy-diffusion regime, lightly damped capsize occurs over the lowered barrier with probability approaching one. A small-bias expansion reveals a sensitivity hierarchy—beam-sea capsize and dead-ship survival are first-order in the heel, and parametric detuning is only second-order—and a two-parameter cusp links the bias to pure loss of stability. All results are validated against safe-basin erosion, Floquet multipliers and Monte-Carlo simulation for a real vessel, offering the International Maritime Organization (IMO) second-generation intact-stability criteria (SGISC) a transparent, analytically based correction for the port–starboard asymmetry their symmetric assumption omits. Full article

(This article belongs to the Section F: Engineering and Materials)

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26 pages, 7158 KB

Open AccessArticle

Performance Improvement of Continuous-Variable Quantum Secret Sharing via Heralded Hybrid Linear Amplifier

by Kunlin Zhou, Yang Yu, Lining Zeng, Shijie Deng and Ying Guo

Symmetry 2026, 18(7), 1214; https://doi.org/10.3390/sym18071214 - 18 Jul 2026

Viewed by 244

Abstract

Continuous-variable quantum secret sharing (CVQSS) distributes a secret key among multiple players while requiring their cooperation for reconstruction. Its performance deteriorates rapidly in sequential multiparty links because optical loss, excess noise, and receiver noise accumulate with the number of players. We investigate a [...] Read more.

Continuous-variable quantum secret sharing (CVQSS) distributes a secret key among multiple players while requiring their cooperation for reconstruction. Its performance deteriorates rapidly in sequential multiparty links because optical loss, excess noise, and receiver noise accumulate with the number of players. We investigate a dealer-side heralded hybrid linear amplifier (HHLA), formed by measurement-based noiseless linear amplification and trusted deterministic preamplification, for mitigating the relative receiver-noise penalty. The accepted data are described by an equivalent Gaussian channel and evaluated with an asymptotic reverse-reconciliation key-rate model against collective Gaussian attacks. We explicitly condition parameter estimation and Eve’s Holevo information on successful heralding, include a cutoff-dependent Gaussian-input estimate of the heralding probability in the rate per transmitted pulse, apply a single calibrated output-quadrature rescaling, and provide parallel heterodyne and homodyne calculations. The homodyne simulation includes detector noise consistently in the mutual information and Holevo bound and enforces the Gaussian-equivalent NLA feasibility constraint. For an excess-noise variance of

0.001

shot-noise units per player, the constrained homodyne optimization gives maximum reported distances of approximately 108, 42, and

25.5

km for 5, 50, and 100 players, respectively, at a reporting floor of

10^{- 6}

bit/pulse. Composable finite-size security, finite-cutoff non-Gaussian corrections, and active-insider verifiability are left for future work. Full article

(This article belongs to the Section A: Computer Science)

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