Symmetry

27 pages, 2524 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

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 Mathematics)

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

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 Computer)

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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

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 Computer)

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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

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 Computer)

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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

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 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

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

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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36 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

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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25 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

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 Engineering and Materials)

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43 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

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 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

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 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

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

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

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

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 Computer)

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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

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

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 Computer)

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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

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 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

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 Mathematics)

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Journal Description

Symmetry

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