Symmetry doi: 10.3390/sym18081322

Authors: Sergey A. Tarasov Anastasia O. Petrova Ekaterina O. Khimich Evgenia S. Nechaeva Galina V. Yarmoschuk Olesya M. Gizitdinova Olga V. Fartushnaia Angelina A. Boriskina Anastasia D. Zatykina Irina V. Molodtsova Kseniya S. Peshketova Liudmila E. Samsonova Maria A. Vershinina Alexey V. Smirnov Polina N. Borisoglebskaya Evgeniy A. Gorbunov Alexander L. Kovalchuk Oleg I. Epstein

For many years, our research group has been studying a phenomenon initially discovered in biological models: a solution of a substance subjected to vibration treatment exerts a modifying effect on an intact sample of the same substance, altering its physicochemical and biological properties. This effect also occurs without direct contact, when the vibration-treated and intact substances are kept in separate vials. Since the emergence of this modifying effect requires minimal energy input, we suggest that it is mediated by pre-existing supramolecular mechanisms. During vibration, the molecules deviate from a specific axis of symmetry (a local stationary state), which likely activates an evolutionarily developed supramolecular control over the molecular spatial structure. While testing this hypothesis, we observed that the modifying effect can be detected at a significant distance. This finding indicates that the phenomenon cannot be explained solely by the direct electromagnetic radiation of the vibrating molecules. Instead, the vibration treatment remotely alters the electromagnetic environment of the intact substance, thereby inducing the modifying effect. While non-trivial quantum effects might underlie this process, we favor the assumption that other fundamental interactions influence the electromagnetic field.

Symmetry doi: 10.3390/sym18081321

Authors: Sibel Cansu Özlem Altunbezel

H-bases provide an attractive alternative to Gröbner bases for the study of polynomial ideals because they are independent of monomial orderings and therefore preserve structural properties that may be obscured in Gröbner basis computations. However, the practical use of H-bases has been limited by the lack of efficient algorithms for their construction, particularly due to the difficulty of computing syzygy modules and eliminating redundant generators. In this paper, we present a new algorithm for computing syzygies and H-bases based on numerical linear algebraic techniques. The proposed approach utilizes QR decomposition to construct syzygies directly from coefficient matrices, avoiding the need for monomial orderings and Gröbner basis computations. A recursive framework derived from QR factorization is developed to compute both syzygy modules and H-bases while simultaneously defining an associated reduction process. In contrast to existing methods based on Gröbner bases, Schreyer’s theorem, or singular value decomposition, the proposed algorithm requires fewer matrix reductions and relies on computationally less demanding operations. Furthermore, its termination criterion is determined by an upper bound independent of Gröbner basis theory. Consequently, the method provides a more streamlined and conceptually simpler framework for the computation of syzygies and H-bases of polynomial ideals.

Symmetry doi: 10.3390/sym18081319

Authors: Huangshui Hu Shuo Liu Qier Kang Suli Zhang Chengshuo Tian

Energy-efficient and robust routing remains a critical problem in wireless sensor networks (WSNs), where limited energy resources and dynamic topologies hinder performance. To address this challenge, a novel cluster-based routing protocol FGOGNN is proposed in this paper, which integrates a Fungal Growth Optimizer (FGO) for adaptive cluster head (CH) selection and a Graph Neural Network (GNN) for inter-cluster routing. The FGO simulates fungal growth processes, ensuring balanced and energy-efficient CH selection while preventing premature convergence. In the routing phase, the GNN dynamically adapts routing paths by leveraging node energy, connectivity, and directional edge features, offering low computational overhead while maintaining accuracy. Extensive simulations show that FGOGNN outperforms existing routing protocols, extending network lifetime by up to 75% and improving throughput by 38%, with a reduction in end-to-end delay. These results demonstrate FGOGNN’s potential for deployment in real-time WSN applications, where energy efficiency and dynamic adaptability are paramount.

Symmetry doi: 10.3390/sym18081320

Authors: Álvaro Antón-Sancho Samer R. Yaseen

Let X be a compact Riemann surface of genus g≥2 equipped with an anti-holomorphic involution τ, and let M(G2) denote the moduli space of stable principal G2-bundles over X. We study the fixed-point loci M(G2)σ of the involutions σ=θ∗∘τ∗ on M(G2), where θ is a Cartan involution of G2 corresponding to one of its two real forms: the compact form G2c and the split form G22,2. We prove that M(G2)σc is connected with Euler characteristic 1, and that M(G2)σs has exactly 2c connected components, where c is the number of connected components of the fixed-point set Xτ. Each component is shown to be a compact, real-analytic, totally real submanifold of M(G2) of real dimension 14(g−1). For the split form, we establish that χ(M(G2)σs)=3g−1 when c=0, where the factor 3=|W(G2)|/|W(SO(4))| arises from the Borel–Hirzebruch formula applied to the symmetric space G22,2/SO(4). We further analyse the monodromy element k0∈SO(4) associated to the non-trivial topological type over the ovals, proving that its centraliser in G22,2 is exactly SO(4) and that its conjugacy class coincides with G22,2/SO(4), with e(O(k0))=3. This supports a conjecture asserting χC(w1,…,wc)=3g−1 for each component when c≥1. Finally, exploiting the Langlands self-duality G2∨≅G2, we show that M(G2)σ is a (B,B,B)-brane in the hyperkähler manifold T∗M(G2), and that it is self-specular: it is preserved by the derived Fourier–Mukai autoequivalence Φ:DbCoh(M(G2))→∼DbCoh(M(G2)) induced by the Langlands self-duality, which restricts to a self-equivalence of DbCoh(M(G2)σ).

Symmetry doi: 10.3390/sym18081318

Authors: Prathima Jayarama Insuk Kim Arjun K. Rathie Sunil D. Purohit

Classical summation theorems for generalized hypergeometric series constitute an important tool in the study of special functions. Motivated by earlier generalizations of these theorems, this paper derives several new summation formulas for generalized hypergeometric functions by employing the generalized classical summation theorems of Lavoie et al. within a well-known hypergeometric identity. The proposed formulas extend previously reported results, including those of Eslahchi and Masjed-Jamei, which arise as special cases of the present work. The obtained identities provide a unified framework for deriving summation formulas and may be useful in further investigations involving generalized hypergeometric functions and related identities.

Symmetry doi: 10.3390/sym18081316

Authors: Yu Lv

Penalty function methods are fundamental for solving constrained optimization problems, yet the widely used l1 exact penalty function is non-differentiable and therefore incompatible with gradient-based algorithms. This paper proposes a continuously differentiable smoothing approximation that preserves the exactness of the original penalty while enabling efficient numerical solution. The proposed smoothing function is shown to be C1 on R, with explicit error bounds and a computable lower bound for the penalty parameter that guarantees exactness under standard constraint qualifications. An iterative algorithm is developed and its convergence is proved. Numerical experiments on benchmark problems validate the effectiveness of the approach. The method is then applied to a policy-driven university financial risk management model with twelve decision variables, three conflicting objectives, and multiple regulatory constraints. Results demonstrate faster convergence and improved objective values compared with the traditional non-smooth penalty method, confirming the practical utility of the proposed smoothing technique.

Symmetry doi: 10.3390/sym18081317

Authors: Liang Zou Cheng Chang Zhiyun Han Kejie Huang Hanwen Ren Rongzhao Jia Zhen Li

Bridge-arm reactors subjected to long-term AC–DC composite currents with multiple harmonics may develop non-uniform winding temperature rise and thermal-expansion mismatch, leading to localized thermal stress concentrations and potential insulation cracking. Unlike previous studies focused mainly on purely AC conditions, this study investigates a ±800 kV dry-type air-core bridge-arm reactor and develops a thermo-mechanical model incorporating AC–DC composite currents and harmonic losses. To mitigate thermal stress concentration, an axially segmented configuration is proposed to relieve the restraint associated with cumulative axial thermal expansion. The results show that a 65% axial segmentation ratio provides the best stress-regulation performance among the investigated cases. Under AC–DC composite conditions containing second- and fifth-order harmonics, the maximum Von Mises stress and maximum first-principal stress decrease by 33.42% and 38.11%, respectively, while the stress distribution becomes more uniform. The analysis is based on a two-dimensional axisymmetric model with one-way thermo-mechanical coupling and excludes long-term cyclic thermal aging and interfacial slip between winding and insulation layers. These findings provide theoretical support for the stress-oriented structural design and reliability assessment of high-capacity bridge-arm reactors.

Symmetry doi: 10.3390/sym18081315

Authors: Yong Ding Wenjie Hou Fan Yang Zhoujian An

A three-dimensional structural model of a cylindrical lithium-ion battery pack incorporating composite phase change material (PCM) is developed in this study, and numerical simulations are conducted using CFD software to investigate the heat dissipation characteristics of the battery pack. The results show that the composite PCM effectively suppresses the temperature rise within the battery pack, maintaining both the temperature and the temperature difference in the battery pack within acceptable ranges. Parameter analysis reveals that increasing the radial thermal conductivity of the battery reduces the heating rate and improves the temperature uniformity of the overall system. Within the investigated parameter range, increasing the thermal conductivity of the composite PCM beyond approximately 1 W/(m·K) results in a region of diminishing improvement in thermal performance. Beyond this range, further increases in thermal conductivity result in only marginal reductions in the maximum temperature, indicating that excessive enhancement of thermal conductivity provides limited thermal benefits and should be balanced with latent heat capacity. An increase in the latent heat of the composite PCM lowers both the maximum temperature and the maximum temperature difference at the end of discharge, thereby enhancing system temperature uniformity. Conversely, enlarging the external air convection heat transfer coefficient yields a limited cooling effect while deteriorating the temperature uniformity within the system. Therefore, on the principle of fully utilizing latent heat and minimizing energy consumption, the external convection heat transfer coefficient should be set as low as possible. This study provides theoretical guidance for the parametric design of PCM-based thermal management systems.

Symmetry doi: 10.3390/sym18081314

Authors: Baidong Zhao Hao Yu Laixing Jing Aman Mussa Chenghan Yang Dingkun Zheng Sholpan Jomartova Huannian Meng

Hyperparameter optimization can reduce validation error while worsening future performance when data are extremely scarce. This study examines that risk in a public railway-mileage implementation workbook containing 17 ordered five-lag input-target rows, of which 14 form the development set and three form a locked outer test. No independently verified 22-point raw series was available; exact archived rows are therefore the primary data, while an overlap-consistent inferred chronology is used only for sensitivity analysis. WOA-CNN and BWO-CNN were evaluated through 10 complete HPO searches under shuffled and rolling-origin validation. Five-search purged runs were used as a stress-test sensitivity, while a fixed Manual CNN served as the matched reference. Median locked-test MAPE was 3.75% for WOA-Random, 2.56% for BWO-Random, 4.16% for WOA-Rolling, and 4.51% for BWO-Rolling, compared with a matched Manual CNN reference mean of 2.54%. The four HPO procedures beat that reference in only 3/10, 5/10, 2/10, and 1/10 searches, respectively. A mean-increment baseline obtained 0.63% MAPE. A separate benchmark of 160 independent simulated series examines sample-size and validation-selection mechanisms under four pre-specified data-generating processes. Taken together, the two analyses characterize how validation design and data availability affect model-selection stability in extremely small time-series settings.

Symmetry doi: 10.3390/sym18081313

Authors: Li Yuan Bo Kong Chenhao Li Anting Guo Wenjiang Huang

Indoor 3D referring segmentation aims to identify and segment the target object in a point cloud according to a natural language expression. Although recent advances in multimodal feature fusion have markedly improved this task, existing methods do not jointly model the structural regularities commonly observed in indoor objects, such as approximate symmetry and repetitive local patterns, in conjunction with the directional constraints conveyed by referring expressions. In this work, we formulate symmetry-aware representation as the extraction of direction-consistent structural responses from both forward and backward traversals of the same spatially ordered sequence while preserving direction-sensitive variations introduced by occlusion, point cloud incompleteness, and cluttered scene layouts. Based on this formulation, we propose PCMTRefer, a symmetry-aware text-guided Point Mamba framework for indoor 3D referring segmentation. The input point cloud is first partitioned using an octree and arranged into a spatially coherent sequence via Z-order (Morton) ordering. A bidirectional state-space encoder then aggregates complementary context from both traversal directions, yielding richer representations of regular boundaries, repetitive structures, and approximately bilateral object geometries. In parallel, an asymmetric text-to-point guidance module injects semantic cues—object categories, attributes, and spatial relationships—into point-wise features, while a Background-Relaxation Token offers an auxiliary matching channel for non-target background regions. A Gumbel-Softmax-based semantic primitive learning module further extracts discriminative cues from referring expressions and integrates language semantics with point-level geometric features through a multi-scale decoder. Experimental results on the ScanRefer benchmark show that PCMTRefer achieves an Overall Acc@0.25 of 58.56%, an Overall Acc@0.5 of 54.19%, and an mIoU of 49.97%. Multi-seed validation further confirms the statistical stability of these results, with standard deviations of less than 0.2% across three independent runs.

Symmetry doi: 10.3390/sym18081312

Authors: Zhenkun Zhang

Cyclic cutwidth problem is a graph layout problem whose goal is to find an embedding ϕ of the vertices of a graph G with n vertices onto a cycle Cn so as to minimize the maximum cutwidth of graph G. According to the literature, this problem is NP-complete, and some exact results regarding cyclic cutwidth have been reported. For an integer k>1, a graph G with cyclic cutwidth k is k-cyclic cutwidth critical if every proper subgraph of G has cyclic cutwidth less than k and G is homeomorphically minimal. In this paper, from the point of view of graph decomposition, we first characterize decomposable structure of some k-cyclic cutwidth critical graphs. We ascertain that a k-cyclic cutwidth critical graph G with a centre u0 has a subgraph decomposition with either two or three members, and we find that each member is a δ-linear cutwidth critical subgraph of G with δ=k−2 or k−1. The result reveals the structural relation between the cyclic cutwidth and the linear cutwidth of graph G.

Symmetry doi: 10.3390/sym18081311

Authors: Mingchuan Wu Jian Lu Yulin Li

Mobile edge computing (MEC) must serve rapidly growing populations of latency-critical and energy-constrained devices, yet distributed offloading faces two coupled problems: learned multi-agent policies depend on the arbitrary numerical ordering of edge servers, which wastes training samples and treats physically equivalent configurations inconsistently, while short-horizon cost minimization overloads attractive servers and destabilizes their queues. This paper presents SEAL-MAC (Symmetry-Equivariant Lyapunov Multi-Agent Actor–Critic), a distributed learning framework that addresses both problems jointly. First, a symmetric resource-set actor with a mirror consistency regularizer enforces server relabeling equivariance of each user’s policy and invariance of its value and Lyapunov critics. Second, a load-symmetric Lyapunov–potential shaping mechanism augments drift-plus-penalty rewards with normalized load-balance signals, coupling queue stability, fairness, and strategic alignment. The shaped interaction is analyzed as a Lyapunov-shaped Markov potential game: exact under orthogonal congestion-separable conditions, and a Markov α-potential game under heterogeneity or interference, yielding conditional finite-time (ϵ+α)-Nash convergence and mean-square queue stability. Each device learns from local observations and O(M) queue broadcasts without exchanging gradients or policies. In simulations with up to 200 users, SEAL-MAC reduces average delay by 9.0%, 95th-percentile delay by 11.8%, energy consumption by 10.6%, and the deadline-violation rate from 3.1% to 1.8% relative to the strongest Lyapunov baseline, halves the empirical one-step deviation gain of an identically shaped Ly-PPO agent (0.048 versus 0.098), and raises the Jain fairness index from 0.88 to 0.94.

Symmetry doi: 10.3390/sym18081310

Authors: Din Prathumwan Sirawit Phakmee Inthira Chaiya Kamonchat Trachoo

Co-infection between influenza and Streptococcus pneumoniae is an important public health concern, since influenza can increase susceptibility to secondary bacterial invasion and enhance bacterial transmission. We develop and analyze a compartmental model of influenza–pneumococcal co-infection with eleven epidemiological compartments, incorporating imperfect vaccination, quarantine, and a three-stage pneumococcal progression from colonization to invasive disease. We establish the positivity and boundedness of solutions, determine the equilibria, and derive, via the next-generation matrix method, the sub-model reproduction numbers R0I and R0P together with the composite threshold R0=max{R0I,R0P}. Each single infection undergoes a forward bifurcation: the disease-free equilibrium is locally asymptotically stable when the corresponding reproduction number is below unity, and a unique endemic equilibrium—globally asymptotically stable in the pneumococcal sub-model, established by a Goh–Volterra Lyapunov function—emerges above it. When both thresholds exceed unity, the two infections compete for the shared susceptible pool and undergo competitive exclusion: one infection persists while the other, together with the co-infected class, is eliminated. We show that the outcome is governed not by the disease-free reproduction numbers but by the invasion reproduction numbers evaluated at the single-infection boundary equilibria, so that even equal disease-free thresholds do not yield coexistence. Numerical simulations confirm the analytical results and quantify the effect of quarantine and vaccination, providing a framework for assessing control of interacting respiratory infections.

Symmetry doi: 10.3390/sym18081309

Authors: Yansong Zhao Qian Xiao Hong Zhu Wenbiao Lu Chunlei Xu Shiwen Su Xiaohui Pan Kai Sun

In medium-voltage DC distribution systems (MVDC-DSs), feeder congestion may occur when multiple converter terminals sharing the same AC feeder experience fast source–load variations. Limited communication further challenges real-time converter coordination and secure system operation. To address these issues, this paper has proposed an adaptive online management strategy for multi-terminal feeder congestion in MVDC-DSs under limited communication. First, the economic operation objective and practical system constraints are formulated within a distributed optimization framework, where feeder congestion limits, voltage security, and load supply requirements are explicitly incorporated. Then, an adaptive online regulation mechanism is embedded into local converter controllers, enabling converter power to be coordinated in real time using only neighboring information. In this way, feeder congestion can be mitigated while reliable load supply and economic operation are maintained. Simulation studies and hardware-in-the-loop experimental results demonstrate the effectiveness, scalability, and real-time applicability of the proposed strategy under dynamic operating conditions.

Symmetry doi: 10.3390/sym18081308

Authors: Weidong Jia Fuzhen Zhou Xiang Dong Wenrui Zhu

The reciprocating cutterbar is evolving from a conventional harvesting mechanism into an intelligent end-effector for crop harvesting, mechanical weeding, and selective cutting. However, plant anisotropy, moisture-dependent fracture, root-soil constraints, vibration, and wear still hinder low-energy cutting, long service life, and robust control. This review integrates harvesting and mechanical weeding within a unified analysis of reciprocating cutterbar technologies. It first links plant tissue structure and dynamic fracture to blade penetration, fiber stretching, crack propagation, and energy dissipation. It then examines how cutting speed, sliding-cut angle, blade clearance, and root-soil anchorage jointly affect performance. Advanced testing, response surface methodology, discrete element method, finite element method, and multiphysics simulations are compared for failure analysis, parameter optimization, and contact modeling. The review further assesses bionic blade design, surface strengthening, composite coatings, novel transmissions, multisource perception, and adaptive control. Key barriers include inconsistent plant-mechanics datasets, computationally intensive models, limited field robustness, and conflicts among performance objectives. We therefore identify digital twins, modular electric cutterbars, and closed-loop control as priorities for translating mechanistic insight into reliable field performance.

Symmetry doi: 10.3390/sym18081307

Authors: Adel Salim Tayyah Sarem H. Hadi Alina Alb Lupaş

This paper presents a new approach to the third-order Hankel determinant problem for the family of starlike functions associated with a symmetric H-domain. Motivated by a recently proposed sharp bound, we develop an analytical method that substantially improves the best known estimate. The obtained upper bound is reduced from 0.104485 to 0.062581, which is remarkably close to the conjectured sharp constant 1/16=0.0625. This significant improvement provides strong evidence supporting the validity of the conjectured bound while introducing a systematic technique for deriving refined coefficient estimates. Furthermore, the proposed approach is sufficiently flexible to be extended to related extremal problems involving Hankel determinants and other subclasses of analytic functions in geometric function theory.

Symmetry doi: 10.3390/sym18081306

Authors: Jian Meng Dacheng Lu Jiawen Liu Han Zhou Jun Fu

During underground mining, stope stability is affected by excavation-induced stress redistribution and nonlinear degradation of rock mass strength. The conventional Mathews stability graph method employs an empirical stress factor A, which does not explicitly consider the nonlinear relationship between stress conditions and rock mass strength. In this study, the generalized Hoek–Brown criterion was introduced to define the maximum stress factor (MSF), and a modified stress factor A′ was developed by considering tensile and shear failure mechanisms. The proposed method incorporates the nonlinear stress–stability relationship of underground stopes and improves the reliability of stability assessment. A copper mine in southwest China was selected as a case study. The rock mass quality indices and stress parameters of ten representative stopes were obtained through field investigations, stope roof stress measurements, discontinuity surveys, and laboratory tests. The modified stress factor A′ was incorporated into the Mathews stability graph to account for excavation-induced stress redistribution and rock mass strength degradation. Compared with the conventional method, the modified approach generally reduced the stability numbers of the investigated stopes, with an average reduction of approximately 25.6% (excluding D1780-1, where the confinement strengthening effect resulted in a slight increase in stability number). The revised stability classifications show good consistency with the FLAC3D simulation results and field observations, providing supporting evidence for the application of the proposed method to underground stope stability assessment in the studied mine.

Symmetry doi: 10.3390/sym18081305

Authors: Xiaofeng Yang Penghui Xue Siyao Liu Zhaoyang Wang Dadong Li Panpan Guo Yixian Wang

Pressure-type rock anchors transfer tendon force to the grouted body through an end bearing plate, placing the grout predominantly in compression; however, their load-transfer mechanisms and axisymmetric stress-field distributions under varying loads remain insufficiently understood. This study combines laboratory physical model testing and three-dimensional numerical simulations to investigate the vertical pull-out performance and load-transfer characteristics of pressure-type rock anchors. Scaled pull-out tests were first conducted to investigate the effects of anchorage length and anchor rebar diameter on load–displacement behavior and ultimate bearing capacity. A finite element model was then established and validated against the experimental results. The results indicate that the axial force of the grout exhibits a highly non-linear distribution, reaching its peak at the borehole bottom. An effective load-transfer length of approximately 3.5 to 3.7 m is identified, within which 90% of the axial force is dissipated. Unlike tension-type anchors, the anchor rebar functions as an integral tension-transmission member, maintaining a near-constant axial force profile outside the bottom boundary zone. Furthermore, the surrounding rock exhibits distinct axisymmetric responses, characterized by funnel-shaped vertical displacement diffusion and bulb-shaped principal stress contours. These findings provide a theoretical framework for optimizing anchorage length and understanding the symmetric mechanical boundary responses of pressure-type anchors in rock mass stabilization.

Symmetry doi: 10.3390/sym18081304

Authors: Xiaofei Du Yifen Liu Weilong Li Rui Wu Qidi Fu

The acoustic black hole (ABH) effect, achieved through a power-law thickness profile, has emerged as a powerful technique for passive vibration and noise control in thin-walled structures by slowing and trapping bending waves. However, prevailing research and designs predominantly focus on concave ABH indentations, which inherently require material removal and can consequently compromise structural strength and load-bearing capacity. To overcome this limitation, this paper introduces a novel symmetrical ring-shaped convex acoustic black hole (CX-ABH) plate. This innovative configuration transitions from the conventional concave geometry to a convex profile, aiming to preserve or even enhance vibration reduction performance while simultaneously improving structural integrity. A finite element model of the proposed ring-shaped CX-ABH plate is established and its calculation accuracy has been numerically validated through mesh independence testing. Vibration response analyses demonstrate its superior performance against a baseline rectangular plate at approximately 1270 Hz, achieving a maximum vibration reduction of 28.26 dB; the kinetic energy density is decreased by up to 2.82 J/m3 and effective energy concentration is achieved within the CX-ABH region. A parametric study was conducted to investigate the influence of key geometric parameters: the ABH radius R, the power exponent m, and the central frustum radius d. To achieve an optimal design, a Kriging surrogate model is constructed based on simulation data and subsequently coupled with a multi-objective genetic algorithm (MOGA) for systematic optimization. The derived optimal parameter set (d = 13.5 mm, m = 2.4, R = 58.9 mm) yields a 16.53% reduction in the mean peak kinetic energy density, validating the effectiveness of the optimization framework. The results conclusively demonstrate that the ring-shaped CX-ABH plate offers a promising and novel structural paradigm, successfully balancing high-efficiency broadband vibration attenuation with a robust structural design.

Symmetry doi: 10.3390/sym18081303

Authors: Nikolai S. Akintsov Artem P. Nevecheria Gaoteng Yuan Vladislav S. Igumnov Stepan N. Andreev Qing-Hua Qin

Standard pushers for the relativistic equations of motion of a charged particle in an electromagnetic field—Boris, Vay, Higuera–Cary—do not, in general, preserve the full symplectic structure of the underlying Hamiltonian system, while high-order non-symplectic schemes such as Runge–Kutta accumulate secular error over long times. We propose a two-stage, symmetry-preserving framework (SP-PINN) for the 3+1-dimensional relativistic dynamics of a charged particle in a prescribed field, including a focused Gaussian laser pulse, that pairs a physics-informed neural network with an explicit symplectic integrator: the network learns a surrogate relativistic Hamiltonian, while the integrator—which is not itself learned—advances it. In Stage 1, an unsupervised physics-informed neural network learns the surrogate from the covariant equations of motion using a Lorentz-invariant loss that enforces the mass-shell constraint H=mc2γ; in Stage 2, the surrogate is advanced with an explicit symplectic map built on Tao’s extended phase space, valid for the non-separable relativistic Hamiltonian. To isolate the geometric integrator from neural-network approximation error, every benchmark figure advances the analytic relativistic Hamiltonian through Stage 2, the learned Stage-1 surrogate being assessed separately. We benchmark against the Boris pusher and Runge–Kutta on three core test problems (adding the Higuera–Cary pusher in the symplecticity diagnostic), supplemented by plane-wave, ensemble, and pulse-family studies, and we measure the first Poincaré–Cartan loop invariant directly as a quantitative diagnostic of symplecticity. The magnetic-field test illustrates the contrast between bounded and secular error growth: Runge–Kutta drifts secularly, the Boris pusher conserves the invariants to machine precision as a volume-preserving gyro-integrator, and the symplectic map keeps the error bounded for all time; on a non-integrable magnetic trap, where no exact volume-preserving rotation exists, the symplectic map alone keeps the energy error bounded. The learned surrogate is the current accuracy bottleneck—not yet competitive with the conventional pushers for the static cases—but for the demanding laser case, a vector-potential light-cone reformulation reduces this surrogate error to (3.0±0.1)×10−4 (three seeds) and yields learned trajectories that remain phase-coherent over essentially the whole interaction. The framework targets laser–plasma acceleration, synchrotron-radiation modeling, and particle tracking.

Symmetry doi: 10.3390/sym18081302

Authors: Francesco Giovanni Celiberto

We present a comprehensive high-energy overview of all-heavy tetraquark production at the High-Luminosity Large Hadron Collider, combining recent advances in heavy-exotic spectroscopy with state-of-the-art fragmentation-based phenomenology. Our study builds upon the newly released TQ4Q2.0 fragmentation framework, which provides a complete leading-power description of fully heavy S-wave tetraquarks with scalar (0++), axial-vector (1+−), and tensor (2++) quantum numbers. The formalism incorporates all active gluon- and heavy-quark-induced production channels within nonrelativistic QCD factorization and implements threshold-aware DGLAP evolution through the HF-NRevo scheme. Particular attention is devoted to the all-bottom sector, whose larger mass scale offers a distinctive laboratory for exploring multiquark formation mechanisms and testing the universality of fragmentation dynamics across heavy-flavor systems. We review the theoretical foundations of the TQ4Q program and discuss the uncertainty budget associated with color-composite long-distance matrix elements and perturbative multiscale variations. Building on this framework, we present precision predictions for bottom tetraquark production in association with jets at HL-LHC energies, obtained within the (sym)JETHAD environment at NLL/NLO+ accuracy. The resulting phenomenology highlights the discovery potential of future high-luminosity measurements and illustrates the impact of modern resummation techniques on rare-hadron observables. This work establishes the TQ4Q2.0 framework as a reliable phenomenological benchmark for collider studies of all-heavy tetraquarks and provides a unified roadmap for future investigations of multiquark dynamics at present and forthcoming hadron facilities.

Symmetry doi: 10.3390/sym18081301

Authors: Dongze Song Zhe Sun

This study addresses the safety and efficiency challenges of manual inspection in complex open-pit mining environments, where terrain steepness, limited coverage, and personnel exposure to hazards render conventional methods inadequate. We propose an integrated UAV inspection framework that combines 3D environmental modeling with a hierarchical optimization paradigm. The framework operates in three sequential stages. First, a high-fidelity 3D terrain model is constructed from point cloud data via skeletal feature extraction, which reduces computational complexity while preserving topographic structure. Second, an upper-layer Traveling Salesman Problem (TSP) solver determines the optimal inspection sequence across mandatory points (loading sites, dump sites, and crushing stations). Third, a lower-layer Chaotic Adaptive Population-based Grey Wolf Optimizer (CAP-GWO) refines the 3D path between consecutive TSP-ordered points, augmented by B-spline smoothing to ensure kinematic feasibility. Key inputs include: (i) raw LiDAR point cloud data of the mining site, (ii) facility coordinates and operational constraints (safety margins, maximum pitch angle, minimum turn radius), and (iii) UAV kinematic parameters. Outputs comprise a smooth, collision-free 3D trajectory with verified constraint satisfaction. Comparative experiments against eight metaheuristic algorithms (PSO, GA, ACO, BA, COA, GWO, SRA, SFOA) demonstrate that the proposed method reduces total path length by 15–20% on synthetic benchmark scenarios while maintaining zero constraint violations. Statistical validation via the Sign Test confirms the significance of these improvements (p < 0.05) across repeated independent trials. The framework is further validated on measured airborne LiDAR data of the Bingham Canyon open-pit copper mine (Utah, USA; USGS 3D Elevation Program), one of the largest operating open-pit mines in the world: on this real terrain, CAP-GWO achieves the best performance among the GWO-family algorithms, with a statistically significant 12.5% improvement over SRA (Wilcoxon p < 0.001) and 24% lower variance than the standard GWO, and all 210 experimental runs produce collision-free trajectories. Notably, the proposed hierarchical optimization framework achieves structural symmetry between the upper-layer sequencing task and the lower-layer path refinement task. This symmetric decomposition significantly reduces computational complexity while preserving solution quality, aligning with the principles of symmetry in engineering optimization. The framework offers a practical solution for autonomous, adaptive inspection scheduling in dynamic mining environments.

Symmetry doi: 10.3390/sym18081300

Authors: Haiwei Lv

This article uses the enhanced direct algebraic method (EDAM) to study the exact traveling wave solutions of the paraxial wave equation with conformable fractional derivative. This method is selected due to its algorithmic simplicity, computational efficiency, and unique capability to yield diverse solution types within a unified algebraic framework. Firstly, the paraxial wave equation with conformable fractional derivative is transformed into an ordinary differential equation through traveling wave transformation. Then, the EDAM is systematically applied to construct accurate traveling wave solutions including twisted solitons, bell-shaped solitons, singular solitons, Weierstrass elliptic function solutions, and Jacobi elliptic function solutions, demonstrating the method’s superiority in handling complex nonlinear structures compared to standard expansion techniques. Finally, based on Matlab software to draw three-dimensional and two-dimensional graphs of partial solutions, these graphs intuitively demonstrate the regulatory effect of the fractional-order parameter α on waveform localization and amplitude. Specifically, as α increases, the soliton localization weakens and the wave packet broadens, providing insights into the dispersion management in optical fibers. The method used in this article is characterized by simple operation and rich solution types, providing an effective approach for studying fractional nonlinear partial-differential equations.

Symmetry doi: 10.3390/sym18081299

Authors: Moaz Tobaigy M. G. Finco

Background: Lower limb prosthesis users commonly exhibit gait asymmetry that may increase metabolic demand. Gait asymmetry, therefore, is frequently targeted in rehabilitation to improve walking efficiency. This review aimed to investigate whether interventions that improve gait asymmetry reduce metabolic demand in unilateral lower limb prosthesis users. Methods: A search of relevant English-language articles was conducted using PubMed and CINAHL, yielding a total of 1067 records. Following title, abstract, and full-text screening, 10 studies met eligibility requirements and were included in the qualitative synthesis. Data extraction focused on participant characteristics, gait asymmetry measures, metabolic outcomes, and intervention characteristics. Results: Four main intervention categories were identified: advanced prosthetic knees and ankles, prosthetic mass manipulation, feedback-based gait interventions, and exercise-based training. Across studies, changes in gait symmetry were not consistently associated with reductions in metabolic demand. Interventions targeting ankle function showed the most consistent reductions in metabolic demand, particularly during demanding walking tasks. In contrast, distal prosthetic mass addition consistently increased metabolic demand and worsened gait symmetry. Conclusion: Current evidence does not support a consistent or direct relationship between gait symmetry and metabolic demand in lower limb prosthesis users. Gait asymmetry may not be inherently metabolically disadvantageous and may represent an energetically optimal adaptation.

Symmetry doi: 10.3390/sym18081298

Authors: Zhenjie Gao Shanhui Liu Gan Shi Yafeng Sun Xinrui Ge Yifan Wang

To address the collaborative optimization problem caused by the strong coupling between 2D sheet metal nesting and flexible shop floor scheduling in the sheet metal manufacturing process for color-separation ovens of satellite-type flexographic printing presses, this paper proposes a collaborative optimization method for nesting and scheduling based on a dual-flow graph attention network and proximal policy optimization. This problem involves inherent structural and resource symmetry in manufacturing operations and is further complicated by process precedence constraints, multi-resource competition, human–machine collaboration, assembly dependencies, and the dynamic coupling between nesting decisions and downstream production takt. Based on the composition of oven components and actual production methods, a collaborative optimization model integrating nesting and scheduling was constructed; a discrete-event simulation-driven joint scheduling environment was established to uniformly model the nesting, cutting, flexible machining, and assembly processes. On this basis, the collaborative nesting–scheduling process was formalized as a Markov decision process, clearly defining the state space, action space, reward function, and state transition mechanism. To enhance the state representation capabilities of the reinforcement learning agent in a high-dimensional discrete action space and under strongly constrained dynamic scheduling scenarios, this paper embeds a dual-stream graph attention network into the PPO framework to develop the DS-GAT-PPO algorithm. This algorithm simultaneously captures spatial nesting relationships among parts as well as temporal dynamic features such as equipment load, worker fatigue, and production takt time, thereby generating feasible and efficient joint scheduling plans. Experimental results show that the proposed method can reduce completion time by approximately 10.9–33.8% while maintaining a high sheet utilization rate; in a comparison of reinforcement learning algorithms, the proposed method achieved better completion times in most test cases and reduced the number of convergence steps by approximately 6.67–66.67%, validating its effectiveness in improving solution quality, convergence efficiency, and scheduling stability. These findings provide a foundation for further research and practical applications of collaborative nesting–scheduling optimization in dynamic manufacturing environments.

Symmetry doi: 10.3390/sym18081297

Authors: Zhonggang Xiong Deqing Liu Mengyi Li Shuai Kang Nan Pang Jingyu Shang Hongyun Wang Youbing Li Weiqing Wang

With the rapid development of modern industry, robotic manipulators are required to achieve increasingly high trajectory-tracking accuracy and robustness in practical applications. To enhance tracking performance under complex operating conditions, this paper proposes an Adaptive Model Predictive Control with Sliding-Mode Robust Compensation (AMPC–SMC) scheme that integrates an adaptive mechanism with sliding-mode control theory. First, a dynamic model of the manipulator is established, and parameter linearization is employed to transform the nonlinear dynamics into a linearly parameterized form with unknown parameters. Second, an adaptive law is derived based on Lyapunov stability theory to update the model parameters online, thereby mitigating the adverse effects of parametric perturbations and external disturbances on tracking accuracy. Building on this, a receding-horizon optimization strategy is introduced by formulating a quadratic cost function that penalizes both tracking errors and control effort, and the optimal control input is obtained by solving the resulting optimization problem. Meanwhile, a sliding-mode term is incorporated as a robust compensator to eliminate residual tracking errors. Finally, the desired trajectory is generated via point-to-point path planning in Cartesian space, and the proposed method is validated on a real six-degree-of-freedom robotic manipulator. Comparative experiments against conventional model predictive control (MPC) and traditional sliding-mode control (SMC) demonstrate that the proposed AMPC-SMC controller achieves remarkably superior tracking performance compared with the conventional MPC and SMC controllers. In terms of tracking accuracy, the mean absolute errors (MAE) of Joint 2, Joint 4 and Joint 5 under AMPC-SMC are reduced by 91.7%, 92.1% and 86.1% respectively relative to MPC, and decreased by 76.1%, 77.3% and 85.1% compared with the standalone SMC controller.

Symmetry doi: 10.3390/sym18081296

Authors: Muhammad Awais Raza Muhammad Khalid Mahmood Ahmad Almutlg

In the original publication [...]

Symmetry doi: 10.3390/sym18081295

Authors: Grigoris Panotopoulos

We model the compact object Cen X-3, which is considered to be a good strange quark star candidate of known mass and radius, incorporating a negative anisotropic factor, and we compute the frequencies of the ten lowest radial oscillation modes. We introduce the anisotropy in three different ways, and we investigate its impact on the spectra.

Symmetry doi: 10.3390/sym18081294

Authors: Edita Mažonienė Dmitrij Šešok

The global prevalence of lung diseases has risen markedly over the past three decades, driven by factors such as population growth, harmful environmental exposures, and increased life expectancy. Epidemiological studies have highlighted not only the growing burden of chronic respiratory disorders but also the trend toward earlier onset across age groups. Early detection and timely treatment of these conditions are essential to improving long-term clinical outcomes and reducing premature mortality. Chest radiography (X-ray) remains the most widely used and cost-effective initial diagnostic tool for respiratory diseases. However, the reliable identification of multiple thoracic abnormalities, such as cardiomegaly, emphysema, hernia, infiltration, mass, nodules, atelectasis, pneumothorax, pleural thickening, pneumonia, fibrosis, consolidation, effusion and edema, poses challenges even for experienced radiologists. Recent advances in deep learning, particularly convolutional neural networks, have improved medical image analysis and classification. In this study, we propose DualPool-DenseNet121, a modified DenseNet121 architecture combining global average and global max pooling, for multi-label classification of thoracic abnormalities on the NIH ChestX-ray14 dataset. Following removal of “No Finding” cases, our analysis used 51,759 pathological radiographs from 14,402 patients, annotated for fourteen pathological conditions, partitioned patient-wise into training, validation, and test sets with verified zero patient overlap. The pipeline incorporates standardized preprocessing, offline augmentation of under-represented classes, staged transfer learning, and per-class decision thresholds selected on the validation set. Model performance was evaluated using per-class AUC together with threshold-based clinical metrics. The proposed model achieved a macro-averaged AUC of 0.803 and a micro-averaged AUC of 0.831 on the held-out test set. Zero-shot external validation on the CheXpert validation set reached a macro-averaged AUC of 0.806 across the five overlapping pathologies with sufficient positive cases. Grad-CAM visualizations indicated that, in correctly classified cases, the model frequently attended to anatomically relevant regions. These results suggest that the proposed approach provides a reliable and reproducible retrospective benchmark for multi-label thoracic abnormality classification. External validation and expert radiological assessment are required before any clinical application can be considered.

Symmetry doi: 10.3390/sym18081293

Authors: Rujie Cao Wenkai Du Guangzhong Gao Lu Yu Hua Bai Jianming Hao Guojun Yang Jiawu Li

Vortex-induced vibration (VIV) poses a significant serviceability concern for wide streamlined box girders of long-span suspension bridges. This study investigates the VIV performance and aerodynamic mitigation of a wide streamlined box girder with a width-to-depth ratio (B/D) of approximately 10 through sectional model wind tunnel testing. The original cross-section was found to exhibit pronounced heaving and torsional VIV at positive wind angles of attack, with amplitudes considerably exceeding the prescribed serviceability limits. A systematic experimental investigation was conducted to evaluate the influence of three geometric parameters, i.e., wind fairing inclination angle, inspection rail position, and pedestrian railing porosity and panel arrangement, on VIV performance. Experimental results demonstrate that reducing the wind fairing inclination angle from 65° to 45° is the most effective mitigation measure. An appropriate porosity of the pedestrian railing is shown to substantially improve VIV performance. Furthermore, under equivalent overall porosity, a uniformly distributed alternation of solid and ventilated panels yields markedly superior VIV suppression compared with continuously sealed arrangements. Subsequent flutter and aerostatic wind tunnel tests confirm that the recommended cross-section preserves the favorable flutter stability and aerostatic performance of the original design. Strouhal number analysis reveals that the VIV lock-in is governed by St ≈ 0.12. Notably, the St number obtained from the pitching moment coefficient is nearly twice that obtained from the lift coefficient.

Symmetry doi: 10.3390/sym18081292

Authors: Saleh Abdulrahman Alkhamis Abdalilah Alhalangy Galal Eldin Abbas Eltayeb Eman Abouelkheir

Unmanned aerial vehicles (UAVs) have become essential cyber-physical platforms for applications such as surveillance, infrastructure inspection, emergency response, and intelligent transportation. However, their tight coupling among sensing, communication, control, actuation, and swarm coordination also exposes them to sophisticated cyber-physical attacks that are difficult to detect using conventional intrusion detection systems. Existing machine learning, deep learning, graph-based, and federated intrusion detection approaches generally rely on statistical feature representations or temporal patterns, providing limited capability to model causal dependencies among interacting UAV subsystems and to generalize across heterogeneous operating environments. To address these limitations, this paper proposes Fed-CGIDS-UAV, a federated causal graph learning framework for cross-domain intrusion detection in cyber-physical UAV networks. The proposed framework models each telemetry window as a typed causal graph in which nodes represent navigation, sensing, communication, control, actuation, and swarm states, while directed edges capture stable operational dependencies. Intrusions are detected by identifying violations of these learned causal relationships, and the framework provides interpretable node-edge explanations to support root-cause analysis. Furthermore, federated learning enables collaborative model training across distributed UAV clients without sharing raw telemetry, thereby preserving data privacy while improving robustness under heterogeneous operating conditions. The proposed framework was implemented and experimentally evaluated in a controlled simulation environment covering four UAV operating domains and six representative attack classes. All experiments were repeated over five independent runs using different random seeds, and the reported results correspond to the measured average performance. The proposed framework was implemented using Python 3.12 (Python Software Foundation, Wilmington, DE, USA) and PyTorch 2.3 (Meta Platforms, Menlo Park, CA, USA). UAV flight data were generated using Microsoft AirSim 1.9.1 (Microsoft Corporation, Redmond, WA, USA), integrated with PX4 Autopilot v1.14 (Dronecode Foundation, San Francisco, CA, USA) and Gazebo Sim 11 (Open Source Robotics Foundation, Mountain View, CA, USA). Within this simulation-based evaluation, Fed-CGIDS-UAV achieved an accuracy of 0.968, an F1-score of 0.956, and an internal–external stability gap (IESG) of 0.028, outperforming conventional machine learning, deep learning, graph-based, and centralized causal baselines while maintaining competitive computational latency. Although these results demonstrate the effectiveness of the proposed framework under controlled simulation conditions, validation using real-flight UAV telemetry remains an important direction for future research. These results demonstrate that integrating causal graph learning with federated optimization provides an effective and interpretable solution for privacy-preserving intrusion detection in heterogeneous cyber-physical UAV environments.

Symmetry doi: 10.3390/sym18081291

Authors: Nikolaos Verykios Christos Gogos

For a semiprime N=sp, the normalized CRT–Bézout map assigns each unit modulo N a unique pair (m,n) in the rectangle {1,…,p−1}×{1,…,s−1}. The coordinates mn, Δ=ms+np, and δ=ms−np satisfy Δ2−δ2=4Nmn and give the geometric diagrams studied here. We describe the bounded affine slices Am+Bn=λ, including their point counts, complement symmetry, vertices, and spacings. We also write modular squaring in these coordinates: the two components evolve independently, while the nonempty fibers are orbits of a four-element Klein group. The complement symmetry of the slices is compatible with these dynamics. Finally, we determine the two admissible points with |δ|=1 and prove an exact count for the points in a central strip |δ|<T. These results describe the geometry attached to a known factorization; they do not give a new factorization algorithm.

Symmetry doi: 10.3390/sym18081290

Authors: Zuhur Alqahtani Ahmed Hagag

This work presents an efficient analytical framework based on the Natural Residual Power Series Method (NRPSM) for solving several forms of the time-fractional Kuramoto–Sivashinsky equation with the Caputo derivative. The proposed method avoids discretization and linearization while producing rapidly convergent analytical series solutions. Earlier residual power series treatments assert convergence under a contractivity assumption without verifying it for the equation at hand. We close this gap by deriving an explicit formula for the contraction constant directly from the problem data, so the convergence criterion is checkable before any computation begins. A rigorous theoretical analysis is established through explicit contraction conditions, convergence proofs in the Sobolev space H4(R), and an explicit geometric-type error estimate that quantifies how the fractional order governs the convergence rate through two competing effects, without presuming a uniform direction of influence. Stability with respect to perturbations in the initial data is also proven using a fractional Gronwall inequality. Numerical results demonstrate excellent agreement with exact and previously published solutions, achieving very small absolute errors using only a few series terms. The obtained results confirm that the NRPSM is an accurate, stable, and computationally efficient approach for nonlinear fractional evolution equations.

Symmetry doi: 10.3390/sym18081289

Authors: Kürşad Nuri Baydili Mehmet Gürcan

Binary logistic regression relies on the symmetric logit link, whose fixed inflection point cannot adapt to asymmetric response mechanisms and so may yield miscalibrated probabilities when the true response curve is asymmetric—a problem that can be particularly consequential in rare event or imbalanced applications. This study proposes a flexible, asymmetric link derived from the von Bertalanffy growth equation and tests a single, calibration-oriented hypothesis: estimating an asymmetric link improves the calibration of the predicted probabilities under asymmetric response mechanisms while leaving discrimination unchanged. The single shape parameter ν (the reparameterized von Bertalanffy shape, ν = m − 1; the growth curve baseline parameter is absorbed into the intercept and is not separately identifiable) is estimated jointly with the regression coefficients by L2-penalized maximum likelihood, giving a coherent generalized linear model rather than a post hoc re-thresholding of logistic scores; the logistic link is nested exactly at ν = 1 (m = 2), so the model reduces to ordinary logistic regression whenever the logit is adequate. The link was benchmarked on identical out-of-sample partitions against four competitors—logistic regression at the 0.5 cut-off, threshold-optimized logistic regression, ridge-penalized logistic regression at the same penalty (which separates the link from the regularization it requires), and Stukel’s generalized logistic model—across a correctly specified logistic mechanism and two asymmetric mechanisms (a complementary log–log mechanism and a von Bertalanffy-type best-case reference), three class imbalance ratios (0.10, 0.25, and 0.50) and five sample sizes (100–10,000), for 45,000 iterations in total. Paired differences were summarized by the Hodges–Lehmann median difference, its 95% confidence interval, and the rank-biserial correlation, with Benjamini–Hochberg control of multiplicity. The results supported the hypothesis. At moderate-to-large sample sizes, and increasingly as prevalence approached 0.50, the proposed link achieved lower Brier score and Log–Loss values than the three logistic competitors under both asymmetric mechanisms—for example, a median Log–Loss reduction against the matched-penalty ridge model of about 0.004 at balanced prevalence and the largest sample (95% confidence interval excluding zero; rank-biserial ≈ −1)—an advantage small in absolute size but, at the larger sample sizes, consistent across essentially every replication, and increasing with sample size and prevalence. Discrimination showed no practically important differences at moderate-to-large sample sizes: the area under the ROC curve and overall accuracy were essentially the same across the five methods, so the flexible link improves the quality of the probabilities without degrading class separation. Under the correctly specified logistic mechanism, the method showed no material deterioration, which is consistent with its exact nesting of the logistic model. The contribution is a growth curve-motivated asymmetric link that improves probability calibration under response asymmetry while preserving discrimination and recovering the logistic model when it is adequate.

Symmetry doi: 10.3390/sym18081288

Authors: Bei Zhang Qiang Liu Mingyu Li Shoudong Xie Weiming Guan Xin Wang Haosen Wang

Weak interlayers in layered rock masses can strongly influence stress-wave propagation, crack development, block motion, and flyrock behavior during bench slope blasting. To clarify the controlling mechanisms, a quasi-two-dimensional single-hole bench-slope model containing a weak interlayer was established using a continuous-discontinuous element method. Three single-factor groups were designed to examine the effects of interlayer thickness, dip angle, and wave impedance on the dynamic response of the slope. The results show that the weak interlayer acts as a geometric and dynamic symmetry-breaking interface. Relative to the locally quasi-radial response around the centrally initiated charge, interface reflection, transmission, and local dissipation produce direction-dependent stress, velocity, crack, and displacement fields. Interlayer thickness controls the spatial extent of this asymmetric response, dip angle determines its preferred direction, and wave impedance governs its dynamic intensity through unequal energy partition across the interface. Within the respective single-factor analyses, relatively larger flyrock responses were observed for the cases with a 2.0 m interlayer thickness, a 70° dip angle, and a low-wave-impedance interlayer. These observations represent comparative trends within the present quasi-two-dimensional single-hole model and should not be regarded as direct predictions of field-scale flyrock risk or a verified combined unfavorable blasting condition. These findings indicate that weak-interlayer geometry and mechanical contrast play important roles in controlling asymmetric blasting responses. Within the assumptions of the present quasi-two-dimensional single-hole model, the results provide mechanistic insights into relative response trends rather than direct field-scale blasting predictions.

Symmetry doi: 10.3390/sym18081287

Authors: Alexander Shepetov Olga Kryakunova Rustam Koichubayev Nikolay Nikolayevskiy Serik Nurakynov Vladimir Ryabov Botakoz Seifullina Irina Tsepakina Ludmila Vildanova Valery Zhukov

Large episodes of solar activity of the 25th cycle, an extreme Forbush decrease event on 19 January 2026, a series of Forbush effects in November 2025, and an event of ground-level enhancement (GLE 77) on 11 November 2025 have left prominent traces in the monitoring data obtained at a height of 3340 m a.s.l. from the detector facilities of the Tien Shan High-Mountain Cosmic Ray Station. The effects these events have caused on the flux of galactic cosmic rays in the several-GeV energy range, as registered with the standard NM64-type neutron supermonitor, are compared here with their influence on the local neutron and gamma radiation background in the high-mountain environment, which was observed in the counting rate records of the thermal neutron and MeV-order-energy gamma radiation detectors also installed at the station.

Symmetry doi: 10.3390/sym18081286

Authors: B. F. L. Ward

We present the origin, development, current status and outlook for the YFS Monte Carlo approach to precision theory for high-energy collider physics. We frame our discussion so that the important contributions of Prof. Stanislaw Jadach arehighlighted.

Symmetry doi: 10.3390/sym18081285

Authors: Yilmaz Ar Nermin Kartli

The Permutation Flow Shop Planning Problem (PFSSP) is fundamental and frequently encountered in manufacturing systems and service operations. This problem is known to be NP-hard for three or more machines. Therefore, various heuristic and metaheuristic algorithms exist to approximate solutions to the problem. The solutions produced by deterministic heuristic algorithms are frequently used as initial solutions for population-based metaheuristic algorithms because they provide feasible schedules of relatively high quality within a short computational time. Heuristic algorithms are also divided into two groups: deterministic and random. In this study, we aim to develop a new deterministic method that improves both solution quality and computational efficiency. Rather than replacing existing deterministic heuristics, we propose a method that aims to enrich the design space of deterministic PFSSP heuristics by introducing two new problem-specific sequence improvement operators inspired by classical sorting principles. The proposed method is based on the integrated use of three complementary components: (i) a Bubble-Swap-based neighborhood structure that increases local search power, (ii) an NEH-style insertion mechanism that uses the strong insertion logic of the classical NEH algorithm, and (iii) a Merge-Global-Swap strategy that provides global optimization based on the completion time value of the entire sequence at each merge step. By integrating these three components, we develop a new deterministic hybrid algorithm called BNMG (Bubble–NEH–Merge–Global). We also call the locally search-enhanced version of our algorithm BNMG-II. Furthermore, we propose a new metric that accounts for computation time to evaluate the performance of the algorithms. When we comprehensively compare the BNMG and BNMG-II algorithms with the classical NEH, the recently developed vN-NEH and NEH-II, vN-NEH+ algorithms in Taillard test problems, we report that they exhibit superior performance according to the mean relative deviation metric (M1/ARPD) and the proposed new metric.

Symmetry doi: 10.3390/sym18081284

Authors: Runjie Shen Chunbiao Li Wangyu Liu Xiaowei Chen

Based on the polarity balance reconstruction, chaotic system with conditional symmetry is coined by introducing absolute value function and trigonometric functions with property of slope polarity inversion. Suitable functions returning the polarity balance based on offset boosting, conditionally symmetric chaotic attractors in the system, can be effectively positioned under different regimes of conditional symmetry. As a result, different amounts of coexisting conditionally symmetric attractors are reproduced even tending to infinity. For comparison, other external functions such as the hyperbolic tangent function can be employed to construct mandatory polarity balance, and thus coexisting attractors of artificial symmetry can be obtained. The evolution of the basin of attraction is analyzed for the observation of offset parameter-dominated multistability. A simplified circuit is designed for verifying the coexisting conditionally symmetric attractors.

Symmetry doi: 10.3390/sym18081283

Authors: Michel Planat

Let Φ be the Riemann theta kernel and D(z)=∫0∞Φ(u)cos(zu) du, so that Ξ(z)=ξ(1/2+iz)=4D(z). The Riemann hypothesis is equivalent to the global growth criterion ∂y|D(x+iy)|2>0 for all real x and all y>0; in the form ∂y|D(x+iy)|2=18ℜ(ξ′(s)ξ(s)¯) with s=1/2+y+ix, this criterion is an identity between entire quantities and remains meaningful at the zeros of ξ. Using the symmetric Hadamard product we close the half-plane σ>1 unconditionally, localising the obstruction to the strip 1/2<σ≤1. In diagonal coordinates a=(u+v)/2, b=(u−v)/2, we derive the exact two-variable representation of the growth derivative and the resulting longitudinal–transverse decomposition ∂y|D|2=2x∫0∞[Qy(a)sin(2xa)+εx(a;y)] da, in which the longitudinal envelope Qy is unconditionally positive. This representation makes it natural to seek positivity blockwise, over the phase-aligned intervals Jm=[mπ/x,(m+1)π/x] on which the oscillation completes a full period. We report that this localisation is obstructed. The companion paper proves that, for each fixed y>0, the globally summed longitudinal and transverse sectors cancel to all algebraic orders as x→∞ (each is of size x−5, their sum is exponentially small, and their ratio tends to −1), and deduces that for every y>0 and all sufficiently large x at least one aligned block is negative. Universal phase-aligned blockwise positivity is therefore impossible, and independent algebraic-order estimates of the two sectors, followed by addition, cannot determine the sign, which survives only in the exponentially small remainder left after the cancellation. Global positivity remains equivalent to the Riemann hypothesis and remains open; what the theta-kernel representation shows is that it cannot be reconstructed from independent local positivity. Positivity here is intrinsically nonlocal. Correlated groupings of blocks, exact resummation, and globally coupled identities are not excluded. No proof of the Riemann hypothesis is claimed.

Symmetry doi: 10.3390/sym18081282

Authors: Siqi Peng Hanshan Li

To address the issues of parameter dependency on empirical settings, insufficient fault feature extraction capability, and limited classification accuracy in rolling bearing fault diagnosis using Variational Mode Decomposition (VMD), a novel fault diagnosis method based on adaptive signal decomposition and intelligent classification integration is proposed. The VMD parameters are adaptively optimized using the Subtraction-Average-Based Optimizer (SABO), and a kurtosis–correlation criterion is introduced to select a single fault-sensitive intrinsic mode function, from which time-domain features are extracted to construct fault feature vectors. The Moth-Flame Optimization Algorithm (MFOA) is employed to optimize the parameters of the Kernel Extreme Learning Machine (KELM) for fault state identification. From the perspective of methodological symmetry, the averaged population update of SABO is invariant to the ordering of search agents, VMD exhibits equivalence under permutation of mode labels, and KELM constructs the sample similarity matrix using a symmetric kernel function. These symmetry-related structures are integrated into the parameter optimization, modal decomposition, and fault classification stages of the proposed method. Experimental validation using the CWRU rolling bearing dataset demonstrates that the proposed method reaches a fault recognition accuracy of 96.73%, outperforming other comparative models and exhibiting superior diagnostic precision and robustness.

Symmetry doi: 10.3390/sym18081281

Authors: Baxian Liu Rui Xie Lei Tao Kuilin Huang

With the continuous expansion of the application of bimetallic tubes in unconventional oil and gas fields due to their excellent corrosion resistance and mechanical properties, the threat of ground settlement to structural integrity is becoming increasingly prominent. The uneven settlement can cause significant additional stress and deformation, while defects in manufacturing, installation, and service can further threaten structural stability. Existing research mainly focuses on defect mechanisms or isolated pipe/soil interactions, with insufficient exploration of the coupling effects between initial defects and loads caused by settlement. This study systematically investigated the buckling propagation behavior of bimetallic pipes considering the coupling effects of geometric defects and soil settlement, using developed finite element models, with particular emphasis on parameter sensitivity analysis. The results indicate that defect location plays a critical role, with upper surface defects exacerbating buckling by forming initial inward indentations. Increased defect depth and area further amplify buckling severity. Internal pressure exerts a partial inhibitory effect, and reducing the diameter/thickness ratio of both the outer and liner pipes effectively lowers the probability of failure. Finally, based on numerical simulations, a BP neural network model and empirical formula are established to predict the ovality of liner pipes under coupled defect and settlement conditions.

Symmetry doi: 10.3390/sym18081280

Authors: Mohammed Brahimi Abderrahmane Belguerna Hamza Daoudi Zouaoui Chikr Elmezouar Fatimah Alshahrani

A robust metaheuristic framework based on a sequential hybrid of Artificial Rabbits Optimisation (ARO) and the Grey Wolf Optimiser (GWO) is proposed for the synthesis of linear (20 elements) and concentric circular (three rings, 4, 6, and 10 elements) antenna arrays. The optimisation strategy is applied to two distinct excitation scenarios—amplitude-only and complex amplitude–phase—to synthesise radiation patterns for both broadside and 30° tilted beams. The hybrid TSARO–GWO algorithm aims to minimise the sidelobe level (SLL) while ensuring compliance with design constraints on HPBW, directivity, and DRR. Experimental results validate the hybrid’s superior efficacy, yielding SLLs as low as −34.8 dB and −38.4 dB for the LAA, and −35.9 dB and −45 dB for the CCAA, respectively, while strictly satisfying half-power beamwidth (HPBW), high directivity, and dynamic range ratio (DRR) constraints. Comparative benchmarks against contemporary optimisers (CTPOA, MVO, FPA, DO, SPS-JADE, AHA, HKOA, SSA, and ALO-SQP) demonstrate that TSARO-GWO achieves SLL values that are competitive with or superior to these recent methods across all six synthesis cases, establishing it as a promising framework for complex array synthesis problems.

Symmetry doi: 10.3390/sym18081279

Authors: Xinyi Yan Jizuo Zhang Jianjun Chen Yaqing Chi Xiao Jiang Tao Chen

The fully depleted silicon on insulator (FDSOI) integrated circuit process has a buried oxygen layer in its structure, which provides many advantages to the FDSOI integrated circuit process, such as isolating the substrate from the conductive channel, reducing leakage current, and lowering the supply voltage. With the vigorous development of space artificial intelligence infrastructure in China and the United States, it is becoming increasingly important to study the single-event upset (SEU) of Static Random-Access Memory (SRAM) cells caused by particle radiation. In this paper, SRAM with different peripheral circuits is designed by domestically produced 22 nm FDSOI CMOS integrated circuit technology. Simulations are used to study the mechanism of single-event effects for SRAM, and single-particle radiation experiments of Kr are used to characterize SEU for SRAM. The results showed that the SRAM without Error Detection and Correction (EDAC) technology had 6486 cell upsets, with incidences of 2-cell upsets occurring 15 times, and no multi-cell flips occurred. For the SRAM using EDAC technology, there was no single-cell upset, 430 incidences of two-cell upsets and one incidence of multi-cell upsets. This provides strong support for fully utilizing EDAC technology to enhance the irradiation-hardening of digital integrated circuits. For metal oxide semiconductor field-effect transistors, the source and drain structures are completely symmetrical, and when in use, they are only connected at different potentials.

Symmetry doi: 10.3390/sym18081277

Authors: Ruoxi Wang Yonghui Qin

With the rapid development of Internet of Things (IoT) technology, the formation control of unmanned surface vehicles (USVs) has attracted increasing attention in marine applications. However, in practical marine missions, the implementation of affine formation maneuver control (AFMC) still faces challenges caused by limited communication resources and restricted interaction frequency of onboard communication devices. To address these issues, this paper proposes a hybrid event-driven interaction mechanism (HEIM) for AFMC of USVs. In this mechanism, a clock variable is first introduced to regulate the interaction process, such that excessively frequent information exchanges can be avoided and the interaction interval can satisfy the minimum communication interval (MCI) required by onboard hardware. However, when the clock variable reaches its lower bound, an interaction may be compulsorily triggered, which introduces a maximum interaction interval constraint. To remove this restriction, an additional threshold is further incorporated as a secondary interaction judgment condition. In this way, the proposed mechanism not only preserves an adjustable MCI but also avoids unnecessary interactions caused solely by the clock variable, thereby further improving communication efficiency. Theoretical analysis and simulation results demonstrate the effectiveness of the proposed method.

Symmetry doi: 10.3390/sym18081278

Authors: Ping Wen Feng Ao Xilin Liu Zhongbin Luo

Symmetry and asymmetry play an important role in image processing and computer vision, particularly when visual structures are corrupted by irregular and spatially heterogeneous degradation. In cultural heritage restoration, ancient murals often contain locally symmetric patterns, repeated decorative motifs, balanced compositions, and style-sensitive contours, while long-term aging introduces asymmetric damage such as cracks, pigment fading, flaking, and partial content loss. Restoring such images therefore requires models that can recover structural regularity from asymmetric degradation while preserving culturally meaningful visual details. In this paper, we propose a symmetry-aware progressive generative framework for non-invasive digital restoration of Dunhuang murals. The proposed model is implemented as a Cauchy–Schwarz-regularized cascading variational autoencoder, which decomposes restoration into three coarse-to-fine stages: global structural recovery, semantic and chromatic refinement, and fine-detail enhancement. To support this progressive process, the latent dimensionality is gradually expanded across stages, enabling the model to move from compact structural abstraction to detail-aware representation learning. Moreover, a Cauchy–Schwarz-divergence-based regularization strategy is introduced to align the aggregated posterior with a mixture-of-Gaussians prior, providing a tractable mechanism for modeling the multi-modal latent structure of mural images. Experiments on the MuralDH benchmark under irregular-mask, crack-like, and mixed degradation settings show that the proposed method achieves competitive restoration quality compared with representative inpainting and diffusion-based baselines, while requiring substantially lower inference cost. Qualitative results further demonstrate improved contour continuity, chromatic coherence, and texture preservation. These results suggest that symmetry-aware progressive generative modeling is a promising tool for sustainable, non-invasive cultural heritage restoration.

Symmetry doi: 10.3390/sym18081276

Authors: Jiayu Zheng Xiangsong Zhang

Hybrid conjugate gradient methods are considered as an efficient family of conjugate gradient (CG) methods used to solve unconstrained optimization problems. In this paper, on account of the outstanding performance of the PRP (Polak–Ribière–Polyak) conjugate gradient method and its exceptional numerical computational stability, we propose a hybrid conjugate gradient method for solving unconstrained optimization problems. By combining two PRP-type directions via convex combination, the proposed search direction dynamically adjusts to gradient change rates and satisfies the sufficient descent property. Under mild conditions, the global convergence of the proposed method is established. Numerical computations are presented to display the efficacy of the proposed algorithm compared to some existing algorithms. It is indicated that the proposed method is more effective in dealing with non-convex optimization problems. Finally, the applicability of the proposed method is shown in image restoration problems with noise, and preliminary experimental results demonstrate its effectiveness compared to some other methods.

Symmetry doi: 10.3390/sym18081275

Authors: Calogero Vetro

This Special Issue of the journal Symmetry is dedicated to recent mathematics investigations, showcasing a collection of twenty research works and four reviews in different fields of mathematics [...]

Symmetry doi: 10.3390/sym18081274

Authors: Mingyuan Hu Xiaodong Qu Jingwen Liu Yang Liu Lei Zhang Ping Wang

High-voltage porcelain column-type equipment systems comprise various equipment units whose porcelain columns generally have circular cross-sections and approximately axisymmetric geometries. Although this geometric symmetry results in nominally equivalent lateral mechanical properties at the component level, differences among equipment types and multiple uncertainty sources may lead to heterogeneous seismic responses at the system level. To investigate the combined effects of ground-motion randomness and uncertainties in the elastic modulus and diameter of porcelain sleeves and damage control indices, Latin hypercube sampling was employed to generate 100 structural models with different modeling parameters. These models were randomly paired one-to-one with 100 selected ground-motion records, and uncertainty in the damage control indices was incorporated into the dynamic response and fragility analyses. The results indicate that the variability in structural seismic responses arises from the combined effects of modeling-parameter uncertainty and ground-motion randomness. These effects are propagated to the fragility curves through changes in the median ground-motion intensity and total logarithmic standard deviation of the fragility functions. The fragility curves accounting for multiple uncertainty sources generally fluctuate around the baseline curves considering ground-motion randomness alone, without exhibiting a consistent upward or downward shift. No strictly monotonic relationship was observed between the coefficients of variation in the uncertain parameters and the total dispersion of the fragility results, although larger parameter variability produced greater deviations from the baseline in some cases. Nevertheless, the overall differences remained limited, with a maximum absolute difference of 0.043. From the perspective of symmetry, the results demonstrate that component-level geometric symmetry coexists with system-level response heterogeneity and cannot alone eliminate the effects of equipment-specific characteristics and stochastic uncertainties on seismic fragility. Within the scope of this study, fragility curves considering ground-motion randomness alone may therefore provide a reasonable approximation for engineering assessment.

Symmetry doi: 10.3390/sym18081273

Authors: Xinghai Hou Donglin Jing Fukun Bi Chenglong He Yong Huang Changjie Wang

Aerial tiny objects naturally possess conjugate symmetry in the frequency domain, yet complex scenarios, heavy clutter, and frequent rotation/occlusion lead to severe detail loss, inaccurate contour modeling, and high false/miss rates in existing detectors. Current Fourier-based methods neither leverage object symmetry nor coordinate with Fourier analysis to jointly enhance contour representation and spatial-frequency feature learning, suffering from weak fusion, phase-sensitive coefficient regression, and poor discriminability. To fill this gap, we propose the Frequency-Aware Fourier Series Detection Network (FAFSDet), which explicitly exploits the inherent symmetry of tiny objects and their frequency-domain representations. Specifically, FAFC (Frequency-Aware Feature Fusion) employs conjugate-symmetry-guided dynamic low-pass filtering, similarity-based rearrangement, and adaptive high-frequency enhancement to recover degraded symmetric patterns. FSPRM (Fourier Series Profile Representation) utilizes the symmetric positive–negative frequency distribution to achieve compact parametric contour encoding and normalized centroid-shape description. FSDIM (Fourier Series Detection Inference) incorporates symmetric multi-scale branches, a rolling-optimization loss that eliminates phase interference while preserving coefficient-regression symmetry, and inverse Fourier transform for precise contour reconstruction and end-to-end detection. Extensive experiments on DOTA, AI-TOD, and UCAS-AOD demonstrate that our method achieves superior performance (mAP 82.18%, 51.2%, and 90.70%, respectively) and strong generalization, particularly in scenarios where symmetry is most severely compromised, confirming that exploiting these symmetry properties substantially boosts detection accuracy.

Symmetry doi: 10.3390/sym18081272

Authors: Pu Yu Yanshan Ma Yuheng Li Chunhao Li

Robust object detection for autonomous driving requires perception models that remain reliable when visible imagery is degraded by darkness, glare, rain, fog, motion blur, or long-range small targets. Visible and thermal infrared cameras provide complementary evidence, yet many RGB–thermal detectors fuse modalities, mainly as aligned tensors, and may underuse relational structure in channel responses, spatial layouts, semantic scales, and modality-specific uncertainty. This paper presents TopoGraph-Fusion, a hierarchical graph-guided dual-modal object detector that formulates fusion as topology-aware reasoning rather than direct feature concatenation. The proposed framework builds a dual-stream backbone for RGB and thermal images, constructs channel-wise topology through a channel-topology graph aggregation module, derives relation-aware spatial and channel global attention from affinity graphs, and replaces fixed feature-pyramid communication with a Graph-Guided Feature-Pyramid Network. A topology-regularized detection objective further encourages stable cross-modal correspondence while suppressing noisy all-to-all connections. Experiments on M3FD, FLIR, RGBTDronePerson, and VEDAI512 cover road scenes, adverse illumination, drone–person perception, and aerial vehicle detection. Within this validation scope, the results and visual analyses indicate that topology-guided fusion improves small-object recall, cross-modal consistency, and robustness under modality imbalance.

Symmetry doi: 10.3390/sym18081271

Authors: Xianjian Jin Huaizhen Lv Jianning Lu Jianbo Lv Nonsly Valerienne Opinat Ikiela

In this paper, the concept of symmetry is applied to design active trajectory tracking control of autonomous distributed drive electric vehicles considering energy efficiency—that is, the construction and solution of active trajectory tracking controllers are symmetrical. This paper proposes a hierarchical control strategy consisting of upper-level control and lower-level control to improve trajectory tracking accuracy of DDEVs considering energy-efficiency characteristics. In the upper-layer control, a sliding mode active disturbance rejection (ADRC) controller is developed to control the front wheel steering angle and active yaw moment to achieve tracking of the desired trajectory, in which an extended state observer (ESO) is synthesized to estimate and compensate for internal model uncertainties and external environmental disturbances. In the lower-layer control, a multi-objective optimization algorithm based on Karush–Kuhn–Tucker (KKT) conditions is designed to realize the torque distribution control for improving energy efficiency and vehicle stability of the distributed drive electric vehicle. Finally, a joint simulation platform based on Matlab/Simulink-CarSim (version 2019) is established for simulation verification. The performances of ADRC, linear quadratic regulator controller (LQR), and model predictive controller (MPC) are compared in snake-like and double-lane-change maneuvers. Simulation results show that the proposed controller can effectively reduce motor energy consumption while maintaining trajectory tracking accuracy and handling stability. This work provides a certain engineering design solution for motion control of intelligent electric vehicles.

Symmetry doi: 10.3390/sym18081270

Authors: Hairong Wang Xinyu Zhang

Seismic trace interpolation reconstructs missing traces caused by incomplete spatial sampling while preserving reflection-event continuity, amplitude fidelity, and waveform character. Randomly missing traces introduce an inherent reliability asymmetry: observed positions contain measured amplitudes, whereas missing positions must be inferred from neighboring seismic events. In U-Net-based encoder–decoder architectures, shallow features extracted around zero-filled missing traces may carry sampling artifacts and unreliable high-frequency details through direct skip connections. To address this limitation, this paper proposes MGF-UNet, a mask-guided gated skip fusion network for 2D seismic interpolation with randomly missing traces. The incomplete seismic patch and binary trace mask are used as a dual-channel input, and mask-guided gates regulate encoder features before decoder fusion. Known-trace preservation is then applied to retain measured traces in the final output, while a hybrid objective supervises missing-trace recovery, full-patch fidelity, and structural coherence. In five-run experiments on Marmousi synthetic data, MGF-UNet achieves the lowest mean MissingRMSE at the 30% and 50% missing ratios and remains competitive in global RMSE, SSIM, and SNR; however, the seed-matched paired comparisons do not reach statistical significance. Field-data comparisons and local waveform analyses further show coherent event reconstruction and reduced residual artifacts. These results suggest that mask-guided skip regulation is a promising strategy for moderate-to-severe random missing-trace interpolation.

Symmetry doi: 10.3390/sym18081269

Authors: Ali Yahya Hummdi Jose Rani Selvakumar Krishnan Junaid Nisar

Let R be a commutative ring. The comaximal graph Γ(R) is defined as an undirected simple graph whose vertices correspond to the elements of R, where two distinct vertices x and y are adjacent if and only if Rx+Ry=R, with Rx denoting the ideal generated by x. For the case R=Zn, the ring of integers modulo n, the comaximal graph is represented as a generalized composition of appropriately chosen graphs. Building on this concept, we demonstrate that the comaximal graph Γ(R) of any commutative Artinian ring with unity can be expressed as an H-join of the complete and null graphs. Utilizing this structural representation, we derive an explicit formula for the Wiener index of the comaximal graph associated with a commutative Artinian ring. Finally, we determine the spectrum of the comaximal graph for such rings.

Symmetry doi: 10.3390/sym18081268

Authors: Waseem Ahmad Khan Oğuz Yağci Khidir Shaib Mohamed Muntasir Suhail Ahmed Himadan Habeeb Ibrahim Naglaa Mohammed

We introduce a bivariate Hahn-factorial q-Laguerre–Tricomi–Appell polynomial class obtained by multiplying a nonsingular Appell factor by a Hahn-factorial deformation of the two-variable q-Laguerre–Tricomi generating kernel. The construction is interpreted formally, coefficientwise, and is not presented as an orthogonality or Laguerre–Hahn functional characterization. Its defining product yields a finite q-binomial convolution, which is the principal mechanism for transferring algebraic and operational properties from the base family to the Appell deformation. We establish direct and inverse connection formulas, a Hessenberg determinant representation, recurrence and higher Hahn-difference formulas, Hahn-shift identities, operational transfer formulas, and quasi-monomiality relations through a basis-defined raising operator. We also derive reductions to the underlying Hahn-factorial q-Laguerre–Tricomi family, one-variable Hahn-Appell and q-Appell families, translated q-Laguerre specializations, admissible Bernoulli–Euler-type subclasses, a singular Genocchi-type convolution, and the joint classical limit q→1−, w→0. A finite coefficient scheme is then used to study representative zero distributions and graphical behavior. The results show that the proposed class is a coherent Appell-type deformation of a Hahn-factorial q-Laguerre–Tricomi kernel and that its structural identities follow from explicitly invertible q-binomial transforms whenever the Appell factor has a nonzero constant term.

Symmetry doi: 10.3390/sym18081267

Authors: Francesco Giovanni Celiberto

All-heavy pentaquarks provide a unique laboratory for exploring the interplay between exotic-hadron structure, heavy-quark dynamics, and perturbative Quantum Chromodynamics (QCD). In this review, we present the completed release of the uncertainty-aware collinear fragmentation-function family PQ5Q1.1, covering both all-charm (P5c) and all-bottom (P5b) states. The extension to the bottom sector follows the resolution of theoretical and numerical issues affecting heavy-bottom fragmentation and completes the public LHAPDF6 release of the PQ5Q1.1 family. The PQ5Q1.1 functions are constructed within a multimodal framework that accounts for both compact multiquark formation and diquark–antiquark–diquark production mechanisms. A replica-based strategy is adopted to quantify perturbative and nonperturbative uncertainties through missing-higher-order variations (F-MHOUs) and controlled modifications of the hadronic wave-function structure (F-NPWF). For phenomenological applications, we employ the data-validated (sym)JETHAD framework to investigate semi-inclusive pentaquark-plus-jet production at NLL/NLO+ accuracy at the High-Luminosity Large Hadron Collider. We present predictions for differential distributions, assessing the impact of fragmentation dynamics, uncertainty propagation, and hadron-structure effects across the charm and bottom sectors. By combining precision fragmentation tools with collider-oriented observables, this review establishes a unified framework connecting heavy-pentaquark spectroscopy, hadronization mechanisms, and high-energy QCD phenomenology.

Symmetry doi: 10.3390/sym18081266

Authors: Kamran Foroutan Farshid Torabi

In this paper, the nonlinear vibration (NV) response of carbon nanotube-reinforced composite (CNTRC) marine panels (MPs) fitted with symmetric multi-vibration absorbers (MVAs) subjected to steady, velocity-dependent hydrodynamic loads is investigated. To model actual marine conditions more realistically, the lift and drag forces varying with flow velocity were taken into account using experimentally supported Matveev-based formulations for a specific ship. Within the shell, three carbon nanotube (CNT) distribution schemes are considered: one uniformly distributed (UD) CNT configuration and two functionally graded (FG) CNT patterns, namely FG-V and FG-X. The analytical framework is further constructed using classical shell theory (CST) by incorporating geometric nonlinear terms, and the Galerkin technique is employed to obtain a reduced-order model. Thereafter, the NV response of the CNTRC-MPs is predicted through the P-T method, which relies on the joint application of the piecewise constant argument and Taylor series expansion. The results indicate that symmetric MVAs can effectively suppress NV behavior and significantly decrease the maximum NV amplitude of the panel. Moreover, the effectiveness of the proposed configuration is shown to depend on both the absorber characteristics and the reinforcement pattern of CNTs. The study demonstrates that the use of symmetric absorber systems offers a practical and efficient passive vibration-control solution for advanced marine composite panels.

Symmetry doi: 10.3390/sym18081264

Authors: Jieping Liu Shixuan Bu Jianghao Wang Xiaojun Chen

Magnetic particle imaging (MPI), as an emerging radiation-free, high-sensitivity molecular imaging technique, holds broad application prospects in fields such as medical diagnosis, angiography, and targeted drug tracking. However, traditional three-dimensional MPI reconstruction algorithms face a problem in balancing reconstruction speed and image resolution. A hybrid reconstruction algorithm (Full Hybrid) based on spatial density constraints and residual iterative optimization is proposed in this work. This paper simulates Lissajous trajectory scanning and the non-linear response of magnetic particles based on the three-dimensional MPI simulation framework. The proposed hybrid method first utilizes the X-space method to obtain a basic spatial prior, then introduces field-free point (FFP) trajectory density to impose spatial weighting constraints on the reconstructed image. Experimental results demonstrated that this hybrid algorithm performs better in the reconstruction of complex three-dimensional topological structures (an H-shaped phantom). Comprehensive evaluation demonstrated that the reconstructed outputs reach a peak signal-to-noise ratio (PSNR) of 12.85 dB, a structural similarity index measure (SSIM) of 0.7321, and a root mean square error (RMSE) of 0.2278. Ablation experiments and comparison experiments further reinforced the advantages of the proposed method. These results demonstrate the numerical feasibility of the proposed reconstruction method for a three-dimensional phantom and provide a basis for further evaluation under multiple simulation conditions and real-scanner measurements.

Symmetry doi: 10.3390/sym18081265

Authors: Roberto Ricci Giuseppe Dattoli

We apply the recently proposed analytic extension of formal indicial umbral calculus to the evaluation and structural interpretation of Bessel moments, replacing formal symbolic constructions with Mellin–Barnes analytic transmutations. The classical umbral representation of J0 converts products of Bessel functions into Gaussian integrals involving sums of independent symbolic operators. This formal mechanism may reproduce correct identities in suitable convergence chambers, but it may also lead to non-admissible hypergeometric expansions at the physically relevant parameter values. The cubic moment already exhibits this difficulty: the formal Appell F4 expansion associated with the equilateral case lies outside its domain of convergence. We address this obstruction by replacing the formal expansion with Mellin–Barnes representations of the corresponding umbral pairings. In this formulation, Ramanujan’s Master Theorem selects the analytic ground state associated with a Bessel product. The factorisation J03=J0J02 fuses the elementary Bessel state with the square state and gives the cubic moment as a one-dimensional Meijer–Barnes function. The same mechanism yields a scaled cubic formula and a fourth-moment Meijer–Barnes representation whose residues give a convergent harmonic-number expansion. The fifth moment marks the first higher-rank case: the natural grouping J05=J02J02J0 leads to a bivariate Barnes transmutation rather than to an ordinary Meijer G-function. Finally, real powers J0α, α>2, are interpreted through Mellin-selected ground states, which need not reduce to finite Gamma products. Thus, Bessel moments provide a concrete hierarchy of analytic umbral representations, from rank-one Meijer–Barnes functions to higher-rank Barnes structures, and distinguish the global analytic meaning of an umbral construction from the local convergence of its residue expansions.

Symmetry doi: 10.3390/sym18081263

Authors: Junwei Wang Wei Dai Konglin Zhu Bo Qu

Exchange-listed medical data products create information and pricing asymmetry because sellers, buyers, and governance reviewers do not observe the same product boundaries, scenario permissions, processing depth, or compliance costs. This study proposes SM-DPF, an auditable pricing reference framework that makes these asymmetric conditions explicit through a reproducible three-layer chain: a public investment anchor, a locked structural scoring model, and a proposed future market learning governance interface that is not implemented or evaluated in the present study. Using 23 de-identified transaction-descriptive records from a single exchange context across insurance claims, model pretraining, and pharmaceutical R&D, the pricing reference outputs y0,i show in-sample diagnostic consistency with de-identified transaction prices yi, with an overall mean absolute percentage error of 4.82% and median absolute percentage error of 4.55%. The same 23 records informed the initial scenario-response calibration and the subsequent diagnostics; the reported errors and correlations are, therefore, in-sample diagnostics only. SM-DPF is a governance-oriented reference tool for transparent listing decisions and audit replay with no transaction price prediction claim.

Symmetry doi: 10.3390/sym18081262

Authors: Zeynep Can

In this paper, we investigate the family of curvature tensors W˜(αi), i=1,2,…,12 under the Schouten–van Kampen connection ∇˜ on Kenmotsu manifolds. Our focus is directed toward evaluating diverse curvature conditions, such as W˜(αj)-flatness, ξ-W˜(αj)-flatness, quasi W˜(αj)-flatness, and ϕ-W˜(αj)-flatness for j=5,8. We show that these curvature restrictions imply that the corresponding Kenmotsu manifolds are Einstein or η-Einstein manifolds. Furthermore, we examine the derivation condition W˜(αj)(Z1,Z2)W˜(αi)=0, j=5,8,i=1,2,…,12 and prove that any Kenmotsu manifold satisfying this condition is an Einstein manifold.

Symmetry doi: 10.3390/sym18081261

Authors: Mingzhe Zhou Jinbao Li Yahong Guo

Accurate volumetric quantification of brain tumor subregions in magnetic resonance imaging (MRI) is sensitive to local prediction bias, because boundary over-expansion, missed enhancing foci, and class confusion at subregion transitions can propagate to case-level volume errors. To address this issue, this study proposes CPLR-Net, a coarse prior-guided logit-space residual refinement framework with functional symmetry between global layout preservation and local error compensation. A Coarse Prior Segmentor first generates coarse logits that encode the global semantic layout of the tumor. These logits are then concatenated with multi-modal MRI and fed into an efficient multi-scale attention (EMA)-enhanced Residual Refiner, where three-dimensional cross-spatial feature recalibration and gated logit residual updating are used to correct ambiguous boundaries and subregion transitions while maintaining consistency with the coarse semantic prior. Experiments on the Brain Tumor Segmentation (BraTS) 2020 and 2021 datasets show that CPLR-Net achieves favorable overall performance in the Dice similarity coefficient, 95th percentile Hausdorff distance, and absolute volume difference compared with representative three-dimensional segmentation methods, supporting accurate segmentation and volumetric quantification. The results suggest that coarse-prior-constrained residual refinement in logit space provides an effective strategy for reliable key-subregion segmentation and quantitative assessment in brain tumor MRI follow-up scenarios.

Symmetry doi: 10.3390/sym18081260

Authors: Hongbo Liu Yingcai Zhang Chen Wu Jing Li

This study investigates the asymmetric impact of Data Elements (DE) on Corporate Environmental Performance (CEP) in China, using a sample of 310 A-share listed firms from 2012 to 2021. The results show that DE significantly enhances CEP, with stronger effects observed in State-Owned Enterprises (SOEs) compared to non-SOEs. Additionally, heavily polluting firms are more responsive to DE than lightly polluting firms, indicating that DE has a stronger effect in industries with greater environmental challenges. The study also highlights regional differences, with firms located in areas with stricter environmental regulations experiencing a more substantial improvement in CEP. Mechanism analysis reveals that DE improves environmental performance through optimizing labor force structure, enhancing management efficiency, and alleviating financing constraints. These findings suggest that the impact of DE on CEP is not uniform, and firms should leverage DE more effectively, particularly in high-pollution industries, regulated regions, and state-owned enterprises, to support green development. The study provides valuable insights for policymakers and business leaders aiming to foster a green transformation in China’s economy.

Symmetry doi: 10.3390/sym18081259

Authors: Khalid Alissa

The Internet of Medical Things (IoMT) has transformed healthcare through real-time patient monitoring, intelligent diagnosis, and seamless exchange of medical data. However, the increasing interconnectivity of IoMT devices has significantly expanded the cybersecurity attack surface, exposing healthcare systems to threats that may compromise patient safety, Data Confidentiality, and service availability. Existing cybersecurity risk assessment methods often face challenges in adequately capturing the ambiguity, incompleteness, and subjectivity inherent in expert-based evaluations of IoMT security risks. To address these limitations, this paper proposes a symmetric hybrid neutrosophic fuzzy-based cybersecurity risk assessment methodology for IoMT environments. The proposed framework integrates Neutrosophic Fuzzy Sets (NFSs), the Analytic Hierarchy Process (AHP), and the Simple Average Method (SAM) to model truth, indeterminacy, and falsity in expert judgments while aggregating multiple expert opinions to produce a comprehensive cybersecurity risk assessment. The framework evaluates cybersecurity risks across seven security dimensions and twenty-eight evaluation sub-factors to identify and prioritize critical IoMT vulnerabilities and risk vectors. The experimental results indicate that Data Protection Assessment is the highest-ranked cybersecurity dimension. At the final evaluation stage, the proposed approach combines normalized criterion weights with expert risk evaluations to produce a cybersecurity risk score of 0.8504, indicating high cybersecurity risk that requires priority mitigation in IoMT situations. The proposed framework is validated by comparing it to traditional multi-criteria decision-making methods and using sensitivity analysis to assess the rankings’ robustness and stability under different decision scenarios and expert preference variations. Results show that the proposed method delivers consistent, accurate, and robust risk prioritization under uncertainty, supporting IoMT cybersecurity decision-making. Healthcare organizations can use the framework to identify cybersecurity threats, prioritize mitigation techniques, and strengthen IoMT systems.

Symmetry doi: 10.3390/sym18081258

Authors: Seda Oğuz Ünal

The Sombor index has recently become a central tool among degree-based graph invariants; however, it does not explicitly isolate degree imbalance along edges. In this work, we introduce the degree-based Contrastive Sombor Index (CSO), which combines endpoint-degree magnitude with local degree imbalance. For a finite simple graph G=(V,E), the index is defined by CSO(G)=∑uv∈E(G)d(u)2+d(v)2−2min{d(u),d(v)}. Unlike the Sombor index, which primarily reflects the magnitude of the endpoint degrees, the CSO contribution vanishes when the endpoint degrees are equal and responds to degree imbalance while retaining degree-scale information. In particular, it can distinguish certain graphs having the same total edgewise irregularity but different endpoint-degree distributions. In this work, we first show that CSO(G)≥0 and prove that CSO(G)=0 if and only if each connected component of G is regular. We also establish general lower and upper bounds for CSO. In addition, we obtain a relation connecting the CSO index with the first Zagreb index and the edgewise degree differences. We also discuss extremal aspects of the index. As an application, we derive an explicit summation formula for CSO on monogenic semigroup graphs. From our computations on Γ(SM), it follows that the asymptotic growth order of the index satisfies CSO(Γ(SM))=Θ(n3). These results show that the CSO index combines degree-magnitude information with sensitivity to unequal endpoint degrees and provides an additional perspective on degree heterogeneity in graphs.

Symmetry doi: 10.3390/sym18081257

Authors: Sasipong Kijsason Sa-Aat Niwitpong Suparat Niwitpong

The coefficient of variation (CV) is a widely used unit-free measure for comparing variability across two or more populations, even when the data are measured on different scales. In this study, we investigate confidence and credible interval estimation for the difference and ratio of CVs from the zero-inflated two-parameter Rayleigh (ZITR) distribution, which accommodates excess zeros and positively skewed observations through a two-component mixture structure. Several interval estimation methods are proposed, including the Bayesian Credible Interval (BCI), Bayesian Highest Posterior Density (HPD), and Approximate Normal (AN) approaches. Their performances are compared with established methods, namely the Generalized Confidence Interval (GCI), Method of Variance Estimates Recovery (MOVER), Percentile Bootstrap (PB), and Bootstrap with Standard Error (BS). A Monte Carlo simulation study is conducted to evaluate the methods in terms of coverage probability (CP), Monte Carlo standard error (MCSE) of the estimated coverage probability, and expected length (EL). The simulation results indicate that the HPD method provides the best overall performance, achieving coverage probabilities close to the nominal level with small MCSE values, indicating stable simulation estimates, while maintaining relatively short interval lengths. The proposed confidence interval methods were further applied to district-level road traffic fatality rates (per 100,000 population) in Uttaradit and Mae Hong Son provinces, Thailand. The road traffic fatality data included fatalities occurring in March 2026. The fatality rates were calculated based on the district-level population size to enable comparisons across districts with different population scales.

Symmetry doi: 10.3390/sym18081256

Authors: Ozturk Tosun Vedat Esen Taner Dindar Ali Samet Sarkın Bekir Gecer Necibe Fusun Oyman Serteller

This study presents a comparison of Double Stator Single Rotor Axial Flux Inner Rotor North–South (DSSR AFIR NS) and Double Stator Single Rotor Axial Flux Inner Rotor North–North (DSSR AFIR NN) configurations based on the magnetic equivalent circuit (MEC) approach. In the first stage, a magnetic equivalent circuit model of the DSSR AFIR NS topology was developed, and the mathematical formulations of the air-gap reluctance, stator tooth reluctance, stator yoke reluctance, and rotor core reluctance constituting the magnetic flux path were derived. Considering the motor’s geometric and electromagnetic characteristics, the magnetic flux and flux density distributions in each region were analytically evaluated. The obtained results were then validated through finite element analysis (FEA). In the second stage, the DSSR AFIR NN topology was investigated using the same methodology, and the electromagnetic performances of the two machines with identical slot numbers, pole numbers, and physical dimensions were compared to determine their respective advantages and limitations. Numerical analyses show the AFIR-NN topology yields higher torque (26.53 Nm) than AFIR-NS (19.43 Nm). Conversely, AFIR-NS exhibits superior magnetic characteristics, with higher back-EMF (34.73 V vs. 19.37 V) and air-gap flux density (0.89 T vs. 0.46 T).

Symmetry doi: 10.3390/sym18081255

Authors: Wiwat Sriphum Thawatchai Chomsiri

The DR Code (Data Restorable Code), originally proposed by Sriphum in 2013, is a two-dimensional barcode that achieves a 33% data-recovery rate against six distinct cases of strip-shaped data loss using simple XOR-based parity. The original work presented a single 3 × 3 arrangement of nine logical blocks (A0, A1, A2, B0, B1, B2, C0, C1, C2) in which each row and each column contain exactly one element from each of the three data classes (A, B, C). This paper systematically enumerates every 3 × 3 arrangement that satisfies this recoverability property and proves, by exhaustive search released as an open-source program (DR15.py), that exactly 2592 such arrangements exist. We then introduce five structural theorems—mirror reflection, vertical flipping, Tetris-style rotation, cyclic column rotation, and cyclic row rotation—and prove that each preserves recoverability. We show that these five generators, viewed as a group action, produce a finite group of order 72 isomorphic to the semi-direct product (C3 × C3) ⋊ D4, which partitions the 2592 patterns into exactly 36 absolute equivalence classes. We further explore the partial-quotient structure under D4 alone (yielding 324 classes, the case the practitioner is most likely to encounter) and under cyclic-only quotient (yielding 288 classes). As practical contributions, we propose (i) a compact equivalence-class encoding that reduces the storage cost of one DR Code template from a naïve 36 bits to 13 bits, and (ii) a canonical-form deduplication scheme suitable for cloud and embedded storage systems. Additionally, we propose two further contributions: a fast pattern-validity oracle based on canonical lookup, and a randomization-friendly DR Code variant for security-aware barcode applications. Empirical results confirm all theorems on the full enumeration of 2592 patterns.

Symmetry doi: 10.3390/sym18081254

Authors: Muhammad Dawood Usman Rasheed Waqas Ahmad Ahsan Bin Tufail Afnan Albahli

Alzheimer’s disease (AD) is a neurodegenerative disorder that causes progressive damage to brain neurons, leading to declines in cognitive and behavioral abilities. This deterioration often results in changes in personality and increasing difficulty in thinking and memory over time. Although there is no cure, early detection is crucial as it allows for more effective management and care. Advances in deep learning have significantly improved the accuracy of brain scan analysis for diagnostic purposes. In this study, we utilized the publicly available Alzheimer’s Disease Neuroimaging Initiative (ADNI) dataset consisting of subjects diagnosed with AD, Mild Cognitive Impairment (MCI), and Normal Control (NC). Each participant has either Magnetic Resonance Imaging (MRI) or Positron Emission Tomography (PET) neuroimaging data, ensuring representation across heterogeneous modalities. The research focuses on comparing gradient-based, edge-based, and segmentation-based data augmentation techniques for early AD detection using neuroimaging and deep learning approaches, particularly 3D Convolutional Neural Networks (3D CNNs). Various augmentation methods were applied, including directional gradient, azimuth gradient direction, numerical gradient, Sobel horizontal edge filter, superpixel oversegmentation, and Canny edge detection. These techniques are evaluated in both binary and multiclass classification tasks involving MRI and PET scans. The results indicate that optimal performance varied depending on the task and modality. For PET-based classification, directional gradient performed the best for AD vs. NC binary classification, achieving an accuracy of 87.24%, while Canny edge detection was most effective for AD vs. MCI binary classification and AD-MCI-NC multiclass classification tasks, achieving accuracies of 72.77% and 59.04%, respectively. For MCI vs. NC, the best result (accuracy = 64.32%) is achieved by combining azimuth gradient direction with Sobel filtering. In contrast, for the MRI-based AD vs. NC classification task, the highest performance is achieved without applying augmentation (balanced accuracy = 60.90%). This research confirms the efficacy of data augmentation methods in the early diagnosis of AD in clinical settings.

Symmetry doi: 10.3390/sym18081253

Authors: Wikanda Phaphan Nattawut Khansai Apitad Kraichok Andrei Volodin

Two-parameter exponential distribution has been widely used in research involving lifetime data, survival analysis, medical research, and reliability assessment. Since variables following this distribution are continuous, the population mean is often regarded as a key parameter of interest. In this paper, confidence intervals for the mean of a two-parameter exponential population are developed based on the joint asymptotic normality of the method of moments estimators, derived using the Delta method. The proposed asymptotic confidence intervals are compared with Wald-type confidence intervals in terms of coverage probability and average interval width. The performance of the proposed intervals is evaluated using Monte Carlo simulation, where particular attention is given to their coverage probability and mean interval width. The performance is further illustrated through a real-data application using daily PM2.5 concentration data from Bangkok, Thailand. Simulation, and the theoretical results indicate that the performance of estimators in the two-parameter exponential distribution improves as the sample size increases. Small samples lead to biased and unstable estimates, with wider confidence intervals and lower coverage probabilities, whereas moderate and large samples yield more accurate and stable results. For large samples, bias becomes negligible, variances approach the corresponding theoretical values, and coverage probabilities and interval widths approach the nominal level, confirming consistency and asymptotic efficiency.

Symmetry doi: 10.3390/sym18081252

Authors: Honghao Xu Peigang Jiao Changhui Zheng Jiaxin Shi Yiheng Zhang

The high-temperature sealing reliability of flange–bolt–gasket systems is governed by the coupled gasket leakage, flange cracking, and bolt yielding. This study investigates a DN200 PN40 (nominal diameter 200 mm and nominal pressure 4.0 MPa) weld-neck flange assembly operating under 300 °C superheated steam at 4 MPa internal pressure. Exploiting the assembly’s 12-fold cyclic rotational symmetry, a 1/12 periodic-sector finite element model with steady-state thermal–structural sequential coupling was developed in ANSYS Workbench and validated against the Omiya–Sawa 3-inch weld-neck flange benchmark at two levels (Level 1: bolt load vs. experiment; Level 2: 250 °C gasket contact pressure vs. reference finite element method (FEM)), with maximum errors below 1.5% in both levels; the benchmark thus establishes the reliability of the modeling procedure rather than constituting a direct experimental validation of the DN200 PN40 configuration. Using a central composite design, second-order response surface models (RSM) and Kriging surrogate models were constructed and compared, followed by Sobol global sensitivity analysis, multi-objective optimization using the non-dominated sorting genetic algorithm II (NSGA-II), and decision-making using the technique for order preference by similarity to ideal solution (TOPSIS), with bolt preload F and gasket width b as design variables. Baseline analysis revealed a differential contact pressure distribution—lower at the inner radius and higher at the outer radius—driven by a −0.308° flange rotation, identifying the inner gasket edge as the critical sealing failure path. RSM outperformed Kriging for the primary objective (mean absolute percentage error (MAPE): 0.72% vs. 3.61%), and the Pareto front collapsed to b = 19 mm. The TOPSIS-recommended optimum (F = 59,942 N, b = 19.00 mm), verified by ANSYS back-substitution, increased the minimum gasket contact pressure by 31.01% while reducing the flange membrane-plus-bending stress by 2.26%, achieving a coordinated improvement of both sealing performance and structural safety.

Symmetry doi: 10.3390/sym18081251

Authors: Zhiyang Cao Sheng Li

The random impact of photovoltaic grid connection on the system has always attracted the attention of the academic community. Therefore, using some theories of stochastic differential equations, especially the stochastic averaging method, to study the bifurcation phenomena caused by random photovoltaic output has become a growing trend. This paper models a two-machine power system excited by stochastic photovoltaic output, based on a quasi-non-integrable Hamiltonian physical model, and analyzes its dynamic behavior from two aspects: stochastic bifurcation and stochastic reliability. First, the two-machine power system is transformed into a quasi-non-integrable Hamiltonian model, through which the system equations are simplified for the following calculations. In the subsequent stochastic bifurcation analysis, we study the effects of damping coefficient and noise intensity on the bifurcation evolution. We find that combined parameter and external excitations lead to more variable bifurcation evolution scenarios. Finally, through stochastic reliability analysis, the stochastic reliability analysis demonstrates the importance of mixed parameter adjustment in maintaining the system’s reliability level. Research shows that the stochastic bifurcation caused by photovoltaic grid connection can occur within a certain range of parameters, and we believe this phenomenon provides valuable insights for subsequent studies.

Symmetry doi: 10.3390/sym18081250

Authors: Dan Su Xiaomei Gao Jun Ji Xuemei Cheng Yinghui Ge Xiaolei Wang Tianrui Zhai

Nanoplasmonic waveguides can efficiently manipulate the propagation characteristics of chiral light and hold great promise for integrated nano optics and on-chip optical information processing. Previous studies have demonstrated that gap plasmon structures composed of gold nanorod and silver nanowire can break the propagation symmetry of chiral light, thereby enabling asymmetric directional propagation. However, there remains considerable scope for enhancing the directional selectivity of these structures. In this work, we systematically investigate the mechanism of how the geometrical parameters of nanostructures regulate the directional propagation of chiral light. Based on the finite-difference time-domain method, the propagation behavior and evolution of directionality of chiral light in nanoplasmonic waveguides are analyzed in detail by changing the morphology and length of gold nanorods. The results show that the geometrically optimized nanostructures can significantly enhance the stability of directional chiral light propagation. At the same light-source position, the directionality of the gold nanorod with tips is approximately 56%, whereas that of the gold nanorod without tips remains approximately 89%. Further investigations show that structural modification of the silver nanowire or geometrical optimization of the structure can effectively reshape the local electromagnetic field distribution, enabling precise control over the propagation direction of chiral light. This work elucidates the physical mechanism underlying geometry-controlled chiral plasmonic propagation and provides a new design strategy for the structural design and performance optimization of high-performance chiral nanophotonic devices, with promising potential applications in chiral optical recognition, on-chip optical information processing, and nanoscale optical communication.

Symmetry doi: 10.3390/sym18081249

Authors: Qianqian Zhang Ting Shu Jinsong Xia

With the continuous evolution of network attack techniques, efficiently selecting the most discriminative feature subset from massive network traffic data has become a key issue for improving the performance of intrusion detection systems. Metaheuristic algorithms, as a core approach for wrapper-based feature selection, directly determine the quality of the selected feature subset through their optimization capability. The Grey Wolf Optimizer (GWO) is popular due to its simple structure and few parameters, where three leader wolves guide the search through weighted cooperation. However, its static weight mechanism cannot adapt to dynamic changes in individual search states and population evolution stages, limiting optimization capability and convergence performance. To address this issue, this study proposes a Deep Reinforcement Learning-based Grey Wolf Optimizer (DGWO), which pre-trains a weight adjustment decision model offline and dynamically adjusts the guiding weights of leader wolves during the online search process, thereby improving the optimization ability of the algorithm. Experimental results on NSL-KDD, UNSW-NB15, and CIC-IDS-2017 datasets show that DGWO outperforms seven comparative feature selection methods. It achieves classification accuracies of 93.59%, 93.40%, and 94.84%, respectively, demonstrating superior performance in accuracy, precision, recall, and F1-score. DGWO promotes symmetry between cybersecurity requirements and reliable intrusion detection.

Symmetry doi: 10.3390/sym18081248

Authors: Song Gao Xinyu Huang Zheng Huang Xinyu Wei

Monocular RGB dense SLAM remains challenging because depth and global metric scale are not directly observable from a single camera. Existing systems often combine optical-flow and monocular-geometry priors predicted by independently trained networks, which can provide inconsistent constraints to bundle adjustment (BA). Our quantitative prior-consistency analysis indicates that this disagreement is an important contributor to unstable local optimization and reconstruction error rather than the sole cause of drift. We propose KP-SLAM, which predicts dense optical flow and paired pointmap priors from a shared representation and incorporates them into the same BA backend. We further introduce a Depth-Scale-Pose-to-Pointmap (DSPP) objective that relates optimized inverse depth, edge-wise relative scale, and camera pose to paired pointmap constraints. Experiments on ScanNet, TUM-RGBD, KITTI, Tanks-and-Temples, and dynamic sequences show improved tracking, depth, and rendering metrics over the compared RGB-only baselines under the reported settings. The results support the usefulness of synchronized priors while also revealing remaining limitations in highly dynamic, weakly textured, and large-scale scenes.

Symmetry doi: 10.3390/sym18081247

Authors: Jetdilog Kotemanee Kittisak Saengsura

Let Xn={1,2,…,n}. Previous work has focused on ordinary semigroup identities and the structural properties of individual transformation monoids. Building on related identity-based work involving one of the present authors, we compare the full transformation semigroup Tn, the order-preserving semigroup On, the order-preserving-or-order-reversing semigroup ODn, the orientation-preserving semigroup OPn, and the anti-cyclic one-line family ORn, treated only as a subset of Tn. For each family S, we determine the least positive exponent ES such that aES is idempotent for every a∈S. This gives zES=z2ES. For n≥2, the exponents for Tn, On, ODn, and OPn are lcm(1,…,n), n−1, 2n−12, and lcm(1,…,n), while E^(ORn)=2n−12 is the subset exponent for ORn. We then study xESyESxES=yESxES. With p=xES and q=yES, it reduces to pqp=qp. This holds exactly when q maps each kernel block of p into a single kernel block of p, and fails exactly when q splits a block. Together with the automatic cases, this test gives a classification of all ordered pairs into automatic, positive, and negative classes.

Symmetry doi: 10.3390/sym18081246

Authors: Cristian Lăzureanu

Dynamical systems are used to describe processes arising in a wide range of fields [...]

Symmetry doi: 10.3390/sym18081245

Authors: Wenyao Li Zhanxiu Wang Zhenghong Jin

Public responses during health crises are shaped by interacting risk perceptions, affect, trust, information needs, overload, misinformation, and protective behavior. Existing applications of the Risk Information Seeking and Processing (RISP) model are largely static and therefore cannot represent stochastic multichannel exposure, delayed correction, or policy feedback. We develop the Stochastic Probabilistic Dynamic RISP (SP-D-RISP) model, which recasts RISP as a bounded stochastic state-space system. Its symmetry structure is explicit: the channel-allocation mechanism is equivariant under simultaneous relabeling of channels and their parameter blocks, while the multi-agent dynamics are invariant to agent relabeling under exchangeable sampling and a label-independent policy. Channel-specific effects, heterogeneous traits, rumor shocks, and interventions generate symmetry breaking. The model combines softmax–multinomial channel competition, discounted Bayesian trust updating, and policy-coupled state transitions. Projection guarantees feasible states by construction, whereas stronger stochastic stability is conditional on a coefficient-level small-gain criterion. For the stationary bounded-memory specification, this criterion is sufficient for Wasserstein contraction, uniqueness of the invariant distribution, and geometric forgetting of initial conditions. The criterion is formulated at the coefficient level and is kept distinct from finite-horizon simulation diagnostics. For the fully disclosed semi-synthetic coefficient vector, the scenario-specific gain matrices have spectral radii between 0.852765 and 0.857123; the worst-case column-sum norm is 0.983948. Thus, the fixed-policy kernels satisfy the stated contraction certificate. For deterministic time-varying paths, the calculation is used only as a common-path one-step certificate, and for the threshold-adaptive rule, it is used only mode by mode rather than as a stationary invariant-law claim. While concentration bounds and Monte Carlo inference quantify population and replication uncertainty, a semi-synthetic experiment with 2500 heterogeneous agents over 90 days examines trust and literacy heterogeneity, clarification delays, communication volume, and intervention portfolios. Within the calibrated SP-D-RISP scenarios, the simulations suggest that higher communication volume may reduce modeled protective behavior when overload effects dominate knowledge gains, delayed clarification may increase transient misinformation, and an integrated portfolio can yield a more favorable simulated outcome profile than the evaluated single-lever strategies.

Symmetry doi: 10.3390/sym18081244

Authors: Narin Petrot

Fixed-point theory and optimization constitute two of the most dynamic and interconnected branches of modern mathematical analysis [...]

Symmetry doi: 10.3390/sym18071243

Authors: Jianfei Zhang Shangyu Wu

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.

Symmetry doi: 10.3390/sym18071242

Authors: Mustafa Ahmed Ali Mehmet Merdan

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.

Symmetry doi: 10.3390/sym18071241

Authors: Kai Cheng Dongkun Li Weidong Tang Lin Liu Xueyu Zhang

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.

Symmetry doi: 10.3390/sym18071240

Authors: Shuangqing Chen Chao Chen Junfei Liu Xingwang Wang Zhe Xu Yongbin Liu Haibin Liang Lulu Zhang Yaqian Liu

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.

Symmetry doi: 10.3390/sym18071239

Authors: Jianxin Wang Runze Zhou Zixuan Wang Lei Zhang Chaoen Xiao Zhao Wang Maosheng He Qian Cheng Kaibo Sun

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.

Symmetry doi: 10.3390/sym18071238

Authors: Avnish Bhowan Magan

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.

Symmetry doi: 10.3390/sym18071237

Authors: Zameddin I. Ismailov Pembe Ipek Al Mohammad Sababheh

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.

Symmetry doi: 10.3390/sym18071236

Authors: James Stavitz

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.

Symmetry doi: 10.3390/sym18071235

Authors: Tea Marasović Vladan Papić

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.

Symmetry doi: 10.3390/sym18071234

Authors: Lingling Yao Zhilin Yang Dongsheng Liang Chenglong Wei

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.

Symmetry doi: 10.3390/sym18071233

Authors: Dong-Gyun Han

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.

Symmetry doi: 10.3390/sym18071232

Authors: Bin Li

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.

Symmetry doi: 10.3390/sym18071231

Authors: Nutchanat Buasri Patiwat Littidej Benjamabhorn Pumhirunroj Donald Slack

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.

Symmetry doi: 10.3390/sym18071230

Authors: Simon Gluzman

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.

Symmetry doi: 10.3390/sym18071229

Authors: Yunpeng Wang Qing Li Chao Zhang Yatong Bai Xiaofei Du Jianfeng Wang Yuda He

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.

Symmetry doi: 10.3390/sym18071228

Authors: Feng Li Yaoyao Feng

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 (S6) 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.

Symmetry doi: 10.3390/sym18071227

Authors: Shaymaa Y. Alkufi Abbas Kareem Wanas Alina Alb Lupas

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×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.

Symmetry doi: 10.3390/sym18071226

Authors: Maksim Iavich Tamari Kuchukhidze Audrius Lopata

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.

Symmetry doi: 10.3390/sym18071225

Authors: Mohammad Saad Alaydaa Gaseb N. Alotibi Ahmad S. Tarawneh Ahmad B. Hassanat

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↦−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.

Symmetry doi: 10.3390/sym18071224

Authors: Mohd Arif Raza Majed Albaity

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 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 αβ=0 or α+β∈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 Γ˜(P) is a line graph of some graph. We then classify the finite commutative rings P such that L(Γ˜(P)) is planar, outerplanar, or 2-outerplanar. Finally, we obtain a complete classification of finite commutative rings P for which L(Γ˜(P)) is toroidal.

Symmetry doi: 10.3390/sym18071223

Authors: Bin He Xurong Teng Long Fan Renlong Liu

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.