Search Results (1,573)

Kaprekar’s routine, i.e., sorting the digits of an integer in ascending and descending order and subtracting the two, defines a finite deterministic map on the state space of fixed-length digit strings. While its attractors (such as 495 for $D=3$ and 6174 for $D=4$) are classical, the global information-theoretic structure of the induced dynamics and its dependence on the digit length D have received little attention. Here an exhaustive analysis is carried out for $D\in \{3,4,5,6\}$. For each D, all states are enumerated and the transition structure is computed numerically; attractors and convergence distances are obtained, and the induced distribution over attractors across iterations is used to construct “entropy funnels”. Despite the combinatorial growth of the state space, average distances remain small and entropy decays rapidly before entering a slow tail. Permutation symmetry is then exploited by grouping states into digit multisets and, in a further reduction, into low-dimensional digit-gap features. On this gap space, a first-order Markov approximation is empirically estimated by counting one-step transitions induced by the exhaustively enumerated deterministic map. From the resulting empirical transition matrix, drift fields and the stationary distribution are computed numerically. These quantities serve as descriptive summaries of the projected dynamics and are not derived in closed form. Full article

Unmanned airships are highly sensitive to parametric uncertainty, persistent wind disturbances, and sensor noise, all of which compromise reliable path-following. Classical control schemes often suffer from chattering and fail to handle index discontinuities on closed-loop paths due to the lack of mechanisms and cannot simultaneously provide formal guarantees on state constraint satisfaction. We address these challenges by developing a unified, constraint-aware guidance and control framework for path-following in uncertain environments. The architecture integrates an extended state observer (ESO) to estimate and compensate lumped disturbances, a barrier Lyapunov function (BLF) to enforce state constraints on tracking errors, and a super-twisting terminal sliding-mode (ST-TSMC) control law to achieve finite-time convergence with continuous, low-chatter control inputs. A constructive Lyapunov-based synthesis is presented to derive the control law and to prove that all tracking errors remain within prescribed error bounds. At the guidance level, a nonlinear curvature-aware line-of-sight (CALOS) strategy with an index-increment mechanism mitigates jump phenomena at loop-closure and segment-transition points on closed yet discontinuous paths. The overall framework is evaluated against representative baseline methods under combined wind and parametric perturbations. Numerical results indicate improved path-following accuracy, smoother control signals, and strict enforcement of state constraints, yielding a disturbance-resilient path-following solution for the cruise of an unmanned airship. Full article

Convergence rate is a key performance index for flight vehicles, and accelerating it remains a critical open issue. In this paper, a fast robust integrated guidance and control scheme for flight vehicles based on convergence rate estimation mechanism is proposed, which improves the control performance and interception accuracy of flight vehicles. In the fast robust control scheme, a convergence rate indicator for integrated guidance and control systems is developed to measure the impact on convergence rate imposed by model nonlinearities and couplings within flight vehicles. Based on the indicator, the influences on convergence rate are transformed and injected into controllers to accelerate the convergence of flight vehicles. The unmatched lumped uncertainties in flight vehicle dynamics are addressed by a backstepping control method and finite-time convergence disturbance observers, which improves the robustness of the vehicle’s control system. Furthermore, the stability analysis of the closed-loop system is performed via the Lyapunov stability theorem. Extensive numerical simulations are conducted to verify the effectiveness and interception performance of the proposed method, and the comparison results confirm that it outperforms three other recently developed robust control methods. Full article

Particle-in-Cell (PIC) simulations require accurate solutions of the electrostatic Poisson equation, yet accuracy often degrades near irregular Dirichlet boundaries on Cartesian meshes. While prior work has focused on left-hand-side (LHS) discretization errors, the impact of right-hand-side (RHS) inaccuracies arising from charge deposition near boundaries remains largely unexplored. This study analyzes numerical errors induced by underestimated RHS values at near-boundary nodes when using embedded finite difference schemes with linear and quadratic boundary treatments. Analytical results in one dimension and truncation error analyses in two dimensions show that RHS inaccuracies affect the two schemes in fundamentally different ways: They reduce boundary-induced errors in the linear scheme but introduce zeroth-order truncation errors in the quadratic scheme, leading to larger global errors. Numerical experiments in one, two, and three dimensions confirm these predictions. In two-dimensional tests, RHS inaccuracies reduce the $L_{\infty }$ error of the linear scheme by a factor of 2–3, while increasing the quadratic-scheme error by several times, and in some cases by nearly an order of magnitude, with both schemes retaining second-order global convergence. A simple $\overline{\delta }$-based RHS calibration is proposed and shown to effectively restore the accuracy of the quadratic scheme. Full article

This paper identifies theoretical vulnerabilities in quantum integrity verification by demonstrating that Bell inequality (BI) violations, central to the detection of quantum entanglement, can align with predictions from hidden variable theories (HVTs) under specific measurement configurations. By invoking a Heisenberg-inspired measurement resolution constraint and finite-resolution positive operator-valued measures (POVMs), we identify “convergence vicinities” where the statistical outputs of quantum and classical models become operationally indistinguishable. These results do not challenge Bell’s theorem itself; rather, they expose a vulnerability in quantum integrity frameworks that treat observed Bell violations as definitive, experiment-level evidence of nonclassical entanglement correlations. We support our theoretical analysis with simulations and experimental results from IBM quantum hardware. Our findings call for more robust quantum-verification frameworks, with direct implications for the security of quantum computing, quantum-network architectures, and device-independent cryptographic protocols (e.g., device-independent quantum key distribution (DIQKD)). Full article

This paper develops a fully discrete finite element scheme for the fractional wave equation with order $\alpha \in (1,2)$, whose solution typically exhibits a weak singularity near the initial time $t=0$. By introducing an auxiliary variable, we first reformulate the fractional wave problem into an equivalent coupled system of two fractional equations. The resulting coupled system is then discretized using the nonuniform Alikhanov formula in time and the standard finite element method on triangular meshes in space. Through rigorous analysis, we establish a pointwise-in-time error estimate for the proposed scheme in the $H^1$ semi-norm. A key advantage of the proposed methodology is its ability to employ a sparser mesh near the initial time to achieve optimal convergence of local errors. In particular, our analysis shows that away from the initial time, the local rate of convergence reaches $O\left(N^{-2}\right)$ in time for $r\ge 2$. Finally, numerical experiments are given to verify the sharpness of the theoretical convergence rates. Full article

To address the conflicts among objectives and the decision-making challenges in the multi-condition adaptive design of battery boxes for new energy vehicles, this study proposes a multi-objective collaborative optimization method based on an improved relaxation factor, aiming to achieve a comprehensive enhancement in both structural lightweighting and mechanical performance. A finite element model of the CTB high-strength steel roll-formed battery box was established and validated through modal testing. According to the Chinese National Standard GB 38031-2025, the mechanical responses of the battery box under random vibration, extreme operating conditions, and impact loads were analyzed to identify performance weaknesses. Sensitivity analysis was conducted to screen the design variables, and an improved relaxation factor strategy based on weight distribution difference information was introduced to construct a multi-objective collaborative optimization model. Furthermore, the entropy-weighted TOPSIS method was employed to enable intelligent decision-making on the Pareto solution set. The results demonstrate that the proposed method outperforms conventional approaches in both convergence speed and solution distribution uniformity. After optimization, the mass of the battery box was reduced by 12.38%, while multiple mechanical performance indicators were simultaneously improved, providing valuable theoretical and engineering guidance for the structural design of power battery systems. Full article

As offshore wind farms expand into deeper waters, the safe transportation of large jacket foundations presents a significant engineering challenge. This study utilizes the SESAM 2022 software suite, based on three-dimensional potential flow theory, to conduct a coupled numerical simulation and parametric analysis of a barge-jacket system. Finite element models of the barge and jacket are established, with mesh convergence verified. The influences of key parameters including wave frequencies (0.4–1.6 rad/s), wave directions (0–180°), forward speeds (0–8 knots) and jacket arrangement (vertical/horizontal) on the six degrees of freedom (6-DOF) dynamic responses of the coupled system are systematically investigated. Based on the observed response characteristics, optimized transportation configurations and practical engineering recommendations are proposed. The findings consolidate previous scattered parametric results into a single, repeatable SESAM-based benchmark data set, offering a reference against which future nonlinear or time-domain models can be validated. Furthermore, this work establishes a systematic parametric basis and offers practical guidance for assessing the safety of offshore wind turbine (OWT) foundation transportation in deep-water environments. Full article

This study investigates the strengthening mechanisms of a Carbon Fiber-Reinforced Polymer (CFRP) grid and Polymer-modified Cement Mortar (PCM) system for damaged reinforced concrete (RC) beams in flexure. Experimental tests were conducted on five short beams to systematically observe the failure modes, load-carrying capacity, strain development, and deflection evolution. A finite element model was established and validated against the experimental results to analyze the effects of key parameters, including the damage degree, number of grid layers, and grid spacing. Theoretical formulas for calculating the ultimate flexural capacity under different failure modes were also derived. The results demonstrate that strengthening undamaged beams yields a more significant improvement in ultimate and cracking loads than strengthening pre-damaged beams. The composite system effectively suppresses crack propagation by enhancing stiffness, albeit at the expense of reduced ductility. The theoretical predictions show good agreement with the experimental data. Parametric analysis reveals that lightly damaged beams exhibit a higher load-bearing potential, whereas severely damaged beams display more ductile behavior. The increase in load capacity converges when the number of grid layers exceeds three. In contrast, reducing the grid spacing significantly enhances flexural capacity due to improved meso-scale structural effects. Full article

Acoustic metamaterials are artificially engineered materials composed of subwavelength structural units, whose effective acoustic properties are primarily determined by structural design rather than intrinsic material composition. By introducing local resonances, these materials can exhibit unconventional acoustic behavior, enabling enhanced sound insulation beyond the limitations of conventional structures. In this study, a thin plate (thin sheet) refers to a structural element whose thickness is much smaller than its in-plane dimensions and can be accurately described using classical thin-plate vibration theory. When resonant mass blocks are attached to a thin plate, a thin-plate acoustic metamaterial is formed through the coupling between plate bending vibrations and local resonances. Thin-plate acoustic metamaterials exhibit excellent sound insulation performance in the low- and mid-frequency ranges. Multilayer configurations and the combination with porous materials can effectively broaden the insulation bandwidth and improve overall performance. However, the large number of structural parameters in multilayer composite thin-plate acoustic metamaterials significantly increases design complexity, making conventional trial-and-error approaches inefficient. To address this challenge, a neural-network-based inverse design framework is proposed for multilayer composite thin-plate acoustic metamaterials. An analytical model of thin-plate metamaterials with multiple attached cylindrical masses is established using the point matching and modal superposition methods and validated by finite element simulations. A multilayer composite unit cell is then constructed, and a dataset of 30,000 samples is generated through numerical simulations. Based on this dataset, a forward prediction network achieves a test error of 1.06%, while the inverse design network converges to an error of 2.27%. The inverse-designed structure is finally validated through impedance tube experiments. The objective of this study is to establish a systematic theoretical and neural-network-assisted inverse design framework for multilayer thin-plate acoustic metamaterials. The main novelties include the development of an accurate analytical model for thin-plate metamaterials with multiple attached masses, the construction of a large-scale simulation dataset, and the proposal of a neural-network-assisted inverse design strategy to address non-uniqueness in inverse design. The proposed approach provides an efficient and practical solution for low-frequency sound insulation design. Full article

This paper investigates the consensus tracking problem for a class of high-order nonlinear multi-agent systems subject to actuator faults, sensor faults, unknown disturbances, and model uncertainties. To effectively address this problem, a hierarchical fault-tolerant control framework with fuzzy adaptive mechanisms is proposed. First, a distributed output predictor based on a finite-time differentiator is constructed for each follower to estimate the leader’s output trajectory and to prevent fault propagation across the network. Second, a novel state and actuator-fault observer is designed to reconstruct unmeasured states and detect actuator faults in real time. Third, a sensor-fault compensation strategy is integrated into a backstepping procedure, resulting in a fuzzy adaptive consensus-tracking controller. This controller guarantees the uniform boundedness of all closed-loop signals and ensures that the tracking error converges to a small neighborhood of the origin. Finally, numerical simulations validate the effectiveness and robustness of the proposed method in the presence of multiple simultaneous faults and disturbances. Full article

This paper presents a finite difference method for solving the Caputo generalized time fractional diffusion equation. The method extends the $L1$ scheme to discretize the time fractional derivative and employs the central difference for the spatial diffusion term. Theoretical analysis demonstrates that the proposed numerical scheme achieves a convergence rate of order $2-\alpha$ in time and second order in space. These theoretical findings are further validated through numerical experiments. Compared to existing methods that only achieve a temporal convergence of order $1-\alpha$, the proposed approach offers improved accuracy and efficiency, particularly when the fractional order $\alpha$ is close to zero. This makes the method highly suitable for simulating transport processes with memory effects, such as oil pollution dispersion and biological population dynamics. Full article

In this work, a high-order meshless framework is developed for the numerical resolution of the temporal–fractional Black–Scholes equation arising in option pricing with long-memory effects. The spatial discretization is carried out with a local radial basis function produced finite difference (RBF–FD) method on seven-node stencils. Analytical differentiation weights are constructed by employing closed-form second integrations of a variant of the inverse multiquadric kernel, which yields sparse differentiation matrices. Explicit formulas are derived for both first- and second-order operators, and a detailed truncation error analysis confirms sixth-order convergence in space. Numerical experiments for European options discuss better accuracy per spatial node than standard finite difference schemes. Full article

We study numerically finite-sample power functions of nonparametric asymptotic tests for at most one change in the mean that are based on convergence in the distribution of sup- and integral functionals of an appropriately weighted and normalized tied-down partial sums process. For each test, a three-way trade-off is observed among its type I errors, power for detecting the change near the beginning or end of the sample, and power for detecting the change in the middle of the sample. By choosing suitable weight functions of a special form, we propose new sup- and integral tests that are shown to be nearly as powerful as the overall most powerful sup-test in the literature, regardless of where the change occurs in the sample. Moreover, the type I errors of the new tests are closer to the asymptotic significance level across various distributions and are lower and converge faster for distributions that are more asymmetric, heavy-tailed, or both. Full article

This expository paper provides a unified and pedagogical introduction to optimal quantization for probability measures supported on spherical curves and discrete subsets of the sphere, emphasizing both continuous and discrete settings. We first present a detailed geometric and analytical foundation for intrinsic quantization on the unit sphere, including definitions of great and small circles, spherical triangles, geodesic distance, Slerp interpolation, the Fréchet mean, spherical Voronoi regions, centroid conditions, and quantization dimensions. Building upon this framework, we develop explicit continuous and discrete quantization models on spherical curves, namely great circles, small circles, and great circular arcs—supported by rigorous derivations and pedagogical exposition. For uniform continuous distributions, we compute optimal sets of n-means and the associated quantization errors on these curves; for discrete distributions, we analyze antipodal, equatorial, tetrahedral, and finite uniform configurations, illustrating convergence to the continuous model. The central conclusion is that for a uniform probability distribution supported on a one-dimensional geodesic subset of total length L, the optimal n-means form a uniform partition and the quantization error satisfies $V_n=L^2/\left(12n^2\right)$.The exposition emphasizes geometric intuition, detailed derivations, and clear step-by-step reasoning, making it accessible to beginning graduate students and researchers entering the study of quantization on manifolds. This article is intended as an expository and tutorial contribution, with the main emphasis on geometric reformulation and pedagogical clarity of intrinsic quantization on spherical curves, rather than on the development of new asymptotic quantization theory. Full article