Search Results (262)

Existing 3D point cloud enhancement methods typically rely on artificially designed geometric transformations or local blending strategies, which are prone to introducing illogical deformations, struggle to preserve global structure, and exhibit insufficient adaptability to diverse degradation patterns. To address these limitations, this paper proposes SFD-ADNet—an adaptive deformation framework based on a dual spatial–frequency domain. It achieves 3D point cloud augmentation by explicitly learning deformation parameters rather than applying predefined perturbations. By jointly modeling spatial structural dependencies and spectral features, SFD-ADNet generates augmented samples that are both structurally aware and task-relevant. In the spatial domain, a hierarchical sequence encoder coupled with a bidirectional Mamba-based deformation predictor captures long-range geometric dependencies and local structural variations, enabling adaptive position-aware deformation control. In the frequency domain, a multi-scale dual-channel mechanism based on adaptive Chebyshev polynomials separates low-frequency structural components from high-frequency details, allowing the model to suppress noise-sensitive distortions while preserving the global geometric skeleton. The two deformation predictions dynamically fuse to balance structural fidelity and sample diversity. Extensive experiments conducted on ModelNet40-C and ScanObjectNN-C involved synthetic CAD models and real-world scanned point clouds under diverse perturbation conditions. SFD-ADNet, as a universal augmentation module, reduces the mCE metrics of PointNet++ and different backbone networks by over 20%. Experiments demonstrate that SFD-ADNet achieves state-of-the-art robustness while preserving critical geometric structures. Furthermore, models enhanced by SFD-ADNet demonstrate consistently improved robustness against diverse point cloud attacks, validating the efficacy of adaptive space-frequency deformation in robust point cloud learning. Full article

Accurate spatiotemporal forecasting underpins high-stakes decision making in smart urban systems, from traffic control and energy scheduling to environment monitoring. Yet two persistent gaps limit current models: (i) spatial modules are often biased toward low-pass smoothing and struggle to reconcile slow global trends with sharp local dynamics; and (ii) the graph structure required for forecasting is frequently latent, while learned graphs can be unstable when built from temporally derived node features alone. We propose SpeQNet, a query-enhanced spectral graph filtering framework that jointly strengthens node representations and graph construction while enabling frequency-selective spatial reasoning. SpeQNet injects global spatial context into temporal embeddings via lightweight learnable spatiotemporal queries, learns a task-oriented adaptive adjacency matrix, and refines node features with an enhanced ChebNetII-based spectral filtering block equipped with channel-wise recalibration and nonlinear refinement. Across twelve real-world benchmarks spanning traffic, electricity, solar power, and weather, SpeQNet achieves state-of-the-art performance and delivers consistent gains on large-scale graphs. Beyond accuracy, SpeQNet is interpretable and robust: the learned spectral operators exhibit a consistent band-stop-like frequency shaping behavior, and performance remains stable across a wide range of Chebyshev polynomial orders. These results suggest that query-enhanced spatiotemporal representation learning and adaptive spectral filtering form a complementary and effective foundation for effective spatiotemporal forecasting. Full article

In the current study, we used Chebyshev’s Pseudospectral Method (CPM), a novel numerical technique, to solve nonlinear third-order Emden–Fowler delay differential (EF-DD) equations numerically. Fractional derivatives are defined by the Caputo operator. These kinds of equations are transformed to the linear or nonlinear algebraic equations by the proposed approach. The numerical outcomes demonstrate the precision and efficiency of the suggested approach. The error analysis shows that the current method is more accurate than any other numerical method currently available. The computational analysis fully confirms the compatibility of the suggested strategy, as demonstrated by a few numerical examples. We present the outcome of the offered method in tables form, which confirms the appropriateness at each point. Additionally, the outcomes of the offered method at various non-integer orders are investigated, demonstrating that the result approaches closer to the accurate solution as a value approaches from non-integer order to an integer order. Additionally, the current study proves some helpful theorems about the convergence and error analysis related to the aforementioned technique. A suggested algorithm can effectively be used to solve other physical issues. Full article

This article proposes single-layer neural network algorithms for solving second-order ordinary differential equations, based on the principles of functional connection. According to this principle, the hidden layer of the neural network is replaced by a functional expansion unit to improve input patterns using orthogonal Chebyshev, Legendre, and Laguerre polynomials. The polynomial neural network algorithms were implemented in the Python programming language using the PyCharm environment. The performance of the polynomial neural network algorithms was tested by solving initial-boundary value problems for the nonlinear Lane–Emden equation. The solution results are compared with the exact solution of the problems under consideration, as well as with the solution obtained using a multilayer perceptron. It is shown that polynomial neural networks can perform more efficiently than multilayer neural networks. Furthermore, a neural network based on Laguerre polynomials can, in some cases, perform more accurately and faster than neural networks based on Legendre and Chebyshev polynomials. The issues of overtraining of polynomial neural networks and scenarios for overcoming it are also considered. Full article

The free vibration of stiffened plates analyzed using classical plate–beam theoretical theory (PBM) simplified the vibrations of stiffeners parallel to the plane of the stiffened plate as the first-order torsional vibration of the stiffener cross-section. This simplification introduces errors in both the natural frequencies and mode shapes of the structure for stiffened plates with relatively tall stiffeners. To mitigate the issue previously described, this paper proposes an enhanced plate–beam theoretical model (EPBM). The EBPM decouples stiffener deformation into two components: (1) bending deformation along the transverse direction of the stiffened plate, governed by Euler–Bernoulli beam theory, and (2) transverse deformation of the stiffeners, modeled using thin plate theory. Virtual torsional springs are introduced at the stiffener–plate and stiffener–stiffener interfaces via penalty function method to enforce rotational continuity. These constraints are transformed into energy functionals and integrated into the system’s total energy. Displacement trial functions constructed from Chebyshev polynomials of the first kind are solved using the Ritz method. Numerical validation demonstrates that the EBPM significantly improves accuracy over the BPM: errors in free-vibration frequency decrease from 2.42% to 0.63% for the first mode and from 9.79% to 1.34% for the second mode. For constrained vibration, the second-mode error is reduced from 4.22% to 0.03%. This approach provides an effective theoretical framework for the vibration analysis of structures with high stiffeners. Full article

In this paper, we establish new hypergeometric representations for two classical integer sequences, namely the Pell and Jacobsthal sequences. Motivated by Dilcher’s hypergeometric formulations of the Fibonacci sequence, we extend this framework to other second-order linear recurrence sequences with distinct characteristic structures. By employing Binet-type formulas, recurrence relations, Chebyshev polynomial connections, and classical transformation properties of Gauss hypergeometric functions, we derive several explicit and alternative representations for the Pell and Jacobsthal numbers. These representations unify known identities, yield new closed-form expressions, and reveal deeper structural parallels between hypergeometric functions and linear recurrence sequences. The results demonstrate that hypergeometric functions provide a systematic and versatile analytical tool for studying special number sequences beyond the Fibonacci case, and they suggest potential extensions to broader families such as Horadam-type sequences and their generalizations. Full article

Bairstow’s method employs synthetic division to express a polynomial $p\left(x\right)$ of degree n in the form $p\left(x\right)=q\left(x\right)(x^2+Bx+C)+R(B,C)x+S(B,C)$, where $q\left(x\right)$ is the quotient polynomial of degree $n-2$, and $R(B,C)$, $S(B,C)$ are the remainder coefficients that depend nonlinearly on the quadratic parameters B and C. The original algorithm proposed by Bairstow uses Newton–Raphson method to solve $R(B,C)=S(B,C)=0$; it requires initial guesses within very narrow attraction basins for ill-conditioned polynomials and fails at singular Jacobian matrices. To address these issues, we reformulate Bairstow’s method as a constrained optimization problem that maximizes $C^2$ as the objective function subject to the constraints $R(B,C)=S(B,C)=0$. While modern, highly optimized, non-linear solvers (available in commercial software like MATLAB) have largely superseded classical iterative polynomial rootfinding techniques, our reformulated Bairstow approach offers distinct advantages for selective root extraction and application-specific constraints. Specifically, the optimization formulation enables the extraction of specific roots of interest rather than computing all roots simultaneously, naturally accommodates additional constraints for application-specific factorization x (such as discriminant conditions for real versus complex root extraction). The $C^2$ objective automatically selects the quadratic factor with the largest root magnitude, enhancing numerical stability during deflation. Numerical experiments validate the approach on polynomials with degree bigger than 10 including cases with simple real roots, multiple roots, mixed real and complex roots, and Chebyshev polynomials, achieving machine precision accuracy with robust handling of the discriminant constraint. Full article

Short-term photovoltaic (PV) power forecasting is pivotal for grid stability and high renewable-energy integration, yet existing hybrid deep-learning models face three unresolved challenges: they fail to balance accuracy, computational efficiency, and interpretability; cannot mitigate iTransformer’s inherent weakness in local feature capture (critical for transient events like minute-level cloud shading); and rely on linear concatenation that mismatches the nonlinear correlations between global multivariate trends and local fluctuations in PV sequences. To address these gaps, this study proposes a novel lightweight hybrid framework—DSC-Chebyshev KAN-iTransformer—for 15-min short-term PV power forecasting. The core novelty lies in the synergistic integration of Depthwise Separable Convolution (DSC) for low-redundancy local temporal pattern extraction, Chebyshev Kolmogorov–Arnold Network (Chebyshev KAN) for adaptive nonlinear fusion and global nonlinear modeling, and iTransformer for efficient capture of cross-variable global dependencies. This design not only compensates for iTransformer’s local feature deficiency but also resolves the linear fusion mismatch issue of traditional hybrid models. Experimental results on real-world PV datasets demonstrate that the proposed model achieves an $R^2$ of 0.996, with root mean square error (RMSE) and mean absolute error (MAE) reduced by 19.6–62.1% compared to state-of-the-art baselines (including iTransformer, BiLSTM, and DSC-CBAM-BiLSTM), while maintaining lightweight characteristics (2.04M parameters, 3.90 GFLOPs) for urban edge deployment. Moreover, Chebyshev polynomial weight visualization enables quantitative interpretation of variable contributions (e.g., solar irradiance dominates via low-order polynomials), enhancing model transparency for engineering applications. This research provides a lightweight, accurate, and interpretable forecasting solution, offering policymakers a data-driven tool to optimize urban PV-infrastructure integration and improve grid resilience amid the global energy transition. Full article

We develop a shared denominator Carathéodory–Fejér (CF) method for efficiently evaluating linear combinations of $\phi$-functions for matrices whose spectrum lies in the negative real axis, as required in exponential integrators for large stiff ODE systems. This entire family is approximated with a single set of poles (a common denominator). The shared pole set is obtained by assembling a stacked Hankel matrix from Chebyshev boundary data for all target functions and computing a single SVD; the zeros of the associated singular-vector polynomial, mapped via the standard CF slit transform, yield the poles. With the poles fixed, per-function residues and constants are recovered by a robust least squares fit on a suitable grid of the negative real axis. For any linear combination of resolvent operators applied to right-hand sides, the evaluation reduces to one shifted linear solve per pole with a single combined right-hand side, so the dominant cost matches that of computing a single $\phi$-function action. Numerical experiments indicate geometric convergence at a rate consistent withHalphen’s constant, and for highly stiff problems our algorithm outperforms existing Taylor and Krylov polynomial-based algorithms. Full article

Background: Quantum Neural Networks (QNNs) combine quantum computing and artificial intelligence to provide powerful solutions for high-dimensional data analysis. In magnetic resonance imaging (MRI), they address the challenges of advanced imaging sequences and data complexity, enabling faster optimization, enhanced feature extraction, and real-time clinical applications. Methods: A literature review using Scopus, PubMed, IEEE Xplore, ACM Digital Library and arXiv identified 84 studies on QNNs in MRI. After filtering for peer-reviewed original research, 20 studies were analyzed. Key parameters such as datasets, architectures, hardware, tasks, and performance metrics were summarized to highlight trends and gaps. Results: The analysis identified datasets supporting tasks like tumor classification, segmentation, and disease prediction. Architectures included hybrid models (e.g., ResNet34 with quantum circuits) and novel approaches (e.g., Quantum Chebyshev Polynomials). Hardware ranged from high-performance GPUs to quantum-specific devices. Performance varied, with accuracy up to 99.5% in some configurations but lower results for complex or limited datasets. Conclusions: The findings provide the first glimpse into the potential of QNNs in MRI, demonstrating accuracy and specificity in diagnostic tasks and biomarker detection. However, challenges such as dataset variability, limited quantum hardware access, and reliance on simulators remain. Future research should focus on scalable quantum hardware, standardized datasets, and optimized architectures to support clinical applications and precision medicine. Full article

In this paper, a new class of fifth-order Chebyshev–Halley-type methods with a single parameter is proposed by using the polynomial interpolation method. The convergence order of the new method is proved. The dynamic behavior of the new method on quadratic polynomials $P\left(x\right)=(x-a)(x-b)$ is analyzed, the strange fixed points and the critical points of the operator are obtained, the corresponding parameter planes and dynamic planes are drawn, the stability and convergence of the iterative method are visualized, and some parameter values with good properties are selected. The fractal results of the new method corresponding to different parameters about polynomial $G\left(x\right)$ are plotted. Numerical results show that the new method has less computing and higher computational accuracy than the existing Chebyshev–Halley-type methods. The fractal results show the new method has good stability and convergence. The numerical results of different iteration methods are compared and agree with the results of dynamic analysis. Full article

Rectangular aqueducts are critical building structures in large-scale water conveyance systems used worldwide. Liquid sloshing can produce hydrodynamic forces that threaten structural safety and long-term performance. This study analytically investigates the vibration characteristics of two-dimensional rectangular cantilever aqueduct systems while accounting for soil–structure interaction (SSI). To reduce sloshing and enhance the performance of the mechanical system, a bottom-mounted vertical baffle is proposed as a hydrodynamic damping solution. Through subdomain analysis, mathematical expressions for liquid potential fields are derived. The continuous liquid is represented through discrete mass–spring elements for dynamic analysis. Horizontal soil impedance is characterized by using Chebyshev orthogonal polynomial approximations with optimized least squares fitting techniques. A dynamic mechanical model for the soil–aqueduct–liquid–baffle coupling system is developed by using the substructure method. Convergence and comparative studies are conducted to validate the reliability of the proposed method. Between the current results and those reported previously, the variation in the first-order sloshing frequency is less than 1.10%. Parametric analyses evaluate how baffle size, baffle position, and soil properties influence sloshing behavior. The presentation of an equivalent analytical model is the novelty of this research. The results can provide the theoretical basis for optimizing anti-sloshing designs in hydraulic building structures, thereby supporting safer and more sustainable engineering practices. Full article

We consider discrete best approximation problems in the setting of tropical algebra, which is concerned with the theory and application of algebraic systems with idempotent operations. Given a set of input–output pairs of an unknown function defined on a tropical semifield, the problem is to determine an approximating rational function formed by two Puiseux polynomials as numerator and denominator. With specified numbers of monomials in both polynomials, the approximation aims at evaluating the exponent and coefficient for each monomial in the polynomials to fit the rational function to the data in the sense of a tropical distance function. To solve the problem, we transform it into an approximation of a vector equation with unknown vectors on both sides, where one side corresponds to the numerator polynomial and the other side to the denominator. Each side involves a matrix with entries dependent on the unknown exponents, multiplied by the vector of unknown coefficients of monomials. We propose an algorithm that constructs a series of approximate solutions by alternately fixing one side of the equation to an already-found result and leaving the other side intact. Each equation obtained is approximated with respect to the vector of coefficients, which yields this vector and approximation error, both parameterized by exponents. The exponents are found by minimizing the error with an optimization procedure based on an agglomerative clustering technique. To illustrate, we present results for an approximation problem in terms of max-plus algebra (a real semifield with addition defined as maximum and multiplication as arithmetic addition), which corresponds to an ordinary problem of piecewise linear approximation of real functions. As our numerical experience shows, the proposed algorithm converges in a finite number of steps and provides a reasonably accurate solution to the problems considered. Full article

In this paper, we present a collocation algorithm for numerically treating the time-fractional Kuramoto–Sivashinsky equation (TFKSE). Certain orthogonal polynomials, which are expressed as combinations of Chebyshev polynomials, and their shifted polynomials are introduced. Some new theoretical formulas regarding these polynomials have been developed, including their operational matrices of both integer and fractional derivatives. The derived formulas will be the foundation for designing the proposed numerical algorithm, which relies on converting the governing problem with its underlying conditions into a nonlinear algebraic system, which can be solved using Newton’s iteration technique. A rigorous error analysis for the proposed combined Chebyshev expansion is presented. Some numerical examples are given to ensure the applicability and efficiency of the presented algorithm. These results demonstrate that the proposed algorithm attains superior accuracy with fewer expansion terms. Full article

The Horadam sequence ${\left\{H_n(a,b;p,q)\right\}}_{n⩾0}$ has been widely studied in combinatorics and number theory. In this paper, we find that the context-free grammar $G=\{x\to px+y,y\to qx\}$ can be used to generate Horadam sequences. Using this grammar, we deduce several identities, including Cassini-like identities. Moreover, we investigate the relationship between two distinct Horadam sequences $H_n(a,b;p,q)$ and $H_n(c,d;p,q)$ with $(a,b)\ne (c,d)$ and provide an approach to derive identities, which can be illustrated by the Fibonacci and Lucas sequences as well as the two kinds of Chebyshev polynomials. Full article