Search Results (260)

Direction-of-arrival (DOA) estimation of cyclostationary signals is an important problem in array signal processing, especially in sensor-limited and underdetermined scenarios. Sparse arrays and cyclostationary statistics can improve virtual degrees of freedom and target selectivity, but incomplete difference coarray information caused by missing lags may degrade virtual covariance reconstruction and reduce the reliability of DOA estimation in closely spaced, coherent, and interference-contaminated environments. To address this issue, this paper proposes a cyclostationary DOA estimation method based on an augmented extended coprime array (AECA), SVT-based hole recovery, and weighted atomic norm minimization (ANM). The proposed method first constructs the cyclic correlation matrix at the target cyclic frequency and maps it into the AECA-based virtual coarray domain. Redundant lag observations are then aggregated, and an iterative hole recovery procedure is applied to obtain an initial structured virtual covariance matrix. On this basis, a weighted ANM-based covariance refinement model is introduced, where directly observed lags and SVT-recovered hole entries are assigned different confidence levels. The final DOA estimates are obtained using MUSIC on the refined virtual covariance matrix. Simulation results under the considered underdetermined, closely spaced, coherent-source, and interference-contaminated scenarios show that the proposed method achieves lower RMSE and clearer spectral responses than the selected baseline methods. Additional ablation, parameter sensitivity, cyclic frequency mismatch, non-Gaussian noise, and runtime analyses further clarify the contribution, robustness range, and computational cost of the proposed framework. Full article

This paper presents a four-component integrable extension of the derivative nonlinear Schrödinger (DNLS) soliton hierarchy, namely, the Kaup–Newell hierarchy of soliton equations. Motivated by a general extension idea for the Kaup–Newell spectral matrix, we propose a specially constructed 4th-order matrix-valued eigenvalue problem involving four potentials and derive the corresponding integrable Hamiltonian hierarchy via the Lax pair framework. A recursion operator and a bi-Hamiltonian structure are established to demonstrate the Liouville integrability of the resulting hierarchy. As an illustrative example, we derive an integrable system of four DNLS equations, each containing two linear dispersion terms, which differs from standard integrable systems. Full article

This paper aims to introduce a real four-component integrable extension of the complex Kaup–Newell soliton hierarchy. Following a general idea for extending the standard Kaup–Newell spectral matrix, we propose a specific matrix eigenvalue problem involving four real potentials and construct the corresponding integrable Hamiltonian hierarchy via the zero-curvature formulation. A recursion operator and a bi-Hamiltonian structure are presented to demonstrate the Liouville integrability of the resulting hierarchy. As an illustrative example, we derive an integrable system of four real derivative nonlinear Schrödinger equations, each containing two linear dispersion terms and generalizing the standard complex derivative nonlinear Schrödinger equations. Full article

This work extends the analytical operation of the Riemann–RHPHilbert approach (RHA) for fractional-order nonlinear integrable systems under the solvable meaning of inverse scattering transform (IST) to variable-coefficient fractional-order nonlinear models. Firstly, based on the matrix spectral problem proposed by Ablowitz, Kaup, Newell, and Segur, this article derives an integer-order integrable system, which is abbreviated as the AKNS hierarchy. Secondly, by taking specific values of the operator in the derived AKNS hierarchy, a variable-coefficient fractional higher-order NLS hierarchy (vfhNLSH) is obtained, and its anomalous dispersion relation (ADR) is derived via formal solution. Significantly, the reductions of the vfhNLSH include three variable-coefficient fractional-order integrable models: the Hirota equation (vfHE), the Lakshmanan–Porsezian–Daniel equation (vfLPDE), and the fifth-order NLS equation (vffNLSE). Finally, we conduct a detailed study on the representative vfHE as an example rather than a special case and construct its explicit N-fold analytical solution based on the extension of the RHA. At the same time, numerical visualization simulations are conducted to demonstrate the waveform structure characteristics of the solutions under $N=1$ and $N=2$ conditions, including solitons, breathers, and their coupled nonlinear waves. The same process is fully applicable to the other two reduced models, with only some differences in the related results and the dynamic behavior of the solutions. It is shown that the temporal part of the Lax pair associated with the vfHE cannot yet be explicitly determined. Therefore, the fractional-order extension of the RHA presented in this article constitutes a formal or RHA-inspired construction, rather than a fully rigorous fractional-order RHA extension. Full article

In this paper, we develop a Legendre spectral operational matrix method for the numerical solution of two-dimensional Volterra–Fredholm integro-differential equations subject to mixed boundary conditions. The proposed approach transforms the physical domain onto a reference square and approximates the unknown solution using a tensor-product Legendre polynomial expansion. Exact operational matrices for differentiation and lower-limit integration are constructed, allowing the original integro-differential problem to be reduced systematically to a finite-dimensional algebraic system for the spectral coefficients. The formulation provides a unified treatment of differential, Volterra, and Fredholm operators within a single spectral framework and avoids complicated discretizations of multidimensional integral terms. For a specialized linear form of the problem, rigorous convergence estimates are established in both $L^2$ and $L^{\infty }$ norms under suitable regularity assumptions on the coefficients and kernels. The analysis shows that the dominant convergence behavior is governed by the differential operator, while the integral terms contribute only higher-order consistency effects. Several benchmark examples involving both linear and nonlinear two-dimensional integro-differential equations are presented to demonstrate the performance of the proposed method. Numerical results exhibit rapid spectral-type error decay as the polynomial degree increases, with the numerical errors approaching machine precision for moderate truncation orders. These results confirm the accuracy, efficiency, and reliability of the proposed Legendre spectral operational matrix framework for solving a broad class of multidimensional integro-differential equations with nonlocal operators. Full article

Solving fractional differential equations using spectral collocation methods based on classical orthogonal polynomials often leads to a reduced convergence rate due to the limited regularity of the solutions. Therefore, spectral collocation methods that employ non-smooth orthogonal functions are frequently preferred for solving various fractional differential equations. This study focuses on solving one- and two-dimensional time-fractional diffusion-wave equations (DWEs). A spectral collocation technique is developed based on fractional-order orthogonal Jacobi functions to approximate the time-fractional derivatives and orthogonal Jacobi polynomials in the spatial directions. For the first time, a fractional-order orthogonal Jacobi functions-based operational matrix is derived and combined with an orthogonal Jacobi polynomials-based operational matrix of second-order derivatives to solve one- and two-dimensional time-fractional DWEs. Three test problems are conducted to evaluate the efficiency of the proposed numerical technique. Full article

In this paper, we discuss extensions of the canonical quantization procedure in quantum field theories. We focus specifically on S-matrix representation as a T-exponent. This extension involves flat bundles on certain infinite dimensional functional manifolds of local time. The motivating problem is first principles treatment of bound states in quantum chromodynamics as well as precision physics of the hydrogen atom and the muonium. Our main results include systematic treatment of flat bundles in an infinite dimensional setting, generalization of Hamiltonian evolution and functional renormalization group evolution equations in quantum field theories. We discuss several results from finite dimensional theory that have analogies in the functional setting. This includes construction of moduli space of flat connections and isomonodromic deformations. One of the outcomes of our analysis is a construction of a rich family of functional flat bundles with rational connections. This class of connections exhibits a rich set of mathematical properties. In particular, we construct examples of the fundamental groups of spaces which have a definable continuum of generators. Physical states correspond to points in the moduli space of bundles on these spaces. On the physics side of things, we conclude that spacetime notions, such as spaces of particle configurations, emerge effectively as spectral sets of functional differential operators. Full article

Existing multi-view clustering approaches based on matrix factorization often fail to jointly capture global high-order correlations and local view-specific characteristics, and they typically suffer from instability in generating final clustering labels. To overcome these limitations, this paper presents a multi-view subspace clustering method termed dual-tensor constrained multi-view subspace clustering (DTCMVSC). Specifically, for each view, we learn an independent latent representation matrix, a projection matrix, and a basis matrix. The latent representations and projection matrices are stacked into third-order tensors, upon which tensor nuclear norm regularization is imposed to simultaneously exploit consensus structures and complementary information across views. Additionally, a consensus regularization term and adaptive view weights are introduced to align the latent representations of different views toward a unified consensus subspace. The resulting optimization problem is efficiently solved under the ADMM framework, after which a similarity matrix is constructed from the consensus representation and spectral clustering is performed to obtain the final labels. Experimental evaluations on six benchmark datasets demonstrate the superiority of DTCMVSC. Specifically, it achieves an ACC of 86.10% on CMU and an NMI of 94.17% on ORL, surpassing even the lowest-performing state-of-the-art baselines by 63.08 and 18.53 percentage points, respectively. Full article

In this paper, a new class of special polynomials, called the Sheffer-type general-$\lambda$-matrix polynomials, is introduced within the framework of the monomiality principle. This family is obtained by combining the structure of Sheffer sequences with the theory of general-$\lambda$ matrix polynomials, which leads to a unified formulation encompassing several polynomial families. Fundamental properties of the proposed polynomials are established, including their generating function, explicit series representation, summation formulas, quasi-monomial structure, differential relations, and determinant representation. The proposed framework addresses an important problem in the theory of special functions: the systematic construction of matrix-valued polynomial families that simultaneously generalize both classical scalar polynomials and existing matrix polynomial hierarchies. Such a unified structure is of broad significance, with applications in quantum mechanics (wave function expansions), mathematical physics (matrix differential equations and spectral problems), approximation theory, and the study of special functions in the matrix domain. Several hybrid forms of the proposed family are derived through appropriate choices of the defining functions, which yield polynomial subclasses related to classical families such as Hermite, Laguerre, Bessel, and Poisson–Charlier polynomials. These subclasses illustrate how the proposed framework provides a systematic approach for constructing and studying generalized polynomial structures. In each case, the matrix parameter L introduces a new layer of structural richness not present in the scalar setting, enabling the modelling of phenomena governed by matrix-valued spectral data. Furthermore, a numerical and graphical investigation of selected hybrid forms is carried out using Mathematica (version 14.3, 2025; Wolfram Research, Inc.). Surface plots, distributions of complex zeros, and real-zero patterns are presented for different parameter values, highlighting the influence of the parameters on the behavior and structural characteristics of the polynomials. Full article

Monocular depth estimation is a central problem in computer vision with applications in robotics, augmented reality, and autonomous driving, yet the self-attention mechanisms used by modern Transformer architectures remain opaque. In this work, we integrate SVD-Inspired Attention (SVDA) into the Dense Prediction Transformer (DPT), introducing a spectrally structured attention formulation for dense prediction that decouples directional alignment from spectral modulation through a learnable diagonal matrix embedded in normalized query–key interactions. Experiments on KITTI and NYU-v2 show that SVDA preserves competitive predictive performance while enabling intrinsic interpretability: on KITTI, AbsRel improves from 0.058 to 0.056 and ${\delta }_1$ from 0.976 to 0.979, while on NYU-v2, AbsRel improves from 0.133 to 0.124 and ${\delta }_1$ from 0.865 to 0.872. This is achieved with only 0.01% additional parameters, at the cost of a measurable runtime overhead associated with the added normalization and spectral modulation. More importantly, SVDA enables six spectral indicators that quantify entropy, rank, sparsity, alignment, selectivity, and robustness, revealing consistent cross-dataset and depth-wise patterns in how attention organizes during training. These properties make the model easier to inspect and better suited to applications where transparency and reliability are important, such as robotics and autonomous navigation. Full article

This paper develops a structured analytical framework and a robust numerical methodology for the spectral stability of time-varying bilateral quaternion differential equations of the form $\dot{q}=A\left(t\right)q+qB\left(t\right)$. By systematically extending classical real matrix theory to non-commutative dynamical systems via exact isometric real representations, this study utilizes the Kronecker product of real adjoint matrices to rigorously elucidate the underlying tensor structure of the bilateral evolution operator. This tensor-based reformulation proves that the Floquet multipliers of the bilaterally coupled system can be strictly decoupled into the product of the spectra corresponding to the left and right unilateral subsystems. Second, a “Scalar-Vector Stability Separation Principle” based on logarithmic norms is proposed, demonstrating that the transient energy evolution of the system is governed exclusively by the Hermitian real parts of the coefficient matrices, remaining entirely independent of the anti-Hermitian imaginary parts (rotation terms). Furthermore, for constant-coefficient and slowly varying systems, the Riesz projection from holomorphic functional calculus is introduced to establish algebraic criteria for exponential dichotomies, thereby revealing a cubic scaling law that relates the robustness threshold to the spectral gap (${\epsilon }_0\propto {\beta }^3$). Numerically, a Quaternion Chebyshev Spectral Collocation Method (Q-CSCM) is embedded within this exact vectorization framework to ensure that the algebraic symmetries of the bilateral system are strictly preserved through the isomorphic mapping. By explicitly constructing the fully discrete Kronecker product matrix via the exact real vectorization isomorphism, discrete energy estimates are utilized to rigorously prove that the numerical scheme successfully inherits the intrinsic spectral accuracy of the Chebyshev approximation. Comprehensive numerical experiments demonstrate that, within the low-dimensional regime, this methodology exhibits substantial temporal approximation efficiency advantages and superior numerical robustness compared to an alternative Legendre spectral baseline, as well as traditional explicit and state-of-the-art implicit symplectic Runge–Kutta methods, particularly when solving stiff and critically stable problems such as nonlinear Riccati oscillators. Full article

This paper investigates the linear stability of Navier–Stokes–Voigt (NSV) fluid flow in a channel filled with a homogeneous porous medium under general asymmetric slip boundary conditions. This study bridges the research gap between idealized theoretical models (uniform coating) and realistic engineering surfaces in superhydrophobic channels. In practice, manufacturing defects often lead to non-uniform slip distributions. By solving the generalized eigenvalue problem using the Chebyshev spectral collocation method, we quantify the sensitivity of the critical Reynolds number to symmetry breaking. The results reveal that symmetric slip achieves optimal stability, whereas symmetry breaking causes a significant destabilizing effect. Energy analysis clarifies the physical origin of this instability. Furthermore, we find that increasing the porous medium permeability parameter or the Voigt regularization parameter effectively counteracts the slip-induced instability. Specifically, flow stability can be restored even under highly asymmetric slip conditions if the porous damping or the viscoelastic regularization effect is sufficiently strong. This implies that inevitable manufacturing defects in engineering can be compensated for by optimizing the porous medium matrix. Full article

Estimating the position and orientation of a camera with respect to an observed scene remains a fundamental problem in computer vision, particularly in calibration procedures and multi-sensor vision systems. This paper revisits the planar Perspective–n–Point (PnP) problem with emphasis on rotation representation, initialization strategy, and optimization behavior. We propose the PnP-ProCay78 algorithm, which combines analytical elimination of translation via quadratic reconstruction error with nonlinear least-squares minimization of projection residuals in Cayley parameter space. A deterministic initialization scheme based on canonical directions of the reconstruction matrix eliminates the need for spectral search over the full solution space. Experimental evaluation on heterogeneous datasets acquired from high-resolution RGB cameras and low-resolution thermal cameras demonstrates that the proposed method achieves reprojection accuracy comparable to state-of-the-art OpenCV implementations such as SQPnP and IPPE. Convergence analysis in Cayley space reveals stable and rapidly contracting optimization trajectories, with consistent behavior across sensors of significantly different resolution and noise characteristics. The results indicate that a carefully chosen rotation parameterization combined with a transparent optimization framework can yield competitive numerical performance while maintaining geometric interpretability and structural simplicity. Full article

Spectral reconstruction is an important way to acquire a high-spatial-resolution multispectral image, and the spectral reconstruction algorithm is the key to its implementation. This work conducts a comparative study of spectral reconstruction algorithms under different asymmetric influencing factors to provide an overview of their performance. Seventeen spectral reconstruction algorithms with different mathematical bases were implemented and compared regarding their adaptability to response format, imaging noise, spectra type, exposure change, and the quality of the reconstructed images. Based on the principles and characteristics of the algorithm, qualitative and quantitative statistical analyses of the results were carried out. Results show that most of the current algorithms: (1) are adaptive to raw and image signal processing (ISP) responses for spectral reconstruction, (2) decrease their spectral reconstruction accuracy with an increase in imaging noise, (3) give poor performance in reconstructing smooth spectra using non-smooth spectra, and (4) exhibit different degrees of sensitivity to exposure changes. In addition, the image quality reconstructed using the raw response is superior to the ISP response. Full article

In order to overcome the computational challenges associated with block preconditioners for Krylov subspace methods, particularly those arising from Schur complement systems, this paper proposes an improved new block (INB) preconditioner for solving 3 × 3 block saddle point problems. A detailed semi-convergence analysis of the iterative scheme induced by the INB preconditioner is provided. Moreover, the spectral properties of the preconditioned matrix are analyzed, revealing strong eigenvalue clustering around one. Efficient formulas for selecting quasi-optimal parameters are derived based on Frobenius-norm minimization. Extensive numerical experiments demonstrate that the proposed INB preconditioner significantly reduces iteration counts and CPU time compared with several existing block preconditioners. Full article