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This letter proposes an efficient tensor model for Direction of Departure (DOD) and Direction of Arrival (DOA) estimation based on bistatic nested Multiple‐Input Multiple‐Output (MIMO) radar. Tensor offers a robust solution in characterizing multi‐dimensional data structures, ensuring data integrity, and has demonstrated remarkable effectiveness in the realm of multi‐dimensional parameter estimation. However, the complexity of traditional methods for constructing tensors is a major obstacle to practical applications. In response, this research introduces a novel approach that utilizes an iterative two‐dimensional Toeplitz matrix reconstruction method to rapidly derive expanded virtual covariance matrices prior to tensor construction. This streamlines the algorithm and improves efficiency. Numerical simulations have verified the superiority of this algorithm. [ABSTRACT FROM AUTHOR]

Using some concepts and tools from Functional Analysis, we express explicit forms of 'eigenvectors' of two Toeplitz operators: An operator of multiplicity one, which appears e.g. in Autoregressive Process, and a banded operator of high multiplicity. Then we investigate the approximate eigenvectors of their associated Toeplitz matrices of large sizes, and after some numerical experiments, the paper concludes with an application in the principal component analysis method for dimension reduction. [ABSTRACT FROM AUTHOR]

In this work, we explore an extension of the Omega calculus in the context of matrix analysis introduced recently by Neto [Matrix analysis and Omega calculus. SIAM Rev. 2020;62(1):264–280]. We obtain Omega representations of analytic functions of three important classes of matrices: companion, tridiagonal, and triangular. Our representation recovers the main results of Chen and Louck [The combinatorial power of the companion matrix. Linear Algebra Appl. 1996;232:261–278] on the powers of the companion matrix. Furthermore, we generalize previous work on the powers of tridiagonal matrices due to Gutiérrez–Gutiérrez in [Powers of tridiagonal matrices with constant diagonals. Appl Math Comput. 2008;206(2):885–891], Öteleş and Akbulak [Positive integer powers of certain complex tridiagonal matrices. Appl Math Comput. 2013;219(21):10448–10455], and triangular matrices following Shur [A simple closed form for triangular matrix powers. Electron J Linear Algebra. 2011;22:1000–1003]. [ABSTRACT FROM AUTHOR]

Tridiagonal Toeplitz P-matrices, M-matrices and totally positive matrices are characterized. For some classes of tridiagonal matrices and tridiagonal Toeplitz matrices, it is shown that many algebraic computations can be performed with high relative accuracy. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the nonlinear Fredholm integro-differential equation of the second kind with singular kernel in two-dimensional NT-DFIDE. Furthermore, we study this new equation numerically. The existence of a unique solution of the equation is proved. The numerical results of NT-DFIDE are obtained by the following methods: Toeplitz matrix method (TMM) and product Nystrom method (PNM). The given applications showed the efficiency of these methods. [ABSTRACT FROM AUTHOR]

Under appropriate technical assumptions, the simple-loop theory allows to derive various types of asymptotic expansions for the eigenvalues of Toeplitz matrices generated by a function f. Independently and under the milder hypothesis that f is even and monotone over [0,π], matrix-less algorithms have been developed for the fast eigenvalue computation of large Toeplitz matrices, within a linear complexity in the matrix order: behind the high efficiency of such algorithms there are the expansions predicted by the simple-loop theory, combined with the extrapolation idea. Here we focus our attention on a change of variable, followed by the asymptotic expansion of the new variable, and we adapt the matrix-less algorithm to the considered new setting. Numerical experiments show a higher precision (till machine precision) and the same linear computation cost, when compared with the matrix-less procedures already presented in the relevant literature. Among the advantages, we concisely mention the following: (a) when the coefficients of the simple-loop function are analytically known, the algorithm computes them perfectly; (b) while the proposed algorithm is better or at worst comparable to the previous ones for computing the inner eigenvalues, it is vastly better for the computation of the extreme eigenvalues; a mild deterioration in the quality of the numerical experiments is observed when dense Toeplitz matrices are considered, having generating function of low smoothness and not satisfying the simple-loop assumptions. [ABSTRACT FROM AUTHOR]

Fractional Black-Scholes (FBS) equation has been widely applied in options pricing problems. However, most of the literatures focus on pricing the single-asset option using the FBS model, and research on time-space FBS with multiple assets has remained vacant. Option pricings are always governed by multiple assets. Under the assumption that fluctuation of asset price is regarded as a fractal transmission system and follows two independent geometric Lévy processes, 2D time-space fractional Black-Scholes equation (TDTSFBSE) is proposed to describe the instantaneous price in the financial market. In this work, we discrete the TDTSFBSE with implicit finite difference and present a fast parallel all-at-once iterative method for the resulting linear system. The proposed method can greatly reduce the storage requirements and computational cost. Theoretically, we discuss the convergence property of the fast iterative method. Numerical examples are given to illustrate the effectiveness and efficiency of our method. [ABSTRACT FROM AUTHOR]

For the well-known exponential complexity, it is a giant challenge to calculate the correlation function for general many-body wave function. We investigate the ground state nth-order correlation functions of the Tonks–Girardeau (TG) gases. Basing on the wavefunction of free fermions and Bose–Fermi mapping method, we obtain the exact ground state wavefunction of TG gases. Utilizing the properties of Vandermonde determinant and Toeplitz matrix, the nth-order correlation function is formulated as (N - n) -order Toeplitz determinant, whose element is the integral dependent on 2 (N - n) sign functions and can be computed analytically. By reducing the integral on domain [ 0 , 2 π ] into the summation of the integral on several independent domains, we obtain the explicit form of the Toeplitz matrix element ultimately. As the applications we deduce the concise formula of the reduced two-body density matrix and discuss its properties. The corresponding natural orbitals and their occupation distribution are plotted. Furthermore, we give a concise formula of the reduced three-body density matrix and discuss its properties. It is shown that in the successive second measurements, atoms appear in the regions where atoms populate with the maximum probability in the first measurement. [ABSTRACT FROM AUTHOR]

This paper is concerned with networks of identical linear quantum stochastic systems which interact with each other and external bosonic fields in a translation invariant fashion. The systems are associated with sites of a multidimensional lattice and are governed by coupled linear quantum stochastic differential equations (QSDEs). The block Toeplitz coefficients of these QSDEs are specified by the energy and coupling matrices which quantify the Hamiltonian and coupling operators for the component systems. We discuss the invariant Gaussian quantum state of the network when it satisfies a stability condition and is driven by statistically independent vacuum fields. A quadratic-exponential functional (QEF) is considered as a risk-sensitive performance criterion for a finite fragment of the network over a bounded time interval. This functional involves a quadratic function of dynamic variables of the component systems with a block Toeplitz weighting matrix. Assuming the invariant state, we study the spatio-temporal asymptotic rate of the QEF per unit time and per lattice site in the thermodynamic limit of unboundedly growing time horizons and fragments of the lattice. A spatio-temporal frequency-domain formula is obtained for the QEF rate in terms of two spectral functions associated with the real and imaginary parts of the invariant quantum covariance kernel of the network variables. A homotopy method and asymptotic expansions for evaluating the QEF rate are also discussed. [ABSTRACT FROM AUTHOR]

Linearly structured covariance matrices are widely used in multivariate analysis. The covariance structure can be chosen from a class of linear structures. Therefore, the optimal structure is identified in terms of minimizing the discrepancy function. In this research, the entropy loss function is used as the discrepancy function. We give a methodology and algorithm for determining the optimal structure from the class of structures under consideration. The accuracy of the proposed method is checked using a simulation study. [ABSTRACT FROM AUTHOR]

Aiming at the problems of poor accuracy, low angular resolution, and low degree of freedom in traditional bistatic nested multiple input multiple output (MIMO) radar for target parameter estimation, a method based on the use of virtual redundant array elements is proposed. The reconstructed Toeplitz matrix algorithm is a method of parameter estimation for the direction of departure (DOD) and direction of arrival (DOA) of the target. Firstly, the two nested sub-arrays are arranged with the zero point as the center to form the receiving array and the transmitting array of the bistatic MIMO radar. The virtual receiving signal processed by the array has a large number of redundant virtual array elements. Second, the covariance values corresponding to the redundant virtual array elements are averaged to replace the original values to form a new virtual received signal, and then two selection matrices are used to construct spatial smoothing sub-arrays in the virtual receiving array and the virtual transmitting array. The method reconstructs the Toeplitz matrix to reconstruct the virtual received signal. Finally, the conventional subspace algorithm is used to estimate the space spectrum of the equivalent virtual signal to realize the mutual matching of DOD and DOA. The proposed algorithm compares better with other algorithms in terms of estimation performance and degree of freedom performance. [ABSTRACT FROM AUTHOR]

A quasi-Toeplitz matrix is a semi-infinite matrix of the form A = T (a) + E , where T (a) is a Toeplitz matrix with entries (T (a)) i , j = a j − i , for a j − i ∈ C , i , j ≥ 1 and E is a compact correction. Quasi-Toeplitz M -matrices are encountered in the study of quadratic matrix equations arising in the analysis of a 2-dimensional Quasi-Birth-Death (QBD) stochastic process. We investigate properties of such matrices and provide conditions under which a quasi-Toeplitz matrix is an M -matrix. We show that under a mild and easy-to-check condition, an invertible quasi-Toeplitz M -matrix has a unique square root that is an M -matrix possessing quasi-Toeplitz structure. The quasi-Toeplitz structure of the square root of M -matrices provides inspirations for proving spectral properties of the quadratic matrix polynomial L (λ) = λ 2 A 1 + λ A 0 + A − 1 having quasi-Toeplitz coefficients where A 1 , A − 1 ≥ 0 and − A 0 in addition is an M -matrix. Some issues concerning the computation of square root of quasi-Toeplitz M -matrices are discussed and numerical experiments are performed. [ABSTRACT FROM AUTHOR]

In this paper we study the eigenvalues of the laplacian matrices of the cyclic graphs with one edge of weight α and the others of weight 1. We denote by n the order of the graph and suppose that n tends to infinity. We notice that the characteristic polynomial and the eigenvalues depend only on Re (α). After that, through the rest of the paper we suppose that 0 < α < 1. It is easy to see that the eigenvalues belong to [ 0 , 4 ] and are asymptotically distributed as the function g (x) = 4 sin 2 ⁡ (x / 2) on [ 0 , π ]. We obtain a series of results about the individual behavior of the eigenvalues. First, we describe more precisely their localization in subintervals of [ 0 , 4 ]. Second, we transform the characteristic equation to a form convenient to solve by numerical methods. In particular, we prove that Newton's method converges for every n ≥ 3. Third, we derive asymptotic formulas for all eigenvalues, where the errors are uniformly bounded with respect to the number of the eigenvalue. [ABSTRACT FROM AUTHOR]

The problem of estimating a large covariance matrix arises in various statistical applications. This paper develops new covariance matrix estimators based on shrinkage regularization. Individually, we consider two kinds of Toeplitz-structured target matrices as the data come from the complex Gaussian distribution. We derive the optimal tuning parameter under the mean squared error criterion in closed form by discovering the mathematical properties of the two target matrices. We get some vital moment properties of the complex Wishart distribution, then simplify the optimal tuning parameter. By unbiasedly estimating the unknown scalar quantities involved in the optimal tuning parameter, we propose two shrinkage estimators available in the large-dimensional setting. For verifying the performance of the proposed covariance matrix estimators, we provide some numerical simulations and applications to array signal processing compared to some existing estimators. [ABSTRACT FROM AUTHOR]

In this paper, we propose an extended aperture direction-of-arrival (DOA) estimation algorithm using a coprime acoustic vector sensor array. Each acoustic vector sensor consists of one pressure sensor and one uniaxial velocity sensor, collocated in a single point in space. By full utilization of both pressure and particle-velocity information in the acoustic wavefield, the whole coarray output is shown to be treated as four pressure–velocity (p–v) components. In this way, array interpolation and Toeplitz matrix reconstruction are performed on each of the four p–v components to exploit the entire coarray aperture. After interpolation, DOA estimates are derived in closed form using the interpolated coarray data. The proposed algorithm allows the inter-sensor spacing of the interpolated coarray to be significantly greater than half a wavelength, thereby achieving aperture extension without needing extra sensors. Moreover, it enables provide the unambiguous DOA estimates with high accuracy, despite the phase aliasing caused by aperture extension. All these advantages are not offered by the existing coprime array DOA estimation algorithms. Finally, computer simulations are carried out to verify the superiority of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we study a fast algorithm for the numerical solution of the 1D distributed-order space-fractional diffusion equation. After discretization by the finite difference method, the resulting system is the symmetric positive definite Toeplitz matrix. The preconditioned conjugate gradient method with a circulant preconditioner is employed to solve the linear system. Theoretically, the spectrum of the preconditioned matrix is proved to be clustered around 1, which can guarantee the superlinear convergence rate of the proposed method. Numerical experiments are carried out to demonstrate the effectiveness of our proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we determine the approximate eigenvalue bound state solution of the radial Schrodinger equation in three-dimension in the presence of the Aharonov-Bohm flux field with two-term diatomic molecular potential in point-like global monopole (PGM) defect. We analyse the effects of factors, such as topological defects, background curvature associated with topological defects as well as the potential. We also show that the energy levels shift due to the presence of the magnetic flux which gives us an analogue of the Aharonov-Bohm effect for the bound state. Finally, we apply the quantum system for some known potential models, such as q-deformed Hulthen-type potential and Manning-Rosen potential, and discussed the effects of various factors on the eigenvalue solutions. [ABSTRACT FROM AUTHOR]

The compressive sensing method is an effective way of under-sampling array signal processing, and the measurement matrix is a key technology of compressive sensing processing, which is of decisive significance to improve the performance of under-sampling array signal processing. In this paper, an under-sampling array signal processing method based on an improved Hadamard measurement matrix was proposed. This method improved the construction of the measurement matrix by a compressing zero method capable of reducing the redundant data, enhancing the non-correlation between columns, improving the RIP condition, and improving the performance of the compressive sensing method. The performance was verified by simulation and real data from the lake, and the results showed that the method proposed in this paper has obvious performance advantages compared with the Toeplitz Circular Matrix, the m-sequence-based matrix, and the partial Hadamard matrix under the same conditions. [ABSTRACT FROM AUTHOR]

A powerful tool for analyzing and approximating the singular values and eigenvalues of structured matrices is the theory of Generalized Locally Toeplitz (GLT) sequences. By the GLT theory one can derive a function, called the symbol, which describes the singular value or the eigenvalue distribution of the sequence, the latter under precise assumptions. However, for small values of the matrix-size of the considered sequence, the approximations may not be as good as it is desirable, since in the construction of the GLT symbol one disregards small norm and low-rank perturbations. On the other hand, Local Fourier Analysis (LFA) can be used to construct polynomial symbols in a similar manner for discretizations, where the geometric information is present, but the small norm perturbations are retained. The main focus of this paper is the introduction of the concept of sequence of "Toeplitz momentary symbols", associated with a given sequence of truncated Toeplitz-like matrices. We construct the symbol in the same way as in the GLT theory, but we keep the information of the small norm contributions. The low-rank contributions are still disregarded, and we give an idea on the reason why this is negligible in certain cases and why it is not in other cases, being aware that in presence of high nonnormality the same low-rank perturbation can produce a dramatic change in the eigenvalue distribution. Moreover, a difference with respect to the LFA symbols is that GLT symbols and Toeplitz momentary symbols are more general - just Lebesgue measurable - and are applicable to a larger class of matrices, while in the LFA setting only trigonometric polynomials are considered and more specifically those related to the approximation stencils. We show the applicability of the approach which leads to higher accuracy in some cases, when approximating the singular values and eigenvalues of Toeplitz-like matrices using Toeplitz momentary symbols, compared with the GLT symbol. Finally, since for many applications and their analysis it is often necessary to consider non-square Toeplitz matrices, we formalize and provide some useful definitions, applicable for non-square Toeplitz momentary symbols. [ABSTRACT FROM AUTHOR]

Structural health monitoring (SHM) of hydraulic steel gates is one of the most important technologies for the safety of water conservancy and agriculture. Among them, the modal parameter identification of SHM can provide the key information for the gate vibration control, model correction, and damage identification. However, the calculation peaks can be generated by the overestimation of the model order, while the noise spikes can be introduced by the measurement noise in the process of modal parameter identification of hydraulic gate operation. There is also a great interference with the modal parameters. Some manual participation can be required in the modal model grading and modal selection during steady-state graph recognition. In this study, an improved potential energy clustering (PHA) covariance-driven stochastic subspace (COV-SSI) hydraulic gate automatic modal parameter identification was proposed to automatically identify the operational modal parameters of hydraulic gates under the flow release excitation. The modal parameters of the hydraulic gate were identified using only the output response signal under the structural drainage excitation. The dynamic characteristics of the structure were also revealed under the real boundary and load working conditions. Firstly, the vibration signals were collected to process by noise reduction using the wavelet threshold method. The spurious modes were then reduced to optimize the quality of vibration signals due to environmental noise. Secondly, the Toeplitz matrix was constructed using the COV-SSI. The response signals were then obtained to decompose by singular value decomposition (SVD). The system order n and the maximum order nmax of the steady-state graph were automatically determined by the singular value weighted judgment index (SWI). The noise mode was also eliminated in the steady-state graph. Then, the PHA was used to realize the automatic identification of structural model parameters. Finally, the accuracy of the improved model was verified by the numerical calculation of the two-degree-of-freedom mass-spring-damping system, followed by an experimental model of an arc gate under drainage excitation, and the comparison with the hammering test. The results show that the new identification was automatically determined the system order and the maximum order of the steady-state diagram without the artificial excitation, and the false poles in the steady-state diagram. An automatic identification was also realized for the model parameters of hydraulic gates under drainage excitation. The maximum and average relative errors of the test were 8.5% and 3.5%, respectively, compared with the hammering method. Therefore, the COV-SSI and potential clustering can be expected to identify the online modal parameters of hydraulic gates. The automatically fixed-order stability diagrams were reduced the influence of human factors, indicating the better identification of modal parameters in the hydraulic gates under the discharge excitation. This finding can provide a promising application for the health monitoring and safety analysis during the hydraulic gates in service. Especially, the artificial forces and excitation signal measurement were greatly reduced for the modal parameters. [ABSTRACT FROM AUTHOR]

In this article, we focus on a two‐dimensional conservative steady‐state Riesz fractional diffusion problem. As is typical for problems in conservative form, we adopt a finite volume (FV)‐based discretization approach. Precisely, we use both classical FVs and the so‐called finite volume elements (FVEs). While FVEs have already been applied in the context of fractional diffusion equations, classical FVs have only been applied in first‐order discretizations. By exploiting the Toeplitz‐like structure of the resulting coefficient matrices, we perform a qualitative study of their spectrum and conditioning through their symbol, leading to the design of a second‐order FV discretization. This same information is leveraged to discuss parameter‐free symbol‐based multigrid methods for both discretizations. Tests on the approximation error and the performances of the considered solvers are given as well. [ABSTRACT FROM AUTHOR]

It is well known that the discretization of fractional diffusion equations with fractional derivatives α∈(1,2)$$ \alpha \in \left(1,2\right) $$, using the so‐called weighted and shifted Grünwald formula, leads to linear systems whose coefficient matrices show a Toeplitz‐like structure. More precisely, in the case of variable coefficients, the related matrix sequences belong to the so‐called generalized locally Toeplitz class. Conversely, when the given FDE has constant coefficients, using a suitable discretization, we encounter a Toeplitz structure associated to a nonnegative function ℱα$$ {\mathcal{F}}_{\alpha } $$, called the spectral symbol, having a unique zero at zero of real positive order between one and two. For the fast solution of such systems by preconditioned Krylov methods, several preconditioning techniques have been proposed in both the one‐ and two‐dimensional cases. In this article we propose a new preconditioner denoted by 풫ℱα which belongs to the τ$$ \tau $$‐algebra and it is based on the spectral symbol ℱα$$ {\mathcal{F}}_{\alpha } $$. Comparing with some of the previously proposed preconditioners, we show that although the low band structure preserving preconditioners are more effective in the one‐dimensional case, the new preconditioner performs better in the more challenging multi‐dimensional setting. [ABSTRACT FROM AUTHOR]

In this paper, the banded Toeplitz matrices generated by f(θ) = (2(1 − cos(θ − θ̃)))d are studied. The function f is a real non-negative function with a zero of order 2d at θ̃ and the generated matrices are ill-conditioned Hermitian positive definite. We show that these banded Toeplitz matrices are similar to the banded real symmetric positive definite Toeplitz matrices that are generated by f(θ) = (2(1−cos(θ)))d . A fast direct solver is proposed to compute the inverse of these real matrices. Numerical experiments show that our proposed method is faster and more stable than the stable Levinson algorithm. [ABSTRACT FROM AUTHOR]

In this note, we revisit the result of the Toeplitz matrix inversion formula proposed by Lv and Huang (2007), then we give a new structured perturbation analysis, which is useful for testing the stability of practical algorithms. It has been shown that our new upper bound is much sharper than the one they have given, especially when the size of the Toeplitz matrix is very large. Numerical experiments illustrate the effectiveness of our theoretical results. [ABSTRACT FROM AUTHOR]

To reduce the complexity of direction-of-arrival (DOA) algorithms in polarization-sensitive arrays and maintain high estimation accuracy while decreasing runtime, this paper proposes a channel compression model based on orthogonal dipole arrays. In addition, a DOA estimation algorithm using atomic norm minimization (ANM) is introduced for this model. This algorithm performs eigenvalue decomposition on the covariance matrix of the data received from the polarization-sensitive channel compressed array. A new observation vector is constructed for the ANM problem under the channel compression model using the obtained eigenvalues and eigenvectors. Subsequently, a Toeplitz matrix is constructed from the optimal solution of the semi-positive definite programming (SDP) problem. The Vandermonde decomposition of the Toeplitz matrix provides estimates of the signal DOA parameters. By combining the vectorization result of the covariance matrix and the least-squares calculation, the polarization auxiliary angle and polarization phase angle information of the incident source are also obtained. Simulation experiments compare the root-mean-squared error (RMSE) of each algorithm at different angular intervals and signal-to-noise ratios (SNR) to confirm the validity of estimating DOA and polarization parameters. [ABSTRACT FROM AUTHOR]

We consider the maximum likelihood estimation of a covariance matrix that is known to be Toeplitz. The corresponding optimization problem is nonconvex, and to compute a local minimizer we propose an efficient implementation of Newton's method. The cost per iteration of a straightforward implementation is of order O (n 4) , where n + 1 is the dimension of the covariance matrix. By exploiting fast Fourier transforms to assemble the gradient and Hessian we reduce the iteration cost to order O (n 3). Through extensive simulations we demonstrate that for direction-of-arrival estimation, the performance of MUSIC is significantly enhanced by using it together with the maximum likelihood Toeplitz covariance estimate. The simulations also demonstrate that our open-source implementation is orders of magnitude faster than two existing Toeplitz covariance estimators. • Maximum likelihood estimation of a Toeplitz covariance matrix can be done efficiently. • The Toeplitz maximum likelihood estimate significantly enhances MUSIC. • We provide an open-source implementation of our method. [ABSTRACT FROM AUTHOR]

We study kill-and-restart and preemptive strategies for the fundamental scheduling problem of minimizing the sum of weighted completion times on a single machine in the non-clairvoyant setting. First, we show a lower bound of 3 for any deterministic non-clairvoyant kill-and-restart strategy. Then, we give for any b>1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$b > 1$$\end{document} a tight analysis for the natural b-scaling kill-and-restart strategy as well as for a randomized variant of it. In particular, we show a competitive ratio of (1+33)≈6.197\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(1+3\sqrt{3})\approx 6.197$$\end{document} for the deterministic and of ≈3.032\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\approx 3.032$$\end{document} for the randomized strategy, by making use of the largest eigenvalue of a Toeplitz matrix. In addition, we show that the preemptive Weighted Shortest Elapsed Time First (WSETF) rule is 2-competitive when jobs are released online, matching the lower bound for the unit weight case with trivial release dates for any non-clairvoyant algorithm. Using this result as well as the competitiveness of round-robin for multiple machines, we prove performance guarantees smaller than 10 for adaptions of the b-scaling strategy to online release dates and unweighted jobs on identical parallel machines. [ABSTRACT FROM AUTHOR]

In this paper, we consider the first Szegő limit theorems on d -torus T d for 1 ≤ d ≤ + ∞. Given φ ∈ L 1 (T d) and a finite subset σ = { ξ 1 ⋯ , ξ n } of the dual group Z d of T d , the truncated Toeplitz matrix with respect to σ is T σ φ = { φ ˆ (ξ j − ξ i) } 1 ≤ i , j ≤ n. For any Følner sequence { σ N } of Z d and φ ∈ L + 1 (T d) , it is shown that lim N → ∞ ⁡ (det ⁡ T σ N φ) 1 | σ N | = exp ⁡ (∫ T d log ⁡ φ d m d). In the case d = + ∞ , we are associated with multiplicative Toeplitz matrix T φ = { φ ˆ (j / i) } i , j ∈ N and the most relevant non-Følner truncation, that is, T N φ = { φ ˆ (j / i) } 1 ≤ i , j ≤ N , where σ N = { 1 , ... , N }. It is shown that for each φ ∈ L R ∞ (T ∞) and f ∈ C [ ess-inf φ , ess-sup φ ] , the limit lim N → ∞ ⁡ 1 N Tr f ( T N φ) exists. Moreover, it is proven that the limit lim N → ∞ ⁡ (det ⁡ T N φ) 1 N exists for any φ ∈ L + 1 (T ∞) with strictly positive essential infimum. These results are directly related to two problems posed by Nikolski and Pushnitski in [30]. [ABSTRACT FROM AUTHOR]

It is known that the solution of a conservative steady-state two-sided fractional diffusion problem can exhibit singularities near the boundaries. As a consequence of this, and due to the conservative nature of the problem, we adopt a finite volume elements discretization approach over a generic non-uniform mesh. We focus on grids mapped by a smooth function which consists in a combination of a graded mesh near the singularity and a uniform mesh where the solution is smooth. Such a choice gives rise to Toeplitz-like discretization matrices and thus allows a low computational cost of the matrix–vector product and detailed spectral analysis. The obtained spectral information is used to develop an ad-hoc parameter-free multigrid preconditioner for GMRES, which is numerically shown to yield good convergence results in presence of graded meshes mapped by power functions that accumulate points near the singularity. The approximation order of the considered graded meshes is numerically compared with the one of a certain composite mesh given in literature that still leads to Toeplitz-like linear systems and is then still well-suited for our multigrid method. Several numerical tests confirm that power-graded meshes result in lower approximation errors than composite ones and that our solver has a wide range of applicability. [ABSTRACT FROM AUTHOR]

To diminish the performance degradation of coprime array beamforming arising from sensor gain and phase uncertainties, we propose a robust beamformer based on covariance matrix modification with augmented instrumental sensors. Initially, the received array data are split into two subarrays. Utilizing the instrumental sensors, we estimate the gain-phase errors of these two subarrays and then recombine them. Subsequently, error compensation is implemented to calibrate the received covariance matrix into a Toeplitz matrix relevant to the coprime array structure, in which the holes need to be further filled by means of some matrix completion operation. Then, the above covariance matrix modification is integrated into the classical beamforming procedure including MUSIC decomposition, spectral peak searching, power estimation, INCM (Interference-plus-Noise Covariance Matrix) reconstruction, thus yielding the final beamformer weight vector. Both theoretical analysis and simulations demonstrate that this method not only can accurately estimate the gain and phase errors, but also can effectively improve the beamformer's output SINR (i.e., the robustness is enhanced), which presents the proposed beamformer with vast potentials in practical sparse-array involved applications. • The designed beamformer can improve the robustness of a coprime array under gain-phase error. • Gain-phase error calibration via setting four instrumental sensors ahead of the coprime array. • Enlarging the continuous range of virtual array and performing matrix completion. [ABSTRACT FROM AUTHOR]

Abstract Coprime arrays (CPAs) have been found to be an effective configuration for adaptive beamforming (ABF). However, most ABF methods for CPAs are embedded with the spatial spectrum estimation technique, and therefore they usually deteriorate in the case of low signal‐to‐noise ratio (SNR), especially without prior information of interferences. To address this issue, a robust ABF method for CPAs via atomic‐norm‐based sparse recovery is proposed. This method begins with initialised virtual array interpolation to avoid aperture loss caused by ‘holes’, which is subsequently transformed to a matrix completion and denoising issue via a Toeplitz step. With the matrix’s Hermitian Toeplitz structure and the intrinsic sparsity of interferences, sparse recovery for the interference covariance matrix is introduced by subtracting the target signal and noise from the received signal and implemented by atomic‐norm minimisation. Concretely, this non‐convex process is decomposed into two convex steps and solved iteratively. Unlike previous methods, this method can interpolate the virtual array and estimate the steering vector of the target and the noise power simultaneously; consequently, the interference‐plus‐noise covariance matrix is reconstructed with merely some trivial entry selections without the need for the spatial spectrum. Furthermore, the optimality conditions and boundedness in this method are proven theoretically. Simulation results demonstrate the effectiveness of the proposed method and its advantages over previous methods under those non‐ideal conditions.

A Pascal-like matrix is constructed whose row entries are the terms of the modified Hermite polynomials of two variables. The multiplication of the two Pascal-like matrices and the power and inverse of the Pascal-like matrix have very similar results to the well-known Binomial Matrix, and the factorization of the Pascal-like matrix has also been given, which is completely similar to the factorization of Binomial Matrix. Furthermore, two simple examples are given to show the application of the power and factorization properties of the Pascal-like matrix. Finally, a generalization of the Pascal-like matrix is given and some combinatorial identities are obtained such as Tepper-like identity. [ABSTRACT FROM AUTHOR]

We investigate the problem of approximating the matrix function f(A) by r(A), with f a Markov function, r a rational interpolant of f, and A a symmetric Toeplitz matrix. In a first step, we obtain a new upper bound for the relative interpolation error 1 − r/f on the spectral interval of A. By minimizing this upper bound over all interpolation points, we obtain a new, simple, and sharp a priori bound for the relative interpolation error. We then consider three different approaches of representing and computing the rational interpolant r. Theoretical and numerical evidence is given that any of these methods for a scalar argument allows to achieve high precision, even in the presence of finite precision arithmetic. We finally investigate the problem of efficiently evaluating r(A), where it turns out that the relative error for a matrix argument is only small if we use a partial fraction decomposition for r following Antoulas and Mayo. An important role is played by a new stopping criterion which ensures to automatically find the degree of r leading to a small error, even in presence of finite precision arithmetic. [ABSTRACT FROM AUTHOR]

In this note, we approximate the von Neumann and Rényi entropies of high-dimensional graphs using the Euler-Maclaurin summation formula. The obtained estimations have a considerable degree of accuracy. The performed experiments suggest some entropy problems concerning graphs whose Laplacians are g -circulant matrices, i.e., circulant matrices with g -periodic diagonals, or quasi-Toeplitz matrices. Quasi means that in a Toeplitz matrix the first two elements in the main diagonal, and the last two, differ from the remaining diagonal entries by a perturbation. [ABSTRACT FROM AUTHOR]

Toeplitz matrices are ubiquitous and play important roles across many areas of mathematics. In this paper, we present some algebraic results concerning block Toeplitz matrices with block entries belonging to a commutative algebra A. The characterization of normal block Toeplitz matrices with entries from a commutative algebra A of normal matrices is also obtained. [ABSTRACT FROM AUTHOR]

The main aim of the present paper is to establish invertibility criteria for Toeplitzplus-Hankel-Bezoutians based on the analysis of the structure of these matrices. In particular, their inverses have an explicit representation as a sum of a Toeplitz and a Hankel matrix whose symbols are the solution of certain linear systems taking the form of generalized resultant equations. These results generalize previous inversion formulas and criteria for the special cases of centrosymmetric or centroskewsymmetric Toeplitz-plus-Hankel-Bezoutians. The inversion of Toeplitz-plus-Hankel-Bezoutians considered here is based on the converse problem, the inversion of Toeplitz-plus-Hankel matrices. Consequently, several modifications of known inversion formulas and new results for Toeplitz-plus-Hankel matrices are developed, which allow a deeper insight into the structure of these matrices, too. [ABSTRACT FROM AUTHOR]

Under the background of spatially correlated color noise, the incidence angle of a jamming signal in a high-speed moving platform rapidly changes, which leads to the degradation of the anti-interference performance and the waveform distortion of the adaptive beamformer. In this paper, a projection-constrained null broadening beamforming algorithm based on the Toeplitz matrix structure is proposed. The algorithm first extracts the subspace of the covariance matrix of the steering vector of the pre-determined extended angle interval and constructs the constraint matrix and the projection transformation matrix. The received signal covariance matrix with a Toeplitz structure is then constructed using the correlation number between each array element and the pre-set reference array element. Finally, the constructed covariance matrix is transformed through projection, and the weight of each array element is constrained by the constraint matrix. The theoretical optimal solution of adaptive wide null beamforming in spatially correlated color noise is obtained. The simulation results show that, compared with the existing robust adaptive beamforming algorithms, the proposed algorithm can efficiently improve the distortion of adaptive anti-jamming beams, and can achieve null broadening in the jamming direction under the condition of spatially correlated color noise, which improves the output signal to the interference-plus-noise ratio (SINR) of the adaptive beamformer. [ABSTRACT FROM AUTHOR]

Stiffened structure-induced gain-phase errors degrade the performance of the high-resolution two-dimensional multiple signal classification (2D-MUSIC) algorithm, which makes it impossible to ensure the high accuracy of impact localization results. To eliminate the localization bias caused by these errors, a calibrated 2D-MUSIC-based impact localization method is first introduced. Firstly, time-frequency characteristics of the non-stationary impact signals are evaluated by experiment to obtain a clear first wave packet or a wave packet that purely corresponds to a single mode through continuous wavelet transform (CWT). Then, the uniform linear array covariance matrix with gain-phase errors is calibrated to be constructed as a Toeplitz structural matrix. By reconstructing covariance matrix R, 2D-MUSIC-based impact localization is calibrated for stiffened curved composite structures. Experimental research on the stiffened curved composite panel is carried out, and these impact localization results demonstrate the validity and effectiveness of the calibrated 2D-MUSIC-based method. [ABSTRACT FROM AUTHOR]

This paper considers an inverse eigenvalue problem for bisymmetric nonnegative matrices. We first discuss the specified structure of the bisymmetric matrices. Then for a given set of real numbers of order maximum five with special conditions, we construct a non-negative bisymmetric matrix such that the given set is its spectrum. Finally, we solve the problem for arbitrary order n in the special case of the spectrum. [ABSTRACT FROM AUTHOR]

The lower triangular Toeplitz matrix of Markov parameters is the finite time, time domain equivalent of the z-transfer function of digital systems. Its properties are therefore of interest, and this paper discovers previously unknown properties. The singular value decomposition (SVD) of the Toeplitz matrix contains the matrix of the singular values, and two orthonormal matrices of input and output singular vectors. An understanding of these three matrices in terms of system frequency response has been available for the limiting case when the Toeplitz matrix tends to infinity. Here we obtain an understanding of what happens to the SVD as the dimension is reduced. A system identification algorithm uses the entries in the input/output singular vectors as samples of a continuous time linear system under free response, with the result that such identified dynamics represent the data. Numerical examples for several different system orders are used to show the characteristics of identified eigenvalues embedded in the singular vectors. Each singular vector contains at least a pure frequency that is close to but not the same as the frequency when the dimension tends to infinity. Other eigenvalues help the output singular vectors to end at zero, and the input singular vectors to start at zero, a property of the finite time matrix. Singular vector dynamics in various Iterative Learning Control (ILC) laws are discussed here and numerically investigated, and the dynamics are relevant to basis function ILC, to finite time low pass filtering design, and to new versions of stable inverse theory, among other applications within the theory of control. [ABSTRACT FROM AUTHOR]

A principal singular vector based on multiple Toeplitz matrices is proposed to solve the accuracy problem of direction-of-arrival (DOA) estimation for coherent signals. First, the data matrix received by uniform linear array (ULA) is transformed into a Toeplitz matrix. An equivalent covariance matrix is obtained by square weighted summation method using the Toeplitz matrix. Then, a polynomial containing DOA information is constructed because the signal space and the steering matrix have the same column space; the Toeplitz matrix is built using polynomial coefficients. The problem is transformed into solving linear equations by establishing the relationship between the Toeplitz matrix and the signal subspace. Furthermore, the weighted least square method is used to obtain multiple candidates for linear equations. Finally, the maximum likelihood (ML) rule is used to select source signal candidates from multiple candidates. In comparison with currently known algorithm, the proposed algorithm has the characteristics of high estimation accuracy, low-complexity, and strong anti-interference ability and resolution. Even when the signal-to-noise ratio (SNR) is low, the snapshot number is small, and multiple signals exist; this method can still provide good estimation performance and resolution, which is more than 90% in most cases. Simulation experiments verify the superiority of the algorithm. [ABSTRACT FROM AUTHOR]

We present an inherently parallel method using a Taylor polynomial as an estimator for several numerical examples of strictly diagonally dominant and banded triangular Toeplitz systems. We demonstrate the effectiveness of the method using multiple precisions for each of four test cases, and compare the results versus classical Jacobi and Newton methods. We also show that the residual norms can be driven down to any arbitrary precision with less hardware requirements than existing methods. Finally, we use truncated Taylor series as initial estimates for the Newton, and in each case, we obtain equal or better accuracy than the Newton method alone, while using less vector computations. [ABSTRACT FROM AUTHOR]

In our present investigation, we extend the idea of q-symmetric derivative operators to multivalent functions and then define a new subclass of multivalent q-starlike functions. For this newly defined function class, we discuss some useful properties of multivalent functions, such as the Hankel determinant, symmetric Toeplitz matrices, the Fekete–Szego problem, and upper bounds of the functional a p + 1 − μ a p + 1 2 and investigate some new lemmas for our main results. In addition, we consider the q-Bernardi integral operator along with q-symmetric calculus and discuss some applications of our main results. [ABSTRACT FROM AUTHOR]

By making use of prestarlike functions, we introduce in this paper a certain family of normalized holomorphic functions defined in the open unit disk, and we establish coefficient estimates for the first four determinants of the symmetric Toeplitz matrices T 2 (2) , T 2 (3) , T 3 (2) and T 3 (1) for the functions belonging to this family. We also mention some known and new results that follow as special cases of our results. [ABSTRACT FROM AUTHOR]

We discuss an explicit algorithm for solving the Wiener–Hopf factorization problem for matrix polynomials. By an exact solution of the problem, we understand the one constructed by a symbolic computation. Since the problem is, generally speaking, unstable, this requirement is crucial to guarantee that the result following from the explicit algorithm is indeed a solution of the original factorization problem. We prove that a matrix polynomial over the field of Gaussian rational numbers admits the exact Wiener–Hopf factorization if and only if its determinant is exactly factorable. Under such a condition, we adapt the explicit algorithm to the exact calculations and develop the ExactMPF package realized within the Maple Software. The package has been extensively tested. Some examples are presented in the paper, while the listing is provided in the electronic supplementary material. If, however, a matrix polynomial does not admit the exact factorization, we clarify a notion of the numerical (or approximate) factorization that can be constructed by following the explicit factorization algorithm. We highlight possible obstacles on the way and discuss a level of confidence in the final result in the case of an unstable set of partial indices. The full listing of the package ExactMPF is given in the electronic supplementary material. [ABSTRACT FROM AUTHOR]

By the principle of differential subordination and the q-derivative operator, we introduce the q-analog SP q snail(; ff; ;) of certain class of analytic functions associated with the generalized Pascal snail. Firstly, we obtain the coefficient estimates and Fekete-Szeg¨o functional inequalities for this class. Meanwhile, we also estimate the corresponding symmetric Toeplitz determinant. Secondly, for all the above results we provide the corresponding results for the reduced classes SP q snail(ff; ;) and RP q snail(ff; ;). Thirdly, we characterize the Bohr radius problems for the function class SP q snail(ff; ;). Lastly, we establish certain results for some new subclasses of functions defined by the neutrosophic Poisson distribution series. [ABSTRACT FROM AUTHOR]