In this paper, we study the matrix period and the competition period of Toeplitz matrices over a binary Boolean ring B={0,1}. Given subsets S and T of {1,…,n−1}, an n×n Toeplitz matrix A=Tn〈S;T〉 is ...defined to have 1 as the (i,j)-entry if and only if j−i∈S or i−j∈T. We show that if maxS+minT≤n and minS+maxT≤n, then A has the matrix period d/d′ and the competition period 1 where d=gcd(s+t|s∈S,t∈T) and d′=gcd(d,minS). Moreover, it is shown that the limit of the matrix sequence {Am(AT)m}m=1∞ is a directed sum of matrices of all ones except zero diagonal. In many literatures we see that graph theoretic method can be used to prove strong structural properties about matrices. Likewise, we develop our work from a graph theoretic point of view.
We compute an asymptotic formula for the supremum of the resolvent norm ||(ζ−T)−1|| over |ζ|≥1 and contractions T acting on an n-dimensional Hilbert space, whose spectral radius does not exceed a ...given r∈(0,1). We prove that this supremum is achieved on the unit circle by an analytic Toeplitz matrix.
A linear code with complementary dual (or an LCD code) is defined to be a linear code which intersects its dual code trivially. Let I be an identity matrix and T be a Toeplitz matrix of the same ...order over a finite field. A Double Toeplitz code (or a DT code) is a linear code generated by a generator matrix of the form (I,T). In 2021, Shi et al. obtained necessary and sufficient conditions for a Double Toeplitz code to be LCD when T is symmetric and tridiagonal. In this paper, by using a result on factoring Dickson polynomials over finite fields, we determine when a Double Toeplitz code is LCD for T being a skew symmetric and tridiagonal matrix. In addition, using a concatenation, we construct LCD codes with arbitrary minimum distance from DT codes over extension fields, provided the length of which is increased if necessary.
In this work, a Sinc-Galerkin method is considered and analyzed for solving the fourth-order partial integro-differential equation with a weakly singular kernel. The time derivative and ...Riemann-Liouville fractional integral term are approximated via the Crank-Nicolson method and the trapezoidal convolution quadrature rule, respectively. Then a fully discrete scheme is formulated via using the Sinc-Galerkin approximation. The exponential convergence rate in space of proposed method are derived. In addition, some properties of the Toeplitz matrix generated by the composite Sinc function at the Sinc node are extended to the cases of arbitrary order in the preliminary knowledge. Finally, some numerical examples are calculated to verify the accuracy and effectiveness of our method.
In this paper, we answer an open conjecture concerning complex symmetric matrices and truncated Toeplitz operators. We study matrix representations of truncated Toeplitz operators with respect to ...orthonormal bases which are invariant under a canonical conjugation map. In particular, we determine necessary and sufficient conditions for when a symmetric matrix is the matrix representation of a truncated Toeplitz operator with respect to a given conjugation invariant orthonormal basis. We specialise our result to the case when the conjugation invariant orthonormal basis is a modified Clark basis. With this specialisation, we answer an open conjecture in the negative, and show not every unitary equivalence between a complex symmetric matrix and a truncated Toeplitz operator arises from modified Clark basis representations. We pose a new refined conjecture for how to realise a model theory for symmetric matrices through the use of truncated Toeplitz operators, and we show this conjecture is equivalent to a specified system of polynomial equations being satisfied.
Efficient long-time integration of nonlinear fractional differential equations is significantly challenging due to the integro-differential nature of the fractional operators. In addition, the ...inherent non-smoothness introduced by the inverse power-law kernels deteriorates the accuracy and efficiency of many existing numerical methods. We develop two efficient first- and second-order implicit-explicit (IMEX) methods for accurate time-integration of stiff/nonlinear fractional differential equations with fractional order α∈(0,1 and prove their convergence and linear stability properties. The developed methods are based on a linear multi-step fractional Adams-Moulton method (FAMM), followed by the extrapolation of the nonlinear force terms. In order to handle the singularities nearby the initial time, we employ Lubich-like corrections to the resulting fractional operators. The obtained linear stability regions of the developed IMEX methods are larger than existing IMEX methods in the literature. Furthermore, the size of the stability regions increase with the decrease of fractional order values, which is suitable for stiff problems. We also rewrite the resulting IMEX methods in the language of nonlinear Toeplitz systems, where we employ a fast inversion scheme to achieve a computational complexity of O(NlogN), where N denotes the number of time-steps. Our computational results demonstrate that the developed schemes can achieve global first- and second-order accuracy for highly-oscillatory stiff/nonlinear problems with singularities.
In this paper, we concentrate on design a bilateral preconditioning for all-at-once system from multidimensional time-space non-local evolution equations with a weakly singular kernel. Firstly, we ...propose an implicit difference scheme for this equation by employing an L2-type formula. Then, we prove the stability and the convergence of the proposed scheme. Based on such a numerical scheme, an L2-type all-at-once system is derived and solved in a parallel-in-space pattern. Based on the special structure of its coefficient matrix, we propose a bilateral preconditioning strategy to accelerate the convergence of Krylov subspace solvers. We theoretically show that the condition number of the preconditioned matrix is uniformly bounded by a constant. Several numerical examples are provided to test the proposed scheme and preconditioning technique.
Coprime array exhibits many advantages over the uniform linear array (ULA) with the same number of physical sensors in resolution performance and interference suppression capability. In this study, ...the authors take the advantages of coprime array to improve the robustness of adaptive beamformer. In the coprime virtual ULA (CV-ULA), they prove that a constructed Toeplitz matrix can be taken as the sample covariance matrix from the perspective of virtual signal characteristics. The CV-ULA Capon spectrum estimator is modified to obtain the directions and powers of all impinging signals. Since the real directions of all impinging signals are located at different angular sectors, they form independent signal subspace for each impinging signal. They also assign independent steering vector mismatches for different impinging signals to obtain their real steering vectors. The steering vector mismatch of each impinging signal is independently obtained by solving its own convex optimisation problem. They reconstruct the interference-plus-noise covariance matrix (INCM) with precise steering vectors and powers of interference signals. The proposed weight vector is computed by combining the desired signal steering vector and the reconstructed INCM. Extensive simulations show that the proposed algorithm provides robustness against many types of model mismatches.
In this paper, we develop a fast numerical method for solving the time-dependent Riesz space fractional diffusion equations with a nonlinear source term in the convex domain. An implicit finite ...difference method is employed to discretize the Riesz space fractional diffusion equations with a penalty term in a rectangular region by the volume-penalization approach. The stability and the convergence of the proposed method are studied. As the coefficient matrix is with the Toeplitz-like structure, the generalized minimum residual method with a preconditioner based on the sine transform is exploited to solve the discretized linear system, where the preconditioner is constructed in view of the combination of two approximate inverse τ matrices, which can be diagonalized by the sine transform. The spectrum of the preconditioned matrix is also investigated. Numerical experiments are carried out to demonstrate the efficiency of the proposed method.
•To fill a low-rank Toeplitz matrix from a small subset of its entries.•Definition of “Toeplitz structure smoothing operator”.•The augmented Lagrange multiplier algorithm with smoothing for TMC ...problem.•Approximation matrices may hold on Toeplitz structure throughout the iteration.
Toplitz matrix completion (TMC) is to fill a low-rank Toeplitz matrix from a small subset of its entries. Based on the augmented Lagrange multiplier (ALM) algorithm for matrix completion, in this paper, we propose a new algorithm for the TMC problem using the smoothing technique of the approximation matrices. The completion matrices generated by the new algorithm are of Toeplitz structure throughout iteration, which save computational cost of the singular value decomposition (SVD) and approximate well the solution. Convergence results of the new algorithm are proved. Finally, the numerical experiments show that the augmented Lagrange multiplier algorithm with smoothing is more effective than the original ALM and the accelerated proximal gradient (APG) algorithms.
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