Linearly connected sequences and spectrally optimal dual frames for erasures

In the case that a frame is prescribed for applications and erasures occur in the process of data transmissions, we examine optimal dual frames for the recovery from single erasures. In contrast to earlier papers, we consider the spectral radius of the error operator instead of its operator norm as a measure of optimality. This notion of optimality is natural when the Neumann series is used to recover the original data in an iterative manner. We obtain a complete characterization of spectrally one-erasure optimal dual frames in terms of the redundancy distribution of the prescribed frame. Our characterization relies on the connection between erasure optimal frames and the linear connectivity property of the frame. We prove that the linear connectivity property is equivalent to the intersection dependent property, and is also closely related to the well-known concept of a k-independent set. Additionally, we also establish several necessary and sufficient conditions for the existence of an alternate dual frame to make the iterative reconstruction work.     

Signal recovery and frames that are robust to erasure

We consider finite frames with high redundancy so that if half the terms transmitted from the sender are randomly deleted during transmission, then on average, the receiver can still recover the signal to within a high level of accuracy. This follows from a result in random matrix theory. We also give an application of the operator Khintchine inequality in the setting of signal recovery when the signal is a matrix with a sparse representation.     

对称r-循环矩阵的快速算法和并行算法

借助于快速付立叶变换(FFT),给出了一种判断对称r-循环线性系统是否有解的快速算法,并且在有解的情况下求出其解,该算法的计算复杂度为O(nlogn),且具有很好的并行性,若使用n台处理机并行处理该算法则只需要O(logn)步.当r=0时,对称r-循环矩阵变成一个上三角型Hankel矩阵,我们也给出了此类矩阵求逆的一种算法.最后将该算法推广到线性同余系统,其运算量仅为O(nlogn).     

On a decomposition for infinite transition matrices

Heyman gives an interesting factorization of I-P, where P is the transition probability matrix for an ergodic Markov chain. We show that this factorization is valid if and only if the Markov chain is recurrent. Moreover, we provide a decomposition result which includes all ergodic, null recurrent as well as the transient Markov chains as special cases. Such a decomposition has been shown to be useful in the analysis of queues. This revised version was published online in June 2006 with corrections to the Cover Date.     

Uncertainty principle for measurable sets and signal recovery in quaternion domains

The classical uncertainty principle of harmonic analysis states that a nontrivial function and its Fourier transform cannot both be sharply localized. It plays an important role in signal processing and physics. This paper generalizes the uncertainty principle for measurable sets from complex domain to hypercomplex domain using quaternion algebras, associated with the quaternion Fourier transform. The performance is then evaluated in signal recovery problems where there is an interplay of missing and time‐limiting data. Copyright © 2017 John Wiley & Sons, Ltd.     

Correction to: On a decomposition for infinite stochastic matrices

An error in the proof to theorem 1 in: On a decomposition for infinite transition matrices, Queueing Systems 27 (1997) 127–130, is corrected. This revised version was published online in June 2006 with corrections to the Cover Date.     

Point estimation for multi-spectral distributed random matrices

In this communication, we consider a p×n random matrix which is normally distributed with mean matrix M and covariance matrix Σ, where the multivariate observation x_i=y_i+?_i with p dimensions on an object consists of two components, the signal y_i with mean vector μ and covariance matrix Σ_s and noise with mean vector zero and covariance matrix Σ_?, then the covariance matrix of x_i and x_j is given by Σ=Cov(x_i,x_j)=Γ⊗(B_|i-j|Σ_s+C_|i-j|Σ_?), where Γ is a correlation matrix; B_|i-j| and C_|i-j| are diagonal constant matrices. The statistical objective is to consider the maximum likelihood estimate of the mean matrix M and various components of the covariance matrix Σ as well as their statistical properties, that is the point estimates of Σ_s,Σ_? and Γ. More importantly, some properties of these estimators are investigated in slightly more general models.     

Universal similarity factorization equalities for commutative quaternions and their matrices

In this work, we have established universal similarity factorization equalities over the commutative quaternions and their matrices. Based on these equalities, real matrix representations of commutative quaternions and their matrices have been derived, and their algebraic properties and fundamental equations have been determined. Moreover, illustrative examples are provided to support our results.     

Symmetry and inverse-closedness of matrix algebras and functional calculus for infinite matrices

We investigate the symbolic calculus for a large class of matrix algebras that are defined by the off-diagonal decay of infinite matrices. Applications are given to the symmetry of some highly non-commutative Banach algebras, to the analysis of twisted convolution, and to the theory of localized frames.

     

An exact global optimization method for deriving weights from pairwise comparison matrices

Some multiple-criteria decision making methods rank actions by associating weights to the different criteria or actions, which are pairwise compared via a positive reciprocal matrix A. There is a vast literature on proposals of different mathematical-programming methods to infer weights from such matrix A. However, it is seldom observed that such optimization problems may be multimodal, thus the standard local-search resolution techniques suggested may be trapped in local optima, yielding a wrong ranking of alternatives. In this note we show that standard tools of global optimization based on interval analysis, lead to globally optimal weights in reasonable time.     

求鳞状因子循环矩阵的逆阵及广义逆阵的快速付氏变换法

借助快速付立叶变换(FFT),本文给出一种求n阶鳞状因子循环矩阵的逆阵、自反g-逆、群逆、Moore-Penrose逆的快速算法,该算法的计算复杂性为O(nlog2n),最后给出的两个数值算例表明了该算法的有效性.     

On universally decodable matrices for space–time coding

The notion of universally decodable matrices (UDMs) was recently introduced by Tavildar and Viswanath while studying slow-fading channels. It turns out that the problem of constructing UDMs is tightly connected to the problem of constructing maximum-distance separable codes. In this paper, we first study the properties of UDMs in general and then we discuss an explicit construction of a class of UDMs, a construction which can be seen as an extension of Reed–Solomon codes. In fact, we show that this extension is, in a sense to be made more precise later on, unique. Moreover, the structure of this class of UDMs allows us to answer some open conjectures by Tavildar, Viswanath, and Doshi in the positive, and it also allows us to formulate an efficient decoding algorithm for this class of UDMs. It turns out that our construction yields a coding scheme that is essentially equivalent to a class of codes that was proposed by Rosenbloom and Tsfasman. Moreover, we point out connections to so-called repeated-root cyclic codes.     

Explicit stationary distributions for compositions of random functions and products of random matrices

If (Y _n) _n ^=1/ is a sequence of i.i.d. random variables onE=(0,+) and iff is positive onE, this paper studies explicit examples of stationary distributions for the Markov chain (W _n) _n=0 defined byW _n=Y _nf(W _n-1). The case wheref is a Moebius function(ax+b)/(cx+d) leads to products of certain random (2,2) matrices and to interesting random continued fractions. These explicit examples are built with a naive idea by considering genral exponential families onE, especially the families of beta distributions of the first and second kind.     

Lower and upper bounds of condition number for Vandermonde-wise matrices and method of fundamental solutions using pseudo radial-lines

Consider the method of fundamental solutions (MFS) for 2D Laplace's equation in a bounded simply connected domain $S$. In the standard MFS, the source nodes are located on a closed contour outside the domain boundary $\mathrm{\Gamma }(=\partial S)$, which is called pseudo-boundary. For circular, elliptic, and general closed pseudo-boundaries, analysis and computation have been studied extensively. New locations of source nodes are proposed along two pseudo radial-lines outside $\mathrm{\Gamma }$. Numerical results are very encouraging and promising. Since the success of the MFS mainly depends on stability, our efforts are focused on deriving the lower and upper bounds of condition number (Cond). The study finds stability properties of new Vandermonde-wise matrices on nodes $x_i\in [a,b]$ with . The Vandermonde-wise matrix is called in this article if it can be decomposed into the standard Vandermonde matrix. New lower and upper bounds of Cond are first derived for the standard Vandermonde matrix, and then for new algorithms of the MFS using two pseudo radial-lines. Both lower and upper bounds of Cond are intriguing in the stability study for the MFS. Numerical experiments are carried out to verify the stability analysis made. Since the fundamental solutions (as $\{\ln |\overline{{PQ}_i}|\}$) are the basis functions of the MFS, new Vandermonde-wise matrices are found. Since the nodes $x_i\in [a,b]$ with may come from approximations and interpolations by the Laurent polynomials with singular part, the conclusions in this article are important not only to the MFS but also to matrix analysis.     

Asymptotics for Lp-norms of Fourier series density estimators

LetX ₁, X₂, , be a sequence of independent, identically distributed bounded random variables with a smooth density functionf. We prove that is asymptotically normal, wheref _{m, n} is the Fourier series density estimator offandw is a nonnegative weight function.Communicated by Edward B. Saff.AMS classification: Primary 60F05, 60F25; Secondary 62G05.     

Cyclic reduction and FACR methods for piecewise hermite bicubic orthogonal spline collocation

Cyclic reduction and Fourier analysis-cyclic reduction (FACR) methods are presented for the solution of the linear systems which arise when orthogonal spline collocation with piecewise Hermite bicubics is applied to boundary value problems for certain separable partial differential equations on a rectangle. On anN×N uniform partition, the cyclic reduction and Fourier analysis-cyclic reduction methods requireO(N ²log₂ N) andO(N ²log₂log₂ N) arithmetic operations, respectively.     

K-functionals and exact values ofn-widths for several classes fromL 2

We obtain exact values of differentn-widths for certain classes of 2π-periodic functions for which theK-functionals of theirrth-order derivatives are majorized by functions satisfying specified restrictions. We also consider some examples of majorants satisfying all the restrictions imposed in the present paper. Translated fromMatematicheskie Zametki, Vol. 66, No. 4, pp. 494–499, October, 1999.