Zero cancellation for general rational matrix functions

The problem of cancelling a specified part of the zeros of a completely general rational matrix function by multiplication with an appropriate invertible rational matrix function is investigated from different standpoints. Firstly, the class of all factors that dislocate the zeros and feature minimal McMillan degree are derived. Further, necessary and sufficient existence conditions together with the construction of solutions are given when the factor fulfills additional assumptions like being J-unitary, or J-inner, either with respect to the imaginary axis or to the unit circle. The main technical tool are centered realizations that deliver a sufficiently general conceptual support to cope with rational matrix functions which may be polynomial, proper or improper, rank deficient, with arbitrary poles and zeros including at infinity. A particular attention is paid to the numerically-sound construction of solutions by employing at each stage unitary transformations, reliable numerical algorithms for eigenvalue assignment and efficient Lyapunov equation solvers.     

Factorization of matrix functions and their inverses via power product expansions

The conversion of a power series with matrix coefficients into an infinite product of certain elementary matrix factors is studied. The expansion of a power series with matrix coefficients as the inverse of an infinite product of elementary factors is also analyzed. Each elementary factor is the sum of the identity matrix and a certain matrix coefficient multiplied by a certain power of the variable. The two expansions provide us with representations of a matrix function and its inverse by infinite products of elementary factors. Estimates on the domain of convergence of the infinite products are given.     

Further inequalities involving the Khatri-Rao product

Styan [G.P.H. Styan, Hadamard products and multivariate statistical analysis. Linear Algebra and Its Appl. 6 (1973) 217-240] established an inequality involving the Hadamard product using statistical reasoning in the context of multivariate analysis. In this paper, the inequality is extended to involve the Khatri-Rao product in the non-negative definite matrix case and in the non-singular Hermitian matrix case. The equality conditions for these extensions are given. Also established are counterpart inequalities in the positive definite matrix case.     

Quasiperiodic spectra and orthogonality for iterated function system measures

We extend classical basis constructions from Fourier analysis to attractors for affine iterated function systems (IFSs). This is of interest since these attractors have fractal features, e.g., measures with fractal scaling dimension. Moreover, the spectrum is then typically quasi-periodic, but non-periodic, i.e., the spectrum is a “small perturbation” of a lattice. Due to earlier research on IFSs, there are known results on certain classes of spectral duality-pairs, also called spectral pairs or spectral measures. It is known that some duality pairs are associated with complex Hadamard matrices. However, not all IFSs X admit spectral duality. When X is given, we identify geometric conditions on X for the existence of a Fourier spectrum, serving as the second part in a spectral pair. We show how these spectral pairs compose, and we characterize the decompositions in terms of atoms. The decompositions refer to tensor product factorizations for associated complex Hadamard matrices. Research supported in part by a grant from the National Science Foundation DMS-0704191.     

Miniaturized matrix solid phase dispersion procedure and solid phase microextraction for the analysis of organochlorinated pesticides and polybrominated diphenylethers in biota samples by gas chromatography electron capture detection

This work has developed a miniaturized method based on matrix solid phase dispersion (MSPD) using C18 as dispersant and acetonitrile–water as eluting solvent for the analysis of legislated organochlorinated pesticides (OCPs) and polybrominated diphenylethers (PBDEs) in biota samples by GC with electron capture (GC-ECD). The method has compared Florisil^®-acidic Silica and C18 as dispersant for samples as well as different solvents. Recovery studies showed that the combination of C18–Florisil^® was better when using low amount of samples (0.1 g) and with low volumes of acetonitrile–water (2.6 mL). The use of SPME for extracting the analytes from the solvent mixture before the injection resulted in detection limits between 0.3 and 7.0 μg kg⁻¹ (expressed as wet mass). The miniaturized procedure was easier, faster, less time consuming than the conventional procedure and reduces the amounts of sample, dispersant and solvent volume by approximately 10 times. The proposed procedure was applied to analyse several biota samples from different parts of the Comunidad Valenciana.     

Computing matrix functions solving coupled differential models

In this paper a modification of the method proposed in [E. Defez, L. Jódar, Some applications of Hermite matrix polynomials series expansions, Journal of Computational and Applied Mathematics 99 (1998) 105–117] for computing matrix sine and cosine based on Hermite matrix polynomial expansions is presented. An algorithm and illustrative examples demonstrate the performance of the new proposed method.     

Some variations of the Bohman–Korovkin theorem

In this paper we develop the main aspects of the Bohman–Korovkin theorem on approximation of continuous functions with the use of A-statistical convergence and matrix summability method which includes both convergence and almost convergence. Since statistical convergence and almost convergence methods are incompatible we conclude that these methods can be used alternatively to get some approximation results.     

Skew-symmetric methods for nonsymmetric linear systems with multiple right-hand sides

By transforming nonsymmetric linear systems to the extended skew-symmetric ones, we present the skew-symmetric methods for solving nonsymmetric linear systems with multiple right-hand sides. These methods are based on the block and global Arnoldi algorithm which is formed by implementing orthogonal projections of the initial matrix residual onto a matrix Krylov subspace. The algorithms avoid the tediously long Arnoldi process and highly reduce expensive storage. Numerical experiments show that these algorithms are effective and give better practical performances than global GMRES for solving nonsymmetric linear systems with multiple right-hand sides.     

Double piling structure of matrix monotone functions and of matrix convex functions

There are basic equivalent assertions known for operator monotone functions and operator convex functions in two papers by Hansen and Pedersen. In this note we consider their results as correlation problem between two sequences of matrix n-monotone functions and matrix n-convex functions, and we focus the following three assertions at each label n among them:

(i) f(0)0 and f is n-convex in [0,α),

(ii) For each matrix a with its spectrum in [0,α) and a contraction c in the matrix algebra M_n,

f(cac)cf(a)c,