Required nonzero patterns for nonsingular sign regular matrices

An n×n real matrix is called sign regular if, for each k(1?k?n), all its minors of order k have the same nonstrict sign. The zero entries which can appear in a nonsingular sign regular matrix depend on its signature because the signature can imply that certain entries are necessarily nonzero. The patterns for the required nonzero entries of nonsingular sign regular matrices are analyzed.     

Numerical ranges of reducible companion matrices

In this paper, we show that a reducible companion matrix is completely determined by its numerical range, that is, if two reducible companion matrices have the same numerical range, then they must equal to each other. We also obtain a criterion for a reducible companion matrix to have an elliptic numerical range, put more precisely, we show that the numerical range of an n-by-n reducible companion matrix C is an elliptic disc if and only if C is unitarily equivalent to A⊕B, where A∈M_n-2, B∈M₂ with σ(B)={aω₁,aω₂}, , ω₁≠ω₂, and .     

A lower bound for the spectral radius of a digraph

We show that the spectral radius ρ(D) of a digraph D with n vertices and c₂ closed walks of length 2 satisfies Moreover, equality occurs if and only if D is the symmetric digraph associated to a -regular graph, plus some arcs that do not belong to cycles. As an application of this result, we construct new sharp upper bounds for the low energy of a digraph, which extends Koolen and Moulton bounds of the energy to digraphs.     

On the signless Laplacian spectral radius of graphs with cut vertices

In this paper, we show that among all the connected graphs with n vertices and k cut vertices, the maximal signless Laplacian spectral radius is attained uniquely at the graph G_n,k, where G_n,k is obtained from the complete graph K_n-k by attaching paths of almost equal lengths to all vertices of K_n-k. We also give a new proof of the analogous result for the spectral radius of the connected graphs with n vertices and k cut vertices (see [A. Berman, X.-D. Zhang, On the spectral radius of graphs with cut vertices, J. Combin. Theory Ser. B 83 (2001) 233-240]). Finally, we discuss the limit point of the maximal signless Laplacian spectral radius.     

Inverting linear combinations of identity and generalized Catalan matrices

We introduce the notion of the generalized Catalan matrix as a kind of lower triangular Toeplitz matrix whose nonzero elements involve the generalized Catalan numbers. Inverse of the linear combination of the Pascal matrix with the identity matrix is computed in Aggarwala and Lamoureux (2002) [1]. In this paper, continuing this idea, we invert various linear combinations of the generalized Catalan matrix with the identity matrix. A simple and efficient approach to invert the Pascal matrix plus one in terms of the Hadamard product of the Pascal matrix and appropriate lower triangular Toeplitz matrices is considered in Yang and Liu (2006) [14]. We derive representations for inverses of linear combinations of the generalized Catalan matrix and the identity matrix, in terms of the Hadamard product which includes the Generalized Catalan matrix and appropriate lower triangular Toeplitz matrix.     

Model selection by sequentially normalized least squares

Model selection by means of the predictive least squares (PLS) principle has been thoroughly studied in the context of regression model selection and autoregressive (AR) model order estimation. We introduce a new criterion based on sequentially minimized squared deviations, which are smaller than both the usual least squares and the squared prediction errors used in PLS. We also prove that our criterion has a probabilistic interpretation as a model which is asymptotically optimal within the given class of distributions by reaching the lower bound on the logarithmic prediction errors, given by the so called stochastic complexity, and approximated by BIC. This holds when the regressor (design) matrix is non-random or determined by the observed data as in AR models. The advantages of the criterion include the fact that it can be evaluated efficiently and exactly, without asymptotic approximations, and importantly, there are no adjustable hyper-parameters, which makes it applicable to both small and large amounts of data.     

Quasicomplete factorizations of rational matrix functions

It is shown that within the class ofn×n rational matrix functions which are analytic at infinity with valueW()=I _n, any rational matrix functionW is the productW=W ₁...W _p of rational matrix functionsW ₁,...,W _p of McMillan degree one. Furthermore, such a factorization can be established with a number of factors not exceeding 2(W)–1, where (W) denotes the McMillan degree ofW.     

Explicit generalized zolotarev polynomials with complex coefficients

We give explicitly a class of polynomials with complex coefficients of degreen which deviate least from zero on [−1, 1] with respect to the max-norm among all polynomials which have the same,m + 1, 2m ≤n, first leading coefficients. Form=1, we obtain the polynomials discovered by Freund and Ruschewyh. Furthermore, corresponding results are obtained with respect to weight functions of the type 1/√ρl, whereρl is a polynomial positive on [−1, 1].     

Small polaron electron transport in reduced CeO2 single crystals

An investigation of the electrical properties of reduced ceria, CeO_2?x, carried out on single crystals, shows that CeO_2?x provides one of the clearest examples of hopping conduction and the small polaron mechanism. Included are conductivity and Seebeck coefficient measurements at constant x, obtained by sealing off the specimen chamber after reduction. The Seebeck coefficient is independent of temperature, suggesting that the number of carriers is constant. On the other hand, the mobility is activated, with activation energy E_h = 0.40 eV at small x and increasing to 0.52 eV at x = 0.25. The results for the mobility preexponential are consistent with the adiabatic theory of small polaron behavior. A puzzling feature of the Seebeck data as a function of x is that, for low x, the data fit the well-known Heikes formula, without a degeneracy factor of 2 for spin. Nevertheless, these data are interpreted to show that the proportion of mobile carriers decreases as x increases, presumably because of the presence of short-range ordered configurations which immobilize some carriers.     

A de bruijn-type formula for enumerating patterns of partitions

We develop a formula for enumerating the partitions of a set up to the action of a group. We give some special cases of this formula and illustrate its use with an example. Some comments are made relating the formula to an earlier formula.This work was supported in part by NSF Contract GP-43010.