An note on the maximization of matrix valued Hankel determinants with application

In this note we consider the problem of maximizing the determinant of moment matrices of matrix measures. The maximizing matrix measure can be characterized explicitly by having equal (matrix valued) weights at the zeros of classical (one dimensional) orthogonal polynomials. The results generalize classical work of Schoenberg (1959) to the case of matrix measures. As a statistical application we consider several optimal design problems in linear models, which generalize the classical weighing design problems. --Matrix measures,Hankel matrix,orthogonal polynomials,approximate optimal designs,spring balance weighing designs

Matrix measures and random walks

In this paper we study the connection between matrix measures and random walks with a tridiagonal block transition matrix. We derive sufficient conditions such that the blocks of the n-step transition matrix of the Markov chain can be represented as integrals with respect to a matrix valued spectral measure. Several stochastic properties of the processes are characterized by means of this matrix measure. In many cases this measure is supported in the interval [-1, 1]. The results are illustrated by several examples including random walks on a grid and the embedded chain of a queuing system. --Markov chain,block tridiagonal transition matrix,spectral measure,matrix measure,quasi birth and death processes,canonical moments

A Note on the Maximization of Matrix Valued Hankel Determinants with Applications

In this note we consider the problem of maximizing the determinant of moment matrices of matrix measures. The maximizing matrix measure can be characterized explicitly by having equal (matrix valued) weights at the zeros of classical (one dimensional) orthogonal polynomials. The results generalize classical work of Schoenberg (1959) to the case of matrix measures. As a statistical application we consider several optimal design problems in linear models, which generalize the classical weighing design problems

Solution of a Generalized Stieltjes Problem

We present the exact solution for a set of nonlinear algebraic equations $\frac{1}{z_l}= \pi d + \frac{2 d}{n} \sum_{m \neq l} \frac{1}{z_l-z_m}$. These were encountered by us in a recent study of the low energy spectrum of the Heisenberg ferromagnetic chain \cite{dhar}. These equations are low $d$ (density) ``degenerations'' of more complicated transcendental equation of Bethe's Ansatz for a ferromagnet, but are interesting in themselves. They generalize, through a single parameter, the equations of Stieltjes, $x_l = \sum_{m \neq l} 1/(x_l-x_m)$, familiar from Random Matrix theory. It is shown that the solutions of these set of equations is given by the zeros of generalized associated Laguerre polynomials. These zeros are interesting, since they provide one of the few known cases where the location is along a nontrivial curve in the complex plane that is determined in this work. Using a ``Green's function'' and a saddle point technique we determine the asymptotic distribution of zeros.Comment: 19 pages, 4 figure

Random walk weakly attracted to a wall

We consider a random walk X_n in Z_+, starting at X_0=x>= 0, with transition probabilities P(X_{n+1}=X_n+1|X_n=y>=1)=1/2-\delta/(4y+2\delta) P(X_{n+1}=X_n+1|X_n=y>=1)=1/2+\delta/(4y+2\delta) and X_{n+1}=1 whenever X_n=0. We prove that the expectation value of X_n behaves like n^{1-(\delta/2)} as n goes to infinity when \delta is in the range (1,2). The proof is based upon the Karlin-McGregor spectral representation, which is made explicit for this random walk.Comment: Replacement with minor changes and additions in bibliography. Same abstract, in plain text rather than Te

Being Focused: When the Purpose of Inference Matters for Model Selection

In contrast to conventional model selection criteria, the Focused Information Criterion (FIC) allows for purpose-specific choice of models. This accommodates the idea that one kind of model might be highly appropriate for inferences on a particular parameter, but not for another. Ever since its development, the FIC has been increasingly applied in the realm of statistics, but this concept appears to be virtually unknown in the economic literature. Using a classical example and data for 35 U.S. industry sectors (1960-2005), this paper provides for an illustration of the FIC and a demonstration of its usefulness in empirical applications

Des langes pour Artémis?

Metal (oxy)­hydroxides (MO_<i>x</i>H_<i>y</i>, M = Fe, Co, Ni, and mixtures thereof) are important materials in electrochemistry. In particular, MO_<i>x</i>H_<i>y</i> are the fastest known catalysts for the oxygen evolution reaction (OER) in alkaline media. While key descriptors such as overpotentials and activity have been thoroughly characterized, the nanostructure and its dynamics under electrochemical conditions are not yet fully understood. Here, we report on the structural evolution of Ni_1−δCo_δO_<i>x</i>H_<i>y</i> nanosheets with varying ratios of Ni to Co, in operando using atomic force microscopy during electrochemical cycling. We found that the addition of Co to NiO_<i>x</i>H_<i>y</i> nanosheets results in a higher porosity of the as-synthesized nanosheets, apparently reducing mechanical stress associated with redox cycling and hence enhancing stability under electrochemical conditions. As opposed to nanosheets composed of pure NiO_<i>x</i>H_<i>y</i>, which dramatically reorganize under electrochemical conditions to form nanoparticle assemblies, restructuring is not found for Ni_1−δCo_δO_<i>x</i>H_<i>y</i> with a high Co content. Ni_0.8Fe_0.2O_<i>x</i>H_<i>y</i> nanosheets show high roughness as-synthesized which increases during electrochemical cycling while the integrity of the nanosheet shape is maintained. These findings enhance the fundamental understanding of MO_<i>x</i>H_<i>y</i> materials and provide insight into how nanostructure and composition affect structural dynamics at the nanoscale

Strong asymptotics for Jacobi polynomials with varying nonstandard parameters

Strong asymptotics on the whole complex plane of a sequence of monic Jacobi polynomials $P_n^{(\alpha_n, \beta_n)}$ is studied, assuming that $$\lim_{n\to\infty} \frac{\alpha_n}{n}=A, \qquad \lim_{n\to\infty} \frac{\beta _n}{n}=B,$$ with $A$ and $B$ satisfying $A > -1$, $B>-1$, $A+B < -1$. The asymptotic analysis is based on the non-Hermitian orthogonality of these polynomials, and uses the Deift/Zhou steepest descent analysis for matrix Riemann-Hilbert problems. As a corollary, asymptotic zero behavior is derived. We show that in a generic case the zeros distribute on the set of critical trajectories $\Gamma$ of a certain quadratic differential according to the equilibrium measure on $\Gamma$ in an external field. However, when either $\alpha_n$, $\beta_n$ or $\alpha_n+\beta_n$ are geometrically close to $\Z$, part of the zeros accumulate along a different trajectory of the same quadratic differential.Comment: 31 pages, 12 figures. Some references added. To appear in Journal D'Analyse Mathematiqu