Large deviation for the empirical eigenvalue density of truncated Haar unitary matrices

Let U_m be an m×m Haar unitary matrix and U_[m,n] be its n×n truncation. In this paper the large deviation is proven for the empirical eigenvalue density of U_[m,n] as m/n→λ and n→∞. The rate function and the limit distribution are given explicitly. U_[m,n] is the random matrix model of quq, where u is a Haar unitary in a finite von Neumann algebra, q is a certain projection and they are free. The limit distribution coincides with the Brown measure of the operator quq.     

Asymptotic zero distribution of a class of 3F2 hypergeometric functions

We study the asymptotic behavior of the zeros of certain families of ₃F₂ functions. Classical tools are used to analyse the asymptotic behavior of the zeros of the polynomial In addition, families of ₃F₂ functions that are connected in a formulaic sense with Gauss hypergeometric polynomials of the form and are investigated. Numerical evidence of the clustering o zeros on certain curves is generated by Mathematica.     

Comment on “The Steiner number of a graph” by G. Chartrand and P. Zhang: [Discrete Mathematics 242 (2002) 41–54]

We show by counterexample that one of the main results in the paper “The Steiner number of a graph” by Chartrand and Zhang (Disc. Math. 242 (2002) 41–54) does not hold. To be more precise, we prove both that not every Steiner set is a geodetic set and that there are connected graphs whose Steiner number is strictly lower than its geodetic number.     

τ-Functions for a two-point version of the KP-hierarchy

In this paper a multipoint version of the linearization of the KP-hierarchy is described. Solutions of this system in the form of wave functions are constructed starting from a suitable Grassmann manifold and a group of commuting flows corresponding to the configuration of n points in . The failure of equivariance at lifting these flows to the determinant bundle over this Grassmann manifold is measured by the determinants of certain Fredholm operators, the so-called τ-functions. In the case of two points an explicit relation between these τ-functions and the wave functions is derived.     

The existence of a positive solution for a singular coupled system of nonlinear fractional differential equations

In this paper we establish the existence of a positive solution to a singular coupled system of nonlinear fractional differential equations. Our analysis rely on a nonlinear alternative of Leray–Schauder type and Krasnoselskii’s fixed point theorem in a cone.     

The effect of photoperiod and prey density on long-term survival of larvae—a data analytic approach where no parametric model is assumed and individual follow-up is not available

A data analytic approach for analysing differences in long-term survival and identifying treatment combinations which provide high survival in a factorial design set-up is presented. The methods are particularly appropriate when individual follow-up data are not available and no simple model fits the data. Upper quantiles of the survival distributions are used as the response variable and within and between-treatment differences are analysed through both a structured ANOVA model and subset-selection procedure. The methods are used to study the effects of several photoperiods and prey density treatments on the long-term survival of the larvae of gilthead seabream, Sparus aurata. The two approaches used are shown to lead to similar conclusions.