Christoffel-Darboux-type formulae and a recurrence for biorthogonal polynomials

It is proved that biorthogonal polynomials obey two different kinds of Christoffel-Darboux-type formulae, one linking polynomials with a different parameter and one combining polynomials with different degrees. This is used to produce a mixed recurrence relation, which is valid for all biorthogonal polynomials. This recurrence relation establishes several results on interlacing property of zeros of successive biorthogonal polynomials and leads to a new result on the interlace of zeros of orthogonal polynomials (of equal degrees) with respect to two distributionsdψ(x) andx ^p dψ(x), 0<p≤1, with support in either [0, 1] or [1, ∞).     

On Periodic Solutions of Degenerate Singularly Perturbed Linear Systems with Multiple Elementary Divisor

We establish sufficient conditions for the existence and uniqueness of a periodic solution of a system of linear differential equations with a small parameter and a degenerate matrix of coefficients of derivatives in the case of a multiple spectrum of a boundary matrix pencil. We construct asymptotics of this solution.     

Structural evolution in amorphous silicon and germanium thin films.

The structural evolution in amorphous silicon and germanium thin films has been investigated by high-resolution transmission electron microscopy (HRTEM) in conjunction with autocorrelation function (ACF) analysis. The results established that the structure of as-deposited semiconductor films is of a high density of nanocrystallites embedded in the amorphous matrix. In addition, from ACF analysis, the structure of a-Ge is more ordered than that of a-Si. The density of embedded nanocrystallites in amorphous films was found to diminish with annealing temperature first, then to increase. The conclusions also corroborate well with the results of diminished medium-range order in annealed amorphous films determined previously by a variable coherence microscopy method.     

Complete ideals and singularities of space curves

We consider complete ideals supported on finite sequences of infinitely near points, in regular local rings with dimensions greater than two. We study properties of factorizations in Lipman special *-simple complete ideals. We relate it to a type of proximity, linear proximity, of the points, and give conditions in order to have unique factorization. Several examples are presented. Received: 2 February 2000 / in final form: 14 March 2001 / Published online: 18 January 2002     

Canonical Modules of Semigroup Rings and a Conjecture of Reiner

We prove that the homotopy types of two naturally defined simplicial complexes are related in the following way: one is homotopy equivalent to a multiple suspension of the canonical Alexander dual of the other. These simplicial complexes arise in free resolutions of semigroup rings and modules. The relation between their homotopy types was conjectured by Reiner, and was suggested by a homological consequence of a result due independently to Danilov and Stanley on canonical modules of normal semigroup rings. Our proof is purely topological, and gives an alternative proof of their result. We also prove a generalization of a result of Hochster saying that these rings are Cohen—Macaulay, and indicate a new proof of Ehrhart's reciprocity law for rational polytopes. Received October 3, 2000, and in revised form April 2, 2001, and May 10, 2001. Online publication November 7, 2001.