George Grossman

On Schur D-stable matrices

Abstract It is shown that vertex stability implies Schur D-stability for real 2 × 2 matrices and real n × n tridiagonal matrices. Additional results describing the class of n × n complex Schur D-stable matrices are given.

Classes of Schur D-stable matrices

Abstract It is shown that vertex stability implies Schur D-stability for real 3×3 matrices. Also, principally nilpotent n×n complex matrices are shown to be perfectly Schur D-stable, and additional characterizations of these matrices are given.

Summation identities for representation of certain real numbers

We present identities used to represent real numbers of the form xum±yvn for appropriately chosen real numbers x, y, u, v and nonnegative integers m and n. We present the proofs of the identities by applying Zeilbergers algorithm.

Limit cycles for successive projections onto hyperplanes in Rn

Abstract In this paper we consider successive orthogonal projections onto m hyperplanes in R n, where m ⩾ 2 and n ⩾ 2. A limit cycle is defined to be a sequence of points formed by projecting onto each of the hyperplanes once in a prescribed order, with the last projection giving the starting point. Several examples, including triangles, quadrilaterals, regular polygons, and arbitrary collections of lines in R 2, are solved for the limit cycle. Limit cycles are found in various ways, including by a limiting process and by solving an mn × mn linear system of equations. The latter approach will produce all the limit cycles for an arbitrary ordered set of m hyperplanes in R n.

Oldroyd's viscosity result extended to circular disk particles dispersed in Newtonian fluids

A viscosity equation is formulated on the basis of Oldroyds theory for elastic and viscous properties of emulsions and suspensions by considering the drops as cylindrical rather than spherical in shape. The problem is formulated in three dimensions using cylindrical coordinates. The result can be considered as applicable to liquid, circular disk particles with negligible thickness, such as platelets, in dilute suspensions. In the present analysis, initially, stress effects are assumed uniform along the length of the cylinder, the z coordinate of velocity decays exponentially with time, and the interactive effects of the particles are assumed negligible