Daniel Drucker

Simplified descriptions of the exceptional bounded symmetric domains

In this paper, we obtain new descriptions of the two exceptional bounded symmetric domains. The descriptions are easy to use because they are phrased in terms of numerical inequalities rather than positive definiteness of operators on ℂ16 or ℂ27. Actually, we obtain simplified descriptions in complex euclidean space forall the natural group orbits in the two compact exceptional hermitian symmetric spaces, not just the two open orbits corresponding to the bounded symmetric domains. The methods used here also lead directly to descriptions of the holomorphic arc components of the orbits, eliminating the need for the special indirect approach used in [2] to handle certain of the holomorphic arc components.

On the discriminant of a trinomial

Abstract The discriminant of a trinomial is obtained by evaluating an associated resultant. In contrast to Swan, who obtained such a formula using algebraic properties of the resultant, we rely on the determinantal formulation of the resultant and invoke a method of calculating determinants due to Drucker and Goldschmidt.

Reflection Properties of Curves and Surfaces

The reflection properties of parabolas, ellipses, and hyperbolas are well known and have many practical applications. What does it mean in general for a curve or surface to have a reflection property? Which curves and surfaces have them? The purpose of this article is to answer both questions. The results show that the conic sections are quite special, and the proofs include a demonstration of the reflection properties of parabolas, ellipses, and hyperbolas all at once, rather than in the usual case-by-case fashion.

A Comparison of Mode-Acceleration and Ritz Vector Reduced Basis Procedures in Transient Analysis

The method of mode-acceleration effectively recovers the first Ritz vector used in Ritz procedures. The theoretical basis for Lanczos algorithms that generate Ritz vectors is explained

Graph theoretic techniques in algebraic geometry II: construction of singular complex surfaces of the rational cohomology type of CP2

Methods of graph theory are used to obtain rational projective surfaces with only rational double points as singularities and with rational cohomology rings isomorphic to that of the complex projective plane. Uniqueness results for such cohomologyCP2s and for rational and integral homologyCP2s are given in terms of the typesAk,Dk, orEk of singularities allowed by the construction.

A comprehensive pythagorean theorem for all dimensions

Abstract We use a 200-year-old theorem on determinants to prove a very general version of the Pythagorean theorem. It relates the square of the n-dimensional volume of an n-parallelotope in ℝm to the sum of the squares of the n-dimensional volumes of the orthogonal projections of the parallelotope onto the k-dimensional coordinate subspaces of ℝm.

A Mathematical Roller Derby

Things to think about: If two of the objects have the same size and shape, does their density affect which one arrives first? If two objects have the same shape and are made of the same material, does their size make a difference? If different solid cylinders of the same weight are rolled down the inclined plane, does the ratio of cylinder length to radius make a difference or do they all arrive at the same time?

A Euclidean interpretation of Dynkin diagrams and its relation to root systems

A simple metric property satisfied by bases of (finite, not necessarily reduced) root systems is used to define sets in Euclidean space that provide models for Dynkin diagrams and their positive semidefinite one-vertex extensions. The theory of root systems can be founded on the study of these ‘Dynkin sets’, and conversely the Dynkin sets representing connected diagrams can be characterized as the bases and extended bases of root systems. (By an ‘extended base’, we mean a base together with the lowest root of a given length.) In this correspondence the role of nonreduced root systems is natural and important.