Sebastian Walcher

The Minnesota notes on Jordan algebras and their applications

Domains of Positivity.- Omega Domains.- Jordan Algebras.- Real and Complex Jordan Algebras.- Complex Jordan Algebras.- Jordan Algebras and Omega Domains.- Half-Spaces.- Appendix: The Bergman kernel function.

ON THE ZEROS OF POLYNOMIALS OVER QUATERNIONS

ABSTRACT We review and expand some results on the number of zeros of polynomials over Hamiltons quaternions, with particular emphasis on those polynomials with coefficients in a degree two subfield.

On algebras of rank three

An algebra of rank three is a commutative, finite dimensional algebra that may be defined by the property that every element generates a subalgebra of dimension not greater than two. In this article we discuss several classes of such algebras, including two classes related to central simple Jordan algebras, and derive some general results which indicate that, with the exception of one pathological class related to nilpotent algebras, every rank three algebra can be constructed either from a quadratic and alternative algebra or from a representation of a Clifford algebra. Among other results, semisimple and simple rank three algebras are characterized, and the radical of an arbitrary rank three algebra is determined.

ON BERNSTEIN ALGEBRAS WHICH ARE TRAIN ALGEBRAS

holds in A. This class of algebras was introduced by Holgate [4], following the original work of Bernstein [2] and subsequent investigations by Lyubich [5] on idempotent quadratic maps from a real simplex into itself. A summary of known results on Bernstein algebras (up to 1980) is given in Worz-Busekros [8], which will also be used as a basic reference on algebras in genetics. All definitions not explicitly stated here can be found in this monograph. Bernstein algebras are not necessarily genetic algebras in the sense of Schafer [6]; indeed there are Bernstein algebras which are not even train algebras, cf. Worz-Busekros [8]. In this note Bernstein algebras which are train algebras are considered. K is assumed to be of characteristic zero throughout, although this hypothesis could be weakened for some of the results. As follows from a result of Outtara [9], the rank polynomial of a Bernstein train algebra is uniquely determined by its degree. This result admits an interpretation as a stationarity principle which is different from (1), and this in turn enables us to solve the differential equation for overlapping generations in the time-continuous model (for K = IR) in closed form. In particular, the long-term behaviour of the solutions can easily be determined. I thank the referee for valuable comments and suggestions.

Multiplier systems for the modular group on the 27-dimensional exceptional domain

We show that the modular group γ3 on the 27-dimensional exceptional domain admits only one multiplier system. At first we demonstrate that ia equals its own commutator subgroup by Jordan theoretic means, where the main tool is Huas identity. Maaβ description of all multiplier systems for the Siegel modular group then leads to the result. As a by-product we can describe the multiplier systems for the modular groups on the 10-dimensional boundary component and the half-spaces of quaternions.

On Sums of Vector Fields

We discuss one case where the integration of a sum of vector fields is reducible to the integration of the summands. Applications include the construction of a class of additive group actions on affine space and a proof that these are stably tame, and also the explicit solution of a class of differential equations from mathematical biology.

Centralizers of locally nilpotent derivations

Locally nilpotent derivations of the polynomial ring in n variables over the complex field, algebraic actions of the additive group Ga of complex numbers on Cn, and vector fields on Cn admitting a strictly polynomial flow, are equivalent objects. The polynomial centralizer of the vector field corresponding to a triangulable locally nilpotent derivation is investigated, yielding a triangulability criterion. Several new examples of nontriangulable Ga actions on Cn are presented.

Algebras of Rank Three

We give a few results on the class of algebras indicated in the title. Among these algebras are Bernstein-Jordan algebras and the pseudo-composition algebras recently investigated by Meyberg and Osborn. Some applications to ordinary differential equations are discussed.

Versuchte Beratung in Aachen. Obligatorische Informationsveranstaltungen and der RWTH für Neueinschreiber Lehramt Mathematik

Für das Lehramt Mathematik gab es an der RWTH in den letzten Jahren sehr hohe Einschreibezahlen, vor allem für das Lehramt Gymnasien/Gesamtschulen. Dieser zunächst erfreulichen Tatsache standen jedoch erschreckend schlechte Ergebnisse bei den Klausuren der Anfangssemester gegenüber. Gespräche und andere Indizien deuteten darauf hin, dass viele Neueinschreiber keine realistische Vorstellung vom Studium, insbesondere von den Mathematik-Anforderungen, hatten.

On the mean value of probability measures on circular graphs

We introduce and discuss a class of difference equations motivated by a problem from combinatorial optimization on graphs. The local behavior at stationary points is investigated in detail, and in the course of this investigation we prove a stability result for certain types of non-isolated stationary points. Results include a complete characterization of the behavior on chain graphs, and a characterization of the local behavior for circular graphs.