We described in [M1] a monoid b G, the face monoid, acting on the integrable highest weight modules of a symmetrizable Kac-Moody algebra. It has similar structural properties as a reductive algebraic monoid whose unit group is a Kac-Moody group G. We found in [M5] two natural extensions of the action of the Kac-Moody group G on its building Ω to actions of the face monoid b G on the building Ω....

We propose a definition for geometric mean of k positive (semi) definite matrices. We show that our definition is the only one in the literature that has the properties that one would expect from a geometric mean, and that our geometric mean generalizes many inequalities satisfied by the geometric mean of two positive semidefinite matrices. We prove some new properties of the geometric mean of ...

We have developed a model for representing the image of segments of blood vessels in x-ray angiograms, and a software package for generating the model from digitized angiograms. The model is based on projections of linearly tapering cylinders and is suitable for the analysis of vessel lumen position, size and shape. This is of interest for research into the disease atherosclerosis. The system h...