Combinatorial properties of f-vectors of convex polytypes
A convex polytype can be defined as a convex hull of a finite numer of points in R^d. The f-vector of a convex polytype is the vector concisting of the number of faces the polytype has in each dimension 0, 1, 2…d-1. An f-vector (or a sequence in general) is unimodal if it first increases to a point and then decreases. This article gives an overview the question of unimodality of f-vectors for different classes of polytypes and gives some new examples polytypes with non unimodal f-vectors.
Numerical Methods in the Calculus of Variations (Norwegian)
The article investigates some of the mathematics involved in the Mars mission of 2004 called the Mars Exploration Rovers mission (MER). MER was the result of many years of work done at NASA's Jet Propulsion Laboratory which serves as the basis of the investigation. The focus of the article is mainly on the use and development of mathematics at the JPL. In particular, the article discusses the possibility of doing mathematical work contributing to basic mathematical research at an institution like the JPL. Many factors limit this possibility. These factors include deadlines, high demand for reliability leading to a lot of reuse from mission to mission, as well as the fact that much of the mathematics used is hidden in software. The article concludes that although the work done at the JPL concerning the Mars mission required a solid mathematical foundation, the work had little impact on basic mathematical research. Instead the JPL plays an equally important role for mathematics - that of a consumer.
The Euler Number (Swedish)
The Euler-number is often defined as the alternating sum of the betti-numbers, or equivalently as the alternate sum of the number of simplices of a given dimension. The problem is that this requires an explicit triangulation. For purposes of actual computation it is much more efficient (and illuminating) to take a more ``functorial'' approach, namely the euler-number which behaves like a cardinal number, but with the difference that we can freely mix dimensions.
In the article this is illustrated by a variety of examples, compact real surfaces, projective spaces and Grassmannians, curves and hypersurfaces. We also show how to relate the Euler-number to the types of singularities of a vector field and to prove the Gauss-Bonnets formula, relating the integral of the Gaussian curvature to the Euler-number of the surface.