Differenzbegriffe zwischen Polytopen und ihre torische Interpretation
The average squared volume of simplices formed by k independent copies from the same probability measure μ on ℝ^d defines an integral measure of dispersion ψ_k(μ), which is a concave functional of μ after suitable normalisation. When k=1 it corresponds to trace(Σ_μ) and when k=d we obtain the usual generalised variance det( Σ_μ), with Σ_μ the covariance matrix of μ. The dispersion ψ_k(μ) generates a notion of simplicial potential at any x in ℝ^d, dependent on μ. We show that this simplicial potential is a quadratic convex function of x, with minimum value at the mean a_μ for μ, and that the potential at a_μ defines a central measure of scatter similar to ψ_k(μ), thereby generalising results by Wilks (1960) and Van der Waart (1965) for the generalised variance. Simplicial potentials define generalised Mahalanobis distances, expressed as weighted sums of such distances in every k-margin, and we show that the matrix involved in the generalised distance is a particular generalised inverse of Σ_μ, constructed from its characteristic polynomial, when k=rank(Σ_μ). Finally, we show how simplicial potentials can be used to define simplicial distances between two distributions, depending on their means and covariances, with interesting features when the distributions are close to singularity.
In the first half of the 19th century Navier and Stokes formulated the equations that describe the flow of water and many other incompressible liquids under standard conditions. These equations now bear the names of their inventors. A century later Leray developed the mathematical foundations of the modern theory of the Navier-Stokes equations both for planar and three-dimensional flows. He introduced the concept of generalized solution to the Cauchy problem and proved its existence for arbitrary (sufficiently regular) data and for an arbitrary time interval. This concept not only reflects the physical assumptions used when deriving the equations but it also forms the basis for the construction of powerful numerical methods. Despite the undeniable success of the Navier-Stokes equations, there are many fluid-like incompressible materials that exhibit phenomena that can not be described by the Navier-Stokes equations. In order to describe these effects a number of models, which are more complicated than the Navier-Stokes equations, have been designed, developed, and used in relevant applications. The aim of this lecture is to survey recent developments, both in the area of theoretical continuum thermodynamics as well as in the field of PDE analysis, which have led to the development of Leray's programme beyond the Navier-Stokes equations.
Datum: 24.10.2018, Raum: G03-106, Zeit: 17:00