Abstract not found

In the following paper we study parametric functionals. First we introduce a generalized mean curvature (so called F-mean curvature). This enables us to describe extremals of parametric funcionals as surfaces of prescribed F-mean curvature. Furthermore we give a di#erential equation for arbitrary immersions generalizing #X = HN and apply this equation to surfaces of vanishing and prescribed F-mean curvature. Especially we prove non-existence results for such surfaces generalizing Theorems by Hildebrandt and Dierkes [3], [6].

. We approximate a hypersurface \Sigma with prescribed anisotropic mean curvature with solutions u ffl to suitable nonlinear elliptic equations depending on a small parameter ffl ? 0 . We work in relative geometry, by endowing IR N with a Finsler norm OE describing the anisotropy. The main result states that \Sigma and fu ffl = 0g are close of order ffl 2 j log fflj 2 , and this estimate is optimal. This is obtained for two different elliptic equations by sub- and supersolutions technique, under smoothness and nondegeneracy assumptions on \Sigma . Basic steps are: (i) an explicit computation of the second variation of the OE -Minkowski content along geodesics; (ii) the definition of a Laplace-Beltrami operator on \Sigma ; (iii ) the expansion of the OE -mean curvature of \Sigma in a suitable tubular neighbourhood. AMS Subject Classification Numbers (1991): 53B40, 53C60, 35B40, 35J60. Key words: Finsler spaces, Surfaces with prescribed mean curvature, Second variation, Asymptoti...

. Value at Risk is a measure of risk exposure of a portfolio and is defined as the maximum possible loss in a certain time frame, typically 1-20 days, and within a certain confidence, typically 95%. Full valuation of a portfolio under a large number of scenarios is a lengthy process. To speed it up, one can make use of the total delta vector and the total gamma matrix of a portfolio and compute a Gaussian integral over a region bounded by a quadric. We use methods from harmonic analysis to find approximate analytic formulas for the Value at Risk as a function of time and of the confidence level. In this framework, the calculation is reduced to the problem of evaluating linear algebra invariants such as traces of products of matrices, which arise from a Feynmann expansion. The use of Fourier transforms is crucial to resum the expansions and to obtain formulas that smoothly interpolate between low and large confidence levels, as well as between short and long time horizons. 1 e-mail: al...

expanding micro universes

Direct ζ-function approach and renormalization of one-loop stress tensors in curved spacetimes