The ratio of the length of the unit circle to the area of the unit disc in Minkowski planes

In their paper ``An Introduction to Finsler Geometry,' J. C. Alvarez and C. Duran asked if there are other Minkowski planes besides the Euclidean for which the ratio of the Minkowski length of the unit ``circle' to the Holmes-Thompson area of the unit disc equals 2. In this paper we show that this ratio is greater than 2, and that the ratio 2 is achieved only for Minkowski planes that are affine equivalent to the Euclidean plane. In other words, the ratio is 2 only when the unit ``circle' is an ellipse.

     

Eventual Right Disfocality on Time Scales

Eventual right disfocality of a dynamic equation y j j ( t )+ p 1 ( t ) y j ( t )+ p 2 ( t ) y ( t )=0, t ] T is established where T is a time scale. The results are obtained by imposing integrability conditions on p 1 and p 2 .     

EVENTUAL STABILITY OF IMPULSIVE DIFFERENTIAL SYSTEMS

In this article, criteria of eventual stability are established for impulsive differential systems using piecewise continuous Lyapunov functions. The sufficient conditions that are obtained significantly depend on the moments of impulses. An example is discussed to illustrate the theorem.     

Representation of projectors involving Minkowski inverse in Minkowski space

A certain class of results about the different representations of Oblique projectors is present in the literature. These results represent Oblique projectors as the functions of orthogonal projectors with given onto and along spaces. But these results are valid under the restriction that the functions of orthogonal projectors involved are invertible. In this paper we extend and generalize these results. The extension lies in making a transition from Euclidean space to Minkowski space M and the generalization is obtain by voiding the invertibility condition and use of the Minkowski inverse. Furthermore, the nobility lies in utilizing the m-projectors instead of the regular orthogonal projectors.     

Eventual regularization for the slightly supercritical quasi-geostrophic equation

We prove that weak solutions of the slightly supercritical quasi-geostrophic equation become smooth for large time. The proof uses ideas from a recent article of Caffarelli and Vasseur and is based on an argument in the style of De Giorgi.     

Eventual convexity of chance constrained feasible sets

In decision-making problems where uncertainty plays a key role and decisions have to be taken prior to observing uncertainty, chance constraints are a strong modelling tool for defining safety of decisions. These constraints request that a random inequality system depending on a decision vector has to be satisfied with a high probability. The characteristics of the feasible set of such chance constraints depend on the constraint mapping of the random inequality system, the underlying law of uncertainty and the probability level. One characteristic of particular interest is convexity. Convexity can be shown under fairly general conditions on the underlying law of uncertainty and on the constraint mapping, regardless of the probability-level. In some situations, convexity can only be shown when the probability-level is high enough. This is defined as eventual convexity. In this paper, we will investigate further how eventual convexity can be assured for specially structured chance constraints involving Copulae. The Copulae have to exhibit generalized concavity properties. In particular, we will extend recent results and exhibit a clear link between the generalized concavity properties of the various mappings involved in the chance constraint for the result to hold. Various examples show the strength of the provided generalization.     

Minkowski bisectors,Minkowski cells and lattice coverings

We prove Barth-type connectedness results for low-codimension smooth subvarieties with good numerical properties inside certain “easy” ambient spaces (such as homogeneous varieties, or spherical varieties). The argument employs some basics from the theory of cones of cycle classes, in particular the notion of bigness of a cycle class.     

Minkowski valuations

Centroid and difference bodies define equivariant operators on convex bodies and these operators are valuations with respect to Minkowski addition. We derive a classification of equivariant Minkowski valuations and give a characterization of these operators. We also derive a classification of contravariant Minkowski valuations and of -Minkowski valuations.

     

Hermann Minkowski

Ohne ZusammenfassungGedächtnisrede, gehalten in der öffentlichen Sitzung der Kgl. Gesellschaft der Wissenschaften zu Göttingen am 1. Mai 1909.     

Computing the Minkowski Sum of Prisms

Within this paper we study the Minkowski sum of prisms (“Cephoids”) in a finite dimensional vector space. For a vector with positive components we write and denote by the associated prism. We provide a representation of a finite sum of prisms in terms of inequalities. Dedicated to the 65th birthday of Alexander Rubinov.