Given a bounded open subset ? of R n , we establish the weak closure affine ball B p A ( ) = { f ? W 0 1 : E ? } with respect to functional introduced by Lutwak, Yang and Zhang in [46] as well its compactness L for any ? . These points use strongly celebrated Blaschke-Santaló inequality. As counterpart, develop basic theory -Rayleigh quotients domains, case, More specifically, -affine versions ...

Affine spheres were introduced by Ţiţeica in [72, 73], and studied later by Blaschke, Calabi, and Cheng-Yau, among others. These are hypersurfaces in affine R which are related to real Monge-Ampère equations, to projective structures on manifolds, and to the geometry of Calabi-Yau manifolds. In this survey article, we will outline the theory of affine spheres their relationships to these topics...

In this paper, an observer based fuzzy adaptive controller (FAC) is designed fora class of large scale systems with non-canonical non-affine nonlinear subsystems. It isassumed that functions of the subsystems and the interactions among subsystems areunknown. By constructing a new class of state observer for each follower, the proposedconsensus control method solves the problem of unmeasured sta...

We give a example of non nilpotent faithful representation of a filiform Lie algebra. This gives one counter-example of the conjecture saying that every affine connection on a filiform Lie group is complete. 1. Affine connection on a nilpotent Lie algebra 1.1. Affine connection on nilpotent Lie algebras. Definition 1. Let g be a n-dimensional Lie algebra over R. It is called affine if there is ...

We introduce the spin Hecke algebra, which is a q-deformation of the spin symmetric group algebra, and its affine generalization. We establish an algebra isomorphism which relates our spin (affine) Hecke algebras to the (affine) Hecke-Clifford algebras of Olshanski and Jones-Nazarov. Relation between the spin (affine) Hecke algebra and a nonstandard presentation of the usual (affine) Hecke alge...

Abstract: An affine Moser-Trudinger inequality, which is stronger than the Euclidean MoserTrudinger inequality, is established. In this new affine analytic inequality an affine energy of the gradient replaces the standard L energy of gradient. The geometric inequality at the core of the affine Moser-Trudinger inequality is a recently established affine isoperimetric inequality for convex bodies...