Given a family of n convex compact sets in the plane, one can always choose n of them which are either pairwise disjoint or pairwise intersecting. On the other hand, there exists a family of n segments in the plane such that the maximum size of a subfamily with pairwise disjoint or pairwise intersecting elements in n*"* ^ n-.

We construct a new family of compact orbifolds O(Θ) with a positive self dual Einstein metric and a one-dimensional group of isometries. Together with another family, introduced in [6] and here denoted by O(Ω), these examples classify all 4-dimensional orbifolds that are quaternion Kähler quotients by a torus of real Grassmannians.

In this paper we survey some recent work directed towards generalizing the fast Fourier transform (FFT). We work primarily from the point of view of group representation theory. In this setting the classical FFT can be viewed as a family of e cient algorithms for computing the Fourier transform of either a function de ned on a nite abelian group, or a bandlimited function on a compact abelian g...

A widely used theorem of analysis asserts that a uniformly bounded, equicontinuous family of functions has a compact closure in the space of continuous functions. This lemma, variously attributed to Arzela, Escoli, Montel, Vitali, and so on, is of importance in the theory of integral equations, conformal mapping, calculus of variations, and so on. In recent years the lemma has been generalized ...

Let F := {fx : x ∈ X} be a family of functionals defined on a Hilbert manifold e E and smoothly depending on a compact n-dimensional manifold X. The aim of this paper is to give a sufficient topological condition on the parameter space X in such a way the family F bifurcate from the trivial branch.

A new family of wavelets is introduced, which is associated with Legendre polynomials. These wavelets, termed spherical harmonic or Legendre wavelets, possess compact support. The method for the wavelet construction is derived from the association of ordinary second order differential equations with multiresolution filters. The low-pass filter associated to Legendre multiresolution analysis is ...

We introduce a new quantum Gromov-Hausdorff distance between C∗-algebraic compact quantum metric spaces. Because it is able to distinguish algebraic structures, this new distance fixes a weakness of Rieffel’s quantum distance. We show that this new quantum distance has properties analogous to the basic properties of the classical Gromov-Hausdorff distance, and we give criteria for when a parame...

Let G be a locally compact topological group. We investigate the type of the classifying space of G for the family of compact subgroups. We give criteria for this space to have a d-dimensional G-CW -model, a finite G-CW -model or a G-CW -model of finite type. Essentially we reduce these questions to discrete groups and to the homological algebra of the orbit category of discrete groups with res...

In this paper, we give a concrete description of the two-fold cover of a simply connected, split real reductive group and its maximal compact subgroup as Chevalley groups. We study the representations of the maximal compact subgroups called pseudospherical representations which appear with multiplicity one in the principal series representation. We introduce a family of canonically defined inte...

In this paper we characterize asymptotic stability via Lyapunov function in general dynamical systems on c-first countable space. We give a family of examples which have first countable but not c-first countable, also c-first countable and locally compact space but not metric space. We obtain several necessary and sufficient conditions for a compact subset M of the phase space X to be asymptoti...