Discrete Aane Spheres 1

Discrete Indeenite Aane Spheres

Non-euclidean Affine Laminations

The purpose of the present paper is to discuss examples of aane Riemann surface laminations which do not admit a leafwise Euclidean structure. The rst example of such a lamination was constructed by Ghys Gh97]. Our discussion is based on the geometric methods developed by Lyubich, Minsky and the author LM97], KL01], which rely on the observation that any aane surface A gives rise in a natural w...

A ne systems in L 2 ( IRd ) : the analysis of the analysis operatorAmos

Discrete aane systems are obtained by applying dilations to a given shift-invariant system. The complicated structure of the aane system is due, rst and foremost, to the fact that it is not invariant under shifts. AAne frames carry the additional diiculty that they are \global" in nature: it is the entire interaction between the various dilation levels that determines whether the system is a fr...

A combinatorial derivation of the Poincare' polynomials of the finite irreducible Coxeter groups

(1) The Poincar e polynomials of the nite irreducible Coxeter groups and the Poincar e series of the aane Coxeter groups on three generators are derived by an elementary combinatorial method avoiding the use of Lie theory and invariant theory. (2) Non-recursive methods for the computation of`standard reduced words' for (signed) permutations are described. The algebraic basis for both (1) and (2...

Extensions of the Heisenberg Group by Dilations and Frames

Two standard tools for signal analysis are the short{time Fourier transform and the continuous wavelet transform. These tools arise as matrix coeecients of square integrable representations of the Heisenberg and aane groups respectively, and discrete frame decompositions of L 2 arise from approximations of corresponding reproducing formulae. Here we study two groups, the so-called aane Weyl-Hei...