Refinement equations are two-scale difference functional equations of the type
\phi(x)=\sum_{k=0}^N ck\phi(2x-k), where c0,...,cN are complex coefficients, N=1,2,... Such equations have been studied in great detail in connections with their role in the wavelets theory, subdivision schemes in the approximation theory and curve design, fractals, probability theory and so on. By the study of spectral properties of the corresponding linear operator Tf(x)=\sum_{k=0}^N ckf(2x-k) (transition operator) we solve several problems in the theory of refinement equations. The properties of the solution \phi will also be discussed.
Each topological group G admits a unique universal minimal dynamical system (M(G),G). For a locally compact non-compact group this is a nonmetrizable system with a rich structure, on which G acts effectively. However there are topological groups for which M(G) is the trivial one point system (groups with the fixed point on compacta property), as well as topological groups G for which M(G) is a metrizable space and for which one has an explicit description. I will survey this new theory as developed by Pestov, Uspenskij and Glasner and Weiss and show how it relies on combinatorial Ramsey type theorems.
We introduce a series of principally different C^\infty -smooth counterexamples to the hypothesis on characterization of the sphere:
If for a smooth convex body K in R^3 and a constant C, in each point of \partial K the principal curvature radii of \partial K are separated by C, then K is a ball.
The hypothesis was proved by A.D.Alexandrov and H.F.Muenzner for analitic bodies. For the general smooth case it remained an open problem for years. Recently, Y.Martinez-Maure presented a C^2-smooth counterexample to the hypothesis.