We study the moments finiteness problem for the class of Lipschitz maps F : [a, b] → R n with images in a compact Lipschitz triangulable curve Γ. We apply the obtained results to the center problem for ODEs describing in some cases (including equations with analytic coefficients) the set of universal centers of such equations by vanishing of finitely many moments from their coefficients.

No abstract

We study a new bi-Lipschitz invariant λ(M ) of a metric space M ; its finiteness means that Lipschitz functions on an arbitrary subset of M can be linearly extended to functions on M whose Lipschitz constants are expanded by a factor controlled by λ(M ). We prove that λ(M ) is finite for several important classes of metric spaces. These include metric trees of arbitrary cardinality, groups of polynomial growth, Gromov-hyperbolic groups, certain classes of Riemannian manifolds of bounded geometry and the finite direct sums of arbitrary combinations of these objects. On the other hand we construct an example of a two-dimensional Riemannian manifold M of bounded geometry for which λ(M ) = ∞. * Research supported in part by NSERC. 2000 Mathematics Subject Classification. Primary 26B35, Secondary 54E35, 46B15.

Abstract. Analogs of almost periodic functions for the unit circle are introduced. Certain uniform algebras generated by such functions are studied, the corona theorems for them are proved, and their maximal ideal spaces are described. §1. Formulation of main results 1.1.The classical almost periodic functions on the real line, as first introduced by H. Bohr in the 1920s, play an important role in various areas of analysis. In the present paper we define analogs of almost periodic functions on the unit circle. We study certain uniform algebras generated by such functions. In particular, in these terms we describe some uniform subalgebras of the algebra H ∞ of bounded holomorphic functions on the open unit disk D ⊂ C that, in a sense, have the weakest possible discontinuities on the boundary ∂D.To formulate the main results of the paper, we start with recalling the definition of an almost periodic function; see [B]. Definition 1.1. A continuous function f : R → C is said to be almost periodic if, for any > 0, there exists l( ) > 0 such that for every t 0 ∈ R the interval [t 0 , t 0 + l( )] contains at least one number τ for whichIt is well known that every almost periodic function f is uniformly continuous and is the uniform limit of a sequence of exponential polynomials {q n } n∈N , where q n (t) := n k=1 c kn e iλ kn t , c kn ∈ C, λ kn ∈ R, 1 ≤ k ≤ n, and i := √ −1. In what follows we consider ∂D with the counterclockwise orientation.as the right or the left endpoint relative to the chosen orientation, respectively. We define almost periodic functions on open arcs of ∂D. 2000 Mathematics Subject Classification. Primary 30H05; Secondary 46J20.

Let R be a unital semi-simple commutative complex Banach algebra, and let M(R) denote its maximal ideal space, equipped with the Gelfand topology. Sufficient topological conditions are given on M(R) for R to be a projective free ring, that is, a ring in which every finitely generated projective R-module is free. Several examples are included, notably the Hardy algebra H ∞ (X) of bounded holomorphic functions on a Riemann surface of finite type, and also some algebras of stable transfer functions arising in control theory.