This paper defines a new notion of bounded computable randomness for certain classes of sub-computable functions which lack a universal machine. In particular, we define such versions of randomness for primitive recursive functions and for PSPACE functions. These new notions are robust in that there are equivalent formulations in terms of (1) Martin-Löf tests, (2) Kolmogorov complexity, and (3)...

A class G of graphs is said to be χ-bounded if there is a function f : N → R such that for all G ∈ G and all induced subgraphs H of G, χ(H) ≤ f(ω(H)). In this paper, we show that if G is a χ-bounded class, then so is the closure of G under any one of the following three operations: substitution, gluing along a clique, and gluing along a bounded number of vertices. Furthermore, if G is χ-bounded...

We give a necessary and a sufficient condition for the boundedness of the Toeplitz product TF TG∗ on the vector valued Bergman space L 2 a(C ), where F and G are matrix symbols with scalar valued Bergman space entries. The results generalize those in the scalar valued Bergman space case [4]. We also characterize boundedness and invertibility of Toeplitz products TF TG∗ in terms of the Berezin t...

We study BMO spaces associated with semigroup of operators on noncommutative function spaces (i.e. von Neumann algebras) and apply the results to boundedness of Fourier multipliers on non-abelian discrete groups. We prove an interpolation theorem for BMO spaces and prove the boundedness of a class of Fourier multipliers on noncommutative Lp spaces for all 1 < p < ∞, with optimal constants in p....