Introduction. Let A be a commutative semi-simple Banach algebra and let A(.4) be the set of nonzero multiplicative functionals on A. Denote by A' the strongly closed span of A(^4) and by A" the Banach space adjoint of A'. Modifying a construction of R. Arens [l] we introduce a multiplication in A" under which A" becomes a commutative Banach algebra. A is algebraically isomorphic to a subalgebra...

In standard kernelization algorithms, the usual goal is to reduce, in polynomial time, an instance (I, k) of a parameterized problem to an equivalent instance (I ′, k′) of size bounded by a function in k. One of the central problems in this area, whose investigation has led to the development of many kernelization techniques, is the Dominating Set problem. Given a graph G and k ∈ N, Dominating ...

We show how to formalize approximate counting via hash functions in subsystems of bounded arithmetic, using variants of the weak pigeonhole principle. We discuss several applications, including a proof of the tournament principle, and an improvement on the known relationship of the collapse of the bounded arithmetic hierarchy to the collapse of the polynomial-time hierarchy.