The purpose of this note is to announce our proof of the AtiyahJones conjecture concerning the topology of the moduli spaces of based su(2)instantons over s4 . Full details and proofs appear in our paper [BHMM1].

In this paper we briefly review and analyze three published proofs of Chaitin’s theorem, the celebrated information-theoretic version of Gödel’s incompleteness theorem. Then, we discuss our main perplexity concerning a key step common to all these demonstrations.

This short paper addresses the open problems left in the paper [5]. Besides giving solutions to these two problems, some clarification concerning the role of the full vocabulary (including functional symbols) in the proofs there given is also discussed.

We shall see what Locke means by this in more detail below, but a brief note is sufficient to show that there is a prima facie conflict. Locke explains that “[w]henever the Mind refers any of its Ideas to any things extraneous to them, they are then capable to be called true or false ... the Mind in such a reference makes a tacit Supposition of their Conformity to that Thing” (2.32.2). That is, an

Williams' research was supported by NSERC of Canada grant A7649. 1991 Mathematics Subject Classi cation: 11Y40, 11Y99 Let p be a prime congruent to 1 modulo 4, n p the Legendre symbol and S(k) = Pp 1 n=1 n k n p . The problem of finding a prime p such that S(3) > 0 was one of the motivating forces behind the development of several of Shanks’ ideas for computing in algebraic number fields, altho...

Motivated by some earlier Diophantine works on triangular numbers by Ljunggren and Cassels, we consider similar problems for general polygonal numbers.

Let S " he the set of all polynomials whose degree does not exceed n and whose all zeros are real but lie outside (-1, 1). Similarly, we say p"E Q " if p,(x) is a real polynomial whose all zeros lie outside the open disk with center at the origin and radius l. Further we will denote by H " the set of all polynomials of degree-n and of the form k=O where q,,k (x) = (1 +x)k (l-x)"-'. Elements of ...

This paper is an appendix to the paper " Random walks on free products of cyclic groups " by J. Mairesse and F. Mathéus. It contains the details of the computations and the proofs of the results concerning the examples treated there.