A constructive proof of Simpson's Rule

For most purposes, one can replace the use of Rolle's theorem and the mean value theorem, which are not constructively valid, by the law of bounded change. The proof of two basic results in numerical analysis, the error term for Lagrange interpolation and Simpson's rule, however seem to require the full strength of the classical Rolle's Theorem. The goal of this note is to justify these two results constructively, using ideas going back to Amp\`ere and Genocchi

Integrals and Valuations

We construct a homeomorphism between the compact regular locale of integrals on a Riesz space and the locale of (valuations) on its spectrum. In fact, we construct two geometric theories and show that they are biinterpretable. The constructions are elementary and tightly connected to the Riesz space structure.Comment: Submitted for publication 15/05/0

Constructive Theory of Banach algebras

We present a way to organize a constructive development of the theory of Banach algebras, inspired by works of Cohen, de Bruijn and Bishop. We illustrate this by giving elementary proofs of Wiener's result on the inverse of Fourier series and Wiener's Tauberian Theorem, in a sequel to this paper we show how this can be used in a localic, or point-free, description of the spectrum of a Banach algebra

Categories of embeddings

AbstractWe present a categorical generalisation of the notion of domains, which is closed under (suitable) exponentiation. The goal was originally to generalise Girard's model of polymorphism to Fω. If we specialise this notion in the poset case, we get new cartesian closed categories of domains

On seminormality

AbstractWe give an elementary and essentially self-contained proof that a reduced ring R is seminormal if and only if the canonical map PicR→PicR[X] is an isomorphism, a theorem due to Swan [R. Swan, On seminormality, J. Algebra 67 (1980) 210–229], generalizing some previous results of Traverso [C. Traverso, Seminormality and the Picard group, Ann. Scuola Norm. Sup. Pisa 24 (1970) 585–595]

A variation of Reynolds-Hurkens Paradox

We present a variation of Hurkens paradox, which can itself be seen as a variation of Reynolds result that there is no set theoretic model of polymorphism