Renormalization and Computation II: Time Cut-off and the Halting Problem

This is the second installment to the project initiated in [Ma3]. In the first Part, I argued that both philosophy and technique of the perturbative renormalization in quantum field theory could be meaningfully transplanted to the theory of computation, and sketched several contexts supporting this view. In this second part, I address some of the issues raised in [Ma3] and provide their development in three contexts: a categorification of the algorithmic computations; time cut--off and Anytime Algorithms; and finally, a Hopf algebra renormalization of the Halting Problem.Comment: 28 page

Proper Functors and Fixed Points for Finite Behaviour

The rational fixed point of a set functor is well-known to capture the behaviour of finite coalgebras. In this paper we consider functors on algebraic categories. For them the rational fixed point may no longer be fully abstract, i.e. a subcoalgebra of the final coalgebra. Inspired by \'Esik and Maletti's notion of a proper semiring, we introduce the notion of a proper functor. We show that for proper functors the rational fixed point is determined as the colimit of all coalgebras with a free finitely generated algebra as carrier and it is a subcoalgebra of the final coalgebra. Moreover, we prove that a functor is proper if and only if that colimit is a subcoalgebra of the final coalgebra. These results serve as technical tools for soundness and completeness proofs for coalgebraic regular expression calculi, e.g. for weighted automata

New Developments in the Search for the Topology of the Universe

Multi-connected Universe models with space idenfication scales smaller than the size of the observable universe produce topological images in the catalogs of cosmic sources. In this review, we present the recent developments for the search of the topology of the universe focusing on three dimensional methods. We present the crystallographic method, we give a new lower bound on the size of locally Euclidean multi-connected universe model of $3000 h^{-1} \hbox{Mpc}$ based on this method and a quasar catalog, we discuss its successes and failures, and the attemps to generalise it. We finally introduce a new statistical method based on a collecting correlated pair (CCP) technique.Comment: 20 pages, 13 figures, Proceedings of the XIXth Texas meeting, Paris 14-18 december 1998, Proceedings of the XIXth Texas meeting, Eds. E. Aubourg, T. Montmerle, J. Paul and P. Peter, article-no: 04/2

Equivariant infinite loop space theory, I. The space level story

We rework and generalize equivariant infinite loop space theory, which shows how to construct G-spectra from G-spaces with suitable structure. There is a naive version which gives naive G-spectra for any topological group G, but our focus is on the construction of genuine G-spectra when G is finite. We give new information about the Segal and operadic equivariant infinite loop space machines, supplying many details that are missing from the literature, and we prove by direct comparison that the two machines give equivalent output when fed equivalent input. The proof of the corresponding nonequivariant uniqueness theorem, due to May and Thomason, works for naive G-spectra for general G but fails hopelessly for genuine G-spectra when G is finite. Even in the nonequivariant case, our comparison theorem is considerably more precise, giving a direct point-set level comparison. We have taken the opportunity to update this general area, equivariant and nonequivariant, giving many new proofs, filling in some gaps, and giving some corrections to results in the literature.Comment: 94 page