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The aim of this note is to find all continuous everywhere defined solutions of the Babbage functional equation stating that the n-th iteration of a function is the identity function.
24 December 2014 /Corrected 29 July 2015

Babbage functional equation

A derivation of Einstein’s vacuum field equations

G.E. Reyes, Embedding manifolds with boundary in smooth toposes, Cahiers de Top. et Geom. Diff. Catég., (48), 83-103, (2007). We embed fully and faithfully the category of manifolds with boundary in some “smooth” toposes. We show that the embedding preserves open coverings and some products. Furthermore, it sends a Weil’s prolongation of a manifold by a Weil algebra into the embedded manifold raised to the associated infinitesimal space of the algebra. Our main tool is to “double” the manifold with boundary to obtain one without boundary.

Reyes G.E. A model of SDG in which only trivial distributions with compact support have a density. The aim of this note is to show that in the classifier of the theory of real analytic (i.e. C ω-)rings, linear functionals defined on the exponential whose base is the ring of “reals” (i.e., the generic model) and whose exponent is a finite power of this ring and having values in the reals, are trivial in a sense to be specified, provided that they have a density. This was conjectured by Anders Kock.

A model of SDG in which only trivial distributions with compact support have a density

This is an improved version (in English) of the Porta/Reyes preprint(in French) in the preceeding entry. Its aim is to embedd manifolds with boundary in smooth toposes defined by closed ideals. To appear in Cahiers de Topologie et Géométrie différentielle catégoriques.

Embedding manifolds with boundary in smooth toposes

A. Kock and G. E. Reyes, Ordinary differential equations and their exponentials, Central European J. of Math. 4 (2006), 64-81

Vector fields or, equivalently, ordinary differential equations have long been considered, heuristically, to be the same as “infinitesimal (pointed) actions” or “infinitesimal flows”, but it is only with the development of Synthetic Differential Geometry (SDG) that we have the tools to formulate these notions and prove their equivalence in a rigourous mathematical way. We exploit this fact to define the exponential of two ordinary differential equations as the exponential of the corresponding infinitesimal actions. The resulting action is seen to be the same as a partial differential equation whose solutions may be obtained by conjugation from the solutions of the differential equations that make up the exponential. Furthermore, we show that this method of conjugation is equivalent, under some conditions, to the method of change of variables, widely used to solve differential equations.

Ordinary differential equations and their exponentials

Anders Kock and Gonzalo Reyes

Cahiers de topologie et géométrie différentielle catégoriques, tome 47, no 1 (2006), p. 2-28.

Cet article expose une théorie synthétique des distributions (qui ne sont pas nécessairement de support compact). On compare cette théorie avec la théorie classique de Schwartz. Cette comparaison s’effectue par un plongement plein de la catégorie des espaces vectoriels convenables (et leurs applications lisses) dans certains gros topos, modèles de la géométrie différentielle synthétique.

Distributions and heat equation in SDG

Porta H. and G.E. Reyes (1980). Variétés à bord et topos lisses. Exposé 7, Séminaire de Géométrie différentielle synthétique, Université de Montréal, 1980. [[Re-typed by the second author in 2004]]

Le but de cet article est de plonger la catégorie des variétés à bord dans le topos de Dubuc (Cahiers topos).

Variétés à bord et topos lisses

Reyes G.E. A mathematical analysis of Masaccio’s Trinity. Preliminary version (February 2004). Published in Categories and Types in Logic, Language, and Physics. Editors C. Casadio et al. Springer LNCS 8222. 2014.

The aim of this note is to study several questions of a mathematical nature suggested by this fresco: (1) How accurate is the use of perspective? (2) What are the dimensions of the chapel? (3) What are the dimensions of the coffers of the vaulted ceiling of the chapel? (4) Where is the point of view situated with respect to the fresco? (5) Where are the different characters situated inside the chapel? (6) What are the “real” heights of the characters portrayed? Questions (1)-(4) admit answers that may be computed starting from the data of the fresco, by using some rules of perspective and simple mathematical facts. This is not true for the others. Nevertheless, we will show that under some reasonable hypotheses estimates may be made. A pictorial reproduction of the Trinity may be unloaded by clicking the next document in Varia: “Masaccio Trinity in the WEB”.

A mathematical analysis of Masaccio’s Trinity

Galli, A., Reyes G.E. and M. Sagastume (2003). Strong amalgamation, Beck-Chevalley for equivalence relations and interpolation in Algebraic Logic. Fuzzy sets and systems 138, 3-23.

We extend Makkai’s proof of strong amalgamation (push-outs of monos along arbitrary maps are monos) from the category of Heyting algebras to a class which includes the categories of symmetric bounded distributive lattices, symmetric Heyting algebras, Heyting modal S4-algebras, Heyting modal bi-S4-algebras, and Lukasiewicz n-valued algebras. We also extend and improve Pitt’s proof that strong amalgamation implies Beck-Chevalley for filters of Heyting algebras to exact categories with certain push-outs. As a consequence, a form of the Interpolation Lemma for some non-classical calculi is proved.

Strong Amalgamation