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Kock A. and G.E. Reyes (2003). Some calculus with extensive quantities: wave equation. arXiv: math.CT/0303297 v1 24 Mar 2003
This paper is a contribution to a synthetic theory of distributions. The sense in which we understand “synthetic” in this context is that we place ourselves in a setting (category) with a ring object having suitable properties, “the smooth reals” and where everything is smooth (differentiable). A main assumption about the category in which we work is that it is cartesian closed, meaning that function “spaces”, and hence some of the methods of functional analysis, are available. The blunt assumption of smoothness allows a simplification of the theory in the sense that distributions are functionals which are linear, continuity comes for free in this smooth universe. We provide topos models which contain both classical smooth manifolds and infinitesimal structures that relate the theory to the classical one. As a pilot project, we show how to construct the fundamental solution of the wave equation: the description of the evolution of a point (Dirac) distribution over time.
Reyes G.E. and A. Royer (2003) On the law of motion in Special Relativiyt. arXiv: physics/0302065 v1 19 Feb 2003
Newton’s law of motion for a particle of a given mass subject to a force at a given time may formulated either as “force = rate of change of the quantity of motion” or, since the mass is constant, as “force = mass times acceleration”, where velocity and acceleration are relative to an inertial frame. This law may be interpreted in either of two ways: (1) The force acting on the particle at the given time during an infinitesimal lapse of time imparts to the laboratory a boost, while the particle maintains its velocity relative to the new frame. (2) The force acting on the particle at a given time during an infinitesimal lapse of time imparts to the particle a boost relative to its proper frame which moves with the same velocity relative to the laboratory. We show that the relativistic law of motion admits both interpretations, the first of which is in fact equivalent to this law. As a consequence, we show that the relativistic law of motion may also be formulated as “force = mass times acceleration” in analogy with Newton’s law, but with a relativistic mass and a relativistic acceleration defined in terms of the relativistic addition law of velocities, rather than ordinary mass and ordinary vectorial addition of velocities that lead to the classical acceleration and to Newton’s law.
Kock A. and G. E. Reyes. Distributions in the Cahiers topos. 34 pages. Version pré́liminaire (February 2002) (SDG)
Reyes G.E. Topos theory in Montréal in the seventies. 18 pages. Preliminary version (September 2001)
Royer A. and G.E. Reyes. Lorentz transformation matrices in (3,1) block form. 18 pages. Version préliminaire (Août 2001)
Reyes G.E. and A. Royer . Forces and the equation of motion in special relativity. 29 pages. Version préliminaire (Août 2001)
Kock A. and G.E. Reyes (2001) Addendum: Aspects of fractional exponent functors.
Theory Appl. Categ, vol. 8. (Electronic)
Kock A. and G.E. Reyes (2001). Some differential equations in SDG. arXiv:math.CT/0104164 17 April 2001
This paper has been superseded. Its contents have been improved and re-written by the authors as three separate papers: “Some calculus with extensive quantities: wave equation”, “Categorical distribution theory; heat equation” and “Ordinary differential equations and their exponentials”. These papers may be unloaded here. I enclose it because it gives a bird’s view on our project.
Royer, A and G.E. Reyes. Relativistic velocity addition as a linear fractional map, and Thomas rotation. 11 pages. Version préliminaire (Avril 2001)
Galli, A., Reyes G.E. and M. Sagastume (2000). Completeness theorems via the double dual functor. Studia Logica 64, pp 61-81.
The aim of this paper is to apply properties of the double dual endofunctor on the category of bounded distributive lattices and some extensions thereof to obtain completeness of certain non-classical propositional logics in a unified way. In particulart, we obtain completeness theorems for Moisil calculus, n-valued Lukasiewicz calculus and Nelson calculus. Furthermore we show some conservativeness results by these methods.
