This paper is partly a survey of this development. In Section 2 we review the construction of the nonstandard universe N . Section 3 discusses the internal sets and functions of N . Here we present two new results for N : the @ 1 -saturation principle and a characterisation of internal functions between nonstandard versions of standard sets. We also briefly indicate how to make the Loeb measure construction over hyperfinite sets. Section 4 discusses the relation between nonstandard real numbers and the canonical real numbers of N . In the final section we exemplify the use of the model to prove results in the calculus of several variables, e.g. the Implicit Function Theorem.

In the context of Synthetic Differential Geometry, we provide, for any manifold, a homotopy equivalence between its deRham complex, and a complex of infinitesimal singular cochains. The equivalence takes wedge product of forms to cup product of singular cochains. The purpose of the present note is to identify the de Rham complex of differential forms on a manifold M with a certain cochain complex related to the singular complex of M . In fact this cochain complex is dual to a certain simplicial subcomplex of the singular complex, consisting of "infinitesimal simplices". The notions make sense in the context of an embedding of the category of smooth manifolds into a suitable topos, more precisely, into a "model for synthetic differential geometry" (SDG). Our comparison is based on some results from [4], and is inspired by Felix and Lavendhomme's [2]. They also provide an identification of the de Rham complex with a complex related to the singular one; they, however, use cubical rather ...

ion problem for PCF (see [BCL86, Cur93, Ong95] for surveys). The importance of full abstraction for the semantics of programming languages is that it is one of the few quality filters we have. Specifically, it provides a clear criterion for assessing how definitive a semantic analysis of some language is. It must be admitted that to date the quest for fully abstract models has not yielded many obvious applications; but it has generated much of the deepest work in semantics. Perhaps it is early days yet. Recently, game semantics has been used to give the first syntax-independent constructions of fully abstract models for a number of programming languages, including PCF [AJM96, HO96, Nic94], richer functional languages [AM95, McC96b, McC96a, HY97], and languages with non-functional features such as reference types and non-local control constructs [AM97c, AM97b, AM97a, Lai97]. A noteworthy feature is that the key definability results for the richer languages are proved by a reduction to...

The rate distortion manifold is considered as a carrier for elements of the theory of information proposed by C. E. Shannon combined with the semantic precepts of F. Dretske’s theory of communication. This type of information space was suggested by R. Wallace as a possible geometric–topological descriptive model for incorporating a dynamic information based treatment of the Global Workspace theory of B. Baars. We outline a more formal mathematical description for this class of information space and further clarify its structural content and overall interpretation within prospectively a broad range of cognitive situations that apply to individuals, human institutions, distributed cognition and massively parallel intelligent machine design. Povzetek: Predstavljena je formalna definicija prostora za opisovanje kognitivnih procesov. 1

this paper, I intend to put together some of the efforts by several people of making aspects of fibre bundle theory into algebra. We prove in particular an "infinitesimal form" of the Gauss-Bonnet Theorem, Corollary 1 below. --- The initiator of these efforts was Charles Ehresmann, who put the notion of

We examine critically some of the existing descriptions of the envelope of a 1-parameter family of surfaces in 3-space. An old, natural, description is

Groupoids provide a more appropriate framework for differential geometry than principal bundles. Synthetic differential geometry is the avantgarde branch of differential geometry, in which nilpotent infinitesimals are available in abundance. The principal objective in this paper is to show within our favorite framework of synthetic differential geometry that the tangent space of the group of bisections of a microlinear groupoid at its identity is naturally a Lie algebra. We give essentially distinct two proofs for its Jacobi identity. Keywords:generalized gauge transformation, infinitesimal gauge transformation, topos-theoretic physics, synthetic differential geometry, groupoid, bisection, Lie algebra, Jacobi identity

W.Lawvere in [4] suggested a approach to differential geometry and to others mathematical disciplines closed to physics, which allows to give definitions of derivatives, tangent vectors and tangent bundles without passages to the limits. This approach is based on a idea of consideration of all settings not in sets but in some cartesian closed category E , particular in some elementary topos. The synthetic differential geometry (SDG) is the theory developed by A.Kock [1] in a context of Lawvere's ideas. In a base of the theory is an assumption of that a geometric line is not a filed of real numbers, but a some nondegenerate commutative ring R of a line type in E . In this work we shall show that SDG allows to develop a Riemannian geometry and write the Einstein's equations of a field on pseudo-Riemannian formal manifold. This give a way for constructing a intuitionistic models of general relativity in suitable toposes. 1 Preliminaries In this paper will be given some metrical notions i...

Traditionally, an infinitesimal quantity is one which, while not necessarily coinciding with zero, is in some sense smaller than any finite quantity. In "practical " approaches to the differential calculus an infinitesimal is a quantity so small that its square and all higher powers can be "neglected". In the theory of limits the term "infinitesimal " is sometimes applied to any sequence whose limit is zero. An infinitesimal magnitude may be regarded as what remains after a continuum (v. the article "Continuity") has been subjected to an exhaustive analysis, in other words, as a continuum "viewed in the small". It is in this sense that continuous curves have sometimes been held to be "composed " of infinitesimal straight lines. Infinitesimals have a long and colourful history. They make an early appearance in the mathematics of the Greek atomist philosopher Democritus (c.450 B.C.), only to be banished by the mathematician Eudoxus (c.350 B.C.) in what was to become official "Euclidean " mathematics. Taking the somewhat obscure form of "indivisibles", they reappear in the mathematics of the late

Introduction The classical synthetic descriptions of the osculating plane of a space curve k, at a point P 2 k, are: 1) it is the plane given by the tangent at P and a neighbour point on k; or 2) it is the plane given by P and a neighbour tangent; or 3) it is the plane given by P and two consecutive neighbours of P on k; or, finally, 4) it is the (common) tangent plane of the surface, swept out by the tangents of the space curve, taken at any point of the tangent through P . Only the last description matches perfectly with the analytic formalism, by means of which the theory is usually made rigourous. We shall present a formalism which a little more directly fits with (or vindicates) the three last of these descriptions; and which is rigourous. It does not vindicate the first description; for, in our formalism, the c