Papers by Claude-Alain Faure
arXiv (Cornell University), Sep 2, 2022
Deterministic evolutions are defined without ad hoc hypotheses but in the context of the axiomati... more Deterministic evolutions are defined without ad hoc hypotheses but in the context of the axiomatic approach due to Aerts and Piron. For systems described by a family of Hilbert spaces one can prove necessary conditions on the resulting flow using the tools of projective geometry. In particular we show that a deterministic evolution is always given by a family of partially defined unitary operators
Dedicated to Francis Buekenhout on the occasion of his 65th birthday Abstract. The fundamental th... more Dedicated to Francis Buekenhout on the occasion of his 65th birthday Abstract. The fundamental theorem of projective geometry is generalized for projective spaces over rings. Let RM and SN be modules. Provided some weak conditions are satisfied, a mor-phism g:PðMÞnE!PðNÞ between the associated projective spaces can be induced by a semilinear map f: M! N. These conditions are satisfied for instance if S is a left Ore domain and if the image of g contains three independent free points. No assumptions are made on the module M, and both modules may have some torsion.
Modern Projective Geometry, 2000
It is well known that the Desargues property for a projective geometry G is equivalent with the e... more It is well known that the Desargues property for a projective geometry G is equivalent with the existence of certain collineations of G. The respective collineations φ have an axis H (i.e. a hyperplane of G such that φx = x for all x ∈ H) and a center z (i.e. a point of G such that l(z, x,φx) for all x ∈ G). These notions axis and center are generalized to the case where φ: G → G is any endomorphism of G and it is shown that also for this case the existence of an axis is equivalent with the existence of a center (provided that φ is non-constant).
Modern Projective Geometry, 2000
Archiv der Mathematik, 2002
Abstract. This paper generalizes former versions of the so-called fundamental theorem of projecti... more Abstract. This paper generalizes former versions of the so-called fundamental theorem of projective geometry. It shows that generalized semilinear maps between vector spaces correspond to certain partial lineations between the associated projective spaces. In particular, it gives a characterization of all nondegenerate lineations between arguesian projective spaces.
Real Analysis Exchange, 2004
Czechoslovak Mathematical Journal, 1995
Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents ... more Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.
Real Analysis Exchange, 1997
This paper gives a descriptive definition of Stieltjes integrals (on a compact interval of the re... more This paper gives a descriptive definition of Stieltjes integrals (on a compact interval of the real line) in the frame of Kurzweil-Henstock integration. Five conditions characterize the functions that are an indefinite integral with respect to some continuous function of generalized bounded variation.
Real Analysis Exchange, 1996
Theorem B For a function f : [a, b] → R the following are equivalent: 1) f is Lebesgue integrable... more Theorem B For a function f : [a, b] → R the following are equivalent: 1) f is Lebesgue integrable, 2) for any ε > 0 there exist two L-integrable functions g ε and h ε such that
Applied Categorical Structures, 1998
After introducing morphisms between projective geometries, some categorical questions are examine... more After introducing morphisms between projective geometries, some categorical questions are examined. It is shown that there are three kinds of embeddings and two kinds of quotients. Furthermore the morphisms decompose in a canonical way into four factors.
Beitrage zur Algebra und Geometrie
Polar spaces are presented from the point of view of paraprojective spaces. Morphisms are introdu... more Polar spaces are presented from the point of view of paraprojective spaces. Morphisms are introduced and some immediate categorical aspects are reviewed. The morphisms of polar spaces are then studied in more details and are shown to preserve the spaces' structure. Finally, it is shown that a morphism of polar spaces can be split into a morphism of nondegenerate polar spaces and a morphism of projective spaces.
Modern Projective Geometry, 2000
Since a morphism from a projective geometry G 1 to another one G 2 is a map that is defined on ce... more Since a morphism from a projective geometry G 1 to another one G 2 is a map that is defined on certain, but in general not on all points of G 1, we begin the chapter with some generalities on the so-called partial maps. A partial map from a set X to a set Y, noted f: X - -→ Y,is a map f from a subset of X, called the domain of f,to Y. The points of X which are not in the domain of f form the so-called kernel of f. The sets together with the partial maps form a category Par. In the first section we give some elementary basic facts concerning this category. One can avoid partial maps by working with the equivalent category Set* of pointed sets. This is sometimes quite useful, but it involves that one has to add to the points of every projective geometry an additional point which can be considered as the zero element of the lattice of subspaces, but has no direct geometric interpretation.
Modern Projective Geometry, 2000
Since the projective geometries together with the morphisms form a category it is natural to exam... more Since the projective geometries together with the morphisms form a category it is natural to examine some categorical properties.
Modern Projective Geometry, 2000
There exist many equivalent axiomatic characterizations of projective geometries. The first two s... more There exist many equivalent axiomatic characterizations of projective geometries. The first two sections of this chapter present two such possibilities. The first one uses the ternary relation collinear on the set of points. The second one works with the operator that associates to each couple of points a, b the line through a and a ≠ b and the singleton {a} if a = b. Another approach, by means of the closure operator C that associates to an arbitrary set A of points the subspace C(A) generated by A, will be studied in Chapter 3. The first section also includes a series of examples. The most important one is the projective geometry associated to an arbitrary vector space V. This geometry will be denoted by P(V). The points of P(V) are the one-dimensional vector subspaces of V, three of them being called collinear if they lie in a vector subspace of dimension ≤ 2.
Modern Projective Geometry, 2000
A geometry is of degree n (where n is any non-negative integer) if one and hence all of the 14 eq... more A geometry is of degree n (where n is any non-negative integer) if one and hence all of the 14 equivalent conditions of 4.4.1 and 4.4.4 are satisfied, and in particular if for any subspaces E, F the equation r(E ∨ F) + r(E ∧ F) = r(E) + r(F) holds provided that r(E ∧ F) ≥ n. The geometries of degree 0 are exactly the projective geometries. Among the geometries of degree 1 one finds the affine geometries and also the projective geometries (we remark that a geometry of degree n is trivially of degree n + 1). Within the geometries of degree 1 the projective geometries can be characterized by an axiom requiring that parallel lines must be equal, the affine geometries by an axiom requiring that for any line δ and any point p there exists a unique line parallel to δ containing p.
Modern Projective Geometry, 2000
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Papers by Claude-Alain Faure