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On a Lie Algebraic Characterization of Vector Bundles Lecomte, Pierre ; Leuther, Thomas ; Zihindula Mushengezi, Elie in Symmetry, Integrability and Geometry: Methods and Applications [=SIGMA] (2012) We prove that a vector bundle E -> M is characterized by the Lie algebra generated by all differential operators on E which are eigenvectors of the Lie derivative in the direction of the Euler vector ... [more ▼] We prove that a vector bundle E -> M is characterized by the Lie algebra generated by all differential operators on E which are eigenvectors of the Lie derivative in the direction of the Euler vector field. Our result is of Pursell-Shanks type but it is remarkable in the sense that it is the whole f ibration that is characterized here. The proof relies on a theorem of [Lecomte P., J. Math. Pures Appl. (9) 60 (1981), 229{239] and inherits the same hypotheses. In particular, our characterization holds only for vector bundles of rank greater than 1. [less ▲] Detailed reference viewed: 57 (10 ULg)Abstract numeration systems (Chapter 3) Lecomte, Pierre ; Rigo, Michel in Berthé, Valérie; Rigo, Michel (Eds.) Combinatorics, Automata and Number Theory (2010) Detailed reference viewed: 85 (9 ULg)Combinatorics, Automata and Number Theory 2006 ; Lecomte, Pierre ; Rigo, Michel in Theoretical Computer Science (2008), 391 Detailed reference viewed: 21 (6 ULg)Real numbers having ultimately periodic representations in abstract numeration systems Lecomte, Pierre ; Rigo, Michel in Information and Computation (2004), 192(1), 57-83 Using a genealogically ordered infinite regular language, we know how to represent an interval of R. Numbers having an ultimately periodic representation play a special role in classical numeration ... [more ▼] Using a genealogically ordered infinite regular language, we know how to represent an interval of R. Numbers having an ultimately periodic representation play a special role in classical numeration systems. The aim of this paper is to characterize the numbers having an ultimately periodic representation in generalized systems built on a regular language. The syntactical properties of these words are also investigated. Finally, we show the equivalence of the classical theta-expansions with our generalized representations in some special case related to a Pisot number theta. (C) 2004 Elsevier Inc. All rights reserved. [less ▲] Detailed reference viewed: 15 (2 ULg)On the representation of real numbers using regular languages Lecomte, Pierre ; Rigo, Michel in Theory of Computing Systems (2002), 35(1, JAN-FEB), 13-38 Using a lexicographically ordered regular language, we show how to represent an interval of R. We determine exactly the possible representations of any element in this interval and study the function ... [more ▼] Using a lexicographically ordered regular language, we show how to represent an interval of R. We determine exactly the possible representations of any element in this interval and study the function which maps a representation onto its numerical value. We make explicit the relationship between the convergence of finite words to an infinite word and the convergence of the corresponding approximations to a real number. [less ▲] Detailed reference viewed: 42 (10 ULg)Numerations systems on a regular language Lecomte, Pierre ; Rigo, Michel in Theory of Computing Systems (2001), 34 Generalizations of positional number systems in which N is recognizable by finite automata are obtained by describing an arbitrary infinite regular language according to the lexicographic ordering. For ... [more ▼] Generalizations of positional number systems in which N is recognizable by finite automata are obtained by describing an arbitrary infinite regular language according to the lexicographic ordering. For these systems of numeration, we show that ultimately periodic sets are recognizable. We also study translation and multiplication by constants as well as the order-dependence of the recognizability. [less ▲] Detailed reference viewed: 35 (2 ULg)Comparison of some modules of the Lie algebra of vector fields Lecomte, Pierre ; Mathonet, Pierre ; in Indagationes Mathematicae (1996), 7(4), 461-471 Detailed reference viewed: 14 (2 ULg) |
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