Gérard Duchamp

NONCOMMUTATIVE SYMMETRIC FUNCTIONS VI: FREE QUASI-SYMMETRIC FUNCTIONS AND RELATED ALGEBRAS

This article is devoted to the study of several algebras related to symmetric functions, which admit linear bases labelled by various combinatorial objects: permutations (free quasi-symmetric functions), standard Young tableaux (free symmetric functions) and packed integer matrices (matrix quasi-symmetric functions). Free quasi-symmetric functions provide a kind of noncommutative Frobenius characteristic for a certain category of modules over the 0-Hecke algebras. New examples of indecomposable Hn(0)-modules are discussed, and the homological properties of Hn(0) are computed for small n. Finally, the algebra of matrix quasi-symmetric functions is interpreted as a convolution algebra.

A multi-step differential transform method and application to non-chaotic or chaotic systems

The differential transform method (DTM) is an analytical and numerical method for solving a wide variety of differential equations and usually gets the solution in a series form. In this paper, we propose a reliable new algorithm of DTM, namely multi-step DTM, which will increase the interval of convergence for the series solution. The multi-step DTM is treated as an algorithm in a sequence of intervals for finding accurate approximate solutions for systems of differential equations. This new algorithm is applied to Lotka-Volterra, Chen and Lorenz systems. Then, a comparative study between the new algorithm, multi-step DTM, classical DTM and the classical Runge-Kutta method is presented. The results demonstrate reliability and efficiency of the algorithm developed.

The lower central series of the free partially commutative group

This paper is devoted to the study of the lower central series of the free partially commutative groupF(A, ϑ) in connection with the associated free partially commutative Lie algebra. Using a convenient Magnus transformation, we show that the quotients of the lower central series ofF(A, ϑ) are free abelian groups and thatF(A, ϑ) can be fully ordered.

Combinatorics and Boson normal ordering: A gentle introduction

We discuss a general combinatorial framework for operator ordering problems by applying it to the normal ordering of the powers and exponential of the boson number operator. The solution of the problem is given in terms of Bell and Stirling numbers enumerating partitions of a set. This framework reveals several inherent relations between ordering problems and combinatorial objects, and displays the analytical background to Wick’s theorem. The methodology can be straightforwardly generalized from the simple example we discuss to a wide class of operators.

Boson normal ordering via substitutions and Sheffer-type polynomials

Abstract We solve the boson normal ordering problem for ( q ( a † ) a + v ( a † ) ) n with arbitrary functions q and v and integer n , where a and a † are boson annihilation and creation operators, satisfying [ a , a † ] = 1 . This leads to exponential operators generalizing the shift operator and we show that their action can be expressed in terms of substitutions. Our solution is naturally related through the coherent state representation to the exponential generating functions of Sheffer-type polynomials. This in turn opens a vast arena of combinatorial methodology which is applied to boson normal ordering and illustrated by a few examples.

Direct and dual laws for automata with multiplicities

We present here theoretical results coming from the implementation of the package called AMULT (automata with multiplicities). We show that classical formulas are optimal for the bounds. Especially they are almost everywhere optimal for the fields R and C. We characterize the dual laws preserving rationality and examine compatibility between the geometry of the K-automata andthese laws. Copyright 2001 Elsevier Science B.V.

The free partially commutative Lie algebra: Bases and ranks

Abstract In this paper, we study the free partially commutative Lie K -algebra L ( A , θ ) defined by a commutation relation θ on an alphabet A . Its behavior is very similar to that of the free Lie algebra. Indeed, we obtain in particular a partially commutative version of Lazards elimination process which allows us to prove that the K -module L ( A , θ ) is free and to construct explicitly K -bases for it. We show also how the classical Witts calculus can be extended to L ( A , θ ).

PARTIALLY COMMUTATIVE MAGNUS TRANSFORMATIONS

Grâce a l’introduction de la notion de forme exponentielle reduite, nous montrons l’injectivite de la transformation de Magnus associee au groupe partiellement commutatif libre F(A, ϑ). Nous en deduisons la separation de la topologie p-adique de F(A, ϑ), l’existence de plus petites racines dans F(A, ϑ) et enfin la structure des centralisateurs dans F(A, ϑ). We introduce the notion of reduced exponential form in the free partially commutative group F(A, ϑ). This allows us to prove that the Magnus transformation associated with F(A, ϑ) is injective. We deduce from this result that the p-adic topology on F(A, ϑ) is Hausdorff, that there exist least roots in F(A, ϑ) and finally the exact structure of centralizers in F(A, ϑ).

ONE-PARAMETER GROUPS AND COMBINATORIAL PHYSICS

In this communication, we consider the normal ordering of sums of elements of the form (a*^r a a*^s), where a* and a are boson creation and annihilation operators. We discuss the integration of the associated one-parameter groups and their combinatorial by-products. In particular, we show how these groups can be realized as groups of substitutions with prefunctions.

ORDERING RELATIONS FOR Q-BOSON OPERATORS, CONTINUED FRACTION TECHNIQUES AND THE Q-CBH ENIGMA

Ordering properties of boson operators have been very extensively studied, and q-analogues of many of the relevant techniques have been derived. These relations have far reaching physical applications and, at the same time, provide a rich and interesting source of combinatorial identities and of their g-analogues. An interesting exception involves the transformation from symmetric to normal ordering, which, for conventional boson operators, can most simply be effected using a special case of the Campbell-Baker-Hausdorff (CBH) formula. To circumvent the lack of a suitable q-analogue of the CBH formula, two alternative procedures are proposed, based on a recurrence relation and on a double continued fraction, respectively. These procedures enrich the repertoire of techniques available in this field. For conventional bosons they result in an expression that coincides with that derived using the CBH formula.