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Copyright of Revue de Synthèse is the property of Brill Academic Publishers and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Revue de Synthèse is the property of Brill Academic Publishers and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Free cumulants were introduced by Speicher as a proper analog of classical cumulants in Voiculescu's theory of free probability. The relation between free moments and free cumulants is usually described in terms of Möbius calculus over the lattice of non-crossing partitions. In this work, we explore another approach to free cumulants and to their combinatorial study using a combinatorial Hopf algebra structure on the linear span of non-crossing partitions. The generating series of free moments is seen as a character on this Hopf algebra. It is characterized by solving a linear fixed-point equation that relates it to the generating series of free cumulants. These phenomena are explained through a process similar to (though different from) the arborification process familiar in the theory of dynamical systems, and originating in Cayley's work. [ABSTRACT FROM AUTHOR]

Invariant theory provides more efficient tools, such as Molien generating functions and integrity bases, than basic group theory, that relies on projector techniques, for the construction of symmetry-adapted polynomials in the symmetry coordinates of a molecular system, because it is based on a finer description of the mathematical structure of the latter polynomials. The present article extends its use to the construction of polynomial bases which span possibly, non-totally symmetric irreducible representations of a molecular symmetry group. Electric or magnetic observables can carry such irreducible representations, a common example is given by the electric dipole moment surface. The elementary generating functions and their corresponding integrity bases, where both the initial and the final representations are irreducible, are the building blocks of the algorithm presented in this article, which is faster than algorithms based on projection operators only. The generating functions for the full initial representation of interest are built recursively from the elementary generating functions. Integrity bases which can be used to generate in the most economical way symmetry-adapted polynomial bases are constructed alongside in the same fashion. The method is illustrated in detail on $$\mathrm {XY_4}$$ type of molecules. Explicit integrity bases for all five possible final irreducible representations of the tetrahedral group have been calculated and are given in the supplemental material associated with this paper. [ABSTRACT FROM AUTHOR]

Motivated by a recent surge of interest for Dynkin operators in mathematical physics and by problems in the combinatorial theory of dynamical systems, we propose here a systematic study of logarithmic derivatives in various contexts. In particular, we introduce and investigate generalizations of the Dynkin operator for which we obtain Magnus-type formulas. [ABSTRACT FROM AUTHOR]

This article aims at advancing the recently introduced exponential method for renormalisation in perturbative quantum field theory. It is shown that this new procedure provides a meaningful recursive scheme in the context of the algebraic and group theoretical approach to renormalisation. In particular, we describe in detail a Hopf algebraic formulation of Bogoliubov's classical R-operation and counterterm recursion in the context of momentum subtraction schemes. This approach allows us to propose an algebraic classification of different subtraction schemes. Our results shed light on the peculiar algebraic role played by the degrees of Taylor jet expansions, especially the notion of minimal subtraction and oversubtractions. [ABSTRACT FROM AUTHOR]

We describe the first representation of the general term of the Rayleigh-Schrödinger series for quasidegenerate systems. Each term of the series is represented by a tree and there is a straightforward relation between the tree and the analytical expression of the corresponding term. The combinatorial and graphical techniques used in the proof of the series expansion allow us to derive various resummation formulas of the series. A relation with several combinatorial objects used for special cases (degenerate or non-degenerate systems) is established. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2011 [ABSTRACT FROM AUTHOR]

Abstract: We tackle the problem of unraveling the algebraic structure of computations of effective Hamiltonians. This is an important subject in view of applications to chemistry, solid state physics or quantum field theory. We show, among other things, that the correct framework for these computations is provided by the hyperoctahedral group algebras. We define several structures on these algebras and give various applications. For example, we show that the adiabatic evolution operator (in the time-dependent interaction representation of an effective Hamiltonian) can be written naturally as a Picard-type series and has a natural exponential expansion. [Copyright &y& Elsevier]

The nuclear potential energy surfaces (PES) used in quantum chemistry inherit the symmetry of the whole molecular system, and are therefore invariant under the action of the nuclear permutation-translation-rotation-inversion group. One can take advantage of this property, both theoretically and numerically. The present article is the prolongation of the works of Schmelzer and Murrell and of Collins and Parsons on the subject. It presents a simplified technique to obtain symmetry-adapted polynomial basis for global PES, together with algorithmic recipes that make the problem computationally tractable. The method is illustrated in detail on XY 4 type of molecules for which a full description of the algebra of invariant polynomials under the full symmetry group of the molecule is given. [ABSTRACT FROM AUTHOR]