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Computer Algebra Methods for Studying and Computing Molecular Conformations
, 1997
"... A relatively new branch of computational biology has been emerging as an effort to supplement traditional techniques of large scale search in drug design by structurebased methods, in order to improve efficiency and guarantee completeness. This paper studies the geometric structure of cyclic molecu ..."
Abstract

Cited by 29 (4 self)
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A relatively new branch of computational biology has been emerging as an effort to supplement traditional techniques of large scale search in drug design by structurebased methods, in order to improve efficiency and guarantee completeness. This paper studies the geometric structure of cyclic molecules, in particular the enumeration of all possible conformations, which is crucial in finding the energetically favorable geometries, and the identification of all degenerate conformations. Recent advances in computational algebra are exploited, including distance geometry, sparse polynomial theory, and matrix methods for numerically solving nonlinear multivariate polynomial systems. Moreover, we propose a complete array of computer algebra and symbolic computational geometry methods for modeling the rigidity constraints, formulating the problems in algebraic terms and, lastly, visualizing the computed conformations. The use of computer algebra systems and of public domain software is illustrated...
Commutation Problems on Sets of Words and Formal Power Series
, 2002
"... We study in this thesis several problems related to commutation on sets of words and on formal power series. We investigate the notion of semilinearity for formal power series in commuting variables, introducing two families of series  the semilinear and the bounded series  both natural generaliza ..."
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Cited by 4 (3 self)
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We study in this thesis several problems related to commutation on sets of words and on formal power series. We investigate the notion of semilinearity for formal power series in commuting variables, introducing two families of series  the semilinear and the bounded series  both natural generalizations of the semilinear languages, and we study their behaviour under rational operations, morphisms, Hadamard product, and difference. Turning to commutation on sets of words, we then study the notions of centralizer of a language  the largest set commuting with a language , of root and of primitive root of a set of words. We answer a question raised by Conway more than thirty years ago  asking whether or not the centralizer of any rational language is rational  in the case of periodic, binary, and ternary sets of words, as well as for rational ccodes, the most general results on this problem. We also prove that any code has a unique primitive root and that two codes commute if and only if they have the same primitive root, thus solving two conjectures of Ratoandromanana, 1989. Moreover, we prove that the commutation with an ccode X can be characterized similarly as in free monoids: a language commutes with X if and only if it is a union of powers of the primitive root of X.
Acknowledgements
, 2002
"... Abstract. We give a theorem of reduction of the structure group of a principal bundle P with regular structure group G. Then, when G is in the classes of regular Lie groups defined by T.Robart in [13], we define the closed holonomy group of a connection as the minimal closed Lie subgroup of G for wh ..."
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Abstract. We give a theorem of reduction of the structure group of a principal bundle P with regular structure group G. Then, when G is in the classes of regular Lie groups defined by T.Robart in [13], we define the closed holonomy group of a connection as the minimal closed Lie subgroup of G for which the previous theorem of reduction can be applied. We also prove an infinite dimensional version of the AmbroseSinger theorem: the Lie algebra of the holonomy group is spanned by the curvature elements.