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1.

Andersson, Fredrik K.

et al.

WorldLight.com AB, Sweden.

Silvestrov, Sergei

Mälardalen University, School of Education, Culture and Communication.

The mathematics of internet search engines2008In: Acta Applicandae Mathematicae - An International Survey Journal on Applying Mathematics and Mathematical Applications, ISSN 0167-8019, E-ISSN 1572-9036, Vol. 104, no 2, p. 211-242Article in journal (Refereed)

Abstract [en]

This article presents a survey of techniques for ranking results in search engines, with emphasis on link-based ranking methods and the PageRank algorithm. The problem of selecting, in relation to a user search query, the most relevant documents from an unstructured source such as the WWW is discussed in detail. The need for extending classical information retrieval techniques such as boolean searching and vector space models with link-based ranking methods is demonstrated. The PageRank algorithm is introduced, and its numerical and spectral properties are discussed. The article concludes with an alternative means of computing PageRank, along with some example applications of this new method.

On the Exel crossed product of topological covering maps2009In: Acta Applicandae Mathematicae - An International Survey Journal on Applying Mathematics and Mathematical Applications, ISSN 0167-8019, E-ISSN 1572-9036, ISSN 0167-8019, Vol. 108, no 3, p. 573-583Article in journal (Refereed)

Abstract [en]

For dynamical systems defined by a covering map of a compact Hausdorff space and the corresponding transfer operator, the associated crossed product C ^{*}-algebras C(X)⋊ _{α,ℒ}ℕintroduced by Exel and Vershik are considered. An important property for homeomorphism dynamical systems is topological freeness. It can be extended in a natural way to in general non-invertible dynamical systems generated by covering maps. In this article, it is shown that the following four properties are equivalent: the dynamical system generated by a covering map is topologically free; the canonical embedding of C(X) into C(X)⋊ _{α,ℒ}ℕis a maximal abelian C ^{*}-subalgebra of C(X)⋊ _{α,ℒ}N; any nontrivial two sided ideal of C(X)⋊ _{α,ℒ}ℕhas non-zero intersection with the embedded copy of C(X); a certain natural representation of C(X)⋊ _{α,ℒ}ℕis faithful. This result is a generalization to non-invertible dynamics of the corresponding results for crossed product C ^{*}-algebras of homeomorphism dynamical systems.

Dynamical systems associated with crossed products2009In: Acta Applicandae Mathematicae - An International Survey Journal on Applying Mathematics and Mathematical Applications, ISSN 0167-8019, E-ISSN 1572-9036, ISSN 0167-8019, Vol. 108, no 3, p. 547-559Article in journal (Refereed)

Abstract [en]

In this paper, we consider both algebraic crossed products of commutative complex algebras A with the integers under an automorphism of A, and Banach algebra crossed products of commutative C*-algebras A with the integers under an automorphism of A. We investigate, in particular, connections between algebraic properties of these crossed products and topological properties of naturally associated dynamical systems. For example, we draw conclusions about the ideal structure of the crossed product by investigating the dynamics of such a system. To begin with, we recall results in this direction in the context of an algebraic crossed product and give simplified proofs of generalizations of some of these results. We also investigate new questions, for example about ideal intersection properties of algebras properly between the coefficient algebra A and its commutant A'. Furthermore, we introduce a Banach algebra crossed product and study the relation between the structure of this algebra and the topological dynamics of a naturally associated system.

Commutativity and ideals in pre-crystalline graded rings2009In: Acta Applicandae Mathematicae - An International Survey Journal on Applying Mathematics and Mathematical Applications, ISSN 0167-8019, E-ISSN 1572-9036, Vol. 108, no 3, p. 603-615Article in journal (Refereed)

Abstract [en]

Pre-crystalline graded rings constitute a class of rings which share many properties with classical crossed products. Given a pre-crystalline graded ring A, we describe its center, the commutant C_{A}(A_{0}) of the degree zero grading part, and investigate the connection between maximal commutativity of A_{0} in A and the way in which two-sided ideals intersect A_{0}.

Commutativity and ideals in strongly graded rings2009In: Acta Applicandae Mathematicae - An International Survey Journal on Applying Mathematics and Mathematical Applications, ISSN 0167-8019, E-ISSN 1572-9036, Vol. 108, no 3, p. 585-602Article in journal (Refereed)

Abstract [en]

In some recent papers by the first two authors it was shown that for any algebraic crossed product A, where A_{0}, the subring in the degree zero component of the grading, is a commutative ring, each non-zero two-sided ideal in A has a non-zero intersection with the commutant C_{A}(A_{0}) of A_{0} in A. This result has also been generalized to crystalline graded rings; a more general class of graded rings to which algebraic crossed products belong. In this paper we generalize this result in another direction, namely to strongly graded rings (in some literature referred to as generalized crossed products) where the subring A_{0}, the degree zero component of the grading, is a commutative ring. We also give a description of the intersection between arbitrary ideals and commutants to bigger subrings than A_{0}, and this is done both for strongly graded rings and crystalline graded rings.