3,145 research outputs found
Finitely generated abelian groups of units
In 1960 Fuchs posed the problem of characterizing the groups which are the
groups of units of commutative rings. In the following years, some partial
answers have been given to this question in particular cases. In this paper we
address Fuchs' question for {\it finitely generated abelian} groups and we
consider the problem of characterizing those groups which arise in some fixed
classes of rings , namely the integral domains, the torsion free
rings and the reduced rings. To determine the realizable groups we have to
establish what finite abelian groups (up to isomorphism) occur as torsion
subgroup of when varies in , and on the other hand, we
have to determine what are the possible values of the rank of when
. Most of the paper is devoted to the study of the class
of torsion-free rings, which needs a substantially deeper study.Comment: 28 page
Finite groups of units of finite characteristic rings
In \cite[Problem 72]{Fuchs60} Fuchs asked the following question: which
groups can be the group of units of a commutative ring? In the following years,
some partial answers have been given to this question in particular cases. The
aim of the present paper is to address Fuchs' question when is a {\it
finite characteristic ring}. The result is a pretty good description of the
groups which can occur as group of units in this case, equipped with examples
showing that there are obstacles to a "short" complete classification. As a
byproduct, we are able to classify all possible cardinalities of the group of
units of a finite characteristic ring, so to answer Ditor's question
\cite{ditor}
A multi-class approach for ranking graph nodes: models and experiments with incomplete data
After the phenomenal success of the PageRank algorithm, many researchers have
extended the PageRank approach to ranking graphs with richer structures beside
the simple linkage structure. In some scenarios we have to deal with
multi-parameters data where each node has additional features and there are
relationships between such features.
This paper stems from the need of a systematic approach when dealing with
multi-parameter data. We propose models and ranking algorithms which can be
used with little adjustments for a large variety of networks (bibliographic
data, patent data, twitter and social data, healthcare data). In this paper we
focus on several aspects which have not been addressed in the literature: (1)
we propose different models for ranking multi-parameters data and a class of
numerical algorithms for efficiently computing the ranking score of such
models, (2) by analyzing the stability and convergence properties of the
numerical schemes we tune a fast and stable technique for the ranking problem,
(3) we consider the issue of the robustness of our models when data are
incomplete. The comparison of the rank on the incomplete data with the rank on
the full structure shows that our models compute consistent rankings whose
correlation is up to 60% when just 10% of the links of the attributes are
maintained suggesting the suitability of our model also when the data are
incomplete
On wild extensions of a p-adic field
In this paper we consider the problem of classifying the isomorphism classes
of extensions of degree pk of a p-adic field, restricting to the case of
extensions without intermediate fields. We establish a correspondence between
the isomorphism classes of these extensions and some Kummer extensions of a
suitable field F containing K. We then describe such classes in terms of the
representations of Gal(F/K). Finally, for k = 2 and for each possible Galois
group G, we count the number of isomorphism classes of the extensions whose
normal closure has a Galois group isomorphic to G. As a byproduct, we get the
total number of isomorphism classes
Block Tridiagonal Reduction of Perturbed Normal and Rank Structured Matrices
It is well known that if a matrix solves the
matrix equation , where is a linear bivariate polynomial,
then is normal; and can be simultaneously reduced in a finite
number of operations to tridiagonal form by a unitary congruence and, moreover,
the spectrum of is located on a straight line in the complex plane. In this
paper we present some generalizations of these properties for almost normal
matrices which satisfy certain quadratic matrix equations arising in the study
of structured eigenvalue problems for perturbed Hermitian and unitary matrices.Comment: 13 pages, 3 figure
A CMV--based eigensolver for companion matrices
In this paper we present a novel matrix method for polynomial rootfinding. By
exploiting the properties of the QR eigenvalue algorithm applied to a suitable
CMV-like form of a companion matrix we design a fast and computationally simple
structured QR iteration.Comment: 14 pages, 4 figure
Estimating an Eigenvector by the power method with a random start
This paper addresses the problem of approximating an eigenvector belonging to the largest eigenvalue of a symmetric positive definite matrix by the power method. We assume that the starting vector is randomly chosen with uniform distribution over the unit sphere. This paper provides lower and upper as well as asymptotic bounds on the randomized error in the ℒp sense, p ∈ [1,+∞]. We prove that it is impossible to achieve sharp bounds that are 1 independent of the ratio between the two largest eigenvalues. This should be contrasted to the problem of approximating the largest eigenvalue, for which Kuczyński and Woźniakowski [SIAM J. Matrix Anal. Appl., 13 (1992), pp. 1094-1122] proved that it is possible to bound the randomized error at the kth step with a quantity that depends only on k and on the size of the matrix. We prove that the rate of convergence depends on the ratio of the two largest eigenvalues, on their multiplicities, and on the particular norm. The rate of convergence is at most linear in the ratio of the two largest eigenvalues
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