916 research outputs found
Factorizations of Elements in Noncommutative Rings: A Survey
We survey results on factorizations of non zero-divisors into atoms
(irreducible elements) in noncommutative rings. The point of view in this
survey is motivated by the commutative theory of non-unique factorizations.
Topics covered include unique factorization up to order and similarity, 2-firs,
and modular LCM domains, as well as UFRs and UFDs in the sense of Chatters and
Jordan and generalizations thereof. We recall arithmetical invariants for the
study of non-unique factorizations, and give transfer results for arithmetical
invariants in matrix rings, rings of triangular matrices, and classical maximal
orders as well as classical hereditary orders in central simple algebras over
global fields.Comment: 50 pages, comments welcom
Poles of regular quaternionic functions
This paper studies the singularities of Cullen-regular functions of one
quaternionic variable. The quaternionic Laurent series prove to be
Cullen-regular. The singularities of Cullen-regular functions are thus
classified as removable, essential or poles. The quaternionic analogues of
meromorphic complex functions, called semiregular functions, turn out to be
quotients of Cullen-regular functions with respect to an appropriate division
operation. This allows a detailed study of the poles and their distribution.Comment: 14 page
Peak reduction technique in commutative algebra
The "peak reduction" method is a powerful combinatorial technique with
applications in many different areas of mathematics as well as theoretical
computer science. It was introduced by Whitehead, a famous topologist and group
theorist, who used it to solve an important algorithmic problem concerning
automorphisms of a free group. Since then, this method was used to solve
numerous problems in group theory, topology, combinatorics, and probably in
some other areas as well.
In this paper, we give a survey of what seems to be the first applications of
the peak reduction technique in commutative algebra and affine algebraic
geometry.Comment: survey; 10 page
Extending structures I: the level of groups
Let be a group and a set such that . We shall describe
and classify up to an isomorphism of groups that stabilizes the set of all
group structures that can be defined on such that is a subgroup of .
A general product, which we call the unified product, is constructed such that
both the crossed product and the bicrossed product of two groups are special
cases of it. It is associated to and to a system called a group extending
structure and we denote it by . There exists a group structure on
containing as a subgroup if and only if there exists an isomorphism of
groups , for some group extending structure
. All such
group structures on are classified up to an isomorphism of groups that
stabilizes by a cohomological type set . A Schreier type theorem is proved and an explicit example is given: it
classifies up to an isomorphism that stabilizes all groups that contain
as a subgroup of index 2.Comment: 17 pages; to appear in Algebras and Representation Theor
Azumaya Objects in Triangulated Bicategories
We introduce the notion of Azumaya object in general homotopy-theoretic
settings. We give a self-contained account of Azumaya objects and Brauer groups
in bicategorical contexts, generalizing the Brauer group of a commutative ring.
We go on to describe triangulated bicategories and prove a characterization
theorem for Azumaya objects therein. This theory applies to give a homotopical
Brauer group for derived categories of rings and ring spectra. We show that the
homotopical Brauer group of an Eilenberg-Mac Lane spectrum is isomorphic to the
homotopical Brauer group of its underlying commutative ring. We also discuss
tilting theory as an application of invertibility in triangulated bicategories.Comment: 23 pages; final version; to appear in Journal of Homotopy and Related
Structure
The homotopy type of the loops on -connected -manifolds
For we compute the homotopy groups of -connected closed
manifolds of dimension . Away from the finite set of primes dividing
the order of the torsion subgroup in homology, the -local homotopy groups of
are determined by the rank of the free Abelian part of the homology.
Moreover, we show that these -local homotopy groups can be expressed as a
direct sum of -local homotopy groups of spheres. The integral homotopy type
of the loop space is also computed and shown to depend only on the rank of the
free Abelian part and the torsion subgroup.Comment: Trends in Algebraic Topology and Related Topics, Trends Math.,
Birkhauser/Springer, 2018. arXiv admin note: text overlap with
arXiv:1510.0519
Noncommutative generalizations of theorems of Cohen and Kaplansky
This paper investigates situations where a property of a ring can be tested
on a set of "prime right ideals." Generalizing theorems of Cohen and Kaplansky,
we show that every right ideal of a ring is finitely generated (resp.
principal) iff every "prime right ideal" is finitely generated (resp.
principal), where the phrase "prime right ideal" can be interpreted in one of
many different ways. We also use our methods to show that other properties can
be tested on special sets of right ideals, such as the right artinian property
and various homological properties. Applying these methods, we prove the
following noncommutative generalization of a result of Kaplansky: a (left and
right) noetherian ring is a principal right ideal ring iff all of its maximal
right ideals are principal. A counterexample shows that the left noetherian
hypothesis cannot be dropped. Finally, we compare our results to earlier
generalizations of Cohen's and Kaplansky's theorems in the literature.Comment: 41 pages. To appear in Algebras and Representation Theory. Minor
changes were made to the numbering system, in order to remain consistent with
the published versio
Molecular evolution of HoxA13 and the multiple origins of limbless morphologies in amphibians and reptiles
Developmental processes and their results, morphological characters, are inherited through transmission of genes regulating development. While there is ample evidence that cis-regulatory elements tend to be modular, with sequence segments dedicated to different roles, the situation for proteins is less clear, being particularly complex for transcription factors with multiple functions. Some motifs mediating protein-protein interactions may be exclusive to particular developmental roles, but it is also possible that motifs are mostly shared among different processes. Here we focus on HoxA13, a protein essential for limb development. We asked whether the HoxA13 amino acid sequence evolved similarly in three limbless clades: Gymnophiona, Amphisbaenia and Serpentes. We explored variation in ω (dN/dS) using a maximum-likelihood framework and HoxA13sequences from 47 species. Comparisons of evolutionary models provided low ω global values and no evidence that HoxA13 experienced relaxed selection in limbless clades. Branch-site models failed to detect evidence for positive selection acting on any site along branches of Amphisbaena and Gymnophiona, while three sites were identified in Serpentes. Examination of alignments did not reveal consistent sequence differences between limbed and limbless species. We conclude that HoxA13 has no modules exclusive to limb development, which may be explained by its involvement in multiple developmental processes
Developmental and evolutionary assumptions in a study about the impact of premature birth and low income on mother–infant interaction
In order to study the impact of premature
birth and low income on mother–infant interaction, four Portuguese samples were gathered: full-term, middle-class (n=99); premature, middle-class (n=63); full-term, low income (n=22); and premature, low income (n=21). Infants were filmed in a free play situation with their mothers, and the results were scored using the CARE Index. By means of multinomial regression analysis, social economic status (SES) was found to be the best predictor of maternal sensitivity and infant cooperative behavior within a set of medical and social factors. Contrary to the expectations of the cumulative risk perspective, two factors of risk (premature birth together with low SES) were as negative
for mother–infant interaction as low SES solely. In this study, as previous studies have shown, maternal sensitivity and infant cooperative behavior were highly correlated, as was maternal control with infant compliance. Our results further indicate that, when maternal lack of responsiveness
is high, the infant displays passive behavior, whereas when the maternal lack of responsiveness is medium, the infant
displays difficult behavior. Indeed, our findings suggest that, in these cases, the link between types of maternal and infant interactive behavior is more dependent on the degree of maternal lack of responsiveness than it is on birth status
or SES. The results will be discussed under a developmental and evolutionary reasonin
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