142,937 research outputs found
Belyi-extending maps and the Galois action on dessins d'enfants
We study the absolute Galois group by looking for invariants and orbits of
its faithful action on Grothendieck's dessins d'enfants. We define a class of
functions called Belyi-extending maps, which we use to construct new Galois
invariants of dessins from previously known invariants. Belyi-extending maps
are the source of the ``new-type'' relations on the injection of the absolute
Galois group into the Grothendieck-Teichmuller group. We make explicit how to
get from a general Belyi-extending map to formula for its associated invariant
which can be implemented in a computer algebra package. We give an example of a
new invariant differing on two dessins which have the same values for the other
readily computable invariants.Comment: 13 pages, 7 figures; submitted for publication; revisions are that
the paper now deals only with Galois invariants of dessins, and that material
is slightly expande
Semistrict models of connected 3-types and Tamsamani's weak 3-groupoids
Homotopy 3-types can be modelled algebraically by Tamsamani's weak
3-groupoids as well as, in the path-connected case, by cat^2-groups. This paper
gives a comparison between the two models in the path-connected case. This
leads to two different semistrict algebraic models of connected 3-types using
the Tamsamani's model. Both are then related to Gray groupoids.Comment: 21 page
An elementary construction of Anick's fibration
Cohen, Moore, and Neisendorfer's work on the odd primary homotopy theory of
spheres and Moore spaces, as well as the first author's work on the secondary
suspension, predicted the existence of a p-local fibration S^2n-1 --> T -->
\Omega S^2n+1 whose connecting map is degree p^r. In a long and complex
monograph, Anick constructed such a fibration for p>= 5 and r>= 1. Using new
methods we give a much more conceptual construction which is also valid for p=3
and r>= 1. We go on to establish several properties of the space T.Comment: 30 page
The inner automorphism 3-group of a strict 2-group
Any group gives rise to a 2-group of inner automorphisms,
. It is an old result by Segal that the nerve of this is the
universal -bundle. We discuss that, similarly, for every 2-group
there is a 3-group and a slightly smaller 3-group
of inner automorphisms. We describe these for
any strict 2-group, discuss how can be
understood as arising from the mapping cone of the identity on and
show that its underlying 2-groupoid structure fits into a short exact sequence
.
As a consequence, encodes the properties of the
universal 2-bundle.Comment: references added, relation to simplicial constructions expanded,
version to appear in JHR
Topological finiteness properties of monoids. Part 1: Foundations
We initiate the study of higher dimensional topological finiteness properties
of monoids. This is done by developing the theory of monoids acting on CW
complexes. For this we establish the foundations of -equivariant homotopy
theory where is a discrete monoid. For projective -CW complexes we prove
several fundamental results such as the homotopy extension and lifting
property, which we use to prove the -equivariant Whitehead theorems. We
define a left equivariant classifying space as a contractible projective -CW
complex. We prove that such a space is unique up to -homotopy equivalence
and give a canonical model for such a space via the nerve of the right Cayley
graph category of the monoid. The topological finiteness conditions
left- and left geometric dimension are then defined for monoids
in terms of existence of a left equivariant classifying space satisfying
appropriate finiteness properties. We also introduce the bilateral notion of
-equivariant classifying space, proving uniqueness and giving a canonical
model via the nerve of the two-sided Cayley graph category, and we define the
associated finiteness properties bi- and geometric dimension. We
explore the connections between all of the these topological finiteness
properties and several well-studied homological finiteness properties of
monoids which are important in the theory of string rewriting systems,
including , cohomological dimension, and Hochschild
cohomological dimension. We also develop the corresponding theory of
-equivariant collapsing schemes (that is, -equivariant discrete Morse
theory), and among other things apply it to give topological proofs of results
of Anick, Squier and Kobayashi that monoids which admit presentations by
complete rewriting systems are left-, right- and bi-.Comment: 59 pages, 1 figur
netgwas: An R Package for Network-Based Genome-Wide Association Studies
Graphical models are powerful tools for modeling and making statistical
inferences regarding complex associations among variables in multivariate data.
In this paper we introduce the R package netgwas, which is designed based on
undirected graphical models to accomplish three important and interrelated
goals in genetics: constructing linkage map, reconstructing linkage
disequilibrium (LD) networks from multi-loci genotype data, and detecting
high-dimensional genotype-phenotype networks. The netgwas package deals with
species with any chromosome copy number in a unified way, unlike other
software. It implements recent improvements in both linkage map construction
(Behrouzi and Wit, 2018), and reconstructing conditional independence network
for non-Gaussian continuous data, discrete data, and mixed
discrete-and-continuous data (Behrouzi and Wit, 2017). Such datasets routinely
occur in genetics and genomics such as genotype data, and genotype-phenotype
data. We demonstrate the value of our package functionality by applying it to
various multivariate example datasets taken from the literature. We show, in
particular, that our package allows a more realistic analysis of data, as it
adjusts for the effect of all other variables while performing pairwise
associations. This feature controls for spurious associations between variables
that can arise from classical multiple testing approach. This paper includes a
brief overview of the statistical methods which have been implemented in the
package. The main body of the paper explains how to use the package. The
package uses a parallelization strategy on multi-core processors to speed-up
computations for large datasets. In addition, it contains several functions for
simulation and visualization. The netgwas package is freely available at
https://cran.r-project.org/web/packages/netgwasComment: 32 pages, 9 figures; due to the limitation "The abstract field cannot
be longer than 1,920 characters", the abstract appearing here is slightly
shorter than that in the PDF fil
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