1,614 research outputs found
Stratifying multiparameter persistent homology
A fundamental tool in topological data analysis is persistent homology, which
allows extraction of information from complex datasets in a robust way.
Persistent homology assigns a module over a principal ideal domain to a
one-parameter family of spaces obtained from the data. In applications data
often depend on several parameters, and in this case one is interested in
studying the persistent homology of a multiparameter family of spaces
associated to the data. While the theory of persistent homology for
one-parameter families is well-understood, the situation for multiparameter
families is more delicate. Following Carlsson and Zomorodian we recast the
problem in the setting of multigraded algebra, and we propose multigraded
Hilbert series, multigraded associated primes and local cohomology as
invariants for studying multiparameter persistent homology. Multigraded
associated primes provide a stratification of the region where a multigraded
module does not vanish, while multigraded Hilbert series and local cohomology
give a measure of the size of components of the module supported on different
strata. These invariants generalize in a suitable sense the invariant for the
one-parameter case.Comment: Minor improvements throughout. In particular: we extended the
introduction, added Table 1, which gives a dictionary between terms used in
PH and commutative algebra; we streamlined Section 3; we added Proposition
4.49 about the information captured by the cp-rank; we moved the code from
the appendix to github. Final version, to appear in SIAG
Infinite loop spaces and nilpotent K-theory
Using a construction derived from the descending central series of the free
groups, we produce filtrations by infinite loop spaces of the classical
infinite loop spaces , , , , , and
. We show that these infinite loop spaces are the zero
spaces of non-unital -ring spectra. We introduce the notion of
-nilpotent K-theory of a CW-complex for any , which extends the
notion of commutative K-theory defined by Adem-G\'omez, and show that it is
represented by , were is the -th term of
the aforementioned filtration of .
For the proof we introduce an alternative way of associating an infinite loop
space to a commutative -monoid and give criteria when it can be
identified with the plus construction on the associated limit space.
Furthermore, we introduce the notion of a commutative -rig and show
that they give rise to non-unital -ring spectra.Comment: To appear in Algebraic and geometric topolog
A little inflation at the cosmological QCD phase transition
We reexamine the recently proposed "little inflation" scenario that allows
for a strong first order phase-transition of QCD at non-negligible baryon
number in the early universe and its possible observable consequences. The
scenario is based on the assumptions of a strong mechanism for baryogenesis and
a quasistable QCD-medium state which triggers a short inflationary period of
inflation diluting the baryon asymmetry to the value observed today. The
cosmological implications are reexamined, namely effects on primordial density
fluctuations up to dark matter mass scales of M_{max} \sim 1 M_{\astrosun},
change in the spectral slope up to M_{max} \sim 10^6 M_{\astrosun},
production of seeds for the present galactic and extragalactic magnetic fields
and a gravitational wave spectrum with a peak frequency around . We discuss the issue of nucleation in more detail and
employ a chiral effective model of QCD to study the impact on small scale
structure formation.Comment: 18 pages, 12 figures, several extensions to the text and structure
formation part was rephrased for better readabilit
Knowledge gaps and acceptability of abbreviated alcohol screening in general practice: A cross-sectional survey of hazardous and non-hazardous drinkers
Background: General practice provides a unique setting where hazardous alcohol consumption can be screened for and behavioural interventions can be implemented in a continuous care model. Our aim was to assess in a general practice population, the prevalence of hazardous drinking, the knowledge and attitudes surrounding alcohol, and the acceptability of brief interventions in alcohol. Methods: A cross-sectional survey in a practice in South London, performed as part of a wider service evaluation. Questionnaires were offered to adult patients awaiting their appointments. Responses were stratified according to hazardous drinking, as per the abbreviated 'Alcohol Use Disorders Identification Test' (AUDIT-C). Results: Of 179 respondents (30 % male), 34 % yielded an AUDIT-C ≥5 and 18 % reported that they never drink alcohol. Male and Caucasian patients were more likely to self-report hazardous drinking, who in turn were more likely to believe in the health benefits of moderate consumption. Little over half of patents thought that alcohol is a risk factor for cancer and were misinformed of its calorific content, suggesting two targets for future improvement. Patients' knowledge about what is a single 'unit' of alcohol was below that expected by random chance 66 % agreed that alcohol screening should feature in all GP consultations. Conclusions: While awareness of alcohol related health risks is generally good, future efforts may benefit from focusing on the association with cancer and calories. Our findings question the utility of the 'unit' system, as well as dissemination of suggested 'health benefits' of moderate consumption. General practice initiatives in screening and brief advice for alcohol deserve further study
Irrational guards are sometimes needed
In this paper we study the art gallery problem, which is one of the
fundamental problems in computational geometry. The objective is to place a
minimum number of guards inside a simple polygon such that the guards together
can see the whole polygon. We say that a guard at position sees a point
if the line segment is fully contained in the polygon.
Despite an extensive study of the art gallery problem, it remained an open
question whether there are polygons given by integer coordinates that require
guard positions with irrational coordinates in any optimal solution. We give a
positive answer to this question by constructing a monotone polygon with
integer coordinates that can be guarded by three guards only when we allow to
place the guards at points with irrational coordinates. Otherwise, four guards
are needed. By extending this example, we show that for every , there is
polygon which can be guarded by guards with irrational coordinates but
need guards if the coordinates have to be rational. Subsequently, we show
that there are rectilinear polygons given by integer coordinates that require
guards with irrational coordinates in any optimal solution.Comment: 18 pages 10 Figure
Converting between quadrilateral and standard solution sets in normal surface theory
The enumeration of normal surfaces is a crucial but very slow operation in
algorithmic 3-manifold topology. At the heart of this operation is a polytope
vertex enumeration in a high-dimensional space (standard coordinates).
Tollefson's Q-theory speeds up this operation by using a much smaller space
(quadrilateral coordinates), at the cost of a reduced solution set that might
not always be sufficient for our needs. In this paper we present algorithms for
converting between solution sets in quadrilateral and standard coordinates. As
a consequence we obtain a new algorithm for enumerating all standard vertex
normal surfaces, yielding both the speed of quadrilateral coordinates and the
wider applicability of standard coordinates. Experimentation with the software
package Regina shows this new algorithm to be extremely fast in practice,
improving speed for large cases by factors from thousands up to millions.Comment: 55 pages, 10 figures; v2: minor fixes only, plus a reformat for the
journal styl
Analysis of contagion maps on a class of networks that are spatially embedded in a torus
A spreading process on a network is influenced by the network's underlying
spatial structure, and it is insightful to study the extent to which a
spreading process follows such structure. We consider a threshold contagion on
a network whose nodes are embedded in a manifold and where the network has both
`geometric edges', which respect the geometry of the underlying manifold, and
`non-geometric edges' that are not constrained by that geometry. Building on
ideas from Taylor et al. \cite{Taylor2015}, we examine when a contagion
propagates as a wave along a network whose nodes are embedded in a torus and
when it jumps via long non-geometric edges to remote areas of the network. We
build a `contagion map' for a contagion spreading on such a `noisy geometric
network' to produce a point cloud; and we study the dimensionality, geometry,
and topology of this point cloud to examine qualitative properties of this
spreading process. We identify a region in parameter space in which the
contagion propagates predominantly via wavefront propagation. We consider
different probability distributions for constructing non-geometric edges ---
reflecting different decay rates with respect to the distance between nodes in
the underlying manifold --- and examine the effect of such choices on the
qualitative properties of the spreading dynamics. Our work generalizes the
analysis in Taylor et al. and consolidates contagion maps both as a tool for
investigating spreading behavior on spatial networks and as a technique for
manifold learning
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