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

An Infeasible-Point Subgradient Method Using Adaptive Approximate Projections

We propose a new subgradient method for the minimization of nonsmooth convex functions over a convex set. To speed up computations we use adaptive approximate projections only requiring to move within a certain distance of the exact projections (which decreases in the course of the algorithm). In particular, the iterates in our method can be infeasible throughout the whole procedure. Nevertheless, we provide conditions which ensure convergence to an optimal feasible point under suitable assumptions. One convergence result deals with step size sequences that are fixed a priori. Two other results handle dynamic Polyak-type step sizes depending on a lower or upper estimate of the optimal objective function value, respectively. Additionally, we briefly sketch two applications: Optimization with convex chance constraints, and finding the minimum l1-norm solution to an underdetermined linear system, an important problem in Compressed Sensing.Comment: 36 pages, 3 figure

Defect topologies in chiral liquid crystals confined to mesoscopic channels

This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in J. Chem. Phys. 142, 194704 (2015) and may be found at https://doi.org/10.1063/1.4920979.We present Monte Carlo simulations in the grand canonical and canonical ensembles of a chiral liquid crystal confined to mesochannels of variable sizes and geometries. The mesochannels are taken to be quasi-infinite in one dimension but finite in the two other directions. Under thermodynamic conditions chosen and for a selected value of the chirality coupling constant, the bulk liquid crystal exhibits structural characteristics of a blue phase II. This is established through the tetrahedral symmetry of disclination lines and the characteristic simple-cubic arrangement of double-twist helices formed by the liquid-crystal molecules along all three axes of a Cartesian coordinate system. If the blue phase II is then exposed to confinement, the interplay between its helical structure, various anchoring conditions at the walls of the mesochannels, and the shape of the mesochannels gives rise to a broad variety of novel, qualitative disclination-line structures that are reported here for the first time.DFG, 65143814, GRK 1524: Self-Assembled Soft-Matter Nanostructures at Interface

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 $BSU$, $BU$, $BSO$, $BO$, $BSp$, $BGL_{\infty}(R)^{+}$ and $Q_0(\mathbb{S}^{0})$. We show that these infinite loop spaces are the zero spaces of non-unital $E_\infty$-ring spectra. We introduce the notion of $q$-nilpotent K-theory of a CW-complex $X$ for any $q\ge 2$, which extends the notion of commutative K-theory defined by Adem-G\'omez, and show that it is represented by $\mathbb Z\times B(q,U)$, were $B(q,U)$ is the $q$-th term of the aforementioned filtration of $BU$. For the proof we introduce an alternative way of associating an infinite loop space to a commutative $\mathbb{I}$-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 $\mathbb{I}$-rig and show that they give rise to non-unital $E_\infty$-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 $\nu_{peak} \sim 4 \cdot 10^{-8} Hz$. 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 $x$ sees a point $y$ if the line segment $xy$ 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 $n$, there is polygon which can be guarded by $3n$ guards with irrational coordinates but need $4n$ 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