An Algebra of Pieces of Space -- Hermann Grassmann to Gian Carlo Rota

We sketch the outlines of Gian Carlo Rota's interaction with the ideas that Hermann Grassmann developed in his Ausdehnungslehre of 1844 and 1862, as adapted and explained by Giuseppe Peano in 1888. This leads us past what Rota variously called 'Grassmann-Cayley algebra', or 'Peano spaces', to the Whitney algebra of a matroid, and finally to a resolution of the question "What, really, was Grassmann's regressive product?". This final question is the subject of ongoing joint work with Andrea Brini, Francesco Regonati, and William Schmitt. The present paper was presented at the conference "The Digital Footprint of Gian-Carlo Rota: Marbles, Boxes and Philosophy" in Milano on 17 Feb 2009. It will appear in proceedings of that conference, to be published by Springer Verlag.Comment: 28 page

Permanents by Möbius inversion

AbstractMöbius inversion techniques developed by Rota [1] are used to justify Ryser's calculation [2, 3] of the permanent of a matrix, and to establish an alternative method of calculation (Proposition 4)

The orbit rigidity matrix of a symmetric framework

A number of recent papers have studied when symmetry causes frameworks on a graph to become infinitesimally flexible, or stressed, and when it has no impact. A number of other recent papers have studied special classes of frameworks on generically rigid graphs which are finite mechanisms. Here we introduce a new tool, the orbit matrix, which connects these two areas and provides a matrix representation for fully symmetric infinitesimal flexes, and fully symmetric stresses of symmetric frameworks. The orbit matrix is a true analog of the standard rigidity matrix for general frameworks, and its analysis gives important insights into questions about the flexibility and rigidity of classes of symmetric frameworks, in all dimensions. With this narrower focus on fully symmetric infinitesimal motions, comes the power to predict symmetry-preserving finite mechanisms - giving a simplified analysis which covers a wide range of the known mechanisms, and generalizes the classes of known mechanisms. This initial exploration of the properties of the orbit matrix also opens up a number of new questions and possible extensions of the previous results, including transfer of symmetry based results from Euclidean space to spherical, hyperbolic, and some other metrics with shared symmetry groups and underlying projective geometry.Comment: 41 pages, 12 figure

Epidemiology, genetics, and subtyping of preserved ratio impaired spirometry (PRISm) in COPDGene.

BackgroundPreserved Ratio Impaired Spirometry (PRISm), defined as a reduced FEV1 in the setting of a preserved FEV1/FVC ratio, is highly prevalent and is associated with increased respiratory symptoms, systemic inflammation, and mortality. Studies investigating quantitative chest tomographic features, genetic associations, and subtypes in PRISm subjects have not been reported.MethodsData from current and former smokers enrolled in COPDGene (n = 10,192), an observational, cross-sectional study which recruited subjects aged 45-80 with ≥10 pack years of smoking, were analyzed. To identify epidemiological and radiographic predictors of PRISm, we performed univariate and multivariate analyses comparing PRISm subjects both to control subjects with normal spirometry and to subjects with COPD. To investigate common genetic predictors of PRISm, we performed a genome-wide association study (GWAS). To explore potential subgroups within PRISm, we performed unsupervised k-means clustering.ResultsThe prevalence of PRISm in COPDGene is 12.3%. Increased dyspnea, reduced 6-minute walk distance, increased percent emphysema and decreased total lung capacity, as well as increased segmental bronchial wall area percentage were significant predictors (p-value <0.05) of PRISm status when compared to control subjects in multivariate models. Although no common genetic variants were identified on GWAS testing, a significant association with Klinefelter's syndrome (47XXY) was observed (p-value < 0.001). Subgroups identified through k-means clustering include a putative "COPD-subtype", "Restrictive-subtype", and a highly symptomatic "Metabolic-subtype".ConclusionsPRISm subjects are clinically and genetically heterogeneous. Future investigations into the pathophysiological mechanisms behind and potential treatment options for subgroups within PRISm are warranted.Trial registrationClinicaltrials.gov Identifier: NCT000608764

Relations between M\"obius and coboundary polynomial

It is known that, in general, the coboundary polynomial and the M\"obius polynomial of a matroid do not determine each other. Less is known about more specific cases. In this paper, we will try to answer if it is possible that the M\"obius polynomial of a matroid, together with the M\"obius polynomial of the dual matroid, define the coboundary polynomial of the matroid. In some cases, the answer is affirmative, and we will give two constructions to determine the coboundary polynomial in these cases.Comment: 12 page

Genome-wide association study of smoking behaviours in patients with COPD

Background Cigarette smoking is a major risk factor for chronic obstructive pulmonary disease (COPD) and COPD severity. Previous genome-wide association studies (GWAS) have identified numerous single nucleotide polymorphisms (SNPs) associated with the number of cigarettes smoked per day (CPD) and a dopamine beta-hydroxylase (DBH) locus associated with smoking cessation in multiple populations. Objective To identify SNPs associated with lifetime average and current CPD, age at smoking initiation, and smoking cessation in patients with COPD. Methods GWAS were conducted in four independent cohorts encompassing 3441 ever-smoking patients with COPD (Global Initiative for Obstructive Lung Disease stage II or higher). Untyped SNPs were imputed using the HapMap (phase II) panel. Results from all cohorts were meta-analysed. Results Several SNPs near the HLA region on chromosome 6p21 and in an intergenic region on chromosome 2q21 showed associations with age at smoking initiation, both with the lowest p=2x10(-7). No SNPs were associated with lifetime average CPD, current CPD or smoking cessation with p<10(-6). Nominally significant associations with candidate SNPs within cholinergic receptors, nicotinic, alpha 3/5 (CHRNA3/CHRNA5; eg, p=0.00011 for SNP rs1051730) and cytochrome P450, family 2, subfamily A, polypeptide 6 (CYP2A6; eg, p=2.78x10(-5) for a non-synonymous SNP rs1801272) regions were observed for lifetime average CPD, however only CYP2A6 showed evidence of significant association with current CPD. A candidate SNP (rs3025343) in DBH was significantly (p=0.015) associated with smoking cessation. Conclusion The authors identified two candidate regions associated with age at smoking initiation in patients with COPD. Associations of CHRNA3/CHRNA5 and CYP2A6 loci with CPD and DBH with smoking cessation are also likely of importance in the smoking behaviours of patients with COPD

Topological Graph Polynomials in Colored Group Field Theory

In this paper we analyze the open Feynman graphs of the Colored Group Field Theory introduced in [arXiv:0907.2582]. We define the boundary graph \cG_{\partial} of an open graph \cG and prove it is a cellular complex. Using this structure we generalize the topological (Bollobas-Riordan) Tutte polynomials associated to (ribbon) graphs to topological polynomials adapted to Colored Group Field Theory graphs in arbitrary dimension

Impact of self-reported Gastroesophageal reflux disease in subjects from COPDGene cohort

Abstract Background The coexistence of gastroesophageal reflux disease (GERD) and COPD has been recognized, but there has been no comprehensive evaluation of the impact of GERD on COPD-related health status and patient-centered outcomes. Methods Cross-sectional and longitudinal study of 4,483 participants in the COPDGene cohort who met GOLD criteria for COPD. Physician-diagnosed GERD was ascertained by questionnaire. Clinical features, spirometry and imaging were compared between COPD subjects without versus with GERD. We evaluated the relationship between GERD and symptoms, exacerbations and markers of microaspiration in univariate and multivariate models. Associations were additionally tested for the confounding effect of covariates associated with a diagnosis of GERD and the use of proton-pump inhibitor medications (PPIs). To determine whether GERD is simply a marker for the presence of other conditions independently associated with worse COPD outcomes, we also tested models incorporating a GERD propensity score. Results GERD was reported by 29% of subjects with female predominance. Subjects with GERD were more likely to have chronic bronchitis symptoms, higher prevalence of prior cardiovascular events (combined myocardial infarction, coronary artery disease and stroke 21.3% vs. 13.4.0%, p < 0.0001). Subjects with GERD also had more severe dyspnea (MMRC score 2.2 vs. 1.8, p < 0.0001), and poorer quality of life (QOL) scores (St. George’s Respiratory Questionnaire (SGRQ) total score 41.8 vs. 34.9, p < 0.0001; SF36 Physical Component Score 38.2 vs. 41.4, p < 0.0001). In multivariate models, a significant relationship was detected between GERD and SGRQ (3.4 points difference, p < 0.001) and frequent exacerbations at baseline (≥2 exacerbation per annum at inclusion OR 1.40, p = 0.006). During a mean follow-up time of two years, GERD was also associated with frequent (≥2/year exacerbations OR 1.40, p = 0.006), even in models in which PPIs, GERD-PPI interactions and a GERD propensity score were included. PPI use was associated with frequent exacerbator phenotype, but did not meaningfully influence the GERD-exacerbation association. Conclusions In COPD the presence of physician-diagnosed GERD is associated with increased symptoms, poorer QOL and increased frequency of exacerbations at baseline and during follow-up. These associations are maintained after controlling for PPI use. The PPI-exacerbations association could result from confounding-by-indication.http://deepblue.lib.umich.edu/bitstream/2027.42/134572/1/12931_2014_Article_1500.pd

Isostatic phase transition and instability in stiff granular materials

In this letter, structural rigidity concepts are used to understand the origin of instabilities in granular aggregates. It is shown that: a) The contact network of a noncohesive granular aggregate becomes exactly isostatic in the limit of large stiffness-to-load ratio. b) Isostaticity is responsible for the anomalously large susceptibility to perturbation of these systems, and c) The load-stress response function of granular materials is critical (power-law distributed) in the isostatic limit. Thus there is a phase transition in the limit of intinitely large stiffness, and the resulting isostatic phase is characterized by huge instability to perturbation.Comment: RevTeX, 4 pages w/eps figures [psfig]. To appear in Phys. Rev. Let

Genetic susceptibility for chronic bronchitis in chronic obstructive pulmonary disease

Background: Chronic bronchitis (CB) is one of the classic phenotypes of COPD. The aims of our study were to investigate genetic variants associated with COPD subjects with CB relative to smokers with normal spirometry, and to assess for genetic differences between subjects with CB and without CB within the COPD population. Methods: We analyzed data from current and former smokers from three cohorts: the COPDGene Study; GenKOLS (Bergen, Norway); and the Evaluation of COPD Longitudinally to Identify Predictive Surrogate Endpoints (ECLIPSE). CB was defined as having a cough productive of phlegm on most days for at least 3 consecutive months per year for at least 2 consecutive years. CB COPD cases were defined as having both CB and at least moderate COPD based on spirometry. Our primary analysis used smokers with normal spirometry as controls; secondary analysis was performed using COPD subjects without CB as controls. Genotyping was performed on Illumina platforms; results were summarized using fixed-effect meta-analysis. Results: For CB COPD relative to smoking controls, we identified a new genome-wide significant locus on chromosome 11p15.5 (rs34391416, OR = 1.93, P=4.99× 10-8) as well as significant associations of known COPD SNPs within FAM13A. In addition, a GWAS of CB relative to those without CB within COPD subjects showed suggestive evidence for association on 1q23.3 (rs114931935, OR = 1.88, P= 4.99 ×10-7). Conclusions: We found genome-wide significant associations with CB COPD on 4q22.1 (FAM13A) and 11p15.5 (EFCAB4A, CHID1 and AP2A2), and a locus associated with CB within COPD subjects on 1q23.3 (RPL31P11 and ATF6). This study provides further evidence that genetic variants may contribute to phenotypic heterogeneity of COPD.publishedVersio