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

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

Rigidity percolation in a field

Rigidity Percolation with g degrees of freedom per site is analyzed on randomly diluted Erdos-Renyi graphs with average connectivity gamma, in the presence of a field h. In the (gamma,h) plane, the rigid and flexible phases are separated by a line of first-order transitions whose location is determined exactly. This line ends at a critical point with classical critical exponents. Analytic expressions are given for the densities n_f of uncanceled degrees of freedom and gamma_r of redundant bonds. Upon crossing the coexistence line, n_f and gamma_r are continuous, although their first derivatives are discontinuous. We extend, for the case of nonzero field, a recently proposed hypothesis, namely that the density of uncanceled degrees of freedom is a ``free energy'' for Rigidity Percolation. Analytic expressions are obtained for the energy, entropy, and specific heat. Some analogies with a liquid-vapor transition are discussed. Particularizing to zero field, we find that the existence of a (g+1)-core is a necessary condition for rigidity percolation with g degrees of freedom. At the transition point gamma_c, Maxwell counting of degrees of freedom is exact on the rigid cluster and on the (g+1)-rigid-core, i.e. the average coordination of these subgraphs is exactly 2g, although gamma_r, the average coordination of the whole system, is smaller than 2g. gamma_c is found to converge to 2g for large g, i.e. in this limit Maxwell counting is exact globally as well. This paper is dedicated to Dietrich Stauffer, on the occasion of his 60th birthday.Comment: RevTeX4, psfig, 16 pages. Equation numbering corrected. Minor typos correcte

Two-Year Pulmonary Safety and Efficacy of Inhaled Human Insulin (Exubera) in Adult Patients With Type 2 Diabetes

OBJECTIVE—The purpose of this study was to evaluate the 2-year pulmonary safety of inhaled human insulin (Exubera [EXU]) in 635 nonsmoking adults with type 2 diabetes

Small grid embeddings of 3-polytopes

We introduce an algorithm that embeds a given 3-connected planar graph as a convex 3-polytope with integer coordinates. The size of the coordinates is bounded by $O(2^{7.55n})=O(188^{n})$. If the graph contains a triangle we can bound the integer coordinates by $O(2^{4.82n})$. If the graph contains a quadrilateral we can bound the integer coordinates by $O(2^{5.46n})$. The crucial part of the algorithm is to find a convex plane embedding whose edges can be weighted such that the sum of the weighted edges, seen as vectors, cancel at every point. It is well known that this can be guaranteed for the interior vertices by applying a technique of Tutte. We show how to extend Tutte's ideas to construct a plane embedding where the weighted vector sums cancel also on the vertices of the boundary face

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

Eosinophil and T Cell Markers Predict Functional Decline in COPD Patients

BACKGROUND. The major marker utilized to monitor COPD patients is forced expiratory volume in one second (FEV1). However, asingle measurement of FEV1 cannot reliably predict subsequent decline. Recent studies indicate that T lymphocytes and eosinophils are important determinants of disease stability in COPD. We therefore measured cytokine levels in the lung lavage fluid and plasma of COPD patients in order to determine if the levels of T cell or eosinophil related cytokines were predictive of the future course of the disease. METHODS. Baseline lung lavage and plasma samples were collected from COPD subjects with moderately severe airway obstruction and emphysematous changes on chest CT. The study participants were former smokers who had not had a disease exacerbation within the past six months or used steroids within the past two months. Those subjects who demonstrated stable disease over the following six months (ΔFEV1 % predicted = 4.7 ± 7.2; N = 34) were retrospectively compared with study participants who experienced a rapid decline in lung function (ΔFEV1 % predicted = -16.0 ± 6.0; N = 16) during the same time period and with normal controls (N = 11). Plasma and lung lavage cytokines were measured from clinical samples using the Luminex multiplex kit which enabled the simultaneous measurement of several T cell and eosinophil related cytokines. RESULTS AND DISCUSSION. Stable COPD participants had significantly higher plasma IL-2 levels compared to participants with rapidly progressive COPD (p = 0.04). In contrast, plasma eotaxin-1 levels were significantly lower in stable COPD subjects compared to normal controls (p < 0.03). In addition, lung lavage eotaxin-1 levels were significantly higher in rapidly progressive COPD participants compared to both normal controls (p < 0.02) and stable COPD participants (p < 0.05). CONCLUSION. These findings indicate that IL-2 and eotaxin-1 levels may be important markers of disease stability in advanced emphysema patients. Prospective studies will need to confirm whether measuring IL-2 or eotaxin-1 can identify patients at risk for rapid disease progression.National Heart, Lung, and Blood Institute (NO1-HR-96140, NO1-HR-96141-001, NO1-HR-96144, NO1-HR-96143; NO1-HR-96145; NO1-HR-96142, R01HL086936-03); The Flight Attendant Medical Research Institute; the Jo-Ann F. LeBuhn Center for Chest Diseas