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

Transition probabilities for general birth-death processes with applications in ecology, genetics, and evolution

A birth-death process is a continuous-time Markov chain that counts the number of particles in a system over time. In the general process with $n$ current particles, a new particle is born with instantaneous rate $\lambda_n$ and a particle dies with instantaneous rate $\mu_n$. Currently no robust and efficient method exists to evaluate the finite-time transition probabilities in a general birth-death process with arbitrary birth and death rates. In this paper, we first revisit the theory of continued fractions to obtain expressions for the Laplace transforms of these transition probabilities and make explicit an important derivation connecting transition probabilities and continued fractions. We then develop an efficient algorithm for computing these probabilities that analyzes the error associated with approximations in the method. We demonstrate that this error-controlled method agrees with known solutions and outperforms previous approaches to computing these probabilities. Finally, we apply our novel method to several important problems in ecology, evolution, and genetics

Precise Critical Exponents for the Basic Contact Process

We calculated some of the critical exponents of the directed percolation universality class through exact numerical diagonalisations of the master operator of the one-dimensional basic contact process. Perusal of the power method together with finite-size scaling allowed us to achieve a high degree of accuracy in our estimates with relatively little computational effort. A simple reasoning leading to the appropriate choice of the microscopic time scale for time-dependent simulations of Markov chains within the so called quantum chain formulation is discussed. Our approach is applicable to any stochastic process with a finite number of absorbing states.Comment: LaTeX 2.09, 9 pages, 1 figur

Aerobic Swim Training Restores Aortic Endothelial Function by Decreasing Superoxide Levels in Spontaneously Hypertensive Rats

OBJECTIVE: We aimed to determine whether aerobic training decreases superoxide levels, increases nitric oxide levels, and improves endothelium-dependent vasodilation in the aortas of spontaneously hypertensive rats. METHODS: Spontaneously hypertensive rats (SHR) and Wistar Kyoto rats (WKY) were distributed into 2 groups: sedentary (SHRsd and WKYsd, n=10 each) and swimming-trained (SHRtr, n=10 and WKYtr, n=10, respectively). The trained group participated in training sessions 5 days/week for 1 h/day with an additional work load of 4% of the animal’s body weight. After a 10-week sedentary or aerobic training period, the rats were euthanized. The thoracic aortas were removed to evaluate the vasodilator response to acetylcholine (10-10 to 10-4 M) with or without preincubation with L-NG-nitro-L-arginine methyl ester hydrochloride (L-NAME; 10-4 M) in vitro. The aortic tissue was also used to assess the levels of the endothelial nitric oxide synthase and nicotinamide adenine dinucleotide oxidase subunit isoforms 1 and 4 proteins, as well as the superoxide and nitrite contents. Blood pressure was measured using a computerized tail-cuff system. RESULTS: Aerobic training significantly increased the acetylcholine-induced maximum vasodilation observed in the SHRtr group compared with the SHRsd group (85.9±4.3 vs. 71.6±5.2%). Additionally, in the SHRtr group, superoxide levels were significantly decreased, nitric oxide bioavailability was improved, and the levels of the nicotinamide adenine dinucleotide oxidase subunit isoform 4 protein were decreased compared to the SHRsd group. Moreover, after training, the blood pressure of the SHRtr group decreased compared to the SHRsd group. Exercise training had no effect on the blood pressure of the WKYtr group. CONCLUSIONS: In SHR, aerobic swim training decreased vascular superoxide generation by nicotinamide adenine dinucleotide oxidase subunit isoform 4 and increased nitric oxide bioavailability, thereby improving endothelial function

A large genome-wide association study of age-related macular degeneration highlights contributions of rare and common variants

Advanced age-related macular degeneration (AMD) is the leading cause of blindness in the elderly with limited therapeutic options. Here, we report on a study of \u3e12 million variants including 163,714 directly genotyped, most rare, protein-altering variant. Analyzing 16,144 patients and 17,832 controls, we identify 52 independently associated common and rare variants (P \u3c 5×10–8) distributed across 34 loci. While wet and dry AMD subtypes exhibit predominantly shared genetics, we identify the first signal specific to wet AMD, near MMP9 (difference-P = 4.1×10–10). Very rare coding variants (frequency \u3c 0.1%) in CFH, CFI, and TIMP3 suggest causal roles for these genes, as does a splice variant in SLC16A8. Our results support the hypothesis that rare coding variants can pinpoint causal genes within known genetic loci and illustrate that applying the approach systematically to detect new loci requires extremely large sample sizes

Cognitive Information Processing

Contains reports on six research projects.National Institutes of Health (Grant 1 PO1 GM-14940-01)National Institutes of Health (Grant 1 PO1 GM-15006-01)Joint Services Electronics Programs (U. S. Army, U. S. Navy, and U. S. Air Force) under Contract DA 28-043-AMC-02536(E)Project MAC, an M. I. T. research programAdvanced Research Projects Agency, Department of Defense, under Office of Naval Research Contract Nonr-4102-(01

On-the-fly Uniformization of Time-Inhomogeneous Infinite Markov Population Models

This paper presents an on-the-fly uniformization technique for the analysis of time-inhomogeneous Markov population models. This technique is applicable to models with infinite state spaces and unbounded rates, which are, for instance, encountered in the realm of biochemical reaction networks. To deal with the infinite state space, we dynamically maintain a finite subset of the states where most of the probability mass is located. This approach yields an underapproximation of the original, infinite system. We present experimental results to show the applicability of our technique

Looking backward: From Euler to Riemann

We survey the main ideas in the early history of the subjects on which Riemann worked and that led to some of his most important discoveries. The subjects discussed include the theory of functions of a complex variable, elliptic and Abelian integrals, the hypergeometric series, the zeta function, topology, differential geometry, integration, and the notion of space. We shall see that among Riemann's predecessors in all these fields, one name occupies a prominent place, this is Leonhard Euler. The final version of this paper will appear in the book \emph{From Riemann to differential geometry and relativity} (L. Ji, A. Papadopoulos and S. Yamada, ed.) Berlin: Springer, 2017

Spectral properties of zero temperature dynamics in a model of a compacting granular column

The compacting of a column of grains has been studied using a one-dimensional Ising model with long range directed interactions in which down and up spins represent orientations of the grain having or not having an associated void. When the column is not shaken (zero 'temperature') the motion becomes highly constrained and under most circumstances we find that the generator of the stochastic dynamics assumes an unusual form: many eigenvalues become degenerate, but the associated multi-dimensional invariant spaces have but a single eigenvector. There is no spectral expansion and a Jordan form must be used. Many properties of the dynamics are established here analytically; some are not. General issues associated with the Jordan form are also taken up.Comment: 34 pages, 4 figures, 3 table