The Discrete Fundamental Group of the Associahedron, and the Exchange Module

The associahedron is an object that has been well studied and has numerous applications, particularly in the theory of operads, the study of non-crossing partitions, lattice theory and more recently in the study of cluster algebras. We approach the associahedron from the point of view of discrete homotopy theory. We study the abelianization of the discrete fundamental group, and show that it is free abelian of rank $\binom{n+2}{4}$. We also find a combinatorial description for a basis of this rank. We also introduce the exchange module of the type $A_n$ cluster algebra, used to model the relations in the cluster algebra. We use the discrete fundamental group to the study of exchange module, and show that it is also free abelian of rank $\binom{n+2}{3}$.Comment: 16 pages, 4 figure

An "Accidental" Symmetry Operator for the Dirac Equation in the Coulomb Potential

On the basis of the generalization of the theorem about K-odd operators (K is the Dirac's operator), certain linear combination is constructed, which appears to commute with the Dirac Hamiltonian for Coulomb field. This operator coincides with the Johnson and Lippmann operator and is intimately connected to the familiar Laplace-Runge-Lenz vector. Our approach guarantees not only derivation of Johnson-Lippmann operator, but simultaneously commutativity with the Dirac Hamiltonian follows.Comment: 6 page

Mod-Gaussian convergence and its applications for models of statistical mechanics

In this paper we complete our understanding of the role played by the limiting (or residue) function in the context of mod-Gaussian convergence. The question about the probabilistic interpretation of such functions was initially raised by Marc Yor. After recalling our recent result which interprets the limiting function as a measure of "breaking of symmetry" in the Gaussian approximation in the framework of general central limit theorems type results, we introduce the framework of $L^1$-mod-Gaussian convergence in which the residue function is obtained as (up to a normalizing factor) the probability density of some sequences of random variables converging in law after a change of probability measure. In particular we recover some celebrated results due to Ellis and Newman on the convergence in law of dependent random variables arising in statistical mechanics. We complete our results by giving an alternative approach to the Stein method to obtain the rate of convergence in the Ellis-Newman convergence theorem and by proving a new local limit theorem. More generally we illustrate our results with simple models from statistical mechanics.Comment: 49 pages, 21 figure

Tamari Lattices and the symmetric Thompson monoid

We investigate the connection between Tamari lattices and the Thompson group F, summarized in the fact that F is a group of fractions for a certain monoid F+sym whose Cayley graph includes all Tamari lattices. Under this correspondence, the Tamari lattice operations are the counterparts of the least common multiple and greatest common divisor operations in F+sym. As an application, we show that, for every n, there exists a length l chain in the nth Tamari lattice whose endpoints are at distance at most 12l/n.Comment: 35page

KP line solitons and Tamari lattices

The KP-II equation possesses a class of line soliton solutions which can be qualitatively described via a tropical approximation as a chain of rooted binary trees, except at "critical" events where a transition to a different rooted binary tree takes place. We prove that these correspond to maximal chains in Tamari lattices (which are poset structures on associahedra). We further derive results that allow to compute details of the evolution, including the critical events. Moreover, we present some insights into the structure of the more general line soliton solutions. All this yields a characterization of possible evolutions of line soliton patterns on a shallow fluid surface (provided that the KP-II approximation applies).Comment: 49 pages, 36 figures, second version: section 4 expande

One Year Clinical Outcomes of Renal Artery Stenting: The Results of ODORI Registry

The safety, efficacy and long term clinical benefits of renal artery revascularization by stenting are still a matter of debate. The aim of our study was to define the safety and efficacy of renal artery stenting with the Tsunami peripheral stent (Terumo Corporation, Tokyo, Japan). The ODORI was a prospective, multicentre registry which enrolled 251 consecutive patients, (276 renal arteries) in 36 centres across Europe. The primary endpoint was acute procedural success defined as <30% residual stenosis after stent placement. Secondary endpoints included major adverse events, blood pressure control, serum creatinine level, and target lesion revascularization (TLR) at 6 and 12 months. Patients were 70 ± 10 years old, 59% were male, 33% had diabetes, and 96% hypertension. The main indications for renal stent implantation were hypertension in 83% and renal salvage in 39%. Direct stent implantation was performed in 76% of the cases. Acute success rate was 100% with residual stenosis of 2.5 ± 5.4%. Systolic/diastolic blood pressure decreased from a mean of 171/89 at baseline to 142/78 mmHg at 6 months (p < 0.0001 vs. baseline), and 141/80 mmHg at 12 months (p < 0.0001 vs. baseline). Mean serum creatinine concentration did not change significantly in the total population. However, there was significant improvement in the highest tercile (from 283 μmol/l at baseline to 205 and 209 μmol/l at 6 and 12 months respectively). At 12-months, rates of restenosis and TLR were 6.6 and 0.8% respectively. The 12 month cumulative rate of all major clinical adverse events was 6.4% while the rate of device or procedure related events was 2.4%. In hypertensive patients with atherosclerotic renal artery stenosis Tsunami peripheral balloon-expandable stent provides a safe revascularization strategy, with a potential beneficial impact on hypertension control and renal function in the highest risk patients

Many non-equivalent realizations of the associahedron

Hohlweg and Lange (2007) and Santos (2004, unpublished) have found two different ways of constructing exponential families of realizations of the n-dimensional associahedron with normal vectors in {0,1,-1}^n, generalizing the constructions of Loday (2004) and Chapoton-Fomin-Zelevinsky (2002). We classify the associahedra obtained by these constructions modulo linear equivalence of their normal fans and show, in particular, that the only realization that can be obtained with both methods is the Chapoton-Fomin-Zelevinsky (2002) associahedron. For the Hohlweg-Lange associahedra our classification is a priori coarser than the classification up to isometry of normal fans, by Bergeron-Hohlweg-Lange-Thomas (2009). However, both yield the same classes. As a consequence, we get that two Hohlweg-Lange associahedra have linearly equivalent normal fans if and only if they are isometric. The Santos construction, which produces an even larger family of associahedra, appears here in print for the first time. Apart of describing it in detail we relate it with the c-cluster complexes and the denominator fans in cluster algebras of type A. A third classical construction of the associahedron, as the secondary polytope of a convex n-gon (Gelfand-Kapranov-Zelevinsky, 1990), is shown to never produce a normal fan linearly equivalent to any of the other two constructions.Comment: 30 pages, 13 figure

Beyond African orality: digital preservation of Mandinka ʿAjamī archives of Casamance

This article focuses on the digital preservation of African sources written in Mandinka ʿAjamī, i.e. the enriched form of the Arabic script used to write the Mandinka language for centuries. ʿAjamī writing has been utilized to document intellectual traditions, histories, belief systems, and cultures of non-Arab Muslims around the world. ʿAjamī texts have played critical roles in the spread of Islam in Africa and continue to be used for both religious and non-religious writings. However, African ʿAjamī texts such as those of the Mandinka people of Casamance in southern Senegal are not well known beyond local communities. ʿAjamī texts in Mandinka and other Mande languages are among the least documented. Only a few Mande ʿAjamī texts are available to scholars. Thanks to the British Library’s Endangered Archives Programme (EAP), Africa’s rich written heritage in ʿAjamī and other scripts previously unavailable to academics is being preserved and made universally accessible.EAP1042 - Endangered Archives Programme, British LibraryAccepted manuscrip

Genome-Wide Association Analysis in Asthma Subjects Identifies SPATS2L as a Novel Bronchodilator Response Gene

Bronchodilator response (BDR) is an important asthma phenotype that measures reversibility of airway obstruction by comparing lung function (i.e. FEV1) before and after the administration of a short-acting β2-agonist, the most common rescue medications used for the treatment of asthma. BDR also serves as a test of β2-agonist efficacy. BDR is a complex trait that is partly under genetic control. A genome-wide association study (GWAS) of BDR, quantified as percent change in baseline FEV1 after administration of a β2-agonist, was performed with 1,644 non-Hispanic white asthmatic subjects from six drug clinical trials: CAMP, LOCCS, LODO, a medication trial conducted by Sepracor, CARE, and ACRN. Data for 469,884 single-nucleotide polymorphisms (SNPs) were used to measure the association of SNPs with BDR using a linear regression model, while adjusting for age, sex, and height. Replication of primary P-values was attempted in 501 white subjects from SARP and 550 white subjects from DAG. Experimental evidence supporting the top gene was obtained via siRNA knockdown and Western blotting analyses. The lowest overall combined P-value was 9.7E-07 for SNP rs295137, near the SPATS2L gene. Among subjects in the primary analysis, those with rs295137 TT genotype had a median BDR of 16.0 (IQR = [6.2, 32.4]), while those with CC or TC genotypes had a median BDR of 10.9 (IQR = [5.0, 22.2]). SPATS2L mRNA knockdown resulted in increased β2-adrenergic receptor levels. Our results suggest that SPATS2L may be an important regulator of β2-adrenergic receptor down-regulation and that there is promise in gaining a better understanding of the biological mechanisms of differential response to β2-agonists through GWAS