The multivariate arithmetic Tutte polynomial

We introduce an arithmetic version of the multivariate Tutte polynomial recently studied by Sokal, and a quasi-polynomial that interpolates between the two. We provide a generalized Fortuin-Kasteleyn representation for representable arithmetic matroids, with applications to arithmetic colorings and flows. We give a new proof of the positivity of the coefficients of the arithmetic Tutte polynomial in the more general framework of pseudo-arithmetic matroids. In the case of a representable arithmetic matroid, we provide a geometric interpretation of the coefficients of the arithmetic Tutte polynomial. \ua9 2012 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France

Zonotopes, toric arrangements, and generalized Tutte polynomials

We introduce a multiplicity Tutte polynomial M(x, y), which generalizes the ordinary one and has applications to zonotopes and toric arrangements. We prove that M(x, y) satisfies a deletion-restriction recurrence and has positive coefficients. The characteristic polynomial and the Poincar\ue9 polynomial of a toric arrangement are shown to be specializations of the associated polynomial M(x, y), likewise the corresponding polynomials for a hyperplane arrangement are specializations of the ordinary Tutte polynomial. Furthermore, M(1, y) is the Hilbert series of the related discrete Dahmen-Micchelli space, while M(x, 1) computes the volume and the number of integral points of the associated zonotope. \ua9 2010 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France

Arithmetic matroids and Tutte polynomials

We introduce the notion of arithmetic matroid, whose main example is provided by a list of elements in a finitely generated abelian group. We study the representability of its dual, and, guided by the geometry of toric arrangements, we give a combinatorial interpretation of the associated arithmetic Tutte polynomial, which can be seen as a generalization of Crapo's formula

On the cohomology of arrangements of subtori

Given an arrangement of subtori of arbitrary codimension in a complex torus, we compute the cohomology groups of the complement. Then, by using the Leray spectral sequence, we describe the multiplicative structure on the associated graded cohomology. We also provide a differential model for the cohomology ring, by considering a toric wonderful model and its Morgan algebra. Finally, we focus on the divisorial case, proving a new presentation for the cohomology of toric arrangements

Universal Tutte characters via combinatorial coalgebras

The Tutte polynomial is the most general invariant of matroids and graphs that can be computed recursively by deleting and contracting edges. We generalize this invariant to any class of combinatorial objects with deletion and contraction operations, associating to each such class a universal Tutte character by a functorial procedure. We show that these invariants satisfy a universal property and convolution formulae similar to the Tutte polynomial. With this machinery we recover classical invariants for delta-matroids, matroid perspectives, relative and colored matroids, generalized permutohedra, and arithmetic matroids, and produce some new convolution formulae. Our principal tools are combinatorial coalgebras and their convolution algebras. Our results generalize in an intrinsic way the recent results of Krajewski--Moffatt--Tanasa.Comment: Accepted version, 51p

Matroids over a ring

We introduce the notion of a matroid M over a commutative ring R, assigning to every subset of the ground set an R-module according to some axioms. When R is a field, we recover matroids. When R D Z, and when R is a DVR, we get (structures which contain all the data of) quasi-arithmetic matroids, and valuated matroids, i.e. tropical linear spaces, respectively. More generally, whenever R is a Dedekind domain, we extend all the usual properties and operations holding for matroids (e.g., duality), and we explicitly describe the structure of the matroids over R. Furthermore, we compute the Tutte-Grothendieck ring of matroids over R. We also show that the Tutte quasi-polynomial of a matroid over Z can be obtained as an evaluation of the class of the matroid in the Tutte-Grothendieck ring

On a Generalization of Zaslavsky's Theorem for Hyperplane Arrangements

We define arrangements of codimension-1 submanifolds in a smooth manifold which generalize arrangements of hyperplanes. When these submanifolds are removed the manifold breaks up into regions, each of which is homeomorphic to an open disc. The aim of this paper is to derive formulas that count the number of regions formed by such an arrangement. We achieve this aim by generalizing Zaslavsky's theorem to this setting. We show that this number is determined by the combinatorics of the intersections of these submanifolds.Comment: version 3: The title had a typo in v2 which is now fixed. Will appear in Annals of Combinatorics. Version. 2: 19 pages, major revision in terms of style and language, some results improved, contact information updated, final versio

Actigraphic Sensors Describe Stroke Severity in the Acute Phase: Implementing Multi-Parametric Monitoring in Stroke Unit

Actigraphy is a tool used to describe limb motor activity. Some actigraphic parameters, namely Motor Activity (MA) and Asymmetry Index (AR), correlate with stroke severity. However, a long-lasting actigraphic monitoring was never performed previously. We hypothesized that MA and AR can describe different clinical conditions during the evolution of the acute phase of stroke. We conducted a multicenter study and enrolled 69 stroke patients. NIHSS was assessed every hour and upper limbs’ motor activity was continuously recorded. We calculated MA and AR in the first hour after admission, after a significant clinical change (NIHSS ± 4) or at discharge. In a control group of 17 subjects, we calculated MA and AR normative values. We defined the best model to predict clinical status with multiple linear regression and identified actigraphic cut-off values to discriminate minor from major stroke (NIHSS ≥ 5) and NIHSS 5–9 from NIHSS ≥ 10. The AR cut-off value to discriminate between minor and major stroke (namely NIHSS ≥ 5) is 27% (sensitivity = 83%, specificity = 76% (AUC 0.86 p < 0.001), PPV = 89%, NPV = 42%). However, the combination of AR and MA of the non-paretic arm is the best model to predict NIHSS score (R2: 0.482, F: 54.13), discriminating minor from major stroke (sensitivity = 89%, specificity = 82%, PPV = 92%, NPV = 75%). The AR cut-off value of 53% identifies very severe stroke patients (NIHSS ≥ 10) (sensitivity = 82%, specificity = 74% (AUC 0.86 p < 0.001), PPV = 73%, NPV = 82%). Actigraphic parameters can reliably describe the overall severity of stroke patients with motor symptoms, supporting the addition of a wearable actigraphic system to the multi-parametric monitoring in stroke units

The risk of stroke recurrence in patients with atrial fibrillation and reduced ejection fraction

Abstract Background: Atrial fibrillation (AF) and congestive heart failure often coexist due to their shared risk factors leading to potential worse outcome, particularly cerebrovascular events. The aims of this study were to calculate the rates of ischemic and severe bleeding events in ischemic stroke patients having both AF and reduced ejection fraction (rEF) (⩽40%), compared to ischemic stroke patients with AF but without rEF. Methods: We performed a retrospective analysis that drew data from prospective studies. The primary outcome was the composite of either ischemic (stroke or systemic embolism), or hemorrhagic events (symptomatic intracranial bleeding and severe extracranial bleeding). Results: The cohort for this analysis comprised 3477 patients with ischemic stroke and AF, of which, 643 (18.3%) had also rEF. After a mean follow-up of 7.5 ± 9.1 months, 375 (10.8%) patients had 382 recorded outcome events, for an annual rate of 18.0%. While the number of primary outcome events in patients with rEF was 86 (13.4%), compared to 289 (10.2%) for the patients without rEF; on multivariable analysis rEF was not associated with the primary outcome (OR 1.25; 95% CI 0.84–1.88). At the end of follow-up, 321 (49.9%) patients with rEF were deceased or disabled (mRS ⩾3), compared with 1145 (40.4%) of those without rEF; on multivariable analysis, rEF was correlated with mortality or disability (OR 1.35; 95% CI 1.03–1.77). Conclusions: In patients with ischemic stroke and AF, the presence of rEF was not associated with the composite outcome of ischemic or hemorrhagic events over short-term follow-up but was associated with increased mortality or disability

THE MULTIVARIATE ARITHMETIC TUTTE POLYNOMIAL

We introduce an arithmetic version of the multivariate Tutte polynomial and a quasi-polynomial that interpolates between the two. A generalized Fortuin-Kasteleyn representation with applications to arithmetic colorings and flows is obtained. We give a new and more general proof of the positivity of the coefficients of the arithmetic Tutte polynomial and (in the representable case) a geometrical interpretation of them