Translation Invariant Filters and van der Waerden’s Theorem

We present a self-contained proof of a strong version of van der Waerden’s Theorem. By using translation invariant filters that are maximal with respect to inclusion, a simple inductive argument shows the existence of “piecewise syndetically”-many monochromatic arithmetic progressions of any length k in every finite coloring of the natural numbers. All the presented constructions are constructive in nature, in the sense that the involved maximal filters are defined by recurrence on suitable countable algebras of sets. No use of the axiom of choice or of Zorn’s Lemma is needed

Infinite monochromatic patterns in the integers

We show the existence of several infinite monochromatic patterns in the integers obtained as values of suitable symmetric polynomials; in particular, we obtain extensions of both the additive and multiplicative versions of Hindman's theorem. These configurations are obtained by means of suitable symmetric polynomials that mix the two operations. The simplest example is the following. For every finite coloring N=C1∪…∪Cr there exists an infinite increasing sequence a<… such that all elements below are monochromatic: a,b,c,…,a+b+ab,a+c+ac,b+c+bc,…,a+b+c+ab+ac+bc+abc,…. The proofs use tools from algebra in the space of ultrafilters βZ

Nonstandard analysis and the sumset phenomenon in arbitrary amenable groups

Beiglboeck, Bergelson and Fish proved that if subsets A,B of a countable discrete amenable group G have positive Banach densities a and b respectively, then the product set AB is piecewise syndetic, i.e. there exists k such that the union of k-many left translates of AB is thick. Using nonstandard analysis we give a shorter alternative proof of this result that does not require G to be countable, and moreover yields the explicit bound that k is not greater than 1/ab. We also prove with similar methods that if {A_i} are finitely many subsets of G having positive Banach densities a_i and G is countable, then there exists a subset B whose Banach density is at least the product of the densities a_i and such that the product B(B^−1) is a subset of the intersection of the product sets A_i(A_i^−1). In particular, the latter set is piecewise Bohr

Quasi-selective ultrafilters and asymptotic numerosities

We isolate a new class of ultrafilters on N, called “quasi-selective” because they are intermediate between selective ultrafilters and P-points. (Under the Continuum Hypothesis these three classes are distinct.) The existence of quasi-selective ultrafilters is equivalent to the existence of “asymptotic numerosities” for all sets of tuples A ⊆ N^k. Such numerosities are hypernatural numbers that generalize finite cardinalities to countable point sets. Most notably, they maintain the structure of ordered semiring, and, in a precise sense, they allow for a natural extension of asymptotic density to all sets of tuples of natural numbers

Proactive risk assessment through failure mode and effect analysis (FMEA) for perioperative management model of oral anticoagulant therapy. A pilot project

Introduction: Correct perioperative management of anticoagulant therapy is essential to prevent thromboembolic events and reduce the risk of bleeding. The lack of universally accepted guidelines makes perioperative anticoagulant therapy management difficult. The present study aims to identify the perioperative risks of oral anticoagulant therapy and to reduce adverse events through Failure Mode and Effect Analysis (FMEA). Materials and Methods: A multidisciplinary working group was set up, and four main phases of the process were identified. Each of these phases was divided into micro-activities to identify the related possible failure modes and their potential consequences. The Risk Priority Number was calculated for each failure mode. Results and Discussion: Seventeen failure modes were identified in the entire perioperative period; those with a higher priority of intervention concern the incorrect timing between therapy suspension and surgery, and the incorrect assessment of the bleeding risk related to the invasive procedure. Conclusion: The FMEA method can help identify anticoagulant therapy perioperative failures and implement the management and patient safety of surgical procedures

Approximate polynomial structure in additively large sets

We show that any subset of the natural numbers with positive logarithmic Banach density contains a set that is within a factor of two of a geometric progression, improving the bound on a previous result of the authors. Density conditions on subsets of the natural numbers that imply the existence of approximate powers of arithmetic progressions are developed and explore

A topological approach to non-Archimedean Mathematics

Non-Archimedean mathematics (in particular, nonstandard analysis) allows to construct some useful models to study certain phenomena arising in PDE's; for example, it allows to construct generalized solutions of differential equations and variational problems that have no classical solution. In this paper we introduce certain notions of non-Archimedean mathematics (in particular, of nonstandard analysis) by means of an elementary topological approach; in particular, we construct non-Archimedean extensions of the reals as appropriate topological completions of $\mathbb{R}$. Our approach is based on the notion of $\Lambda$-limit for real functions, and it is called $\Lambda$-theory. It can be seen as a topological generalization of the $\alpha$-theory presented in \cite{BDN2003}, and as an alternative topological presentation of the ultrapower construction of nonstandard extensions (in the sense of \cite{keisler}). To motivate the use of $\Lambda$-theory for applications we show how to use it to solve a minimization problem of calculus of variations (that does not have classical solutions) by means of a particular family of generalized functions, called ultrafunctions.Comment: 22 page

Hypernatural Numbers as Ultrafilters

In this paper we present a use of nonstandard methods in the theory of ultrafilters and in related applications to combinatorics of numbers

Suitability of switchgrass (Panicum virgatum L.) as a forage crop in the Mediterranean area

In Mediterranean rainfed cropping systems, drought-resistant crops can increase yield and availability of forage during the summer period. In North America, switchgrass (Panicum virgatum L.) has been used for decades as pasture and fodder. In Europe, switchgrass has been investigated mainly for its potential as an energy crop. e overall aim of the present study was to analyse the suitability of switchgrass as a forage crop in a Mediterranean environment. A eld trial was carried out in Central Italy (Pisa) to evaluate the productivity and nutritive value in mature stands of two switchgrass varieties (Alamo and Blackwell). Alamo reached the maturity for hay harvest (boot stage) in August, about one-month delay with respect to Blackwell. At this stage, the biomass was 13.3 and 7.5 t ha-1 of dry matter (DM) in Alamo and Blackwell, respectively. Both varieties produced a summer regrowth harvested in autumn. Nutritive value declined during the growing season due to the increase of bre and the reduction of protein content. Saponin content signi cantly di ered between varieties and according to the growth stage, ranging from 1.8 to 4.5 mg g-1 DM. is study provides useful knowledge to favour the introduction of perennial grasses as forage crops in the Mediterranean, leading to several environmental bene ts when compared with the annual species that currently cover half of the forage cropland in Tuscany

Non-sutureless minimally invasive aortic valve replacement: mini-sternotomy versus mini-thoracotomy: a series of 1130 patients

OBJECTIVES: Aortic valve replacement through conventional sternotomy still represents the gold-standard surgical approach for aortic valve disease. However, given the increasing number of patients with comorbidities, strategies that can improve operative results are always sought. Minimally invasive aortic valve surgery, although related to a steep learning curve, might be associated with improved post- operative outcomes. The main aim of this study was to assess whether significant differences exist in terms of operative and early results between a mini-sternotomy and a right mini-thoracotomy approach for isolated aortic valve replacement without sutureless technologies. METHODS: This is an observational retrospective multicentre study from nine Italian cardiac centres that analyses prospectively collected data of patients who underwent isolated minimally invasive aortic valve replacement between January 2010 and December 2014. Two approaches are considered (mini-sternotomy and mini-thoracotomy) and compared in terms of operative and early outcomes. RESULTS: After interrogation of the centralized database, a total of 1130 patients were retrieved (854 mini-sternotomy and 276 mini-thoracotomy). Patients in the mini-sternotomy group had a higher risk profile. There was no difference in terms of early mortality; cardiopulmonary bypass and cross-clamp time did not differ significantly between the groups; and a significantly higher number of reoperations for bleeding was observed in the right mini-thoracotomy group. CONCLUSIONS: Both mini-sternotomy and mini-thoracotomy could be performed safely, with low mortality and postoperative morbidity. The mini-thoracotomy approach was associated with a significantly higher rate of reoperation for bleeding. Uptake among cardiac centres was low. Sutureless technologies could potentially increase surgical volume by simplifying the mini-thoracotomy procedure