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

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

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

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

Inhibition of interleukin-6-induced matrix metalloproteinase-2 expression and invasive ability of lemon peel polyphenol extract in human primary colon cancer cells

Among matrix metalloproteinases (MMPs), MMP-9/2 are key enzymes involved in the proteolysis of extracellular matrices in the inflammatory process and in cancer. Since MMP-9/2 expression levels, activity, and secretion is up-regulated during inflammation in response to pro-inflammatory cytokines, such as interleukin-6 (IL-6), many efforts have been devoted to identifying factors that could inhibit the IL-6-induced MMP-9/2 expression. Up to now, several reports in-dicated that polyphenols from fruits and vegetables are among the major components of health promotion for their antioxidant properties and also for their anti-inflammatory and anti-cancer agents. Among plant derived polyphenols, lemon (Citrus limon) peel extract (LPE) shows anti-cancer properties in various cancer types. In our previous work, we demonstrated that LPE can reduce IL-6-induced migration/invasiveness and MMP-9/2 up-regulation in some gastric cancer cell lines. This study aims to exploit the anti-cancer properties of LPE using an in vitro system model of inflam-mation, consisting of IL-6-exposed human primary colon cancer cells. We first analyzed the effect of LPE on IL-6-induced cell migration and invasiveness by wound healing and Boyden chamber assay, respectively. The MMP-2 mRNA expression levels and gelatinolytic activity in the cell culture media were determined by q-PCR analysis and gelatin zymography, respectively, and finally, the effects of LPE on IL-6-induced JAK2/STAT3 signaling pathways have been investigated by Western blotting analysis. Our results show that LPE is able to inhibit the IL-6-dependent cell migration and invasiveness associated with the up-regulation of MMP-2 expression levels and that these effects are correlated to the STAT3 phosphorylation in human primary T88 and T93 colon cancer cells

Executive function abnormalities in pathological gamblers

Background: Pathological gambling (PG) is an impulse control disorder characterized by persistent and maladaptive gambling behaviors with disruptive consequences for familial, occupational and social functions. The pathophysiology of PG is still unclear, but it is hypothesized that it might include environmental factors coupled with a genetic vulnerability and dysfunctions of different neurotransmitters and selected brain areas. Our study aimed to evaluate a group of patients suffering from PG by means of some neuropsychological tests in order to explore the brain areas related to the disorder. Methods: Twenty outpatients (15 men, 5 women), with a diagnosis of PG according to DSM-IV criteria, were included in the study and evaluated with a battery of neuropsychological tests: the Wisconsin Card Sorting Test (WCST), the Wechsler Memory Scale revised (WMS-R) and the Verbal Associative Fluency Test (FAS). The results obtained in the patients were compared with normative values of matched healthy control subjects. Results: The PG patients showed alterations at the WCST only, in particular they had a great difficulty in finding alternative methods of problem-solving and showed a decrease, rather than an increase, in efficiency, as they progressed through the consecutive phases of the test. The mean scores of the other tests were within the normal range. Conclusion: Our findings showed that patients affected by PG, in spite of normal intellectual, linguistic and visual-spatial abilities, had abnormalities emerging from the WCST, in particular they could not learn from their mistakes and look for alternative solutions. Our results would seem to confirm an altered functioning of the prefrontal areas which might provoke a sort of cognitive "rigidity" that might predispose to the development of impulsive and/or compulsive behaviors, such as those typical of PG. © 2008 Marazziti et al; licensee BioMed Central Ltd