A purely algebraic characterization of the hyperreal numbers

The hyperreal numbers of nonstandard analysis are characterized in purely algebraic terms as homomorphic images of a suitable class of rings of functions.

     

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 indicated 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 inflammation, 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.     

Alpha-theory: An elementary axiomatics for nonstandard analysis

The methods of nonstandard analysis are presented in elementary terms by postulating a few natural properties for an infinite “ideal” number . The resulting axiomatic system, including a formalization of an interpretation of Cauchy's idea of infinitesimals, is related to the existence of ultrafilters with special properties, and is independent of ZFC. The Alpha-Theory supports the feeling that technical notions such as superstructure, ultrapower and the transfer principle are definitely not needed in order to carry out calculus with actual infinitesimals.     

Bimodal system (luminophore and paramagnetic contrastophore) derived from Ln(III) complexes based on a bipyridine-containing macrocyclic ligand

The synthesis of a new 15-membered polyaza-macrocyclic ligand L3H3, which is based on a 2,2'-bipyridine moiety and a diethylenetriaminetriacetic acid core, is reported. The lanthanide chelates of this octadentate ligand were programmed for bimodal probes, luminescent agents (Sm, Eu, Tb, Dy), and magnetic resonance imaging agents (Gd3+). The neutral 1:1 complexes with these Ln3+ ions were prepared and studied in aqueous solution by luminescence and NMR techniques. The main photophysical characteristics of these complexes (i.e., the absorption and luminescence spectra, the metal-centered lifetimes, and the overall luminescence yields, Phi) were measured. In addition, the role played by nonradiative pathways (vibrational energy transfer involving coordinated water molecules, involvement of ligand-to-metal charge-transfer excited states, or metal --> ligand back transfer) is discussed. The L3.Eu and L3.Tb complexes show very bright luminescence when photoexcited from the lowest-energy absorption band of the bipyridine chromophore. The luminescence quantum yields in an air-equilibrated water solution at room temperature are 0.10 and 0.21, respectively, despite the presence of one water molecule in the first coordination sphere of the metal ion. NMR data show that L3.Gd contains also one H2O molecule in the inner sphere. The proton longitudinal relaxivity, r1, of this complex is 3.4 s(-1) mM(-1) (0.47 T, 310 K) and the rotational correlation time, tau(R), is 57 ps (310 K). These values are comparable to those of the clinically used Gd-DTPA. Interestingly, the water exchange rate between the coordination site and the bulk solvent is slow (tau(M) = 3.5 micros at 310 K). The presence of water molecules in the second sphere and in rapid exchange with the solvent is discussed. Finally, it was found by luminescence and NMR experiments that these lanthanide complexes are stable versus transmetalation by several cations (especially Ca2+ and Zn2+) at physiological pH and have no interaction with blood proteins.     

Topological and Nonstandard Extensions

We introduce a notion of topological extension of a given set X. The resulting class of topological spaces includes the Stone-ech compactification X of the discrete space X, as well as all nonstandard models of X in the sense of nonstandard analysis (when endowed with a natural topology). In this context, we give a simple characterization of nonstandard extensions in purely topological terms, and we establish connections with special classes of ultrafilters whose existence is independent of ZFC.     

Topological and Nonstandard Extensions

We introduce a notion of topological extension of a given set X. The resulting class of topological spaces includes the Stone-ech compactification X of the discrete space X, as well as all nonstandard models of X in the sense of nonstandard analysis (when endowed with a natural topology). In this context, we give a simple characterization of nonstandard extensions in purely topological terms, and we establish connections with special classes of ultrafilters whose existence is independent of ZFC.     

Area-filling curves

Archiv der Mathematik - In this paper, we study area-filling curves, i.e. continuous and injective mappings defined on [0, 1] whose graph has positive measure. Current literature calls...     

Hausdorff ultrafilters

We give the name Hausdorff to those ultrafilters that provide ultrapowers whose natural topology (-topology) is Hausdorff, e.g. selective ultrafilters are Hausdorff. Here we give necessary and sufficient conditions for product ultrafilters to be Hausdorff. Moreover we show that no regular ultrafilter over the ``small' uncountable cardinal can be Hausdorff. ( is the least size of an ultrafilter basis on .) We focus on countably incomplete ultrafilters, but our main results also hold for -complete ultrafilters.

     

Linearly Stratified Models for the Foundations of Nonstandard Mathematics

Assuming the existence of an inaccessible cardinal, transitive full models of the whole set theory, equipped with a linearly valued rank function, are constructed. Such models provide a global framework for nonstandard mathematics.