Hausdorff dimension of three-period orbits in Birkhoff billiards

We prove that the Hausdorff dimension of the set of three-period orbits in classical billiards is at most one. Moreover, if the set of three-period orbits has Hausdorff dimension one, then it has a tangent line at almost every point.Comment: 10 pages, 1 figur

Eigenfunctions for smooth expanding circle maps

We construct a real-analytic circle map for which the corresponding Perron-Frobenius operator has a real-analytic eigenfunction with an eigenvalue outside the essential spectral radius when acting upon $C^1$-functions.Comment: 10 pages, 2 figure

Algebraic entropy for algebraic maps

We propose an extension of the concept of algebraic entropy, as introduced by Bellon and Viallet for rational maps, to algebraic maps (or correspondences) of a certain kind. The corresponding entropy is an index of the complexity of the map. The definition inherits the basic properties from the definition of entropy for rational maps. We give an example with positive entropy, as well as two examples taken from the theory of Backlund transformations

Conjugating Biotin to Ruthenium(II) Arene Units via Phosphine Ligand Functionalization

Two-step functionalization of 4-diphenylphosphino benzoic acid with biotin afforded 2-(biotinyloxy)ethyl 4-(diphenylphosphanyl)benzoate (LP), that was subsequently used to synthesize the Ru(II) arene complexes [RuCl2(η6-p-cymene)(LP)] (1), [Ru(C2O4)(η6-p-cymene)(LP)] (2) and [Ru(curc)(η6-p-cymene)(LP)]NO3 ([3]NO3), the latter incorporating curcumin (curcH) as an additional bioactive fragment. [Ru(curc)(η6-p-cymene)(PPh3)]NO3 ([4]NO3) was also prepared as a reference compound. Compounds 2 and [3]NO3 exhibited excellent stability in water/DMSO solution while being slowly activated in the cell culture medium over 72 hours. Together with LP, they were therefore assessed for their antiproliferative activity towards a panel of cancer cell lines, with different levels of biotin transporter expression. The apparent affinity of the compounds towards avidin varies, and their antiproliferative activity does not correlate with biotin transporter expression, although it is systematically enhanced when biotin-free cell culture medium is used

The compound Poisson limit ruling periodic extreme behaviour of non-uniformly hyperbolic dynamics

We prove that the distributional limit of the normalised number of returns to small neighbourhoods of periodic points of non-uniformly hyperbolic dynamical systems is compound Poisson. The returns to small balls around a fixed point in the phase space correspond to the occurrence of rare events, or exceedances of high thresholds, so that there is a connection between the laws of Return Times Statistics and Extreme Value Laws. The fact that the fixed point in the phase space is a repelling periodic point implies that there is a tendency for the exceedances to appear in clusters whose average sizes is given by the Extremal Index, which depends on the expansion of the system at the periodic point. We recall that for generic points, the exceedances, in the limit, are singular and occur at Poisson times. However, around periodic points, the picture is different: the respective point processes of exceedances converge to a compound Poisson process, so instead of single exceedances, we have entire clusters of exceedances occurring at Poisson times with a geometric distribution ruling its multiplicity. The systems to which our results apply include: general piecewise expanding maps of the interval (Rychlik maps), maps with indifferent fixed points (Manneville-Pomeau maps) and Benedicks-Carleson quadratic maps.Comment: To appear in Communications in Mathematical Physic

Large deviations for non-uniformly expanding maps

We obtain large deviation results for non-uniformly expanding maps with non-flat singularities or criticalities and for partially hyperbolic non-uniformly expanding attracting sets. That is, given a continuous function we consider its space average with respect to a physical measure and compare this with the time averages along orbits of the map, showing that the Lebesgue measure of the set of points whose time averages stay away from the space average decays to zero exponentially fast with the number of iterates involved. As easy by-products we deduce escape rates from subsets of the basins of physical measures for these types of maps. The rates of decay are naturally related to the metric entropy and pressure function of the system with respect to a family of equilibrium states. The corrections added to the published version of this text appear in bold; see last section for a list of changesComment: 36 pages, 1 figure. After many PhD students and colleagues having pointed several errors in the statements and proofs, this is a correction to published article answering those comments. List of main changes in a new last sectio

Simultaneous determination of Deoxynivalenol, Deoxynivalenol-3-Glucoside and Nivalenol in wheat grains by HPLC-PDA with immunoaffinity column cleanup

Deoxynivalenol-3-glucoside (D3G) is a modified mycotoxin formed by the metabolism of plants through the conjugation of deoxynivalenol (DON) with glucose. Toxicology studies of D3G for human and animal health are still under investigation, and the development of practical and reliable methods for its direct determination, especially in cereal matrices, is of great importance. In the present study, a methodology for simultaneous determination of D3G, DON, and nivalenol (NIV) in wheat grains, using immunoaffinity column (IAC) cleanup, separation by C18 column and detection by ultraviolet (UV) absorption, was optimized and in-house validated. The results demonstrated adequate values of D3G recovery from IAC and spiked samples. Intraday precision, linearity, limit of detection and limit of quantification (LOQ) were also adequate for the determination of these mycotoxins. Range of applicability varied from 47.1 to 1000 g/kg for D3G and from 31.3 to 1000 g/kg for DON and NIV, with recovery ranging from 84.7±7.2 % to 112.3±8.1Felipe Trombete is grateful for a doctoral fellowship provided by the Brazilian Federal Agency for Support and Evaluation of Graduate Education (CAPES)

Stable isotope dilution assay for the accurate determination of mycotoxins in maize by UHPLC-MS/MS

A fast, easy-to-handle and cost-effective analytical method for 11 mycotoxins currently regulated in maize and other cereal-based food products in Europe was developed and validated for maize. The method is based on two extraction steps using different acidified acetonitrile–water mixtures. Separation is achieved using ultrahigh-performance liquid chromatography (UHPLC) by a linear water–methanol gradient. After electrospray ionisation, tandem mass spectrometric detection is performed in dynamic multiple reaction monitoring mode. Since accurate mass spectrometric quantification is hampered by matrix effects, uniformly [13C]-labelled mycotoxins for each of the 11 compounds were added to the sample extracts prior to UHPLC-MS/MS analysis. Method performance parameters were obtained by spiking blank maize samples with mycotoxins before as well as after extraction on six levels in triplicates. The twofold extraction led to total recoveries of the extraction steps between 97% and 111% for all target analytes, including fumonisins. The [13C]-labelled internal standards efficiently compensated all matrix effects in electrospray ionisation, leading to apparent recoveries between 88% and 105% with reasonable additional costs. The relative standard deviations of the whole method were between 4% and 11% for all analytes. The trueness of the method was verified by the measurement of several maize test materials with well-characterized concentrations. In conclusion, the developed method is capable of determining all regulated mycotoxins in maize and presuming similar matrix effects and extraction recovery also in other cereal-based foods