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

Toksikološke metode otkrivanja opojnih droga u tragovima: kromatografska, spektroskopska i biološka karakterizacija derivata ecstasyja

Analysis often reveals variability in the composition of ecstasy pills from pure 3,4-methylenedioxymethamphetamine (MDMA) to mixtures of MDMA derivatives, amphetamine, and other unidentifi ed substances. For a comprehensive toxicological analysis one needs to know all steps to MDMA synthesis which may originate impurities. The aim of this study was to synthesise and determine the chemical-physical and in vitro biological properties of a series of MDMA derivatives. 3,4-methylendioxyphenyl-2-nitropropene (MDNP) was obtained by condensation of piperonal with an excess of nitroethane in the presence of ammonium acetate. MDNP was then reduced to methylenedioxyamphetamine (MDA) by LiAlH3. All compounds were analysed using HPLC and spectroscopic technique [Raman, nuclear magnetic resonance (NMR), or infrared (IR)] at all the steps of synthesis. In addition, we assessed the biological potentials of these compounds by measuring in vitro their (i) blood cell/whole blood partition coeffi cient, (ii) binding to plasmatic proteins (Fbp), and (iii) membrane adsorption. Chemical structure was determined with antibody fl uorescence polarisation immunoassay (FPIA). This study showed the presence of solid impurities, particularly of a neurotoxic compound of Al3+ in the fi nal products. FPIA identifi ed the aminoethane group close to the substituted benzene ring, but did not detect the two major precursors of MDMA: MDNP and piperonal. Raman spectroscopy is an attractive alternative technique to characterise ecstasy pills and it can identify stereoisomeric forms such as cis-MDNP and trans-MDNP, which exhibit signals at 1650 cm-1 and 1300 cm-1, respectively.Analize često otkriju neujednačenost sastava tableta ecstasyja od čistoga 3,4-metilendioksimetamfetamina (MDMA) do mješavina njegovih derivata, amfetamina i drugih neutvrđenih tvari. Stoga je za kvalitetnu toksikološku analizu potreban uvid u sve korake sinteze MDMA, s obzirom na to da se ondje vjerojatno kriju izvori nečistoće (prekursori, katalizatori). Cilj ovog ispitivanja bio je sintetizirati derivate MDMA te napraviti njihovu kemijsko-fi zikalnu i biološku in vitro karakterizaciju. 3,4-metilendioksifenil-2-nitropropen (MDNP) dobiven je kondenzacijom piperonala u suvišku nitroetana uz dodatak amonijeva acetata. Njegovom redukcijom s pomoću LiAlH3 dobiven je 3,4-metilendioksiamfetamin (MDA). Svi spojevi iz pojedinih koraka sinteze karakterizirani su s pomoću tekućinske kromatografi je visoke djelotvornosti (HPLC) i spektroskopskih tehnika [Ramanove spektroskopije, nuklearne magnetske rezonancije (NMR-a) te infracrvene spektroskopije (IR-a)]. Usto je ocijenjen i njihov biološki učinak in vitro mjerenjem (i) koefi cijenta raspodjele krvna stanica/puna krv, (ii) vezanja za bjelančevine u plazmi (Fbp) te (iii) adsorpcije na membranu. Kemijska je struktura utvrđena s pomoću fl uorescentnoga polarizacijskog imunokemijskog testa (FPIA). Analiza je u konačnim proizvodima utvrdila prisutnost krutih nečistoća, napose spojeva neurotoksičnog aluminija (Al3+). FPIA je prepoznao aminoetansku skupinu blizu supstituiranoga benzenskog prstena, ali ne i dva glavna prekursora za MDMA: MDNP i piperonal. Posebno je zanimljiva Ramanova spektroskopija budući da (i) pruža privlačnu alternativu za karakterizaciju sastava tableta ecstasyja te (ii) može otkriti stereoizomerne cis/trans-oblike spoja poput cis-MDNP-a odnosno trans-MDNP-a, čiji se signal vidi na 1650 cm-1 odnosno 1300 cm-1

Extreme value statistics for dynamical systems with noise

We study the distribution of maxima ( extreme value statistics ) for sequences of observables computed along orbits generated by random transformations. The underlying, deterministic, dynamical system can be regular or chaotic. In the former case, we show that, by perturbing rational or irrational rotations with additive noise, an extreme value law appears, regardless of the intensity of the noise, while unperturbed rotations do not admit such limiting distributions. In the case of deterministic chaotic dynamics, we will consider observables specially designed to study the recurrence properties in the neighbourhood of periodic points. Hence, the exponential limiting law for the distribution of maxima is modified by the presence of the extremal index , a positive parameter not larger than one, whose inverse gives the average size of the clusters of extreme events. The theory predicts that such a parameter is unitary when the system is perturbed randomly. We perform sophisticated numerical tests to assess how strong the impact of noise level is when finite time series are considered. We find agreement with the asymptotic theoretical results but also non-trivial behaviour in the finite range. In particular, our results suggest that, in many applications where finite datasets can be produced or analysed, one must be careful in assuming that the smoothing nature of noise prevails over the underlying deterministic dynamics