Central Limit Theorems for the Shrinking Target Problem

Suppose B _i :=B(p,r _i ) are nested balls of radius r _i about a point p in a dynamical system (T,X,μ). The question of whether T ⁱ x∈B _i infinitely often (i.o.) for μ a.e. x is often called the shrinking target problem. In many dynamical settings it has been shown that if $E_{n}:=\sum_{i=1}^{n} \mu(B_{i})$ diverges then there is a quantitative rate of entry and $\lim_{n\to\infty} \frac{1}{E_{n}} \sum_{j=1}^{n} 1_{B_{i}} (T^{i} x) \to1$ for μ a.e. x∈X. This is a self-norming type of strong law of large numbers. We establish self-norming central limit theorems (CLT) of the form $\lim_{ n\to\infty} \frac{1}{a_{n}} \sum_{i=1}^{n} [1_{B_{i}} (T^{i} x)-\mu(B_{i})] \to N(0,1)$ (in distribution) for a variety of hyperbolic and non-uniformly hyperbolic dynamical systems, the normalization constants are $a^{2}_{n} \sim E [\sum_{i=1}^{n} 1_{B_{i}} (T^{i} x)-\mu(B_{i})]^{2}$ . Dynamical systems to which our results apply include smooth expanding maps of the interval, Rychlik type maps, Gibbs-Markov maps, rational maps and, in higher dimensions, piecewise expanding maps. For such central limit theorems the main difficulty is to prove that the non-stationary variance has a limit in probability.     

Small values of signed harmonic sums

For every $\tau \in \mathbb{R}$ and every integer N, let ${\mathfrak{m}}_N(\tau )$ be the minimum of the distance of τ from the sums ${\sum }_{n=1}^N{s}_n/n$, where $s_1,\dots ,s_n\in \{?1,+1\}$. We prove that ${\mathfrak{m}}_N(\tau )<\exp ?(?C{(\log ?N)}^2)$, for all sufficiently large positive integers N (depending on C and τ), where C is any positive constant less than $1/\log ?4$.     

A conjectural extension of Hecke’s converse theorem

By using specific subsequences of two different types of generalized Stern polynomials, we obtain several related classes of finite and infinite continued fractions involving a single term \(z^{t^j}\) in their partial numerators, where z is a complex variable and t is a positive integer. This approach is extended to other, sparser, subsequences of Stern polynomials, based on certain Lucas functions; this then leads to further infinite classes of continued fractions.     

Dynamic uniqueness for stochastic chains with unbounded memory

We say that a probability kernel exhibits dynamic uniqueness (DU) if all the stochastic chains starting from a fixed past coincide on the future tail $\sigma$-algebra. Our first theorem is a set of properties that are pairwise equivalent to DU which allow us to understand how it compares to other more classical concepts. In particular, we prove that DU is equivalent to a weak-${?}^2$ summability condition on the kernel. As a corollary to this theorem, we prove that the Bramson–Kalikow and the long-range Ising models both exhibit DU if and only if their kernels are ${?}^2$ summable. Finally, if we weaken the condition for DU, asking for coincidence on the future $\sigma$-algebra for almost every pair of pasts, we obtain a condition that is equivalent to $\beta$-mixing (weak-Bernoullicity) of the compatible stationary chain. As a consequence, we show that a modification of the weak-${?}^2$ summability condition on the kernel is equivalent to the $\beta$-mixing of the compatible stationary chain.     

Extreme value theory for singular measures

In this paper, we perform an analytical and numerical study of the extreme values of specific observables of dynamical systems possessing an invariant singular measure. Such observables are expressed as functions of the distance of the orbit of initial conditions with respect to a given point of the attractor. Using the block maxima approach, we show that the extremes are distributed according to the generalised extreme value distribution, where the parameters can be written as functions of the information dimension of the attractor. The numerical analysis is performed on a few low dimensional maps. For the Cantor ternary set and the Sierpinskij triangle, which can be constructed as iterated function systems, the inferred parameters show a very good agreement with the theoretical values. For strange attractors like those corresponding to the Lozi and He?non maps, a slower convergence to the generalised extreme value distribution is observed. Nevertheless, the results are in good statistical agreement with the theoretical estimates. It is apparent that the analysis of extremes allows for capturing fundamental information of the geometrical structure of the attractor of the underlying dynamical system, the basic reason being that the chosen observables act as magnifying glass in the neighborhood of the point from which the distance is computed.     

Surfactant design for the 1,1,1,2-tetrafluoroethane-water interface: ab initio calculations and in situ high-pressure tensiometry

In situ high-pressure tensiometry and ab initio calculations were used to rationally design surfactants for the 1,1,1,2-tetrafluoroethane-water (HFA134a|W) interface. Nonbonded pair interaction (binding) energies (E(b)) of the complexes between HFA134a and candidate surfactant tails were used to quantify the HFA-philicity of selected moieties. The interaction between HFA134a and an ether-based tail was shown to be predominantly electrostatic in nature and much more favorable than that between HFA134a and a methyl-based fragment. The interfacial activity of (i) amphiphiles typically found in FDA-approved pressurized metered-dose inhaler (pMDI) formulations, (ii) a series of nonionic surfactants with methylene-based tails, and (iii) a series of nonionic surfactants with ether-based tails was investigated at the HFA134a|W interface using in situ tensiometry. This is the first time that the tension of the surfactant-modified HFA134a|W interface has been reported in the literature. The ether-based surfactants were shown to be very interfacially active, with tension decreasing by as much as 27 mN.m(-)(1). However, the methyl-based surfactants, including those from FDA-approved formulations, did not exhibit high activity at the HFA134a|W interface. These results are in direct agreement with the E(b) calculations. Significant differences in interfacial activity are noted for surfactants at the 2H,3H-perfluoropentane (HPFP)|water and HFA134a|W interfaces. Care should be taken, therefore, when results from the mimicking solvent (HPFP) are extrapolated to HFA134a-based systems. The results shown here are of relevance in the selection of surfactants capable of forming and stabilizing reverse aqueous aggregates in HFA-based pMDIs, which are promising formulations for the systemic delivery of biomolecules to and through the lungs.     

Multivariate optimization of a solid phase microextraction-headspace procedure for the determination of benzene, toluene, ethylbenzene and xylenes in effluent samples from a waste treatment plant

A method based on solid-phase microextraction (SPME) and gas chromatography with flame ionization detection (GC-FID) has been optimized for the determination of benzene, toluene, ethylbenzene and xylenes (BTEX) in water released from a waste treatment plant. The extraction step was optimized using fractional factorial and central composite designs including the following experimental factors: saline concentration; extraction time; desorption time; agitation velocity; headspace volume. A multiple function was used to describe the experimental conditions for simultaneous extraction of the compounds. The procedure, based on direct SPME at 50 degrees C, using a polydimethylsiloxane fiber, showed good linearity (r>0.997 over a concentration range 2-200 microg L(-1)) and repeatability (relative standard deviation (RSD)<4.23%) for all compounds, with limits of detection ranging from 0.05 to 0.28 microg L(-1), and limits of quantification ranging from 0.14 to 0.84 microg L(-1). Concentrations of the target compounds in these samples were between 145.8 and 1891 microg L(-1).     

X-ray and mössbauer studies of tricyclohexyltin(IV) halides. The crystal structures of (cyclo-C6H11)3SnX (X = F,Br and I)

The structures of tricyclohexyltin fluoride (I), bromide (II) and iodide (III) have been determined by X-ray analysis. Compound I crystallizes in the space group P2₁/m with a = 10.422(6), b = 17.238(9), c = 5.769(3) Å, β = 104.6(1)° and Z = 2. Compounds II and III crystallize in the space group Pcmn with a = 10.427(6), b = 16.914(9), c = 11.366(6) Å, Z = 4; and a = 10.400(6), b = 16.900(10), c = 11.400(4) Å, Z = 4, respectively. All three structures consist of discrete tetrahedral (cyclo-C₆H₁₁)₃SnX units.The temperature dependence of the Mössbauer resonance areas has been examined in order to obtain information about the relationship between chemical structure and lattice dynamics.