Asymptotic aspects of Schreier graphs and Hanoi Towers groups

We present relations between growth, growth of diameters and the rate of vanishing of the spectral gap in Schreier graphs of automaton groups. In particular, we introduce a series of examples, called Hanoi Towers groups since they model the well known Hanoi Towers Problem, that illustrate some of the possible types of behavior. To cite this article: R. Grigorchuk, Z. ?unik´, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Metal Ion Interaction of Two New Tritopic N2O13 Macrobicycles with B15C5 Moieties in Acetonitrile Solution

The synthesis of two new tritopic crown ligands (L1 and L2) bearing two benzo-15-crown-5 lateral moieties linked through a dibenzo-trioxa chain together with their interaction with metal ions, in acetonitrile and acetonitrile–water (50%, v/v) solutions is reported. The influence of K⁺, Na⁺, Li⁺, Ca²⁺, Ba²⁺, Cu²⁺, Zn²⁺, Cd²⁺, Pb²⁺ and Al³⁺, on the spectroscopic properties of these diaza-polyoxa ligands was investigated by absorption spectrophotometry and in some cases by fluorescence emission spectroscopy. Coordination with alkaline (Na⁺, K⁺ and Li⁺) and alkaline earth (Ca²⁺and Ba²⁺) metal ions is assumed to be weak with both macrobicyclic ligands, while the interaction with both imine and amine derivatives causes a minor effect in the absorption spectra. Coordination with Cu²⁺, Zn²⁺ and Pb²⁺ in acetonitrile solution causes a major change in the absorption spectra of the chromophores. In the case of Cu²⁺, addition of the metal to L1 or L2 leads to a blue–violet complex in solution with an absorbance maximum centred at 590 nm. Interaction of the Schiff-base L1 with Pb²⁺ leads to a short wavelength shift in the absorption bands, comparable with the ZnL1 complex. Presence of transition metal ions such as Co²⁺, Ni²⁺and Cd²⁺ do not remarkably affect the absorption spectra of L1 and L2 in solution. Trivalent aluminium has a modest effect in the absorption bands of both N₂O₁₃ donor set bismacrocyclic ligands. The fluorescence study of L2 in the presence of Na⁺, K⁺, Ca²⁺, Ba²⁺, Co²⁺, Cu²⁺, Ni²⁺, Pb²⁺ and Al³⁺shows that Cu²⁺, Pb²⁺ and Al³⁺ complexes form non-fluorescent complexes.     

Colorimetric determination of iron in infant fortified formulas by sequential injection analysis

 A procedure is described for the colorimetric determination of iron in infant fortified formulas based on sequential injection analysis (SIA). Iron(III) complexation with thiocyanate is used as colour developing reaction. The system enables the determination of iron in the samples (after digestion by dry ashing and treatment with 0.2 mol/L nitric acid in the range of 0.50–20.0 mg/L, consuming 140 μL of the sample and 8 mg thiocyanate per determination. The reactor geometry and the adjustment of the ionic content of the calibration solutions is important for the accuracy of the results. A regression line according to the equation [Fe(III) (mg/L)]_SIA=−0.3(±0.4)+1.03(±0.04) [Fe(III) (mg/L)]_FAAS was obtained after comparative analysis of a set of 12 samples. The measurement rate was 34 s, thus allowing to analyze 100 samples per hour with a relative standard deviation lower than 2%. Received: 30 July 1996/Revised: 1 October 1996/Accepted: 4 October 1996     

Electron diffraction study on the molecular structure of methane sulphonyl fluoride

The molecular structure of methane sulphonyl fluoride in the vapour state was studied by electron diffraction. Assuming a value of 2.480A?for the distance between the oxygen atoms from a microwave determination, the following geometrical parameters (r_a structure) have been obtained: r(C-H) = 1.093±0.010Å, r(S-O) = 1.410±0.003Å, r(S-F) = 1.561 ±0.004Å, r(S-C) = 1.759±0.006Å, ∠F-S-C = 98.2±1.5°, ∠-S-F = 106.2±0.4°, ∠-O-S-O = 123.1 ±1.5° and ∠H-C-H = 112.9±1.9°. All the observed variations in the molecular geometries of (CH₃)₂SO₂, CH₃SO₂Cl, CH₃SO₂F and SO₂F₂ may be accounted for by the valence shell electron pair repulsion theory. It is particularly advantageous to combine electron diffraction and microwave data in studying sulphone molecular geometries.     

The Distribution of the Sojourn Time for the Brownian Excursion

The solutions of various problems in the theories of queuing processes, branching processes, random graphs and others require the determination of the distribution of the sojourn time (occupation time) for the Brownian excursion. However, no standard method is available to solve this problem. In this paper we approximate the Brownian excursion by a suitably chosen random walk process and determine the moments of the sojourn time explicitly. By using a limiting approach, we obtain the corresponding moments for the Brownian excursion. The moments uniquely determine the distribution, enabling us to derive an explicit formula.     

Stopping Rules for the Stochastic Nested Partitions Method

We have recently developed a global optimization methodology for solving combinatorial problems with either deterministic or stochastic performance functions. This method, the Nested Partitions (NP) method has been shown to generate a Markov chain and with probability one to converge to a global optimum. In this paper, we study the rate of convergence of the method through the use of Markov Chain Monte Carlo (MCMC) methods, and use this to derive stopping rules that can be applied during simulation-based optimization. A numerical example serves to illustrate the feasibility of our approach.     

Versatile Sugar Derivatives for the Synthesis of Potential Degradable Hydrophilic‐Hydrophobic Polyurethanes and Polyureas

Biodegradable polymers obtained from renewable natural sources are currently receiving increasing attention because they are an alternative to the traditional petroleum‐based plastics. In the present communication we describe the synthesis of the diol monomers 2,3,4‐tri‐O‐benzyl‐L‐arabinitol (ABnOH) and 2,3,4‐tri‐O‐benzylxylitol (XBnOH), and the diamino monomers 1,5‐diamino‐1,5‐dideoxy‐2,3,4‐tri‐O‐benzyl‐L‐arabinitol (ABnNH ₂ ) and 1,5‐diamino‐1,5‐dideoxy‐2,3,4‐tri‐O‐benzylxylitol (XBnNH ₂ ), which can be used in the preparation of new potentially biodegradable sugar‐based polymers. As an example, we describe the synthesis and characterization of a polyurethane [PU‐(ABnOH‐HMDI)] and a polyurea [PUR‐(ABnNH₂‐HMDI)] by poly addition reaction of ABnOH and ABnNH ₂ with 1,6‐hexamethylene diisocyanate.