Statistical properties of subgroups of free groups

The usual way to investigate the statistical properties of finitely generated subgroups of free groups, and of finite presentations of groups, is based on the so‐called word‐based distribution: subgroups are generated (finite presentations are determined) by randomly chosen k ‐tuples of reduced words, whose maximal length is allowed to tend to infinity. In this paper we adopt a different, though equally natural point of view: we investigate the statistical properties of the same objects, but with respect to the so‐called graph‐based distribution, recently introduced by Bassino, Nicaud and Weil. Here, subgroups (and finite presentations) are determined by randomly chosen Stallings graphs whose number of vertices tends to infinity. Our results show that these two distributions behave quite differently from each other, shedding a new light on which properties of finitely generated subgroups can be considered frequent or rare. For example, we show that malnormal subgroups of a free group are negligible in the graph‐based distribution, while they are exponentially generic in the word‐based distribution. Quite surprisingly, a random finite presentation generically presents the trivial group in this new distribution, while in the classical one it is known to generically present an infinite hyperbolic group. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

Carbon Participation in the Solvolysis of 6-exo-substituted 2-exo- and 2-endo-Norbornyl p-Toluenesulfonates. Norbornanes part 5

The solvolysis rates and products of the 6-exo-substituted 2-exo- 1a - 1u , and 2-endo-norbornyl p-toluenesulfonates 2a - 2u , have been determined. In general, the rate constants for 1 and 2 (log k) correlate well with the inductive constants σ of the substitutents at C(6); however, their sensitivity to σ is much larger in the 2-exo-series 1 than in the 2-endo-series 2 . This differential transmission of polar effects is the cause of decreasing 2-exo/2-endo rate ratios from 2388 for R = t-C₄H₉ to 0.37 for R = Br, i. e. with increasing electron attraction by the substituent. The high sensitivity of the rate constants for the 2-exo-p-toluenesulfonates 1 to σ indicates an unusually strong inductive interaction between C(6) and the incipient cationic center at C(2). This interaction is ascribed to the participation of the pentacoordinate C(6)-atom, i. e. to 1,3-bridging, a consequence of steric hindrance of nucleophilic solvent participation in norbornanes. Donor substituents enhance 1,3-bridging, lead to faster reactions and to the formation of 2-exo substitution products. Conversely, acceptor substituents reduce 1,3-bridging, decrease rates and facilitate the formation of 2-endo substitution products. Graded 1,3-bridging is discussed in the light of Winstein's nonclassical ion concept.     

Backbone curves,Neimark-Sacker boundaries and appearance of quasi-periodicity in nonlinear oscillators: application to 1:2 internal resonance and frequency combs in MEMS

Meccanica - Quasi-periodic solutions can arise in assemblies of nonlinear oscillators as a consequence of Neimark-Sacker bifurcations. In this work, the appearance of Neimark-Sacker bifurcations is...     

Shadowgraph, Schlieren and interferometry in a 2D cavitating channel flow

Cavitation plays an important role in fuel atomization mechanisms, but the physics of cavitation and its impact on spray formation and injector efficiency are not well documented yet. Experimental investigations are required to support the development and the validation of numerical models and the design of tomorrow??s injectors, in the context of pollutant and fuel consumption reduction. The complexity of modern injectors and the extreme conditions of injection do not facilitate experimental investigations. In this paper, experiments are conducted in a simplified geometry. The model nozzle consists of a transparent 2D micro-channel supplied with a test oil (ISO 4113). Three different optical techniques are proposed to investigate the channel flow, with the pressure drop between upstream and downstream chambers as a parameter. A shadowgraph-like imaging technique allows the observation of cavitation inception and vapor cavities development throughout the channel. The technique also reveals the presence of density gradients (pressure or temperature) in the channel flow. However, this additional information is balanced by difficulties in image interpretation, which are discussed in the paper. In addition, a combination of Schlieren technique and interferometric imaging is used to measure the density fields inside the channel. The three techniques results are carefully analyzed and confronted. These results reveal a wealth of information on the flow, with pressure waves generated by bubble collapses, turbulence in the wake of vapor cavities and bubble survival in flow regions of high pressure. Our results also show that cavitation inception is located in the shear layers between the recirculation zones and the main flow, relatively far from the inlet corner, where the pressure is minimum in average. To explain this behavior, we propose a scenario of cavitation inception based on the occurrence and the growing of instabilities in the shear layers.     

Pd-catalysed asymmetric mono- and bis-alkoxycarbonylation of vinylarenes

The asymmetric alkoxycarbonylation of vinylarenes catalysed by palladium complexes bearing chiral phosphine ligands has attracted much attention over the last decades. The products of both mono- and bis(alkoxycarbonylation) reactions are important intermediates in the syntheses of pharmaceuticals such as 2-arylpropionic acids, the most important class of non-steroidal anti-inflammatory drugs. In this article, a general overview of the topics will be presented and the recent advances in this field will be particularly detailed. Besides the term alkoxycarbonylation, hydroesterification and hydroalkoxycarbonylation are also used in the literature to describe this reaction. Furthermore, more specific terms such as methoxycarbonylation can be found. In this report, the term alkoxycarbonylation will be used as the general term, and specific terms will be used to unambiguously define which reaction is meant.     

The remarkable effect of the 7-substituent in the diastereoselective oxidative rearrangement of indoles: asymmetric synthesis of 3,3-disubstituted oxindoles

The nature of the 7-substituent has a remarkable effect on the diastereoselectivity of the oxidative rearrangement of indole-2-carboxamides derived from (S)-2-methoxymethylpyrrolidine into chiral 3,3-disubstituted oxindoles.     

Optical properties of erbium-doped fluorochlorozirconate glasses

We report the optical properties of a fluorochlorozirconate (FCZ) glass with the composition 53% ZrF₄, 20% NaF, 3.5% AlF₃, 3% LaF₃, 0.5% InF₃, (20 ? x)% BaCl₂, x% BaF₂ with x varying from 0% to 2%, and doped with various amounts of trivalent erbium by the addition of ErCl₃. Annealing of the as-prepared glass in inert (N₂) or reducing (5%H₂ + 95%Ar) atmospheres at temperatures that ensure the conversion of the glass into a glass-ceramic by the nucleation of BaCl₂ nanocrystals, does not significantly change any of Er³⁺ related absorption and photoluminescence (PL) characteristics. We have carried out a Judd–Ofelt analysis of the absorption spectra and obtained Ω₂ = (1.92 ± 0.3) × 10^? 20 cm², Ω₄ = (0.88 ± 0.16) × 10^? 20 cm² and Ω₆ = (0.59 ± 0.08) × 10^? 20 cm², and also the radiative lifetimes of the ⁴I_13/2→⁴I_15/2, ⁴I_11/2→⁴I_15/2 and ⁴S_3/2→⁴I_15/2 bands. The radiative lifetime from the Judd–Ofelt analysis for the ⁴I_13/2→⁴I_15/2 band is in good agreement with the experimentally measured PL decay time. The examination of the optical properties of powdered samples with different average particle size does not show any photon trapping effects. We have determined the spectral absorption and emission cross-sections and then estimated the possible spectral optical gain for varying degrees of relative populations of the ⁴I_13/2 and ⁴I_15/2 manifolds.