A Simple Procedure for the Preparation of Chiral ‘Tris(hydroxymethyl)methane’ Derivatives

The aldol adducts 1a – 13a of R,R-2(tertbutyl)-6-methyl-1,3-dioxan-4-one (from 3-hydroxybutanoic acid) to aldehydes, single diastereoisomers obtained as described previously, are acetylated or benzoylated to the corresponding esters 1b – 5b and 6c – 13c , respectively, which in turn are reduced with LiAlH₄ to the title compounds 14 – 24 . The enantiomerically pure triols thus available may be useful as chiral building blocks, as auxiliaries for enantioselective reactions, and as center pieces for chiral dendrimers.     

Periodic Duffing delay equations with state dependent impulses

Conditions are obtained for the Duffing equation with delay and state dependent impulses to admit a periodic solution.     

Metallic glasses: Mössbauer contribution, physical properties and applications

We report first on the essential contributions of Mössbauer spectrometry to investigate iron-based metallic glasses. In addition to the analysis of the distributions of hyperfine data in terms of Fe coordination and neighbours, Mössbauer spectrometry is able to estimate the magnetic anisotropy from the magnetic texture, which can be correlated to the solidification process. After an overview of the different physical properties of the amorphous metallic alloys we also present their most advanced applications.     

A Frequency Selection Criterion in Spatially Developing Flows

The possible existence of global modes or self-excited linear resonances in spatially developing systems is explored within the framework of the WKBJ approximation. It is shown that the existence and properties of the dominant global mode may be deduced from the variations of the local absolute frequency ω₀ with distance X. The main results are summarized in two theorems: (1) A system with no region of absolute instability does not sustain temporally growing global modes with an O(1) growth rate. (2) If the singularity X, closest to the real X-axis of the complex function ω₀(X) is a saddle point, the most unstable global mode has, to leading order in the WKBJ approximation, a complex frequency ω₀(X_s). Thus, it will be temporally growing only if ω₀(X_s) is positive.     

Existence of Equilibria in Economies with Externalities and Nonconvexities

We consider a general equilibrium model of an economy in which the production possibilities, the consumption sets and the preferences of the consumers are represented by set-valued mappings which depend on the environment to take into account the possibility of external effect. In order to encompass all kinds of nonconvexities, we do not put any convexity assumption either on the graph of the set-valued mapping which describes the technological possibilities or on the production set for a given environment. The firms are instructed to set their prices according to general pricing rules which may depend on the production plans of other producers and on consumption plans.We report an existence result of general equilibria. As in the model without external effects, the key hypotheses are bounded loss and survival assumptions. Nevertheless, we also assume that the set-valued mappings which describe the fundamentals of the economy are lower semi-continuous and have a closed graph.Our framework is sufficiently large to generalize previous works on the existence of competitive equilibria with externalities when the firms have convex production sets and on the existence of equilibria with general pricing rule without externality.