On Subsets with Small Product in Torsion-Free Groups

G be a nonabelian torsion-free group. Let C be a finite generating subset of G such that . We prove that, for all subsets B of G with , we have . In particular, a finite subset X with cardinality satisfies the inequality if and only if there are elements , such that the following two conditions hold: (i) . (ii) where . Received: October 13, 1997/Revised: Revised August 18, 1998     

Synthesis of Branched Polymers under Continuous‐Flow Microprocess: An Improvement of the Control of Macromolecular Architectures

Polymerization reactions can benefit from continuous‐flow microprocess in terms of kinetics control, reactants mixing or simply efficiency when high‐throughput screening experiments are carried out. In this work, we perform for the first time the synthesis of branched macromolecular architecture through a controlled/‘living' polymerization technique, in tubular microreactor. Just by tuning process parameters, such as flow rates of the reactants, we manage to generate a library of polymers with various macromolecular characteristics. Compared to conventional batch process, polymerization kinetics shows a faster initiation step and more interestingly an improved branching efficiency. Due to reduced diffusion pathway, a characteristic of microsystems, it is thus possible to reach branched polymers exhibiting a denser architecture, and potentially a higher functionality for later applications.     

Diffusion Crosslinking Kinetics of Thermoset Polymers

The problem of the delayed crosslinking arising due to the diffusion of a crosslinker molecule from a vector particle within a thermoset polymer matrix is discussed. The crosslinker molecules, initially embedded in the vector particles, diffuse from the particle to the polymer matrix and react with the functional groups grafted to polymer chains. The two‐step diffusion crosslink kinetics is studied theoretically on the example of polyethylene functionalized with anhydride groups reacting with 1,4‐butanediol as a crosslinker molecule. This provides the independent of time limiting distribution functions of the concentration of crosslinks, unreacted and intermediate functional groups, which are shown to be rather sensitive to system parameters as rates of chemical reactions, ratio of diffusion coefficients of matrix to vector particles, stoichiometric composition of the reactive groups, etc. The nonmonotonic distribution of the crosslink concentration around the vector particles at certain range of the parameters is revealed.

     

Subsurface Carbon: A General Feature of Noble Metals

Carbon moieties on late transition metals are regarded as poisoning agents in heterogeneous catalysis. Recent studies show the promoting catalytic role of subsurface C atoms in Pd surfaces and their existence in Ni and Pt surfaces. Here energetic and kinetic evidence obtained by accurate simulations on surface and nanoparticle models shows that such subsurface C species are a general issue to consider even in coinage noble‐metal systems. Subsurface C is the most stable situation in densely packed (111) surfaces of Cu and Ag, with sinking barriers low enough to be overcome at catalytic working temperatures. Low‐coordinated sites at nanoparticle edges and corners further stabilize them, even in Au, with negligible subsurface sinking barriers. The malleability of low‐coordinated sites is key in the subsurface C accommodation. The incorporation of C species decreases the electron density of the surrounding metal atoms, thus affecting their chemical and catalytic activity.     

Extreme statistics of complex random and quantum chaotic states

Complex random states have the statistical properties of the Gaussian and circular unitary ensemble eigenstates of random matrix theory. Even though their components are correlated by the normalization constraint, it is nevertheless possible to derive compact analytic formulas for their extreme values' statistical properties for all dimensionalities. The maximum intensity result slowly approaches the Gumbel distribution even though the variables are bounded, whereas the minimum intensity result rapidly approaches the Weibull distribution. Since random matrix theory is conjectured to be applicable to chaotic quantum systems, we calculate the extreme eigenfunction statistics for the standard map with parameters at which its classical map is fully chaotic. The statistical behaviors are consistent with the finite-N formulas.     

Electrically Switchable Magnetic Molecules: Inducing a Magnetic Coupling by Means of an External Electric Field in a Mixed‐Valence Polyoxovanadate Cluster

Herein we evaluate the influence of an electric field on the coupling of two delocalized electrons in the mixed‐valence polyoxometalate (POM) [GeV₁₄O₄₀]^8? (in short V₁₄) by using both a t‐J model Hamiltonian and DFT calculations. In absence of an electric field the compound is paramagnetic, because the two electrons are localized on different parts of the POM. When an electric field is applied, an abrupt change of the magnetic coupling between the two delocalized electrons can be induced. Indeed, the field forces the two electrons to localize on nearest‐neighbors metal centers, leading to a very strong antiferromagnetic coupling. Both theoretical approaches have led to similar results, emphasizing that the sharp spin transition induced by the electric field in the V₁₄ system is a robust phenomenon, intramolecular in nature, and barely influenced by small changes on the external structure.