Scaling properties of ℤ k- 1 actions on the circle

We derive universal scaling properties for ^k–1 actions on the circle whose generators have rotation numbers algebraic of degreek. As fork=2 these properties can be explained for arbitraryk in terms of a renormalization group transformation. It has at least one trivial fixed point corresponding to an action whose generators are pure rotations. The spectrum of the linearized transformation in this fixed point is analyzed completely. The fixed point is hyperbolic with a (k–1)-dimensional unstable manifold. In the casek=2 the known results are therefore recovered.     

The direct conversion of carbamates to ureas using aluminum amides

The conversion of carbamates into ureas using aluminum amide complexes is reported. This reaction is a convenient method to prepare bi-, tri- and tetra-substituted ureas from carbamate-protected primary or secondary amines by reaction with primary or secondary amines in the presence of stoichometric quantities of trimethylaluminum. A reactivity trend of the various carbamates was observed and methyl and benzyl carbamates were reacted selectively in the presence of t-butyl carbamates.     

2,6-Lutidine-N-oxide complexes of rare-earth bromides

2,6-Lutidine-N-oxide (LNO) complexes of rare-earth bromides of the composition wheren = l for M = La, Pr, Nd, Sm, Gd, Ho, Er; andn = 0 for M = Y have been prepared and characterised by analyses, conductance and infrared data. Infrared spectra of the complexes indicate that the coordination of ligand to the metal ion takes place through the oxygen of the ligand, and the water molecule in the complexes present is coordinated to the metal. A coordination number of seven has been suggested to all the rare-earth metal ions.     

Fractal behavior in trapping and reaction: A random walk study

We investigate the trapping of a random walker in fractal structures (Sierpinski gaskets) with randomly distributed traps. The survival probability is determined from the number of distinct sites visited in the trap-free fractals. We show that the short-time behavior and the long-time tails of the survival probability are governed by the spectral dimensiond. We interpolate between these two limits by introducing a scaling law. An extension of the theory, which includes a continuous-time random walk on fractals, is discussed as well as the case of direct trapping. The latter case is shown to be governed by the fractal dimensiond.     

Cover Picture: Trimeric Perfluoro‐ortho‐phenylenemercury: A Versatile Lewis Acidic Host (Chem. Eur. J. 21/2003)

The cover picture shows how trimeric perfluoro‐ortho‐phenylene mercury (center), one of the simplest trifunctional Lewis acidic hosts, interacts with organic molecules to afford various adducts. The planarity of this trinuclear complex as well as its overall polarizability compounded with relativistic effects at mercury permits the occurrence of noncovalent interactions and accounts for the tendency of this compound to form cofacial dimers as observed in the structure of the acetone adduct (top left). With a triply coordinated acetone molecule, the structure of this adduct also substantiates the cooperative effects that arise from the proximity and accessibility of the mercury centers. The complexation of arenes, such as benzene (bottom left) and naphthalene (bottom right), also occurs and leads to the formation of binary stacks in which the arene establishes multiple linkages with the mercury centers of the trifunctional Lewis acids. In addition to displaying unusual coordination environment, the arene units present in those stacks exhibit remarkable phosphorescent properties. For more details, see the paper by F. P. Gabbaï, R. E. Taylor, and M. R. Haneline on p. 5188 ff.     

Aza-Morita-Baylis-Hillman reactions of N-(benzylidene)polyfluoroanilines with methyl acrylate and acrylonitrile

Aza-Morita-Baylis-Hillman reactions of N-(benzylidene)polyfluoroanilines 1 with methyl acrylate or acrylonitrile were studied. It was found that Lewis base, solvent and reaction temperature can significantly affect the reaction. Using 3-hydroxyquinuclidine (3-HQD) as a Lewis base in the reactions of 1 with methyl acrylate in DMF, the normal aza-Morita-Baylis-Hillman adducts 3 were formed in moderate to excellent yields. For the reactions of 1 with acrylonitrile, 1,4-diazabicyclo[2.2.2]octane (DABCO) is the best Lewis base giving the corresponding aza-Morita-Baylis-Hillman adducts 4 as the sole product in good to moderate yield. However, upon treatment of 1 with acrolein 2c, the corresponding reaction did not occur even in the presence of a variety of catalysts.     

Cation-crown ether complex formation in water. II. Alkali and alkaline earth cations and 12-crown-4, 15-crown-5, and 18-crown-6

The stability constants and the partial molal volume and isentropic partial molal compressibility changes of complex formation between cations and crown ethers in water at 25°C are presented. The cations involved are Na⁺, K⁺, Rb⁺, Cs⁺, Ca²⁺, and Ba²⁺, and the crown ethers are 12-crown-4, 15-crown-5, and 18-crown-6. Values of V of complex formation have been discussed in terms of two simple models, one based on the scaled particle theory, and the others on the Drude-Nernst continuum model. The results indicate that the charge of the potassium cation in 18-crown-6 is especially well screened from the water. On this basis hydration numbers of complexed cations have been calculated. This shows that the size of the cation compared to the crown ether hole is important for the contacts between complexed cations and water.     

Random walks on inhomogeneous lattices

For lattices with two kinds of points (black and white), distributed according to a translation-invariant joint probability distribution, we study statistical properties of the sequence of consecutive colors encountered by a random walker moving through the lattice. The probability distribution for the single steps of the walk is considered to be independent of the colors of the points. Several exact results are presented which are valid in any number of dimensions and for arbitrary probability distributions for the coloring of the points and the steps of the walk. They are used to derive a few general properties of random walks on lattices containing traps.Presented at the Symposium on Random Walks, Gaithersburg, MD, June 1982.     

A note on differentiability of the cluster density for independent percolation in high dimensions

The cluster density function of independent percolation in ad-dimensional lattice is considered. For eachn, it is shown that(p) has finitenth leftderivative at critical probabilityp _c ifd is sufficiently large. This result agrees with the Bethe lattice approximation, where thenth one-sided derivative of(p) is bounded atp _c for alln.     

Transference numbers for ultra concentrated solutions of aqueous hydrochloric acid at 25°C

Transference numbers of HCl(aq) solutions at 25°C, from 0.01 to 13.6 mol-kg^–1(m) have been obtained by measuring the emf of cells with transference using hydrogen gas/platinum electrodes. Good agreement is obtained at concentrations up to 1 m with all previous data, and our results strongly corroborate those of King and Spiro over the 2–8m concentration range. The transference numbers of the hydronium ion fit the empirical equation, _H ^HCl = 0.821 + 0.0457m ^1/2 – 2.476×10^–2m – 1.90×10^–4 m ² – 1.45×10^–5 m ³ the maximum deviation in T _H ^HCl being 0.003.