An exact formula for the measure dimensions associated with a class of piecewise linear maps

An exact formula for the various measure dimensions of attractors associated with contracting similitudes is given. An example is constructed showing that for more general affine maps the various measure dimensions are not always equal.Communicated by Michael F. Barnsley.     

A novel approach to Morita-Baylis-Hillman (MBH) lactones via the Lewis acid-promoted couplings of α,β-unsaturated lactone with aldehydes

The Morita-Baylis-Hillman (MBH)-type reaction of α,β unsaturated δ-lactone with various aldehydes has been achieved without the direct use of a Lewis base. The new MBH process can be conveniently carried out by the slow addition of the diethylaluminum iodide into the solution of lactone and aldehyde in dichloromethane at 0°C. Modest to good yields were obtained (50-65%) for eight examples.     

Model for rebound reactions: M+RI

A simple model, consisting of ‘two-body’ reactant and product trajectories joined by an instantaneous switch, is proposed for the alkali atom-alkyl iodide rebound reactions. The electronic potential surfaces governing these trajectories are justified in terms of the electron jump model. Encouraging overall agreement is found with extensive molecular beam data for the K, Cs + CH₃I reactions.     

On the spectral representation of symmetric stable processes

The so-called spectral representation theorem for stable processes linearly imbeds each symmetric stable process of index p into L^p (0 < p ≤ 2). We use the theory of L^p isometries for 0 < p < 2 to study the uniqueness of this representation for the non-Gaussian stable processes. We also determine the form of this representation for stationary processes and for substable processes. Complex stable processes are defined, and a complex version of the spectral representation theorem is proved. As a corollary to the complex theory we exhibit an imbedding of complex L^q into real or complex L^p for 0 < p < q ≤ 2.     

Refinable subspaces of a refinable space

Local refinable finitely generated shift-invariant spaces play a significant role in many areas of approximation theory and geometric design. In this paper we present a new approach to the construction of such spaces. We begin with a refinable function which is supported on . We are interested in spaces generated by a function built from the shifts of .

     

Mesh ratios for best-packing and limits of minimal energy configurations

For N-point best-packing configurations ω _N on a compact metric space (A,ρ), we obtain estimates for the mesh-separation ratio γ(ω _N ,A), which is the quotient of the covering radius of ω _N relative to A and the minimum pairwise distance between points in ω _N . For best-packing configurations ω _N that arise as limits of minimal Riesz s-energy configurations as s→∞, we prove that γ(ω _N ,A)≦1 and this bound can be attained even for the sphere. In the particular case when N=5 on S ² with ρ the Euclidean metric, we prove our main result that among the infinitely many 5-point best-packing configurations there is a unique configuration, namely a square-base pyramid $\omega_{5}^{*}$ , that is the limit (as s→∞) of 5-point s-energy minimizing configurations. Moreover, $\gamma(\omega_{5}^{*},S^{2})=1$ .