Some estimates of theH ∞-uniform distribution

The sequences (y _n )=(a+blogn+ _n ) are uniformly distributed in the sense of the summation methodH . In the present paper the speed of convergence of this procedure is estimated for these, sequences and for some other sequences. For the sequences (a+blogn) theH -means converge considerably faster than logarithmic means.     

On the structure of a random sum-free set

Summary There is a natural probability measure on the set of all sum-free sets of natural numbers. If we represent such a set by its characteristic functions, then the zero-one random variabless(i) are far from independent, and we cannot expect a law of large numbers to hold for them. In this paper I conjecture a decomposition of into countably many more tractible pieces (up to a null set). I prove that each piece has positive measure, and show that, within each piece, a random set almost surely has a density which is a fixed rational number depending only on the piece. For example, the first such piece is made up of sets consisting entirely of odd numbers; it has probability 0.218 ..., and its members almost surely have density 1/4.     

On quadratic and sesquilinear functionals

LetX be a leftA-module, whereA is either a complex Banach *-algebra with an identity element or the field of quaternions. The main result of this note is that forQ, anA-quadratic functional defined onX, there exists a sesquilinear functionalB such thatB(x,x)=Q(x) holds for allxX.     

Local limits and harmonic functions for nonisotropic random walks on free groups

Summary Nearest neighbour random walks on the homogeneous tree representing a free group withs generators (2s) are investigated. By use of generating functions and their analytic properties a local limit theorem is derived. A study of the harmonic functions corresponding to the random walk leads to properties that characterize ther-harmonic function connected with the local limits.     

Coloring of universal graphs

LetG be a graph andr a cardinal number. Extending the theorem of J. Folkman we show that if eitherr or clG are finite then there existsH with clH = clG andH (G) _r ¹ . Answering a question of A. Hajnal we show that countably universal graphU ₃ satisfiesU ₃ (U₃) _r ¹ for every finiter.     

Sets of tiles with a prescribed number of tilings

Summary For every k2 and r1 there exists a set of k prototiles that admits exactly r distinct tilings. All the tilings obtained are periodic.     

Comparative studies of static dipole polarizabilities of ionic crystals on the basis of different self-consistent potentials

The static dipole polarizabilities, ^D , have been studied for the ions O^2–, F^–, Na⁺, Mg²⁺, Cl^–, K⁺ and Ca²⁺ in the crystals NaF, KF, NaCl, KC1, MgO and CaO. The starting zero-order wave functions have been generated using various exchange- and exchange-correlation potentials in order to study the effect of these potentials on ^D . The direct contribution to the dipole polarizability, _o ^D , has been determined by the uncoupled Hartree-Fock method. Self-consistency effects have been included by the geometric approximation. The crystal potential is incorporated using the Watson sphere model. Good agreement between theoretical and experimental results are found for those self-consistent potentials which exclude self-interaction.     

Automated carbohydrate synthesis to drive chemical glycomics

This feature article describes the development of the first automated solid-phase oligosaccharide synthesizer. A series of chemical challenges had to be addressed to accomplish this breakthrough and provide rapid access to oligosaccharides of biological significance. Accelerated synthesis of glycoconjugates promises to greatly impact the emerging field of glycobiology. Chemical glycomics uses synthetic carbohydrates and analogs to study their role in recognition, signal transduction pathways and other events of fundamental biomedical significance and shapes up to become the next major wave in biomedical research. The automated synthesis of a novel malaria vaccine candidate is discussed to illustrate the medical potential of chemical glycomics.     

Dimethylzinc-mediated addition of alkenylzirconocenes to alpha-keto and alpha-imino esters

[reaction: see text] Hydrozirconation of alkynes followed by in situ transmetalation to dimethylzinc and 1,2-addition to activated ketones and N-diphenylphosphinoylimines leads to tertiary allylic alcohols and amines in high overall yield. With 8-phenylmenthol as the chiral auxiliary, si-face attack proceeds in good to excellent diastereoselectivities.