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.     

Conversion of clavulanic acid into thiadeoxa nuclear analogues

1,4-Addition of sulphur nucleophiles to the diene ($\text{12}$) derived $\text{via}$ the pen-2-em ($\text{5}$) from clavulanic acid provides the thiadeoxa analogues ($\text{14}$–$\text{15}$). X-ray analysis of the ester ($\text{14}$) shows the thermodynamically stable isomers to have the same relative stereochemistry as clavulanic acid.     

Field ionization mass spectrometry—IV. Isotopically labeled heptanals

The field ionization (FI) mass spectra of n-heptanal and a series of deuterium labeled analogs have been studied, with the objectives of initiating systematic investigations of reaction mechanisms of FI produced ions and to permit comprison with those found for other ionization processes. It is now recognized that FI ions have: (a) lower average internal energies and (b) shorter residence times than similar ions generated by electron-impact (EI), and the possibility exists of H/D-randomization occuring in ions formed by desorption from the emitter, by unimolecular decomposition close to the emitter and by either ‘fast’ or ‘slow’ metastable decompositions. In this study only the peak shifts of normal ions could be utilized; accurate mass measurements of all major ions revealed elemental compositions similar to EI. A site-specific McLafferty rearrangement gave the base peak at m/e 44 ([C₂H₄O]^+.), although the apparently complementary ion at m/e 70 ([C₅H₁₀]^+.) arose in a less specific process. Ions at m/e 43 ([C₃H₇]⁺) and 71 ([C₅H₁₁]⁺ 80%; [C₄H₇O]⁺ 20%) were apparantly generated without significant H/D-scrambling. Of special interest was the observation of the rearrangement ion at m/e 86 ([C₅H₁₀O]^+.) caused by loss of C-2 and C-3 as C₂H₄, as found for EI. It is concluded that at least in this system, decomposing molecular ions formed: (a) in the gas phase extremely close to the emitter and/or (b) on the emitter surface, have lifetimes sufficiently short to preclude complete H/D randomization. The results also provide evidence for common fragmentation mechanisms for heptanal molecular ions at both the low end and the high end of the energy distribution.     

Isomerisation in the chemistry of platinacyclobutane complexes

The platinacyclobutane complexes PtCl₂L₂(C₃H₅Me)], L  pyridine, CD₃CN, or tetrahydrofuran, exist as mixtures of isomers containing P$\text{tCH}{}_2\text{CHMeC}$H₂ or P$\text{tCHMeCH}{}_2\text{CH}{}_2$ groups in rapid equilibrium. Decomposition occurs in some cases to give [PtCl₂L(CH₃CH₂CHCH₂)]. Stereospecific skeletal isomerisation also occurs in metallocyclobutanes containing the groups P$\text{tCHRCHRC}$H₂  P$\text{tCHRCH}{}_2\text{C}$HR, when R  aryl further decomposition gives ν-allylplatinum complexes.