The Ramifications of the Centres: Quantised Function Algebras at Roots of Unity

This paper continues the study of quantised function algebrasO[G] of a semisimple group G at an lth root of unity . Thesealgebras were introduced by De Concini and Lyubashenko in 1994,and studied further by De Concini and Procesi and by Gordon,amongst others. Our main purpose here is to increase understandingof the finite-dimensional factor algebras O[G](g), for g G.We determine the representation type and block structure ofthese factors, and (for many g) describe them up to isomorphism.A series of parallel results is obtained for the quantised Borelalgebras and . 2000 Mathematical Subject Classification: 16W35,17B37.     

Intramolecular carbolithiation reactions of chiral alpha-amino-organolithium species

Enantiomerically enriched alpha-amino-organolithium species, in which the lithium atom is attached to a stereogenic carbon centre, have been found to be chemically stable at room temperature in a solvent of very low polarity and undergo intramolecular carbolithiation onto an unactivated alkene. The configurational stability of the chiral organolithium species, bearing a variety of N-alkenyl substituents, was probed by studying the enantiomeric purity of the cyclization products. With N-but-3-enyl-2-lithiopyrrolidine, cyclization to the five-membered ring is more rapid than racemization and a high yield of the pyrrolizidine alkaloid (+)-pseudoheliotridane was obtained with no loss of optical purity. In contrast, with N-pent-4-enyl-2-lithiopyrrolidine, cyclization to the six-membered ring was found to occur with significant loss of optical purity. The cyclization to the six-membered ring was determined to occur with a half-life, t(1/2) approximately 90 min at 23 degrees C. The epimerization of this organolithium species in hexane/Et2O 4:1 was calculated to have a half-life, t(1/2) approximately 30 min at 23 degrees C. Enhanced levels of enantioselectivity for the formation of the indolizidine ring system were obtained using an alkene bearing a terminal phenylthio substituent. With N-[(3-phenylthio)-prop-2-enyl]-2-lithiopyrrolidine, cyclization to the four-membered ring occurs with poor enantioselectivity at low temperature in THF but is highly enantioselective at room temperature in a solvent of very low polarity.     

Synthesis of a giant 222 carbon graphite sheet

In this paper we present the synthesis and characterization of the so far largest polycyclic aromatic hydrocarbon (PAH), containing 222 carbon atoms or 37 separate benzene units. First a suitable three-dimensional oligophenylene precursor molecule is built up by a sequence of Diels-Alder and cyclotrimerization reactions and then planarized in the final step by oxidative cyclodehydrogenation to the corresponding hexagonal PAH. Structural proof is based on isotopically resolved MALDI-TOF mass spectra and electronic characteristics are studied by UV/Vis spectroscopy.     

Improved synthesis of (+/-)-4,12-dihydroxy[2.2]paracyclophane and its enantiomeric resolution by enzymatic methods: planar chiral (R)- and (S)-phanol

(+/-)-4,12-Dihydroxy[2.2]paracyclophane [(+/-)-PHANOL] is readily prepared from [2.2]paracyclophane by an improved synthetic protocol. Enzymatic kinetic resolution of its bis-acetate proceeds with good enantioselection. Separation, hydrolysis, and recrystallization provides both enantiomers of PHANOL in high enantiopurity.     

Fusion and planarization of a quinoidal porphyrin dimer

The crystal structure, near-infrared spectrum and electrochemistry of a quinoidal triply-linked porphyrin dimer are compared with those of its singly-linked precursor; fusing the two porphyrins planarizes the pi-system and reduces the optical HOMO-LUMO gap while increasing the gap between the first oxidation and reduction potentials.     

Symmetric inertial-confinement-fusion-capsule implosions in a double-z-pinch-driven hohlraum

An inertial-confinement-fusion (ICF) concept using two 60-MA Z pinches to drive a cylindrical hohlraum to 220 eV has been recently proposed. The first capsule implosions relevant to this concept have been performed at the same physical scale with a lower 20-MA current, yielding a 70+/-5 eV capsule drive. The capsule shell shape implies a polar radiation symmetry, the first high-accuracy measurement of this type in a pulsed-power-driven ICF configuration, within a factor of 1.6-4 of that required for scaling to ignition. The convergence ratio of 14-21 is to date the highest in any pulsed-power ICF system.     

A Semigroup Crossed Product Arising in Number Theory

Recently Bost and Connes have studied an interesting C*-algebraicHecke algebra arising in number theory. Here it is shown thatthis algebra can be realised as a semigroup crossed product,and be profitably studied using methods developed by the authorsfor analysing Toeplitz algebras. One main result is a characterisationof faithful representations of the Hecke algebra.     

Regular Subgraphs in Graphs and Rooted Graphs and Definability in Monadic Second - Order Logic

We investigate the definability in monadic ∑₁¹ and monadic Π₁¹ of the problems REG_k, of whether there is a regular subgraph of degree k in some given graph, and XREG_k, of whether, for a given rooted graph, there is a regular subgraph of degree k in which the root has degree k, and their restrictions to graphs in which every vertex has degree at most k, namely REG_k^k and XREG_k^k, respectively, for k ≥ 2 (all our graphs are undirected). Our motivation partly stems from the fact (which we prove here) that REG_k^k and XREG_k^k are logspace equivalent to CONN and REACH, respectively, for k ≥ 3, where CONN is the problem of whether a given graph is connected and REACH is the problem of whether a given graph has a path joining two given vertices. We use monadic first - order reductions, monadic ∑₁¹ games and a recent technique due to Fagin, Stockmeyer and Vardi to almost completely classify whether these problems are definable in monadic ∑₁¹ and monadic Π₁¹, and we compare the definability of these problems (in monadic ∑₁¹ and monadic Π₁¹ with their computational complexity (which varies from solvable using logspace to NP - complete).