Commutative Schur Rings of Maximal Dimension

A commutative Schur ring over a finite group G has dimension at most s _G  = d ₁ + … +d _r , where the d _i are the degrees of the irreducible characters of G. We find families of groups that have S-rings that realize this bound, including the groups SL(2, 2 ⁿ ), metacyclic groups, extraspecial groups, and groups all of whose character degrees are 1 or a fixed prime. We also give families of groups that do not realize this bound. We show that the class of groups that have S-rings that realize this bound is invariant under taking quotients. We also show how such S-rings determine a random walk on the group and how the generating function for such a random walk can be calculated using the group determinant.     

An extragradient method for relaxed cocoercive variational inequality and equilibrium problems

The purpose of this paper is to investigate the problem of finding the common element of the set of common fixed points of a countable family of nonexpansivemappings, the set of an equilibrium problem and the set of solutions of the variational inequality problem for a relaxed cocoercive and Lipschitz continuous mapping in Hilbert spaces. Then, we show that the sequence converges strongly to a common element of the above three sets under some parameter controlling conditions, which are connected with Yao, Liou, Yao, Takahashi and many others.     

Platinum-catalysed allylic alkylation: reactivity, enantioselectivity, and regioselectivity

The use of platinum complexes as catalysts for allylic substitution has been studied. A variety of different complexes catalyse the reaction, and several substrates have been tested. In the alkylation of mono(alkyl)-substituted allylic acetates, regioselectivity is highly dependent on ligand choice. By using tricyclohexylphosphine as the ligand, almost complete formation of branched products is observed. The development of a highly enantioselective (ca. 80-90% ee) reaction that makes use of chiral diphenylphosphinooxazoline ligands (abbreviated as (S)-PN) is also described. The enantioselectivity is highly dependent on the ratio of ligand to platinum (when the ratio ligand/Pt is greater than 1:1, the ee drops off dramatically). This is in contrast to palladium and is interpreted in terms of differing coordination chemistry for the two metals ((S)-PN is hemilabile when complexed to platinum) and should be of significance to future systems that utilise heterobidentate ligands. The crystal structures of two isoelectronic platinum and palladium complexes [[(S)-PN]MCl2] are also described.     

Action of braid groups on determinantal ideals, compact spaces and a stratification of Teichmüller space

We show that for n≥3 there is an action of the braid group B _n on the determinantal ideals of a certain n×n symmetric matrix with algebraically independent entries off the diagonal and 2s on the diagonal. We show how this action gives rise to an action of B _n on certain compact subspaces of some Euclidean spaces of dimension . These subspaces are real semi-algebraic varieties and include spheres of dimension -1 on which the kernel of the action of B _n is the centre of B _n . We investigate the action of B _n on these subspaces. We also show how a finite number of disjoint copies of the Teichmüller space for the n-punctured disc is naturally a subset of this ℝ and how this cover (in the broad sense) of Teichmüller space is a union of non-trivial B _n -invariant subspaces. The action of B _n on this cover of Teichmüller space is via polynomial automorphisms. For the case n=3 we show how to define modular forms on the 3-dimensional Teichmüller space relative to the action of B ₃. Oblatum 29-V-2000 & 7-XI-2000?Published online: 5 March 2001     

Spatial disorder in a smectic A mesophase

Starting with the equations of motion for a stiff chain, a projection operator approach is utilized to develop diffusional equations for the dynamics of the end-to-end distance. The diffusion equation resulting has a spatial-dependent diffusion coefficient calculable from equilibrium properties of the chain, and a frequency-dependent part which requires dynamical information. The analysis is applied, in so far as the spatial dependence of D is determined, for three and four bond chains. A critique of this procedure is provided.     

Modulation of cartilage differentiation by melanoma inhibiting activity/cartilage-derived retinoic acid-sensitive protein (MIA/CD-RAP)

Melanoma inhibiting activity/cartilage-derived retinoic acid-sensitive protein (MIA/CD-RAP) is a small soluble protein secreted from malignant melanoma cells and from chondrocytes. Recently, we revealed that MIA/CD-RAP can modulate bone morphogenetic protein (BMP)2-induced osteogenic differentiation into a chondrogenic direction. In the current study we aimed to find the molecular details of this MIA/CD-RAP function. Direct influence of MIA on BMP2 by protein-protein-interaction or modulating SMAD signaling was ruled out experimentally. Instead, we revealed inhibition of ERK signaling by MIA/CD-RAP. This inhibition is regulated via binding of MIA/CD-RAP to integrin α5 and abolishing its activity. Active ERK signaling is known to block chondrogenic differentiation and we revealed induction of aggrecan expression in chondrocytes by treatment with MIA/CD-RAP or PD098059, an ERK inhibitor. In in vivo models we could support the role of MIA/CD-RAP in influencing osteogenic differentiation negatively. Further, MIA/CD-RAP-deficient mice revealed an enhanced calcified cartilage layer of the articular cartilage of the knee joint and disordered arrangement of chondrocytes. Taken together, our data indicate that MIA/CD-RAP stabilizes cartilage differentiation and inhibits differentiation into bone potentially by regulating signaling processes during differentiation.     

Fusions of character tables III: Fusions of cyclic groups and a generalisation of a condition of Camina

In this paper we characterise finite groups whose character tables fuse from the character table of a cyclic group. We also show a connection with a generalisation of the Camina pair condition introduced by Camina in [Ca].     

Enantioselective total synthesis of (-)-dactylolide

[reaction: see text] The enantioselective total synthesis of (-)-dactylolide is reported. The absolute stereochemistry of the tetrahydropyran was established by catalytic asymmetric Jacobsen hetero-Diels-Alder reaction. The remote C19 stereocenter was introduced by a sequence of chelation-controlled Grignard addition and Ireland-Claisen rearrangement.