On outer automorphism groups of coxeter groups

Summary It is shown that the outer automorphism group of a Coxeter groupW of finite rank is finite if the Coxeter graph contains no infinite bonds. A key step in the proof is to show that if the group is irreducible andΠ ₁ andΠ ₂ any two bases of the root system ofW, thenΠ ₂ = ±ωΠ ₁ for some ω εW. The proof of this latter fact employs some properties of the dominance order on the root system introduced by Brink and Howlett. This article was processed by the author using the Springer-Verlag TEX PJour1g macro package 1991.     

THE MOLECULAR WEIGHT OF HAEMATOPORPHYRIN DERIVATIVE, ITS GEL COLUMN FRACTIONS and SOME OF ITS COMPONENTS IN AQUEOUS SOLUTION

Abstract— The average molecular weights of haematoporphyrin derivative (HPD), the fractions of HPD that can he obtained by gel chromatography and of purified haematoporphyrin and protoporphyrin in aqueous solution have been determined by ultracentrifugation. The results show that HPD contains polymeric material with the excluded fraction from the gel column (HPD aggregate) having an average molecular weight of greater than 20000. The two remaining gel column fractions of HPD have lower molecular weights and their similarity indicates that these fractions do not separate because of molecular weight differences. Purified haematoporphyrin has a comparatively low molecular weight in aqueous solution but the data is not capable of discriminating between monomer, dimer or slightly higher oligomer. In contrast, protoporphyrin sediments to the bottom of the centrifuge tube under the conditions of sedimentation equilibrium indicating that it has an average molecular weight considerably greater than that of HPD aggregate.     

Ground vs. excited state electron transfer: adsorbed monolayers and trimers in solution

Transient emission spectroscopy has been used to probe the rate of photoinduced electron transfer between metal centres within a novel trimeric complex [[Os(bpy)2(bpe)2][Os(bpy)2Cl]2]4+, where bpy is 2,2'-bipyridyl and bpe is trans-1,2-bis-(4-pyridyl)ethylene. Transient emission experiments on the trimer, and on [Os(bpy)2 (bpe)2]2+ in which the [Os(bpy)2 Cl]+ quenching moieties are absent, reveal that the rate of photoinduced electron transfer (PET) across the bpe bridge is 1.3 +/- 0.1 x 10(8) s(-1). Investigations into the driving forces for oxidation and reduction of the electronically excited state within the trimer indicate that quenching of the [Os(bpy)2 (bpe)2]2+ centre within the trimer involves electron transfer from the [bpe Os(bpy)2 Cl]+ centres to the electronically excited state with a driving force of -0.3 eV. Monolayers of the complex, [Os(bpy)2 bpe pyridine]2+, have been formed by spontaneous adsorption onto platinum microelectrodes and used to probe the dynamics of electron transfer across the trans-1,2-bis-(4-pyridyl)ethylene bridge in the ground state. These monolayers are stable and exhibit well defined voltammetric responses for the Os2+/3+ redox reaction. Cyclic voltammograms recorded at high scan rates can be accurately modelled according to a non-adiabatic electron transfer model based on the Marcus theory using a standard heterogeneous electron transfer rate constant, k(o), of 3.1 +/- 0.2 x 10(4) s(-1) and a reorganization energy of 0.4 +/- 0.1 eV. This rate constant is a factor of approximately two orders of magnitude smaller than that found for photoinduced electron transfer across the same bpe bridge for identical driving forces. This significant difference is interpreted in terms of both the nature of the orbitals involved in electrochemically and optically driven electron transfer, as well as the strength of electronic coupling between two molecular components as opposed to a molecular component and a metal electrode.     

A methodology for the constructive approximation of nonlinear operators defined on noncompact sets

We consider the constructive approximation of a non-linear operator that is known on a bounded but not necessarily compact set. Our main result can be regarded as an extension of the classical Stone-Weierstrass Theorem and also shows that the approximation is stable to small

disturbances.

This problem arises in the modelling of real dynamical systems where an input-output mapping is known only on some bounded subset of the input space. In such cases it is desirable to construct a model of the real system with a complete input-output map that preserves, in some approximate sense, the known mapping. The model is normally constructed from an algebra of elementary continuous functions.

We will assume that the input space is a separable Hilbert space. To solve the problem we introduce a special weak topology and show that uniform continuity of the given operator in the weak topology provides an alternative compactness condition that is sufficient to justify the desired approximation.     

Toward Best Approximation of Nonlinear Systems: A Case of Models with Memory

The problem of nonlinear dynamical system modeling, considered in this paper, is motivated by restrictions arising in real-world tasks. The restrictions are that first, a system input cannot be entirely observed for one trial. Second, the system model must be subjected to the causality principle. Third, the input is corrupted by noise so that no relationship between the reference input and noise is known. Fourth, the model should have some degrees of freedom so that the associated accuracy can be regulated by a variation of these freedom degrees. We propose and justify new procedures for the nonlinear system modeling that are initialized by these motivations. The models are nonlinear and given by so called r-degree operators that can be reduced to a matrix form presentation. To satisfy the restrictions above, the matrices have special structures that we call the lower p-band matrices. The degree r of the models is the required degree of freedom. The rigorous analysis of errors associated with the presented techniques is given. Numerical experiments with real data demonstrate the efficiency of the proposed approach.     

Steric trigger as a mechanism for CB1 cannabinoid receptor activation

To determine the moiety that behaves as the steric trigger to activate the CB(1) cannabinoid receptor, conformational properties of the nonclassical cannabinoid CP55244, one of the most potent CB(1) receptor agonists, were characterized by conformational analysis, rotational barrier calculations, and molecular dynamics (MD) simulations. It was shown from the present MD simulations that the torsion angles phi1 and phi4 of the C3 side chain showed the most dramatic change when compared with the ground-state receptor-bound conformation, indicating that rotation around these torsion angles is responsible for releasing the ligand strain energy. Multiple stages would be involved in the ligand conformational change. As a molecular mechanism for the ligand-induced CB(1) receptor conformational change, we propose that the C3 side chain serves as the steric trigger, while the ACD-ring moiety of CP55244 serves as the plug. Steric clash with helices within the binding pocket would induce microconformational adaptation within the protein. This mechanism would suggest that rotational flexibility in a ligand may be as important a determinant of agonist activity as the pharmacophoric elements that can be identified.