Frobenius manifolds and Frobenius algebra-valued integrable systems

The notion of integrability will often extend from systems with scalar-valued fields to systems with algebra-valued fields. In such extensions the properties of, and structures on, the algebra play a central role in ensuring integrability is preserved. In this paper, a new theory of Frobenius algebra-valued integrable systems is developed. This is achieved for systems derived from Frobenius manifolds by utilizing the theory of tensor products for such manifolds, as developed by Kaufmann (Int Math Res Not 19:929–952, 1996), Kontsevich and Manin (Inv Math 124: 313–339, 1996). By specializing this construction, using a fixed Frobenius algebra \({\mathcal {A}},\) one can arrive at such a theory. More generally, one can apply the same idea to construct an \({\mathcal {A}}\)-valued topological quantum field theory. The Hamiltonian properties of two classes of integrable evolution equations are then studied: dispersionless and dispersive evolution equations. Application of these ideas are discussed, and as an example, an \({\mathcal {A}}\)-valued modified Camassa–Holm equation is constructed.     

Synthetic methods for polyamine linkers and their application to combinatorial chemistry

Summary Polyamines and polyamine conjugates display a diverse range of important biological functions, ranging from antibiotics to immunosuppressants and glutamate receptor antagonists. For these reasons, polyamines provide an excellent template/scaffold for combinatorial chemistry. In this paper we present methods for the solid-phase immobilisation of polyamines for use in synthetic and combinatorial chemistry and describe how they have been employed in the preparation of a number of important polyamine conjugates and polyamine libraries. Thus, we have designed, synthesised and utilised a number of polyamine linkers for both solution and resin screening combinatorial application.     

Mod pq Galois representations and Serre's conjecture

We consider linear representations of the Galois groups of number fields in two different characteristics and examine conditions under which they arise simultaneously from a motive.     

Modelling self-ordering behaviour on Ag covered Pt vicinal surfaces

We use computer simulation to explore the formation process of a monolayer of Ag on a stepped Pt(111) substrate and the formation of 3D Pt nanostructures on an Ag covered (111) and (100) Pt substrate. We show that broken lines of Pt nanostructures are preferred at the step edges on the (111) substrate while continuous lines of Pt nanowires are preferred at the step edge on the (100) substrate. This different behaviour is due to the exposed front facet of the nanostructures running along the step, specifically for the (100) stepped substrate a nanowire grown on the step edge has a stable (111) exposed front facet, whereas a nanowire grown on the (111) substrate would have an unstable (100) front facet (depending on the direction of the step). For the Pt nanowires grown on the (100) substrate we show how arriving Pt dimers (and monomers) preferentially move up off the Ag substrate onto the nanowire's (111) facet where they undergo fast diffusion. We also show that these Pt dimers (and monomers) move up and down the nanowire's facet until a vacancy or defect is encountered.     

SIMS of organic layers with unknown matrix parameters: Locating the interface in dual beam argon gas cluster depth profiles

Four simple methods are evaluated to determine their accuracies for establishing the interface location in secondary ion mass spectrometry intensity depth profiles of organic layers where matrix effects have not been measured. Accurate location requires the separate measurement of each ion's matrix factor. This is often not possible, and so estimates using matrix-less methods are required. Six pure organic material interfaces are measured using many secondary ions to compare their locations from the four methods with those from full evaluation with matrix terms. For different secondary ions, matrix effects cause the apparent interface positions to vary over 20 nm. The shifts in the intensity profiles on going from a layer of P into a layer of Q are in the opposite direction to that for going from Q into P, so doubling layer thickness errors. The four methods are as follows: M1, use of the median interface position in the intensity profiles for the five lightest ions for 15 ≤ m/z ≤ 150; M2, extrapolation of the position for each ion to m/z = 0 for ions with m/z ≤ 150; M3, as M2 but for m/z ≤ 300; and M4, the extreme positions for all m/z ≤ 100. Comparison with the location using matrix terms shows their ranking, from best to worst, to be M4, M3, M1, and M2 with average errors of 10%, 12%, 14%, and 17%, respectively, of the profile interface full widths at half maximum. Use of pseudo-molecular ions is very much poorer, exceeding 50%, and should be avoided.     

In-situ detection of drugs-of-abuse on clothing using confocal Raman microscopy

This study describes the application of confocal Raman microscopy to the detection and identification of drugs-of-abuse in situ on undyed natural synthetic fibres, and coloured textile specimens. Raman spectra were obtained from drug particles trapped between the fibres of the specimens. Pure samples of cocaine hydrochloride and N-methyl-3,4-methylenedioxy-amphetamine HCl (MDMA-HCl) were used in this study. Raman spectra were collected from drug particles of an average size in the range 5-15 μm. Despite the presence of spectral bands arising from the natural and synthetic polymer and dyed textiles, the drugs could be identified by their characteristic Raman bands. If necessary, interfering bands could be successfully removed by spectral subtraction. Furthermore, Raman spectra were recorded from drug particles trapped between the fibres of highly fluorescent specimens. Interference from the fibres, including background fluorescence, was overcome by careful focusing of the confocal beam and the resulting spectra allow ready differentiation from interference from the fibres substrate bands. Spectra of several drugs-of-abuse on dyed and undyed clothing substrates were readily obtained within 3 min with little or no sample preparation and with no alteration of the evidential material.