Irregular abelian semigroups with weakly amenable semigroup algebra

In this paper we give counterexamples for the open problem, posed by Blackmore (Semigroup Forum 55:359–377, 1987) of whether weak amenability of a semigroup algebra ℓ ¹(S) implies complete regularity of the semigroup S. We present a neat set of conditions on a commutative semigroup (involving concepts well known to those working with semigroups, e.g. the counterexamples are nil and 0-cancellative) which ensure that S is irregular (in fact, has no nontrivial regular subsemigroup), but ℓ ¹(S) is weakly amenable. Examples are then given.     

The dimer problem for narrow rectangular arrays: A unified method of solution,and some extensions

In this paper we consider the dimer problem forM×N rectangular arrays, whereM andN are positive integers,M being small. A unified method for solving such problems is given, and is applied to the casesM=2 (the solution of which is already known (see [1, 4]) andM=3 which, it seems, has not previously been solved. The method is also applicable to a wider class of problems, and some examples of such applications are given. In theory it is always possible to obtain a closed solution to these problems in the form of rational generating functions. In practice this is feasible only for very small values ofM, but the methods described will enable numerical results for larger values ofM to be found by means of a computer program.     

A comparison of analytic series methods for Laplacian free boundary problems

Solutions to Laplace's equation are required for a wide range of problems. Arguably, the most difficult class of problems involves a “free” boundary, where the location of one (or more) of the boundaries is initially unknown. Analytical solutions for these problems were restricted to regular boundary geometries. However, recently the classical series method has been modified, to cater for arbitrary boundary geometries, using least squares methods. For free boundary problems, solutions can be obtained by solving a sequence of known boundary problems—at each iteration, the series coefficients can be estimated. Efficient calculation of the series coefficients becomes very important, particularly when the number of iterations is relatively high. In this paper, three methods for estimating the series coefficients will be described, in the context of a free boundary problem. The computational cost of each method will be analysed and compared, and the most appropriate method for this class of problem is indicated.     

Molecular and atomic oxygen adsorption on the kinked Pt(321) surface

Oxygen adsorption and desorption were characterized on the kinked Pt(321) surface using high resolution electron energy loss spectroscopy, thermal desorption spectroscopy and Auger electron spectroscopy. Some dissociation of molecular oxygen occurs even at 100 K on the (321) surface indicating that the activation barrier for dissociation is smaller on the Pt(321) surface than on the Pt(111) surface. Molecular oxygen can be adsorbed at 100 K but only in the presence of some adsorbed atomic oxygen. The dominance of the v(OO) molecular oxygen stretching mode in the 810 to 880 cm^?1 range indicates that the molecular oxygen adsorbs as a peroxo-like species with the OO axis parallel or nearly parallel to the surface, as observed previously on the Pt(111) surface [Gland et al., Surface Sci. 95 (1980) 587]. The existence of at least two types of peroxo-like molecular oxygen is suggested by both the unusual breadth of the v(OO) stretching mode and breadth of the molecular oxygen desorption peak. Atomic oxygen is adsorbed more strongly on the rough step sites than on the smooth (111) terraces, as indicated by the increased thermal stability of atomic oxygen adsorbed along the rough step sites. The two forms of adsorbed atomic oxygen can be easily distinguished by vibrational spectroscopy since oxygen adsorbed along the rough step sites causes a v(PtO) stretching mode at 560 cm^?1, while the v(PtO) stretching mode for atomic oxygen adsorbed on the (111) terraces appears at 490 cm^?1, a value typical of the (111) surface. Two desorption peaks are observed during atomic oxygen recombination and desorption from the Pt(321) surface. These desorption peaks do not correlate with the presence of the two types of adsorbed atomic oxygen. Rather, the first order low temperature peak is a result of the fact that about three times more atomic oxygen can be adsorbed on the Pt(321) surface than on the Pt(111) surface (where only a second order peak is observed). The heat of desorption for atomic oxygen decreases from about 290kJ/mol (70 kcal/mol) to about 196 kJ/mol (47 kcal/mol) with increasing coverage. Preliminary results concerning adsorption of molecular oxygen from the gas phase in an excited state are also briefly discussed.     

Foraging society organization: A simple model of a complex transition

The evolutionary development of the hominids that culminated in the appearance of Homo sapiens included the subdivision of the species into societies on the basis of culturally, instead of biologically, constructed differentiation. It is argued that this change must have occurred after the mental ability to formulate and culturally express conceptual structures of extended relationships had been been biologically introduced, and that intergroup competition within a species provided the selective impetus for this more complex form of organization. The combination of conceptual structures for organization at a more extensive scale and the effects of intergroup competition would lead to a restructuring of the whole species into society like groups.     

Extending an operator from a Hilbert space to a larger Hilbert space,so as to reduce its spectrum

In our earlier paper [1] we showed that given any elementx of a commutative unital Banach algebraA, there is an extensionA′ ofA such that the spectrum ofx inA′ is precisely the essential spectrum ofx inA. In [2], we showed further that ifT is a continuous linear operator on a Banach spaceX, then there is an extensionY ofX such thatT extends continuously to an operatorT ⁻ onY, and the spectrum ofT ⁻ is precisely the approximate point spectrum ofT. In this paper we take the second of these results, and show further that ifX is a Hilbert space then we can ensure thatY is also a Hilbert space; so any operatorT on a Hilbert spaceX is the restriction to one copy ofX of an operatorT ⁻ onX ⊕X, whose spectrum is precisely the approximate point spectrum ofT. This result is “best possible” in the sense that if isany extension to a larger Banach space of an operatorT, it is a standard exercise that the approximate point spectrum ofT is contained in the spectrum of .