This papergives examples to show that a polar operator is not necessarily AC andan AC operator is not necessarilypolar.

We consider the classic shortest queue problem in the heavy traffic limit. We assume that the second server works slowly and that the service rate of the first server is nearly equal to the arrival rate. Solving for the (asymptotic) joint steady state queue length distribution involves analyzing a backward parabolic partial differential equation, together with appropriate side conditions. We explicitly solve this problem. We thus obtain a two-dimensional approximation for the steady state queue length probabilities.

We give thefirst examples of contact metric spaces which are weakly locally$\varphi$-symmetric (that is, $\nabla R=0$ on horizontal vectors), butnot strongly (that is, not all reflections with respect to thecharacteristic lines are local isometries). These examples arethree-dimensional non-unimodular Lie groups with a left-invariantcontact metric structure. We exhibit additional symmetries on thesespaces.

1991 Mathematics Subject Classification: 53B20,53C15, 53C25.

Boundary value problems for linear transport equations sometimes require the explicit construction of solutions when boundary conditions are prescribed only on parts of the boundary and thus necessitate the construction of half-range expansions. In contrast to the standard eigenfunction expansion, a half-range expansion of a given function must be reconstructed just over half the domain, using just half of the eigenfunctions. The difficulty of such expansions arises because the eigenfunctions are not orthogonal, though they are complete, over half the domain, and there is no obvious method of obtaining the expansion coefficients. Here we use complex variable techniques to find explicit formulas for the coefficients of half-range expansions for regular, negative definite Sturm–Liouville operators. We prove that the half-range expansion formula is unique, and find the corresponding half-range Green functions.

LetM be a real hypersurface of the complex projective spacePn(C). The Ricci tensor Sof M is recurrent if there exists a 1-form $\alpha$such that $\nablaS=S\otimes\alpha;$. In this paper we show that thereare no real hypersurfaces with recurrent Ricci tensor ofPn(C) under the condition that$\xi$ is a principal curvature vector.

1991 MathematicsSubject Classification 53C40(53C25).

Denoteby $(R,\cdot)$ the multiplicative semigroup of an associative algebra$R$ over an infinite field, and let $(R,\circ)$ represent $R$ whenviewed as a semigroup via the circle operation $x\circy=x+y+xy$. In thispaper we characterize the existence of an identity in these semigroupsin terms of the Lie structure of $R$. Namely, we prove that thefollowing conditions on $R$ are equivalent: the semigroup $(R,\circ)$satisfies an identity; the semigroup $(R,\cdot)$ satisfies a reducedidentity; and, the associated Lie algebra of $R$ satisfies the Engelcondition. When $R$ is finitely generated these conditions are eachequivalent to $R$ being upper Lienilpotent.

1991 Mathematics Subject Classification 16R40, 20M07, 20M25

Although the European Journal of Applied Mathematics is a journal that has no official affiliation with any particular academic organisation, its aims overlap those of many institutes, centres, societies and, especially, conferences. Indeed, in our first ten years we have published three special issues related to conferences on topics that have been timely and important for the development of new mathematical ideas relevant to practical problems. Thus, when ICIAM, the world's largest industrial-and-applied mathematics conference, took place in Edinburgh, Scotland, in July 1999, we were only too happy to be associated with a gathering of so many people who all worked in the style of EJAM, and the organisers kindly allowed us to co-sponsor the conference reception. A highlight of this evening was Professor Graham Wilks' dramatic portrayal of Sir Isaac Newton, complete with apple. Accompanied by his wife, Graham sang the following verse.

It is known that the class of mils generalizes that of pramarts and martingales in the limit. Also every Banach space-valued mil (Xn) with lim infnE(‖Xn‖)<∞ can be written in a unique form: $X_n=M_n+P_n(n\in\rm{N})$, where $(M_n)$ is a uniformly integrable martingale and $(P_n)$ converges to zero a.s. in norm. We shall show that this result still holds for a class which essentially generalizes that of mils. Another class of Banach space-valued martingale-like sequences, still containing all pramarts is defined and shown to have the decomposition above under the following much weaker condition: $\rm{lim inf}_{r\inT}E(\VertX_{\tau}\Vert)<\infty$, where T denotes the set of all bounded stopping times.

1991 Mathematics Subject Classification. 60G48, 60B11.

In thispaper we give a characterization of the central elements of the algebraA** for a class of weakly sequentially completeBanach algebras A and present a detailed study of severalquestions related to Arens regularity of the algebraA.

1991 Mathematics Subject Classification Primary: 43A10, 46B20, 46J99.

Let A be a UHF-algebra andK the C*-algebra of all compact operators on a countablyinfinite-dimensional Hilbert space. In this note we shall find allprojections p in A with $pAp\congA$ and, using theseprojections, we shall determine the group of automorphisms of$K_0(A\otimes$K) induced by those of $A\otimes$K insome cases.

1991 Mathematics Subject Classification 46L40.

In thispaper G denotes a non-identity finite soluble group. IfA is an irreducible G-module,EndGA is a division ring by Schur's Lemma,actually a field, since G finite forces A to befinite. Moreover A is a vector space overEndGA with respect to$\alphaa:=\alpha(a),\alpha\in\rm{End}_GA,a\inA$. We let$\varphi_G(A):=\rm{dim}_{\rm{End}_GA}A$. Any chief factor of Gis an irreducible G-module via the conjugation action, and itis central precisely when it is a trivial G-module. By arefined version of the Theorem of Jordan-Hölder [1, p. 33]the number $\delta_G(A)$ of complemented chief factors of G,which are G-isomorphic to a given A, is constant forany chief series of G. We say that A iscomplemented, as aG-module, if$\delta_G(A)>0$.

We present a general procedure for recursively improving the invariance of a numerical integrator under a symmetry group. If h is a symmetry, we construct the adjoint method h−1h. In each time step we apply either the original method or the adjoint method, according to a prescription based on the Thue–Morse sequence. The outcome is a solution sequence which displays progressively smaller symmetry errors, to any desired order in the time-step. The method can also be used to force the solution to stay close to a desired submanifold of phase space, while retaining structural properties of the original method.

Thenonabelian tensor square G[otimes] G of a group G is generated by the symbolsg[otimes] h, g,h ∈ G, subject to the relations$$gg\prime\otimesh=(^gg\prime\otimes^gh)(g\otimesh) andg\otimeshh\prime-(g\otimesh)(^hg\otimes^hh\prime),$$ for all $g,g\prime,h,h\prime\in G< / f>, where $^gg\prime=gg\primeg^{−1}$. The nonabelian tensor squareis a special case of the nonabelian tensor product which has its origins inhomotopy theory. It was introduced by R. Brown and J.-L. Loday in [4]and [5], extending ideas of J.H.C. Whitehead in [10]. The topicof this paper is the classification of 2-generator 2-groups of class two up toisomorphism and the determination of nonabelian tensor squares for thesegroups.

The bifurcation of symmetric superconducting solutions from the normal solution is considered for the one-dimensional Ginzburg–Landau equations by the methods of formal asymptotics. The behaviour of the bifurcating branch depends upon the parameters d, the size of the superconducting slab, and κ, the Ginzburg–Landau parameter. It was found numerically by Aftalion & Troy [1] that there are three distinct regions of the (κ, d) plane, labelled S1, S2 and S3, in which there are at most one, two and three symmetric solutions of the Ginzburg–Landau system, respectively. The curve in the (κ, d) plane across which the bifurcation switches from being subcritical to supercritical is identified, which is the boundary between S2 and S1∪S3, and the bifurcation diagram is analysed in its vicinity. The triple point, corresponding to the point at which S1, S2 and S3 meet, is determined, and the bifurcation diagram and the boundaries of S1, S2 and S3 are analysed in its vicinity. The results provide formal evidence for the resolution of some of the conjectures of Aftalion & Troy [1].

The‘canonical embedding approach’ was introduced by the secondauthor and, subsequently, it has been applied several times to prove theembeddability of certain regular extensions by groups into semidirectproducts by groups. In the present paper this technique is generalizedso that it is suitable to handle regular extensions by inversesemigroups. As an application, B. Billhardt's embedding theorem onregular extensions of semilattices by inverse semigroups isreproved.