We would like to obtain the transmutation operator V, associated with the self-adjoint operators −d2/(w(x) dx2) and (−d2/(w(x) dx2)) + h(x), where w(x) ≎ xa as x → 0. We shall show that V = 1 + K, where K is a lower triangular Volterra operator.

A class of nonlinear Hill's equations on ℝ is considered, where the nonlinearity is concentrated on a compact interval [−N, N]. For values of the parameter λ not in the spectrum of the linearised equation (which is purely continuous) an equivalent nonlinear Sturm–Liouville problem on [−N, N] with parameter-dependent boundary conditions at x = ± N is given. Extending this problem to all real values of the parameter in a suitable way makes it possible to prove the existence of unbounded solution components for both the extended Sturm–Liouville problem and the original problem. The complicated structure of the extended problem results in new phenomena. For example, the number of zeros of different functions in the same solution component may be different.

Forms of the colour algebra introduced by Domokos and Kövesi-Domokos are studied by relating them to the well-known Cayley–Dickson algebras. Automorphisms groups and derivation algebras of these algebras are also determined.

We treat several classes of Riemannian manifolds whose shape operators of geodesic spheres or Jacobi operators share some properties with the ones on symmetric spaces.

We consider a dissipative reaction–diffusion equation on a thin L-shaped domain (with the thinness measured by a parameter ε); we determine the limit equation for ε = 0 and prove the upper semicontinuity of the global attractors at ε = 0. We also state a lower semicontinuity result. When the limit equation is one-dimensional, we prove convergence of any orbit to a singleton.

Group actions on compact surfaces have received considerable attention during the past century. The surface has often carried an analytic structure and been considered a Riemann surface or, equivalently, a complex algebraic curve.

The scalar nonlinear convection-diffusion equation

is considered, for given initial data and zero Dirichlet boundary conditions, in a smooth bounded domain Ω⊂ℝn. The homogeneous viscous Burgers' equation in one dimension is well-known to possess a unique, exponentially attracting equilibrium. These properties are shown to be preserved in the generalisation considered. Furthermore, the equilibrium is shown to be bounded in the maximum norm independently of the function a. The main methods used are maximum principles, and a variational method due to Stampacchia.

Transition probabilities are calculated which make the construction of diffusions on finitely ramified fractals straightforward. In contrast to former approaches using Brouwer's Fixed Point Theorem, we consider an approximation procedure based on the iteration of a nonlinear map L. Physically, this is done by ‘coarse-graining-renormalisation of finite electric resistor networks’. Mathematically, it is a convergence problem for quotients of Dirichlet forms on finite graphs. These graphs approximate finitely ramified fractals. The basic tool is a contraction theorem for the renormalisation map L which allows the use of known results about nested fractals for non-nested (p.c.f. self-similar) ones. In general, the above contraction is not strict because several linear independent fixed points occur.

A construction is given for a trace function on the semigroup algebra of a certain type of E-unitary inverse semigroup over any subfield of the complex field that is closed under complex conjugation. In particular, the method applies to the semigroup algebras of free inverse semigroups of arbitrary rank.

This paper shows how to construct Galois field extensions of Hilbertian fields with a given group out of some subclass (called ‘semiabelian groups’ by Matzat [2]) of all soluble groups as Galois group. This is done in a fairly explicit way by constructing polynomials whose Galois groups are universal in the sense that every group in the above subclass is obtained as a quotient of some of them.

We derive asymptotic formulae for the distribution functions of the real parts of the eigenvalues of an oblique derivative problem involving an indefinite weight function.

The objective is to derive a variation of constants formula for systems of functional differential equations (or delay differential equations) coupled with functional equations (or difference equations). The difficulties arise because of the constraints imposed by the functional equations.

A correspondence of a semigroup S is any subsemigroup of S × S, and the set of all correspondences of S, with the operations of composition and involution and the relation of set-theoretic inclusion, forms the bundle of correspondences of S, denoted by (S). For semigroups S and T, any isomorphism of (S) onto (T) is called a -isomorphism of S upon T. Similar notion can be introduced for other types of algebras and in the general frame of category theory. The principal goal of this paper is to study -isomorphisms of completely regular semigroups (that is, unions of groups) and of one other interesting class of semigroups.

Brian Hartley asked me whether a free (nilpotent of class 2 and exponent p2)-group of countable rank has a faithful linear representation of finite degree, p here being a prime of course. The answer is yes. The point is that this then yields via work of F. Leinen and M. J. Tomkinson, see [3,3.6] an image of a linear p-group, which is not even finitary linear. The question of which relatively free groups have faithful linear representations dates back at least to work of W. Magnus in the 1930's, see [4, pp. 33, 34 and the final comment on p. 40] for a discussion of this. Our construction, which works more generally, is a further contribution. We write ℜc for the variety of nilpotent groups of class at most c and (ℭ9 for the variety of groups of exponent dividing q.

Let M be a smooth m-dimensional submanifold in (m + d)-dimensional Euclidean space ℝm+d For x ∊ M and a non-zero vector X in TXM, we define the (d + l)-dimensional affine subspace E(x, X) ofℝm+d by

In this paper we show that each quasiperiodic standing wave solution of the real Ginzburg–Landau equation which is on the global branch emanating from the Eckhaus unstable periodic orbit is itself unstable. A rigorous proof of the instability is given by showing that the linearised operator about such a solution has spectrum which contains an interval along the unstable axis of the spectral plane. The proof employs some geometric and topological methods arising from a dynamical systems approach to the analysis of the eigenvalue problem for the linearised operator.

Presentations of Coxeter type are defined for semigroups. Minimal right ideals of a semigroup defined by such a presentation are proved to be isomorphic to the group with the same presentation. A necessary and sufficient condition for these semigroups to be finite is found. The structure of semigroups defined by Coxeter-type presentations for the symmetric and alternating groups is examined in detail.

We investigate the Cauchy problem for a hydrodynamic model for semiconductors. An existence theorem of global weak solutions with large initial data is obtained by using the fractional step Lax—Friedrichs scheme and Godounov scheme.

Let R be a commutative ring (with non-zero identity) and let M be an R-module.