In a forthcoming paper, N. M. Khan gives a condition for a variety of commutative semigroups V to be saturated in the sense of Howie and Isbell (1967) (i.e. epis are onto for each S ∈ V). We show the necessity of the condition by constructing a non-saturated semigroup which is a member of every commutative variety not satisfying Khan's condition. This determination of the saturated varieties of commutative semigroups enables us then to prove that these varieties form a sublattice of the lattice of varieties of all commutative semigroups.

A subalgebra M of a Lie algebra L is called modular in L if M is a modular element in the lattice of the subalgebras of L. Our aim is to study the finite-dimensional Lie algebras all of whose maximal subalgebras are modular. We characterize these algebras over any field of characteristic zero.

Characteristic initial value problems associated with hyperbolic equations of the form uxy + g(x, y)u = 0 are considered for (x, y)∈ℝ+× ℝ+. New criteria for the existence of a nodal line asymptotic to the axes are established, as are criteria for the existence of a zero beyond such a nodal line. Some numerical solutions are presented in graphical form and discussed relative to what is known about oscillation properties of such problems.

Small cancellation theory has been extended to symmetrized subsets of free products, amalgamated free products and Higman-Neumann-Neumann (H.N.N.) extensions. We though that it was possible to obtain results on decision problems if we could define small cancellation conditions for finite subsets.

Sacerdote and Schupp (1974) defined the small cancellation condition C'(l/6) for symmetrized subsets of an H.N.N. extension. We define this condition for finite subsets, with the following properties:

For each finite subset X, there is a symmetrized subset X1 with the same normal closure and, if X1 satisfies C'(l/6), then X satisfies C'(l/6).

For some H.N.N. extensions, we can decide whether any finite subset satisfies C'(l/6), and, in this case, we can solve the word problem for the corresponding quotient.

In this paper entire solutions of differential equations with polynomial coefficients are considered and bounds on the maximum modulus and the index are obtained, when the equation is of second order and the coefficients are of second degree.

We prove the absence of singular continuous spectra for Schrödinger operators − Δ + V with long-range potential V such that V and (1 + r)1+ε(∂/∂r) V is (− Δ)-compact by using a modified “Mourre type” estimate and by Kato-Lavine's H-smoothness theory.

A spin factor is a JW-factor of type I2. It is shown that certain automorphisms of finite dimensional spin factors extend to extremal positive linear maps on complex matrix algebras which are not decomposable, and hence, do not preserve extreme rays of the positive cone.

A notable achievement in the algebraic theory of semigroups has been the discovery by Nambooripad of the natural order on a regular semigroup. He has shown that this order is compatible with multiplication if and only if the semigroup is locally inverse, in the sense that every local submonoid is an inverse semigroup. In this paper we determine precisely when the natural order is compatible on the right (respectively left) with multiplication; this is so if and only if every local submonoid is ℒ-unipotent (respectively ℛ-unipotent).

Linearized local disturbances on a vortex sheet are known to develop singularities after a finite term in some cases but not in others. A simple test for the appearance of such singularities is given in terms of the Fourier transform of the initial disturbance. Such singularities are a consequence of the artificiality of the vortex sheet model and should not be regarded as physically meaningful.

A standard method of constructing Steiner triple systems of order 19 from the Steiner triple system of order 9 gives rise to 212 different such systems. It is shown that there are just three isomorphism classes amongst these systems. Representatives of each isomorphism class are described and the orders of their automorphism groups are determined.

The first non-arbitrary coefficient, α12, of the Buckmaster expansions is evaluated in the context of the extended Goldstein-Stewartson theory. Leading terms of the next order contributions to the skin friction and heat transfer coefficients are also obtained.

The Hilbert boundary value problem for a first order nonlinear elliptic system in the plane with linear boundary conditions of nonnegative index is (under suitable side conditions uniquely) solved by use of the Newton imbedding method. This constructive method is based on an a priori estimate which arises from an integral representation formula for C1-functions first developed by Haack and Wendland. The approximation procedure yields an error estimate too.

A theorem of Rudin states that if B is the open unit ball in ℂN, N > 1, if 0<ρ < 1, if is the family of all complex lines in ℂN at a distance ρ from the origin and if f ∈ C(∂B) is such that for every Λ∈ the function f|Λ∂B has a continuous extension to Λ ∩ B which is holomorphic in Λ ∩ B, then f has a continuous extension to B which is holomorphic in B. In this paper we show that when N = 2, the theorem still holds if is replaced by a considerably smaller family.

We give a new perturbation theorem for symmetric differential expressions (relatively bounded perturbations, with relative bound 1) and prove with this theorem a new limit-point criterion generalizing earlier results of Schultze. We also obtain some new results in the fourth-order case.

In this paper, the existence of global smooth solutions and the formation of singularities of solutions for strictly hyperbolic systems with general eigenvalues are discussed for the Cauchy problem with essentially periodic small initial data or nonperiodic initial data. A result of Klainerman and Majda is thus extended to the general case.

Duchon (1978) considered interpolation in ℝn by “Dm-splines”, which are interpolating functions having, in a sense, minimum energy. The purpose of this paper is to consider the analogous interpolation at the lattice of points in ℝn with integer co-ordinates, generalising aspects of Schoenberg's (1973) theory of cardinal spline interpolation. Following Schoenberg, we prove that higher order “basic” splines can be written as convolutions of lower order ones, using a new notion of convolution due to Jones (1982).

The exterior Dirichlet and Neumann problems can be treated very satisfactorily by using a fundamental solution which is modified by adding radiating spherical wave functions. It has been shown [3], that the coefficients of these added terms can be chosen to ensure that the associated boundary integral equation formulation of the problem was uniquely solvable and, in addition, that the modified Green's function was a least squares best approximation to the exact Green's function for the problem. Here we show that the coefficients can be chosen to ensure not only unique solvability but also minimization of the norm of the modified integral operator. This leads to a constructive method of solution. The theory is illustrated when the boundary is a sphere and when it is a perturbation of a sphere.