The invariants of pencils of binary cubics were described in an earlier paper by the author. In this note, we give a complete set of covariants for such pencils, and show how each type of pencil may be denned by the identical vanishing of covariants.

We consider elliptic operators of the form , on L2(Rn), and establish conditions under which T is essentially self-adjoint on , and self-adjoint on H2m(Rn)∩D(q).

In this paper the adjoint operator is derived for a multi-point differential operator with a varying structure in a suitably chosen Hilbert space. The formal differential operator is given by different differential expressions in the adjoining intervals. This form of adjoint operator is used to characterize self-adjointness.

The class of M-nilpotent rings is defined as a generalisation of the class of T-nilpotent rings. Certain results for radicals of T-nilpotent rings are shown to hold also in this larger class of rings.

Under appropriate conditions on b and a function g with 2k +1 simple zeros, the equation

has a maximal compact invariant set Ab,g in C([−1,0]R), consisting of the zeros of g and the one-dimensional unstable manifolds of these zeros. For k =2, it is shown that there may be a saddle connection in the flow on Ab,g for some g. This implies that the zeros of g as elements of the flow on Ab,g cannot be given the natural order of the reals.

The weight functions w(x) for which the Riemann fractional integral operator Iα is bounded from the Lebesgue space Lp(wp) into Lq(wq), l/q = l/p −, have been characterized by Muckenhoupt and Wheeden. In this paper, we prove an inversion formula for Iα in the context of these weighted spaces and we also characterize the range of Iα as a subset of Lq(wq) Similar results are proved for other fractional integrals. These results may be viewed as weighted analogues of certain results of Stein and Zygmund, Herson and Heywood, Heywood, and Kober who considered the unweighted case, w(x) = l.

It is shown that the equation (p2y”)”–(p1y’)’+ p0y = 0 has exactly two linearly independent solutions on [0,∞) with finite Dirichlet integral when the coefficients are nonnegative and p2 satisfies a condition which includes all nondecreasing functions. An inequality for the Dirichlet form is derived and used to extend characterizations of the domains of certain self-adjoint operations associated with the differential expression to arbitrary symmetric boundary conditions at 0.

Liouville theorems are obtained for fourth order elliptic systems of the form Δ2U + B Δ U + AU = 0 and for fourth order nonlinear equations of the form Δ2u−q(x) Δu + p(x)f(u) = 0 as a consequence of two related subharmonic functions, the mean value property of subharmonic functions, and a basic Green identity.

The paper considers a linear non-homogeneous boundary value problem for a class of neutral type functional differential equations. A necessary and sufficient condition for the existence of a unique solution of that problem is obtained.

One possible generalisation of homological algebra to non-abelian structures is suggested.

The Fibonacci groups F(r, n) have been studied by various authors, chiefly in order to determine which ones are finite. This article contains a summary of the known results about this problem, followed by some further results obtained by the author. In particular, the orders of the groups F(r, 3) for r ≡ 2 (mod 3) and F(r, 4) for r ≡ 2 (mod 4) are determined, and various other Fibonacci groups are proved infinite by methods similar to those of Chalk and Johnson.

In this paper we discuss the existence and multiplicity of solutions to some perturbed bifurcation problems. By using sub and supersolution techniques along with an anti-maximum principle, simple proofs of some “well known” local results of perturbed bifurcation theory are obtained. The existence of global continua of solutions is proved by using degree theory arguments.

Given and similar , modelled on radial Laplace-Beltrami operators (ρp = , in this paper we begin the study of transmutations which leads to elliptic equations Working with and transmutations Qm → −D2 for m > −½ and −D2 → for m < −½, we obtain a transmutation formulation and derivation of many results of generalized axially symmetric potential theory in the first case and in both cases generalized Hilbert transforms (different). Canonical generalizations are then automatic using general transmutation theory.

In this paper the authors obtain sufficient conditions for the existence and uniqueness of the initial value problem of functional differential equations of neutral type with infinite delays, making use of some earlier results of the present authors.

Following an earlier paper by Akinrelere (1981), we consider a laminar boundary layer at low speeds in which density is sensibly constant and frictional heating is neglected. Also following the approach of Goldstein (1948) and Stewartson (1958), a singularity is established at separation for the thermal fields. The heat transfer is determined as a function of ξ = xs – x¼/l where xs is the separation point and l in a characteristic length.

The results are for arbitrary Prandtl number σ. The results of Curle (1979) that the heat transfer near separation varies as σ¼ (at least for the first four terms) are confirmed.

Let S be an inverse semigroup and F a field. It is shown that if F has characteristic 0 and is not algebraic over its prime subfield then the algebra of S over F is semiprimitive (i.e. Jacobson semisimple). This generalises a well-known theorem on group algebras due to Amitsur. Similar results for the case in which F has prime characteristic are obtained under the additional hypotheses that S is completely semisimple or that S is E-unitary with a totally ordered semilattice.

We continue with the work of an earlier paper concerning the use of partial differential equations to prove the uniform convergence of the eigenfunction expansion associated with a left definite two-parameter system of ordinary differential equations of the second order.

In the paper the existence is proved of a solution of a non-linear Goursat problem for a 4th order partial differential equation with the boundary conditions given on four curves emanating from a common point. The problem is reduced to a system of integro-functional equations and then Schauder's fixed point theorem is applied.