In this paper a class of weighted Sobolev spaces defined in terms of square integrability of the gradient multiplied by a weight function, is studied. The domain of integration is either the space Rn or a half-space of Rn. Conditions on the weight functions that will ensure density of classes of smooth functions or functions with compact support, and compact embedding theorems, are derived. Finally the results are applied to a class of isoperimetrical problems in the calculus of variations in which the domain of integration is unbounded.

We discuss convex l-subgroups of an l-group G in their role as direct summands, not so much of G as of each other. This is done by writing A ≥dB for subgroups A, B to mean that A is a direct summand of B, and studying the properties of the resulting poset. It is shown to be a hypolattice, that is, to have local lattice properties in a certain sense. However it need not be a lattice; and there may exist meets of pairs of elements, outside the hypolattice structure. It need not be conditionally complete even when G is conditionally complete. We look also at the map which sends a subgroup to its lattice-closure; the lattice-closed subgroups also form a hypolattice. Our main result asserts that this hypolattice is conditionally complete if G is. The paper ends with some examples and counter examples in C(X).

By use of the theory of asymptotic expansions for first-order linear systems of ordinary differential equations, asymptotic formulas are obtained for the solutions of an nth order linear homogeneous ordinary differential equation with complex coefficients having asymptotic expansions in a sector of the complex plane. These asymptotic formulas involve the roots of certain polynomials whose coefficients are obtained from the asymptotic expansions of the coefficients of the differential operator.

It was proved by Howie in 1966 that , the semigroup of all singular mappings of a finite set X into itself, is generated by its idempotents. Implicit in the method of proof, though not formally stated, is the result that if |X| = n then the n(n – 1) idempotents whose range has cardinal n – 1 form a generating set for. Here it is shown that if n ≧ 3 then a minimal set M of idempotent generators for contains ½n(n–1) members. A formula is given for the number of distinct sets M.

Groups having two generators and two relations are studied. The Reidemeister-Schreier algorithm is used to determine presentations for their derived groups. This enables the orders of the groups to be found. Necessary and sufficient conditions are given for the groups to be metabelian. Certain classes closely related to the class are also discussed.

We consider the set S(f) of λ-values for which there exists a function g ∊ L(t 0, ∞) such that

has a solution not tending to 0; here f is a fixed function which is positive, non-decreasing and tends to ∞ with t. It is shown that if the jumps of logf(t) at its discontinuities are uniformly bounded, then S(f) is an additive group. This group is determined in some cases; some related groups are noted, which may coincide with S(f).

Comparisons have been made of the eigenvalues and the corresponding eigenfunctions of the eigenvalue problems

and

with φ ∈ C(-∞, +∞) and 0≦φ(x)≦C|x|i+1(1+|x|1), −∞<x<+∞ where i and l are arbitrary positive numbers with i≧2k≧2, k integer. In first approximation the eigenvalues λ and λ− and the corresponding eigenfunctions ψ and ψ are the same for ε→0; the error decreases whenever the exponent i increases.

This paper proves some inequalities for the imaginary part of the transcendental function in a simply connected sector of the complex z-plane, where 0 < v < 1, and part of the boundary depends on v. These inequalities arose in a work of Everitt and Jones [1] which was on a general integral inequality. We give an alternative method of proving these Bessel function inequalities.

We consider differential-delay equations which can be written in the form

The functions fi and gk are all assumed odd. The equation

is a special case of such equations with q = N + 1 (assuming f is an odd function). We obtain an essentially best possible theorem which ensures the existence of a non-constant periodic solution x(t) with the properties (1) x(t)≧0 for 0≦t≦q, (2) x(–t) = –x(t) for all t and (3) x(t + q) = –x(t) for all t. We also derive uniqueness and constructibility results for such special periodic solutions. Our theorems answer a conjecture raised in [8].

In answer to a question raised by John Barrett in 1961, conditions are established under which the existence of a systems-conjugate point of a fourth-order equation also assures the existence of a 2–2 conjugate point. These results lead to new conjugacy criteria and apply to certain non-selfadjoint equations more general than those previously considered.

The symmetric differential expression M determined by Mf = − Δf;+qf on G, where Δ is the Laplacian operator and G a region of n-dimensional real euclidean space Rn, is said to be separated if qfϵL2(G) for all f ϵ Dt,; here D1 ⊂ L2(G) is the maximal domain of definition of M determined in the sense of generalized derivatives. Conditions are given on the coefficient q to obtain separation and certain associated integral inequalities.

Classically the tangent sphere bundles have formed a large class of contact manifolds; their contact structures are not in general regular, however. Specifically we prove that the natural contact structure on the tangent sphere bundle of a compact Riemannian manifold of non-positive constant curvature is not regular.

This paper is concerned with finding upper bounds on the set of eigenvalues of self-adjoint differential operators generated in the Hilbert space L2[0, ∞) by the differential expression

on [0,∞), together with a real homogeneous boundary condition at t = 0.

The existence of a classical solution to the initial boundary value problem for a semi-infinite extensible string is proved. The result is obtained by using a Galerkin procedure on a semi-infinite interval.

We consider the equation Lφ − λpφ + γqφ2 = f on a bounded domain in Rn with homogeneous Neumann-Dirichlet boundary conditions. L is a negative definite uniformly elliptic differential operator, while, p, q and f are positive functions. We show that there exists exactly one positive solution for each λ ∈ R and γ > 0. This solution can be analytically continued throughout Re γ > 0: it is a Laplace transform of a positive measure. The measure is bounded prior to the bifurcation point of the associated “homogeneous” equation and unbounded after. Noting that any Laplace transform of positive measure has associated with it a natural sequence of Tchebycheff systems, it now follows that one can obtain monotonically converging upper and lower bounds which are provided by the generalized Padé approximants generated from the Tchebycheff systems.

Formulas are determined for the deficiency numbers of a formally symmetric ordinary differential operator with complex coefficients which have asymptotic expansions of a prescribed type on a half-axis. An implication of these formulas is that for any given positive integer there exists a formally symmetric ordinary differential operator whose deficiency numbers differ by that positive integer.

We classify group-theoretically all separable coordinate systems for the eigenvalue equation of the Laplace-Beltrami operator on the hyperboloid = 1, finding 71 orthogonal and 3 non-orthogonal systems. For a number of cases the explicit spectral resolutions are worked out. We show that our results have application to the problem of separation of variables for the wave equation and to harmonic analysis on the hyperboloid and the group manifold SL(2, R). In particular, most past studies of SL(2, R) have employed only 6 of the 74 coordinate systems in which the Casimir eigenvalue equation separates.

Within recent years considerable attention has been devoted to extensions of the classical Sturmian theory of real linear homogeneous differential equations of the second order. In particular, such extensions have included not only self-adjoint systems of differential equations, but also higher order self-adjoint differential and integro-differential equations. For problems in these latter categories, however, only limited attention has been given to detailed application of the general oscillation and comparison criteria. The present paper is devoted to this area, and, in particular, it is shown how existing criteria may be exploited to obtain comparison theorems between equations of different orders. Although the presented results have ready extensions to vector differential and integro-differential equations of higher order, [see, for example, 5,6,7], for simplicity attention is restricted to scalar equations. Section 2 is devoted to the statement of known general criteria of oscillation for self-adjoint equations of higher order, with special applications of these criteria presented in Section 3. Finally, Section 4 sketches the framework of corresponding applications for self-adjoint higher order integro-differential equations.