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).

General formulae are obtained for the reflection and transmission of harmonic acoustic waves by a curved interface between two media when the frequency is high. In addition to refracted rays there turn up tunnelling rays, if the surface is concave to the source, which are emitted from an evanescent region when the phenomenon of total internal reflection would be anticipated. Uniformly valid formulae dealing with the transition from refraction to tunnelling in both transmission and reflection are derived.

The theory is applied to the circular cylinder and to the top-hat circular jet. In the latter case it is suggested that radiation may tend to be more significant at inclinations of 50°-65° (downstream) and 25°-40° (upstream) to the axis of the cylinder. The augmentation due to tunnelling rays in propagation upstream is mentioned.

Let S be a Clifford semigroup with identity. The weak containment question is posed for S, and answered affirmatively when each of the maximal groups Se in S is amenable. The amenability of S itself is characterised in terms of PL(S), the set of normalised positive definite functions on S arising from the left regular representation of S. A type of mean associated with PL(S) and satisfying a condition weaker than left invariance is introduced.

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.

This paper deals with the oscillatory and asymptotic behaviour of all solutions of a class of nth order (n > 1) non-linear differential equations with deviating arguments involving the so called nth order r-derivative of the unknown function x defined by

where r1, (i = 0,1,…, n – 1) are positive continuous functions on [t0, ∞). The results obtained extend and improve previous ones in [7 and 15] even in the usual case where r0 = r1 = … = rn–1 = 1.

Using an algebraic approach to the nth order linear differential equations (see [3] for n = 2 and [13 and 14] for n ≧ 2) it is shown the essence of the relation between the limit circle classification and boundedness of solutions of y″ = q(t)y. On this example it is demonstrated that if a problem can be formulated in the rank of the approach, then it is only a technical matter to answer it, e.g. the relationship studied here is based on a standard fact from the theory of functions.

A generic bifurcation theory is developed which is somewhat different from the approach in [4]. We put special emphasis on equations satisfying additional symmetry properties and on the non-generic bifurcation sets arising in this context. We apply our results on the von Kármán equations for the buckling of a rectangular plate under a compressive thrust and a normal load.

In this paper we study diffusion-reaction equations arising in the theory of chemical reactions. We prove the global existence and uniqueness of a solution without any restriction for the Lewis number and the Biot numbers. In addition, it is shown that there is at least one stationary state which is globally asymptotically stable (in a rather strong sense) provided the Thiele number is sufficiently small. These results are obtained as special cases from much more general results on semilinear parabolic systems, derived below.

In the coordinatization of lattices by Baer semigroups, two notable gaps that remain to be filled concern the coordinatization of modular and distributive lattices. In this paper we present coordinatizations of modular lattices. In a subsequent paper we shall deal with the distributive case. Here we show that a bounded lattice L is modular if and only if L can be coordinatized by a Baer semigroup S such that if eS, fS ∊ R(S) then there exist idempotents ē, ∊ S such that ēS = eS, S = fS and e¯, commute; equivalently, if and only if L can be coordinatized by a Baer semigroup S such that if eS, fS ∊ R(S) with e idempotent then there is an idempotent such that S = fS and e = ee.

M. G. Krein's method of directing functional is generalized in a straight-forward way to symmetric linear relations. Applications to Stieltjes differential boundary problems are indicated.

In this paper we study the stabilization problem for non autonomous control processes in Hilbert spaces. We prove that a stabilizing feedback exists if and only if an associated Riccati equation has a bounded solution which is symmetric and positive definite.

An application to control processes with delays in control is presented.

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 is known that the class of all reachable states in boundary control of systems described by parabolic equations in one space dimension is independent of the time during which control is applied. This result is generalized here to systems governed by the heat equation in an arbitrary number of space variables.

We present an example (section 3) which serves two purposes: it is a counterexample to a conjecture of Weidmann (see section 1) and it clarifies the relations between different properties related to A-compactness (see section 2).

Techniques from the theory of partial differential equations are employed to prove the uniform convergence of the eigenfunction expansion associated with a two-parameter system of ordinary differential equations of the second order.

Let Vrs, s = 1 …, K be Hermitian operators on Hilbert spaces Hr, r = 1, …, k. For x = x1⊗…⊗xk ∊ H1⊗…⊗Hk we define Δx by the formal determinantal expansion Δx = ⊗det{Vrsxr}. Δ is extended to all of H by linearity and continuity. The paper presents results concerning positivity properties of Δ on decomposable tensors.

In previous papers [11,12,13], certain spaces of generalized functions were studied from the point of view of fractional calculus. In this paper, we show how a Hankel transform Hv of order v can be defined on for all complex numbers v except for those lying on a countable number of lines of the form Re v = constant in the complex v-plane. The mapping properties of Hv on are obtained. Various connections between Hv (or modifications of Hv) and operators of fractional integration are examined.

Two closely related classes X(n) and Y(n) of two generator two relation groups are studied. The group presentations arise from an investigation of a Fibonacci type group of order 1512. The Reidemeister-Schreier algorithm is used to show that the groups X(n) are finite and not metabelian. The orders of these groups are determined and shown to be divisible by powers of Fibonacci numbers or by powers of Lucas numbers. In addition these groups add to the relatively few examples of non-metabelian two generator two relation groups whose orders are known precisely.

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.

The connection between the structure of a near-ring and that of the group on which it acts is used to obtain results concerning the structure of near-rings. A generalized R series is defined for an R module, where R is a zero-symmetric left near-ring, and it is shown that all R modules have maximal R series. The idea of a near-ring which annihilates a series is introduced and some easy consequences of the definition are pointed out. Semi-primitive near-rings are introduced and a general structural result connecting the last two ideas is given. Some special cases which generalize earlier results on endomorphism near-rings are stated. Finally some of the limitations of the idea of semi-primitive near-rings are shown, and some applications are given, in particular to the endomorphism near-rings of soluble groups and of the symmetric groups.