We obtain an explicit formula for the essential norm of a Hankel operator with its symbol in the space PC, which is the closure in L∞ of the space of piecewise continuous functions on the unit circle . It follows from this formula that functions in PC can be approximated as closely by functions in C, the continuous functions on the circle, as by functions in the much larger space H∞ + C. This is an example of the way in which properties of the Hardy spaces can be derived from properties of Hankel operators.

In this paper, a class of Boolean rings containing the class discussed in papers by Seever (1968) and Faires (1976), is defined in such a way that an extension of the classical Vitali–Hahn–Saks theorem holds for exhausting additive set functions. Some new compact topological spaces K for which C(K) is a Grothendieck space are constructed and a Nikodym type theorem is deduced from it. The Boolean algebras of Seever and Faires and those we study here are defined by ‘interpolation properties’ between disjoint sequences in the algebra. We give an example at the end of the paper that illustrates the difficulties arising when we try to find a larger class of Boolean algebras, defined in terms of such properties, for which the Vitali–Hanh–Saks theorem holds.

Consider solutions 〈H(x, ε), G(x, ε)〉 of the von Kármán equations for the swirling flow between two rotating coaxial disks

and

We assume that |H(x, ε)| + |Hʹ(x, ε)| + |G(x, ε)|≦B. This work considers shapes and asymptotic behaviour as ε→0+. We consider the type of limit functions 〈H(x), G(x)〉 that are permissible. In particular, if 〈H(x, ε), G(x, ε)〉 also satisfy the boundary conditions H(0, ε)=H(1, ε)=0, Hʹ(0, ε)=Hʹ(1, ε)=0 then H(x) has no simple zeros. That is, there does not exist a point Z ε [0, 1] such that H(x)=0, Hʹ(z)≠0. Moreover, the case of “cells” which oscillate is studied in detail.

A sufficient condition on the angles of a bounded open subset of ℝn is given, ensuring the best regularity of solutions of a class of elliptic problems with non-linear mixed boundary conditions.

The behaviour of heat transfer and skin friction is analysed in a compressible laminar boundary layer with external velocity Ue(x)(l + α sin ωt The Mach number M is assumed small but finite so that high frequency flows (s ≫1) in which c =αγM2s/2 = O(1) are considered. Solutions, obtained by matching in the Stokes and Prandtl layers, involve summation of Fourier-like series to give the dominant terms in the heat transfer and skin friction. Results, for c =½, verify that a previous approximate method gives a reasonable description of unsteady heat transfer and skin friction; forc =1 there is a substantial increase in amplitude of heat transfer but little change of phase.

The present note is concerned to develop the principle of limiting absorption for the Laplacian Δ on a two-point homogeneous noncompact space M = G/H subject to a real-valued potential perturbation V. Such a property depends on the detailed structure of the Laplacian in a suitable coordinate system while V is assumed to satisfy a short-range condition.

An interval-type LP criterion for

is derived in which “positive” coefficients play a prominent role. When pn = 1 and all the other pi are zero this reduces to a result of Ismagilov (1962). Successive specializations are obtained with the growth of the pi constrained by monomials in x. Previous LP criteria of Everitt (1968) and Hinton (1972, 1974) are shown to be special cases.

We give a simple description of the wave operators appearing in the Lax-Phillips scattering theory. This is used to derive a relation between the scattering matrix and a kind of time delay operator and to characterize all scattering systems having the same scattering operator.

This note presents some integrodifferential inequalities in n-independent variables which are generalizations of the integrodifferential inequalities recently established by Pachpatte in two independent variables.

Dual similarity solutions in the context of mixed convection are presented. In contrast to the Falkner–Skan solutions the bifurcation point is found to be distinct from the point of vanishing skin friction. The eigenvalue problem arising out of a stability analysis of these solutions is examined numerically. The numerical evidence would seem to indicate that the margin of stability is associated with the onset of reverse flow as opposed to the bifurcation point, as conjectured by Banks and Drazin in 1973.

In this paper the semi-discrete Galerkin approximation of initial boundary value problems for Maxwell's equations is analysed. For the electric field a hyperbolic system of equations is first derived. The standard Galerkin method is applied to this system and a priori error estimates are established for the approximation.

An asymptotic theory is developed for linear differential equations of odd order. The theory is applied to the evaluation of the deficiency indices N+ and N− associated with symmetric differential expressions of odd order. General conditions on the coefficients are given under which all possible values of N+ and N subject to | N+ − N | ≦ 1 are realized.

We obtain an asymptotic estimate for the number of eigenvalues less than λ of an operator of Schrödinger type

defined on an unbounded domain.

In this paper we obtain some new results on a nonlinear parabolic system related to the equations of the nematic liquid crystals and introduced in earlier papers by J. P. Dias.

These results mainly concern the existence and uniqueness of generalized solutions for discontinuous data and also their asymptotic behaviour in various cases.

In this paper real numbers A1, A2 are obtained which are such that for λ > A1 or λ < A2 there are no non-trivial solutions of the equation

which are in L2(a,∞) for some a>0. Precise values for A1 and A2 are obtained in certain cases.

A comprehensive appraisal of the title problem is presented in terms of a characterizing nondimensional co-ordinate ξ which is based upon the half excess of the momentum of the jet, J. Perturbation features of the problem appear as regular and singular boundary conditions in ξ upstream and downstream respectively. The conservation of momentum excess provides a monitor on the consistency of regular and singular perturbation series solutions. In particular the conservation constraint on the downstream singular perturbation solution confirms the inadequacy of expansions in inverse half powers of ξ and justifies formally the introduction of logarithmic terms.

The formulation provides the basis for a complete numerical integration over the semi-infinite region. Accordingly detailed knowledge of velocity excess along the axis of the jet is obtained and an undetermined coefficient in the asymptotic downstream perturbation solution may be estimated.

The groups of units of indefinite ternary quadratic forms with rational integer coefficients contain subgroups of index two which are isomorphic to Fuchsian groups and which, for zero forms, are commensurable with the classical modular group. This is used to obtain a family of forms whose groups are representatives of the conjugacy classes of maximal groups associated with zero forms. The signatures of the groups of the forms in this family are determined and it is shown that the group associated to any zero form is isomorphic to a subgroup of finite index in the group of one of three particular forms. This last result should be compared with the corresponding result by Mennicke on non-zero forms.

The paper presents sufficient conditions on the coefficients of second and fourth order differential equations to ensure that there exists at least one pair of conjugate points on an interval (a, b), −∞≦ a <b ≦ ∞. Oscillation criteria related to the equation (p(x)y″)″ + q(x)y = 0, 0 < x < ∞, are proved with no sign restrictions on q(x).