The relationship between weak and sequential weak continuity for mappings between Banach spaces and semigroups on Banach space is studied.

As is well known, the deficiency indices of the minimal symmetric differential operator of even order 2m with real coefficients on the half-line lie within the limits m and 2m, and are equal. I. M. Glazman [6] first showed, by examples, that for such operators the deficiency indices can actually take all the intermediate values m ≦ P ≦ 2m. Subsequently fresh examples of operators with real coefficients, whose deficiency indices take all possible values, were constructed by various authors (S. A. Orlov [11], M. V. Fedoryuk [4], and others).

This paper sets out to study the spectrum of self-adjoint extensions of the minimal operator associated with the third-order formally symmetric differential expression. The technique employed is the method of singular sequences. Sufficient conditions are established on the coefficients of the differential expression in order that the spectrum should cover the entire real axis. Particular cases in which the coefficients behave roughly as powers of x as the magnitude of x becomes large are then considered, and certain conclusions are drawn regarding the spectra under different restrictions on these powers of x.

The paper discusses the nature of the singularities at any finite point t = t0 of real solutions x(t), y(t) of the system of differential equations x′ = P(x, y), y′ = Q(x, y) in which P, Q are polynomials in both x and y. Its main interest is in cases when the leading terms of P, Q are of the same degree n. Conditions are given under which the only possible singularities are algebraic poles and pseudo-poles. Conditions are also given under which the only possible singularities are of wildly oscillatory type.

For a class of Fuchsian groups, which includes integral automorphs of quadratic forms and unit groups of indefinite quaternion algebras, it is shown that the geometry of a suitably chosen fundamental region leads to explicit bounds for a complete set of generators.

1. In this paper we study the asymptotic behaviour of solutions of algebraic equations with real functions as coefficients, using mainly algebraic properties of the class to which the coefficients belong. To that end we introduce the notion of an m-group of functions and prove the main theorem by a procedure originated in [1]. As a corollary we obtain sufficient conditions for a class F of functions to possess the property that solutions of algebraic equations with coefficients in F are again members of F. We conclude by applying these results to the classical Hardy's logarithmico-exponential class ℋ [2]

It is well known that the existence of transcendental meromorphic solutions of non-linear ordinary differential equations puts severe restrictions on the equation, the most striking example being the theorem of Malmquist [3]. The value distribution theory of R. Nevanlinna was applied to such questions by K. Yosida [8] who used it to prove Malmquist's theorem as well as important generalisations. An alternate approach was given by H. Wittich [4,5,6] and in his argument the finiteness of the order played an essential role. Wittich estimated the corresponding enumerative and proximity functions via the calculus of residues. In this note a geometric argument is proposed instead (closest packing of small discs in a bigger circle or on its rim). This method seems to generalise more readily to Yosida's extensions.

The diffraction of a line source of sound by an absorbing semi-infinite half plane in the presence of a fluid flow is examined. It is found that the radiated sound intensity, in the half space in which the source is located, can be considerably reduced by a suitable choice of the absorption parameter. For subsonic flow the system exhibits no acoustic instabilities.

The principal results of this paper concern the spectral properties of the maximal realisation Pp in Lp(Rn) of a formally self adjoint constant coefficient strongly elliptic partial differential operator P(D), assumed to be homogeneous of order 2m, for 1 ≦ p ≦ ∞ and n ≧ 2. If we assume that , for 2n/n + l ≦ p ≦ 2n/n−1, together with certain assumptions on the associated real zero surfaces P(ξ) = λ, λ > 0, then σ(Pp) = σc(Pp) = [0, ∞). We obtain an estimate on the norm of the resolvent of Pp for points near the real axis, which allows us to establish the existence of a generalised resolution of the identity in the sense of Kocan.