On page 674 of the present volume of the Proceedings (Part 3), in line 3 of the heading:

for University of Princeton, U.S.A.

read Institute for Defense Analysis, Princeton, N.J.

The purpose of this paper is to prove Theorems 1 and 2 below by exhibiting them as special cases of a more general result, Theorem 3, which admits of a simple proof.

In (7), we investigated some elements in the Adams spectral sequence for the stable homotopy groups of spheres, and proved that they were never boundaries, for any differential. This paper extends and generalizes these results: we consider more elements than in (7), and also prove that many of them do not survive to the E∞ term, so that they fail to be cycles for some dr, and this differential is therefore non-zero.

The idea of exact filling bundle may be described roughly as follows. Suppose that ξk is a vector bundle with fibre Rk, total space E(ξk) and base X. We say that ξk is a real k-plane bundle on X. Let in be the trivial n-plane bundle on X so that E(in) = X × Rn. A bundle monomorphism j: ξk → in defines a map : E(ξk)→Rn obtained by composition of the embedding E(ξk)→E(in) and the product projection E(in) → Rn. The map represents each fibre of ξk as a k-plane in Rn.

There is a close parallelism between the theories of convergence of directed nets and of filters, in which ‘subnet’ corresponds, in general, to ‘refinement’. With the standard definitions, however (1), pages 65 et seq., this correspondence is not exact, as there is no coarsest net converging to x0 of which all other nets with the same limit are subnets. (Suppose, for example, that a net X = {xj,: j ∈ J} in R1 has both the sequence-net S = {n−1; n = 1, 2,…} and the singleton-net {0} as subnets. Then (with an obvious notation), there exist

such that j0 ≥ jn for all n, while jn ≥ j0 for all n ≥ n0 say. But, given any j ∈ J, there exists n with jn ≥ j: it follows that jn ∈ j for all n ≥ n0 (independent of j); thus X cannot converge to 0. Even if nets with a last member are excluded, a similar result can be obtained by considering the net Y = {yθ; Θ an ordinal less than ω1}, where yθ = 0 for all Θ. If X has both Y and S as subnets we can show that (with a similar notation) there exists Θ0 such that Θ ≥ Θ0 implies jθ ≥ all jn, but also n0 such that n ≥ n0 implies ; the rest is as before.) Moreover, the theory of convergence classes, (l), pages 73 et seq., contains a condition (Kelley's condition (c)) whose analogue need not be separately stated for filters. These differences can be removed by adopting a wider definition of subnet, a course which does not seem unnatural, inasmuch as the standard definition is already wider than the ‘obvious’ one, and our proposed definition is equivalent to the standard one in the special case of sequences.

It is well known that a path-connected Hausdorff space is arc-connected. Indeed, given a path P, that is, a continuous map. f of the unit interval I0(0 ≤ t ≤ 1) into the Hausdorff space X, there is an arc joining the points f(0), f(1) (supposed distinct) which is obtained by ‘cutting out loops in P’. More precisely, there exist a continuous increasing map α of I0 onto itself and a homeomorphism φ of I0 onto φ(I0) ⊂ X, such that if [t0, t1] is any maximal interval of constancy of α (including the case t0 = t1) then f(t0) = f(t1) = φοα(t0). The function φ, α, can be defined by a constructive process.

It has been known for some time that the product of a non-metrizable Hausdorff space and any (non-trivial) Hausdorff space cannot be the continuous image of an ordered continuum. (For a survey of this and related problems, see Mardešić and Papić ((1)).) Further, it has been shown by Treybig ((2)) (and independently by the present author) that the product of two Hausdorff spaces cannot even be the continuous image of an ordered compactum unless both the spaces are metrizable or one is finite. It is therefore of some interest to give a simple example of a space X which is the continuous image of an ordered compactum K and contains the product of a non-metrizable space and an infinite discrete space, imbedded in such a way as to form a sequence of homeomorphic subsets with a connected (non-trivial) topological limit.

This paper deals with the existence of a quadratic form as a Liapunov function for a linear homogeneous vector differential equation with constant coefficient matrix such that the total derivative of the Liapunov function is strictly negative semi-definite and not identically equal to 0 for every non-trivial solution of the given equation. A theorem is proved which guarantees the existence of a strictly semi-definite quadratic form which is not identically equal to 0 for every non-trivial solution if and only if the coefficient matrix of the equation is not proportional to the unit matrix. Furthermore, it is proved that if the equilibrium of the given linear equation is asymptotically stable and if the coefficient matrix is not proportional to the unit matrix, then there exists a positive definite quadratic form as a Liapunov function whose total derivative is strictly negative semi-definite and not identically equal to 0 for every non-trivial solution of the given equation. It is also shown how this theorem fits into the system of already known theorems concerned with quadratic forms as Liapunov functions for a linear equation with constant coefficient matrix.

For an n-vector x = (xi) and n × n matrix A = (aij) with complex elements, let |x|2 = Σi|xi|2,|A|2 = ΣiΣj|aij|2. Also, M(A), m(A) denoteℜλ1,ℜλn, respectively, where λ1,…,λA are the eigenvalues of A arranged so that ℜλ1 ≥ … ≥ ℜλn. Throughout this paper A(t) denotes a matrix whose elements aij(t) are complex valued Lebesgue integrable functions of t in (0, T) for all T > 0. Then M(A(t)), m(A(t)) are also Lebesgue integrable in (0, T) for all T > 0. The characteristic exponent μ of a non-zero solution x(t) of the n × n system of differential equations

can be defined, following Perron ((12)), as

where ℒ denotes lim sup as t → + ∞. When |A(t)| is bounded in (0,∞), μ is finite; in other cases it could be ± ∞.

In the usual notation let

where (a)n = a(a + 1)(a + 2)…(a + n − 1), (a)0 = 1.

Dr L. J. Slater ((3), page 628) gave an expansion of the form

for all values of n and t, real or complex.

Isotropic ionospheric propagation of electromagnetic waves yields refractive index profiles via the standard transcendental equations. Some transformations of these equations have long been regarded as standard. Here, every possible transformation is derived by a systematic analysis, yielding all possible refractive index profiles satisfying certain stated criteria regarding the wave frequency.

The Navier–Stokes equations governing the motion of viscous compressible fluids are

where ρ is the density, q is the velocity vector, V is the scalar potential of the external force field, I is the idemfactor and Φ is the stress-tensor

Milne-Thomson ((l)) has given a general solution of (1) and (2) in the form

The two formulations of quantum mechanics in q-space only are compared by the construction of Schrödinger's operator identity. The examination in detail of the examples of the harmonic oscillator and free particle shows clearly the role of the re-normalization term in both formulations.

The investigation of the isotropic scattering of radiation in an infinite spherically symmetric atmosphere, which was carried out in a previous paper, is extended to a finite atmosphere bounded by a totally absorbing shell.

It is shown that the solution may be expressed as a series which will certainly converge when the radius of the shell is large enough, and whose terms are found by iteration.

Numerical investigation shows that even for quite modest values of the radius, the convergence is so fast that for most practical purposes all but the first term can be neglected. Except at the very ends of the range, a very simple expression then gives an excellent approximation.

The combined effects of uniform thermal and magnetic fields on the propagation of plane waves in a homogeneous, initially unstressed, electrically conducting elastic medium have been investigated.

When the magnetic field is parallel to the direction of wave propagation, the compression wave is purely thermo-elastic and the shear wave is purely magneto-elastic in nature. For a transverse magnetic field, the shear waves remain elastic whereas the compression wave assumes magneto-thermo-elastic character due to the coupling of all the three fields—mechanical, magnetic and thermal. In the general case, the waves polarized in the plane of the direction of wave propagation and the magnetic field are not only coupled but are also influenced by the thermal field, once again exhibiting the coupling of the three fields. The shear wave polarized transverse to the plane retains its magneto-elastic character.

Notation.

Hi = primary magnetic field components,

ht = induced magnetic field components,

To = initial thermal field,

θ = induced thermal field,

C = compression wave velocity.

S = shear wave velocity,

ui = displacement components,

cv = specific heat at constant volume,

k = thermal conductivity,

η = magnetic diffusivity,

μe = magnetic permeability,

λ, μ = Lamé's constants,

β = ratio of coefficient of volume expansion to isothermal compressibility.

In this paper we consider the problem of distribution of stress in an infinite wedge of homogeneous elastic isotropic solid under the usual assumptions pertaining to plane strain in classical (infinitesimal) theory of elasticity. The wedge is supposed to occupy the region 0 ≤ ρ ≤ ∞, − α ≤ θ ≤ α (in plane polar coordinates) where the pole is taken on the apex of the wedge with the line bisecting the wedge angle as the initial line. We report here the investigations of two types of stress fields: (i) the stress field which is set up by the application of known pressure to inner surfaces of a crack situated on the bisector of the wedge angle, and (ii) the stress field generated by the indentation of the plane faces of the wedge by the rigid punch. The corresponding boundary-value problems are shown to be equivalent to the problem of solving dual integral equations involving inverse Mellin-transforms. Similar equations have been discussed by Srivastav ((l)) in a recent paper. The method of (1) is easily modified to reduce the dual equations to single Fredholm equations of the second kind which are best solved numerically. The boundary-value problems discussed here are of the mixed type and appear to have been considered only recently by Matczynski ((2)) who has investigated a contact problem. He identifies the problem with a Wiener-Hopf integral equation and solves the integral equation approximately using a method due to Koiter ((3)). The method presented here seems to have the advantage that it requires no elaborate tools of analysis which are necessary for resolving the problem of Wiener-Hopf and numerical results may be obtained comparatively easily.

The forced torsional motion of a semi-infinite isotropic right-circular elastic cylinder is studied in this paper, the vibrations being excited by a harmonically oscillating rigid disc, securely attached to the plane face of the cylinder. Two distinctsituations are considered, namely, when the curved surface of the cylinder is clamped and when it is stress free. In both cases the problem is reduced to the solution of a Fredholm integral equation of the second kind which can be solved iteratively when the frequency of oscillation is small and the cylinder radius large. Using the iterative solutions, approximate expressions are deduced for the distribution of shear stress under the disc and the driving couple required. In an appendix a discussion is given of a mathematically similar problem in the theory of the swirling flow of a perfect fluid.

It is shown that, when a plane harmonic P or S wave is incident upon a two-or three-dimensional obstacle in an infinite elastic solid, the scattering cross-section of the obstacle can be calculated from an appropriate far-field scattering amplitude.