A simple and direct variational method is described for finding both complex and real eigenvalues of the wave equation of anomalous propagation in a horizontally stratified atmosphere. It may be looked upon as an extension of Rayleigh's method to complex eigenvalues. In this paper it is illustrated by an example, taken from the duct theory of super-refraction, in which the refractive index of the air varies with height according to a power law. Numerical agreement in the values for the lowest order eigenvalues with those obtained by the differential analyser is better than ½%.

In the normal procedure for establishing linearized stability criteria a mean steady state which we denote by U is perturbed by a small additional disturbance u in the Eulerian sense. In the linearization it is usual to regard the disturbance as resolved into Fourier components which are additive so far as the first order goes. The linearized problem will then determine a time amplification factor eiσr so that the linearized disturbance is of the form uexp{i(αx + σt)} and the whole motion U + uexp{i(αx + σt)}. As usual it is implied that the real part of complex functions such as uexp{i(αx + σt)} be taken. In cases of instability (imaginary part of σ negative), where the magnitude of the disturbance increases with the time, we are prompted to ask how energy is supplied to the disturbance. This question is especially relevant to many of the classical problems of fluid motion instability in which the systems are self-contained in the sense that energy is neither supplied nor withdrawn by an outside agency. If such a system is non-dissipative we expect that the total energy of the organized motion and the electromagnetic field, if present, shall be conserved, whilst such energy in a dissipative system will be gradually reduced by heat losses. In either case energy which is taken into a growing disturbance must be provided from the mean energy stored in the system in its initial state. One of the simplest cases of instability to examine from this point of view is the Helmholtz instability of a plane in viscid vortex sheet. This system is self-contained and conservative in the energy of the organized motion, and moreover the only form of energy with which we have to deal is kinetic energy. In the first part of this note we examine how disturbance kinetic energy is provided in this case by the break up of the organized motion of the mean stream. In the second part we consider what happens to the energy balance of a current-vortex sheet in the non-dissipative case. There are here two forms of energy, kinetic and magnetic, and it is shown that when the interface is unstable the overall magnetic energy increases at the expense of the kinetic energy.

Consider a queueing system M/G/s with the arrival intensity λ, the service time distribution function B(t) (B(0) < 1) having a finite mean and the waiting room size N ≤ ∞. If s < ∞ and N = ∞, we shall also assume that its relative traffic intensity is less than 1. Since the arrival process of this system is Poisson, it is immediate that in this case the distribution of the number of arrivals during an interval is infinitely divisible.

This article is concerned with the bifurcation of critical points of a smooth real-valued function on a Banach space. There are two somewhat different aspects considered. The first, referred to as reduction, is a refinement of the Liapunov-Schmidt procedure by incorporating techniques from the theory of determinate germs and their universal unfoldings. A definite reduction procedure is given, whereby a germ or unfolding given in a Banach space may be replaced by a germ or unfolding in a finite-dimensional space in such a way that the geometry of the bifurcation set and the overlying critical point set is preserved. The aim is to provide a practicable tool, not an exhaustive theoretical discussion. How practicable it is may be seen from another paper (Magnus and Poston(7)) in which it is applied to a problem in elasticity; indeed, the present paper in part was originally the ‘machinery’ section of that paper. The second aspect, stability, is of more theoretical interest. The well known structural stability of universal unfoldings (see (5)) is extended to infinite-dimensions. Only a local theory is given.

In this paper, together with [7] and [8], we shall be concerned with estimating the number of solutions of the inequality

for almost all α (in the sense of Lebesgue measure on Iℝ), where , and both m and n are restricted to sets of number-theoretic interest. Our aim is to prove results analogous to the following theorem (an improvement given in [2] of an earlier result of Khintchine [10]) and its quantitative developments (for example, see [11, 12,6]):

Let ψ(n) be a non-increasing positive function of a positive integer variable n. Then the inequality (1·1) has infinitely many, or only finitely many, solutions in integers to, n(n > 0) for almost all real α, according to whether the sum

diverges, or converges, respectively.

After the development of the modular theory one can construct non-commutative Lp-spaces associated with a von Neumann algebra M which is not necessarily semifinite (see [4], [12], etc.).

Here we answer a question raised by Zaleskii (5) and reiterated by Segal (3) of whether a finitely generated nilpotent group can have primitive irreducible representations.

Will J. Merry computed Rabinowitz Floer homology above Mañé's critical value in terms of loop space homology in [14] by establishing an Abbondandolo–Schwarz short exact sequence. The purpose of this paper is to provide an alternative proof of Merry's result. We construct a continuation homomorphism for symplectic deformations which enables us to reduce the computation to the untwisted case. Our construction takes advantage of a special version of the isoperimetric inequality which above Mañé's critical value holds true.

The main result of [15] is that the classifying space of the stable mapping class group after plus construction BΓ+∞ is an infinite loop space. This result is used to show that, localized away from two, a connected component of the stable homotopy groups of spheres QS0 splits off BΓ+∞. The splitting is a splitting of infinite loop spaces. It follows immediately that the homology with coefficients in ℤ[½] of the infinite symmetric group is a direct summand of the homology of the stable mapping class group.

By solving the fundamental integral equation an expression is obtained for the electrical resistance which takes into account, to the first approximation, the mutual influence of the impurities and the lattice vibrations. It is found that deviations from Matthiessen's rule are to be expected and that these deviations are surprisingly large. The formulae derived only indicate the trend of the resistance curve, but this trend is not confirmed by experiment. Whereas the theory indicates that the mutual influence of the impurities and the lattice vibrations should decrease the electrical resistance, the experimental results of Grüneisen on copper show that, when deviations from Matthiesen's rule occur, the resistance is increased.

1. Mr J. P. Gabbatt has discussed in the most recent Part of the Proceedingsof this Society the Pedal locus of a simplex in hyperspace. It is, however, possible to regard the pedal property of the circumcircle somewhat differently and so to seek other extensions. Given a circle, any three points on it are vertices of an inscribed triangle, and the feet of the perpendiculars on the sides from any fourth point of the circle are collinear. Is there any curve in space on which an analogous property holds for any five points, viz. that the feet of the perpendiculars from any one upon the faces of the tetrahedron formed by the other four are coplanar?

It will be shown that curves of order n exist in Euclidean space of n dimensions on which any n + 2 points have such a property; but that the curves cannot be real if n is odd.

We shall consider a non-stationary Markov chain on a countable state space E. The transition probabilities {P(s, t), 0 ≤ s ≤ t <t0 ≤ ∞} are assumed to be continuous in (s, t) uniformly in the state i ε E.

A sinusoidal travelling wave-train is at normal incidence upon a two-dimensional finite dock fixed on the surface of a body of water of great depth, and the problem investigated herein is that of finding the limiting form of the induced velocity potential for short waves. Of particular interest are the amplitudes of the wave-trains reflected and transmitted towards infinity by such an obstacle. The potential is expressed as a sum of coupled semi-infinite dock potentials, whence results a pair of weakly coupled integral equations for the solution. This formulation of the problem is shown to be amenable to an approximate solution for large wave-numbers, and the first few terms are derived in formal expansions for the reflection and transmission coefficients.

We prove that every flat tensor-idempotent in the module category Mod- of a tensor-triangulated category comes from a unique smashing ideal in . We deduce that the lattice of smashing ideals forms a frame.

It is conjectured that there exist only finitely many isomorphism classes of endomorphism algebras of abelian varieties of bounded dimension over a number field of bounded degree. We explore this conjecture when restricted to quaternion endomorphism algebras of abelian surfaces of $\mathrm{GL}_2$-type over $\mathbb{Q}$ by giving a moduli interpretation which translates the question into the diophantine arithmetic of Shimura curves embedded in Hilbert surfaces. We address the resulting problems on these curves by local and global methods, including Chabauty techniques on explicit equations of Shimura curves.

We establish Whitehead and Ganea characterizations for proper LS-category. We use the embedding of the proper category into the exterior category, and construct in the latter a suitable closed model structure of Strøm type. Then, from the axiomatic LS-category arising from the exterior homotopy category we can recover the corresponding proper LS invariants. Some applications are given.

At the 1977 Georgia Topology Conference, Cappell and Shaneson(6) announced the following theorem.

Let M be a closed oriented smooth four-manifold.

An invariant of a three-manifold with boundary is derived from topological quantum field theory. This invariant is then used as an obstruction to embedding one 3-manifold into another.

Recently, Dasbach, Futer, Kalfagianni, Lin and Stoltzfus extended the notion of a Tait graph by associating a set of ribbon graphs (or, equivalently, cellularly embedded graphs) to a link diagram. Here we focus on Seifert graphs, which are the ribbon graphs of a knot or link diagram that arise from Seifert states. We provide a characterization of Seifert graphs in terms of Eulerian subgraphs. This characterization can be viewed as a refinement of the fact that Seifert graphs are bipartite. We go on to examine the family of ribbon graphs that arises by forming the parallels of a link diagram and determine how the genus of the ribbon graph of a r-fold parallel of a link diagram is related to that of the original link diagram.

A method is described for measuring the heat of adsorption of hydrogen on a clean tungsten filament. The value obtained is 2·8 × 104 calories per mol.