Existence and uniqueness theorems are proved for the solution of a Dirichlet-Neumann-Third mixed boundary value problem for the Helmholtz equation in ℝ3. The proofs make use of an equivalent system of two integral equations of the second kind.

We study, in unbounded domains Ω⊂Rn, an elliptic semilinear problem with homogeneous boundary conditions. We assume that the nonlinear term f(x, u, Du) satisfies some condition of quadratic growth with respect to Du. We prove, in the framework of weighted Sobolev spaces, that, if and are respectively a subsolution and a supersolution of our problem, then there exists a least solution ū and a greatest solution û in the ordered interval and we obtain some multiplicity results.

An E-unitary inverse semigroup, S, has the property that, if x=S, and e2 = e=S, then (xe)2 = xe implies that x2 = x. As a consequence of this, we can see that S is an extension of its semilattice of idempotents, E, by its maximal group morphic image, G. Thus, following McAlister (1974), we attempt to describe S in terms of E and G. If we extend the semilattice E to a larger semilattice F, we are able to describe S in terms of a semi-direct product of F and G, giving a new interpretation to the approach of Schein (1975).

This paper gives sufficient conditions ensuring that a non-linear control system of the form

is controllable by means of control functions u(t), such that each ui(t) only takes two values, with a finite number of switches. It is assumed that the ‘unperturbed’ system ẋ = A(t)x + B(t)u is controllable in the usual sense, i.e. by measurable and bounded controls.

We first study the Poisson equation Δu = f in Ώω, and , where Ωω = {(r cos θ, r sin θ): 0<r<1, θ ∈(0,ω)} is a sector in ℝ2, ω ∈ (0, 2π), Г0 = {(cos θ, sin θ): θ ∈ (0, ω)} and Г1 = ∂Ωω − Г0,b and λ are in ℝ1. We obtain Schauder-type estimates and Fredholm alternative theory for the problem. We then study the symmetry breaking problem for the Gel'fand equation Δu + λeu = 0 in Ωω and obtain a complete picture about the relationships among three parameters λ, b, and ω in the problem.

The theories of fractional calculus and of the Hankel transform developed in for the spaces F′p, μ of generalised functions are used to study distributional analogues of dual integral equations of Titchmarsh type. These are shown to have infinitely many solutions in F′p, μ under very general conditions on the parameters involved. These results are used to study the corresponding classical problem in weighted Lp spaces. Existence and uniqueness of classical solutions are investigated and examples given of both uniqueness and non-uniqueness for the classical problem.

Since the year 1889, when the last of a series of six papers with a title similar to the above appeared, further research has led to the discovery of a number of writings belonging to the period then dealt with, viz., 1693–1844. Of those an account is now given before proceeding to the papers of later date than 1844.

If X is a compact Hausdorff space and E a dual Banach space, let C(X, Eσ*) denote the Banach space of continuous functions F from X to E when the latter space is provided with its weak * topology, normed by . It is shown that if X and Y are extremally disconnected compact Hausdorff spaces and E is a uniformly convex Banach space, then the existence of an isometry between C(X, Eσ*) and C(Y, Eσ*) implies that X and Y are homeomorphic.

For a Dirac system on the line, we study the Jost solutions and the existence of half-bound states. A characterisation of this phenomenon is given in terms of the M(λ)-function, which supplements a well-known spectral characterisation for the system. As a corollary, we deduce the spectral density behaviour at the spectral gap endpoints, which is shown to depend on the possible existence of half-bound states.

In the general theory of non-selfadjoint elliptic boundary value problems involving an indefinite weight function, there arises the problem of obtaining a priori estimates for solutions about points of discontinuity of the weight function. Here we deal with this problem for the case where the weight function vanishes on a set of positive measure.

The aim of this paper is to characterise sets of anisotropic weighted capacity zero. In this we generalise previous known results for the isotropic equivalent. A particular case of this zero capacity set is used to generalise removable singularity results for weak solutions of degenerate quasilinear parabolic equations and for their elliptic equivalent when its structure is still essentially isotropic, with the anisotropy confined to the mixed norms of the generalised Lebesgue spaces involved.

In this paper are described some new spinning-tops, gyrostatic apparatus, and illustrations of gyrostatic action devised by the author in the course of the last nine months. The apparatus has been built in the workshops of the Natural Philosophy Institute of the University of Glasgow, and has been employed in illustrating the lectures on rotational motion delivered in the Natural Philosophy classes in the course of the session just closed.

Fig. 1 shows a model aeroplane fitted with a small gyrostat in the position of the propeller. The gyrostat is therefore fixed on the front of the aeroplane with the plane of its flywheel at right angles to the length of the machine. The flywheel of the gyrostat, so fitted, represents exactly the rotor of the engine and the propeller of an actual aeroplane.

On Monday, 3rd December 1917, in brilliant weather, a meteor was observed by many persons in S.E. Scotland. Its luminous course lasted apparently for some seconds. It is described as extremely bright, like limelight, or a length of wire at white heat; it left a trail, part of which became detached but continued to follow it, and it was accompanied by two thin streamers. It was seen from Hexham, Northumberland, 120 miles from where it fell, low down on the horizon. From Edinburgh several observers noticed it at an angle which may be put at 45° from the horizon and a bearing somewhat to the east of north. From Dunfermline it was reported at “no great altitude” and travelling in a line inclined about 15° to the horizon.

1. If for aα+bβ+cγ+…we put

the the product of |a1b2c3| and |f1g2h3| may be written

Impressions of the tracks of large birds from this locality have lately been objects of attraction to visitors to the museum at Wellington, New Zealand. To these Dr Hector, F.R.S., has affixed a label, stating that they are from the “Sea shore sand” at Poverty bay, a harbour on the east coast of the north island. “Sand rock” would have been a preferable term, as to most observers the description is calculated to convey the idea that these footprints are but of yesterday's date.

THE subject of the “negative” viscosity of solutions has of late attracted considerable attention. Since the publication of the paper by one of us with Ranken in the Transactions of the Royal Society of Edinburgh, there have appeared a paper by Jones and Veazey, and two by Getman.

Since the latter of the papers by Getman contains a comprehensive review of the older literature on the subject, it is unnecessary for us to do more than indicate those points which are of fundamental importance. First noticed by Hübner in 1873, “negative” viscosity was observed by Sprung in the course of his extensive researches to occur in solutions of several potassium and ammonium salts, while corresponding sodium salts do not cause it. Later investigations added to the salts which are known to give rise to it; mention may be made of hydrobromic acid, in which it occurs at low temperatures but not at higher temperatures, whilst it does not occur at all in the case of hydrochloric acid.

In the microphone transmitter, as usually employed in circuit with a battery and a Bell telephone, we have essentially two pieces of carbon resting lightly against each other, through which the current passes. That the instrument may work effectively two things are requisite,—first, that the carbons be always in contact, or at least sufficiently near for the current to pass between them; and, secondly, that they be not pressed together so tightly as to prevent any motion of the one relatively to the other. This state of things is sufficiently well described by the term “loose contact,” first used, I believe, by Professor Stokes.

The radioactivity of potassium was discovered by Campbell and Wood (1907). Since the β-ray activity shown by all the potassium salts investigated was very small, the suspicion naturally arose that this activity was due to a trace of radioactive impurity and not to the potassium itself. All tests for the presence of an emanation gave, however, negative results, and it became necessary to conclude that the observed emission of the β-radiation was indeed a property of the potassium atom. The subsequent investigations of Campbell (1907) and of Hoffman (1924), whose attempts to alter the activity by chemical operations were unsuccessful, supported this conclusion, as did also the work of Biltz and Marcus, who prepared specimens of potassium sulphate from potassium minerals of different geological ages and found that the activity of the preparations per atom of potassium was always the same.

In this paper the authors investigate two classes of triple series equations of trigonometric type. Closed form solutions are given for eight triple series in each class.