Some axially symmetric boundary value problems of potential theory are formulated as integral equations of the first kind. In each case the kernel admits an expansion, for small values of a parameter of the problem, that leads to an approximate integral equation whose solution provides a direct asymptotic estimate for the physical quantity of primary interest. A manipulation of the original and modified integral equations provides an efficient formula for calculating higher order terms in the asymptotic expansion.

The classical von Neumann–Oxtoby–Ulam Theorem states the following:

Given non-atomic Borel probability measures μ, λ on In such that

there exists a homeomorphism h of In onto itself fixing the boundary pointwise such that for any λ-measurable set S

It is known that the above theorem remains valid if In is replaced by any compact finite dimensional manifold [2], [4] or with I∞, the Hilbert cube, [8].

The object of this note is to show that, in applying the methods of hypercomplex numbers to the theory of determinants, there is, for many purposes, no gain in using a particular number system.

In this note we consider the question: If R is a right Noetherian ring and I is an invertible ideal of R, how do the Krull dimensions of various modules, factor rings and over-rings of R, connected with I, compare with the Krull dimension of R? This question is prompted by results in (5) and (6). In comparing the Krull dimension of the ring R with that of the ring R/I, the best result would be that the Krull dimension of the ring R is exactly one greater than that of the ring R/I. This result is not true in general; however, we see, in Theorem 2.4, that if the invertible ideal is contained in the Jacobson radical the result holds. In the general case we find it is necessary to introduce an over-ring T of R generated by the inverse I−1 of I. We then see that the Krull dimension of R is the larger of two possibilities: (a) Krull dimension of R/I plus one or (b) Krull dimension of T. In order to prove this result we construct a strictly increasing map from the poset of right ideals of R to the cartesian product of the poset of right ideals of T with a poset of certain infinite sequences of right ideals of R/I.

Let S be the focus of the conic, PX the directrix, e the eccentricity.

Let V be the middle point of PP′.

Draw VK perpendicular to the directrix, and with centre V describe a circle, radius equal to e.VK.

Join SP, SP′ and draw radii Vp, Vp′, parallel to SP′, SP, and let PP′ meet the directrix in L. Then p, p′ are on the line SL.

The main aim of the present note is to compare C(X) and C(υX), the spaces of real-valued continuous functions on a completely regular space X and its real 1–1 compactification υX, with regard to weak compactness and weak countable compactness. In a sense to be made precise below, it is shown that C(X) and C(υX) have the same absolutely convex weakly countably compact sets. In certain circumstances countable compactness may be replaced by compactness, in which case one obtains a nice representation of the Mackey completion of the dual space of C(X) (Theorems 5, 6, 7).

The study of periodic, irrotational waves of finite amplitude in an incompressible fluid of infinite depth was reduced by Levi-Civita (1) to the determination of a function

regular analytic in the interior of the unit circle ρ = 1 and which satisfies the condition

The Euler characteristic of an even dimensional submanifold in a space of constant curvature is given in terms of Weyl's curvature invariants. A derivation of Chern's kinematic formula in non-Euclidean space is completed. As an application of above results Weyl's tube formula about an odd-dimensional submanifold in a space of constant curvature is obtained.

The greater part of this paper consists of generalisations of well-known theorems regarding perpendiculars to the sides of a triangle, or other base-lines, the perpendiculars being replaced by isoclinals.

Much has been written in recent years on the foundations of geometry, chiefly in Germany and Italy, and the relations of the various Non-Euclidean geometries to the Euclidean system are now more generally known among mathematicians. But most of these writings involve a knowledge of more advanced mathematics, while it has been found difficult to represent even the simplest Non-Euclidean geometry—that of Bolyai-Lobatschewsky—in an elementary manner.

The generation of acoustic disturbances in a fluid of semi-infinite extent by the motion of a circular piston surrounded by a plane rigid baffle has been studied quite extensively (see (1), (2), (3), (4) and further references given in these papers). Attention has been devoted mainly to the case in which the piston executes a harmonic oscillation of small amplitude, and only comparatively recently has Oberhettinger (2) demonstrated how the time-harmonic solution can be used to solve the more general problem in which the normal velocity of the piston is an arbitrary function of time. The purpose of the present paper is two-fold. Firstly, we point out that for arbitrary normal motion of the piston the " baffled piston problem " can be solved directly, and in a particularly simple manner, by means of a technique involving integral transforms which has been applied by Mitra (5) and Eason (6) to the study of shear wave propagation in an elastic half-space. Secondly, we give a more detailed account than appears to be available in the literature of the structure of the sound pulse generated by the arbitrary normal motion of a baffled piston.

In [4] Elmer Rees proves that the symplectic group Sp(n) can be smoothly embedded in Euclidean space with codimension 3n, and the unitary group U(n) with codimension n. These are special cases of a result he obtains for a compact connected Lie group G. The general technique is first to embed G/T, where T is a maximal torus, as a maximal orbit of the adjoint representation of G, and then to extendto an embedding of G by using a maximal orbit of a faithful representation of G. In thisnote, we observe that in the cases G = Sp(n) or SU(n) an improved result is obtained byusing the “symplectic torus” S3 x … x S3 in place of T = S1 x … x S1. As in Rees's construction, the normal bundle of the embedding of G is trivial.

Menelaus's theorem is

If a transversal DEF cut the sides of triangle ABC at the points D B F, then

and conversely.

where (Am m) represents a square of m rows and m columns of a's; (Bm r) represents a rectangle of m rows and r columns of b's; (Cr m) represents a rectangle of r rows and m columns of c's;, and (Orr) represents a square of r rows and r columns of zeros.

Let be a linear differential expression involving n independent variables xi the coefficients Aik Bi, and C being functions of the independent variables but not involving the dependent variable u. Associated with F(u) is the adjoint expression