In this paper we derive an expression for the point source Green's function for the reduced wave equation, valid in an angular sector whose angle is equal to a rational multiple of π. This Green's function can be used to find new expressions for the field produced by the diffraction of a spherical wave by a wedge whose angle can be expressed as a rational multiple of π. The expressions obtained will be in the form of source terms and real integrals representing the diffracted field. The general result obtained is used to derive a new representation for the solution of the problem of diffraction by a mixed hard–soft half plane.

Let Aθ be the universal C*-algebra generated by a pair U, V of unitaries satisfying VU = e2πiθUV where θ = p/q with p, q∈ℤ, 0 < p < q and (p, q) = 1. It is shown that every endomorphism of Aθ is unital, injective and centre preserving. The corresponding surjections of T2 are characterized and it is shown that the determinants of the associated endomorphisms of K1(Aθ)≅ℤ2 are congruent to 1 modulo q.

A famous chain of theorems, due originally to de Longchamps (l) and afterwards rediscovered by Pesci(2), Morley(3) and Grace(4), goes as follows:

(1) Given four lines in a plane, the four circumcentres O3 of the triangles formed by omitting each one of the lines in turn lie all on the same circle C4 with centre O4 say.

(2)Given five lines in a plane, the centres O4 of the five circles C4 obtained by omitting each of the five lines in turn lie all on the same circle C6 with centre O5 say.

And in general

(3) Given (n+1) lines in a plane, the (n+1) centres On of the circles Cn+1 formed by omitting each of the lines in turn lie all on the same circle Cn+1 with centre On+1.

The investigation of a recently devised model in which the permittivity profile takes the form of an asymmetrical Epstein layer is adapted for use in a gyrational medium. Reflexion and transmission coefficients for normally incident electromagnetic waves are obtained and resonance properties are discussed.

Since the publication of the fundamental papers by Lifshitz (1, 2) and Montroll and Potts (3, 4) many authors have investigated the effect of an isotopic impurity on the lattice vibrations of a harmonic crystal at zero temperature. A fairly broad knowledge is now available on scattering amplitudes, localized modes and resonance modes (6, 7). Nevertheless, as pointed out by Maradudin and Montroll (see (7), p. 430), a closed form solution to the problem has been found only for the one-dimensional crystal, the work done on two and three-dimensional crystals being predominantly numerical. Unfortunately the one-dimensional crystal, as an approximation for a real crystal is an oversimplified model, incapable as it is of exhibiting resonance modes. To the author's knowledge the most significant exact result concerning the classical behaviour at zero temperature of crystals having a dimensionality higher than one is the connexion, calculated by Mahanty et al. (5) between localized mode frequency and impurity mass for the case of a square lattice undergoing planar vibrations.

The general problem can be stated as follows. Take a Brownian motion Bt started at − x < 0, and consider the additive functional At = ∫L(a,t)m(da), where L(a, t) is the Brownian local time. We suppose that m = m+ — m−, where these are positive measures supported respectively on (0, ∞) and (— ∞, 0). Then, with the equalization time defined by T = inf {t > 0: At = 0}, we ask for an explicit evaluation of the law π (x,dy) = P−x[BT∈dy]. In [8, 9] we showed how π (x,dy) can be obtained by solving an integral convolution equation of Wiener-Hopf type. The method used there exploits a technique of Ray [10].

In this paper, using the results of the preceding paper, I develop a modification of Hahn's method (1) for calculating the electromagnetic field inside a type of axially symmetric resonator. A process of successive approximation is given for solving the equation which leads to the resonant wave-length λ, both in the case where all the resonator parameters are given and we wish to find λ, and also in the case where λ is given and one of the parameters (the outer radius) is to be found. Some numerical results are given.

One page in Ramanujan's lost notebook is devoted to claims about a certain integral with two parameters. One claim gives an inversion formula for the integral that is similar to the transformation formula for theta functions. Other claims are remindful of Gauss sums. In this paper we prove all the claims made by Ramanujan about this integral.

In this paper a new generalization of the Meijer transform is given. It also generalizes the Mainra transform and the transform due to Varma. An inversion theorem is established and the result obtained has been illustrated by several examples.

Let G be a matrix. If there does not exist an infinite set of non-zero elements such that no two are on the same line (row or column), the matrix is said to be of finite term-rank. A set of lines such that each non-zero element belongs to some line of the set is said to be a covering of G.

The experiments described in this paper were carried out in the University of Sydney during November 1924 and, owing to the short time which was available, are of a purely preliminary character. The following account is given as they are not likely to be resumed in the immediate future.

In this paper we generalise the main result of [4] for manifolds that are not necessarily Einstein. In fact, we obtain an upper bound for the volume of a locally volume-minimising closed hypersurface Σ of a Riemannian 5-manifold M with scalar curvature bounded from below by a positive constant in terms of the total traceless Ricci curvature of Σ. Furthermore, if Σ saturates the respective upper bound and M has nonnegative Ricci curvature, then Σ is isometric to 𝕊4 up to scaling and M splits in a neighbourhood of Σ. Also, we obtain a rigidity result for the Riemannian cover of M when Σ minimises the volume in its homotopy class and saturates the upper bound.

Hua domains, generalized Hua domains and Hua constructions, named after the great Chinese mathematician Luogeng Hua (Loo-Keng Hua), are generalizations of Cartan–Hartogs domains introduced by Weiping Yin around the end of the 20th century. In this paper, we give a complete description of automorphism groups of generalized Hua domains. We also discuss the corresponding problem for Hua constructions.

If a second order ordinary differential equation has a simple or a double pole at a point zfl, then the standard Liouville-Green approximation could sometimes be valid near that point. In this paper we present an asymptotic series solution that is always valid near a double pole. A solution for that of a simple pole is also indicated. Asymptotic validity is proved.

1. In a recent paper I have given some definite integrals involving Legendre functions which, as a limiting case, give known results involving Bessel functions. In another paper I have shown how some integrals involving Bessel functions can be obtained from Bateman's integral

and the well-known expansion

We prove a theorem describing the limiting fine-scale statistics of orbits of a point in hyperbolic space under the action of a discrete subgroup. Similar results have been proved only in the lattice case with two recent infinite-volume exceptions by Zhang for Apollonian circle packings and certain Schottky groups. Our results hold for general Zariski dense, non-elementary, geometrically finite subgroups in any dimension. Unlike in the lattice case orbits of geometrically finite subgroups do not necessarily equidistribute on the whole boundary of hyperbolic space. But rather they may equidistribute on a fractal subset. Understanding the behavior of these orbits near the boundary is central to Patterson–Sullivan theory and much further work. Our theorem characterises the higher order spatial statistics and thus addresses a very natural question. As a motivating example our work applies to sphere packings (in any dimension) which are invariant under the action of such discrete subgroups. At the end of the paper we show how this statistical characterization can be used to prove convergence of moments and to write down the limiting formula for the two-point correlation function and nearest neighbor distribution. Moreover we establish a formula for the 2 dimensional limiting gap distribution (and cumulative gap distribution) which also applies in the lattice case.

In what follows (E, p, B) will always denote a fibering in the sense of Serre, † with arc-wise connected base B, projection p:E → B, and fibre F = p−1(b), b ɛ B. Coefficients will always be taken in a commutative ring A with a unit element, and the cohomology will be singular (cubical), as in (3).

We construct free abelian subgroups of the group U(AΓ) of untwisted outer automorphisms of a right-angled Artin group, thus giving lower bounds on the virtual cohomological dimension. The group U(AΓ) was studied in [5] by constructing a contractible cube complex on which it acts properly and cocompactly, giving an upper bound for the virtual cohomological dimension. The ranks of our free abelian subgroups are equal to the dimensions of principal cubes in this complex. These are often of maximal dimension, so that the upper and lower bounds agree. In many cases when the principal cubes are not of maximal dimension we show there is an invariant contractible subcomplex of strictly lower dimension.

Mislin and Talelli showed that a torsion-free group in with periodic cohomology after some steps has finite cohomological dimension. In this note we look at similar questions for groups with torsion by considering Bredon cohomology. In particular we show that every elementary amenable group acting freely and properly on some × Sm admits a finite dimensional model for G.