The Schwarzschild Interior Solution represents a static sphere the proper density of which has the same value throughout. Though it is sometimes referred to as an “incompressible” sphere it is physically unacceptable since (formally) the speed of sound within it is infinite. Perhaps the most natural analogue of the classical incompressible sphere is therefore a sphere such that the speed of sound is everywhere just equal to the speed of light. This paper investigates spheres of this kind in some detail.

Recall that the spectrum, σ(T), of a linear operator T in a complex Banach space is the set of complex numbers λ such that T—λI does not have a densely defined bounded inverse. It is known [7, § 5.1] that σ(T) is a closed subset of the complex plane C. If T is not bounded, σ(T) may be empty or the whole of C. If σ(T) ≠ C and T is closed the spectral mapping theorem, is valid for complex polynomials p(z) [7, §5.7]. Also, if T is closed and λ ∉ σ(T), (T–λI)−1 is everywhere defined.

Let X be a topological space equipped with a binary relation R; that is, R is a subset of the Cartesian square X×X. Following Wallace [5], we write Deviating from [7], we shall follow Wallace [4] to call the relation R continuous if RA*⊂(RA)* for each A⊂X, where * designates the topological closure. Borrowing the language from the Ordered System, though our relation R need not be any kind of order relation, we say that a subset S of X is R-decreasing (R-increasing) if RS ⊂ S(SR ⊂ S), and that S is Rmonotone if S is either R-decreasing or R-increasing. Two R-monotone subsets are of the same type if they are either both R-decresaing or both Rincreasing.

Recently a power series representation of Hypergeometric functions with matrix argument has been established. This representation involves a special type of spherical function from the theory of semi-simple Lie groups, called the zonal polynomials. A general theory of these polynomials is well established; however an explicit representation of them is lacking. This paper considers two integrals which are related to this explicit representation. The final paragraph considers a third integral which gives an application of a result from a previous paper of the author.

The lower radical of a module type. For a ring R with unit, the module type t(R) was defined in [6] as follows: t(0) = 0; t(R) = d if every free R-module has invariant rank; t(R) = (c, k) for integers c, k ≧ 1 if every free R-module of rank < c has invariant rank, while a free module of rank h ≧ c has rank h + nk for any integer n ≧ 0. The module types form a lattice under the ordering 0 < (c, k) < d and (c', k') ≦ (c, k) if and only if c ' ≦ c and k' Ⅰ k. Two of the basic theorems on types are:

A. [6; Theorem 2, p. 115] If R → R' is a unit-preserving homomorphism, then t(R')≦ t(R).

Various semigroups of partial transformations (and more generally, semigroups of binary relations) on a set have been studied by a number of Soviet mathematicians; to mention only a few: Gluskin [2], Ljapin [4], Shutov [6], Zaretski [7], [8]. In their study the densely embedded ideal of a semigroup introduced by Ljapin [4] plays a central role. In fact, a concrete semigrou Q is described in several instances by its abstract characteristic, namely either by a set of postulates on an abstract semigroup or by a set of postulates (which are usually much simpler) on an abstract semigroup S which is a densely embedded ideal of a semigroup T isomorphic to Q. In many cases, the densely embedded ideal S is a completely 0-simple semigroup. The following theorem [3, 1.7.1] reduces the study of a semigroup Q with a weakly reductive densely embedded ideal S to the study of the translational hull of S:

Theorem (Gluskin). If S is a weakly reductive densely embedded ideal of a semigroup Q, then Q is isomorphic to the translational hull ω(S) of S.

Unsuccessful efforts to synthesize benzo[a]biphenylene and dibenzo-biphenylenes by the dehydrohalogenation of suitable intermediates are described. 2-o-Chlorophenyl-2-phenylpropionitrile with potassamide in liquid ammonia failed to give 8-cyano-7-phenylbicyclo[4,2,o]octa-I,3,5-triene, while diethyl 2-bromobenzylphenylmalonate with sodium in dry ethanol yielded I-phenylisochroman-3-one after hydrolysis. Bromination of 3,3′,4,4′-tetrahydro-2,2′-binaphthyl was accompanied by spontaneous dehydrobromination to yield 2,2′-binaphthyl, and interaction of phenyl-o-chlorophenylbromomethane with phenyllithium followed by chloracetic acid gave 1,2-di-o-chlorophenylethane.

The problem of determining, within the limits of the classical theory of elasticity, the displacements and stresses in the interior of a semi-infinite solid (z ≧ 0) when a part of the boundary surface (z = 0) is forced to rotate through a given angle ω about an axis which is normal to the undeformed plane surface of the solid, has been discussed by several authors [7, 8, 9, 1, 11, and others]. All of this work is concerned with rotating a circular area of the boundary surface and the field equation to be solved is, essentially, J. H. Mitchell's equation for the torsion of bars of varying circular cross-sections.

It is the custom of our Society that the President of the day should at some period in his term of office address the Fellows. The manner in which he interprets this charge will, of course, vary according to individual preference, but for obvious reasons, he usually elects to give his personal interpretation of that area of science in which he is himself most deeply concerned.

Our Society was founded in the days before science was split into highly specialized sections and long before the advent of the specialist societies restricted to one scientific discipline, such as the Chemical Society, the Biochemical Society or the Society for Experimental Biology. This, in the opinion of many of our Fellows, is its main strength, since, as a general Society concerned with all aspects of science, it can exert an integrative function; in fact, it serves many of the purposes of the Academies which are so well known in other parts of Europe, as is made clear in the Letters Patent granting our Armorial Bearings.

The translational hull Ω(S) of a semigroup S plays an important role in the theory of ideal extensions of semigroups. In fact, every ideal extension of S by a semigroup T with zero can be constructed using a certain partial homomorphism of T\0 into Ω(S); a particular case of interest is when S is weakly reductive (see §4.4 of [3], [2], [7]). A theorem of Gluskin [6, 1.7.1] states that if S is a weakly reductive semigroup and a densely embedded ideal of a semigroup Q, then Q and Ω(S) are isomorphic. A number of papers of Soviet mathematicians deal with the abstract characteristic (abstract semigroup, satisfying certain conditions, isomorphic to the given semigroup) of various classes of (partial) transformation semigroups in terms of densely embedded ideals (see, e.g., [4]). In many of the cases studied, the densely embedded ideal in question is a completely 0-simple semigroup, so that Gluskin's theorem mentioned above applies. This enhances the importance of the translational hull of a weakly reductive, and in particular of a completely 0-simple semigroup. Gluskin [5] applied the theory of densely embedded ideals (which are completely 0-simple semigroups) also to semigroups and rings of endomorphisms of a linear manifold and to certain classes of abstract rings.

In recent years there have appeared solutions of several integral equations of the type

in which the kernel K(x) contains (as a factor) one of the classical orthogonal polynomials or a hypergeometric function.

Let S be a semigroup whose set E of idempotents is non-empty. We define a partial ordering ≧ on E by the rule that e ≧ f and only if ef = f = fe. If E = {ei: i∈ N}, where N denotes the set of all non-negative integers, and if the elements of E form the chain

then S is called an ω-semigroup.

It is the purpose of this note to discuss the solution of the pair of series

where F(x) and G(x) are given and the coefficients an are to be determined.

By means of a generalized ring of quotients multiplicative ideal theory is studied in an arbitrary (associative) ring. A suitable generalization of the concept of maximal order is given and factorization theorems are obtained for the nonsingular (two sided) ideals, which generalize the theorems of Artin and E. Noether.

This paper studies two particular cases of the general 2-parameter eigenvalue problem namely

where A, B, B1, B2, C, C1, C2 are self-adjoint operators in Hilbert space, all except A being bounded. The disposable parameters λ and μ have to be determined so that the equations have non-trivial solutions x, y.

On the assumption that the solution is known for ∊ = o, solutions are constructed in the form of series for λ, μ, x, y as power series in ∊ with finite radius of convergence.