Complex (associative) H*-algebras were introduced and studied in detail by Ambrose[1]; it was proved that every complex H*-algebra with zero annihilator is the l2-sum of a suitable family of topologically simple complex H*-algebras and that the H*-algebras (H) of all Hilbert-Schmidt operators on any complex Hilbert space H are the only topologically simple complex H*-algebras. In a recent paper [2] Balachandran and Swaminathan observe that the reduction of the theory of real H*-algebras to the topologically simple case follows easily with minor changes of the complex argument, and they prove a theorem describing topologically simple real H*-algebras. This theorem can be equivalently reformulated as follows.

In (5), Peter Freyd recently raised the question of whether every Grothendieck topos could be obtained from the topos of sets by means of the two constructions of taking sheaves on a locale and of taking continuous actions of a topological group (i.e. the topos-theoretic analogues of the set-theorists' techniques of forcing extensions and permutation models). He showed that these two constructions do suffice to within epsilon; provided we allow ourselves the freedom to take exponential varieties (4) (which do not change the internal logic of the topos) we can obtain every Grothendieck topos in this way.

1·1. Groups of the first kind. In [11], Patterson proved a hyperbolic space analogue of Khintchine's theorem on simultaneous Diophantine approximation. In order to state Patterson's theorem, some notation and terminology are needed. Let ‖x‖ denote the usual Euclidean norm of a vector x in k+1, k + 1-dimensional Euclidean space, and letbe the unit ball model of k + 1-dimensional hyperbolic space with Poincaré metric ρ. A non-elementary geometrically finite group G acting on Bk + 1 is a discrete subgroup of Möb (Bk+l), the group of orientation preserving Mobius transformations preserving Bk + 1, for which there exists some convex fundamental polyhedron with finitely many faces. Since G is non-elementary, the limit set L(G) of G – the set of limit points in the unit sphere Sk of any orbit of G in Bk+1 – is uncountable. The group G is said to be of the first kind if L(G) = Sk and of the second kind otherwise.

The technique adopted in some experiments employing Geiger-Mueller counters is described. The behaviour of a counter in which the extinction of the discharge is external is contrasted with behaviour of a counter in which extinction is internal and dependent on the addition to the gas of some organic vapour such as alcohol. It is shown that the internally extinguished type of counter may be used with a resistance of the order 50,000 ohms in series with it while still retaining all its useful features.

Methods of construction of internally extinguished counters are described, particular attention being paid to the question of developing a thin-walled type suitable for experiments employing the method of coincidences. The effect of using different wall materials and gases is investigated with a view to finding those most suitable.

The measurement of the quantum energy of hard γ-radiation by the method of coincidences is discussed and details of a reliable two-stage amplifying circuit with a resolving time T of 10−6 sec. are given. Various factors which may influence the value of T are noted.

Suppose is a symmetric square integrable kernel on the unit square [0, 1]2. Then

is a compact symmetric operator on the Hilbert space L2[0, 1]. H. Weyl (see [2]) has shown that, if then the eigenvalues

of T satisfy as n → ∞. We prove a related result. Let W12[0, 1]2 denote the space of all K(x, t) ε L2[0, 1]2 which are absolutely continuous in x for each t and absolutely continuous in t for each x, and the partial derivatives ∂K/∂x(x, t), ∂K/∂t(x, t) are both in L2[0, 1]2. We slow that the eigenvalues of any satisfy .

It is known as a result of various experiments on slow neutrons (1) that a heavy nucleus possesses an enormous number of energy levels which are very closely spaced if the nucleus is highly excited. Strong theoretical reasons for the existence of this great number of levels were given by Bohr (2) and since then various attempts have been made to calculate the number of energy levels of a heavy nucleus. In the first of these, due to Bethe(3), it was assumed that the interaction between the nucleons was small so that the nucleus could be treated as a gas. The alternative assumption, that we may consider the interaction to be large in comparison with the kinetic energy of the nucleons, was proposed by Bohr and Kalckar(4). In the present note we assume as our model a nucleus consisting of neutrons and protons independent of one another; we then have a neutron-gas in equilibrium with a proton-gas, Fermi-Dirac statistics being applied to both.

Although there are many variations of finite-difference methods of obtaining approximate numerical solutions to ordinary differential equations they share the common feature that they tend to treat an equation of a given type as a standard problem and take no account of any special characteristics the wanted solution may have. We here suggest an alternative procedure when the wanted solution exhibits exponential characteristics. In essence the idea is that if a differential equation has an exponential type solution y(x) it is useful to solve numerically, instead of the equation for y, the equation for u = logey. The error-building and stability characteristics are then those of u rather than y and consequently the accuracy of the solutions may be improved. Although there is nothing basically new in this, of course, the point that we demonstrate is that the differential equation in y can be solved numerically in such a manner that the transformation from y to u is not actually carried out, i.e. we retain the original dependent variable but take account of the exponential variation by modifying the integration formula. Consider for example, in the usual notation, the first-order equation

with a given initial condition y(x0) = y0. If x0, x1, …, xr, xn is a set of pivotal values of x;, usually assumed equally spaced so that xr+1 − xr = h, the usual approach replaces (1) by the formula

which, once the integral is expressed in terms of pivotal values of f using a difference series, represents a step-by-step formula for constructing successive values of y.

In this note we describe the natural coordinatizations of a Delzant space defined as a reduced phase space (symplectic geometry view-point) and give explicit formulas for the coordinate transformations. For each fixed point of the torus action on the Delzant polytope, we have a maximal coordinatization of an open cell in the Delzant space which contains the fixed point. This cell is equal to the domain of definition of one of the natural coordinatizations of the Delzant space as a toric variety (complex algebraic geometry view-point), and we give an explicit formula for the toric variety coordinates in terms of the reduced phase space coordinates. We use considerations in the maximal coordinate neighborhoods to give simple proofs of some of the basic facts about the Delzant space, as a reduced phase space, and as a toric variety. These can be viewed as a first application of the coordinatizations, and serve to make the presentation more self-contained.

In the proof of the Freudenthal–Weil theorem in, for example (5), essential use is made of the fact that if G and H are compact analytic groups and ø: G → H is a continuous epimorphism then ø(Z(G)0) = Z(H)0 where the subscript 0 denotes the identity component of a topological group G and Z(G) its centre. Although this is sufficient for the proof of the Freudenthal–Weil theorem it raises the interesting question as to whether actually ø(Z(G)) = Z(H) (from which the above would follow) and, if so, in what generality this can be expected. The present paper deals with this question, in more general form, as well as certain of its structural consequences.

Local cohomology functors are constructed for the category of cohomological functors on an essentially small triangulated category ⊺ equipped with an action of a commutative noetherian ring. This is used to establish a local-global principle and to develop a notion of stratification, for ⊺ and the cohomological functors on it, analogous to such concepts for compactly generated triangulated categories.

A matrix A is said to be tripotent whenever A3 = A. The study of tripotent matrices is of statistical interest since if the n × 1 real random vector X follows an N(0, I) distribution and A is a symmetric matrix then the real quadratic form X′AX is distributed as the difference of two independently distributed X2 variates if and only if A3 = A. In fact, a necessary and sufficient condition that A is tripotent is that there exist two idempotent matrices B and C such that A = B – C, and BC = 0. Using properties of diagonalizable matrices, we will prove several algebraic characterizations of r-potent matrices that extend the known results for tripotent matrices. Our first result will be to obtain an analogous decomposition for an arbitrary r-potent matrix.

In the theory of the iteration of a rational function or transcendental entire function R(z) of the complex variable z we study the sequence of natural iterates, {Rn(z):n = 0, 1,…}, of R, where

The domain of definition of the iterates is , the extended complex plane (if R is rational), and (if R is entire transcendental) with the topology of the chordal metric and euclidean metric respectively. Fatou(5) and Julia(9) developed a global theory of the iteration of a rational function. In (6) Fatou extended the theory of (5) to transcendental entire functions. A central role is played in the theory by the F-set, F(R), of R, R rational or entire, which is defined to be the set of points at which the family of iterates do not form a normal family in the sense of Montel.

The problem of whether there exist static solutions of Einstein's equations for two extended bodies surrounded by vacuum is investigated. Equilibrium conditions for the general static situation are derived and their physical meaning discussed. Special cases of the problem are solved by using these equilibrium conditions. In one of these cases, the essential assumptions are that the space time is axisymmetric and that the matter of which the two bodies are composed consists of perfect fluid. Solu tions are also obtained in which these restrictions are weakened.

From the fact that Hausdorff s-dimensional measure is a regular Carathéodory outer measure follows (see Saks (3), ch. II, §§ 6, 8) the standard result:Theorem A. If {En} is any increasing sequence of sets, then ∧sEn → ∧s(ΣEn) as n → ∞.

Since ∧s X is denned (for every set X) as , the problem arises whether for every positive δ and every increasing sequence of sets one can prove

Now there are four possible ways of defining ; it is the lower bound of taken over coverings ΣUr of X by convex sets, but in most applications it is not important whether the restriction on the convex sets is that they shall be(i) open and of diameter less than δ,(ii) closed and of diameter less than δ,(iii) open and of diameter not exceeding δ, or(iv) closed and of diameter not exceeding δ.

In this note a simple principle is explained for constructing a transformation group which is a free product of given transformation groups. The principle does not seem to have been formulated explicitly, though it has been used in a more or less vague form in the theory of discontinuous groups (see, for instance, L. R. Ford, Automorphic functions, vol. I, pp. 56–59). It is perhaps of interest that the formulation given here is purely set-theoretic, without any topology, and that it can apply to any free product, whatever the cardinal number of the set of factors. The principle is used to establish the closure under the formation of countable free product of the family of groups which can be represented as discontinuous subgroups of a certain group of rational projective transformations. (The word ‘discontinuous’ is used here in a weak sense, defined later.) Finally, these results are applied to give a new proof of the theorem of Gruenberg that a free product of residually finite groups is itself residually finite (K. W. Gruenberg: Residual properties of groups, Proc. London Math. Soc. (3), 7 (1957), 29–62. See Corollary (ii) of Theorem 4.1, p. 44). The present proof is completely different from Gruenberg's and seems to be of interest for its own sake, though it does not appear to lead to Gruenberg's other results in this connexion. I have to thank Mr W. J. Harvey and Mr C. Maclachlan for checking a first draft of this paper and pointing out a few errors.

An element B ∈ Bn, the braid group on n strings, which can be written as

in terms of the standard generators of Bn is called a split braid. It is easy to see that the resulting closed braid is the connected sum of the closed braids Ĉ and on k + 1 and n − k strings respectively (figure 1).

For a strut under thrust when buckling is resisted by a force proportional to the displacement, it is shown that for both a clamped and a pinned strut there is stability for any length when the thrust is less than a certain critical value; and the relation is found between the length and the first value of the thrust above this that will give buckling. A notion of the wave-length and number of nodes after buckling is obtained.

The results are applied to the theory of the formation of mountains by horizontal compression, and it is shown that the crust of the earth would be able to transmit, without buckling, stresses right up to the breaking stress across regions of continental extent, unless the depth down to the level of no strain were less than 16 metres.

An attempt is made to employ the results also to consider the stability of solid surface films under compression, and it is seen that other factors, making for stability, must enter besides rigidity and gravity. This is supplied by surface tension, and it then appears that collapse due to a weakness in the film must precede buckling.

A rough calculation is given of the frequency of vibration of an atom in a solid film.

We prove the result stated in the title, which answers the equicharacteristic case of a question of Vasconcelos.

By investigating the relation between growth and value distribution A. Melas established a qualitative version of a Picard type theorem for universal Taylor series, that is: every universal Taylor series on the open unit disk $D$, assumes every complex value with at most one exception on infinite subsets of $D$ that approach the boundary of $D$ rather slowly. On the other hand, we show that there are universal Taylor series on $D$ such that the infinite subset of $D$ on which exactly one value is assumed, can approach the boundary of $D$ arbitrarily fast. Hence in view of Melas' work our result is the best possible. We also study the problem of interpolation by universal Taylor series.