Several years ago, Fisher and Fuller(1) proved that if P is a real square matrix with all members of its ‘nested set’ of principal minors non-zero, then there exists a real diagonal matrix, D, such that the characteristic roots of DP are all real, negative and distinct. This interesting and powerful result, used by the authors to derive further results concerning convergence of linear iterative processes, has since also proved of interest to economists studying the stability of economic general equilibrium.

Abelian theorems for the H-transform of functions and generalized functions are obtained as the complex variable of the transform approaches zero or infinity in a wedge domain in the right half plane. Quasi-asymptotic behaviour (q.a.b.) of the H-transformable generalized functions is defined. A structure theorem for generalized functions possessing q.a.b. is proved and is applied to obtain the asymptotic behaviour of the H-transform of generalized functions having q.a.b. The theorems are illustrated by examples.

The basic equations of non-relativistic magnetohydrodynamics are briefly reviewed. Beginning with the energy-momentum tensor for the coupled electromagnetic and fluid fields relativistic energy and momentum equations are formulated. The effect of heat conduction is included and relativistic constitutive equations are discussed. A relativistic first law of thermodynamics is developed in terms of internal energy and entropy. Finally, the second law and entropy production rate are derived in a covariant fashion.

1. Introduction. The group U(H) of unitary transformations of a complex Hilbert space H, endowed with its strong operator topology, is of interest in the study of unitary representations of a topological group. The unitary transformations of H induce a group U(Ĥ) of transformations of the associated projective space Ĥ. The projective unitary group U(Ĥ) with its strong operator topology is used in the study of projective (ray) representations. U(Ĥ) is, as a group, the quotient of U(H) by the subgroup S1 of scalar multiples of the identity. In this paper we prove that the strong operator toplogy of U(Ĥ) is in fact the quotient of the strong operator topology on U(H). This is related to the fact that U(H) is a principal bundle over U(Ĥ) with fibre S.

Let Qn be a real indefinite quadratic form in n variables x1, x2,…, xn, of determinant D ≠ 0 and of type (r, s), 0 < r < n, n = r + s. Let σ denote the signature of Qn so that σ = r − s. It is known (see e.g. Blaney(4)) that, given any real numbers c1 c2, …, cn, there exists a constant C depending upon n and σ only such that the inequality

has a solution in integers x1, x2, …, xn. Let Cr, s denote the infimum of all such constants. Clearly Cr, s = Cs, r, so we need consider non-negative signatures only. For n = 2, C1, 1 = ¼ follows from a classical result of Minkowski on the product of two linear forms. When n = 3, Davenport (5) proved that C2, 1 = 27/100. For all n and σ = 0, Birch (3) proved that Cr, r = ¼. In 1962, Watson(18) determined the values of Cr, s for all n ≥ 21 and for all signatures σ. He proved that

Watson also conjectured that (1·2) holds for all n ≥ 4. Dumir(6) proved Watson's conjecture for n = 4. For n = 5, it was proved by Hans-Gill and Madhu Raka(7, 8). The author (12) has proved the conjecture for σ = 1 and all n. In the preceding paper (13) we proved that C5, 1 = 1. In this paper we prove Watson's conjecture for σ = 2, 3 and 4.

We investigate four properties related to an elliptic curve Et in Legendre form with parameter t: the curve Et has complex multiplication, E−t has complex multiplication, a point on Et with abscissa 2 is of finite order, and t is a root of unity. Combining all pairs of properties leads to six problems on unlikely intersections. Using a variety of techniques we solve these problems with varying degrees of effectivity (and for three of them we even present the list of all possible t).

1. A method has lately been developed by N. Muschelišvili (1) for the solution of problems of the slow two-dimensional motion of viscous liquid and of the corresponding problems of plane stress and plane strain, in cases in which the area in the x, y-plane that is concerned can be represented conformally on the interior of the circle |ζ| = 1 in the ζ-plane by a relation of the form z = x + iy = r(ζ), where r(ζ) is a rational function of ζ. In most problems in which the method has been used the function r(ζ) has been a simple one, but it is of importance to consider a rational function of as general a form as possible since, given any relation z = f(ζ), it will usually be possible to find a rational function that approximates to f(ζ) throughout the circle |ζ| = 1 and for a close approximation a complicated function r(ζ) will in general be required.

In a Newtonian fluid, in which the body forces are conservative, in which the pressure is a function only of density and in which the kinematic viscosity v is uniform, the vorticity ω satisfies the equation

where u is the velocity field, so that ω = ▿ × u. This equation is normally derived by taking the curl of the Navier–Stokes equations. However, the vorticity has many interpretations in terms of the angular velocity of elements of fluid and it is natural to expect that (1) can be derived by equating the rate of change of the angular momentum of a small material volume element about its centre of mass with the moment of the forces acting on the element. Such a derivation is presented here in the hope that it may be of pedagogic interest.

Cusp singularities were introduced and described in detail in Hirzebruch's fundamental paper [3] (se recall some of the basic results in § 1 below). They form a natural and well-behaved class, included in Laufer's ‘minimally elliptic’ singularities [5]. Those which occur on hypersurfaces appear also as hyperbolic singularities in Arnold's [1] classification of 1-modal singularities.

Conrad ((2)), has shown that any lattice group which obeys (C.F.) each strictly positive element exceeds at most a finite number of pairwise orthogonal elements may be constructed, from a family of simply ordered groups, by carrying out, alternately, the operations of forming finite direct sums and lexico extensions, at most a countable number of times. The main result of this paper, Theorem 3.1, gives necessary and sufficient conditions for a multilattice group, which obeys (ℋ*), to be isomorphic to a multilattice group which is constructed from a family of almost ordered groups, by carrying out, alternately, the operations of forming arbitrary direct sums and lexico extensions, any number of times; we call such a group a lexico sum of the almost ordered groups.

The famous conjecture of Watson(11) on the minima of indefinite quadratic forms in n variables has been proved for n ≤ 5, n ≥ 21 and for signatures 0 and ± 1. For the details and history of the conjecture the reader is referred to the author's paper(8). In the succeeding paper (9), we prove Watson's conjecture for signature ± 2 and ± 3 and for all n. Thus only one case for n = 6 (i.e. forms of type (1, 5) or (5, 1)) remains to he proved which we do here; thereby completing the case n = 6. This result is also used in (9) for proving the conjecture for all quadratic forms of signature ± 4. More precisely, here we prove:

Theorem 1. Let Q6(x1, …, x6) be a real indefinite quadratic form in six variables of determinant D ( < 0) and of type (5, 1) or (1, 5). Then given any real numbers ci, 1 ≤ i ≤ 6, there exist integers x1,…, x6such that

Multilattice groups have been introduced by Benado ((1)) and considered by Vaida ((9)), who has initiated the study of the structure of a class of these groups. The purpose of this paper is to further the investigation begun by Vaida.

Basic in the work contained here is the set H consisting of the differences between the minimal upper bounds of pairs of elements of a multilattice group G. By means of conditions imposed on G and based on H, we determine the structure of the congruence relations on a multilattice group and study direct sums and lexicographic products and extensions of such groups (sections 2, 3).

It is well known that a path-connected Hausdorff space is arc-connected. Indeed, given a path P, that is, a continuous map. f of the unit interval I0(0 ≤ t ≤ 1) into the Hausdorff space X, there is an arc joining the points f(0), f(1) (supposed distinct) which is obtained by ‘cutting out loops in P’. More precisely, there exist a continuous increasing map α of I0 onto itself and a homeomorphism φ of I0 onto φ(I0) ⊂ X, such that if [t0, t1] is any maximal interval of constancy of α (including the case t0 = t1) then f(t0) = f(t1) = φοα(t0). The function φ, α, can be defined by a constructive process.

The present note was inspired by the desire to see whether the exposition of the theory of the Segre cubic primal of ten nodes was simplified by using only five coordinates instead of the six redundant coordinates introduced by Stéphanos and Castelnuovo. But the simple remark that the ten nodes may be separated into two simplexes which are polars of one another in regard to a quadric—which may or may not be novel—suggests the comparison of the theory of five associated lines in space of four dimensions with familiar properties of eight associated points in three dimensions; especially as it appears (§ 8) that the ten nodes do not form an associated set. With the repetition, for the sake of clearness, of several results which are familiar in general terms, there seems enough novelty to make the note of some utility. The form found for the equation of the Segre primal in five coordinates (§ 5) seems also noticeable.

In (2) I described a canonical isometric representation of an arbitrary real Banach space X by vector-valued functions (with the uniform norm) on a compact Hausdorif space ω with the following properties: (1) the representing function space is invariant under multiplications by continuous real functions on ω; (2) the norm of each representing function, as a real non-negative function on ω, is upper semicontinuous; and (3) this decomposition of X is maximally fine. I called attention to the class of spaces X for which at every point of ω the component space of this representation is one-dimensional or 0, so that the representing functions are in effect real valued. I propose to call such Banach spaces square, because of the shape of the unit ball in the two-dimensional case. In (2) I stated without proof, erroneously as it turns out, that the class of square spaces coincides with what Lindenstrauss in (4) called G-spaces. The primary purpose of this paper is to show that the class of square spaces is actually properly contained in that of G-spaces. It is known ((2), p. 620, Example 1) that it contains properly the class of continuous function spaces C(ω). Among G-spaces are the M-spaces treated by Kakutani as Banach lattices (3). I shall show further that neither class, square spaces or M-spaces (regarded now purely as Banach spaces), contains the other.

This paper treats two related sets of problems in the theory of random graphs. In Sections 2 and 3 we study random spanning subtrees of a complete graph (or, equivalently, random labelled trees). It is shown that the number of common edges of two such random trees asymptotically has a Poisson distribution with expectation 2. Similar results are obtained for the number of edges in the intersection or union of more than two random trees.

The main result of this paper is an oscillation theorem on linear self-adjoint differential systems and a corresponding eigenvalue problem. It establishes a formula between the number of focal points of a so-called conjoined basis of the differential system on a given compact interval and the number of eigenvalues which are less than the given eigenvalue parameter. It extends an earlier result of the author and generalizes an oscillation theorem of M. Morse. Among others the proof of the theorem requires a formula on the index of the difference of symmetric solutions of a corresponding Riccati matrix differential equation. This index formula is the other new result presented.

We propose an exceptionally short and effortless approach to the fundamental facts of the representation theory of compact groups on Hilbert spaces and expand our methods to prove some generalizations concerning representations on arbitrary locally convex vector spaces.

A repetition has been made, with the introduction of electrical counting methods, of the experiment of Bothe and Geiger on the simultaneous production of a scattered quantum and of a recoil electron in the scattering of X-rays. Evidence has been obtained that such coincidences do actually occur, in agreement with the results of the original experiment, and with the Compton theory of scattering.

It is pointed out that the free particle, the harmonic oscillator and certain other systems are such that the Feynman summation over histories is independent of the class of histories summed over. These examples are therefore useless as tests of methods of defining and carrying out the summation.