1. Introduction. The problem of extending Dirac's equation of the electron to general relativity has been attacked by many authors, by methods which fall roughly into either of two classes according as the formulation does or does not require the introduction of a local Galilean system of coordinates at each point of space-time. As examples of the former class we mention the methods of Fock (1929) and of Cartan (1938), and as representing the latter class the method described by Ruse (1937). Also, Whittaker (1937) discovered a vector whose vanishing is completely equivalent to the Dirac equations, but this method, unlike the others in the second category, does not apply the Riemannian technique to spinors but only to vectors and tensors derived from these. Now Cartan has denied the possibility of fitting a spinor into Riemannian Geometry if his point of view of spinors is adhered to, and this he argues accounts for the “choquant” properties with which they have been endowed by the geometricians in order to enable them to write down an expression of the usual form for the covariant derivative of a spinor. Consequently, doubt has been cast on the compatibility of the various methods, so in this paper an attempt is made to clarify the matter by working out explicitly the case of the general metric by some of the more important of these methods.

1. The following condition may be new; it does not appear in any of the books:—

The condition that the straight line

should be a normal to the general conic,

referred to rectangular axes, is

where Σ is written for

and Δ, A, B, C, F, G, H have their usual meanings.

Let S be a compact Hausdorff space and let Φ: C(S)→E be a linear operator defined on the space of real-valued continuous functions on S and taking values in a (real) topological vector space E. Then Φ is called exhaustive (7) if given any sequence of functions fn ∈ C(S) such that fn ≧ 0 and

then Φ(fn)→0 If E is complete then it was shown in (7) that exhaustive maps are precisely those which possess regular integral extensions to the space of bounded Borel functions on S; this is equivalent to possessing a representation

where μ is a regular countably additive E-valued measure defined on the σ-algebra of Borel subsets of S.

In an earlier paper (2) we considered the following question “If S is a cyclic subgroup of a finite group G and S ∩ F(G) = 1, where F(G) is the Fitting subgroup of G, does there necessarily exist a conjugate Sx of S in G with S ∩ Sx = l?” and we gave an affirmative answer for G simple or soluble. In this paper we answer the question affirmatively in general (in fact we prove a somewhat stronger result (Theorem 3)). We give an example of a group G with a cyclic subgroup S such that (i) no nontrivial subgroup of S is normal in G and (ii) no x exists for which S ∩ Sx = 1.

Let the Laurent expansion of the Riemann zeta function ξ(s) about s=1 be written in the form

It has been discovered independently by many authors that, in terms of this notation, the coefficient

The following is an elementary discussion of certain propositions in solid geometry which are commonly left to a later stage.

Let ABCD be a quadrilateral formed of rods freely jointed, let points E, F, G, H be taken on AB, BC, CD, DA respectively and let EG, FH be joined by rods freely jointed to the former. Let us investigate whether the resulting framework is rigid.

In a recent paper Professor A. C. Aitken directed attention to a little known type of inverse factorial series, which he calls inverse central factorial series. He showed that this type of series is a very powerful means of asymptotic representation of functions of a certain type, and concluded his note with the-remark that there is here scope for considerable investigation. As a modest contribution to this investigation, the asymptotic theory, foreshadowed in the last paragraph of Aitken's paper, will be given here: it is hoped to give the convergence theory of inverse central factorial series later.

In some chemical reaction–diffusion processes, the reaction takes place only at some local sites, due to the presence of a catalyst. In this paper we study the well-posedness of a model problem of this type. Sufficient conditions are found to ensure global existence and finite time blowup. The blowup rate and the blowup set are also investigated in the case of special nonlinearity.

Most lecturers on Analytical Statics only consider the following catenaries: the common catenary, the parabolic catenary of the suspension bridge, and, less frequently, the catenary of uniform strength. Of these the second only represents the form of the suspension bridge chain when the weight of the chain is neglected in comparison with that of the roadway, while the first represents the opposite extreme when the weight of the chain alone is taken into account.

In [3] Laffey has shown that if Z is a cyclic subgroup of a finite subgroup G, then either a nontrivial subgroup of Z is normal in the Fitting subgroup F(G) or there exists a g in G such that Zg∩Z = 1. In this note we offer a simple proof of the following generalisation of that result:

Theorem. Let G be a finite group and X and Y cyclic subgroups of G. Then there exists a g in G such that Xg∩Y⊴F(G).

In each metric space (X, d) there is defined the space Lip X of complex-valued, bounded, and uniformly Lipschitzian functions. In the algebra Lip X, it is natural to ask for ideals closed in various notions of convergence, and also to identify the invertible elements. In particular, are the invertible elements exactly those with no zero in X? Wiener's Tauberian Theorem in Fourier analysis is the first and most remarkable example of this harmonious state of affairs. A moment's reflection confirms that, for the algebra Lip X, this is true only for compact metric spaces X, the trivial examples in our investigation. We therefore introduce a type of convergence weaker than convergence in norm; it has already proved useful in some problems in descriptive set theory and reflects in a subtle way the metric properties of X. A sequence (fn) in Lip X converges strongly to g, written s – limfn=g, if ∥fn∥≦C in the Banach space Lip X and lim fn(x)=g(x) for each element x of X. In Section 3 we explain how this is really a type of convergence in the dual space of a certain Banach space . This brings us to the edge of some recondite questions about iterated (or even transfinite) limits, and we have adhered to the notion of strong limits to avoid these questions. To illustrate the differences between these two approaches, we mention this problem: which maximal ideals of Lip X are closed with respect to strong convergence of sequences? This is not the problem studied in Section 1.

The object of the present note is to obtain a number of infinite integrals involving Struve functions and parabolic cylinder functions. 1. G. N. Watson(1) has proved that

From (1)

follows provided that the integral is convergent and term-by-term integration is permissible. A great many interesting particular cases of (2) are easily deducible: the following will be used in this paper.