Hughes and Mann(1) have shown that, under certain conditions, the distribution curve of the energies of electrons scattered inelastically by an atom or molecule has the same shape as that of the X-ray Compton line. Hughes and his collaborators (1–5) have applied the method to an examination of a number of atoms and molecules in the gaseous state under which condition it can be assumed that each atom or molecule acts as an independent scattering centre. Coulson and the author (6–11) have applied quantum methods to derive a theoretical shape of the Compton line for certain of the molecules for which wave functions are available and have discussed the discrepancy between the experimental and the theoretical results(10, 11).

Let {kr} (r = 0, 1, 2, …; 1 ≤ kr ≤ h) be a positively regular, finite Markov chain with transition matrix P = (pjk). For each possible transition j → k let gjk(x)(− ∞ ≤ x ≤ ∞) be a given distribution function. The sequence of random variables {ξr} is defined where ξr has the distribution gjk(x) if the rth transition takes the chain from state j to state k. It is supposed that each distribution gjk(x) admits a two-sided Laplace-Stieltjes transform in a real t-interval surrounding t = 0. Let P(t) denote the matrix {Pjkmjk(t)}. It is shown, using probability arguments, that I − sP(t) admits a Wiener-Hopf type of factorization in two ways for suitable values of s where the plus-factors are non-singular, bounded and have regular elements in a right half of the complex t-plane and the minus-factors have similar properties in an overlapping left half-plane (Theorem 1).

Ascending series. Let G be a group and λ an ordinal number. An ascending series of G of type λ is a set of subgroups Gα of G, defined for all α ≤ λ, and such that (i) G0 = 1, Gλ = G; (ii)Gα is a proper normal subgroup of Gα+1 for all α < λ (iii) for all limit ordinals μ ≤ λ.

A simple sufficient condition is given for the existence of a lacuna in two-dimensional wave propagation governed by an equation of the second order. This work was inspired by Petrowsky's very general work (Petrowsky (3)) but for waves in two space dimensions we do not need the sophisticated machinery developed by him.

In a previous paper (4) central limit theorems were obtained for sequences of m-dependent random variables (r.v.'s) asymptotically stationary to second order, the sufficient conditions being akin to the Lindeberg condition (3). In this paper similar theorems are obtained for sequences of m-dependent r.v.'s with bounded variances and with the property that for large n, where s′n is the standard deviation of the nth partial sum of the sequence. The same basic ideas as in (4) are used, but the proofs have been simplified. The results of this paper are examined in relation to earlier ones of Hoeffding and Robbins(5) and of the author (4). The cases of identically distributed r.v.'s and of vector r.v.'s are mentioned.

The aim of this paper is twofold: first, to construct the compact regular coreflection of uniform frames, that is, the frame counterpart of the Samuel compactification of uniform spaces (Samuel [10]), and then to use this for a new description of the completion of a uniform frame, as an alternative to those previously given by Isbell [6] on the one hand and Kříž [8] on the other. In addition, we present a few further results, as well as new proofs of known ones, that are naturally connected with completions and arise particularly easily from our approach to them. Most prominently among these, we identify the uniform space of minimal Cauchy filters of a uniform frame as the spectrum of its completion.

We denote by b(n) the number of ways of expressing the positive integer n as the sum of powers of 2 and we call b(n) ‘the binary partition function’. This function has been studied by Euler (1), Tanturri (2–4), Mahler (5), de Bruijn(6) and Pennington (7). Euler and Tanturri were primarily concerned with deriving formulae for the precise calculation of b(n), whereas Mahler deduced an asymptotic formula for log b(n) from his analysis of the functions satisfying a certain class of functional equations. De Bruijn and Pennington extended Mahler's work and obtained more precise results.

This paper concerns the existence and uniqueness of non-negative solutions of non-linear Volterra integral equations of the type

and

where the kernel function k(.,.) is non-negative and sufficiently smooth, and either 0 < p < 1 or – 1 < p < 1. We will consider also the corresponding Fredholm equations

and

We present a contribution to the structure theory of locally compact groups. The emphasis is on compactly generated locally compact groups which admit no infinite discrete quotient. It is shown that such a group possesses a characteristic cocompact subgroup which is either connected or admits a non-compact non-discrete topologically simple quotient. We also provide a description of characteristically simple groups and of groups all of whose proper quotients are compact. We show that Noetherian locally compact groups without infinite discrete quotient admit a subnormal series with all subquotients compact, compactly generated Abelian, or compactly generated topologically simple.

Two appendices introduce results and examples around the concept of quasi-product.

If Tαβ is the energy-tensor, then, for any vector field ξα, the equation

follows from the energy equation

Suppose first that the physical system considered is complete, i.e. that the energytensor vanishes beyond some world tube Z. Let L be a time-like world line running inside the tube. We integrate both sides of (a) over a portion of Z, and we transform an integral over a four-dimensional region into a linear integral over L. We obtain the variational equation

in which the m's are tensors characteristic of the physical system. An essential feature of the m's is that they are symmetrical in their two last superscripts. Equation (c) has to be satisfied by every field ξα, provided that the ξ's vanish, with all their derivatives, at the ends of the integration path.

For any real function h(t) which is continuous and monotonic increasing for t > 0 with , Hausdorff (10) in 1918 denned a Carathéodory measure with respect to h(t) which has subsequently been known as Hausdorff measure. For analysing sets in Euclidean space, these measures have proved both useful and interesting. Given a real function Φ(t) which is continuous and monotonic decreasing for t > 0 with , Frostman(9) in 1935 denned capacity with respect to Φ(t). Lebesgue measure in Euclidean k-space is a special case of Hausdorff measure, and capacity with respect to Φ(t) becomes logarithmic capacity or Newtonian capacity in the cases , Φ(t)=1/t, respectively. The interrelationship between h-measure and Φ-capacity has been of interest in both directions: (i) in applications to function theory one may be able to determine whether or not a set has positive capacity by examining the h-measure for suitable h(t) (see, for example, (5)); (ii) it may be possible to determine the measure properties of a set from knowledge of its capacity (see, for example, (7) and (17)).

Dvoretsky, Erdös and Kakutani (3), showed that Brownian paths in 4 dimensions have zero 2-dimensional capacity, and their method gives the same result for Brownian paths in n-space whenever n ≥ 3. However, if we apply the method indicated by Hausdorff (4) for constructing a linear set having measure 1 with respect to a given measure function h(x), the Cantor-type set we obtain when h(x) = xα log log 1/x, where 0 < α < 1, is easily seen to have zero α-capacity but infinite α-measure; and similar methods apply for other values of α. Thus the result mentioned above does not imply that Brownian paths in n-space (n ≥ 3) have zero 2-measure. The other relevant result is due to Lévy (6), who showed that Brownian paths in the plane have zero Lebesgue measure (and therefore zero Hausdorff 2-measure) with probability 1. However, his method of proof cannot be extended to deal with Brownian paths in n-space.

The conditions under which bound states associated with atoms in the surface of a metal may exist are investigated, using the tight-binding approximation. These states arise as a result of modifications in the parameters of certain atoms. The modifications required to produce (a) bound states associated with all the atoms in the surface (surface states) and (b) bound states associated with particular small groups of atoms are found for the simple cubic lattice. It is also shown that most of the simpler crystal structures do not exhibit surface states without such modifications; in the graphite and diamond lattices, however, surface states exist solely by virtue of the lattice geometry.

I. It is shown that a general argument can be given for omitting the diverging parts of any quantized field interacting with a particle. The field equations thus freed from all singularities still contain the reaction of the field on the particle to the same extent as the reaction can be derived classically from the conservation of energy. The new field equations can be solved generally in terms of a simultaneous set of integral equations and do not give rise to any fundamental difficulties whatsoever.

II. The new field equations are applied to the multiple processes of the meson theory which occur, for instance, when a meson collides with a nuclear particle, the meson splitting up into a number n of secondary mesons. The cross-sections γn are worked out for n = 1, 2, 3, 4. γn has a maximum at some energy εn which increases with n. For ε ≫ εn, γn decreases like ε−2n−4 except for n = 1, when γ1 ∞ ε−2. γ1 (ordinary scattering) is larger than all the other γ's at all energies, but the probability of high multiplicities is comparable with that for low multiplicities or even greater in some energy regions.

The theorem which is still known as Nakayama's Conjecture shows how the modular characters of the symmetric group Sn can be divided into blocks of various weights, those with the same weight having similar properties. In fact, all blocks of weight one have essentially the same decomposition numbers and these are easy to describe. In recent work, Scopes [16, 17] has shown that there are essentially only finitely many possibilities for the decomposition numbers for blocks of any given weight, and has given a bound for the number. We develop the combinatorics implicit in her work, and so establish an improved bound.

Functional data analysis, or FDA, is a relatively new and rapidly growing area of statistics. A substantial part of the interest in the field derives from new types of data that are generated through the application of new technologies. Statistical methodologies, such as linear regression, which are effectively finite-dimensional in conventional statistical settings, become infinite-dimensional in the context of functional data. As a result, the convergence rates of estimators based on functional data can be relatively slow, and so there is substantial interest in methods for dimension reduction, such as principal components analysis (PCA). However, although the statistical development of PCA for FDA has been underway for approximately two decades, relatively high-order theoretical arguments have been largely absent. This makes it difficult to assess the impact that, for example, eigenvalue spacings have on properties of eigenvalue estimators, or to develop concise first-order limit theory for linear functional regression. This paper shows how to overcome these hurdles. It develops rigorous arguments that underpin stochastic expansions of estimators of eigenvalues and eigenfunctions, and shows how to use them to answer statistical questions. The theory is based on arguments from operator theory, made more challenging by the requirement of statisticians that closeness of functions be measured in the L∞, rather than L2, metric. The statistical implications of the properties we develop have been discussed elsewhere, but the theoretical arguments that lie behind them have not been presented before.

1. The problem investigated in this note was suggested by a study of the mathematical basis of change ringing. Its campanological application is discussed in § 4.

In this paper we carry out a systematic study of generalized maximal surfaces in Lorentz–Minkowski space L3, with emphasis on their branch points. Roughly speaking, such a surface is given by a conformal mapping from a Riemann surface S in L3. In the last years, several authors [1, 2, 5, 6] have dealt with regular maximal surfaces in L3, i.e. with isometric immersions, with zero mean curvature, of Riemannian 2-manifolds M in L3. So, the term ‘regular’ means free of branch points. As in the minimal case, a conformal structure is naturally induced on M, which becomes a Riemann surface S. The corresponding isometric immersion is then conformal on S, and it does not have any singular points on S (i.e. points on which the differential of the mapping is not one-to-one). This is the way in which generalized maximal surfaces include regular ones. Moreover, branch points are the singular points of the conformal mapping on S. Whereas branch points of generalized minimal surfaces are isolated, we shall show in Section 2 that, in addition to isolated branch points, a generalized maximal surface in L3. may have non-isolated ones, in fact they constitute a 1-dimensional submanifold in a certain open subset of S (see Section 2). So our purpose is two-fold, firstly to study and explain in detail the branch points, and secondly to state several geometric results involving prescribed behaviour of those points on the surface.

The object of this paper is to compare some of the wave functions which have been suggested for the normal helium atom, including one calculated in the first section of the paper, as regards energy, magnetic susceptibility and electric polarizability. Since this atom provides the simplest two-electron problem, such a comparison is of special interest and may indicate the direction in which improvements might be made in the calculation of wave functions for more complicated atoms.

In the integral the functions g(z), f(z, α) are analytic functions of their arguments, and N is a large positive parameter. When N tends to ∞, asymptotic expansions can usually be found by the method of steepest descents, which shows that the principal contributions arise from the saddle-points, i.e. the values of z at which ∂f/∂z = 0. The position of the saddle-points varies with α, and if for some α (say α = 0) two saddle-points z1(α), z2(α) coincide (say at z = 0) the ordinary method of steepest descents gives expansions which are not uniformly valid for small α. In an earlier paper (Chester, Friedman and Ursell (1)), a uniform asymptotic expansion of the form

was obtained, where Ai and Ai' are the Airy function and its derivative respectively, and where the regular functions A(α) and ζ(α) are given by

The coefficient functions As(α), Bs(α) are also regular functions of a for which, however, no explicit expressions are known. This Airy-function expansion was shown to be valid in a circle |α| ≥ Rα, independent of N. Airy-function expansions of the same form but involving slightly different argument functions instead of our A(α) and ζ(α) have long been known but are valid only in a region which shrinks to the point α = 0 as N → ∞. This improvement in the region of validity greatly simplifies the matching of steepest-descents and Airy-function expansions across the common region of validity.

In the present paper a further improvement is obtained. The validity of our Airyfunction expansion is extended to a still larger region which may be unbounded and which in many practical cases covers the whole region of interest, so that no matching with other expansions is needed. For this purpose the relation between steepestdescents and Airy-function expansions is investigated.

It is easy to see that by a process of matching the steepest-descents coefficients can be expressed in terms of the Airy-function coefficients. It is now shown that conversely the Airy-function coefficients can be expressed in terms of the steepestdescents coefficients, and that they involve the two saddle-points symmetrically.It is thus possible to infer desired properties of the Airy-function coefficients (e.g.analytic continuation and boundedness) from the corresponding properties of the steepest-descents coefficients, and hence to infer the equivalence of the two expansions (except near α = 0).

The following result is typical. Suppose that the steepest-descents expansion can be shown to be valid in a region (excluding a neighbourhood of α = 0) of the α-plane, and suppose further that the steepest-descents coefficients and the functions ζ(α) and A (α) can be shown to satisfy certain simple conditions of regularity and boundedness.(In practice it is usually not difficult to verify these.) Then it is shown that our Airy-function expansions can be continued into the same region and that it is a valid asymptotic expansion of the integral there.

These conditions of boundedness are not satisfied in a certain integral arising in the study of Kelvin's ship-wave pattern, where the coefficients at one saddle-point become unbounded near the track of the disturbance. The argument is modified to show that, nevertheless, the Airy-function expansion holds uniformly up to the track of the disturbance.