The ringoids discussed in the preceding paper of this series(3) are characterized by two axioms which permit the operations of an exact category to be carried out within the structure of the ringoid. These operations do not in general include the formation of direct sums or products, for which new axioms are required. For a finite number of constituents these axioms are equivalent to the corresponding axiom in Buchsbaum's paper(1).

The following conjecture was apparently first enunciated by Catalan (3) in 1844 but has never been proved.

Einleitung. Man verdankt Richard Dedekind (siehe etwa(1)) das erste Beispiel einer Umkehrungsformel einer endlichen Reihe; dabei spielt, wie man weiβ, die sogenannte Möbius'sche Funktion (der Zahlentheorie) eine groβe Rolle. Auch die sogenannten Aufzählungsmethoden der Gruppentheorie, usw., die mit der soeben erwähnten Dedekindschen Umkehrungsformel eng verbunden sind, stützen sich, wie hauptsächlich von Herren Louis Weisner(2) und Philip Hall (3) gezeigt wurde, auf eine passende Verallgemeinerung der gewöhnlichen Möbius'schen Funktion, die für alle teilweise geordneten Mengen einen Sinn hat, deren Quotienten endlich sind. Siehe auch Ward (4).

The group of the title is one of the classical groups and can be represented—will indeed be represented here—as the group of those projectivities in a certain finite plane that leave invariant the set of zeros of a Hermitian ternary form. Its properties in the large, so to speak, are common knowledge, being particular instances of those of unitary finite groups; the conjugate sets, and sufficient information to determine, or perhaps nearly to determine, the group characters are given in (7). The methods to be used here are elementary; they take advantage of the base field, and consequently the numbers of points in the geometry, being relatively small. Much detailed information emerges that seems worth putting on record; it does not occupy a disproportionate amount of room and might elude the coarser net of an investigation based on generalities.

Consider a lattice L in the Cartesian plane consisting of all points (x, y) such that either x or y is an integer. Points with integer coordinates (positive, negative, or zero) are called vertices and the sides of the unit squares (including endpoints) are called links. Each link of L is assigned the designation active with probability p or passive with probability 1 − p, independently of all other links. To avoid trivial cases, we shall always assume 0 < p < 1. The lattice L, with the designations active or passive attached to the links, is called a random maze. A set of links is called connected if the points comprising the links (including endpoints) form a connected point set in the plane.

An extension of Bellman's analogue of Wald's identity is proved for Markov processes with a finite number of states, in which (a) the diagonal matrices of the state functions and the transition matrices may be heterogeneous but are so related that the product of the two matrices which apply at any stage has at least one modal vector in common with all of the corresponding products which apply at other stages, and (b) the probability distribution of the duration of any one realization of the process satisfies certain conditions. The condition (a) is called equimodality. The conditions for (b) are satisfied when there is asymptotically a sufficiently abrupt exponential upper bound to the probability distribution of the duration, and extends simply to validate analogues involving latent roots other than the one which appears in Bellman's formula. A certain generalization of Wald's stopping rule produces a bound of this kind.

The determination of a pair of fundamental units in a totally real cubic field involves two operations—finding a pair of independent units (i.e. such that neither is a power of the other) and from these a pair of fundamental units (i.e. a pair ε1; ε2 such that every unit of the field is of the form with rational integral m, n). The first operation may be accomplished by exploring regions of the integral lattice in which two conjugates are small or else by factorizing small primes and comparing different factorizations—a trial-and-error method, but often a quick one. The second operation is accomplished by obtaining inequalities which must be satisfied by a fundamental unit and its conjugates and finding whether or not a unit exists satisfying these inequalities. Recently Billevitch ((1), (2)) has given a method, of the nature of an extension of the first method mentioned above, which involves less work on the second operation but no less on the first.

Let denote the polynomial algebra over the integers in countably many variables ui (i ≥ 1). Let ∂ be the derivation of defined on the generators by. Thus if is graded by dim ui=i, then∂ is homogeneous, of degree − 1. The result is that ∂ is onto, and that its kernel is a polynomial algebra in , where is homogeneous of degree i and the coefficient of ui in is p if i is a power of a prime p, and 1 if i is not a prime power.

Andrews and Curtis have shown (1) that one can embed two Sn's in En+2 for n = 2, in such a way that one sphere cannot be shrunk to a point in the residue space of the other. In this paper the result is shown to be true for any n ≥ 1. (The result is obvious for n = 1.) The method is to calculate the appropriate homotopy group of the residue space of one sphere, and to show that the embedding of the other sphere represents a non-zero element of the group. The two spheres can both be embedded analytically.

In this note the problem of random flights is considered for the case of a small number of vectors, each vector possessing the same constant magnitude. A formula is developed which permits the calculation of the probability function of the sum of N + 1 such vectors if the function of the sum of N vectors is known. The formulae for the probability distributions up to N = 8 are quoted and some of their practical applications mentioned.

B. H. Neumann (5) has shown, by using free products with an amalgamated subgroup, that every group can be embedded in a divisible group. This theorem can also be proved, in an entirely different way, by making use of the wreath product (Baumslag(1)). In view of this connexion between roots in groups and the wreath product, a study is made here of the conditions which ensure that the wreath product (and also the unrestricted wreath product) shares certain properties regarding the existence and uniqueness of roots with the groups from which it is composed.

Let be a free additive abelian group, and let be a basis of , so that every element of can be expressed in a unique way as a (finite) linear combination with integral coefficients of elements of . We shall be concerned with the ring of endomorphisms of , the sum and product of the endomorphisms φ, χ being defined, in the usual manner, by the equations

A permutation of a set will be called restricted if it moves only a finite number of elements. We call an endomorphism of a permutation endomorphism if it induces a restricted permutation of the basis .

If {λ;n}, {bn} are sequences of complex numbers, and we consider the series ∑bn exp (−λnx), given as convergent in (0, 1) (i.e. the open invertal (0,1)) to f(x)∈L, then, writing

(if λn = 0 the corresponding term is ½bnx2) where the series is supposed is to be uniformly convergent in (0, 1), we have

for 0<h<h(x).If we know that the second member of (2) tends to f(x) as h → +0, it will follow that F(x) is a repeated integral of f(x) ((1), 671). If there is a sequence {φv(x)} of integrable functions with the property that

then, on multiplying (1) by φv(x) and integrating over (0,1), we obtain a formula for bv in terms of F(x). On integrating by parts twice, bv will be expressed in terms of f(x), and this will constitute a uniqueness theorem for the series ∑bn exp (−λnx).

Imagine a nearly horizontal, statistically uniform, random surface ζ(x, y), Gaussian in the sense that the second derivatives , , have a normal joint distribution. The problem considered is the statistical distribution of the quantity

where J and Ω denote the mean curvature and total curvature of the surface, respectively, and ν is a constant parameter.

In papers I(1) and II(2), the author has investigated the Stokes phenomenon associated with the arbitrary constants that occur in the asymptotic expressions of solutions of the nth-order differential equation

and has provided rules for tracing a given linear combination of n such asymptotic expressions across certain defined lines, designated as Stokes lines, anti-Stokes lines and branch-cuts. The object of these investigations is to apply the results to the approximate solution of more complicated linear differential equations of the nth order, and to this end it is an advantage to develop the theory using matrix techniques. In the present paper, the equations are recast into a special matrix form, and it is shown how the results obtained in II may be used in a systematic manner.

The behaviour of the Chern classes or of the canonical classes of an algebraic variety under a dilatation has been studied by several authors (Todd (8)–(11), Segre (5), van de Ven (12)). This problem is of interest since a dilatation is the simplest form of birational transformation which does not preserve the underlying topological structure of the algebraic variety. A relation between the Chern classes of the variety obtained by dilatation of a subvariety and the Chern classes of the original variety has been conjectured by the authors cited above but a complete proof of this relation is not in the literature.

In a recent series of papers (10), (11), (12), I. M. James has made an illuminating study of Stiefel manifolds. We shall begin by describing his results (for the complex case). Let Wn, k, for k > 1, denote the complex Stiefel manifold U(n)/U(n − k), where U(n) is the unitary group in n variables. Then we have a natural fibre map Wn, k → Wn, 1 = S2n−1, where Sr denotes the r-dimensional sphere. Let Pn, k, for k ≥ 1, denote the ‘stunted complex projective space’ obtained from the (n − 1)-dimensional complex projective space† Pn by identifying to a point a subspace Pn−k. Then we have a natural ‘cofibre map’ Pn, k → Pn, 1 = S2n−2. The space Pn, k is said to be S-reducible if some suspension of the map Pn, k → S2n−2 has a right homotopy inverse. The results of James can then be summarized as follows.

In two recent papers ((l), (2)) I investigated inclusion relations between IF (integral function) methods of summability. In the present paper the investigation is continued with the aid of standard Mellin transform theory and results due to Rogosinski and to Kuttner.

An expansion of the Whittaker function Wk, m(z) in terms of Bessel functions is given for 2m an integer or zero. Simple formulae are derived for the cases of m = 0, ½, 1 and . The case m = 0 provides expansions for the exponential, sine, cosine and similar integrals. The cases m = ½ and provide expansions for the irregular Coulomb wave functions having angular momenta zero and one.

Self-consistent fields without exchange have been calculated for the ground state of Pr3+ and Tm3+. The spin-orbit coupling constant ζ(4f) is found to have the values 785 and 2742 cm−1 in Pr3+ and Tm3+, respectively. The corresponding values of are 29·4 and 77·5 Å−3. Values of the Slater integrals F2(4f, 4f), F4(4f, 4f) and F6(4f, 4f) are also given for each structure.