1. Let f(x) be a polynomial of degree n ≥ 2 with integral coefficients, the highest coefficient being positive. It is well known that if f(x) is an exact k-th power for all sufficiently large integers x, where k ≥ 2, then f(x) = g(x)k identically, where g(x) is another polynomial with integral coefficients. (See Pólya and Szegö (4), section 8, problems 114, 190; also Davenport, Lewis and Schinzel(1), where other references are given.) The main purpose of this note is to prove that if we suppose only that

In a recent paper of the same title (2), Conway considered the problem of dividing a square of side N, into squares of integral side. The case when we demand that all these squares are of unequal size has been considered in some detail by various authors ((l),(4),(6),(7)). Conway considered the case with the condition relaxed. Dudeney ((3), Problem 173) has considered the same problem for the special case N = 13; hence the title of this paper.

A betweenness semigroup is a semigroup possessing a ternary relation, ‘b lies between a and c’, which is invariant under the semigroup operation. We take as our axioms of betweenness those suggested by Shepperd in (2) and given below in 1·01–1·04. It was shown in (2) that these axioms lead to just two types of betweenness; namely a linear betweenness corresponding to the order of points on an undirected straight line or a cyclic betweenness of just four points on a circle. In (3), Shepperd characterized betweenness groups and we here generalize his results to betweenness semigroups.

A normed algebra A is a pre-B*-algebra if its norm satisfies ∥x*x∥= ∥x∥2 for all elements x ∈ A; if A is also complete in its norm, then A is a B*-algebra (see (l), page 180). In the study of certain locally convex algebras, the problem arose of expressing the condition that an algebra be a pre-B*-algebra in terms of its properties as a locally convex algebra, rather than in terms of the norm. A solution to this problem is presented in this note; the application to the theory of locally convex algebras will appear elsewhere.

. Let Q be a well-quasi-ordered set, i.e. a set on which a reflexive and transitive relation ≤ is defined and such that, for every infinite sequence q1,q2,… of elements of Q, there exist i and j such that i < j and qi ≤ qi. A restricted transfinite sequence on Q is a function from a well-ordered set onto a finite subset of Q. If f, g are restricted transfinite sequences on Q with domains A, B respectively and there exists a one-to-one order-preserving mapping μ of A into B such that f(α) ≤ h(μ(α)) for every α ∈ A, we write f ≤ g. It is proved that this rule well-quasi-orders the set of restricted transfinite sequences on Q. The proof uses the following subsidiary theorem, which is a generalization of a classical theorem of Ramsey (4). Let P be the set of positive integers, and A(I) denote the set of ascending finite sequences of elements of a subset I of P. If s, t∈A(P), write s ≺ t if, for some m, the terms of s are the first m terms of t. Let T1,…,Tn be disjoint subsets of A(P) whose union T does not include two distinct sequences s, t such that s ≺ t. Then there exists an infinite subset I of P such that T ∩ A(I)is contained in a single Tj.

Let G be a group, and let φ (x1, …, xn) be any word in the variables x1, …, xn. We denote by φ(G) and φ*(G) respectively the corresponding verbal and marginal subgroups of G, which are defined as follows.

(i) φ(G)is the subgroup generated by all elements of the form φ(a1; …, an) with ai in G(for i = 1, 2, …, n);

(ii) an element b of G lies in φ*(G) if and only if

for all choices of a1, …, an in G and i= 1, 2, …,n.

In the development of homological algebra, one has to prove at some point that, in defining the derived functors of ⊗ and of Hom, it makes no difference whether we resolve both variables or only one of them. Taking ⊗ ( = ⊗R) as a typical example, what has to be proved is

(A) If the complex F is projective, or even flat, as a right R-module, and if f: P → A is a projective resolution of the left R-module A, then

is an isomorphism.

In (8), we have shown how, in general, the problem of identifying the differentials in the Adams spectral sequence (see(1),(4)) is equivalent to that of calculating certain higher-order cohomology operations (in the sense of (6)). However, we propose to investigate here a slightly different method, based on the naturality of the spectral sequence, which can be used to show that certain elements are never boundaries, for any differential.

The purpose of this note is to answer a question asked in an earlier paper ((2), page 315). Let X be a closed, connected, non-orientable 2-manifold and f: X → X a homeomorphism. It was shown in (2) that f is isotopic to a product of homeomorphisms of two basic types, the c-homeomorphism, which will now be called a twist, and the Y-homeomorphism (see (2) for definitions). It was also proved that f cannot, in general, be expressed as a product of twists alone. The following result is proved here:

Notations. If G denotes an Abelian group and L a locally compact group topology on G, G(L) will denote the resulting topological group, and will denote the character group with the usual Pontrjagin topology (which is locally compact). Rn will denote the real n-dimensional vector group. Finally, for any set X, |X| will denote its cardinal number.

Our motivation for this paper is to be found in (2) and (3). In (2) Varopoulos considers inductive limits of topological groups, in particular what he calls ‘ℒ∞’. (He calls a topology an ℒ∞-topology when it is the inductive limit of a decreasing sequence of locally compact Hausdorff topologies.) In (2) he proves that much of the classical theory of locally compact Abelian groups also goes through for Abelian ℒ∞-groups, in particular Pontrjagin duality.

We are concerned in this work with the following question:

Suppose that i is a continuous algebraic isomorphism from the topological group H onto a subgroup of the topological group G and suppose that the image i(H) is not closed in G; then what can we say about the cardinal number

We observe two easy results.

We shall study the following complex Stiefel fibring:

.

In particular we shall study the problem: for what values of n and k does the fibring(1.1) admit a cross-section? A necessary condition for the existence of a cross-section has been found by Atiyah and Todd (8). We shall show (Theorem 1.1 below) that the condition of Atiyah and Todd is sufficient (as well as necessary) for the existence of a cross-section. The problem stated above is therefore completely solved.

Some time ago Besicovitch(l) gave an example of a linear Gδ-set E of Besicovitch dimension 1, with the property that E – A also has dimension 1, for each Fσ.-set A contained in E. This shows that an Fσ-set contained in E cannot be a very close approximation to E, but it leaves plenty of scope for Fσ-sets contained in E to be reasonably good approximations to E.

In this note we shall consider the series

where lim sup |an| = 1 and

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.

Bickley and McNamee (1) describe techniques for obtaining the solution of finite difference equations, arising from partial differential equations, making extensive use of matrix methods. In all cases solutions are obtained by solving algebraic equations as distinct from differential equations. For example, in order to solve

the second space derivative is replaced by finite differences and the time derivative is replaced either by substituting the backward finite difference form or by using the Laplace transformation.

The most general differential equation to which the dynamical description of the title applies is

where dots denote differentiation with respect to t. The essential problem for this equation is to determine the behaviour of solutions as t → ∞. When we attack this problem, the most obvious question is whether, under reasonable conditions on p(t), every solution is bounded as t → ∞ this question is open except when g(x) is linear. In the special case when p(t) is periodic, (1·1) may have periodic solutions; it is clear that any such solution is bounded, and it is worth mentioning that finding periodic solutions is the easiest way of finding particular bounded ones. So long as the bounded-ness problem is unsolved, there is a special interest in finding a large class of particular bounded solutions: if we know such a class then, although we cannot say whether the general solution is bounded or not, we can make the imprecise comment that either the general solution is in fact bounded or the structure of the whole set of solutions is quite complicated.

An important unsolved problem in the theory of non-linear oscillations is to establish the boundedness or unboundedness of the general solution of

where dots denote differentiation with respect to t. When p(t) is periodic, we may seek periodic solutions. This search is interesting for its own sake, and of course leads us to special bounded solutions. In three previous papers (2, 3, 4) I have exhibited the equation

as tractable: on the assumption that e(t) is even and periodic, it was shown that the equation has an infinity of periodic solutions.

1. A topological game is a game in which the positions are the points of a topological space; the set of possible moves from any such point varies continuously with the point. Play can start from any point of the space and at each point it is specified which of the players has the initiative. Play ends when a position is encountered at which the set of positions from which the player with the initiative can choose is empty. The payoff to each player depends on the set of positions met in the play. The notion of topological game is due to Berge (1), (2).