The author wishes to make the following corrections to his paper, entitled ‘On the relative merits of correlated and importance sampling for Monte Carlo integration’, which appeared in Proc. Cambridge Philos. Soc. 61 (1965), 497–498

Let

be a series of type Ω of the group G. By this we mean that Ω is an ordered set (we shall always understand ‘ordered’ to mean ‘totally ordered’) and Λσ and Vσ are subgroups of G satisfying

(i) Vσ ◃ Λσ for all σεΩ,

(ii) Λσ ≤ Vτ if σ < τ

(iii) , where G−1 denotes the set of elements ≠ 1 of G and Λσ−Vσ the set of elements of Λσ not belonging to Vσ,

(iv) Vσ ≠ Λσ for all σ ε Ω.

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

On page 674 of the present volume of the Proceedings (Part 3), in line 3 of the heading:

for University of Princeton, U.S.A.

read Institute for Defense Analysis, Princeton, N.J.

The infinite group

is the group of direct symmetry operations of the tessellation {3,7} of the hyperbolic plane ((3), chapter 5). This has the smallest fundamental region of any such tessellation, and related to this property is the fact that the group (2, 3, 7) has a remarkable wealth of interesting finite factor groups, corresponding to the finite maps obtained by identifying the results of suitable translations in the hyperbolic plane. The simplest example of this is the group LF(2,7), which is Klein's simple group of order 168. I have studied this group in an earlier paper ((4)), showing inter alia that the group is obtained as a factor group of (2,3,7) by adjoining any one of the relations

each of which implies the others. The method used was to find a set of generators for the normal subgroup with quotient group LF(2,7) and, working entirely within this subgroup, to exhibit that any one of these relations implies its collapse. The technique of working with this subgroup had been developed earlier and applied in (6) to prove that the factor group

is finite and of order 10,752.

The purpose of this paper is to prove Theorems 1 and 2 below by exhibiting them as special cases of a more general result, Theorem 3, which admits of a simple proof.

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 lattice ring is a lattice group ((l), page 214) and a ring in which ab ≥ 0 whenever a ∧ b ≥ 0.

In any lattice group (commutative or not) we define a+ = a ∨ 0, a− = (−a) ∨ 0 and |a| = a+ + a−. Itisknown((1). pages 219,220) that a+ ∧ a− = 0, a = a+ − a−, |a| = a+ ∨ a−, and that a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c), and a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c). For a non-empty subset M of a lattice group we define

In (7), we investigated some elements in the Adams spectral sequence for the stable homotopy groups of spheres, and proved that they were never boundaries, for any differential. This paper extends and generalizes these results: we consider more elements than in (7), and also prove that many of them do not survive to the E∞ term, so that they fail to be cycles for some dr, and this differential is therefore non-zero.

In this paper we obtain results on equicontinuity and apply them to certain recursive properties of topological transformation groups (X, T, π) with uniform phase space X. For example, in the special case that each transition πt is uniformly continuous we consider the transformation group (X,Ψ,ρ), where Ψ is the closure of {πt|t∈T} in the space of all unimorphisms of X onto X with the topology of uniform convergence (space index topology) and p(x, φ) = φ(x) for (x, φ)∈ X × Ψ. (See (1), page 94, 11·18.) If π is a mapping on X × T we usually write ‘xt’ for ‘π(x, t)’ and ‘AT’ for ‘π(A × T)’ where A ⊂ X. In this case we obtain the following results:

I. If (X, T, π) is almost periodic [regularly almost periodic] and πxis equicontinuous, then (X,Φ,ρ) is almost periodic [regularly almost periodic]

II. Let A be a compact subset of X such that. If the left and right uniformities of T are equal and (X, T, π) is almost periodic [regularly almost periodic], then (X, T, π) is almost periodic [regularly almost periodic].

Introduction. Spectral synthesis is the study of whether functions in a certain set, usually a translation invariant subspace (a variety), can be synthesized from certain simple functions, exponential monomials, which are contained in the set. This problem is transformed by considering the annihilator ideal in the dual space, and after taking the Fourier transform the problem becomes one of deciding whether a function is in a certain ideal, that is, we have a ‘division problem’. Because of this we must take into consideration the possibility of the Fourier transforms of functions having zeros of order greater than or equal to 1. This is why, in the original situation, we study whether varieties are generated by their exponential monomials, rather than just their exponential functions. This viewpoint of the problem as a division question, of course, perhaps throws light on why Wiener's Tauberian theorem works, and is implicit in the construction of Schwartz's and Malliavin's counter examples to spectral synthesis in L1(G) (cf. Rudin ((4))).

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.

The idea of exact filling bundle may be described roughly as follows. Suppose that ξk is a vector bundle with fibre Rk, total space E(ξk) and base X. We say that ξk is a real k-plane bundle on X. Let in be the trivial n-plane bundle on X so that E(in) = X × Rn. A bundle monomorphism j: ξk → in defines a map : E(ξk)→Rn obtained by composition of the embedding E(ξk)→E(in) and the product projection E(in) → Rn. The map represents each fibre of ξk as a k-plane in Rn.

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).

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.

A central role in the theory of smoothing combinatorial manifolds is played by the Cairns–Hirsch Theorem, which may be expressed (in a weak form) as follows:

If M is a combinatorial manifold and if M × R has a differentiable structure a, compatible with its combinatorial structure then M has a differentiable structure λ, such that (M × R)α is diffeomorphic with Mλ × R.

. 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.

We introduce the concepts of a local seminorm on a topological group and of a locally convex group, showing that discrete groups, locally compact Abelian groups and compact groups are locally convex, and that a topological vector space is locally convex as a linear space if and only if it is locally convex as a group. We show that notions of differentiability, analyticity and derivability can be defined for locally convex groups and that these notions are suitably related and well behaved. We prove that for a locally compact Abelian group G the Fourier transforms of measures of compact support on the character group Ĝ are analytic, and for G compact the coefficients of continuous irreducible unitary representations are. Using these spaces of analytic functions we define the basic concepts of a differential geometry.

There is a close parallelism between the theories of convergence of directed nets and of filters, in which ‘subnet’ corresponds, in general, to ‘refinement’. With the standard definitions, however (1), pages 65 et seq., this correspondence is not exact, as there is no coarsest net converging to x0 of which all other nets with the same limit are subnets. (Suppose, for example, that a net X = {xj,: j ∈ J} in R1 has both the sequence-net S = {n−1; n = 1, 2,…} and the singleton-net {0} as subnets. Then (with an obvious notation), there exist

such that j0 ≥ jn for all n, while jn ≥ j0 for all n ≥ n0 say. But, given any j ∈ J, there exists n with jn ≥ j: it follows that jn ∈ j for all n ≥ n0 (independent of j); thus X cannot converge to 0. Even if nets with a last member are excluded, a similar result can be obtained by considering the net Y = {yθ; Θ an ordinal less than ω1}, where yθ = 0 for all Θ. If X has both Y and S as subnets we can show that (with a similar notation) there exists Θ0 such that Θ ≥ Θ0 implies jθ ≥ all jn, but also n0 such that n ≥ n0 implies ; the rest is as before.) Moreover, the theory of convergence classes, (l), pages 73 et seq., contains a condition (Kelley's condition (c)) whose analogue need not be separately stated for filters. These differences can be removed by adopting a wider definition of subnet, a course which does not seem unnatural, inasmuch as the standard definition is already wider than the ‘obvious’ one, and our proposed definition is equivalent to the standard one in the special case of sequences.

Introduction. Let K be a connected simplicial complex, finite or infinite, its polyhedron ((2), page 45) being the space X. Then X is connected. Suppose further that X is simply connected. For any group G of simplicial transformations of X, H will denote the normal subgroup generated by elements which have a non-empty fixed-point set.