Let K > ω and . It is known that there then exists a non-trivial k-additive 2-valued measure, Vk, on the set [K]k of subsets of k of cardinality K..

In this paper we define two order relations between every two partizan games with possible draw positions. We use these relations to prove two ‘simplest form’ theorems, and to describe an algorithm for computing the simplest form of the ‘finite games’ which are an extension of Conway's short games.

Let g(y) denote an arbitrary real-valued function satisfying g(y) → ∞ as y → ∞. One expects, and, subject to certain conjectures, Heath-Brown(3) has proved, that for all y in (1, X], apart from a set of measure o(X), the interval (y, y + g(y) log y] contains primes.

A set X in Euclidean space is Valentine n–convex, or simply n–convex' if it has the following property. If X contains a subset Y consisting of n distinct points then X also contains the points of at least one segment with end points in Y. We show here that the vector sum of two plane compact 3-convex sets is 5-convex (which complements the result of I. D. Calvert(1) that the intersection of two plane compact 3-convex sets is 5-convex) and that the vector sum of a plane connected compact 3-convex set with itself is 4-convex. These results are not true in 4 dimensional space. It is an open question whether or not they are true in 3-dimensional space.

The nonsingular model of Kummer's surface, or its birational equivalent the Weddle surface, is an octavic surface F in [5], projective space of dimension 5 (11), ((10), p. 53), ((1), pp. 218, 219). F is the base variety of a net N of quadrics with a common self-polar simplex S, and has on it 32 lines. These form a single orbit under the group G, of order 32, consisting of the harmonic homologies fixing S.

This paper improves a method of Godwin (4) for finding a pair of fundamental units in a totally real cubic field. The determination of such a unit pair is a well known computational problem. There is an old algorithm (circa 1896) of Voronoi which solves this problem, but the algorithm is quite complicated (an account of it is given in the book of Delone and Faddeev ((3), chapter IV, part A)). The method of Godwin is, in principle, much simpler. However, this method also has its drawbacks (more is said about this in Section 4 below). Indeed, when Godwin's student Angell produced his large table (see (1)) of totally real cubic fields some 15 years after (4) appeared, Voronoi's algorithm was used to compute the pairs of fundamental units.

In (2) Lennox and Stonehewer proved: Let G be a group with subnormal subgroups H and K, and let J = 〈H, K〉. Suppose that J/J′ has finite rank. Then

(i) given positive integers a and b, there exists c such that γc(J) ≤ γa(H). γb(K) and

(ii) J is subnormal in G.

In this paper we prove a lifting result for local cohomology. As a special case we get the following result for the Serre-cohomology over a projective variety:

Proposition (1·1). Let ℱ be a coherent sheaf over X, where X is a projective variety over an algebraically closed field k. Let i ≽ 0 and assume that there is a pencil P of hyper-plane sections which is in general position with respect to ℱ (which means that x ∉ H, ∀x ∈ Ass(ℱ), ∀H∈p), and such that for each H ∈ P Hi(X, ℱ│H(n)) = 0, ∀n ≪ 0. Then Hi + 1(X, ℱ) = 0, ∀n ≪ 0.

The achromatic number ψ(G) of a graph G is the greatest number of colours in a proper colouring of the vertices of G such that for every pair of colours some vertex of the first colour and some vertex of the second colour are adjacent. We prove that almost all graphs Gn with n vertices satisfy n/(k+l) < ψ(Gn) < n/(k–1), where k = k(n) = (log2n)½. We show also that the achromatic number is a ‘global’ invariant.

The object of this paper is to prove a necessary and sufficient condition on two groups H, K for their join always to be subnormal in a group G whenever they are embedded subnormally in G.

It is an old problem whether two binary quadratic forms representing the same numbers are equivalent. Legendre asserted without proof that two positive definite forms representing the same numbers are equivalent. In 1895, Bauer(1) proved Legendre's assertion for primitive forms with the same discriminant. In 1938 Delone (2) proved that two positive definite binary forms with real coefficients are equivalent if they represent the same numbers, with the sole exception of the pair equivalent to

and their scalings. Watson (3) rediscovered this result in 1979 and asked the same question for indefinite binary forms.

Let G be an affine, connected, semisimple, simply connected algebraic group over an algebraically closed field k of non-zero characteristic p and, for a positive integer n, let Gn be the nth infinitesimal subgroup (the kernel of the nth power of the Frobenius morphism). The purpose of this note is to point out that, for certain rational G-modules V, the natural homomorphism

is the zero map, for m ≽ 0. For such modules V, taking m = 0, we see from the 5-term exact sequence of Lyndon-Hochschild-Serre that the inflation-restriction sequence

is exact and thus the dimension of H1(G, V), when finite, may be calculated by adding the dimensions of the outer terms. The practical value of this result seems to lie in the special case n = 1 and V = L(λ) ⊗ L(µ), the tensor product of simple rational G-modules. In this case H1(G/G1) VG1 is H1(G, L(λ′) ⊗ L(¯′)) for λ′, ¯′ ‘smaller’ than λ µ and the above exact sequence gives an inductive procedure for computing Ext1G between simple rational (G-modules once the (G-module structure of Ext1G1 (equi valently Ext1u1 where% is the restricted enveloping algebra of the Lie algebra of G), for simple G1-modules, is known. Using the rather complicated structure of Weyl modules Cline, in (4), calculated Ext1G between simple modules for G = SL2(k) (this process was reversed in (5)). The present note has been used by S. EIB. Yehia, (17), in finding the corresponding result for SL3(k). We believe these Ext groups to be of increasing importance in view of their connection with Lusztig's conjecture, (14), on the characters of the simple rational G-modules and the related conjecture, 7–2 of (l), of Andersen.

Let G be a simply primitive permutation group on a set Ω of order p2, where p is a prime (necessarily odd). In theorem 27·2 of (9), Wielandt states without proof:

Theorem A. (i) ¦G¦ is not divisible by p3;

(ii) if G has a pair of Sylow p-subgroups with nontrivial intersection, then G has an imprimitive subgroup of index 2 which is the direct product of two intransitive groups.

Let H and K be topological groups, and let HⓈ K denote the semidirect product determined by a homomorphism (η): H → A(K), where A(K) is the automorphism group of K. In this paper we consider two restricted types of semidirect products. We say that HⓈ K is a semidirect product of type I if η(h) is the identity on Z(K), the centre of K, for each hє H, and of type II if η(H) є I(K), where I(K) is the group of inner automorphisms of K. We obtain conditions under which a type II semidirect product of two groups with equal uniformities has equal uniformities, and conditions under which a type I (hence type II) product of two central groups is central. A group G is central if G/Z(G) is compact, where Z(G) is the centre of G.

In (1) and (2) we studied a lattice of extension rings associated with a commutative ring R with identity. When R, M is a one-dimensional Cohen-Macaulay local ring the elements of are just those integral extensions of R contained in the total quotient ring T(R) and such that lengthR(S/R) is finite. Experiments with local rings of singular points on algebraic curves indicate that only the simplest singularities give rise to finite lattices. So the problem arises as to which local rings R give rise to which finite lattices. In later papers this problem will be investigated in detail, at least when R is of low embedding dimension. The purpose of the present note is to establish some general results which indicate the size of the problem.

The braid group on n + 1 strings has the presentation

where xi corresponds to the single crossing of the i string over the i + 1 string, and rij is a commutator when i < j − 1, and a braid relator when i = j − 1. B1 is the trivial group and B2 is infinite cyclic; we shall assume n ≥ 2 throughout.

All rings considered in this paper are non-trivial commutative algebras over a field k; unless indicated otherwise, all tensor products are understood to be taken over k. (Some of the results and concepts extend to general rings, but it is not worth while noting such generalizations.)

1. Introduction and statement of results. Let f: Mn-1 → ℝn be an immersion into Euclidean space ℝn. Each unit vector v to ℝn determines a height function bv: ℝn → ℝ. The corresponding half-space Lv = b-1([0, ∞) has boundary Hv = (b−1). and L is a (globally) supporting half-space for M at m є M if (m) є H and f(M) ∩ Lv = f(M) ∩ Hv.

We derive a family of lower bounds for the Perron root of a non-negative irreducible matrix. These lower bounds are better than certain lower bounds of the Rayleigh quotient type, also derived in this paper. For the particular case of a symmetric non-negative irreducible matrix, our lower bound is always better than a corresponding Rayleigh quotient and, as shown in example 4, it can be infinitely better.

Introduction. The embeddability of l1(Γ), for uncountable sets Γ, into subspaces of Banach spaces of the form C(Ω) was investigated first by Hagler in (6) and subsequently by Haydon in (7), (8) and Argyros and Negrepontis in (1). An important role in the development of the above subject is played by a lemma of Rosenthal (12) that translates the functional analytic problem of finding a family {fξ: ξ Γ} of elements of C(Ω) equivalent to the usual basis of l1(Γ) into the problem of the existence of an independent family {(Aξ, Bξ,): ξ є Γ} of closed subsets.