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.

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.

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.

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 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.

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.

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.

The construction of Dyer-Lashof operations in K-theory outlined in (6) and refined in (12) depends in an essential way on the descriptions of the mod-p K-theory of EZp, ×ZpXp and EΣ ×σ p Xp given there. Unfortunately, these descriptions are incorrect when p is odd except in the case where the Bockstein β is identically zero in K*(X; Zp), and even in this case the methods of proof used in (6) and (12) are not strong enough to show that the answer given there is correct. In this paper we repair this difficulty, obtaining a complete corrected description of K*(EZp ×ZpXp; Zp) and K*(EΣp) (theorem 3·1 below, which should be compared with ((12); theorems 3·8 and 3·9) and ((6); theorem 3)). Because of the error, the method used in (6) and (12) to construct Dyer-Lashof operations fails to go through for odd primes when non-zero Bocksteins occur, and it is not clear that this method can be repaired. We shall not deal with the construction of Dyer-Lashof operations in this paper. Instead, the first author will give a complete treatment of these operations in (5), using our present results and the theory of H∞-ring spectra to obtain strengthened versions of the results originally claimed in ((12); theorem 5·1). There is also a minor error in the mod-2 results of (12) (namely, the second formula in (12), theorem 3·8 (a) (ii)) should read

where B2 is the second mod-2 Bockstein, and a similar change is necessary in the second formula of ((12), theorem 3·8(b) (ii)). The correction of this error requires the methods of (5) and will not be dealt with here; fortunately, the mod-2 calculations of ((12), §6–9), (10) and (11) are unaffected and remain true as stated.

This paper is a continuation of (3), in which we proved extensions, to generalized Stieltjes transforms, of some inversion theorems of Widder(l) for ordinary Stieltjes transforms. The numbering of formulae, lemmas and theorems continues on consecutively from that in (3). The last formula in (3) is numbered (32), and the last lemma or theorem is Theorem 18.

Those mixed boundary-value problems which can usefully be treated analytically often lead to the following mathematical problem. Two functions u(x), σ(x), defined over the interval ([0, ∞), take prescribed values over complementary portions of that interval; specifically, let

where p(x) is usually a simple function, for example a constant or a power of x. There exists a relation between u(x) and σ(x) which can be most simply expressed as a relation between their Hankel transforms. Using a circumflex to denote the Hankel transform, for example with

where Jv denotes as usual the Bessel function of the first kind of order v, we can state that relation between u and σ as follows:

where A(ξ) is a known function, determined at an earlier stage of the analysis. The problem is to derive u(x) for (xє [ 0, a), or σ(x) for x є (a, ∞).

An extremum variational principle for boundary value problems described by a non-linear fourth order differential equation qIV = f(q, x) is constructed. For approximate solutions an error estimate procedure, based on the value of the functional, is developed and applied to two concrete problems.

Introduction. In (9) Newman and Penrose introduced a differential operator which they denoted ð, the phonetic symbol edth. This operator acts on spin weighted, or spin and conformally weighted functions on the two-sphere. It turns out to be very useful in the theory of relativity via the isomorphism of the conformal group of the sphere and the proper inhomogeneous Lorentz group (11, 4). In particular, it can be viewed (2) as an angular momentum lowering operator for a suitable representation of SO(3) and can be used to investigate the representations of the Lorentz group (4). More recently, edth has appeared in the good cut equation describing Newman's ℋ-space for an asymptotically flat space-time (10). This development is closely related to Penrose's theory of twistors and, in particular, to asymptotic twistors (14).

The non-linear graviton construction of Penrose (9) and the ℋ-space construction of Newman (6) are two complementary techniques for constructing complex four dimensional space-times with quadratic metric and anti-self-dual curvature tensor.

In the former, the space-time is the space of holomorphic sections of a complex fibre space obtained by deforming part of flat twistor space. In the latter the space-time is the space of regular solutions of a differential equation, the good cut equation.

Pathologies arise in the non-linear graviton construction when the normal bundle of a holomorphic section changes. This is reflected in the ℋ-space construction by a change in the character of the solutions of the linearized good cut equation, the Newman equation.

Approximation by the Poisson distribution arises naturally in the theory of random graphs, as in many other fields, when counting the number of occurrences of individually rare and unrelated events within a large ensemble. For example, one may be concerned with the number of times that a particular small configuration is repeated in a large graph, such questions being considered, amongst others, in the fundamental paper of Erdös and Rényi (4). The technique normally used to obtain such approximations in random graph theory is based on showing that the factorial moments of the quantity concerned converge to those of a Poisson distribution as the size of the graph tends to infinity. Since the rth factorial moment is just the expected number of ordered r-tuples of events occurring, it is particularly well suited to evaluation by combinatorial methods. Unfortunately, such a technique becomes very difficult to manage if the mean of the approximating Poisson distribution is itself increasing with the size of the graph, and this limits the scope of the results obtainable.

Let K ⊂ ℝ3 be an open bounded strictly convex domain with smooth connected compact boundary ∂K. Set

We wish to study the filtered scattering amplitude, related to the transmission problem

Here w± are the limiting values of w on ∂Ω; from the Ω;± side, while ∂w±/∂n are the corresponding limiting values of the normal derivative ∂w?/∂n on ∂Ω;. The function α(x) ∈ C∞(K¯), called the index of refraction, has the property α(x) > 0 for x ∈ K¯.

In this note we examine the influence of surface tension on surface waves incident against a fixed vertical plane wall. The motion is time harmonic and is determined by making the initial assumption that the free-surface slope at the wall is prescribed. From the unique solution obtained for the velocity potential, the parameter involved in this specification can be determined, for small laboratory-scale waves at least, using some longstanding experimental results on meniscus behaviour at a moving contact line. The effect of surface tension is to produce a motion wherein reflexion from the wall is not complete and there is a local disturbance, in contrast to the classical standing-wave motion in the absence of surface tension.

N. Martin. ‘Cobordism of homology manifolds’

A. W. Jessop(10) has pointed out that the proof of Lemma 4·5 is wrong as the N D condition enabling the inductive use of Theorem 4·3 will not hold in the case where r = 2m + 3.

In what follows Theorem 4·3 and Lemma 4·5 are restated with slightly different conditions but with essentially similar conclusions. References to diagrams are those in the original. Paper references are those of the original together with others listed at the end here.

N. Martin. ‘Transverse regularity for maps of homology manifolds’

An example given by Galewski and Stern (l) showed that Theorem A of the original paper was false as stated. The error is traceable to the codimension three condition which is not sufficient here.

E. M. Wright. ‘A quadratic recurrence of Faltung type.’

Middle of p. 197. The value of x( − ⅓) should be

x (− ⅓) = 0·351298985… = ei−1 × 0·95492965…

The error was mine and not Dr Ponting's.