Although the classification of affine cubic curves was undertaken by Newton(4), in one of the first major exercises ever in coordinate geometry (see Cayley(2) for a fuller account), a parallel study of cubic functions seems not to have been contemplated till recently. The essential difference is that a function f defines a pencil of curves – its level curves – which have to be considered simultaneously. The author's interest in the subject arose from problems in singularity theory (concerning canonical stratifications), and a later paper in the series will have applications of this kind. Here we study the simplest case as an introduction. Our techniques are entirely classical, but the results are hard to find elsewhere. In later papers we intend to study cubic functions on ℝ2, and on ℂ3.

P. R. Scott (1) has asked which two-dimensional closed convex set E, centro-symmetric in the origin O, and containing no other Cartesian lattice-point in its interior, maximizes the ratio A/P, where A, P are the area, perimeter of E; he conjectured that the answer is the ‘rounded square’ (‘cushion’ in what follows), described below. We shall prove this, indeed in a more general setting, by seeking to maximize

where κ is a parameter (0 < κ < 2); the set of admissible E is those E centro-symmetric in 0 that do not contain in their interior certain fixed lattice-points. There are two problems, the unrestricted one , where there is no given upper bound on A (it will become apparent that this problem only has a finite answer when κ ≥ 1) and the restricted one , when one is given a bound B and we must have A ≤ B. Special interest attaches to the case B = 4, both because of Minkowski's theorem: any E symmetric in O and containing no other lattice-point has area at most 4; and because it turns out that it is a ‘natural’ condition: the algebraic expressions simplify to a remarkable extent. Hence in what follows, the ‘restricted case ’ shall mean A ≤ 4.

Mathematical literature contains a number of theorems which assert the existence of a structure ‘universal’ in relation to some given system of objects. Two examples will suffice to indicate the nature of these theorems.

The richness of the theory of functions over the complex field makes it natural to look for a similar theory for the only other non-trivial real associative division algebra, namely the quaternions. Such a theory exists and is quite far-reaching, yet it seems to be little known. It was not developed until nearly a century after Hamilton's discovery of quaternions. Hamilton himself (1) and his principal followers and expositors, Tait(2) and Joly(3), only developed the theory of functions of a quaternion variable as far as it could be taken by the general methods of the theory of functions of several real variables (the basic ideas of which appeared in their modern form for the first time in Hamilton's work on quaternions). They did not delimit a special class of regular functions among quaternion-valued functions of a quaternion variable, analogous to the regular functions of a complex variable.

It was noted by Klein(4), part I, ch. II, §9, that if a polyhedral group G acts on the 2-sphere identified with P1(ℂ), then the polynomials cΠg∈G(λ − ag μ) form, as a∈ℂ varies, a vector space of dimension 2 over ℂ. G is generated by elements T1, T2 fixing respectively a vertex and a face; T1T2 fixes an edge and these elements have orders p, q, 2 where 1/p + 1/q > ½ Taking a to be a vertex or the mid-point of a face or an edge yields polynomials of the form and which are thus linearly dependent. The ring of all (relatively) invariant polynomials is generated by F1F2 and F3 subject to this single relation.

Let H be a finite dimensional real or complex Hilbert space. We denote by Λ(x, y, z) the area of the triangle with vertices x, y, z ∈ H. A map f: H → H is triangle contractive TC if 0 < α < 1 and for each x, y, z ∈ H either

or

and

and

We prove that if f is TC either there is a fixed point w = f(w) or a fixed line L = ⊃ f(L) We characterize the f which are TC and continuous but have no fixed point.

Recently (3) Mazur proved that if N is a prime number such that some elliptic curve E over Q admits a Q-rational isogeny then N is one of 2, 3, 5, 7, 11, 13, 17, 19, 37, 43, 67 or 163.

This paper presents an algebraic model for a general class of two-person perfect-information partizan games which may contain cycles, i.e. sequences of positions that may be assumed more than once. An algorithm is given for determining the winning, losing and tie-positions of disjunctive compounds of partizan games.

Using arguments involving combinatorial enumeration and asymptotics we compute the probability that a point of a random tree is fixed. The method is also applied to homeomorphically irreducible trees to illustrate how it works for other species of trees. To the nearest per cent, the limiting probability of a fixed point in a randomtree is 70%, and for homeomorphically irreducible trees it is 20%.

(1) In this paper we are concerned with normalized integer-valued length functions on a group G, that is, mappings l:G→ℤ satisfying three axioms:

(A1′)l(1) = 0

(A2) l(x) = l(x−1)for all x∈G

(A4) d(x, y) > d(x, z) implies that d(x, z) = d(y, z) for all x, y and z in G, where d(x, y) = ½(l(x) + l(y)−l(xy−1)).

These axioms were first considered by Lyndon(6), where their significance is discussed. Lyndon's axiom A1, which we shall not use, stated that l(x) = 0 if and only if x = 1. His axiom A 3 was that d(x, y) ≥ 0 for all x and y in G, but it was noted in (2) that this follows from A1′, A2 and A4. In particular, taking x = y, we find that l(x) ≥ 0 for all x in G. The other major axioms used in (6) were:

(A0) l(x2) > l(x), provided that x ≠ 1, and (A5) d(x, y) + d(xl, y1) > l(x) = l(y) implies that x = y.

Let G be a group with the lower central series

Let

where Σ runs over all non-negative integers a1, a2,…, an such that and is the aith symmetric power of the abelian group Gi/Gi+1 where

Let I (G) be the augmentation ideal of G in , the group ring of G over . Define the additive group Qn (G) = In (G) / In+1 (G) for any n ≥ 1. Then it is well known that Q1(G) ≅ W1(G) for any group G. Losey (4,5) proved that Q2(G) ≅ W2(G) for any finitely generated group G. Furthermore recently Tahara(12) proved that Q3(G) is a certain precisely defined quotient of W3(G) for any finite group G.

Let S0 be a complex projective surface with only isolated Gorenstein singularities (see Introduction to (12)). By Serre's criterion ((4), p. 185) this is equivalent to saying that S0 is normal and Gorenstein. By an algebraic smooth deformation of S0, we shall mean a flat, proper morphism of varieties, ρ: say, with fibre ρ−1(y0) = S0 for some y0 ∈ Y and with the general fibre ρ−1(y) = S being a smooth surface. In the paper (12), we studied such smooth deformations of S0 and in particular the behaviour of the plurigenera Pn of the surfaces in the family. The main result of (12) was the fact that Pn(S0) ≤ Pn(S) for all positive integers n, where the choice of the particular smooth surface was irrelevant by a result of Iitaka(5). To prove the above result we introduced what were called the arithmetic plurigenera of S0, which we define again below. In this paper we shall study more closely these arithmetic quantities, and in the process answer some of the questions posed in (11).

The purpose of the present note is to investigate the closure operations common to both the operations of forming classes and formations. Furthermore, new descriptions of formations and saturated formations in terms of closure operations are given.

In (11) and (12), a comparatively elementary approach to the use of dualizing complexes in commutative algebra has been developed. Dualizing complexes were introduced by Grothendieck and Hartshorne in (2) for use in algebraic geometry; the approach to dualizing complexes in (11) and (12) differs from that of Grothendieck and Hartshorne in that it avoids use of the concepts of triangulated category, derived category, and localization of categories, and instead places great emphasis on the concept of quasi-isomorphism of complexes of modules over a commutative Noetherian ring.

The following result has at least the appeal of intuitive plausibility. Let U and V be subvarieties of an algebraic variety X; let x ∈ X be an isolated point of the intersection of U and V, and let I(X, U. V, x) denote the intersection multiplicity (in some sense to be made precise) of U and V at x. Now let Tx be the tangent space to X at x, within which lie the tangent cones Cu, Cv to U and V at x. Then

Here we consider the problem of Whitney stratifying some very simple jet spaces. This type of problem has also been considered by Lander, Bedford and Gibson, (see (7), (1), (2)). Following Bedford and Gibson, we shall consider complex jet spaces.

Let Γ be an undirected graph and G a subgroup of aut (Γ) acting transitively on the vertex set V(Γ) of Γ. Let x be an arbitrary vertex of Γ. We denote by T(x) the set of vertices adjacent to x and by G(x)Γ(x) the permutation group induced by the stabilizer G(x) of x in G on Γ(x); G(x)Γ(x) is called the subconstituent of G (with respect to Γ). Let G1(x) = {a ∈ G(x)|a ∈ G(y) for each y ∈ Γ(x)}. For each y ∈ Γ(x), let G(x, y) = G(x) ∩ G(y) and G1(x, y) = G1(x) ∩ G1(y). An s-path is an (s+ l)-tuple (x0, x1, …, xs) of vertices such that xi−1 ∈ Γ(xi) if 1 ≤ i ≤ s and xi−2 ≠ xi if 2 ≤ i ≤ s. Γ is called (G, s)-transitive if G acts transitively on the set of all s-paths but intransitively on the set of all (S+1)-paths in Γ.

1. Let G be a group, ZG its integral group ring and Δ = ΔG the augmentation ideal ZG By an augmentation quotient of G we mean any one of the ZG-modules

where n, r ≥ 1. In recent years there has been a great deal of interest in determining the abelian group structure of the augmentation quotients Qn(G) = Qn,1(G) and

(see (1, 2, 7, 8, 9, 12, 13, 14, 15)). Passi(8) has shown that in order to determine Qn(G) and Pn(G) for finite G it is sufficient to assume that G is a p-group. Passi(8, 9) and Singer(13, 14) have obtained information on the structure of these quotients for certain classes of abelian p-groups. However little seems to be known of a quantitative nature for nonabelian groups. In (2) Bachmann and Grünenfelder have proved the following qualitative result: if G is a finite group then there exist natural numbers n0 and π such that Qn(G) ≅ Qn+π (G) for all n ≥ n0; if Gω is the nilpotent residual of G and G/Gω has class c then π divides l.c.m. {1, 2, …, c}. There do not appear to be any examples in the literature of this periodic behaviour for c > 1. One of goals here is to present such examples. These examples will be from the class of finite p-groups in which the lower central series is an Np-series.

Let M be a von Neumann algebra acting on a Hilbert space , and let G be a locally compact group. We consider an extension of G by , the unitary group of M. If the triple satisfies an additional axiom, we say that it is an extended covariant system. We define a Hilbert space and operators , acting on . The von Neumann algebra is then the covariance algebra of the extended covariant system , denoted by .

In (1) the authors introduced the idea of a syntopogenous pre-ordered space and, in (2), discussed the more general notion of a convex syntopogenous space. The former generalizes Nachbin's ‘uniform preordered space’ (4) and Singal and Lal's ‘proximity preordered space’ (7); the latter does the same for Nachbin's ‘convex topological space’ (4) and Redfield's ‘nearly uniform ordered space’ (5).