The classification of affine cubic functions in the real case is a fairly easy corollary of that in the complex case (9). However as the results can be easily interpreted by diagrams, one can obtain a much richer understanding. For example, the question of which types of cubic curve occur as level curves of which types of function is now much less trivial. This will lead us first to re-examine the classification of cubic curves going back to Newton (4). Next the ‘dynamic’ approach of considering these curves as members of families leads to the diagrams associated with the umbilic catastrophes of Thorn (8). However the consideration of functions rather than of curves gives a 1-dimensional foliation of these diagrams which we describe next. We conclude by placing the results back in a protective setting.

Let N be an integer ≥ 1. The affine modular curve Y0(N) parameterizes isomorphism classes of pairs (E; F), where E is an elliptic curve defined over ℂ, the field of complex numbers, and F is a cyclic subgroup of order N. The compacti-fication X0(N) is an algebraic curve defined over ℚ.

We show that if {Gi}J ε I is a generating set for an (elementary) topos ℰ then {P(Gi)}iεI is a cogenerating set for x2130;. From this we show that if topos ℰ contains an object G whose subobjects generate ℰ, then ΩG is a cogenerator for ℰ. Let denote the topos of finite sets and functions. We also show that if ℰ1 is a topos and ℰ2 is a bounded -topos then every geometric morphism ℰ1 → ℰ2 is essential.

The purpose of this note is to prove the following result.

Theorem 1. let M be a connected matroid. Suppose that C is a circuit of M and p and q are elements of M. Then M has circuits cp and Cq such that .

Knuth (12) seems to have been the first to carry out non-trivial matroid operations on a computer. However, as he remarks, there are considerable computational difficulties. In this paper we examine in detail the computational complexity of some fundamental matroid properties. The model of computation is the Hausmann–Korte oracle machine introduced in (10), (11) to deal with properties of independence spaces and fixed point problems. It was these papers (10), (11) and (12) which motivated the present work. Basically, the problems we will be discussing are matroid analogues of low-level complexity problems for graphs. For an excellent survey of this we refer to the recent monograph of Bollobés ((3), chapter 8). We shall use the approach of (3), and Bollobaé;s and Eldridge(4) rather than the more general treatment of Hausmann and Korte (11).

Let J be a complex Banach space and a complex Jordan algebra equipped with an algebra involution *. Then J is a Jordan C*-algebra if the following conditions are satisfied:

(Ua is defined on page 3).

In this paper we extend a result of Johnson and Lahr(3), which characterizes the multiplier algebra of L1(a, b) (the algebra of Lebesgue integrable functions on the interval of real numbers from a to b, under order convolution) to the L1 algebra of a general totally ordered semigroup. Similar work has been done in (l), but under more restrictive conditions.

Let Γ be a discrete subgroup of G = PSL(2,ℝ) and p be the canonical map from the universal covering group on to PSL(2,ℝ). By a continuity argument on a fundamental polygon for Γ acting on the hyperbolic plane 2, Milnor (4) obtained a presentation for p−1(Γ) whenever 2/Γ is compact and of genus zero. Using Teichmüller theory and a double Reidemeister-Schreier process, Macbeath(2) showed that the general compact case can be deduced from Milnor's result. We give here a method of obtaining a presentation which uses only the geometry of a Dirichlet region and which applies equally to the non-compact case.

If a function and all its derivatives and integrals are absolutely uniformly bounded, then the function is a sine function with period 2π.

Let be the circle group, M() the set of bounded Borel measures on and ℤ the additive group of integers. If μ ∈ M() and n ∈ ℤ, define

A well-known result of Rajchman states that

The following quantitative generalization of this result has been given in (2) by K. de Leeuw and Y. Katznelson.

Let H(t, θ) be the hyperplane in Rn (n ≥ 2) which is perpendicular to the unit vector θ, and distant t from the origin; that is H(t, θ) = {x ε Rn: x.θ = t}. (Note that H(t, θ) and H(−t, − θ) are the same hyperplane.) If f(x) εℒ1(Rn), we will denote the integral of f with respect to (n − 1)-dimensional Lebesgue measure over H(t, θ) by F(t, θ), termed the projection or sectional integral of f over H(t,θ). By Fubini's theorem, F(t, θ) exists for almost all t for any θ. Throughout this paper we will assume that f(x) has support in X, a compact convex subset of Rn. In Section 2 we examine some of the topologies that may be defined on functions on X in terms of the F(t, θ), and in the remainder of the paper we examine the extremal problem suggested by Croft (4), that of maximizing the integral of f over the set X with the constraint that the F(t, θ) are uniformly bounded above. We examine in particular how the extremal values depend on the convex set X. In the final section the extremal problem is related to a generalization of Bang's plank theorem and the theory of capacities, and several conjectures are proposed.

We say that a real vector bundle ξ over a finite C.W. complex X is stably trivial of type (n, k) or, simply, of type (n, k) if ξ ⊕ kε ≅ nε, where ε denotes a trivial line bundle. The following theorem is an immediate corollary (see (12)) of a theorem of T. Y. Lam ((9), theorem 2).

We prove the following generalized version of the complex Adams conjecture (see Theorem 10·4), as announced in (5).

A vector identity associated with the Dirichlet tessellation is proved as a corollary of a more general result. The identity has applications in interpolation and smoothing problems in data analysis, and may be of interest in other areas.

Let be a standard d-dimensional Brownian motion and L a smooth functional on the space C = Cd[0, 1] of continuous functions from [0,1] to Rd, such that Denote Then

is a square integrable martingale which can be represented as a stochastic integral; consequently the random variable has the representation

For Fréchet differentiable functions L, Clark(1) gave the following formula for the integrand e:

This paper is concerned with a particular method used in the calculation of asymptotic expansions of solutions to problems which contain a small parameter ∈. The physical idea underlying the method is that the parameter introduces new scales (of length and/or time) into the problem, and that the solution should incorporate these (see, for example, Cole (1)). This may be done in various ways: two-timing (two-scaling, multiple-scaling) does it by demanding that the solution is a function of all relevant scales. Consider, for example, a linear oscillator whose frequency is slowly adjusted, governed by

and subject to some initial condition. Account of the slowly varying frequency is taken by the dependence of Ω on ∈t, where 0 < ∈ ≪ 1, and we assume that Ω > 1 for all arguments. The rapid oscillatory behaviour anticipated in the solution is customarily described by a phase function Θ(isin;t)/∈, a generalization of Ωt the case that Ω is constant, while slow changes due to frequency variation occur on a scale ∈t.

Let {Xn} be a sequence of independent random variables each having a common d.f. V1. Suppose that V1 belongs to the domain of normal attraction of a stable d.f. V0 of index α 0 ≤ α ≤ 2. Here we prove that, if the c.f. of X1 is absolutely integrable in rth power for some integer r > 1, then for all large n the d.f. of the normalized sum Zn of X1, X2, …, Xn is absolutely continuous with a p.d.f. vn such that

as n → ∞, where v0 is the p.d.f. of Vo.