One of the most studied parameters in all of graph theory is the chromatic number. Undoubtedly, its popularity as a subject for research is due to its intimate relationship with the famous Four Colour Problem.

Notation: Let M and Q be manifolds. The embedding f: M → Q is said to be proper if f−1∂Q = ∂M.

Let be the complex vector space consisting of all complex-valued functions of a non-negative real variable t. For each positive number u, the shift operator Iu is the mapping of into itself defined by the formula

A linear operator T which maps a subspace of into itself is said to be a V-operator (13) if:

(a) for each x in , the complex-conjugate function x* is in ;

(b) both and \ are invariant under the shift operators;

(c) every shift operator commutes with T.

(Property (a) ensures that every function x in can be uniquely expressed as x1 + ix2, where x1 and x2 are real functions in .)

We recall that a matrix A is said to sum a sequence x if Axε c, the space of all convergent sequences, and that A is conservative if it sums every convergent sequence. If A is conservative, A defines a continuous linear operator on c. Berg (2), Crawford (3)and Whitley (9) have proved the following theorem:

Theorem 1. A conservative matrix sums no bounded divergent sequence if and only if,considered as an operator on c, it is range closed and has finite-dimensional null-spac

Kwapień(2) and Schwartz (3) have shown that the inclusion mapping I: ι1→ι2 is 0-radonifying. We shall give another proof of this, using an inequality due to Khint-chine(1).

Let 0 ≤ λ0 < λ1 < … < λn → ∞, and let

If as ω → ∞, we write

Suppose that

converges for σ > 0,

where τp(h) is finite for |h| < δ. If

we write

1. The classical Kneser conjecture states that if M is a closed 3-manifold and π1(M) ≈ A * B, a free product, then there exist 3-manifolds MA, MB such that

and π1(MA) ≈ A, π1(MB) ≈ B. This has been proved by John Stallings(6).

It is the object of this paper to prove a transversality theorem for homology manifolds which is strong enough to be then able to prove a Thom theorem for the cobordism ring of homology manifolds. The new bundle theory necessary for this follows on naturally from that developed in ‘Homology Cobordism Bundles’(2).

A general method is given and illustrated by application to particular cases for obtaining subdominant solutions of stiff difference and differential equations, i.e. when rapidly varying solutions – transients or otherwise – are possible but are in fact excluded by the initial conditions.

In this paper G is a locally compact Abelian group, φ a complex-valued function defined on the dual Γ, Lp(G) (1 ≤ p ≤ ∞) the usual Lebesgue space of index p formed with respect to Haar measure, C(G) the set of all bounded continuous complex-valued functions on G, and C0(G) the set of all f ∈ C(G) which vanish at infinity.

1. Let A be a summability method given by the sequence-to-sequence transformation

We suppose throughout that, for each n

converges; this is a much weaker assumption than the regularity of A. Then we define

We also suppose throughout that the sequence {sk} is formed by taking the partial sums of the series Σak; that is to say that

Let A' denote the summability method given by the series-to-sequence transformation

Following Lorent and Zeller (4), (5), we describe A, A' as dual summability methods. We recall that formally,

Let X be a non-negative discrete random variable with distribution {Px} and Y be a random variable denoting the undestroyed part of the random variable X when it is subjected to a destructive process such that

This paper is concerned with the symplectic bordism ring MSp*, whose structure is still largely unknown despite the partial results of Liulevicius(3), Novikov(4) and Stong(lO). It is the first of two elaborating(8), and it relies completely on (7) for notation and prerequisites. In particular we assume the reader to be familiar with (7), section 8, which establishes the relation between the Sp Hattori-Stong conjecture (that KO decides MSp) and the spectral sequence

We are here concerned entirely with computations in this spectral sequence.

Let f(x) and g(x) be rational functions of x with integer coefficients, p an odd prime, e(x) = e(2πix/p), e(u/v) = e(w), where u = vw (mod p)† and χ(x) is a character mod p. It is well known that very abstruse methods are required to find an estimate for sums such as

where in the summation, ‡ the values of x are omitted for which eitherf(x) or g(x) is not defined.

Several years ago, Fisher and Fuller(1) proved that if P is a real square matrix with all members of its ‘nested set’ of principal minors non-zero, then there exists a real diagonal matrix, D, such that the characteristic roots of DP are all real, negative and distinct. This interesting and powerful result, used by the authors to derive further results concerning convergence of linear iterative processes, has since also proved of interest to economists studying the stability of economic general equilibrium.

This paper is designed to pave the way for the author's work on the symplectic bordism ring MSp*(12), (13). We here discuss all the notation used, and collect together all the theorems quoted therein. At the same time, I hope that some of the results presented here might be of interest in their own right. Our central theme is the study of the hurewicz map e: S*(X) → E*(X), both in general and certain specific cases.

The problem is governed by a Jones integral equation and the solution is shown to depend upon a single function which occurs naturally after a contour deformation has produced extensive cancellation in the work. The far-scattered field off the axis of symmetry is found in detail, yielding terms which are believed to be new.

1. Professor J. W. S. Cassels (in (l)) recently enunciated a conjecture concerning Kummer sums. His conjecture suggests that there is a close connexion between the Kummer sum and a certain product of Weierstrass elliptic functions.

In this paper we consider the biquadratic Gauss sum. A theory similar to that used in (1) and in (4) suggests that there is a corresponding connexion between the biquadratic sum and a product of Jacobian elliptic functions. The enunciation of this conjecture appears in section 6. But first of all we require some preliminary results.

This is the second of two papers elaborating (4), and is concerned with the more geometrical aspects of the symplectic bordism ring in low dimensions. As such, it is a natural sequel to (2). However, the reader who does not care for the algebra therein need only consult (3), in which appears all the notation and prerequisites for this paper.

Initially a general incident field is considered and the equations are split into Fourier components. Each Fourier component gives rise to an integral equation similar to that obtained when investigating diffraction of a normally incident plane wave. After the oblique wave has been specified an analysis similar to that used for the normal wave leads to a solution of the problem containing an infinite number of constants. It is shown, however, that these constants do not affect the leading terms of the high frequency scattering coefficient.