Showing papers in "Bulletin of The London Mathematical Society in 1973"

Spectral Asymmetry and Riemannian Geometry

Abstract: This has an analytic continuation to the whole s-plane as a meromorphic function of 5 and s = 0 is not a pole: moreover CA(o) can be computed as an explicit integral over the manifold [9]. In this note we shall introduce a refinement of this invariant when A is no longer positive and we shall study its geometrical significance for an important class of operators (first order systems) arising from Riemannian geometry. A full exposition will be given elsewhere. Suppose therefore that A is self-adjoint and elliptic but no longer positive. The eigenvalues are now real but can be positive or negative. We define, for Re(s) large,

A theorem about random fields

Abstract: Averintsev [1] and Spitzer [2] proved that the class of Markov fields is identical to the class of Gibbs ensembles when the domain is a finite subset of the cubic lattice and each site may be in either of two given states. Hammersley and Clifford [3] proved the same result for the more general case when the domain is the set of sites of an arbitrary finite graph and the number of possible states for each site is finite. In order to show this, they extended the notion of a Gibbs ensemble to embrace more complex interactions than occur on the cubic lattice. Their method was circuitous and showed merely the existence of a potential function for a Markov field with little indication of its form. In [4], Preston gives a more direct approach to the two-state problem and presents an explicit formula for the potential. We show here that the equivalence of Markov fields and Gibbs ensembles follows immediately from a very simple application of the Mobius inversion theorem of [5] which allows us to construct a natural expression for the potential function of a Markov field. We confine our attention to the set of sites of an arbitrary finite graph and allow each site to be in any one of a countable set of states. The two-state solution of Preston emerges as a corollary.

On the Structure of Edge Graphs

Abstract: Every graph appearing in this note is a finite edge graph without loops and multiple edges. Denote by G(n, m) a graph with n vertices and ni edges . K r (t) denotes a graph with r groups of t vertices each, in which two vertices are connected if and only if they belong to different groups . By dividing n vertices into r-1 almost equal groups and connecting the points in different groups one obtains a graph on n vertices with ((r 2)/2(r 1) + u (1)) 11 2 edges which does not contain a Kr(l) . On the other hand, it was shown by Erdős and Stone [7] that ((r 2)/2(r 1) + s) n2 (e > 0) edges assure already the existence of a Kr(t), where t -~ oo as n -p co . This result is the inost essential part of the theorems on the structure of extremal graphs, see e .g. [3], [4], [6], [9] . Let us formulate the result of Erdős and Stone more precisely . Given n, r and e, put rn = [((r-2)/2(r-1)+e)n2] ([x] denotes the integer part of x) and define g(n, r, e) = min {t : every G(n, in) contains a K r (t)} .

A Table of Complex Cubic Fields

Abstract: A table of real cubic fields with discriminant less than 20,000 is given by Godwin and Samet [6]. This paper describes the construction of a complementary table of the 3169 non-conjugate complex cubic fields with discriminants greater than -20,000, which extends the existing table for discriminants greater than -1000 given by Delone and Faddeev [3]. The basis of this calculation is the following theorem, similar to one used by Godwin [5].