1. Introduction 514

2. Eigenvalues and eigenfunctions 516

2.1 Spectral asymptotics 516

2.2 The isoperimetric problem 518

2.3 The Rellich inequality 519

2.4 Numerical studies of the rectangular plate 519

2.5 The ground state 521

2.6 Stability properties 522

2.7 The ideal column 523

3. Solving elliptic equations 524

3.1 One-dimensional results 524

3.2 Constant-coefficient operators 524

3.3 Green functions of bounded regions 525

3.4 The Poisson problem 526

3.5 The Dirichlet problem 526

3.6 Conical points 527

4. The semigroup e−Ht 528

4.1 Elliptic systems 528

4.2 Extensions to Lp 529

4.3 Generalized Schrödinger operators 530

4.4 Kato class potentials 530

5. Heat kernel bounds 531

5.1 General properties of heat kernels 532

5.2 Gaussian upper bounds 532

5.3 Sharp constants 533

5.4 Complex time bounds 534

5.5 Pointwise lower bounds 535

6. Lp spectral theory 536

6.1 Functional calculus 536

6.2 Helffer–Sjöstrand calculus 537

6.3 Spectral independence 538

7. Non-self-adjoint operators 538

7.1 Second-order operators 538

7.2 Lp multiplier theory 539

7.3 H∞ functional calculus 540

7.4 Applications of the H∞ functional calculus 540

References 541

Answering a long-standing question in the theory of torsion modules, we show that weakly productively bounded domains are necessarily productively bounded. (See the Introduction for definitions.) Moreover, we prove a twin result for the ideal lattice L of a domain equating weak and strong global intersection conditions for families (Xi)i∈I of subsets of L with the property that ∩i)i∈IAi≠0 whenever Ai∈Xi. Finally, we show that for domains with Krull dimension (and countably generated extensions thereof), these lattice-theoretic conditions are equivalent to productive boundedness.

A remarkable theorem of Birch [2] shows that a system of homogeneous polynomials with rational coefficients has a non-trivial zero, provided only that these polynomials are of odd degree, and the system has sufficiently many variables in terms of the number and degrees of these polynomials. Despite four decades of effort, the problem of obtaining a reasonable bound for the latter number of variables has proved to be one of great difficulty. When the system consists of a single cubic form, Davenport [4] has succeeded in showing that 16 variables suffice, and Schmidt [17, 18, 19, 20] has devoted a series of papers to systems of cubic forms, showing in particular that 5140 variables suffice for pairs of cubic forms, and that (10r)5 variables suffice for systems of r cubic forms. The current state of knowledge for forms of higher degree is, by comparison, extremely weak (but see [21, 22]), and so it seems worthwhile expending further effort on the case of systems of cubic forms. In this paper we improve on Schmidt's result for pairs of cubic forms. In contrast with the sophisticated versions of the Hardy–Littlewood method employed by Davenport and Schmidt, our approach is based on an elementary idea of Lewis [12], and is applicable in arbitrary number fields. This method also has consequences for the existence of linear spaces of rational solutions on cubic hypersurfaces, thereby improving on work of Lewis and Schulze-Pillot [14] on this topic.

For n a positive integer, a group G is called core-n if H/HG has order at most n for every subgroup H of G (where HG is the normal core of H, the largest normal subgroup of G contained in H). It is proved that a locally finite core-n group G has an abelian subgroup whose index in G is bounded in terms of n.

Let G be a finite simple group. A conjecture of J. D. Dixon, which is now a theorem (see [2, 5, 9]), states that the probability that two randomly chosen elements x, y of G generate G tends to 1 as [mid ]G[mid ]→∞. Geoff Robinson asked whether the conclusion still holds if we require further that x, y are conjugate in G. In this note we study the probability Pc(G) that 〈x, xy〉=G, where x, y∈G are chosen at random (with uniform distribution on G×G). We shall show that Pc(G)→1 as [mid ]G[mid ]→∞ if G is an alternating group, or a projective special linear group, or a classical group of bounded dimension. In fact, some (but not all) of the exceptional groups of Lie type will also be dealt with.

We describe an uncountable family of compact group automorphisms with entropy log 2. Each member of the family has a distinct dynamical zeta function, and the members are parametrised by a probability space. A positive proportion of the members have positive upper growth rate of periodic points, and almost all of them have an irrational dynamical zeta function.

If infinitely many Mersenne numbers have a bounded number of prime divisors, then a typical member of the family has upper growth rate of periodic points equal to log 2, and lower growth rate equal to zero.

Let G be a finite group, let V be an [ ]G-module of finite dimension d, and denote by β(V, G) the minimal number m such that the invariant ring S(V)G is generated by finitely many elements of degree at most m. A classical result of E. Noether says that β(V, G)[les ][mid ]G[mid ] provided that char [ ] is coprime to [mid ]G[mid ]!. If char [ ] divides [mid ]G[mid ], then no bounds for β(V, G) are known except for very special choices of G. In this paper we present a constructive proof of the following. If H[les ]G with [G[ratio ]H]∈[ ]*, and if the restriction V[mid ]H is a permutation module (for example, if V is a projective [ ]G-module and H∈Sylp(G)), then β(V, G)[les ]max{[mid ]G[mid ], d([mid ]G[mid ]−1)} regardless of char [ ].

In [1], Beardon introduced the Apollonian metric α defined for any domain D in ℝ by

formula here

This metric is Möbius invariant, and for simply connected plane domains it satisfies the inequality αD[les ]2ρD, where ρD denotes the hyperbolic distance in D, and so gives a lower bound on the hyperbolic distance. Furthermore, it is shown in [1, Theorem 6.1] that for convex plane domains, the Apollonian metric satisfies ρD[les ]4 sinh [½αD], and, by considering the example of the infinite strip {x+iy[ratio ][mid ]y[mid ]<1}, that the best possible constant in this inequality is at least π. In this paper we make the following improvements.

We consider the functional equation F∘T−F=f, where T is a measure-preserving transformation and f is a continuous function. We show that if there is an L∞ function F which satisfies this equation, then F is constrained to satisfy a number of regularity conditions, and, in particular, if T is a one-sided Bernoulli shift, then we show that there is a continuous function F satisfying this equation. We show that this is not the case for the two-sided shift.

It is proved that every commutative, nilpotent extension of a commutative C*-algebra splits topologically.

In [2], H. Furstenberg studied a distal action of a locally compact group G on a compact metric space X, and established a structure theorem. As a consequence, he showed that if G is abelian, then a simply connected space X does not admit a minimal distal G-action.

In this paper we concern ourselves with a nonsingular flow ϕ={ϕt} on a closed 3-manifold M. Recall that ϕ is called distal if for any distinct two points x, y ∈ M, the distance d(ϕtx, ϕty) is bounded away from 0. The distality depends strongly upon the time parametrization. For example, there exists a time parametrization of a linear irrational flow on T2 which yields a nondistal flow [4, 6].

John Leslie Britton died in a climbing accident on the Isle of Skye on 13 June 1994 at the age of 66. He became a member of the London Mathematical Society in 1957, served the society as Meetings and Membership Secretary from 1973 to 1976, and at the same time was Editor of the Society's Newsletter.