Let G be a regular graph with even girth g = 2r ≥ 4 and valency k ≥ 3. It is well known, and easy to prove, that G must have at least n0 = n0(g, k) vertices, where

By considering the action of the Hecke operators on the logarithm of the Dedekind eta function together with the modular transformation formula for this function, Knopp (8) proved an extension of an identity of Dedekind for the classical Dedekind sums first mentioned by H. Petersson. By looking at the action of the Hecke operators on certain Lambert series studied by Apostol(l) together with the transformation formulae for these series, Parson and Rosen (9) established an analogous identity for a type of generalized Dedekind sum. A special case of this identity was initially proved by Carlitz(6). In this note an elementary proof of these identities is given. The Hecke operators are applied directly to the Dedekind sums without invoking the transformation formulae for the logarithm of the eta function or for the Lambert series. (Recently, L. Goldberg has given another elementary proof of Knopp's identity.)

In an earlier paper (5) we showed that a finitely generated nilpotent group which is not abelian-by-finite has a primitive irreducible representation of infinite dimension over any non-absolute field. Here we are concerned primarily with the converse question: Suppose that G is a polycyclic-by-finite group with such a representation, then what can be said about G?

Two measures are strongly equivalent if they have the same sets of zero measure and the same sets of infinite measure. Given a group G of strongly non-singular measurable transformations of a non-atomic positive measure space (X, β, p), if G is amenable, then a necessary and sufficient condition for there to be a G-invariant positive measure on (X, β) which is strongly equivalent to p is that p(E) > 0 implies inf p(gE) > 0 and also p(E) < ∞ implies

Surface integrals of curvature arise naturally in integral geometry and geometrical probability, most often in connection with the Quermassintegrale or cross-section integrals of convex bodies. They enjoy many desirable properties, such as the ability to be determined by summing or averaging over lower-dimensional sections or projections. In fact the Quermassintegrale are the only functionals of convex bodies to meet certain, quite reasonable, requirements. The conclusion has often been drawn, especially in practical applications, that the Quermassintegrale and their associated curvature integrals have a canonical status to the exclusion of all other quantities.

We give sufficient conditions on the potential V(x) which ensure that the Schrödinger operator (1 · 1) of quantum mechanics has no singular continuous spectrum This generalizes previous results.

The theory of evolution equations in Hamiltonian form is developed by use of some differential complexes arising naturally in the formal theory of partial differential equations. The theory of integral invariants is extended to these infinite-dimensional systems, providing a natural generalization of the notion of a conservation law. A generalization of Noether's theorem is proved, giving a one-to-one correspondence between one-parameter (generalized) symmetries of a Hamiltonian system and absolute line integral invariants. Applications include a new solution to the inverse problem of the calculus of variations, an elementary proof and generalization of a theorem of Gel'fand and Dikiî on the equality of Lie and Poisson brackets for Hamiltonian systems, and a new hierarchy of conserved quantities for the Korteweg–de Vries equation.

For proving the existence of periodic orbits of autonomous ordinary differential equations, three different methods are available, namely, the Hopf bifurcation theorem, the torus principle and the Poincaré–Bendixson theorem. Until recently, the Poincaré–Bendixson theorem was applicable only to equations of the second order. However, it was extended in (13, 14) to certain equations of higher order by means of a plane projection technique. In §§ 2, 3 of the present paper this technique is adapted to prove analogues of the Poincaré–Bendixson theorem for a class of autonomous retarded functional differential equations. For such equations the orbits lie in a Banach space studied by Hale ((7), p. 43). The main hypothesis of this theorem is that the equation has a bounded semi-orbit whose ω-limit set contains no critical points. The problem of finding such a semi-orbit is solved in § 4 for a class of dissipative equations. The practical application of these results requires the construction of certain functionals which are similar to the Liapunov functionals of stability theory. In § 5, these functionals are constructed for a large class of retarded feedback control equations and explicit conditions for the existence of periodic orbits are deduced. Some practical details are illustrated in § 6 by applying the theory to certain scalar delay-differential equations.

Perturbations of nonlinear equations exhibiting bifurcation from a simple eigenvalue, can either preserve or destroy the bifurcation. Both types of perturbation are investigated, and some of the interesting degenerate cases are explored in some detail.

Let X, Y be complex manifolds with smooth (C∞) boundaries ∂X, ∂Y. We give conditions which ensure that a smooth map Φ: ∂X → ∂Y has an extension to a holomorphic map X → Y. Let J denote the complex structure on X. We say that Φ satisfies the ‘tangential Cauchy-Riemann equation ∂¯bΦ = 0’if the differential dΦ restricted to the complex subspace Tp(∂X) ∩ JTp(∂X) of the tangent space Tp(∂X) is complex linear at all points p ∈ ∂X. Clearly this is a necessary condition for the existence of a holomorphic extension. A further necessary condition is that there exists no topological obstruction to extension, hence we assume that a smooth extension φ: X → Y is given and we shall look for a holomorphic map f: X → Y with the same boundary values.

The purpose of this note is to show that, for any algebraic element a of a Banach algebra and certain analytic functions f, one can give an upper bound for ‖f(a)‖ in terms of ‖a‖ and the spectral radius ρ(a) of a. To illustrate the nature of the result, consider the norms of powers of an element a of unit norm. In general, the spectral radius formula

contains all that can be said (that is, the limit ρ(a) can be approached arbitrarily slowly). If we have the additional information that a is algebraic of degree n we can say a good deal more. In the case of a C*-algebra we have the neat result that, if ‖a‖ ≤ 1,

(see Theorem 2), while for a general Banach algebra we have at least

In (7) we proved maximal and pointwise ergodic theorems for transformations a of a von Neumann algebra which are linear positive and norm-reducing for both the operator norm ‖ ‖∞ and the integral norm ‖ ‖1 associated with a normal trace ρ on . Here we introduce a class of Banach spaces of unbounded operators, including the Lp spaces defined in (6), in which the transformations α reduce the norm, and in which the mean ergodic theorem holds; that is the averages

converge in norm.

V. P. Snaith's theorem giving splittings of the suspension spectrum of ΩnΣnX for a connected space X (8) has been exploited in several areas of homotopy theory. (See (2), (5) and (6), for instance.) Although this is a theorem about spectra, Snaith's proof (as well as a subsequent proof of Cohen, May, and Taylor (4)) proceeds unstably, on the space level. While the methods employed in these proofs are useful, they are somewhat complicated. The purpose of this note is to give a simple proof of this theorem, using only spectrum level arguments.

1. Few mod 2 finite simply-connected spaces are known to support an H-structure. These examples arise from products of Lie groups and seven spheres. It has been shown in (12) and (14) that the space of a mod 2 finite H-space of given rank can belong to only a finite number of different homotopy types. The situation at odd primes is very different; many examples of mod p finite H-spaces are known, such as p-localized odd-dimensional spheres. Therefore there are infinitely many homotopy types of prescribed rank. If one insists that the mod pH-space has a strictly associative multiplication, or equivalently has the homotopy type of a topological group, the picture is less clear. Examples have been constructed which do not arise from Lie groups and spheres but there are strong restrictions on just what can occur by these techniques, see (5). There are only finitely many known examples in each rank. We establish that there can in fact only be a finite number of possible homotopy types for the spaces.

Let , where the Xi, i = 1, 2, … are independent identically distributed random variables. Classical extreme value theory, described for example in the books of do Haan(6) and Galambos(3) gives conditions under which there exist constants an > 0 and bn such that

where G(x) is taken to be one of the extreme value distributions G1(x) = exp (− e−x), G2(x) = exp (− x−a) (x > 0, α > 0) and G3(x) = exp (−(− x)α) (x < 0, α > 0).

Let G be an infinite partially directed graph of finite outgoing degree. Thus G consists of an infinite set of vertices, together with a set of edges between certain prescribed pairs of vertices. Each edge may be directed or undirected, and the number of edges from (but not necessarily to) any given vertex is always finite (though possibly unbounded). A path on G from a vertex V1 to a vertex Vn (if such a path exists) is a finite sequence of alternate edges and vertices of the form E12, V2, E23, V3, …, En − 1, n, Vn such that Ei, i + 1 is an edge connecting Vi and Vi + 1 (and in the direction from Vi to Vi + 1 if that edge happens to be directed). In mixed Bernoulli percolation, each vertex Vi carries a random variable di, and each edge Eij carries a random variable dij. All these random variables di and dij are mutually independent, and take only the values 0 or 1; the di take the value 1 with probability p, while the dij take the value 1 with probability p. A path is said to be open if and only if all the random variables carried by all its edges and all its vertices assume the value 1. Let S be a given finite set of vertices, called the source set; and let T be the set of all vertices such that there exists at least one open path from some vertex of S to each vertex of T. (We imagine that fluid, supplied to all the source vertices, can flow along any open path; and thus T is the random set of vertices eventually wetted by the fluid). The percolation probability

is defined to be the probability that T is an infinite set.

Let (S, d) be a separable metric space equipped with its Borel σ field . Let {Yn, n ≥ 1} be a sequence of S-valued random elements defined on a probability space (Ω, , p). Assume Yn ⇒ Y converges weakly to an S-valued random element Y. Let {Nn, n ≥ 1} be a sequence of positive integer-valued random variables defined on the same probability space (Ω, , p).

This paper is concerned with geometrical properties of null hypersurfaces in Lorentzian manifolds. Null hypersurfaces have metrics with vanishing determinants and this degeneracy of these metrics leads to several difficulties. First, the contravariant metric cannot immediately be defined, so the connection cannot be specified uniquely in the normal way. Secondly, the normal is a null vector lying in the tangent plane, which makes it necessary to look for some other vector to rig the hypersurface, and makes it impossible to normalise the normal in the usual way. These problems are considered in this paper.

In this paper it is shown that a class of linearized interface-wave problems for two superposed inviscid liquids of unequal densities occupying regions which are symmetric about the interface can be reduced to a surface-wave problem in the lower region together with a classical hydrodynamical problem for potential flow in the lower region under a plane lid. The effect of interfacial tension is included. Examples of fundamental singularities in two semi-infinite liquids are given.