Let G be a nilpotent, connected and simply connected real Lie group and let Γ be a lattice in G. A necessary and sufficient condition is given for a Lie subgroup of G to act ergodically on Γ[setmn ]G. As an application, we characterize the irreducible unitary representations of G for which the restriction to Γ remains irreducible.

Suppose M is an oriented 3-manifold. A Dehn surgery on M (defined below) is a process by which M is altered by deleting a tubular neighbourhood of an embedded circle and replacing it again via some diffeomorphism of the boundary torus. It was shown by Lickorish [Li] and Wallace [Wa] that any closed oriented connected 3-manifold can be obtained from any other such manifold by a finite sequence of Dehn surgeries. Thus under this equivalence relation all closed oriented 3-manifolds are equivalent. We shall investigate this same question for more restricted classes of surgeries. In particular we shall insist that our Dehn surgeries preserve the integral (or rational) homology groups. Specifically, if M0 and M1 have isomorphic integral (respectively rational) homology groups, is there a sequence of Dehn surgeries, each of which preserves integral (respectively rational) homology, that transforms M0 to M1? What is the situation if we further restrict the Dehn surgeries to preserve more of the fundamental group? Is there a difference if we require ‘integral’ surgeries? We also show that these Dehn surgery relations are strongly connected to the following questions concerning another point of view towards understanding 3-manifolds. Is there a Heegard splitting of M0, M0 = H1 ∪fH2 (Hi are handlebodies of genus g and f is a homeomorphism of their common boundary surface), and a homeomorphism g of ∂H1 such that M1 has a Heegard splitting using g ∘ f as the identification? Since there are many natural subgroups of the mapping class group, such as the Torelli subgroup and the ‘Johnson subgroup’, one can ask the same question where g is restricted to lie in one of these subgroups. This is related to work of Morita on Casson's invariant for homology 3-spheres [Mo1]. Even under these restrictions it has been known for some time that any homology 3-sphere is related to S3. This fact has been used to define, calculate and understand invariants of homology 3-spheres (such as Casson's invariant) by choosing such a ‘path to S3’ in the ‘space’ of 3-manifolds.

Several authors [1, 5, 9] have investigated the algebraic and transcendental values of the Gaussian hypergeometric series

(formula here)

for rational parameters a, b, c and algebraic and rational values of z ∈ (0, 1). This led to several new identities such as

(formula here)

and

(formula here)

where Γ(x) denotes the gamma function. It was pointed out by the present authors [6] that these results, and others like it, could be derived simply by combining certain classical F transformation formulae with the singular values of the complete elliptic integral of the first kind K(k), where k denotes the modulus.

Here, we pursue the methods used in [6] to produce further examples of the type (1·2) and (1·3). Thus, we find the following results:

(formula here)

The result (1·6) is of particular interest because the argument and value of the F function are both rational.

It is well known that the Wallman-type compactifications of a Tychonoff space X can be obtained as spaces of all regular zero-one measures on suitable lattices of subsets of X (see [1, 2, 4, 12]). Using the technique developed in [5, 6], we find for any Tychonoff space X a Boolean algebra [Bscr ]X and a set [Lscr ]X of sublattices of [Bscr ]X having the following property: for any Hausdorff compactification cX of X there exists a (unique) LcX ∈ [Lscr ]X such that the maximal spectrum of LcX and the space of all u-regular zero-one measures on the Boolean subalgebra b(LcX) of [Bscr ]X, generated by LcX, are Hausdorff compactifications of X equivalent to cX. Let us give more details now.

We show that the asymptotic behaviour of the quantization error allows the definition of dimensions for probability distributions, the upper and the lower quantization dimension. These concepts fit into standard geometric measure theory, as the upper quantization dimension is always between the packing and the upper box-counting dimension, whereas the lower quantization dimension is between the Hausdorff and the lower box-counting dimension.

In 1922–23, Julia and Fatou proved that any 2 rational functions f and g of degree at least 2 such that f(g(z)) = g(f(z)), have the same Julia set. Baker then asked whether the result remains true for nonlinear entire functions. In this paper, we shall show that the answer to Baker's question is true for almost all nonlinear entire functions. The method we use is useful for solving functional equations. It actually allows us to find out all the entire functions g which permute with a given f which belongs to a very large class of entire functions.

This paper is concerned with the exponential sums

(formula here)

and with Heilbronn's type exponential sums

(formula here)

where (and below) p is prime, m, n, a and b are integers with m > n > 0 and p [nmid ] ab, and er(x) = exp(2πix/r) as usual.

We study boundary value problems for a linear evolution equation with spatial derivatives of arbitrary order, on the domain 0 < x < L, 0 < t < T, with L and T positive finite constants. We present a general method for identifying well-posed problems, as well as for constructing an explicit representation of the solution of such problems. This representation has explicit x and t dependence, and it consists of an integral in the k-complex plane and of a discrete sum. As illustrative examples we solve some two-point boundary value problems for the equations iqt + qxx = 0 and qt + qxxx = 0.

In this paper we shall consider differential operators that admit a periodic structure (cf. [1, 2])

(formula here),

where

(formula here),

are Hölder continuous for some α > 0 and periodic, so that, say, without loss of generality, we have

(formula here).

We shall assume that Q is elliptic, i.e. that the matrix A = (aij) satisfies

(formula here),

for some

(formula here).

For a general family of weight functions, a relation between orthogonal polynomials on the unit sphere in ℝd+m and orthogonal polynomials on the unit ball in ℝd is discussed and used to show that the summability of the Fourier orthogonal series on the unit ball follows from the summability on the unit sphere. Convergence of the Cesàro summability is established for a number of weight functions on the sphere and on the ball.

The principal result of the paper reduces the study of certain weakly compact sets in Banach spaces of measurable operators to that of the corresponding sets of generalized singular value functions. In particular, under natural conditions, it is shown that the orbit of a relatively weakly compact subset of the Köthe dual of a symmetric space of measurable operators affiliated with some semi-finite von Neumann algebra is again relatively weakly compact.

In this paper we apply pointwise subsequence ergodic theory to show that certain sequences of integers constructed randomly are multiply intersective. Analogous results for sequences constructed from Hardy fields are also indicated.

Let [Efr ] be a Banach space with a normalized, 1-unconditional basis. Each operator on the [Efr ]-direct sum of a sequence ([Xfr ]i)i∈ℕ of Banach spaces corresponds to an infinite matrix. We study whether this correspondence is multiplicative, in which case we say that matrix multiplication works. We prove that matrix multiplication works if at least one of the following two conditions is satisfied:

(i) for each i ∈ ℕ, each operator from [Xfr ]i to [Efr ] is compact;

(ii) the basis of [Efr ] is shrinking and, for each i ∈ ℕ, each operator from [Efr ] to [Xfr ]i is compact.

In the case where [Efr ] is either c0 or [lscr ]p, where 1 [les ] p < ∞, the converse also holds.

We show that, in some cases, the projective and the injective tensor products of two Banach spaces do not have the Dunford–Pettis property (DPP). As a consequence, we obtain that (c0 &[otimes ]circ;πc0)** fails the DPP. Since (c0 &[otimes ]circ;πc0)* does enjoy it, this provides a new space with the DPP whose dual fails to have it. We also prove that, if E and F are [Lscr ]1-spaces, then E &[otimes ]circ;ε has the DPP if and only if both E and F have the Schur property. Other results and examples are given.