Willis has established a structure theory of locally compact totally disconnected groups. An important feature of this theory is the notion of a tidy subgroup. In this paper we provide several results regarding these subgroups.

A proof is given that the improper Riemann integral of ζ(s, a) with respect to the real parameter a, taken over the interval (0, 1], vanishes for all complex s with ℜ(s) < 1. The integral does not exist (as a finite real number) when ℜ(s) ≥ 1.

This paper considers random fractals generated by contractive transformations which satisfy a separation condition weaker than the open set condition. This condition allows overlaps in the iterations. Estimates of the Hausdorff dimension of this kind of random fractals are obtained.

We study the boundedness of singular integral operators that are imaginary powers of the Laplace operator in Rn, especially from weighted Hardy spaces to weighted Lebesgue spaces where 0 < p ≤ 1. In particular, we prove some estimates for these operators when 0 < p ≤ 1 and w is in the Muckenhoupt's class Aq, for some q > 1.

An infinite family of finite semigroups is studied. It is shown that most of them do not generate a quasivariety which admits a natural duality.

Steady waves at the free surface of an incompressible fluid passing over a depression are considered. By studying a KdV equation with negative forcing term, new types of solutions are discovered numerically and a new cut-off value of the Froude number, above which unsymmetric solitary-wave-like wave solutions exist, is also found.

Given two different positive integers k and l, a (k, l)-free set of some group (G, +) is defined as a set  ⊂ G such that k∩l = ∅. This paper is devoted to the complete determination of the structure of (k, l)-free sets of ℤ/pℤ (p an odd prime) with maximal cardinality. Except in the case where k = 2 and l = 1 (the so-called sum-free sets), these maximal sets are shown to be arithmetic progressions. This answers affirmatively a conjecture by Bier and Chin which appeared in a recent issue of this Bulletin.

Let rk (n) denote the number of representations of n as the sum of k squares. We give elementary proofs of relations between rk (n) and rk (m) where n = 4λm and 4 ∤ m, when k = 5, 7, 9 and 11. The relations, which were first stated without proof by Stieltjes, are of the form rk (n) = Crk (m) where C depends on λ and on the residue of m modulo 8. They have recently been include by S. Cooper in a more complete description of the relations between rk (n) and rk (n′) where n′ is the squarefree part of n, when k = 5, 7, 9 and 11.

In this note, we consider multipliers on weighted function spaces over totally disconnected locally compact Abelian groups (Vilenkin groups). First we present an multiplier result. Then we give an multiplier result under a similar condition of Lu-Yang type. In Section 3, we obtain a result about the boundedness of multipliers on weighted Besov spaces.

Suppose that A is a finite direct product of commutative rings. We show from first principles that a Gröbner basis for an ideal of A[x1,…,xn] can be easily obtained by ‘joining’ Gröbner bases of the projected ideals with coefficients in the factors of A (which can themselves be obtained in parallel). Similarly for strong Gröbner bases. This gives an elementary method of constructing a (strong) Gröbner basis when the Chinese Remainder Theorem applies to the coefficient ring and we know how to compute (strong) Gröbner bases in each factor.

A Banach space X has the average distance property if there exists a unique real number r such that for each positive integer n and all x1,…,xn in the unit sphere of X there is some x in the unit sphere of X such that

We show that lp does not have the average distance property if p > 2. This completes the study of the average distance property for lp spaces.

Let G be the group of isometries of a homogeneous tree . In the first part of this paper we decompose G in terms of certain subgroups N, ℤ, and K to obtain the related integral formula

Then, by using ideas of A. Ionescu and the formula above, we prove that and that a related maximal operator on  is bounded from L2, 1() to L2,∞(). We finally show that Lp, 1(K\G/K) is a commutative Banach algebra of convolutors for Lp(G) and give an explicit description of its Gelfand spectrum.

A new class of bilinear permutation polynomials was recently identified. In this note we determine the class of permutation polynomials which represents the functional inverse of the bilinear class.