Commutative rings form a very special subclass of rings, which shows quite different behaviour from the general case. For example, in a (non-trivial) commutative ring, the absence of zero-divisors is sufficient as well as necessary for the existence of a field of fractions, whereas for general rings, another infinite set of conditions is needed to characterize subrings of skew fields. This suggests the study of a class of rings which includes all commutative rings as well as all integral domains: reversible rings, where a ring is called reversible if ab = 0 implies ba = 0. It turns out that this condition helps to simplify other ring conditions, as we shall see in Section 2, although most of these results are at a somewhat superficial level. We therefore introduce a more technical notion, full reversibility, in Section 3, and show that this is the precise condition for the least matrix ideal to be proper and consist entirely of non-full matrices. Further, we show in Section 4 that a fully reversible ring is embeddable in a skew field if and only if it is an integral domain.

In what follows, all rings are associative, with a unit element 1 which is preserved by ring homomorphisms, inherited by subrings and acts unitally on modules.

I am grateful to V. de O. Ferreira for his comments, in particular the suggestion of using the notion of unit-stable rings.

For a wide class of set ideals (including, for example, all uniform ideals), a criterion of completeness of their symmetry groups is provided in terms of ideal quotients (polars). We apply it to partition ideals, and derive the extended Sierpiński–Erdös duality principle. We demonstrate that if the measure and category ideals I0 and I1 on the real line ℝ are partition (or, equivalently, if just I0 is partition), then not only are they isomorphic via an involution, but they also have complete (and distinct) symmetry groups coinciding, respectively, with the symmetry groups of the polars I⊥0 and I⊥1. The measure and category ideals on ℝ (and in more general spaces) are partition (Oxtoby) ideals assuming Martin's Axiom. In this case their polars are, respectively, the ideals generated by c-Sierpiński and c-Lusin sets. It is well known that the isomorphism of the measure and category ideals is not provable in ZFC. We show that the isomorphism of their symmetry groups is likewise unprovable.

We prove that Yuzvinskii's entropy addition formula cannot be extended for the action of certain finitely generated nonamenable groups.

Let K be a compact subset of ℝn, 0[les ]s[les ]n. Let Ps0, [Pscr ]s denote s-dimensional packing premeasure and measure, respectively. We discuss in this paper the relation between Ps0 and [Pscr ]s. We prove: if Ps0(K)<∞, then [Pscr ]s(K) = Ps0(K); and if Ps0(K) = ∞, then for any ε>0, there exists a compact subset F of K such that [Pscr ]s(F) = Ps0(F) and [Pscr ]s(F)[ges ] [Pscr ]s(K)−ε.

In this paper the authors develop a new approach to the problem of ‘propagation of smallness’ for harmonic functions in arbitrary domains, in ℝn (n[ges ]2). The main result of this paper is a certain logarithmic-convexity relation for the L2-norms of harmonic functions. As a consequence, new kinds of uniqueness results for harmonic functions are obtained. The method works also for analytic functions in [Copf ], with Lp-norms (p>0).

The following result is established.

THEOREM. Let G be a periodic, residually finite group with all subgroups sub-normal. Then G is nilpotent.

The well-known groups of Heineken and Mohamed [1] show that the hypothesis of residual finiteness cannot be omitted here, while examples in [5] show that a residually finite group with all subgroups subnormal need not be nilpotent. The proof of the Theorem will use the results of Möhres that a group with all subgroups subnormal is soluble [3] and that a periodic hypercentral group with all subgroups subnormal is nilpotent [4]. Borrowing an idea from [2], the plan is to construct certain subgroups H and K that intersect trivially, and to show that the subnormality of both leads to a contradiction.

We study the Čerednik–Drinfeld p-adic uniformization of certain Atkin–Lehner quotients of Shimura curves over ℚ. We use it to determine over which local fields they have rational points and divisors of a given degree. Using a criterion of Poonen and Stoll, we show that the Shafarevich–Tate group of their jacobians is not of square order for infinitely many cases.

We show that every μ-constant family of isolated hypersurface singularities of type F(x, t) =f(x)+tg(x), where t is a parameter, is topologically trivial. The proof uses only the curve selection lemma, and hence, for an appropriately translated statement, also works over the reals and for some families of non-isolated singularities. Some applications to study the singularities at infinity of complex polynomials are given.

In 1938, I. J. Schoenberg asked for which positive numbers p is the function exp(−∥x∥p) positive definite, where the norm is taken from one of the spaces [lscr ]np, q>2. The solution of the problem was completed in 1991, by showing that for every p∈(0, 2], the function exp(−∥x∥p) is not positive definite for the [lscr ]nq norms with q>2 and n[ges ]3. We prove a similar result for a more general class of norms, which contains some Orlicz spaces and q-sums, and, in particular, present a simple proof of the answer to Schoenberg's original question. Some consequences concerning isometric embeddings in Lp spaces for 0<p[les ]2 are also discussed.

Throughout this paper, we shall let Σ be a subset of [0, 1] having cardinality N. We shall consider Σ to be a set of slopes, and for any s∈Σ, we shall let es be the unit vector of slope s in ℝ2. Then, following [7], we define the maximal operator on ℝ2 associated with the set Σ by

formula here

The history of the bounds obtained on [Mfr ]0Σ is quite curious. The earliest study of related operators was carried out by Cordoba [2]. He obtained a bound of C√(1+logN) on the L2 operator norm of the Kakeya maximal operator over rectangles of length 1 and eccentricity N. This operator is analogous to [Mfr ]0Σ with

formula here

However, for arbitrary sets Σ, the best known result seems to be C(1+logN). This follows from Lemma 5.1 in [1], but a point of view which produces a proof appears already in [8]. However, in this paper, we prove the following.

In 1980, Robertson, Tweddle and Yeomans solved the metrizable BCE problem in certain special cases, for example, in the normable case under assumption of the Continuum Hypothesis (CH). A complete positive solution was recently given by the author and L. M. Sánchez Ruiz under an assumption, weaker than CH, that scales exist. A better partial solution for the general BCE problem obviates scales or other extra-ZFC assumptions, and entirely lays to rest the metrizable BCE problem. The proof stimulates thought about how the least infinite-dimensionality of normed barrelled spaces impinges on the separable quotient problem for Banach spaces.

It is well known that there exist infinite closed subsets E of [ ] such that A(E) = C(E) (see, for example, [3]). Such sets are called Helson sets. Let E be a closed subset of [ ], let 0 < α < 1, and let Aα(E) be the restriction of the Beurling algebra Aα([ ]). Then Aα(E) ⊂ lipαE. We shall show that Aα(E) = lipαE if and only if E is finite. This answers a question raised by Pedersen [5], where partial results were obtained.

Let a be an element in a complex Banach algebra with unit, and let r be a nonnegative number. The Gelfand spectral radius formula implies that the spectrum of a is included in the disk {z∈C:[mid ]z[mid ][les ]r} if and only if

formula here

We work in the smooth category.

An (oriented) (ordered) m-component n-(dimensional) link is a smooth oriented submanifold L = {K1, …, Km} of Sn+2 which is the ordered disjoint union of m manifolds, each PL-homeomorphic to the standard n-sphere. If m = 1, then L is called a knot.

We say that m-component n-dimensional links L0 and L1 are (link-)concordant or (link-)cobordant if there is a smooth oriented submanifold C˜ = {C1, …, Cm} of Sn+2 × [0, 1] which meets the boundary transversely in ∂C˜, is PL-homeomorphic to L0 × [0, 1], and meets Sn+2 × {l} in Ll (l = 0, 1). If m = 1, then we say that n-knots L0 and Ll are (knot-)concordant or (knot-)cobordant. Then we call C a concordance-cylinder of the two n-knots L0 and Ll.

If an n-link L is concordant to the trivial link, then we call L a slice link.

If an n-link L = {K1, …, Km} ⊂ Sn+2 = ∂Bn+3 ⊂ Bn+3 is slice, then there is a disjoint union of (n + 1)-discs Dn+11 [amalg ] … [amalg ] Dn+1m in Bn+3 such that Dn+1i ∩ Sn+2 = ∂Dn+1i = Ki. (Dn+11, …, Dn+1m) is called a set of slice discs for L. If m = 1, then Dn+11 is called a slice disc for the knot L.

Basil Rennie was born in London on 24 December 1920. He came from a long line of engineers, a family tradition that surfaced in much of his later mathematical work. He attended the University College School in London, where he obtained a Mathematical Scholarship at Peterhouse, Cambridge. After graduating in 1941, he found employment first with the Rolls Royce Aero Engine division, then with Austin Motor Works. In 1943 he joined the Fleet Air Arm of the Royal Navy as a radio mechanic, and he served in the Pacific Fleet until the end of the war. This was his first contact with Australia, and he seems to have liked what he saw.

After his service with the Navy, Rennie resumed his studies at Peterhouse and received a PhD in 1949. Given his strong practical bent, it is perhaps surprising that he chose lattice theory as the subject of his thesis; apart from an article [1] in the Proceedings of the London Mathematical Society (he became a member in 1947) and a small booklet [2] published at his own expense, he never touched lattice theory again. It was at Peterhouse that he took up rowing, an activity which became a life-long interest.

In 1950 Rennie accepted an offer of a senior lectureship at the University of Adelaide in South Australia. This was a time of considerable post-war expansion at the University, and its forward-looking Vice-Chancellor A. P. Rowe recruited a number of young and promising staff from overseas, some to leading positions. For instance, he established a Mathematical Physics department (unique in Australia) with the 30-year-old H. S. Green as its head, which became one of the most active research departments in Australia.