In this paper, we show first that any prime (or semiprime) element of a commutative Banach algebra must have closed range. As a corollary, we find that in a commutative radical Banach algebra, all primes are zero divisors; indeed, all semiprimes are zero divisors (see below for the definition of semiprimeness). Our result is also true of a semiprime that is in the centre of a noncommutative Banach algebra.

The proof is fairly simple and entertaining, and we obtain a result that is helpful for the ambitious classification of elements in commutative radical Banach algebras being attempted by Marc Thomas. It is also related to the unbounded Kleinecke–Shirov conjecture.

The main theorem of this paper generalizes a classic Aumann's characterization of compact convex sets, via the acyclicity of their hypersections, to arbitrary weakly closed subsets of a locally convex linear space.

Using the twistorial approach and some previous results, we prove the conjecture that the dimension of the moduli space of harmonic maps of area $4 \pi d$ from the 2-sphere to the $2n$-sphere is $2 d+n^2$ for the particular case $n=3$.

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.

Let $f \in \Z[x]$ be a polynomial of degree d. The paucity of non-trivial positive integer solutions to the equation $f(x_1)+f(x_2)=f(x_{3})+f(x_{4})$ is established, provided that $d \geq 7$. Also the corresponding situation is investigated for equal sums of three like polynomials.

Let p be a prime number, and let G be a p-group which is not elementary abelian. For every mod-p cohomology class ξ of G which restricts trivially to all proper subgroups, we show that ξp = 0. This gives upper bounds for nilpotency degrees of such classes of G and of nilpotent mod-p cohomology classes of finite groups.

This paper proves that a subgroup of finite index in a positively finitely generated profinite group has maximal subgroup growth at most $n^{\log (n\!)}$. In particular such a subgroup cannot be free, answering a question by L. Pyber.

Let V be a finite-dimensional vector space over a finite field. The general and special linear groups, GL(V) and SL(V), act on the exterior algebras $\Lambda^*V$ and $\Lambda^*V^*$ of V and its dual $V^*$, and on the symmetric algebra $\S^*V$. The subring of SL(V)-invariants of $\Lambda^*V\otimes\S^*V$ was determined by Dickson and Mui. This paper describes the equivalent, but simpler, calculation of the invariant subring of $\Lambda^*V^*\otimes\S^*V$ as a representation of GL(V)/SL(V).

Power series $\sum_{n=0}^\infty f(n)x^n$ with non-zero convergence radius $R(f)$ are considered, and the arithmetical nature (that is, irrationality, or even transcendence) of the corresponding multivariate series $\smash{\sum_{i_1,\ldots,i_m\!=0}^\infty f(i_1+\ldots +i_m)x_1^{i_1}\cdot\ldots \cdot x_m^{i_m}}$ is studied if $x_1,\ldots,x_m$ and the sequence $(f(n))$ satisfy appropriate arithmetical conditions. It follows that such arithmetical results can be written down easily if linear independence results on the function $F(x)$, defined in $|x|\,{<}\,R(f)$ by the original one-dimensional power series, and possibly its derivatives at the points $x_\mu$ are known. Some typical applications are explicitly stated.

The existence of periodic solutions for the nonlinear asymmetric oscillator $ x''{+}\alpha x^+{-}\beta x^- {=} h(t),\qquad (' = d/dt) $ is discussed, where $\alpha, \beta$ are positive constants satisfying $ {1}/{\sqrt{\alpha}} + {1}/{\sqrt{\beta}} = {2}/{n} $ for some positive integer $n\in {\bf \mathbb{N}}$ and $h(t)\in L^\infty(0,2\pi)$ is $2\pi$-periodic with $x^{\pm}=\max\{\pm x, 0\}$.

For any positive integer n we let $P(n)$ be the largest prime factor of n. We improve and generalize several results of P. Erdős and C. Stewart on $P(n!+1)$. In particular, we show that $\limsup_{n \to \infty}P(n!+1)/n \ge 2.5$, which improves their lower bound of $\limsup_{n \to \infty} P(n!+1)/n >2$.

The action of translation operators on wavelet subspaces in higher dimensions is investigated. This action defines an equivalence relation on the set of single wavelets of $L^2(\mathbb R^n)$ associated with an arbitrary dilation matrix. The corresponding equivalence classes are characterized in terms of the support of the Fourier transform of the wavelets. Further, examples of wavelets in each of these classes are constructed. This construction shows the existence of wavelets for which the associated wavelet subspaces are invariant under various groups of translation operators.

The context of this note is as follows. One considers a connected reductive group G and a Frobenius endomorphism F[ratio ]G→G defining G over a finite field of order q. One denotes by GF the associated (finite) group of fixed points.

Let [lscr ] be a prime not dividing q. We are interested in the [lscr ]-blocks of the finite group GF. Such a block is called unipotent if there is a unipotent character (see, for instance, [6, Definition 12.1]) among its representations in characteristic zero. Roughly speaking, it is believed that the study of arbitrary blocks of GF might be reduced to unipotent blocks (see [2, Théorème 2.3], [5, Remark 3.6]). In view of certain conjectures about blocks (see, for instance, [9]), it would be interesting to further reduce the study of unipotent blocks to the study of principal blocks (blocks containing the trivial character). Our Theorem 7 is a step in that direction: we show that the local structure of any unipotent block of GF is very close to that of a principal block of a group of related type (notion of ‘control of fusion’, see [13, §49]).

Let G be a transitive permutation group on a set Ω such that, for ω∈Ω, the stabiliser Gω induces on each of its orbits in Ω\{ω} a primitive permutation group (possibly of degree 1). Let N be the normal closure of Gω in G. Then (Theorem 1) either N factorises as N=GωGδ for some ω, δ∈Ω, or all unfaithful Gω-orbits, if any exist, are infinite. This result generalises a theorem of I. M. Isaacs which deals with the case where there is a finite upper bound on the lengths of the Gω-orbits. Several further results are proved about the structure of G as a permutation group, focussing in particular on the nature of certain G-invariant partitions of Ω.

A new inequality concerning generalized characters of $p$-groups is obtained and applications to bounding the number of irreducible characters in blocks of finite groups are given.

We give a general method for expressing various characteristic numbers of the multiple-point manifolds of an immersion.

We prove tight closure analogues of results of Watanabe about chains and families of integrally closed ideals.

Let M and N be closed non-positively curved manifolds, and let f[ratio ]M→N be a smooth map. Results of Eells and Sampson [1] show that f is homotopic to a harmonic map ϕ, and Hartman [6] showed that this ϕ is unique when N is negatively curved and f∗(π1M) is not cyclic. Lawson and Yau conjectured that if f[ratio ]M→N is a homotopy equivalence, where M and N are negatively curved, then the unique harmonic map ϕ homotopic to f would be a diffeomorphism. Counterexamples to this conjecture appeared in [2], and later in [7] and [5].

There remains the question of whether a ‘topological’ Lawson–Yau conjecture holds.

Let V be a commutative valuation domain of arbitrary Krull-dimension (rank), with quotient field F, and let K be a finite Galois extension of F with group G, and S the integral closure of V in K. If, in the crossed product algebra K [midast ] G, the 2-cocycle takes values in the group of units of S, then one can form, in a natural way, a ‘crossed product order’ S [midast ] G ⊆ K [midast ] G. In the light of recent results by H. Marubayashi and Z. Yi on the homological dimension of crossed products, this paper discusses necessary and/or sufficient valuation-theoretic conditions, on the extension K/F, for the V-order S [midast ] G to be semihereditary, maximal or Azumaya over V.

It is shown that a rational function of degree [ges ] 2 admits an invariant line field with respect to some measure μ, which is an equilibrium state of a Hölder continuous potential whose topological pressure is greater than its supremum, only in very special cases when the Julia set is either a geometric circle or an interval, or totally disconnected and contained in a real-analytic curve.