The purpose of this paper is to present a proof of the following theorem of Maclagan-Wedderburn.*

Every finite skew-field† is a field.

The proof depends on group theory and on the properties of Galois fields. As an introduction, §§1–4 are devoted to a systematic and self-contained account of the theory of Galois fields.

We give an explicit description of the stable reduction of superelliptic curves of the form yn=f(x) at primes $\mathfrak{p}$ whose residue characteristic is prime to the exponent n. We then use this description to compute the local L-factor and the exponent of conductor at $\mathfrak{p}$ of the curve.

Biharmonic Lagrangian surfaces of constant mean curvature in complex space forms are classified. A further important point is that new examples of marginally trapped biharmonic Lagrangian surfaces in an indefinite complex Euclidean plane are obtained. This fact suggests that Chen and Ishikawa's classification of marginally trapped biharmonic surfaces [6] is not complete.

We present a systematic construction of the discrete KP hierarchy in terms of Sato–Wilson-type shift operators. Reductions of the equations in this hierarchy to 1+1-dimensional integrable lattice systems are considered, and the problems that arise with regard to the symmetry algebra underlying the reduced systems as well as the ultradiscretizability of these systems are discussed. A scheme for constructing ultradiscretizable reductions that give rise to Yang–Baxter maps is explained in two explicit examples.

We prove that the class number of the imaginary quadratic field is divisible by n for any positive integers k and n with 22k < 3n, by using Y. Bugeaud and T. N. Shorey's result on Diophantine equations.

The set Dn of all n × n doubly-stochastic matrices is a semigroup with respect to ordinary matrix multiplication. This note is concerned with the determination of the maximal subgroups of Dn. It is shown that the number of subgroups is finite, that each subgroup is finite and is in fact isomorphic to a direct product of symmetric groups. These results are applied in § 3 to yield information about the least number of permutation matrices whose convex hull contains a given doubly-stochastic matrix.

In the earlier article [7], I began the study of rational period functions for the modular group Γ(l) = SL(2, Z) (regarded as a group of linear fractional transformations) acting on the Riemann sphere. These are rational functions q(z) which occur in functional equations of the form

where k∈Z and F is a function meromorphic in the upper half-plane ℋ, restricted in growth at the parabolic cusp ∞. The growth restriction may be phrased in terms of the Fourier expansion of F(z) at ∞:

with some μ∈Z. If (1.1) and (1.2) hold, then we call F a modular integral of weight 2k and q(z) the period of F.

1. It is known that any polynomial in μ. can be expanded as a linear function of Legendre polynomials [1]. In particular, we have

The earlier coefficients, say A0, A2, A4 may easily be found by equating the coefficients of μp+q, μp+q-2, μp+q-4 on the two sides of (1). The general coefficient A2k might then be surmised, and the value verified by induction. This may have been the method followed by Ferrers, who stated the result as an exercise in his Spherical Harmonics (1877). A proof was published by J. C. Adams [2]. The proof now to be given follows different lines from his.

It is shown that ${\mathcal{Z}}$-stable projectionless C*-algebras have the property that every element is a limit of products of two nilpotents. This is then used to classify the approximate unitary equivalence classes of positive elements in such C*-algebras using traces.

We explicitly classify helicoidal and translational constant mean curvature surfaces in ${\mathbb H}^2\times {\mathbb R}$.

We consider a one-dimensional weakly damped wave equation, with a damping coefficient depending on the displacement. We prove the existence of a regular connected global attractor of finite fractal dimension for the associated dynamical system, as well as the existence of an exponential attractor.

Let Mn(ℂ) be the algebra of n × n complex matrices, and let be its unitary group. Given A, B ε Mn(ℂ), the A-numerical radius of B is the nonnegative quantity

In particular, for A = diag(1, 0, …, 0) it reduces to the classical numerical radius r(B) = max||x*Bx|:x*x = 1}. In [1] Goldberg and Straus proved that rA is a generalized matrix norm (i.e. a positive definite seminorm) on Mn(ℂ) if and only if A is nonscalar and tr A ≠ 0. This result agrees with the well-known fact that the classical numerical radius r is a generalized matrix norm. The nontrivial part of the proof is to show that if A is nonscalar and tr A ≠ 0 then rA is positive definite; that is, for any B ε Mn(ℂ), tr(AU*BU) = 0 for all U ε implies B = 0. The proof given in [1] is computational and involves the use of differentiation on matrices. Later Marcus and Sandy [2] gave three elementary proofs of the result. Their proofs are still computational in nature and two of them need knowledge of multilinear algebra.

The nonabelian tensor square G[otimes ] G of a group G is generated by the symbols g[otimes ] h, g,h ∈ G, subject to the relations $$gg\prime\otimesh=(^gg\prime\otimes^gh)(g\otimesh) and g\otimeshh\prime-(g\otimesh)(^hg\otimes^hh\prime),$$ for all $g,g\prime,h,h\prime \in G< / f>, where $^gg\prime=gg\primeg^{−1}$. The nonabelian tensor square is a special case of the nonabelian tensor product which has its origins in homotopy theory. It was introduced by R. Brown and J.-L. Loday in [4] and [5], extending ideas of J.H.C. Whitehead in [10]. The topic of this paper is the classification of 2-generator 2-groups of class two up to isomorphism and the determination of nonabelian tensor squares for these groups.

The main result of this series of papers is a theorem on the free product of groups (Theorem 1) which formed part of a doctoral thesis. This theorem has an immediate application to the word problem (Theorem 2). Usually the word problem refers to a finite system of generators and a finite number of defining relations, but in this context it is more natural to allow an infinite number of generators and defining relations. This (infinite) word problem is not solvable in general (Example 2).

Let S and T be inverse semigroups. Their free product S inv T is their coproduct in the category of inverse semigroups, defined by the usual commutative diagram. Previous descriptions of free products have been based, like that for the free product of groups, on quotients of the free semigroup product S sgp T. In that framework, a set of canonical forms for S inv T consists of a transversal of the classes of the congruence associated with the quotient. The general result [4] of Jones and previous partial results [3], [5], [6] take this approach.

The development of the theory of local rings has been greatly stimulated by the importance of the applications to algebraic geometry, but it is none the less true that this stimulus has produced a theory which, on aesthetic grounds, is somewhat unsatisfactory. In the first place, if a local ring Q arises in the ordinary way from a geometric problem, then Qwill have the same characteristic as its residue field. It is partly for this reason that our knowledge of equicharacteristic local rings is much more extensive than it is of those local rings which present the case of unequal characteristics. Again, in the geometric case, the integral closure of Q in its quotient field will be a finite Q-module. Here, once more, we have a special situation which it would be desirable to abandon from the point of view of a general abstract theory.

Certain generalized Samelson products of SO(3) are calculated and applications to the homotopy of gauge groups are given.

J. Berman [2] initiated the study of a variety k of bounded distributive lattices endowed with a dual homomorphic operation paying particular attention to certain subvarieties km, n. Subsequently, A. Urquhart [8] named the algebras in k distributive Ockham algebras, and developed a duality theory, based on H. A. Priestley's order-topological duality for bounded distributive lattices [6], [7]. Amongst other things, Urquhart described the ordered spaces dual to the subdirectly irreducible algebras in Sif. This work was developed further still by M. S. Goldberg in his thesis and the paper [5]. Recently, T. S. Blyth and J. C. Varlet [3], in abstracting de Morgan and Stone algebras, studied a subvariety MS of the variety k1.1. The main result in [3]is that there are, up to isomorphism, nine subdirectly irreducible algebras in MS and their Hasse diagrams are exhibited. The methods employed in [3] are purely algebraic and can be generalized to show that, up to isomorphism, there are twenty subdirectly irreducible algebras in k1.1. In section 3 of this paper, we take a short cut to this result by utilizing the results of Urquhart and Goldberg. Our basic method is simple: the results of Goldberg [5] are applied to k1,1 to produce a certain eight-element algebra B1 in k1,1, whose lattice reduct is Boolean and whose subalgebras are, up to isomorphism, precisely the subdirectly irreducibles in k1.1. We then pick out of the list of twenty such algebras those belonging to the variety MS. In section 4, we sketch a purely algebraic proof along the lines followed by Blyth and Varlet in [3].

We obtain a new characterisation for weighted Bergman spaces Apα on the unit ball n of ℂn in terms of a double integral of the functions |f(z) − f(w)|/|z − w| and |f(z) − f(w)|/|1 − 〈 z, w〉|.

Let w be a strictly positive function on ℂ and let , respectively denote the Banach spaces of those entire functions φ(z) with ∣φ(z)∣= O(w(z)) and ∣φ(z)∣ = o(w(z)). In this generality, these spaces may contain only constants, but for many functions w(z) these will be interesting Banach spaces with norm

.

We study two specific problems.