We define a new height function $\mathcal{R}(\alpha)$, the Remak height of an algebraic number $\alpha$. We give sharp upper and lower bounds for $\mathcal{R}(\alpha)$ in terms of the classical Mahler measure $M(\alpha)$. Study of when one of these bounds is exact leads us to consideration of conjugate sets of algebraic numbers of norm $\pm 1$ lying on two circles centred at 0. We give a complete characterization of such conjugate sets. They turn out to be of two types: one related to certain cubic algebraic numbers, and the other related to a non-integer generalization of Salem numbers which we call extended Salem numbers.

AMS 2000 Mathematics subject classification: Primary 11R06

Let $A$ be a graded, commutative Hopf algebra. We study an action of the symmetric group $\sSi_n$ on the tensor product of $n-1$ copies of $A$; this action was introduced by the second author in 1 and is relevant to the study of commutativity conditions on ring spectra in stable homotopy theory 2.

We show that for a certain class of Hopf algebras the cohomology ring $H^*(\sSi_n;A^{\otimes n-1})$ is independent of the coproduct provided $n$ and $(n-2)!$ are invertible in the ground ring. With the simplest coproduct structure, the group action becomes particularly tractable and we discuss the implications this has for computations.

AMS 2000 Mathematics subject classification: Primary 16W30; 57T05; 20C30; 20J06; 55S25

Whereas pseudovarieties of commutative semigroups are known to be finitely based, the globals of monoidal pseudovarieties of commutative semigroups are shown to be finitely based (or of finite vertex rank) if and only if the index is 0, 1 or $\omega$. Nevertheless, on these pseudovarieties, the operation of taking the global preserves decidability. Furthermore, the gaps between many of these globals are shown to be big in the sense that they contain chains which are order isomorphic to the reals.

AMS 2000 Mathematics subject classification: Primary 20M07; 20M05

Let $X$ be a Banach space and $\xi$ an ordinal number. We study some isomorphic classifications of the Banach spaces $X^\xi$ of the continuous $X$-valued functions defined in the interval of ordinals $[1,\xi]$ and equipped with the supremum norm. More precisely, first we use the continuum hypothesis to give an isomorphic classification of $C(I)^\xi$, $\xi\geq\omega_1$. Then we present a characterization of the separable Banach spaces $X$ that are isomorphic to $X^\xi$, $\forall\xi$, $\omega\leq\xi lt \omega_1$. Finally, we show that the isomorphic classifications of $(C(I)\oplus F^*)^\xi$ and $\ell_\infty(\N)^\xi$, where $F$ is the space of Figiel and $\omega\leq\xi lt \omega_1$ are similar to that of $\R^\xi$ given by Bessaga and Pelczynski.

AMS 2000 Mathematics subject classification: Primary 46B03; 46B20; 46E15

The aim of this paper is to prove the existence and uniqueness of mild and classical solutions of the non-local Cauchy problem for a semilinear integrodifferential equation with deviating argument. The results are established by using the method of semigroups and the contraction mapping principle. The paper generalizes certain results of Lin and Liu.

AMS 2000 Mathematics subject classification: Primary 34K05; 34K30

A harmonic morphism defined on $\mathbb{R}^3$ with values in a Riemann surface is characterized in terms of a complex analytic curve in the complex surface of straight lines. We show how, to a certain family of complex curves, the singular set of the corresponding harmonic morphism has an isolated component consisting of a continuously embedded knot.

AMS 2000 Mathematics subject classification: Primary 57M25. Secondary 57M12; 58E20

The purpose of this work is to establish a priori $C^{2,\alpha}$ estimates for mesh function solutions of nonlinear difference equations of positive type in fully nonlinear form on a uniform mesh, where the fully nonlinear finite difference operator $\F$ is concave in the second-order variables. The estimate is an analogue of the corresponding estimate for solutions of concave fully nonlinear elliptic partial differential equations. We use the results for the special case that the operator does not depend explicitly upon the independent variables (the so-called frozen case) established in part I to approach the general case of explicit dependence upon the independent variables. We make our approach for the diagonal case via a discretization of the approach of Safonov for fully nonlinear elliptic partial differential equations using the discrete linear theory of Kuo and Trudinger and an especially agreeable mesh function interpolant provided by Kunkle. We generalize to non-diagonal operators using an idea which, to the author’s knowledge, is novel. In this paper we establish the desired Hölder estimate in the large, that is, on the entire mesh $n$-plane. In a subsequent paper a truly interior estimate will be established in a mesh $n$-box.

AMS 2000 Mathematics subject classification: Primary 35J60; 35J15; 39A12. Secondary 39A70; 39A10; 65N06; 65N22; 65N12

It is known that ℚ-derived univariate polynomials (polynomials defined over ℚ, with the property that they and all their derivatives have all their roots in ℚ) can be completely classified subject to two conjectures: that no quartic with four distinct roots is ℚ-derived, and that no quintic with a triple root and two other distinct roots is ℚ-derived. We prove the second of these conjectures.

AMS 2000 Mathematics subject classification: Primary 11G30

Let $H$ be a Hall $\pi$-subgroup of a finite $\pi$-separable group $G$, and let $\alpha$ be an irreducible Brauer character of $H$. If $\alpha(x)=\alpha(y)$ whenever $x,y \in H$ are $p$-regular and $G$-conjugate, then $\alpha$ extends to a Brauer character of $G$.

AMS 2000 Mathematics subject classification: Primary 20C15; 20C20

Given finite sets of cyclic words $\{u_1,\dots,u_k\}$ and $\{v_1,\dots,v_k\}$ in a finitely generated free group $F$ and two finite groups $A$ and $B$ of outer automorphisms of $F$, we produce an algorithm to decide whether there is an automorphism which conjugates $A$ to $B$ and takes $u_i$ to $v_i$ for each $i$. If $A$ and $B$ are trivial, this is the classic algorithm due to Whitehead. We use this algorithm together with Cohen and Lustig’s solution to the conjugacy problem for Dehn twist automorphisms of $F$ to solve the conjugacy problem for outer automorphisms which have a power which is a Dehn twist. This settles the conjugacy problem for all automorphisms of $F$ which have linear growth.

AMS 2000 Mathematics subject classification: Primary 20F32. Secondary 57M07

We describe the structure of finitely generated cotorsion modules over commutative noetherian rings. Also we characterize the so-called covering morphisms between finitely generated modules over these rings.

AMS 2000 Mathematics subject classification: Primary 16D10. Secondary 13C11

If $T:A_{0}\rightarrow B$ boundedly and $T:A_{1}\rightarrow B$ compactly, then a result of Lions–Peetre shows that $T:A\rightarrow B$ compactly for a certain class of spaces $A$ which are intermediate with respect to $A_{0}$ and $A_{1}$. We investigate to what extent such results can hold for arbitrary intermediate spaces $A$. The ‘dual’ case of an operator $S$ such that $S:X\rightarrow Y_{0}$ boundedly and $S:X\rightarrow Y_{1}$ compactly, is also considered, as well as similar questions for other closed operator ideals.

AMS 2000 Mathematics subject classification: Primary 46B70; 47D50

McAlister proved that every regular locally inverse semigroup can be covered by a regular Rees matrix semigroup over an inverse semigroup by means of a homomorphism which is locally an isomorphism. We generalize this result to the class of semigroups with local units whose local submonoids have commuting idempotents and possessing what we term a ‘McAlister sandwich function’.

AMS 2000 Mathematics subject classification: Primary 20M10. Secondary 20M17

Let $E$ be a real uniformly smooth Banach space and let $A$ be a nonlinear $\phi$-strongly quasi-accretive operator with range $R(A)$ and open domain $D(A)$ in $E$. For a given $f\in E$, let $A$ satisfy the evolution system $\rd u(t)/\rd t+Au(t)=f$, $u(0)=u_0$. We establish the strong convergence of the Ishikawa and Mann iterative methods with appropriate error terms recently introduced by Xu to the equilibrium points of this system. Related results deal with the strong convergence of the iterative methods to the fixed points of $\phi$-strong pseudocontractions defined on open subsets of $E$.

AMS 2000 Mathematics subject classification: Primary 47H06; 47H15; 47H17

Herstein showed that the conjugacy class of a non-central element in the multiplicative group of a division ring is infinite. We prove similar results for units in algebras and orders and give applications to group rings.

AMS 2000 Mathematics subject classification: Primary 16U60. Secondary 16H05; 16S34; 20F24; 20C05

Let $G$ be a locally compact group, $A$ a continuous trace $C^*$-algebra, and $\alpha$ a pointwise unitary action of $G$ on $A$. It is a result of Olesen and Raeburn that if $A$ is separable and $G$ is second countable, then the crossed product $A\times_\alpha G$ has continuous trace. We present a new and much more elementary proof of this fact. Moreover, we do not even need the separability assumptions made on $A$ and $G$.

AMS 2000 Mathematics subject classification: Primary 46L55