Let 0 → N → E → F → 0 be a short exact sequence of torsion-free Kronecker modules. Suppose that N and F have rank one. The module F is classified by a height function h defined on the projective line. If N is finite-dimensional, h is supported on a set of cardinality less than that of its domain and h takes on the value ∞, then E embeds into F. The converse holds if all such E embed into F. This embeddability is in contrast to the situation with other rings such as commutative domains, where it never occurs.

Let N(n) be the set of all integers that can be written in the form where ∊(n) → 0 as n → ∞, answering a question of P. Erdös and R. L. Graham.

We consider the Banach algebra consisting of linear operators T which are defined on the simple functions and have bounded extensions Tp on LP for all values of p ∊ [1, ∞]. We show that the 'integral' operators in this algebra form a right ideal, and that each Tp associated to an integral T is regular. When the underlying measure is finite or special discrete we show further that every Tp is regular for every T in the algebra. Algebraic techniques together with interpolation results are then used to get relationships between the spectrum and the order spectrum of the associated Tp 's.

Denote by g(n) the largest integer m such that every set of integers of size n contains a subset of size m whose pairwise sums are distinct. It is shown that g(n) > cn1/2 for any constant c < 2/25 and all sufficiently large n.

A necessary and sufficient condition to obtain a volumelike form from an affine connection is given in terms of the Čech cohomology, after the volume-like form is naturally defined without a Riemannian metric. A necessary condition for an affine connection to be a Riemannian connection for some metric is also given.

The general problem is to express an element of the derived group of a free group as a product of a minimal number of commutators. An old conjecture is settled in the negative, and a new conjecture and a number of related questions are posed.

Analytic multivalued functions are a powerful tool in spectral theory. In this paper we study analytic multivalued functions λ → K(λ) whose values are segments. Under the additional assumption that K is continuous, we prove that the endpoints of K vary locally holomorphically.

In this brief note, we will show how in principle to find all units in the integral group ring ZG, whenever G is a finite group such that and Z(G) each have exponent 2, 3, 4 or 6. Special cases include the dihedral group of order 8, whose units were previously computed by Polcino Milies [5], and the group discussed by Ritter and Sehgal [6]. Other examples of noncommutative integral group rings whose units have been computed include , but in general very little progress has been made in this direction. For basic information on units in group rings, the reader is referred to Sehgal [7].

One of the questions concerning the Hexagonal Packing Lemma ([1], [3], [4]) is the rate of convergence of Sn . It was suggested in [3] and [4] that Sn = 0(1/n). In the following we prove this conjecture under the additional condition of some "nice" behaviour of the "circle function".

We give a simple construction of non-Noetherian Hilbert domains whose maximal ideals are all finitely generated. Such domains we call unruly Hilbert domains.

We characterize when the product of two (unbounded) derivations of a C*-algebra is a derivation.

For completely distributive CSL algebras, hyper-reflexivity is equivalent to a description of the compact perturbation of the algebra analogous to the Fall-Arveson-Muhly Theorem for nest algebras.

Consider a covering p : X → B of connected topological spaces. IfB is a compact polyhedron, a classical result of H. Hopf [4] says that the end spaceE(X) of X is an invariant of the group G of covering transformations. Thus it becomesmeaningful to define the end space of the finitely generated group G as E(G) := E(X).

Finite factor groups of are called Hurwitz groups. Here we prove that apart from 2G2(3), G2(2), G2(3) and G2(4), all the groups 2G2(32n+1) and G2(q), q = pn, p € P, are Hurwitz groups. For the proof, (2, 3, 7) structure constants are calculated from the character tables [2], [7], and then with the lists of maximal subgroups [8], [5], the existence of generating triples is deduced.

We show that is an integer. Special cases include the theorems of Wilson and Fermat.

The limiting behavior of sequences of quasiconformal homeomorphisms of the n-sphere Sn is studied using a substitute to the Poincaré extension of Möbius transformations introduced by Tukia. Adapted versions of the limit set and the conical limit set known in the theory of Kleinian groups are utilized. Most of the results also hold for families of homeomorphisms of Sn with the convergence property introduced by Gehring and Martin.

Let G be a finite group. A natural invariant c(G) of G has been defined by W.J. Ralph, as the order (possibly infinite) of a distinguished element of a certain abelian group associated to G. Ralph has shown that c(Zn ) = 1 and c(Z 2 ⴲ Z2 ) = 2. In the present paper we show that c(G) is finite whenever G is a dihedral group or a 2-group, and obtain upper bounds for c(G) in these cases.

We give conditions under which a vector valued Ornstein Uhlenbeck process has continuous sample paths in lp for 1 ≦ p < ∞. We also show when the space lp is not entered at all, i.e., when it has zero capacity.

We propose the following problem. Given an array of vertical mirrors that simultaneously transmit and reflect, and a single incoming ray of light, describe the configuration of all light rays that are generated. We solve the problem here for a certain infinite configuration of mirrors; the solution involves the winning positions (a(n), b(n)) of Wythoff's game.

The interesting question of the title was posed by J. Bergen and this note answers it in the negative. The main result characterizes rings which can be a finite union of proper right annihilators, and shows that any such commutative ring must have a total annihilator.