We give a positive answer to a question of Horst Tietz. A theorem of his that is related to the Mittag-Leffler theorem looks like a duality restult under some locally convex topology on the space of meromorphic functions. Tietz has posed the problem of finding such a topology. It is shown that a topology introduced by Holdguün in 1973 solves the problem. The main tool in the study of this topology is a projective description of it that is derived here. We also argue that Holdgrün's topology is the natural locally convex topology on the space of meromorphic functions.

The exponential sum S(x) = Σe(f(m + x)) has mean square size O(M), when m runs through M consecutive integers, f(x) satisfies bounds on the second and third derivatives, and x runs from 0 to 1.

For a compact group G, we compute the Kazhdan constants κ(G, G) obtained by taking G itself as a generating subset. We get κ(G, G) = if G is finite of order n, and κ(G, G) = if G is infinite.

In this paper we study categorical compactness with respect to a class of objects F being motiveated by examples arising from modules, abelian groups, and various classes of non-abelian groups. This theory is then applied to the category of not necessarily associative rings. In particular, we study the example arising from the class of all torsion-free rings. This work extends some recent results of B. J. Gardner for associative rings and radical classes.

We measure the separation of the zeros of the polynomial f(x) = by δ(f) = mini(ai+1 − ai). We establish a bound for the amount by which the ratio δ(f′ − kf)/δ(f) exceeds 1.

Let E(0, ∞) be a separable symmetric function space, let M be a semifinite von Neumann algebra with normal faithful semifinite trace μ, and let E(M, μ) be the symmetric operator space associated with E(0, ∞). If E(0, ∞) has the uniform Kadec-Klee property with respect to convergence in measure then E(M, μ) also has this property. In particular, if LΦ(0, ∞) (ϕ(0, ∞)) is a separable Orlicz (Lorentz) space then LΦ(M, μ) (Λϕ (M, μ)) has the uniform Kadec-Klee property with respect to convergence in measure on sets of finite measure if and only if the norm of E(0, ∞) satisfies G. Birkhoff's condition of uniform monotonicity.

We study commutators in pseudo-orthogonal groups O2nR (including unitary, symplectic, and ordinary orthogonal groups) and in the conformal pseudo-orthogonal groups GO2nR. We estimate the number of commutators, c(O2nR) and c(GO2nR), needed to represent every element in the commutator subgroup. We show that c(O2nR) ≤ 4 if R satisfies the ∧-stable condition and either n ≥ 3 or n = 2 and 1 is the sum of two units in R, and that c(GO2nR) ≤ 3 when the involution is trivial and ∧ = R∈. We also show that c(O2nR) ≤ 3 and c(GO2nR) ≤ 2 for the ordinary orthogonal group O2nR over a commutative ring R of absolute stable rank 1 where either n ≥ 3 or n = 2 and 1 is the sum of two units in R.

For a module M Over an Artin algebra R, we discuss the question of whether the Yoneda extension algebra Ext(M, M) is finitely generated as an algebra. We give an answer for bounded modules M. (These are modules whose syzygies have direct summands of bounded lengths.)

Roth's Theorem says that given ρ < 2 and an algebraic number α, all but finitely many rational numbers x/y satisfy |α - (x/y)|< |y|-ρ. We give upper bounds for the number of these exceptional rationals when 3 ≤ ρ ≤ d, where d is the degree of α. Our result suplements bounds given by Bombieri and Van der Poorten when 2 > ρ ≤ 3; naturally the bounds become smaller as ρ increases.

A group G has all of its subgroups normal-by-finite if H / coreG(H) is finite for all subgroups H of G. These groups can be quite complicated in general, as is seen from the so-called Tarski groups. However, the locally finite groups of this type are shown to be abelian-by-finite; and they are then boundedly core-finite, that is to say, there is a bound depending on G only for the indices | H: coreG(H)|.

In 1960, Trevor Evans gave a best possible embedding of a partial latin square of order n in a latin square of order t, for any t ≥ 2n. A latin square of order n is equivalent to a 3-cycle system of Kn, n, n, the complete tripartite graph. Here we consider a small embedding of partial 3k-cycle systems of Kn, n, n of a certain type which generalizes Evans' Theorem, and discuss how this relates to the embedding of patterned holes, another recent generalization of Evans' Theorem.

We introduce a natural notion of full coactions of a locally compact group on a Hilbert C*-module, and associate each full coaction in a natural way to an ordinary coaction. We also introduce a natural notion of strong Morita equivalence of full coactions which is sufficient to ensure strong Morita equivalence of the corresponding crossed product C*-algebras.