For a complete local ring R with maximal ideal m, we define a linear topology on a Matlis reflexive R-module M which coincides with the m-adic topology on M in case M is finitely generated. We show that a Matlis reflexive module is complete in this topology.

We study the distribution of the imaginary parts of zeros near the real axis of quadratic L-functions. More precisely, let K(s) be chosen so that |K(1/2 ± it)| is rapidly decreasing as t increases. We investigate the asymptotic behaviour of

as D → ∞. Here denotes the sum over the non-trivial zeros p = 1/2 + iγ of the Dirichlet L-function L(s, χd), and χd = () is the Kronecker symbol. The outer sum is over all fundamental discriminants d that are in absolute value ≤ D. Assuming the Generalized Riemann Hypothesis, we show that for

In this note we obtain a convolution identity for the coefficients Bn(α, θ, q) defined by

using Ramanujan's 1Ψ1 summation. The identity contains as special cases convolution identities of Kung-Wei Yang and a few more interesting analogue.

In 1991 Renaud defined a boundary function φ(n) for union-closed sets, and evaluated it to n = 17. Also in 1991, Mallows examined a sequence a(n) defined recursively by Conway in 1988.

Investigation of some properties of strictly reduced ordered power sets, a class of union-closed sets, leads to the conclusion that a(n + 1) is an upper bound for φ(n), and the union-closed sets conjecture holds if the conjecture φ(n) = a(n + 1) is valid.

Sufficient conditions are given for the existence of an mth power root of one polynomial modulo another, over the complexes or the reals. Examples show the non-necessity of the conditions. In particular cases there can exist infinitely many square roots.

It is known that, under suitable assumptions, the subdifferential ∂(f □ g) of the infimal convolution of two convex functions f and g coincides with the parallel sum of ∂ f and ∂ g. We prove that a similar formula holds for the subdifferential of the deconvolution of two convex functions: under appropriate hypothesis it coincides with the parallel star-difference of the sub-differentials of the functions.

Let G be a separable locally compact group which admits a non-trivial compact normal subgroup. It is shown that the group C*-algebra C*(G) of G may be decomposed as a direct sum of ideals whose structure is determined up to *-isomorphism. Applications are given to Type 1, [FD]− groups; in particular it is shown that the group C*-algebra of such a group is a direct sum of homogeneous C*-algebras.

Let f(z) be an n-valued algebroid function of finite lower order μ. In this paper, we give some further results on the deficiencies of f(z). Particularly if 0 ≤ μ ≤ 1/2, the corresponding result is best possible.