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.

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.

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.

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.