Let A(n) be the number of partitions of { 1 , … , n} into two sets A, B of cardinality n/2 such that . Then there is the asymptotic result

The following problem arose in a conversation with Abraham Zaks: “Suppose R is an associative ring with identity such that every finitely generated left ideal is generated by idempotents. Is R von-Neumann regular?” In the literature the “s” in “idempotents” is missing, and is replaced by “an idempotent”. The answer is, “Yes!”

If n is a natural number, let A(n) be the number of integers, k, such that 0 < k < n and n divides . Then ϕ(n) ≤ A(n) ≤ n - 1 - 2ω(n) + ɛ, where ?(n) denotes the number of distinct prime factors of n, and ɛ = 0 unless n is twice a prime, in which case ɛ = 1.

A theorem is proved which gives a new test for singular points on the circle of convergence of the function f(z) defined by a power series ∑anzn finite non-zero radius of convergence.

In this note a formula analogous to Eisenstein's well known formula is presented for Fp-ε/p, where Fn is the nth Fibonacci number (F0 = 0, F1 = 1), p an odd prime, and

This formula is:

The purpose of this paper is to prove the following result:

Theorem. Let T be a directed tree with k arcs and with no directed path of length 2. Then if G is any directed graph with n points and at least 4kn arcs, T is a subgraph of G.

It would be appropriate to call T an antidirected tree or a source-sink tree, since every point either has all its arcs directed outward or all inward. As N. G. de Bruijn has noted (personal communication), such a linear bound in n cannot hold if T is replaced by any directed graph other than a union of such trees. The above theorem strengthens one of Graham [1], where an implicit bound of c(k)n is obtained, where c(k) is exponentially large. The proof we give here is also shorter. We first give two simple lemmas. Both are essentially due to ErdÄs, but it is not clear where either first appeared. Their proofs are easy, so we give them here for completeness; in neither case do we state quite the best possible result.

A subset of a topological space is called discrete iff every point in the space has a neighborhood which meets the set in at most one point. Discrete sets are useful for decomposing the images of certain maps and for generalizing closed maps. All discrete sets are closed iff the space is T1. As a result of characterizing discrete and countably discrete maps, theorems due to Vaĭnšteĭn and Engelking are extended to these maps.

The authors have found that the formula for U(t)— the probability generating function (p.g.f) of a return to the starting position, obtained by Jain [1, p. 344] is incorrect and have obtained its correct formulae.

L’auteur démontre la presque-périodicité des solutions à trajectoires relativement compactes de l’Équation différentielle abstraite dans un espace de Hilbert: [(d/dt)−A]x = 0 où A =A+ + A, avec A+ un opérateur linéaire symétrique et A_antisymétrique.

The sole purpose of this paper is to prove that if R1 and R2 are algebras with 1 over a common field F, with each (Ri:F) > 1 and at least one (Ri:F) > 2, then the extended center of the coproduct R1∐R2 is equal to F.

In the spirit of our previous paper (Math. Z. 156, 265 - 277 (1977)) we present a functional analytic proof of the following result of M. A. Akcoglu: Every positive contraction on a reflexive Lp-space has a lattice dilation.

In this note the stability theorem of Albert and Baker concerning the n-th difference equation is proved by using invariant means.

An example is given of a surjective map τ: [0,1] → [0,1] which takes every interval of [0,1] onto [0,1] eventually, but does not do so for certain other sets of positive measure.

Under appropriate conditions it is shown that the best interpolating approximation to a given function in the uniform norm is a limit of best unconstrained approximations with respect to a certain sequence of discontinuous weight functions.

We give an example of a p-convex quasi-Banach space E with 0 < p < 1 such that every bilinear map B: E × E →F into a p-convex quasi-Banach space F is identically zero. This resolves a question of Waelbroeck.