In this note I investigate smoothness and regularity of the ring the ring of integers. These results do not seem to be well known, especially those dealing with regularity. At any rate, several of my colleagues knew that was not smooth, but thought that it might be regular. I was surprised by the number of possibilities that can occur, as well as by the fact that A can be regular but not smooth. As expected the prime 2 plays a special role. Smoothness depends on a, b, c mod 2, and regularity depends on a, b, c mod 4.

A function M is defined which maps the plane onto a square region in such a way that the planar graphs In, exp, X, —X, 1/X, and all compositions formed from them are transformed into straight lines. One can then solve for their intersections. It also provides a natural definition for the repeated composition of In with itself t times, where t can be a non-integer.

Let , where ai, s, and x are real, and II denotes the product over cyclic rearrangements of the subscripts. We show that, in five special cases, Dn, s(x)Dn, s(y) is greater than a fixed multiple of Dn, s(x + y).

It is shown that if 1<n ≡ 0 or 1 (mod 2 m), then the edges of Kn may be partitioned into isomorphic copies of a graph D3(m) and also of a graph D4(m), graphs consisting respectively of a triangle with an attached path of m - 3 edges or a quadrilateral with an attached path of m - 4 edges. If m is a power of 2 then the above condition is shown to be necessary and sufficient for the existence of such a partition.

Let (n|k) denote the number of k-choices 1≤x1<x2<…<xk≤n satisfying xi-xi-1≥2, i = 2,…, k, n + x1-xk≥2; let (m, n | k) = Σi+j=k (m | i)(n | j). Several elementary proofs of the new identity (m, n|k) = (m + n | k) if 0≤k<m≤n. and

if 0≤m≤n, m≤k, are given. Generalizations and applications are considered.

Several characterizations for prime semilattices are obtained. Prime semilattices that are compactly packed by filters have been characterized. Solution to the problem, “Find a condition on a semilattice by which every filter can be expressed as the intersection of all prime filters containing it”, is furnished.

It is well known and immediate that in an associative ring a nilpotent one-sided ideal generates a nilpotent two-sided ideal. The corresponding open question for alternative rings was raised by M. Slater [6, p. 476]. Hitherto the question has been answered only in the case of a trivial one-sided ideal J (i.e., in case J2 = 0) [5]. In this note we solve the question in its entirety by showing that a nilpotent one-sided ideal K of an alternative ring generates a nilpotent two-sided ideal. In the process we find an upper bound for the index of nilpotency of the ideal generated. The main theorem provides another proof of the fact that a semiprime alternative ring contains no nilpotent one-sided ideals. Finally we note the analogous result for locally nilpotent one-sided ideals.

In this paper, we prove that if S is a locally compact semigroup and T a locally compact Borel measurable subsemigroup of S, then T has a topological left invariant mean if and only if there is a topological left T-invariant mean M on S such that M(xT) = 1, where xT is the characteristic functional of T in S.

A finite group is said to have deficiency zero if it can be presented with an equal number of generators and relations. Finite metacyclic groups of deficiency zero have been classified, see [1] or [6]. Finite non-metacyclic groups of deficiency zero, which we denote by FD0-groups, are relatively scarce. In [3] I. D. Macdonald introduced a class of nilpotent FD0-groups all having nilpotent class≤8. The largest nilpotent class known for a Macdonald group is 7 [4]. Only a finite number of nilpotent FD0-groups, other than the Macdonald groups, seem to be known [5], [7]. In this note we exhibit a class of FD0-groups which contains nilpotent groups of arbitrarily large nilpotent class.

In a recent paper [1] I. B. Lazarevic announced an extension of results of L. Tornheim [2; Theorems 2 & 3] concerning points of contact between two distinct members of an n-parameter family and between a member of an n-parameter family and a corresponding convex function. In the proofs of these extensions [1; Theorems 3.1 & 3.2] use is made of Tornheim′s Convergence Theorem [2; Theorem 5]; however this theorem is not correctly applied in [1] since it requires distinct limiting nodes, and that hypothesis necessarily fails in the approach used in [1], In this note proofs of results more general than those in [1] are given independent of convergence theorems.

For l≤p<∞ μ∈ℝ, let ℒμ.p denote the collection of functions f, measurable on (0, ∞) and such that

Let C0 be the collection of functions continuous and compactly supported on (0, ∞); it is known that C0 is dense in ℒμ.p—see [2; Lemma 2.2]. If X and Y are Banach spaces, denote by [X, Y] the collection of bounded linear operators from X into Y, abbreviating [X, X] to [X].

A variety of applications depend on the number of solutions of polynomial equations over finite fields. Here the usual situation is reversed and we show how to use geometrical methods to estimate the number of solutions of a non-homogeneous symmetric equation in three variables.

In this paper we show how to compute the probability that a sequence Zn = X0 + … + Xn of partial sums of dependent random variables, each taking on the values ±1, will first leave an interval (a, b) at a; and how to compute the expected time it takes for the partial sums to leave the interval (a, b).

A theorem on the barycentre of a measure is proven which leads to generalization of Carathéodory′s theorem and to extension of various results. A mathematical programming problem is examined in application.

The geometric notions of a gap and gap points between “concentric” spheres in a normed linear space are introduced and studied. The existence of gap points characterizes finitedimensional spaces. General conditions are given under which an infinite-dimensional normed linear space admits concentric spheres such that both these spheres and their dual spheres fail to have gap points.

Let be the least positive residue modulo 2tk of (2j- l)h. Define ut to be the number of with l≤j≤2t-2k such that . At the Special Session in Combinatorial Number Theory at the 1977 Summer AMS Meeting Szekeres [2] asked for a simple proof that if (h, 2k)=1, then

Here a simple proof will be given for the following equivalent result.

With the aid of a recent result obtained by the first author, an expression is derived which unifies the well-known Jensen and Gould formulas.

Conditions for the existence and conjugacy of complements of certain minimal ideals of solvable Lie algebras over a Noetherian ring R are considered. Let L be a solvable Lie algebra and A be a minimal ideal of L. If L/A is nilpotent and L is not nilpotent then A has a complement in L, all such complements are conjugate and self-normalizing and if C is a complement then there exists an x∈L such that C = {y∈L; yadnx = 0 for some n = 1, 2,…}. A similar result holds if A is self-centralizing and a finitely generated R-module.

If M is a simply-connected 4-manifold with boundary, let D(M) denote its double MU∂M(-M). If M is closed, let D(M) denote M#-M. In either case, D(M) is a simply-connected 4-manifold of index zero, and so by a theorem of Wall [8], M#k(S2xS2) must be standard for k sufficiently large, where by standard we mean diffeomorphic to the connected sum of copies of S2 x S2 and S2×S2, the non-trivial S2 bundle over S2 (which is itself diffeomorphic to ℂP2#-ℂP2 [7]). In this paper we give abound on k, in the case where M has no 3-handles.

In a paper with the above title, T. M. Apostol and S. Chowla [1] proved the following result:

Theorem 1.For relatively prime integers h and k, l ≤ h ≤ k, the series

does not admit of an Euler product decomposition, that is, an identity of the form

1

except when h = k = l; fc = 1, fc = 2. The series on the right is extended over all primes p and is assumed to be absolutely convergent forR(s)>1.