Let p denote a prime and n a positive integer ≥ 2. Let Nn(p) denote the number of polynomials xn + x + a, a = 1, 2,…, p-l, which are irreducible (mod p). Chowla [5] has made the following two conjectures:

Conjecture 1. There is a prime p0(n), depending only on n, such that for all primes p ≥ p0(n)

If x and y are independent p component column vectors, and the conditional distribution of x, given x+y = z, is known, what can be said about the distributions of x and y? This problem has been solved by Seshadri (1966) in the particular case when the conditional distribution of x, given x+y = z, is multivariate normal. In fact Seshadri′s paper implicitly contains a characterization of the multivariate linear exponential distribution

(1)

where A(x) is a function of x not involving the p component column vector w of constant terms.

The application of Borel summability to Fourier series has been discussed by Takahashi and Wang [8] and Sahney [5]. Sahney [6] and Sinvhal [7] obtained sufficient conditions for the Borel summability of the derived Fourier series and its conjugate series, respectively. Kathal [3] obtained different conditions in the case of the conjugate series. In this paper we give a new criterion for Borel summability of Fourier series.

In a recent note [1], M. R. Embry proved that if A is an operator on a Banach space then, under a certain condition on the spectrum of A, each operator commuting with An also commutes with A, where n is a fixed positive integer. It turns out that, when A is a finite matrix, Embry′s conditions imply that A is a polynomial in An and hence plainly each operator commuting with An also commutes with A.

Let X1, X2,…, Xk, denote k ≥ 2 indeterminates and let f(X1,…, Xk) be a homogeneous polynomial, in GF(pn) [X1,…, Xk], which is irreducible but not absolutely irreducible over GF(pn). Thus f is irreducible in GF(pn)[X1,…, Xk] but reducible in some GF(pnm) [X1,…, Xk], m > 1. For any polynomial h(X1,…, Xk) in GF(pnℓ)[X1,…, Xk, ℓ ≥ 1, let Npn(h) denote the number of (x1,…, xk) ∈ GF(pn)×…× GF(pn) such that (hx1…, xk) = 0.

In this note we present an extremely short proof of Goldie's theorem on the structure of semiprime Noetherian rings [1]. The outline of the proof was given by Procesi and Small in [4]. By utilizing the concept of the singular ideal of a ring we have been able to weaken the hypotheses of many of the steps in [4]. Most significantly, we are able to avoid a reduction to the case of prime rings, and in Lemma 5 we give an informative list of the relationship between regular elements and essential ideals of semiprime rings.

An nth order homogeneous linear differential equation is said to be disconjugate on the interval I of the real line in case no non-trivial solution of the equation has more than n - 1 zeros (counting multiplicity) on I. It is the purpose of this paper to establish several necessary and sufficient conditions for disconjugacy of the third order linear differential equation

(1.1)

where pi(t) is continuous on the compact interval [a, b], i = 0, 1, 2.

The generating series for the elementary symmetric function Er, the complete symmetric function Hr, are defined by

respectively.

A celebrated theorem of Birkhoff ([1], [6]) states that the set of n × n doubly stochastic matrices is identical with the convex hull of the set of n × n permutation matrices. Birkhoff [2, p. 266] proposed the problem of extending his theorem to the set of infinite doubly stochastic matrices. This problem, which is often known as Birkhoffs Problem III, was solved by Isbell ([3], [4]), Rattray and Peck [7], Kendall [5] and Révész [8].

R. L. Graham has posed the following question:

Given n positive integers a1, < a2 <… < an, does there exists a pair of indices i, j such that ai/(ai, aj) ⩾ n? ((ai, aj) = g.c.d. of ai and aj).

If A is a bounded linear operator on an infinite-dimensional complex Hilbert space H, let lat A denote the collection of all subspaces of H that are invariant under A; i.e., all closed linear subspaces M such that x ∈ M implies (Ax) ∈ M. There is very little known about the question: which families F of subspaces are invariant subspace lattices in the sense that they satisfy F = lat A for some A? (See [5] for a summary of most of what is known in answer to this question.) Clearly, if F is an invariant subspace lattice, then {0} ∈ F, H ∈ F and F is closed under arbitrary intersections and spans. Thus, every invariant subspace lattice is a complete lattice.

In what follows the notation and terminology of [7] are used and all rings are assumed to have a unity element.

The purpose of this note is to give some partial answers to the question: under which conditions on a ring A and a group G is the group ring AG semi-perfect?

For the convenience of the reader a few definitions and results will be reviewed. A ring R is called semi-perfect if R/RadR (Jacobson radical) is completely reducible and idempotents can be lifted modulo RadR (i.e., if x is an idempotent of R/RadR there is an idempotent e of R so that e + RadR = x).

Let F be a free group of rank ⩾ 2, let F/R ≅ π, and let F0 = F/[R, R]. Auslander and Lyndon showed that the center of Fo is a subgroup of R/[R, R] = Ro, and that it is non-trivial if and only if π is finite [1, corollary 1.3 and theorem 2]. In this paper it will be shown that there is a canonically defined (and not always trivial) quotient group of the center of F which depends only on π.

Let R be a commutative domain, I an ideal of R and write E1 for the infective envelope of R / I. In this note the following theorem will be proved:

Theorem. For a prime ideal P of a commutative domain R the following are equivalent:

1. (i) Every factor module of Ep is an indecomposable infective module;

1. (ii) Every non-zero prime ideal P′ ⊆ P is contained in only one maximal ideal M of R, and RM is an almost maximal valuation ring.

In the theory of surface area one meets situations where a function z = f(x, y) which is defined and continuous on a closed rectangle E, is partially differentiable on E except on a subset of E of Lebesgue measure zero.

The following theorem appears in [1].

Let R be a closed simply connected region of the inversive plane bounded by a Jordan curve J, and let J be divided into three closed arcs A1, A2, A3. Then there exists a circle contained in R and having points in common with all three arcs.

Le but de cette note est de donner une caractérisation purement catégorique des catégories de modules semi-simples sur un anneau. Nous allons voir que ces catégories sont précisément celles qui sont localement artiniennes au sens de P. Gabriel et dont le radical au sens de G. M. Kelly est nul.

A near-ring is a triple (R, +,.) such that (R, +) is a group, (R,.) is a semigroup and. is left distributive over +; i.e. w(x + z) = wx + wz for each w, x, z in R. A near-field is a nearring such that the nonzero elements form a group under multiplication. Zassenhaus [3] showed that if R is a finite near-field, then (R, + ) is abelian. B.H. Neumann [1] extended this result to all near-fields. Recently another proof of this important result was given by Zemmer [4]. The purpose of this note is to give another generalization of the Zassenhaus theorem. In fact, we shall prove the following.

For p a prime ≡ 1 (mod n), where n is an odd positive integer, let k(p, n) denote the least integer k such that the numbers xn and (-x)n, where x = 1,2,…, k, yield all the non-zero n-th power residues (mod p) (possibly with repetitions). Clearly