It is well known that complex analytic functions and harmonic functions of two real variables are closely related, so that from methods and results in complex function theory one can easily obtain theorems on those harmonic functions. This is the prototype of a relation between complex analysis and a partial differential equation (Laplace's equation in two variables). In the case of more general linear partial differential equations, one can establish similar relations.

This note considers the question of the connectedness of the set of limit points of the A-transforms of a sequence, where A is a conservative Hausdorff, quasi-Hausdorff or Meyer-König- Ramanujan type of matrix. New proofs of some known results, as well as some new results are obtained.

La portée et les buts de cet article sont signalés au début des différents paragraphes et dans les remarques qui suivent le théorème 3, le lemme 5 et le théorème 21. Limitons-nous à mentionner ici que le sujet traité est à la fois en relation avec la théorie des systèmes d'inéquations linéaires (théorie des polyèdres coniques), avec le concept de point Λ-extrême, avec les diverses notions d'efficacité par rapport à un système d'objectifs linéaires et avec la programmation à objectifs linéaires multiples.

Let R be a ring with identity. The study of rings in which every left (right) ideal is quasi-injective was begun by Jain, Mohamed, and Singh (3). They called these rings left (right) q-rings. A number of structure theorems have been proved for q-rings. See, for example, (1), (2), and (5). A ring with the dual property (rings in which every homomorphic image of R as a left (right) R-module is quasi-projective) is called left (right) q*. These rings were first studied by Koehler (4), where some results connecting q* -rings with q-rings were obtained.

In this paper the notion of inverse cluster set, which was recently introduced and studied for functions by T. R. Hamlett and P. E. Long (Proc. Amer. Math. Soc, 53 (1975), 470-476), is extended to and investigated for multifunctions. We generalize the notion of inverse cluster set, extend to multifunctions and generalize some known results for inverse cluster sets of functions and offer some new results. In the latter sections, compactness generalizations are characterized in terms of inverse cluster sets and some results on connected and conectivity functions are extended to multifunctions.

Let R be an associative ring with identity. If R is von- Neumann regular of a left v-ring, then for each left ideal, I, we have I 2 = I. In this note we study rings such that for each left ideal L there exists an integer n = n(L)>0 such that Ln = Ln+1 . We call such rings stable rings. We completely describe the stable commutative rings. These descriptions give rise to concepts related to, but more general than, finite Goldie dimension and T-nilpotence, and a notion of power pure.

Let H be a real Hilbert space with its dual space H'. The norm and inner product in H are denoted by ||.|| and 〈.,.〉 respectively. We denote by 〈.,.〉, the pairing between H' and H.

If a(u, v) is a bilinear form and F is a real-valued continuous functional on H, then we consider I[v], a functional defined by

This paper considers what conditions on the order structure of a Riesz space will insure that one locally solid topology is finer than another, or when does one topology induce a finer topology than another on the order bounded subsets. The basic tool employed for the comparisons will be the carrier of a locally solid topology. We shall deal mainly with topologies whose carriers are order dense; a locally solid topology with order dense carrier will be called entire.

Two fixed point theorems for multi-valued mappings in a complete, ε-chainable metric space are proved. The theorems, thus established, extend result of M. Edelstein, Peter K. F. Kuhfittig, Hwei-mei Ko and Yueh-hsia Tsai, S. B. Nadler, Jr. and S. Reich.

Let

1

be the simple continued fraction (SCF) of an irrational number x. The partial quotients ai which we shall sometimes refer to as the "terms" of the SCF are integers and, for i ≥ 2, they are positive. If x is a quadratic irrationality then, by Lagrange's Theorem, the right side of (1) becomes periodic from some point on.

Dans ce qui suit f(X) désigne un polynöme irréductible de degré n de et un polynöme de tel que a≠0 et r désigne un entier non nul.

On montre dans cet article que les polynÔmes ne sont pas irréductibles sur pour m≥3.

We study here the smoothness of solutions of the Dirichlet problem for elliptic equations in a region G with a piece-wise smooth boundary. The smoothness of the solution given depends on the smoothness of the coefficients of the equation, the boundary, the boundary function and the values of the angles on the boundary and the values of the coefficients of the second derivatives at the corners.

A von Neumann algebra is said to be approximately finite-dimensional if it is of the form

where Mn⊆Mn+1 for each n and each Mn is a finite-dimensional matrix algebra. A factor is said to be ITPFI if it is of the form

A partially ordered set P has the fixed point property if every orderpreserving mapping f of P to P has a fixed point, that is, f(a) = a for some aϵP; call P fixed point free if P does not have the fixed point property.

A regular Borel measure μ is said to be r*-invariant on a locally compact semigroup if μ(Ba -1) = μ(B) for all Borel sets B and points a of S, where Ba -1 ={xϵS, xaϵB}. In [1] Argabright conjectured that the support of an r*-invariant measure on a locally compact semigroup is a left group, Mukherjea and Tserpes [4] proved this conjecture in the case that the measure is finite; however their method of proof fails when the measure is infinite.

Let D be a division ring with center Z. Suppose for all xϵD, there exists a monic polynomial, fx(t), with integer coefficients such that fx(x)ϵZ. Then D is commutative.

The following question has been floating around for some time now and is also stated as Research Problem 26 in [4]:

Let R, S be unital rings and let 〈x〉 be an infinite cyclic group. Does R〈x〉≃S〈x〉 imply R≃S?

In this note, we present a collection of examples which answer the question in the negative. However, all of these examples consist of non-commutative rings, and the problem is still open in the case where R and S are assumed to be commutative.