Given an arbitrary function g :X → (-∞, +∞] and a lowersemicontinuous convex function h:X → (-∞, +∞], we give the general expression of the conjugate (g — h)* of g - h in terms of g* and h*. As a consequence, we get Toland's duality theorem:

Let R be a ring with identity not equal to zero. A right R-module is said to be quasi-injective (pseudo-injective) if for every submodule N of M, every R-homomorphism (R-monomorphism) of N into M can be extended to an R-endomorphism of M [7] ([13]).

Let P - {p1,p2,. . . ,pn} be an independent point-set in ℝd (i.e., there are no d + 1 on a hyperplane). A simplex determined by d + 1 different points of P is called empty if it contains no point of P in its interior. Denote the number of empty simplices in P by fd(P). Katchalski and Meir pointed out that . Here a random construction Pn is given with , where K(d) is a constant depending only on d. Several related questions are investigated.

We denote by ZG the integral group ring of the finite group G. We call ±g, for g in G, a trivial unit of ZG. For G abelian, Higman [4] (see also [3, p. 262 ff]) showed that every unit of finite order in ZG is trivial. For arbitrary finite G (indeed, for a torsion group G, not necessarily finite), Higman [4] showed that every unit in ZG is trivial if and only if G is

1. (i) abelian and the order of each element divides 4, or

2. (ii) abelian and the order of each element divides 6, or

3. (iii) the direct product of the quaternion group of order 8 and an abelian group of exponent 2.

Let Q denote the field of rational numbers. If m, n are distinct squarefree integers the field formed by adjoining √m and √n to Q is denoted by Q(√m, √n). Since Q(√m, √n) = Q(√m, √n) and √m + √n has for its unique minimal polynomial x4 —2(m + n)x2 + (m - n)2, Q(√m, √n) is a biquadratic field over Q. The elements of Q(√m, √n) are of the form a0 + a1√m + a2√n + a3√mn, where a1, a2, a3 ∊ Q. Any element of Q(√m, √n) which satisfies a monic equation of degree ≥ 1 with rational integral coefficients is called an integer of Q(√m, √n). The set of all these integers is an integral domain. In this paper we determine the explicit form of the integers of Q(√m, √n) (Theorem 1), an integral basis for Q(√m, √n) (Theorem 2), and the discriminant of Q(√m, √n) (Theorem 3).

Let

It is proved that the function f reaches its maximum for n = 6 983 776 800, and that maxn≥2f(n) = 1.5379. The proof deals with superior highly composite numbers introduced by Ramanujan.

The relation between the categories of Fuzzy Sets and that of Sheaves is explored and the precise connection between them is explicated. In particular, it is shown that if the notion of fuzzy sets is further fuzzified by making equality (as well as membership) fuzzy, the resultant categories are indeed toposes.

The concept of the M-radical of a submodule B of an R-module A is discussed (R is a commutative ring with identity and A is a unitary fl-module). The M-radical of B is defined as the intersection of all prime submodules of A containing B. The main result of the paper is that if denotes the ideal radical of (B:A), then M-rad B = provided that A is a finitely generated multiplication module. Additionally, it is shown that if A is an arbitrary module, where for some

We give a new measure-theoretical proof of the uniform distribution property of points in model sets (cut and project sets). Each model set comes as a member of a family of related model sets, obtained by joint translation in its ambient (the ‘physical’) space and its internal space. We prove, assuming only that the window defining the model set is measurable with compact closure, that almost surely the distribution of points in any model set from such a family is uniform in the sense of Weyl, and almost surely the model set is pure point diffractive.

Throughout this note, we adopt the graph-theoretical terminology and notation of Harary [3]. A graph G is hypohamiltonianif G is not hamiltonian but the deletion of any point u from G results in a hamiltonian graph G-u. Gaudin, Herz, and Rossi [2] proved that the smallest hypohamiltonian graph is the Petersen graph. Using a computer for a systematic search, Herz, Duby, and Vigué [4] found that there is no hypohamiltonian graph with 11 or 12 points. However, they found one with 13 and one with 15 points. Sousselier [4] and Lindgren [5] constructed independently the same sequence of hypohamiltonian graphs with 6k+10 points. Moreover, Sousselier found a cubic hypohamiltonian graph with 18 points. This graph and the Petersen graph were the only examples of cubic hypohamiltonian graphs until Bondy [1] constructed an infinite sequence of cubic hypohamiltonian graphs with 12k+10 points. Bondy also proved that the Coxeter graph [6], which is cubic with 28 points, is hypohamiltonian.

The object of this paper is to present results concerning the structure of 3-connected graphs and of 5-chromatic and 6-chromatic graphs and also a theorem on contraction and a theorem of Turan type. The Axiom of Choice is assumed.

A new class of 3-dimensional contact metric manifolds is found. Moreover it is proved that there are no such manifolds in dimensions greater than 3.

In a previous note on derivations [1] we determined the structure of a prime ring R which has a derivation d≠0 such that the values of d commute, that is, for which d(x) d(y) = d(y) d(x) for all x, y∈R. Perhaps even more natural might be the question: what elements in a prime ring commute with all the values of a non-zero derivation? We address ourselves to this question here, and settle it.

An ellipsoid G is associated uniquely with a positive definite matrix A via

Note that all ellipsoids discussed here are centred at 0. Given G1, and G2 we seek another ellipsoid circumscribed about G1 ∩ G2. It is easy to see that

if and only if x'hx ≤ maxi x'aix for all vectors x.

In this paper, we prove that if an almost co-Kähler manifold of dimension greater than three satisfying $\unicode[STIX]{x1D702}$-Einstein condition with constant coefficients is a Ricci soliton with potential vector field being of constant length, then either the manifold is Einstein or the Reeb vector field is parallel. Let $M$ be a non-co-Kähler almost co-Kähler 3-manifold such that the Reeb vector field $\unicode[STIX]{x1D709}$ is an eigenvector field of the Ricci operator. If $M$ is a Ricci soliton with transversal potential vector field, then it is locally isometric to Lie group $E(1,1)$ of rigid motions of the Minkowski 2-space.

Le but de ce travail est de prolonger les travaux de Y. Utumi sur les anneaux continus.

Nous appelons module quasi-continu un module stable par tout projecteur de son enveloppe injective. Un anneau unitaire A est dit quasi-continu à gauche (resp. à droite) si le A-module à gauche (resp. à droite ) est quasi-continu. Cette définition est plus générale que celle d'anneau continu à gauche donnée par Yuzo Utumi, et quand l'anneau est régulier, elles coïncident (§111).

Let R be a prime ring and d≠0 a derivation of R. We examine the relationship between the structure of R and that of d(R). We prove that if R is an algebra over a commutative ring A such that d(R) is a finitely generated submodule then R is an order in a simple algebra finite dimensional over its center.

The aim of this paper is to give a number of characterizations of the rings of the title. In particular, these turn out to be precisely those exchange rings whose idempotents are all central. They are also those rings in which every element is the sum of a unit and a central idempotent.

This is an expository note to show how an “infinite abacus” (to be defined presently) can be programmed to compute any computable (recursive) function. Our method is probably not new, at any rate, it was suggested by the ingenious technique of Melzak [2] and may be regarded as a modification of the latter.

By an infinite abacus we shall understand a countably infinite set of locations (holes, wires etc.) together with an unlimited supply of counters (pebbles, beads etc.). The locations are distinguishable, the counters are not. The confirmed finitist need not worry about these two infinitudes: To compute any given computable function only a finite number of locations will be used, and this number does not depend on the argument (or arguments) of the function.

This paper is concerned with the study of the regularity for the multisublinear maximal operator. It is proved that the multisublinear maximal operator is bounded on first-order Sobolev spaces. Moreover, two key point-wise inequalities for the partial derivatives of the multisublinear maximal functions are established. As an application, the quasi-continuity on the multisublinear maximal function is also obtained.