Let R be a commutative ring with non-zero identity and let K be the total quotient ring of R. We call R a G-ring if K is finitely generated as a ring over R. This generalizes Kaplansky′s definition of G-domain [5].

Let Z(R) be the set of zero divisors in R. Following [7] elements of R—Z(R) and ideals of R containing at least one such element are called regular. Artin-Tate's characterization of Noetherian G-domains [1, Theorem 4] carries over with a slight adjustment to characterize a Noetherian G-ring as being semi-local in which every regular prime ideal has rank one.

The Levitzki radical, which is fundamental in the study of algebras satisfying a polynomial identity, has been shown to exist in the varieties of alternative and Jordan algebras (see Zhevlakov [8], Zwier [9], and Tsai [7]— for an important application of this radical to alternative algebras satisfying a polynomial identity, see Slater [6]). In fact, Hartley [4] even investigated local nilpotence for Lie algebras, though this property can not be radical in the sense of Kurosh-Amitsur [3] for these algebras.

This paper deals with the notion of equational compactness and related concepts in the special case of G-sets for an arbitrary group G. It provides characterizations of pure extensions, pure-essential extensions, and equational compactness in terms of the stability groups of a G-set, proves the general existence of equationally compact hulls, and gives an explicit description of these. Further, it establishes, among other results, that all G-sets are equationally compact iff all subgroups of the group G are finitely generated, that every equationally compact G-set is a retract of a topologically compact one, and that for free groups G with infinite basis there are homogeneous G-sets which are not equationally compact.

The complementarity problem is defined and studied for cases where the constraints involve convex cones, thus extending the real and complex complementarity problems. Special cases of the problem are equivalent to dual, linear or quadratic, programs over polyhedral cones.

A previous result of the author concerning the parametric representation of infinitely many solutions of the title equation is strongly improved. New classes each containing infinitely many solutions of the equation for specified values of d are stated explicitly. The method of solution hinges heavily on solving the generalized Pell’s equation x2—Dy2=c.

Rings (all of which are assumed to be associative) with no non-zero nilpotent elements will be called reduced rings; R is a reduced ring if and only if x2=0 implies x=0, for all x∈R. In 2. we prove that the following conditions on an annihilator ideal I of a reduced ring are equivalent: I is a maximal annihilator, I is prime, I is a minimal prime, I is completely prime. A characterization of reduced rings with the maximum condition on annihilators is given in 3.

Let f∈Lp (1≤p<∞) 2π-periodic,

(1)

and let us consider the rth order moduli of continuity of f

(2)

R. de Vore stated the following conjectures:

(a) Let ηv→0 decreasingly,

(3)

It is well known that in the Euclidean plane there are seven distinct frieze patterns, i.e. seven ways to generate an infinite design bounded by two parallel lines. In the hyperbolic plane, this can be generalized to two types of frieze patterns, those bounded by concentric horocycles and those bounded by concentric equidistant curves. There are nine such frieze patterns; as in the Euclidean case, their symmetry groups are and

Let α be a non-negative real number, λ≡{λ,n}(n≥0) a strictly increasing unbounded sequence with λ0≥0 and let be an arbitrary series with partial sums s≡{sn}. Write

where s(t)=sn for λn<t≤λn+1, s(t)=0 for 0≤t≤λ0. The series ∑ an or the sequence of partial sums s={sn} is summable to ṡ by the Riesz method (R, λ, α) if

as ω→∞.

In a recent paper [1] MM. Arnal and Pinczon have classified all complex irreducible representations (ρ, V) of sl(2) having the property (P) that there exists a non-zero element x∈sl(2) such that ρ(x) admits an eigenvalue. It is the purpose of this note to demonstrate, by example, that there exist irreducible representations of sl(2) which do not have property (P). As usual, we consider sl(2) embedded in its universal enveloping algebra U and identify the representations of sl(2) and U.

Let X and Y be two Banach spaces and let B(X, Y) denote the set of bounded linear operators with domain X and range in 7. For T∈B(X, Y), let N(T) denote the null space and R(T) the range of T. J. I. Nieto [5, p. 64] has proved the following two interesting results. An operator T∈B(X, Y) has a left regularizer, i.e., there exists a Q∈B(Y, X) such that QT=I+A, where I is the identity on X and A∈B(X, X) is a compact operator, if and only if dim N(T)<∞ and R(T) has a closed complement.

In this Bulletin Ligh [2] generalized to finite near rings with identity a theorem Zassenhaus [5] used to prove every finite near field has abelian addition. B. H. Neumann [4] extended Zassenhaus’ result, using similar techniques and showed that all near fields are abelian. It has been an open question whether Ligh’s generalization could be carried out to infinite near rings with identity. The purpose of this paper is to show that Ligh’s theorem cannot be so extended. In particular, it cannot be extended even to distributively generated near rings, a type of near ring which has been useful in studying endomorphism rings of non-abelian groups [1,3].

Vasil’eva, [2], demonstrates a close connection between the explicit formulae for solutions to the linear difference equation with constant coefficients

(1.1)

where z is an n-vector, A an n×n constant matrix, τ>0, and a corresponding differential equation with constant coefficients

(1.2)

(1.2) is obtained from (1.1) by replacing the difference z(t—τ) by the first two terms of its Taylor Series expansion, combined with a suitable rearrangement of the terms.

Every ⊥-symmetric relatively semi-orthocomplemented lattice is M-symmetric. This answers the Problem 1 in [2] in the affirmative and provides a new proof to a result on ⊥-symmetric lattices proved in [2] (Corollary below). The notation and terminology are as in [2].

Let 〈L; ⋀, V〉 be a lattice. Two elements a and b of L are said to form a modular pair, in symbols aMb, if

The relation aM✶b is defined dually.

The results described in this paper were obtained in the study of analytic maps between flat two-tori. It is felt that analogous results will be obtained from a similar study of the analytic maps between higher-dimensional flat tori.

It is well-known that the multiplicative group GL+(2, Q) acts on the upper halfplane as a group of linear fractional transformations, i.e. for each element

An element x in a lattice L is join-reducible (meet-reducible) in L if there exist y, z∈L both distinct from x such that x=y⋁z (x=y⋀z); x is join-irreducible (meet-irreducible) in L if it is not join-reducible (meet-reducible) in L; x is doubly irreducible in L if it is both join- and meet-irreducible in L. Let J(L), M(L), and Irr(L) denote the set of all join-irreducible elements in L, meet-irreducible elements in L, and doubly irreducible elements in L, respectively, and ℓ(L) the length of L, that is, the order of a maximum-sized chain in L minus one.

Let X be a topological space and G a subgroup of the homeomorphism group H(X) with the topology of point-wise convergence. It is well-known that if X is uniformizable and G is equicontinuous with respect to a compatible uniformity then G is a topological group. In this paper we show that essentially this same result applies when X is only an R0-space (and hence in particular if X is T1 or regular). A corresponding result for regular spaces has been proved [2].

The purpose of this paper is to define partially closed braids (§3) and to prove that every partially closed braid has a canonical form easily obtainable (§5). These objects are of interest because they can be used to represent knots tied in a string.

Braids have an obvious intuitive meaning to which we shall refer. Braids are also elements of the braid groups of E. Artin [1], defined for each integer n greater than one by the presentation

In [3], Niven proved that for any positive integer k, the density of the set of positive integers n for which (n, (φ(n))≤k is zero (where φ is the Euler to tient function). In this paper, we prove a related result—namely if k and j are any positive integers, then the density of the set of positive integers n for which (n,σj(n))≤k is zero (where σj(n) is the sum of the jth powers of the positive divisors of n). We will borrow from Niven’s technique, but we must make some crucial modifications.

Si S est un ensemble dénombrable sur lequel porte une chaîne de Markov, si pn(x, y) est la probabilité de transition de x à y au ne instant, on posera An(x) la totalité des y de S tels que pn(x, y)>0, ce sont les états accessibles à partir de x à l’instant n. Dans le cas de deux types de chaînes de Markov, nous recherchons des informations sur les états accessibles; ce sera premièrement les marches aléatoires sur Z et deuxièmement les processus de Galton-Watson sur ℕ. Nous recherchons en particulier les obstructions arithmétiques à ce qu’un état soit accessible.