Throughout this paper all near-rings will be zero-symmetric and left distributive. A near-ring with minimal condition on right N-subgroups will be called an near-ring. It is well known (see (1), 3.40, p. 90)) that a nil right N-subgroup of an near-ring is nilpotent. However, in a deeper study of near-rings a stronger result than this is sometimes required (2, p. 77).

In previous work of the author and M. Culler, contractible simplicial complexes were constructed on which the group of outer automorphisms of a free group of finite rank acts with finite stabilizers and finite quotient. In this paper, it is shown that these complexes are Cohen-Macauley, a property they share with buildings. In particular, the link of a vertex in these complexes is homotopy equivalent to a wedge of spheres of codimension 1.

In 1909 Dr Thomas Muir, in a paper on the above topic, gave several theorems involving the derivation of a circulant, and it is the writer's purpose in this paper to extend these investigations with a number of other results.

Let M be a surface immersed in an m-dimensional space form Rm(c) of curvature c = 1, 0 or −1. Let h be the second fundamental form of this immersion; it is a certain symmetric bilinear mapping for x ∈ M, where Tx is the tangent space and the normal space of M at x. Let H be the mean curvature vector of M in Rm(c) and 〈, 〉 the scalar product on Rm(c). If there exists a function λ on M such that 〈h(X, Y), H〉 = λ〈X, Y〉 for all tangent vectors X, Y, then M is called a pseudo-umbilical surface of Rm(c). Let D denote the covariant differentiation of Rm(c) and η be a normal vector field. If we denote by D*η the normal component of Dη, then D* defines a connection in the normal bundle. A normal vector field η is said to be parallel in the normal bundle if Dη = 0. The length of mean curvature vector is called the mean curvature.

In this paper, we prove that if is an increasing sequence of strictly positive and continuous functions on a locally compact Hausdorff space X such that then the Fréchet space C(X) is distinguished if and only if it satisfies Heinrich's density condition, or equivalently, if and only if the sequence satisfies condition (H) (cf. e.g.‵[1] for the introduction of (H)). As a consequence, the bidual λ∞(A) of the distinguished Köthe echelon space λ0(A) is distinguished if and only if the space λ1(A) is distinguished. This gives counterexamples to a problem of Grothendieck in the context of Köthe echelon spaces.

The class of prime Noetherian v-H orders is a class of Noetherian prime rings including the commutative integrally closed Noetherian domains, and the hereditary Noetherian prime rings, and designed to mimic the latter at the level of height one primes. We continue recent work on the structure of indecomposable injective modules over Noetherian rings by describing the structure of such a module E over a prime Noetherian v-H order R in the case where the assassinator P of E is a reflexive prime ideal. This description is then applied to a problem in torsion theory, so generalising work of Beck, Chamarie and Fossum.

The purpose of this note is to describe some algebraic conditions on a Banach algebra which force it to be finite dimensional. We shall assume throughout that we are dealing with Banach algebras over the field of complex numbers, C.

A 2-complex K is called almost-acyclic if H2(K) = 0 and H1(K) is torsion-free. This class of complexes was introduced in a previous paper (2), and applied to a problem of J. H. C. Whitehead concerning aspherical 2-complexes. In this note, the methods developed in (2) are used to study the finitely-generated subgroups of the fundamental group of an almost-acyclic 2-complex.

In the two parts of this investigation previously published it has been shown that the solution in terms of elliptic functions represents the motion of the particular dynamical system under consideration throughout the whole range of values of s and g for which a real solution exists, except for those values for which s = 2g and k = 1, but that, on the other hand, the series solution is convergent and represents the motion only so long as

for values of s and g for which the sign of this inequality is reversed the trigonometric series representing the solution are divergent. It is of importance to investigate what discontinuities, if any, of the system correspond to values of s and g which lie on the boundary of the region of convergence; the present part is concerned primarily with showing that under such circumstances no discontinuity of the system exists, thus confirming the suggestions made in Part I., § 12.

Much has been written, from the algebraical as well as from the geometrical standpoint, on the subject of pencils of quadrics: algebraically the problem consists of the canonical reduction of a pencil of quadratic forms, and the classical paper on the subject is by Weierstrass. But among the different kinds of pencils of quadratic forms there is the “singular pencil,” in which the discriminant of every form belonging to the pencil is zero; interpreted geometrically this means that every quadric belonging to the pencil is a cone. This case was expressly excluded from consideration by Weierstrass, and the canonical reduction was only accomplished later by Kronecker. But, although Weierstrass and Kronecker together solved completely the problem of the canonical reduction of a pencil of quadratic forms, a much clearer insight into the nature of the problem was gained when Segre gave the geometrical solution. He published two papers, one dealing with the non-singular pencils and the other with the singular pencil.

Let co, c1, …, cn-1 be the nonzero complex numbers and let C = (cu+1,v+1) = (cn+u-v), O≦u,v≦n — 1, be a cyclic matrix, where n + u — v is taken modulo n. In this paper we shall give the solution of the linear equations

where Lu (0≦u≦n —1) is a fixed complex number. In Theorem 1 weshall give a necessary and sufficient condition for (1) to have an integral solution.

Fibonacci algebras are groups equipped with an extra unary operation φ that satisfies a Fibonacci-type law. We described in an earlier paper the free objects in the resulting varieties, and here we do the same in the case when φ is assumed to be periodic. They turn out to be central extensions of Burnside groups with finite kernels whose orders can be expressed in terms of the resultants of certain polynomials.

The generalization of the Magnus embedding [7] proved by Smelkin [9] may bestated as follows. Let L be a free group freely generated by the set xi(i∈I), and let R be a normal subgroup of L with G = L/R. If V is any variety of groups and ∏ is the V-freegroup with free generating set the symbols [g, xi] (g∈G, i∈I), then L/V(R) is embeddedin the semidirect product ∏ ⋊ G (where the action of G on ∏ is given by h · [g, xi] = [hg, xi], for h, g ∈ G).

The object of this paper is to remove the difficulty that arises in giving a general proof by projection methods of this theorem, without in any way interfering with the single-valuedness of the position of a radius vector tracing out angles from a given initial position, when the values of the trigonometrical ratios are given.

Implicit operations (new operations commuting with all old homomorphisms) on pseudovarieties have been shown to play an important role in the study of these classes. They may be used to axiomatize sub-pseudovariaties and to describe recognizable subsets of (relatively) free objects. This paper presents a case study for the pseudovariety CS consisting of all finite simple semigroups. Based on a result of profinite group theory, a structural description of semigroups of implicit operations on finite simple semigroups is used to deduce that CS is join-irreducible.

In this paper the factorization of arithmetical numbers of the form , where x; is a rational number such that kx is a perfect square, is investigated by means of a trigonometrical transformation. The number k will be taken to be prime for the present.