A permutation group A of degree n acting on a set X has a certain number of orbits, each a subset of X. More generally, A also induces an equivalence relation on X(k) the set of all k subsets of X, and the resulting equivalence classes are called k orbits of A, or generalized orbits. A self-complementary k-orbit is one in which for every k-subset S in it, X—S is also in it. Our main results are two formulas for the number s(A) of self-complementary generalized orbits of an arbitrary permutation group A in terms of its cycle index. We show that self-complementary graphs, digraphs, and relations provide special classes of self-complementary generalized orbits.

The purpose of this note is to offer a equalization of the concept of Goldie dimension and to prove a structure theorem (Theorem 3.1) for modules satisfying the conditions of this dualization. In this paper, all rings considered are associative with unit and all modules are unital.

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 [5] N. D. Lane and P. Scherk discuss arcs in the conformai (inversive) plane which are met by every circle at not more than three points; i.e., arcs of cyclic order three. This paper is concerned with the analysis of normal arcs of cyclic order four in the conformai plane.

Some results generalizing a fixed point theorem due to R. Kannan are presented.

0. Let be a family of self-mappings of a metric space (X, d).

Let {Xa}a∈A be a non-empty family of compact spaces and let F be a subset of the m-product P{Xa:a∈A}. The Tychonoff problem is the determination of the sets F which are pointwise compact. It is known that P{Xa: a ∈ A} is pointwise compact [5, p. 400]. Also the set F of all point-closed multifunctions of P{Xa:a∈ A) is pointwise compact.

This paper gives several monotonicity properties of all oscillatory solutions of equations with separable and nonseparable nonlinearities which are more general than the Emden- Fowler equations

*

Principally, if x(t) is an oscillatory solution, conditions are given such that; if a(t)↑ ∞ as t → ∞, then x(t) → 0; and, if a(t) ↓ 0 as t → ∞, then lim sup | x(t) | = ∞.

Cette note a pour but d’expliciter une structure mathématique spéciale: . Cette structure sera aux fonctions finies (i.e. fonctions d’un ensemble fini dans un autre) ce que l’anneau ℤ des entiers est aux ensembles finis. Nous ne nous servirons des résultats de la théorie des catégories et des topos que comme outils de base pour les constructions considérées. Le lecteur ne doit donc pas s′attendre à des résultats catégoriques fondamentaux mais bien à un nouvel anneau pouvant servir en théorie des nombres, en combinatoire et peut-être même en physique.

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 be a non-empty set and f:X→X. A point x ∈ X is (i) a fixed point off f(x)=x, and (ii) a periodic point of f iff there is a positive integer N such that fN(x)=x. Also a periodic orbit of f is the (finite) set {x, f(x), f2(x),…} where x is a periodic point of f.

Let A be a complex sequentially complete commutative locally m-convex topological algebra which is symmetric with continuous involution. The purpose of this note is to prove that every multiplicative linear functional on A is bounded (Theorem 3). In fact, we prove a more general result for operators on real algebras (Theorem 1) from which we derive the above result.

In his paper [1] on homogeneous spaces W. L. Chow states that “ Any one-toone adjacency preserving transformation of the Grassmann space of all the [r] of Sn(0 < r < n— 1) onto itself is a transformation of the basic group of the space.” In the proof both the transformation and its inverse are assumed to be adjacency preserving. See also Dieudonne [2] p. 81.

Let S be a non-empty family of real valued continuous functions on [a, b]. Diaz and McLaughlin [1], [2], and Dunham [3] have considered the problem of simultaneously approximating two continuous functions f1 and f2 by elements of S. If || • || denotes the supremum norm, then the problem is to find an element * ∈ S if it exists, for which

The purpose of this note is to investigate boundary points of the numerical range of an operator in terms of inner and outer center points. Some applications on commutators are given.

Throughout this note, an operator will always mean a bounded linear operator on a Hilbert space X.

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.

Let Un(f) denote the class of all n × n (0, 1)-matrices with precisely r-ones, r≥3, in each row and column. Then

Boundary value problems for ordinary differential equations have long been the subject of extensive research activity. In particular, questions concerning the existence and uniqueness of solutions for these problems have received much attention, and algebraic fixed-point theorems have served as important tools in such investigations. For example Picard [8] based his pioneering work in this area on the use of successive approximation techniques, and recently his classical methods have been refined and extended to more general nonlinear problems (see [1] and [4]). The standard procedure for applying these techniques requires that the boundary value problem under consideration first be converted into an equivalent integral equation through the choice of a suitable Green’s function. The resulting theory is consequently limited to problems for which such a formulation is possible.

Let E be a real Banach space with norm ρ. Let S={x ∊ E: ρ(x) = 1}. A norm on E is admissible if it generates the same topology as ρ.

The normρ is Gateaux differ-entiable if for each x ∊ S and u ∊ E

exists.

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).

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].