Showing papers in "Bulletin of The Australian Mathematical Society in 1974"
163 citations
TL;DR: In this paper, a connected account of matrix quadratic equations is given, and some new results and new proofs of known results are given, as well as new proofs for known results.
Abstract: Matrix quadratic equations have found the most diverse applications. The present article gives a connected account of their theory, and contains some new results and new proofs of known results.
118 citations
TL;DR: In this article, the authors studied common fixed points of mappings of a complete metric space into itself, and obtained generalizations of Ray Theorems, which they used to obtain the results.
Abstract: The object of this paper is to study common fixed points of mappings of a complete metric space into itself. The results obtained are generalizations of Ray Theorems.
37 citations
TL;DR: In this article, a class of multifunctions with measurable selectors is introduced, which is both well-supplied with a measurable selector and yet is comprehensive enough to include those kinds of multifunction which have been most commonly studied before.
Abstract: Let S and X be any two sets; then a mapping Γ which assigns to each point t in S a set Γ(t) of points in X is called a multifunction from S into X. A selector for Γ is a function f from S into X such that f(t) ∈ Γ(t) for each t. We introduce here a class of multifunctions which is both well-supplied with measurable selectors and yet is comprehensive enough to include those kinds of multifunction which have been most commonly studied before. Hence in order to show that a multifunction with non-empty values, which may arise naturally in an implicit function problem, has a measurable selector, it is sufficient to show that it is of Souslin type.
35 citations
TL;DR: In this article, necessary and sufficient conditions on G and S are given in order that there should exist a Dedekind domain D with class group G with the property that S is the set of classes that contain maximal ideals of D.
Abstract: Let G be an abelian group, and let S be a subset of G. Necessary and sufficient conditions on G and S are given in order that there should exist a Dedekind domain D with class group G with the property that S is the set of classes that contain maximal ideals of D. If G is a torsion group, then S is the set of classes containing the maximal ideals of D if and only if S generates G. These results are used to determine necessary and sufficient conditions on a family {Hλ} of subgroups of G in order that there should exist a Dedekind domain D with class group G such that {G/Hλ} is the family of class groups of the set of overrings of D. Several applications are given.
35 citations
TL;DR: In this article, the two generator restricted Burnside group of exponent five was shown to have order 5 3lt and class 12 by two independent methods, and a consistent commutator power presentation for the group was given.
Abstract: The two generator restricted Burnside group of exponent five is shown to have order 5 3lt and class 12 by two independent methods. A consistent commutator power presentation for the group is given.
30 citations
TL;DR: In this article, it was shown that complete graphs on rm and rm + 1 vertices, r > 1, can be decomposed into stars with m edges, if and only if r is even or m is odd.
Abstract: A star is a connected graph in which every vertex but one has valency 1. This paper concerns the question of when complete graphs can be decomposed into stars, all of the same order, which have pairwise disjoint edge-sets. It is shown that the complete graphs on rm and rm + 1 vertices, r > 1, can be decomposed into stars with m edges, if and only if r is even or m is odd.
30 citations
TL;DR: In this paper, the degree of a matrix of rational functions is obtained in a simplified way, which enables them to be generalised to matrices whose elements are not necessarily rational functions.
Abstract: The properties of the degree of a matrix of rational functions are obtained in a simplified way, which enables them to be generalised to matrices whose elements are not necessarily rational functions. On the basis of these results a theory of realisations is developed, which similarly generalises the theory of state space realisations of a matrix of rational functions.
28 citations
TL;DR: In this article, it was shown that pn(G ) ≥ max{0, od(v )-id( v )} the sum taken over all vertices v of G, with equality holding if G is acyclic.
Abstract: A path decomposition of a digraph G (having no loops or multiple arcs) is a family of simple paths such that every arc of G lies on precisely one of the paths of the family. The path number, pn ( G ) is the minimal number of paths necessary to form a path decomposition of G . We show that pn ( G ) ≥ max{0, od( v )-id( v )} the sum taken over all vertices v of G , with equality holding if G is acyclic. If G is a subgraph of a tournament on n vertices we show that pn ( G ) ≤ with equality holding if G is transitive. We conjecture that pn ( G ) ≤ for any digraph G on n vertices if n is sufficiently large, perhaps for all n ≥ 4.
23 citations
TL;DR: In this paper, the connection between the concepts of isoclinism and covering groups for finite groups is examined and the main result is that all covering groups of a given finite group are mutually isOClinic.
Abstract: The article examines the connection between the concepts of isoclinism and covering groups for finite groups. The main result is that all covering groups for a given finite group are mutually isoclinic. The converse is false.
20 citations
TL;DR: In this paper, it was shown that if a variety V of universal algebras, defined by a set Σ of identities, is closed under the construction of power algesbras then V can be defined by the subset Σ′ of Σ consisting of those identities ν = w of the set of identities such that every variable in w occurs exactly once on both sides.
Abstract: It is shown that if a variety V of (universal) algebras, defined by a set Σ of identities, is closed under the construction of power algebras then V can be defined by the subset Σ′ of Σ consisting of those identities ν = w of Σ such that every variable in ν = w occurs exactly once on both sides.
TL;DR: In this article, necessary and sufficient conditions for the order completeness of the Banach lattices C(X, E) and L1(μ, E), in terms of the compactness of the order intervals in E, are given.
Abstract: Let E be a Banach lattice. Necessary and sufficient conditions are given for the order completeness of the Banach lattices C(X, E) and L1(μ, E) in terms of the compactness of the order intervals in E. The results have interpretations in terms of spaces of compact and nuclear operators.
TL;DR: In this article, it was shown that if n ≡ 0 (mod 4) then there exist n × n weighing matrices of every degree k ≤ n, and if n is not a power of 2, then there exists an integer t n for which there are weight matrices with every degree ≤ t.
Abstract: A weighing matrix is an n × n matrix W = W ( n , k ) with entries from {0, 1, −1}, satisfying = WW t = KI n . We shall call k the degree of W . It has been conjectured that if n ≡ 0 (mod 4) then there exist n × n weighing matrices of every degree k ≤ n . We prove the conjecture when n is a power of 2. If n is not a power of two we find an integer t n for which there are weighing matrices of every degree ≤ t .
TL;DR: In this paper, it was shown that the groups F(2, 8) and F( 2, 10) are infinite, and that the only Fibonacci group whose finiteness or infiniteness has not been determined has not yet been determined.
Abstract: Fibonacci groups are the groupswhere r is a natural number. The groups F(2, 8) and F(2, 10) are shown to he infinite, thus leaving F(2, 9) as the only Fibonacci group whose finiteness or infiniteness has not been determined.
TL;DR: An algorithm for computing the unique stationary distribution of an infinite regular stochastic matrix of a structural form subsuming both upper-Hessenberg and generalized renewal matrices of this kind is presented in this paper.
Abstract: An algorithm is presented for computing the unique stationary distribution of an infinite regular stochastic matrix of a structural form subsuming both upper-Hessenberg and generalized renewal matrices of this kind. Convergence is elementwise, monotone from above, from information within finite truncations, of increasing order.
TL;DR: In this article, it was shown that the sum of the absolute values of the coefficients of the mth transformation polynomial Fm (u, v) of the Weber modular function j(ω) of level 1 is not greater than 2(36n+57)2n when m = 2n is a power of 2.
Abstract: In a previous paper (Acta Arith. 21 (1972), 89–97), I had proved that the sum of the absolute values of the coefficients of the mth transformation polynomial Fm (u, v) of the Weber modular function j(ω) of level 1 is not greater than
2(36n+57)2n
when m = 2n is a power of 2. The aim of the present paper is to give an analogous bound for the case of general m. This upper bound is much less good and of the form where c > 0 is an absolute constant which can be determined effectively. It seems probable that also in the general case an upper bound of the form
eO(m10gm)
should hold, but I have not so far succeeded in proving such a result.
TL;DR: The recent work of Ostrom and others has made it clear that the class of finite translation planes is quite large and by no means completely determined as yet as mentioned in this paper, and it is thus important that various types of finite translational planes be subjected to detailed study.
Abstract: The study of finite translation planes is central to the study of finite projective planes. Almost all the known finite planes are either translation planes or their duals or appear in a derivable chain of planes based on a dual translation plane. The recent work of Ostrom and others has made it clear that the class of finite translation planes is quite large and by no means completely determined as yet. It is thus important that various types of finite translation plane be subjected to detailed study. Such studies must inevitably lead to a more complete theory of translation planes and a more intimate understanding of related classes of planes.
TL;DR: In this paper, a regular completion with the universal property is obtained for each member of a certain class of Cauchy spaces by embedding the space in a complete function algebra with the continuous convergence structure.
Abstract: A regular completion with the universal property is obtained for each member of a certain class of Cauchy spaces by embedding the Cauchy space in a complete function algebra with the continuous convergence structure.
TL;DR: In this article, it was shown that every equidistant set is closed in the bw-topology of a normed linear space when X = lp (1 < p < ∞).
Abstract: In this note some results concerning the equidistant set E(−x, x) and the kernel Mθ of the metric projection PM, where M is a Chebyshev subspace of a normed linear space X, have been obtained. In particular, when X = lp (1 < p < ∞), it has been proved that every equidistant set is closed in the bw-topology of the space. In c0 no equidistant set has this property.
TL;DR: In this article, a lower bound for the smallest positive value of a convex distance function on the lattice of all points x with integral coordinates on a suitable hyperplane with integral coefficients u 1, u 2, un not all zero.
Abstract: Since Minkowski's time, much progress has been made in the geometry of numbers, even as far as the geometry of numbers of convex bodies is concerned. But, surprisingly, one rather obvious interpretation of classical theorems in this theory has so far escaped notice. Minkowski's basic theorem establishes an upper estimate for the smallest positive value of a convex distance function F(x) on the lattice of all points x with integral coordinates. By contrast, we shall establish a lower estimate for F(x) at all the real points X on a suitable hyperplane with integral coefficients u1, …, un not all zero. We arrive at this estimate by means of applying to Minkowski's Theorem the classical concept of polarity relative to the unit hypersphere This concept of polarity allows generally to associate with known theorems on point lattices analogous theorems on what we call hyperplane lattices. These new theorems, although implicit in the old ones, seem to have some interest and perhaps further work on hyperplane lattices may lead to useful results. In the first sections of this note a number of notations and results from the classical theory will be collected. The later sections deal then with the consequences of polarity.
TL;DR: In this paper, an extension of Banach's contraction mapping principle to Hausdorff spaces is presented, in fact to the larger class of topological spaces in which convergent sequences have unique limits.
Abstract: This paper presents an extension of Banach's contraction mapping principle to Hausdorff spaces, in fact to the larger class of topological spaces in which convergent sequences have unique limits. This is achieved by considering topologies on X generated by families of quasi-pseudo-metrics on X. An extension of the concept of Cauchy sequence to this non-metric setting is given.
TL;DR: In this article, the authors present the solution of a long-standing problem, namely, the determination of a set of polynomials in two independent variables which are biorthogonal over a triangular region.
Abstract: We present the solution of a long-standing problem, namely, the determination of a set of polynomials in two independent variables which are biorthogonal over a triangular region to a set of polynomials previously introduced by Appell. Some elementary properties of our polynomials are investigated.
TL;DR: In this paper, the usual expansions of the resolvent and Fredholm determinant are shown to hold for an n × n symmetric kernel N(x, y) with arbitrary domain in Rp under weakened continuity conditions.
Abstract: Multivariate versions of Mercer's Theorem and the usual expansions of the resolvent and Fredholm determinant are shown to hold for an n × n symmetric kernel N(x, y) with arbitrary domain in Rp under weakened continuity conditions. Further, the resolvent and determinant of N(x, y) − a(x)b(y) are given in terms of those of N(x, y).
TL;DR: In this paper, the existence of global solution branches for positive mappings was proved for boundary value problems for systems of ordinary differential equations, and a related result for mappings in wedges.
Abstract: We prove the existence of global solution branches for positive mappings. This improves an earlier result of the author. We also prove a related result for mappings in wedges. We then use these two results to prove the existence of solutions for boundary-value problems for systems of ordinary differential equations.
TL;DR: In this article, the orientation-preserving equivariant homeomorphism of a compact, connected, oriented topological G-manifold has been shown to be even.
Abstract: Let X be a compact, connected, oriented topological G-manifold, where G is a compact connected Lie group. Assume that the fixed point set is finite but nonempty, the action is otherwise free, and the orbit space is a manifold. It follows that either G = U(1) = S1 and dimX =4 or G = Sp(1) = S3 and dimX = 8, and the number of fixed points is even. The authors prove that these ∪(1)-manifolds (respectively, Sp(1)-manifolds) are classified up to orientation-preserving equivariant homeomorphism by (1) the orientation-preserving homeomorphism type of their orbit 3-manifolds (respectively, 5-manifolds), and(2) the (even) number of fixed points.Both the homeomorphism type in (1) and the even number in (2) are arbitrary, and all the examples are constructed. The smooth analog for U(1) is also proved.
TL;DR: In this paper, the authors construct infinitely many nonisomorphic finitely presented metabelian groups with the same finite quotients, using modules over a suitably chosen ring, and give an example of infinitely many split extensions of a fixed finitely-presented metabilian group by a fixed finite abelian group.
Abstract: Let F(G) denote the set of isomorphism classes of finite quotients of the group G. Two groups G and H are said to have the same finite quotients if F(G) = F(H). We construct infinitely many nonisomorphic finitely presented metabelian groups with the same finite quotients, using modules over a suitably chosen ring. These groups also give an example of infinitely many nonisomorphic split extensions of a fixed finitely presented metabelian group by a fixed finite abelian group, all having the same finite quotients.
TL;DR: In this article, a method is given for constructing commutative Banach algebras which admit two inequivalent complete norm topologies and the result is applied to show that the action of any non-algebraic analytic function may fail to be uniquely defined.
Abstract: It has recently been shown that discontinuous functional calculi exist for certain commutative Banach algebras Such an algebra thus possesses two distinct calculi so that there exist analytic functions whose action on the algebra is not uniquely determined In this note a method is given for constructing commutative Banach algebras which admit two inequivalent complete norm topologies and the result is applied to show that the action of any non-algebraic analytic function may fail to be uniquely defined
TL;DR: In this article, it was shown that the continuity of the metric projection supported by a Chebyshev set does not imply that the set is approximatively compact, and it is indeed so in a large class of Banach spaces, including the locally uniformly convex spaces.
Abstract: In the paper “Some remarks on approximative compactness”, Rev. Roumaine Math. Pures Appl . 9 (1964), Ivan Singer proved that if K is an approximatively compact Chebyshev set in a metric space, then the metric projection onto K is continuous. The object of this paper is to show that though, in general, the continuity of the metric projection supported by a Chebyshev set does not imply that the set is approximatively compact, it is indeed so in a large class of Banach spaces, including the locally uniformly convex spaces. It is also proved that in such a space X the metric projection onto a Chebyshev set is continuous on a set dense in X .
TL;DR: Kaplansky's observation, that a commutative ring R is (von Neumann) regular if and only if each simple R-module is injective, is generalized to projective modules over a projective ring as discussed by the authors.
Abstract: Kaplansky's observation, namely, a commutative ring R is (von Neumann) regular if and only if each simple R-module is injective, is generalized to projective modules over a commutative ring.
TL;DR: In this paper, a monodiffric analogue of a discrete analytic function of the first kind (or monodifield function) is found. But the problem of finding a suitable analogue of the function is not solved.
Abstract: An unsolved problem in discrete analytic function theory has been to find a suitable analogue of the function . An analogue z(α), of the function zα, is found here for discrete analytic functions of the first kind (or monodiffric functions). This function resolves a conjecture of Isaacs in the negative, and at the same time it introduces multi-valued functions into the discrete analytic theory.