Showing papers in "Bulletin of The Australian Mathematical Society in 1987"
TL;DR: In this article, a finite sum of −-valued eigenfuctions of the Laplacian A of M acting on E -valued functions is expressed as a finite submanifolds of a hypersphere.
Abstract: ca E"n be expressedas a finite sum of ^-valued eigenfuctions of the Laplacian A of M ,acting on E -valued functions. Minimal submanifolds of a hypersphereReceived 12 March 1986. This work was done while the first authorwas a C.N.R. visiting professor at the University of Rome. He would liketo take this opportunity to express his thanks to the Consiglio Nazionaledelle Ricerche of Italy for the invitation and financial support, andalso to express his many thanks to his colleagues" at Rome for theirhospitality.Copyright Clearance Centre, Inc. Serial-fee code: 0004-9729/87$A2.00 + 0.00.161
144 citations
TL;DR: In this article, it was shown that in both knot groups and one-relator groups with center, the finitely generated normal subgroups are finitely separable, which is the only subgroup of a 3-manifold group of finite rank that is not separable.
Abstract: An example is given of an infinite cyclic extension of a free group of finite rank in which not every finitely generated subgroup is finitely separable. This answers negatively the question of Peter Scott as to whether in all finitely generated 3-manifold groups the finitely generated subgroups are finitely separable. In the positive direction it is shown that in knot groups and one-relator groups with centre, the finitely generated normal subgroups are finitely separable.
72 citations
TL;DR: Semigroups consisting of composition operators on weighted Bergman spaces are considered in this article, where one specific semigroup is used to obtain information on an averaging operator, and their infinitesimal generators are identified.
Abstract: Semigroups consisting of composition operators are considered on weighted Bergman spaces. They are strongly continuous, and their infinitesimal generators are identified. One specific semigroup is used to obtain information on an averaging operator.
51 citations
Abstract: We establish derivative estimates and existence theorems for the Dirichlet and Neumann problems for nonlinear, degenerate elliptic equations of the form F (D2u) = g in balls. The degeneracy arises through the possible vanishing of the function g and the degenerate Monge-Ampere equation is covered as a special case.
49 citations
TL;DR: In this paper, it was shown that the relation F = Lim Fn for sequences of closed sets is equivalent to the pointwise convergence of ǫ(d (.,Fn) to d(.,F).
Abstract: A metric space 〈X,d〉 is said to have nice closed balls if each closed ball in X is either compact or the entire space. This class of spaces includes the metric spaces in which closed and bounded sets are compact and those for which the distance function is the zero-one metric. We show that these are the spaces in which the relation F = Lim Fn for sequences of closed sets is equivalent to the pointwise convergence of 〈d (.,Fn)〉 to d (.,F). We also reconcile these modes of convergence with three other closely related ones.
42 citations
TL;DR: In this article, the authors considered the two cases K ≥ 1 and K 1, and presented certain generalizations of these results, and showed that these results can be generalized to the case K = 1.
Abstract: Let P ( z ) be a polynomial of degree n which does not vanish in the disk | z | K . For K = 1 , it is known that In this paper we consider the two cases K ≥ 1 and K 1 , and present certain generalizations of these results.
29 citations
TL;DR: On etudie des equations paraboliques avec des termes de forcage and on donne des conditions suffisantes for que toutes les solutions des problemes aux valeurs limites soient oscillatoires dans un domaine cylindrique as mentioned in this paper.
Abstract: On etudie des equations paraboliques avec des termes de forcage et on donne des conditions suffisantes pour que toutes les solutions des problemes aux valeurs limites soient oscillatoires dans un domaine cylindrique
29 citations
TL;DR: In this article, a partie du program relatif a la determination de tous les graphes primitifs transitifs suivant la distance (Γ) is presented.
Abstract: Presentation d'une partie du programme relatif a la determination de tous les graphes primitifs transitifs suivant la distance. Classification de tous les graphes transitifs suivant la distance (Γ) tels que An≤Aut Γ≤Aut. An pour un groupe alterne An et Aut Γ opere primitivement sur les sommets de Γ
22 citations
TL;DR: In this paper, an asymptotic formula for the number of partitions of a large positive integer n into r unequal positive integer parts and maximal summand k is derived, and the number has a normal distribution about its maximum, the largest summand an extreme value distribution.
Abstract: An asymptotic formula is derived for the number of partitions of a large positive integer n into r unequal positive integer parts and maximal summand k. The number of parts has a normal distribution about its maximum, the largest summand an extreme-value distribution. For unrestricted partitions the two distributions coincide and both are extreme-valued. The problem of joint distribution of unrestricted partitions with r parts and largest summand k remains unsolved.
20 citations
TL;DR: In this article, it is shown that if on an Asplund space there exists an equivalent Gâteaux differentiable norm, then its dual has a projectional resolution of identity.
Abstract: If au Asplund space is weakly countably determined, then it admits an equivalent Fréchet differentiable norm and is weakly compactly generated. If on an Asplund space there exists an equivalent Gâteaux differentiable norm, then its dual has a projectional resolution of identity.
20 citations
TL;DR: In this article, it was shown that a continuous mapping f of a compact real interval I to itself which is chaotic in the sense of Li and Yorke, cannot have a scrambled set residual in an subinterval of I.
Abstract: A continuous mapping f of a compact real interval I to itself which is chaotic in the sense of Li and Yorke, cannot have a scrambled set residual in an subinterval of I. This shows that chaos in this case cannot be large from a topological point of view.
TL;DR: The fonction elliptique de Weierstrass verifie an equation differentielle non lineaire du premier ordre and an equation non-lineaire du second ordre.
Abstract: La fonction elliptique de Weierstrass verifie une equation differentielle non lineaire du premier ordre et une equation differentielle non lineaire du second ordre. Ces deux equations peuvent etre discretisees de telle sorte que les solutions des equations aux differences qui en resultent coincident exactement avec les valeurs correspondantes de la fonction elliptique
TL;DR: In this article, it was shown that every compact convex set in a Banach space X is an intersection of balls provided the cone generated by the set of all extreme points of the dual unit ball of X* is dense in X* in the topology of uniform convergence on compact sets in X.
Abstract: We show that every compact convex set in a Banach space X is an intersection of balls provided the cone generated by the set of all extreme points of the dual unit ball of X* is dense in X* in the topology of uniform convergence on compact sets in X. This allows us to renorm every Banach space with transfinite Schauder basis by a norm which shares the mentioned intersection property.
TL;DR: In this paper, several common fixed point theorems for four continuous mappings in Menger and metric spaces are proved and these mappings are assumed to satisfy some generalizations of the contraction condition.
Abstract: In this paper several common fixed point theorems for four continuous mappings in Menger and metric spaces are proved. These mappings are assumed to satisfy some generalizations of the contraction condition.
TL;DR: In this paper, it was shown that even when A is complete and cocomplete, well-powered and cowellpowered, the intersection of two reflective subcategories need not be reflective.
Abstract: Given a category A, we consider the (often large) set Ref A of its reflective (full, replete) subcategories, ordered by inclusion. It is known that, even when A is complete and cocomplete, wellpowered and cowellpowered, the intersection of two reflective subcategories need not be reflective. Supposing that A admits (i) small limits and (ii) arbitrary (even large) intersections of strong subobjects, we prove that an infimum ∧iCi in Ref A must necessarily be the intersection ∩iCi. Accordingly Ref A is not in general, even for good A, a complete lattice. We show, however, under the same conditions on A, that Ref A does admit small suprema ∨iCi, given by the closure in A of the union ∪iCi under the limits of type (i) and (ii) above.
TL;DR: In this paper, the authors give an exposition of some work on the power structure of p-groups, especially work of the author's, and the titles of the three sections are: Section 1. P-groups which have regular power structure; Section 2. Some weaker power structure properties; Section 3.
Abstract: In this survey article we give an exposition of some work on the power structure of p-groups; especially work of the author's. The titles of the three sections are: Section 1. p-groups which have regular power structure; Section 2. Some weaker power structure properties; Section 3. p-central series of p-groups.
TL;DR: In this paper, it was shown that the hypercone over S2 × S4 is strictly area-minimizing in ℝ8 and the existence of smooth embedded stable hypersurfaces over ℓ 8 which are not areaminimising.
Abstract: In this paper we show that the hypercone over S2 × S4 is strictly area-minimizing in ℝ8 We also show the existence of smooth embedded stable hypersurfaces in ℝ8 which are not area-minimizing
TL;DR: Holomorphic curves in a complex quadric arise naturally as the complex conjugate of the Gauss map of a minimal surface in Euclideanspace as mentioned in this paper, and play a central role ingenerating a special class of harmonic maps of surfaces into spheres, complex projective space, and complex Grassmannians.
Abstract: Holomorphic curves in a complex quadric arise naturally as thecomplex conjugate of the Gauss map of a minimal surface in Euclideanspace. In addition, such holomorphic curves play a central role ingenerating a special class of harmonic maps of surfaces into spheres,complex projective space, and the complex Grassmannians. (See Eells-Wood[6], Bryan [2]t , Chern-Wolfson [51, Ramanathan L141, and many referencescited in these papers.)Received 11 March 1986.Copyright Clearance Centre, Inc. Serial-fee code: 0004-9729/87$A2.00 + 0.00.
TL;DR: In this article, it was shown that I admit no continuous Gateaux differentiable function with bounded nonempty support and with norm-weak continuous derivative for a Banach space X. The proof is easy and selfcontained and does not use any complicated geometrical lemma.
Abstract: A consequence of the main proposition includes results of Tacon, and John and Zizler and says: If a Banach space X possesses a continuous Gateaux differentiable function with bounded nonempty support and with norm-weak continuous derivative, then its dual X* admits a projectional resolution of the identity and a continuous linear one-to-one mapping into a (T) . The proof is easy and selfcontained and does not use any complicated geometrical lemma. If the space X is in addition weakly countably determined, then X* has an equivalent dual locally uniformly rotund norm. It is also shown that I admits no continuous Gateaux differentiable 00
TL;DR: In recent years, there has been a revival of interest in the method of integral equations for the solution of boundary value problems as mentioned in this paper, and the boundary element method has become an important tool for the computational solution of partial differential equations.
Abstract: In recent years there has been a revival of interest in the method of integral equations for the solution of boundary value problems. On the theoretical side, classical results have been extended by relaxing assumptions on the smoothness of the boundary, and in applications the boundary element method has become an important tool for the computational solution of partial differential equations. Both these developments are obviously of great interest to numerical analysts, especially since the integral equation formulations lead to numerical schemes which are completely independent of finite difference and finite element methods.
TL;DR: In this article, the implicit and explicit complementarity problems are studied in the context of fixed point theory in Hilbert space. But the implicit complementarity problem is not considered in this paper.
Abstract: In this paper we study both the implicit and the explicit complementarity problem using some special and interesting connections between the complementarity problem and fixed point theory in Hilbert space.
TL;DR: In this paper, it was shown that if the right hand side of the equation is an element of H − 1, n (Ω), then u belongs to the Orlicz space L Φ where Φ( t ) = exp(| t | n /( n − 1) ) − 1.
Abstract: Let u be a week solution of homogeneous Dirichlet problem for a second order elliptic equation of divergence form, in a bounded open subset of ℝ n . We prove, that if the right hand side of the equation is an element of H −1, n (Ω), then u belongs to the Orlicz space L Φ where Φ( t ) = exp(| t | n /( n −1) ) − 1. We employ the properties of the Schwartz symmetrization thus obtaining the “best” constant of the estimate.
TL;DR: In this paper, the problem of describing the set of rings containing a given ring as an essential ideal is addressed, and the results obtained are applied to some problems of ring theory and radicals.
Abstract: This paper concerns the problem of description of the set of rings containing a given ring as an essential ideal. The results obtained are applied to some problems of ring theory and radicals.
TL;DR: In this article, the authors present a complete description of the closed primary (respectively prime) ideals contained in a maximal ideal of the Shilov boundary of H∞, where H ∞ is the Banach algebra of all bounded analytic functions in the unit disc.
Abstract: Let H∞ be the Banach algebra of all bounded analytic functions in the unit disc. We present a complete description of the closed primary (respectively prime) ideals contained in a maximal ideal of the Shilov boundary of H∞. The paper is also concerned with chains of prime ideals in H∞.
TL;DR: In this article, the smallest size of a set for which a finite fundamental inverse semigroup can be faithfully represented by partial transformations of that set is the number of join irreducible elements of its semilattice of idempotents.
Abstract: This paper shows that the smallest size of a set for which a finite fundamental inverse semigroup can be faithfully represented by partial transformations of that set is the number of join irreducible elements of its semilattice of idempotents.
TL;DR: In this paper, Baumslag, B.H. Neumann, Hanna Neumann and Peter M.Neumann introduced the concept of strong discrimination in such a way that for a fixed finite cardinal r the following two statements are equivalent: (2) F r (V) strongly discriminates V.
Abstract: Gilbert Baumslag, B.H. Neumann, Hanna Neumann, and Peter M. Neumann successfully exploited their concept of discrimination to obtain generating groups of product varieties via the wreath product construction. We have discovered this same underlying concept in a somewhat different context. Specifically, let V be a non-trivial variety of algebras. For each cardinal α let F α ( V ) be a V -free algebra of rank α. Then for a fixed cardinal r one has the equivalence of the following two statements: (1) F r (V) discriminates V . (1*) The F s (V) satisfy the same universal sentences for all s ≥ r . Moreover, we have introduced the concept of strong discrimination in such a way that for a fixed finite cardinal r the following two statements are equivalent: (2) F r (V) strongly discriminates V . (2*) The F s (V) satisfy the same universal formulas for all s ≥ r whenever elements of F r (V) are substituted for the unquantified variables. On the surface (2) and (2*) appear to be stronger conditions than (1) and (1*). However, we have shown that for particular varieties (of groups) (2) and (2*) are no stronger than (1) and (1*).
TL;DR: In this article, theoreme de type Hukuhara for des equations integrales non lineaires de Volterra dans des espaces localement convex was presented.
Abstract: On donne un theoreme de type Hukuhara pour des equations integrales non lineaires de Volterra dans des espaces localement convexes
TL;DR: In this article, a class of convex cones, which is the largest possible with regard to existence of cone-maximal points in abstract vector optimization problems, is studied, and the authors present an independent study of the class.
Abstract: The aim of the present note is an independent study of a class of convex cones, which is the largest possible with regard to existence of cone-maximal points in abstract vector optimization problems.
TL;DR: In this paper, a tangential fc-block is defined as a simple matroid representable over GF(q) with critical exponent k+1 for which every loopless minor has critical exponent as most k. The main contribution of the thesis is to develop methods by which one begins with one tangential k-block and constructs several others.
Abstract: The critical problem for matroids, introduced by Crapo and Rota in [2] provides a natural and unified setting for a broad range of extremal combinatorial problems. A tangential fc-block over GF(q) is a simple matroid representable over GF(.q) with critical exponent k+1 for which every loopless minor has critical exponent as most k . Such matroids are of central importance in the critical problem, (see for example [3] or [4]). The thesis is primarily concerned with a study of tangential k-blocks. The main contribution of the thesis is to develop methods by which one begins with one tangential k-block and constructs several others. Other topics (initially studied for their role in the critical problem) develop their own momentun. Brown truncations and Dilworth truncations of matroids receive attention as objects of interest in their own right.
TL;DR: In this article, it was shown that a stable (or minimizing) hypercone in a Euclidian space can be approximated by smooth hypersurfaces of positive mean curvature, provided that the given minimal cone satisfies some additional hypothesis.
Abstract: Here we initiate the study of the following problem. Let Ω be a compact domain in a Riemannian manifold such that ∂Ω is of minimum area for the contained volume. Can ∂Ω be approximated by smooth hypersurfaces of positive mean curvature? It reduces to the question of whether or not a stable (or minimizing) hypercone in a Euclidian space can be approximated by smooth hypersurfaces of positive mean curvature. The positive solution to the problem may be useful for studying the curvature and topology of Ω.We show in this paper that such approximation is possible provided that the given minimal cone satisfies some additional hypothesis.