Showing papers in "Bulletin of The Australian Mathematical Society in 1994"
TL;DR: In this paper, the authors prove certain inequalities for elementary symmetric funtions that are relevant to the study of partial differential equations associated with curvature problems, which is relevant to our work.
Abstract: In this note, we prove certain inequalities for elementary symmetric funtions that are relevant to the study of partial differential equations associated with curvature problems.
129 citations
TL;DR: In this article, it was shown that a Banach space enjoys the weak fixed point property if it is e0-inquadrate for some e0 < 2 and has WORTH; that is, if then, ║xn + x → 0, for all x.
Abstract: It has recently been shown that a Banach space enjoys the weak fixed point property if it is e0-inquadrate for some e0 < 2 and has WORTH; that is, if then, ║xn — x║ — ║xn + x║ → 0, for all x. We establish the stronger conclusion of weak normal structure under the substantially weaker assumption that the space has WORTH and is ‘e0-inquadrate in every direction’ for some e0 < 2.
63 citations
TL;DR: In this paper, the authors studied the unique range set of meromorphic functions and showed that the set S = {w | w13 + w11 + 1 = 0} is the unique set with 13 elements.
Abstract: This paper studies the unique range set of meromorphic functions and shows that the set S = {w | w13 + w11 + 1 = 0} is unique range set of meromorphic functions with 13 elements.
48 citations
TL;DR: In this paper, the authors investigated the situation where a group G has an abelian subgroup H with connected transversals and showed that if H is finite then G is solvable.
Abstract: In this paper we investigate the situation where a group G has an abelian subgroup H with connected transversals. We show that if H is finite then G is solvable. We also investigate some special cases where the structure of H is very close to the structure of a cyclic group. Finally we apply our results to loop theory and we show that if the inner mapping group of a finite loop Q is abelian then Q is centrally nilpotent.
38 citations
TL;DR: In this article, the existence of solutions for the elliptic systems on ℝN was established, such that u, v ∈ W1, 2(ℝ N), where with q(x)→∞ as |x| → ∞ and (x, u, V) being superlinear or sublinear as.
Abstract: We establish the existence of solutions for the elliptic systems on ℝN: such that u, v ∈ W1, 2(ℝN), where with q(x)→∞ as |x| → ∞ and (x, u, v) being superlinear or sublinear as .
26 citations
TL;DR: In this article, a new coincidence theorem for a Browder type set-valued mapping and an upper semi-continuous setvalued mapping with compact acyclic values in an H-space was established.
Abstract: In this paper, we establish a new coincidence theorem for a Browder type set-valued mapping and an upper semi-continuous set-valued mapping with compact acyclic values in an H-space which generalises some recent results in the literature. As applications we obtain two H a type coincidence theorems and existence theorems of maximal elements for preference correspondences.
26 citations
TL;DR: The uniqueness of limit cycles for quadratic systems with an invariant parabola and for cubic systems with four real line invariants was proved in this paper, and a new simple proof was given for limit cycles occurring in unfoldings of certain vector fields with codimension two singularities.
Abstract: The uniqueness of limit cycles is proved for quadratic systems with an invariant parabola and for cubic systems with four real line invariants. Also a new, simple proof is given of the uniqueness of limit cycles occurring in unfoldings of certain vector fields with codimension two singularities.
25 citations
TL;DR: In this article, the Cahn-Hilliard system was shown to generate a dissipative semigroup on the phase space ℋ = [H2(Ω)]m.
Abstract: The Cahn-Hilliard system, a natural extension of the single Cahn-Hilliard equation in the case of multicomponent alloys, will be shown to generate a dissipative semigroup on the phase space ℋ = [H2(Ω)]m. Following Hale's ideas and based on the existence and form of the Lyapunov functional, our main result will be the existence of a global attractor on a subset of ℋ. New difficulties specific to the system case make our problem interesting.
25 citations
TL;DR: In this paper, the authors provided irrationality measures for certain values of binomial functions and definite integrals of rational functions using Jacobi type polynomials and divisibility considerations of their coefficients.
Abstract: The paper provides irrationality measures for certain values of binomial functions and definite integrals of some rational functions. The results are obtained using Jacobi type polynomials and divisibility considerations of their coefficients.
20 citations
TL;DR: In this paper, the authors present in the reflexive Banach space setting a natural and simple proof of the formula of the approximate subdifferential of a composite function, and prove that it is a simple formula.
Abstract: The aim of this note is to present in the reflexive Banach space setting a natural and simple proof of the formula of the approximate subdifferential of a composite function.
19 citations
TL;DR: In this article, the authors studied almost sure convergence for arrays of independent and identically distributed random variables and obtained a condition under which Marcinkiewicz's strong law holds and a rate analogous to the law of the iterated logarithm under a condition weaker than Hu and Weber's.
Abstract: In this paper we study almost sure convergence for arrays of independent and identically distributed random variables. We obtain a condition under which Marcinkiewicz's strong law holds and get a rate analogous to the law of the iterated logarithm under a condition weaker than Hu and Weber's.
TL;DR: In this article, the authors study optimality conditions for an efficient solution in various senses of a general multiobjective optimisation problem in abstract spaces and derive necessary and sufficient conditions for a feasible solution to be a weak minimum (a minimum, a strong minimum or a proper minimum).
Abstract: In this paper we study optimality conditions for an efficient solution in various senses of a general multiobjective optimisation problem in abstract spaces. We utilise properties of the Clarke's generalised differential and properties of a conesubconvexlike function to derive a few necessary and/or sufficient conditions for a feasible solution to be a weak minimum (a minimum, a strong minimum or a proper minimum) of the vector optimisation problem. The results in this paper are extensions and refinements of some known results in vector optimisation.
TL;DR: In this paper, it was shown that G has a finite covering by nilpotent subgroups if and only if any infinite set of elements of G contains a pair {x, y} such that [x, ny] = 1 for some integer n = n(x, Y) ≥ 0.
Abstract: Let G be a finitely generated soluble group. Lennox and Wiegold have proved that G has a finite covering by nilpotent subgroups if and only if any infinite set of elements of G contains a pair {x, y} such that (x, y) is nilpotent. The main theorem of this paper is an improvement of the previous result: we show that G has a finite covering by nilpotent subgroups if and only if any infinite set of elements of G contains a pair {x, y} such that [x, ny] = 1 for some integer n = n(x, y) ≥ 0.
TL;DR: In this paper, the symmetric powers of quantum GLn are studied and a complete description of the subcomodule structure is given, including sub-comodules and sub-cliques.
Abstract: In this paper we study the symmetric powers of quantum GLn. These are comodules and we give a complete description of the subcomodule structure.
TL;DR: In this article, it was shown that at a local minimum point of the composition of nonsmooth vector valued and real valued functions, the generalised gradient admits a special, relatively sharp chain rule, that yields sharper results than have been seen before in the context of the generalized gradient.
Abstract: Nonsmooth calculus using the approximate subdifferential of Mordukhovich and loffe admits a sharper chain rule, hence sharper applications in optimisation, than does the generalised gradient of Clarke. We observe, however, that at a local minimum point of the composition of nonsmooth vector valued and real valued functions, the generalised gradient admits a special, relatively sharp chain rule, that yields sharper results than have been seen before in the context of the generalised gradient.
TL;DR: For an ideal I in a commutative ring R, the authors showed that if n! is a unit in R, then In = In, and gave an example of a doubly generated ideal I with Is not finitely generated.
Abstract: For an ideal I in a commutative ring R we consider the ideal In = ({in | i ∈I}). We show that if n! is a unit in R, then In = In. We give an example of a doubly generated ideal I with Is not finitely generated.
TL;DR: In this paper, the diagonal form of the braid generator is derived from which a general closed formula is obtained for link polynomials corresponding to certain induced representations of the symmetric group and its subgroups.
Abstract: Unitary representations of the braid group and corresponding link polynomials are constructed corresponding to each irreducible representation of a quantum double finite group algebra. Moreover the diagonal form of the braid generator is derived from which a general closed formula is obtained for link polynomials. As an example, link polynomials corresponding to certain induced representations of the symmetric group and its subgroups are determined explicitly.
TL;DR: In this article, it was shown that any extension of a finitely presented infinite group by a finite group is qsf, i.e., the universal cover of the complex can be approximated by simply connected finite complexes.
Abstract: A finitely presented group G is quasi-simply-filtrated (abbreviated qsf) if, given a finite complex with fundamental group G, the universal cover of the complex can be “approximated” by simply connected finite complexes. This notion is a generalisation of a concept of Casson's used in the study of three-manifolds.In this paper we show that any extension of a finitely presented infinite group by a finitely presented infinite group is qsf.
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TL;DR: In this article, it was shown that monomorphisms and epimorphisms are dual to each other via Morita duality, and a left artinian ring R with Jacobson radical J has a MORITA duality if either J/J2 is a central bimodule or R is artinian right duo and R/J is commutative.
Abstract: Modules whose nonzero endomorphisms are epimorphisms and modules whose nonzero endomorphisms are monomorphisms are considered in this paper. We prove that these two classes of modules are dual to each other via Morita duality. We also prove that a left artinian ring R with Jacobson radical J has a Morita duality if either (1) J/J2 is a central bimodule; or (2) R is artinian right duo and R/J is commutative.
TL;DR: In this article, it was shown that non-zero nilpotent ideals in amenable Banach algebras cannot even have the approximation property, under certain additional hypotheses.
Abstract: It is well known that non-zero nilpotent ideals in amenable Banach algebras must be infinite-dimensional. We show that under certain additional hypotheses such ideals cannot even have the approximation property.
TL;DR: The main result in this paper is that if D satisfies a non-trivial generalised power central rational identity, then D is finite dimensional over K. As a corollary, we obtain the following result.
Abstract: Let D be a skew field with uncountable centre K. The main result in the present paper is as follows: If D satisfies a non-trivial generalised power central rational identity, then D is finite dimensional over K. As a corollary we obtain the following result. Let a be an element of D such that (a−1x−1ax)q(x) ∈ K for all x ∈ D ﹨ {0} where q(x) is a positive integer depending on x. Then a ∈ K.
TL;DR: In this paper, a semigroup S is called E-solid if and only if for all idempotents e, f, g ∈ S such that e L f R g there exists an idemepotent h ∈ s such that R h L g.
Abstract: A semigroup S is called E-solid if and only if for all idempotents e, f, g ∈ S such that e L f R g there exists an idempotent h ∈ S such that e R h L g . Each completely regular semigroup is E-solid. We characterise the idempotents of an arbitrary E-solid regular semigroup as a set with a binary operation on it satisfying a given finite set of identities.
TL;DR: A generalised polynomial formalism for algebra of piece-wise defined polynomials has been developed in this article, which is based on the Boolean power construction of universal algebra.
Abstract: We develop a generalised polynomial formalism which captures the concept of an algebra of piece-wise defined polynomials. The formalism is based on the Boolean power construction of universal algebra. A generalisation of the theory of substitution homomorphisms is developed. The abstract operation of composition of generalised polynomials in one variable is defined and shown to correspond to function composition. On one level, a polynomial may be viewed as an algorithm, giving a sequence of operations to be performed on a finite set of elements of an algebra. We consider an extension of this notion which also captures the important algorithmic notion of branching. The emphasis is entirely algebraic, and the role of substitution is as significant as for standard polynomials. We refer the reader to Lausch and Nobauer [1] for the background theory of generalised polynomials for single sorted universal algebras. We build on the many sorted case, although the basic concepts are essentially the same. The
TL;DR: In this article, it was proved that under the following condition, the sum f of the double trigonometric series with coefficients cjk is integrable and the rectangular partial sums smn(f, x, y) converge to f in L 1 norm:.
Abstract: It is proved that under the following condition, the sum f of the double trigonometric series with coefficients cjk is integrable and the rectangular partial sums smn(f, x, y) converge to f in L1 norm: .
TL;DR: In this article, the authors show that a Banach space has modulus of convexity of power type p if and only if best approximants to points from straight lines are uniformly strongly unique of order p. Assuming that the space is smooth, they derive a characterisation of the best simultaneous approximant to two elements, and use the characterisation to prove that p-type modulus implies order p strong unicity of the simultaneous approximation.
Abstract: We show that a Banach space has modulus of convexity of power type p if and only if best approximants to points from straight lines are uniformly strongly unique of order p. Assuming that the space is smooth, we derive a characterisation of the best simultaneous approximant to two elements, and use the characterisation to prove that p –type modulus of convexity implies order p strong unicity of the simultaneous approximant.
TL;DR: In this paper, a general class of variational inequalities for lower semicontinuous functions due to A. Szulkin was investigated and existence and multiplicity results were obtained by using minimax principles.
Abstract: In this paper, we investigate a general class of variational inequalities. Existence and multiplicity results are obtained by using minimax principles for lower semicontinuous functions due to A. Szulkin.
TL;DR: The concept of superspace is fundamental for some recent physical theories, notably supersymmetry, and a mathematical feedback for it is provided by superanalysis and supergeometry.
Abstract: The concept of superspace is fundamental for some recent physical theories, notably supersymmetry, and a mathematical feedback for it is provided by superanalysis and supergeometry. We survey the state of affairs in superanalysis, shifting our attention from supermanifold theory to “plain” superspaces. The two principal existing approaches to superspaces are sketched and links between them discussed. We examine a problem by Manin of representing even geometry (analysis) as a collective effect in infinite-dimensional purely odd geometry (analysis), by applying the technique of nonstandard (infinitesimal) analysis.
TL;DR: In this paper, the authors prove an abstract theorem for solutions of a variational inequality on a cone and use it to study the free boundary problem of elastohydrodynamic lubrication from mechanical engineering.
Abstract: In this paper the authors prove an abstract theorem for solutions of a variational inequality on a cone and use it to study the free boundary problem of elastohydrodynamic lubrication from mechanical engineering. The mathematical model is set in a one-dimensional geometry. The existence of a solution for every non-negative lubricant viscosity is proved, and some properties useful for the numerical analysis are obtained.
TL;DR: In this article, the authors studied the class of groups having no nontrivial linear representations over certain fields and showed the class to be closed under perfect extensions with locally soluble kernel, by applying to both binate groups and the acyclic automorphism groups of de la Harpe and McDuff.
Abstract: We study the class of groups having no nontrivial linear representations over certain fields. After showing the class to be closed under perfect extensions with locally soluble kernel, we expand considerably the number of acyclic groups known to be in the class, by application to both binate groups and the acyclic automorphism groups of de la Harpe and McDuff.
TL;DR: The abstract triangle groups Δ(2, 4, r) can be defined for any positive integer r by Δ( 2, 4 r) = 〈x, y | x2 = y4 = (xy)r = 1〉.
Abstract: The abstract triangle groups Δ(2, 4, r) can be defined for any positive integer r by Δ(2, 4, r) = 〈x, y | x2 = y4 = (xy)r = 1〉. In this paper we show that for every r ≥ 6, all but finitely many of the alternating groups An can be obtained as quotients of Δ(2, 4, r).