Showing papers in "Bulletin of The Australian Mathematical Society in 2006"
TL;DR: In this article, the existence of positive solutions to the class of nonlocal boundary value problems of the p-Kirchhoff type was studied. But the problem of finding positive solutions was not studied.
Abstract: This paper is concerned with the existence of positive solutions to the class of nonlocal boundary value problems of the p-Kirchhoff type and where Ω is a bounded smooth domain of ℝN, 1 < p < N, s ≥ p* = (pN)/(N – p) and M and f are continuous functions.
207 citations
TL;DR: In this paper, new inequalities for the general case of convex functions defined on linear spaces which improve the famous Jensen's inequality are established for complex and real n-tuples.
Abstract: New inequalities for the general case of convex functions defined on linear spaces which improve the famous Jensen's inequality are established. Particular instances in the case of normed spaces and for complex and real n-tuples are given. Refinements of Shannon's inequality and the positivity of Kullback-Leibler divergence are obtained.
170 citations
TL;DR: In this article, the authors discuss boundedness and compactness of the products of composition and differentiation between Hardy spaces, and discuss the relation of Hardy spaces to Hardy spaces with Hardy spaces.
Abstract: We shall discuss boundedness and compactness of the products of composition and differentiation between Hardy spaces.
83 citations
TL;DR: In this paper, a new concept of generalised f-projection operator is introduced, which extends the generalised projection operator πK : B* → K, where B is a reflexive Banach space with dual space B* and K is a nonempty, closed and convex subset of B. As an application, the existence of solution for a class of variational inequalities in Banach spaces is studied.
Abstract: In this paper, we introduce a new concept of generalised f-projection operator which extends the generalised projection operator πK : B* → K, where B is a reflexive Banach space with dual space B* and K is a nonempty, closed and convex subset of B. Some properties of the generalised f-projection operator are given. As an application, we study the existence of solution for a class of variational inequalities in Banach spaces.
69 citations
TL;DR: In this paper, the authors study Lancret type problems for curves in Sasakian 3-manifolds and prove that a curve in Euclidean 3-space is of constant slope if and only if its ratio of curvature and torsion is constant.
Abstract: A classical theorem by Lancret says that a curve in Euclidean 3-space is of constant slope if and only if its ratio of curvature and torsion is constant. In this paper we study Lancret type problems for curves in Sasakian 3-manifolds.
69 citations
TL;DR: In this paper, the authors established several theorems concerning properly embedded constant mean curvature surfaces (cmc-surfaces) in homogeneously regular 3-manifolds.
Abstract: We establish several theorems concerning properly embedded constant mean curvature surfaces (cmc-surfaces) in homogeneously regular 3-manifolds, when the mean curvature H is large
54 citations
TL;DR: This work investigates methods for computing explicitly with homomorphisms (and particularly endomorphisms) of Jacobian varieties of algebraic curves, and describes several families of hyperelliptic curves whose Jacobians have complex multiplication and/or real multiplication.
Abstract: In this work, we investigate methods for computing explicitly with homomorphisms (and particularly endomorphisms) of Jacobian varieties of algebraic curves. Jacobians of hyperelliptic curves have been a subject of considerable interest in recent years since their proposal as a source of groups for cryptography by Koblitz. While efficient algorithms and compact representations for computing with Jacobians are well-known, methods for explicitly computing with homomorphisms between them are relatively undeveloped. The intent of this thesis is to contribute to the understanding of constructing explicit homomorphisms of Jacobians, and to provide a variety of efficient and practical examples and applications of the theory. We begin by discussing the theory of correspondences on curves. This geometric approach to homomorphisms of Jacobians is analogous to the study of a function by analysis of its graph. The advantage of correspondences is that they permit the expression of any homomorphism of Jacobians in a compact way, as a divisor on a product of curves. In particular, correspondences do not require the construction of the Jacobians themselves. While correspondences are a classical subject, we treat the theory in a constructive, algorithmic fashion. We then give a series of explicit examples of correspondences inducing non-trivial homomorphisms. Extending work of Cassou-Nogues and Couveignes, we give families of hyperelliptic curves of genus three, five, six, seven, ten and fifteen, defined over number fields of low degree, whose Jacobians have isogenies to other hyperelliptic Jacobians. For each family, we construct a correspondence to making these isogenies explicitly computable. We describe several families of hyperelliptic curves whose Jacobians have complex multiplication and/or real multiplication, including families described by Mestre and by Tautz, Top and Verberkmoes. Again, we construct correspondences to make the complex and real multiplication explicit. We then use the correspondences to derive efficiently Received 23rd August, 2006 Thesis submitted to The University of Sydney, December 2005. Degree approved, July 2006. Supervisors: Dr David R. Kohel and Professor John Cannon. Copyright Clearance Centre, Inc. Serial-fee code: 0004-9727/06 SA2.00+0.00.
48 citations
TL;DR: In this paper, the convolution sums are evaluated for all n ∈ ℕ and evaluated for n ∆ ∆ for all ∆ ≥ 0.5 ∆.
Abstract: The convolution sums and are evaluated for all n ∈ ℕ.
39 citations
TL;DR: In this article, it was shown that a retarded delay differential equation has a global attractor in the space C ([τ, ], ℝd) for a given nonzero constant delay τ 0.
Abstract: It is shown that if a retarded delay differential equation has a global attractor in the space C ([—τ0, ], ℝd) for a given nonzero constant delay τ0, then the equation has an attractor Aτ in the space C ([—τ, 0], ℝd) for nearby constant delays τ. Moreover the attractors Aτ converge upper semi continuously to in C ([—τ0, 0], ℝd) in the sense that they are identified through corresponding segments of entire trajectories in ℝd with nonempty compact subsets of C ([—τ0, 0], ℝd) which converge upper semi continuously to in C ([—τ0, 0], ℝd).
37 citations
TL;DR: In this paper, the Fourier transforms on the real positive line were studied, and conditions on a real function u(x) in x > 0 were obtained that its Fourier cosine transform v(t) = u (x) cos xt dx is positive.
Abstract: This note concerns Fourier transforms on the real positive line. In particular, we seek conditions on a real function u(x) in x > 0, that ensure that its Fourier-cosine transform v(t) = u(x) cos xt dx is positive. We prove first that this is so for all t > 0, if u"(x) > 0 for all x > 0, that is, that everywhere-convex functions have everywhere-positive Fourier-cosine transforms. We then obtain a complex-plane criterion for some types of non-convex u(x). Finally we consider criteria on u(x) that imply positivity of v(t) for t > t0, for some t0 > 0.
36 citations
TL;DR: The theory of infinite binary sequences called Sturmian words has been studied from combinatorial, algebraic, and geometric points of view as mentioned in this paper, with the most well-known example being the ubiquitous Fibonacci word.
Abstract: Combinatorics on words plays a fundamental role in various fields of mathematics, not to mention its relevance in theoretical computer science and physics. Most renowned among its branches is the theory of infinite binary sequences called Sturmian words, which are fascinating in many respects, having been studied from combinatorial, algebraic, and geometric points of view. The most well-known example of a Sturmian word is the ubiquitous Fibonacci word, the importance of which lies in combinatorial pattern matching and the theory of words. Properties of the Fibonacci word and, more generally, Sturmian words have been extensively studied, not only because of their significance in discrete mathematics, but also due to their practical applications in computer imagery (digital straightness), theoretical physics (quasicrystal modelling) and molecular biology.
TL;DR: In this article, it was shown that in a real smooth and uniformly convex Banach space, appropriately constructed approximating fixed point sequences can be strongly convergent, even in a Hilbert space.
Abstract: Consider a nonexpansive self-mapping T of a bounded closed convex subset of a Banach space. Banach's contraction principle guarantees the existence of approximating fixed point sequences for T. However such sequences may not be strongly convergent, in general, even in a Hilbert space. It is shown in this paper that in a real smooth and uniformly convex Banach space, appropriately constructed approximating fixed point sequences can be strongly convergent.
TL;DR: In this paper, the authors considered the logistic equation −Δu = a (x) u − b (x uq) uq on all of RN with a γ > −2, τ ∈ (−∞, ∞) and showed that this problem has a unique positive solution.
Abstract: We consider the logistic equation −Δu = a (x) u − b (x) uq on all of RN with a (x)/|x|γ and b (x)/|x|τ bounded away from 0 and infinity for all large |x|, where γ > −2, τ ∈ (−∞, ∞). We show that this problem has a unique positive solution. This considerably improves some earlier results. The main new technique here is a Safonov type iteration argument. The result can also be proved by a technique introduced by Marcus and Veron, and the two different techniques are compared.
TL;DR: In this paper, the authors studied multiplicative character sums taken on the values of a nonhomogeneous Beatty sequence where α,β ∈ ℝ, and α is irrational.
Abstract: We study multiplicative character sums taken on the values of a non-homogeneous Beatty sequence where α,β ∈ ℝ, and α is irrational. In particular, our bounds imply that for every fixed e > 0, if p is sufficiently large and p½+e ≤ N ≤ p, then among the first N elements of ℬα,β, there are N/2+o(N) quadratic non-residues modulo p. When more information is available about the Diophantine properties of α, then the error term o(N) admits a sharper estimate.
TL;DR: In this paper, approximate convex functions are characterised in terms of Clarke generalised gradient, and optimality conditions for quasi efficient solutions of nonsmooth vector optimisation problems are derived.
Abstract: Approximate convex functions are characterised in terms of Clarke generalised gradient. We apply this characterisation to derive optimality conditions for quasi efficient solutions of nonsmooth vector optimisation problems. Two new classes of generalised approximate convex functions are defined and mixed duality results are obtained.
TL;DR: In this paper, the distance of a Bloch function to the little Bloch space, β 0, was studied. But the distance was not defined. And it was not shown that the distance formulas in β 0 have Bloch type spaces analogues.
Abstract: Motivated by a formula of P. Jones that gives the distance of a Bloch function to BMOA, the space of bounded mean oscillations, we obtain several formulas for the distance of a Bloch function to the little Bloch space, β0. Immediate consequences are equivalent expressions for functions in β0. We also give several examples of distances of specific functions to β0. We comment on connections between distance to β0 and the essential norm of some composition operators on the Bloch space, β. Finally we show that the distance formulas in β have Bloch type spaces analogues.
TL;DR: In this article, a new system of variational inclusions involving (H, η)-monotone operators in Hilbert spaces is introduced and studied, and the authors prove the existence and uniqueness of solutions and the convergence of some new three-step iterative algorithms for this system and its special cases.
Abstract: In this paper, We introduce and study a new system of variational inclusions involving(H, η)-monotone operators in Hilbert spaces. By using the resolvent operator method associated with (H, η)-monotone operators, we prove the existence and uniqueness of solutions and the convergence of some new three-step iterative algorithms for this system of variational inclusions and its special cases. The results in this paper extends and improves some results in the literature.
TL;DR: In this article, the existence of positive periodic solutions to the equation x = f (t, x) = f is proved based on a fixed point theorem in cones, where the nonlinearity changes sign.
Abstract: In this paper, we study the existence of positive periodic solutions to the equation x″ = f (t, x). It is proved that such a equation has more than one positive periodic solution when the nonlinearity changes sign. The proof relies on a fixed point theorem in cones.
TL;DR: New reverses of the Schwarz inequality in inner product spaces that incorporate the classical Klamkin-McLenaghan result for the case of positive n-tuples are given in this paper.
Abstract: New reverses of the Schwarz inequality in inner product spaces that incorporate the classical Klamkin-McLenaghan result for the case of positive n-tuples are given. Applications for Lebesgue integrals are also provided.
TL;DR: In this article, the authors present some of the group theoretic properties of reversing symmetry groups, and classify their structure in simple cases that occur frequently in several well-known groups of dynamical systems.
Abstract: We present some of the group theoretic properties of reversing symmetry groups, and classify their structure in simple cases that occur frequently in several well-known groups of dynamical systems.
TL;DR: In this article, the boundary spectrum of a Banach algebra is investigated, where ∂S denotes the topological boundary of the set S of all non-invertible elements of all elements of A and where a is an element of A.
Abstract: We investigate some properties of the set S∂(a) = {λ ∈ ℂ: λ – a ∈ ∂S} (which we call the boundary spectrum of a) where ∂S denotes the topological boundary of the set S of all non-invertible elements of a Banach algebra A, and where a is an element of A.
TL;DR: The length of every pair { A, B } of n × n complex matrices is at most 2 n − 2, if n ≤ 5 as mentioned in this paper, and for n ≥ 5, the (possibly empty) words in a pair {A, B} of length at most Θ(n − 2 ) span the unital algebra generated by A and B.
Abstract: The length of every pair { A, B } of n × n complex matrices is at most 2 n − 2, if n ≤ 5. That is, for n ≤ 5, the (possibly empty) words in A, B of length at most 2 n − 2 span the unital algebra generated by A, B . For every positive integer m there exist m × m complex matrices C, D such that the length of the pair {C, D} is 2 m – 2.
TL;DR: In this paper, the existence and regularity of solutions to semilinear elliptic Neumann problems are investigated, motivated by the Poisson-Boltzmann equation of biophysics and semiconductor modeling.
Abstract: The existence and regularity of solutions to semilinear elliptic Neumann problems are investigated. Motivated by the Poisson–Boltzmann equation of biophysics and semiconductor modeling, the nonlinearity is assumed to be a continuous, strictly monotone increasing function that passes through the origin with asymptotically superlinear and unbounded growth. Pseudomonotone operator theory is utilised to establish the existence and uniqueness of a weak solution in the Sobolev space W1,2. With an additional assumption on the nonlinearity, we show that this weak solution belongs to .
TL;DR: An integral representation formula in terms of the normal Gauss map for minimal surfaces in 3-dimensional solvable Lie groups with left invariant metric is obtained in this paper, where the representation is based on the left invariance metric.
Abstract: An integral representation formula in terms of the normal Gauss map for minimal surfaces in 3-dimensional solvable Lie groups with left invariant metric is obtained.
TL;DR: In this article, the authors characterize locally convex spaces whose (weakly) precompact (respectively, compact) subsets are metrisable, and apply this approach to get Cascales-Orihuela's, Valdivia's and Robertson's metrisation theorems for (pre)compact sets.
Abstract: This self-contained paper characterises those locally convex spaces whose (weakly) precompact (respectively, compact) subsets are metrisable. Applications and examples are provided. Our approach also applies to get Cascales-Orihuela's, Valdivia's and Robertson's metrisation theorems for (pre)compact sets.
TL;DR: In this paper, a nonlinear elliptic equation involving p(x)-growth conditions on a bounded domain having cylindrical symmetry was studied, and existence and multiplicity results using as main tools the mountain pass theorem of Ambosetti and Rabinowitz and Ekeland's variational principle.
Abstract: In this paper we study a nonlinear elliptic equation involving p(x) -growth conditions on a bounded domain having cylindrical symmetry. We establish existence and multiplicity results using as main tools the mountain pass theorem of Ambosetti and Rabinowitz and Ekeland's variational principle.
TL;DR: In this article, the authors characterized the n-dimensional complete spacelike hypersurfaces Mn in a de Sitter space with constant scalar curvature and two distinct principal curvatures, and showed that if the multiplicities of such principal curvature are greater than 1, then Mn is isometric to Hk (sinh r) × Sn−k (cosh r), 1 < k < n − 1.
Abstract: In this paper, we characterise the n-dimensional (n ≥ 3) complete spacelike hypersurfaces Mn in a de Sitter space with constant scalar curvature and with two distinct principal curvatures. We show that if the multiplicities of such principal curvatures are greater than 1, then Mn is isometric to Hk (sinh r) × Sn−k (cosh r), 1 < k < n − 1. In particular, when Mn is the complete spacelike hypersurfaces in with the scalar curvature and the mean curvature being linearly related, we also obtain a characteristic Theorem of such hypersurfaces.
TL;DR: In this paper, the exact value of packing measure for homogeneous Cantor sets has not yet been calculated even though that of Hausdorff measures was evaluated by Qu, Rao and Su in (2001).
Abstract: For a class of homogeneous Cantor sets, we find an explicit formula for their packing dimensions. We then turn our attention to the value of packing measures. The exact value of packing measure for homogeneous Cantor sets has not yet been calculated even though that of Hausdorff measures was evaluated by Qu, Rao and Su in (2001). We give a reasonable lower bound for the packing measures of homogeneous Cantor sets. Our results indicate that duality does not hold between Hausdorff and packing measures.
TL;DR: In this article, the authors studied the relation between Enochs' notion of envelopes and the notion of a hull of an R-module and its relation with a monomorphism.
Abstract: Let R be a ring and ℒ a class of R-modules. An R-module N is called ℒ-injective if for all L ∈ ℒ. An ℒ-injective hull of an R-module M is defined to be a homomorphism φ: M → F with F ℒ-injective such that for any monomorphism f: M → F′ with F′ ℒ-injective, there is a monomorphism g: F → F′ satisfying gφ = f. The aim of this paper is to study ℒ-injective hulls and their relations with ℒ-injective envelopes in Enochs' sense.