Showing papers in "Bulletin of The Australian Mathematical Society in 2016"
TL;DR: In this article, the endpoint regularity of the uncentred Hardy-Littlewood maximal function was proved for the local maximum points of the corresponding maximal functions, which are of interest in their own right.
Abstract: In this note we give a simple proof of the endpoint regularity for the uncentred Hardy–Littlewood maximal function on . Our proof is based on identities for the local maximum points of the corresponding maximal functions, which are of interest in their own right.
41 citations
TL;DR: In this paper, the authors give sufficient conditions for a graph to be traceable and Hamiltonian in terms of the Wiener index and the complement of the graph, which correct and extend the result of Yang.
Abstract: We give sufficient conditions for a graph to be traceable and Hamiltonian in terms of the Wiener index and the complement of the graph, which correct and extend the result of Yang [‘Wiener index and traceable graphs’, Bull. Aust. Math. Soc. 88 (2013), 380–383]. We also present sufficient conditions for a bipartite graph to be traceable and Hamiltonian in terms of its Wiener index and quasicomplement. Finally, we give sufficient conditions for a graph or a bipartite graph to be traceable and Hamiltonian in terms of its distance spectral radius.
28 citations
TL;DR: For any positive integer, this article showed that there are infinitely many real quadratic fields that do not admit -ary universal quadrastic forms (without any restriction on the parity of their cross coefficients).
Abstract: For any positive integer we show that there are infinitely many real quadratic fields that do not admit -ary universal quadratic forms (without any restriction on the parity of their cross coefficients).
25 citations
TL;DR: In this paper, the rank and idempotent rank of a semigroup E(X,P) generated by the idempots of the semigroup T (X and P) were derived.
Abstract: We calculate the rank and idempotent rank of the semigroup E(X,P) generated by the idempotents of the semigroup T (X,P), which consists of all transformations of the finite set X preserving a non-uniform partition P. We also classify and enumerate the idempotent generating sets of this minimal possible size. This extends results of the first two authors in the uniform case.
19 citations
TL;DR: In this article, the density of the set of coprime pairs of algebraic integers in a number field with ring of integers is shown to be 1/{it\\zeta}_{K}(m), where k is the Dedekind zeta function of the number field.
Abstract: Let $K$ be a number field with ring of integers ${\\mathcal{O}}$. After introducing a suitable notion of density for subsets of ${\\mathcal{O}}$, generalising the natural density for subsets of $\\mathbb{Z}$, we show that the density of the set of coprime $m$-tuples of algebraic integers is $1/{\\it\\zeta}_{K}(m)$, where ${\\it\\zeta}_{K}$ is the Dedekind zeta function of $K$. This generalises a result found independently by Mertens [‘Ueber einige asymptotische Gesetze der Zahlentheorie’, J. reine angew. Math. 77 (1874), 289–338] and Cesàro [‘Question 75 (solution)’, Mathesis 3 (1883), 224–225] concerning the density of coprime pairs of integers in $\\mathbb{Z}$.
18 citations
TL;DR: For each positive, let denote the identity obtained from the Adjan identity by substituting and. as discussed by the authors showed that every monoid which satisfies for each positive and generates a variety containing the bicyclic monoid is non-finitely based.
Abstract: For each positive , let denote the identity obtained from the Adjan identity by substituting and . We show that every monoid which satisfies for each positive and generates a variety containing the bicyclic monoid is nonfinitely based. This implies that the monoid (respectively, ) of two-by-two upper triangular tropical matrices over the tropical semiring (respectively, ) is nonfinitely based.
17 citations
TL;DR: In this paper, it was shown that each 3-Jordan homomorphism between Banach algebras is a 3-homomorphism, under special hypotheses, and that any 3-jordanomorphism is a 2-JH.
Abstract: We show that, under special hypotheses, each 3-Jordan homomorphism ${\\it\\varphi}$ between Banach algebras ${\\mathcal{A}}$ and ${\\mathcal{B}}$ is a 3-homomorphism.
15 citations
TL;DR: In this paper, the authors give three identities involving multiple zeta values of height one and of maximal height; an explicit formula for the height-one multiple-zeta values, a regularized sum formula, and a sum formula for maximal height.
Abstract: We give three identities involving multiple zeta values of height one and of maximal height; an explicit formula for the height-one multiple zeta values, a regularized sum formula, and a sum formula for the multiple zeta values of maximal height.
15 citations
TL;DR: In this article, the authors examined topological states of matter from the perspective of non-commutative geometry and KK-theory, including the quantum Hall effect and topological insulators, and showed that the Hall conductance can be expressed in terms of the index pairing of the Fermi projection of disordered Hamiltonians with a spectral triple encoding the geometry of the sample's momentum space.
Abstract: This thesis examines topological states of matter from the perspective of noncommutative geometry and KK-theory. Examples of such topological states of matter include the quantum Hall effect and topological insulators. For the quantum Hall effect, we consider a continuous model and show that the Hall conductance can be expressed in terms of the index pairing of the Fermi projection of a disordered Hamiltonian with a spectral triple encoding the geometry of the sample’s momentum space. The presence of a magnetic field means that noncommutative algebras and methods must be employed. Higher dimensional analogues of the quantum Hall system are also considered, where the index pairing produces the ‘higher-dimensional Chern numbers’ in the continuous setting. Next we consider a discrete quantum Hall system with an edge. We show that topological properties of observables concentrated at the boundary can be linked to invariants from a boundary-free model via the Kasparov product. Hence we obtain the bulk-edge correspondence of the quantum Hall effect in the language of KK-theory. Finally we consider topological insulators, which come from imposing (possibly anti-linear) symmetries on condensed-matter systems and studying the invariants that are protected by these symmetries. We show how symmetry data can be linked to classes in real or complex KK-theory. Finally we prove the bulk-edge correspondence for topological insulator systems by linking bulk and edge systems using the Kasparov product in KKO-theory.
14 citations
TL;DR: In this paper, the authors deal with Sobolev inequalities for Riesz potentials of functions in Lebesgue spaces of variable exponents near Soboleve's exponent over non-dubling metric measure spaces.
Abstract: Our aim in this paper is to deal with Sobolev inequalities for Riesz potentials of functions in Lebesgue spaces of variable exponents near Sobolev’s exponent over nondoubling metric measure spaces.
13 citations
TL;DR: In this article, it was shown that the finite classical simple groups up to dimension 7 can be generated for all values of $q$€€€ 1/$€ 2€ 0.
Abstract: In this paper, we prove that the finite simple groups $\text{PSp}_{6}(q)$
, ${\rm\Omega}_{7}(q)$
and $\text{PSU}_{7}(q^{2})$
are $(2,3)$
-generated for all $q$
. In particular, this result completes the classification of the $(2,3)$
-generated finite classical simple groups up to dimension 7.
TL;DR: In this article, a class of nonlinear mappings which is properly larger than the class of nonexpansive mappings is defined, and a fixed point theorem for this new class of mappings has been proved.
Abstract: We define a class of nonlinear mappings which is properly larger than the class of nonexpansive mappings. We also give a fixed point theorem for this new class of mappings.
TL;DR: In this article, the authors proved the Brück conjecture for a nonconstant entire function with hyper-order (i.e., the derivative of the function is a small function with respect to the entire function) by studying the infinite hyperorder solutions of the linear differential equations.
Abstract: The Brück conjecture states that if a nonconstant entire function $f$ with hyper-order ${\\it\\sigma}_{2}(f)\\in [0,+\\infty )\\setminus \\mathbb{N}$ shares one finite value $a$ (counting multiplicities) with its derivative $f^{\\prime }$, then $f^{\\prime }-a=c(f-a)$, where $c$ is a nonzero constant. The conjecture has been established for entire functions with order ${\\it\\sigma}(f)<+\\infty$ and hyper-order ${\\it\\sigma}_{2}(f)<{\\textstyle \\frac{1}{2}}$. The purpose of this paper is to prove the Brück conjecture for the case ${\\it\\sigma}_{2}(f)=\\frac{1}{2}$ by studying the infinite hyper-order solutions of the linear differential equations $f^{(k)}+A(z)f=Q(z)$. The shared value $a$ is extended to be a ‘small’ function with respect to the entire function $f$.
TL;DR: In this article, the authors studied the Mazur-Ulam property of the sum of two strictly convex Banach spaces and gave an equivalent form of the isometric extension problem and two equivalent conditions to decide whether all such spaces admit the Ulam property.
Abstract: In this article, we study the Mazur–Ulam property of the sum of two strictly convex Banach spaces. We give an equivalent form of the isometric extension problem and two equivalent conditions to decide whether all strictly convex Banach spaces admit the Mazur–Ulam property. We also find necessary and sufficient conditions under which the -sum of two strictly convex Banach spaces admit the Mazur–Ulam property.
TL;DR: In this paper, the authors used three different types of deterministic compartmental epidemic models to investigate the transmission dynamics of Severe Acute Respiratory Syndrome (SARS), with the inclusion of a reaction-diffusion system.
Abstract: This thesis is based on using three different types of deterministic compartmental epidemic models to investigate the transmission dynamics of Severe Acute Respiratory Syndrome (SARS). These models are represented by ordinary and partial differential equations, with the inclusion of a reaction-diffusion system. The first model assumes Susceptible-Exposed-Infected-Diagnosed-Recovered (SEIJR) populations, whereas the second and third models are extensions of the first model, with treatment and quarantine compartments added. For different initial population distributions the system of differential equations, representing different compartments, has been solved in the presence of diffusion in the first three compartments i.e susceptible, exposed and infected. In this study the effects of diffusion on SARS transmission are investigated as are the effects of some intervention strategies. It is shown that diffusion and initial population distribution play crucial roles in disease transmission. Then, the same system is solved numerically with diffusion in the susceptible, exposed and infected compartments and with cross-diffusion in the susceptible and exposed compartments for different cases. Using clinical and demographic information for SARS, the SEIJR model is further extended to a Susceptible-Exposed-Infected-Diagnosed-Treated-Recovered (SEIJTR) model. The SEIJTR model’s parameters are estimated, again using available field data on the 2003 SARS epidemic in Hong Kong. After that, model parameters are analysed for sensitivity and uncertainty. Three different techniques are used to perform the sensitivity analysis. The effect of the treatment compartment on SARS transmission is then numerically studied. Stability analyses of steady state and treatment-reduced basic reproduction number are performed. Studies show that availability of treatment can reduce infection significantly. Finally, a quarantine compartment is added to the SEIJTR model in order to study the effects of quarantine on SARS transmission. The results for this extended SEQIJTR model are compared with those for the SEIJTR model in which only isolation and treatment but no quarantine measures, are used as intervention. The investigations show that the presence of quarantine measures effectively reduces disease transmission.
TL;DR: In this paper, a uniqueness theorem for finite-order meromorphic solutions to a class of difference equations of the Malmquist type was proved, and the uniqueness theorem was later proved for finite order meromorphic solution to a set of differentials of the same type.
Abstract: In this note, we prove a uniqueness theorem for finite-order meromorphic solutions to a class of difference equations of Malmquist type. Such solutions .
TL;DR: In this article, an explicit piecewise-polynomial closed form for the probability density function of the distance traveled by a uniform random walk in an odd-dimensional space was derived.
Abstract: We derive an explicit piecewise-polynomial closed form for the probability density function of the distance traveled by a uniform random walk in an odd-dimensional space, based on recent work of Borwein, Straub, and Vignat [1] and by R. Garcia-Pelayo [3].
TL;DR: In this article, the authors proved an asymptotic formula for the sum of the number of divisors of a divisor in the problem of finding a solution to a given problem.
Abstract: We prove an asymptotic formula for the sum $\sum_{n \leq N} d(n^2 - 1)$, where $d(n)$ denotes the number of divisors of $n$. During the course of our proof, we also furnish an asymptotic formula for the sum $\sum_{d \leq N} g(d)$, where $g(d)$ denotes the number of solutions $x$ in $\mathbb{Z}_d$ to the equation $x^2 \equiv 1 \mod d$.
TL;DR: In this article, the theory of reproducing kernel Hilbert spaces and the pair correlation formula constructed by Chandee et al. were used to construct simple zeros of primitive Dirichlet functions under the assumption of generalised Riemann hypothesis.
Abstract: In this paper, by using the theory of reproducing kernel Hilbert spaces and the pair correlation formula constructed by Chandee et al. [‘Simple zeros of primitive Dirichlet -functions are simple in a proper sense, under the assumption of the generalised Riemann hypothesis.
TL;DR: In this article, the authors count a dual set of Stirling permutations by the number of alternating runs and study properties of the generating functions, including recurrence relations, grammatical interpretations and convolution formulas.
Abstract: In this paper, we count a dual set of Stirling permutations by the number of alternating runs and study properties of the generating functions, including recurrence relations, grammatical interpretations and convolution formulas.
TL;DR: In this article, it was shown that finite subset spaces of a metric space can form a nested sequence under natural isometric embeddings, and that this sequence admits Lipschitz retractions when the space is a Hilbert space.
Abstract: Finite subset spaces of a metric space $X$
form a nested sequence under natural isometric embeddings $X=X(1)\subset X(2)\subset \cdots \,$
. We prove that this sequence admits Lipschitz retractions $X(n)\rightarrow X(n-1)$
when $X$
is a Hilbert space.
TL;DR: This work uses steady-state analysis to investigate how the amount of sludge formed depends upon the residence time and the use of a settling unit, and shows that there are five generic response diagrams, depending upon the value of the effective recycle parameter, which are desirable.
Abstract: The activated sludge process is one of the major aerobic processes used in the biological treatment of wastewater. A significant drawback of this process is the production of excess sludge, the disposal of which can account for 50-60% of the running costs of a plant. Thus there is a growing interest in methods that reduce the volume and mass of excess sludge produced as part of biological wastewater treatment processes. In practice a target value is often set for the sludge content inside the bioreactor. If the sludge content is higher than the target value, the process is stopped and the reactor is cleaned. This is undesirable as it increases running costs. In chapter 2 we investigate a simple model for the activated sludge process in which the influent contains a mixture of soluble and biodegradable particulate substrate. Within the bioreactor the biodegradable particulate substrate is hydrolyzed to form soluble substrate. The soluble organics are used for energy and growth by the biomass. Biomass decay produces soluble substrate in addition to inert material. We use steady-state analysis to investigate how the amount of sludge formed depends upon the residence time and the use of a settling unit. We show that when the steady-state sludge content is plotted as a function of the residence time that there are five generic response diagrams, depending upon the value of the effective recycle parameter. Four of them are desirable because the sludge content is below the target value if the residence time is higher than some critical value that is not ‘too large’ in practice. In chapter 3 we investigate how the volume and mass of excess sludge produced by the activated sludge process can be reduced by coupling the bioreactor used in the process to a sludge disintegration unit. In chapter 4 a seemingly minor modification is made to the model in chapter 2.
TL;DR: In this paper, it was shown that the generalized Lau product is isomorphic as a Banach algebra to the usual direct product, and also correct some misleading claims made about the relationship between this generalized product and an older construction of Monfared (Studia Mathematica, 2007).
Abstract: Several papers have, as their raison d'etre, the exploration of the generalized Lau product associated to a homomorphism $T:B\to A$ of Banach algebras. In this short note, we demonstrate that the generalized Lau product is isomorphic as a Banach algebra to the usual direct product $A\oplus B$. We also correct some misleading claims made about the relationship between this generalized Lau product, and an older construction of Monfared (Studia Mathematica, 2007).
TL;DR: Projection and reflection algorithms as mentioned in this paper exploit relatively simpler structure in each of the individual constraint sets from the collection, which can be used to solve the feasibility problem which asks for a point in the intersection of a collection of constraint sets.
Abstract: This thesis investigates the family of so-called projection and reflection methods. These methods form the basis for a class of iterative algorithms which can be used to solve the feasibility problem which asks for a point in the intersection of a collection of constraint sets. Many optimisation and reconstruction problems can be profitably modelled within this framework, although the formulation is not always immediately obvious. In a typical feasibility problem the target intersection set is difficult to deal with directly. Projection and reflection algorithms overcome this difficulty by exploiting relatively simpler structure in each of the individual constraint sets from the collection.
TL;DR: In this paper, the number of points on the Dwork hypersurface is expressed in terms of McCarthy's $p$ -adic hypergeometric function for any odd prime.
Abstract: We express the number of points on the Dwork hypersurface $X_{\\unicode[STIX]{x1D706}}^{d}:x_{1}^{d}+x_{2}^{d}+\\cdots +x_{d}^{d}=d\\unicode[STIX]{x1D706}x_{1}x_{2}\\cdots x_{d}$ over a finite field of order $q\
ot \\equiv 1\\,(\\text{mod}\\,d)$ in terms of McCarthy’s $p$ -adic hypergeometric function for any odd prime $d$ .
TL;DR: In this paper, it was shown that a non-cyclotomic totally complex number field without a unit primitive element can be generated by a reciprocal integer if and only if the Galois group of the normal closure is contained in the hyperoctahedral group $B_{d}$¯¯¯¯.
Abstract: We characterise number fields without a unit primitive element, and we exhibit some families of such fields with low degree. Also, we prove that a noncyclotomic totally complex number field $K$
, with degree $2d$
where $d$
is odd, and having a unit primitive element, can be generated by a reciprocal integer if and only if $K$
is not CM and the Galois group of the normal closure of $K$
is contained in the hyperoctahedral group $B_{d}$
.
TL;DR: Bourgain and Shparlinski as mentioned in this paper showed that for almost all primes, there is a multiple that can be written in binary as with (corresponding to Hamming weight seven) and infinitely many primes with a multiplicative subgroup, for some, of size, where the sumproduct set does not cover completely.
Abstract: We improve recent results of Bourgain and Shparlinski to show that, for almost all primes , there is a multiple that can be written in binary as with (corresponding to Hamming weight seven). We also prove that there are infinitely many primes with a multiplicative subgroup , for some , of size , where the sum–product set does not cover completely.
TL;DR: In this article, the authors introduce some refinements of numerical radius inequalities for Hilbert space invertible operators, and prove that if is an invertable operator then, then.
Abstract: We introduce some new refinements of numerical radius inequalities for Hilbert space invertible operators. More precisely, we prove that if is an invertible operator, then .