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 Bulletin of the Australian Mathematical SocietyJournal Prestige (SJR): 0.44 Number of Followers: 2      Subscription journal ISSN (Print) 0004-9727 - ISSN (Online) 1755-1633 Published by Cambridge University Press  [387 journals]
• BAZ volume 102 Issue 1 Cover and Front matter
• PubDate: 2020-08-01T00:00:00.000Z
DOI: 10.1017/S0004972719001527
Issue No: Vol. 102, No. 1 (2020)

• BAZ volume 102 Issue 1 Cover and Back matter
• PubDate: 2020-08-01T00:00:00.000Z
DOI: 10.1017/S0004972719001539
Issue No: Vol. 102, No. 1 (2020)

• A NOTE ON NORMALISED HEAT DIFFUSION FOR GRAPHS
• Authors: BOGDAN NICA
Pages: 1 - 6
Abstract: We show that, on graphs which have precisely three distinct Laplacian eigenvalues, heat diffusion enjoys a monotonic behaviour.
PubDate: 2020-08-01T00:00:00.000Z
DOI: 10.1017/S0004972719001102
Issue No: Vol. 102, No. 1 (2020)

• SINGULARITY OF ORIENTED GRAPHS FROM SEVERAL CLASSES
• Authors: XIAOXUAN CHEN; JING YANG, XIANYA GENG, LONG WANG
Pages: 7 - 14
Abstract: A digraph is called oriented if there is at most one arc between two distinct vertices. An oriented graph $D$ is nonsingular if its adjacency matrix $A(D)$ is nonsingular. We characterise all nonsingular oriented graphs from three classes: graphs in which cycles are vertex disjoint, graphs in which all cycles share exactly one common vertex and graphs formed by cycles sharing a common path. As a straightforward corollary, the singularity of oriented bicyclic graphs is determined.
PubDate: 2020-08-01T00:00:00.000Z
DOI: 10.1017/S0004972719001151
Issue No: Vol. 102, No. 1 (2020)

• FROM TOPOLOGIES OF A SET TO SUBRINGS OF ITS POWER SET
• Authors: ALI JABALLAH; NOÔMEN JARBOUI
Pages: 15 - 20
Abstract: Let $X$ be a nonempty set and ${\mathcal{P}}(X)$ the power set of $X$ . The aim of this paper is to identify the unital subrings of ${\mathcal{P}}(X)$ and to compute its cardinality when it is finite. It is proved that any topology $\unicode[STIX]{x1D70F}$ on $X$ such that $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}^{c}$ , where $\unicode[STIX]{x1D70F}^{c}=\{U^{c}\mid U\in \unicode[STIX]{x1D70F}\}$ , is a unital subring of ${\mathcal{P}}(X)$ . It is also shown that $X$ is finite if and only if any unital subring of ${\mathcal{P}}(X)$ is a topology
PubDate: 2020-08-01T00:00:00.000Z
DOI: 10.1017/S0004972720000015
Issue No: Vol. 102, No. 1 (2020)

• SOME REMARKS ON MINIMAL ASYMPTOTIC BASES OF ORDER THREE
• Authors: DENGRONG LING; MIN TANG
Pages: 21 - 30
Abstract: We study a question on minimal asymptotic bases asked by Nathanson [‘Minimal bases and powers of 2’, Acta Arith. 49 (1988), 525–532].
PubDate: 2020-08-01T00:00:00.000Z
DOI: 10.1017/S0004972719001345
Issue No: Vol. 102, No. 1 (2020)

• LOWER BOUNDS FOR PERIODS OF DUCCI SEQUENCES
• Authors: FLORIAN BREUER; IGOR E. SHPARLINSKI
Pages: 31 - 38
Abstract: A Ducci sequence is a sequence of integer $n$ -tuples obtained by iterating the map $$\begin{eqnarray}D:(a_{1},a_{2},\ldots ,a_{n})\mapsto (|a_{1}-a_{2}|,|a_{2}-a_{3}|,\ldots ,|a_{n}-a_{1}|).\end{eqnarray}$$ Such a sequence is eventually periodic and we denote by $P(n)$ the maximal period of such sequences for given odd $n$ . We prove a lower bound for $P(n)$ by counting certain partitions. We then estimate the size of these partitions via the multiplicative order of two modulo $n$ .
PubDate: 2020-08-01T00:00:00.000Z
DOI: 10.1017/S0004972719001187
Issue No: Vol. 102, No. 1 (2020)

• ++++++++ ++++++++ ++++++++++++ ++++++++++++++++$n$ ++++++++++++ ++++++++ +++++BY+LINEAR+COMBINATIONS+OF+FOUR+TRIANGULAR+NUMBERS&rft.title=Bulletin+of+the+Australian+Mathematical+Society&rft.issn=0004-9727&rft.date=2020&rft.volume=102&rft.spage=39&rft.epage=49&rft.aulast=SUN&rft.aufirst=ZHI-HONG&rft.au=ZHI-HONG+SUN&rft_id=info:doi/10.1017/S0004972719001400">TRANSFORMATION FORMULAS FOR THE NUMBER OF REPRESENTATIONS OF $n$ BY LINEAR
COMBINATIONS OF FOUR TRIANGULAR NUMBERS
• Authors: ZHI-HONG SUN
Pages: 39 - 49
Abstract: Let $\mathbb{Z}$ and $\mathbb{Z}^{+}$ be the set of integers and the set of positive integers, respectively. For $a,b,c,d,n\in \mathbb{Z}^{+}$ , let $t(a,b,c,d;n)$ be the number of representations of $n$ by $\frac{1}{2}ax(x+1)+\frac{1}{2}by(y+1)+\frac{1}{2}cz(z+1)+\frac{1}{2}dw(w+1)$ with $x,y,z,w\in \mathbb{Z}$ . Using theta function identities we prove 13 transformation formulas for $t(a,b,c,d;n)$ and evaluate $t(2,3,3,8;n)$ , $t(1,1,6,24;n)$ and $t(1,1,6,8;n)$ .
PubDate: 2020-08-01T00:00:00.000Z
DOI: 10.1017/S0004972719001400
Issue No: Vol. 102, No. 1 (2020)

• ++++++++ ++++++++ ++++++++++++ ++++++++++++++++$B(q)$ ++++++++++++ ++++++++ ++++&rft.title=Bulletin+of+the+Australian+Mathematical+Society&rft.issn=0004-9727&rft.date=2020&rft.volume=102&rft.spage=50&rft.epage=58&rft.aulast=MAO&rft.aufirst=RENRONG&rft.au=RENRONG+MAO&rft_id=info:doi/10.1017/S0004972719001175">ARITHMETIC PROPERTIES OF COEFFICIENTS OF THE MOCK THETA FUNCTION $B(q)$
• Authors: RENRONG MAO
Pages: 50 - 58
Abstract: We investigate the arithmetic properties of the second-order mock theta function $B(q)$ and establish two identities for the coefficients of this function along arithmetic progressions. As applications, we prove several congruences for these coefficients.
PubDate: 2020-08-01T00:00:00.000Z
DOI: 10.1017/S0004972719001175
Issue No: Vol. 102, No. 1 (2020)

• INFINITE FAMILIES OF CONGRUENCES FOR OVERPARTITIONS WITH RESTRICTED ODD
DIFFERENCES
• Authors: BERNARD L. S. LIN; JIAN LIU, ANDREW Y. Z. WANG, JIEJUAN XIAO
Pages: 59 - 66
Abstract: Let $\overline{t}(n)$ be the number of overpartitions in which (i) the difference between successive parts may be odd only if the larger part is overlined and (ii) if the smallest part is odd then it is overlined. Ramanujan-type congruences for $\overline{t}(n)$ modulo small powers of $2$ and $3$ have been established. We present two infinite families of congruences modulo $5$ and $27$ for $\overline{t}(n)$ , the first of which generalises a recent result of Chern and Hao [‘Congruences for two restricted overpartitions’, Proc. Math. Sci. 129 (2019), Article 31].
PubDate: 2020-08-01T00:00:00.000Z
DOI: 10.1017/S0004972719001254
Issue No: Vol. 102, No. 1 (2020)

• AXIOMATISABILITY OF THE CLASS OF MONOLITHIC GROUPS IN A VARIETY OF
NILPOTENT GROUPS
• Authors: JOSHUA T. GRICE
Pages: 67 - 76
Abstract: The class of all monolithic (that is, subdirectly irreducible) groups belonging to a variety generated by a finite nilpotent group can be axiomatised by a finite set of elementary sentences.
PubDate: 2020-08-01T00:00:00.000Z
DOI: 10.1017/S0004972719001631
Issue No: Vol. 102, No. 1 (2020)

• ON A CONJECTURE ON THE PERMUTATION CHARACTERS OF FINITE PRIMITIVE GROUPS
• Authors: PABLO SPIGA
Pages: 77 - 90
Abstract: Let $G$ be a finite group with two primitive permutation representations on the sets $\unicode[STIX]{x1D6FA}_{1}$ and $\unicode[STIX]{x1D6FA}_{2}$ and let $\unicode[STIX]{x1D70B}_{1}$ and $\unicode[STIX]{x1D70B}_{2}$ be the corresponding permutation characters. We consider the case in which the set of fixed-point-free elements of $G$ on $\unicode[STIX]{x1D6FA}_{1}$ coincides with the set of fixed-point-free elements of $G$ on $\unicode[STIX]{x1D6FA}_{2}$ , that is, for every $g\in G$ , $\unicode[STIX]{x1D70B}_{1}(g)=0$ if and only if
PubDate: 2020-08-01T00:00:00.000Z
DOI: 10.1017/S0004972719001060
Issue No: Vol. 102, No. 1 (2020)

• CHARACTER GRAPHS WITH NONBIPARTITE HAMILTONIAN COMPLEMENT
• Authors: MAHDI EBRAHIMI
Pages: 91 - 95
Abstract: For a finite group $G$ , let $\unicode[STIX]{x1D6E5}(G)$ denote the character graph built on the set of degrees of the irreducible complex characters of $G$ . In this paper, we obtain a necessary and sufficient condition which guarantees that the complement of the character graph $\unicode[STIX]{x1D6E5}(G)$ of a finite group $G$ is a nonbipartite Hamiltonian graph.
PubDate: 2020-08-01T00:00:00.000Z
DOI: 10.1017/S0004972719001163
Issue No: Vol. 102, No. 1 (2020)

• GROUPS WHOSE NONNORMAL SUBGROUPS ARE METAHAMILTONIAN
• Authors: DARIO ESPOSITO; FRANCESCO DE GIOVANNI, MARCO TROMBETTI
Pages: 96 - 103
Abstract: If $\mathfrak{X}$ is a class of groups, we define a sequence $\mathfrak{X}_{1},\mathfrak{X}_{2},\ldots ,\mathfrak{X}_{k},\ldots$ of group classes by putting $\mathfrak{X}_{1}=\mathfrak{X}$ and choosing $\mathfrak{X}_{k+1}$ as the class of all groups whose nonnormal subgroups belong to $\mathfrak{X}_{k}$ . In particular, if $\mathfrak{A}$ is the class of abelian groups,  $\mathfrak{A}_{2}$ is the class of metahamiltonian groups, that is, groups whose nonnormal subgroups are abelian. The aim of this paper is to study the structure of $\mathfrak{X}_{k}$ -groups, with special emphasis on the case  $\mathfrak{X}=\mathfrak{A}$ . Among other results, it will be proved that a group has a finite commutator subgroup if and only if it is locally graded and belongs to $\mathfrak{A}_{k}$ for some positive integer
PubDate: 2020-08-01T00:00:00.000Z
DOI: 10.1017/S0004972719001047
Issue No: Vol. 102, No. 1 (2020)

• ODD–EVEN DECOMPOSITION OF FUNCTIONS
• Authors: IOSIF PINELIS
Pages: 104 - 108
Abstract: The main result of this note implies that any function from the product of several vector spaces to a vector space can be uniquely decomposed into the sum of mutually orthogonal functions that are odd in some of the arguments and even in the other arguments. Probabilistic notions and facts are employed to simplify statements and proofs.
PubDate: 2020-08-01T00:00:00.000Z
DOI: 10.1017/S0004972719001394
Issue No: Vol. 102, No. 1 (2020)

• QUASICONFORMAL HARMONIC MAPPINGS BETWEEN DOMAINS CONTAINING INFINITY
• Authors: DAVID KALAJ
Pages: 109 - 117
Abstract: Assume that $\unicode[STIX]{x1D6FA}$ and $D$ are two domains with compact smooth boundaries in the extended complex plane $\overline{\mathbf{C}}$ . We prove that every quasiconformal mapping between $\unicode[STIX]{x1D6FA}$ and $D$ mapping $\infty$ onto itself is bi-Lipschitz continuous with respect to both the Euclidean and Riemannian metrics.
PubDate: 2020-08-01T00:00:00.000Z
DOI: 10.1017/S0004972719001278
Issue No: Vol. 102, No. 1 (2020)

• ON A LATTICE GENERALISATION OF THE LOGARITHM AND A DEFORMATION OF THE
DEDEKIND ETA FUNCTION
• Authors: LAURENT BÉTERMIN
Pages: 118 - 125
Abstract: We consider a deformation $E_{L,\unicode[STIX]{x1D6EC}}^{(m)}(it)$ of the Dedekind eta function depending on two $d$ -dimensional simple lattices $(L,\unicode[STIX]{x1D6EC})$ and two parameters $(m,t)\in (0,\infty )$ , initially proposed by Terry Gannon. We show that the minimisers of the lattice theta function are the maximisers of $E_{L,\unicode[STIX]{x1D6EC}}^{(m)}(it)$ in the space of lattices with fixed density. The proof is based on the study of a lattice generalisation of the logarithm, called the lattice logarithm, also defined by Terry Gannon. We also prove that the natural logarithm is characterised by a variational problem over a class of one-dimensional lattice logarithms.
PubDate: 2020-08-01T00:00:00.000Z
DOI: 10.1017/S000497272000012X
Issue No: Vol. 102, No. 1 (2020)

• ON THE HYPERSTABILITY OF THE DRYGAS FUNCTIONAL EQUATION ON A RESTRICTED
DOMAIN
• Authors: JEDSADA SENASUKH; SATIT SAEJUNG
Pages: 126 - 137
Abstract: We prove hyperstability results for the Drygas functional equation on a restricted domain (a certain subset of a normed space). Our results are more general than the ones proposed by Aiemsomboon and Sintunavarat [‘Two new generalised hyperstability results for the Drygas functional equation’, Bull. Aust. Math. Soc.95 (2017), 269–280] and our proof does not rely on the fixed point theorem of Brzdęk as was the case there. A characterisation of the Drygas functional equation in terms of its asymptotic behaviour is given. Several examples are given to illustrate our generalisations. Finally, we point out a misleading statement in the proof of the second result in the paper by Aiemsomboon and Sintunavarat and propose its correction.
PubDate: 2020-08-01T00:00:00.000Z
DOI: 10.1017/S0004972719001096
Issue No: Vol. 102, No. 1 (2020)

• ARENS PRODUCTS, ARENS REGULARITY AND RELATED PROBLEMS
• Authors: RUKI MATSUI; YUJI TAKAHASHI
Pages: 138 - 150
Abstract: We study the second dual algebra of a Banach algebra and related problems. We resolve some questions raised by Ülger, which are related to Arens products. We then discuss a question of Gulick on the radical of the second dual algebra of the group algebra of a discrete abelian group and give an application of Arens regularity to Fourier and Fourier–Stieltjes transforms.
PubDate: 2020-08-01T00:00:00.000Z
DOI: 10.1017/S0004972719001114
Issue No: Vol. 102, No. 1 (2020)

• ++++++++ ++++++++ ++++++++++++ ++++++++++++++++$L^{p}$ ++++++++++++ ++++++++ ++++-SPACES&rft.title=Bulletin+of+the+Australian+Mathematical+Society&rft.issn=0004-9727&rft.date=2020&rft.volume=102&rft.spage=151&rft.epage=161&rft.aulast=LO&rft.aufirst=CHING-ON&rft.au=CHING-ON+LO&rft.au=ANTHONY+WAI-KEUNG+LOH&rft_id=info:doi/10.1017/S000497271900159X">COMPACT WEIGHTED COMPOSITION OPERATORS BETWEEN $L^{p}$ -SPACES
• Authors: CHING-ON LO; ANTHONY WAI-KEUNG LOH
Pages: 151 - 161
Abstract: We provide complete characterisations for the compactness of weighted composition operators between two distinct $L^{p}$ -spaces, where $1\leq p\leq \infty$ . As a corollary, when the underlying measure space is nonatomic, the only compact weighted composition map between $L^{p}$ -spaces is the zero operator.
PubDate: 2020-08-01T00:00:00.000Z
DOI: 10.1017/S000497271900159X
Issue No: Vol. 102, No. 1 (2020)

• ANCIENT SOLUTIONS OF CODIMENSION TWO SURFACES WITH CURVATURE PINCHING
• Authors: ZHENGCHAO JI
Pages: 162 - 171
Abstract: We prove rigidity theorems for ancient solutions of geometric flows of immersed submanifolds. Specifically, we find conditions on the second fundamental form that characterise the shrinking sphere among compact ancient solutions for the mean curvature flow in codimension two surfaces.
PubDate: 2020-08-01T00:00:00.000Z
DOI: 10.1017/S0004972720000179
Issue No: Vol. 102, No. 1 (2020)

• ASSESSING HEALTHCARE PROVIDERS’ PERFORMANCE WITH AND WITHOUT RISK
• Authors: DANIEL MAURICIO MORALES-SILVA
Pages: 172 - 173
PubDate: 2020-08-01T00:00:00.000Z
DOI: 10.1017/S0004972720000398
Issue No: Vol. 102, No. 1 (2020)

• FINITE-ELEMENT METHODS AND MULTI-FIELD APPLICATIONS
Pages: 174 - 176
PubDate: 2020-08-01T00:00:00.000Z
DOI: 10.1017/S0004972720000416
Issue No: Vol. 102, No. 1 (2020)

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