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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  [386 journals]
• BAZ volume 101 Issue 2 Cover and Front matter
• PubDate: 2020-04-01T00:00:00.000Z
DOI: 10.1017/S0004972719001485
Issue No: Vol. 101, No. 2 (2020)

• BAZ volume 101 Issue 2 Cover and Back matter
• PubDate: 2020-04-01T00:00:00.000Z
DOI: 10.1017/S0004972719001497
Issue No: Vol. 101, No. 2 (2020)

• ++++++++ ++++++++ ++++++++++++ ++++++++++++++++$K_{16}$ ++++++++++++ ++++++++ ++++&rft.title=Bulletin+of+the+Australian+Mathematical+Society&rft.issn=0004-9727&rft.date=2020&rft.volume=101&rft.spage=177&rft.epage=185&rft.aulast=GILL&rft.aufirst=MICHAEL&rft.au=MICHAEL+J.+GILL&rft.au=IAN+M.+WANLESS&rft_id=info:doi/10.1017/S0004972719000856">PERFECT 1-FACTORISATIONS OF $K_{16}$
• Authors: MICHAEL J. GILL; IAN M. WANLESS
Pages: 177 - 185
Abstract: We report the results of a computer enumeration that found that there are 3155 perfect 1-factorisations (P1Fs) of the complete graph  $K_{16}$ . Of these, 89 have a nontrivial automorphism group (correcting an earlier claim of 88 by Meszka and Rosa [‘Perfect 1-factorisations of $K_{16}$ with nontrivial automorphism group’, J. Combin. Math. Combin. Comput. 47 (2003), 97–111]). We also (i) describe a new invariant which distinguishes between the P1Fs of  $K_{16}$ , (ii) observe that the new P1Fs produce no atomic Latin squares of order 15 and (iii) record P1Fs for a number of large orders that exceed prime powers by one.
PubDate: 2020-04-01T00:00:00.000Z
DOI: 10.1017/S0004972719000856
Issue No: Vol. 101, No. 2 (2020)

• ++++++++ ++++++++ ++++++++++++ ++++++++++++++++$W$ ++++++++++++ ++++++++ ++++-OPERATOR+AND+NONCROSSING PARTITIONS&rft.title=Bulletin+of+the+Australian+Mathematical+Society&rft.issn=0004-9727&rft.date=2020&rft.volume=101&rft.spage=186&rft.epage=200&rft.aulast=SUN&rft.aufirst=HAO&rft.au=HAO+SUN&rft_id=info:doi/10.1017/S0004972719001084">DEGREE OF THE $W$ -OPERATOR AND NONCROSSING PARTITIONS
• Authors: HAO SUN
Pages: 186 - 200
Abstract: The $W$ -operator, $W([n])$ , generalises the cut-and-join operator. We prove that $W([n])$ can be written as the sum of $n!$ terms, each term corresponding uniquely to a permutation in $S_{\!n}$ . We also prove that there is a correspondence between the terms of $W([n])$ with maximal degree and noncrossing partitions.
PubDate: 2020-04-01T00:00:00.000Z
DOI: 10.1017/S0004972719001084
Issue No: Vol. 101, No. 2 (2020)

• THE NUMBER OF CYCLIC SUBGROUPS OF FINITE ABELIAN GROUPS AND MENON’S
IDENTITY
• Authors: MARIUS TĂRNĂUCEANU
Pages: 201 - 206
Abstract: We give a new formula for the number of cyclic subgroups of a finite abelian group. This is based on Burnside’s lemma applied to the action of the power automorphism group. The resulting formula generalises Menon’s identity.
PubDate: 2020-04-01T00:00:00.000Z
DOI: 10.1017/S0004972719000601
Issue No: Vol. 101, No. 2 (2020)

• CLOSED FORMS FOR DEGENERATE BERNOULLI POLYNOMIALS
• Authors: LEI DAI; HAO PAN
Pages: 207 - 217
Abstract: Qi and Chapman [‘Two closed forms for the Bernoulli polynomials’, J. Number Theory159 (2016), 89–100] gave a closed form expression for the Bernoulli polynomials as polynomials with coefficients involving Stirling numbers of the second kind. We extend the formula to the degenerate Bernoulli polynomials, replacing the Stirling numbers by degenerate Stirling numbers of the second kind.
PubDate: 2020-04-01T00:00:00.000Z
DOI: 10.1017/S0004972719001266
Issue No: Vol. 101, No. 2 (2020)

• ++++++++ ++++++++ ++++++++++++ ++++++++++++++++$U$ ++++++++++++ ++++++++ ++++-NUMBERS+IN+FIELDS+OF+FORMAL+POWER+SERIES+OVER+FINITE+FIELDS&rft.title=Bulletin+of+the+Australian+Mathematical+Society&rft.issn=0004-9727&rft.date=2020&rft.volume=101&rft.spage=218&rft.epage=225&rft.aulast=KEKEÇ&rft.aufirst=GÜLCAN&rft.au=GÜLCAN+KEKEÇ&rft_id=info:doi/10.1017/S0004972719000832">$U$ -NUMBERS IN FIELDS OF FORMAL POWER SERIES OVER FINITE FIELDS
• Authors: GÜLCAN KEKEÇ
Pages: 218 - 225
Abstract: In the field $\mathbb{K}$ of formal power series over a finite field $K$ , we consider some lacunary power series with algebraic coefficients in a finite extension of $K(x)$ . We show that the values of these series at nonzero algebraic arguments in $\mathbb{K}$ are $U$ -numbers.
PubDate: 2020-04-01T00:00:00.000Z
DOI: 10.1017/S0004972719000832
Issue No: Vol. 101, No. 2 (2020)

• ON LITTLEWOOD’S PROOF OF THE PRIME NUMBER THEOREM
• Authors: ALEKSANDER SIMONIČ
Pages: 226 - 232
Abstract: In this note we examine Littlewood’s proof of the prime number theorem. We show that this can be extended to provide an equivalence between the prime number theorem and the nonvanishing of Riemann’s zeta-function on the one-line. Our approach goes through the theory of almost periodic functions and is self-contained.
PubDate: 2020-04-01T00:00:00.000Z
DOI: 10.1017/S0004972719000881
Issue No: Vol. 101, No. 2 (2020)

• GENERALISATION OF A RESULT ON DISTINCT PARTITIONS WITH BOUNDED PART
DIFFERENCES
• Authors: RUNQIAO LI; BERNARD L. S. LIN, ANDREW Y. Z. WANG
Pages: 233 - 237
Abstract: We generalise a result of Chern [‘A curious identity and its applications to partitions with bounded part differences’, New Zealand J. Math. 47 (2017), 23–26] on distinct partitions with bounded difference between largest and smallest parts. The generalisation is proved both analytically and bijectively.
PubDate: 2020-04-01T00:00:00.000Z
DOI: 10.1017/S0004972719000820
Issue No: Vol. 101, No. 2 (2020)

• ON THE EULER CHARACTERISTICS OF SIGNED SELMER GROUPS
• Authors: SUMAN AHMED; MENG FAI LIM
Pages: 238 - 246
Abstract: Let $p$ be an odd prime number and $E$ an elliptic curve defined over a number field $F$ with good reduction at every prime of $F$ above $p$ . We compute the Euler characteristics of the signed Selmer groups of $E$ over the cyclotomic $\mathbb{Z}_{p}$ -extension. The novelty of our result is that we allow the elliptic curve to have mixed reduction types for primes above $p$ and mixed signs in the definition of the signed Selmer groups.
PubDate: 2020-04-01T00:00:00.000Z
DOI: 10.1017/S0004972719000704
Issue No: Vol. 101, No. 2 (2020)

• PST-GROUPS&rft.title=Bulletin+of+the+Australian+Mathematical+Society&rft.issn=0004-9727&rft.date=2020&rft.volume=101&rft.spage=247&rft.epage=254&rft.aulast=CHI&rft.aufirst=ZHANG&rft.au=ZHANG+CHI&rft.au=ALEXANDER+N.+SKIBA&rft_id=info:doi/10.1017/S0004972719000741">ON A LATTICE CHARACTERISATION OF FINITE SOLUBLE        class="italic">PST-GROUPS
• Authors: ZHANG CHI; ALEXANDER N. SKIBA
Pages: 247 - 254
Abstract: Let $\mathfrak{F}$ be a class of finite groups and $G$ a finite group. Let ${\mathcal{L}}_{\mathfrak{F}}(G)$ be the set of all subgroups $A$ of $G$ with $A^{G}/A_{G}\in \mathfrak{F}$ . A chief factor $H/K$ of $G$ is $\mathfrak{F}$ -central in $G$ if $(H/K)\rtimes (G/C_{G}(H/K))\in \mathfrak{F}$ . We study the structure of
PubDate: 2020-04-01T00:00:00.000Z
DOI: 10.1017/S0004972719000741
Issue No: Vol. 101, No. 2 (2020)

• ++++++++ ++++++++ ++++++++++++ ++++++++++++++++$p$ ++++++++++++ ++++++++ ++++-GROUPS+ALL+OF+WHOSE+NONNORMAL+SUBGROUPS+HAVE+BOUNDED+NORMAL+CORES&rft.title=Bulletin+of+the+Australian+Mathematical+Society&rft.issn=0004-9727&rft.date=2020&rft.volume=101&rft.spage=255&rft.epage=265&rft.aulast=YANG&rft.aufirst=DONGFANG&rft.au=DONGFANG+YANG&rft.au=LIJIAN+AN,+HENG+LV&rft_id=info:doi/10.1017/S0004972719000753">FINITE $p$ -GROUPS ALL OF WHOSE NONNORMAL SUBGROUPS HAVE BOUNDED NORMAL
CORES
• Authors: DONGFANG YANG; LIJIAN AN, HENG LV
Pages: 255 - 265
Abstract: Given a positive integer $m$ , a finite $p$ -group $G$ is called a $BC(p^{m})$ -group if $|H_{G}|\leq p^{m}$ for every nonnormal subgroup $H$ of $G$ , where $H_{G}$ is the normal core of $H$ in $G$ . We show that $m+2$ is an upper bound for the nilpotent class of a finite
PubDate: 2020-04-01T00:00:00.000Z
DOI: 10.1017/S0004972719000753
Issue No: Vol. 101, No. 2 (2020)

• INTERSECTIONS OF SUBGROUPS IN VIRTUALLY FREE GROUPS AND VIRTUALLY FREE
PRODUCTS
• Authors: ANTON A. KLYACHKO; ANASTASIA N. PONFILENKO
Pages: 266 - 271
Abstract: This note contains a (short) proof of the following generalisation of the Friedman–Mineyev theorem (earlier known as the Hanna Neumann conjecture): if $A$ and $B$ are nontrivial free subgroups of a virtually free group containing a free subgroup of index $n$ , then $\text{rank}(A\cap B)-1\leq n\cdot (\text{rank}(A)-1)\cdot (\text{rank}(B)-1)$ . In addition, we obtain a virtually-free-product analogue of this result.
PubDate: 2020-04-01T00:00:00.000Z
DOI: 10.1017/S0004972719000765
Issue No: Vol. 101, No. 2 (2020)

• CONJUGATING AUTOMORPHISMS OF GRAPH PRODUCTS: KAZHDAN’S PROPERTY (T)
AND SQ-UNIVERSALITY
• Authors: ANTHONY GENEVOIS; OLGA VARGHESE
Pages: 272 - 282
Abstract: An automorphism of a graph product of groups is conjugating if it sends each factor to a conjugate of a factor (possibly different). In this article, we determine precisely when the group of conjugating automorphisms of a graph product satisfies Kazhdan’s property (T) and when it satisfies some vastness properties including SQ-universality.
PubDate: 2020-04-01T00:00:00.000Z
DOI: 10.1017/S0004972719000844
Issue No: Vol. 101, No. 2 (2020)

• ON DISCRETENESS OF SUBGROUPS OF QUATERNIONIC HYPERBOLIC ISOMETRIES
Pages: 283 - 293
Abstract: Let $\mathbf{H}_{\mathbb{H}}^{n}$ denote the $n$ -dimensional quaternionic hyperbolic space. The linear group $\text{Sp}(n,1)$ acts on $\mathbf{H}_{\mathbb{H}}^{n}$ by isometries. A subgroup $G$ of $\text{Sp}(n,1)$ is called Zariski dense if it neither fixes a point on $\mathbf{H}_{\mathbb{H}}^{n}\cup \unicode[STIX]{x2202}\mathbf{H}_{\mathbb{H}}^{n}$ nor preserves a totally geodesic subspace of $\mathbf{H}_{\mathbb{H}}^{n}$ . We prove that a Zariski dense subgroup $G$ of $\text{Sp}(n,1)$ is discrete if for every loxodromic element $g\in G$ the two-generator subgroup
PubDate: 2020-04-01T00:00:00.000Z
DOI: 10.1017/S000497271900087X
Issue No: Vol. 101, No. 2 (2020)

• ++++++++ ++++++++ ++++++++++++ ++++++++++++++++$q$ ++++++++++++ ++++++++ ++++-ANALOGUE+OF+A+HYPERGEOMETRIC+CONGRUENCE&rft.title=Bulletin+of+the+Australian+Mathematical+Society&rft.issn=0004-9727&rft.date=2020&rft.volume=101&rft.spage=294&rft.epage=298&rft.aulast=GU&rft.aufirst=CHENG-YANG&rft.au=CHENG-YANG+GU&rft.au=VICTOR+J.+W.+GUO&rft_id=info:doi/10.1017/S0004972719000777">A $q$ -ANALOGUE OF A HYPERGEOMETRIC CONGRUENCE
• Authors: CHENG-YANG GU; VICTOR J. W. GUO
Pages: 294 - 298
Abstract: We give a $q$ -analogue of the following congruence: for any odd prime $p$ , $$\begin{eqnarray}\mathop{\sum }_{k=0}^{(p-1)/2}(-1)^{k}(6k+1)\frac{(\frac{1}{2})_{k}^{3}}{k!^{3}8^{k}}\mathop{\sum }_{j=1}^{k}\biggl(\frac{1}{(2j-1)^{2}}-\frac{1}{16j^{2}}\biggr)\equiv 0\;(\text{mod}\;p),\end{eqnarray}$$ which was originally conjectured by Long and later proved by Swisher. This confirms a conjecture of the second author [‘A $q$ -analogue of the (L.2) supercongruence of Van Hamme’, J. Math. Anal. Appl. 466 (2018), 749–761].
PubDate: 2020-04-01T00:00:00.000Z
DOI: 10.1017/S0004972719000777
Issue No: Vol. 101, No. 2 (2020)

• ON APPROXIMATELY ADDITIVE MAPPINGS IN 2-BANACH SPACES
• Authors: JANUSZ BRZDĘK; EL-SAYED EL-HADY
Pages: 299 - 310
Abstract: We show how some Ulam stability issues can be approached for functions taking values in 2-Banach spaces. We use the example of the well-known Cauchy equation $f(x+y)=f(x)+f(y)$ , but we believe that this method can be applied for many other equations. In particular we provide an extension of an earlier stability result that has been motivated by a problem of Th. M. Rassias. The main tool is a recent fixed point theorem in some spaces of functions with values in 2-Banach spaces.
PubDate: 2020-04-01T00:00:00.000Z
DOI: 10.1017/S0004972719000868
Issue No: Vol. 101, No. 2 (2020)

• EMBEDDINGS OF FREE TOPOLOGICAL VECTOR SPACES
• Authors: ARKADY LEIDERMAN; SIDNEY A. MORRIS
Pages: 311 - 324
Abstract: It is proved that the free topological vector space $\mathbb{V}([0,1])$ contains an isomorphic copy of the free topological vector space $\mathbb{V}([0,1]^{n})$ for every finite-dimensional cube $[0,1]^{n}$ , thereby answering an open question in the literature. We show that this result cannot be extended from the closed unit interval $[0,1]$ to general metrisable spaces. Indeed, we prove that the free topological vector space $\mathbb{V}(X)$ does not even have a vector subspace isomorphic as a topological vector space to $\mathbb{V}(X\oplus X)$ , where $X$ is a Cook continuum, which is a one-dimensional compact metric space. This is also shown to be the case for a rigid Bernstein set, which is a zero-dimensional subspace of the real line.
PubDate: 2020-04-01T00:00:00.000Z
DOI: 10.1017/S000497271900090X
Issue No: Vol. 101, No. 2 (2020)

• ON MODULATED TOPOLOGICAL VECTOR SPACES AND APPLICATIONS
• Authors: WOJCIECH M. KOZLOWSKI
Pages: 325 - 332
Abstract: We introduce a notion of modulated topological vector spaces, that generalises, among others, Banach and modular function spaces. As applications, we prove some results which extend Kirk’s and Browder’s fixed point theorems. The theory of modulated topological vector spaces provides a very minimalist framework, where powerful fixed point theorems are valid under a bare minimum of assumptions.
PubDate: 2020-04-01T00:00:00.000Z
DOI: 10.1017/S0004972719000716
Issue No: Vol. 101, No. 2 (2020)

• ON ALMOST STABLE CMC HYPERSURFACES IN MANIFOLDS OF BOUNDED SECTIONAL
CURVATURE
• Authors: JULIEN ROTH; ABHITOSH UPADHYAY
Pages: 333 - 338
Abstract: We show that almost stable constant mean curvature hypersurfaces contained in a sufficiently small ball of a manifold of bounded sectional curvature are close to geodesic spheres.
PubDate: 2020-04-01T00:00:00.000Z
DOI: 10.1017/S0004972719000935
Issue No: Vol. 101, No. 2 (2020)

• THEORY AND STATISTICS OF LONG-RANGE DEPENDENT RANDOM PROCESSES
Pages: 339 - 341
PubDate: 2020-04-01T00:00:00.000Z
DOI: 10.1017/S0004972719001308
Issue No: Vol. 101, No. 2 (2020)

• THE PERFORMANCE OF SOME STATISTICAL PROCEDURES USED IN CASE-CONTROL
STUDIES AND METHYLOMICS
• Authors: RUPERT E. H. KUVEKE
Pages: 342 - 344
PubDate: 2020-04-01T00:00:00.000Z
DOI: 10.1017/S0004972719001369
Issue No: Vol. 101, No. 2 (2020)

• STATISTICAL ANALYSIS OF LONG-RANGE DEPENDENT RANDOM PROCESSES AND FIELDS
• Authors: HUDA MOHAMMED ALOMARI
Pages: 345 - 347
PubDate: 2020-04-01T00:00:00.000Z
DOI: 10.1017/S0004972719001588
Issue No: Vol. 101, No. 2 (2020)

• CONTRIBUTIONS TO COMPUTATIONAL BAYESIAN STATISTICS
• Authors: LEAH SOUTH
Pages: 348 - 349
PubDate: 2020-04-01T00:00:00.000Z
DOI: 10.1017/S0004972720000088
Issue No: Vol. 101, No. 2 (2020)

• TEACHING ALGEBRA WITH DIGITAL TECHNOLOGY: FACTORS INFLUENCING SECONDARY
MATHEMATICS TEACHERS’ TASK DEVELOPMENT AND IMPLEMENTATION
• Authors: IRESHA GAYANI RATNAYAKE
Pages: 350 - 352
PubDate: 2020-04-01T00:00:00.000Z
DOI: 10.1017/S0004972719001436
Issue No: Vol. 101, No. 2 (2020)

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