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 Bulletin of the Australian Mathematical SocietyJournal Prestige (SJR): 0.44 Number of Followers: 1      Subscription journal ISSN (Print) 0004-9727 - ISSN (Online) 1755-1633 Published by Cambridge University Press  [387 journals]
• BAZ volume 100 Issue 2 Cover and Front matter
• PubDate: 2019-10-01T00:00:00.000Z
DOI: 10.1017/S0004972718001557
Issue No: Vol. 100, No. 2 (2019)

• BAZ volume 100 Issue 2 Cover and Back matter
• PubDate: 2019-10-01T00:00:00.000Z
DOI: 10.1017/S0004972718001569
Issue No: Vol. 100, No. 2 (2019)

• ALTERNATING COLOURINGS OF THE VERTICES OF A REGULAR POLYGON
• Authors: SHIVANI SINGH; YULIYA ZELENYUK
Pages: 177 - 181
Abstract: Let $n,r,k\in \mathbb{N}$ . An $r$ -colouring of the vertices of a regular $n$ -gon is any mapping $\unicode[STIX]{x1D712}:\mathbb{Z}_{n}\rightarrow \{1,2,\ldots ,r\}$ . Two colourings are equivalent if one of them can be obtained from another by a rotation of the polygon. An $r$ -ary necklace of length $n$ is an equivalence class of $r$ -colourings of $\mathbb{Z}_{n}$ . We say that a colouring is $k$ -alternating if all $k$ consecutive vertices have pairwise distinct colours. We compute the smallest number $r$ for which there exists a
PubDate: 2019-10-01T00:00:00.000Z
DOI: 10.1017/S0004972719000157
Issue No: Vol. 100, No. 2 (2019)

• ++++++++ ++++++++ ++++++++++++ ++++++++++++++++$P_{5}$ ++++++++++++ ++++++++ ++++,+GEM)-FREE+GRAPHS&rft.title=Bulletin+of+the+Australian+Mathematical+Society&rft.issn=0004-9727&rft.date=2019&rft.volume=100&rft.spage=182&rft.epage=188&rft.aulast=CAMERON&rft.aufirst=KATHIE&rft.au=KATHIE+CAMERON&rft.au=SHENWEI+HUANG,+OWEN+MERKEL&rft_id=info:doi/10.1017/S0004972719000352">A BOUND FOR THE CHROMATIC NUMBER OF ( $P_{5}$ , GEM)-FREE GRAPHS
• Authors: KATHIE CAMERON; SHENWEI HUANG, OWEN MERKEL
Pages: 182 - 188
Abstract: As usual, $P_{n}$ ( $n\geq 1$ ) denotes the path on $n$ vertices. The gem is the graph consisting of a $P_{4}$ together with an additional vertex adjacent to each vertex of the $P_{4}$ . A graph is called ( $P_{5}$ , gem)-free if it has no induced subgraph isomorphic to a $P_{5}$ or to a gem. For a graph $G$ , $\unicode[STIX]{x1D712}(G)$ denotes its chromatic number and $\unicode[STIX]{x1D714}(G)$ denotes the maximum size of a clique in $G$ . We show that
PubDate: 2019-10-01T00:00:00.000Z
DOI: 10.1017/S0004972719000352
Issue No: Vol. 100, No. 2 (2019)

• A NOTE ON THE SUM OF RECIPROCALS
• Authors: YUCHEN DING; YU-CHEN SUN
Pages: 189 - 193
Abstract: We prove that, given a positive integer $m$ , there is a sequence $\{n_{i}\}_{i=1}^{k}$ of positive integers such that $$\begin{eqnarray}m=\frac{1}{n_{1}}+\frac{1}{n_{2}}+\cdots +\frac{1}{n_{k}}\end{eqnarray}$$ with the property that partial sums of the series $\{1/n_{i}\}_{i=1}^{k}$ do not represent other integers.
PubDate: 2019-10-01T00:00:00.000Z
DOI: 10.1017/S0004972719000558
Issue No: Vol. 100, No. 2 (2019)

• NOTE ON SUMS INVOLVING THE EULER FUNCTION
• Authors: SHANE CHERN
Pages: 194 - 200
Abstract: In this note, we provide refined estimates of two sums involving the Euler totient function, $$\begin{eqnarray}\mathop{\sum }_{n\leq x}\unicode[STIX]{x1D719}\biggl(\biggl[\frac{x}{n}\biggr]\biggr)\quad \text{and}\quad \mathop{\sum }_{n\leq x}\frac{\unicode[STIX]{x1D719}([x/n])}{[x/n]},\end{eqnarray}$$ where $[x]$ denotes the integral part of real $x$ . The above summations were recently considered by Bordellès et al. [‘On a sum involving the Euler function’, Preprint, 2018, arXiv:1808.00188] and Wu [‘On a sum involving the Euler totient function’, Preprint, 2018, hal-01884018].
PubDate: 2019-10-01T00:00:00.000Z
DOI: 10.1017/S0004972719000066
Issue No: Vol. 100, No. 2 (2019)

• PERIODS OF DUCCI SEQUENCES AND ODD SOLUTIONS TO A PELLIAN EQUATION
• Authors: FLORIAN BREUER
Pages: 201 - 205
Abstract: A Ducci sequence is a sequence of integer $n$ -tuples generated 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 $n$ . We prove a new upper bound in the case where $n$ is a power of a prime $p\equiv 5\hspace{0.6em}({\rm mod}\hspace{0.2em}8)$ for which $2$ is a primitive root and the Pellian equation $x^{2}-py^{2}=-4$ has no solutions in odd integers $x$ and $y$ .
PubDate: 2019-10-01T00:00:00.000Z
DOI: 10.1017/S0004972719000212
Issue No: Vol. 100, No. 2 (2019)

• ++++++++ ++++++++ ++++++++++++ ++++++++++++++++$x^{2}+D$ ++++++++++++ ++++++++ ++++&rft.title=Bulletin+of+the+Australian+Mathematical+Society&rft.issn=0004-9727&rft.date=2019&rft.volume=100&rft.spage=206&rft.epage=215&rft.aulast=GHADERMARZI&rft.aufirst=AMIR&rft.au=AMIR+GHADERMARZI&rft_id=info:doi/10.1017/S0004972719000625">ON THE FACTORISATION OF $x^{2}+D$
Pages: 206 - 215
Abstract: Let $D$ be a positive nonsquare integer, $p$ a prime number with $p\nmid D$ and $0C_{2}$ with $x^{2}+D=p^{n}m$ we have $m>x^{\unicode[STIX]{x1D70E}}$ . As an application, we show that for
PubDate: 2019-10-01T00:00:00.000Z
DOI: 10.1017/S0004972719000625
Issue No: Vol. 100, No. 2 (2019)

• VANISHING COEFFICIENTS IN FOUR QUOTIENTS OF INFINITE PRODUCT EXPANSIONS
• Authors: DAZHAO TANG
Pages: 216 - 224
Abstract: Motivated by Ramanujan’s continued fraction and the work of Richmond and Szekeres [‘The Taylor coefficients of certain infinite products’, Acta Sci. Math. (Szeged)40(3–4) (1978), 347–369], we investigate vanishing coefficients along arithmetic progressions in four quotients of infinite product expansions and obtain similar results. For example, $a_{1}(5n+4)=0$ , where $a_{1}(n)$ is defined by $$\begin{eqnarray}\displaystyle {\displaystyle \frac{(q,q^{4};q^{5})_{\infty }^{3}}{(q^{2},q^{3};q^{5})_{\infty }^{2}}}=\mathop{\sum }_{n=0}^{\infty }a_{1}(n)q^{n}. & & \displaystyle \nonumber\end{eqnarray}$$
PubDate: 2019-10-01T00:00:00.000Z
DOI: 10.1017/S0004972719000327
Issue No: Vol. 100, No. 2 (2019)

• A LOWER BOUND FOR THE LARGE SIEVE WITH SQUARE MODULI
• Authors: STEPHAN BAIER; SEAN B. LYNCH, LIANGYI ZHAO
Pages: 225 - 229
Abstract: We prove a lower bound for the large sieve with square moduli.
PubDate: 2019-10-01T00:00:00.000Z
DOI: 10.1017/S0004972719000224
Issue No: Vol. 100, No. 2 (2019)

• ON THE DISTRIBUTION OF THE RANK STATISTIC FOR STRONGLY CONCAVE
COMPOSITIONS
• Authors: NIAN HONG ZHOU
Pages: 230 - 238
Abstract: A strongly concave composition of $n$ is an integer partition with strictly decreasing and then increasing parts. In this paper we give a uniform asymptotic formula for the rank statistic of a strongly concave composition introduced by Andrews et al. [‘Modularity of the concave composition generating function’, Algebra Number Theory7(9) (2013), 2103–2139].
PubDate: 2019-10-01T00:00:00.000Z
DOI: 10.1017/S0004972719000169
Issue No: Vol. 100, No. 2 (2019)

• A BRIEF NOTE ON SOME INFINITE FAMILIES OF MONOGENIC POLYNOMIALS
• Authors: LENNY JONES
Pages: 239 - 244
Abstract: Suppose that $f(x)=x^{n}+A(Bx+C)^{m}\in \mathbb{Z}[x]$ , with $n\geq 3$ and $1\leq m PubDate: 2019-10-01T00:00:00.000Z DOI: 10.1017/S0004972719000182 Issue No: Vol. 100, No. 2 (2019) • ++++++++ ++++++++ ++++++++++++ ++++++++++++++++$p$++++++++++++ ++++++++ ++++-ADIC+LIE EXTENSION&rft.title=Bulletin+of+the+Australian+Mathematical+Society&rft.issn=0004-9727&rft.date=2019&rft.volume=100&rft.spage=245&rft.epage=255&rft.aulast=BHAVE&rft.aufirst=AMALA&rft.au=AMALA+BHAVE&rft.au=LACHIT+BORA&rft_id=info:doi/10.1017/S0004972719000108">ON THE SELMER GROUP OF A CERTAIN$p$-ADIC LIE EXTENSION • Authors: AMALA BHAVE; LACHIT BORA Pages: 245 - 255 Abstract: Let$E$be an elliptic curve over$\mathbb{Q}$without complex multiplication. Let$p\geq 5$be a prime in$\mathbb{Q}$and suppose that$E$has good ordinary reduction at$p$. We study the dual Selmer group of$E$over the compositum of the$\text{GL}_{2}$extension and the anticyclotomic$\mathbb{Z}_{p}$-extension of an imaginary quadratic extension as an Iwasawa module. PubDate: 2019-10-01T00:00:00.000Z DOI: 10.1017/S0004972719000108 Issue No: Vol. 100, No. 2 (2019) • ON THE DIMENSION OF PERMUTATION VECTOR SPACES • Authors: LUCAS REIS Pages: 256 - 267 Abstract: Let$K$be a field that admits a cyclic Galois extension of degree$n\geq 2$. The symmetric group$S_{n}$acts on$K^{n}$by permutation of coordinates. Given a subgroup$G$of$S_{n}$and$u\in K^{n}$, let$V_{G}(u)$be the$K$-vector space spanned by the orbit of$u$under the action of$G$. In this paper we show that, for a special family of groups PubDate: 2019-10-01T00:00:00.000Z DOI: 10.1017/S0004972719000340 Issue No: Vol. 100, No. 2 (2019) • A NEW SUM–PRODUCT ESTIMATE IN PRIME FIELDS • Authors: CHANGHAO CHEN; BRYCE KERR, ALI MOHAMMADI Pages: 268 - 280 Abstract: We obtain a new sum–product estimate in prime fields for sets of large cardinality. In particular, we show that if$A\subseteq \mathbb{F}_{p}$satisfies$|A|\leq p^{64/117}$then$\max \{|A\pm A|,|AA|\}\gtrsim |A|^{39/32}.$Our argument builds on and improves some recent results of Shakan and Shkredov [‘Breaking the 6/5 threshold for sums and products modulo a prime’, Preprint, 2018, arXiv:1806.07091v1] which use the eigenvalue method to reduce to estimating a fourth moment energy and the additive energy$E^{+}(P)$of some subset$P\subseteq A+A$. Our main novelty comes from reducing the estimation of$E^{+}(P)$to a point–plane incidence bound of Rudnev [‘On the number of incidences between points and planes in three dimensions’, Combinatorica 38(1) (2017), 219–254] rather than a point–line incidence bound used by Shakan and Shkredov. PubDate: 2019-10-01T00:00:00.000Z DOI: 10.1017/S0004972719000303 Issue No: Vol. 100, No. 2 (2019) • A NILPOTENCY CRITERION FOR SOME VERBAL SUBGROUPS • Authors: CARMINE MONETTA; ANTONIO TORTORA Pages: 281 - 289 Abstract: The word$w=[x_{i_{1}},x_{i_{2}},\ldots ,x_{i_{k}}]$is a simple commutator word if$k\geq 2,i_{1}\neq i_{2}$and$i_{j}\in \{1,\ldots ,m\}$for some$m>1$. For a finite group$G$, we prove that if$i_{1}\neq i_{j}$for every$j\neq 1$, then the verbal subgroup corresponding to$w$is nilpotent if and only if$|ab|=|a||b|$for any$w$-values$a,b\in G$of coprime orders. We also extend the result to a residually finite group PubDate: 2019-10-01T00:00:00.000Z DOI: 10.1017/S0004972719000054 Issue No: Vol. 100, No. 2 (2019) • A NOTE ON THE PERIODICITY OF ENTIRE FUNCTIONS • Authors: KAI LIU; PEIYONG YU Pages: 290 - 296 Abstract: We give some sufficient conditions for the periodicity of entire functions based on a conjecture of C. C. Yang, using the concepts of value sharing, unique polynomial of entire functions and Picard exceptional value. PubDate: 2019-10-01T00:00:00.000Z DOI: 10.1017/S0004972719000030 Issue No: Vol. 100, No. 2 (2019) • EXTENDABLE TEMPERATURES • Authors: NEIL A. WATSON Pages: 297 - 303 Abstract: Let$E$and$D$be open subsets of$\mathbb{R}^{n+1}$such that$\overline{D}$is a compact subset of$E$, and let$v$be a supertemperature on$E$. We call a temperature$u$on$D$extendable by$v$if there is a supertemperature$w$on PubDate: 2019-10-01T00:00:00.000Z DOI: 10.1017/S0004972719000248 Issue No: Vol. 100, No. 2 (2019) • A SYSTEM OF FUNCTIONAL EQUATIONS SATISFIED BY COMPONENTS OF A QUADRATIC FUNCTION AND ITS STABILITY • Authors: KANET PONPETCH; VICHIAN LAOHAKOSOL, SUKRAWAN MAVECHA Pages: 304 - 316 Abstract: A system of functional equations satisfied by the components of a quadratic function is derived via their corresponding circulant matrix. Given such a system of functional equations, general solutions are determined and a stability result for such a system is established. PubDate: 2019-10-01T00:00:00.000Z DOI: 10.1017/S000497271900025X Issue No: Vol. 100, No. 2 (2019) • A GENERALISATION OF THE FROBENIUS RECIPROCITY THEOREM • Authors: H. KUMUDINI DHARMADASA; WILLIAM MORAN Pages: 317 - 322 Abstract: Let$G$be a locally compact group and$K$a closed subgroup of$G$. Let$\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D70B}$be representations of$K$and$G$respectively. Moore’s version of the Frobenius reciprocity theorem was established under the strong conditions that the underlying homogeneous space$G/K$possesses a right-invariant measure and the representation space$H(\unicode[STIX]{x1D6FE})$of the representation$\unicode[STIX]{x1D6FE}$of$K$is a Hilbert space. Here, the theorem is proved in a more general setting assuming only the existence of... PubDate: 2019-10-01T00:00:00.000Z DOI: 10.1017/S0004972719000042 Issue No: Vol. 100, No. 2 (2019) • A NOTE ON RADIAL SYMMETRY FOR AN INTEGRAL EQUATION OF WOLFF TYPE • Authors: YUN WANG; LIXIN TIAN Pages: 323 - 327 Abstract: We prove that positive solutions of an integral equation of Wolff type are radially symmetric and decreasing about some point in$R^{n}$. The hypotheses allow a wider range of exponents and are easier to apply than those in previous work. PubDate: 2019-10-01T00:00:00.000Z DOI: 10.1017/S0004972719000121 Issue No: Vol. 100, No. 2 (2019) • APPROXIMATIONS OF SUBHOMOGENEOUS ALGEBRAS • Authors: TATIANA SHULMAN; OTGONBAYAR UUYE Pages: 328 - 337 Abstract: Let$n$be a positive integer. A$C^{\ast }$-algebra is said to be$n$-subhomogeneous if all its irreducible representations have dimension at most$n$. We give various approximation properties characterising$n$-subhomogeneous$C^{\ast }$-algebras. PubDate: 2019-10-01T00:00:00.000Z DOI: 10.1017/S000497271900008X Issue No: Vol. 100, No. 2 (2019) • A QUANTITATIVE EXTENSION OF SZLENK’S THEOREM • Authors: ANDRZEJ KRYCZKA Pages: 338 - 343 Abstract: We show that for a bounded subset$A$of the$L_{1}(\unicode[STIX]{x1D707})$space with finite measure$\unicode[STIX]{x1D707}$, the measure of weak noncompactness of$A$based on the convex separation of sequences coincides with the measure of deviation from the Banach–Saks property expressed by the arithmetic separation of sequences. A similar result holds for a related quantity with the alternating signs Banach–Saks property. The results provide a geometric and quantitative extension of Szlenk’s theorem saying that every weakly convergent sequence in the Lebesgue space$L_{1}$has a subsequence whose arithmetic means are norm convergent. PubDate: 2019-10-01T00:00:00.000Z DOI: 10.1017/S0004972719000170 Issue No: Vol. 100, No. 2 (2019) • RECOVERY OF THE TEMPERATURE DISTRIBUTION IN A MULTILAYER FRACTIONAL DIFFUSION EQUATION • Authors: KHIEU T. TRAN; LUAN N. TRAN, HONG B. Q. NGUYEN, KHANH Q. TRA Pages: 344 - 352 Abstract: We study the inverse boundary value problem for fractional diffusion in a multilayer composite medium. Given data in the right boundary of the second layer, the problem is to recover the temperature distribution in the first layer, which is inaccessible for measurement. The problem is ill-posed and we propose a Fourier spectral approach to achieve Hölder approximations. The convergence analysis is performed in both the$L^{2}$- and$L^{\infty }\$ -settings.
PubDate: 2019-10-01T00:00:00.000Z
DOI: 10.1017/S000497271900011X
Issue No: Vol. 100, No. 2 (2019)

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