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Publisher: Cambridge University Press   (Total: 372 journals)

 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  [372 journals]
• BAZ volume 99 Issue 1 Cover and Front matter
• PubDate: 2019-02-01T00:00:00.000Z
DOI: 10.1017/S0004972718001478
Issue No: Vol. 99, No. 1 (2019)

• BAZ volume 99 Issue 1 Cover and Back matter
• PubDate: 2019-02-01T00:00:00.000Z
DOI: 10.1017/S000497271800148X
Issue No: Vol. 99, No. 1 (2019)

• NOTE ON THE NUMBER OF DIVISORS OF REDUCIBLE QUADRATIC POLYNOMIALS
• Authors: ADRIAN W. DUDEK; ŁUKASZ PAŃKOWSKI, VICTOR SCHARASCHKIN
Pages: 1 - 9
Abstract: Lapkova [‘On the average number of divisors of reducible quadratic polynomials’, J. Number Theory 180 (2017), 710–729] uses a Tauberian theorem to derive an asymptotic formula for the divisor sum $\sum _{n\leq x}d(n(n+v))$ where $v$ is a fixed integer and $d(n)$ denotes the number of divisors of  $n$ . We reprove this result with additional terms in the asymptotic formula, by investigating the relationship between this divisor sum and the well-known sum $\sum _{n\leq x}d(n)d(n+v)$ .
PubDate: 2019-02-01T00:00:00.000Z
DOI: 10.1017/S0004972718000734
Issue No: Vol. 99, No. 1 (2019)

• REPRESENTATION FUNCTIONS ON ABELIAN GROUPS
• Authors: WU-XIA MA; YONG-GAO CHEN
Pages: 10 - 14
Abstract: Let $G$ be a finite abelian group, $A$ a nonempty subset of $G$ and $h\geq 2$ an integer. For $g\in G$ , let $R_{A,h}(g)$ denote the number of solutions of the equation $x_{1}+\cdots +x_{h}=g$ with $x_{i}\in A$ for $1\leq i\leq h$ . Kiss et al. [‘Groups, partitions and representation functions’, Publ. Math. Debrecen85(3) (2014), 425–433] proved that (a) if $R_{A,h}(g)=R_{G\setminus A,h}(g)$ for all $g\in G$ , then
PubDate: 2019-02-01T00:00:00.000Z
DOI: 10.1017/S0004972718000783
Issue No: Vol. 99, No. 1 (2019)

• ON A PROBLEM OF CHEN AND LEV
• Authors: SHI-QIANG CHEN; MIN TANG, QUAN-HUI YANG
Pages: 15 - 22
Abstract: For a given set $S\subset \mathbb{N}$ , $R_{S}(n)$ is the number of solutions of the equation $n=s+s^{\prime },sr\geq 0$ and that $A$ and $B$ are sets with $A\cup B=\mathbb{N}$ and $A\cap B=\{r+mk:k\in \mathbb{N}\}$ . We prove that if $R_{A}(n)=R_{B}(n)$ for all positive integers
PubDate: 2019-02-01T00:00:00.000Z
DOI: 10.1017/S0004972718001107
Issue No: Vol. 99, No. 1 (2019)

• ++++++++ ++++++++ ++++++++++++ ++++++++++++++++$n$ ++++++++++++ ++++++++ +++++WITH+POLYNOMIAL+IMAGE+COPRIME+WITH+THE+ ++++++++ ++++++++ ++++++++++++ ++++++++++++++++$n$ ++++++++++++ ++++++++ ++++TH+TERM+OF+A+LINEAR+RECURRENCE&rft.title=Bulletin+of+the+Australian+Mathematical+Society&rft.issn=0004-9727&rft.date=2019&rft.volume=99&rft.spage=23&rft.epage=33&rft.aulast=MASTROSTEFANO&rft.aufirst=DANIELE&rft.au=DANIELE+MASTROSTEFANO&rft.au=CARLO+SANNA&rft_id=info:doi/10.1017/S0004972718000606">ON NUMBERS $n$ WITH POLYNOMIAL IMAGE COPRIME WITH THE $n$ TH TERM OF A
LINEAR RECURRENCE
• Authors: DANIELE MASTROSTEFANO; CARLO SANNA
Pages: 23 - 33
Abstract: Let $F$ be an integral linear recurrence, $G$ an integer-valued polynomial splitting over the rationals and $h$ a positive integer. Also, let ${\mathcal{A}}_{F,G,h}$ be the set of all natural numbers $n$ such that $\gcd (F(n),G(n))=h$ . We prove that ${\mathcal{A}}_{F,G,h}$ has a natural density. Moreover, assuming that $F$ is nondegenerate and $G$ has no fixed divisors, we show that the density of ${\mathcal{A}}_{F,G,1}$ is 0 if and only if ${\mathcal{A}}_{F,G,1}$ is finite.
PubDate: 2019-02-01T00:00:00.000Z
DOI: 10.1017/S0004972718000606
Issue No: Vol. 99, No. 1 (2019)

• PERFECT POWERS THAT ARE SUMS OF TWO POWERS OF FIBONACCI NUMBERS
• Authors: ZHONGFENG ZHANG; ALAIN TOGBÉ
Pages: 34 - 41
Abstract: In this paper, we consider the Diophantine equations $$\begin{eqnarray}\displaystyle F_{n}^{q}\pm F_{m}^{q}=y^{p} & & \displaystyle \nonumber\end{eqnarray}$$ with positive integers $q,p\geq 2$ and $\gcd (F_{n},F_{m})=1$ , where $F_{k}$ is a Fibonacci number. We obtain results for $q=2$ or $q$ an odd prime with $q\equiv 3\;(\text{mod}\;4),3 PubDate: 2019-02-01T00:00:00.000Z DOI: 10.1017/S0004972718000916 Issue No: Vol. 99, No. 1 (2019) • LINEAR INDEPENDENCE OF POWERS OF SINGULAR MODULI OF DEGREE THREE • Authors: FLORIAN LUCA; ANTONIN RIFFAUT Pages: 42 - 50 Abstract: We show that two distinct singular moduli$j(\unicode[STIX]{x1D70F}),j(\unicode[STIX]{x1D70F}^{\prime })$, such that for some positive integers$m$and$n$the numbers$1,j(\unicode[STIX]{x1D70F})^{m}$and$j(\unicode[STIX]{x1D70F}^{\prime })^{n}$are linearly dependent over$\mathbb{Q}$, generate the same number field of degree at most two. This completes a result of Riffaut [‘Equations with powers of singular moduli’, Int. J. Number Theory, to appear], who proved the above theorem except for two explicit pairs of exceptions consisting of numbers of degree three. The purpose of this article is to treat these two remaining cases. PubDate: 2019-02-01T00:00:00.000Z DOI: 10.1017/S0004972718000965 Issue No: Vol. 99, No. 1 (2019) • PARITY RESULTS FOR PARTITIONS WHEREIN EACH PART APPEARS AN ODD NUMBER OF TIMES • Authors: MICHAEL D. HIRSCHHORN; JAMES A. SELLERS Pages: 51 - 55 Abstract: We consider the function$f(n)$that enumerates partitions of weight$n$wherein each part appears an odd number of times. Chern [‘Unlimited parity alternating partitions’, Quaest. Math. (to appear)] noted that such partitions can be placed in one-to-one correspondence with the partitions of$n$which he calls unlimited parity alternating partitions with smallest part odd. Our goal is to study the parity of$f(n)$in detail. In particular, we prove a characterisation of$f(2n)$modulo 2 which implies that there are infinitely many Ramanujan-like congruences modulo 2 satisfied by the function$f.$The proof techniques are elementary and involve classical generating function dissection tools. PubDate: 2019-02-01T00:00:00.000Z DOI: 10.1017/S0004972718001041 Issue No: Vol. 99, No. 1 (2019) • IDEAL CHAINS IN RESIDUALLY FINITE DEDEKIND DOMAINS • Authors: YU-JIE WANG; YI-JING HU, CHUN-GANG JI Pages: 56 - 67 Abstract: Let$\mathfrak{D}$be a residually finite Dedekind domain and let$\mathfrak{n}$be a nonzero ideal of$\mathfrak{D}$. We consider counting problems for the ideal chains in$\mathfrak{D}/\mathfrak{n}$. By using the Cauchy–Frobenius–Burnside lemma, we also obtain some further extensions of Menon’s identity. PubDate: 2019-02-01T00:00:00.000Z DOI: 10.1017/S0004972718001090 Issue No: Vol. 99, No. 1 (2019) • BETTI NUMBERS FOR CERTAIN COHEN–MACAULAY TANGENT CONES • Authors: MESUT ŞAHİN; NİL ŞAHİN Pages: 68 - 77 Abstract: We compute Betti numbers for a Cohen–Macaulay tangent cone of a monomial curve in the affine$4$-space corresponding to a pseudo-symmetric numerical semigroup. As a byproduct, we also show that for these semigroups, being of homogeneous type and homogeneous are equivalent properties. PubDate: 2019-02-01T00:00:00.000Z DOI: 10.1017/S0004972718000898 Issue No: Vol. 99, No. 1 (2019) • PROJECTIVE CHARACTERS WITH PRIME POWER DEGREES • Authors: YANG LIU Pages: 78 - 82 Abstract: We consider the relationship between structural information of a finite group$G$and$\text{cd}_{\unicode[STIX]{x1D6FC}}(G)$, the set of all irreducible projective character degrees of$G$with factor set$\unicode[STIX]{x1D6FC}$. We show that for nontrivial$\unicode[STIX]{x1D6FC}$, if all numbers in$\text{cd}_{\unicode[STIX]{x1D6FC}}(G)$are prime powers, then$G$is solvable. Our result is proved by classical character theory using the bijection between irreducible projective representations and irreducible constituents of induced representations in its representation group. PubDate: 2019-02-01T00:00:00.000Z DOI: 10.1017/S0004972718000825 Issue No: Vol. 99, No. 1 (2019) • OVERGROUPS OF WEAK SECOND MAXIMAL SUBGROUPS • Authors: HANGYANG MENG; XIUYUN GUO Pages: 83 - 88 Abstract: A subgroup$H$is called a weak second maximal subgroup of$G$if$H$is a maximal subgroup of a maximal subgroup of$G$. Let$m(G,H)$denote the number of maximal subgroups of$G$containing$H$. We prove that$m(G,H)-1$divides the index of some maximal subgroup of$G$when$H$is a weak second maximal subgroup of$G$. This partially answers a question of Flavell [‘Overgroups of second maximal subgroups’, Arch. Math.64(4) (1995), 277–282] and extends a result of Pálfy and Pudlák [‘Congruence lattices of finite algebras and intervals in sub... PubDate: 2019-02-01T00:00:00.000Z DOI: 10.1017/S0004972718000904 Issue No: Vol. 99, No. 1 (2019) • ON OZAKI CLOSE-TO-CONVEX FUNCTIONS • Authors: VASUDEVARAO ALLU; DEREK K. THOMAS, NIKOLA TUNESKI Pages: 89 - 100 Abstract: Let$f$be analytic in$\mathbb{D}=\{z\in \mathbb{C}:|z|
PubDate: 2019-02-01T00:00:00.000Z
DOI: 10.1017/S0004972718000989
Issue No: Vol. 99, No. 1 (2019)

• MULTIPLICITIES OF EIGENVALUES OF THE DIFFUSION OPERATOR WITH RANDOM JUMPS
FROM THE BOUNDARY
• Authors: JUN YAN; GUOLIANG SHI
Pages: 101 - 113
Abstract: This paper deals with a non-self-adjoint differential operator which is associated with a diffusion process with random jumps from the boundary. Our main result is that the algebraic multiplicity of an eigenvalue is equal to its order as a zero of the characteristic function $\unicode[STIX]{x1D6E5}(\unicode[STIX]{x1D706})$ . This is a new criterion for determining the multiplicities of eigenvalues for concrete operators.
PubDate: 2019-02-01T00:00:00.000Z
DOI: 10.1017/S0004972718001120
Issue No: Vol. 99, No. 1 (2019)

• A COMPACT QUALITATIVE UNCERTAINTY PRINCIPLE FOR SOME NONUNIMODULAR GROUPS
• Authors: WASSIM NASSERDDINE
Pages: 114 - 120
Abstract: Let $G$ be a separable locally compact group with type $I$ left regular representation, $\widehat{G}$ its dual, $A(G)$ its Fourier algebra and $f\in A(G)$ with compact support. If $G=\mathbb{R}$ and the Fourier transform of $f$ is compactly supported, then, by a classical Paley–Wiener theorem, $f=0$ . There are extensions of this theorem for abelian and some unimodular groups. In this paper, we prove that if $G$ has no (nonempty) open compact subsets, $\hat{f}$ , the regularised Fourier cotransform of  $f$ , is compactly supported and
PubDate: 2019-02-01T00:00:00.000Z
DOI: 10.1017/S0004972718001119
Issue No: Vol. 99, No. 1 (2019)

• A CHARACTERISATION OF EUCLIDEAN NORMED PLANES VIA BISECTORS
• Authors: JAVIER CABELLO SÁNCHEZ; ADRIÁN GORDILLO-MERINO
Pages: 121 - 129
Abstract: Our main result states that whenever we have a non-Euclidean norm $\Vert \cdot \Vert$ on a two-dimensional vector space $X$ , there exists some $x\neq 0$ such that for every $\unicode[STIX]{x1D706}\neq 1$ , $\unicode[STIX]{x1D706}>0$ , there exist $y,z\in X$ satisfying $\Vert y\Vert =\unicode[STIX]{x1D706}\Vert x\Vert$ , $z\neq 0$ and $z$ belongs to the bisectors $B(-x,x)$ and $B(-y,y)$ . We also give several results about the geometry of the unit sphere of strictly convex planes.
PubDate: 2019-02-01T00:00:00.000Z
DOI: 10.1017/S0004972718000758
Issue No: Vol. 99, No. 1 (2019)

• THE ROSENTHAL–SZASZ INEQUALITY FOR NORMED PLANES
• Authors: VITOR BALESTRO; HORST MARTINI
Pages: 130 - 136
Abstract: We study the classical Rosenthal–Szasz inequality for a plane whose geometry is determined by a norm. This inequality states that the bodies of constant width have the largest perimeter among all planar convex bodies of given diameter. In the case where the unit circle of the norm is given by a Radon curve, we obtain an inequality which is completely analogous to the Euclidean case. For arbitrary norms we obtain an upper bound for the perimeter calculated in the anti-norm, yielding an analogous characterisation of all curves of constant width. To derive these results, we use methods from the differential geometry of curves in normed planes.
PubDate: 2019-02-01T00:00:00.000Z
DOI: 10.1017/S0004972718000813
Issue No: Vol. 99, No. 1 (2019)

• ENTIRE SOLUTIONS OF A CURVATURE FLOW IN AN UNDULATING CYLINDER
• Authors: LIXIA YUAN; BENDONG LOU
Pages: 137 - 147
Abstract: We consider a curvature flow $V=\unicode[STIX]{x1D705}+A$ in a two-dimensional undulating cylinder $\unicode[STIX]{x1D6FA}$ described by $\unicode[STIX]{x1D6FA}:=\{(x,y)\in \mathbb{R}^{2}\mid -g_{1}(y) PubDate: 2019-02-01T00:00:00.000Z DOI: 10.1017/S0004972718000795 Issue No: Vol. 99, No. 1 (2019) • OPTIMAL STOPPING FOR THE LAST EXIT TIME • Authors: DAN REN Pages: 148 - 160 Abstract: Given a one-dimensional downwards transient diffusion process$X$, we consider a random time$\unicode[STIX]{x1D70C}$, the last exit time when$X$exits a certain level$\ell$, and detect the optimal stopping time for it. In particular, for this random time$\unicode[STIX]{x1D70C}$, we solve the optimisation problem$\inf _{\unicode[STIX]{x1D70F}}\mathbb{E}[\unicode[STIX]{x1D706}(\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70C})_{+}+(1-\unicode[STIX]{x1D706})(\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D70F})_{+}]$over all stopping times$\unicode[STIX]{x1D70F}$. We show that the process should stop optimally when it runs below some fixed level$\unicode[STIX]{x1D705}_{\ell }$for the first time, where$\unicode[STIX]{x1D705}_{\ell }$is the unique solution in the interval$(0,\unicode[STIX]{x1D706}\ell )\$ of an explicitly defined equation.
PubDate: 2019-02-01T00:00:00.000Z
DOI: 10.1017/S0004972718000990
Issue No: Vol. 99, No. 1 (2019)

• A SEPARATION PRINCIPLE FOR THE STABILISATION OF A CLASS OF FRACTIONAL
ORDER TIME DELAY NONLINEAR SYSTEMS
• Authors: NADHEM ECHI; IMED BASDOURI, HANEN BENALI
Pages: 161 - 173
Abstract: We establish a separation principle for a class of fractional order time-delay nonlinear differential systems. We show that a nonlinear time-delay observer is globally convergent and give sufficient conditions under which the observer-based controller stabilises the system.
PubDate: 2019-02-01T00:00:00.000Z
DOI: 10.1017/S0004972718000837
Issue No: Vol. 99, No. 1 (2019)

• DETECTION OF LONGITUDINAL BRAIN ATROPHY PATTERNS CONSISTENT WITH
PROGRESSION TOWARDS ALZHEIMER’S DISEASE
• Authors: MARCELA I. CESPEDES
Pages: 174 - 176
PubDate: 2019-02-01T00:00:00.000Z
DOI: 10.1017/S0004972718001089
Issue No: Vol. 99, No. 1 (2019)

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