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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  [400 journals]
• BAZ volume 104 issue 2 Cover and Front matter

PubDate: 2021-10-01T00:00:00.000Z
DOI: 10.1017/S0004972720000866
Issue No: Vol. 104, No. 2 (2021)

• BAZ volume 104 issue 2 Cover and Back matter

PubDate: 2021-10-01T00:00:00.000Z
DOI: 10.1017/S0004972720000878
Issue No: Vol. 104, No. 2 (2021)

• BIASES IN INTEGER PARTITIONS

Authors: BYUNGCHAN KIM; EUNMI KIM
Pages: 177 - 186
Abstract: We show that there are biases in the number of appearances of the parts in two residue classes in the set of ordinary partitions. More precisely, let $p_{j,k,m} (n)$ be the number of partitions of n such that there are more parts congruent to j modulo m than parts congruent to k modulo m for $m \geq 2$. We prove that $p_{1,0,m} (n)$ is in general larger than $p_{0,1,m} (n)$. We also obtain asymptotic formulas for $p_{1,0,m}(n)$ and $p_{0,1,m}(n)$ for $m \geq 2$.
PubDate: 2021-10-01T00:00:00.000Z
DOI: 10.1017/S0004972720001495
Issue No: Vol. 104, No. 2 (2021)

• ASKING QUESTIONS TO DETERMINE THE PRODUCT OF CIRCULARLY ARRANGED NUMBERS

Pages: 187 - 195
Abstract: Fix positive integers k and n with $k \leq n$. Numbers $x_0, x_1, x_2, \ldots , x_{n - 1}$, each equal to $\pm {1}$, are cyclically arranged (so that $x_0$ follows $x_{n - 1}$) in that order. The problem is to find the product $P = x_0x_1 \cdots x_{n - 1}$ of all n numbers by asking the smallest number of questions of the type $Q_i$: what is $x_ix_{i + 1}x_{i + 2} \cdots x_{i+ k -1}$? (where all the subscripts are read modulo n). This paper studies the problem and some of its generalisations.
PubDate: 2021-10-01T00:00:00.000Z
DOI: 10.1017/S0004972721000216
Issue No: Vol. 104, No. 2 (2021)

Authors: PU QIAO; XINGZHI ZHAN
Pages: 196 - 202
Abstract: A graph is called radially maximal if it is not complete and the addition of any new edge decreases its radius. Harary and Thomassen [‘Anticritical graphs’, Math. Proc. Cambridge Philos. Soc. 79(1) (1976), 11–18] proved that the radius r and diameter d of any radially maximal graph satisfy $r\le d\le 2r-2.$ Dutton et al. [‘Changing and unchanging of the radius of a graph’, Linear Algebra Appl. 217 (1995), 67–82] rediscovered this result with a different proof and conjectured that the converse is true, that is, if r and d are positive integers satisfying $r\le d\le 2r-2,$ then there exists a radially maximal graph with radius r and diameter $d.$ We prove this conjecture and a little more.
PubDate: 2021-10-01T00:00:00.000Z
DOI: 10.1017/S0004972720001471
Issue No: Vol. 104, No. 2 (2021)

• $\boldsymbol{\mathcal{CF}}$ -CONNECTED+GRAPHS+FOR+ $\boldsymbol{K}_{\boldsymbol{m,n}}$ &rft.title=Bulletin+of+the+Australian+Mathematical+Society&rft.issn=0004-9727&rft.date=2021&rft.volume=104&rft.spage=203&rft.epage=210&rft.aulast=STAŠ&rft.aufirst=MICHAL&rft.au=MICHAL+STAŠ&rft.au=JURAJ+VALISKA&rft_id=info:doi/10.1017/S000497272000129X">ON PROBLEMS OF $\boldsymbol{\mathcal{CF}}$ -CONNECTED GRAPHS FOR
$\boldsymbol{K}_{\boldsymbol{m,n}}$

Authors: MICHAL STAŠ; JURAJ VALISKA
Pages: 203 - 210
Abstract: A connected graph G is $\mathcal {CF}$-connected if there is a path between every pair of vertices with no crossing on its edges for each optimal drawing of G. We conjecture that a complete bipartite graph $K_{m,n}$ is $\mathcal {CF}$-connected if and only if it does not contain a subgraph of $K_{3,6}$ or $K_{4,4}$. We establish the validity of this conjecture for all complete bipartite graphs $K_{m,n}$ for any $m,n$ with $\min \{m,n\}\leq 6$, and conditionally for $m,n\geq 7$ on the assumption of Zarankiewicz’s conjecture that $\mathrm {cr}(K_{m,n})=\big \lfloor \frac {m}{2} \big \rfloor \big \lfloor \frac {m-1}{2} \big \rfloor \big \lfloor \frac {n}{2} \big \rfloor \big \lfloor \frac {n-1}{2} \big \rfloor$.
PubDate: 2021-10-01T00:00:00.000Z
DOI: 10.1017/S000497272000129X
Issue No: Vol. 104, No. 2 (2021)

• ON ASYMPTOTIC BASES WHICH HAVE DISTINCT SUBSET SUMS

Authors: SÁNDOR Z. KISS; VINH HUNG NGUYEN
Pages: 211 - 217
Abstract: Let k and l be positive integers satisfying $k \ge 2, l \ge 1$. A set $\mathcal {A}$ of positive integers is an asymptotic basis of order k if every large enough positive integer can be represented as the sum of k terms from $\mathcal {A}$. About 35 years ago, P. Erdős asked: does there exist an asymptotic basis of order k where all the subset sums with at most l terms are pairwise distinct with the exception of a finite number of cases as long as $l \le k - 1$? We use probabilistic tools to prove the existence of an asymptotic basis of order $2k+1$ for which all the sums of at most k elements are pairwise distinct except for ‘small’ numbers.
PubDate: 2021-10-01T00:00:00.000Z
DOI: 10.1017/S0004972721000174
Issue No: Vol. 104, No. 2 (2021)

• SUMS OF FOUR SQUARES WITH A CERTAIN RESTRICTION

Authors: YUE-FENG SHE; HAI-LIANG WU
Pages: 218 - 227
Abstract: Z.-W. Sun [‘Refining Lagrange’s four-square theorem’, J. Number Theory 175 (2017), 169–190] conjectured that every positive integer n can be written as $x^2+y^2+z^2+w^2\ (x,y,z,w\in \mathbb {N}=\{0,1,\ldots \})$ with $x+3y$ a square and also as $n=x^2+y^2+z^2+w^2\ (x,y,z,w \in \mathbb {Z})$ with $x+3y\in \{4^k:k\in \mathbb {N}\}$. In this paper, we confirm these conjectures via the arithmetic theory of ternary quadratic forms.
PubDate: 2021-10-01T00:00:00.000Z
DOI: 10.1017/S0004972720001501
Issue No: Vol. 104, No. 2 (2021)

• n+INTO+DISTINCT+PARTS&rft.title=Bulletin+of+the+Australian+Mathematical+Society&rft.issn=0004-9727&rft.date=2021&rft.volume=104&rft.spage=228&rft.epage=237&rft.aulast=MERCA&rft.aufirst=MIRCEA&rft.au=MIRCEA+MERCA&rft_id=info:doi/10.1017/S0004972720001422">ON THE SUM OF PARTS IN THE PARTITIONS OF n
INTO DISTINCT PARTS

Authors: MIRCEA MERCA
Pages: 228 - 237
Abstract: We investigate the sum of the parts in all the partitions of n into distinct parts and give two infinite families of linear inequalities involving this sum. The results can be seen as new connections between partitions and divisors.
PubDate: 2021-10-01T00:00:00.000Z
DOI: 10.1017/S0004972720001422
Issue No: Vol. 104, No. 2 (2021)

• $\textbf{2}$ +AND+ $\textbf{3}$ &rft.title=Bulletin+of+the+Australian+Mathematical+Society&rft.issn=0004-9727&rft.date=2021&rft.volume=104&rft.spage=238&rft.epage=248&rft.aulast=SINGH&rft.aufirst=AJIT&rft.au=AJIT+SINGH&rft.au=RUPAM+BARMAN&rft_id=info:doi/10.1017/S0004972720001513">DIVISIBILITY OF CERTAIN SINGULAR OVERPARTITIONS BY POWERS OF $\textbf{2}$
AND $\textbf{3}$

Authors: AJIT SINGH; RUPAM BARMAN
Pages: 238 - 248
Abstract: Andrews introduced the partition function $\overline {C}_{k, i}(n)$, called the singular overpartition function, which counts the number of overpartitions of n in which no part is divisible by k and only parts $\equiv \pm i\pmod {k}$ may be overlined. We prove that $\overline {C}_{6, 2}(n)$ is almost always divisible by $2^k$ for any positive integer k. We also prove that $\overline {C}_{6, 2}(n)$ and $\overline {C}_{12, 4}(n)$ are almost always divisible by $3^k$. Using a result of Ono and Taguchi on nilpotency of Hecke operators, we find infinite families of congruences modulo arbitrary powers of $2$ satisfied by $\overline {C}_{6, 2}(n)$.
PubDate: 2021-10-01T00:00:00.000Z
DOI: 10.1017/S0004972720001513
Issue No: Vol. 104, No. 2 (2021)

• STRICTLY REAL FUNDAMENTAL THEOREM OF ALGEBRA USING POLYNOMIAL INTERLACING

Authors: SOHAM BASU
Pages: 249 - 255
Abstract: Without resorting to complex numbers or any advanced topological arguments, we show that any real polynomial of degree greater than two always has a real quadratic polynomial factor, which is equivalent to the fundamental theorem of algebra. The proof uses interlacing of bivariate polynomials similar to Gauss’s first proof of the fundamental theorem of algebra using complex numbers, but in a different context of division residues of strictly real polynomials. This shows the sufficiency of basic real analysis as the minimal platform to prove the fundamental theorem of algebra.
PubDate: 2021-10-01T00:00:00.000Z
DOI: 10.1017/S0004972720001434
Issue No: Vol. 104, No. 2 (2021)

• FIXED POINTS OF POLYNOMIALS OVER DIVISION RINGS

Pages: 256 - 262
Abstract: We study the discrete dynamics of standard (or left) polynomials $f(x)$ over division rings D. We define their fixed points to be the points $\lambda \in D$ for which $f^{\circ n}(\lambda )=\lambda$ for any $n \in \mathbb {N}$, where $f^{\circ n}(x)$ is defined recursively by $f^{\circ n}(x)=f(f^{\circ (n-1)}(x))$ and $f^{\circ 1}(x)=f(x)$. Periodic points are similarly defined. We prove that $\lambda$ is a fixed point of $f(x)$ if and only if $f(\lambda )=\lambda$, which enables the use of known results from the theory of polynomial equations, to conclude that any polynomial of degree $m \geq 2$ has at most m conjugacy classes of fixed points. We also show that in general, periodic points do not behave as in the commutative case. We provide a sufficient condition for periodic points to behave as expected.
PubDate: 2021-10-01T00:00:00.000Z
DOI: 10.1017/S0004972721000113
Issue No: Vol. 104, No. 2 (2021)

• $\textbf{2}$ -TRANSITIVE+NONNORMAL+CAYLEY+GRAPHS+OF+FINITE+SIMPLE+GROUPS&rft.title=Bulletin+of+the+Australian+Mathematical+Society&rft.issn=0004-9727&rft.date=2021&rft.volume=104&rft.spage=263&rft.epage=271&rft.aulast=FANG&rft.aufirst=XIN&rft.au=XIN+GUI+FANG&rft.au=JIE+WANG,+SANMING+ZHOU&rft_id=info:doi/10.1017/S0004972720001446">CLASSIFICATION OF TETRAVALENT $\textbf{2}$ -TRANSITIVE NONNORMAL CAYLEY
GRAPHS OF FINITE SIMPLE GROUPS

Authors: XIN GUI FANG; JIE WANG, SANMING ZHOU
Pages: 263 - 271
Abstract: A graph $\Gamma$ is called $(G, s)$-arc-transitive if $G \le \text{Aut} (\Gamma )$ is transitive on the set of vertices of $\Gamma$ and the set of s-arcs of $\Gamma$, where for an integer $s \ge 1$ an s-arc of $\Gamma$ is a sequence of $s+1$ vertices $(v_0,v_1,\ldots ,v_s)$ of $\Gamma$ such that $v_{i-1}$ and $v_i$ are adjacent for
PubDate: 2021-10-01T00:00:00.000Z
DOI: 10.1017/S0004972720001446
Issue No: Vol. 104, No. 2 (2021)

• ON ANALOGUES OF HUPPERT’S CONJECTURE

Authors: YONG YANG
Pages: 272 - 277
Abstract: Let G be a finite group and $\chi$ be a character of G. The codegree of $\chi$ is ${{\operatorname{codeg}}} (\chi ) ={|G: \ker \chi |}/{\chi (1)}$. We write $\pi (G)$ for the set of prime divisors of $|G|$, $\pi ({{\operatorname{codeg}}} (\chi ))$ for the set of prime divisors of ${{\operatorname{codeg}}} (\chi )$ and $\sigma ({{\operatorname{codeg}}} (G))= \max \{|\pi ({{\operatorname{codeg}}} (\chi ))| : \chi \in {\textrm {Irr}}(G)\}$. We show that $|\pi (G)| \leq ({23}/{3}) \sigma ({{\operatorname{codeg}}} (G))$. This improves the main result of Yang and Qian [‘The analog of Huppert’s conjecture on character codegrees’, J. Algebra 478 (2017), 215–219].
PubDate: 2021-10-01T00:00:00.000Z
DOI: 10.1017/S0004972720001409
Issue No: Vol. 104, No. 2 (2021)

• AN ANALOGUE OF HUPPERT’S CONJECTURE FOR CHARACTER CODEGREES

Authors: A. BAHRI; Z. AKHLAGHI, B. KHOSRAVI
Pages: 278 - 286
Abstract: Let G be a finite group, let ${\mathrm{Irr}}(G)$ be the set of all irreducible complex characters of G and let $\chi \in {\mathrm{Irr}}(G)$. Define the codegrees, ${\mathrm{cod}}(\chi ) = |G: {\mathrm{ker}}\chi |/\chi (1)$ and ${\mathrm{cod}}(G) = \{{\mathrm{cod}}(\chi ) \mid \chi \in {\mathrm{Irr}}(G)\}$. We show that the simple group ${\mathrm{PSL}}(2,q)$, for a prime power $q>3$, is uniquely determined by the set of its codegrees.
PubDate: 2021-10-01T00:00:00.000Z
DOI: 10.1017/S0004972721000046
Issue No: Vol. 104, No. 2 (2021)

• ON THE PRONORM OF A GROUP

Authors: MATTIA BRESCIA; ALESSIO RUSSO
Pages: 287 - 294
Abstract: The pronorm of a group G is the set $P(G)$ of all elements $g\in G$ such that X and $X^g$ are conjugate in ${\langle {X,X^g}\rangle }$ for every subgroup X of G. In general the pronorm is not a subgroup, but we give evidence of some classes of groups in which this property holds. We also investigate the structure of a generalised soluble group G whose pronorm contains a subgroup of finite index.
PubDate: 2021-10-01T00:00:00.000Z
DOI: 10.1017/S0004972720001549
Issue No: Vol. 104, No. 2 (2021)

• Z-GROUP&rft.title=Bulletin+of+the+Australian+Mathematical+Society&rft.issn=0004-9727&rft.date=2021&rft.volume=104&rft.spage=295&rft.epage=301&rft.aulast=COSTANZO&rft.aufirst=DAVID&rft.au=DAVID+G.+COSTANZO&rft.au=MARK+L.+LEWIS,+STEFANO+SCHMIDT,+EYOB+TSEGAYE,+GABE+UDELL&rft_id=info:doi/10.1017/S0004972720001318">THE CYCLIC GRAPH OF A Z-GROUP

Authors: DAVID G. COSTANZO; MARK L. LEWIS, STEFANO SCHMIDT, EYOB TSEGAYE, GABE UDELL
Pages: 295 - 301
Abstract: For a group G, we define a graph $\Delta (G)$ by letting $G^{\scriptsize\#}=G\setminus {\{\,1\,\}}$ be the set of vertices and by drawing an edge between distinct elements $x,y\in G^{\scriptsize\#}$ if and only if the subgroup $\langle x,y\rangle$ is cyclic. Recall that a Z-group is a group where every Sylow subgroup is cyclic. In this short note, we investigate $\Delta (G)$ for a Z-group G.
PubDate: 2021-10-01T00:00:00.000Z
DOI: 10.1017/S0004972720001318
Issue No: Vol. 104, No. 2 (2021)

• A BIJECTION OF INVARIANT MEANS ON AN AMENABLE GROUP WITH THOSE ON A
LATTICE SUBGROUP

Authors: JOHN HOPFENSPERGER
Pages: 302 - 307
Abstract: Suppose G is an amenable locally compact group with lattice subgroup $\Gamma$. Grosvenor [‘A relation between invariant means on Lie groups and invariant means on their discrete subgroups’, Trans. Amer. Math. Soc. 288(2) (1985), 813–825] showed that there is a natural affine injection $\iota : {\text {LIM}}(\Gamma )\to {\text {TLIM}}(G)$ and that $\iota$ is a surjection essentially in the case $G={\mathbb R}^d$, $\Gamma ={\mathbb Z}^d$. In the present paper it is shown that $\iota$ is a surjection if and only if $G/\Gamma$ is compact.
PubDate: 2021-10-01T00:00:00.000Z
DOI: 10.1017/S0004972720001525
Issue No: Vol. 104, No. 2 (2021)

• p-BANACH+BEURLING+ALGEBRAS&rft.title=Bulletin+of+the+Australian+Mathematical+Society&rft.issn=0004-9727&rft.date=2021&rft.volume=104&rft.spage=308&rft.epage=319&rft.aulast=DABHI&rft.aufirst=PRAKASH&rft.au=PRAKASH+A.+DABHI&rft.au=DARSHANA+B.+LIKHADA&rft_id=info:doi/10.1017/S0004972720001392">ON ALGEBRA ISOMORPHISMS BETWEEN p-BANACH
BEURLING ALGEBRAS

Authors: PRAKASH A. DABHI; DARSHANA B. LIKHADA
Pages: 308 - 319
Abstract: Let $(G_1,\omega _1)$ and $(G_2,\omega _2)$ be weighted discrete groups and \$0
PubDate: 2021-10-01T00:00:00.000Z
DOI: 10.1017/S0004972720001392
Issue No: Vol. 104, No. 2 (2021)

• ON A FABRIC OF KISSING CIRCLES

Authors: VIERA ČERŇANOVÁ
Pages: 320 - 329
Abstract: Applying circle inversion on a square grid filled with circles, we obtain a configuration that we call a fabric of kissing circles. We focus on the curvature inside the individual components of the fabric, which are two orthogonal frames and two orthogonal families of chains. We show that the curvatures of the frame circles form a doubly infinite arithmetic sequence (bi-sequence), whereas the curvatures in each chain are arranged in a quadratic bi-sequence. We also prove a sufficient condition for the fabric to be integral.
PubDate: 2021-10-01T00:00:00.000Z
DOI: 10.1017/S0004972720001367
Issue No: Vol. 104, No. 2 (2021)

• EXACT LOWER BOUND ON AN ‘EXACTLY ONE’ PROBABILITY

Authors: IOSIF PINELIS
Pages: 330 - 336
Abstract: We obtain the exact lower bound on the probability of the occurrence of exactly one of n random events each of probability p.
PubDate: 2021-10-01T00:00:00.000Z
DOI: 10.1017/S0004972721000071
Issue No: Vol. 104, No. 2 (2021)

• NONNEGATIVE POLYNOMIALS, SUMS OF SQUARES AND THE MOMENT PROBLEM

Authors: ABHISHEK BHARDWAJ
Pages: 337 - 341
PubDate: 2021-10-01T00:00:00.000Z
DOI: 10.1017/S0004972721000058
Issue No: Vol. 104, No. 2 (2021)

• BEYOND SLOW-FAST: RELAXATION OSCILLATIONS IN SINGULARLY PERTURBED
NONSMOOTH SYSTEMS

Authors: SAMUEL JELBART
Pages: 342 - 343
PubDate: 2021-10-01T00:00:00.000Z
DOI: 10.1017/S0004972721000459
Issue No: Vol. 104, No. 2 (2021)

• THE RESOLVENT AND RIESZ TRANSFORM ON CONNECTED SUMS OF MANIFOLDS WITH
DIFFERENT ASYMPTOTIC DIMENSIONS

Authors: DANIEL NIX
Pages: 344 - 345
PubDate: 2021-10-01T00:00:00.000Z
DOI: 10.1017/S0004972721000496
Issue No: Vol. 104, No. 2 (2021)

• INVESTIGATING ATTRIBUTE RISKS AND CONSTRUCTING LINKAGE ERROR MODELS FOR

Authors: Y. MA
Pages: 346 - 348
PubDate: 2021-10-01T00:00:00.000Z
DOI: 10.1017/S0004972721000289
Issue No: Vol. 104, No. 2 (2021)

• MODELLING OF WITHDRAWAL OF A STRATIFIED FLUID FROM A POROUS MEDIUM

Authors: SUHA IBRAHIM SALIH AL-ALI
Pages: 349 - 350
PubDate: 2021-10-01T00:00:00.000Z
DOI: 10.1017/S0004972721000150
Issue No: Vol. 104, No. 2 (2021)

• APPLICATIONS OF SANDPILE ALGORITHMS TO MODELLING EDGE LOCALISED MODE
PHENOMENOLOGY IN MAGNETICALLY CONFINED FUSION PLASMAS

Authors: CRAIG BOWIE
Pages: 351 - 352
PubDate: 2021-10-01T00:00:00.000Z
DOI: 10.1017/S0004972721000460
Issue No: Vol. 104, No. 2 (2021)

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