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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 100 Issue 3 Cover and Front matter
• PubDate: 2019-12-01T00:00:00.000Z
DOI: 10.1017/S0004972718001570
Issue No: Vol. 100, No. 3 (2019)

• BAZ volume 100 Issue 3 Cover and Back matter
• PubDate: 2019-12-01T00:00:00.000Z
DOI: 10.1017/S0004972718001582
Issue No: Vol. 100, No. 3 (2019)

• BISECTORS IN VECTOR GROUPS OVER INTEGERS
• Authors: SHENG BAU; YIMING LEI
Pages: 353 - 361
Abstract: We present an example of an isometric subspace of a metric space that has a greater metric dimension. We also show that the metric spaces of vector groups over the integers, defined by the generating set of unit vectors, cannot be resolved by a finite set. Bisectors in the spaces of vector groups, defined by the generating set consisting of unit vectors, are completely determined.
PubDate: 2019-12-01T00:00:00.000Z
DOI: 10.1017/S0004972719000807
Issue No: Vol. 100, No. 3 (2019)

• APPLICATIONS OF LERCH’S THEOREM TO PERMUTATIONS OF QUADRATIC
RESIDUES
• Authors: LI-YUAN WANG; HAI-LIANG WU
Pages: 362 - 371
Abstract: Let $n$ be a positive integer and $a$ an integer prime to $n$ . Multiplication by $a$ induces a permutation over $\mathbb{Z}/n\mathbb{Z}=\{\overline{0},\overline{1},\ldots ,\overline{n-1}\}$ . Lerch’s theorem gives the sign of this permutation. We explore some applications of Lerch’s result to permutation problems involving quadratic residues modulo $p$ and confirm some conjectures posed by Sun [‘Quadratic residues and related permutations and identities’, Preprint, 2018, arXiv:1809.07766]. We also study permutations involving arbitrary $k$ th power residues modulo $p$ and primitive roots modulo a power of  $p$ .
PubDate: 2019-12-01T00:00:00.000Z
DOI: 10.1017/S000497271900073X
Issue No: Vol. 100, No. 3 (2019)

• A NOTE ON MATRIX APPROXIMATION IN THE THEORY OF MULTIPLICATIVE DIOPHANTINE
APPROXIMATION
• Authors: YUAN ZHANG
Pages: 372 - 377
Abstract: We prove the Hausdorff measure version of the matrix form of Gallagher’s theorem in the inhomogeneous setting, thereby proving a conjecture posed by Hussain and Simmons [‘The Hausdorff measure version of Gallagher’s theorem—closing the gap and beyond’, J. Number Theory186 (2018), 211–225].
PubDate: 2019-12-01T00:00:00.000Z
DOI: 10.1017/S000497271900039X
Issue No: Vol. 100, No. 3 (2019)

• POSITIVE PROPORTION OF SHORT INTERVALS CONTAINING A PRESCRIBED NUMBER OF
PRIMES
• Authors: DANIELE MASTROSTEFANO
Pages: 378 - 387
Abstract: We prove that for every $m\geq 0$ there exists an $\unicode[STIX]{x1D700}=\unicode[STIX]{x1D700}(m)>0$ such that if $0 PubDate: 2019-12-01T00:00:00.000Z DOI: 10.1017/S0004972719000467 Issue No: Vol. 100, No. 3 (2019) • PRIMES IN ARITHMETIC PROGRESSIONS AND NONPRIMITIVE ROOTS • Authors: PIETER MOREE; MIN SHA Pages: 388 - 394 Abstract: Let$p$be a prime. If an integer$g$generates a subgroup of index$t$in$(\mathbb{Z}/p\mathbb{Z})^{\ast },$then we say that$g$is a$t$-near primitive root modulo$p$. We point out the easy result that each coprime residue class contains a subset of primes$p$of positive natural density which do not have$g$as a$t$-near primitive root and we prove a more difficult variant. PubDate: 2019-12-01T00:00:00.000Z DOI: 10.1017/S0004972719000443 Issue No: Vol. 100, No. 3 (2019) • ON COMPLEX HOMOGENEOUS SINGULARITIES • Authors: QUY THUONG LÊ; LAN PHU HOANG NGUYEN, DUC TAI PHO Pages: 395 - 409 Abstract: We study the singularity at the origin of$\mathbb{C}^{n+1}$of an arbitrary homogeneous polynomial in$n+1$variables with complex coefficients, by investigating the monodromy characteristic polynomials$\unicode[STIX]{x1D6E5}_{l}(t)$as well as the relation between the monodromy zeta function and the Hodge spectrum of the singularity. In the case$n=2$, we give a description of$\unicode[STIX]{x1D6E5}_{C}(t)=\unicode[STIX]{x1D6E5}_{1}(t)$in terms of the multiplier ideal. PubDate: 2019-12-01T00:00:00.000Z DOI: 10.1017/S0004972719000613 Issue No: Vol. 100, No. 3 (2019) • ++++++++ ++++++++ ++++++++++++ ++++++++++++++++$d3$++++++++++++ ++++++++ ++++&rft.title=Bulletin+of+the+Australian+Mathematical+Society&rft.issn=0004-9727&rft.date=2019&rft.volume=100&rft.spage=410&rft.epage=418&rft.aulast=LATERVEER&rft.aufirst=ROBERT&rft.au=ROBERT+LATERVEER&rft_id=info:doi/10.1017/S0004972719000273">A REMARK ON THE CHOW RING OF KÜCHLE FOURFOLDS OF TYPE$d3$• Authors: ROBERT LATERVEER Pages: 410 - 418 Abstract: We prove that a Küchle fourfold$X$of type d3 has a multiplicative Chow–Künneth decomposition. We present some consequences for the Chow ring of$X$. PubDate: 2019-12-01T00:00:00.000Z DOI: 10.1017/S0004972719000273 Issue No: Vol. 100, No. 3 (2019) • DIAMETER OF COMMUTING GRAPHS OF SYMPLECTIC ALGEBRAS • Authors: XIANYA GENG; LITING FAN, XIAOBIN MA Pages: 419 - 427 Abstract: Let$F$be an algebraically closed field of characteristic$0$and let$\operatorname{sp}(2l,F)$be the rank$l$symplectic algebra of all$2l\times 2l$matrices$x=\big(\!\begin{smallmatrix}A & B\\ C & -A^{t}\end{smallmatrix}\!\big)$over$F$, where$A^{t}$is the transpose of$A$and$B,C$are symmetric matrices of order$l$. The commuting graph PubDate: 2019-12-01T00:00:00.000Z DOI: 10.1017/S0004972719000583 Issue No: Vol. 100, No. 3 (2019) • ON THE REGULARITY OF CHARACTER DEGREE GRAPHS • Authors: Z. SAYANJALI; Z. AKHLAGHI, B. KHOSRAVI Pages: 428 - 433 Abstract: Let$G$be a finite group and let$\text{Irr}(G)$be the set of all irreducible complex characters of$G$. Let$\unicode[STIX]{x1D70C}(G)$be the set of all prime divisors of character degrees of$G$. The character degree graph$\unicode[STIX]{x1D6E5}(G)$associated to$G$is a graph whose vertex set is$\unicode[STIX]{x1D70C}(G)$, and there is an edge between two distinct primes$p$and$q$if and only if$pq$divides PubDate: 2019-12-01T00:00:00.000Z DOI: 10.1017/S0004972719000315 Issue No: Vol. 100, No. 3 (2019) • MONOLITHIC BRAUER CHARACTERS • Authors: XIAOYOU CHEN; MARK L. LEWIS Pages: 434 - 439 Abstract: Let$G$be a group,$p$be a prime and$P\in \text{Syl}_{p}(G)$. We say that a$p$-Brauer character$\unicode[STIX]{x1D711}$is monolithic if$G/\ker \unicode[STIX]{x1D711}$is a monolith. We prove that$P$is normal in$G$if and only if$p\nmid \unicode[STIX]{x1D711}(1)$for each monolithic Brauer character$\unicode[STIX]{x1D711}\in \text{IBr}(G)$. When$G$is PubDate: 2019-12-01T00:00:00.000Z DOI: 10.1017/S0004972719000364 Issue No: Vol. 100, No. 3 (2019) • ON THE PROBABILITY DISTRIBUTION OF THE PRODUCT OF POWERS OF ELEMENTS IN COMPACT LIE GROUPS • Authors: VU THE KHOI Pages: 440 - 445 Abstract: In this paper, we study the probability distribution of the word map$w(x_{1},x_{2},\ldots ,x_{k})=x_{1}^{n_{1}}x_{2}^{n_{2}}\cdots x_{k}^{n_{k}}$in a compact Lie group. We show that the probability distribution can be represented as an infinite series. Moreover, in the case of the Lie group$\text{SU}(2)$, our computations give a nice convergent series for the probability distribution. PubDate: 2019-12-01T00:00:00.000Z DOI: 10.1017/S0004972719000492 Issue No: Vol. 100, No. 3 (2019) • SCHUR’S COLOURING THEOREM FOR NONCOMMUTING PAIRS • Authors: TOM SANDERS Pages: 446 - 452 Abstract: For$G$a finite non-Abelian group we write$c(G)$for the probability that two randomly chosen elements commute and$k(G)$for the largest integer such that any$k(G)$-colouring of$G$is guaranteed to contain a monochromatic quadruple$(x,y,xy,yx)$with$xy\neq yx$. We show that$c(G)\rightarrow 0$if and only if$k(G)\rightarrow \infty$. PubDate: 2019-12-01T00:00:00.000Z DOI: 10.1017/S0004972719000406 Issue No: Vol. 100, No. 3 (2019) • A REMARK ON THE SEPARABLE QUOTIENT PROBLEM FOR TOPOLOGICAL GROUPS • Authors: SIDNEY A. MORRIS Pages: 453 - 457 Abstract: The Banach–Mazur separable quotient problem asks whether every infinite-dimensional Banach space$B$has a quotient space that is an infinite-dimensional separable Banach space. The question has remained open for over 80 years, although an affirmative answer is known in special cases such as when$B$is reflexive or even a dual of a Banach space. Very recently, it has been shown to be true for dual-like spaces. An analogous problem for topological groups is: Does every infinite-dimensional (in the topological sense) connected (Hausdorff) topological group$G$have a quotient topological group that is infinite dimensional and metrisable? While this is known to be true if$G$is the underlying topological group of an infinite-dimensional Banach space, it is shown here to be false even if$G$is the underlying topological group of an infinite-dimensional locally convex space. Indeed, it is shown that the free topological vector space on any countably infinite$k_{\unicode[STIX]{x1D714}}$-space is an infinite-dimensional toplogical vector space which does not have any quotient topological group that is infinite dimensional and metrisable. By contrast, the Graev free abelian topological group and the Graev free topological group on any infinite connected Tychonoff space, both of which are connected topological groups, are shown here to have the tubby torus$\mathbb{T}^{\unicode[STIX]{x1D714}}$, which is an infinite-dimensional metrisable group, as a quotient group. PubDate: 2019-12-01T00:00:00.000Z DOI: 10.1017/S0004972719000571 Issue No: Vol. 100, No. 3 (2019) • ESTIMATES OF THE SECOND DERIVATIVE OF BOUNDED ANALYTIC FUNCTIONS • Authors: GANGQIANG CHEN Pages: 458 - 469 Abstract: Assume a point$z$lies in the open unit disk$\mathbb{D}$of the complex plane$\mathbb{C}$and$f$is an analytic self-map of$\mathbb{D}$fixing 0. Then Schwarz’s lemma gives$|f(z)|\leq |z|$, and Dieudonné’s lemma asserts that$|f^{\prime }(z)|\leq \min \{1,(1+|z|^{2})/(4|z|(1-|z|^{2}))\}$. We prove a sharp upper bound for$|f^{\prime \prime }(z)|$depending only on$|z|$. PubDate: 2019-12-01T00:00:00.000Z DOI: 10.1017/S0004972719000595 Issue No: Vol. 100, No. 3 (2019) • BOUNDARY SCHWARZ LEMMA FOR SOLUTIONS TO NONHOMOGENEOUS BIHARMONIC EQUATIONS • Authors: MANAS RANJAN MOHAPATRA; XIANTAO WANG, JIAN-FENG ZHU Pages: 470 - 478 Abstract: We establish a boundary Schwarz lemma for solutions to nonhomogeneous biharmonic equations. PubDate: 2019-12-01T00:00:00.000Z DOI: 10.1017/S0004972719000947 Issue No: Vol. 100, No. 3 (2019) • ESSENTIAL AMENABILITY OF DUAL BANACH ALGEBRAS • Authors: MOHSEN ZIAMANESH; BEHROUZ SHOJAEE, AMIN MAHMOODI Pages: 479 - 488 Abstract: We show that an essentially amenable Banach algebra need not have an approximate identity. This answers a question posed by Ghahramani and Loy [‘Generalized notions of amenability’, J. Funct. Anal. 208 (2004), 229–260]. Essentially Connes-amenable dual Banach algebras are introduced and studied. PubDate: 2019-12-01T00:00:00.000Z DOI: 10.1017/S0004972719000510 Issue No: Vol. 100, No. 3 (2019) • LIPSCHITZ-TYPE INEQUALITIES FOR ANALYTIC FUNCTIONS IN BANACH ALGEBRAS • Authors: SILVESTRU SEVER DRAGOMIR Pages: 489 - 497 Abstract: In this paper we provide some bounds for the quantity$\Vert f(y)-f(x)\Vert$, where$f:D\rightarrow \mathbb{C}$is an analytic function on the domain$D\subset \mathbb{C}$and$x$,$y\in {\mathcal{B}}$, a Banach algebra, with the spectra$\unicode[STIX]{x1D70E}(x)$,$\unicode[STIX]{x1D70E}(y)\subset D$. Applications for the exponential and logarithmic functions on the Banach algebra${\mathcal{B}}$are also given. PubDate: 2019-12-01T00:00:00.000Z DOI: 10.1017/S000497271900042X Issue No: Vol. 100, No. 3 (2019) • ++++++++ ++++++++ ++++++++++++ ++++++++++++++++$R$++++++++++++ ++++++++ ++++-BOUNDEDNESS&rft.title=Bulletin+of+the+Australian+Mathematical+Society&rft.issn=0004-9727&rft.date=2019&rft.volume=100&rft.spage=498&rft.epage=506&rft.aulast=ARNOLD&rft.aufirst=LORIS&rft.au=LORIS+ARNOLD&rft.au=CHRISTIAN+LE+MERDY&rft_id=info:doi/10.1017/S0004972719000431">NEW COUNTEREXAMPLES ON RITT OPERATORS, SECTORIAL OPERATORS AND$R$-BOUNDEDNESS • Authors: LORIS ARNOLD; CHRISTIAN LE MERDY Pages: 498 - 506 Abstract: Let${\mathcal{D}}$be a Schauder decomposition on some Banach space$X$. We prove that if${\mathcal{D}}$is not$R$-Schauder, then there exists a Ritt operator$T\in B(X)$which is a multiplier with respect to${\mathcal{D}}$such that the set$\{T^{n}:n\geq 0\}$is not$R$-bounded. Likewise, we prove that there exists a bounded sectorial operator$A$of type$0$on$X$which is a multiplier with respect to PubDate: 2019-12-01T00:00:00.000Z DOI: 10.1017/S0004972719000431 Issue No: Vol. 100, No. 3 (2019) • ++++++++ ++++++++ ++++++++++++ ++++++++++++++++$F(G\times+\mathbb{R}^{n},2)$++++++++++++ ++++++++ ++++&rft.title=Bulletin+of+the+Australian+Mathematical+Society&rft.issn=0004-9727&rft.date=2019&rft.volume=100&rft.spage=507&rft.epage=517&rft.aulast=ZAPATA&rft.aufirst=CESAR&rft.au=CESAR+A.+IPANAQUE+ZAPATA&rft_id=info:doi/10.1017/S0004972719000479">CATEGORY AND TOPOLOGICAL COMPLEXITY OF THE CONFIGURATION SPACE$F(G\times
\mathbb{R}^{n},2)$• Authors: CESAR A. IPANAQUE ZAPATA Pages: 507 - 517 Abstract: The Lusternik–Schnirelmann category cat and topological complexity TC are related homotopy invariants. The topological complexity TC has applications to the robot motion planning problem. We calculate the Lusternik–Schnirelmann category and topological complexity of the ordered configuration space of two distinct points in the product$G\times \mathbb{R}^{n}$and apply the results to the planar and spatial motion of two rigid bodies in$\mathbb{R}^{2}$and$\mathbb{R}^{3}\$ respectively.
PubDate: 2019-12-01T00:00:00.000Z
DOI: 10.1017/S0004972719000479
Issue No: Vol. 100, No. 3 (2019)

• WEAK SUBORDINATION OF MULTIVARIATE LÉVY PROCESSES
• Authors: KEVIN W. LU
Pages: 518 - 519
PubDate: 2019-12-01T00:00:00.000Z
DOI: 10.1017/S0004972719000662
Issue No: Vol. 100, No. 3 (2019)

• STREAMFLOW AND SOIL MOISTURE FORECASTING WITH HYBRID DATA INTELLIGENT
MACHINE LEARNING APPROACHES: CASE STUDIES IN THE AUSTRALIAN
MURRAY–DARLING BASIN
Pages: 520 - 522
PubDate: 2019-12-01T00:00:00.000Z
DOI: 10.1017/S0004972719000893
Issue No: Vol. 100, No. 3 (2019)

• MATHEMATICAL MODELLING OF THE EVOLUTION OF HUMAN BEHAVIOURS AND STRATEGIC
CHOICE
• Authors: SARA L. LOO
Pages: 523 - 524
PubDate: 2019-12-01T00:00:00.000Z
DOI: 10.1017/S0004972719000984
Issue No: Vol. 100, No. 3 (2019)

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