• Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2021-01-14
BYUNGCHAN KIM; EUNMI KIM

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

更新日期：2021-01-14
• Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2021-01-14
AJIT SINGH; RUPAM BARMAN

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 $\overline {C}_{6, 2}(n)$

更新日期：2021-01-14
• Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2021-01-14
YUE-FENG SHE; HAI-LIANG WU

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

更新日期：2021-01-14
• Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2021-01-11
XIN GUI FANG; JIE WANG; SANMING ZHOU

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 $1 \le i \le s$ and $v_{i-1}\ne v_{i+1}$ for $1 \le i \le s-1$

更新日期：2021-01-11
• Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2021-01-11
PU QIAO; XINGZHI ZHAN

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

更新日期：2021-01-11
• Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2021-01-11
MICHAEL J. J. BARRY

Let p be a prime and let $J_r$ denote a full $r \times r$ Jordan block matrix with eigenvalue $1$ over a field F of characteristic p. For positive integers r and s with $r \leq s$, the Jordan canonical form of the $r s \times r s$ matrix $J_{r} \otimes J_{s}$ has the form $J_{\lambda _1} \oplus J_{\lambda _2} \oplus \cdots \oplus J_{\lambda _{r}}$. This decomposition determines a partition $\lambda 更新日期：2021-01-11 • Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2021-01-11 PRAKASH A. DABHI; DARSHANA B. LIKHADA Let$(G_1,\omega _1)$and$(G_2,\omega _2)$be weighted discrete groups and$0\lt p\leq 1$. We characterise biseparating bicontinuous algebra isomorphisms on the p-Banach algebra$\ell ^p(G_1,\omega _1)$. We also characterise bipositive and isometric algebra isomorphisms between the p-Banach algebras$\ell ^p(G_1,\omega _1)$and$\ell ^p(G_2,\omega _2)$and isometric algebra isomorphisms between$\ell

更新日期：2021-01-11
• Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2021-01-08
BIN LIU

We characterise bounded and compact generalised weighted composition operators acting from the weighted Bergman space $A^p_\omega$, where $0 更新日期：2021-01-08 • Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2021-01-08 VIERA ČERŇANOVÁ Applying circle inversion on a square grid filled with circles, we obtain a configuration that we call a fabric of kissing circles. We focus on the curvature inside the individual components of the fabric, which are two orthogonal frames and two orthogonal families of chains. We show that the curvatures of the frame circles form a doubly infinite arithmetic sequence (bi-sequence), whereas the curvatures 更新日期：2021-01-08 • Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-12-18 CHING-ON LO; ANTHONY WAI-KEUNG LOH We investigate the boundedness, compactness, invertibility and Fredholmness of weighted composition operators between Lorentz spaces. It is also shown that the classes of Fredholm and invertible weighted composition maps between Lorentz spaces coincide when the underlying measure space is nonatomic. 更新日期：2020-12-18 • Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-12-14 DAVID G. COSTANZO; MARK L. LEWIS; STEFANO SCHMIDT; EYOB TSEGAYE; GABE UDELL For a group G, we define a graph$\Delta (G)$by letting$G^{\scriptsize\#}=G{\setminus} \lbrace 1\rbrace $be the set of vertices and by drawing an edge between distinct elements$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)$for a Z-group 更新日期：2020-12-18 • Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-12-14 F. E. BROCHERO MARTÍNEZ; DANIELA OLIVEIRA; LUCAS REIS Let n be a positive integer and let$\mathbb{F} _{q^n}$be the finite field with$q^n$elements, where q is a prime power. We introduce a natural action of the projective semilinear group on the set of monic irreducible polynomials over the finite field$\mathbb{F} _{q^n}$. Our main results provide information on the characterisation and number of fixed points. 更新日期：2020-12-18 • Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-12-14 RONGSHENG WU; MINJIA SHI We study the k-Galois linear complementary dual (LCD) codes over the finite chain ring$R=\mathbb {F}_q+u\mathbb {F}_q$with$u^2=0$, where$q=p^e$and p is a prime number. We give a sufficient condition on the generator matrix for the existence of k-Galois LCD codes over R. Finally, we show that a linear code over R (for$k=0, q> 3$) is equivalent to a Euclidean LCD code, and a linear code over R 更新日期：2020-12-18 • Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-12-11 MIRCEA MERCA We investigate the sum of the parts in all the partitions of n into distinct parts and give two infinite families of linear inequalities involving this sum. The results can be seen as new connections between partitions and divisors. 更新日期：2020-12-18 • Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-12-04 JING TIAN; KEXIANG XU; SANDI KLAVŽAR The general position number of a connected graph is the cardinality of a largest set of vertices such that no three pairwise-distinct vertices from the set lie on a common shortest path. In this paper it is proved that the general position number is additive on the Cartesian product of two trees. 更新日期：2020-12-18 • Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-12-02 W. E. LONGSTAFF We introduce the$\textbf{h}$-minimum spanning length of a family${\mathcal A}$of$n\times n$matrices over a field$\mathbb F$, where$\textbf{h}$is a p-tuple of positive integers, each no more than n. For an algebraically closed field$\mathbb F$, Burnside’s theorem on irreducibility is essentially that the$(n,n,\ldots ,n)$-minimum spanning length of${\mathcal A}$exists if${\mathcal A}$更新日期：2020-12-18 • Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-12-01 NGUYEN XUAN THO Let K be an algebraic number field. We investigate the K-rational distance problem and prove that there are infinitely many nonisomorphic cubic number fields and a number field of degree n for every$n\geq 2$in which there is a point in the plane of a unit square at K-rational distances from the four vertices of the square. 更新日期：2020-12-18 • Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-12-01 MICHAL STAŠ; JURAJ VALISKA A connected graph G is$\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

更新日期：2020-12-18
• Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-11-06
ANDREW LI

Aigner showed in 1934 that nontrivial quadratic solutions to $x^4 + y^4 = 1$ exist only in $\mathbb Q(\sqrt {-7})$ . Following a method of Mordell, we show that nontrivial quadratic solutions to $x^4 + 2^ny^4 = 1$ arise from integer solutions to the equations $X^4 \pm 2^nY^4 = Z^2$ investigated in 1853 by V. A. Lebesgue.

更新日期：2020-12-18
• Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-11-20
RICHARD P. BRENT; DAVID J. PLATT; TIMOTHY S. TRUDGIAN

We consider the sum $\sum 1/\gamma$ , where $\gamma$ ranges over the ordinates of nontrivial zeros of the Riemann zeta-function in an interval $(0,T]$ , and examine its behaviour as $T \to \infty$ . We show that, after subtracting a smooth approximation $({1}/{4\pi }) \log ^2(T/2\pi ),$ the sum tends to a limit $H \approx -0.0171594$ , which can be expressed as an integral. We calculate H to high

更新日期：2020-11-21
• Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-11-18
MAHDI EBRAHIMI

For a finite group G, let $\Delta (G)$ denote the character graph built on the set of degrees of the irreducible complex characters of G. A perfect graph is a graph $\Gamma$ in which the chromatic number of every induced subgraph $\Delta$ of $\Gamma$ equals the clique number of $\Delta$ . We show that the character graph $\Delta (G)$ of a finite group G is always a perfect graph. We also prove

更新日期：2020-11-18
• Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-11-09
VYACHESLAV M. ABRAMOV

For an infinite Toeplitz matrix T with nonnegative real entries we find the conditions under which the equation $\boldsymbol {x}=T\boldsymbol {x}$ , where $\boldsymbol {x}$ is an infinite vector column, has a nontrivial bounded positive solution. The problem studied in this paper is associated with the asymptotic behaviour of convolution-type recurrence relations and can be applied to problems arising

更新日期：2020-11-09
• Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-11-09
NGUYEN CONG MINH; LUU BA THANG; TRAN NAM TRUNG

Let I be a zero-dimensional ideal in the polynomial ring $K[x_1,\ldots ,x_n]$ over a field K. We give a bound for the number of roots of I in $K^n$ counted with combinatorial multiplicity. As a consequence, we give a proof of Alon’s combinatorial Nullstellensatz.

更新日期：2020-11-09
• Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-11-09
CLEMENS FUCHS; SEBASTIAN HEINTZE

Let $(G_n)_{n=0}^{\infty }$ be a nondegenerate linear recurrence sequence whose power sum representation is given by $G_n = a_1(n) \alpha _1^n + \cdots + a_t(n) \alpha _t^n$ . We prove a function field analogue of the well-known result in the number field case that, under some nonrestrictive conditions, $|{G_n}| \geq ( \max _{j=1,\ldots ,t} |{\alpha _j}| )^{n(1-\varepsilon )}$ for $n$ large

更新日期：2020-11-09
• Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-11-09
JOACHIM KÖNIG

We investigate unramified extensions of number fields with prescribed solvable Galois group G and certain extra conditions. In particular, we are interested in the minimal degree of a number field K, Galois over $\mathbb {Q}$ , such that K possesses an unramified G-extension. We improve the best known bounds for the degree of such number fields K for certain classes of solvable groups, in particular

更新日期：2020-11-09
• Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-11-09
REZA ESMAILVANDI; MEHDI NEMATI

Let $H$ be a compact subgroup of a locally compact group $G$ . We first investigate some (operator) (co)homological properties of the Fourier algebra $A(G/H)$ of the homogeneous space $G/H$ such as (operator) approximate biprojectivity and pseudo-contractibility. In particular, we show that $A(G/H)$ is operator approximately biprojective if and only if $G/H$ is discrete. We also show that $A(G/H)^{**}$

更新日期：2020-11-09
• Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-11-06
KIJTI RODTES

We provide a generalised Laplace expansion for the permanent function and, as a consequence, we re-prove a multinomial Vandermonde convolution. Some combinatorial identities are derived by applying special matrices to the expansion.

更新日期：2020-11-06
• Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-10-21
THOMAS MORRILL

We examine a recursive sequence in which $s_n$ is a literal description of what the binary expansion of the previous term $s_{n-1}$ is not. By adapting a technique of Conway, we determine the limiting behaviour of $\{s_n\}$ and dynamics of a related self-map of $2^{\mathbb {N}}$ . Our main result is the existence and uniqueness of a pair of binary sequences, each the complement-description of the other

更新日期：2020-10-30
• Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-10-21
YASUTSUGU FUJITA; MAOHUA LE

Jeśmanowicz conjectured that $(x,y,z)=(2,2,2)$ is the only positive integer solution of the equation $(*)\; ((\kern1.5pt f^2-g^2)n)^x+(2fgn)^y=((\kern1.5pt f^2+g^2)n)^x$ , where n is a positive integer and f, g are positive integers such that $f>g$ , $\gcd (\kern1.5pt f,g)=1$ and $f \not \equiv g\pmod 2$ . Using Baker’s method, we prove that: (i) if $n>1$ , $f \ge 98$ and $g=1$ , then $(*)$ has no

更新日期：2020-10-30
• Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-10-08
E. P. COJUHARI; B. J. GARDNER

Ryabukhin showed that there is a correspondence between elementary radical classes of rings and certain filters of ideals of the free ring on one generator, analogous to the Gabriel correspondence between torsion classes of left unital modules and certain filters of left ideals of the coefficient ring. This correspondence is further explored here. All possibilities for the intersection of the ideals

更新日期：2020-10-08
• Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-10-05
RONG CHEN; F. G. GARVAN

We find and prove a class of congruences modulo 4 for eta-products associated with certain ternary quadratic forms. This study was inspired by similar conjectured congruences modulo 4 for certain mock theta functions.

更新日期：2020-10-05
• Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-10-05
HUO-JUN RUAN; JIAN-CI XIAO; BING YANG

The notion of recurrent fractal interpolation functions (RFIFs) was introduced by Barnsley et al. [‘Recurrent iterated function systems’, Constr. Approx.5 (1989), 362–378]. Roughly speaking, the graph of an RFIF is the invariant set of a recurrent iterated function system on $\mathbb {R}^2$ . We generalise the definition of RFIFs so that iterated functions in the recurrent system need not be contractive

更新日期：2020-10-05
• Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-10-05
DRAGOS GHIOCA; DAC-NHAN-TAM NGUYEN

We provide a direct proof of a Bogomolov-type statement for affine varieties V defined over function fields K of finite transcendence degree over an arbitrary field k, generalising a previous result (obtained through a different approach) of the first author in the special case when K is a function field of transcendence degree $1$ . Furthermore, we obtain sharp lower bounds for the Weil height of

更新日期：2020-10-05
• Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-10-05
JANGWON JU; DAEJUN KIM; KYOUNGMIN KIM; MINGYU KIM; BYEONG-KWEON OH

A (positive definite and integral) quadratic form is said to be prime-universal if it represents all primes. Recently, Doyle and Williams [‘Prime-universal quadratic forms $ax^2+by^2+cz^2$ and $ax^2+by^2+cz^2+dw^2$ ’, Bull. Aust. Math. Soc.101 (2020), 1–12] classified all prime-universal diagonal ternary quadratic forms and all prime-universal diagonal quaternary quadratic forms under two conjectures

更新日期：2020-10-05
• Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-10-05
W. E. LONGSTAFF

We introduce the notion of the slot length of a family of matrices over an arbitrary field ${\mathbb {F}}$ . Using this definition it is shown that, if $n\ge 5$ and A and B are $n\times n$ complex matrices with A unicellular and the pair $\{A,B\}$ irreducible, the slot length s of $\{A,B\}$ satisfies $2\le s\le n-1$ , where both inequalities are sharp, for every n. It is conjectured that the slot length

更新日期：2020-10-05
• Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-10-05
GIORGIO GUBBIOTTI; NALINI JOSHI

We compactify and regularise the space of initial values of a planar map with a quartic invariant and use this construction to prove its integrability in the sense of algebraic entropy. The system has certain unusual properties, including a sequence of points of indeterminacy in $\mathbb {P}^{1}\!\times \mathbb {P}^{1}$ . These indeterminacy points lie on a singular fibre of the mapping to a corresponding

更新日期：2020-10-05
• Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-10-05
PENG-FEI ZHANG; XIN-HAN DONG

For $n\geq 3$ , let $Q_n\subset \mathbb {C}$ be an arbitrary regular n-sided polygon. We prove that the Cauchy transform $F_{Q_n}$ of the normalised two-dimensional Lebesgue measure on $Q_n$ is univalent and starlike but not convex in $\widehat {\mathbb {C}}\setminus Q_n$ .

更新日期：2020-10-05
• Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-10-02

Suppose that $\mathcal {A}$ is a unital subhomogeneous C*-algebra. We show that every central sequence in $\mathcal {A}$ is hypercentral if and only if every pointwise limit of a sequence of irreducible representations is multiplicity free. We also show that every central sequence in $\mathcal {A}$ is trivial if and only if every pointwise limit of irreducible representations is irreducible. Finally

更新日期：2020-10-02
• Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-10-02
HAZEL BROWNE

We present several results on the connectivity of McKay quivers of finite-dimensional complex representations of finite groups, with no restriction on the faithfulness or self-duality of the representations. We give examples of McKay quivers, as well as quivers that cannot arise as McKay quivers, and discuss a necessary and sufficient condition for two finite groups to share a connected McKay quiver

更新日期：2020-10-02
• Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-10-02
ANUP B. DIXIT; KAMALAKSHYA MAHATAB

We study lower bounds of a general family of L-functions on the $1$ -line. More precisely, we show that for any $F(s)$ in this family, there exist arbitrarily large t such that $F(1+it)\geq e^{\gamma _F} (\log _2 t + \log _3 t)^m + O(1)$ , where m is the order of the pole of $F(s)$ at $s=1$ . This is a generalisation of the result of Aistleitner, Munsch and Mahatab [‘Extreme values of the Riemann zeta

更新日期：2020-10-02
• Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-10-02
NEIL A. WATSON

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. A temperature u on D is called extendable by v if there is a supertemperature w on E such that $w=u$ on D and $w=v$ on $E\backslash \overline D$ . From earlier work of N. A. Watson, [‘Extendable temperatures’, Bull. Aust. Math. Soc.100 (2019), 297–303], either

更新日期：2020-10-02
• Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-09-22
SHI-QIANG CHEN; XIAO-HUI YAN

For a given set $S\subseteq \mathbb {Z}_m$ and $\overline {n}\in \mathbb {Z}_m$ , let $R_S(\overline {n})$ denote the number of solutions of the equation $\overline {n}=\overline {s}+\overline {s'}$ with ordered pairs $(\overline {s},\overline {s'})\in S^2$ . We determine the structure of $A,B\subseteq \mathbb {Z}_m$ with $|(A\cup B)\setminus (A\cap B)|=m-2$ such that $R_{A}(\overline {n})=R_{B}(\overline 更新日期：2020-09-22 • Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-09-21 BENJAMIN SAMBALE Linckelmann and Murphy have classified the Morita equivalence classes of p-blocks of finite groups whose basic algebra has dimension at most$12$. We extend their classification to dimension$13$and$14$. As predicted by Donovan’s conjecture, we obtain only finitely many such Morita equivalence classes. 更新日期：2020-09-21 • Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-09-21 FARIDEH SHAFIEI; MOHAMMAD REZA DARAFSHEH; FARROKH SHIRJIAN A finite group whose irreducible complex characters are rational-valued is called a rational group. The aim of this paper is to determine the rational almost simple and rational quasi-simple groups. 更新日期：2020-09-21 • Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-09-17 SHANE CHERN; DAZHAO TANG We construct eta-quotient representations of two families of q-series involving the Rogers–Ramanujan continued fraction by establishing related recurrence relations. We also display how these eta-quotient representations can be utilised to dissect certain q-series identities. 更新日期：2020-09-18 • Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-09-09 ELCHIN HASANALIZADE In this paper we sharpen Hildebrand’s earlier result on a conjecture of Erdős on limit points of the sequence${\{d(n)/d(n+1)\}}$. 更新日期：2020-09-10 • Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-09-09 SHAMIK DAS; ANUPAM SAIKIA The notion of$\theta $-congruent numbers is a generalisation of congruent numbers where one considers triangles with an angle$\theta $such that$\cos \theta $is a rational number. In this paper we discuss a criterion for a natural number to be$\theta $-congruent over certain real number fields. 更新日期：2020-09-10 • Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-06-18 THEERAYOOT PHOCHAI; SATIT SAEJUNG Zhang [‘On hyperstability of generalised linear functional equations in several variables’, Bull. Aust. Math. Soc.92 (2015), 259–267] proved a hyperstability result for generalised linear functional equations in several variables by using Brzdęk’s fixed point theorem. We complete and extend Zhang’s result. We illustrate our results for general linear equations in two variables and Fréchet equations 更新日期：2020-08-21 • Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-07-17 XIAOXIA WANG; MINGBING YUE By making use of the ‘creative microscoping’ method, Guo and Zudilin [‘Dwork-type supercongruences through a creative$q$-microscope’, Preprint, 2020, arXiv:2001.02311] proved several Dwork-type supercongruences, including some conjectures of Swisher. In this paper, we apply the Guo–Zudilin method to prove a new Dwork-type supercongruence, which uniformly generalises several conjectures of Swisher 更新日期：2020-07-17 • Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-07-13 BURCU ÇINARCI; TEMHA ERKOÇ Isaacs and Seitz conjectured that the derived length of a finite solvable group$G$is bounded by the cardinality of the set of all irreducible character degrees of$G$. We prove that the conjecture holds for$G$if the degrees of nonlinear monolithic characters of$G$having the same kernels are distinct. Also, we show that the conjecture is true when$G$has at most three nonlinear monolithic characters 更新日期：2020-07-13 • Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-07-03 SAI WANG; DEIN WONG; FENGLEI TIAN An undirected graph$G$is determined by its$T$-gain spectrum (DTS) if every$T$-gain graph cospectral to$G$is switching equivalent to$G$. We show that the complete graph$K_{n}$and the graph$K_{n}-e$obtained by deleting an edge from$K_{n}$are DTS, the star$K_{1,n}$is DTS if and only if$n\leq 2$, and an odd path$P_{2m+1}$is not DTS if$m\geq 2$. We give an operation for constructing 更新日期：2020-07-03 • Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-07-03 YUEYU WU; YUNQING ZHANG; YAOJUN CHEN An edge-coloured graph$G$is called properly connected if any two vertices are connected by a properly coloured path. The proper connection number,$pc(G)$, of a graph$G$, is the smallest number of colours that are needed to colour$G$such that it is properly connected. Let$\unicode[STIX]{x1D6FF}(n)$denote the minimum value such that$pc(G)=2$for any 2-connected incomplete graph$G$of order 更新日期：2020-07-03 • Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-07-03 FLORICA C. CÎRSTEA Urysohn’s lemma is a crucial property of normal spaces that deals with separation of closed sets by continuous functions. It is also a fundamental ingredient in proving the Tietze extension theorem, another property of normal spaces that deals with the existence of extensions of continuous functions. Using the Cantor function, we give alternative proofs for Urysohn’s lemma and the Tietze extension 更新日期：2020-07-03 • Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-07-01 SHANE CHERN 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 更新日期：2020-07-01 • Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-06-24 AKSHAT MUDGAL Given$d\in \mathbb{N}$, we establish sum-product estimates for finite, nonempty subsets of$\mathbb{R}^{d}$. This is equivalent to a sum-product result for sets of diagonal matrices. In particular, let$A$be a finite, nonempty set of$d\times d$diagonal matrices with real entries. Then, for all$\unicode[STIX]{x1D6FF}_{1}<1/3+5/5277$, $$\begin{eqnarray}|A+A|+|A\cdot A|\gg _{d}|A|^{1+\unicode 更新日期：2020-06-24 • Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-06-18 YOUNG JAE SIM; DEREK K. THOMAS Let f be analytic in the unit disk \mathbb{D}=\{z\in \mathbb{C}:|z|<1\} and {\mathcal{S}} be the subclass of normalised univalent functions given by f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n} for z\in \mathbb{D} . We give sharp upper and lower bounds for |a_{3}|-|a_{2}| and other related functionals for the subclass {\mathcal{F}}_{O}(\unicode[STIX]{x1D706}) of Ozaki close-to-convex functions 更新日期：2020-06-18 • Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-06-18 HUAIBIN LI Consider an intermittent map f_{\unicode[STIX]{x1D705}}:[0,1]\rightarrow [0,1] and a Hölder continuous potential \unicode[STIX]{x1D711}:[0,1]\rightarrow \mathbb{R} . We show that \unicode[STIX]{x1D719} is stochastic for f_{\unicode[STIX]{x1D705}} if and only if the topological pressure P(f_{\unicode[STIX]{x1D705}},\unicode[STIX]{x1D711}) satisfies P(f_{\unicode[STIX]{x1D705}},\unicode[ 更新日期：2020-06-18 • Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-06-11 JEAN-CHRISTOPHE BOURIN; EUN-YOUNG LEE We obtain several norm and eigenvalue inequalities for positive matrices partitioned into four blocks. The results involve the numerical range W(X) of the off-diagonal block X , especially the distance d from 0 to W(X) . A special consequence is an estimate,$$\begin{eqnarray}\text{diam}\,W\left(\left[\begin{array}{@{}cc@{}}A & X\\ X^{\ast } & B\end{array}\right]\right)-\text{diam}\,W\bi 更新日期：2020-06-11 • Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-06-10 MARTIN BUNDER; PETER NICKOLAS; JOSEPH TONIEN For a positive real number$t\$ , define the harmonic continued fraction $$\begin{eqnarray}\text{HCF}(t)=\biggl[\frac{t}{1},\frac{t}{2},\frac{t}{3},\ldots \biggr].\end{eqnarray}$$ We prove that $$\begin{eqnarray}\text{HCF}(t)=\frac{1}{1-2t(\frac{1}{t+2}-\frac{1}{t+4}+\frac{1}{t+6}-\cdots \,)}.\end{eqnarray}$$

更新日期：2020-06-10
• Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-06-10
MIN CHEN

We derive estimates relating the values of a solution at any two points to the distance between the points for quasilinear parabolic equations on compact Riemannian manifolds under Ricci flow.

更新日期：2020-06-10
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