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BEING CAYLEY AUTOMATIC IS CLOSED UNDER TAKING WREATH PRODUCT WITH VIRTUALLY CYCLIC GROUPS Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2021-04-13 DMITRY BERDINSKY, MURRAY ELDER, JENNIFER TABACK
We extend work of Berdinsky and Khoussainov [‘Cayley automatic representations of wreath products’, International Journal of Foundations of Computer Science 27(2) (2016), 147–159] to show that being Cayley automatic is closed under taking the restricted wreath product with a virtually infinite cyclic group. This adds to the list of known examples of Cayley automatic groups.
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ON ASYMPTOTIC BASES WHICH HAVE DISTINCT SUBSET SUMS Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2021-04-12 SÁNDOR Z. KISS, VINH HUNG NGUYEN
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
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A NEW REFINEMENT OF FINE’S PARTITION THEOREM Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2021-03-29 JIAYU KANG, RUNQIAO LI, ANDREW Y. Z. WANG
We find a new refinement of Fine’s partition theorem on partitions into distinct parts with the minimum part odd. As a consequence, we obtain two companion partition identities. Both analytic and combinatorial proofs are provided.
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ON GRAEV’S THEOREM FOR FREE PRODUCTS OF HAUSDORFF TOPOLOGICAL GROUPS Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2021-03-29 GURAM SAMSONADZE, DALI ZANGURASHVILI
The paper gives a simple proof of Graev’s theorem (asserting that the free product of Hausdorff topological groups is Hausdorff) for a particular case which includes the countable case of $k_\omega $-groups and the countable case of Lindelöf P-groups. For this it is shown that in these particular cases the topology of the free product of Hausdorff topological groups coincides with the $X_0$-topology
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REFLEXIVITY INDEX AND IRRATIONAL ROTATIONS Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2021-03-29 BINGZHANG MA, K. J. HARRISON
We determine the reflexivity index of some closed set lattices by constructing maps relative to irrational rotations. For example, various nests of closed balls and some topological spaces, such as even-dimensional spheres and a wedge of two circles, have reflexivity index 2. We also show that a connected double of spheres has reflexivity index at most 2.
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ADDITIVE BASES AND NIVEN NUMBERS Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2021-03-25 CARLO SANNA
Let $g \geq 2$ be an integer. A natural number is said to be a base-g Niven number if it is divisible by the sum of its base-g digits. Assuming Hooley’s Riemann hypothesis, we prove that the set of base-g Niven numbers is an additive basis, that is, there exists a positive integer $C_g$ such that every natural number is the sum of at most $C_g$ base-g Niven numbers.
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CONNECTED COMPONENTS IN THE INVARIABLY GENERATING GRAPH OF A FINITE GROUP Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2021-03-25 DANIELE GARZONI
We prove that the invariably generating graph of a finite group can have an arbitrarily large number of connected components with at least two vertices.
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A NOTE ON SPIRALLIKE FUNCTIONS Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2021-03-25 Y. J. SIM, D. K. THOMAS
Let f be analytic in the unit disk $\mathbb {D}=\{z\in \mathbb {C}:|z|<1 \}$ and let ${\mathcal S}$ be the subclass of normalised univalent functions with $f(0)=0$ and $f'(0)=1$ , given by $f(z)=z+\sum _{n=2}^{\infty }a_n z^n$ . Let F be the inverse function of f, given by $F(\omega )=\omega +\sum _{n=2}^{\infty }A_n \omega ^n$ for $|\omega |\le r_0(f)$ . Denote by $ \mathcal {S}_p^{* }(\alpha )$ the
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ON A WEIGHTED SUM OF MULTIPLE -VALUES OF FIXED WEIGHT AND DEPTH Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2021-03-19 YOSHIHIRO TAKEYAMA
The multiple T-value, which is a variant of the multiple zeta value of level two, was introduced by Kaneko and Tsumura [‘Zeta functions connecting multiple zeta values and poly-Bernoulli numbers’, in: Various Aspects of Multiple Zeta Functions, Advanced Studies in Pure Mathematics, 84 (Mathematical Society of Japan, Tokyo, 2020), 181–204]. We show that the generating function of a weighted sum of multiple
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FIXED POINTS OF POLYNOMIALS OVER DIVISION RINGS Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2021-03-01 ADAM CHAPMAN, SOLOMON VISHKAUTSAN
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
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EXACT LOWER BOUND ON AN ‘EXACTLY ONE’ PROBABILITY Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2021-02-23 IOSIF PINELIS
We obtain the exact lower bound on the probability of the occurrence of exactly one of n random events each of probability p.
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A SPARSITY RESULT FOR THE DYNAMICAL MORDELL–LANG CONJECTURE IN POSITIVE CHARACTERISTIC Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2021-02-23 DRAGOS GHIOCA, ALINA OSTAFE, SINA SALEH, IGOR E. SHPARLINSKI
We prove a quantitative partial result in support of the dynamical Mordell–Lang conjecture (also known as the DML conjecture) in positive characteristic. More precisely, we show the following: given a field K of characteristic p, a semiabelian variety X defined over a finite subfield of K and endowed with a regular self-map $\Phi :X{\longrightarrow } X$ defined over K, a point $\alpha \in X(K)$ and
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CHARACTERISTIC POLYNOMIALS OF SIMPLE ORDINARY ABELIAN VARIETIES OVER FINITE FIELDS Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2021-02-19 LENNY JONES
We provide an easy method for the construction of characteristic polynomials of simple ordinary abelian varieties ${{\mathcal A}}$ of dimension g over a finite field ${{\mathbb F}}_q$, when $q\ge 4$ and $2g=\rho ^{b-1}(\rho -1)$, for some prime $\rho \ge 5$ with $b\ge 1$. Moreover, we show that ${{\mathcal A}}$ is absolutely simple if $b=1$ and g is prime, but ${{\mathcal A}}$ is not absolutely simple
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FUSION 2-CATEGORIES WITH NO LINE OPERATORS ARE GROUPLIKE Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2021-02-19 THEO JOHNSON-FREYD, MATTHEW YU
We show that if ${\mathcal C}$ is a fusion $2$-category in which the endomorphism category of the unit object is or , then the indecomposable objects of ${\mathcal C}$ form a finite group.
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STRONGLY BOUNDED LOCALLY INDICABLE GROUPS Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2021-02-19 SAMUEL M. CORSON
We give the construction of some locally indicable groups which are strongly bounded (every abstract action on a metric space has bounded orbits).
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AN ANALOGUE OF HUPPERT'S CONJECTURE FOR CHARACTER CODEGREES Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2021-02-08 A. BAHRI, Z. AKHLAGHI, B. KHOSRAVI
Let G be a finite group, let ${\text{Irr}}(G)$ be the set of all irreducible complex characters of G and let $\chi \in {\text{Irr}}(G)$. Define the codegrees, ${\text{cod}}(\chi ) = |G: {\text{ker}}\chi |/\chi (1)$ and ${\text{cod}}(G) = \{{\text{cod}}(\chi ) \mid \chi \in {\text{Irr}}(G)\} $. We show that the simple group ${\text{PSL}}(2,q)$, for a prime power $q>3$, is uniquely determined by the
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SOME OBSERVATIONS AND SPECULATIONS ON PARTITIONS INTO d-TH POWERS Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2021-01-28 MACIEJ ULAS
The aim of this article is to provoke discussion concerning arithmetic properties of the function $p_{d}(n)$ counting partitions of a positive integer n into dth powers, where $d\geq 2$. Apart from results concerning the asymptotic behaviour of $p_{d}(n)$, little is known. In the first part of the paper, we prove certain congruences involving functions counting various types of partitions into dth
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TWO SUPERCONGRUENCES RELATED TO MULTIPLE HARMONIC SUMS Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2021-01-28 ROBERTO TAURASO
Let p be a prime and let x be a p-adic integer. We prove two supercongruences for truncated series of the form $$\begin{align*}\sum_{k=1}^{p-1} \frac{(x)_k}{(1)_k}\cdot \frac{1}{k}\sum_{1\le j_1\le\cdots\le j_r\le k}\frac{1}{j_1^{}\cdots j_r^{}}\quad\mbox{and}\quad \sum_{k=1}^{p-1} \frac{(x)_k(1-x)_k}{(1)_k^2}\cdot \frac{1}{k}\sum_{1\le j_1\le\cdots\le j_r\le k}\frac{1}{j_1^{2}\cdots j_r^{2}}\end{align*}$$which
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ON GOOD APPROXIMATIONS AND THE BOWEN–SERIES EXPANSION Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2021-01-25 LUCA MARCHESE
We consider the continued fraction expansion of real numbers under the action of a nonuniform lattice in $\text {PSL}(2,{\mathbb R})$ and prove metric relations between the convergents and a natural geometric notion of good approximations.
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TRIPLET INVARIANCE AND PARALLEL SUMS Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2021-01-25 TSIU-KWEN LEE, JHENG-HUEI LIN, TRUONG CONG QUYNH
Let R be a semiprime ring with extended centroid C and let $I(x)$ denote the set of all inner inverses of a regular element x in R. Given two regular elements $a, b$ in R, we characterise the existence of some $c\in R$ such that $I(a)+I(b)=I(c)$. Precisely, if $a, b, a+b$ are regular elements of R and a and b are parallel summable with the parallel sum ${\cal P}(a, b)$, then $I(a)+I(b)=I({\cal P}(a
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COPRIME COMMUTATORS IN THE SUZUKI GROUPS Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2021-01-20 GIOVANNI ZINI
In this note we show that every element of a simple Suzuki group $^2B_2(q)$ is a commutator of elements of coprime orders.
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ON THE PRONORM OF A GROUP Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2021-01-20 MATTIA BRESCIA, ALESSIO RUSSO
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.
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A BIJECTION OF INVARIANT MEANS ON AN AMENABLE GROUP WITH THOSE ON A LATTICE SUBGROUP Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2021-01-18 JOHN HOPFENSPERGER
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
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ON ANALOGUES OF HUPPERT'S CONJECTURE Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2021-01-18 YONG YANG
Let G be a finite group and $\chi $ be a character of G. The codegree of $\chi $ is $\text{codeg} (\chi ) ={|G: \ker \chi |}/{\chi (1)}$. We write $\pi (G)$ for the set of prime divisors of $|G|$, $\pi (\text{codeg} (\chi ))$ for the set of prime divisors of $\text{codeg} (\chi )$ and $\sigma (\text{codeg} (G))= \max \{|\pi (\text{codeg} (\chi ))| : \chi \in {\textrm {Irr}}(G)\}$. We show that $|\pi
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STRICTLY REAL FUNDAMENTAL THEOREM OF ALGEBRA USING POLYNOMIAL INTERLACING Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2021-01-18 SOHAM BASU
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
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A VARIANT OF CAUCHY'S ARGUMENT PRINCIPLE FOR ANALYTIC FUNCTIONS WHICH APPLIES TO CURVES CONTAINING ZEROS Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2021-01-18 MAHER BOUDABRA, GREG MARKOWSKY
The standard version of Cauchy's argument principle, applied to a holomorphic function f, requires that f has no zeros on the curve of integration. In this note, we give a generalisation of such a principle which covers the case when f has zeros on the curve, as well as an application.
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BIASES IN INTEGER PARTITIONS 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
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DIVISIBILITY OF CERTAIN SINGULAR OVERPARTITIONS BY POWERS OF 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 $\overline {C}_{6, 2}(n)$
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SUMS OF FOUR SQUARES WITH A CERTAIN RESTRICTION 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
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CLASSIFICATION OF TETRAVALENT -TRANSITIVE NONNORMAL CAYLEY GRAPHS OF FINITE SIMPLE GROUPS 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 $1 \le i \le s-1$
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THE DIAMETER AND RADIUS OF RADIALLY MAXIMAL GRAPHS 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
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A NEW ALGORITHM FOR DECOMPOSING MODULAR TENSOR PRODUCTS 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
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ON ALGEBRA ISOMORPHISMS BETWEEN p-BANACH BEURLING ALGEBRAS 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
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GENERALISED WEIGHTED COMPOSITION OPERATORS ON BERGMAN SPACES INDUCED BY DOUBLING WEIGHTS 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
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ON A FABRIC OF KISSING CIRCLES 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
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WEIGHTED COMPOSITION OPERATORS BETWEEN LORENTZ SPACES 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.
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THE CYCLIC GRAPH OF A Z-GROUP 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
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MÖBIUS–FROBENIUS MAPS ON IRREDUCIBLE POLYNOMIALS 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.
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A NOTE ON k-GALOIS LCD CODES OVER THE RING 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
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ON THE SUM OF PARTS IN THE PARTITIONS OF n INTO DISTINCT PARTS 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.
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THE GENERAL POSITION NUMBER OF THE CARTESIAN PRODUCT OF TWO TREES 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.
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-MINIMUM SPANNING LENGTHS AND AN EXTENSION TO BURNSIDE’S THEOREM ON IRREDUCIBILITY 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 ${\mathcal A}$
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NOTES ON THE K-RATIONAL DISTANCE PROBLEM 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.
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ON PROBLEMS OF -CONNECTED GRAPHS FOR 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
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THE DIOPHANTINE EQUATION IN QUADRATIC NUMBER FIELDS 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.
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A HARMONIC SUM OVER NONTRIVIAL ZEROS OF THE RIEMANN ZETA-FUNCTION 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
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THE CHARACTER GRAPH OF A FINITE GROUP IS PERFECT 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
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FIXED POINT THEOREM FOR AN INFINITE TOEPLITZ MATRIX 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
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THE NUMBER OF ROOTS OF A POLYNOMIAL SYSTEM 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.
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ON THE GROWTH OF LINEAR RECURRENCES IN FUNCTION FIELDS 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
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ON UNRAMIFIED SOLVABLE EXTENSIONS OF SMALL NUMBER FIELDS 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
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SOME HOMOLOGICAL PROPERTIES OF FOURIER ALGEBRAS ON HOMOGENEOUS SPACES 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 $A(G/H)^{**}$
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MULTINOMIAL VANDERMONDE CONVOLUTION VIA PERMANENT 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.
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A NOTE ON JEŚMANOWICZ’ CONJECTURE CONCERNING NONPRIMITIVE PYTHAGOREAN TRIPLES 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
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LOOK, KNAVE 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
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FURTHER REMARKS ON ELEMENTARY RADICALS AND ASSOCIATED FILTERS OF IDEALS 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
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CONGRUENCES MODULO 4 FOR WEIGHT $\textbf{3/2}$ ETA-PRODUCTS 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.
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POINTS OF SMALL HEIGHT ON AFFINE VARIETIES DEFINED OVER FUNCTION FIELDS OF FINITE TRANSCENDENCE DEGREE 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
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PRIME-UNIVERSAL DIAGONAL QUADRATIC FORMS 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
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SPACE OF INITIAL VALUES OF A MAP WITH A QUARTIC INVARIANT 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