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Showing papers in "Bulletin of The Australian Mathematical Society in 2019"


Journal ArticleDOI
TL;DR: In this article, the authors apply the Schwarz lemma to find general formulas for the third coefficient of Caratheodory functions dependent on a parameter in the closed unit polydisk.
Abstract: We apply the Schwarz lemma to find general formulas for the third coefficient of Caratheodory functions dependent on a parameter in the closed unit polydisk. Next we find sharp estimates of the Hankel determinant , that is, the subclass of the class of functions convex in the direction of the imaginary axis.

30 citations


Journal ArticleDOI
TL;DR: The domination game and the game domination number for split graphs were investigated in this paper, and it was shown that the domination number is n/2 for any isolate-free split graph.
Abstract: We investigate the domination game and the game domination number $\\unicode[STIX]{x1D6FE}_{g}$ in the class of split graphs. We prove that $\\unicode[STIX]{x1D6FE}_{g}(G)\\leq n/2$ for any isolate-free $n$ -vertex split graph $G$ , thus strengthening the conjectured $3n/5$ general bound and supporting Rall’s $\\lceil n/2\\rceil$ -conjecture. We also characterise split graphs of even order with $\\unicode[STIX]{x1D6FE}_{g}(G)=n/2$ .

20 citations



Journal ArticleDOI
Kai Liu1, Peiyong Yu1
TL;DR: In this article, the authors give sufficient conditions for the periodicity of entire functions based on a conjecture of C. C. Yang, using the concepts of value sharing, unique polynomial of the entire functions and Picard exceptional value.
Abstract: We give some sufficient conditions for the periodicity of entire functions based on a conjecture of C. C. Yang, using the concepts of value sharing, unique polynomial of entire functions and Picard exceptional value.

11 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that the maximum size of a clique in a graph G is bounded by the chromatic number of the clique and the maximum number of cliques in G.
Abstract: As usual, $P_{n}$ ( $n\\geq 1$ ) denotes the path on $n$ vertices. The gem is the graph consisting of a $P_{4}$ together with an additional vertex adjacent to each vertex of the $P_{4}$ . A graph is called ( $P_{5}$ , gem)-free if it has no induced subgraph isomorphic to a $P_{5}$ or to a gem. For a graph $G$ , $\\unicode[STIX]{x1D712}(G)$ denotes its chromatic number and $\\unicode[STIX]{x1D714}(G)$ denotes the maximum size of a clique in $G$ . We show that $\\unicode[STIX]{x1D712}(G)\\leq \\lfloor \\frac{3}{2}\\unicode[STIX]{x1D714}(G)\\rfloor$ for every ( $P_{5}$ , gem)-free graph $G$ .

8 citations


Journal ArticleDOI
TL;DR: In this paper, vanishing Taylor coefficients along arithmetic progressions in four quotients of infinite product expansions were investigated and similar results were obtained for the case where the product expansion is a product expansion.
Abstract: Motivated by Ramanujan’s continued fraction and the work of Richmond and Szekeres [‘The Taylor coefficients of certain infinite products’, Acta Sci. Math. (Szeged) 40(3–4) (1978), 347–369], we investigate vanishing coefficients along arithmetic progressions in four quotients of infinite product expansions and obtain similar results. For example, $a_{1}(5n+4)=0$ , where $a_{1}(n)$ is defined by $$\\begin{eqnarray}\\displaystyle {\\displaystyle \\frac{(q,q^{4};q^{5})_{\\infty }^{3}}{(q^{2},q^{3};q^{5})_{\\infty }^{2}}}=\\mathop{\\sum }_{n=0}^{\\infty }a_{1}(n)q^{n}. & & \\displaystyle \ onumber\\end{eqnarray}$$

8 citations


Journal ArticleDOI
TL;DR: In this paper, the Variance Gamma model is extended with Thorin subordinators, which is an extension of both univariate and multivariate subordination and provides two applications: a weak formulation of Variance-alpha-Gamma process that exhibits a wider range of dependence than using traditional subordination, and fit this model to a S&P500-FTSE100 data set.
Abstract: Subordinating a multivariate L\\'evy process, the subordinate, with a univariate subordinator gives rise to a pathwise construction of a new L\\'evy process provided the subordinator and the subordinate are independent processes. The Variance Gamma model in finance was generated accordingly from a Brownian motion and a Gamma process. Alternatively, multivariate subordination can be used to create L\\'evy processes, but this requires the subordinate to have independent or indistinguishable components. In this paper, we show that there exists another operation acting on pairs (T,X) of L\\'evy processes which creates a L\\'evy process X \\.o T. Here, T is a subordinator, but X is an arbitrary L\\'evy process with possibly dependent coordinates. We show that this method is an extension of both univariate and multivariate subordination and provide two applications. Firstly, we give a weak formulation of the Variance-alpha-Gamma-process that exhibits a wider range of dependence than using traditional subordination, and fit this model to a S&P500-FTSE100 data set. Secondly, the Variance Generalised Gamma Convolution class of L\\'evy processes formed by subordinating Brownian motion with Thorin subordinators is further extended using weak subordination.

8 citations


Journal ArticleDOI
TL;DR: In this article, the authors investigated whether the property of having linear quotients is inherited by ideals generated by multigraded shifts of a Borel ideal and a square-free Borel Ideal.
Abstract: We investigate whether the property of having linear quotients is inherited by ideals generated by multigraded shifts of a Borel ideal and a squarefree Borel ideal. We show that the ideal generated by the first multigraded shifts of a Borel ideal has linear quotients, as do the ideal generated by the . Furthermore, we show that equigenerated squarefree Borel ideals share the property of being squarefree Borel with the ideals generated by multigraded shifts.

8 citations


Journal ArticleDOI
TL;DR: In this article, Lerch's theorem is applied to permutation problems involving quadratic residues modulo $p$ and confirm some conjectures posed by Sun and Gurewitz.
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 $p$ .

7 citations


Journal ArticleDOI
TL;DR: In this paper, the authors obtained a new sum-product estimate in prime fields for sets of large cardinality using a point-plane incidence bound rather than the point-line incidence bound used by Shakan and Shkredov.
Abstract: We obtain a new sum–product estimate in prime fields for sets of large cardinality. In particular, we show that if to a point–plane incidence bound of Rudnev [‘On the number of incidences between points and planes in three dimensions’, Combinatorica 38(1) (2017), 219–254] rather than a point–line incidence bound used by Shakan and Shkredov.

7 citations


Journal ArticleDOI
TL;DR: A subset -sum-free subset of an abelian group is defined in this paper, where a subset is defined as a subset of a set of subsets of a group.
Abstract: A subset -sum-free subset of an abelian group’, Int. J. Number Theory 5(6) (2009), 953–971].

Journal ArticleDOI
TL;DR: In this paper, the convolution of two right halfplane mappings was reformulated correctly and provided a solution to it in a more general form, where the normalization of the functions was considered incorrectly.
Abstract: Dorff et al. [‘Convolutions of harmonic convex mappings’, Complex Var. Elliptic Equ. 57(5) (2012), 489–503] formulated a question concerning the convolution of two right half-plane mappings, where the normalisation of the functions was considered incorrectly. In this paper, we reformulate the problem correctly and provide a solution to it in a more general form. We also obtain two new theorems which correct and improve related results.

Journal ArticleDOI
TL;DR: In this paper, the authors give sharp bounds for the initial coefficients of the Taylor expansion of strongly Ozaki close-to-convex functions, and of the inverse function, together with some growth estimates.
Abstract: Let . We give sharp bounds for the initial coefficients of the Taylor expansion of such functions in the class of strongly Ozaki close-to-convex functions, and of the initial coefficients of the inverse function, together with some growth estimates.

Journal ArticleDOI
TL;DR: In this paper, the authors characterise finite unitary rings such that all Sylow subgroups of the group of units are cyclic, up to isomorphism, and they show that up to cyclic isomorphisms, a ring of odd cardinality is one of the three types of rings in the unitary ring hierarchy.
Abstract: We characterise finite unitary rings $R$ such that all Sylow subgroups of the group of units $R^{\\ast }$ are cyclic. To be precise, we show that, up to isomorphism, $R$ is one of the three types of rings in $\\{O,E,O\\oplus E\\}$ , where $O\\in \\{GF(q),\\mathbb{Z}_{p^{\\unicode[STIX]{x1D6FC}}}\\}$ is a ring of odd cardinality and $E$ is a ring of cardinality $2^{n}$ which is one of seven explicitly described types.

Journal ArticleDOI
TL;DR: In this article, it was shown that if a subgroup of a finite group is a quasimax-quasiormal subgroup, then for every chief factor (H/K) of the semidirect product (G/C), the subgroup is a σ-primary subgroup.
Abstract: Let $G$ be a finite group and $\sigma =\{\sigma_{i} | i\in I\}$ some partition of the set of all primes $\Bbb{P}$, that is, $\sigma =\{\sigma_{i} | i\in I \}$, where $\Bbb{P}=\bigcup_{i\in I} \sigma_{i}$ and $\sigma_{i}\cap \sigma_{j}= \emptyset $ for all $i e j$. We say that $G$ is $\sigma$-primary if $G$ is a $\sigma _{i}$-group for some $i$. A subgroup $A$ of $G$ is said to be: ${\sigma}$-subnormal in $G$ if there is a subgroup chain $A=A_{0} \leq A_{1} \leq \cdots \leq A_{n}=G$ such that either $A_{i-1}\trianglelefteq A_{i}$ or $A_{i}/(A_{i-1})_{A_{i}}$ is $\sigma$-primary for all $i=1, \ldots, n$, modular in $G$ if the following conditions hold: (i) $\langle X, A \cap Z \rangle=\langle X, A \rangle \cap Z$ for all $X \leq G, Z \leq G$ such that $X \leq Z$, and (ii) $\langle A, Y \cap Z \rangle=\langle A, Y \rangle \cap Z$ for all $Y \leq G, Z \leq G$ such that $A \leq Z$. In this paper, a subgroup $A$ of $G$ is called $\sigma$-quasinormal in $G$ if $L$ is modular and ${\sigma}$-subnormal in $G$. We study $\sigma$-quasinormal subgroups of $G$. In particular, we prove that if a subgroup $H$ of $G$ is $\sigma$-quasinormal in $G$, then for every chief factor $H/K$ of $G$ between $H^{G}$ and $H_{G}$ the semidirect product $(H/K)\rtimes (G/C_{G}(H/K))$ is $\sigma$-primary.

Journal ArticleDOI
TL;DR: In this article, two conjectural congruences on the th Apery number were proved, which were recently proposed by Z.-H. Sun and Z.-C. Sun.
Abstract: We prove two conjectural congruences on the th Apery number, which were recently proposed by Z.-H. Sun.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the ring of integers is irreducible over the integers of the set of integers of $mathbb{Q}(n,mB), where the discriminant of $f(x) = 0.
Abstract: Suppose that $f(x)=x^{n}+A(Bx+C)^{m}\\in \\mathbb{Z}[x]$ , with $n\\geq 3$ and $1\\leq m

Journal ArticleDOI
TL;DR: In this paper, it was shown that a character degree graph associated to a graph whose vertex set is, and there is an edge between two distinct primes is -regular for some natural number if and only if is a regular bipartite graph.
Abstract: Let be a finite group and let be the set of all irreducible complex characters of . Let be the set of all prime divisors of character degrees of . The character degree graph associated to is a graph whose vertex set is , and there is an edge between two distinct primes and if and only if divides for some . We prove that is -regular for some natural number if and only if is a regular bipartite graph.


Journal ArticleDOI
TL;DR: In this paper, it was shown that there exists a Ritt operator that is a multiplier with respect to a Schauder decomposition on a Banach space, such that the set of the set is not $R$-bounded.
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 $\mathcal D$, such that the set $\{e^{-tA}\, : \, t\geq 0\}$ is not $R$-bounded.

Journal ArticleDOI
TL;DR: In this article, the authors study best proximity points in the framework of metric spaces with -distances and extend, generalise and unify several well-known fixed point results in the literature.
Abstract: We study best proximity points in the framework of metric spaces with -distances. The results extend, generalise and unify several well-known fixed point results in the literature.

Journal ArticleDOI
TL;DR: In this paper, it was shown that two distinct singular moduli can generate the same number field of degree at most two, except for two explicit pairs of exceptions consisting of numbers of degree three.
Abstract: We show that two distinct singular moduli , 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.

Journal ArticleDOI
TL;DR: In this article, the boundary Schwarz lemma for solutions to nonhomogeneous biharmonic equations is established for non-homogeneous non-convex solutions to bi-harmonic problems.
Abstract: We establish a boundary Schwarz lemma for solutions to nonhomogeneous biharmonic equations.

Journal ArticleDOI
TL;DR: Chen and Lev as mentioned in this paper proved that for all positive integers with identical representation functions, there exists an integer with 2 2 l 2 l − 1 − 1/1/1 − 1− 1/2 l 2 − 1-1/2l − 1.
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 $n$ , then there exists an integer $l\geq 1$ such that $r=2^{2l}-1$ and $m=2^{2l+1}-1$ . This solves a problem of Chen and Lev [‘Integer sets with identical representation functions’, Integers 16 (2016), A36] under the condition $m>r$ .

Journal ArticleDOI
TL;DR: In this article, it was shown that a Brauer character is monolithic if and only if it is solvable in a group of Brauer characters, and if the characters are irreducible.
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 $p\ mid \\unicode[STIX]{x1D711}(1)$ for each monolithic Brauer character $\\unicode[STIX]{x1D711}\\in \\text{IBr}(G)$ . When $G$ is $p$ -solvable, we also prove that $P$ is normal in $G$ and $G/P$ is nilpotent if and only if $\\unicode[STIX]{x1D711}(1)^{2}$ divides $|G:\\ker \\unicode[STIX]{x1D711}|$ for all monolithic irreducible $p$ -Brauer characters $\\unicode[STIX]{x1D711}$ of $G$ .

Journal ArticleDOI
TL;DR: In this paper, the authors consider the problem of generating function dissection using classical dissection tools, and present proof techniques that involve classical static and dynamic dissection techniques, as well as static dissection.
Abstract: We consider the function The proof techniques are elementary and involve classical generating function dissection tools.

Journal ArticleDOI
TL;DR: The Banach-Mazur separable quotient problem as mentioned in this paper asks whether every infinite-dimensional Banach space can be represented as a metrisable group, which is a quotient group.
Abstract: The Banach–Mazur separable quotient problem asks whether every infinite-dimensional Banach space , which is an infinite-dimensional metrisable group, as a quotient group.

Journal ArticleDOI
TL;DR: In this article, it was shown that for all irreducible monomial $p$ -Brauer characters of a finite group, the problem is solvable if and only if (1) √ √ 2 √ 1/1/2) = 0.
Abstract: Let $G$ be a finite group and let $p$ be a prime factor of $|G|$ . Suppose that $G$ is solvable and $P$ is a Sylow $p$ -subgroup of $G$ . In this note, we prove that $P{\\vartriangleleft}G$ and $G/P$ is nilpotent if and only if $\\unicode[STIX]{x1D711}(1)^{2}$ divides $|G:\\ker \\unicode[STIX]{x1D711}|$ for all irreducible monomial $p$ -Brauer characters $\\unicode[STIX]{x1D711}$ of $G$ .

Journal ArticleDOI
TL;DR: A Ducci sequence is a sequence of integer tuples generated by iterating the map as discussed by the authors, which is eventually periodic and the maximal period of such sequences for given $n$.
Abstract: A Ducci sequence is a sequence of integer $n$ -tuples generated by iterating the map $$\\begin{eqnarray}D:(a_{1},a_{2},\\ldots ,a_{n})\\mapsto (|a_{1}-a_{2}|,|a_{2}-a_{3}|,\\ldots ,|a_{n}-a_{1}|).\\end{eqnarray}$$ Such a sequence is eventually periodic and we denote by $P(n)$ the maximal period of such sequences for given $n$ . We prove a new upper bound in the case where $n$ is a power of a prime $p\\equiv 5\\hspace{0.6em}({\\rm mod}\\hspace{0.2em}8)$ for which $2$ is a primitive root and the Pellian equation $x^{2}-py^{2}=-4$ has no solutions in odd integers $x$ and $y$ .

Journal ArticleDOI
TL;DR: In this paper, the authors investigate several quantitative properties of meromorphic solutions to some differential-difference equations and generalised delay differential-differenterence equations, and they show that meromorphic solution to these problems can be found in a certain sense as illustrated by several examples.
Abstract: We investigate several quantitative properties of entire and meromorphic solutions to some differential-difference equations and generalised delay differential-difference equations. Our results are sharp in a certain sense as illustrated by several examples.