样式: 排序: IF: - GO 导出 标记为已读
-
THE SUMMED PAPERFOLDING SEQUENCE Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2024-03-25 MARTIN BUNDER, BRUCE BATES, STEPHEN ARNOLD
The sequence $a( 1) ,a( 2) ,a( 3) ,\ldots, $ labelled A088431 in the Online Encyclopedia of Integer Sequences, is defined by: $a( n) $ is half of the $( n+1) $ th component, that is, the $( n+2) $ th term, of the continued fraction expansion of $$ \begin{align*} \sum_{k=0}^{\infty }\frac{1}{2^{2^{k}}}. \end{align*} $$ Dimitri Hendriks has suggested that it is the sequence of run lengths of the paperfolding
-
CHOQUET INTEGRALS, HAUSDORFF CONTENT AND FRACTIONAL OPERATORS Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2024-03-19 NAOYA HATANO, RYOTA KAWASUMI, HIROKI SAITO, HITOSHI TANAKA
We show that the fractional integral operator $I_{\alpha }$ , $0<\alpha
-
A CHARACTERISATION OF SOLUBLE -GROUPS Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2024-03-15 ZHIGANG WANG, A-MING LIU, VASILY G. SAFONOV, ALEXANDER N. SKIBA
Let G be a finite group. A subgroup A of G is said to be S-permutable in G if A permutes with every Sylow subgroup P of G, that is, $AP=PA$ . Let $A_{sG}$ be the subgroup of A generated by all S-permutable subgroups of G contained in A and $A^{sG}$ be the intersection of all S-permutable subgroups of G containing A. We prove that if G is a soluble group, then S-permutability is a transitive relation
-
A CLASS OF SYMBOLS THAT INDUCE BOUNDED COMPOSITION OPERATORS FOR DIRICHLET-TYPE SPACES ON THE DISC Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2024-03-14 ATHANASIOS BESLIKAS
We study the problem of determining the holomorphic self maps of the unit disc that induce a bounded composition operator on Dirichlet-type spaces. We find a class of symbols $\varphi $ that induce a bounded composition operator on the Dirichlet-type spaces, by applying results of the multidimensional theory of composition operators for the weighted Bergman spaces of the bi-disc.
-
NON-PÓLYA FIELDS WITH LARGE PÓLYA GROUPS ARISING FROM LEHMER QUINTICS Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2024-03-11 NIMISH KUMAR MAHAPATRA, PREM PRAKASH PANDEY
We construct a new family of quintic non-Pólya fields with large Pólya groups. We show that the Pólya number of such a field never exceeds five times the size of its Pólya group. Finally, we show that these non-Pólya fields are nonmonogenic of field index one.
-
WEIERSTRASS ZETA FUNCTIONS AND p-ADIC LINEAR RELATIONS Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2024-03-11 DUC HIEP PHAM
We discuss the p-adic Weierstrass zeta functions associated with elliptic curves defined over the field of algebraic numbers and linear relations for their values in the p-adic domain. These results are extensions of the p-adic analogues of results given by Wüstholz in the complex domain [see A. Baker and G. Wüstholz, Logarithmic Forms and Diophantine Geometry, New Mathematical Monographs, 9 (Cambridge
-
A NOTE ON JUDICIOUS BISECTIONS OF GRAPHS Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2024-03-08 SHUFEI WU, XIAOBEI XIONG
Let G be a graph with m edges, minimum degree $\delta $ and containing no cycle of length 4. Answering a question of Bollobás and Scott, Fan et al. [‘Bisections of graphs without short cycles’, Combinatorics, Probability and Computing27(1) (2018), 44–59] showed that if (i) G is $2$ -connected, or (ii) $\delta \ge 3$ , or (iii) $\delta \ge 2$ and the girth of G is at least 5, then G admits a bisection
-
ON SOME CONGRUENCES INVOLVING CENTRAL BINOMIAL COEFFICIENTS Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2024-03-08 GUO-SHUAI MAO
We prove the following conjecture of Z.-W. Sun [‘On congruences related to central binomial coefficients’, J. Number Theory13(11) (2011), 2219–2238]. Let p be an odd prime. Then $$ \begin{align*} \sum_{k=1}^{p-1}\frac{\binom{2k}k}{k2^k}\equiv-\frac12H_{{(p-1)}/2}+\frac7{16}p^2B_{p-3}\pmod{p^3}, \end{align*} $$ where $H_n$ is the nth harmonic number and $B_n$ is the nth Bernoulli number. In addition
-
ON THE SET OF KRONECKER NUMBERS Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2024-03-08 SAYAN GOSWAMI, WEN HUANG, XIAOSHENG WU
A positive even number is said to be a Maillet number if it can be written as the difference between two primes, and a Kronecker number if it can be written in infinitely many ways as the difference between two primes. It is believed that all even numbers are Kronecker numbers. We study the division and multiplication of Kronecker numbers and show that these numbers are rather abundant. We prove that
-
ON THE DIOPHANTINE EQUATION Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2024-03-06 ELCHIN HASANALIZADE
A generalisation of the well-known Pell sequence $\{P_n\}_{n\ge 0}$ given by $P_0=0$ , $P_1=1$ and $P_{n+2}=2P_{n+1}+P_n$ for all $n\ge 0$ is the k-generalised Pell sequence $\{P^{(k)}_n\}_{n\ge -(k-2)}$ whose first k terms are $0,\ldots ,0,1$ and each term afterwards is given by the linear recurrence $P^{(k)}_n=2P^{(k)}_{n-1}+P^{(k)}_{n-2}+\cdots +P^{(k)}_{n-k}$ . For the Pell sequence, the formula
-
IDEMPOTENT GENERATORS OF INCIDENCE ALGEBRAS Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2024-03-05 N. A. KOLEGOV
The minimum number of idempotent generators is calculated for an incidence algebra of a finite poset over a commutative ring. This quantity equals either $\lceil \log _2 n\rceil $ or $\lceil \log _2 n\rceil +1$ , where n is the cardinality of the poset. The two cases are separated in terms of the embedding of the Hasse diagram of the poset into the complement of the hypercube graph.
-
GENERATING FUNCTIONS FOR THE QUOTIENTS OF NUMERICAL SEMIGROUPS Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2024-02-29 FEIHU LIU
We propose generating functions, $\textrm {RGF}_p(x)$ , for the quotients of numerical semigroups which are related to the Sylvester denumerant. Using MacMahon’s partition analysis, we can obtain $\textrm {RGF}_p(x)$ by extracting the constant term of a rational function. We use $\textrm {RGF}_p(x)$ to give a system of generators for the quotient of the numerical semigroup $\langle a_1,a_2,a_3\rangle
-
A NOTE ON PROJECTIONS IN ÉTALE GROUPOID ALGEBRAS AND DIAGONAL-PRESERVING HOMOMORPHISMS Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2024-02-29 BENJAMIN STEINBERG
Carlsen [‘ $\ast $ -isomorphism of Leavitt path algebras over $\Bbb Z$ ’, Adv. Math.324 (2018), 326–335] showed that any $\ast $ -homomorphism between Leavitt path algebras over $\mathbb Z$ is automatically diagonal preserving and hence induces an isomorphism of boundary path groupoids. His result works over conjugation-closed subrings of $\mathbb C$ enjoying certain properties. In this paper, we characterise
-
PROJECTIVE CHARACTER VALUES ON REAL AND RATIONAL ELEMENTS Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2024-02-29 R. J. HIGGS
Let $\alpha $ be a complex-valued $2$ -cocycle of a finite group G with $\alpha $ chosen so that the $\alpha $ -characters of G are class functions and analogues of the orthogonality relations for ordinary characters are valid. Then the real or rational elements of G that are also $\alpha $ -regular are characterised by the values that the irreducible $\alpha $ -characters of G take on those respective
-
DIOPHANTINE TRANSFERENCE PRINCIPLE OVER FUNCTION FIELDS Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2024-02-28 SOURAV DAS, ARIJIT GANGULY
We study the Diophantine transference principle over function fields. By adapting the approach of Beresnevich and Velani [‘An inhomogeneous transference principle and Diophantine approximation’, Proc. Lond. Math. Soc. (3)101 (2010), 821–851] to function fields, we extend many results from homogeneous to inhomogeneous Diophantine approximation. This also yields the inhomogeneous Baker–Sprindžuk conjecture
-
GRAPH CHARACTERISATION OF THE ANNIHILATOR IDEALS OF LEAVITT PATH ALGEBRAS Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2024-02-15 LIA VAŠ
If E is a graph and K is a field, we consider an ideal I of the Leavitt path algebra $L_K(E)$ of E over K. We describe the admissible pair corresponding to the smallest graded ideal which contains I where the grading in question is the natural grading of $L_K(E)$ by ${\mathbb {Z}}$ . Using this description, we show that the right and the left annihilators of I are equal (which may be somewhat surprising
-
EDGE WEIGHTING FUNCTIONS ON THE SEMITOTAL DOMINATING SET OF CLAW-FREE GRAPHS Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2024-02-12 JIE CHEN, HONGZHANG CHEN, SHOU-JUN XU
In an isolate-free graph G, a subset S of vertices is a semitotal dominating set of G if it is a dominating set of G and every vertex in S is within distance 2 of another vertex of S. The semitotal domination number of G, denoted by $\gamma _{t2}(G)$ , is the minimum cardinality of a semitotal dominating set in G. Using edge weighting functions on semitotal dominating sets, we prove that if $G\neq
-
ON THE LIMIT SET OF A COMPLEX HYPERBOLIC TRIANGLE GROUP Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2024-02-02 MENGQI SHI, JIEYAN WANG
Let $\Gamma =\langle I_{1}, I_{2}, I_{3}\rangle $ be the complex hyperbolic $(4,4,\infty )$ triangle group with $I_1I_3I_2I_3$ being unipotent. We show that the limit set of $\Gamma $ is connected and the closure of a countable union of $\mathbb {R}$ -circles.
-
GENERALISED MUTUALLY PERMUTABLE PRODUCTS AND SATURATED FORMATIONS, II Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2024-01-30 ADOLFO BALLESTER-BOLINCHES, SESUAI Y. MADANHA, TENDAI M. MUDZIIRI SHUMBA, MARÍA C. PEDRAZA-AGUILERA
A group $G=AB$ is the weakly mutually permutable product of the subgroups A and B, if A permutes with every subgroup of B containing $A \cap B$ and B permutes with every subgroup of A containing $A \cap B$ . Weakly mutually permutable products were introduced by the first, second and fourth authors [‘Generalised mutually permutable products and saturated formations’, J. Algebra595 (2022), 434–443]
-
CONGRUENCES FOR RANKS OF PARTITIONS Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2024-01-29 RENRONG MAO
Ranks of partitions play an important role in the theory of partitions. They provide combinatorial interpretations for Ramanujan’s famous congruences for partition functions. We establish a family of congruences modulo powers of $5$ for ranks of partitions.
-
HOMOLOGICAL LINEAR QUOTIENTS AND EDGE IDEALS OF GRAPHS Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2024-01-29 NADIA TAGHIPOUR, SHAMILA BAYATI, FARHAD RAHMATI
It is well known that the edge ideal $I(G)$ of a simple graph G has linear quotients if and only if $G^c$ is chordal. We investigate when the property of having linear quotients is inherited by homological shift ideals of an edge ideal. We will see that adding a cluster to the graph $G^c$ when $I(G)$ has homological linear quotients results in a graph with the same property. In particular, $I(G)$ has
-
MAXIMAL SUBSEMIGROUPS OF INFINITE SYMMETRIC GROUPS Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2024-01-29 SUZANA MENDES-GONÇALVES, R. P. SULLIVAN
Brazil et al. [‘Maximal subgroups of infinite symmetric groups’, Proc. Lond. Math. Soc. (3)68(1) (1994), 77–111] provided a new family of maximal subgroups of the symmetric group $G(X)$ defined on an infinite set X. It is easy to see that, in this case, $G(X)$ contains subsemigroups that are not groups, but nothing is known about nongroup maximal subsemigroups of $G(X)$ . We provide infinitely many
-
MULTIPLE SOLUTIONS FOR -LAPLACIAN EQUATIONS WITH NONLINEARITY SUBLINEAR AT ZERO Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2024-01-29 SHIBO LIU
We consider the Dirichlet problem for $p(x)$ -Laplacian equations of the form $$ \begin{align*} -\Delta_{p(x)}u+b(x)\vert u\vert ^{p(x)-2}u=f(x,u),\quad u\in W_{0}^{1,p(x)}(\Omega). \end{align*} $$ The odd nonlinearity $f(x,u)$ is $p(x)$ -sublinear at $u=0$ but the related limit need not be uniform for $x\in \Omega $ . Except being subcritical, no additional assumption is imposed on $f(x,u)$ for $|u|$
-
ON THE CUMULATIVE DISTRIBUTION FUNCTION OF THE VARIANCE-GAMMA DISTRIBUTION Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2024-01-29 ROBERT E. GAUNT
We obtain exact formulas for the cumulative distribution function of the variance-gamma distribution, as infinite series involving the modified Bessel function of the second kind and the modified Lommel function of the first kind. From these formulas, we deduce exact formulas for the cumulative distribution function of the product of two correlated zero-mean normal random variables.
-
COMPLETE EMBEDDINGS OF GROUPS Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2024-01-26 MARTIN R. BRIDSON, HAMISH SHORT
Every countable group G can be embedded in a finitely generated group $G^*$ that is hopfian and complete, that is, $G^*$ has trivial centre and every epimorphism $G^*\to G^*$ is an inner automorphism. Every finite subgroup of $G^*$ is conjugate to a finite subgroup of G. If G has a finite presentation (respectively, a finite classifying space), then so does $G^*$ . Our construction of $G^*$ relies
-
NEAR OPTIMAL THRESHOLDS FOR EXISTENCE OF DILATED CONFIGURATIONS IN Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2024-01-26 PABLO BHOWMIK, FIRDAVS RAKHMONOV
Let $\mathbb {F}_q^d$ denote the d-dimensional vector space over the finite field $\mathbb {F}_q$ with q elements. Define for $\alpha = (\alpha _1, \dots , \alpha _d) \in \mathbb {F}_q^d$ . Let $k\in \mathbb {N}$ , A be a nonempty subset of $\{(i, j): 1 \leq i < j \leq k + 1\}$ and $r\in (\mathbb {F}_q)^2\setminus {0}$ , where $(\mathbb {F}_q)^2=\{a^2:a\in \mathbb {F}_q\}$ . If $E\subset \mathbb {F}_q^d$
-
ON THE CHARACTERISATION OF ALTERNATING GROUPS BY CODEGREES Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2024-01-26 MALLORY DOLORFINO, LUKE MARTIN, ZACHARY SLONIM, YUXUAN SUN, YONG YANG
Let G be a finite group and $\mathrm {Irr}(G)$ the set of all irreducible complex characters of G. Define the codegree of $\chi \in \mathrm {Irr}(G)$ as $\mathrm {cod}(\chi ):={|G:\mathrm {ker}(\chi ) |}/{\chi (1)}$ and let $\mathrm {cod}(G):=\{\mathrm {cod}(\chi ) \mid \chi \in \mathrm {Irr}(G)\}$ be the codegree set of G. Let $\mathrm {A}_n$ be an alternating group of degree $n \ge 5$ . We show that
-
BOUNDING ZETA ON THE 1-LINE UNDER THE PARTIAL RIEMANN HYPOTHESIS Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2024-01-10 ANDRÉS CHIRRE
We provide explicit bounds for the Riemann zeta-function on the line $\mathrm {Re}\,{s}=1$ , assuming that the Riemann hypothesis holds up to height T. In particular, we improve some bounds in finite regions for the logarithmic derivative and the reciprocal of the Riemann zeta-function.
-
COUNTING UNIONS OF SCHREIER SETS Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2023-12-27 KEVIN BEANLAND, DMITRIY GOROVOY, JĘDRZEJ HODOR, DANIIL HOMZA
A subset of positive integers F is a Schreier set if it is nonempty and $|F|\leqslant \min F$ (here $|F|$ is the cardinality of F). For each positive integer k, we define $k\mathcal {S}$ as the collection of all the unions of at most k Schreier sets. Also, for each positive integer n, let $(k\mathcal {S})^n$ be the collection of all sets in $k\mathcal {S}$ with maximum element equal to n. It is well
-
INTERSECTING THE TORSION OF ELLIPTIC CURVES Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2023-12-27 NATALIA GARCIA-FRITZ, HECTOR PASTEN
Bogomolov and Tschinkel [‘Algebraic varieties over small fields’, Diophantine Geometry, U. Zannier (ed.), CRM Series, 4 (Scuola Normale Superiore di Pisa, Pisa, 2007), 73–91] proved that, given two complex elliptic curves $E_1$ and $E_2$ along with even degree- $2$ maps $\pi _j\colon E_j\to \mathbb {P}^1$ having different branch loci, the intersection of the image of the torsion points of $E_1$ and
-
SOLVABLE GROUPS WHOSE NONNORMAL SUBGROUPS HAVE FEW ORDERS Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2023-12-27 LIJUAN HE, HENG LV, GUIYUN CHEN
Suppose that G is a finite solvable group. Let $t=n_c(G)$ denote the number of orders of nonnormal subgroups of G. We bound the derived length $dl(G)$ in terms of $n_c(G)$ . If G is a finite p-group, we show that $|G'|\leq p^{2t+1}$ and $dl(G)\leq \lceil \log _2(2t+3)\rceil $ . If G is a finite solvable nonnilpotent group, we prove that the sum of the powers of the prime divisors of $|G'|$ is less
-
DIVISIBILITY OF SUMS OF PARTITION NUMBERS BY MULTIPLES OF 2 AND 3 Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2023-12-22 NAYANDEEP DEKA BARUAH
We show that certain sums of partition numbers are divisible by multiples of 2 and 3. For example, if $p(n)$ denotes the number of unrestricted partitions of a positive integer n (and $p(0)=1$ , $p(n)=0$ for $n<0$ ), then for all nonnegative integers m, $$ \begin{align*}\sum_{k=0}^\infty p(24m+23-\omega(-2k))+\sum_{k=1}^\infty p(24m+23-\omega(2k))\equiv 0~ (\text{mod}~144),\end{align*} $$ where $\omega
-
2-LOCAL ISOMETRIES OF SOME NEST ALGEBRAS Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2023-12-18 BO YU, JIANKUI LI
Let H be a complex separable Hilbert space with $\dim H \geq 2$ . Let $\mathcal {N}$ be a nest on H such that $E_+ \neq E$ for any $E \neq H, E \in \mathcal {N}$ . We prove that every 2-local isometry of $\operatorname {Alg}\mathcal {N}$ is a surjective linear isometry.
-
A NOTE ON THE ZEROS OF L-FUNCTIONS ASSOCIATED TO FIXED-ORDER DIRICHLET CHARACTERS Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2023-12-18 C. C. CORRIGAN
We use the Weyl bound for Dirichlet L-functions to derive zero-density estimates for L-functions associated to families of fixed-order Dirichlet characters. The results improve on previous bounds given by the author when $\sigma $ is sufficiently distant from the critical line.
-
ANY DUAL OPERATOR SPACE IS WEAKLY LOCALLY REFLEXIVE Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2023-12-12 ZHE DONG, JINZE JIANG, YAFEI ZHAO
We introduce the notion of weakly local reflexivity in operator space theory and prove that any dual operator space is weakly locally reflexive.
-
ON A CONJECTURE OF LENNY JONES ABOUT CERTAIN MONOGENIC POLYNOMIALS Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2023-11-21 SUMANDEEP KAUR, SURENDER KUMAR
Let $K={\mathbb {Q}}(\theta )$ be an algebraic number field with $\theta $ satisfying a monic irreducible polynomial $f(x)$ of degree n over ${\mathbb {Q}}.$ The polynomial $f(x)$ is said to be monogenic if $\{1,\theta ,\ldots ,\theta ^{n-1}\}$ is an integral basis of K. Deciding whether or not a monic irreducible polynomial is monogenic is an important problem in algebraic number theory. In an attempt
-
ADDITIVE COMPLETION OF THIN SETS Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2023-11-15 JIN-HUI FANG, CSABA SÁNDOR
Two sets $A,B$ of positive integers are called exact additive complements if $A+B$ contains all sufficiently large integers and $A(x)B(x)/x\rightarrow 1$ . For $A=\{a_1
-
ON ENDOMORPHISMS OF EXTENSIONS IN TANNAKIAN CATEGORIES Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2023-11-07 PAYMAN ESKANDARI
We prove analogues of Schur’s lemma for endomorphisms of extensions in Tannakian categories. More precisely, let $\mathbf {T}$ be a neutral Tannakian category over a field of characteristic zero. Let E be an extension of A by B in $\mathbf {T}$ . We consider conditions under which every endomorphism of E that stabilises B induces a scalar map on $A\oplus B$ . We give a result in this direction in the
-
AN ALGEBRAIC INTERPRETATION OF THE SUPER CATALAN NUMBERS Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2023-11-06 KEVIN LIMANTA
We extend the notion of polynomial integration over an arbitrary circle C in the Euclidean geometry over general fields $\mathbb {F}$ of characteristic zero as a normalised $\mathbb {F}$ -linear functional on $\mathbb {F}[\alpha _1, \alpha _2]$ that maps polynomials that evaluate to zero on C to zero and is $\mathrm {SO}(2,\mathbb {F})$ -invariant. This allows us to not only build a purely algebraic
-
AN EFFECTIVE BOUND FOR GENERALISED DIOPHANTINE m-TUPLES Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2023-11-06 SAUNAK BHATTACHARJEE, ANUP B. DIXIT, DISHANT SAIKIA
For $k\geq 2$ and a nonzero integer n, a generalised Diophantine m-tuple with property $D_k(n)$ is a set of m positive integers $S = \{a_1,a_2,\ldots , a_m\}$ such that $a_ia_j + n$ is a kth power for $1\leq i< j\leq m$. Define $M_k(n):= \text {sup}\{|S| : S$ having property $D_k(n)\}$. Dixit et al. [‘Generalised Diophantine m-tuples’, Proc. Amer. Math. Soc. 150(4) (2022), 1455–1465] proved that $M_k(n)=O(\log
-
THE DIFFERENCE ANALOGUE OF THE TUMURA–HAYMAN–CLUNIE THEOREM Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2023-11-06 MINGLIANG FANG, HUI LI, XIAO YAO
We prove a difference analogue of the celebrated Tumura–Hayman–Clunie theorem. Let f be a transcendental entire function, let c be a nonzero constant and let n be a positive integer. If f and $\Delta _c^n f$ omit zero in the whole complex plane, then either $f(z)=\exp (h_1(z)+C_1 z)$ , where $h_1$ is an entire function of period c and $\exp (C_1 c)\neq 1$ , or $f(z)=\exp (h_2(z)+C_2 z)$ , where $h_2$
-
PARTITIONS OF NATURAL NUMBERS AND THEIR WEIGHTED REPRESENTATION FUNCTIONS Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2023-10-27 SHUANG-SHUANG LI, YU-QING SHAN, XIAO-HUI YAN
For any positive integers $k_1,k_2$ and any set $A\subseteq \mathbb {N}$ , let $R_{k_1,k_2}(A,n)$ be the number of solutions of the equation $n=k_1a_1+k_2a_2$ with $a_1,a_2\in A$ . Let g be a fixed integer. We prove that if $k_1$ and $k_2$ are two integers with $2\le k_1
-
LINEAR INDEPENDENCE OF VALUES OF THE q-EXPONENTIAL AND RELATED FUNCTIONS Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2023-10-23 ANUP B. DIXIT, VEEKESH KUMAR, SIDDHI S. PATHAK
We establish the linear independence of values of the q-analogue of the exponential function and its derivatives at specified algebraic arguments, when q is a Pisot–Vijayaraghavan number. We also deduce similar results for cognate functions, such as the Tschakaloff function and certain generalised q-series.
-
NOWHERE-ZERO -FLOWS IN CAYLEY GRAPHS OF ORDER Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2023-10-20 JUNYANG ZHANG, HANG ZHOU
It is proved that Tutte’s $3$ -flow conjecture is true for Cayley graphs on groups of order $8p$ where p is an odd prime.
-
ON THE EXCEPTIONAL SET OF TRANSCENDENTAL ENTIRE FUNCTIONS IN SEVERAL VARIABLES Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2023-10-20 DIEGO ALVES, JEAN LELIS, DIEGO MARQUES, PAVEL TROJOVSKÝ
We prove that any subset of $\overline {\mathbb {Q}}^m$ (closed under complex conjugation and which contains the origin) is the exceptional set of uncountably many transcendental entire functions over $\mathbb {C}^m$ with rational coefficients. This result solves a several variables version of a question posed by Mahler for transcendental entire functions [Lectures on Transcendental Numbers, Lecture
-
FINITE BASIS PROBLEM FOR INVOLUTION MONOIDS OF ORDER FIVE Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2023-10-09 BIN BIN HAN, WEN TING ZHANG, YAN FENG LUO
An example of a nonfinitely based involution monoid of order five has recently been discovered. We confirm that this example is, up to isomorphism, the unique smallest among all involution monoids.
-
SOME COUNTING QUESTIONS FOR MATRIX PRODUCTS Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2023-10-09 MUHAMMAD AFIFURRAHMAN
Given a set X of $n\times n$ matrices and a positive integer m, we consider the problem of estimating the cardinalities of the product sets $A_1 \cdots A_m$ , where $A_i\in X$ . When $X={\mathcal M}_n(\mathbb {Z};H)$ , the set of $n\times n$ matrices with integer elements of size at most H, we give several bounds on the cardinalities of the product sets and solution sets of related equations such as
-
A NOTE ON NORMALISED GROUND STATES FOR THE TWO-DIMENSIONAL CUBIC-QUINTIC NONLINEAR SCHRÖDINGER EQUATION Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2023-10-09 DEKE LI, QINGXUAN WANG
We consider the two-dimensional minimisation problem for $\inf \{ E_a(\varphi ):\varphi \in H^1(\mathbb {R}^2)\ \text {and}\ \|\varphi \|_2^2=1\}$ , where the energy functional $ E_a(\varphi )$ is a cubic-quintic Schrödinger functional defined by $E_a(\varphi ):=\tfrac 12\int _{\mathbb {R}^2}|\nabla \varphi |^2\,dx-\tfrac 14a\int _{\mathbb {R}^2}|\varphi |^4\,dx+\tfrac 16a^2\int _{\mathbb {R}^2}|\varphi
-
THE LIFTING PROBLEM FOR UNIVERSAL QUADRATIC FORMS OVER SIMPLEST CUBIC FIELDS Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2023-10-06 DANIEL GIL-MUÑOZ, MAGDALÉNA TINKOVÁ
The lifting problem for universal quadratic forms over a totally real number field K consists of determining the existence or otherwise of a quadratic form with integer coefficients (or $\mathbb {Z}$ -form) that is universal over K. We prove the nonexistence of universal $\mathbb {Z}$ -forms over simplest cubic fields for which the integer parameter is big enough. The monogenic case is already known
-
ON EXTERIOR POWERS OF REFLECTION REPRESENTATIONS Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2023-10-06 HONGSHENG HU
In 1968, Steinberg [Endomorphisms of Linear Algebraic Groups, Memoirs of the American Mathematical Society, 80 (American Mathematical Society, Providence, RI, 1968)] proved a theorem stating that the exterior powers of an irreducible reflection representation of a Euclidean reflection group are again irreducible and pairwise nonisomorphic. We extend this result to a more general context where the inner
-
TRIPLE-PRODUCT-FREE SETS Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2023-10-06 PRECIOUS U. AGIGOR-MIKE, SARAH B. HART, MARTIN C. OBI
In this paper, we study triple-product-free sets, which are analogous to the widely studied concept of product-free sets. A nonempty subset S of a group G is triple-product-free if $abc \notin S$ for all $a, b, c \in S$ . If S is triple-product-free and is not a proper subset of any other triple-product-free set, we say that S is locally maximal. We classify all groups containing a locally maximal
-
SUMSETS CONTAINING A TERM OF A SEQUENCE Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2023-09-18 MIN CHEN, MIN TANG
Let $S=\{s_{1}, s_{2}, \ldots \}$ be an unbounded sequence of positive integers with $s_{n+1}/s_{n}$ approaching $\alpha $ as $n\rightarrow \infty $ and let $\beta>\max (\alpha , 2)$ . We show that for all sufficiently large positive integers l, if $A\subset [0, l]$ with $l\in A$ , $\gcd A=1$ and $|A|\geq (2-{k}/{\lambda \beta })l/(\lambda +1)$ , where $\lambda =\lceil {k}/{\beta }\rceil $ , then $kA\cap
-
ON THE N-POINT CORRELATION OF VAN DER CORPUT SEQUENCES Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2023-09-15 CHRISTIAN WEIß
We derive an explicit formula for the N-point correlation $F_N(s)$ of the van der Corput sequence in base $2$ for all $N \in \mathbb {N}$ and $s \geq 0$ . The formula can be evaluated without explicit knowledge about the elements of the van der Corput sequence. This constitutes the first example of an exact closed-form expression of $F_N(s)$ for all $N \in \mathbb {N}$ and all $s \geq 0$ which does
-
THE MORDELL–LANG CONJECTURE FOR SEMIABELIAN VARIETIES DEFINED OVER FIELDS OF POSITIVE CHARACTERISTIC Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2023-09-08 DRAGOS GHIOCA, SHE YANG
Let G be a semiabelian variety defined over an algebraically closed field K of prime characteristic. We describe the intersection of a subvariety X of G with a finitely generated subgroup of $G(K)$.
-
AN IMPROVEMENT TO A THEOREM OF LEONETTI AND LUCA Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2023-09-01 TRAN NGUYEN THANH DANH, HOANG TUAN DUNG, PHAM VIET HUNG, NGUYEN DINH KIEN, NGUYEN AN THINH, KHUC DINH TOAN, NGUYEN XUAN THO
Leonetti and Luca [‘On the iterates of the shifted Euler’s function’, Bull. Aust. Math. Soc., to appear] have shown that the integer sequence $(x_n)_{n\geq 1}$ defined by $x_{n+2}=\phi (x_{n+1})+\phi (x_{n})+k$ , where $x_1,x_2\geq 1$ , $k\geq 0$ and $2 \mid k$ , is bounded by $4^{X^{3^{k+1}}}$ , where $X=(3x_1+5x_2+7k)/2$ . We improve this result by showing that the sequence $(x_n)$ is bounded by
-
A NOTE ON BRØNDSTED’S FIXED POINT THEOREM Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2023-08-31 OLEG ZUBELEVICH
We show that for the case of uniformly convex Banach spaces, the conditions of Brøndsted’s fixed point theorem can be relaxed.
-
ON SUMS INVOLVING THE EULER TOTIENT FUNCTION Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2023-08-24 ISAO KIUCHI, YUKI TSURUTA
Let $\gcd (n_{1},\ldots ,n_{k})$ denote the greatest common divisor of positive integers $n_{1},\ldots ,n_{k}$ and let $\phi $ be the Euler totient function. For any real number $x>3$ and any integer $k\geq 2$ , we investigate the asymptotic behaviour of $\sum _{n_{1}\ldots n_{k}\leq x}\phi (\gcd (n_{1},\ldots ,n_{k})). $
-
ON THE EXPECTED UNIFORM ERROR OF BROWNIAN MOTION APPROXIMATED BY THE LÉVY–CIESIELSKI CONSTRUCTION Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2023-08-24 BRUCE BROWN, MICHAEL GRIEBEL, FRANCES Y. KUO, IAN H. SLOAN
The Brownian bridge or Lévy–Ciesielski construction of Brownian paths almost surely converges uniformly to the true Brownian path. We focus on the uniform error. In particular, we show constructively that at level N, at which there are $d=2^N$ points evaluated on the Brownian path, the uniform error and its square, and the uniform error of geometric Brownian motion, have upper bounds of order $\mathcal
-
A COUNTEREXAMPLE TO A RESULT OF JABERI AND MAHMOODI Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2023-08-10 A. SAHAMI, S. F. SHARIATI
We show that $\ell ^1(\mathbb {N}_\wedge )$ is $\varphi $ -amenable for each multiplicative linear functional $\varphi :\ell ^1(\mathbb {N}_\wedge )\rightarrow \mathbb {C}.$ This is a counterexample to the final corollary of Jaberi and Mahmoodi [‘On $\varphi $ -amenability of dual Banach algebras’, Bull. Aust. Math. Soc.105 (2022), 303–313] and shows that the final theorem in that paper is not valid
-
GROUPS WITH FEW NONPOWER SUBGROUPS Bull. Aust. Math. Soc. (IF 0.7) Pub Date : 2023-08-10 JIWEI ZHENG, WEI ZHOU, D. E. TAYLOR
For a group G and $m\ge 1$ , let $G^m$ denote the subgroup generated by the elements $g^m$ , where g runs through G. The subgroups not of the form $G^m$ are the nonpower subgroups of G. We classify the groups with at most nine nonpower subgroups.