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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

    更新日期:2021-01-14
  • 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 $\overline {C}_{6, 2}(n)$

    更新日期:2021-01-14
  • 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

    更新日期:2021-01-14
  • 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 $1 \le i \le s-1$

    更新日期:2021-01-11
  • 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

    更新日期:2021-01-11
  • 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

    更新日期:2021-01-11
  • 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

    更新日期:2021-01-11
  • 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

    更新日期:2021-01-08
  • 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

    更新日期:2021-01-08
  • 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.

    更新日期:2020-12-18
  • 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

    更新日期:2020-12-18
  • 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.

    更新日期:2020-12-18
  • 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

    更新日期:2020-12-18
  • 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.

    更新日期:2020-12-18
  • 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.

    更新日期:2020-12-18
  • -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 ${\mathcal A}$

    更新日期:2020-12-18
  • 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.

    更新日期:2020-12-18
  • 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

    更新日期:2020-12-18
  • 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.

    更新日期:2020-12-18
  • 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

    更新日期:2020-11-21
  • 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

    更新日期:2020-11-18
  • 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

    更新日期:2020-11-09
  • 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.

    更新日期:2020-11-09
  • 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

    更新日期:2020-11-09
  • 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

    更新日期:2020-11-09
  • 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 $A(G/H)^{**}$

    更新日期:2020-11-09
  • 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.

    更新日期:2020-11-06
  • 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

    更新日期:2020-10-30
  • 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

    更新日期:2020-10-30
  • 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

    更新日期:2020-10-08
  • 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.

    更新日期:2020-10-05
  • EXISTENCE AND BOX DIMENSION OF GENERAL RECURRENT FRACTAL INTERPOLATION FUNCTIONS
    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
  • 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

    更新日期:2020-10-05
  • 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

    更新日期:2020-10-05
  • THE SLOT LENGTH OF A FAMILY OF MATRICES
    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
  • 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

    更新日期:2020-10-05
  • STARLIKENESS AND CONVEXITY OF CAUCHY TRANSFORMS ON REGULAR POLYGONS
    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
  • CENTRAL SEQUENCES IN SUBHOMOGENEOUS UNITAL C*-ALGEBRAS
    Bull. Aust. Math. Soc. (IF 0.542) Pub Date : 2020-10-02
    DON HADWIN; HEMANT PENDHARKAR

    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
  • CONNECTIVITY PROPERTIES OF MCKAY QUIVERS
    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
  • LARGE VALUES OF L-FUNCTIONS ON THE 1-LINE
    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
  • UNIQUENESS OF EXTENDABLE TEMPERATURES
    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
  • PARTITIONS OF $\mathbb {Z}_m$ WITH IDENTICAL REPRESENTATION FUNCTION
    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
  • BLOCKS WITH SMALL-DIMENSIONAL BASIC ALGEBRA
    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
  • RATIONAL NEARLY SIMPLE GROUPS
    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
  • THE ROGERS–RAMANUJAN CONTINUED FRACTION AND RELATED ETA-QUOTIENT REPRESENTATIONS
    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
  • DISTRIBUTION OF THE DIVISOR FUNCTION AT CONSECUTIVE INTEGERS
    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
  • ON $\boldsymbol{\theta} $ -CONGRUENT NUMBERS OVER REAL NUMBER FIELDS
    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
  • HYPERSTABILITY OF GENERALISED LINEAR FUNCTIONAL EQUATIONS IN SEVERAL VARIABLES
    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
  • A $q$ -ANALOGUE OF A DWORK-TYPE SUPERCONGRUENCE
    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
  • A NOTE ON DERIVED LENGTH AND CHARACTER DEGREES
    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
  • GRAPHS DETERMINED BY THEIR $T$ -GAIN SPECTRA
    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
  • NOTE ON MINIMUM DEGREE AND PROPER CONNECTION NUMBER
    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
  • PROOFS OF URYSOHN’S LEMMA AND THE TIETZE EXTENSION THEOREM VIA THE CANTOR FUNCTION
    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
  • ‘NOTE ON SUMS INVOLVING THE EULER FUNCTION’
    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
  • SUM-PRODUCT ESTIMATES FOR DIAGONAL MATRICES
    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
  • ON THE DIFFERENCE OF COEFFICIENTS OF OZAKI CLOSE-TO-CONVEX FUNCTIONS
    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
  • STOCHASTIC POTENTIALS OF INTERMITTENT MAPS
    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
  • NUMERICAL RANGE AND POSITIVE BLOCK MATRICES
    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
  • AN EXACT FORMULA FOR THE HARMONIC CONTINUED FRACTION
    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
  • GRADIENT ESTIMATES VIA TWO-POINT FUNCTIONS FOR PARABOLIC EQUATIONS UNDER RICCI FLOW
    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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