Showing papers in "Journal of The Australian Mathematical Society in 2012"
TL;DR: In this paper, the authors provide evaluations of several recently studied higher and multiple Mahler measures using log-sine integrals, complemented with an analysis of generating functions and identities for log-Sine Integrals which allows the evaluations to be expressed in terms of zeta values or more general polylogarithmic terms.
Abstract: We provide evaluations of several recently studied higher and multiple Mahler measures using log-sine integrals. This is complemented with an analysis of generating functions and identities for log-sine integrals which allows the evaluations to be expressed in terms of zeta values or more general polylogarithmic terms. The machinery developed is then applied to evaluation of further families of multiple Mahler measures.
28 citations
TL;DR: The existence of infinitely many composite numbers simultaneously passing all elliptic curve primality tests assuming a weak form of a standard conjecture on the bound on the least prime in (special) arithmetic progressions was shown in this paper.
Abstract: In 1987, Gordon gave an integer primality condition similar to the familiar test based on Fermat’s little theorem, but based instead on the arithmetic of elliptic curves with complex multiplication. We prove the existence of infinitely many composite numbers simultaneously passing all elliptic curve primality tests assuming a weak form of a standard conjecture on the bound on the least prime in (special) arithmetic progressions. Our results are somewhat more general than both the 1999 dissertation of the first author (written under the direction of the third author) and a 2010 paper on Carmichael numbers in a residue class written by Banks and the second author.
17 citations
TL;DR: In this article, the factors of products of at most two binomial coefficients were studied for higher-order Catalan numbers, motivated by higher order Catalan numbers and higher order Spanish numbers.
Abstract: In this paper, motivated by Catalan numbers and higher-order Catalan numbers, we study factors of products of at most two binomial coefficients.
17 citations
TL;DR: In this article, it was shown that the exponent of a subgroup generated by a multilinear commutator word is bounded in terms of e and w only, where e is a positive integer and G is a finite group.
Abstract: Let w be a multilinear commutator word. We prove that if e is a positive integer and G is a finite group in which any nilpotent subgroup generated by w-values has exponent dividing e then the exponent of the corresponding verbal subgroup w(G) is bounded in terms of e and w only.
17 citations
TL;DR: In this paper, the authors investigated the behaviour of the Liouville function with respect to the Riemann hypothesis and other properties of the zeros of the riemann zeta function.
Abstract: Abstract We investigate the behaviour of the function $L_{\\alpha }(x) = \\sum _{n\\leq x}\\lambda (n)/n^{\\alpha }$, where $\\lambda (n)$ is the Liouville function and $\\alpha $ is a real parameter. The case where $\\alpha =0$ was investigated by Pólya; the case $\\alpha =1$, by Turán. The question of the existence of sign changes in both of these cases is related to the Riemann hypothesis. Using both analytic and computational methods, we investigate similar problems for the more general family $L_{\\alpha }(x)$, where $0\\leq \\alpha \\leq 1$, and their relationship to the Riemann hypothesis and other properties of the zeros of the Riemann zeta function. The case where $\\alpha =1/2$is of particular interest.
17 citations
TL;DR: In this article, the authors give an explicit form of the group that is a subgroup of the Hilbert modular group of its trace field and provide an interval map that is piecewise linear fractional, given in terms of group elements.
Abstract: We give continued fraction algorithms for each conjugacy class of triangle Fuchsian group of signature (3; n;1), with n 4. In particular, we give an explicit form of the group that is a subgroup of the Hilbert modular group of its trace field and provide an interval map that is piecewise linear fractional, given in terms of group elements. Using natural extensions, we find an ergodic invariant measure for the interval map. We also study Diophantine properties of approximation in terms of the continued fractions and show that these continued fractions are appropriate to obtain transcendence results.
17 citations
TL;DR: In this article, the authors used p-adic methods, primarily Hensel's lemma and padic interpolation, to count fixed points, two-cycles, collisions, and solutions to related equations modulo powers of a prime p.
Abstract: Brizolis asked for which primes p greater than 3 there exists a pair (g,h) such that h is a fixed point of the discrete exponential map with base g, or equivalently h is a fixed point of the discrete logarithm with base g. Various authors have contributed to the understanding of this problem. In this paper, we use p-adic methods, primarily Hensel’s lemma and p-adic interpolation, to count fixed points, two-cycles, collisions, and solutions to related equations modulo powers of a prime p.
15 citations
TL;DR: In this paper, a lower bound for the number of positive square-free integers n, up to x, for which the products Pi(p vertical bar n) (p + a) (over primes p) are perfect rth powers for all the integers a in A.
Abstract: Let r be an integer greater than 1, and let A be a finite, nonempty set of nonzero integers. We obtain a lower bound for the number of positive squarefree integers n, up to x, for which the products Pi(p vertical bar n)(p + a) (over primes p) are perfect rth powers for all the integers a in A. Also, in the cases where A = {-1} and A = {+1}, we will obtain a lower bound for the number of such n with exactly r distinct prime factors.
13 citations
TL;DR: In this paper, the authors study when weighted composition operators C; acting between weighted Bergman spaces of innite order are power bounded resp. uniformly mean ergodic, i.e.
Abstract: We study when weighted composition operators C; acting between weighted Bergman spaces of innite order are power bounded resp. uniformly mean ergodic.
13 citations
TL;DR: In this article, the von Mangoldt function has been used to give bounds on sums of the form P n N (n) exp(2 iag n =m) and P n n N(n) (g n + a), where m is a natural number, a and g are integers coprime to m, and is a multiplicative character modulo m.
Abstract: We give new bounds on sums of the form P n N (n) exp(2 iag n =m) and P n N (n) (g n + a), where is the von Mangoldt function, m is a natural number, a and g are integers coprime to m, and is a multiplicative character modulo m. In particular, our results yield bounds on the sums P p N exp(2 iaMp=m) and P p N (Mp) with Mersenne numbers Mp = 2 p 1, where p is prime.
10 citations
TL;DR: In this paper, it was shown that for real algebraic numbers, the boundedness of regular and nearest integer partial quotients is equivalent, and that rationals and quadratic irrationals have bounded partial quotient in the Hurwitz expansion.
Abstract: Conjecturally, the only real algebraic numbers with bounded partial quotients in their regular continued fraction expansion are rationals and quadratic irrationals. We show that the corresponding statement is not true for complex algebraic numbers in a very strong sense, by constructing, for every even degree that have bounded complex partial quotients in their Hurwitz continued fraction expansion. The Hurwitz expansion is the complex generalization of the nearest integer continued fraction for real numbers. In the case of real numbers the boundedness of regular and nearest integer partial quotients is equivalent.
TL;DR: A variant of Fermat's factoring algorithm which is competitive with SQUFOF in practice but has heuristic run time complexity O(n1/3) and a sparse class of integers for which the algorithm is particularly effective.
Abstract: We describe a variant of Fermat’s factoring algorithm which is competitive with SQUFOF in practice but has heuristic run time complexity O(n1/3) as a general factoring algorithm. We also describe a sparse class of integers for which the algorithm is particularly effective. We provide speed comparisons between an optimised implementation of the algorithm described and the tuned assortment of factoring algorithms in the Pari/GP computer algebra package.
TL;DR: In this paper, the authors investigated the number of symmetric matrices of nonnegative integers with zero diagonal such that each row sum is the same and determined the asymptotic value of this number as the size of the matrix tends to infinity.
Abstract: We investigate the number of symmetric matrices of nonnegative integers with zero diagonal such that each row sum is the same. Equivalently, these are zero-diagonal symmetric contingency tables with uniform margins, or loop-free regular multigraphs. We determine the asymptotic value of this number as the size of the matrix tends to infinity, provided the row sum is large enough. We conjecture that one form of our answer is valid for all row sums. An example appears in Figure 1.
TL;DR: In this article, the existence of primes and primitive divisors in function field analogues of classical divisibility sequences was studied under various hypotheses, and it was shown that Lucas sequences and elliptic divisability sequences over function fields defined over number fields contain infinitely many irreducible elements.
Abstract: In this note we study the existence of primes and of primitive divisors in function field analogues of classical divisibility sequences. Under various hypotheses, we prove that Lucas sequences and elliptic divisibility sequences over function fields defined over number fields contain infinitely many irreducible elements. We also prove that an elliptic divisibility sequence over a function field has only finitely many terms lacking a primitive divisor.
TL;DR: In this paper, the authors considered subgroups X of full wreath products Sym(Gamma) wr Sym(Delta) in product action and showed that in a suitable conjugate of X, the subgroup induced by a stabilizer of a coordinate delta in Delta only depends on the induced action of X on Delta.
Abstract: The wreath product of two permutation groups G < Sym(Gamma) and H < Sym(Delta) can be considered as a permutation group acting on the set Pi of functions from Delta to Gamma. This action, usually called the product action, of a wreath product plays a very important role in the theory of permutation groups, as several classes of primitive or quasiprimitive groups can be described as subgroups of such wreath products. In addition, subgroups of wreath products in product action arise as automorphism groups of graph products and codes. In this paper we consider subgroups X of full wreath products Sym(Gamma) wr Sym(Delta) in product action. Our main result is that, in a suitable conjugate of X, the subgroup of Sym(Gamma) induced by a stabilizer of a coordinate delta in Delta only depends on the orbit of delta under the induced action of X on Delta. Hence, if the action of X on Delta is transitive, then X can be embedded into a much smaller wreath product. Further, if this X-action is intransitive, then X can be embedded into a direct product of such wreath products where the factors of the direct product correspond to the X-orbits in Delta. We offer an application of the main theorems to error-correcting codes in Hamming graphs.
TL;DR: In this article, an upper bound on the least positive integer T = T(f) such that no quotient of two distinct Tth powers of roots of f is a root of unity is given.
Abstract: Let K be a number field. For f∈K[x], we give an upper bound on the least positive integer T=T(f) such that no quotient of two distinct Tth powers of roots of f is a root of unity. For each e>0 and each f∈ℚ[x] of degree d≥d(e) we prove that . In the opposite direction, we show that the constant 2 cannot be replaced by a number smaller than 1 . These estimates are useful in the study of degenerate and nondegenerate linear recurrence sequences over a number field K.
TL;DR: In this article, a matrix-variate generalization of the Gauss hypergeometric distribution was proposed and several properties of its properties were studied. And the probability density functions of the product of two independent random matrices when one of them is Gauss-hypergeometric were derived.
Abstract: In this paper, we propose a matrix-variate generalization of the Gauss hypergeometric distribution and study several of its properties. We also derive probability density functions of the product of two independent random matrices when one of them is Gauss hypergeometric. These densities are expressed in terms of Appell’s first hypergeometric function F1 and Humbert’s confluent hypergeometric function Φ1of matrix arguments.
TL;DR: In this article, a systematic treatment of caloric measure for arbitrary open sets is given, where the caloric measure is defined only on the essential boundary of the set, and criteria for the resolutivity of essential boundary functions are given.
Abstract: Abstract We give a systematic treatment of caloric measure for arbitrary open sets. The caloric measure is defined only on the essential boundary of the set. Our main result gives criteria for the resolutivity of essential boundary functions, and their integral representation in terms of caloric measure. We also characterize the caloric measure null sets in terms of the boundary singularities of nonnegative supertemperatures.
TL;DR: In this paper, a variant of Hofstadter's $Q$ -sequence is analyzed and its frequency sequence is 2-automatic, and an automaton for computing the sequence is explicitly given.
Abstract: Following up on a paper of Balamohan et al . [‘On the behavior of a variant of Hofstadter’s $q$
-sequence’, J. Integer Seq. 10 (2007)], we analyze a variant of Hofstadter’s $Q$
-sequence and show that its frequency sequence is 2-automatic. An automaton computing the sequence is explicitly given.
TL;DR: In this article, the authors consider a multi-parameter family of canonical coordinates and mirror maps originally introduced by Zudilin and prove that all coefficients in their Taylor expansions at 0 are positive.
Abstract: We consider a multi-parameter family of canonical coordinates and mirror maps originally introduced by Zudilin. This family includes many of the known one-variable mirror maps as special cases, in particular many of modular origin and the celebrated ‘quintic’ example of Candelas, de la Ossa, Green and Parkes. In a previous paper, we proved that all coefficients in the Taylor expansions at 0 of these canonical coordinates (and, hence, of the corresponding mirror maps) are integers. Here we prove that all coefficients in their Taylor expansions at 0 are positive. Furthermore, we provide several results about the behaviour of the canonical coordinates and mirror maps as complex functions. In particular, we address their analytic continuation, points of singularity, and radius of covergence. We present several very precise conjectures on the radius of covergence of the mirror maps and the sign pattern of the coefficients in their Taylor expansions at 0.
TL;DR: In this article, a property weaker than weak uniform distribution is studied for polynomial-like multiplicative functions, in particular for varphi (n) and sigma (n).
Abstract: Abstract For a class of multiplicative integer-valued functions $f$ the distribution of the sequence $f(n)$ in restricted residue classes modulo $N$ is studied. We consider a property weaker than weak uniform distribution and study it for polynomial-like multiplicative functions, in particular for $\varphi (n)$ and $\sigma (n)$.
TL;DR: In this paper, the lattice of compatible quasi-orders is characterized by describing its join-and meet-irreducible elements, and it is shown that the unary operation has finitely many nontrivial kernel classes and its graph is a binary tree.
Abstract: Rooted monounary algebras can be considered as an algebraic counterpart of directed rooted trees. We work towards a characterization of the lattice of compatible quasiorders by describing its join- and meet-irreducible elements. We introduce the limit . For a partial order, it is known that complete meet-irreducibility means that the corresponding partially ordered structure is subdirectly irreducible. For a rooted monounary algebra it is shown that this property implies that the unary operation has finitely many nontrivial kernel classes and its graph is a binary tree.
TL;DR: In this article, the authors show how to compute ℚ(α)-irrationality measures of a number ξ∉𝕂, and ǫ-non-quadraticity measures if [ǫ(ǫ):ǫ>2.
Abstract: Abstract Let 𝕂⊂ℂ be a number field. We show how to compute 𝕂-irrationality measures of a number ξ∉𝕂, and 𝕂-nonquadraticity measures of ξ if [𝕂(ξ):𝕂]>2. By applying the saddle point method to a family of double complex integrals, we prove ℚ(α)-irrationality measures and ℚ(α)-nonquadraticity measures of log α for several algebraic numbers α∈ℂ, improving earlier results due to Amoroso and the second-named author.
TL;DR: In this article, the authors studied the differentiability of the limiting distribution function associated to the normalized Euler function defined on the shifted primes and showed that it is differentiable.
Abstract: We study the differentiability of the limiting distribution function associated to the normalized Euler function defined on the shifted primes.
TL;DR: In this paper, the authors give exact formulae for the quantities Dn:=qnα−pn in several typical types of Tasoev continued fractions, and a simple example of the type of continued fraction considered is α=[0;ua,ua2,ua3, etc.
Abstract: Denote the nth convergent of the continued fraction α=[a0;a1,a2,…] by pn/qn=[a0;a1,…,an]. In this paper we give exact formulae for the quantities Dn:=qnα−pn in several typical types of Tasoev continued fractions. A simple example of the type of Tasoev continued fraction considered is α=[0;ua,ua2,ua3,…].
TL;DR: In this article, a family of ideals representing ideal classes of order 2 in quadratic number fields is constructed and relations between their ideal classes are governed by certain cyclic quartic extensions of the rationals.
Abstract: We construct a family of ideals representing ideal classes of order 2 in quadratic number fields and show that relations between their ideal classes are governed by certain cyclic quartic extensions of the rationals.
TL;DR: In this paper, the authors introduce a new construction involving Rees matrix semigroups and max-plus algebras that is very convenient for generating sets of centroids.
Abstract: Abstract We introduce a new construction involving Rees matrix semigroups and max-plus algebras that is very convenient for generating sets of centroids. We describe completely all optimal sets of centroids for all Rees matrix semigroups without any restrictions on the sandwich matrices.
TL;DR: In this article, the authors discuss the following general question and some of its extensions: Is it true that at least one among ξ and ξ′ satisfies 𝒫?
Abstract: We discuss the following general question and some of its extensions. Let (ek)k≥1 be a sequence with values in {0,1}, which is not ultimately periodic. Define ξ:=∑ k≥1ek/2k and ξ′:=∑ k≥1ek/3k. Let 𝒫 be a property valid for almost all real numbers. Is it true that at least one among ξ and ξ′ satisfies 𝒫?
TL;DR: In this article, a family of radical convolution Banach algebras on intervals (0,a] which are of Sobolev type are defined in terms of derivatives.
Abstract: We present a family of radical convolution Banach algebras on intervals (0,a] which are of Sobolev type; that is, they are defined in terms of derivatives. Among other properties, it is shown that all epimorphisms and derivations of such algebras are bounded. Also, we give examples of nontrivial concrete derivations.
TL;DR: In this article, it was shown that this equation has no solutions if deg f ≥ 3 and if deg g ≥ 2, the polynomial g(x) has nonzero derivative g′ (x)≠0 in K[x] and the integer m≥2 is not divisible by the characteristic of the field K.
Abstract: In this paper we solve the equation f(g(x))=f(x)hm(x) where f(x), g(x) and h(x) are unknown polynomials with coefficients in an arbitrary field K, f(x) is nonconstant and separable, deg g≥2, the polynomial g(x) has nonzero derivative g′(x)≠0 in K[x] and the integer m≥2 is not divisible by the characteristic of the field K. We prove that this equation has no solutions if deg f≥3 . If deg f=2 , we prove that m=2 and give all solutions explicitly in terms of Chebyshev polynomials. The Diophantine applications for such polynomials f(x) , g(x) , h(x) with coefficients in ℚ or ℤ are considered in the context of the conjecture of Cassaigne et al. on the values of Liouville’s λ function at points f(r) , r∈ℚ.