https://ogma.newcastle.edu.au/vital/access/ /manager/Index ${session.getAttribute("locale")} 5 https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:26658 Wed 11 Apr 2018 17:08:06 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:30912 r₁,..., r_s: Z_n≥0 → C be linearly recurrent sequences whose associated eigenvalues have arguments in πQ and let F(z) := Σ_n ≥ 0 f(n)zn, where f(n) ∈ {r1(n),..., rs(n)} for each n ≥ 0. We prove that if F(z) is bounded in a sector of its disk of convergence, then it is a rational function. This extends a very recent result of Tang and Wang, who gave the analogous result when the sequence f(n) takes on values of finitely many polynomials.]]> Wed 11 Apr 2018 16:49:35 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:14415 Wed 11 Apr 2018 15:47:25 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:23973 2)F(z⁴) +z⁴F(z¹⁶)=0. Specifically, we prove that the functions F(z), F(z⁴), F′(z), and F′(z⁴) are algebraically independent over ℂ(z). An application of a celebrated result of Ku. Nishioka then allows one to replace ℂ(z) by ℚ when evaluating these functions at a nonzero algebraic number α in the unit disc.]]> Wed 11 Apr 2018 15:22:41 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:30913 z, but also over Cz(M), where (M) is the set of meromorphic functions. Several examples and corollaries are given, with special attention to nonnegative regular functions.]]> Wed 11 Apr 2018 15:20:26 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:31692 k(z) := Σ^∞_n=0 z^kn (1 − z^kn)⁻¹ is transcendental for all algebraic numbers z with |z| < 1. We give a similar result for F_k(z) := Σ^∞n=0 z^kn)(1 + z^kn)⁻¹. These results were known to Mahler, though our proofs of the function transcendence are new and elementary; no linear algebra or differential calculus is used.]]> Wed 11 Apr 2018 15:07:27 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:30911 Wed 11 Apr 2018 12:44:14 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:34337 F of a Mahler function F(z) and develop a quick test for the transcendence of F(z) over ℂ(z), which is determined by the value of the eigenvalue λ_F. While our first test is quick and applicable for a large class of functions, our second test, while a bit slower than our first, is universal; it depends on the rank of a certain Hankel matrix determined by the initial coefficients of F(z). We note that these are the first transcendence tests for Mahler functions of arbitrary degree. Several examples and applications are given.]]> Wed 04 Sep 2019 10:06:10 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:41410 Wed 03 Aug 2022 11:31:20 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:47643 Tue 24 Jan 2023 14:45:00 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:39110 kⁿ, where k is the base of the Mahler function, as well as some applications concerning transcendence and algebraic independence. For example, we show that the generating function of the Thue–Morse sequence and any Mahler function (to the same base) which has a nonzero Mahler eigenvalue are algebraically independent over C(z). Finally, we discuss asymptotic bounds towards generic points on the unit circle.]]> Tue 10 May 2022 14:26:34 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:40110 Thu 28 Jul 2022 15:35:25 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:31577 Sat 24 Mar 2018 08:45:03 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:31341 n) denote the number of prime divisors of n counting multiplicity. One can show that for any positive integer m and all j = 0,1,…,m – 1, we have #{n ≤ x : Ω(n) ≡ j(modm} = x/m + o(x^α), with α = 1. Building on work of Kubota and Yoshida, we show that for m > 2 and any j = 0,1,…,m – 1, the error term is not o(x^α) for any α < 1.]]> Sat 24 Mar 2018 08:44:37 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:31335 f : ℕ → K be a multiplicative function taking values in a field K of characteristic 0, and write F(z) = Σ_n≥1f(n)zⁿ ∈ K[[z]] for its generating series. If F(z) is algebraic, then either there is a natural number k and a periodic multiplicative function χ(n) such that f(n) = n^kχ(n) for all n or f(n) is eventually zero. In particular, the generating series of a multiplicative function taking values in a field of characteristic zero is either transcendental or rational. For K = ℂ, we also prove that if the generating series of a multiplicative function is D-finite, then it must either be transcendental or rational.]]> Sat 24 Mar 2018 08:44:34 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:31836 f(z) ∈ ℂ [[z]] satisfying a functional equation of the form [formula could not be replicated] where A_k(z) ∈ ℂ [z]. In particular, we show that if f(z) satisfies a minimal functional equation of the above form with n ≥ 2, then f(z) is necessarily transcendental. Towards a more complete classification, the case n = 1 is also considered.]]> Sat 24 Mar 2018 08:44:04 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:20778 0 such that ∣f(n)∣ > c log n infinitely often. We end our paper by answering a question of Borwein, Choi and Coons on the sums of completely multiplicative automatic functions.]]> Sat 24 Mar 2018 08:06:02 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:17309 Sat 24 Mar 2018 08:01:48 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:19433 Sat 24 Mar 2018 07:51:57 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:20656 m. As a consequence of this result, we prove two recent conjectures of Aragón Artacho et al. [‘Walking on real numbers’, Math. Intelligencer 35(1) (2013), 42–60] concerning the base-b expansion of Stoneham numbers.]]> Sat 24 Mar 2018 07:49:58 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:29468 Sat 24 Mar 2018 07:41:28 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:28287 Sat 24 Mar 2018 07:41:23 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:28086 m for each positive integer m, the set of points {(n, 2ⁿ mod 3^m): n = 0}, viewed as a subset of Z=0 ×Z=0 is bi-periodic, with minimal periods f(3^m) (horizontally) and 3^m (vertically). We show that if one considers the classes of n modulo 6, one obtains a finer structural classification. This result is presented within the context of the question of strong normality of Stoneham numbers.]]> Sat 24 Mar 2018 07:39:48 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:30904 F(z) ϵ C[[z]] is a power series with coefficients from a finite set, then F(z) is either rational or it is transcendental over the field of meromorphic functions.]]> Sat 24 Mar 2018 07:30:39 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:30905 k-regular sequence based on information from its k-kernel. In order to provide such a classification, we introduce the notion of a growth exponent for k-regular sequences and show that this exponent is equal to the base-k logarithm of the joint spectral radius of any set of a special class of matrices determined by the k-kernel.]]> Sat 24 Mar 2018 07:30:39 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:23671 Sat 24 Mar 2018 07:16:51 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:24900 1 be a real number and define the β-transformation on [0,1] by T_β : x ↦ βx mod 1. Further, define: [formula could not be replicated] and [formula could not be replicated], where Ψ : ℕ → ℝ>₀ is a positive function such that Ψ(n) → 0 as n → ∞. In this paper, we show that each of the above sets obeys a Jarník-type dichotomy, that is, the generalized Hausdorff measure is either zero or full depending upon the convergence or divergence of a certain series. This work completes the metrical theory of these sets.]]> Sat 24 Mar 2018 07:14:52 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:23925 A be a finite set of d x d matrices with integer entries and let m_n(Α) be the maximum norm of a product of n elements of A. In this paper, we classify gaps in the growth m_n(Α); specifically, we prove that lim_n→∞log m_n(A)/log n ∈ℤ≥₀⋃{∞}. This has applications to the growth of regular sequences as defined by Allouche and Shallit.]]> Sat 24 Mar 2018 07:10:14 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:46217 Mon 14 Nov 2022 12:04:51 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:53042 Fri 17 Nov 2023 11:30:53 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:47228 Fri 16 Dec 2022 11:54:28 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:44095 Fri 07 Oct 2022 08:29:20 AEDT ]]>