https://ogma.newcastle.edu.au/vital/access/ /manager/Index ${session.getAttribute("locale")} 5 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:28734 Wed 11 Apr 2018 12:10:08 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:26737 n/2 be the ratio of D(n) to the Hadamard upper bound. Using the probabilistic method, we prove new lower bounds on D(n) and R(n) in terms of d = n - h, where h is the order of a Hadamard matrix and h is maximal subject to h ≤ n. For example, [forumal cannot be replicated]. By a recent result of Livinskyi, d²/h^1/2 → 0 as n → 8, so the second bound is close to (πe/2)^-d/2 for large n. Previous lower bounds tended to zero as n → ∞ with d fixed, except in the cases d ∈ {0, 1}. For d ≥ 2, our bounds are better for all sufficiently large n. If the Hadamard conjecture is true, then d ≤ 3, so the first bound above shows that R(n) is bounded below by a positive constant (πe/2)^-3/2 > 0.1133.]]> Wed 11 Apr 2018 10:30:42 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:44432 0 from O(exp(−πt)) to O(exp(−2πt)). We discuss a similar example due to Olver [‘Error bounds for asymptotic expansions, with an application to cylinder functions of large argument’, in: Asymptotic Solutions of Differential Equations and Their Applications (ed. C. H. Wilcox) (Wiley, New York, 1964), 16–18], and a connection with the Stokes phenomenon.]]> Tue 28 Nov 2023 15:44:34 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:13816 Tue 24 Aug 2021 14:28:05 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:31017 computational and experimental mathematics. is now a full-fledged discipline with mathematics, and the larger field of computational science is now taking its place as an experimental discipline on a par with traditional experimental fields. In this new realm, reproducibility comes to the forefront as an essential part of the computational research enterprise, and establishing procedures to ensure and facilitate reproducibility is now a central focus of researchers in the field. In this study, we describe our attempts to reproduce the results of a recently published article by Reinhard Ganz, who concluded that the decimal expansion of p is not statistically random, based on an analysis of several trillion decimal digits provided by Yee and Kondo. While we are able to reproduce the specific findings of Ganz, additional statistical analysis leads us to reject his overall conclusion.]]> Sat 24 Mar 2018 07:34:53 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:28706 n has no proper Hadamard submatrix of order m > n/2. We generalize this result to maximal determinant submatrices of Hadamard matrices, and show that an interval of length ~ n/2 is excluded from the allowable orders. We make a conjecture regarding a lower bound for sums of squares of minors of maximal determinant matrices, and give evidence to support it. We give tables of the values taken by the minors of all maximal determinant matrices of orders ≤ 21 and make some observations on the data. Finally, we describe the algorithms that were used to compute the tables.]]> Sat 24 Mar 2018 07:30:08 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:26948 Sat 24 Mar 2018 07:27:02 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:24828 Sat 24 Mar 2018 07:15:13 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:40448 Fri 29 Jul 2022 08:26:00 AEST ]]>