https://novaprd-lb.newcastle.edu.au/vital/access/manager/Index ${session.getAttribute("locale")} 5 https://novaprd-lb.newcastle.edu.au/vital/access/manager/Repository/uon:12967 Wed 11 Apr 2018 17:22:28 AEST ]]> https://novaprd-lb.newcastle.edu.au/vital/access/manager/Repository/uon:7550 Wed 11 Apr 2018 17:17:34 AEST ]]> https://novaprd-lb.newcastle.edu.au/vital/access/manager/Repository/uon:13492 1,then the number #(|y|,N) of 1-bits in the expansion of |y| through bit position N satisfies #(|y|,N) > CN^1/D for a positive number C (depending on y) and sufficiently large N. This in itself establishes the transcendency of a class of reals ∑_n≥o 1/2^f(n) where the integer-valued function f grows sufficiently fast; say, faster than any fixed power of n. By these methods we re-establish the transcendency of the Kempner–Mahler number ∑_n≥o 1/2²ⁿ, yet we can also handle numbers with a substantially denser occurrence of 1’s. Though the number z = ∑_n≥o 1/2ⁿ² has too high a 1’s density for application of our central result, we are able to invoke some rather intricate number-theoretical analysis and extended computations to reveal aspects of the binary structure of z².]]> Wed 11 Apr 2018 17:14:51 AEST ]]> https://novaprd-lb.newcastle.edu.au/vital/access/manager/Repository/uon:12975 Wed 11 Apr 2018 16:51:30 AEST ]]> https://novaprd-lb.newcastle.edu.au/vital/access/manager/Repository/uon:6446 Wed 11 Apr 2018 16:50:58 AEST ]]> https://novaprd-lb.newcastle.edu.au/vital/access/manager/Repository/uon:13576 Wed 11 Apr 2018 16:50:54 AEST ]]> https://novaprd-lb.newcastle.edu.au/vital/access/manager/Repository/uon:10338 Wed 11 Apr 2018 16:28:22 AEST ]]> https://novaprd-lb.newcastle.edu.au/vital/access/manager/Repository/uon:10649 Wed 11 Apr 2018 15:40:07 AEST ]]> https://novaprd-lb.newcastle.edu.au/vital/access/manager/Repository/uon:13549 Wed 11 Apr 2018 14:54:32 AEST ]]> https://novaprd-lb.newcastle.edu.au/vital/access/manager/Repository/uon:13529 Wed 11 Apr 2018 11:29:42 AEST ]]> https://novaprd-lb.newcastle.edu.au/vital/access/manager/Repository/uon:12923 b,c = Σ_n≥1 1/(cⁿb^cn), for coprime integers b ≥ 2 and c ≥ 2. These are of interest because, according to previous studies, α_b,c is known to be b-normal, meaning that every m-long string of base-b digits appears in the base-b expansion of the constant with precisely the limiting frequency b^-m. So, for example, the constant α_2,3 = Σn≥1 1/(3ⁿ2³ⁿ) is 2-normal. More recently it was established that α_b,c is not bc-normal, so, for example,α_2,3 is provably not 6-normal. In this paper, we extend these findings by showing that α_b,c is not B-normal, where B = b^pc^q r, for integers b and c as above, p, q, r ≥ 1, neither b nor c divide r, and the condition D=c^q/pr^1/p/b^c-1 < 1 is satisfied. It is not known whether or not this is a complete catalog of bases to which α_b,c is nonnormal. We also show that the sum of two B-nonnormal Stoneham constants as defined above, subject to some restrictions, is B-nonnormal.]]> Wed 11 Apr 2018 10:56:19 AEST ]]> https://novaprd-lb.newcastle.edu.au/vital/access/manager/Repository/uon:13722 Wed 11 Apr 2018 09:55:47 AEST ]]> https://novaprd-lb.newcastle.edu.au/vital/access/manager/Repository/uon:12969 0, x, y are real numbers and Odenotes the odd integers. In this paper we first survey the earlier work and then discuss the sum (1) more completely. Conclusions: As in the previous study, we find some surprisingly simple closed-form evaluations of these sums. In particular, we find that in some cases these sums are given by 1/π logA, where A is an algebraic number. These evaluations suggest that a deep theory interconnects all such summations.]]> Wed 11 Apr 2018 09:47:17 AEST ]]> https://novaprd-lb.newcastle.edu.au/vital/access/manager/Repository/uon:10339 Wed 11 Apr 2018 09:34:30 AEST ]]> https://novaprd-lb.newcastle.edu.au/vital/access/manager/Repository/uon:13500 Wed 11 Apr 2018 09:19:32 AEST ]]> https://novaprd-lb.newcastle.edu.au/vital/access/manager/Repository/uon:32052 m, n oddcos (mπx)cos (nπy)/(m² + n²) = log A, where  A is an algebraic number, and explicit minimal polynomials associated with A were computed for a few specific rational arguments x and y. Based on these results, one of us (Kimberley) conjectured a number-theoretic formula for the degree of A in the case x = y = 1/s for some integer s. These earlier studies were hampered by the enormous cost and complexity of the requisite computations. In this study, we address the Poisson polynomial problem with significantly more capable computational tools. As a result of this improved capability, we have confirmed that Kimberley’s formula holds for all integers s up to 52 (except for s = 41, 43, 47, 49, 51, which are still too costly to test), and also for s = 60 and s = 64. As far as we are aware, these computations, which employed up to 64,000-digit precision, producing polynomials with degrees up to 512 and integer coefficients up to 10²²⁹, constitute the largest successful integer relation computations performed to date. By examining the computed results, we found connections to a sequence of polynomials defined in a 2010 paper by Savin and Quarfoot. These investigations subsequently led to a proof, given in the Appendix, of Kimberley’s formula and the fact that when s is even, the polynomial is palindromic (i.e., coefficient a^k = a_{m − k}, where m is the degree).]]> Thu 26 Apr 2018 11:36:50 AEST ]]> https://novaprd-lb.newcastle.edu.au/vital/access/manager/Repository/uon:39623 Thu 16 Jun 2022 11:25:13 AEST ]]> https://novaprd-lb.newcastle.edu.au/vital/access/manager/Repository/uon:12958 Sat 24 Mar 2018 10:37:17 AEDT ]]> https://novaprd-lb.newcastle.edu.au/vital/access/manager/Repository/uon:12928 Sat 24 Mar 2018 10:37:10 AEDT ]]> https://novaprd-lb.newcastle.edu.au/vital/access/manager/Repository/uon:12944 Sat 24 Mar 2018 10:36:36 AEDT ]]> https://novaprd-lb.newcastle.edu.au/vital/access/manager/Repository/uon:11558 Sat 24 Mar 2018 10:33:09 AEDT ]]> https://novaprd-lb.newcastle.edu.au/vital/access/manager/Repository/uon:15946 0, every m-long string of digits in the base-b expansion of α appears, in the limit, with frequency b^-m. Although almost all reals in [0, 1] are b-normal for every b, it has been rather difficult to exhibit explicit examples. No results whatsoever are known, one way or the other, for the class of “natural” mathematical constants, such as π,e,2√ and log2. In this paper, we summarize some previous normality results for a certain class of explicit reals and then show that a specific member of this class, while provably 2-normal, is provably not 6-normal. We then show that a practical and reasonably effective pseudorandom number generator can be defined based on the binary digits of this constant and conclude by sketching out some directions for further research.]]> Sat 24 Mar 2018 08:23:41 AEDT ]]> https://novaprd-lb.newcastle.edu.au/vital/access/manager/Repository/uon:14064 Sat 24 Mar 2018 08:22:33 AEDT ]]> https://novaprd-lb.newcastle.edu.au/vital/access/manager/Repository/uon:12922 n = ∫₀¹logⁿΓ(x)dx for 1≤n≤4 and make some comments regarding the general case. The subsidiary computational challenges are substantial, interesting and significant in their own right]]> Sat 24 Mar 2018 08:18:13 AEDT ]]> https://novaprd-lb.newcastle.edu.au/vital/access/manager/Repository/uon:19060 Sat 24 Mar 2018 08:05:25 AEDT ]]> https://novaprd-lb.newcastle.edu.au/vital/access/manager/Repository/uon:21382 iα_ix_i with α_i rational and x_i complex. The included algorithms, called _SimplifySum² and implemented in Mathematica, remove redundant terms, attempt to make terms and the full expression real, and remove terms using repeated application of the multipair PSLQ integer relation detection algorithm. Also included are facilities for making substitutions according to user-specified identities. We illustrate this toolset by giving some real-world examples of its usage, including one, for instance, where the tool reduced a symbolic expression of approximately 100 000 characters in size enough to enable manual manipulation to one with just four simple terms.]]> Sat 24 Mar 2018 08:04:58 AEDT ]]> https://novaprd-lb.newcastle.edu.au/vital/access/manager/Repository/uon:20686 Sat 24 Mar 2018 07:55:38 AEDT ]]> https://novaprd-lb.newcastle.edu.au/vital/access/manager/Repository/uon:20685 Sat 24 Mar 2018 07:55:38 AEDT ]]> https://novaprd-lb.newcastle.edu.au/vital/access/manager/Repository/uon:6476 Sat 24 Mar 2018 07:47:10 AEDT ]]> https://novaprd-lb.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://novaprd-lb.newcastle.edu.au/vital/access/manager/Repository/uon:23327 Sat 24 Mar 2018 07:13:30 AEDT ]]> https://novaprd-lb.newcastle.edu.au/vital/access/manager/Repository/uon:22779 Sat 24 Mar 2018 07:12:15 AEDT ]]>