https://ogma.newcastle.edu.au/vital/access/ /manager/Index en-au 5 https://ogma.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://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:12986 n^(−a)(−z)~C(a,z)n^−a/2−1/4e2^√nz. We introduce a computationally motivated contour integral that allows efficient numerical Laguerre evaluations yet also leads to the complete asymptotic series over the full parameter domain of subexponential behavior. We present a fast algorithm for symbolic generation of the rather formidable expansion coefficients. Along the way we address the difficult problem of establishing effective (i.e., rigorous and explicit) error bounds on the general expansion. A primary tool for these developments is an “exp-arc” method giving a natural bridge between converging series and effective asymptotics.]]> Wed 11 Apr 2018 15:46:16 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:12970 Wed 11 Apr 2018 15:20:08 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:13500 Wed 11 Apr 2018 09:19:32 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:13068 Sat 24 Mar 2018 08:15:39 AEDT ]]>