2,531 research outputs found
Fast computation of Bernoulli, Tangent and Secant numbers
We consider the computation of Bernoulli, Tangent (zag), and Secant (zig or
Euler) numbers. In particular, we give asymptotically fast algorithms for
computing the first n such numbers in O(n^2.(log n)^(2+o(1))) bit-operations.
We also give very short in-place algorithms for computing the first n Tangent
or Secant numbers in O(n^2) integer operations. These algorithms are extremely
simple, and fast for moderate values of n. They are faster and use less space
than the algorithms of Atkinson (for Tangent and Secant numbers) and Akiyama
and Tanigawa (for Bernoulli numbers).Comment: 16 pages. To appear in Computational and Analytical Mathematics
(associated with the May 2011 workshop in honour of Jonathan Borwein's 60th
birthday). For further information, see
http://maths.anu.edu.au/~brent/pub/pub242.htm
On irregular prime power divisors of the Bernoulli numbers
Let () denote the usual -th Bernoulli number. Let
be a positive even integer where or . It is well known
that the numerator of the reduced quotient is a product of powers of
irregular primes. Let be an irregular pair with B_l/l \not\equiv
B_{l+p-1}/(l+p-1) \modp{p^2}. We show that for every the congruence
B_{m_r}/m_r \equiv 0 \modp{p^r} has a unique solution where m_r \equiv
l \modp{p-1} and . The sequence
defines a -adic integer which is a zero of a certain
-adic zeta function originally defined by T. Kubota and H. W.
Leopoldt. We show some properties of these functions and give some
applications. Subsequently we give several computations of the (truncated)
-adic expansion of for irregular pairs with
below 1000.Comment: 42 pages; final accepted paper, slightly revised and extended, to
appear in Math. Com
Visual word recognition in bilinguals: Phonological priming from the second to the first language
In this study, the authors show that cross-lingual phonological priming is possible not only from the 1st language (L1) to the 2nd language (L2), but also from L2 to L1. In addition, both priming effects were found to have the same magnitude and to not be related to differences in word naming latencies between L1 and L2. The findings are further evidence against language-selective access models of bilingual word processing and are more in line with strong phonological models of visual word recognition than with the traditional dual-route models
Explicit isogeny descent on elliptic curves
In this note, we consider an l-isogeny descent on a pair of elliptic curves
over Q. We assume that l > 3 is a prime. The main result expresses the relevant
Selmer groups as kernels of simple explicit maps between finite- dimensional
F_l-vector spaces defined in terms of the splitting fields of the kernels of
the two isogenies. We give examples of proving the l-part of the Birch and
Swinnerton-Dyer conjectural formula for certain curves of small conductor.Comment: 17 pages, accepted for publication in Mathematics of Computatio
Universal characteristics of fractal fluctuations in prime number distribution
The frequency of occurrence of prime numbers at unit number spacing intervals exhibits self-similar fractal fluctuations concomitant with inverse power law form for power spectrum generic to dynamical systems in nature such as fluid flows, stock market fluctuations and population dynamics. The physics of long-range correlations exhibited by fractals is not yet identified. A recently developed general systems theory visualizes the eddy continuum underlying fractals to result from the growth of large eddies as the integrated mean of enclosed small scale eddies, thereby generating a hierarchy of eddy circulations or an inter-connected network with associated long-range correlations. The model predictions are as follows: (1) The probability distribution and power spectrum of fractals follow the same inverse power law which is a function of the golden mean. The predicted inverse power law distribution is very close to the statistical normal distribution for fluctuations within two standard deviations from the mean of the distribution. (2) Fractals signify quantum-like chaos since variance spectrum represents probability density distribution, a characteristic of quantum systems such as electron or photon. (3) Fractal fluctuations of frequency distribution of prime numbers signify spontaneous organization of underlying continuum number field into the ordered pattern of the quasiperiodic Penrose tiling pattern. The model predictions are in agreement with the probability distributions and power spectra for different sets of frequency of occurrence of prime numbers at unit number interval for successive 1000 numbers. Prime numbers in the first 10 million numbers were used for the study
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