1,697 research outputs found

    A Fibonacci analogue of the two's complement numeration system

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    Using the classic two's complement notation of signed integers, the fundamental arithmetic operations of addition, subtraction, and multiplication are identical to those for unsigned binary numbers. We introduce a Fibonacci-equivalent of the two's complement notation and we show that addition in this numeration system can be performed by a deterministic finite-state transducer. The result is based on the Berstel adder, which performs addition of the usual Fibonacci representations of nonnegative integers and for which we provide a new constructive proof. Moreover, we characterize the Fibonacci-equivalent of the two's complement notation as an increasing bijection between Z\mathbb{Z} and a particular language.Comment: v3: 21 pages, 3 figures, 3 tables. v4: 24 pages, added a new section characterizing the Fibonacci's complement numeration system as an increasing bijection. v5: changes after revie

    Redundancy of minimal weight expansions in Pisot bases

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    Motivated by multiplication algorithms based on redundant number representations, we study representations of an integer nn as a sum n=∑kϵkUkn=\sum_k \epsilon_k U_k, where the digits ϵk\epsilon_k are taken from a finite alphabet Σ\Sigma and (Uk)k(U_k)_k is a linear recurrent sequence of Pisot type with U0=1U_0=1. The most prominent example of a base sequence (Uk)k(U_k)_k is the sequence of Fibonacci numbers. We prove that the representations of minimal weight ∑k∣ϵk∣\sum_k|\epsilon_k| are recognised by a finite automaton and obtain an asymptotic formula for the average number of representations of minimal weight. Furthermore, we relate the maximal order of magnitude of the number of representations of a given integer to the joint spectral radius of a certain set of matrices

    A Comparative Study of Some Pseudorandom Number Generators

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    We present results of an extensive test program of a group of pseudorandom number generators which are commonly used in the applications of physics, in particular in Monte Carlo simulations. The generators include public domain programs, manufacturer installed routines and a random number sequence produced from physical noise. We start by traditional statistical tests, followed by detailed bit level and visual tests. The computational speed of various algorithms is also scrutinized. Our results allow direct comparisons between the properties of different generators, as well as an assessment of the efficiency of the various test methods. This information provides the best available criterion to choose the best possible generator for a given problem. However, in light of recent problems reported with some of these generators, we also discuss the importance of developing more refined physical tests to find possible correlations not revealed by the present test methods.Comment: University of Helsinki preprint HU-TFT-93-22 (minor changes in Tables 2 and 7, and in the text, correspondingly

    On the character variety of the three-holed projective plane

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    We study the (relative) SL(2,C) character varieties of the three-holed projective plane and the action of the mapping class group on them. We describe a domain of discontinuity for this action, which strictly contains the set of primitive stable representations defined by Minsky, and also the set of convex-cocompact characters. We consider the relationship with the previous work of the authors and S. P. Tan on the character variety of the four-holed sphere.Comment: 27 page
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