21 research outputs found
Robot's hand and expansions in non-integer bases
We study a robot hand model in the framework of the theory of expansions in
non-integer bases. We investigate the reachable workspace and we study some
configurations enjoying form closure properties.Comment: 22 pages, 10 figure
Parallel addition in non-standard numeration systems
We consider numeration systems where digits are integers and the base is an
algebraic number such that and satisfies a
polynomial where one coefficient is dominant in a certain sense. For this class
of bases , we can find an alphabet of signed-digits on which addition is
realizable by a parallel algorithm in constant time. This algorithm is a kind
of generalization of the one of Avizienis. We also discuss the question of
cardinality of the used alphabet, and we are able to modify our algorithm in
order to work with a smaller alphabet. We then prove that satisfies
this dominance condition if and only if it has no conjugate of modulus 1. When
the base is the Golden Mean, we further refine the construction to
obtain a parallel algorithm on the alphabet . This alphabet cannot
be reduced any more
-block parallel addition versus -block parallel addition in non-standard numeration systems
Parallel addition in integer base is used for speeding up multiplication and
division algorithms. -block parallel addition has been introduced by
Kornerup in 1999: instead of manipulating single digits, one works with blocks
of fixed length . The aim of this paper is to investigate how such notion
influences the relationship between the base and the cardinality of the
alphabet allowing parallel addition. In this paper, we mainly focus on a
certain class of real bases --- the so-called Parry numbers. We give lower
bounds on the cardinality of alphabets of non-negative integer digits allowing
block parallel addition. By considering quadratic Pisot bases, we are able to
show that these bounds cannot be improved in general and we give explicit
parallel algorithms for addition in these cases. We also consider the
-bonacci base, which satisfies the equation . If in a base being a -bonacci number -block parallel
addition is possible on the alphabet , then ; on the other hand, there exists a such that -block
parallel addition in this base is possible on the alphabet , which
cannot be reduced. In particular, addition in the Tribonacci base is -block
parallel on alphabet .Comment: 21 page
Shift Radix Systems - A Survey
Let be an integer and . The {\em shift radix system} is defined by has the {\em finiteness
property} if each is eventually mapped to
under iterations of . In the present survey we summarize
results on these nearly linear mappings. We discuss how these mappings are
related to well-known numeration systems, to rotations with round-offs, and to
a conjecture on periodic expansions w.r.t.\ Salem numbers. Moreover, we review
the behavior of the orbits of points under iterations of with
special emphasis on ultimately periodic orbits and on the finiteness property.
We also describe a geometric theory related to shift radix systems.Comment: 45 pages, 16 figure
Efficient Algorithms for gcd and Cubic Residuosity in the Ring of Eisenstein Integers
We present simple and efficient algorithms for computing gcd and cubic residuosity in the ring of Eisenstein integers, Z[zeta] , i.e. the integers extended with zeta , a complex primitive third root of unity. The algorithms are similar and may be seen as generalisations of the binary integer gcd and derived Jacobi symbol algorithms. Our algorithms take time O(n^2) for n bit input. This is an improvement from the known results based on the Euclidian algorithm, and taking time O(n· M(n)), where M(n) denotes the complexity of multiplying n bit integers. The new algorithms have applications in practical primality tests and the implementation of cryptographic protocols. The technique underlying our algorithms can be used to obtain equally fast algorithms for gcd and quartic residuosity in the ring of Gaussian integers, Z[i]
Geometrical aspects of expansions in complex bases
We study the set of the representable numbers in base
with and and with digits in
a arbitrary finite real alphabet . We give a geometrical description of the
convex hull of the representable numbers in base and alphabet and an
explicit characterization of its extremal points. A characterizing condition
for the convexity of the set of representable numbers is also shown.Comment: 23 pages, 5 figure
Rhapsody in Fractional
This paper studies several topics related with the concept of “fractional”
that are not directly related with Fractional Calculus, but can help
the reader in pursuit new research directions. We introduce the concept of
non-integer positional number systems, fractional sums, fractional powers
of a square matrix, tolerant computing and FracSets, negative probabilities,
fractional delay discrete-time linear systems, and fractional Fourier
transform