View More View Less
  • 1 Dipartimento di Scienze di Base e Applicate per L'ingegneria, Sapienza Università di Roma, Via A. Scarpa 16, 00161 Roma, Italy
Restricted access

Abstract

We study the set of the representable numbers in base with ρ>1 and n∊ℕ and with digits in an arbitrary finite real alphabet A. We give a geometrical description of the convex hull of the representable numbers in base q and alphabet A and an explicit characterization of its extremal points. A characterizing condition for the convexity of the set of representable numbers is also shown.

  • [1] Akiyama, S. Thuswaldner, J. M. 2004 A survey on topological properties of tiles related to number systems Geom. Dedicata 109 89105 .

  • [2] Daróczy, Z. Kátai, I. 1988 Generalized number systems in the complex plane Acta Math. Hungar. 51 409416 .

  • [3] Dimitrov, V. A. Jullien, G. A. Miller, W. C. 1985 Complexity and fast algorithms for multiexponentiation Computers, IEEE Transactions 31 141147.

    • Search Google Scholar
    • Export Citation
  • [4] Frougny, Ch. and Surarerks, A., On-line multiplication in real and complex base, Computer Arithmetic, IEEE Symposium (2003).

  • [5] Gilbert, W. J. 1981 Geometry of radix representations The Geometric Vein: The Coxeter Festschrift Springer New York 129139.

  • [6] Gilbert, W. J. 1984 Arithmetic in complex bases Math. Mag. 57 7781 .

  • [7] Gilbert, W. J. 1987 Complex bases and fractal similarity Ann. Sci. Math. Québec 11 6577.

  • [8] Heckbert, P. S. 1994 Graphics Gems IV Academic Press.

  • [9] Kátai, I. Kovács, B. 1981 Canonical number systems in imaginary quadratic fields Acta Math. Acad. Sci. Hungar. 37 159164 .

  • [10] Kátai, I. Szabó, J. 1975 Canonical number systems for complex integers Acta Sci. Math. (Szeged) 37 255260.

  • [11] Knuth, D. E. 1960 An imaginary number system Comm. ACM 3 245247 .

  • [12] Knuth, D. E. 1971 The Art of Computer Programming 1 2 Addison-Wesley Publishing Co. Reading, Mass.

  • [13] Komornik, V. Loreti, P. 2007 Expansions in complex bases Canad. Math. Bull. 50 399408 .

  • [14] Pedicini, M. 2005 Greedy expansions and sets with deleted digits Theoret. Comput. Sci. 332 313336 .

  • [15] Penney, W. 1965 A “binary” system for complex numbers J. ACM 12 247248 .

  • [16] Piché, D., Complex bases, number systems and their application to the fractal wavelet image coding, Ph.D. Thesis, Univ. Waterloo, Ontario, 2002.

    • Search Google Scholar
    • Export Citation
  • [17] Pineda, J., A parallel algorithm for polygon rasterization, in: Proc. 15th Conference on Computer Graphics and Interactive Techniques (1988), pp. 1720.

    • Search Google Scholar
    • Export Citation
  • [18] Solinas, J. A. 2000 Efficient arithmetic on Koblitz curves Des. Codes Cryptogr. 19 195249 .

Monthly Content Usage

Abstract Views Full Text Views PDF Downloads
Nov 2020 0 2 1
Dec 2020 0 0 0
Jan 2021 0 0 0
Feb 2021 0 0 0
Mar 2021 0 0 0
Apr 2021 0 0 0
May 2021 0 0 0