Numsys

Introduction

The generalization of positional number representations to a wide range of digit sets or to higher dimensions is a fascinating story. Grünwald (1885) investigated negative-based, Kempner (1936), Knuth (1960), Khmelnik (1964), Penney (1965) complex-based systems. From the 70's Kátai, B.Kovács, Környei, Pethő (the "Hungarian school") and Gilbert examined systematically the radix extensions in algebraic number fields. In the 90's the topological aspects of radix representations was studied by Bandt, Indlekofer, Járai, Kátai, Lagarias, Wang, Vince, and later by Akiyama, Thuswaldner and others. The canonical number representation was generalized to arbitrary polynomial systems by Pethő (1989), and investigated later extensively by many authors (incl. Akiyama, Brunotte, Kovács, Pethő, Rao, Scheicher, Thuswaldner). The number system concept in general lattices was investigated first by Vince (1993). The algorithmical aspects of canonical (polynomial) systems was initiated by Brunotte (2001) and for general lattices by Kovács (2000). A special type of radix systems (SRS) is lengthly studied by Thuswaldner and his co-workers (the "Austrian school").

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Downloads

In this section you can download multiple things that helps you to analyze generalized number systems.

Sage NumSysAnylzer

You can download here the Sage version of the NumSysAnylzer. Read the README.MD in the compressed folder for more information.

Sample inputs

At this link you can download list of expansive polynomials, that can be loaded easily by the systems.

Python NumSysAnylzer

You can download here the Python version of the NumSysAnylzer. Read the README.MD in the compressed folder for more information.

Relevant publications

We are trying to collect every topic related publications. This listing isn't complete yet, if you want to share your paper here, please send it to the info@numsys.info.

Akiyama, S., Borbély, T., Brunotte, H., Pethő, A., Thuswaldner, J., On a generalization of the radix representation - a survey, in High Primes and Misdemeanours: lectures in honour of the 60th birthday of Hugh Cowie Williams, Fields Inst. Commun., 41, 2004, 19-27. Akiyama, S., Borbély, T., Brunotte, H., Pethő, A., Thuswaldner, J., Generalized radix representations and dynamical systems I., Acta Math. Hungar., 108, 2005, 207-238, Akiyama, S., Pethő, A., On canonical number systems, Theoretical Computer Science, 270, 1-2, 2002, 921-933, Akiyama, S., Rao, H., New criteria for canonical number systems, Acta Arithm., 111, 1, 2004, 5-25, Akiyama, S., Thuswaldner, J. M., Topological properties of two dimensional number systems, Journal de Theorie des Nombres de Bordeaux, 12, 2000, 69-79, Bandt, C., Self-similar sets V., Integer matrices and fractal tilings of Rn, Am. Math. Soc., 112, 1991, 549-562, Bandt, C., Classification of self-affine lattice tilings, J. London Math. Soc, 50, 3, 1994, 581-593, Bar Barbé, A., von Haeseler, F., Binary number systems for Zk, J. Number Theory, 117, 1, 2006, 14-30, Brunotte, H., Huszti, A., Pethő, A., Bases of canonical number systems in quartic algebraic number fields, J. Théorie Nombres de Bordeaux, 18, 2006, 537-557, Brunotte, H., On trinomial bases of radix representations of algebraic integers, Acta Sci. Math. (Szeged), 67, 2001, 521-527, Brunotte, H., Characterization of CNS polynomials, Acta Sci. Math., (Szeged), 68, 2002, 673-679, Burcsi, P., Kovács, A., Papp-Varga, Zs., Decision and Classification Algorithms for Generalized Number Systems, Ann. Univ. Sci. Budapest. Sect. Comput., 28, 2008, 141-156, Daróczy, Z., Kátai, I., Generalized number systems in the complex plane, Acta Math. Hung., 51, 3-4, 1988, 409-416, Fraenkel, A. S., Systems of numeration, Amer. Math. Monthly, 92, 1985, 105-114, Germán, L., Kovács, A., On number system constructions, Acta Math. Hungar, 115, 1-2, 2007, 155-167, Gilbert, W. J., Arithmetic in Complex Bases, Math. Mag., 57, 2, 1984, 77-81, Gilbert, W. J., Radix representation of quadratic fields, J. Math. Anal. Appl., 83, 1991, 264-274, Gilbert, W. J., Fractal geometry derived from complex basis, Math. Intell., 4, 1982, 78-86, Gilbert, W. J., Geometry of radix representation, The Geometric Vein, Springer, 1981, 129-139, Gilbert, W. J., Complex numbers with three radix expansions, Can. J. Math., 34, 1982, 1335-1348, Gilbert, W. J., The fractal dimension of sets derived from complex bases, Can. Math. Bull., 29, 1986, 495-500, Gilbert, W. J., The division algorithm in complex bases, Can. Math. Bull., 39, 1, 1996, 47-54, Gilbert, W. J., Gaussian integers as bases for exotic number systems, The Mathematial Heritage of C. F. Gauss, (ed. by Rassias, G. M.), World Scientific Publ. Co., 1994, Grossman, E. H., Number bases in quadratic fields, Studia Sci. Math. Hung., 20, 1985, 55-58, Grüwald, V., Intorno all'aritmetica dei sistemi numerici a base negativa con particolare riguardo al sistema numerico a base negativo-decimale per lo studio delle sue analogie coll'aritmetica ordinaria (decimale), Giornale di matematiche di Battaglini, 23, 1885, 203-221, Hudoba, P., Kovács, A., Some improvements on number expansion computations, Annales Univ. Sci. Budapest, Sect. Comp., 46, 2017, 81-96, Indlekofer, K.-H., Kátai, I., Racskó, P., Some remarks on generalized number systems, Acta Sci. Math., 37, 1993, 543-553, Indlekofer, K.-H., Kátai, I., Racskó, P., Number systems and fractal geometry, Probability Theory and its Appliations (Eds. Galambos, J. and Kátai, I.), Kluwer Publ., 1992, 319-334, Indlekofer, K.-H., Járai, A., Kátai, I., On some properties of attractors generated by iterated function systems, Acta Sci. Math., 60, 1995, 411-427, Kátai, I., Number systems in imaginary quadratic fields, Annales Univ. Si. Bud., Sect. Comp., 14, 1994, 91-103, Kátai, I., Construction of number systems in algebraic number fields, Annales Univ. Si. Bud., Set. Comp., 18, 1999, 103-107, Kátai, I., Generalized number systems and fractal geometry, Janus Pannonius University, 1995, 40, Kátai, I., Kovács, B., Kanonische Zahlensysteme bei reellen quadratischen algebraischen Zahlen, Acta Sci. Math., 42, 1980, 99-107, Kátai, I., Kovács, B., Canonical number systems in imaginary quadratic fields, Acta Math. Hung., 37, 1981, 159-164, Kátai, I., Környei, I., On number systems in algebraic number fields, Publ. Math. Debrecen, 41, 1992, 289-294, Kátai, I., Szabó, J., Canonical number systems for complex integers, Acta Sci. Math., 37, 1975, 255-260, Kempner, A.J., Anormal system of numeration, Amer. Math. Monthly, 43, 1936, 610-617, Kenyon, R., Self-replicating tilings, in Symbolic Dynamics and Its Applications (Walters, P. Ed.), Contemporary Mathematics, Amer. Math. Soc., 135, 1992, 239-264, Khmelnik, S.I., Specialized digital computer for operations with complex numbers, Questions of Radio Electronics (in Russian) , XII (2), 1964, Knuth, D.E., An Imaginary Number System, Communication of the ACM, 3, 4, 1960, Komornik, V., Pethő, A., Common expansions in non-integer bases, Publ. Math. Debrecen, 85, 2014, Kovács, A., Expansions of gaussian integers with non-negative digits, Math. Pannonica, 10, 2, 1999, 177-191, Kovács, A., On computation of attractors for invertible expanding linear operators in Zk, Publ. Math. Debrecen, 56, 1-2, 2000, 97-120, Kovács, A., Generalized binary number systems, Annales Univ. Sci. Budapest, Sect. Comp., 20, 2001, 195-206, Kovács, A., Canonical expansions of integers in imaginary quadratic fields, Acta Math. Hung., 93, 4, 2001, 347-357, Kovács, A., Number Systems in Lattices, PhD thesis, Eötvös Loránd University, Budapest, 2001, 1-98, Kovács, A., On number expansions in lattices, Math. and Comp. Modelling, 8, 2003, 909-915, Kovács, B., Canonical number systems in algebraic number fields, Acta Math. Hung., 37, 4, 1981, 405-407, Kovács, B., Integral domains with canonical number systems, Publ. Math. Debrecen, 36, 1989, 153-156, Kovács, B., Representation of complex numbers in number systems, Acta Math. Hung., 58, 1991, 113-120, Kovács, B., Number systems, Computational Number Theory, Walter de Gruyter Publ., 1991, 21-25, Kovács, B., Környei, I., On the periodicity of the radix representation, Annales Univ. Sci. Bud., Sect. Comp., 13, 1992, 129-133, Kovács, B., Pethő, A., Canonical number systems in the ring of integers, Publ. Math. Debrecen, 30, 1983, 39-44, Kovács, B., Pethő, A., Number systems in integral domains, especially in orders of algebraic number fields, Acta Sci. Math., 55, 1991, 287-299, Kovács, B., Pethő, A., On a representation of algebraic integers, Studia Sci. Math. Hung., 27, 1992, 169-172, Körmendi, S., Canonical number systems in Q( √3 2), Acta Sci. Math., 50, 1986, 351-357, Lagarias, J. C., Wang, Y., Integral self-affine tiles in R n I., Standard and non-standard digit sets, J. London Math. Soc., 54, 2, 1996, 161-179, Lagarias, J. C., Wang, Y., Self-affine tiles in Rn, Adv. in Math., 121, 1996, 21-49, Lehmer, D.H., A machine method for solving polynomial equations, J. ACM, 2, 1961, 151-162, Matula, D. W., Basic digit sets for radix representation, J. ACM, 29, 1982, 1131-1143, Mauldin, R. D., Williams, S. C., Hausdorff dimension in graph directed constructions, Transaction of AMS, 309, 2, 1988, 811-829, Odlyzko, A.M., Non-negative digit sets in positional number systems, London Math. Soc., 37, 1978, 213-229, Penney, W., A binary system for complex numbers, J. ACM, 12, 1965, 247-248, Pethő, A., On a polynomial transformation and its application to the construction of a public key cryptosystem, Computational Number Theory, Walter de Gruyter Publ., 1991, 31-44, Pethő, A., On the periodic expansion of algebraic numbers, Annales Univ. Sci. Bud., Sect. Comp., 18, 1999, 167-174, Pethő, A., Connections between power integral bases and radix representations in algebraic number fields, Proc. 2003 Nagoya Conf. "Yokoi-Chowla Conjecture and Related Problems", Eds.: S-i. Katayama, C. Levesque and T. Nakahara, Furukawa Total Pr., 2004, 115-125, Robert, A., A good basis for the computing with complex numbers, Elemente der Mathematik, 49, 1994, 111-117, Safer, T., Polygonal radix representation of complex numbers, Theor. Comp. Sci., 210, 1999, 159-171, Scheicher, K., Thuswaldner, J.M., On the characterization of canonical number systems, Osaka J. Math., 41, 2, 2004, 327-351, Steidl, G., On symmetric radix representation of gaussian integers, BIT, 29, 1989, 563-571, Tátrai, A., Parallel implementations of Brunotte's algorithm, Journal of Parallel and Distributed Computing, 71, 4, 2011, 565-572, Thuswaldner, J., Fractal dimension of sets induced by bases of imaginary quadratic fields, Math. Slovaca, 48, 1998, 365-371, Thuswaldner, J., Fractals and number systems in real quadratic number fields, Acta Math. Hung., 90, 3, 2001, 253-269, Thuswaldner, J., Attractors for Invertible Expanding Linear Operators and Number Systems in Z2, Publ. Math. Debrecen, 58, 2001, 423-440, Vince, A., Radix representation and rep-tiling, 24th Southeastern Intern. Conf. on Combinatorics, Graph Theory and Computing, Boca Raton, FL, 1993, Congr. Numer., 98, 1993, 199-212, Vince, A., Replicating tessellations, SIAM J. Discrete Math., 6, 1993, 501-521, Vince, A., Rep-tiling euclidean space, Aequationes Math., 50, 1995, 191-213, Vince, A., Self-replicating tiles and their boundary, Discrete Comp. Geom., 21, 3, 1999, 463-476,

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