Weak and strong orders of linear recurring sequences
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Weak and strong orders of linear recurring sequences Zenonas Navickas1 · Minvydas Ragulskis2 · Dovile Karaliene3 · Tadas Telksnys2
Received: 5 April 2016 / Revised: 23 October 2017 / Accepted: 7 November 2017 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2017
Abstract The concept of the strong order of linear recurring sequence (LRS) is introduced in this paper. Necessary and sufficient conditions for the existence of the strong LRS order are derived. The strong LRS order is exploited for the formalization of the problem of the extension of a sequence from the available fragment (fragments) of that sequence. The definition of the strong LRS order opens new possibilities for formal sequence analysis whenever the weak LRS order of that sequence exists. Computational experiments with discrete iterative maps are used to illustrate the applicability of the strong LRS order in nonlinear system analysis. Keywords Linear recurring sequence · Strong and weak orders · Sequence extrapolation Mathematics Subject Classification 65Q30 · 37M10 · 37G35
1 Introduction The theory and applications of linear recurrence sequences have been addressed in a large variety of publications during the recent decades. Mikhalev et al. discuss linear recurring sequences over modules (Mikhalev and Nechaev 1996); Kurakin et al. (1995) consider kdimensional linear recurring sequences over rings and modules; Everest et al. (2003) analyze
Communicated by Jose Alberto Cuminato.
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Tadas Telksnys [email protected]
1
Department of Mathematical Modeling, Kaunas University of Technology, Studentu 50-325, Kaunas, Lithuania
2
Research Group for Mathematical and Numerical Analysis of Dynamical Systems, Kaunas University of Technology, Studentu 50-147, Kaunas, Lithuania
3
Department of Applied Mathematics, Kaunas University of Technology, Studentu 50-325, Kaunas, Lithuania
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the number-theoretic properties of linear recurrence sequences and their generalizations. Linear recurring sequence subgroups in finite fields are considered in Brison and Nogueira (2003); exponential sums for nonlinear recurring sequences are discussed in Niederreiter and Winterhof (2008); the ABC conjecture in binary recurring sequences is investigated in Ribenboim and Walsh (1999); a completely automated approach to identifying recurring local sequence motifs in proteins is presented in Han and Baker (1995); canonical forms for recurring sequences over Galois field are derived in Singh and Al-Zaid (1991). Recent developments in error correcting codes and cryptography in general has attracted a new wave of attention to linear recurring sequences (Fillmore and Marx 1968). Sequences defined by periodic recursive relations are investigated in Tam (1997). Linear recurring arrays are used to define and characterize n-dimensional cyclic codes and their dual codes over quasiFrobenius rings in Lu et al. (2004). The relationship of linear recurrences with polynomial coefficients to sequences generated by recurrences with constant c
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