-
Previous Article
Lyapunov exponents of cocycles over non-uniformly hyperbolic systems
- DCDS Home
- This Issue
-
Next Article
Chemotaxis model with nonlocal nonlinear reaction in the whole space
Characterization of noncorrelated pattern sequences and correlation dimensions
1. | Department of Mathematics, Huazhong University of Science and Technology, Wuhan 430074, China |
2. | School of Information and Mathematics, Yangtze University, Jingzhou 434023, China |
3. | Advanced Mathematical Institute, Osaka City University, Osaka, 558-8585, Japan |
We consider the correlation functions of binary pattern sequences of degree 3 as well as those with general degrees and special patterns and obtain necessary and sufficient conditions to be noncorrelated. We also obtain the correlation dimensions for those with degree 2.
References:
[1] |
J.-P. Allouche and P. Liardet,
Generalized Rudin-Shapiro sequences, Acta Arith., 60 (1991), 1-27.
doi: 10.4064/aa-60-1-1-27. |
[2] |
D. W. Boyd, J. H. Cook and P. Morton, On sequences of $±$1's defined by binary patterns,
Dissertationes Math., 283 (1989), 64pp. |
[3] |
J. Coquet, T. Kamae and M. Mendès France,
Sur la mesure spectrale de certaines suites arithmétiques, Bull. Soc. Math. France, 105 (1977), 369-384.
|
[4] |
N. P. Fogg,
Substitutions in Dynamics, Arithmetics and Combinatorics, Springer-Verlag, Berlin, 2002. |
[5] |
C. Godrèche and J. M. Luck,
Multifractal analysis in reciprocal space and the nature of the Fourier transform of self-similar structures, J. Phys. A, 23 (1990), 3769-3797.
doi: 10.1088/0305-4470/23/16/024. |
[6] |
P. Morton,
Connections between binary patterns and paperfolding, Sém. Théor. Nombres Bordeaux(2), 2 (1990), 1-12.
doi: 10.5802/jtnb.16. |
[7] |
P. Morton and W. J. Mourant,
Paper folding, digit patterns and groups of arithmetic fractals, Proc. London Math. Soc.(3), 59 (1989), 253-293.
doi: 10.1112/plms/s3-59.2.253. |
[8] |
M. Niu and Z. X. Wen,
Correlation dimension of the spectral measure for m-multiplicative sequences, (Chinese), Acta Math. Sci. Ser. A (Chin. Ed.), 27 (2007), 862-870.
|
[9] |
L. Peng and T. Kamae,
Spectral measure of the Thue-Morse sequence and the dynamical system and random walk related to it, Ergodic Theory Dynam. Systems, 36 (2016), 1247-1259.
doi: 10.1017/etds.2014.121. |
[10] |
K. Petersen,
Ergodic Theory, Cambridge University Press, Cambridge, 1983. |
[11] |
M. Queffélec,
Substitution Dynamical Systems-Spectral Analysis, Springer-Verlag, Berlin, 1987. |
[12] |
M. A. Zaks, A. S. Pikovsky and J. Kurths,
On the correlation dimension of the spectral measure for the Thue-Morse sequence, J. Statist. Phys., 88 (1997), 1387-1392.
doi: 10.1007/BF02732440. |
show all references
References:
[1] |
J.-P. Allouche and P. Liardet,
Generalized Rudin-Shapiro sequences, Acta Arith., 60 (1991), 1-27.
doi: 10.4064/aa-60-1-1-27. |
[2] |
D. W. Boyd, J. H. Cook and P. Morton, On sequences of $±$1's defined by binary patterns,
Dissertationes Math., 283 (1989), 64pp. |
[3] |
J. Coquet, T. Kamae and M. Mendès France,
Sur la mesure spectrale de certaines suites arithmétiques, Bull. Soc. Math. France, 105 (1977), 369-384.
|
[4] |
N. P. Fogg,
Substitutions in Dynamics, Arithmetics and Combinatorics, Springer-Verlag, Berlin, 2002. |
[5] |
C. Godrèche and J. M. Luck,
Multifractal analysis in reciprocal space and the nature of the Fourier transform of self-similar structures, J. Phys. A, 23 (1990), 3769-3797.
doi: 10.1088/0305-4470/23/16/024. |
[6] |
P. Morton,
Connections between binary patterns and paperfolding, Sém. Théor. Nombres Bordeaux(2), 2 (1990), 1-12.
doi: 10.5802/jtnb.16. |
[7] |
P. Morton and W. J. Mourant,
Paper folding, digit patterns and groups of arithmetic fractals, Proc. London Math. Soc.(3), 59 (1989), 253-293.
doi: 10.1112/plms/s3-59.2.253. |
[8] |
M. Niu and Z. X. Wen,
Correlation dimension of the spectral measure for m-multiplicative sequences, (Chinese), Acta Math. Sci. Ser. A (Chin. Ed.), 27 (2007), 862-870.
|
[9] |
L. Peng and T. Kamae,
Spectral measure of the Thue-Morse sequence and the dynamical system and random walk related to it, Ergodic Theory Dynam. Systems, 36 (2016), 1247-1259.
doi: 10.1017/etds.2014.121. |
[10] |
K. Petersen,
Ergodic Theory, Cambridge University Press, Cambridge, 1983. |
[11] |
M. Queffélec,
Substitution Dynamical Systems-Spectral Analysis, Springer-Verlag, Berlin, 1987. |
[12] |
M. A. Zaks, A. S. Pikovsky and J. Kurths,
On the correlation dimension of the spectral measure for the Thue-Morse sequence, J. Statist. Phys., 88 (1997), 1387-1392.
doi: 10.1007/BF02732440. |
| pattern set | spectrum | |
| | ||
| | | noncorrelated sequence |
| | | |
| | ||
| | singular spectrum | |
| | | |
| | periodic sequence | |
|
| pattern set | spectrum | |
| | ||
| | | noncorrelated sequence |
| | | |
| | ||
| | singular spectrum | |
| | | |
| | periodic sequence | |
|
[1] |
Zilong Wang, Guang Gong. Correlation of binary sequence families derived from the multiplicative characters of finite fields. Advances in Mathematics of Communications, 2013, 7 (4) : 475-484. doi: 10.3934/amc.2013.7.475 |
[2] |
Aixian Zhang, Zhengchun Zhou, Keqin Feng. A lower bound on the average Hamming correlation of frequency-hopping sequence sets. Advances in Mathematics of Communications, 2015, 9 (1) : 55-62. doi: 10.3934/amc.2015.9.55 |
[3] |
Hua Liang, Wenbing Chen, Jinquan Luo, Yuansheng Tang. A new nonbinary sequence family with low correlation and large size. Advances in Mathematics of Communications, 2017, 11 (4) : 671-691. doi: 10.3934/amc.2017049 |
[4] |
Ferruh Özbudak, Eda Tekin. Correlation distribution of a sequence family generalizing some sequences of Trachtenberg. Advances in Mathematics of Communications, 2021, 15 (4) : 647-662. doi: 10.3934/amc.2020087 |
[5] |
Michael Barnsley, James Keesling, Mrinal Kanti Roychowdhury. Special issue on fractal geometry, dynamical systems, and their applications. Discrete and Continuous Dynamical Systems - S, 2019, 12 (8) : i-i. doi: 10.3934/dcdss.201908i |
[6] |
Wenbing Chen, Jinquan Luo, Yuansheng Tang, Quanquan Liu. Some new results on cross correlation of $p$-ary $m$-sequence and its decimated sequence. Advances in Mathematics of Communications, 2015, 9 (3) : 375-390. doi: 10.3934/amc.2015.9.375 |
[7] |
Lassi Roininen, Markku S. Lehtinen, Sari Lasanen, Mikko Orispää, Markku Markkanen. Correlation priors. Inverse Problems and Imaging, 2011, 5 (1) : 167-184. doi: 10.3934/ipi.2011.5.167 |
[8] |
Zhenyu Zhang, Lijia Ge, Fanxin Zeng, Guixin Xuan. Zero correlation zone sequence set with inter-group orthogonal and inter-subgroup complementary properties. Advances in Mathematics of Communications, 2015, 9 (1) : 9-21. doi: 10.3934/amc.2015.9.9 |
[9] |
Xiaohui Liu, Jinhua Wang, Dianhua Wu. Two new classes of binary sequence pairs with three-level cross-correlation. Advances in Mathematics of Communications, 2015, 9 (1) : 117-128. doi: 10.3934/amc.2015.9.117 |
[10] |
Huaning Liu, Xi Liu. On the correlation measures of orders $ 3 $ and $ 4 $ of binary sequence of period $ p^2 $ derived from Fermat quotients. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2021008 |
[11] |
Limengnan Zhou, Daiyuan Peng, Hongyu Han, Hongbin Liang, Zheng Ma. Construction of optimal low-hit-zone frequency hopping sequence sets under periodic partial Hamming correlation. Advances in Mathematics of Communications, 2018, 12 (1) : 67-79. doi: 10.3934/amc.2018004 |
[12] |
Vladimír Špitalský. Local correlation entropy. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5711-5733. doi: 10.3934/dcds.2018249 |
[13] |
Ming Su, Arne Winterhof. Hamming correlation of higher order. Advances in Mathematics of Communications, 2018, 12 (3) : 505-513. doi: 10.3934/amc.2018029 |
[14] |
Yuhua Sun, Zilong Wang, Hui Li, Tongjiang Yan. The cross-correlation distribution of a $p$-ary $m$-sequence of period $p^{2k}-1$ and its decimated sequence by $\frac{(p^{k}+1)^{2}}{2(p^{e}+1)}$. Advances in Mathematics of Communications, 2013, 7 (4) : 409-424. doi: 10.3934/amc.2013.7.409 |
[15] |
Nian Li, Xiaohu Tang, Tor Helleseth. A class of quaternary sequences with low correlation. Advances in Mathematics of Communications, 2015, 9 (2) : 199-210. doi: 10.3934/amc.2015.9.199 |
[16] |
Xin-Guo Liu, Kun Wang. A multigrid method for the maximal correlation problem. Numerical Algebra, Control and Optimization, 2012, 2 (4) : 785-796. doi: 10.3934/naco.2012.2.785 |
[17] |
Kaitlyn (Voccola) Muller. SAR correlation imaging and anisotropic scattering. Inverse Problems and Imaging, 2018, 12 (3) : 697-731. doi: 10.3934/ipi.2018030 |
[18] |
Nasab Yassine. Quantitative recurrence of some dynamical systems preserving an infinite measure in dimension one. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 343-361. doi: 10.3934/dcds.2018017 |
[19] |
Hua Liang, Jinquan Luo, Yuansheng Tang. On cross-correlation of a binary $m$-sequence of period $2^{2k}-1$ and its decimated sequences by $(2^{lk}+1)/(2^l+1)$. Advances in Mathematics of Communications, 2017, 11 (4) : 693-703. doi: 10.3934/amc.2017050 |
[20] |
Houduo Qi, ZHonghang Xia, Guangming Xing. An application of the nearest correlation matrix on web document classification. Journal of Industrial and Management Optimization, 2007, 3 (4) : 701-713. doi: 10.3934/jimo.2007.3.701 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]