• Previous Article
    Exact and WKB-approximate distributions in a gene expression model with feedback in burst frequency, burst size, and protein stability
  • DCDS-B Home
  • This Issue
  • Next Article
    Multiple solutions for Schrödinger lattice systems with asymptotically linear terms and perturbed terms
April  2022, 27(4): 2115-2128. doi: 10.3934/dcdsb.2021125

Singular function emerging from one-dimensional elementary cellular automaton Rule 150

Department of Mathematics, Kyoto University of Education, Kyoto, 612-8522, Japan

Received  October 2020 Revised  March 2021 Published  April 2022 Early access  April 2021

This paper presents a singular function on the unit interval $ [0, 1] $ derived from the dynamics of one-dimensional elementary cellular automaton Rule $ 150 $. We describe the properties of the resulting function, which is strictly increasing, uniformly continuous, and differentiable almost everywhere, and show that it is not differentiable at dyadic rational points. We also derive functional equations that this function satisfies and show that this function is the only solution of the functional equations.

Citation: Akane Kawaharada. Singular function emerging from one-dimensional elementary cellular automaton Rule 150. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2115-2128. doi: 10.3934/dcdsb.2021125
References:
[1]

G. Cantor, De la puissance des ensembles parfaits de points: Extrait d'une lettre adressée à l'éditeur [the power of perfect sets of points: Extract from a letter addressed to the editor], Acta Mathematica, 4 (1884), 381-392.  doi: 10.1007/BF02418423.

[2]

J. C. Claussen, Time evolution of the rule 150 cellular automaton activity from a Fibonacci iteration, Journal of Mathematical Physics, 49 (2008), 062701, 12 pp. doi: 10.1063/1.2939398.

[3]

D. L. Cohn, Measure Theory, Birkhäuser Basel, second edition, 2013. doi: 10.1007/978-1-4614-6956-8.

[4]

K. Culik Ⅱ and S. Dube, Fractal and recurrent behavior of cellular automata, Complex Systems, 3 (1989), 253-267. 

[5]

G. de Rham, Sur quelques courbes definies par des equations fonctionnelles, Rendiconti del Seminario Matematico Università e Politecnico di Torino, 16 (1957), 101–113.

[6]

G. A. Edgar, editor, Classics on Fractals, Studies in Nonlinearity, Addison-Wesley Publishing Company, 1993.

[7]

F. v. HaeselerH.-O. Peitgen and G. Skordev, Cellular automata, matrix substitutions and fractals, Annals of Mathematics and Artificial Intelligence, 8 (1993), 345-362.  doi: 10.1007/BF01530797.

[8]

M. Hata and M. Yamaguti, The Takagi function and its generalization, Japan Journal of Applied Mathematics, 1 (1984), 183-199.  doi: 10.1007/BF03167867.

[9]

A. Kawaharada and T. Namiki, Cumulative distribution of rule 90 and Lebesgue's singular function, Proceedings of Automata, 2014 (2014), 165-169. 

[10]

A. Kawaharada, Fractal patterns created by Ulam's cellular automaton, 2014 Second International Symposium on Computing and Networking, (2014), 484–486. doi: 10.1109/CANDAR.2014.51.

[11]

A. Kawaharada and T. Namiki, Fractal structure of a class of two-dimensional two-state cellular automata, Proceedings of International Workshop on Applications and Fundamentals of Cellular Automata 2017, (2017), 205–208. doi: 10.1109/CANDAR.2017.89.

[12]

A. Kawaharada and T. Namiki, Relation between spatio-temporal patterns generated by two-dimensional cellular automata and a singular function, International Journal of Networking and Computing, 9 (2019), 354-369.  doi: 10.15803/ijnc.9.2_354.

[13]

A. Kawaharada and T. Namiki, Number of nonzero states in prefractal sets generated by cellular automata, Journal of Mathematical Physics, 61 (2020), 092702, 17 pp. doi: 10.1063/5.0004652.

[14]

K. Kawamura, On the set of points where Lebesgue's singular function has the derivative zero, Proceedings of the Japan Academy, Series A, Mathematical Sciences, 87 (2011), 162-166.  doi: 10.3792/pjaa.87.162.

[15]

K. Kobayashi, On the critical case of Okamoto's continuous non-differentiable functions, Proceedings of the Japan Academy, Series A, Mathematical Sciences, 85 (2009), 101-104.  doi: 10.3792/pjaa.85.101.

[16]

Z. Łomnicki and S. Ulam, Sur la théorie de la mesure dans les espaces combinatoires et son application au calcul des probabilités i. variables indépendantes, Fundamenta Mathematicae, 23 (1934), 237-278.  doi: 10.4064/fm-23-1-237-278.

[17]

H. Okamoto, A remark on continuous, nowhere differentiable functions, Proceedings of the Japan Academy, Series A, Mathematical Sciences, 81 (2005), 47-50.  doi: 10.3792/pjaa.81.47.

[18]

H. Okamoto and M. Wunsch, A geometric construction of continuous, strictly increasing singular functions, Proceedings of the Japan Academy, Series A, Mathematical Sciences, 83 (2007), 114-118.  doi: 10.3792/pjaa.83.114.

[19]

R. Salem, On some singular monotonic functions which are strictly increasing, Transactions of the American Mathematical Society, 53 (1943), 427-439.  doi: 10.1090/S0002-9947-1943-0007929-6.

[20]

T. Takagi, A simple example of the continuous function without derivative, Proceedings of the Physico-Mathematical Society of Japan, 1 (1903), 176-177.  doi: 10.11429/subutsuhokoku1901.1.F176.

[21]

S. Takahashi, Self-similarity of linear cellular automata, Journal of Computer and System Sciences, 44 (1992), 114-140.  doi: 10.1016/0022-0000(92)90007-6.

[22]

K. Weierstrass, Über continuirliche functionen eines reellen arguments, die für keinen werth des letzeren einen bestimmten differentialquotienten besitzen, Mathematische Werke, 2 (1872), 71–74. (English translation in [6], pages 3–9). doi: 10.1007/978-3-322-91273-2_5.

[23]

S. J. Willson, Cellular automata can generate fractals, Discrete Applied Mathematics, 8 (1984), 91-99.  doi: 10.1016/0166-218X(84)90082-9.

[24]

M. Yamaguti, M. Hata and J. Kigami, Mathematics of Fractals, Translations of Mathematical Monographs, American Mathematical Society, 1997. (translated by K. Hudson). doi: 10.1090/mmono/167.

show all references

References:
[1]

G. Cantor, De la puissance des ensembles parfaits de points: Extrait d'une lettre adressée à l'éditeur [the power of perfect sets of points: Extract from a letter addressed to the editor], Acta Mathematica, 4 (1884), 381-392.  doi: 10.1007/BF02418423.

[2]

J. C. Claussen, Time evolution of the rule 150 cellular automaton activity from a Fibonacci iteration, Journal of Mathematical Physics, 49 (2008), 062701, 12 pp. doi: 10.1063/1.2939398.

[3]

D. L. Cohn, Measure Theory, Birkhäuser Basel, second edition, 2013. doi: 10.1007/978-1-4614-6956-8.

[4]

K. Culik Ⅱ and S. Dube, Fractal and recurrent behavior of cellular automata, Complex Systems, 3 (1989), 253-267. 

[5]

G. de Rham, Sur quelques courbes definies par des equations fonctionnelles, Rendiconti del Seminario Matematico Università e Politecnico di Torino, 16 (1957), 101–113.

[6]

G. A. Edgar, editor, Classics on Fractals, Studies in Nonlinearity, Addison-Wesley Publishing Company, 1993.

[7]

F. v. HaeselerH.-O. Peitgen and G. Skordev, Cellular automata, matrix substitutions and fractals, Annals of Mathematics and Artificial Intelligence, 8 (1993), 345-362.  doi: 10.1007/BF01530797.

[8]

M. Hata and M. Yamaguti, The Takagi function and its generalization, Japan Journal of Applied Mathematics, 1 (1984), 183-199.  doi: 10.1007/BF03167867.

[9]

A. Kawaharada and T. Namiki, Cumulative distribution of rule 90 and Lebesgue's singular function, Proceedings of Automata, 2014 (2014), 165-169. 

[10]

A. Kawaharada, Fractal patterns created by Ulam's cellular automaton, 2014 Second International Symposium on Computing and Networking, (2014), 484–486. doi: 10.1109/CANDAR.2014.51.

[11]

A. Kawaharada and T. Namiki, Fractal structure of a class of two-dimensional two-state cellular automata, Proceedings of International Workshop on Applications and Fundamentals of Cellular Automata 2017, (2017), 205–208. doi: 10.1109/CANDAR.2017.89.

[12]

A. Kawaharada and T. Namiki, Relation between spatio-temporal patterns generated by two-dimensional cellular automata and a singular function, International Journal of Networking and Computing, 9 (2019), 354-369.  doi: 10.15803/ijnc.9.2_354.

[13]

A. Kawaharada and T. Namiki, Number of nonzero states in prefractal sets generated by cellular automata, Journal of Mathematical Physics, 61 (2020), 092702, 17 pp. doi: 10.1063/5.0004652.

[14]

K. Kawamura, On the set of points where Lebesgue's singular function has the derivative zero, Proceedings of the Japan Academy, Series A, Mathematical Sciences, 87 (2011), 162-166.  doi: 10.3792/pjaa.87.162.

[15]

K. Kobayashi, On the critical case of Okamoto's continuous non-differentiable functions, Proceedings of the Japan Academy, Series A, Mathematical Sciences, 85 (2009), 101-104.  doi: 10.3792/pjaa.85.101.

[16]

Z. Łomnicki and S. Ulam, Sur la théorie de la mesure dans les espaces combinatoires et son application au calcul des probabilités i. variables indépendantes, Fundamenta Mathematicae, 23 (1934), 237-278.  doi: 10.4064/fm-23-1-237-278.

[17]

H. Okamoto, A remark on continuous, nowhere differentiable functions, Proceedings of the Japan Academy, Series A, Mathematical Sciences, 81 (2005), 47-50.  doi: 10.3792/pjaa.81.47.

[18]

H. Okamoto and M. Wunsch, A geometric construction of continuous, strictly increasing singular functions, Proceedings of the Japan Academy, Series A, Mathematical Sciences, 83 (2007), 114-118.  doi: 10.3792/pjaa.83.114.

[19]

R. Salem, On some singular monotonic functions which are strictly increasing, Transactions of the American Mathematical Society, 53 (1943), 427-439.  doi: 10.1090/S0002-9947-1943-0007929-6.

[20]

T. Takagi, A simple example of the continuous function without derivative, Proceedings of the Physico-Mathematical Society of Japan, 1 (1903), 176-177.  doi: 10.11429/subutsuhokoku1901.1.F176.

[21]

S. Takahashi, Self-similarity of linear cellular automata, Journal of Computer and System Sciences, 44 (1992), 114-140.  doi: 10.1016/0022-0000(92)90007-6.

[22]

K. Weierstrass, Über continuirliche functionen eines reellen arguments, die für keinen werth des letzeren einen bestimmten differentialquotienten besitzen, Mathematische Werke, 2 (1872), 71–74. (English translation in [6], pages 3–9). doi: 10.1007/978-3-322-91273-2_5.

[23]

S. J. Willson, Cellular automata can generate fractals, Discrete Applied Mathematics, 8 (1984), 91-99.  doi: 10.1016/0166-218X(84)90082-9.

[24]

M. Yamaguti, M. Hata and J. Kigami, Mathematics of Fractals, Translations of Mathematical Monographs, American Mathematical Society, 1997. (translated by K. Hudson). doi: 10.1090/mmono/167.

Figure 1.  Spatio-temporal pattern of Rule $ 150 $, $ \{T_{150} x_o\}_{n = 0}^{31} $
Figure 2.  Limit set of Rule $ 150 $ from the single site seed $ x_o $
Figure 3.  $ \{cum_{150}(n)\} $ of Rule $ 150 $ for $ 0 \leq n < 256 $
Figure 4.  $ F(x) $ and the limit set of Rule $ 150 $
Table 1.  Local rule of Rule $ 150 $
$ x_{i-1} x_i x_{i+1} $ $ 111 $ $ 110 $ $ 101 $ $ 100 $ $ 011 $ $ 010 $ $ 001 $ $ 000 $
$ (T_{150} x)_i $ $ 1 $ $ 0 $ $ 0 $ $ 1 $ $ 0 $ $ 1 $ $ 1 $ $ 0 $
$ x_{i-1} x_i x_{i+1} $ $ 111 $ $ 110 $ $ 101 $ $ 100 $ $ 011 $ $ 010 $ $ 001 $ $ 000 $
$ (T_{150} x)_i $ $ 1 $ $ 0 $ $ 0 $ $ 1 $ $ 0 $ $ 1 $ $ 1 $ $ 0 $
[1]

Petr Kůrka. On the measure attractor of a cellular automaton. Conference Publications, 2005, 2005 (Special) : 524-535. doi: 10.3934/proc.2005.2005.524

[2]

Gelasio Salaza, Edgardo Ugalde, Jesús Urías. Master--slave synchronization of affine cellular automaton pairs. Discrete and Continuous Dynamical Systems, 2005, 13 (2) : 491-502. doi: 10.3934/dcds.2005.13.491

[3]

Yusra Bibi Ruhomally, Muhammad Zaid Dauhoo, Laurent Dumas. A graph cellular automaton with relation-based neighbourhood describing the impact of peer influence on the consumption of marijuana among college-aged youths. Journal of Dynamics and Games, 2021, 8 (3) : 277-297. doi: 10.3934/jdg.2021011

[4]

Hasib Khan, Cemil Tunc, Aziz Khan. Green function's properties and existence theorems for nonlinear singular-delay-fractional differential equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : 2475-2487. doi: 10.3934/dcdss.2020139

[5]

Seick Kim, Longjuan Xu. Green's function for second order parabolic equations with singular lower order coefficients. Communications on Pure and Applied Analysis, 2022, 21 (1) : 1-21. doi: 10.3934/cpaa.2021164

[6]

Manuel Fernández-Martínez. Theoretical properties of fractal dimensions for fractal structures. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1113-1128. doi: 10.3934/dcdss.2015.8.1113

[7]

T.K. Subrahmonian Moothathu. Homogeneity of surjective cellular automata. Discrete and Continuous Dynamical Systems, 2005, 13 (1) : 195-202. doi: 10.3934/dcds.2005.13.195

[8]

Achilles Beros, Monique Chyba, Oleksandr Markovichenko. Controlled cellular automata. Networks and Heterogeneous Media, 2019, 14 (1) : 1-22. doi: 10.3934/nhm.2019001

[9]

Uta Renata Freiberg. Einstein relation on fractal objects. Discrete and Continuous Dynamical Systems - B, 2012, 17 (2) : 509-525. doi: 10.3934/dcdsb.2012.17.509

[10]

Umberto Mosco, Maria Agostina Vivaldi. Vanishing viscosity for fractal sets. Discrete and Continuous Dynamical Systems, 2010, 28 (3) : 1207-1235. doi: 10.3934/dcds.2010.28.1207

[11]

Marcus Pivato. Invariant measures for bipermutative cellular automata. Discrete and Continuous Dynamical Systems, 2005, 12 (4) : 723-736. doi: 10.3934/dcds.2005.12.723

[12]

Yongjie Wang, Nan Gao. Some properties for almost cellular algebras. Electronic Research Archive, 2021, 29 (1) : 1681-1689. doi: 10.3934/era.2020086

[13]

Oliver Penrose, John W. Cahn. On the mathematical modelling of cellular (discontinuous) precipitation. Discrete and Continuous Dynamical Systems, 2017, 37 (2) : 963-982. doi: 10.3934/dcds.2017040

[14]

Eduardo Ibarguen-Mondragon, Lourdes Esteva, Leslie Chávez-Galán. A mathematical model for cellular immunology of tuberculosis. Mathematical Biosciences & Engineering, 2011, 8 (4) : 973-986. doi: 10.3934/mbe.2011.8.973

[15]

Michael L. Frankel, Victor Roytburd. Fractal dimension of attractors for a Stefan problem. Conference Publications, 2003, 2003 (Special) : 281-287. doi: 10.3934/proc.2003.2003.281

[16]

Palle E. T. Jorgensen and Steen Pedersen. Orthogonal harmonic analysis of fractal measures. Electronic Research Announcements, 1998, 4: 35-42.

[17]

Abdon Atangana, Ali Akgül. On solutions of fractal fractional differential equations. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3441-3457. doi: 10.3934/dcdss.2020421

[18]

Seheon Ham, Hyerim Ko, Sanghyuk Lee. Circular average relative to fractal measures. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022100

[19]

Claude-Michel Brauner, Michael L. Frankel, Josephus Hulshof, Alessandra Lunardi, G. Sivashinsky. On the κ - θ model of cellular flames: Existence in the large and asymptotics. Discrete and Continuous Dynamical Systems - S, 2008, 1 (1) : 27-39. doi: 10.3934/dcdss.2008.1.27

[20]

Mostafa Adimy, Fabien Crauste, Laurent Pujo-Menjouet. On the stability of a nonlinear maturity structured model of cellular proliferation. Discrete and Continuous Dynamical Systems, 2005, 12 (3) : 501-522. doi: 10.3934/dcds.2005.12.501

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (298)
  • HTML views (364)
  • Cited by (0)

Other articles
by authors

[Back to Top]