Algebraic and Analytic Randomness

Allouche, Jean‐Paul

doi:10.1007/3-540-45463-2_17

Cited by 8 publications

(7 citation statements)

References 22 publications

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“…In conclusion, we have presented a detailed statistical analysis of the blocks appearing in the binary expansions of the Feigenbaum constants a and d for the logistic map [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39]. Significant deviations from randomness are revealed, leading to the possibility that these constants correspond to non-normal binary numbers.…”

Section: Discussionmentioning

confidence: 96%

Statistical analysis of the first digits of the binary expansion of Feigenbaum constants α and δ

Karamanos

Kotsireas

2005

Journal of the Franklin Institute

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Section: Discussionmentioning

confidence: 96%

Statistical analysis of the first digits of the binary expansion of Feigenbaum constants α and δ

Karamanos

Kotsireas

2005

Journal of the Franklin Institute

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“…One must noted that the previous remark is in fact valid for all kind of model which can be constructed from a given set of experimental data as, in practice, this is impossible to have access to the full scale sequence (2) . A limit model can not and will never represent the reality of a phenomenon as we have no possibilities to select between different admissible models up to a given scale.…”

Section: 12mentioning

confidence: 99%

“…Using these notions, it is possible to select from a given symbol sequence a natural candidate with low complexity and representing the associated multiscale Okamoto's function in one of the previous family. We refer to the report of J-P. Allouche in [2] for more details and precise definition of these notions.…”

Section: Complexity Of Multiscalementioning

confidence: 99%

Multiscale functions, scale dynamics, and applications to partial differential equations

Cresson

Pierret

2016

Journal of Mathematical Physics

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Abstract. -Modeling phenomena from experimental data, always begin with a choice of hypothesis on the observed dynamics such as determinism, randomness, derivability etc. Depending on these choices, different behaviors can be observed. The natural question associated to the modeling problem is the following : "With a finite set of data concerning a phenomenon, can we recover its underlying nature ? From this problem, we introduce in this paper the definition of multi-scale functions, scale calculus and scale dynamics based on the time-scale calculus (see [3]). These definitions will be illustrated on the multi-scale Okamoto's functions. IntroductionThis article deals with new mathematical tools to deal with scale phenomena and applications to partial differential equations. The framework that we have developed can be read from a mathematical point of view following each definitions and theorems. However, this framework can be seen as a synthesis of different tentative of one of the authors in order to deal with scales in geometry and analysis in the context of different physical problems (see [5,7,6,12,10]), in particular the scale relativity theory developed by L. Nottale [20,21,22], and more generally modelling problems. As a consequence, before coming to more mathematical considerations, we picture some important problems in modelling in Physics which are underlying our framework.Modelling a given phenomenon from experimental data using classical mathematical tools always assume, sometimes implicitly, a given framework hypothesis on the real nature of the phenomenon which can be also called the texture of reality. As an example, classical mechanics is developed using the classical differential calculus to write speed and acceleration MULTISCALE FUNCTIONS, SCALE DYNAMICS AND APPLICATIONS 3 of particle and implicitly assuming that the behaviour of these particles can be described using smooth curves on a given space. Depending on this framework, different behaviors will be predicted or not and will be confronted to reality. However, this assumption about the real nature of a phenomenon is in general not so easy to decide and in some sense depends on philosophical considerations (positivism, etc) which can not be proved. The classical debate between A. Einstein and N. Bohr about the nature of quantum physics is a famous example.The previous problem can be handled using a different approach, looking at the way mathematical models for a given phenomenon are constructed. Indeed, the framework question is in fact related to two different facts which are in general mixed in the literature. In order to put in evidence these points, we first remind very roughly the usual way to construct a model for a given phenomenon :-Acquiring experimental data.-Computations of relevant quantities (velocity, acceleration, etc).-Functional relation between these quantities (at a discrete level).-Asymptotic passage to a continuous model under a specific choice of hypothesis.-Comparison to reality using numerical simulations Putting ...

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“…Definition [89]: the block complexity of a sequence with values in a finite alphabet is the function , where is the number of different blocks of length that occur in the sequence.…”

Section: Conservation Of Information In This Randommentioning

confidence: 99%

An Information Theoretic Analysis of Random Number Generator based on Cellular Automaton

Nayyeri¹,

Dastghaibyfard²

2018

ijacsa

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Abstract-Realization of Randomness had always been a controversial concept with great importance both from theoretical and practical Perspectives. This realization has been revolutionized in the light of recent studies especially in the realms of Chaos Theory, Algorithmic Information Theory and Emergent behavior in complex systems. We briefly discuss different definitions of Randomness and also different methods for generating it. The connection between all these approaches and the notion of Normality as the necessary condition of being unpredictable would be discussed. Then a complex-system-based Random Number Generator would be introduced. We will analyze its paradoxical features (Conservative Nature and reversibility in spite of having considerable variation) by using information theoretic measures in connection with other measures. The evolution of this Random Generator is equivalent to the evolution of its probabilistic description in terms of probability distribution over blocks of different lengths. By getting the aid of simulations we will show the ability of this system to preserve normality during the process of coarse graining.

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Algebraic and Analytic Randomness

Cited by 8 publications

References 22 publications

Statistical analysis of the first digits of the binary expansion of Feigenbaum constants α and δ

Statistical analysis of the first digits of the binary expansion of Feigenbaum constants α and δ

Multiscale functions, scale dynamics, and applications to partial differential equations

An Information Theoretic Analysis of Random Number Generator based on Cellular Automaton

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