Thursday, April 25, 2019

Some Fun with Gentle Chaos, the Golden Ratio, and Stochastic Number Theory

So many fascinating and deep results have been written about the number (1 + SQRT(5)) / 2 and its related sequence - the Fibonacci numbers - that it would take years to read all of them. This number has been studied both for its applications (population growth, architecture) and its mathematical properties, for over 2,000 years. It is still a topic of active research.
Lag-1 auto-correlation in digit distribution of good seeds, for b-processes
I show here how I used the golden ratio for a new number guessing game (to generate chaos and randomness in ergodic time series) as well as new intriguing results, in particular:
  • Proof that the rabbit constant it is not normal in any base; this might be the first instance of a non-artificial mathematical constant for which the normalcy status is formally established.
  • Beatty sequences, pseudo-periodicity, and infinite-range auto-correlations for the digits of irrational numbers in the numeration system derived from perfect stochastic processes
  • Properties of multivariate b-processes, including integer or non-integer bases.
  • Weird behavior of auto-correlations for the digits of normal numbers (good seeds) in the numeration system derived from stochastic b-processes
  • A strange recursion that generates all the digits of the rabbit constant
Content of this article
1. Some Definitions
2. Digits Distribution in b-processes
3. Strange Facts and Conjectures about the Rabbit Constant
4. Gaming Application
  • De-correlating Using Mapping and Thinning Techniques
  • Dissolving the Auto-correlation Structure Using Multivariate b-processes
5. Related Articles
Read full articles, here

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