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. 

New Stock Trading and Lottery Game Rooted in Deep Math

I describe here the ultimate number guessing game, played with real money. It is a new trading and gaming system, based on state-of-the-art mathematical engineering, robust architecture, and patent-pending technology. It offers an alternative to the stock market and traditional gaming. This system is also far more transparent than the stock market, and can not be manipulated, as formulas to win the biggest returns (with real money) are made public. Also, it simulates a neutral, efficient stock market. In short, there is nothing random, everything is deterministic and fixed in advance, and known to all users. Yet it behaves in a way that looks perfectly random, and public algorithms offered to win the biggest gains require so much computing power, that for all purposes, they are useless -- except to comply with gaming laws and to establish trustworthiness.

We use private algorithms to determine the winning numbers, and while they produce the exact same results as the public algorithms (we tested this extensively), they are incredibly more efficient, by many orders of magnitude. Also, it can be mathematically proved that the public and private algorithms are equivalent, and we actually proved it. We go through this verification process for any new algorithm introduced in our system. 

In the last section, we offer a competition: can you use the public algorithm to identify the winning numbers computed with the private (secret) algorithm? If yes, the system is breakable, and a more sophisticated approach is needed, to make it work. I don't think anyone can find the winning numbers (you are welcome to prove me wrong), so the award will be offered to the contestant providing the best insights on how to improve the robustness of this system. And if by chance you manage to identify those winning numbers, great, you'll get a bonus! But it is not a requirement to win the award.

Read the full article

Content

1. Description, Main Features and Advantages

2. How it Works: the Secret Sauce

• Public Algorithm
• The Winning Numbers
• Using Seeds to Find the Winning Numbers
• ROI Tables

3. Business Model and Applications

• Managing the Money Flow

4. Challenge and Statistical Results

• Data Science / Math Competition
• Controlling the Variance of the Portfolio Value
• Probability of Cracking the System

Most Popular Content on DSC

We have been in existence for over 10 years now, with content in many different places, lists, categories, and databases. This is an attempt to put everything together in one place, and help our readers (re-)discover some great articles and resources that were lost on the Internet over the years, but still sit on our web servers. We are making them come back to life. We are in the process of organizing it in a way that is user-friendly.  Some of the resources below are very recent, and some are pretty old, but we only kept what is still relevant and useful today. To not miss this type of content in the future, subscribe to our newsletter.

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Long-range Correlations in Time Series: Modeling, Testing, Case Study

We investigate a large class of auto-correlated, stationary time series, proposing a new statistical test to measure departure from the base model, known as Brownian motion. We also discuss a methodology to deconstruct these time series, in order to identify the root mechanism that generates the observations. The time series studied here can be discrete or continuous in time, they  can have various degrees of smoothness (typically measured using the Hurst exponent) as well as long-range or short-range correlations between successive values. Applications are numerous, and we focus here on a case study arising from some interesting number theory problem. In particular, we show that one of the times series investigated in my article on randomness theory [see here, read section 4.1.(c)] is not Brownian despite the appearance. It has important implications regarding the problem in question. Applied to finance or economics, it makes the difference between an efficient market, and one that can be gamed.

This article it accessible to a large audience, thanks to its tutorial style, illustrations, and easily replicable simulations. Nevertheless, we discuss modern, advanced, and state-of-the-art concepts. This is an area of active research. 

 Content

1. Introduction and time series deconstruction

• Example
• Deconstructing time series
• Correlations, Fractional Brownian motions

2. Smoothness, Hurst exponent, and Brownian test

• Our Brownian tests of hypothesis
• Data

3. Results and conclusions

• Charts and interpretation
• Conclusions

Read the full article, here.