Quantum Dynamics, Logistic Map, And Digit Distribution Of Special Math Constants

Using nan logistic representation alternatively of nan guidelines quadratic strategy arsenic successful insubstantial 53 (here), I get very akin quantum dynamics, this clip for nan usability sin2(√x) alternatively of exp(x). When x is simply a mini integer aliases a merchandise of consecutive primes, my model reveals caller insights connected nan digit distribution of awesome mathematics constants. I besides talk heavy findings astir nan chaotic quality of dynamical systems and respective applications including successful AI.

Contents

1. Introduction.
2. Logistic representation and nan digit sum function
   . . .  Model comparison, pinch illustrations
   . . .  Normality of typical mathematics constants
   . . .  Applications and references
3. Re-balancing an uneven digit distribution
   . . .  Digit-balancing transforms
   . . .  Digit artifact balancing
4. Conclusion
5. Main Python code

Summary

This article is nan 4th successful this series. The erstwhile ones attraction connected nan simplest quadratic map, while this 1 shows really we tin attain akin results pinch nan astir well-known dynamical system: nan logistic map.

Orbits successful chaotic dynamical systems are very delicate to nan seed. Starting pinch adjacent seeds S0 = x · 2−2n and S′0 = 0, I computed ∆k = |Sk − S′k| and showed that ∆n ≈ sin2(√x) pinch a precision of astir 2n bits, besides getting bully approximations to ∆k erstwhile k ≤ n. In nan logistic map, S′0 is simply a fixed point.

I besides discussed nan unsocial quantic behaviour of nan digit sum usability erstwhile utilizing typical values of x. This opens up caller directions to study nan digit distribution of typical mathematics constants, e successful particular. Most importantly, it leads to galore applications including cryptography, synthetic data, shape discovery aliases impervious automation pinch LLMs, agent-based modeling, and more.

The worldly presented present tin besides beryllium utilized to complement a people connected dynamical systems, for technological research, aliases to commencement a PhD thesis connected nan subject.

Download method insubstantial and information generator (Python)

The PDF pinch galore illustrations is disposable for free arsenic insubstantial 54, here. It besides features accelerated Python codification (with nexus to GitHub) to woody pinch gigantic numbers. The underlying mentation is explained successful detail, pinch respective modern references. The bluish links successful nan PDF are clickable erstwhile you download nan archive from GitHub and position it successful immoderate browser but whitethorn not beryllium clickable successful nan GitHub “view mode”. I dream GitHub hole this rumor successful nan future!

About nan Author

Vincent Granville is simply a pioneering GenAI scientist, co-founder at BondingAI.io, nan LLM 2.0 level for hallucination-free, secure, in-house, lightning-fast Enterprise AI astatine standard pinch zero weight and nary GPU. He is besides writer (Elsevier, Wiley), publisher, and successful entrepreneur pinch multi-million-dollar exit. Vincent’s past firm acquisition includes Visa, Wells Fargo, eBay, NBC, Microsoft, and CNET. He completed a post-doc successful computational statistic astatine University of Cambridge.