Mesmerizing Video that will Make you Love Math, Engineering, and Probability
Featuring the roots of various polynomials whose coefficients are the binary digits of numbers like π. Complex roots cluster around the circle, and the few real ones are on the horizontal X-axis (positive on the right, negative on the left). Download the free paper to see how to interpret the colors and get the […]
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Quantum, chaotic and fractal types of algorithmic convergence
For centuries, mathematicians worked on problems where the convergence is either smooth or does not happen. Now the concept of chaotic convergence is mainstream, popularized by the stochastic gradient descent in deep neural networks, central to LLMs. In my most recent book here, I discuss many cases involving various types of chaotic convergence. In some […]
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Pi Day in the age of AI — The Missing $1m Millenium Prize
Pi Day is celebrated every year on 3/14. Enjoy and share the video I created with our “Pi Day AI agent” at BondingAI, generating hundreds of webpages, turning each one in a screenshot (a frame in the video). All done in Python with source code available here. The video is also on YouTube, here. For […]
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xLLM Version 2.0: GitHub Repository with Innovative AI Agents
I am putting all the new code and documentation about xLLM v 2.0 on GitHub, starting with various AI agents. At least, what is open-source and public (there is a lot more behind the public material). All home-made from scratch with radically different technology. You can check the new repository, here. Start with the README […]
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Checking for Randomness: Replacing Test Batteries with a Single Test
In cybersecurity applications where replicability is critical, or when building pseudo-random number generators, it is typical to perform a large number of various tests to check if a sequence of bits is random enough for practical purposes. This is also true in scientific research, to assess whether or not the digits of π or other […]
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Spectacular New Discovery about the Digits of π
Everyone believes that the digits of constants such as π or √2 cannot be distinguished from a sequence of random bits. The first few trillion successfully pass all tests of randomness. However, proving that they indeed behave perfectly randomly is arguably one of the oldest and most difficult unsolved math conjectures. So far, nobody succeeded […]
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New Book: Breakthroughs on the Digit Distribution of Classic Constants
Since the first edition entitled “0 and 1 — From Elemental Math to Quantum AI” and released in early 2025, a lot of progress has been made. Fascinating new results have been uncovered and proved by the author, many still leading to interesting quantum dynamics. In 100 pages, the new material presented here goes far […]
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Short Introduction to Signal Processing and Convolution
In less than 3 pages, this tutorial covers signal processing and convolution quite thoroughly, even advanced concepts. I illustrate the techniques with the Riemann zeta function and the kernel method, along with short, home-made Python code that shows all the detailed steps, rather than based on Blackbox Python libraries. The document looks like a cheat […]
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xLLM: 30 Articles Shaping the Future of Enterprise AI in 2026
Over several decades, I unlearned everything that I learned in college classes, and built a new discipline from scratch, as much different from traditional AI than it is from standard machine learning, statistics and computer science. Outside academia with a focus on practical applications. Item #2 in the list below is the culmination of this […]
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Simple Normality Test with Application to Random Number Generation
Numbers such as π, e, log 2 or √2 have binary digits (bits) that look randomly distributed. They are very good candidates to generate randomness especially in cryptography. One way to assess their randomness is by proving that they are normal numbers. Such a proof has remained elusive for centuries. Here I focus on a […]
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