A neuron is a thing of beauty. Ever since Santiago Ramón y Cajal stained them with silver nitrate to make them visible under the microscopes of the 1880s (see drawing above), their ramifications have fired the scientific imagination. Ramón y Cajal called them the butterflies of the soul. Your browser does not support the audio element. Those ramifications--dendrites by the dozen to collect incoming signals, called action potentials, from other neurons, and a single axon to pass on the summed wisdom of those signals in the form of another action potential, turn neurons into parts of far bigger structures known as neural networks.

The methodology described here has broad applications, leading to new statistical tests, new type of ANOVA (analysis of variance), improved design of experiments, interesting fractional factorial designs, a better understanding of irrational numbers leading to cryptography, gaming and Fintech applications, and high quality random number generators (and when you really need them). It also features exact arithmetic / high performance computing and distributed algorithms to compute millions of binary digits for an infinite family of real numbers, including detection of auto- and cross-correlations (or lack of) in the digit distributions. The data processed in my experiment, consisting of raw irrational numbers (described by a new class of elementary recurrences) led to the discovery of unexpected apparent patterns in their digit distribution: in particular, the fact that a few of these numbers, contrarily to popular belief, do not have 50% of their binary digits equal to 1. It turned out that perfectly random digits simulated in large numbers, with a good enough pseudo-random generator, also exhibit the same strange behavior, pointing to the fact that pure randomness may not be as random as we imagine it is. Ironically, failure to exhibit these patterns would be an indicator that there really is a departure from pure randomness in the digits in question. In addition to new statistical / mathematical methods and discoveries and interesting applications, you will learn in my article how to avoid this type of statistical traps that lead to erroneous conclusions, when performing a large number of statistical tests, and how to not be misled by false appearances. I call them statistical hallucinations and false outliers. This article has two main sections: section 1, with deep research in number theory, and section 2, with deep research in statistics, with applications. You may skip one of the two sections depending on your interests and how much time you have. Both sections, despite state-of-the-art in their respective fields, are written in simple English. It is my wish that with this article, I can get data scientists to be interested in math, and the other way around: the topics in both cases have been chosen to be exciting and modern.

Despite the promising results obtained with [representations developed from Web image], the experiments demonstrate that object classification with real-life robotic data is far from being solved."

In 1948, Claude Shannon published a paper called A Mathematical Theory of Communication[1]. This paper heralded a transformation in our understanding of information. Before Shannon's paper, information had been viewed as a kind of poorly defined miasmic fluid. But after Shannon's paper, it became apparent that information is a well-defined and, above all, measurable quantity. Indeed, as noted by Shannon, A basic idea in information theory is that information can be treated very much like a physical quantity, such as mass or energy.

Despite the promising results obtained with [representations developed from Web image], the experiments demonstrate that object classification with real-life robotic data is far from being solved."

Pick up a favorite CD. Then slide it into the slot on the player-and listen as the music comes out just as crystal clear as the day you first opened the plastic case. Before moving on with the rest of your day, give a moment of thought to the man whose revolutionary ideas made this miracle possible: Claude Elwood Shannon. Shannon, who died in February after a long illness, was one of the greatest of the giants who created the information age. John von Neumann, Alan Turing and many other visionaries gave us computers that could process information. But it was Claude Shannon who gave us the modern concept of information-an intellectual leap that earns him a place on whatever high-tech equivalent of Mount Rushmore is one day established. The entire science of information theory grew out of one electrifying paper that Shannon published in 1948, when he was a 32-year-old researcher at Bell Laboratories.

Quantum information science is a young field, its underpinnings still being laid by a large number of researchers [see "Rules for a Complex Quantum World," by Michael A. Nielsen; Scientific American, November 2002]. Classical information science, by contrast, sprang forth about 50 years ago, from the work of one remarkable man: Claude E. Shannon. In a landmark paper written at Bell Labs in 1948, Shannon defined in mathematical terms what information is and how it can be transmitted in the face of noise. What had been viewed as quite distinct modes of communication--the telegraph, telephone, radio and television--were unified in a single framework. Shannon was born in 1916 in Petoskey, Michigan, the son of a judge and a teacher.

Not long after his birth on April 30, 1916, it became clear that Claude Shannon was good with gadgets. As a youth, he fixed radios for nearby stores and converted barbed-wire fences into a telegraph line, through which he communicated with a friend. After graduating from the University of Michigan in 1936, Shannon took a job as a research assistant at MIT, where he turned that talent toward research that would change the course of history. It was at MIT that he worked on a machine called the "differential analyzer" -- then the world's leading computer but by modern standards a clumsy monolith of gears and motors that took a whole week to solve a single equation. There had to be a better way, and Shannon found it.

But the point I wish to make is that we can now calculate many thousands of times as fast as we could in 1953 and at least a million times as fast as we could three hundred years ago. Now this change is quite extraordinary, if one compares it for example with the increase in the speed of travel. A satellite orbiting the earth or moving towards the planets is unlikely to go much faster than twenty-five or thirty thousand miles an hour. An ordinary man can usually do two and a half or three, so that the satellite is perhaps ten thousand times as fast as a walking man. The enormous increase in speed of travel has changed our world and our ideas of the potentially possible. We don't use satellites to go from Manchester to Edinburgh in a few minutes, but we hope to explore the solar system.