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An international collaboration between researchers at the RIKEN Center for Brain Science (CBS) in Japan, the University of Tokyo, and University College London has demonstrated that self-organization of neurons as they learn follows a mathematical theory called the free energy principle.

The principle accurately predicted how real neural networks spontaneously reorganize to distinguish incoming information, as well as how altering neural excitability can disrupt the process. The findings thus have implications for building animal-like artificial intelligences and for understanding cases of impaired learning. The study was published August 7 in Nature Communications.

When we learn to tell the difference between voices, faces, or smells, networks of neurons in our brains automatically organize themselves so that they can distinguish between the different sources of incoming information. This process involves changing the strength of connections between neurons, and is the basis of all learning in the brain.

An interdisciplinary team of mathematicians, engineers, physicists, and medical scientists has discovered a surprising connection between pure mathematics and genetics. This connection sheds light on the structure of neutral mutations and the evolution of organisms.

Number theory, the study of the properties of positive integers, is perhaps the purest form of mathematics. At first sight, it may seem far too abstract to apply to the natural world. In fact, the influential American number theorist Leonard Dickson wrote “Thank God that number theory is unsullied by any application.”

And yet, again and again, number theory finds unexpected applications in science and engineering, from leaf angles that (almost) universally follow the Fibonacci sequence, to modern encryption techniques based on factoring prime numbers. Now, researchers have demonstrated an unexpected link between number theory and evolutionary genetics.

If two statisticians were to lose each other in an infinite forest, the first thing they would do is get drunk. That way, they would walk more or less randomly, which would give them the best chance of finding each other. However, the statisticians should stay sober if they want to pick mushrooms. Stumbling around drunk and without purpose would reduce the area of exploration, and make it more likely that the seekers would return to the same spot, where the mushrooms are already gone.

Such considerations belong to the statistical theory of “random walk” or “drunkard’s walk,” in which the future depends only on the present and not the past. Today, random walk is used to model share prices, molecular diffusion, neural activity, and population dynamics, among other processes. It is also thought to describe how “genetic drift” can result in a particular gene—say, for blue eye color—becoming prevalent in a population. Ironically, this theory, which ignores the past, has a rather rich history of its own. It is one of the many intellectual innovations dreamed up by Andrei Kolmogorov, a mathematician of startling breadth and ability who revolutionized the role of the unlikely in mathematics, while carefully negotiating the shifting probabilities of political and academic life in Soviet Russia.

O.o!!!

This holographic concept could explain a mystery about black holes, but the math may not represent reality.

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Until the early 20th century, the question of whether light is a particle or a wave had divided scientists for centuries. Isaac Newton held the former stance and advocated for his “corpuscular” theory. But by the early 19th century, the wave theory was making a comeback, thanks in part to the work of a French civil engineer named Augustin-Jean Fresnel.

Born in 1,788 to an architect, the young Fresnel had a strict religious upbringing, since his parents were Jansenists — a radical sect of the Catholic Church that embraced predestination. Initially he was home-schooled, and did not show early academic promise; he could barely read by the time he was eight. Part of this may have been due to all the political upheaval in France at the time. Fresnel was just one year old when revolutionaries stormed the Bastille in 1,789, and five when the Reign of Terror began.

Eventually the family settled in a small village north of Caen, and when Fresnel was 12, he was enrolled in a formal school. That is where he discovered science and mathematics. He excelled at both, so much so that he decided to study engineering, first at the École Polytechnique in Paris, and then at the École Nacionale des Ponts et Chaussées.

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Kamilla Cziráki, a geophysics student at the Faculty of Science of Eötvös Loránd University (ELTE), has taken a new approach to researching the navigation systems that can be used on the surface of the moon to plan future journeys.

Working with Professor Gábor Timár, head of the Department of Geophysics and Space Sciences, Cziráki calculated the parameters used in the Earth’s GPS system for the moon using the method of mathematician Fibonacci, who lived 800 years ago. Their findings have been published in the journal Acta Geodaetica et Geophysica.

Now, as humanity prepares to return to the moon after half a century, the focus is on possible methods of lunar navigation. It seems likely that the modern successors to the lunar vehicles of the Apollo missions will now be assisted by some form of satellite navigation, similar to the GPS system on Earth. In the case of Earth, these systems do not take into account the actual shape of our planet, the geoid, not even the surface defined by sea level, but a rotating ellipsoid that best fits the geoid.

Knowing a little about the local connections on flight maps and other networks can reveal a lot about a system’s global structure.