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An interdisciplinary team of mathematicians, engineers, physicists, and medical scientists have uncovered an unexpected link between pure mathematics and genetics, that reveals key insights into 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.

There is increasing talk of quantum computers and how they will allow us to solve problems that traditional computers cannot solve. It’s important to note that quantum computers will not replace traditional computers: they are only intended to solve problems other than those that can be solved with classical mainframe computers and supercomputers. And any problem that is impossible to solve with classical computers will also be impossible with quantum computers. And traditional computers will always be more adept than quantum computers at memory-intensive tasks such as sending and receiving e-mail messages, managing documents and spreadsheets, desktop publishing, and so on.

There is nothing “magic” about quantum computers. Still, the mathematics and physics that govern their operation are more complex and reside in quantum physics.

The idea of quantum physics is still surrounded by an aura of great intellectual distance from the vast majority of us. It is a subject associated with the great minds of the 20^th century such as Karl Heisenberg, Niels Bohr, Max Planck, Wolfgang Pauli, and Erwin Schrodinger, whose famous hypothetical cat experiment was popularized in an episode of the hit TV show ‘The Big Bang Theory’. As for Schrodinger, his observations of the uncertainty principle, serve as a reminder of the enigmatic nature of quantum mechanics. The uncertainty principle holds that the observer determines the characteristics of an examined particle (charge, spin, position) only at the moment of detection. Schrödinger explained this using the theoretical experiment, known as the paradox of Schrödinger’s cat. The experiment’s worth mentioning, as it describes one of the most important aspects of quantum computing.

Structured light waves with spiral phase fronts carry orbital angular momentum (OAM), attributed to the rotational motion of photons. Recently, scientists have been using light waves with OAM, and these special “helical” light beams have become very important in various advanced technologies like communication, imaging, and quantum information processing. In these technologies, it’s crucial to know the exact structure of these special light beams. However, this has proven to be quite tricky.

Interferometry—superimposing a light field with a known reference field to extract information from the interference—can retrieve OAM spectrum information using a camera. As the camera only records the intensity of the interference, the measurement technique encounters additional crosstalk known as “signal-signal beat interference” (SSBI), which complicates the retrieval process. It’s like hearing multiple overlapping sounds, making it difficult to distinguish the original notes.

In a recent breakthrough reported in Advanced Photonics, researchers from Sun Yat-sen University and École Polytechnique Fédérale de Lausanne (EPFL) used a powerful mathematical tool called the Kramers-Kronig (KK) relation, which helps with understanding and solving the problem. This tool enabled them to untangle the complex helical light pattern from the camera’s intensity-only measurements for single-shot retrieval in simple on-axis interferometry. Exploring the duality between the time-frequency and azimuth-OAM domains, they apply the KK approach to investigate various OAM fields, including Talbot self-imaged petals and fractional OAM modes.

David Chalmers is a philosopher at New York University and the Australian National University. He is Professor of Philosophy and co-director of the Center for Mind, Brain, and Consciousness at NYU, and also Professor of Philosophy at ANU.

Chalmers works in the philosophy of mind and in related areas of philosophy and cognitive science. He is especially interested in consciousness, but am also interested in all sorts of other issues in the philosophy of mind and language, metaphysics and epistemology, and the foundations of cognitive science.

From an early age, he excelled at mathematics, eventually completing his undergraduate education at the University of Adelaide with a Bachelor’s degree in Mathematics and Computer Science. He then briefly studied at Lincoln College at the University of Oxford as a Rhodes Scholar before receiving his PhD at Indiana University Bloomington under Douglas Hofstadter. He was a Postdoctoral Fellow in the Philosophy-Neuroscience-Psychology program directed by Andy Clark at Washington University in St. Louis from 1993 to 1995, and his first professorship was at UC Santa Cruz, from August 1995 to December 1998.

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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