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Category: mathematics

The words “optimal” and “optimize” derive from the Latin “optimus,” or “best,” as in “make the best of things.” Alessio Figalli, a mathematician at the university ETH Zurich, studies optimal transport: the most efficient allocation of starting points to end points. The scope of investigation is wide, including clouds, crystals, bubbles and chatbots.

Dr. Figalli, who was awarded the Fields Medal in math that is motivated by concrete problems found in nature. He also likes the discipline’s “sense of eternity,” he said in a recent interview. “It is something that will be here forever.” (Nothing is forever, he conceded, but math will be around for “long enough.”) “I like the fact that if you prove a theorem, you prove it,” he said. “There’s no ambiguity, it’s true or false. In a hundred years, you can rely on it, no matter what.”

The study of optimal transport was introduced almost 250 years ago by Gaspard Monge, a French mathematician and politician who was motivated by problems in military engineering. His ideas found broader application solving logistical problems during the Napoleonic Era — for instance, identifying the most efficient way to build fortifications, in order to minimize the costs of transporting materials across Europe.

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A team of mathematicians and statisticians from the University of Wisconsin-La Crosse, the University of Tennessee and Valparaiso University, all in the U.S., has found new evidence that wolves had ample time to self-domesticate and evolve into modern dogs. In their study published in the journal Proceedings of the Royal Society B, the group developed a computer simulation showing the evolution process.

Prior research has suggested that the process of self-domesticating and then slowly evolving into modern dogs would have taken too long. Additionally, researchers believe that humans and dogs have been living in close proximity for approximately 30,000 years and that for the past 15,000 years, humans have been breeding them to perform certain tasks. But what happened in the first 15,000 years is less clear.

Some have suggested that humans may have begun encouraging the friendliest to hang around by adopting their puppies, finding their presence advantageous. Others have suggested that wolves moved ever closer to groups of humans for access to leftover food. But this , others have noted, would take more than 15,000 years to reach the point where humans began breeding them.

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Standing at the intersection between mathematics and the tiler’s trade is the so-called einstein problem. Despite its name, this mathematical question has nothing to do with the Nobel Prize winner Albert Einstein. It asks: Can you seamlessly tile an endless surface with a single shape (an “einstein”) in such a way that the resulting pattern is never repeated? Such a “proto-tile” was first discovered in 2022 by the English amateur mathematician David Smith.

Empa researcher Karl-Heinz Ernst is neither a mathematician nor a tiler. As a chemist, he researches the crystallization of molecules on . He never expected to deal with the einstein problem in his professional life—until his doctoral student Jan Voigt approached him with the unusual results of an experiment.

When a certain molecule crystallized on a , instead of the expected regular structure, irregular patterns were formed that never seemed to repeat themselves. Even more surprising: Each time he repeated the experiment, different aperiodic patterns emerged.

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A game of chess requires its players to think several moves ahead, a skill that computer programs have mastered over the years. Back in 1996, an IBM supercomputer famously beat the then world chess champion Garry Kasparov. Later, in 2017, an artificial intelligence (AI) program developed by Google DeepMind, called AlphaZero, triumphed over the best computerized chess engines of the time after training itself to play the game in a matter of hours.

More recently, some mathematicians have begun to actively pursue the question of whether AI programs can also help in cracking some of the world’s toughest problems. But, whereas an average game of chess lasts about 30 to 40 moves, these research-level math problems require solutions that take a million or more steps, or moves.

In a paper appearing on the arXiv preprint server, a team led by Caltech’s Sergei Gukov, the John D. MacArthur Professor of Theoretical Physics and Mathematics, describes developing a new type of machine-learning algorithm that can solve math problems requiring extremely long sequences of steps. The team used their to solve families of problems related to an overarching decades-old math problem called the Andrews–Curtis conjecture. In essence, the algorithm can think farther ahead than even advanced programs like AlphaZero.

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What can a moon’s tidal friction teach us about its formation and evolution? This is what a recent study published in Science Advances hopes to address as a team of researchers at the University of California Santa Cruz investigated a connection between the spin rate and tidal energy on Saturn’s moon, Titan, to determine more about Titan’s interior. This study has the potential to help researchers better understand the internal processes of Titan, leading to better constraints on the existence of a subsurface ocean.

For the study, the researchers used a combination of data obtained by NASA’s now-retired Cassini spacecraft and a series of mathematical calculations to determine Titan’s tidal dissipation, which is the amount of tidal energy lost in an object from friction and other processes, and for which the only moons in the solar system this has been successfully been accomplished being the Earth’s Moon and Jupiter’s volcanic moon, Io. Better understanding a moon’s tidal dissipation helps researchers better understand its formation and evolution, which the researchers successfully estimated for Titan.

“Tidal dissipation in satellites affects their orbital and rotational evolution and their ability to maintain subsurface oceans,” said Dr. Brynna Downey, who is a postdoctoral researcher at the Southwest Research Institute in Colorado and lead author of the study. “Now that we have an estimate for the strength of tides on Titan, what does it tell us about how quickly the orbit is changing? What we discovered is that it’s changing very quickly on a geologic timescale.”

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For centuries, the I-Ching, or Book of Changes, has fascinated scholars, mystics, and seekers alike. It is often considered a mere divination tool, a mystical means of interpreting the world through the casting of hexagrams.

But what if the I-Ching is something more? What if it operates as a structured probability space, exhibiting patterns and behaviors reminiscent of quantum mechanics?

Our latest research suggests that the I-Ching might not be a random oracle but instead a system governed by deep mathematical structures.

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Quantum physics, space documentary, and the fabric of reality—these are not just abstract ideas but the keys to unlocking the mysteries of existence. What is reality? Is it an illusion, a simulation, or something far beyond our comprehension? In this mind-expanding documentary, we explore the very fabric of the universe, from the bizarre behavior of quantum mechanics to the cosmic forces shaping space and time.

The universe is a grand puzzle, and science has only begun to unravel its secrets. Quantum physics reveals a world where particles exist in multiple states at once, where time behaves unpredictably, and where observation itself shapes reality. But how does this strange quantum realm connect to the vast expanse of space? Is the fabric of reality woven with unseen forces that govern everything, from black holes to the flow of time itself?

This space documentary takes you on a journey through the cutting-edge theories that challenge our understanding of the cosmos. Could our universe be a hologram? Is time an illusion? Do parallel realities exist beyond our perception? With stunning visuals, expert insights, and mind-bending concepts, we push the boundaries of what we know about existence.

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Laser diodes are semiconductors that generate light and amplify it using repeated reflection or “optical feedback.” Once the light has achieved desirable optical gain, laser diodes release it as powerful laser beams.

Photonic crystal surface-emitting lasers (PCSELs) are advanced where the optical gain is typically distributed laterally to the propagating light within a photonic crystal (PC) structure. They differ from traditional lasers by separating gain, feedback, and emission functions, offering scalable single-mode power and innovative designs. This leads to enhanced performance and new application possibilities.

In a paper that was published in the IEEE Journal of Selected Topics in Quantum Electronics on 20 November 2024, researchers have developed a method to numerically simulate the interaction of light waves within PCSELs.

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A theoretical particle that travels faster than light, the tachyon has long intrigued physicists and fueled decades of speculation. Initially conceived as a possible solution to quantum and relativity paradoxes, tachyons remain purely hypothetical. Despite the lack of experimental evidence, they continue to serve as a thought-provoking concept in modern physics.

A recent study by an international team of researchers has reignited interest in tachyons, suggesting they might be possible within the framework of Einstein’s special theory of relativity. This bold claim challenges conventional understandings of causality and time, raising fundamental questions about the structure of reality. If confirmed, it could lead to a radical shift in how scientists perceive the limits of physical laws.

Physicist Gerald Feinberg introduced the idea of tachyons in 1962, proposing that such particles could always travel faster than light without ever slowing down to subluminal speeds. His argument was based on the concept of imaginary mass, a theoretical construct involving the square root of a negative number. This allowed for the mathematical possibility of faster-than-light motion without explicitly violating relativity.

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