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Category: mathematics

It has puzzled scientists for years whether and how bacteria, that live from dissolved organic matter in marine waters, can carry out N2 fixation. It was assumed that the high levels of oxygen combined with the low amount of dissolved organic matter in the marine water column would prevent the anaerobic and energy consuming N2 fixation.

Already in the 1980s it was suggested that aggregates, so-called “marine snow particles,” could possibly be suitable sites for N2 fixation, and this was recently confirmed. Still, it has been an open question why the carrying out this N2 fixation can be found worldwide in the ocean. Moreover, the global magnitude and the distribution of the activity have been unknown… until now.

In a new study, researchers from the Leibniz Centre for Tropical Marine Research in Germany, Technical University of Denmark, and the University of Copenhagen demonstrate, by use of mechanistic mathematical models, that bacteria attached to marine snow particles can fix N2 over a wide range of temperatures in the global oceans, from the tropics to the poles, and from the surface to the abyss.

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We say a message is incoherent when we can’t make it out, or when it doesn’t make sense. A scribbled note, a drunken argument or a conversation taking place five tables down in a crowded cafe might all be incoherent. In general, “coherent” means the opposite—consistent, connected, clear.

In science, the word coherence takes on more specific, mathematical definitions, but they all get at a similar concept: Something is coherent if it can be understood, if it forms a unified whole and if those first two qualities persist.

Scientists originally developed the concept of coherence to understand and describe the wave-like behavior of light. Since then, the concept has been generalized to other systems involving waves, such as acoustic, electronic and quantum mechanical systems.

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Physicists have found a simple and effective way to skip over an energy level in a three-state system, potentially leading to increased quantum computational power with fewer qubits.

Nearly a century ago, Lev Landau, Clarence Zener, Ernst Stückelberg, and Ettore Majorana found a mathematical formula for the probability of jumps between two states in a system whose energy is time-dependent. Their formula has since had countless applications in various systems across physics and chemistry.

Now physicists at Aalto University’s Department of Applied Physics have shown that the jump between different states can be realized in systems with more than two via a virtual transition to an intermediate state and by a linear chirp of the drive frequency. This process can be applied to systems where it is not possible to modify the energy of the levels.

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Elon Musk’s AI startup xAI has introduced Grok 3, the latest version of its chatbot model, which Musk describes as the most advanced AI system yet.

XAI claims Grok 3 outperforms rival AI models from Alphabet’s Google Gemini, DeepSeek’s V3, Anthropic’s Claude, and OpenAI’s GPT-4o in benchmarks for math, science, and coding.

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A strange molecular pattern, first mistaken for an error, led researchers to an unexpected discovery: molecules forming non-repeating structures similar to the einstein tiling problem.

This phenomenon, driven by chirality and energy balance, could pave the way for novel insights into molecular physics.

At the crossroads of mathematics and tiling lies the einstein problem—a puzzle that, despite its name, has nothing to do with Albert Einstein. The question is simple yet profound: Can a single shape tile an infinite surface without ever creating a repeating pattern? In 2022, English amateur mathematician David Smith discovered such a shape, known as a “proto-tile.”

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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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