In mathematics, a Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. ^{[ 1 ]} More precisely, given any small positive distance, all excluding a finite number of elements of the sequence are less than that given distance from each other. Cauchy sequences are named after Augustin-Louis Cauchy; they may occasionally be known as fundamental sequences. ^{[ 2 ]}.

Caltech scientists have found a fast and efficient way to add up large numbers of Feynman diagrams, the simple drawings physicists use to represent particle interactions. The new method has already enabled the researchers to solve a longstanding problem in the materials science and physics worlds known as the polaron problem, giving scientists and engineers a way to predict how electrons will flow in certain materials, both conventional and quantum.

In the 1940s, physicist Richard Feynman first proposed a way to represent the various interactions that take place between electrons, photons, and other fundamental particles using 2D drawings that involve straight and wavy lines intersecting at vertices. Though they look simple, these Feynman diagrams allow scientists to calculate the probability that a particular collision, or scattering, will take place between particles.

Since particles can interact in many ways, many different diagrams are needed to depict every possible interaction. And each diagram represents a mathematical expression. Therefore, by summing all the possible diagrams, scientists can arrive at quantitative values related to particular interactions and scattering probabilities.

Comparing our findings across species with those across growth led us to a final question. If the purpose of sleep shifts from being about neural reorganisation as children to being about repair once we’re grown, when exactly does that transition occur, and how sudden is it? Armed with our new theory plus human developmental data, we could answer this question with surprising accuracy: the transition occurs when we’re extremely young – at about 2.5 years of age – and it happens extremely abruptly, like water freezing at 0°C.

We were delighted with this stunning result. First, it gave us an even greater appreciation for the critical importance of sleep: never again would we underestimate its importance for our children, especially in their first few years of life when their sleep is doing something so fundamentally different and extraordinarily important, something that seemingly can’t be made up for later in life. Second, we had discovered that these two states of sleep, while they looked remarkably similar from the outside, are actually analogous to completely different states of matter before and after the stark dividing line of 2.5 years of age. Before 2.5 years, our brains are more fluid and plastic, enabling us to learn and adapt quickly, similar to the state of water flowing around obstacles. After 2.5 years, our brains are much more crystalline and frozen, still capable of learning and adapting but more like glaciers slowly pushing across a landscape.

Many questions still remain. How much does sleep vary across humans and across species? Can this early fluid phase of sleep be extended? Is this phase already extended or shortened in some individuals, and what costs or benefits are associated with that? What other functions of sleep have piggybacked on to the primary functions of repair and neural reorganisation? How do the different reasons for sleep compete for or share sleep time, either across ages or even within a single night? It will take much more work to fully unravel the mysteries of sleep, but our recent insights – about age-based shifts in the purpose of sleep and the mathematical, predictive theories that quantify them – represent an essential tool to plumb these depths even further.

For decades, scientists have looked to light as a way to speed up computing. Photonic neural networks—systems that use light instead of electricity to process information—promise faster speeds and lower energy use than traditional electronics.

But despite their potential, these systems have struggled to match the accuracy of digital neural networks. A key reason: most photonic systems still mimic the structure and training methods of digital models, introducing errors when translating from software to hardware.

Now, a research team from Northwestern Polytechnical University and Southeast University in China has developed a new kind of photonic neural network that breaks free from this digital imitation. Their design, published in Advanced Photonics Nexus, uses physical transformations of light to process information directly, without relying on mathematical models. This approach not only improves accuracy but also highlights a new direction for building smarter, faster AI hardware.

Scientists have used ultracold atoms to successfully demonstrate a novel method of particle acceleration that could unlock a new understanding of how cosmic rays behave, a new study reveals.

More than 70 years after its formulation, researchers have observed the Fermi acceleration mechanism in a laboratory by colliding ultracold atoms against engineered movable potential barriers—delivering a significant milestone in high-energy astrophysics and beyond.

Fermi acceleration is the mechanism responsible for the generation of cosmic rays, as postulated by physicist Enrico Fermi in 1949. The process itself also features some universal properties that have spawned a wide range of mathematical models, such as the Fermi-Ulam model. Until now, however, it has been difficult to create a reliable Fermi accelerator on Earth.

Some numbers are so unimaginably large that they defy the bounds of modern mathematics, and now mathematicians are closing in on a number that may mark the edge of this bizarre abyss

You’re not bad at math. You’ve just not been zapped enough.

D.P. Kroese, Z.I. Botev, T. Taimre, R. Vaisman. Data Science and Machine Learning: Mathematical and Statistical Methods, Chapman and Hall/CRC, Boca Raton, 2019.

The purpose of this book is to provide an accessible, yet comprehensive textbook intended for students interested in gaining a better understanding of the mathematics and statistics that underpin the rich variety of ideas and machine learning algorithms in data science.

Inspired by the “Harry Potter” stories and the Disney Channel show “Wizards of Waverly Place,” 7-year-old Sabrina Corsetti emphatically declared to her parents one afternoon that she was, in fact, a wizard.

“My dad turned to me and said that, if I really wanted to be a wizard, then I should become a physicist. Physicists are the real wizards of the world,” she recalls.

That conversation stuck with Corsetti throughout her childhood, all the way up to her decision to double-major in physics and math in college, which set her on a path to MIT, where she is now a graduate student in the Department of Electrical Engineering and Computer Science.

While her work may not involve incantations or magic wands, Corsetti’s research centers on an area that often produces astonishing results: integrated photonics. A relatively young field, integrated photonics involves building computer chips that route light instead of electricity, enabling compact and scalable solutions for applications ranging from communications to sensing.

MIT graduate student Sabrina Corsetti is exploring the cutting edge of integrated photonics, which involves building computer chips that route light instead of electricity. Her projects have included a chip-sized 3D printer and miniaturized optical systems for quantum computing.

Researchers have used machine learning to dramatically speed up the processing time when simulating galaxy evolution coupled with supernova explosion. This approach could help us understand the origins of our own galaxy, particularly the elements essential for life in the Milky Way.

The findings are published in The Astrophysical Journal.

The team was led by Keiya Hirashima at the RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS) in Japan, along with colleagues from the Max Planck Institute for Astrophysics (MPA) and the Flatiron Institute.