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Category: mathematics – Page 65

Scientists use AI to investigate structure and long-term behavior of galaxies

Bayreuth scientists are investigating the structure and long-term behavior of galaxies using mathematical models based on Einstein’s theory of relativity. Their innovative approach uses a deep neural network to quickly predict the stability of galaxy models. This artificial intelligence-based method enables efficient verification or falsification of astrophysical hypotheses in seconds.

The research objective of Dr. Sebastian Wolfschmidt and Christopher Straub is to investigate the structure and long-term behavior of galaxies. “Since these cannot be fully analyzed by , we use mathematical models of galaxies,” explains Christopher Straub, a doctoral student at the Chair of Mathematics VI at the University of Bayreuth.

“In order to take into account that most galaxies contain a black hole at their center, our models are based on Albert Einstein’s general theory of relativity, which describes gravity as the curvature of four-dimensional spacetime.”

Mathematical model reveals how a pit viper is able to find its dinner in the dead of night

In the animal kingdom, there are many grand examples of species that make sense of their world by expertly deciphering even weak signals from their surroundings.

An eagle soaring above the ground spies a river fish down below, about to swallow a bug; a hungry black bear smells a morsel of food two miles away in a dense thicket; a duck-billed platypus, swimming in a freshwater creek, closes its eyes and detects the electric impulses of a tasty tadpole nearby.

Then there are the pit vipers.

AI-Powered Proof Generator Helps Debug Software

Not all software is perfect—many apps, programs, and websites are released despite bugs. But the software behind critical systems like cryptographic protocols, medical devices, and space shuttles must be error-free, and ensuring the absence of bugs requires going beyond code reviews and testing. It requires formal verification.

Formal verification involves writing a mathematical proof of your code and is “one of the hardest but also most powerful ways of making sure your code is correct,” says Yuriy Brun, a professorat the University of Massachusetts Amherst.

To make formal verification easier, Brun and his colleagues devised a new AI-powered method called Baldur to automatically generate proofs. The accompanying paper, presented in December 2023 at the ACM Joint European Software Engineering Conference and Symposium on the Foundations of Software Engineering in San Francisco, won a Distinguished Paper award. The team includes Emily First, who completed the study as part of her doctoral dissertation at UMass Amherst; Markus Rabe, a former researcher at Google, where the study was conducted; and Talia Ringer, an assistant professor at the University of Illinois Urbana-Champaign.

Quasi-integrable Arrays: The Family Grows

A new approach to solving arrays of two-dimensional differential equations may allow researchers to go beyond the one-dimensional oscillator paradigm.

A frictionless pendulum and a pendulum clock behave alike, but they belong to different worlds: Hamiltonian systems and dissipative systems, respectively. In the Hamiltonian world, completely integrable—that is, solvable—systems serve as a mathematical basis for dealing with more general cases that aren’t integrable. An analogous strategy doesn’t work for nonlinear non-Hamiltonian dissipative systems, however. In that case, the best researchers can achieve is partial integrability. Until recently, it was thought that an array of globally coupled oscillators could be partially integrable only if each oscillator has only one degree of freedom. Now Rok Cestnik and Erik Martens, both at Lund University in Sweden, report on a quasi-integrable system consisting of N two-dimensional oscillators described by ordinary differential equations (ODEs) [1].

New method flips the script on topological physics

The branch of mathematics known as topology has become a cornerstone of modern physics thanks to the remarkable—and above all reliable—properties it can impart to a material or system. Unfortunately, identifying topological systems, or even designing new ones, is generally a tedious process that requires exactly matching the physical system to a mathematical model.

Researchers at the University of Amsterdam and the École Normale Supérieure of Lyon have demonstrated a model-free method for identifying topology, enabling the discovery of new topological materials using a purely experimental approach. The research is published in the journal Proceedings of the National Academy of Sciences.

Topology encompasses the properties of a system that cannot be changed by any “smooth deformation.” As you might be able to tell from this rather formal and abstract description, topology began its life as a branch of mathematics. However, over the last few decades physicists have demonstrated that the mathematics underlying topology can have very real consequences. Topological effects can be found in a wide range of physical systems, from individual electrons to large-scale .

What is time? An astronomer explains

It wasn’t until Albert Einstein that we developed a more sophisticated mathematical understanding of time and space that allowed physicists to probe deeper into the connections between them. In their endeavors, physicists also discovered that seeking the origin of time forces us to confront the origins of the universe itself.

What exactly is time, and how did it come into being? Did the dimension of time exist from the moment of the Big Bang, or did time emerge as the universe evolved? Recent theories about the quantum nature of gravity provide some unique and fantastic answers to these millennia-old questions.

Chemists use blockchain to simulate more than 4 billion chemical reactions essential to origins of life

Cryptocurrency is usually “mined” through the blockchain by asking a computer to perform a complicated mathematical problem in exchange for tokens of cryptocurrency. But in research appearing in the journal Chem a team of chemists has repurposed this process, asking computers to instead generate the largest network ever created of chemical reactions which may have given rise to prebiotic molecules on early Earth.

This work indicates that at least some primitive forms of metabolism might have emerged without the involvement of enzymes, and it shows the potential to use blockchain to solve problems outside the financial sector that would otherwise require the use of expensive, hard to access supercomputers.

“At this point we can say we exhaustively looked for every possible combination of chemical reactivity that scientists believe to had been operative on primitive Earth,” says senior author Bartosz A. Grzybowski of the Korea Institute for Basic Science and the Polish Academy of Sciences.

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