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Category: mathematics – Page 67
Mathematical model connects innovation and obsolescence to unify insights across diverse fields
In Lewis Carroll’s Through the Looking-Glass, the Red Queen tells Alice, “It takes all the running you can do, to keep in the same place.” The race between innovation and obsolescence is like this.
Recent evidence about the slowing of technological and scientific progress in contrast to the accelerating epidemiological risks in a globalized world—in the opposite direction—indicates the importance of the relative rates of innovation and obsolescence.
When does innovation outpace, or fail to outpace, obsolescence? Understanding this dynamic is nascent, and the way that innovation is discussed is largely fragmented across fields. Despite some qualitative efforts to bridge this gap, insights are rarely transferred.
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 astronomical observations, 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 ocean currents.