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Category: mathematics – Page 34

The original version of this story appeared in Quanta Magazine.

In October, a Falcon Heavy rocket is scheduled to launch from Cape Canaveral in Florida, carrying NASA’s Europa Clipper mission. The $5 billion mission is designed to find out if Europa, Jupiter’s fourth-largest moon, can support life. But because Europa is constantly bombarded by intense radiation created by Jupiter’s magnetic field, the Clipper spacecraft can’t orbit the moon itself. Instead, it will slide into an eccentric orbit around Jupiter and gather data by repeatedly swinging by Europa—53 times in total—before retreating from the worst of the radiation. Every time the spacecraft rounds Jupiter, its path will be slightly different, ensuring that it can take pictures and gather data from Europa’s poles to its equator.

To plan convoluted tours like this one, trajectory planners use computer models that meticulously calculate the trajectory one step at a time. The planning takes hundreds of mission requirements into account, and it’s bolstered by decades of mathematical research into orbits and how to join them into complicated tours. Mathematicians are now developing tools which they hope can be used to create a more systematic understanding of how orbits relate to one another.

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Researchers have discovered what they’re calling a “cosmic glitch” in gravity, which could potentially help explain the universe’s strange behavior on a cosmic scale.

As detailed in a new paper published in the Journal of Cosmology and Astroparticle Physics, the team from the University of Waterloo and the University of British Columbia in Canada posit that Albert Einstein’s theory of general relativity may not be sufficient to explain the accelerating expansion of the universe.

Einstein’s “model of gravity has been essential for everything from theorizing the Big Bang to photographing black holes,” said lead author and Waterloo mathematical physics graduate Robin Wen in a statement about the research. “But when we try to understand gravity on a cosmic scale, at the scale of galaxy clusters and beyond, we encounter apparent inconsistencies with the predictions of general relativity.”

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A novel mathematical technique from the University of Surrey now simplifies space mission planning by mapping efficient routes, akin to a subway map, potentially revolutionizing travel to the Moon and beyond.

Just as sat-nav did away with the need to argue over the best route home, scientists from the University of Surrey have developed a new method to find the optimal routes for future space missions without the need to waste fuel.

The new method uses mathematics to reveal all possible routes from one orbit to another without guesswork or using enormous computer power.

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Researchers have successfully used 40-year-old mathematics to explain quantum tunneling, providing a unified approach to diverse quantum phenomena.

Quantum mechanical effects such as radioactive decay, or more generally: ‘tunneling’, display intriguing mathematical patterns. Two researchers at the University of Amsterdam now show that a 40-year-old mathematical discovery can be used to fully encode and understand this structure.

Quantum Physics – Easy and Hard.

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Quantum mechanics, the most potent theory physicists have developed, doesn’t make sense. What I mean by that statement is that quantum mechanics — which was developed to describe the microworld of molecules, atoms, and subatomic particles — leaves its users without a common-sense picture of what it describes. Full of what seem to be paradoxes and puzzles, quantum physics demands, for most scientists, an interpretation: a way of making sense of its mathematical formalism in terms of a concrete description of what exists in the world and how we interact with it. Unfortunately, after a century not one but a basketful of “quantum interpretations” have been proposed. Which one is correct? Which one most clearly understands what quantum physics has been trying to tell us these past 100 years?

In light of these questions, I’m beginning a series that explores the most radical of all the quantum interpretations, the one I think gets it right, or at least is pointed in the right direction. It is a relative newcomer to the scene, so you may not have heard of it. But it has been gaining a lot of attention recently because it doesn’t just ask us to reimagine how we view the science of atoms; it asks us to reimagine the process of science itself.

The term “QBism” was shorthand for “Quantum Bayesianism” when this idea/theory/interpretation was first proposed in the late 1990s and early 2000s. The name hit the nail on the head because “Bayesianism” is a radical way of interpreting probabilities. The Bayesianist approach to what we mean by probability differs strongly from what you learned in school about coin flips and dice rolls and how frequently a particular result can be expected to appear. Since probabilities lie at the heart of quantum mechanics, QBism zeroed in on a key aspect of quantum formalism — one that other interpretations had missed or swept under the rug — because it focused squarely on how we interpret probabilities. We’re going to dig deep into all of this as we go along in this series, but since today’s column is supposed to be the introduction, let’s start with a 10,000-foot view of what’s at stake in the great “Quantum Interpretation Wars” so we can see where QBism fits in.

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Now Princeton researchers have sparked new life into static. Using millions of hours of computational time to run detailed simulations, the researchers found a way to describe static charge atom-by-atom with the mathematics of heat and work. Their paper appeared in Nature Communications on March 23.

The study looked specifically at how charge moves between materials that do not allow the free flow of electrons, called insulating materials, such as vinyl and acrylic. The researchers said there is no established view on what mechanisms drive these jolts, despite the ubiquity of static: the crackle and pop of clothes pulled from a dryer, packing peanuts that cling to a box.

“We know it’s not electrons,” said Mike Webb, assistant professor of chemical and biological engineering, who led the study. “What is it?”

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Theoretical physicists employ their imaginations and their deep understanding of mathematics to decipher the underlying laws of the universe that govern particles, forces and everything in between. More and more often, theorists are doing that work with the help of machine learning.

As might be expected, the group of theorists using machine learning includes people classified as “computational” theorists. But it also includes “formal” theorists, the people interested in the self-consistency of theoretical frameworks, like string theory or quantum gravity. And it includes “phenomenologists,” the theorists who sit next to experimentalists, hypothesizing about new particles or interactions that could be tested by experiments; analyzing the data the experiments collect; and using results to construct new models and dream up how to test them experimentally.

In all areas of theory, machine-learning algorithms are speeding up processes, performing previously impossible calculations, and even causing theorists to rethink the way theoretical physics research is done.

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Black holes, white holes, wormholes, anti-universes, and all kinds of awesome relativity weirdness:

Einstein was wrong about black holes, what else? Use code veritasium at the link below to get an exclusive 60% off an annual Incogni plan: https://incogni.com/veritasium.

A massive thank you to Prof. Geraint F. Lewis and Prof. Juan Maldacena for their expertise and help with this video.

A huge thank you to those who helped us understand this complicated topic: Dr. Suddhasattwa Brahma, Prof. Carlo Rovelli, Dr. Hal Haggard, Prof. Martin Bojowald, Dr. Francesca Vidotto, Prof. Andrew Hamilton, and Dr. Carl-Fredrik Nyberg Brodda.

A special thanks to Alessandro Roussel from ScienceClic for his spectacular simulations and feedback on the video. Check out his channel here: https://ve42.co/ScienceClic.

Continue reading “Something Strange Happens When You Follow Einstein’s Math” | >

1/ Researchers have found that AI models can solve complex tasks like “3SUM” by using simple dots like “…” instead of sentences.

Researchers have found that specifically trained LLMs can solve complex problems just as well using dots like “…” instead of full sentences. This could make it harder to control what’s happening in these models.

The researchers trained Llama language models to solve a difficult math problem called “3SUM”, where the model has to find three numbers that add up to zero.

Usually, AI models solve such tasks by explaining the steps in full sentences, known as “chain of thought” prompting. But the researchers replaced these natural language explanations with repeated dots, called filler tokens.

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