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partial differential equations

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A GIF of a spinning vortex made up of fluid equations
fluid dynamics

Computer Proof ‘Blows Up’ Centuries-Old Fluid Equations

By Jordana Cepelewicz
November 16, 2022
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For more than 250 years, mathematicians have wondered if the Euler equations might sometimes fail to describe a fluid’s flow. A new computer-assisted proof marks a major breakthrough in that quest.

A GIF of ice melting in a glass of water
mathematical physics

Mathematicians Prove Melting Ice Stays Smooth

By Mordechai Rorvig
October 6, 2021
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After decades of effort, mathematicians now have a complete understanding of the complicated equations that model the motion of free boundaries, like the one between ice and water.

A GIF showing increasingly more twists being added to the curves on a sphere.
geometry

Mathematicians Identify Threshold at Which Shapes Give Way

By Mordechai Rorvig
June 3, 2021
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A new proof establishes the boundary at which a shape becomes so corrugated, it can be crushed.

An illustration showing pink and purple particles flowing around a geometric computer chip.
artificial intelligence

Latest Neural Nets Solve World’s Hardest Equations Faster Than Ever Before

By Anil Ananthaswamy
April 19, 2021
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Two new approaches allow deep neural networks to solve entire families of partial differential equations, making it easier to model complicated systems and to do so orders of magnitude faster.

An abstract black ball on a blue background.
Abstractions blog

In Violation of Einstein, Black Holes Might Have ‘Hair’

By Jonathan O'Callaghan
February 11, 2021
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A new study shows that extreme black holes could break the famous “no-hair” theorem, and in a way that we could detect.

Illustration of a robot translating calculus into branching, tree-like structures
artificial intelligence

Symbolic Mathematics Finally Yields to Neural Networks

By Stephen Ornes
May 20, 2020
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After translating some of math’s complicated equations, researchers have created an AI system that they hope will answer even bigger questions.

Animation of black hole formation.
mathematical physics

New Math Proves That a Special Kind of Space-Time Is Unstable

By Steve Nadis
May 11, 2020
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Einstein’s equations describe three canonical configurations of space-time. Now one of these three — important in the study of quantum gravity — has been shown to be inherently unstable.

Photo of black and white swirls of mixing paint
fluid dynamics

Mathematicians Prove Universal Law of Turbulence

By Kevin Hartnett
February 4, 2020
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By exploiting randomness, three mathematicians have proved an elegant law that underlies the chaotic motion of turbulent systems.

Abstractions blog

For Fluid Equations, a Steady Flow of Progress

By Kevin Hartnett
January 13, 2020
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A startling experimental discovery about how fluids behave started a wave of important mathematical proofs.


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