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Navier-Stokes 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 ring of bubbles in blue water.
fluid dynamics

Mathematicians Coax Fluid Equations Into Nonphysical Solutions

By Leila Sloman
May 2, 2022
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The famed Navier-Stokes equations can lead to cases where more than one result is possible, but only in an extremely narrow set of situations.

Simulated water particles swirl in an upward vortex.
mathematical physics

Deep Learning Poised to ‘Blow Up’ Famed Fluid Equations

By Jordana Cepelewicz
April 12, 2022
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For centuries, mathematicians have tried to prove that Euler’s fluid equations can produce nonsensical answers. A new approach to machine learning has researchers betting that “blowup” is near.

Illustration of a figure working on a laptop surrounded by flasks and liquids evaporating into discrete shapes in the air.
Quantized Columns

The Uselessness of Useful Knowledge

By Robbert Dijkgraaf
October 20, 2021
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Today’s powerful but little-understood artificial intelligence breakthroughs echo past examples of unexpected scientific progress.

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.

Multimedia

The Map of Mathematics

By Kevin Hartnett
February 13, 2020
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Explore our surprisingly simple, absurdly ambitious and necessarily incomplete guide to the boundless mathematical universe.

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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