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partial differential equations
Computer Proof ‘Blows Up’ Centuries-Old Fluid Equations
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.
Mathematicians Prove Melting Ice Stays Smooth
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.
Mathematicians Identify Threshold at Which Shapes Give Way
A new proof establishes the boundary at which a shape becomes so corrugated, it can be crushed.
Latest Neural Nets Solve World’s Hardest Equations Faster Than Ever Before
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.
In Violation of Einstein, Black Holes Might Have ‘Hair’
A new study shows that extreme black holes could break the famous “no-hair” theorem, and in a way that we could detect.
Symbolic Mathematics Finally Yields to Neural Networks
After translating some of math’s complicated equations, researchers have created an AI system that they hope will answer even bigger questions.
New Math Proves That a Special Kind of Space-Time Is Unstable
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.
Mathematicians Prove Universal Law of Turbulence
By exploiting randomness, three mathematicians have proved an elegant law that underlies the chaotic motion of turbulent systems.
For Fluid Equations, a Steady Flow of Progress
A startling experimental discovery about how fluids behave started a wave of important mathematical proofs.