The Perona-Malik equation is a celebrated example of nonlinear forward-backward diffusion, introduced in the context of image denoising. It can be viewed as the formal gradient-flow of a functional with a convex-concave integrand. In spite of its mathematical ill-posedness, numerical experiments exhibit better than expected behavior of its solutions.
After a general introduction to the equation itself, we present a few approximation schemes, some classical and some more recent. The approximating solutions show distinct behavior on three time scales (called fast, standard, and slow time). We provide a rigorous explanation for the slow time behavior of the different approximations.
In order to carry out this analysis, we prove an abstract result about passing to the limit in gradient-flows (in the more general context of the theory of maximal slope curves in metric spaces). We are guided by the general principle that "the limit of a family of gradient-flows is the gradient-flow of the limiting functional".

Abstract: The first personal computing revolution took place not in Silicon Valley in the 1980s but in Pisa in the 13th Century. The medieval counterpart to Steve Jobs was a young Italian called Leonardo, better known today by the nickname Fibonacci. Thanks to a recently discovered manuscript in a library in Florence, the story of how this little known genius came to launch the modern commercial world can now be told.

Based on Devlin’s latest book The Man of Numbers: Fibonacci’s Arithmetical Revolution (Walker & Co, July 2011) and his co-published companion e-book Leonardo and Steve: The Young Genius Who Beat Apple to Market by 800 Years.

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We consider the time for extinction for a contact process on a tree of bounded degree as the number of vertices tends to infinity. We show that
uniformly over all such trees the extinction time tends to infinity as the
exponential of the number of vertices if the infection parameter is strictly above the critical value for the one dimensional contact process.
An application to the contact process on NSW graphs is given.