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Discretised wave equations

The wave equation is usually expressed in the form

where is a function of both time and space , with being the Laplacian operator. One can generalise this equation in a number of ways, for instance by replacing the spatial domain with some other manifold and replacing the Laplacian with the Laplace-Beltrami operator or adding lower order terms (such as a potential, or a coupling with a magnetic field). 1,149 more words


A Banach algebra proof of the prime number theorem

The prime number theorem can be expressed as the assertion

as , where

is the von Mangoldt function. It is a basic result in analytic number theory, but requires a bit of effort to prove. 1,967 more words