Boundary value problems, medical imaging and the asymptotics of Riemann's zeta function

摘要

For many years, the use of the Wiener-Hopf technique in acoustics and other physical problems was the only manifestation of an application of the Riemann-Hilbert formalism. However, in the last 50 years, this formalism and its natural generalization called the ?-bar formalism have appeared in a large number of problems in mathematics and mathematical physics. In this paper, the impact of these formalisms in three separate areas is reviewed: first, the development of a novel, hybrid numerical-analytical method for solving boundary value problems for linear and integrable nonlinear PDEs, known as the unified transform or the Fokas method (www. wikipedia.org/wiki/Fokas_method). Second, the introduction of a new algorithm in nuclear medical imaging called the attenuated spline reconstruction technique (aSRT). Third, a novel approach to the Lindel?f hypothesis, which is a close relative of the Riemann hypothesis.