A brief historical perspective of the Wiener-Hopf technique

It is a little over 75 years since two of the most important mathematicians of the 20th century collaborated on finding the exact solution of a particular equation with semi-infinite convolution type integral operator. The elegance and analytical sophistication of the method, now called the Wiener–Hopf technique, impress all who use it. Its applicability to almost all branches of engineering, mathematical physics and applied mathematics is borne out by the many thousands of papers published on the subject since its conception. The Wiener–Hopf technique remains an extremely important tool for modern scientists, and the areas of application continue to broaden. This special issue of the Journal of Engineering Mathematics is dedicated to the work of Wiener and Hopf, and includes a number of articles which demonstrate the relevance of the technique to a representative range of model problems

Numerical solution of scattering problems using a Riemann--Hilbert formulation

A fast and accurate numerical method for the solution of scalar and matrix Wiener--Hopf problems is presented. The Wiener--Hopf problems are formulated as Riemann--Hilbert problems on the real line, and a numerical approach developed for these problems is used. It is shown that the known far-field behaviour of the solutions can be exploited to construct numerical schemes providing spectrally accurate results. A number of scalar and matrix Wiener--Hopf problems that generalize the classical Sommerfeld problem of diffraction of plane waves by a semi-infinite plane are solved using the approach

Scattering by a semi-infinite lattice and the excitation of Bloch waves

The interaction of a time-harmonic plane wave with a semi-infinite lattice of identical circular cylinders is considered. No assumptions about the radius of the cylinders, or their scattering properties, are made. Multipole expansions and Graf’s addition theorem are used to reduce the boundary value problem to an infinite linear system of equations. Applying the z transform and disregarding interaction effects due to certain strongly damped modes then leads to a matrix Wiener–Hopf equation with rational elements. This is solved by a straightforward method that does not require matrix factorisation. Implementation of the method requires that the zeros of the matrix determinant be located numerically, and once this is achieved, all far field quantities can be calculated. Numerical results that show the proportion of energy reflected back from the edge are presented for several different lattice geometries. 1

Mathematically modelling the deformation of frictional elastic half-spaces in contact with a rolling rigid cylinder

In this thesis we derive an analytical model of the deformation of an elastic half-space caused by a cylindrical roller. The roller is considered rigid, and is forced into the half-space and rolls across its surface, with contact modelled by Coulomb friction. In general, portions of the surface of the roller in contact with the half-space may slip across the surface of the half-space, or may stick to it. In this thesis, we consider the contact surface to have a central sticking region as well as a simplifying regime where the entire contact surface is fully slipping. This results in two mixed boundary value problem, which are formulated into a 4_4 matrix Wiener{Hopf problem for the stick-slip regime and a 2_2 matrix Wiener{Hopf problem for the full-slip regime. The exponential factors in the Wiener{Hopf matrix allows a solution by following the iterative method of Priddin, Kisil, and Ayton (Phil. Trans. Roy. Soc. A 378, p. 20190241, 2020) which is implemented numerically by computing Cauchy transforms using a spectral method following Slevinsky and Olver (J. Comput. Phys. 332, pp. 290{315, 2017). The limits of the contact region and stick-slip transitions are located a posteriori by applying an free-boundary method based on the secant method. The solution is illustrated with several examples, and the frictional regimes are analysed

From Physics to Number Theory via Noncommutative Geometry, Part II: Renormalization, the Riemann-Hilbert correspondence, and motivic Galois theory

We establish a precise relation between Galois theory in its motivic form with the mathematical theory of perturbative renormalization (in the minimal subtraction scheme with dimensional regularization). We identify, through a Riemann-Hilbert correspondence based on the Birkhoff decomposition and the t'Hooft relations, a universal symmetry group (the "cosmic Galois group" suggested by Cartier), which contains the renormalization group and acts on the set of physical theories. This group is closely related to motivic Galois theory. We construct a universal singular frame of geometric nature, in which all divergences disappear. The paper includes a detailed overview of the work of Connes-Kreimer and background material on the main quantum field theoretic and algebro-geometric notions involved. We give a complete account of our results announced in math.NT/0409306.Comment: 97 pages LaTeX, 17 eps figure