Algebraic Boundary Problems and Integral Transforms

Many of the problems usually considered in science and engineering but also in economics and actuarial mathematics find their adequate expression in the language of so-called boundary problems. One translates the process of interest into a differential equation along with additional boundary conditions; the former typically describes a law of nature or some model relation while the latter corresponds to measurements or adjustments that identify the process uniquely. There is a vast array of powerful numerical methods for solving boundary problems. The numerical treatment necessarily involves approximations and fixing a single numerical instantiation out of an infinite manifold of possible ones. For investigating solutions algebraically like studying dependence on parameters or possible decomposition schemes (so-called factorizations) or a suitable transformation to a simpler domain one has to represent the problem as well as the solution in an algebraic manner, even if this needs simplification of the model (for example linearization, which in practical terms usually means limitation to small regions or oscillations). In this project we would like to propose such algebraic methods that will allow us to directly represent, decompose and analyze the solution operators. Here these methods should be understood from a practical perspective, meaning as accessible computer programs that we shall provide in the frame of a suitable computer algebra package. In the long run we have the vision of an integrated workbench that will bring together numerical and algebraic methods with modern visualization techniques. There will be two concrete applications that we want to study in this project: (1) A key problem in actuarial mathematics concerns the risk of accrued premium capital in the view of incoming insurance claims from the clients. For measuring the probability of ruin and other crucial stochastic parameters of the risk process one utilizes the so-called Gerber-Shiu function, which in turn can be characterized by a boundary problem. While we have investigated the latter with algebraic operator methods in a simplified model, leading to the great satisfaction of actuarial mathematicians, we hope to incorporate tax payments in a more accurate model that we will subject to the new methods of his project. (2) A typical problem in engineering mechanics consists in determining the distribution of stress in Kirchhoff plates, a certain model of elastic strain often applicable in technical materials. We have treated such a case under simplifying symmetry assumptions but would like to cover the full 2D model case with the tools to be developed in this project.

Modeling technical and economic processes typically leads to differential equations, which are investigated and solved by a variety of mathematical tools - some focused on various structural relations (especially symbolic-algebraic methods), some on actual numbers in the models (mainly numerical-analytic methods). Naturally, the most effective strategy is to combine the power of both approaches. While this has been achieved in some areas, one of the most vital tools for tackling linear differential equations - an important subclass - is the Fourier transform, but it has been employed almost exclusively in the numerical-analytic realm. We have set up the first symbolic-algebraic theory of Fourier transforms, as a bedrock for future developments not only in linear differential equations but also in other mathematical branches such as signal processing, cryptography, econometry, number theory, statistics and in physical application areas such as acoustics, optics, astrophysics. We have set up the new theory in such a way that it kills several birds with one stone. Founding it on a suitable generic mathematical framework (usually referred to as Pontryagin duality), we can treat all the important variations of Fourier operators occurring in practical work, in particular Fourier integrals, Fourier series, and discrete Fourier transforms. While Fourier series are used for treating periodic signals (such as sound waves in music or light waves in optics), one has to resort to Fourier integrals for modeling aperiodic phenomena (such as the instantaneous temperature profile of a blast furnace). In the digital world, both are replaced by the discrete Fourier transforms, which represent signals by finitely many values. In either case, the same algebraic theory can be used for analyzing the structure of the signal space and the hierarchy of its possible extensions. This is especially important for clarifying which Fourier transforms can be written in closed form and thus are represented within the chosen signal space. As an interesting spin-off, we have also found new results in a related branch of algebra, the so-called cohomology theory of nilpotent groups, where research is focused on identifying the basic building blocks of various algebraic structures such as Heisenberg groups. One of the important applications of the research in this project - and this was our original motivation - is to represent the solution operators (named Green's operators in the application domain) in terms of Fourier transforms. Using the algebraic framework set up in this project, it is now possible to build up the algebraic texture needed for this purpose (an operator ring or an operator quiver, as it is called in the jargon). This would be a worthwhile subject for future research work and applications.