Concrete Mathematics: Fractals, Dynamics and Distributions

START project Y 96 Concrete Mathematics: Fractals, Dynamics and Distributions Peter GRABNER 19.06.1998 The proposed project covers several areas of research, which are connected by their methodology and their motivation from mathematical physics: diffusion on fractals, point distribution on spheres and digital sequences. It is a major aim of this project to establish new interrelations between these areas. Brownian motion on fractals is a rather new area of research, which developed rapidly in the last years. Initiated by recent results of the proposer the project intends to find more explicit information on parameters of diffusion processes on p.c.f. self-similar fractals. Later this shall be extended to non-p.c.f. fractals, which require a totally different technical approach. The Laplace operator as the infinitesimal generator of the diffusion and its eigenfunctions and eigenvalues shall be studied. Especially, the interrelation to digital functions shall be investigated. Furthermore, numerical simulations of the diffusion process and the heat kernel shall be performed. Discrete point distributions have been studied intensively from different points of view since the end of the last century. The motivations for these studies come from geometry, potential theory, numerical mathematics, and many other fields. In this project different methods for distributing points on spheres shall be investigated: number theoretical constructions based on modular forms, minimal energy concepts, and spherical designs. Several concepts of discrepancy shall be studied for these constructions and shall be compared in numerical experiments. These notions of discrepancy shall be compared also with respect to their applications to numerical integration. The study of digital sequences has its origins in number theory and ergodic theory. Such sequences have found applications in the construction of well-distributed point sets, the description of fractal sets, and even mathematical physics. Here it could be applied to various discrete models in: the Ising-model for magnetism, discrete Schrödinger equations, and quasi-crystals. Digital description of fractals mirror the self-similar structure and give new possibilities of explaining periodicity phenomena encountered in the study of diffusions on fractal state spaces. Several problems, which remained open in the study of dynamical properties of digital expansions, shall be attacked in this project. Influence of the proposed work on the development of the field The point of view of concrete and explicit mathematics gives new and more applicable insights into the areas covered by the project. The major aim is to make as much information explicit as possible. - A more explicit knowledge of the behaviour of Brownian motion on fractals is important for the application of this theory to mathematical physics, and also is of interest on its own. - Explicit constructions of well-distributed point sets on the sphere have applications to numerical analysis; explicit error bound for numerical integration will be a result of a thorough investigation of the distribution properties of these point sets. - Digital constructions will be one of the tools used in the study of Brownian motion on fractals and its explicit description. Furthermore, the study of new dynamical systems related to various digital expansions will yield further insight into this subject. In his previous work the proposer has shown that the methods of concrete mathematics have their applications to many more areas of research than combinatorics and discrete mathematics. This project intends to contribute to the popularization of concrete mathematics and to present the power of its methods in different areas of mathematics.

The proposed project covers several areas of research, which are connected by their methodology and their motivation from mathematical physics: diffusion on fractals, point distribution on spheres and digital sequences. It is a major aim of this project to establish new interrelations between these areas. Brownian motion on fractals is a rather new area of research, which developed rapidly in the last years. Initiated by recent results of the proposer the project intends to find more explicit information on parameters of diffusion processes on p.c.f. self-similar fractals. Later this shall be extended to non-p.c.f. fractals, which require a totally different technical approach. The Laplace operator as the infinitesimal generator of the diffusion and its eigenfunctions and eigenvalues shall be studied. Especially, the interrelation to digital functions shall be investigated. Furthermore, numerical simulations of the diffusion process and the heat kernel shall be performed. Discrete point distributions have been studied intensively from different points of view since the end of the last century. The motivations for these studies come from geometry, potential theory, numerical mathematics, and many other fields. In this project different methods for distributing points on spheres shall be investigated: number theoretical constructions based on modular forms, minimal energy concepts, and spherical designs. Several concepts of discrepancy shall be studied for these constructions and shall be compared in numerical experiments. These notions of discrepancy shall be compared also with respect to their applications to numerical integration. The study of digital sequences has its origins in number theory and ergodic theory. Such sequences have found applications in the construction of well-distributed point sets, the description of fractal sets, and even mathematical physics. Here it could be applied to various discrete models in: the Ising-model for magnetism, discrete Schrödinger equations, and quasi-crystals. Digital description of fractals mirror the self-similar structure and give new possibilities of explaining periodicity phenomena encountered in the study of diffusions on fractal state spaces. Several problems, which remained open in the study of dynamical properties of digital expansions, shall be attacked in this project. Influence of the proposed work on the development of the field The point of view of concrete and explicit mathematics gives new and more applicable insights into the areas covered by the project. The major aim is to make as much information explicit as possible. - A more explicit knowledge of the behaviour of Brownian motion on fractals is important for the application of this theory to mathematical physics, and also is of interest on its own. - Explicit constructions of well-distributed point sets on the sphere have applications to numerical analysis; explicit error bound for numerical integration will be a result of a thorough investigation of the distribution properties of these point sets. - Digital constructions will be one of the tools used in the study of Brownian motion on fractals and its explicit description. Furthermore, the study of new dynamical systems related to various digital expansions will yield further insight into this subject. In his previous work the proposer has shown that the methods of concrete mathematics have their applications to many more areas of research than combinatorics and discrete mathematics. This project intends to contribute to the popularization of concrete mathematics and to present the power of its methods in different areas of mathematics.