Generalized fuzzy topographic topological mapping

Abstract

Fuzzy Topographic Topological Mapping (FTTM) is a model for solving neuromagnetic inverse problem. FTTM consists of four topological spaces which are connected by three homeomorphisms. FTTM 1 and FTTM 2 were designed to present 3-D view of an unbounded single current and bounded multicurrent sources, respectively. It has been showed that FTTM 1 and FTTM 2 are homeomorphic and this homeomorphism will generate another 14 FTTM. There is a conjecture stated that if there exist n numbers of FTTM, then they will generate another n4 ? n new FTTM. In this thesis, the conjecture is proven by using geometrical features of FTTM. In the process, several definitions such as sequence of FTTM, sequence of polygon, sequence of cube with combination of two, three and four FTTM are developed. Some geometrical and algebraic properties of sequences of FTTM are identified and proven. A new conjecture is also proposed in this thesis which states that the number of generating Fkn if there are k components and n models of Fk is nk ? n . Surprisingly, the nonzero sequence of cube with combination of two, three and four FTTM appeared in Pascal’s Triangle.