External direct sum of fuzzy topographic topological mapping

Abstract

Fuzzy Topographic Topological Mapping (FTTM) is a novel mathematical model for solving neuromagnetic inverse problem. At this moment, Fuzzy Topographic Topological Mapping 1 (FTTM 1), which is used to solve the inverse problem for determining single current source and Fuzzy Topographic Topological Mapping 2 (FTTM 2), which is used to solve the inverse problem for determining multiple current sources have been developed. The purpose of this research is to construct the external direct sum of both FTTM 1 and FTTM 2. Firstly, the external direct sum of FTTM 1 was established then followed by establishment of external direct sum of FTTM 2. Next, the other purpose of this research is to construct linear transformations that map every component in FTTM 1 to a corresponding component of FTTM 2. In the process, several definitions and theorems were adopted in particular and the theorem of linear transformation with proving technique by construction was highlighted. Besides, the first isomorphism theorem for FTTM was built. Finally, these structures were analyzed in order to study the information gathered the inverse problem for determining single and multiple current sources