Graph of pseudo degree zero generated by sequence of fuzzy topographic topological mapping

Abstract

Fuzzy topological topographic mapping (FTTM) is a set of topological spaces and algorithms used in solving neuro magnetic inverse problem. The original FTTM consists of four topological spaces and three algorithms. Previous study has led to a new version, namely FTTM 2 which has been used to transform magnetic data to image data. It has been known that FTTMs have special properties, namely the homeomorphism between its components as well as homeomorphism between components of different FTTM versions. Through this property new FTTM can be generated by combining FTTM components of different versions which can be depicted in a form of graphs. A special type of FTTM graph called FTTM graph of pseudo degree zero can be obtained by generating an FTTM where each adjacent components belong to different versions. Furthermore, FTTM can be generalized to include n number of components and k number of versions in a sequence of *FTTM????. Investigation by previous research has yielded a conjecture on *FTTM??3, named the Elsafi’s Conjecture. This study focuses on the number of FTTM graph of pseudo degree zero generated from the sequence *FTTM????, specifically by *FTTM??3. This study proves the Elsafi’s Conjecture analytically and it generalizes the conjecture to include k number of components. A simulation is developed to determine the number of FTTM graph of pseudo degree zero. The algorithm is implemented using the C++ programming language and further improved by parallelizing the algorithm. This implementation yields two new conjectures. Further, a novel concept to depict FTTM namely FTTM grid is introduced for sequence of FTTM and FTTM path. Using this newly concept, the Elsafi’s Conjecture is finally.