Abstract
Fuzzy topological topographic mapping (FTTM) is a mathematical model which consists of a set of homeomorphic topological spaces designed to solve the neuro magnetic inverse problem. A sequence of FTTM, FTTMn, is an extension of FTTM that is arranged in a symmetrical form. The special characteristic of FTTM, namely the homeomorphisms between its components, allows the generation of new FTTM. The generated FTTMs can be represented as pseudo graphs. A graph of pseudo degree zero is a special type of graph where each of the FTTM components differs from the one adjacent to it. Previous researchers have investigated and conjectured the number of generated FTTM pseudo degree zero with respect to n number of components and k number of versions. In this paper, the conjecture is proven analytically for the first time using a newly developed grid-based method. Some definitions and properties of the novel grid-based method are introduced and developed along the way. The developed definitions and properties of the method are then assembled to prove the conjecture. The grid-based technique is simple yet offers some visualization features of the conjecture.
Highlights
Introduction assembled to prove the conjectureThe grid-based technique is simple yet offers some visualizationFuzzy topographic topological mapping (FTTM) [1] was introduced to s features of the conjecture.neuro magnetic inverse problem, with regards to the sources of el Keywords: Fuzzy topological topographic mapping (FTTM); graph; pseudo degree; sequence from epileptic patients
[1] was introduced to solve the neuro formally as follows (see magnetic inverse problem, with regards to the sources of electroencephalography (EEG) signals recorded from epileptic patients
∗ FTTMnk is an extended generalization of FTTM that is represented polygon is aarranged back to frontwith where first polygon represen by a graph of sequence offrom k number of polygons n sidesthe or vertices
Summary
≅ Aas well, denoted as FTTMnk. Without the loss of generality, the collection of the k version of FTTM, is called as a sequence of FTTM unless. A new FTTM can be generated from a combination of components from different versions of FTTM due to∗ their homeomorphisms. Theorem 1 is extended to include n number of FTTM components. [2] The number of generated FTTM that can be created from ∗ FTTMnk is is presented. ∗ FTTMnk is an extended generalization of FTTM that is represented polygon is aarranged back to frontwith where first polygon represen by a graph of sequence offrom k number of polygons n sidesthe or vertices. An FTTM edge nis added polygon represents FTTMn2 and so forth.
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