Extended Graph of the Fuzzy Topographic Topological Mapping Model
Fuzzy topological topographic mapping (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mi>T</mi><mi>T</mi><mi>M</mi></mrow></semantics>...
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| Autores principales: | , , , , |
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| Formato: | article |
| Lenguaje: | EN |
| Publicado: |
MDPI AG
2021
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| Materias: | |
| Acceso en línea: | https://doaj.org/article/377aca7a54df40908229f39545c39ef8 |
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| Sumario: | Fuzzy topological topographic mapping (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mi>T</mi><mi>T</mi><mi>M</mi></mrow></semantics></math></inline-formula>) is a mathematical model which consists of a set of homeomorphic topological spaces designed to solve the neuro magnetic inverse problem. A sequence of <i>FTTM</i>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mi>T</mi><mi>T</mi><msub><mi>M</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula>, is an extension of <i>FTTM</i> that is arranged in a symmetrical form. The special characteristic of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mi>T</mi><mi>T</mi><mi>M</mi></mrow></semantics></math></inline-formula>, namely the homeomorphisms between its components, allows the generation of new <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mi>T</mi><mi>T</mi><mi>M</mi></mrow></semantics></math></inline-formula>. The generated <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mi>T</mi><mi>T</mi><mi>M</mi></mrow></semantics></math></inline-formula>s can be represented as pseudo graphs. A graph of pseudo degree zero is a special type of graph where each of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mi>T</mi><mi>T</mi><mi>M</mi></mrow></semantics></math></inline-formula> components differs from the one adjacent to it. Previous researchers have investigated and conjectured the number of generated <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mi>T</mi><mi>T</mi><mi>M</mi></mrow></semantics></math></inline-formula> pseudo degree zero with respect to <i>n</i> number of components and <i>k</i> 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. |
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