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Index Matrices: Towards an Augmented Matrix Calculus

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"Index Matrices: Towards an Augmented Matrix Calculus" (ISBN 978-3-319-10945-9) is a monograph by Krassimir Atanassov, published in 2014 in the Springer series "Studies in Computational Intelligence", Vol. 573. This is the first monograph that presents the very concept of an index matrix and its related augmented matrix calculus in a comprehensive form.

It mostly illustrates the exposition with examples related to the generalized nets and intuitionistic fuzzy sets which are examples of an extremely wide array of possible application areas. The present book contains the basic results of the author over index matrices and some of its open problems with the aim to stimulating more researchers to start working in this area.

Book contents

  1. Index Matrices: Definitions, Operations, Relations
    1. Definitions of an Index Matrix with Real Number Elements and (0; 1)-index Matrix
    2. Operations Over R-IMs and (0; 1)-IMs
    3. Relations Over R-IM and (0; 1)-IM
    4. Index Matrices with Elements Logical Variables, Propositions or Predicates
    5. Relations Over L-IMs
    6. Operations “Reduction” Over an R-IM, (0; 1)-IM and L-IM
    7. Operation “Projection” Over an R-IM, (0; 1)-IM and L-IM
    8. Operation “Substitution” Over an R-IM, (0; 1)-IM and L-IM
    9. An Example from Number Theory
    10. An Example from Graph Theory
  2. Intuitionistic Fuzzy IMs
    1. Short Remarks on Intuitionistic Fuzziness
    2. IFIMs and EIFIMs
    3. Standard Operations Over EIFIMs
    4. Relations Over EIFIMs
    5. Level Operators Over EIFIMs
    6. Aggregation Operations Over EIFIMs
    7. Extended Modal Operators Defined Over EIFIMs
    8. An Example with Intuitionistic Fuzzy Graphs
  3. Extended Index Matrices
    1. Definition of an Extended Index Matrix
    2. Operations Over EIMs
    3. Relations Over EIMs
    4. Hierarchical Operators Over EIMs
    5. New Operations Over EIMs
    6. EIMs, Determinants and Permanents
    7. Transposed EIM
    8. An Example: An Intercriteria Decision Making Method
  4. Temporal IFIMs
    1. Operations Over ETIFIMs
    2. Relations Over ETIFIMs
    3. Specific Operations Over ETIFIMs
    4. An Example: Temporal Intuitionistic Fuzzy Cognitive Map
  5. Index Matrices with Function-Type of Elements
    1. Definition of the Index Matrix with Function-Type of Elements
    2. Standard Operations Over IMFEs
    3. Relations Over IMFEs
    4. Operations Over IMFEs and IMs
    5. An Example: IM-Interpretation of a Multilayer Perceptron
  6. Three Dimensional IMs
    1. Definition of a Three Dimensional IMs
    2. Operations Over 3D-IMs
    3. Relations Over 3D-IMs
    4. Operations “Reduction” Over an 3D-IM
    5. Operation “Projection” Over an IM
    6. Operation “Substitution” Over an 3D-IM
    7. An Example with Bookshops

External links