Index Matrices: Towards an Augmented Matrix Calculus by Krassimir T. Atanassov PDF

By Krassimir T. Atanassov

ISBN-10: 3319109448

ISBN-13: 9783319109442

ISBN-10: 3319109456

ISBN-13: 9783319109459

This ebook offers the very proposal of an index matrix and its comparable augmented matrix calculus in a finished shape. It more often than not illustrates the exposition with examples relating to the generalized nets and intuitionistic fuzzy units that are examples of an incredibly large choice of attainable program parts. the current e-book includes the elemental result of the writer over index matrices and a few of its open issues of the purpose to stimulating extra researchers to begin operating during this area.

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Additional resources for Index Matrices: Towards an Augmented Matrix Calculus

Example text

Let X be a fixed set of some objects. In the particular cases, they can be either real numbers, or only the numbers 0 or 1, or logical variables, propositions or predicates, etc. Let operations ◦, ∗ : X × X → X be fixed. An Extended IM (EIM) with index sets K and L (K , L ⊂ I ∗ ) and elements from set X is called the object (see, [18]): l1 k1 ak1 ,l1 .. . [K , L , {aki ,l j }] ≡ ki aki ,l1 .. . km akm ,l1 . . l j . . ln . . ak1 ,l j . . ak1 ,ln . . . . . , . . aki ,l j . . aki ,ln .

K1 ,ln , νk1 ,ln .. ... . μki ,ln , νki ,ln .. ... , km μkm ,l1 , νkm ,l1 . . μkm ,l j , νkm ,l j . . μkm ,ln , νkm ,ln where for every 1 ≤ i ≤ m, 1 ≤ j ≤ n: 0 ≤ μki ,l j , νki ,l j , μki ,l j + νki ,l j ≤ 1. For brevity, we can mention the above object by [K , L , { μki ,l j , νki ,l j }], where K = {k1 , k2 , . . , km }, L = {l1 , l2 , . . , ln }, for 1 ≤ i ≤ m, and 1 ≤ j ≤ n: μki ,l j , νki ,l j , μki ,l j + νki ,l j ∈ [0, 1]. 1 (continued) 2b 3−2b ¬41 3 , 3 b+λ−1 a+λ ¬42,λ 2λ , 2λ , where λ ≥ 1 b+γ a+γ ¬43,γ 2γ +1 , 2γ +1 , where γ ≥ 1 ¬44,α,β ¬45,ε,η b+α−1 a+β α+β , α+β , where α ≥ 1, β ∈ [0, α] min(1, ν A (x) + ε), max(0, μ A (x) − η) Now, for above sets K and L, the EIFIM is defined by: [K ∗ , L ∗ , { μki ,l j , νki ,l j }] α1k , β1k k1 , ..

5 Level Operators Over EIFIMs Let the EIFIM A = [K ∗ , L ∗ , { μki ,l j , νki ,l j }] be given. Let for i = 1, 2, 3 : ρi , σi , ρi + σi ∈ [0, 1] be fixed numbers. In [7,13], several level operators are defined. One of them, for a given IFS X = { x, μ X (x), ν X (x) |x ∈ E} is Nα,β (X ) = { x, μ X (x), ν X (x) |x ∈ E & μ X (x) ≥ α & ν X (x) ≤ β}, where α, β ∈ [0, 1] are fixed and α + β ≤ 1. Here, its analogues are introduced. They are three: Nρ11 ,σ1 , Nρ22 ,σ2 , Nρ33 ,σ3 and affect the K -, L-indices and μki ,l j , νki ,l j -elements, respectively.

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Index Matrices: Towards an Augmented Matrix Calculus by Krassimir T. Atanassov


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