# Allard W., Almgren J.Frederick Geometric Measure Theory and by William K. Allard, Frederick J., Jr. Almgren

By William K. Allard, Frederick J., Jr. Almgren

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**Extra info for Allard W., Almgren J.Frederick Geometric Measure Theory and the Calculus of Variations**

**Sample text**

Nevertheless, in practice, real number indices are needed for fuzzy numbers ordering. For this purpose, some characteristic numbers of a fuzzy set [18] could be used. It seems, however, more natural to substitute the obtained discrete set I of α-levels with a real number: ˜ = P( B˜ > A) αP α ( B˜ α > A˜ α )/ α∈I α. 4) emphasises that the contribution of the α-level to the overall probability estimation is increasing with an increase in its number. 4) instead of α. 1 Let the following four triangular fuzzy numbers, depicted in Fig.

The total dominance of A˜ i over A˜ j with the index of optimism β ∈ [0, 1] is defined as follows. 42) The index of optimism is used to reflect a decision maker’s degree of optimism [28]. The total dominance index is used to define the rules of comparison of two fuzzy numbers. They are the following: If DiTj < 0 Then If DiTj > 0 Then If DiTj = 0 Then A˜ i < A˜ j , A˜ i > A˜ j , A˜ i = A˜ j . 1. 8. This gives the ordering A˜ 3 < A˜ 1 < A˜ 2 < A˜ 4 , which exactly the same as the one obtained using the previous methods.

It is also time consuming and requires to use additional tools, such as numerical integration and solving systems of nonlinear equations. It also requires the pairwise comparison of fuzzy numbers to be ordered. 37) They are used to determine the comparison rules: ˜ A) ˜ > R( A, ˜ B) ˜ iff R( B, ˜ ˜ ˜ ˜ iff R( B, A) = R( A, B) A˜ < A˜ ≈ ˜ B, ˜ B. 38) can be written in the following form: ˜ B) ˜ > R( A, ˜ B) ˜ iff −R( A, ˜ ˜ ˜ ˜ iff −R( A, B) = R( A, B) A˜ < A˜ = ˜ B, ˜ B. 1. 7. This gives exactly the same order as the one obtained using the methods described so far.