By Prof. Dr. Jacques Octave Dubois, Prof. Dr. Alexei Gvishiani (auth.)

The ebook comprises major components dealing either with nonlinear techniques within the examine of complicated usual structures. In its first half it introduces a brand new building of development attractiveness difficulties appropriate to quite a lot of geophysical and environmental items. The type in such difficulties depends upon time which permits one to formulate the situation of balance of ultimate classifications for verification of its reliability. a brand new vast set of development popularity algorithms with studying ("Voting by way of Set of Features") is brought. Theoretical and algorithmical elements are illustrated through a few examples of purposes to normal threat assessment.

In the second one a part of the publication, an alternate method of the geophysical functions is given by way of dynamic structures and corresponding instruments. Dynamic structures reviews are worthy to appreciate nonlinear time sequence. Self geared up Criticality and multifractal research are strong new keys for realizing many typical phenomena. most of these equipment are of significant curiosity in any lengthy information dossier processing.

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Let us show that the algorithms VEF-O, VEF-1 and VEF-2 are included in the class VSF for some values of the weights, N = n, and U = Ur = {Y 1 , ... ) YN}, where Y i = {w: w(i) = O}, i = 1, ... ) n. 23) i=l where we introduce the following symbols: a+ = (I a1 I, ... , 1an I); sgn(a) = (sgn(ad, ... , sgn(an )) , with, sgn(x) = 1, if X 2: 0; sgn(x) = 0, if x

By the intersection of any (finite or infinite) number of sets A", we call the set A", of the elements that belong to each of the sets A",. g. 4) The equality A = B means that A c Band B c A. g. x E (AUB) nCo Therefore x belongs to C and, in addition, at least to one of the sets An C or B n C. 3). On the other hand let x E (AnC) U(B nC). g. x E AUB. 3) is proved. 4) Another operation in the set theory is substraction. 3). Here, generally speaking, we do not assume that A::::> B. Sometimes the symbolization A - 6 is used for the substraction.

Therefore f(x) E AUB and x E f- 1(A)U(B). 2 The co-image of the intersection of two sets is equal to the intersection of their co-images. f-1(AnB) = f-1(A) Uf- 1(B). Proof If x E f- 1(A)n(B) then f(x) E f(x) E B. Therefore x E f-1(A) and x x E f-1(A) n f-1(B). g Therefore f(x) E A and f(x) E B or f(x) x E f-1(AnB). g. E f-1(B). f(x) E A and In other words x E f-1(A) and x E f-1(B). E AnB. 3 The image of the union of two sets is equal to the union of their images J(AUB) = J(A)UJ(B). ProoJIf Y E J(AUB) then y = J(x), where x belongs to at least one of the sets A or E.