PSI - Issue 40

A.G. Chentsov et al. / Procedia Structural Integrity 40 (2022) 105–111 Chentsov A.G., Chentsov P.A./ Structural Integrity Procedia 00 (2022) 000 – 000

107

3

In connection with (3), we introduce two unrestrictive conditions. But, at first, we agree about the following designations; if h is an ordered pair (see[13, c.67]), then by pr 1 ( h ) and pr 2 ( h ) we denote the first and the second elements of h ; h = ( pr 1 ( h ) , pr 2 ( h )) . By (3) elements of K 1 and K 2 are address ordered pair. So, for ℳ 1 -problem, we require that ∀ 0 , 0 ⊂ K 1 , 0 ≠ ∅, ∃ z 0 ∈ K 0 : pr 1 (z 0 ) ≠ pr 2 (z) ∀ z ∈ K 0 . Analogously, for ℳ 2 -problem, we suppose that ∀ K 0 ǡ 0 ⊂ K 2 , 0 ≠ ∅, ∃ z 0 ∈ K 0 : pr 1 (z 0 ) ≠ pr 2 (z) ∀ z ∈ K 0 . (two above-mentioned conditions are fulfilled in applied problems typically; these conditions exclude the route looping). By ℙ 1 and ℙ 2 we denote the sets of all permutations of indexes of ̅1̅,̅̅N̅ and ̅1̅,̅ ̅̅̅−̅̅̅N̅ respectively. Then, by [6, (4.4.6)] 1 ≜ { α ∈ P 1 | α − 1 (pr 1 (z)) < α − 1 (pr 2 (z)) ∀ z ∈ K 1 } ≠ ∅ , (4) 2 ≜ { α ∈ P 2 | α − 1 (pr 1 (z)) < α − 1 (pr 2 (z)) ∀ z ∈ K 2 } ≠ ∅ . (5) As 1 (as 2 ), we obtain the set of all admissible by precedence routes for ℳ 1 -problem (for ℳ 2 -problem). By ℙ we denote the set of all permutations of indexes of 1̅̅,̅̅ ̅ ; this set is connected with ℙ 1 and ℙ 2 . For establishment of this connection, we introduce ℙ̃ 2 ≜ { ( ( – ) + ) ∈ ̅̅̅+̅̅1̅̅,̅ ̅ : α ∈ ℙ 2 }; (6) so, (6) is a nonempty set of injective mappings operating in ̅N̅̅+̅̅̅1̅̅, ̅ ̅ . Now, we introduce gluing of mappings from ℙ 1 and ℙ 2 ̃ : under α 1 ∈ ℙ 1 and α 2 ∈ ℙ 2 ̃ , in the form of α 1 ⋄ α 2 : 1̅̅,̅̅ ̅ → 1̅̅,̅̅ ̅ , we obtain the mapping for which (( α 1 ⋄ α 2 )(k) ≜ α 1 (k) ∀ k ∈ 1̅̅,̅̅ ̅ ) & (( α 1 ⋄ α 2 )(k) ≜ α 2 (k) ∀ k ∈ ̅ ̅̅̅+̅̅̅1̅̅, ̅ ̅ ). (7) In connection with (6), we introduce auxiliary transformation for permutations of ℙ 2 : if α ∈ ℙ 2 , then tr( α ) : ̅1̅̅+̅̅̅ ̅̅̅, ̅ ̅ → 1̅̅̅+̅̅̅ ̅̅̅, ̅ ̅ is defined by the rule: tr ( α )( k ) ≜ α ( k − N ) + N under k ∈ ̅N̅̅+̅̅̅1̅̅, ̅ ̅ ) . . Then, for α ∈ ℙ 1 and β ∈ ℙ 2 we suppose that α ♦ β ≜ α ⋄ tr( β ); (8) then we obtain that α ♦ β ∈ P is defined by the following rule: (( α ♦ β )(k) = α (k) ∀ k ∈ 1, N ) & (( α ♦ β )(l) = tr( β )(l) = β (l − N ) + N ∀ l ∈ ̅ ̅̅̅+̅̅̅1̅̅, ̅ ̅ ). (9) In particular, α ♦ β ∈ ℙ ∀ α ∈ 1 ∀ β ∈ 2 . In this connection, we introduce the set P ≜ { α ♦ β : α ∈ 1 , β ∈ 2 }; (10)