In this study, we adopted the second method and transformed the two nominal variables into landslide density, which is defined as: Landslide density = B i B i A i A i B i B i A i A i ∑ i = 1 N B i B i A i A i ∑ i = 1 N B i B i A i A i. Assuming one variable has N classes.
We call the set Γ n ( A ) = ⋃ i = 1 n [ A i A i + 1 ). an n-polygon, or a polygon if no confusion is caused. The angleaat A i and the angle ∠A are defined as ∠ A i : = ∠ ( A i − A i − 1, A i + 1 − A i ), i = 1, 2, …, n and ∠ A : = min 1 ⩽ i ⩽ n { ∠ A i }.
The derivative of f(A i ) with respect to A i is given by ∂f ( A i ) ∂ A i = - ∂ ∂ A i t r E i E i H G i - 1 = ∂ ∂ A i t r G i - 1 E i E i H G i - 1 Z i + U r, i Λ r, i A i Λ s, i V s, i H E i H + Y i + U r, i Λ r, i A i A i H Λ r, i U r, i H - ∂ ∂ A i t r E i H G i - 1 U r, i Λ r, i A i Λ s, i V s, i H. (36).
After selecting animal i, the entire relationship matrix was made conditional on the genotype of animal i, A(1) = A - A: i A i :/ A ii where A: i is the vector of relationships of i with all individuals in the pedigree.
Proof To show (i), it is obvious that I − ρ ∑ i = 1 N a i A i is a positive linear bounded operator on H, which yields that ∥ I − ρ ∑ i = 1 N a i A i ∥ = sup { | 〈 I − ρ ∑ i = 1 N a i A i x, x 〉 | : x ∈ H, ∥ x ∥ = 1 }.
Hence, I − ρ ∑ i = 1 N a i A i is a nonexpansive mapping.
which implies that ∑ i = 1 N a i A i is a γ ¯ -strongly positive operator.
Next, we show that Q C ( I − λ ∑ i = 1 N a i A i ) is a nonexpansive mapping.
Next, we show that I − ρ ∑ i = 1 N a i A i is a nonexpansive mapping.
Applying (2.3), we can conclude that J M, λ ( I − λ ∑ i = 1 N a i A i ) is a nonexpansive mapping for all i = 1, 2, …, N. □.
Moreover, J M, λ ( I − λ ∑ i = 1 N a i A i ) is a nonexpansive mapping, for all 0 < λ < 2 η.
