A-Cross Product for Autocorrelated Fuzzy Processes: The Hutchinson Equation

Abstract

This paper presents a new operation of multiplication between linearly correlated fuzzy numbers based on the concept of cross product, set for fuzzy numbers in general. It is proved that this operation is closed in the set of linearly correlated fuzzy numbers. Some properties of the multiplication are listed, and an application on the delayed logistic model, the Hutchinson equation, is provided when considering the population an autocorrelated fuzzy process. Lastly, an analysis of the stability of the solution and biological interpretations are established.

Fuzzy stationary Schrödinger equation with correlated fuzzy boundaries

“…For a given asymmetric fuzzy number A, Longo et al defined the crossproduct on R F (A) by replacing the standard operations by those from the real Banach space R F (A) with addition and scalar product induced by the bijection ψ 1 [26]. As we mentioned in Remark 3, these operations can be rewritten in terms of the operations on C F (A) as in Equation 9.…”

“…. The A-cross product between B and C is defined as the fuzzy number W given by According to [26], B ⊙ C is a fuzzy number of the space R F (A) and is defined for all B, C ∈ R F (A) . In addition, the A-cross product B ⊙ C can be written in terms of the coefficients (p, r), (q, s) ∈ R 2 such that B = pA + r and C = qA + s as follows:…”

“…Furthermore, if A is an non-symmetrical fuzzy number, we have all quantities to obtain Ψ I (x) and Ψ II (x) and, consequently, the fuzzy wave function given by (26) which can be written as follow:…”

“…Therefore, an boundary condition given by a real number implies a real boundary condition for Ψ ′ II (0) = Ψ ′ II (0), with a proportionality factor equal to −ρ. Therefore, the fuzzy wave function (26) with Ψ I (x) and Ψ II (x) given by (47) can be written in the form By choosing E = 0.95 we observe classical quantum physics for E ∼ = V 0 . It is observed that the diameter of the fuzzy wave function does not depend on energy, as expected.…”