By using the hypotheses of the mappings B, g, G, T, N, η 2 and R ρ 2 n, N B, η 2 in Theorem 5.1, and the same method as the one above, we can get.
In this section, we present results obtained with the proposed system and with a dual background-based system that does not use an FSM (pixels are classified by using the hypotheses shown in Table 1 and an evidence value in a similar way as proposed in [5]), which we use as reference system.
Let us denote by ( u i, φ i, ω, θ ). the difference of two solutions, where u i = u i 1 − u i 2, φ i = φ i 1 − φ i 2, ω = ω 1 − ω 2, θ = θ 1 − θ 2. If we apply (41) for the difference, we are lead to ∫ B ( ϱ u i u i + I i j φ i φ j + J ω 2 ) d v + 1 T 0 ∫ 0 t ∫ B k i j θ, j θ, i d v d s = 0. From this equality, by using the hypotheses (ii) and (iv) of Theorem 2, we obtain u i = 0, φ i = 0, ω = 0.
In this case we can apply Theorem 3.1 by using the hypothesis., the function is nonnegative such that (3.10). is finite for every, and the improper integral is finite whenever is oscillatory.
That the sum ∑ i = 1 t I ( w i ) is direct follows by again using the hypothesis that the w i are sinks.
This conclusion can be reached by using the null hypothesis and parsimony arguments.
By using the induction hypothesis, we have H n - 1 ( F n - 1 ) = ( J n - 1 ⋊ ⋯ ⋊ J 1 ) ⋊ G n - 1 η n - 1 g e o ( k ).
If there is some α i = 0, then delete the number α i and for the remaining n − 1 number, inequality (3.1) is obvious by using the inductive hypothesis.
First, by using the simplifying hypothesis that the mode shapes of the plates in contact with the liquid (wet modes) are the samein vacuo, the approach based on the non-dimensionalized added virtual mass incremental (NAVMI) factor is applied, so that all numerical computations can be made non-dimensional.
