By the comparison theorem, for the minimal solution of the Riccati equation related to (2) and (17), we have 0 < h ′ ( t ) h ( t ) ≤ h ˜ ′ ( t ) h ˜ ( t ). for large t; see, e.g., [11].
It determines the minimal solution that accounts for every event-pair change and yields a consensus that contains all hypotheses of movement that must necessarily form part of any equally most parsimonious solution to the observed event-pair changes [[ 19]: page 239].
The minimal solution space was constrained for 10%.
We don't prove this result, but the convergence of the sequence of bounds for the minimal solution follows immediately by using the same arguments considered in [1] for the case of Modified Bessel functions of the first kind.
Two necessary and sufficient conditions for the minimal solution existence are given.
Similarly, the same analysis holds for the minimal solution (X_{S}).
To search for the "minimal solution" in the variables' space, we defined the squared distance Z between two solutions as the sum of the square of the relative changes for all variables Xi: <img src="http://journals.plos.org/plosone/article/asset?id=info?doi/10.1371/journal.pone.0016362.e045.PNG" class= inline-graphic"/>.
}t in J, x (t_{0}) = x_{0} inmathbb{R}, end{cases} where (J=[t_{0},t_{0}+a)), in (mathbb{R}) for some fixed (t_{0},a inmathbb {R}) with (a > 0), and (f, g inmathcal{C}(Jtimesmathbb{R}, mathbb{R})). They proved the existence of the maximal and minimal solution for this equation.
Among all solutions w with this property, there is the so-called minimal solution w ̃ for which w ̃ k < w k on some interval [m, ∞), where r k + w k > 0 and r k + w ̃ k > 0. The minimal solution w ̃ can be constructed as follows. Let (1) be disconjugate on [n, ∞) and let N > n.
