In the above examples, f, g, ϕ, φ, ρ, ϱ, χ, and ψ are singular at 0. It is easily seen that the results in [1, 8, 24 27, 29 31] cannot be applied to solve BVP (4.1), BVP (4.2), and BVP (4.3).
More general models are possible, for example [ 20] model f g with a random forest [ 29] and score a predictor s g (t) with a variable importance measure specific to this model.
For example, if we model f g as a linear function (2) f g (X T g ) = ∑ t ∈ T g β t, g X t, then the score s g (t) should typically assess the probability that β t, g is non-zero [ 23].
Example 2 Let continuous functions f, g : [ − 1, 1 ] → R + be given.
Example: E, F, F#, G (breath); G#, A, B♭, B (breath), and so on.
For example, f ≥ g and g ▷ S h imply that f ▷ S h.
Concrete examples of mappings f, g : R + → E are presented for above results when E is the space c 0 of those sequences of real numbers which converge to 0. In every example, the mapping g has the discontinuity of second kind at every rational point of R +.
Melodies are generally built out of a pentatonic system (for example, C, D, F, G, A) to which two auxiliary tones (E, B) may be added to make other pentatonic melodies.
In this example the mappings f, g, and T do not satisfy the general Meir-Keeler contractive condition.
Example 11 Let F, G : R × R → R be defined by F ( x, y ) = x y and G ( x, y ) = 2 3 ( x + y ) for all ( x, y ) ∈ X × X.
The following is an example of mappings f, g, h, R, S and T for which all ordered pairs ( f, g ), ( g, h ) and ( h, f ) are partially weakly increasing with respect to R, S and T but not weakly increasing with respect to them.
