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End-point results are obtained when the amplitude is either smooth or satisfies a homogeneity condition in the frequency variable.
Following Lemma is for the property of ϕ and ψ with asymptotic homogeneity condition (Φ1) and (Φ2), which is very useful for our analysis.
Noting ( a f ) ∗ ( t ) = | a | f ∗ ( t ), t > 0, the homogeneity condition follows ∥ a f ∥ M p, q ; λ loc = | a | ∥ f ∥ M p, q ; λ loc.
Thus, in realistic examples of assemblages of statistically relevant factors from biomedicine and social science, the objective homogeneity condition is unlikely to be satisfied, or in any practical sense, satisfiable.
Second, and relatedly, while explanation in QM satisfies the objective homogeneity condition, it is dubious that the sorts of "statistical explanations" found in the social and biomedical sciences do so.
By contrast, it seems quite unlikely that this homogeneity condition will be satisfied in most (indeed, in any) of the biomedical and sociological illustrations that have figured in the literature on statistical explanation.
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It is possible to impose restrictions on the parameters (homogeneity conditions).
Spatial homogeneity conditions are applicable inside the frame, namely, the spatial domain.
The elasticities of the factor material can be calculated by using the adding-up and homogeneity conditions.
where f : [0,1] × ℝ × ℝ → ℝ is continuous and ϕ, ψ : ℝ → ℝ are odd increasing homeomorphisms of ℝ, when ϕ, ψ satisfy the asymptotic homogeneity conditions.
This idea was further supported in [14] and extended in [26] who considered parametrized redistribution kernels of arbitrary shape that meet homogeneity conditions in both space and time.
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com