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Fig. 1 The estimated smooth terms of the unconstrained model without varying coefficients, h1. a the smooth function of age, (b) the smooth function of relative diameter at breast height, (c) the smooth function of topex index, (d) the smooth function of temperature sum during the vegetation period and (e) the smooth function of aridity index.
Since is a smooth function of for each,, it follows from (3.11) that (3.13). is a smooth function of for each,.
The proposed method can optimize any arbitrary smooth function of states and input.
Suppose that u x, t) is a smooth function, of bounded variation, in (Omega).
and is smooth function of, and at least of order three in variables.
The nonlinear term f ( v ) is a given smooth function of v ∈ R.
Then, (3.8). is a smooth function of for each,, and is linearly independent.
where is smooth function of, and at least of three order in variables.
where is smooth function of, and at least of order three in variables and (3.24).
By the induction argument, we have for each, (3.21). is a smooth function of for each,.
( NPL_{t}^{e}) represents a smooth function of expected losses through the business cycle.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com