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Letcbe a function defined on the unit circle with Fourier coefficients {cn}∞n=−∞.
The state of the system can thus be viewed as a function defined on a high dimensional space, associated with the fluctuations of the underlying process.
A function defined on a Banach space X is called Δ-convex if it can be represented as a difference of two continuous convex functions.
For a function defined on as (46).
where is a function defined on.
Let be a function defined on.
and let be a function defined on a neighborhood of.
Let be a function defined on, valued in.
Let f be a function defined on (mathbb{N}_{0}).
Let f be a function defined on (mathbb{N}_{a}).
Let f be a function defined on ([0,1]).
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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