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First, each P i acts on the space of functions having continuous derivatives - but we need functions having p discontinuities.
The fact that the eigenvalues of ad g ( K ~ ) ( v ~ i ) belong to k show that the c j i necessarily belong to k. Indeed v ~ i acts on g ( K ~ ) α j as multiplication by the scalar c j i.
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"I act on autopilot".
I act on impulse.
"I acted on impulse," Mr. Osbourne told reporters.
"I act on the leadership principal.
So I act on instinct and learn something each time.
Sometimes (most times), I act on irrational impulses.
"I acted on the advice of executives and lawyers".
"I acted on complete impulse in my remarks," Ford said.
Why hadn't I acted on Mr X years ago?
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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