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The phrase "acknowledgment of how" is correct and usable in written English.
It can be used when discussing recognition or awareness of a particular method or manner in which something occurs or is done.
Example: "The acknowledgment of how the team collaborated effectively led to a successful project outcome."
Alternatives: "recognition of how" or "realization of how".
Exact(48)
That is acknowledged by the OBR, although there is no acknowledgment of how this will be funded.
There's no acknowledgment of how lucky you've been if you break up bitterly.
"This is an acknowledgment of how serious a problem it is".
For AMC, an Emmy for "Breaking Bad" would be a welcome acknowledgment of how it, like Netflix, has changed television.
The speech was in effect an acknowledgment of how completely Hezbollah depends on the Assad government's survival.
That seems like more of an acknowledgment of how fragile that support is than any poll yet taken.
Similar(9)
Still, there were acknowledgments of how the rest of the country may feel.
Interestingly, most couch their feelings in acknowledgments of how backwards their insecurities are.
For India, which established diplomatic relations with Israel in 1992, Mr. Sharon's plans for a visit -- scheduled for Monday, though it is not yet known how today's developments in the Middle East will affect his plans -- will be the most public acknowledgment yet of how far its foreign policy has shifted from its once unequivocal support for Palestinian self-determination.
But few offer discerning analyses, evidence-based recommendations or even the simple acknowledgment of just how hard believing in oneself can sometimes be.
That's where the plentiful humor comes from in "Abundance" — the casual acknowledgment of just how difficult Western rural life was 150 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