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The phrase "with a known" is correct and usable in written English.
This phrase can be used when referring to something that is known by people in general. For example, "Charles Dickens is a renowned author, with a known body of classic works, such as Great Expectations and A Tale of Two Cities."
Exact(60)
I started with a known quantity.
The response variables are correlated with a known covariance matrix.
"We now have a very robust model with a known cause for autistic-like behaviors.
Methods: Rothemund begins with a solution containing long strands of DNA with a known sequence.
The show became a conspicuous success with a known celebrity at its helm.
Presidential elections in Uzbekistan have always come with a known outcome.
The Challenger Drag Pak will also enter competition with a known quantity behind the wheel.
Of the women and girls with a known country of birth, 90% were born in Africa.
Quantification was accomplished by comparing with a known hexadecane standard curve.
In patients with a known cancer, the drug pinpoints additional tumors in bone.
Why collaborate in conspiracy with a known actor?
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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