1a.
Any of the children did
n't turn in their homework. (Not OK)
1b.
Any of the children
never turned in their homework. (Not OK)
Any has a wide scope, so it's not compatible with words that express a narrow scope: 'not' or 'never'. To narrow the scope, use None:
1c.
None of the children turned in the work. (OK)
Below,
Never modifies the auxiliary 'did'. The phrase 'Never did' is outside of the sentence proper (
any of the children turn in the work), so it doesn't narrow the scope of
any.
any remains in full-scope:
1d.
Never did
any of the children turn in the work. (OK)
1b.
Any of the children never turned in their homework. (Not OK)
Would you have an example or two for your second question?
By the way, did you feel the earthquake the other night? I did. It shook me out of my bed in Toyoda-cho!
We're supposed to have another one pretty soon, so says the news. It's supposed to be a 5! I'm scared.
