About the usage of "with" in "There are grey goos everywhere, with many consuming the haemocoels of tardigrades"

Question

Is the sentence below grammatical?

There are a lot of people, with many wearing tuxedos.

There are some cute teenage girls in the club, with many of them being PhD graduates in topological field theory.

I check through my dictionary and don't see a definition of with that makes the above sentence legit.

with wɪð preposition preposition: with

1. accompanied by (another person or thing). "a nice steak with a bottle of red wine" synonyms: accompanied by, in the company of, escorted by "she's gone out with her boyfriend"

2. having or possessing (something). "a flower-sprigged blouse with a white collar" wearing or carrying. "a small man with thick glasses"

3. indicating the instrument used to perform an action. "cut the fish with a knife" indicating the material used for a purpose. "fill the bowl with water"

4. in opposition to. "a row broke out with another man"

5. indicating the manner or attitude in which a person does something. "the people shouted with pleasure"

6. indicating responsibility. "leave it with me"

7. in relation to. "my father will be angry with me" affected by (a particular fact or condition). "he's in bed with the flu" indicating the cause of (a condition). "he was trembling with fear" because of (something) and as it happens. "wisdom comes with age"

8. employed by. "she's with the Inland Revenue now" using the services of. "I bank with the TSB"

9. in the same direction as. "marine mammals generally swim with the current"

10. indicating separation or removal from something. "to part with one's dearest possessions"

cf. ODO

So my questions are:

1] can I use "with" this way?

There are partial differential equations on this paper, with many of them having no real solutions.

2] or with an adjective?

There are deuterium atoms in this jar, with many/some of them ionized.

3] or without "of them"?

There are grey goos everywhere, with many/some consuming the haemocoels of tardigrades.

Answer 1

While RHK Webster's lists the above usage of 'with' as a prepositional one, it flags the peculiar usage, calling it a 'function word'. Though this is rather misleading (all prepositions lie somewhere along the lexical word --- function word continuum, their actual location dependent on the actual context they're used in), RHKW are right to imply that this is a very marginal use even for a preposition. They don't even attempt to give a definition of this sense of 'with' other than an explanation of the use to which it is put in a context:

(17.) (used as a function word to specify [introduce] an additional circumstance or condition): We climbed the hill, with Jeff following behind.

It is almost always more idiomatic to drop the 'with' and use the remaining absolute construction. The RHKW example is unusual in that the inclusion of the 'with' sounds slightly more natural, if anything; this is probably due to the lack of semantic cohesion between the main clause and the absolute clause. Two sentences would (unless previous context linked the clauses more closely) be equally appropriate, with little change even in emphasis: We climbed the hill. Jeff followed behind. Of course, stylistically, this is more staccato.

Answer 2

The manner of with used in your sentences is

2. having or possessing (something). "a flower-sprigged blouse with a white collar" wearing or carrying. "a small man with thick glasses"

You write:

There are a lot of people, with many wearing tuxedos.

I find this construction a little bit odd, but I doubt that it's ungrammatical. More commonly I'd expect to hear

There are a lot of people, many in tuxedos.