English grammar experts!

"After considering a few names beginning with H" is an adverb clause containing the PARTICIPIAL phrase "considering a few names beginning with H." "Beginning with H" is a participial phrase modifying "names."

#10 -- a participle functions as an adjective. A gerund functions as a noun. (Do you agree so far?)

What noun or pronoun is "considering" modifying in that sentence?

What noun or pronoun is the object of the preposition "after"?

#11 - In this sentence, "after" is not used as a preposition; it is used as a conjunction.

ergo, neither are gerunds; both are participles

Then what's the finite verb in the clause introduced by the conjunction "after"? What tense is it? What's it's subject?

E.g. in the clause "After they considered a few names beginning with H, . . . ", "After" is a conjunction; "considered" is the finite verb; it's in the past tense; and "they" is its subject.

no, i take it back. "after" is a preposition in that sentence, and "considering" is a gerund. vinny is right

I'm with Vinny, too. "After (preposition) + considering (gerund) a few names (rest of noun phrase) beginning (participle) with H, ..."

Participles function as adjectives and verbs, never nouns.

If you want to get precise, "a few names" is the object of the gerund.

For what it's worth...

"After considering a few names beginning with H"
("After (we) considered a few names (that) begin with H")

I would consider this to be two reduced participle clauses, the first being adverbial and the second adjectival. "After" is a preposition in this case (although it can be a conjunction, but not here) as there are no complete (i.e. finite) clauses following it.

Both "considering" and "beginning" are present participles, then. Participles may not be nouns per se, but they can be the object of a preposition in participle clauses (like "on arriving, we went straight to the bar").

#18 -- This is getting tiresome. II "on arriving", "arriving" is a gerund. How do I know that? Because it's the object of a preposition. In "The train arriving on track 9 is the Panama Limited", "arriving" is a participle, modifying a noun.

You're right that the prepositional phrase "After considering" functions adverbially.