Adverbial Phrases

From Ch. 18-1 of Azar's Understanding and Using English Grammar:
"A modifying adverbial phrase that is the reduction of an adverb clause modifies the subject of the main clause."
Question 1:
Shouldn't the modifying adverbial phrase, being "adverbial", modify the verb of the main clause instead of the subject?
Question 2:
So would it be correct to say that the only difference between participial phrases and these "modifying adverbial phrases" is that the adverbial phrases are introduced by subordinating conjunctions while participial phrases are not?
I.e.
While walking to class, I ran into an old friend. --> modifying adverbial phrase
Walking to class, I ran into an old friend. --> participial phrase
Thanks.

[Soup]

If the modifying phrase is headed by an adverb then its form is an adverbial phrase:

Adverbial phrase
While walking to class, I ran into an old friend.
=> Here 'while' modifies 'walking to class'.

Reduced
Walking to class, I ran into an old friend.
=> Here 'walking to class' modifies 'I'.

Underlying structure
While I was walking into class, I ran into an old friend.
=> Here 'while I was walking to class' functions as a dependent clause.

Note that walking is a present participle.

[Clark]

My personal opinion of the grammar textbook you've mentioned is 'they have a right to say whatever they want to'.

[Soup]

Originally Posted by Clark

My personal opinion of the grammar textbook you've mentioned is 'they have a right to say whatever they want to'.

What does that mean?

[Clark]

Meaning that sometimes they don't know what they are saying. I've just opened it at random and read on page 92: Degrees of certainty. e.g. Sam must not be hungry. ??? How do you like it?

[Soup]

Originally Posted by Clark

Meaning that sometimes they don't know what they are saying. I've just opened it at random and read on page 92: Degrees of certainty. e.g. Sam must not be hungry. ??? How do you like it?

Clark, sorry, but I am having quite the difficult time this evening understanding your prose. What are you talking about?

[Clark]

Probably I had a hard day
Doesn't it strike as something odd that 'must' is used in its negative form to indicate supposition? In all grammars we find that 'mustn't' can mean only prohibition.

[Soup]

Originally Posted by Clark

Probably I had a hard day
Doesn't it strike as something odd that 'must' is used in its negative form to indicate supposition? In all grammars we find that 'mustn't' can mean only prohibition.

That's a different topic. You'll need to start a new post.

Good night.