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1. Speak out paying attention to the intonation.

I think the structure of #1 is a verb + another verb+ing.

Are the same pattern of sentences like these?

2. Remember doing it.

3. He spends too much time excercising.
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I think it needs a comma at least, for clarity:

Speak out, paying attention to the intonation. -- A conjunction is elided, I think, which the comma stands in for: Speak out while paying attention to the intonation.

I think the structure of #1 is a verb + another verb+ing.-- No, I don't think it is a single structure.

Are the same pattern of sentences like these?-- No.

2. Remember doing it.

3. He spends too much time excercising.
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Do you happen to know the same pattern of verbs(sentences) like #1?
Any time a reduced adverbial clause of the same sort appears:

He arrived late, still chewing gum.

I sat quite still, fearing that the tiger would notice me.

etc.
"Speak out, paying attention to the intonation. -- A conjunction is elided, I think, which the comma stands in for: Speak out while paying attention to the intonation.(#1)"

You said "A conjunction is elided." We can say #1 like you said. And can we say "Speak out and pay attention to the intonation."?

2. He arrived late, still chewing gum.
3. I sat quite still, fearing that the tiger would notice me.


Likewise, can I say "He arrived late, and still chewed gum." for #2? and "I sat quite still and feared that the tiger would notice me." for #3?
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You said "A conjunction is elided." We can say #1 like you said. And can we say "Speak out and pay attention to the intonation."?-- Yes, that would be OK, too.

Likewise, can I say "He arrived late, and was still chewing gum." for #2? and "I sat quite still and feared that the tiger would notice me." for #3?-- Yes, as amended (you must consider the nature of the verb).