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'Sitting at the cafe with my friends, I suddenly realised that I had left the oven on at home.'

Like this, participle have an adverbial usage sometimes.

But how come? I heard from my english teacher there is omitting cojunction or something. But I am not sure actually.

Is there anyone knowing about history or origin about how participle can be used like that?

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'Sitting at the cafe with my friends, I suddenly realised that I had left the oven on at home.'

The underlined clause modifies the subject, I. It is not adverbial.
The clause can be converted to an adverbial of time by adding a preposition.

While sitting at the cafe with my friends, I suddenly realised that I had left the oven on at home.'

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anonymousparticiple have an adverbial usage sometimes.

Participle clauses are almost always adverbial in some sense of the word "adverbial" because they give information about condition, reason, result or time, but they are not truly adverbs. This one is about time. As Alphecca Stars has shown, the missing conjunction is "while", and you might say that the underlying subordinate clause of time is "While I was sitting at the cafe with my friends".

The implied subject of these kinds of participle clauses is the same as the subject of the main clause. In this case that subject is "I". I was sitting at the cafe and I suddenly realised ....

You can paraphrase the following sentence in the same way:

Balancing a bucket of water on her head, the acrobat rode a unicycle around the circus ring.

CJ

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Comments  

By the way, the phrase should be essential to the meaning if it doesn't have the adverb with it. His remembering is not a consequence of his sitting in a cafe. "Sitting at the cafe with my friends, I realized that I didn't want to move to Nebraska after all.", not "Sitting at the cafe with my friends, I took my phone out of my pocket.", but "While sitting at the cafe with my friends, I took my phone out of my pocket."

The meaning intended, that it was only after he had left his digs and gone to the cafe that he remembered, needs more words.

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