I would be pleased if someone could explain to me the participle in this sentence :
James has made a breakthrough in his math this year, doing excellent work in comparison with last year
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James has made a breakthrough in his math this year, [doing excellent work in comparison with last year].
The bracketed element is a gerund-participial (non-finite) clause functioning as a supplementary adjunct.
Supplements are not modifiers, but loosely attached expressions set off by intonation (and usually punctuation) presenting supplementary, non-integrated content.
Although supplements are not syntactically dependent on a head, they are semantically related to an 'anchor' (sometimes called a 'host'). In your example, the anchor is the subject of the clause which the supplement follows, i.e. "James".