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Last semester I took a functional analysis course and I read about sesquilinear forms, I have later seen the term sesquiregular for measures and graphs. Looking at wikipedia it seems that sesqui is a synonym for semi. Why is this term preferred over semi sometimes? What are the origins of the usage of this prefix?

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It's Latin and it means "one-and-a-half times."

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"Sesqui-" as a prefix means "one and a half", "more by half" (so $1.5$ times).

"Semi-", on the other hand, means "half" (so $0.5$ times).

It is Latin, likely a contraction of something like "semis-qui".