4

"The" and "the" would be natural. The set up is rather odd. You talk about "the two teachers", but there is only one teacher. There were two meetings, but one teacher. And, of course, why would you need to explain that "the one teacher was one teacher". If the teacher had been wearing a disguise... perhaps - but ...


2

In these examples, "the" is on the boundary between a title and indicating uniqueness. Bagheera the Black Panther (with the capitals) is indicating that there is only one Black Panther, and that it is Bagheera. Since it is from The Jungle Book, it might be that there is only one black panther that is in the area right now. Or that he is the only ...


2

The name "Kilimanjaro" is being used as a modifier in this phrase. It modifies the word "volcano". "The" is used, because the word "volcano" needs an article. It would need an article even if the modifier was removed: The last eruption of the volcano was 200 years ago. The presence of a name doesn't change the need ...


2

‘The’ should be used, firstly because ‘The’ is used to talk about specific things in this context it is referring to a university. Also from my understanding I believe ‘The University of California’ is the full name of the university as opposed to simply ‘University of California’.


1

Including the article in OP's context more strongly implies all [of the] wetlands were drained. Without it, we might well suppose the optional but unspecified "determiner" here could have been, for example, some. Most likely all of the wetlands haven't been drained, so it's probably better not to include it.


1

Assuming there is only one Lebesgue measure, the "the Lebesgue measure" would be correct, and you should not drop the definite article. If there are many different measures then you could say "a Lebesgue measure". In either case you shouldn't omit the article, as the mathematical sense of "measure" is countable.


1

English is probably different from your native/other language(s) in this way: It's common to use the word "possession" in grammar to mean a more general kind of relationship indicated by a "possessive pronoun". For example, I don't own "my sister". (I did previously, but I sold her because she is somewhat impolite.) Both of the ...


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