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"Start of term" and "end of term" are British English idioms, used with the special meaning of "term" in education (roughly corresponding to American "semester"). As far as I can think, they are not u …

This is not a grammatical question, but one of usage. "In the USA" sounds natural to me, "In USA" not (British English Speaker). The iWeb corpus has 141 175 instances of "In/to the USA", against 35 …

The syntax of things like titles and headlines is different from normal speech. One the distinctions is that articles are often omitted. In most contexts "Time dependence of ... " would require an a …

I don't think I would use class that way (without an article) myself, but I wouldn't find it odd to hear it. It follows a pattern of referring to places or events that are (examples of) an institution …

No, he means "the" because the use contains a (reduced) defining (or restrictive) relative clause. The legal test used to decide ... is exactly equivalent to The legal test which is used …

Well, like most "why" questions about language, the real answer is "because that's what we say". But we can delve into the patterns of English a bit. "Shopping" in "go shopping" is not a noun like …

The meaning is quite different: opportunity implies the time and physical ability - it could also include permission, but that is rather lower on the list of elements that might contribute to it than …

I think signal might be used as a mass noun by some communications technologists, but in normal English it is a singular count noun, and so needs an article in most circumstances. Here it needs the.

While analysis can be a mass-noun, in this context it is obviously referring to a particular piece or instance of analysis, and so is a singular count noun and should have an article. Therefore the an …

In your first example explanation is countable, so it requires an article; but either article is possible. If you use the it implicitly says that there is only one possible explanation; if you use a i …

In the absence of any further context, "A" implies that there are several standard methods of k-decomposition, and you are using one of them but not, at this point, specifying which; "the" implies tha …

I don't know the answer, but I think it's unlikely that Chester is a first name here. I think it is probably a double-barrelled surname "Mr Fred Chester Becker and family". But it could be the Becke …

This is absolute standard use of definite and indefinite articles. When a subject is first introduced, we use the indefinite article, because either there isn't a specific one in mind, or the speake …

This is the use of the for "the prototype", "the abstract", "X-s in general". We see it also in the phrase that used to be common in the middle of the last century "splitting the atom". It used to …

Both are grammatical. They have different meanings, and either might be appropriate in context. By far the more likely is the king left an heir, because "an heir" is new in the discourse, and is most …