“Specialize” as a transitive verb and an antonym for “generalize”

Question

In mathematical writing, I would like to find a transitive verb that means to "apply a general theory to a special case in order to get a theory for this special case". In other words, it should be an antonym for "generalize" when it means to "make a theory widely applicable". For example,

"One of the most important branches of mathematics is the study of objects known as manifolds, which results from generalizing these ideas to three or more dimensions."

"Caution should be used in generalizing this study's results to other student populations."

I wanted to use "specialize". For example,

"We specialize this theorem to the case that ..."

Here is more or less the real situation. We have a theory that concerns a sequence of functions {f_k} and a sequence of matrices {M_k}. In the theory, under assumption A({f_k}, {M_k}), we have conclusion C({f_k}, {M_k}). This is our general theory. Meanwhile, we have another sequence of functions {F_k} and a sequences of vectors {v_k}; defining f_k = f(F_k, v_k) and M_k=M(F_k, v_k), and apply the aforementioned theory to this special configuration of {f_k} and {M_k}, we get our special theory. In this process, we do nothing but expressing A({f_k}, {M_k}) and C({f_k}, {M_k}) in terms of {F_k} and {v_k} (with reformulations and simplifications).

However, "specialize" seems to be only an intransitive verb, and it means to concentrate on and become an expert in a particular subject or skill. For example, see

https://www.oxfordlearnersdictionaries.com/definition/english/specialize?q=specialize, https://www.lexico.com/definition/specialize , https://dictionary.cambridge.org/dictionary/english/specialize , and https://www.ldoceonline.com/dictionary/specialize .

Fortunately, in the 2nd item of https://www.oed.com/view/Entry/185980?redirectedFrom=specialize#eid , "specialize" can mean "to make special, specific, or narrower in scope; to invest with a special character or function", which is what I want. But it is mentioned to be "now somewhat rare", which explains why this meaning is not mentioned in the aforementioned dictionaries.

Question:

1. What is the best word to work as the antonym of "generalize" in the above-mentioned scenario?

2. Is it good to use "specialize" as a transitive verb in this scenario?

3. (new) Is "particularize" of "specify" a good choice?

Any comments or criticism will be highly appreciated. Thank you very much.

Answer 1

If all you are doing is expressing the theory in some specified fixed-dimensional domain, you can say "the theory resolves to or particularizes to X in three dimensions" (since you aren't really doing anything to change the theory or its strengths and weaknesses.)

But if you are able to exploit the theory in special ways in this domain, and you incorporate those exploits, then I think specialize is appropriate—as in "the n-dimensional wave equation can be specialized for odd-dimensioned and even-dimensioned domains." This is not transitive, though; so I would tend to stick with intransitive constructions.

(And this is a good time to use the passive voice.)

Answer 2

"Restrict" is in common usage in the mathematical literature for some of the situations you describe. For example, "We restrict consideration of the theorem to three dimensions".