Thanks for your comments. I was merely curious as to whether anyone might say that they have seen or used that format before. I have been given a series of bibliographies to edit that were translated into English from Japanese. The client generally prefers that I keep the translation as close to the Japanese as possible and wants to leave that format as is, but I am arguing that I have never seen it used in English and can't find any style guide that recommends it.

Yes, I expect that this expression is the result of confusing two different idioms. The speaker is a college-educated professional from the Jacksonville FL area. According to rational-madness.blogspot.com/2006/07, Miss USA 2006 also used this expression. I believe Miss USA 2006 is from Alabama, so perhaps this saying has some currency in the south. In many of the examples I found on the Web, it seems to refer to having a tattoo of a heart on the shoulder, so context is also important. But apparently there are people who speak this way.

Although not everyone will agree with me, I think I would edit this to read "...where c is a constant, f(.) is a monotonic function, and x and y are random variables."

So, Jim, I take it you would object to anyone referring to their birthplace as their "native soil?" I agree that there are times when minor distinctions in the meanings of words can be important, but insofar as Merriam-Webster defines arid as "having insufficient rainfall to support agriculture," I think the OP's sentence would be fine in the vernacular.

I wouldn't say that either 1 or 4 "sound" fine. If I were editing a paper and came across any of these, I would make significant changes. But I do consider them all grammatically well formed, since I don't consider grammar to be prescriptive.