# Three sentences regarding the definitive article in technical/mathematical writing

I am writing a 40-page (technical) paper about mathematics. Most grammar advice is easy to find. But, the hardest part I have been struggling the past days is when to use the definitive article.

The reason is that I would like to have my paper written as close as possible to what a native speaker would write (not caring whether it is British or American). My native language is German and a particular difficult problem is the use, or the omission, of the definitive article "the". I Did quite some research that is got me to my guess formulated below.

Sentence 1: This sounds like native English:

Using the variance 'v' of inductive type 'I' in its argument 'a', we can finish the proof.

In the above sentence I highlighted the mathematical variables with ' '.

My interpretation/guess: Since "the variance" and "the inductive type" is a repetition, we are allowed to omit the second "the". Thus, my guess, if we do not omit the second "the", the sentence sounds redundant and less like native English:

Using the variance 'v' of the inductive type 'I' in its argument 'a', we can finish the proof.

Sentence 2: Which version sound more native:

To obtain the right hand side of required type A, we map the function F over the list L.

OR:

To obtain the right hand side of the required type A, we map the function F over the list L.

Sentence 3: Here, I have no clue, which version sounds more like native English:

Furthermore, monotonicity of G gives us the isomorphism I.

OR:

Furthermore, the monotonicity of G gives us the isomorphism I.

Finally, what grammar rules am I missing? Something I can google in the future?

• I wouldn't omit the article in any of your sentences. – Centaurus Aug 1 '16 at 22:29
• Agree with Centaurus. 'The' may not be needed to understand, but in an academic setting writing for an English speaking audience, the article should be there. Without it, these sentences read more like computer code than written English. – USER_8675309 Aug 1 '16 at 22:33
• Thank you both very much! This is a very valuable feedback, as I would have used not-so-optimal variants! – mrsteve Aug 1 '16 at 22:52
• I'd write "monotonicity" instead "the monotonicity" (also based on most examples I've seen in papers, though my sample's biased)—"the X" implies there's a definite "X" (in English), often a unique one (in math), so "the monotonicity" feels a bit weird. Clearly wrong would be "a monotonicity" or "two monotonicities" (as long as monotonicity is a property also in your field), suggesting "monotonicity" is uncountable—in which case it doesn't take any articles (think "the bread"). – Blaisorblade Jul 9 '17 at 21:08