Is an article used before a noun that references a variable?

Question

I illustrated two examples because it is difficult to formally explain my question. Can someone tell which ones of the examples below are correct?

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1. We denote the position of a node v by xv, and a set of the locations of all nodes in a set V by XV.
2. We denote the position of node v by xv, and a set of the locations of all nodes in set V by XV.

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1. The reward function for a mission m is defined as [ f_m := ~~~~ ].
2. The reward function for mission m is defined as [ f_m := ~~~~ ].

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Note that I know that articles should be used if mathematical variable symbols are not used.

Answer 1

Both conditions are grammatically correct. You would use "a node" to describe any node and "node" to describe a specific node. Though a bit awkward, your second example (in both cases) could be rewritten:

We denote the position of the node v by xv, and the set of the locations of all nodes in the set V by XV.

(Please note that in this example you should have matching articles, so "the position" should be followed by "the set." This is not universally true. It is here due to the mathematical context. The specific node must belong to the specific set. It's because of the association of V➞v, but the specifics of how English is used in mathematics (or law, or medicine) is outside the scope of ELL.)

Colloquially, the article "the" can be dropped (and therefore implied) when a compound noun is involved. Thus, "the node v" becomes "node v" and "the Princess Margaret" becomes "Princess Margaret." You shouldn't do this with simple nouns. "The tree" cannot become "tree."