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I am writing a paper and don't know whether I need to use "the", or no article in the following cases.

1) I write "Let M be a matrix such as...". Then I need to refer to this matrix further in the text. Which one is correct:

  • Consider the element M(i,j) of the matrix M

  • Consider the element M(i,j) of matrix M

  • Consider element M(i,j) of the matrix M

  • Consider element M(i,j) of matrix M

2) Again, I define an algorithm, and name it "algorithm A". Which one is correct:

  • Applying the algorithm A, one gets...

  • Applying algorithm A, one gets...

3) Finally, I need to refer to an algorithm from another paper. This paper contains several algorithms, none of them has a unique name, but from the text it’s clear which of the algorithms is meant. Which one is correct:

  • This procedure is based on the algorithm from [3]

  • This procedure is based on algorithm from [3]

  • It has nothing to do with math. Just follow the English grammar rules for the definite article. Good Luck. – Kris Sep 3 '19 at 9:52
  • #1 is unclear without context. Both are grammatical. Which is correct depends on your intent. #2 no article. #3 use the article. – Lawrence Sep 3 '19 at 17:27
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The custom in mathematical writing is that once an item has been named, you don't use the article.

Consider element M(i,j) of the matrix M

would be appropriate in the beginning, where you're starting to talk about the matrix M.

After you've outlined or described Algorithm A, then you can talk about it without an or the.

Look at some published articles in math or computer science and then follow the rule, When in Rome, do as the Romans do.

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