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In complexity theory, we sometimes have to speak about Knuth’s big omega notation. Or about Knuth’s big Omega notation? Here the confusion starts. I can imagine all four variants of "big/Big omega/Omega notation" and, if you like hyphens, all four variants of "big/Big-omega/Omega notation". I am typesetting a main-body text (not a paper's title).

  1. Which versions are correct, which are preferrable without a hyphen?

  2. Which versions are correct, which are preferrable with a hypen?

For those who are interested: the notation, in Knuth's variant, is

Ω(f) ≝ { g: naturals→naturals | ∃ real c>0 ∃ natural n ∀ natural m≥n : g(m) ≥ cf(m) } .

An aside: the little omega / little Omega function also exists, defined as

ω(f) ≝ { g: naturals→naturals | ∀ real c>0 ∃ natural n ∀ natural m≥n : g(m) > cf(m) } .

  • When they are used in the title of a notation, book, film, or theory, &c., it is customary to capitalize all words except interior articles. The hyphens, it seems to me, are a matter of convention in the particular field. – P. E. Dant Reinstate Monica Nov 11 '16 at 20:52
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    When I write "title", I don't mean the "title" of a paper. I mean the name of a thing. So every use of the name Big Omega should be capitalized. The title of Steinbeck's book is The Grapes of Wrath, and every use should be capitalized; so also for Big Omega. – P. E. Dant Reinstate Monica Nov 11 '16 at 20:55
  • @LeonMeier, hyphenated or unhyphenated, you should capitalize a name, for example Little-Endian and Big-Endian. However, if you look around the internet you'll find little consistency in when to capitalize or not. So go with what seems best to you. – Andrew Nov 11 '16 at 20:57
  • The hyphens are a matter either of convention in the field or editor's choice. I'm not familiar with complexity except insofar as it applies to bass fishing. – P. E. Dant Reinstate Monica Nov 11 '16 at 20:58
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    I would capitalize both, but it's a matter of taste. Whatever you choose, though, be consistent. – P. E. Dant Reinstate Monica Nov 11 '16 at 21:05
  • @P.E.Dant -- Your comments would make a good answer. – Jasper Nov 11 '16 at 21:33

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In general, it is a good practice to treat the name of a thing—a notation, book, film, or theory, &c.—as a proper name. We capitalize such things in English, e.g.:

The Grapes of Wrath
Star Wars: The Wrath of Khan
Theory of General Relativity
Big Omega notation

Note that here, it may be the convention within the field to also capitalize the class of thing named, in this case "notation".

Interior "small" words such as "the", "and", and "of" are usually not capitalized, but they are capitalized when they come first, e.g. "The Grapes of Wrath" above.

There are many examples of hyphenated titles of this kind, and the usual convention seems to be that both terms are capitalized, as in Big-Endian.

Where hyphenation is concerned, that is also usually a matter of convention within a given field. If the field provides no such guidance, the writer or editor must decide; but any such decision to hypenate a term should always be applied consistently throughout the work.

P. E. Dant Reinstate Monica
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