Wikipedia:Reference desk/Archives/Mathematics/2022 March 22

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March 22

Are there Greek letters analogs of Σ notation and Π notation for other operations?

My whole life I thought the sigma with stuff around it was complex calculus when it was just a slightly more feature-rich version of 1+2+...x. Just like when school first taught algebra, very anticlimactic. Sagittarian Milky Way (talk) 02:28, 22 March 2022 (UTC)

None that are generally understood. Σ and Π generalize algebraic operations that are associative. There are other, non-algebraic operations that are associative and have notations for aggregative application, such as $\forall$ generalizing $\land ,$ but these do not use Greek letters.  --Lambiam 08:46, 22 March 2022 (UTC)

(edit conflict) I don't think there are any more Greek letters, but there are other similar symbols. Some common ones are $\bigcap$ and $\bigcup$ used in set theory. You can find others on help:math, including:

$\coprod \,\bigoplus \,\bigotimes \,\bigodot \,\bigsqcup \,\biguplus \,\bigvee \,\bigwedge$

--RDBury (talk) 08:48, 22 March 2022 (UTC)

I think I saw that on Stargate SG-1. --Trovatore (talk) 20:11, 22 March 2022 (UTC)

LOL. I must have missed that episode :) Wikipedia has a chart of stargate glyphs because of course it does. For some reason they don't seem to be representable in Unicode or LaTeX. RDBury (talk) 08:51, 24 March 2022 (UTC)

There is also Gauss's big-K notation for continued fractions, though it appears to be very uncommon. (It seems impossible to draw an actual big K with LaTeX - or at least with the Wikimedia LaTeX-subset.) catslash (talk) 11:44, 22 March 2022 (UTC)

I proofread part of a highly advanced math textbook aimed at introducing professors and grad students to a specialized field (way, way, way beyond my pay grade). I can't tell you how many different Greek symbols, symbols on symbols, symbols on symbols on symbols and so on the author used. Clarityfiend (talk) 19:18, 25 March 2022 (UTC)

Every specialized field within mathematics has its own notational conventions – although often less standardized than those of more general and basic concepts. These specialized notations may not be understood outside that field. For example, the article (or rather stub) Symplectic frame bundle is, I suppose, abracadabra to also seasoned mathematicians who have no prior familiarity with symplectic geometry. Additionally, a mathematical author may arbitrarily introduce new notations on the fly, provided that these do not introduce ambiguity and are defined in terms of notations that are already known at that point. Their scope is then limited to the work in which they are defined. Selecting an apt notation can be a challenge, and some authors are more adept at that than others. Using capital Greek letters for these new notations may help to avoid clashes with earlier notations. The complexity of the notation may of course also reflect the inherent complexity of the subject matter it is used for.  --Lambiam 23:48, 25 March 2022 (UTC)

LaTeX defines, in addition to the Greek alphabet, Hebrew, Blackboard bold, Sans serif, Calligraphy, Fraktur, and various combinations of these with italics, bold, etc. One use of these variations is to distinguish types of objects, for example an author might use lower case for integers, upper case for rationals, lowercase Greek for reals, and upper case Greek for complex numbers. This might seem unnecessarily complicated, but it can help make mathematical text easier to understand when there several types of mathematical objects in play. --RDBury (talk) 06:50, 26 March 2022 (UTC)

A long-standing wish of mine is to see the article Mathematical notation expanded with an extensive – even if far from exhaustive – list of such and other conventions. Now there are a few hints in Variable (mathematics) and a brief section on Non-Latin-based mathematical notation. There is also a stubby and strangely selective article entitled Typographical conventions in mathematical formulae. What I know about the topic is based on exposure; writing it down will constitute OR. (Judging by the table of contents, the book The Language of Mathematics: A Linguistic and Philosophical Investigation appears fairly comprehensive, but GBS offers only a very partial view.)  --Lambiam 15:05, 26 March 2022 (UTC)

It's a little more than "slightly more feature rich": the operators allow infinite sums/products to be unambiguously expressed, and to do this the syntax supports bound variables, which is famously tricky. — Charles Stewart (talk) 22:05, 26 March 2022 (UTC)

So this (already more feature-rich than simple factorial) isn't assumed to be increasing by one each time unless specified?: 5*6*7*...999 And of course besides complete unambiguousness pi notation also has the added feature of being able to say you're multiplying not x's but 5^eπ^eπx^2.99's or whatever is starboard of the pi. Sagittarian Milky Way (talk) 01:48, 28 March 2022 (UTC)

I'm not sure what this means. (I am actually sure I don't know what it means.) To denote $5{\times }6{\times }7{\times }\dots {\times }999$ without ellipsis I'd write $\textstyle \prod _{i=5}^{999}i.$ For the conventional use of the $\textstyle \prod$ notation, see Multiplication § Product of a sequence and Infinite product. However, the bound variable (here $i$) may also range over an unordered set; see Iterated binary operation § Notation. This is only useful if you have a concise definition of the set; "$\textstyle \prod _{i\in \{5^{e},\pi ^{e^{\pi }},x^{2.99}\}}i$" is not better than "$5^{e}\pi ^{e^{\pi }}x^{2.99}.$" --Lambiam 02:52, 28 March 2022 (UTC)

It is said that the most fundamental questions in mathematics are 'so what' and 'who cares'. I don't think substantially many people really care about fixing notation for comparatively less used operations. The ones mentioned above are really the most commonly understood. Of course, some symbols are field (and even author) specific.-- Abdul Muhsy talk 03:11, 28 March 2022 (UTC)

Minimal game schedule

Suppose that eight people play each other in two teams of four. Can they arrange the teams so that in four games each person opposes each of the others at least once? There are $8X7=56$ pairings to cover, and each game offers $4X4=16$, so $56/16=3.5$ means that at least four games are necessary. The first grouping can be arbitrary, ABCD v EFGH say, but I am unable to find exactly three more to cover the requirement, and suspect that it may not be possible. Can anybody have more success?→2A00:23C6:AA07:4C00:C811:D89C:1720:BB00 (talk) 16:20, 22 March 2022 (UTC)

I think that ABCD v. EFGH, ACEG v. BDFH, and ABEF v. CDGH should suffice (although there is a good chance I messed up somewhere, if so please do let me know), assuming the ordering of the pairing doesn't matter (e.g. person A being on team 1 and person B being on team 2 is the same thing as A on 2 and B on 1.). GalacticShoe (talk) 16:27, 22 March 2022 (UTC)

There are only 8 × 7 / 2 = 28 pairings to cover with this reasonable assumption. I thought of the same solution in seconds (except swapping the second and third game, and it took longer to check the solution). It's not possible in two games since none of ABCD could be on the same team in the second game. PrimeHunter (talk) 16:42, 22 March 2022 (UTC)

I think that, indeed, if one uses "alternating blocks" of size $2^{k}$ where $k\in [[0,n-1]]$ (i.e., the first $2^{k}$ people play for the first team, then the next $2^{k}$ play for the second team, and they alternate for the remaining players) then one can arrive at minimal game schedule of $n$ games for $2^{n}$ people playing each other in teams of $2^{n-1}$; I'm fairly sure this is optimal for the particular case of powers of $2$ but I am currently not in prime thinking condition to prove this. In general, too, I think this could probably be extended to a graph-theoretic problem involving cliques. GalacticShoe (talk) 17:09, 22 March 2022 (UTC)

(OP) Apologies, I messed up the requirements in translating the problem to a more familiar format. Actually, it involves eight chefs who cook meals for each other in groups of four. Do four meals suffice for each chef to sample the cooking of each of the others? So it is $56$ experiences to be covered $16$ at a time. Again, the first meal may as well be ABCD cooking for EFGH. Being unable to derive a schedule by hand, I simulated several million random groupings for the final three meals without getting a complete coverage, but wondered if anyone could get an analytic solution.→2A00:23C6:AA07:4C00:5C25:1FA4:961F:F9F2 (talk) 08:07, 23 March 2022 (UTC)

Ah, understood.

First thing we can notice is that if four meals were to suffice, each cook would have to be on each side twice; this is the only way they can cook for everyone and be cooked for by everyone. If we assume WLOG that cook A cooks in round 1 and round 2 for the other cooks, then since there are only 6 other cooks allowed to cook in round 1 and 2, while there are 7 other cooks total, we know that some cook (WLOG, let it be cook B) must cook in specifically round 3 and 4.

Now, since in rounds 1 and 2 A cooks for B twice, leaving 6 empty slots for cooks to be cooked for by A, we know that 3 of the 6 other cooks is cooked for in round 1 and the other 3 are cooked for in round 2. Let's say that the chefs doing the cooking are, in round 1, ACDE, and in round 2, AFGH.

Now, we can notice that C, D, and E must cook for each other in the next 2 rounds, as must F, G, and H, but this is impossible, since this requires that each cook in the grouping of 3 cook once for both other chefs and be cooked for once by both other chefs, but this prevents both of those other chefs from cooking for each other. So a four-round schedule is impossible. GalacticShoe (talk) 15:36, 23 March 2022 (UTC)

Thanks for the confirmation of impossibility. →2A00:23C6:AA07:4C00:C88D:649D:FEBE:8E9B (talk) 23:08, 23 March 2022 (UTC)