Wikipedia:Reference desk/Archives/Mathematics/2017 May 2

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May 2

Algebraic topology and climate change

What, if any, applications does algebraic topology have to the study of climate change?--129.97.125.27 (talk) 03:23, 2 May 2017 (UTC)

I imagine that persistent homology and other types of fancy topological data analysis could be used to study climate data. Googling "topological data analysis climate change" gives some results. Staecker (talk) 11:11, 3 May 2017 (UTC)

Harmonic number by LCD

What expression can be found for the n-th harmonic number using the lesat common denominator method?(Thanks) Also with the same method for a similar situation involving a sum of reciprocals of n arbitrary integers a, b, c, d, e, f, g......a(n)?--193.231.19.53 (talk) 12:03, 2 May 2017 (UTC)

• I have a feeling that if you know the term "harmonic number", either you should be able to answer that question yourself, or you just read it in a homework paper. Hint: start by reducing all fractions in the sum $H_{n}=\sum _{k=1}^{n}{\frac {1}{k}}$ to the same denominator. Tigraan^{Click here to contact me} 17:35, 4 May 2017 (UTC)
  • If we search the common denominator for the reciprocals of the first n integers, how would it be influenced by the distribution of primes among the first n pozitive integers? (I see it says in the intro of harmonic number that Euler founded a connection between the divergence of the harmonic series and the infinity of primes!).--82.137.14.110 (talk) 22:38, 8 May 2017 (UTC)

LaTeX and the empty set

Resolved

LaTeX has two symbols for the empty set, $\emptyset$ and $\varnothing$, with second one more closely resembling the html/unicvode symbol ∅. When, if ever, would you use the first symbol? Why does the second symbol, which I would think would be preferred, have the more obscure name \varnothing rather than what you would expect, \empty? I'm more interested in the first question since presumably you could sum up the answer to the second as "historical reasons." --RDBury (talk) 21:49, 2 May 2017 (UTC)

\varnothing isn't strictly LaTeX - it was added by the American Mathematical Society in AMS-LaTeX. Donald Knuth apparently prefers the slashed-zero form - there's a certain logic to it (and my impression of Knuth is that he is a big fan of anything with "a certain logic to it"), since the empty set is the additive identity. A little bit of history about it here. Smurrayinchester 08:33, 3 May 2017 (UTC)

I'm not quite sure what you mean by it being the "additive identity". Are you taking set union to be addition?

Of course, there's a literal-minded sense in which it's the additive identity on the natural numbers, since by the usual implementation of mathematics in set theory, the natural number 0 just is the empty set. (Personally I would prefer to say that it is "coded by the empty set", but the simplest exposition is to take it as literally identical to the empty set.) Note that this is not true of the integer 0, the rational number 0, or the real number 0, which are coded by three other distinct sets. --Trovatore (talk) 08:59, 3 May 2017 (UTC)

Yes, should probably have been clearer I meant "analogous to the additive identity", sorry. Smurrayinchester 09:45, 3 May 2017 (UTC)

Thanks, that was helpful. So the upshot, I take it, is that if you're coming from pure math then you'd probably use $\varnothing$, especially if you're using HTML But if your background is computer science or for some reason you don't have access to the AMS flavor of LaTeX then you'd more likely use $\emptyset$. Actually the identification 0→∅ is fairly arbitrary. For example if you like the definition of a number as an equivalence class under equipotence then 0 is actually {∅}. The mapping 0→∅, 1→{∅}, ... gives a particularly nice statement of the Axiom of Infinity, so I can see why it's often used. --RDBury (talk) 02:12, 4 May 2017 (UTC)

Well, I agree it's (a bit) arbitrary. That's why I said I prefer to say "coded".

It turns out to be quite a nice coding, though, in all sorts of ways, viewing natural numbers as a special case of ordinal numbers, and coding each ordinal by the set of all ordinals less than it. Just makes a lot of arguments go a little bit smoother.

The equivalence-class coding doesn't work without some elaboration. It's true that zero would be {∅}, because the empty set is the unique set with zero elements. However there are already a proper class of sets with a single element, so you can't even get to the next step, naively. You can do it if you use the Scott trick. --Trovatore (talk) 03:17, 4 May 2017 (UTC)

I assume Russel & Whitehead had a work-around for the "1 not a set" issue, but I don't know the details. Bourbaki gets around it, at least as far as I can tell, by using epsilon calculus (=tau calculus in their notation) -- basically just choose some random, but forever fixed, representative of each class. Actually, the construction for N I was thinking of was 0=∅, 1={0}, 2={1}, 3={2}, ... which is even simpler, but then you don't get card(n)=n so you get complications down the road when you try to connect numbers with cardinality. --RDBury (talk) 20:15, 4 May 2017 (UTC)

I couldn't tell you about Russell and Whitehead. Principia Mathematica is basically a dead letter, of historical interest only. It was absurdly complicated, and used its own notation that did not catch on and which hardly anyone can read.

My vague notions about it, which take for what they're worth, are that it was part of the logicist program that foundered on the shoals of Gödel. So maybe they tried to replace 1 with something like the second-order predicate "exactly one element satisfies predicate P"? I'm really sort of guessing here. But it wouldn't have been an equivalence class in the modern sense, because they didn't work with sets in the modern sense. --Trovatore (talk) 20:39, 4 May 2017 (UTC)

Well, the grand exception to the irrelevance of Principia Mathematica might be its proof of 1 + 1 = 2, which gets mentioned over and over again, although it too is mostly only of historical interest. ^_^ Double sharp (talk) 04:53, 7 May 2017 (UTC)