Timeline for Are all smooth functions composites of 0-, 1-, and 2-ary functions?
Current License: CC BY-SA 3.0
10 events
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Feb 1, 2015 at 10:24 | comment | added | Paul Taylor | Wikipedia is not very informative on (Kolmogorov and) Arnold's work on this topic. Do you have references to (the original papers and) some "popular" account of the ideas that are involved? (The question should perhaps be tagged as real analysis.) | |
Feb 1, 2015 at 8:50 | history | edited | David Spivak | CC BY-SA 3.0 |
wanted to make the title question more searchable.
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Feb 1, 2015 at 8:45 | vote | accept | David Spivak | ||
Feb 1, 2015 at 7:52 | history | edited | David Spivak | CC BY-SA 3.0 |
used Todd Trimble's comment to redesign my definition of truncated algebraic theories.
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Feb 1, 2015 at 7:42 | history | edited | David Spivak | CC BY-SA 3.0 |
typo
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Feb 1, 2015 at 1:32 | answer | added | Eric Wofsey | timeline score: 28 | |
Feb 1, 2015 at 1:05 | comment | added | Theo Johnson-Freyd | My strong recollection is that you get all functions $\mathbb R^n \to \mathbb R$ by composing functions $\mathbb R \to \mathbb R$ and addition $+ : \mathbb R^2 \to \mathbb R$. My recollection is that this is true in both the continuous and smooth cases. But Qiaochu's comment makes me worried that perhaps I'm remembering Arnold's solution to Hilbert's problem for continuous functions, and perhaps my recollections are wrong for the smooth category. | |
Jan 31, 2015 at 23:38 | comment | added | Todd Trimble | Might you mean by $\mathcal{C}_{\leq n}$ the smallest Lawvere subtheory contained in $\mathcal{C}$ and with the same $k$-ary operations for $k = 0, \ldots, n$? | |
Jan 31, 2015 at 23:31 | comment | added | Qiaochu Yuan | Vladimir Arnold answered this question for continuous functions, it seems: en.wikipedia.org/wiki/Hilbert%27s_thirteenth_problem | |
Jan 31, 2015 at 23:17 | history | asked | David Spivak | CC BY-SA 3.0 |