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Aug
13
comment On counting and generating all $k$-permutations of a multiset
@hardmath: I need to think about it... Naively, I would have expected that such an algorithm would, in general, produce some configurations more than once. On the other hand, if the generation happens in lexicographic order, I suppose that it would be relatively inexpensive to weed these duplicates out. I need to work out the details. Thanks for your suggestion.
Aug
13
asked On counting and generating all $k$-permutations of a multiset
Aug
12
awarded  Popular Question
Aug
5
awarded  Stellar Question
Jul
26
accepted On the importance of natural transformations
Jul
23
awarded  Yearling
Jul
14
comment On the importance of natural transformations
Also, as I think more about your answer, I'd say that, in a way, it just reiterates what I wrote in my question: the definition of natural transformation adds nothing to what I already knew. (It's like learning that one has been "speaking prose all one's life without knowing it". What does one do with this fact?) I don't think I ever saw the proof of the fact that the complex eigenvalues of a matrix in $\mathrm{GL}_n\mathbb{R}$ are also eigenvalues of the same matrix when viewed as belonging to $\mathrm{GL}_n\mathbb{C}$, but this fact strikes me as more or less obvious.
Jul
14
comment On the importance of natural transformations
Thanks. I think I get the gist of your answer, but I'm confused by your use of the expression $\mathbb{R}[t] \hookrightarrow \mathbb{C}[t]$, and more specifically, the trailing $[t]$'s. If I were to write out how I understand your answer, I would have written something beginning with: "let $f$ be the insertion of $\mathbb{R} \hookrightarrow \mathbb{C}$", etc. Am I right?
Jul
14
revised On the importance of natural transformations
added 2 characters in body
Jul
14
revised On the importance of natural transformations
added 10 characters in body
Jul
14
revised On the importance of natural transformations
added 113 characters in body
Jul
14
revised On the importance of natural transformations
added 113 characters in body
Jul
14
revised On the importance of natural transformations
deleted 72 characters in body
Jul
14
asked On the importance of natural transformations
Jul
11
awarded  Disciplined
Jul
6
comment Looking for elementary proof of “for a circle, $C^2/A = 4 \pi$”
The suitability of this approach, then, hinges on the details of the proof of $A = \frac{1}{2} rC$... Since you don't give them, I assume that they are very straightforward, but I can't find a particularly simple proof of this assertion. In any case, thanks for posting this strategy.
Jul
6
revised What's the name given to the ratio $P^2/A$ for a closed figure in the Euclidean plane?
added 19 characters in body
Jul
6
accepted What's the name given to the ratio $P^2/A$ for a closed figure in the Euclidean plane?
Jul
5
asked What's the name given to the ratio $P^2/A$ for a closed figure in the Euclidean plane?
Jul
5
revised Looking for elementary proof of “for a circle, $C^2/A = 4 \pi$”
fixed typo in (3)