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Jan
18
comment Laplace, Legendre, Fourier, Hankel, Mellin, Hilbert, Borel, Z…: unified treatment of transforms?
I owe everyone an apology for taking so long to accept an answer. They are all very good, as well as difficult to digest.
Jan
18
comment Laplace, Legendre, Fourier, Hankel, Mellin, Hilbert, Borel, Z…: unified treatment of transforms?
@CameronWilliams: encyclopediaofmath.org/index.php/Idempotent_analysis
Dec
28
comment Looking for an “irreducible equivalence predicate” on $\mathbb{Z}$
@yoyo311: I plead guilty on all counts. Indeed my attempt to sharpen/formalize the informal description (of the post's first part) is pretty hopeless. I probably should get rid of it altogether... Maybe I should phrase it like this: I'm looking for a binary predicate that is "easy to compute" (as in the product of two large primes is "easy to compute"), and for which any known factoring classifier is "hard to compute" (as in factors of the product of two large primes are "hard to compute").
Dec
28
comment Looking for an “irreducible equivalence predicate” on $\mathbb{Z}$
@AndréNicolas: I knew that choosing the [computability] tag for this question was a bad idea...
Oct
4
comment On the identity map requirement in the definition of category
@EricWofsey: I repeat: no one is proposing "to define isomorphisms without identities." The proposal is to discard the requirement that all objects have identities, and doing so has no effect whatsoever on the definition of an isomorphism. It remains exactly as it was before. Of course, if there is an isomorphism between two objects, then such objects must have identities, ipso facto, but this can't be the reason for requiring that every object have an identity, since there's no requirement that every object participate in at least one isomorphism.
Oct
4
comment On the identity map requirement in the definition of category
@EricWofsey: as I already pointed out in my original post, dropping from the definition of a category the requirement that every object have an identity arrow does not in any way preclude the definition of "isomorphism", "inverse morphism", etc. Of course, these definitions do depend on the concept of an "identity arrow". What seems to me gratuitous is not the concept of an identity arrow per se, but rather the requirement that every object have one.
Oct
4
comment On the identity map requirement in the definition of category
@MaliceVidrine: Maybe so, but the Yoneda lemma dates from 1954, whereas Eilenberg and Mac Lane invented category theory in the early 1940's, therefore it seems to me unlikely that they included the identity arrow requirement in order to preserve the Yoneda lemma. My understanding is that Eilenberg and Mac Lane were primarily after capturing the notion of natural transformation, which, AFAICT, does not need the identity arrow requirement at all.
Sep
19
comment Six object classes | products | co-products in search of a category
OK, I see. The bit I missed was the part about taking "the full subcategory of the representatives." Thanks for the clarification.
Sep
19
comment Six object classes | products | co-products in search of a category
My confusion stems from my failure to see how one can define the desired category without specifying the functor from the original category to it. IOW, the two problems you mention seem coextensive to me. The only way out I can come up with is that maybe we know (somehow) that the skeleton category exists, even if a construction of it (i.e. the functor from the original category) cannot be easily specified. If this interpretation is correct, I'm still a bit mystified; I will have to venture into the pages you linked to to dispel my lingering confusion. Thanks for your patience!
Sep
19
comment Six object classes | products | co-products in search of a category
@KevinCarlson: Thanks. Just now I was having thoughts in that general direction.
Sep
19
comment Six object classes | products | co-products in search of a category
Would the fact that the construction of skeletons cannot be made functorial imply that in fact no way to define morphisms to get a category from the first case in my question? (See also the addendum I made to my question a few minutes ago.)
Sep
19
comment Six object classes | products | co-products in search of a category
Thanks for your answer! It arrived while I was writing some additional thoughts on the first case (though originally I incorrectly wrote $\mathbf{Set}$ instead of $\mathbf{FinSet}$). Those remarks already suggest that the non-negative integers ($\mathbb{N}_0$) can be viewed as the quotient of $\mathrm{ob}(\mathbf{FinSet})$ relative to $\mathrm{card}$. What I still don't have is a clear idea of how exactly how one extends this "quotienting" $\mathrm{ob}(\mathbf{FinSet}) \to \mathbb{N}_0$ to its morphisms (and thereby define the morphisms between elements of the quotient, $\mathbb{N}_0$).
Aug
15
comment On pentagonal tilings
Thanks, although the answers posted now prompt the question: what exactly is the difference between, e.g., patterns 4 (dark blue) and 8 (medium gray)?
Aug
14
comment On counting and generating all $k$-permutations of a multiset
Thanks! In the meantime I realized that the generation problem can be "factored" into two generation problems, arranged as an outer loop and an inner loop; the outer loop is the generation of all $k$-combinations (i.e. $k$-submultisets) from the multiset; the inner one is the generation of all the permutations of the current $k$-submultiset. There are many non-recursive algorithms for the inner loop generation. I have not found a non-recursive algorithm for the outer loop one, but I guess it would not be difficult to convert one of these to a relatively simple iteration, as you did here.
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.
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
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
5
comment Looking for elementary proof of “for a circle, $C^2/A = 4 \pi$”
Thanks, that's an elegant rendition.
Jul
5
comment Looking for elementary proof of “for a circle, $C^2/A = 4 \pi$”
Thank you. Even though, as you point out, this proof is not complete, and therefore (rigorously speaking) it isn't rigorous, it contains the essentials of what I was after. It does a very good job of demarcating those bits that may really require non-elementary methods from those that don't.