max taldykin
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 Sep1 comment Constructive proof of pigeonhole principle Sorry for delay. I just completed the proof, you can see it here. It is still hairy and badly structured. I'll try to simplify it and then will post it here as a separate answer. Thanks again for the detailed explanation. Aug27 comment Constructive proof of pigeonhole principle I succeeded formalizing everything except $(\forall f)(\forall k)(\forall m)[(\exists i < m)[f(i) = k] \lor \lnot (\exists i < m)[f(i) = k]]$. It is easy to prove it by classic axiom but I still trying to understand how to prove it by induction. Aug27 comment Constructive proof of pigeonhole principle I'll try to formalize this proof an then will accept your answer. Aug27 comment Constructive proof of pigeonhole principle Thank you very much. The proof is really easy and straightforward, I even thought about applying inductive hypothesis to some new function but then got distracted and switched to googling for hints. Aug27 comment Constructive proof of pigeonhole principle Yep, sorry. I added my definition (failed to TeXify it, but hope it is clear as it is). Nov27 comment Expected value of minimum distance I need some time to understand this but accepting it for now. Thanks for answering. Nov27 comment Expected value of minimum distance @Yury, Wow. Can you please post this as an answer with little more explanations? Nov27 comment Expected value of minimum distance @Yury, I think we can assume only $n > m$ not $n >> m$. But if this somehow simplifies solution, you are welcome: solution to some similar problem is better than no solution at all. Nov27 comment Expected value of minimum distance @mike, thanks. reading about order stats now, seems relevant. Nov27 comment Expected value of minimum distance @Henry, yes integers are not necessarily distinct Nov27 comment Expected value of minimum distance @Yury, closed form is preferable Nov26 comment Expected value of minimum distance @user1551, fixed, thanks for pointing. Nov26 comment Expected value of minimum distance @tohecz, hmm.. good point. I'm not sure, but for now I think mean value of $x$ (as a function of $m$ and $n$) will be enough. Aug26 comment What numerical methods can solve $\sin(x) + \sin(y) = \sin(xy)$ it seems that this method is somehow similar to the second method suggested by Robert Israel. thanks for your answer, it gives some perspective from analytical viewpoint. Aug26 comment What numerical methods can solve $\sin(x) + \sin(y) = \sin(xy)$ @lhf, great links! (definitely I need to understand interval arithmetic) Aug26 comment What numerical methods can solve $\sin(x) + \sin(y) = \sin(xy)$ @J. M.: thanks, it seems quite useful (but first I need to read hundred of pages to understand interval arithmetic). I wish I could accept your answer. Aug26 comment What numerical methods can solve $\sin(x) + \sin(y) = \sin(xy)$ sorry, but I don't see how this answers my question. As I understand you are trying to represent original equation in parametric form by comparing two Taylor expansions. But wolframalpha draws something completely different for resulting equation. Aug25 comment Do objects in category form a set? btw, it's possible to define categories without notion of object, with arrows only Aug25 comment Do objects in category form a set? here is nice implementation by Edward Kmett hackage.haskell.org/package/categories Aug25 comment What numerical methods can solve $\sin(x) + \sin(y) = \sin(xy)$ thanks! can you give some references to more detailed description of these methods, or maybe their names (which I can google for)?