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stats | profile views | 122 |
Jun 30 |
comment |
Linear transformation of a polyhedron
Indeed, 'dualization' (polar reciprocation) provides a geometric proof of the equivalence, albeit a somewhat involved one. Off the top of my head, I think Stoer and Witzgall present that approach in their book on "Convexity and optimization" but I do not have it handy to check at the moment. |
Jun 29 |
comment |
Linear transformation of a polyhedron
What's wrong with the following? (Probably very close to what @Misha had in mind.) View a (not necessarily bounded) polytope as $\text{Conv}(\{u_i\}_{1 \le i \le k})+\text{Cone}(\{v_i\}_{1 \le i \le m})$ where $\text{Cone}(\{v_i\}_{1 \le i \le m}) = \left\{ \sum_{i=1}^m t_i v_i, \, t_i \ge 0 \right\}$. Now, if $L \in \text{End}(\mathbb{R}^n)$, then $L(P) = \text{Conv}(\{L(u_i)\}_{1 \le i \le k})+\text{Cone}(\{L(v_i)\}_{1 \le i \le m})$, which yields (1) (and so (2)). |
Oct 13 |
awarded | Caucus |
Sep 6 |
comment |
Generalized quasi-perfect numbers
I wondered about this a while ago and dblues had some nice preliminary observations in the very last post of this AoPS thread: artofproblemsolving.com/Forum/viewtopic.php?f=57&t=83696 |
Jul 17 |
awarded | Commentator |
Jul 17 |
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Integer-distance sets
@Joseph O'Rourke: The answer is the same in $\mathbb{R}^d$ with $d \ge 3$. See e.g. the original paper by N. H. Anning and P. ErdÅ‘s: ams.org/journals/bull/1945-51-08/S0002-9904-1945-08407-9/… |
Jul 17 |
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Integer-distance sets
@Carl: It is not an unsolved problem anymore. Such a set has been constructed by T. Kreisel and S. Kurz in 2008: arxiv.org/abs/0804.1303v1 |
Jul 16 |
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Generalization of Rado's Single Equation theorem
See also this nicely presented expository note by T. Tao (p. 12 in particular): math.ucla.edu/~tao/preprints/Expository/ramsey.dvi |
Jun 6 |
awarded | Nice Answer |
Apr 2 |
comment |
sum of digits in different bases
For 1000: our maximal $n$ must satisfy $(s_10(n)=) s_2(n) \le 9$. There are only 5 numbers with 9 ones in binary --- none of them works. Then check those numbers whose decimal sum is 8 (800, 710, etc.). None of them work until 503, which is the answer in this case. Granted, this is still somewhat brute force... |
Mar 22 |
answered | Rational points on a sphere in $\mathbb{R}^d$ |
Jun 16 |
awarded | Enthusiast |
Jun 7 |
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determinants and polynomials in matrices
For discussions of it, I highly recommend the following three threads on the AoPS forum: artofproblemsolving.com/Forum/viewtopic.php?f=349&t=43292 artofproblemsolving.com/Forum/viewtopic.php?f=349&t=218429 artofproblemsolving.com/Forum/viewtopic.php?f=349&t=279631 |
Jun 4 |
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Formula in common: How to search for same/similar equations in other knowledge domains?
There was a long discussion touching on that issue over at Tim Gowers's blog: gowers.wordpress.com/2012/03/21/… |
May 29 |
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Bounding a signed sum of complex numbers
There is no need to (re)post it on the AoPS forum: artofproblemsolving.com/Forum/viewtopic.php?f=42&t=386586 |
May 20 |
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Estimate for products of integers that are relatively prime with $N$
Yes, mod N indeed, thank you. |
May 20 |
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Elementary applications of Krein-Milman
A reference for that proof is the book "A Problem Seminar" by Donald J. Newman. |
May 20 |
answered | Estimate for products of integers that are relatively prime with $N$ |
May 19 |
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Primes with more ones than zeroes in their Binary expansion
The thread mathoverflow.net/questions/22629/… contains an excellent answer of Ben Green's giving a very rough overview of the method used in the paper by M. Drmota, C. Mauduit and J. Rivat, as well as other interesting comments related to Gil's question. |
May 18 |
awarded | Supporter |