Daniel m3
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 Apr 6 comment theta functions and Brownian motion Have a look at the following papers: -W. B. Jurkat and J. W. van Horne, The proof of the central limit theorem for theta sums; -J. Marklof, Limit theorems for theta sums; -F. Cellarosi and J. Marklof, Quadratic Weyl sums, automorphic functions, and invariance principles. Nov 25 comment Adelic integral factorization This is treated in A. Weil's book on 'Adeles and Algebraic Groups'. In particular, see section(s up to) 3.4. Aug 20 awarded Yearling Aug 7 comment When is $(q^k-1)/(q-1)$ a perfect square? I seemed to recall it being proved in Ribenboim's book about Catalan's conjecture (published before Mihailescu solved it), and, indeed, Shorey writes that it is (on p. 111, apparently) in his MathSciNet review of the book: ams.org/mathscinet-getitem?mr=95a:11029. 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 comment 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 comment 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 comment 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 comment 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 comment 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 comment 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 comment Estimate for products of integers that are relatively prime with $N$ Yes, mod N indeed, thank you.