October 2008 - mathfun@mailman.xmission.com

Re: [math-fun] what's the best way of finding n unknown integers from their sums taken k at a time
by Dan Asimov 14 Oct '08

14 Oct '08

A better statement of one exceptional example: * The vector space R^n has no real division algebra structure on it for all n except 2^0, 2^1, 2^2, 2^3. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] Financial Crisis = Mathematical Malpractise ?
by Steve Witham 14 Oct '08

14 Oct '08

>Steve Witham wrote: > > > > Models can often be very fruitful even when they highly oversimplify > > things. >From: Robert Baillie <rjbaillie(a)frii.com> > >granted, but even there, economics falls on its face: unlike physicists, >economists cannot predict the future. I wouldn't call that falling on its face, I'd call it not being physics. Recently I've been inspired to learn more about economics and its status. One book high on my list is Mises "The Ultimate Foundation of Economic Science," which seems to be about why trying to emulate physics is inappropriate for economics. It's free online in both html and pdf: http://mises.org/books/ufofes --Steve

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Re: [math-fun] what's the best way of finding n unknown integers from their sums taken k at a time
by Michael Reid 14 Oct '08

14 Oct '08

thanks R! very informative. in particular, the example answers part (c*) of the proposed problem. my only suggestion is that the text does not make it explicitly clear if you are considering multi-sets (or usual sets). i infer that you are considering multi-sets (i checked that the example works on the level of multi-sets), so perhaps you could state that explicitly. is the usual-set case of any (further) interest? mike > I wonder if the `book at home' is UPINT? > Section C5 currently reads as follows: > > > \usection{C5}{Sums determining members of a set.} > > \hGidx{Moser, Leo} Leo Moser asked, and Selfridge, > Straus and others largely settled, to what extent > the sums of all the pairs of numbers in a set > determine the set. They show that if the cardinality > is not a power of two, then the members are > determined. Suppose that $y_1$, $y_2$, $\ldots$, $y_s$ > are the sums $x_i+x_j$ ($i\neq j$)of the numbers > $x_1$, $x_2$, $\ldots$, $x_{2^k}$, so that > $s=2^{k-1}(2^k-1)$. Are there more than two sets > $\{x_i\}$ which give rise to the same set $\{y_j\}$\,? > If $k=3$ there may be three such sets, for example. > $$\{\pm1,\pm9,\pm15,\pm19\},\quad\{\pm2,\pm6,\pm12,\pm22\}, > \quad\{\pm3,\pm7,\pm13,\pm21\}$$ > but there can't be more than three. For $k>3$ the > problem is open. > > The corresponding problem where sums of {\em triples} > of elements of a set are given has been settled by > Boman \& Linusson. The exceptions are precisely > 3, 6, 27, 486. For $n=27$ they give five examples of > which the simplest is $\{-4,-1^{10},2^{16}\}$ and > its negative, where exponents denote repetitions. > For $n=486$ they give > $\{-7,-4^{56},-1^{231},2^{176},5^{22}\}$ and its > negative. > > The corresponding problem for sums of {\em four} > distinct elements was settled by Ewell. > > \medskip > > \small > > \Aidx{BakerI} > I.~N.~Baker, Solutions of the functional equation > $(f(x))^2-f(x^2)=h(x)$, {\it Canad.\ Math.\ Bull.}, > {\bf3}(1960) 113--120. > > \Aidx{Bolker} > E.~Bolker, The finite Radon transform, > {\it Contemp.\ Math.}, {\bf63}(1987) 27--50; > {\it MR} {\bf88b}:51009. > > \Aidx{Boman}\Aidx{ONeil} > J.~Boman, E.~Bolker \& P.~O'Neil, The combinatorial > Radon transform modulo the symmetric group, > {\it Adv.\ Appl.\ Math.}, {\bf12}(1991) 400--411; > {\it MR} {\bf92m}:05014. > > \Aidx{Linus} > Jan Boman \& Svante Linusson, Examples of non-uniqueness for the > combinatorial Radon transform modulo the symmetric group, > {\it Math.\ Scand.}, {\bf 78}(1996) 207--212; > {\it MR} {\bf97g}:05014. > > \Aidx{Ewell} > John A.~Ewell, On the determination of sets by sets of sums of > fixed order, {\it Canad.\ J.\ Math.}, {\bf20}(1968) 596--611. > > \Aidx{Gordon}\Aidx{Fraen}\Aidx{Strau} > B.~Gordon, A.~S.~Fraenkel \& E.~G.~Straus, On the determination > of sets by the sets of sums of a certain order, {\it Pacific > J.\ Math.}, {\bf12}(1962) 187--196; {\it MR} {\bf27} \#3576. > > \Aidx{Honsb} > Ross A.~Honsberger, A gem from combinatorics, > {\it Bull.\ ICA}, {\bf1}(1991) 56--58. > > \Aidx{Lambek}\Aidx{Moser} > J.~Lambek \& L.~Moser, On some two way classifications > of the integers, {\it Canad.\ Math.\ Bull.}, > {\bf2}(1959) 85--89. > > \Aidx{LiuB}\Aidx{gX} > B.~Liu \& X.~Zhang, On harmonious labelings of graphs, {\it Ars > Combin.}, {\bf36}(1993) 315--326. > > L.~Moser, Problem E1248, {\it Amer.\ Math.\ Monthly}, {\bf64}(1957) > 507. > > \Aidx{Ossow} > J.~Ossowski, On a problem of Galvin, {\it Congressus Numerantium}, > {\bf96}(1993) 65--74. > > \Aidx{RogerD} > D.~G.~Rogers, A functional equation: solution to Problem 89-19*, > {\it SIAM Review}, {\bf32}(1990) 684--686. > > \Aidx{Selfr} > J.~L.~Selfridge \& E.~G.~Straus, On the determination of numbers > by their sums of a fixed order, {\it Pacific J.\ Math.}, > {\bf8}(1958) 847--856; {\it MR} {\bf22} \#4657. > > \normalsize > > Suggestions for improvement wd be much appreciated. > > Best, R. > > On Tue, 14 Oct 2008, Michael Reid wrote: > > > > >> Interestingly enough, the 2-at-a-time sum problem for n=4 isn't necessarily > >> unique! For example, {1, 2, 3, 6} and {0, 3, 4, 5} produce the same sum > >> set, {3, 4, 5, 7, 8, 9}, when the sums are taken 2 at a time. > > > > and there's an easy bootstrapping to produce such an example for > > n = 2^k+1 , starting from an example for n = 2^k . this is basically > > part (a) of the proposed theorem. it appears that the bootstrapping > > can be arranged to get the starting multisets to be actual sets > > (no repeats). i do not see if it is possible to also get the same for > > the 2-at-a-time sums multiset by the bootstrapping method i have in mind. > > perhaps another method can accomplish that. > > > > part (c*) of the problem, which is starred, meaning its solution is > > unknown, asks if there are 3 distinct multisets that have the same > > 2-at-a-time sums multiset. > > > > > >> I have a book (at home, alas) that talks about this problem. Basically, for > >> n a power of 2, the numbers are not necessarily determined by the sums taken > >> two at a time. If n is not a power of 2, then the numbers are uniquely > >> determined. I can dig up the reference later, if anyone is interested. > > > > count me interested! > > > > mike > >

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[math-fun] the sequence fans mailing list has been reborn
by N. J. A. Sloane 14 Oct '08

14 Oct '08

Dear Math Fun, The sequence fans mailing list has started up again, and already has 60 members. Lots of interesting discussions about sequences! Go to http://groups.yahoo.com/group/seqfan to sign up. You can choose to receive postings via email, or not. Neil

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Re: [math-fun] what's the best way of finding n unknown integers from their sums taken k at a time
by Michael Reid 14 Oct '08

14 Oct '08

there is a problem related to this (for the 2-at-a-time sums) in the problem section of the most recent american mathematical monthly. in the problem, they use multisets for both the original collection and the collection of 2-at-a-time sums. i haven't worked through it yet, but it's a starting point. the reference is: problem 11389, proposed by elizabeth r. chen and jeffrey c. lagarias, american mathematical monthly 115 (october 2008), no. 8, p. 758. mike > In WW Rouse Ball's A short account of the history of mathematics he > gives the following "characteristic problem from Diophantus": > > Find four numbers, the sum of every arrangement 3 at a time being > given; say 22, 24, 27, 20. > > What about > > Find four numbers, the sum of every arrangement two at a time being > given, say 14, 12, 23, 16, 27, 25 ? > > I'm interested in what work might have been done on problems like > this. I'm particularly interested in the > good algorithms for such problems. > > Hard, easy, or what? >

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[math-fun] what's the best way of finding n unknown integers from their sums taken k at a time
by Thane Plambeck 14 Oct '08

14 Oct '08

In WW Rouse Ball's A short account of the history of mathematics he gives the following "characteristic problem from Diophantus": Find four numbers, the sum of every arrangement 3 at a time being given; say 22, 24, 27, 20. What about Find four numbers, the sum of every arrangement two at a time being given, say 14, 12, 23, 16, 27, 25 ? I'm interested in what work might have been done on problems like this. I'm particularly interested in the good algorithms for such problems. Hard, easy, or what? -- Thane Plambeck tplambeck(a)gmail.com http://www.plambeck.org/ehome.htm

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Re: [math-fun] Financial Crisis = Mathematical Malpractise ?
by Steve Witham 13 Oct '08

13 Oct '08

>From: Robert Baillie <rjbaillie(a)frii.com> >However, >the axioms of economics, especially free-market economics, are false >and are known by economists to be false. Models can often be very fruitful even when they highly oversimplify things. Point masses and Newtonian physics. So the question is not one of absolute truth or falsity but of things like accuracy, applicability, fruitfulness, respect for limitations vs. exaggerated claims, and so forth. Questions beyond the scope of the current volume. --Steve

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Re: [math-fun] mysterious tree and fruit
by Dan Asimov 12 Oct '08

12 Oct '08

Just one thing: << Although called the "Osage orange" it is not related to the fruit that we eat called orange. >> In fact the name "Osage orange" refers also to the tree, and that's why it's called "orange": the wood is bright orange. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] mysterious tree and fruit
by Simon Plouffe 12 Oct '08

12 Oct '08

Hello, as you may know, most of the plants, fruits, etc do have a growing pattern when they grow, there are 3 sections in a banana, 5 sections in an apple, pineapple do have 5 and 8 spirals, daisies have 21 petals (most of the time), this is not knew material at all. But here is one that I have found in Nantes, it is a tree in a park, just beside a street. When I saw these <fruits> the first time, I thought these were tennis balls in a fluorescent green color. Not at all : these things grows on a tree : http://www.lacim.uqam.ca:16080/~plouffe/fruit/ They do NOT have any growing pattern at all. Does anyone has any idea what tree and fruit this could be ?? I asked many peoples and nobody came with an answer, I went to see wikipedia for exotic fruits as well without success. Simon Plouffe

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Re: [math-fun] Sum z^2^n (Was: Theta_3(0,q) near the unit circle)
by Dan Asimov 11 Oct '08

11 Oct '08

Christian Bower wrote: << Gene wrote: << Here's a nice challenge problem. Let f(z) = sum( z^(2^n) / n!, n=0..infinity). We know that f(z) is analytic on the open unit disk and that f and all its derivatives are continuous and bounded on the closed unit disk. Find a function g(z) analytic on the open complement of the unit disk (possibly including infinity), with g and its derivatives continuous and bounded on the closed complement, and that agrees with f and its derivatives on the unit circle. The obvious extension method, Schwartz reflection, does not work, since f does not map the unit circle into itself. >> Can't we just use f(1/z)? >> Trouble is, for z = i, say, we have f(1/z) = f(-i) = f(z) - 2i != f(z). --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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