Re: Re: algebraic numbers

*To*: mathgroup at smc.vnet.net*Subject*: [mg106176] Re: [mg106152] Re: algebraic numbers*From*: DrMajorBob <btreat1 at austin.rr.com>*Date*: Mon, 4 Jan 2010 05:58:37 -0500 (EST)*References*: <hhc7a1$2o2$1@smc.vnet.net> <200912300912.EAA17052@smc.vnet.net>*Reply-to*: drmajorbob at yahoo.com

> The issue here is > whether the student has enough common culture with the test writer to > find the same answer. And that's *always* an issue. So those are cultural conformity questions?!? That's even worse than I thought! Bobby On Sun, 03 Jan 2010 02:40:36 -0600, Noqsi <jpd at noqsi.com> wrote: > On Jan 2, 3:05 am, DrMajorBob <btre... at austin.rr.com> wrote: >> When I clicked on the link below, the search field was already filled >> with >> the sequence >> >> target = {1, 2, 3, 6, 11, 23, 47, 106, 235}; >> >> Searching yielded "A000055 Number of trees with n unlabeled >> nodes." >> >> I tried a few Mathematica functions on it: >> >> FindLinearRecurrence@target >> >> FindLinearRecurrence[{1, 2, 3, 6, 11, 23, 47, 106, 235}] >> >> (fail) >> >> FindSequenceFunction@target >> >> FindSequenceFunction[{1, 2, 3, 6, 11, 23, 47, 106, 235}] >> >> (fail) >> >> f[x_] = InterpolatingPolynomial[target, x] >> >> 1 + (1 + (1/ >> 3 + (-(1/ >> 12) + (7/ >> 120 + (-(1/ >> 60) + (1/144 - (41 (-8 + x))/20160) (-7 + x)) (-6 + >> x)) (-5 + x)) (-4 + x)) (-3 + x) (-2 + x)) (-1 + x) >> >> and now the next term: >> >> Array[f, 1 + Length@target] >> >> {1, 2, 3, 6, 11, 23, 47, 106, 235, 322} >> >> But, unsurprisingly, the next term in A000055 is 551, not 322. >> >> A000055 actually starts with another three 1s, but that doesn't change >> things much: >> >> target = {1, 1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235}; >> >> FindLinearRecurrence@target >> >> FindLinearRecurrence[{1, 1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235}] >> >> (fail) >> >> FindSequenceFunction@target >> >> FindSequenceFunction[{1, 1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235}] >> >> (fail) >> >> f[x_] = InterpolatingPolynomial[target, x] >> >> 1 + (1/24 + (-(1/ >> 40) + (1/ >> 90 + (-(1/ >> 280) + (1/ >> 1008 + (-(43/ >> 181440) + (191/3628800 - (437 (-11 + x))/ >> 39916800) (-10 + x)) (-9 + x)) (-8 + x)) (-7 + >> x)) (-6 + x)) (-5 + x)) (-4 + x) (-3 + x) (-2 + x) (-1 + >> x) >> >> Array[f, 1 + Length@target] >> >> {1, 1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235, -502} >> >> So I ask you, from the data alone: what's the next term? > > It's the sort of question where one might expect a specialist to > recognize a familiar sequence. It's all context. > > Consider that in a narrow mathematical sense, spectroscopy is an > utterly ambiguous, "ill conditioned" problem. But show me a gigagauss > cyclotron spectrum, and I'll recognize it as such (see the > acknowledgment at the end of arxiv.org/pdf/astro-ph/0306189: the > authors were struggling to contrive an interpretation from atomic > physics before one of them showed the spectrum to me). But I expect > very few could do this, since few have the background. > >> >> If one had the Encyclopedia of Integer Sequences handy, those SAT >> questions could be interesting. But they'd still be nonsense. > > No they are not. Remember that the SAT isn't about the ability of a > student to function in some ideal abstract world of infinite > possibility. In the real world of academia, every single question they > will encounter will be ambiguous in some sense. The issue here is > whether the student has enough common culture with the test writer to > find the same answer. And that's *always* an issue. > >> >> Bobby >> >> >> >> On Fri, 01 Jan 2010 04:32:58 -0600, Noqsi <j... at noqsi.com> wrote: >> > On Dec 31, 1:16 am, DrMajorBob <btre... at austin.rr.com> wrote: >> >> >> This is a little like those idiotic SAT and GRE questions that ask >> >> "What's >> >> the next number in the following series?"... where any number will >> do. >> >> Test writers don't seem to know there's an interpolating polynomial >> (for >> >> instance) to fit the given series with ANY next element. >> >> > Explanations in terms of epicycles may be mathematically adequate in a >> > narrow sense, but an explanation in terms of a single principle >> > applied repeatedly is to be preferred in science. The ability to >> > recognize such a principle is important. >> >> > And my mathematical logician son (who's looking over my shoulder) >> > directed me tohttp://www.research.att.com/~njas/sequences/for >> > research on this topic. When he encounters such a sequence in his >> > research, he finds that knowledge of a simple genesis for the sequence >> > can lead to further insight. >> >> -- >> DrMajor... at yahoo.com > > -- DrMajorBob at yahoo.com