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Re: algebraic numbers

  • To: mathgroup at smc.vnet.net
  • Subject: [mg106152] Re: algebraic numbers
  • From: Noqsi <jpd at noqsi.com>
  • Date: Sun, 3 Jan 2010 03:40:36 -0500 (EST)
  • References: <hhc7a1$2o2$1@smc.vnet.net> <200912300912.EAA17052@smc.vnet.net>

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



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