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Topic: algebraic numbers
Replies: 17   Last Post: Jan 8, 2010 4:16 AM

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DrMajorBob

Posts: 1,448
Registered: 11/3/08
Re: algebraic numbers
Posted: Jan 4, 2010 6:00 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

> 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@noqsi.com> wrote:

> On Jan 2, 3:05 am, DrMajorBob <btre...@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...@noqsi.com> wrote:

>> > On Dec 31, 1:16 am, DrMajorBob <btre...@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...@yahoo.com

>
>



--
DrMajorBob@yahoo.com




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