Wikipedia:Reference desk/Archives/Mathematics/2014 July 30

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July 30

General Term

Let ${\color {white}.}\quad a_{n}={\frac {(-1)^{n}}{\zeta (n)}}\cdot \int _{0}^{1}{\frac {\ln ^{n}x}{(1+x)^{2}}}dx.\quad {\color {white}.}$ Then the first few terms are:

a2	a3	a4	a5	a6	a7	a8	a9	a10
1	9/2	21	225/2	1395/2	19845/4	40005	722925/2	7243425/2

I was not able to find this sequence at the OEIS, and I was wondering whether
someone might help me find a general formula or algorithm for generating it. —
86.125.202.236 (talk) 01:22, 30 July 2014 (UTC)

Hint: ${\frac {\ln ^{n}x}{(1+x)^{2}}}=\sum _{k=0}^{\infty }\ln ^{n}(x)(-1)^{k}(k+1)x^{k}$. If you work through the sums you should end up with $a_{n}=n!\cdot 2^{-n}\cdot (2^{n}-2)$. DTLHS (talk) 03:02, 30 July 2014 (UTC)

Resolved

 – - Thank you! :-)

Interestingly, this sequence is intimately related to the enumeration of functions which map a discrete space to itself. RomanSpa (talk) 08:02, 31 July 2014 (UTC)

Spectrum Problem in FMT

So, mentioning 0-1 laws in an above question caused me to end up rereading [1] from Fagin, which made me get reinterested in spectrums. I noticed [2] from 2009, which is interesting. I was wondering if anyone had any other reading suggestions on the topic; or if any further significant progress has been made since that was published? --Also, it mentions that the twin primes (and a few other sets of primes) are spectrums, it mentions that this because each such set is rudimentary, and all rudimentary sets are spectra; however, I was curious if anyone was aware of what FO sentences they are spectra of? More generally, given a set S that is a spectrum, is it possible to work backwards and get a sentence it corresponds with - for example, finite fields give that prime powers are a spectrum, could we run this backwards knowing that prime powers are a spectrum? Thank you for any help:-) By the way, I notice that we have Spectrum of a theory, but this does not discuss the FMT notion of spectrum, nor anything relating to it - do we have a page that does, or is this something that is not yet covered by Wikipedia?Phoenixia1177 (talk) 05:16, 30 July 2014 (UTC)

FMT = Finite model theory. -- ToE 15:05, 31 July 2014 (UTC)

Some thoughts: I can't find anything specific to spectra in the sense of finite model theory described in your first link on WP, but perhaps I haven't looked hard enough. As you likely know, the word "spectrum" is semantically overloaded in math and physics, but there are usually some similarities. E.g. Spectrum_(functional_analysis) talks about the generalization of eigenvalues of bounded linear operators. I'm unfamiliar with model theory, but there might be some connections there... Edit: that is probably nonsense. My dim memories of Decomposition_of_spectrum_(functional_analysis)#Decomposing_the_spectrum and Essential_spectrum made them seem more similar to the FMT definition.

As for finding a sentence that 'generates' a certain spectrum, that sounds like an inverse problem, and those are notoriously ill posed. Like a good counter example, my hunch is that there is not a general purpose algorithm for constructing such sentences, it would seem to require a clever human familiar with the concepts. As for further reading, lacking any better suggestions, I would just skim through the works that cite the Fagin paper on Google scholar. So, no answers from me, but I thought the links might help. SemanticMantis (talk) 21:18, 31 July 2014 (UTC)