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

= July 30 =

General Term
Let $${\color{white}.}\quad a_n=\frac{(-1)^n}{\zeta(n)}\cdot\int_0^1\frac{\ln^nx}{(1+x)^2}dx.\quad{\color{white}.}$$ Then the first few terms are:

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: $$\frac{\ln^nx}{(1+x)^2} = \sum_{k=0}^\infin\ln^n(x) (-1)^k (k+1) x^k$$. 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)


 * 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 from Fagin, which made me get reinterested in spectrums. I noticed 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) 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)