June 2014 - mathfun@mailman.xmission.com

[math-fun] neutron star materials
by Henry Baker 29 Jun '14

29 Jun '14

The recent discussions about neutron stars provoked the following question. This question isn't about neutron stars, per se, but about "nuclear" materials made from the basic stuff of neutrons & protons. We've been assuming that materials such as this can only exist inside neutron stars which are held together by extremely high gravity, but I'm not 100% convinced. For example, we used to think that diamonds could only be made under extreme pressure, but this has been found to be false; diamond coatings can be made with with chemical vapor deposition (CVD) under relatively low pressure. http://en.wikipedia.org/wiki/Synthetic_diamond. What I'm thinking of is a kind of "foam" which is a more-or-less fractal material made out of nuclear "soup" (protons-cum-neutrons). Think Sierpinski pyramids or Apollonian foams. Since the material is mostly empty space, it could be considerably less dense than the stuff of neutron stars. What I'm hoping for is some sort of quantum effect that enables this "foam" to keep from collapsing, and hence have some rigidity, but perhaps also have some decent strength, so that it could exist in macroscopic form. (There is also the problem of "nuclear" stability, but perhaps there are wave functions which are stable due to the geometric form of the structure.) Now the biggest problem is why we haven't seen this stuff already. The fact that we haven't seen it before merely indicates that it may take very special conditions to form it -- e.g., a _manufactured material_ or some sort of collision of two neutron stars. I don't know enough physics to guess whether such a material could really exist, which is why I'm posing the question.

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[math-fun] Pascal triangle row: shift-and-add
by Fred Lunnon 29 Jun '14

29 Jun '14

What happens when row n of the Pascal triangle is shifted by k then added to itself? Or in continuous terms, two binomial distributions, identical apart from the shift along the x-axis, are summed? Consider k fixed and n increasing: initially n_C_i + n_C_j where i+j = n+k is bimodal; but at some point n_0 it becomes unimodal and remains so for n > n_0. Numerical experiments suggest that n_0 ~ k^2 - 5/3 ; indeed this bound estimate is remarkably good, being only <0.1 too large at k = 2 , and <0.001 too large at k = 20 . No doubt probability wonks know all about this stuff already: so can anyone point me at a reference to a proof of the bound; or alternatively explain why it's perfectly obvious? Fred Lunnon

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[math-fun] Hanna's A195865 and A198518 apparently count certain trees
by Joerg Arndt 28 Jun '14

28 Jun '14

Request to people into enumeration of trees, please eyeball https://oeis.org/draft/A198518 and https://oeis.org/draft/A195865 and tell me the "Apparently"s can be removed! Best, jj [Apologies for starting two threads within a day!]

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[math-fun] Aitken-compositions? (A011972)
by Joerg Arndt 28 Jun '14

28 Jun '14

Recap: the compositions of n can be generated by always moving one unit of the rightmost part to the left and spreading its remaining units out to the left. Example (n=5): [ 1 1 1 1 1 ] [ 1 1 1 2 ] [ 1 1 2 1 ] [ 1 1 3 ] [ 1 2 1 1 ] [ 1 2 2 ] [ 1 3 1 ] [ 1 4 ] [ 2 1 1 1 ] [ 2 1 2 ] [ 2 2 1 ] [ 2 3 ] [ 3 1 1 ] [ 3 2 ] [ 4 1 ] [ 5 ] Now slightly change the above recipe by moving that "one unit" by _two_ places. Example (n=9): [ 1 1 1 1 1 1 1 1 1 ] [ 1 1 1 1 1 1 2 1 ] [ 1 1 1 1 1 2 2 ] [ 1 1 1 1 2 2 1 ] [ 1 1 1 1 3 2 ] [ 1 1 1 2 3 1 ] [ 1 1 1 3 3 ] [ 1 1 2 3 1 1 ] [ 1 1 2 4 1 ] [ 1 1 3 4 ] [ 1 2 3 1 1 1 ] [ 1 2 3 2 1 ] [ 1 2 4 2 ] [ 1 3 4 1 ] [ 1 4 4 ] [ 2 4 1 1 1 ] [ 2 4 2 1 ] [ 2 5 2 ] [ 3 5 1 ] [ 4 5 ] The number of such compositions is given by http://oeis.org/A011972 (note that such compositions of the triangular numbers give http://oeis.org/A000110 ). Can anyone shed some light on this? This riddle has been bugging me for a long time, and I still cannot see anything pertinent. Best, jj

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[math-fun] Overview of sequence arising from binary quadratic forms
by Neil Sloane 27 Jun '14

27 Jun '14

Since Warren has mentioned some interesting questions concerning numbers represented by quadratic forms, let me mention that for the past month I've been surveying the sequences in the OEIS that arise from binary quadratic forms ax^2+bxy+cy^2. The result can be seen here: <a href=" https://oeis.org/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a> Here's a summary. A binary quadratic form f(x,y) = ax^2+bxy+cy^2 has discriminant d = b^2-4ac, and is positive definite if d < 0, or indefinite if d > 0. f(x,y) represents an integer n if there are integers x and y such that f(x,y)=n. This web page gives an index to the following sequences in the OEIS: - numbers (or primes) represented by positive definite binary quadratic forms (Section 3), - numbers (or primes) represented by indefinite binary quadratic forms (Section 4), This page also lists programs for computing these sequences (in Section 5) and a list of references and links (in Section 6) There is a lot still to be done, in case people would like to help: - there are several cases where a sequence appears to arise in several different ways, not all of which have been proved to give the same sequence (see for example the multiple meanings listed in A033212 and A141184). Once these have been resolved, the entries can probably be merged. - not all the sequences presently in the OEIS have been included, and - there are many more that could be added (both to the OEIS and here). - Furthermore, hundreds of the sequences mentioned here were computed using a program QuadPrimes (see Section 5 and A106856) that unfortunately contains a bug, which can occasionally cause it to produce incorrect answers. These sequences are listed in Section 7 and need to be rechecked. - Many of the sequences arising from indefinite quadratic forms (see Section 4) were computed by "brute force", which is a notoriously unreliable method (as one knows from studying Pell's equation). These also need to be checked. Neil

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[math-fun] some conjectures (theorems?) about quadratic forms
by Warren D Smith 27 Jun '14

27 Jun '14

Jeff D Caldwell: How about ax^2 + bxy + cy^2 with a = -1, b = 1 and c = 0 for conjecture #2? The form generates every natural exactly the number of times as the number of factors of that natural, counting such that a prime squared has 3 factors. (The form also generates integers less than 0.) On Fri, Jun 27, 2014 at 11:03 AM, Warren D Smith <warren.wds at gmail.com> wrote: > D.Lehmer showed that there is an infinite set of numbers > representable as a sum of 4 squares in exactly 1 way (up to re-ordering), > http://oeis.org/A006431 > > 1. I conjecture there does NOT exist any integer ternary quadratic form Q, > for which there is an infinite set of numbers > representable as Q(a,b,c) but only in a bounded nonzero number of ways. > In fact, if N is representable N=Q(a,b,c), then I conjecture > N is representable in at least N^0.499 ways (for all sufficiently large N). --WDS: JDC's example looks good, except it is a binary quadratic form Q(x,y), and I wanted ternary Q(x,y,z).

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[math-fun] Pascal triangle row: shift-and-add
by Warren D Smith 27 Jun '14

27 Jun '14

--I won't explain "why this is perfectly obvious"... but the fact k^2 is the correct leading term can be seen as "obvious" to some degree essentially because maxima behave like quadratics. More precisely, the binomial function has a max and mean at n/2 and it has std deviation of order n (exactly n? anyhow exactly known) and the quadratic approximation that agrees with those facts plus the fact the binomial is always integer valued is enough to see you aren't going to be able to get any valley between the peaks if n>LargeEnoughConst*k^2. (The sum of two concave-down quadratics, or indeed of any two concave-down functions, is concave-down. The binomial function is concave down if we stay in a region near enough to its max, and is approximately quadratic there.) Now I would think by using stirling series you could probably with enough effort extract all the higher order terms you want, given that you found 3 terms using numerical hogwash.

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[math-fun] some conjectures (theorems?) about quadratic forms
by Warren D Smith 27 Jun '14

27 Jun '14

D.Lehmer showed that there is an infinite set of numbers representable as a sum of 4 squares in exactly 1 way (up to re-ordering), http://oeis.org/A006431 1. I conjecture there does NOT exist any integer ternary quadratic form Q, for which there is an infinite set of numbers representable as Q(a,b,c) but only in a bounded nonzero number of ways. In fact, if N is representable N=Q(a,b,c), then I conjecture N is representable in at least N^0.499 ways (for all sufficiently large N). 2. I conjecture (actually, for this one I have a proof sketch, but I have not checked its details carefully) that there does not exist any integer quadratic form, in any finite number of variables, for which both (a) every natural number is representable (b) but only in a bounded number of ways. In fact I think my proof will show that if (a), then the number of representations must, for an infinite subset of N, grow faster than some positive power of N. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] another bit of craziness re neutron stars -- new regime which ain't there
by Warren D Smith 27 Jun '14

27 Jun '14

If you really, really had a large neutron star -- just a smidgen short of what would be required to turn it into a black hole -- then might we enter a new regime? Specifically, imagine a neutron decaying, releasing a neutrino (or antineutrino, whatever). If the escape velocity for the star is larger than the velocity of that neutrino (which will be damn near lightspeed) then one could argue that in a sense, neutron decay has now become impossible. Anyhow, cooling via the mechanism of neutron decay / re-synthesis cycle (where neutrinos released in both cases) is now impossible. So: new regime for neutron stars. However... I believe this actually cannot happen, because I think that the escape velocity for a stable neutron star cannot be brought arbitrarily near lightspeed -- if you try, then it'd collapse once it reached some threshold like 0.6c, or something like that. In which case, this "new regime" is the empty set.

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Re: [math-fun] Neutron stars as atoms
by Henry Baker 27 Jun '14

27 Jun '14

What insight would be gained from such a definition? Neutron stars are presumably held together by gravity, not by quantum forces, so nuclear "chemistry" isn't implicated. Neither is normal chemistry implicated, as I see no possibility of multiple neutron stars getting together as molecules. You raise an interesting possibility of a substantial net charge on such an object, but wouldn't such a charge allow a huge amount of energy to bleed off in the form of moving charges generating electromagnetic waves ? Neutron stars appear to be a totally different kind of matter, with the possibility of all kinds of bizarre behaviors. At 08:46 AM 6/25/2014, Whitfield Diffie wrote: > I recall reading the intreaguing observation that stable neutron >structures run up to around atomic wieght 300 and then their is a long >gap before you get to the neutron stars with atomic weights around >10^50. Perhaps we should regard the neutron stars as isotopes of very >heavy elements. The atomic numbers would be miniscule by comparison >with the atomic weights but a mildely positively charged neutron star >doesn't seem impossible: would adding a billion protons make a >difference? > > Whit

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