October 2014 - mathfun@mailman.xmission.com

Re: [math-fun] Logic in (old) pop music
by Henry Baker 07 Oct '14

07 Oct '14

The Monkees' singing is a denial-of-service attack on the Grelber insult server ? "Grelber also appeared as a computer program on old Unix systems. Typing the command 'Grelber' would cause the computer to insult the user." https://en.wikipedia.org/wiki/Broom-Hilda The Monkee's invented "resource starvation" due to unfair scheduling ? https://en.wikipedia.org/wiki/Resource_starvation At 10:47 AM 10/7/2014, James Propp wrote: >Or that the Monkees' claim that "We're too busy singing to put anybody down"

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[math-fun] Happy birthday Neil and the OEIS!
by Alex Bellos 07 Oct '14

07 Oct '14

Here's my Guardian story today on the OEIS. Featuring funsters. The man who loved only integer sequences http://www.theguardian.com/science/alexs-adventures-in-numberland/2014/oct/…

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[math-fun] Asimov quadratic forms "density" question
by Warren D Smith 07 Oct '14

07 Oct '14

> From: Dan Asimov <dasimov(a)earthlink.net> > Given arbitrary integers A, B, C they determine the quadratic form > Q(x,y) = Ax^2 + Bxy + Cy^2 > * Does the limiting density dens(A,B,C) of the set of integers represented > by Q always exist??? > * If so, can one determine its value??? > Here dens(A,B,C) is defined as follows: > Define the set > Q_R := {Q(x,y) | (x,y) \in Z^2 with x^2+y^2 <= R^2} > Note that any repeated value of Q(x,y) appears in Q_R with multiplicity = 1 > here. Now let > dens(A,B,C) := limit as R -> oo of (card(Q_R) / (pi R^2)), > if it exists. (Here pi R^2 is a stand-in for the number of lattice points > lying inside the disk > of radius R about the origin. But the two expressions are asymptotic to > each other, so this > should not be a problem.) > > Problem: Given any integers A, B, C does the limit dens(A,B,C) exist? > And if so, what is its value? --yes, this density always exists because its value always is always zero! More interesting question would be, the "density ratio" for Q versus some fixed forms such as x^2+y^2, x^2+xy+y^2, or, for indefinite forms, xy. Does that always exist, and if so what is its value. (I believe this answer is known to be "yes"... you'd need to search the literature.)

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Re: [math-fun] FullSimplify embarrassment contest
by Bill Gosper 07 Oct '14

07 Oct '14

JBuddenhagen>Don't know mathematica. Perhaps of interest, wolframalpha also fails to simplify to 0, see http://www.wolframalpha.com/input/?i=arccos%28-%287%2F9%29%29+-+2*arcsec%28… However my old maple 7 outputs 0 for simplify(arccos(-(7/9)) - 2*arcsec(3)); <JB GACK! Instead Malpha expresses the 0 in terms inverse Jacobi elliptic functions! And then asks me to pay for additional computation time. My Macsyma hasn't yet adapted to Mavericks, but I can pretty much predict the outcome. Macsyma's closest (not very) equivalents to the FullSimplify shotgun are rat, ratsimp, and radcan, which are for canonicalizing certain classes of expressions, e.g. rational and algebraic functions, logs, powers, etc. They wouldn't apply here, at least directly. Likewise trigsimp, which isn't for arctrigs. As with Mathematica, trigexpand(cos(%)), or ratsimp thereof, would surely work. But to get 0 directly, I would try radcan(logarc(%)), converting the arctrigs to complex logs. In fact, Mathematica's equivalent is In[99]:= TrigToExp[%97] Out[99]= -(\[Pi]/2) + I Log[-((7 I)/9) + (4 Sqrt[2])/9] - 2 I Log[I/3 + (2 Sqrt[2])/3] In[100]:= FullSimplify[%] Out[100]= 0 So it's pretty much a bug that FullSimplify forgot to try TrigToExp, or got scared by the intermediate swell. I'll bet I have somewhere some more egregious cases. Anyone else? --rwg On Sun, Sep 28, 2014 at 6:13 PM, Bill Gosper <billgosper(a)gmail.com> wrote: What is the "simplest" 0 FullSimplify can't see? Extra credit if FunctionExpand fails too. I'll start the bidding with In[97]:= FullSimplify[FunctionExpand[Subtract @@ %]] Out[97]= ArcCos[-(7/9)] - 2 ArcSec[3] In[98]:= TrigExpand[Cos[%]] Out[98]= 1 --rwg ________________________________

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[math-fun] Li(x), Ramanujan series
by Bill Gosper 07 Oct '14

07 Oct '14

From: Bill Gosper <billgosper(a)gmail.com> Bisecting the sum, plus minor fooling around in Mma gave Out[446]= LogIntegral[x] == CoshIntegral[Log[x]] + SinhIntegral[Log[x]] In[457]:= FullSimplify[%446, x > 1] Out[457]= True WDS> Ramanujan [notebook]: Li(x) = EulerMascheroniGamma + lnln(x) - sqrt(x) * SUM(n>0) a[n] * ln(x)^n where a[n] = (-1/2)^n / (n!) * SUM(0<=m<=floor((n-1)/2)) 2/(2*m+1) which I suppose has a nicer matrix product reformulation. <WDS Triangular 3x3, confluent. I seem to recall the nonexistence of confluent 2x2 path-invariant systems (except CF), but maybe luck is better with 3x3. WDS>--"confluent" means what?< Nothing like it sounds. It just means pFq when p≠q. They are limiting cases of p+1Fp: add √t to two numerator parameters, and t/z to a denominator parameter, and let t blow up. wds> --The matrix product reformulation has the advantage that it runs faster than the naive double sum. Of course, I admit, we can just be a teeny bit smarter about the double sum to evaluate N terms of the outer sum in O(N) steps. Also, I'd like to understand where this identity came from and if there are more.<wds You should at least get something nice from CoshIntegral[Log[x]] + SinhIntegral[Log[x]] It looks tedious to work the Mma derivation backwards due to intermediate swell. wds> The naive series for Li(x) is Li(expX) = Ei(X) = EulerMascheroniGamma + ln(X) + SUM(k>0) X^k / (k*k!) oho I see http://en.wikipedia.org/wiki/Exponential_integral mentions both this and Ramanujan's faster series, except they seem to have a sign error in the latter. Example questions: for which rationals R do we get nice series of the form Li(x) = EulerMascheroniGamma + lnln(x) + x^R * SUM(n>0) c[n] * ln(x)^n ? <wds I probably won't bite on this, except to say your coauthor Conway claims Mascheroni lacks the claim to fame.

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[math-fun] Li(x), Ramanujan formula
by Warren D Smith 07 Oct '14

07 Oct '14

Incomplete investigation suggests that Ei(x) = EulerMascheroniGamma + ln(x) + exp(R*x) * SUM(n>0) c[n] * x^n converges fastest (in the sense the |c[n]| shrink the fastest asymptotically) when R=1/2, Ramanujan's choice -- or if 1/2 is not optimum, it seems nearly optimum. How could that be proven? -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Li(x), Ramanujan series
by Warren D Smith 06 Oct '14

06 Oct '14

> From: Bill Gosper <billgosper(a)gmail.com> > Bisecting the sum, plus minor fooling around in Mma gave > Out[446]= LogIntegral[x] == CoshIntegral[Log[x]] + SinhIntegral[Log[x]] > > In[457]:= FullSimplify[%446, x > 1] > > Out[457]= True > WDS> > > Ramanujan [notebook]: > > Li(x) = EulerMascheroniGamma + lnln(x) - sqrt(x) * SUM(n>0) a[n] * ln(x)^n > > where > > a[n] = (-1/2)^n / (n!) * SUM(0<=m<=floor((n-1)/2)) 2/(2*m+1) > > which I suppose has a nicer matrix product reformulation. > <WDS > > Triangular 3x3, confluent. I seem to recall the nonexistence of confluent > 2x2 path-invariant systems (except CF), but maybe luck is better with 3x3. --"confluent" means what? --The matrix product reformulation has the advantage that it runs faster than the naive double sum. Of course, I admit, we can just be a teeny bit smarter about the double sum to evaluate N terms of the outer sum in O(N) steps. Also, I'd like to understand where this identity came from and if there are more. The naive series for Li(x) is Li(expX) = Ei(X) = EulerMascheroniGamma + ln(X) + SUM(k>0) X^k / (k*k!) oho I see http://en.wikipedia.org/wiki/Exponential_integral mentions both this and Ramanujan's faster series, except they seem to have a sign error in the latter. Example questions: for which rationals R do we get nice series of the form Li(x) = EulerMascheroniGamma + lnln(x) + x^R * SUM(n>0) c[n] * ln(x)^n ?

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[math-fun] Tall towers -- out of humans!
by Henry Baker 06 Oct '14

06 Oct '14

I wonder if any of these folks are civil engineers? http://www.theguardian.com/world/gallery/2014/oct/06/human-towers-of-tarrag…

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[math-fun] Li(x), Ramanujan formula
by Bill Gosper 06 Oct '14

06 Oct '14

Bisecting the sum, plus minor fooling around in Mma gave Out[446]= LogIntegral[x] == CoshIntegral[Log[x]] + SinhIntegral[Log[x]] In[457]:= FullSimplify[%446, x > 1] Out[457]= True WDS> Ramanujan [notebook]: Li(x) = EulerMascheroniGamma + lnln(x) - sqrt(x) * SUM(n>0) a[n] * ln(x)^n where a[n] = (-1/2)^n / (n!) * SUM(0<=m<=floor((n-1)/2)) 2/(2*m+1) which I suppose has a nicer matrix product reformulation. <WDS Triangular 3x3, confluent. I seem to recall the nonexistence of confluent 2x2 path-invariant systems (except CF), but maybe luck is better with 3x3. If so, we'd get identities, but of what use? Even if the new matrices are triangular, it's unlikely they'd reveal any big surprises. And since Ramanujan's series is already confluent, there isn't much need for convergence speedup. --rwg

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[math-fun] Li(x), Ramanujan formula
by Warren D Smith 06 Oct '14

06 Oct '14

Ramanujan [notebook]: Li(x) = EulerMascheroniGamma + lnln(x) - sqrt(x) * SUM(n>0) a[n] * ln(x)^n where a[n] = (-1/2)^n / (n!) * SUM(0<=m<=floor((n-1)/2)) 2/(2*m+1) which I suppose has a nicer matrix product reformulation.

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