December 2015 - mathfun@mailman.xmission.com

[math-fun] Periodic functions & Fourier transforms ?
by Henry Baker 22 Dec '15

22 Dec '15

I learned about periodic functions & Fourier transforms while an undergraduate. In particular, I learned that the Fourier transform of a periodic function was a countable set of complex numbers representing the Fourier coefficients of evenly spaced impulse "functions". These coefficients represented the amplitude and phase of each of the "overtones" of the fundamental frequency of the periodic function. The only problem with this analysis is that we have taken an *uncountable* set of real numbers -- i.e., the periodic function values over one period -- and represented them by a *countable* set of complex numbers -- the Fourier coefficients. I would guess that the problem is that nearly all such periodic functions *can't* be represented by such a Fourier transform, perhaps because "almost all" such functions have too many *discontinuities*; i.e., they aren't sufficiently "smooth". If there were a countable number of discontinuities, that function could still conceivably be represented by a Fourier series. But once the number of discontinuities becomes uncountable, we're out of luck? Would this explanation be a reasonable intuition? (No wonder Cantor went mad!)

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[math-fun] Periodic functions & Fourier transforms ?
by Warren D Smith 22 Dec '15

22 Dec '15

It is the smoothness. Any fourier series that converges everywhere can be regarded as a maclaurin series of an function that's analytic and bounded within the unit disk (fourier is when restrict to the disk boundary). Well anyhow for the right kind of fourier series. Analytic functions are described by a countable number of coefficients. It is a delusion to think there is an uncountable number of degrees of freedom. So to make that clear, what we want is some theorem saying: Specifying the value of the analytic function at a countable set of points within the disk, will suffice to specify the function. Is such a theorem known? I can't think of one off the top of my head. But here, voila, I proved it: use monte carlo integration to compute the maclaurin series coefficients. With probability=1, any particular series coefficient will get computed in this way with error that in the infinite-points limit goes to zero. That proves that a countable set of points must exist, that works to compute any finite-numbered coefficient exactly; and hence, theorem proven. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Crosses along the highway of math
by James Propp 21 Dec '15

21 Dec '15

Has anyone compiled a list of disasters caused by errors in basic math? I don't mean things like the vibration-induced collapse of the Tacomah Narrows Bridge; that involved a fairly sophisticated mistake and a lack of imagination. I mean things like the Hubble Telescope debacle, which I gather in part stemmed from a failure to keep track of units and do necessary conversions. I'm asking because on their final exams, many of my students changed 3^n + (-2)^n to 3^n - 2^n. Next time I teach the course, I plan to explicitly remind them about this common mistake (and to exhort them not to commit it), but even more than that, I'd like to tell them about some memorably terrible thing that happened as a result of somebody somewhere neglecting to use parentheses where they were needed, or misusing them in some fashion. And even more, I'd like to see a compendium of such adverse outcomes, so that any time I want to warn the students away from a particular kind of mistake, I can say something like "If you make this mistake on my exam, you might lose points. And if you make this mistake after you graduate, you might kill hundreds of people." (I don't know if crosses-along-highways are a universal thing, so I'll pause to explain that here in the U.S. they signify places along a roadway where a fatal accident occurred.) Jim Propp

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[math-fun] Periodic functions & Fourier transforms ?
by Warren D Smith 21 Dec '15

21 Dec '15

To Andy Latto, you are misunderstanding -- I did not claim EVERY countable set of points worked, I was claiming MANY countable sets of points work (in fact random points work with probability 1). Anyhow, my theorem seems obsoleted by Gareth McC's theorem, which shows any everywhere-dense countable point set works, and for merely-continuous functions. Finally, Andy Latto's sort of counterexample DOES work to kill some variants of Eugene Salamin's claimed theorem. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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Re: [math-fun] Crosses along the highway of math
by Hilarie Orman 21 Dec '15

21 Dec '15

I suppose one might argue that any mistake in a computer program is an arithmetic error. And if we are in some sense digital entities, then any human mistake is an arithmetic error. It's zeroes and ones all the way down. Hilarie

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[math-fun] Cyclic convolution
by Henry Baker 20 Dec '15

20 Dec '15

I'm trying to make the following theorem as blindingly obvious to a layperson as possible. We have a periodic function f(x)=f(x+P), e.g., P=2pi. We'd like to cyclicly convolve this periodic function f(x) over the period P with a constant function c0(x). It is obvious to most mathematicians/engineers that the result is another constant function c1(x). One way to see this is that a constant function factors out of the convolution integral, leaving the integral to simply compute the mean of f(x) over the period. Another way to see this uses Fourier transforms: Let F(z) be the Fourier transform of f(x), and let C0(Z) be the Fourier transform of c0(x). But the Fourier transform C0(z) of a constant function c0(x) is an impulse at the origin, whose coefficient value is the mean (or "DC") value of c0(x) over one period. When we point-wise multiply F(z)*C0(z), every coefficient but the DC coefficient of F(z) is multiplied by zero, so we end up with F(0)*C0(0) as the only coefficient in the result, so once we inverse transform this result, we once again get a constant function c1(x) -- whose value is now mean(f(x))*mean(c0(x)), all over one period. QED Since the audience isn't composed of mathematicians, is there any way to make this obvious point "trivially obvious" ? The best intuition I can come up with is that if f(x) averages to 1 -- i.e., F(0)=1 -- and if c0(x)=1 -- i.e., C0(0)=1 -- then c0(x) acts like a kind of "zero" for the convolution operation, in that c0(x) (*) f(x) = f(x) (*) c0(x) = c0(x). (Here, we denote convolution by "(*)".) ------- If we consider a discrete domain, then we can represent a function by a Z-transform: i.e., f(i) by f(i)*z^i, and c0(i)=c*z^i. Let F(z)=sum(f(i)*z^i,i,0,N) and C0(z)=sum(c*z^i,i,0,N), so to perform a convolution, we compute F(z)*C0(z). Finally, to get the cyclic convolution we do (mod (Z^N)-1). Once again, I'd love to make this idea blindingly obvious to a lay audience.

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[math-fun] a circle packing theorem
by Erich Friedman 20 Dec '15

20 Dec '15

back in undergraduate complex analysis, i learned the following theorem, whose proof is easy enough using conformal mappings. does this theorem have a name? Let D be a disk entirely inside the unit circle. Consider all collections of non-overlapping disks C so that each member of C is: a) inside the unit circle and tangent to the unit circle, b) doesn't overlap D but is tangent to D, and c) tangent to exactly two other members of C. (Thus the collection C forms a tangent ring around D.) For a given D, the collection is either empty or infinite. erich

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[math-fun] upper limit to black hole size?
by Warren D Smith 20 Dec '15

20 Dec '15

> From: Henry Baker <hbaker1(a)pipeline.com> > Schwarzschild radius ~ 1000 AU > ~ 139 light-hours > ~ 5.8 light-days > ~ 1 light-week. > > So, this black hole is bigger than our solar system > almost out to the Oort cloud. > > http://hyperphysics.phy-astr.gsu.edu/hbase/Astro/blkhol.html > > http://arxiv.org/pdf/1511.08502v2.pdf > > How Big Can a Black Hole Grow? > > Andrew King > > Department of Physics & Astronomy, University of Leicester, Leicester LE1 > 7RH UK > > Astronomical Institute Anton Pannekoek, University of Amsterdam, Science > Park 904, NL-1098 XH Amsterdam, The Netherlands > > 10 December 2015 > > ABSTRACT > > I show that there is a physical limit to the mass of a black hole, above > which it cannot > grow through luminous accretion of gas, and so cannot appear as a quasar or > active > galactic nucleus. The limit is Mmax ~ 5*10^10 Msun for typical parameters, > but can reach > Mmax ~ 2.7*10^11 Msun in extreme cases (e.g. maximal prograde spin). The > largest black > hole masses so far found are close to but below the limit. The Eddington > luminosity > ~ 6.5 * 10^48 ergs^-1 corresponding to Mmax is remarkably close to the > largest AGN > bolometric luminosity so far observed. --his title is very deceptive. His claim is, if a black hole were too big, then any infalling "accretion disk" pattern of gas, would not be "luminous" which would tend both to make the black hole invisible, and would tend to slow down its rate of growth. The reason is any accretion disk would become unstable and break up, so the black hole could not grow steadily in an accretion-disk style. But: that in no way stops a black hole from being heavier than King's so-called upper limit, and it also in no way stops a heavier hole from being detected. It could grow by nonluminous accretion such as swallowing stars, and such a swallow by the way would be observable. This growth mechanism might be slow, but is not zero. Also, it could be detected by its gravitational effects. http://www.space.com/28664-monster-black-hole-largest-brightest-ever.html claims the largest known hole is 1.2*10^10 solar masses, not too far below King's upper limit. I think this whole claim by King is important in the sense it helpfully prevents black holes from growing too big and swallowing whole galaxy... or at least that would take much longer than the current age of the universe. It thus helps explain why universe is the way it currently is. QUESTION: Express King's mass upper limit in terms of fundamental physical constants. (He didn't, but it ought to be possible.)

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[math-fun] Biggest black hole ~ 5x10^10 suns
by Henry Baker 20 Dec '15

20 Dec '15

FYI -- Schwarzschild radius ~ 1000 AU ~ 139 light-hours ~ 5.8 light-days ~ 1 light-week. So, this black hole is bigger than our solar system almost out to the Oort cloud. http://hyperphysics.phy-astr.gsu.edu/hbase/Astro/blkhol.html http://arxiv.org/pdf/1511.08502v2.pdf How Big Can a Black Hole Grow? Andrew King Department of Physics & Astronomy, University of Leicester, Leicester LE1 7RH UK Astronomical Institute Anton Pannekoek, University of Amsterdam, Science Park 904, NL-1098 XH Amsterdam, The Netherlands 10 December 2015 ABSTRACT I show that there is a physical limit to the mass of a black hole, above which it cannot grow through luminous accretion of gas, and so cannot appear as a quasar or active galactic nucleus. The limit is Mmax ~ 5Ã10^10 Msun for typical parameters, but can reach Mmax ~ 2.7Ã10^11 Msun in extreme cases (e.g. maximal prograde spin). The largest black hole masses so far found are close to but below the limit. The Eddington luminosity ~ 6.5 Ã 10^48 ergs^-1 corresponding to Mmax is remarkably close to the largest AGN bolometric luminosity so far observed.

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[math-fun] Ptolemaic synthesis of a triangle-valued step function,
by Bill Gosper 20 Dec '15

20 Dec '15

in case you haven't seen 2D Gibbs ringing: gosper.org/gibbstriangle.gif --rwg

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