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Re: [math-fun] Curve-fitting methods ?
by Keith F. Lynch 26 Sep '18

26 Sep '18

Andy Latto <andy.latto(a)pobox.com> wrote: > Keith F. Lynch <kfl(a)keithlynch.net> wrote: >> One of the most astonishing (to me) calculus theorems is that >> any smooth curve (in the sense that all of its derivatives are >> continuous) can only be extended in one way. Any non-zero-length >> segment of such a curve, no matter how short, uniquely determines >> the rest of the curve. Also, if you know all the derivatives at >> any one point, then you know the value at every point. > What's most astonishing to me is the fact that calculus classes, > while usually not actually making any false statements, seem > designed to mislead people into thinking that this statement is > true. It's completely false. Thanks for the correction. I don't know where I learned it. I own several calculus books. Most of them are older than I am (and I am not young), but presumably this has been known to be false for centuries. What I do know is I didn't learn it in any calculus class, as I've never taken a calculus class. I picked up my math on the street. That's what comes from hanging out with the wrong crowd, I guess. > and for x < 1 with the function > f(x) = -47 e(- (1/x^2)) Why 47? Will other numbers work? > Why do so many people think the false "theorem" is true? One reason > is that when we consider functions from the complex numbers to the > complex numbers, rather than from the reals to the reals, the > theorem is true. At least that rescues the Riemann zeta function. Nobody ever seems to suggest that it has more than one possible analytic continuation. What about from the reals to the complex numbers or vice versa? > I think the real reason the false theorem is believed is that > calculus texts place great emphasis on the notion of the Taylor > series of a function, and want to emphasize its importance. Its > importance would seem diminished if the texts let out the dirty > secret that even if the Taylor series converges, there is no reason > to expect that it converges to the original function, rather than > to some completely different function, even on a neighborhood of 0. That's disturbing. I'll admit to worshiping at the altar of the power series. Mainly because what else is there? At least I didn't (directly) use them when I had to raise e to the power of a matrix. Instead I divided the matrix by 2^n, then added the identity matrix to it, then squared it n times. I tried several values for n, and used the one that had results closest to its neighbors. (Is this a known algorithm, or is it original with me?) >> Since it seems plausible that my own motion is smooth, i.e. >> I never experience infinite anything (distance, velocity, >> acceleration, jerk, snap, crackle, or pop (yes, those are the >> official terms for successive derivatives of position)), that >> implies that all my past and future travels are predetermined. > Only if you think it plausible that your motion is well-defined and > smooth for complex values of t, and that there is reason that the > resulting function is differentiable in the strong form of being > differentiable as a function from C to C, rather than as a function > from R^2 to R^2. Does it help that my position, velocity, etc., are all three-vectors? Probably not. For one thing, if my position is defined as my center of gravity, whenever I trim a fingernail there's an instant when the clipping ceases to be part of me. At that instant my center of gravity instantaneously shifts. It's arbitrary exactly when in the process of trimming a nail that happens, but I don't think there's anything arbitrary about it happening, or about it being instantaneous.

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Re: [math-fun] fine structure constant, Pi, euler formula and approximations : Atiyah paper on the F.S. constant
by Henry Baker 26 Sep '18

26 Sep '18

How can it be possible for a proof of RH to be so short? As an aside, these Todd polynomials have some pretty interesting properties for approximating stuff, so they must have been utilized for other things already, right?? At 10:43 AM 9/25/2018, Fred Lunnon wrote: > The text of MFA's Monday lecture (weirdly unreadable on YouTube) is at >https://drive.google.com/file/d/17NBICP6OcUSucrXKNWvzLmrQpfUrEKuY/view > >Generally speaking, even when a paper is concerned with a topic of which >I know next to nothing, I can very quickly form an impression of whether >it is ground-breaking, authoritative, pedestrian, amateurish, crackpot, etc. > > But not on this occasion ... WFL > >On 9/25/18, Simon Plouffe <simon.plouffe(a)gmail.com> wrote: >> Hello, >> >> I have found the article of Atiyah about the Todd function >> and the fine structure constant, in this paper >> Atiyah reinvent Euler's Formula : exp(I*Pi) + 1 = 0 in >> a very surprising way. >> >> https://drive.google.com/file/d/1WPsVhtBQmdgQl25_evlGQ1mmTQE0Ww4a/view >> >> If you can't get it, I made a copy here: >> plouffe.fr/2018-The_Fine_Structure_Constant.pdf >> >> Bonne lecture, best regards, >> >> Simon Plouffe

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[math-fun] "hate speech" coding, aka "escape sequence" coding
by Henry Baker 26 Sep '18

26 Sep '18

(Regardless of the subject heading, this is not a political post.) Governments around the world, most obviously the EU, China and Thailand, require the "scrubbing"/"deletion" of certain "speech", which boils down to the scrubbing/elision of certain subsequences. The recent EU decision also requires the scrubbing/elision of all *copyrighted* subsequences, thus vastly increasing the % of subsequences prohibited from transmission and processing on networked computer systems. Computer scientists have long had to deal with this sort of thing on an ad hoc basis -- e.g., how to represent the quote character within a quoted string, or how to represent the *preamble* sequence of bits for a block on a disk drive, or the *preamble* sequence of bits for a data block sent on the radio waves. Indeed, in the case of disk blocks and radio blocks, one can still find sequences that inadvertently are treated as preamble sequences which reset the disk/radio circuitry. This error has been used to successfully hack these systems. However, with a large and growing database of prohibited subsequences, the requirement exists for coding systems that can somehow avoid all the prohibited subsequences, while permiting non-prohibited subsequences. Leaving aside the issue of whether the set of prohibited subsequences is decidable -- let's assume that it is -- what sort of encoding schemes are possible in a world where the database of prohibited sequences is not constant, but is ever growing ??

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Re: [math-fun] fine structure constant, Pi, euler formula and approximations : Atiyah paper on the F.S. constant
by Dan Asimov 25 Sep '18

25 Sep '18

I believe that the Chern character and the Todd class play an important role in the Atiyah-Singer index theorem* (c. 1963), for which among other things Atiyah won a Fields medal in 1966. —Dan ————————————————————————————————————————————————————————— * https://www.maths.ed.ac.uk/~v1ranick/papers/gilkey3.pdf Henry Baker wrote: ----- As an aside, these Todd polynomials have some pretty interesting properties for approximating stuff, so they must have been utilized for other things already, right?? -----

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Re: [math-fun] fine structure constant, Pi, euler formula and approximations : Atiyah paper on the F.S. constant
by Dan Asimov 25 Sep '18

25 Sep '18

Here is a write-up Atiyah posted about his work on the Riemann hypothesis: http://vixra.org/pdf/1809.0481v1.pdf —Dan

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[math-fun] fine structure constant, Pi, euler formula and approximations : Atiyah paper on the F.S. constant
by Simon Plouffe 25 Sep '18

25 Sep '18

Hello, I have found the article of Atiyah about the Todd function and the fine structure constant, in this paper Atiyah reinvent Euler's Formula : exp(I*Pi) + 1 = 0 in a very surprising way. https://drive.google.com/file/d/1WPsVhtBQmdgQl25_evlGQ1mmTQE0Ww4a/view If you can't get it, I made a copy here: plouffe.fr/2018-The_Fine_Structure_Constant.pdf Bonne lecture, best regards, Simon Plouffe

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[math-fun] For lovers of Pi
by françois mendzina essomba2 25 Sep '18

25 Sep '18

Hello, For all x, sum((2*n)!/(2^(3*n+1/2))/n!^2/(2*n+1)*(1/cosh(2*x)^(1/2))^(2*n+1)*cosh((2*n+1)*x),n = 0 .. infinity) = 1/4*Pi FME...

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Re: [math-fun] Curve-fitting methods ?
by Henry Baker 24 Sep '18

24 Sep '18

Blame Newton. This inventor of "calculus" was fond of doing everything with power series, so complex numbers inadvertently got dragged into the picture. Post-Newton, engineers also found power series to be incredibly useful, so they defined "calculus" as "stuff you can do with power series". Most of my MIT undergraduate engineering courses were heavily wedded to power series & complex analysis. It took *fractals* to finally move engineering a bit away from power series (and topics like the Newton's (!) N-body problem, where the power series may exist, but are useless because they converge almost nowhere). At 07:16 AM 9/24/2018, Andy Latto wrote: >Why do so many people think the false "theorem" is true? One reason is >that when we consider functions from the complex numbers to the >complex numbers, rather than from the reals to the reals, the theorem >is true. The statement that a function from C to C is differentiable >is a much stronger statement than the statement that it is >differentiable as a function from R^2 to R^2. The latter statement >says that the function is well approximated locally by a function in >the family of linear functions from R^2 to R^2, a space with real >dimension 4. The former statement says that this function is one of >the ones that corresponds to multiplication by a complex number, a >linear subspace of dimension 2. The statement that a function is >differentiable from C to C doesn't just say it's smooth; it says it's >conformal, a much more rigid criterion. The fact that the theorem is >true in complex analysis has little bearing on the kind of >curve-fitting we are talking about here, since there is no meaningful >extension of the functions in question to complex arguments and >values, and no reason to expect that the resulting function would be >complex-analytic if we did extend it.

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Re: [math-fun] Curve-fitting methods ?
by Keith F. Lynch 24 Sep '18

24 Sep '18

A few random thoughts on curve fitting: At a job some years ago in which I had to fit a curve to some data which showed how much hotter something got in response to how much radiant energy, I was criticized for first adding an additional data point representing no increase in temperature with no radiant energy. I added it because it was obviously true, and would help to make the curve more realistic. I was criticized because that experiment wasn't actually performed. When I first learned of Fourier transforms, I dreamed of getting rich by using them to project stock data ahead. I was disappointed when I discovered that all I got were the same sets of historic stock values repeated over and over again. Another learning experience. I had a similar experience after I learned that any N points can be fitted exactly onto an order N-1 polynomial. I was disappointed when I saw that the curve took absurd values, not just when extrapolated, but also when interpolated. One of the most astonishing (to me) calculus theorems is that any smooth curve (in the sense that all of its derivatives are continuous) can only be extended in one way. Any non-zero-length segment of such a curve, no matter how short, uniquely determines the rest of the curve. Also, if you know all the derivatives at any one point, then you know the value at every point. Since it seems plausible that my own motion is smooth, i.e. I never experience infinite anything (distance, velocity, acceleration, jerk, snap, crackle, or pop (yes, those are the official terms for successive derivatives of position)), that implies that all my past and future travels are predetermined. Eventually, especially after studying radio theory, I realized that that's just an instance of the many paradoxes that come from assuming an infinite signal-to-noise ratio. Or from assuming infinite anything. There's *always* noise in any physical signal, even if it's only from quantum fluctuations. A radio receiver is basically a device which fits a curve to a noisy signal in an attempt to reconstruct the intended signal. One of my projects for satisfying extreme skeptics is a way to construct a crude picture of the Earth from years of data on the brightness of the Earth-lit part of the crescent moon as seen from one's backyard. Of course the skeptic would have to trust my software, so there's little point in my writing such software unless I am the skeptic. Still, it's an interesting project. A much easier project is using a table of distances between cities to calculate the radius of the Earth. (Has anyone done that?) Any four cities will suffice, assuming a spherical Earth. With more cities you can tell more about Earth's shape. Or you can just try more subsets of four, and average all of the resulting values of radius to best approximate the correct radius of the spherical Earth. Should all sets of four cities be equally weighted? If not, what should determine their relative weightings, assuming all the distances between cities listed in the table are equally accurate and equally precise?

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Re: [math-fun] Curve-fitting methods ?
by Henry Baker 23 Sep '18

23 Sep '18

At 08:24 AM 9/23/2018, Richard Howard wrote: >If the physics of the problem leads one to suspect a power law (or >exponential), it is tempting to fit a straight line on a log-log (or >log-linear) plot in Excel. > >This does not weight all points (and their uncertainties) properly. I can imagine that the error bounds on measurement errors don't scale properly after taking logs, but if the measurement errors are small compared with the underlying process variation (assuming that it is, indeed, a power law), then the weighting shouldn't be that far off. Or is there some other issue that I'm missing here?

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