September 2018 - mathfun@mailman.xmission.com

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 24 Sep '18

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

24 Sep '18

The following xkcd comic (#2048) is funny, but it brings up a good point: are there any good methods for deciding when a particular curve fits a particular set of data points? Should statistics & spreadsheet programs utilize such a method to warn users that the fit they've chosen isn't very good? I've thought of somehow using color along the curve to indicate where the fit is good (perhaps "green") and portions along the curve where the fit isn't so good (perhaps "red"). A *spline* curve could also indicate its tension by means of color, for example. (Of course, none of this is going to help those who are color blind!) https://xkcd.com/2048/

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

23 Sep '18

A basic technique in statistics is "regression", which just means fitting a curve (or surface) to data points. This has been studied extensively for 200 years since Gauss invented least squares. Statisticians have very precise measures of how well a curve fits the data. (E.g., "A Second Course in Statistics: Regression Analysis" (7th Edition) by Mendenhall and Sincich.) —Dan Henry Baker wrote: ----- Should statistics & spreadsheet programs utilize such a method to warn users that the fit they've chosen isn't very good? -----

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[math-fun] For the logarithm of 2
by françois mendzina essomba2 23 Sep '18

23 Sep '18

Hello... log(2)=2*sum(((-1)^(n+1)*cosh(Pi*n))/((sqrt(cosh(Pi)^2+1)+cosh(Pi))^n*n),n=1..infinity); for all t log(2)=2*sum(((-1)^(n+1)*cosh(n*t))/(n*(sqrt(cosh(t)^2+1)+cosh(t))^n),n=1..infinity);

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

23 Sep '18

Hi Richard: Uh, ok. What would be the correct weighting in this instance? (I've actually worked with such power laws before -- e.g., for caches hits/misses.) 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. > >Definitely need a warning. > >--R > >On Sun, Sep 23, 2018 at 10:34 AM Henry Baker <hbaker1(a)pipeline.com> wrote: >> The following xkcd comic (#2048) is funny, but it brings up a good point: >> are there any good methods for deciding when a particular curve fits a >> particular set of data points? >> >> Should statistics & spreadsheet programs utilize such a method to warn >> users that the fit they've chosen isn't very good? >> >> I've thought of somehow using color along the curve to indicate where the >> fit is good (perhaps "green") and portions along the curve where the fit >> isn't so good (perhaps "red"). A *spline* curve could also indicate its >> tension by means of color, for example. (Of course, none of this is going >> to help those who are color blind!) >> >> https://xkcd.com/2048/

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Re: [math-fun] Atiyah RH claim?
by Dan Asimov 22 Sep '18

22 Sep '18

The abstract someone posted (if genuine) reads: ----- Title: The Riemann Hypothesis Abstract: The Riemann Hypothesis is a famous unsolved problem dating from 1859. I will present a simple proof using a radically new approach. It is based on work of von Neumann (1936), Hirzebruch (1954) and Dirac (1928). ----- I note only that neither the title nor the abstract state unambiguously that there will be a proof *of the Riemann hypothesis* per se. —Dan ----- Anyone have opinions, rumors, hearsay, etc. about Atiyah's claimed Riemann Hypothesis proof talk on Monday? -----

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Re: [math-fun] How many pieces? puzzle
by Dan Asimov 22 Sep '18

22 Sep '18

That sounds like the checkerboard lattice D_n in n-space, D_n = {(x_1, ..., x_n) in Z^n | x_1 + ... + x_n == 0 mod 2} —Dan ----- the unit-sphere packing where centres have evenly many odd integer Cartesian components. -----

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[math-fun] Atiyah RH claim?
by Michael Kleber 22 Sep '18

22 Sep '18

Anyone have opinions, rumors, hearsay, etc. about Atiyah's claimed Riemann Hypothesis proof talk on Monday? Circulating rumor: https://mobile.twitter.com/troublewithecon/status/1042611713839751169 Confirmation of what's in the abstracy by Heidelberg Laureate Forum: https://twitter.com/HLForum/status/1042793378629541889 John Baez's counter, that Aityah's other recent big claims have not held up: https://mobile.twitter.com/johncarlosbaez/status/1042609961216335874 "Everyone who knows him well has been too embarrassed to publicly discuss the reasons." --Michael -- Forewarned is worth an octopus in the bush.

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