September 2018 - mathfun@mailman.xmission.com

Re: [math-fun] Curve-fitting methods ?
by Keith F. Lynch 27 Sep '18

27 Sep '18

Brent Meeker <meekerdb(a)verizon.net> wrote: > What you do is iterate. You can take three points and using > spherical geometry determine a radius and a center. But any three points that obey the triangle inequality are compatible with any radius of curvature whatsoever. (Except for a radius less than that which makes them coplanar with the center of the circle they're on.)

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Re: [math-fun] Curve-fitting methods ?
by Keith F. Lynch 27 Sep '18

27 Sep '18

Henry Baker <hbaker1(a)pipeline.com> wrote: > Keith F. Lynch wrote: >> 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. > I'll bite. Have you constructed this crude picture of Earth, or are > you suggesting that it could be done? The latter, as I thought would be clear from, "there's little point in my writing such software...." In addition to the software, I'd need hardware with which I could measure the brightnesses of the earthlit part of the moon, the sunlit part of the moon, and an equal area of the sky near the moon. I'd subtract out the brightness of the sky near the moon so I'm only paying attention to light from the moon. I'd look at the ratio of brightnesses of the sunlit and earthlit parts (as adjusted for lunar phase and for how the lunar albedo varies due to lunar topography) to compensate for absorption as the light approaches me through Earth's atmosphere. Ideally this would be done in full color, so as to get a full color image of Earth. > I would be interested in such a picture, as it is an analogy to the > "1 pixel camera" discussed here previously. Years ago, I saw images of Pluto based entirely on its brightness measured from Earth. The north-south resolution was almost nil, presumably because all the observations were made at nearly the same Pluto latitude. Since Pluto has a high obliquity, this can be corrected by watching it for a whole orbit, 248 years. I doubt anyone will bother, since today we have high resolution images from New Horizons, a Pluto flyby probe. Indeed, I can't find those old low-res images online, since nearly any request for Pluto images returns images from that space probe. We can do better with the moon, since its terrestrial latitude varies from about -30 to +30 degrees. And we only have to wait 19 years for the moon to be in every possible state relative to the sun and our planet, not 248. That should give enough data that averaging would remove the clouds from the image and let you clearly see the continents, oceans, ice caps, and major islands. I'm a fairly radical skeptic. I believe trusted authorities are often lying to us, are mistaken, or are repeating others' lies or innocent mistakes. (As I myself recently did here with a mistaken claim about smooth functions. Sorry.) So I prefer to observe, measure, calculate, reason out, and prove as many things for myself as possible. If anyone's curious why I believe this, please ask me off-list. Thanks.

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

26 Sep '18

Brent Meeker <meekerdb(a)verizon.net> wrote: > Keith F. Lynch wrote: >> A much easier project is using a table of distances between cities >> to calculate the radius of the Earth. (Has anyone done that?) > I'm sure it's been done. I've seen it illustrated in textbooks. I'm curious what the best way is. Here are my first thoughts: The first city can be given any coordinates. The second city can be placed anywhere on a sphere around the first city with a radius equal to their distance. The third city can be placed anywhere on the circle where the spheres of the correct distances from the first two cities intersect. The fourth city can be placed on either of the two points where the three correct-distance spheres intersect. But what about the fifth city? Unless the data are of infinite precision and accuracy, there probably won't be an exact intersection point of the correct-distance spheres around the first four cities. You could do a least-squares fit, but why should all of the error budget go to the fifth city? You could do simulated annealing, or something like it, where all the cities are free to wiggle around by smaller and smaller amounts so as to minimize the sum of the squares of the errors in their pairwise distances. Anyhow, after you have N cities, you simply find a point equidistant from all of them, or as close to it as possible. That will be the center of the Earth. Again, you'd probably want to do simulated annealing to distribute the error budget. Maybe combine it with the first annealing. What is wrong with this picture? The main problem is that the distances from the table were never intended to be straight line distances, but distances along the curve of the Earth. You can convert from one to the other easily enough, but only if you already know the Earth's radius, which is what we set out to find out. Maybe if you repeated the process several times, using the radius from the previous run to get the curvatures for the current run, it would converge to the correct value? > Of course cities which are far from all the others provide better > estimates, since the proportional distance error is smaller. Another approach is to stop with four cities. Which four? As you said, the ones that are furthest from each other. (Assuming all table entries are equally accurate and precise.) But four cities have six mutual distances. Which one(s) should be maximized? The longest? The shortest? The sum of all six? The product of all six? The volume of the resulting tetrahedron? If I recall correctly, just as any three points (if not all on a line) are on a circle, i.e. are equidistant from a unique point, any four points (if not all on a plane) are on a sphere, i.e. are equidistant from a unique point, the center of that sphere. Of course you'd still need to repeat the process with successive estimates of the curvature. And you'd want to do it for more than one set of four cities. Then there's the simplest approach of all: Find the maximum distance between any two cities, add a little, and call that half the circumference of the Earth.

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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 26 Sep '18

26 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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