March 2018 - mathfun@mailman.xmission.com

[math-fun] Vacationing in France
by David Wilson 08 Apr '18

08 Apr '18

The wife and I are thinking about a vacation to Paris, which is one of her bucket list items. The only people in France with whom I have correspondence are on seqfan, e.g. M. Gerard or Dr. Plouffe, others? I was hoping I might get some inside information on when, why and how to do Paris.

8 10

[math-fun] Structured light (was Single pixel cameras)
by Keith F. Lynch 01 Apr '18

01 Apr '18

Henry Baker <hbaker1(a)pipeline.com> wrote: > A classical (1930's-1990's) TV camera *raster scans* an image (which > hopefully remains still for the duration of the scan) with a single > point spot, so one could -- at least in theory -- dispense with all > of the lenses, stage lights, etc., and simply have a single-pixel > sensor which receives light that was produced by a laser beam > producing a single point scanned over the scene whose image is > being captured. I came up with almost the same idea decades ago. My idea was a "poor man's video camera." A black and white TV would be adjusted to display a bright blank raster. A lens would be used to project this raster onto the scene you want to get a video image of. The signal would be picked up by a photocell. I realized that the image would look as if the camera was where the projecting lens was and as if the light source was where the photocell was. You could aim the photocell at the ceiling to get more uniform illumination rather than a spotlight effect. You could have multiple photocells with color filters in front of them to get a color image. You'd need to exclude light from all other sources, as it would tend to wash out the image, reducing the contrast. I later realized that the image quality would be poor due to the slow phosphor on the TV. But I'm sure it would have worked. I wonder if anyone ever did it. I'm curious how you'd scan the laser over the scene. Mechanically? For a color image you'd need at least three lasers, one for each primary color. I've given a lot of thought to what would be possible with what I call structured light, light whose properties (wavelength, polarization, duration, direction, etc.) are carefully controlled, together with detectors that are sensitive to those properties. It might be possible to replace x-ray machines for subjects that aren't totally opaque to visible light. Generate a very brief (picosecond) flash of light on one side of the subject, and watch for just the *first* light to pass through the subject. That would be light that took a direct path rather than being scattered. A variation of that is to take the difference between the brightness from each direction light that's reflected after slightly different periods of time. For instance if someone is buried in an avalanche, it should be possible to look successively deeper through the snow, and notice a difference when the time allowed is just barely long enough for the pulse of light to reach the subject before being reflected back. Since the pulse would be very brief and a very narrow wavelength, background light wouldn't interfere significantly, unless you were looking through something that was almost opaque. Another idea that I had years ago was that it would be possible to reconstruct what's on a video screen if you can see a surface that's illuminated by the screen. Someone eventually did it; see https://dl.acm.org/citation.cfm?id=830537 Then there's the laser microphone, a way of listening to whatever you can illuminate with a laser. And you can read all the data being transmitted over a modem if you have a good view of the modem lights -- or even of a surface that's illuminated by them. And there's a way of seeing around corners using lasers: https://www.youtube.com/watch?v=JWDocXPy-iQ With some effort, it ought to be possible to play a CD or DVD without taking it out of its box, even without taking its box down from the shelf. Or to read closed books on a shelf. Party trick: Consider a set of six wavelengths, red wavelengths R1 and R2 which are indistinguishable to the eye, green wavelengths G1 and G2 which are indistinguishable to the eye, and blue wavelengths B1 and B2 which are indistinguishable to the eye. Consider a pigment or dye which reflects R1, G1, and B1, but not R2, G2, or B2. By switching the LEDs illuminating a room between R1 and R2, between G1 and G2, and between B1 and B2, you can make a surface covered with that pigment or dye any color you like. Wearing what looks like an ordinary cotton t-shirt which keeps changing color will make you the hit of the party. Ultimately, if you can capture and reproduce the instantaneous amplitude of light with a space resolution small compared to the wavelength of light and a time resolution short compared to the frequency of light, it should be possible to re-create any visual experience whatsoever, including a "TV screen" you can use a telescope, spectroscope, or microscope on, or an invisibility cloak. Of course this would require processors millions of times faster than today's. Good luck hiding the waste heat. > Of course, there's nothing special about the raster scanning > pattern; indeed, *any* pattern which covers the image will work, > so long as the reconstruction laser follows the same pattern. > I believe that some of the alternatives to 1920's and 1930's > TV systems used other nonraster scanning schemes. Yes. Mechanical systems work better with smooth curves than with the sawtooth waveform used with electronic TVs. > But wait -- there's more! > There's also nothing special about using a *point* of light! One > could illuminate the picture to be scanned with a long sequence of > random 2D patterns of light, with the single-pixel sensor converting > its *average* taken over the entire scene into a time-varying signal > whose next value would be the *average over the entire scene* of the > light pattern reflected from the next random pattern, and so on. For greatest efficiency, the 2D patterns should be Costas arrays. If I recall correctly, an image from a multiple-pinhole camera can be disambiguated only if the pinholes form a Costas array.

3 4

[math-fun] Two sums
by françois mendzina essomba2 31 Mar '18

31 Mar '18

hello I noticed that: for some values of a and b... theta=(1/b)*sum(((-1)^(n+1)/n)*(sin(b*theta)/sin(a*theta))^n*sin(n*theta*(b+a)),n=1..inf) ; I have deducted: theta=(sum(((-1)^(n+1)*(sinh(b*theta))^n*sinh((b+a)*n*theta))/(n*(sinh(a*theta))^n),n=1..inf))/b; an example : a=1/15 ; b= 1/30 ; theta = Pi FME...

2 1

Re: [math-fun] Structured light (was Single pixel cameras)
by rcs＠xmission.com 30 Mar '18

30 Mar '18

[from Bill Ackerman -- Rich] Date: Fri, 30 Mar 2018 00:53:11 -0400 Subject: Re: [math-fun] Structured light (was Single pixel cameras) From: wba <wbackerman(a)gmail.com> Actually, the "P4" phosphor of old televisions was extremely fast. Much faster than the "P1" phosphor of oscilloscope tubes, for example. It was so fast that, if you took a picture with a sufficiently fast shutter, you would see only a single dot. When you watched TV, your eyes were seeing nothing but a moving dot of light. This is the principle of the "flying spot scanner", which was a "poor man's TV camera", back when actual image orthicons cost over $1000 in 1960 dollars. You arrange for the raster of a TV (with no modulation on the signal) to go through a transparency or equivalent arrangement, into a photocell. There was a cute instance of this, for fairly sophisticated electronics hobbyists. One can scan, for example, 35mm slides (the way serious photography enthusiasts took their pictures back then) by wiring up a slide projector with a good-quality photocell where the projection lamp should be, and aim and focus the projector at the TV, with its video modulation shut off. Use the slide projector mechanism in the usual way, send the output to your ham radio 450MHz transmitter or whatever, and bore your ham radio friends with pictures of your vacation, the same way you do when you invite them over. ---- On 03/30/2018 12:21 AM, Keith F. Lynch wrote: Henry Baker <hbaker1(a)pipeline.com> wrote: A classical (1930's-1990's) TV camera *raster scans* an image (which hopefully remains still for the duration of the scan) with a single point spot, so one could -- at least in theory -- dispense with all of the lenses, stage lights, etc., and simply have a single-pixel sensor which receives light that was produced by a laser beam producing a single point scanned over the scene whose image is being captured. I came up with almost the same idea decades ago. My idea was a "poor man's video camera." A black and white TV would be adjusted to display a bright blank raster. A lens would be used to project this raster onto the scene you want to get a video image of. The signal would be picked up by a photocell. I realized that the image would look as if the camera was where the projecting lens was and as if the light source was where the photocell was. You could aim the photocell at the ceiling to get more uniform illumination rather than a spotlight effect. You could have multiple photocells with color filters in front of them to get a color image. You'd need to exclude light from all other sources, as it would tend to wash out the image, reducing the contrast. I later realized that the image quality would be poor due to the slow phosphor on the TV. But I'm sure it would have worked. I wonder if anyone ever did it. I'm curious how you'd scan the laser over the scene. Mechanically? For a color image you'd need at least three lasers, one for each primary color. I've given a lot of thought to what would be possible with what I call structured light, light whose properties (wavelength, polarization, duration, direction, etc.) are carefully controlled, together with detectors that are sensitive to those properties. It might be possible to replace x-ray machines for subjects that aren't totally opaque to visible light. Generate a very brief (picosecond) flash of light on one side of the subject, and watch for just the *first* light to pass through the subject. That would be light that took a direct path rather than being scattered. A variation of that is to take the difference between the brightness from each direction light that's reflected after slightly different periods of time. For instance if someone is buried in an avalanche, it should be possible to look successively deeper through the snow, and notice a difference when the time allowed is just barely long enough for the pulse of light to reach the subject before being reflected back. Since the pulse would be very brief and a very narrow wavelength, background light wouldn't interfere significantly, unless you were looking through something that was almost opaque. Another idea that I had years ago was that it would be possible to reconstruct what's on a video screen if you can see a surface that's illuminated by the screen. Someone eventually did it; see https://dl.acm.org/citation.cfm?id=830537 Then there's the laser microphone, a way of listening to whatever you can illuminate with a laser. And you can read all the data being transmitted over a modem if you have a good view of the modem lights -- or even of a surface that's illuminated by them. And there's a way of seeing around corners using lasers: https://www.youtube.com/watch?v=JWDocXPy-iQ With some effort, it ought to be possible to play a CD or DVD without taking it out of its box, even without taking its box down from the shelf. Or to read closed books on a shelf. Party trick: Consider a set of six wavelengths, red wavelengths R1 and R2 which are indistinguishable to the eye, green wavelengths G1 and G2 which are indistinguishable to the eye, and blue wavelengths B1 and B2 which are indistinguishable to the eye. Consider a pigment or dye which reflects R1, G1, and B1, but not R2, G2, or B2. By switching the LEDs illuminating a room between R1 and R2, between G1 and G2, and between B1 and B2, you can make a surface covered with that pigment or dye any color you like. Wearing what looks like an ordinary cotton t-shirt which keeps changing color will make you the hit of the party. Ultimately, if you can capture and reproduce the instantaneous amplitude of light with a space resolution small compared to the wavelength of light and a time resolution short compared to the frequency of light, it should be possible to re-create any visual experience whatsoever, including a "TV screen" you can use a telescope, spectroscope, or microscope on, or an invisibility cloak. Of course this would require processors millions of times faster than today's. Good luck hiding the waste heat. Of course, there's nothing special about the raster scanning pattern; indeed, *any* pattern which covers the image will work, so long as the reconstruction laser follows the same pattern. I believe that some of the alternatives to 1920's and 1930's TV systems used other nonraster scanning schemes. Yes. Mechanical systems work better with smooth curves than with the sawtooth waveform used with electronic TVs. But wait -- there's more! There's also nothing special about using a *point* of light! One could illuminate the picture to be scanned with a long sequence of random 2D patterns of light, with the single-pixel sensor converting its *average* taken over the entire scene into a time-varying signal whose next value would be the *average over the entire scene* of the light pattern reflected from the next random pattern, and so on. For greatest efficiency, the 2D patterns should be Costas arrays. If I recall correctly, an image from a multiple-pinhole camera can be disambiguated only if the pinholes form a Costas array.

1 0

[math-fun] Single pixel cameras
by Henry Baker 30 Mar '18

30 Mar '18

There have been a number of recent articles (from MIT and others) about "single pixel" cameras and "ghost imaging". But IMHO most of these descriptions confuse rather than promote understanding. Here's my take: A classical (1930's-1990's) TV camera *raster scans* an image (which hopefully remains still for the duration of the scan) with a single point spot, so one could -- at least in theory -- dispense with all of the lenses, stage lights, etc., and simply have a single-pixel sensor which receives light that was produced by a laser beam producing a single point scanned over the scene whose image is being captured. The classical TV camera thus converts a 2D image into a "1D" intensity v. time signal. The 2D image can be reconstructed by the same scanning pattern from the same laser, at least in theory, by varying the laser intensity with the original captured intensity values as this laser scans a *blank* screen. Of course, there's nothing special about the raster scanning pattern; indeed, *any* pattern which covers the image will work, so long as the reconstruction laser follows the same pattern. I believe that some of the alternatives to 1920's and 1930's TV systems used other nonraster scanning schemes. But wait -- there's more! There's also nothing special about using a *point* of light! One could illuminate the picture to be scanned with a long sequence of random 2D patterns of light, with the single-pixel sensor converting its *average* taken over the entire scene into a time-varying signal whose next value would be the *average over the entire scene* of the light pattern reflected from the next random pattern, and so on. We can now reconstruct the picture in a manner reminiscent of the laser point scanner by projecting the same sequence of random 2D patterns onto a blank screen and modulating their intensity of the (single) projector bulb with the time-varying signal we just captured. With modern stage lighting coming from large arrays of LED's, whose intensities can be varied at very high frequencies, it should be possible to go through a pattern sequence fast enough to capture a scene in pretty decent resolution relatively quickly. Now we could also choose 2D illuminating light patterns that cycle through various horizontal and vertical wave patterns with various spatial wavelengths and phases. Voila! We have just approximated a 2D Fourier transform of the 2D image.

2 1

Re: [math-fun] Infinitely long de Bruijn-like chains ?
by Henry Baker 29 Mar '18

29 Mar '18

OK, every random bitstring "eventually" contains as a substring every 2^k-bit integer, for every k. It should be relatively easy to compute the length of a random string necessary to make sure that the probability of seeing *every* 2^k-bit integer is > 1/2 (or > epsilon, for some a priori epsilon). Suppose I want to check this particular property of a RNG; how long should I wait -- i.e., how many bits -- before becoming suspicious that the RNG is somehow *censoring* certain subsequences? If I want to "censor" a RNG, I could simply keep a k-bit shift-register and look for the sequence I want to censor and simply flip the last bit in order to disagree. If want to censor 2 different substrings, I could keep 2 different shift-registers and comparators. I could also use the various sequence comparison hacks -- e.g., comparing the searched sequences with themselves to optimize how much state to keep around when checking for multiple censored subsequences. Suppose someone gives me a censored RNG, but no indication of k -- the number of bits in the censored sequence. How should I organize my computation to check for censoring for every k<n for some a priori maximum n that I'm willing to check for? At 02:05 PM 3/29/2018, Adam P. Goucher wrote: >I've wondered the same question before. My not-particularly-elegant solution >is to take, for all n, the de Bruijn sequence of length 2^n, cyclically >permuted such that it begins with a maximal string of zeroes, and concatenate >them. So we get something like this: > >01 0011 00010111 0000... > >It's half as efficient as a single de Bruijn code, because you might have >to go up to 2^(n+1) to find your favourite length-n bitstring. > >Best wishes, > >Adam P. Goucher > >> Sent: Thursday, March 29, 2018 at 1:36 AM >> From: "Fred Lunnon" <fred.lunnon(a)gmail.com> >> To: math-fun <math-fun(a)mailman.xmission.com> >> Subject: Re: [math-fun] Infinitely long Gray-like codes ? >> >> de Bruijn cycles might be a better starting point --- see OEIS A080679, and >> https://en.wikipedia.org/wiki/De_Bruijn_sequence >> >> WFL >> >> On 3/29/18, Henry Baker <hbaker1(a)pipeline.com> wrote: >> > Most of the applications in the Wikipedia article on Gray codes >> > consider *fixed length* codes for some length n. >> > >> > One of the properties of binary gray codes is that substrings of them >> > contain all of the n-bit binary integers, but far more efficiently >> > than simply concatenating all 2^n of them. >> > >> > I'm interested in a slightly different *infinite* sequence of >> > bits, which contain these n-bit substrings as early as possible, >> > s.t., for any n, there is a function f(n) that tells my how >> > large of an initial substring of this infinite sequence I have >> > to generate in order to make sure that all 2^n binary integers >> > appear at least once. >> > >> > Furthermore, I may want to "tune" this infinite sequence in >> > order to change the statistics of how frequently all of the >> > different k-bit integers appear, how frequently all of the >> > (k+1)-bit integers appear, etc. >> > >> > Has anyone studied such sequences?

1 0

[math-fun] Sprained Wankels
by Bill Gosper 29 Mar '18

29 Mar '18

Brent, I'm not sure that was for math-fun, but you MOG guys might like http://toobnix.org/?cat=62 particularly the possibility of compression via non-constant pitch. --rwg

1 0

[math-fun] Infinitely long Gray-like codes ?
by Henry Baker 29 Mar '18

29 Mar '18

Most of the applications in the Wikipedia article on Gray codes consider *fixed length* codes for some length n. One of the properties of binary gray codes is that substrings of them contain all of the n-bit binary integers, but far more efficiently than simply concatenating all 2^n of them. I'm interested in a slightly different *infinite* sequence of bits, which contain these n-bit substrings as early as possible, s.t., for any n, there is a function f(n) that tells my how large of an initial substring of this infinite sequence I have to generate in order to make sure that all 2^n binary integers appear at least once. Furthermore, I may want to "tune" this infinite sequence in order to change the statistics of how frequently all of the different k-bit integers appear, how frequently all of the (k+1)-bit integers appear, etc. Has anyone studied such sequences?

3 2

[math-fun] Infinitely long *chain-like* codes ?
by Bill Gosper 29 Mar '18

29 Mar '18

This one' s not full period, but here's a Golly visualization of a shift register sequence. --rwg # Pseudorandom LWSS generator with p46 logic # A lightweight spaceship stream is emitted representing a # pseudorandom binary sequence satisfying the recursion # a[n] = a[n-1] XOR a[n-19]. (By spreading out the pattern, # the 19 can be changed.) The sequence has period 413385, # so the gun has period 46*413385 = 19015710. # Built by Bill Gosper, October 1989 or earlier # (Header by Dean Hickerson, 4/11/93) x = 167, y = 139, rule = B3/S23 77b2o7b2o$77b2o6bo2b2o$76bo2bo4b6o$77b2obo5b4o$77b2obo2$77b2obo$77b2ob o5b4o$74b2ob3o4b6o11b2o$74bobobo6bo2b2o11b2o$75b4o7b2o$76b2o40$22b2o$ 22b2o5$157b2o$157b2o5$23b2o3b2o5$21b2obo3bob2o$21bo2bo3bo2bo$22b3o3b3o 126b2o5b2o$156bobo5bobo$156bo2bo3bo2bo$157b2o5b2o5$22b2o$22b2o5$157b2o $157b2o9$2b2o$2b2o6$2b3o3b3o$bo2bo3bo2bo$bo3bobo3bo$2obobobobob2o$2ob 2o3b2ob2o69b2o$b3o5b3o69bo2b2o$49b2o30bo3bo$48bo2bo30bo2bo$48bo2bo30b 3o$49b3o$82b3o$9bo72bo2bo$8bobo58b2o10bo3bo10b2o$7bo3bo37b3o17b2o10bo 2b2o10b2o$7b5o24b2o10bo2bo11b2o17b2o$6bobobobo23b2o10bo2bo11b2o$7bo3bo 37b2o2$7bo3bo$6bobobobo$2b2o3b5o76b2o$2b2o3bo3bo75bobo$8bobo75b2obo$9b o38b3o3bo32b2o26b2o$46b5o2bobo32bo25b2ob2o$41bo3b2o3b2o3bo59bo2bo$36bo 3bo4b3o3bo2bo33bo26bo2bo$35bo8bob2o3b3o33b2o27b2o$35bo2b2o5bo32b2o6b2o bo$36b2o5bo2b2o3b3o24b2o7bobo26b2o$37b3o3b5o3bo2bo33b2o25bo2bo$45b2o3b 2o3bo12b2o45bo2bo6b2o$37b3o3b8o2bobo12b2o44b2ob2o6b2o$36b2o5bo2bob3o3b o60b2o$21b2o12bo2b2o5bo$21b2o12bo8b2o$36bo3bo$41bo! Subject: Re: [math-fun] Infinitely long Gray-like codes ? Date: 2018-03-28 17:36 From: Fred Lunnon <fred.lunnon(a)gmail.com> To: math-fun <math-fun(a)mailman.xmission.com> Reply-To: math-fun <math-fun(a)mailman.xmission.com> de Bruijn cycles might be a better starting point --- see OEIS A080679, and https://en.wikipedia.org/wiki/De_Bruijn_sequence WFL On 3/29/18, Henry Baker <hbaker1(a)pipeline.com> wrote: > Most of the applications in the Wikipedia article on Gray codes > consider *fixed length* codes for some length n. > > One of the properties of binary gray codes is that substrings of them > contain all of the n-bit binary integers, but far more efficiently > than simply concatenating all 2^n of them. > > I'm interested in a slightly different *infinite* sequence of > bits, which contain these n-bit substrings as early as possible, > s.t., for any n, there is a function f(n) that tells my how > large of an initial substring of this infinite sequence I have > to generate in order to make sure that all 2^n binary integers > appear at least once. > > Furthermore, I may want to "tune" this infinite sequence in > order to change the statistics of how frequently all of the > different k-bit integers appear, how frequently all of the > (k+1)-bit integers appear, etc. > > Has anyone studied such sequences?

1 0

[math-fun] For a dance...
by françois mendzina essomba2 29 Mar '18

29 Mar '18

I have no worries, the Cameroonian law is not interested in Poligamists... (Gamma(1/4))^2/(8*sqrt(Pi))=(((sqrt(17)+2^(5/2))*sqrt(257)+2^(9/2)*sqrt(17))*sqrt(337)+2^(9/2)*sqrt(17)*sqrt(257))/(2^(5/2)*sqrt(17)*sqrt(257)*sqrt(337)) + sum(S(n),n=1..infinity) ; With S(n)=2^(-n-2)*sum(1/sqrt((2*k-1)^4*2^(-4*n-8)+1)-1/sqrt(k^4*2^(-4*n-4)+1),k=1..2^(n+1)); FME...

1 0