September 2008 - mathfun@mailman.xmission.com

[math-fun] Density of {n > 0 | phi(n)/n <= c}
by Dan Asimov 05 Sep '08

05 Sep '08

Just saw this 1928 theorem discussed on sci.math.research: given any c, 0 < c < 1, the set S(c) of positive integers n satisfying phi(n)/n <= c has a well-defined density D(c) in {1,2,3,...}. And, D(c) is a continuous function of c. It's believed to have an infinite derivative at some points, and a (finite) one-sided derivative at the others, but these things seem to be unproven. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] a=18; 2a=36; 3*6=a
by Eric Angelini 05 Sep '08

05 Sep '08

Hello Math-Fun, are there many integers "a" having the property that the digit-product of "2a" equals "a"? (e.g. a = 18, 2a = 36, 3*6 = a) Best, É.

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Re: [math-fun] Reflection from a moving mirror
by Eugene Salamin 04 Sep '08

04 Sep '08

At 05:47 PM 9/3/2008, Eugene Salamin wrote: >Test your physical intuition. We know that when light reflects from a stationary mirror > >(1) The frequency of the reflected light equals the frequency of the incident light, > >(2) The direction of the incident ray, the direction of the reflected ray, and the mirror normal are coplanar, and > >(3) The angle of reflection equals the angle of incidence. > >Suppose however, that the mirror is moving parallel to its surface. Do these three principles of reflection continue to hold? > >Gene _______________________________________________ The links to papers reporting experimental tests give only the abstracts. The Nature paper describes an experiment with frequency resolution of one part in 10^16. Unfortunately, the authors, in their excitement, forgot to include the conclusions in the abstract. Can someone tell us what were the measured results? Meanwhile, I will stick out my neck, and present my analysis of the problem. Let's solve the problem two ways, first by Lorentz transforming from the rest frame of the mirror, and second by a clever group theoretic trick. A plane wave of light is is described by a wave vector k which points in the direction of propagation and has length 2 pi/lambda, lambda being the wavelength. The angular frequency (= 2 pi frequency) is c |k| = ck. In relativity, this 3-vector is combined with angular frequency to form the 4-vector which gets Lorentz transformed. For simplicity assume units with c = 1. In the rest frame of the mirror, which is taken to be the xy-plane, the wave vectors can be resolved in direction cosines, and then the incident and reflected wave 4-vectors are ki = k (1, cos a, cos b, -cos c), kr = k (1, cos a, cos b, cos c). If we now transform to a frame moving with velocity v/c = tanh s along the x-axis, the Lorentz transformation matrix is block diagonal. It is the identity in the yz-plane, and in the tx plane is [[cosh s, sinh s], [sinh s, cosh s]]. In the frame in which the mirror is moving, we thus have ki'[t] = kr'[t] = k (cosh s + sinh s cos a) = k', ki'[x] = kr'[x] = k (sinh s + cosh s cos a) = k' cos a', ki'[y] = kr'[y] = k cos b = k' cos b', - ki'[z] = kr'[z] = k cos c = k' cos c'. Although there is a Doppler shift, from k to k', between the two frames, in the one moving frame, the frequencies of the incident and reflected light are the same. Also in the one moving frame, direction cosines of the two light rays are the same parallel to the mirror, and reversed perpendicular to the mirror, just as for a fixed mirror. Thus all three reflection principles hold for a mirror moving parallel to its surface. There is one caveat. The frequency in the rest frame of the mirror is different from that applied in the lab frame. If the material no longer behaves as a mirror in its rest frame at the shifted frequency, then it will not behave as a mirror in the lab frame. Now let's solve the reflection problem the clever way. The incident plane wave has a space-time variation given by (again with c = 1) exp(i (ki[x] x + ki[y] y + ki[z] z - ki t)). The physical arrangement is invariant under translation in the xy-plane. Hence the factors exp(i ki[x] x) and exp(i k[y] y), which constitute a one-dimensional representation of the translation group, should continue to describe the reflected wave. Likewise, the situation is invariant under translation in time, so the factor exp(- i ki t) should also persist. So kr[x] = ki[x], kr[y] = ki[y], kr = ki. We also must have k[x]^2 + k[y]^2 + k[z]^2 = k^2. So we can solve for kr[z]: k[z] = +/- ki[z]. The incident wave goes toward the mirror, the reflected wave away, so kr[z] = - ki[z]. This demonstrates the three stated principles of reflection. The proof remains valid whether the mirror is stationary or moving uniformly at constant velocity parallel to its surface. [What happens if light reflects from a rotating disk?] The principle of translation invariance implies that the wavelength is sufficiently long that it cannot sense the microcrystalline or atomic structure of the mirror material. The same idea can be used to derive Snell's law for refraction for a stationary medium of refractive index n. It is a consequence of Maxwell's equations that the transmitted wave vector kt has length n times longer than in vacuum. This is clear since the frequency stays the same (time translation invariance), the speed of light goes down by n, the wavelength goes down by n, and the wave vector goes up by n. For the transmitted wave vector, by xy translation invariance, kt[x] = ki[x], kt[y] = ki[y]. Thus kt[x]^2 + kt[y]^2 + kt[z]^2 = kt^2, ki[x]^2 + ki[y]^2 + kt[z]^2 = n^2 ki^2, ki^2 sin(theta[i])^2 + n^2 ki^2 cos(theta[t])^2 = n^2 ki^2, sin(theta[i])^2 = n^2 sin(theta[t])^2, sin(theta[i]) = n sin(theta[t]), Snell's law. There is a physical interpretation of the relation between translation invariance and the preservation of the exp(i k[x] x) factors. For a wave exp(i k.r), the momentum of a photon is p = hbar k. The preservation of exp(i k[x] x) is the preservation of the momentum component p[x]. A system that is invariant under translation in a certain direction cannot alter the component of photon momentum along that direction. Gene

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Re: [math-fun] Reflection from a moving mirror
by Eugene Salamin 04 Sep '08

04 Sep '08

At 05:47 PM 9/3/2008, Eugene Salamin wrote: >Test your physical intuition. We know that when light reflects from a stationary mirror > >(1) The frequency of the reflected light equals the frequency of the incident light, > >(2) The direction of the incident ray, the direction of the reflected ray, and the mirror normal are coplanar, and > >(3) The angle of reflection equals the angle of incidence. > >Suppose however, that the mirror is moving parallel to its surface. Do these three principles of reflection continue to hold? > >Gene _______________________________________________ The links to papers reporting experimental tests give only the abstracts. The Nature paper describes an experiment with frequency resolution of one part in 10^16. Unfortunately, the authors, in their excitement, forgot to include the conclusions in the abstract. Can someone tell us what were the measured results? Meanwhile, I will stick out my neck, and present my analysis of the problem. Let's solve the problem two ways, first by Lorentz transforming from the rest frame of the mirror, and second by a clever group theoretic trick. A plane wave of light is is described by a wave vector k which points in the direction of propagation and has length 2 pi/lambda, lambda being the wavelength. The angular frequency (= 2 pi frequency) is c |k| = ck. In relativity, this 3-vector is combined with angular frequency to form the 4-vector which gets Lorentz transformed. For simplicity assume units with c = 1. In the rest frame of the mirror, which is taken to be the xy-plane, the wave vectors can be resolved in direction cosines, and then the incident and reflected wave 4-vectors are ki = k (1, cos a, cos b, -cos c), kr = k (1, cos a, cos b, cos c). If we now transform to a frame moving with velocity v/c = tanh s along the x-axis, the Lorentz transformation matrix is block diagonal. It is the identity in the yz-plane, and in the tx plane is [[cosh s, sinh s], [sinh s, cosh s]]. In the frame in which the mirror is moving, we thus have ki'[t] = kr'[t] = k (cosh s + sinh s cos a) = k', ki'[x] = kr'[x] = k (sinh s + cosh s cos a) = k' cos a', ki'[y] = kr'[y] = k cos b = k' cos b', - ki'[z] = kr'[z] = k cos c = k' cos c'. Although there is a Doppler shift, from k to k', between the two frames, in the one moving frame, the frequencies of the incident and reflected light are the same. Also in the one moving frame, direction cosines of the two light rays are the same parallel to the mirror, and reversed perpendicular to the mirror, just as for a fixed mirror. Thus all three reflection principles hold for a mirror moving parallel to its surface. There is one caveat. The frequency in the rest frame of the mirror is different from that applied in the lab frame. If the material no longer behaves as a mirror in its rest frame at the shifted frequency, then it will not behave as a mirror in the lab frame. Now let's solve the reflection problem the clever way. The incident plane wave has a space-time variation given by (again with c = 1) exp(i (ki[x] x + ki[y] y + ki[z] z - ki t)). The physical arrangement is invariant under translation in the xy-plane. Hence the factors exp(i ki[x] x) and exp(i k[y] y), which constitute a one-dimensional representation of the translation group, should continue to describe the reflected wave. Likewise, the situation is invariant under translation in time, so the factor exp(- i ki t) should also persist. So kr[x] = ki[x], kr[y] = ki[y], kr = ki. We also must have k[x]^2 + k[y]^2 + k[z]^2 = k^2. So we can solve for kr[z]: k[z] = +/- ki[z]. The incident wave goes toward the mirror, the reflected wave away, so kr[z] = - ki[z]. This demonstrates the three stated principles of reflection. The proof remains valid whether the mirror is stationary or moving uniformly at constant velocity parallel to its surface. [What happens if light reflects from a rotating disk?] The principle of translation invariance implies that the wavelength is sufficiently long that it cannot sense the microcrystalline or atomic structure of the mirror material. The same idea can be used to derive Snell's law for refraction for a stationary medium of refractive index n. It is a consequence of Maxwell's equations that the transmitted wave vector kt has length n times longer than in vacuum. This is clear since the frequency stays the same (time translation invariance), the speed of light goes down by n, the wavelength goes down by n, and the wave vector goes up by n. For the transmitted wave vector, by xy translation invariance, kt[x] = ki[x], kt[y] = ki[y]. Thus kt[x]^2 + kt[y]^2 + kt[z]^2 = kt^2, ki[x]^2 + ki[y]^2 + kt[z]^2 = n^2 ki^2, ki^2 sin(theta[i])^2 + n^2 ki^2 cos(theta[t])^2 = n^2 ki^2, sin(theta[i])^2 = n^2 sin(theta[t])^2, sin(theta[i]) = n sin(theta[t]), Snell's law. There is a physical interpretation of the relation between translation invariance and the preservation of the exp(i k[x] x) factors. For a wave exp(i k.r), the momentum of a photon is p = hbar k. The preservation of exp(i k[x] x) is the preservation of the momentum component p[x]. A system that is invariant under translation in a certain direction cannot alter the component of photon momentum along that direction. Gene

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Re: [math-fun] Reflection from a moving mirror
by Eugene Salamin 04 Sep '08

04 Sep '08

----- Original Message ---- From: Henry Baker <hbaker1(a)pipeline.com> To: Eugene Salamin <gene_salamin(a)yahoo.com> Cc: math-fun <math-fun(a)mailman.xmission.com> Sent: Wednesday, September 3, 2008 6:04:02 PM Subject: Re: [math-fun] Reflection from a moving mirror If this experiment occurred in a vacuum, and the wavelength of the light far exceeded the surface roughness of the mirror, how would the light even know that the mirror was moving? -------------------- Consider the propagation of light in a medium of refractive index n. When the medium is at rest, the velocity is c/n. If the medium moves with velocity v in the same direction as the light, the velocity of the light follows from the relativistic velocity addition formula: v' = (c/n + v)/(1 + (v/c)/n) = c/n + (1 - n^-2) v, approximately for v << c. This was measured interferometrically in flowing water by Fizeau in 1851, and the (1 - n^-2) "aether drag" coefficient was confirmed. Modern electronics permits the experiment to be done by timing pulses of light. In this case, the group index replaces the refractive index in the above equations. How does the light even know that the water is moving? Remark about the velocity addition formula. Parametrize the velocity by v/c = tanh s. Then v1 "+" v2 = v3 becomes s1 + s2 = s3. This s is the natural group parameter for Lorentz boosts, just as angle is for rotations. Gene At 05:47 PM 9/3/2008, Eugene Salamin wrote: >Test your physical intuition. We know that when light reflects from a stationary mirror > >(1) The frequency of the reflected light equals the frequency of the incident light, > >(2) The direction of the incident ray, the direction of the reflected ray, and the mirror normal are coplanar, and > >(3) The angle of reflection equals the angle of incidence. > >Suppose however, that the mirror is moving parallel to its surface. Do these three principles of reflection continue to hold? > >Gene _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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Re: [math-fun] Reflection from a moving mirror
by Dan Asimov 04 Sep '08

04 Sep '08

Gene wrote: << We know that when light reflects from a stationary mirror (1) The frequency of the reflected light equals the frequency of the incident light, (2) The direction of the incident ray, the direction of the reflected ray, and the mirror normal are coplanar, and (3) The angle of reflection equals the angle of incidence. Suppose however, that the mirror is moving parallel to its surface. Do these three principles of reflection continue to hold? >> Aside: --------------------------------------------------------------- [[[[[Above all, this reminded me that I have never understood why mirror reflection occurs. It's tempting to think it's just like a ball bouncing off a hard surface, but since a photon is much smaller than an atom (and certainly than an atom of, say, silver), I'd expect that even the smoothest mirror humans can make would have a rather irregular surface when measured on the scale of one photon.]]]]] --------------------------------------------------------------- Amiddle: But let's ignore all this stuff and assume mirror reflection from a stationary mirror is just like an ideal projectile's bouncing off a hard surface (but relativistically). In this case I'd start by using the frame of reference of the mirror, with the light source doing the moving. Since the speed of light does not depend on the motion of the light source, all three laws of reflection should hold *in the frame of reference of the mirror*. To get the final answer -- what an observer in the reference frame of the light source would see -- we'd then just need to express what we know about the mirror's frame of reference in terms of the light source's. We could simplify things by assuming the motion of the mirror is slow enough compared to c that we can ignore relativity. Assuming the mirror is moving at constant speed, this should put the kibosh to 1) and 3) above, but 2) should still be true. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] Reflection from a moving mirror
by Eugene Salamin 04 Sep '08

04 Sep '08

Test your physical intuition. We know that when light reflects from a stationary mirror (1) The frequency of the reflected light equals the frequency of the incident light, (2) The direction of the incident ray, the direction of the reflected ray, and the mirror normal are coplanar, and (3) The angle of reflection equals the angle of incidence. Suppose however, that the mirror is moving parallel to its surface. Do these three principles of reflection continue to hold? Gene

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Re: [math-fun] Nested Klein bottles
by Dan Asimov 02 Sep '08

02 Sep '08

Joshua wrote: << http://www.sciencemuseum.org.uk/images/I046/10314758.aspx Nice photo, nice piece of glass, and some nice humor: "It consists of three Klein bottles, one inside another." LOL! >> Interesting. (Of course to a topologist a "real" Klein bottle is a surface that can't be modeled in 3-space without self-intersection, but be that as it may.) They kinda fail to be nested along the thin tubes that curve away from the "small ends" of the bottles and re-enter along the sides. *****>>>>>So, suppose you made this standard picture of a Klein bottle in 3-space out of a 5-ply sheeting, where plies 1,3,5 are glass and plies 2,4 are some volatile solid that sublimates shortly after the model is completed. (Something like taking three stacked strips of paper and going through the usual construction of a Moebius band with them.) QUESTION: When this model is completed, what do you have, topologically? --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] Nested Klein bottles
by Joshua Zucker 01 Sep '08

01 Sep '08

http://www.sciencemuseum.org.uk/images/I046/10314758.aspx Nice photo, nice piece of glass, and some nice humor: "It consists of three Klein bottles, one inside another." LOL! --Joshua Zucker

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[math-fun] Smallest magic squares of powers
by Christian Boyer 01 Sep '08

01 Sep '08

Oct 7th, 2007: the smallest known magic square of 4th powers is now a 44x44 square constructed by Jaroslaw Wroblewski (Poland) and Hugo Pfoertner (Germany). Jaroslaw built the semi-magic square in June, then Hugo needed a very impressive computing job of 3 months, on 57 processors of a SGI computer http://www.sgi.com/products/remarketed/origin3000/, in order to find two magic diagonals in this square! A big progress, because the previous smallest known was a 243x243 square, coming from the 243x243 tetramagic square of Pan Fengchu (China). The best known results, and their first authors, are now: Magic square Smallest known ------------ --------------- Of squares 4x4 (Euler, 1770) Of cubes 9x9 (Boyer, 2006) Of 4th powers 44x44 (Wroblewski-Pfoertner, 2007) Of 5th powers 729x729 (from the pentamagic square of Li Wen, 2003) ON ANY POWER, WE DO NOT KNOW WHAT THE SMALLEST POSSIBLE MAGIC SQUARE IS! For example, as you know, we still do not know if a 3x3 magic square of squares is possible... 4x4, 5x5, 6x6, 7x7, and 8x8 magic squares of cubes are unknown... And the problem of the diagonals is extremely difficult, as proved by the impressive computing search of Hugo. It remains a big gap between the smallest known magic and semi-magic squares (=with or without magic diagonals) of 4th powers: 44x44 versus 4x4... Here is the 4x4 semi-magic square of 4th powers (Morgenstern-Boyer, 2006): 14101^4 938^4 931^4 37762^4 35866^4 20881^4 21038^4 13393^4 24806^4 30191^4 30418^4 9263^4 413^4 32026^4 31787^4 1106^4 Christian.

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