Wikipedia:Reference desk/Archives/Mathematics/2012 July 13

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July 13

Confusion about orbit of PSL(2,5)/A_5 on unordered triples (10/20?)

Hello,

I am confused about the action of the projective special linear (simple) group $PSL(2,5)$ on unordered triples. This group is isomorphic to the alternating group $A_{5}$. Hence we obtain an action of $A_{5}$ on the six points of a projective line.

I read that there should be 2 orbits of size 10 on unordered triples, both of them 2-designs.

On the other hand, I seem to obtain that the stabilizer of an unordered triple under PSL(2,5) has size 3: for the stabilizer of ${0,1,\infty }$ I find the projective transformations $x,1/(1-x),(x-1)/x$. Using the stabilizer theorem, I then obtain that one orbit should have size 60/3=20.

I must have made a silly mistake, but I can't see it? Evilbu (talk) 06:28, 13 July 2012 (UTC)

I haven't worked with these things for a while, but I think I can see what the problem is. The reciprocals of your three transformations ${\frac {1}{x}},1-x,{\frac {x}{x-1}}$ also stabilize the unordered triple $\{0,1,\infty \}$. They look like they shouldn't be in $PSL(2,5)$ because the corresponding matrices have determinant -1, but since 2 is a square root of -1 modulo 5 we can multiply the numerators and denominators by 2 and obtain ${\frac {2}{2x}},{\frac {2-2x}{2}},{\frac {2x}{2x-2}}$ for which the corresponding matrices have determinant 1. 60.234.242.206 (talk) 12:24, 14 July 2012 (UTC)

Thank you very much! For some reason I was thinking -1 was not a square, since I was also working modulo 3. I did have a reason for this: over a field of size three, and using well-chosen triples for coordinates using PSL(2,5), one can construct the Gewirtz graph on 56 vertices in a surprisingly explicit way. (Perhaps I could add that to the article on that graph)Evilbu (talk) 09:47, 15 July 2012 (UTC)

As interesting, and surprising as it may be. We have a strict policy on no original research. — Fly by Night (talk) 16:24, 16 July 2012 (UTC)

If only it were!:) It's not original research, it follows from a description of the Hill cap, due to Calderbank and Kantor. Evilbu (talk) 16:36, 16 July 2012 (UTC)

What's the correct word for "periodizing" the input to a periodic function?

In math we often "-ize" things when we transform them to a more convenient or easily comparable form (e.g. "normalizing" a vector so it has unit length, "standardizing" a normal random variable so it has mean 0 and SD 1).

Suppose I am working with a real function f(x) which is periodic along the real line with period P. I can start counting my periods from x = 0 (so the first runs from 0 to P, the second from P to 2P etc) - this is arbitrary but convenient. For ease of computation I might produce a lookup table for the first period, or may be able to write an asymptotic expansion or easy explicit formula for f(x) there (e.g. for the sawtooth function with period 1 and amplitude 10, very f(x) = 10x works in the first period, but the general formula involves floor functions).

The crux is: to compute f(x) for general x, it is often easier to transform x to t = x + nP (for some integer n) such that t lies in [0, P). Then f(t) is cheap/easy to calculate, and is equal to f(x). Explicit example: let f be the sawtooth function with period 1 and amplitude 10, then to calculate f(6.8) I note that f(6.8) = f(0.8) and since 0.8 lies in the first period, f(0.8) = 8. What is the name of such a transformation from x to t "modulo P"? Is it "periodizing"? This must be a common transformation in many applications.

More generally if f(x) is periodic with period P then it is equivalent to a function g(x) with period 1 that has been stretched by scale factor P in the x direction. If I want to work with a whole set of such functions with different periods, I can focus on working with the unit-periodic g(x) (e.g. obtain an explicit formula or lookup table on [0, 1)). Then for f(x) with arbitrary period P and arbitrary x, I can easily map x to t as before, then calculate u = t/P in [0, 1). I have f(x) = f(t) = g(u) which is easily computed. In some sense u represents "what fraction into a complete cycle does x lie?" For instance if f(x) is the sawtooth function with period 2 and amplitude 10 then to calculate f(9.5) I can observe f(9.5) = f(1.5 + 4*2) = f(1.5) = g(0.75) = 7.5. We can see that x = 9.5 is "three-quarters of the way through the cycle from x = 8 to x = 10" and that's why we evaluate g(0.75). Does this transformation of x to "x modulo P over P" have a name? More generally is there a name for transformations from the real line to [0, 1)? ManyQuestionsFewAnswers (talk) 14:20, 13 July 2012 (UTC)

Just adding to say these transformations can obviously be viewed in terms of mapping to an equivalence class. Perhaps there is a generic name for a mapping to a equivalence class? ManyQuestionsFewAnswers (talk) 23:13, 13 July 2012 (UTC)

Considered as a a mapping to an equivalence class, I believe quotient map is an acceptable term, although that redirects to the topological case. Straightontillmorning (talk) 11:34, 14 July 2012 (UTC)

Thanks, I suspected this might be the case. ManyQuestionsFewAnswers (talk) 13:49, 14 July 2012 (UTC)

Relevant article is periodic summation, also known as periodization. Sławomir Biały (talk) 19:57, 14 July 2012 (UTC)

Thanks. "Periodization" in the textbooks seems to refer to the mapping from aperiodic functions to periodic functions. ManyQuestionsFewAnswers (talk) 21:06, 14 July 2012 (UTC)

Any real number x=IP(x)+FP(x) is uniquely split into an integer part IP(x) and a fractional part FP(x), where IP(x) is an integer and 0≤FP(x)<1. If a function f is periodic with period P, then f(x)=f(P×FP(x/P)). Bo Jacoby (talk) 12:21, 14 July 2012 (UTC).

Yes, this is the correct formula, is there an English name for the mapping x -> P * FP(x/P)? ManyQuestionsFewAnswers (talk) 13:49, 14 July 2012 (UTC)

Phase. Bo Jacoby (talk) 19:02, 14 July 2012 (UTC).

Yes, it would have been clearer if I had said mentioned phase in my question. But is the mapping itself, or its result, called the phase? In a function with period 2, do we say that x = 9.5 has "a phase of 0.75"? I am more used to seeing the phase expressed as an angle, although obviously this is not the only way to do it. ManyQuestionsFewAnswers (talk) 21:06, 14 July 2012 (UTC)

There are several measuring units for angle. "a phase of 0.75 periods" is OK. So is "a phase of 270 degrees". The phase of the year may be measured in months. The phase of the month may be measured in days. The phase of the day may be measured in hours and minutes. Bo Jacoby (talk) 02:11, 15 July 2012 (UTC).

Brilliant, thanks, that's very clear. And matches the article "Phase in sinusoidal functions or in waves has two different, but closely related, meanings. One is the initial angle of a sinusoidal function at its origin and is sometimes called phase offset. Another usage is the fraction of the wave cycle which has elapsed relative to the origin" (I presume that second definition extends to non-sinusoidal functions). Could you just clarify something about the terminology. "Phase" can clearly be used to represent the output of of the mapping 9.5 -> 0.75 I defined above. Can the word also be used to describe the mapping itself? The mapping is definitely not the "phase function", which is to with reflected intensities. ManyQuestionsFewAnswers (talk) 02:38, 15 July 2012 (UTC)

A wave of length λ (meter per wave) and period P (second per wave) has at point x and time t the phase φ=φ₀+x/λ−t/P waves. The number 2πφ is the phase measured in radian. The wave function is 1^φ = e^2πiφ, using the convenient but nonstandard notation 1^φ. It depends only on the fractional part of φ because e^2πi=1. It is also convenient, but nonstandard, to introduce tau = τ = 2π naming the unit of wave length: m/τ, and the unit of period: s/τ, and the unit of phase: τ. With this convention your function could be called 'modulo tau'. Bo Jacoby (talk) 13:33, 15 July 2012 (UTC).

Many thanks, you've been very helpful! Really dumb follow up question based on your answer: I often slip into calling the function that describes a waveform, e.g. sawtooth or a sine offset by a phase angle, as a "wave function" (like you've just done!) and then "correct myself" because wave function seems to be most often used in the quantum mechanical sense! But in fact, is "wave function" a permissible term for waves in general? ManyQuestionsFewAnswers (talk) 01:01, 16 July 2012 (UTC)

Yes, "wave function" is a permissible term for waves in general. The wave function of a pressure wave is called the pressure, and the wave function of an electro-magnetic wave is called the electro-magnetic field, and the wave function of a violin string is called the displacement, but as the quantum mechanical wave function had no classical name it was simply called the wave function, I guess. Bo Jacoby (talk) 09:00, 16 July 2012 (UTC).

Fourier Analysis

How do most people learn Fourier analysis? I want to learn fourier analysis and I wanted to know what book I should buy. — Preceding unsigned comment added by Thepurplelefant (talk • contribs) 16:35, 13 July 2012 (UTC)

Do you have some more information on your background, goals, and current knowledge? What book to recommend depends a bit on that (and on what you want to learn Fourier analysis for). Do you know Lebesgue integration? Are you interested in numerical methods? Etc. —Kusma (t·c) 16:51, 13 July 2012 (UTC)

Begin by reading our article on discrete fourier transform. Bo Jacoby (talk) 12:29, 14 July 2012 (UTC).

Dr. Euler's Fabulous Formula by Paul J. Nahin is a good introduction to Fourier series and Fourier integrals (and a lot more besides). Or you could try this Stanford University lecture series on YouTube. For a more advanced treatment, there is a Schaum's Outline. Gandalf61 (talk) 13:51, 14 July 2012 (UTC)