Wikipedia:Reference desk/Archives/Mathematics/2014 August 5

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August 5

Associated Spherical Functions ?

My reference book is N. Vilenkin, Special Functions and the Theory of Group representations. American Mathematical Society., 1968, reprinted in 1988. On page 553 there is sub-paragraph:

3. Associated Spherical Functions. Let us consider the associated spherical functions $t_{MO}^{R}$(g) of the representations $T^{R}$(g). Previously (on page 549) the author defines $T^{R}$(g) as a irreducible representations of Class 1 of the group M(n) where M(n) is the group of motions in n-dimensional Euclidean space.

What are the Associated Spherical Functions? I could find them neither in Wikipedia nor in Wolfram. Thanks you. --AboutFace 22 (talk) 00:14, 5 August 2014 (UTC)

I don't have the book, but in the special case you are describing the spherical functions are ordinary spherical harmonics. (For more general groups, there are other associated spherical functions that involve zonal spherical functions, but that level of generality may not be important here.) Sławomir Biały (talk) 01:19, 5 August 2014 (UTC)

Thank you for the answer, however I doubt that what you said is correct. This book has a chapter on rotations in 2-sphere (group SO(3)) and over there the spherical harmonics appear naturally. Has it been the case that Associated Spherical harmonics appeared to be identical with ordinary Spherical harmonics it would have been indicated in the text. This is clearly not the case. The author presents a number of steps in formulas conversion and eventually the functions in question appear in this form:

$t_{MO}^{iR}(g_{r})$ = $i^{m}$Γ(${n \over 2}$)${\sqrt {\Gamma (n+m-2)(2m+n-2) \over {m!\Gamma (n-1)}}}$ ${\frac {\mathbf {J_{m+{\frac {n-2}{2}}^{(Rr)}}} }{({\frac {Rr}{2}})^{\frac {n-2}{2}}}}$

"where we recall, $g_{r}$ is a (parallel) translation by r along the axis O$x_{n}$" The last phrase is a direct quote from the book. $\mathbf {J}$ in the formula is a Bessel function, and Γ is Gamma function.

This chapter in the book appears skimpy and is difficult for me to understand in details but I need it. I am wondering if you can point to me other books where translational transformations in E(n) appear in Lie Group context: M(n). I need representations of translations in 3-D space primarily. Thanks, --AboutFace 22 (talk) 17:09, 5 August 2014 (UTC)

I don't clearly understand what's going on there, but I think they are making something rather simple more complicated than it needs to be. The standard Fourier decomposition of $L^{2}(\mathbb {R} ^{n})$ is into spaces H_k consisting of products of radial functions by the (solid) spherical harmonics of degree k. The Fourier representation of radial functions involves Bessel functions, so these appear naturally as coefficients. A good source on Fourier analysis on Euclidean space is the book by that title written by Elias Stein and Guido Weiss, and includes the full details of this decomposition. Hopefully that helps. Sławomir Biały (talk) 18:52, 5 August 2014 (UTC)

Thank you for the reference. Do they talk about translations? This is the most important aspect for me now. --AboutFace 22 (talk) 21:33, 5 August 2014 (UTC)

No, that reference does not discuss translations in this context. However, I believe there is a canonical way to lift these functions up to the group. For the group of rotations, the zonal spherical harmonics depend on a pair of points x and y of the sphere. You can associate to these two points a rotation that moves the one point into the other, and define the zonal harmonic of that rotation. The "zonalness" of the harmonic means that the value does not depend on which rotation one uses, and so this does give a well-defined function on the group. The same kind of thing is going to be true for these "associated spherical functions", although I'm not quite clear on what the precise setup is. Sławomir Biały (talk) 12:19, 6 August 2014 (UTC)

I want to thank you again for your contribution/posts. They are very important to me because the subject is very difficult. I also want to say that my interest is not spurious. I have a specific task on hand and am trying to find the most suitable set of operators to solve it. Reading through your notes and some of what I could find about the reference you posted I am coming to the conclusion that instead of motions in M(3) I should focus on dilations on 2-sphere. Until yesteday I had no idea that dilations could be defined in Hilbert space. Perhaps later today I will post a few questions about how I can use dilations in my work.

I am very comfortable with rotations though. I have no dark or unresolved aspects to them. It is a Lie group well represented in Hilbert space. My case is simple, it is not a quantum mechanical situation. I don't have to consider spinors or whatnot.

Thank you much. --AboutFace 22 (talk) 13:23, 6 August 2014 (UTC)

Ring of square matrices --- subrings with interesting center

Part of my problem is that I can't pinpoint what I'm exactly looking for. The ring of square matrices (of order n) has the trivial ring of scalar matrices as center. The ring of diagonal matrices is commutative, but the maximal subring (I suspect, no proof) for which it is the center is the ring itself. Are there interesting subrings that have interesting center? With interesting I like to denote rings that are not trivial or nearly trivial, unlike the scalar or diagonal matrices are.

95.112.234.56 (talk) 16:33, 5 August 2014 (UTC)

One that does not fit into any of the categories you list is the matrix representation of complex numbers, although still fairly "trivial". Here the entire subring is its own centre. —Quondum 17:02, 5 August 2014 (UTC)

That was exactly what got me start thinking. 95.112.234.56 (talk) 17:05, 5 August 2014 (UTC)

Did you think about quaternions too? RomanSpa (talk) 17:07, 5 August 2014 (UTC)

Yes, but I stopped as they are not commutative. What got me wondering was such: the derivative of a mapping of real numbers to real numbers (if it exists) is again a mapping from real numbers to real numbers.

Not so for $R^{n}\to R^{n}$. But then, with some restrictions, $R^{2}\to R^{2}$ is essentially $C\to C$ where the derivative is in the same class. Are there restrictions on $R^{n}\to R^{n}$, too, such that the derivative is in the same class? 95.112.234.56 (talk) 17:47, 5 August 2014 (UTC)

Derivatives of Rⁿ → R^m do exist (see Generalizations of the derivative and Fréchet derivative). And the derivative of a mapping C → C in complex analysis only exists on a very constrained set of functions on R² – essentially it must satisfy the Cauchy–Riemann equations. There is the directional derivative that maps (Rⁿ → Rⁿ) → (Rⁿ → Rⁿ) and is thus to the same class, though this is a derivative that needs a parameter (a vector field). The derivative in general is often of a different class from the mapping. Why would you want to make this restriction? And this seems quite removed from the question of the centre of a matrix subring. —Quondum 18:48, 5 August 2014 (UTC)

YES, I am aware they do exist, but in different classes. And I did not want to overload the question, perhaps this was a mistake.

I want to see something like a Taylor expansion, which depends on derivatives within the same class, or else (much worse) an inflation of dimensions. Hope you get my intention. 95.112.234.56 (talk) 19:37, 5 August 2014 (UTC)

The example of exp(M), defined in terms a series for an arbitrary square matrix M (and indeed any associative R-algebra), gives some ideas. One might restrict the functions under consideration to those that can be expressed as a power series, in which the coefficient must be from the centre of the algebra, and this may be where you're coming from. This cannot be done with arbitrary functions – with a noncommutative associative algebra, a simple power series (with whatever left-coefficient you use for each term) does not cover the space of continuous functions within the algebra. To take a simple example, the space of homogeneous polynomials of degree 1 over one variable in the quaternions H needs four separate coefficients drawn from H. —Quondum 22:47, 5 August 2014 (UTC)

A couple of comments. (1) The Wedderburn-Artin theorem should be useful. A semisimple algebra over the reals is a direct sum of complete matrix algebras over the reals, complex numbers, and quaternions. Each of these has a center that is easy to describe. (2) For an algebra that is not semisimple, there are non-trivial invariant subspaces by Nakayama's lemma. It seems likely that an element of the center is determined by its restriction to the minimal invariant subspaces. (3) The model case that you should probably think of in regards to point (2) are the parabolic algebras (block upper triangular matrices). The minimal invariant subspaces correspond to the first diagonal block in that case. Sławomir Biały (talk) 20:21, 6 August 2014 (UTC)

Is there a name for this computation?

Hello. I would like to ask if there is a more mathematical concise name for the following:

When you have a set of real numbers, the "Average of the distance of each point from the mean of the set".

In mathematical terms: ${\frac {1}{n}}(\sum _{i=0}^{n}x_{i}-{\overline {X}})$

I realize this isn't quite the standard deviation for populations, as there are terms that are squared in that. Does THIS specific computation have a name besides the really long explanation I've placed in quotes?

Thanks for your help!

216.173.144.188 (talk) 21:35, 5 August 2014 (UTC)

Your formula actually gives a value of zero for any (finite) set of real numbers, more or less from the definition of the mean. If instead you take average of the absolute values of the distances between the $x_{i}$ and the mean then you get the mean absolute deviation. AndrewWTaylor (talk) 21:53, 5 August 2014 (UTC)

Thanks. This makes sense now that i think about it! 216.173.144.188 (talk) 22:10, 5 August 2014 (UTC)

And for a reasonably normal distribution, the MAD is about 80% of the SD.→86.146.61.61 (talk) 12:15, 6 August 2014 (UTC)