Wikipedia:Reference desk/Archives/Mathematics/2011 June 13

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June 13

Ogden's lemma examples

Our article on Ogden's lemma (and many other pages I've found on the web) use the example {aⁱb^jc^kd^l : i = 0 or j = k = l} but this language can be proven non-context-free using the pumping lemma and closure of CFLs under intersection by regular languages (take ab^*c^*d^*). Are there examples of languages which can be proven to be non-CFLs with Ogden's lemma, but not with the pumping lemma and closure of CFLs under regular intersection and gsm mappings? --146.141.1.96 (talk) 10:51, 13 June 2011 (UTC)

nilpotent cone of a Lie algebra

The article nilpotent cone claims that "The nilpotent cone...is invariant under the adjoint action of ${\mathfrak {g}}$ on itself." Take the example of ${\mathfrak {g}}={\mathfrak {sl}}_{2}=\langle E,F,H\rangle$ as in the article, so the nilcone N is spanned by E and F. Then $\operatorname {ad} (E)(F)=H$ which is not in the nilcone, so N is not ad-invariant. Have I misunderstood something, or is the article wrong? Tinfoilcat (talk) 11:15, 13 June 2011 (UTC)

If g is a Lie algebra and x is an element of g, then the adjoint action ad_x : g → g is defined by ad_x(y) = [x,y] for all y in g. To prove that the nilpotent cone, say N, is invariant under the adjoint action you just need to show that for all x and y in N, the Lie bracket [x,y] is again in N. — Fly by Night (talk) 14:47, 13 June 2011 (UTC)

FbN, I know what the adjoint map is and what it means to be ad-invariant (it's not quite what you say - ad-invariant means a Lie ideal, not a Lie subalgebra). The example I gave in my post seems to show that the nilcone is not ad-invariant since $[E,F]=H$ which is not in the nilcone, whereas E and F are. Tinfoilcat (talk) 14:54, 13 June 2011 (UTC)

I don't recall mentioning subalgebras or ideas. I'm sorry I wasn't able to help. — Fly by Night (talk) 15:50, 13 June 2011 (UTC)

FbN, I was very bitey above. Sorry. When you say "for all x and y in N, the Lie bracket [x,y] is again in N " that's equivalent to it being a subalgebra. Being an ideal (= being ad-invariant) is that for n in N, x in the Lie algebra, $[x,n]\in N$ - a little stronger. Tinfoilcat (talk) 15:59, 13 June 2011 (UTC)

No problem. Proving it's a subalgebra means that it's automatically an ideal too, by definition (because of the way the nilpotent cone is defined in the first place). Or at least I think so… — Fly by Night (talk) 16:06, 13 June 2011 (UTC)

The statement is wrong. The nilpotent cone is invariant under the action on Int(g) on g (interior auromorphisms). Sławomir Biały (talk) 15:53, 13 June 2011 (UTC)

Thanks. I'll fix the page (if you haven't got there first). Tinfoilcat (talk) 15:59, 13 June 2011 (UTC)

Normalizing Associated Legendre Polynomials to get Spherical Harmonics

Hi all. I've been working on this problem for like weeks, and I can't seem to figure it out. I'm trying to normalize Associated Legendre polynomials to turn them into Spherical harmonics. The integral comes out to:

$\int _{-1}^{1}P_{\ell _{1}}^{m}(x)P_{\ell _{2}}^{m}(x)dx={\frac {1}{A^{2}}}$

where $A$ is the normalization constant. $A$ can be found in Spherical harmonics#Orthogonality and normalization. I know that it involves integrating by parts $\ell +m$ times, and that the boundary terms vanish in each case, but I'm not sure why they vanish. Can anyone point me to a (very detailed) discussion of how to actually do the integral, or maybe a better way than by parts? Thanks!--Dudemanfellabra (talk) 23:32, 13 June 2011 (UTC)

Hi, I had a quick look in Boas - 'Mathematical Methods in the Physical Sciences'. She has a problem (prob 10.10 in second edition) about the normalisation, first some results:

a) $P_{l}^{m}(x)={\frac {1}{2^{l}l!}}(1-x^{2})^{m/2}{\frac {d^{l+m}}{dx^{l+m}}}(x^{2}-1)^{l}$

This comes from substituting in Rodriques' formula (see Legendre_polynomials) for the Legendre polynomials. Next Boas says derive

b) $P_{l}^{m}(x)=(-1)^{m}{\frac {(l+m)!}{(l-m)!}}{\frac {(1-x^{2})^{-m/2}}{2^{l}l!}}{\frac {d^{l-m}}{dx^{l-m}}}(x^{2}-1)^{l}$

Multiply these two results together (this forms your normalisation integrand) then integrate by parts repeatedly until both l+m and l-m derivatives are just l derivatives. Then use

$\int _{-1}^{1}[P_{l}(x)]^{2}dx={\frac {2}{2l+1}}$ - I assume because you end up having an integrand that looks like two Legendre polynomials multiplied together.

Now, the derivation of result (b) is a task in itself. Apparently one can show that

${\frac {d^{l-m}}{dx^{l-m}}}(x^{2}-1)^{l}={\frac {(l-m)!}{(l+m)!}}(x^{2}-1)^{m}{\frac {d^{l+m}}{dx^{l+m}}}(x^{2}-1)^{l}$

by starting with $(x^{2}-1)^{l}=(x-1)^{l}(x+1)^{l}$ and finding derivatives using Leibniz' rule. Good luck, I'd find a copy of Boas and work from that if I was you. Christopherlumb (talk) 19:55, 14 June 2011 (UTC)

I found a copy of Boas, but unfortunately the method for part b) is not given. After applying Leibniz' rule, I get

${\frac {d^{l-m}}{dx^{l-m}}}(x^{2}-1)^{l}=\sum _{i=0}^{l-m}{\binom {l-m}{i}}{\frac {d^{l-m-i}}{dx^{l-m-i}}}(x+1)^{l}{\frac {d^{i}}{dx^{i}}}(x-1)^{l}$

I have no idea where to go from here...--Dudemanfellabra (talk) 23:13, 14 June 2011 (UTC)

I'm really struggling myself, I think you have to get expressions for the $l+m$th and $l-m$th derivatives and compare them term-by-term, so

${\frac {d^{l+m}}{dx^{l+m}}}(x^{2}-1)^{l}=\sum _{i=0}^{l+m}{\binom {l+m}{i}}{\frac {d^{l+m-i}}{dx^{l+m-i}}}(x+1)^{l}{\frac {d^{i}}{dx^{i}}}(x-1)^{l}$

we note that the derivatives on the right-hand side are just acting on $(x\pm 1)^{l}$ so that any derivative higher than $l$ differentiates to zero. This means the sum gets truncated: we must have $i\leq l$ (else the d/dx^i goes to zero) and $i\geq m$ (else the d/dx^(l+m-i) will go to zero). So:

${\frac {d^{l+m}}{dx^{l+m}}}(x^{2}-1)^{l}=\sum _{i=m}^{l}{\binom {l+m}{i}}{\frac {d^{l+m-i}}{dx^{l+m-i}}}(x+1)^{l}{\frac {d^{i}}{dx^{i}}}(x-1)^{l}$

now we have no-more terms in this sum than for the $l-m$ sum, next evaluate the derivatives:

${\frac {d^{l+m}}{dx^{l+m}}}(x^{2}-1)^{l}=\sum _{i=m}^{l}{\binom {l+m}{i}}l(l-1)\ldots (1-l-m+i){\frac {(x+1)^{l}}{(x+1)^{l+m-i}}}$

$\times l(l-1)\ldots (1-i){\frac {(x-1)^{l}}{(x-1)^{i}}}$

I've written the derivatives as fractions where you'd normally just write $(x^{2}-1)^{l-i}$. Next job is to make this look like the l-m derivative:

${\frac {d^{l-m}}{dx^{l-m}}}(x^{2}-1)^{l}=\sum _{j=0}^{l-m}{\binom {l-m}{j}}{\frac {d^{l-m-j}}{dx^{l-m-j}}}(x+1)^{l}{\frac {d^{j}}{dx^{j}}}(x-1)^{l}$

${\frac {d^{l-m}}{dx^{l-m}}}(x^{2}-1)^{l}=\sum _{j=0}^{l-m}{\binom {l-m}{j}}l(l-1)\ldots (1-l+m+j){\frac {(x+1)^{l}}{(x+1)^{l-m-j}}}$

$\times l(l-1)\ldots (1-j){\frac {(x-1)^{l}}{(x-1)^{j}}}$

Let $j=k-m$ then our sum runs from $k=m$ to $k=l$:

${\frac {d^{l-m}}{dx^{l-m}}}(x^{2}-1)^{l}=\sum _{k=m}^{l}{\binom {l-m}{k-m}}l(l-1)\ldots (1-l+k){\frac {(x+1)^{l}}{(x+1)^{l-k}}}$

$\times l(l-1)\ldots (1-k+m){\frac {(x-1)^{l}}{(x-1)^{k-m}}}$

Pull a factor of (x^2-1)^m out the front:

${\frac {d^{l-m}}{dx^{l-m}}}(x^{2}-1)^{l}=(x^{2}-1)^{m}\sum _{k=m}^{l}{\binom {l-m}{k-m}}l(l-1)\ldots (1-l+k){\frac {(x+1)^{l}}{(x+1)^{l+m-k}}}$

$\times l(l-1)\ldots (1-k+m){\frac {(x-1)^{l}}{(x-1)^{k}}}$

It's looking very similar to the l+m version above, I haven't the energy to go through to check the factorials all work out but I think this is on the right track 213.249.173.33 (talk) 21:09, 15 June 2011 (UTC)

AH! THANK YOU THANK YOU THANK YOU THANK YOU! Haha I had figured out what the (l+m-i)th and (l-m-i)th derivatives were (btw, the factorials in your above derivation are a little off. I'll show the correct one below.), but I didn't think about shifting the index of one of the sums so that they can be compared properly. By shifting that index and using falling factorials for the derivatives, I was able to show the correct relationship. I'll show it below and continue to work on the rest of the problem. Thanks for all your help!

Derivation showing alternative way to write Plm(x)

${\frac {d^{l+m}}{dx^{l+m}}}(x^{2}-1)^{l}=\sum _{i=0}^{l+m}{\binom {l+m}{i}}{\frac {d^{l+m-i}}{dx^{l+m-i}}}(x+1)^{l}{\frac {d^{i}}{dx^{i}}}(x-1)^{l}$

$(x\pm 1)^{l}$ has only l non-vanishing derivatives, so $i\geq m,i\leq l$. Therefore, the sum can be written from i=m to l. Using falling factorial notation for the derivatives and expanding the binomial coefficients in factorial form yields

${\frac {d^{l+m}}{dx^{l+m}}}(x^{2}-1)^{l}=\sum _{i=m}^{l}{\frac {(l+m)!}{i!(l+m-i)!}}{\frac {l!}{(i-m)!}}(x+1)^{i-m}{\frac {l!}{(l-i)!}}(x-1)^{l-i}$

Doing the Leibniz formula for the l-m derivative yields

${\frac {d^{l-m}}{dx^{l-m}}}(x^{2}-1)^{l}=\sum _{j=0}^{l-m}{\binom {l-m}{j}}{\frac {d^{l-m-j}}{dx^{l-m-j}}}(x+1)^{l}{\frac {d^{j}}{dx^{j}}}(x-1)^{l}$

Again, expanding the coefficients and using falling factorials,

${\frac {d^{l-m}}{dx^{l-m}}}(x^{2}-1)^{l}=\sum _{j=0}^{l-m}{\frac {(l-m)!}{j!(l-m-j)!}}{\frac {l!}{(m+j)!}}(x+1)^{m+j}{\frac {l!}{(l-j)!}}(x-1)^{l-j}$

Let $k=m+j\Rightarrow k\in [m,l]$ Plugging in:

${\frac {d^{l-m}}{dx^{l-m}}}(x^{2}-1)^{l}=\sum _{k=m}^{l}{\frac {(l-m)!}{(k-m)!(l-k)!}}{\frac {l!}{k!}}(x+1)^{k}{\frac {l!}{(l-k+m)!}}(x-1)^{l-k+m}$

Now that the two sums have the same indices (i and k are just dummy variables, so we may as well say they're the same thing; let's choose i), we can divide the two sums to find the ratio of the l+m and l-m derivatives. That is,

${\frac {{\frac {d^{l+m}}{dx^{l+m}}}(x^{2}-1)^{l}}{{\frac {d^{l-m}}{dx^{l-m}}}(x^{2}-1)^{l}}}=\sum _{i=m}^{l}{\frac {(l+m)!}{i!(l+m-i)!}}{\frac {l!}{(i-m)!}}{\frac {l!}{(l-i)!}}{\frac {(i-m)!(l-i)!}{(l-m)!}}{\frac {i!}{l!}}{\frac {(l-i+m)!}{l!}}(x+1)^{-m}(x-1)^{-m}$

which, after cancelling and observing that all i's disappear so the sum can be dropped, yields:

${\frac {{\frac {d^{l+m}}{dx^{l+m}}}(x^{2}-1)^{l}}{{\frac {d^{l-m}}{dx^{l-m}}}(x^{2}-1)^{l}}}={\frac {(l+m)!}{(l-m)!}}(x^{2}-1)^{-m}$

This gives the relationship desired in part b) of Christopherlumb's comment above (with the exception of the $(-1)^{m}$ factor... I'm hoping that doesn't matter though haha). I'll now attempt to work through the integral using this identity. Again, thanks for all the help!--Dudemanfellabra (talk) 23:34, 15 June 2011 (UTC)