Wikipedia:Reference desk/Archives/Mathematics/2007 June 22

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June 22

Coupled differential equation

Just for kicks, I'd like to solve this coupled set of differential equations

${\frac {df}{dx}}=\alpha g(x)-xf(x)$

${\frac {dg}{dx}}=\alpha f(x)+xg(x)$

Is there a closed form power series for the answer? --HappyCamper 02:08, 22 June 2007 (UTC)

I think so, and it doesn't look too bad. If we let $f(x)=a_{0}+a_{1}x+a_{2}x^{2}+\ldots$ and $g(x)=b_{0}+b_{1}x+b_{2}x^{2}+\ldots$, then you can substitute that all into the DE to get recurrence relations for $a_{n}$ and $b_{n}$, which are coupled in two groups - the $a_{2m}$ and $b_{2m+1}$, and the $a_{2m+1}$ and $b_{2m}$. Working them out for the first few terms in each series suggests that you could probably find a closed formula, since each one comes in a form of either $p(\alpha )a_{0}$ or $p(\alpha )b_{0}$, where p is a polynomial of degree n (when looking at the nth coefficient). Confusing Manifestation 03:34, 22 June 2007 (UTC)

Mathematica produces a direct closed form solution:

${\begin{cases}f(x)=C_{2}\,e^{{\frac {x^{2}}{2}}-ax+\ln(e^{2ax}-e^{C_{1}})}\\g(x)=f(x){\frac {e^{2ax}+e^{C_{1}}}{e^{2ax}-e^{C_{1}}}}\end{cases}}$

which you could calculate the power series for if you want. --mglg_(talk) 21:43, 22 June 2007 (UTC)

Wow - even better! Thanks! --HappyCamper 23:19, 22 June 2007 (UTC)

Obviously, I could have re-written that answer more simply as:

${\begin{cases}f(x)=C_{2}\,e^{{\frac {x^{2}}{2}}-ax}(e^{2ax}-C_{3})\\g(x)=C_{2}\,e^{{\frac {x^{2}}{2}}-ax}(e^{2ax}+C_{3})\end{cases}}$

--mglg_(talk) 00:13, 23 June 2007 (UTC)

Both were very useful to me. Thanks again!! --HappyCamper 15:05, 23 June 2007 (UTC)

Oops, I was full of it (as usual?): my expressions above are not solutions to your problem but to the much simpler problem without the minus sign. With the correct sign, I fail to find anything very useful (Mathematica spits out integrals of complicated sums of hypergeometric functions). --mglg_(talk) 02:08, 24 June 2007 (UTC)

Why does the extra minus sign complicate things so much? Is it because the symmetry is sort of removed from the system? --HappyCamper 17:59, 24 June 2007 (UTC)

Do maximal ideals of the homogeneous coordinate ring correspond to the points on it?

Hello,

let $Y$ be a projective variety, in $\mathbb {P} ^{n}$ . Let$S$ be the ring of polynomials in the variables $x_{0},\ldots ,x_{n}$. Let $I(Y)$ be the homogeneous ideal, generated by the homogeneous polynomials vanishing on $Y$. Let $S(Y)=S/I(Y)$ be the homogeneous coordinate ring.

Now, how much is this "like the affine case"? I mean, does every (homogeneous?) maximal ideal of $S(Y)$ correspond to a specific point on the projective variety $Y$ ?

Note that I am cautious when I write homogeneous, since I'm not even sure whether or not$S(Y)$ is a graded ring.

Many thanks Evilbu 12:02, 22 June 2007 (UTC)

There is an analog of the Nullstellensatz for projective varieties [1]. Also, for a graded ring A and homogeneous ideal I, $A/I$ will always be graded. See Graded algebra. nadav (talk) 06:26, 24 June 2007 (UTC)

Thank you very much. That's a very clear explanation, but unfortunately it doesn't really goes into depth about the maximal ideals. Do you know something about it. My instructor tried to convince me that points correspond to maximal ideals, but I don't see it?Evilbu 10:38, 24 June 2007 (UTC)

Applications of Pure Math

I've heard several times that lots of things that were once thought the silly, idealistic, nigh-childish fancy of pure mathematicians have found actual uses. I fact, I've heard there's some famous quote along the lines of "All results in mathematics, no matter how abstract, are in danger of being applied." The only examples I can think of are certain parts of number theory in the form of RSA (and related cryptographic systems), and nonEuclidean geometry in the form of relativity. Does anyone know of others? Black Carrot 17:16, 22 June 2007 (UTC)

Maybe non-Archimedean metrics have a place in quantum physics? (Google p-adic analysis physics or see p-adic and adelic physics) iames 19:34, 22 June 2007 (UTC)

Our article on group theory lists, next to cryptography, applications in the sciences. There is also the use of category theory in physics; see for example http://math.ucr.edu/home/baez/categories.html and http://math.ucr.edu/home/baez/quantum/. The concept of Von Neumann algebras can be considered to belong to abstract algebra, but was of course motivated by the possible application to quantum mechanics. See also Wikipedia:Reference desk/Archives/Mathematics/2007 June 18#Useless mathematics for an almost perfectly complementary question.  --Lambiam^Talk 06:22, 23 June 2007 (UTC)

If uses of category theory in theoretical physics (also computer science) are considered applications, then perhaps uses of foundations of mathematics topics in philosophy also qualify. Boolean logic was famously applied as a model for digital electronics. nadav (talk) 06:44, 23 June 2007 (UTC)

Thanks, that's a good start. Lot of good suggestions. Yeah, I remember that discussion. It looks like the only example he got was hyper-inaccessible cardinals, which someone else said actually do have applications. Real shame. Let me know if you think of any more. Black Carrot 04:23, 24 June 2007 (UTC)