Luboš Motl

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bio website motls.blogspot.com location Czech Republic age 40 member for 2 years, 11 months seen Mar 12 at 7:48 profile views 2,486

Hi, I am a string theorist and a publicist.

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 May21 revised Complex Exponential Expansion added 16 characters in body May20 answered What are the finite subgroups of $SU_2(C)$? May20 revised I can't seem to simplify this trigonometric differential into the required form added 68 characters in body May20 answered I can't seem to simplify this trigonometric differential into the required form May20 comment If $A\in M_{n}(\mathbb{C})$ is a Hermitian matrix so $I-iA$ is Invertible Dear Nir, Hermitian matrices have purely real eigenvalues, that's why they're used by the physicists to express all real observables such as position or angular momentum. Compute $b(v,Av)$ for any normalized eigenvector $v$ - it's real because of hermiticity which makes it equal to $b(Av,v)=b(v,Av)^*$. This inner product is the eigenvalue, too. So it must be real. May19 revised Additive group of rational numbers added 31 characters in body May19 awarded Mortarboard May19 comment Additive group of rational numbers You're right, it's a direct product. May19 answered Integrate in Mathematica takes forever May19 comment When does variété mean manifold? I just want to confirm Ryan, click at "English" in the left column of these French Wikipedia articles: fr.wikipedia.org/wiki/Vari%C3%A9t%C3%A9_alg%C3%A9brique and fr.wikipedia.org/wiki/Vari%C3%A9t%C3%A9_(g%C3%A9om%C3%A9trie) - add the end-parenthesis by hand to the URL - Variete without adjectives is mostly manifold, but with the algebraic adjective it is mostly variety. In Czech, the difference is too subtle for all but 5-10 people haha but "manifold" is translated as "varieta", too. May19 answered How can there be multiple irreducible representations of a group each having distinct dimension? May19 answered Sturm-Liouville Problem May19 answered Additive group of rational numbers May19 answered How to project the surface of a hypersphere into the full volume of a sphere? May19 comment Is $\frac{m-1}{x}$ an unbiased estimator of $\theta$ for given pdf? To prove that the Euler integral $\int_0^\infty dt\exp(-t)t^n$ is equal to $n!=\Gamma(n+1)$ for integer $n$, just integrate it by parts several times, so that the exponent above $t$ is always reduced by one. In this way, you will collect factors of $n, n-1$, and so on, and finally you collect the whole $n!$ and the remaining integral will be $\int_0^\infty dt \exp(-t)$ which is easily integrated to one. May19 comment Is $\frac{m-1}{x}$ an unbiased estimator of $\theta$ for given pdf? I didn't do it with Mathematica first, it's just an ordinary Euler integral! Cardinal didn't do any extra steps to calculate the Euler integral, either. He just wrote that it's an elementary integral just like I did. May19 comment Is $\frac{m-1}{x}$ an unbiased estimator of $\theta$ for given pdf? Apologies but how is it different, if it is, from a copy of my answer that was posted 1 hour earlier? ;-) May19 awarded Supporter May19 comment How can I solve this non-linear differential equation? Dear @Hannesh, it's not only legal but mandatory to allow all integration constants throughout the calculation being arbitrary complex numbers. Solving equations - algebraic or differential - in the reals isn't simpler than in complex numbers. Quite on the contrary, it's more complicated because you must solve it using all possible complex values of the parameters, and at the very end, you must do an extra job of filtering out the solutions that are not real. See the exchanges right under your question. May19 comment How can I solve this non-linear differential equation? Exactly, this is the right compact form of the solution. tanh is sinh/cosh so its derivative is $(\cosh^2 t - \sinh^2 t)/\cosh^2 t = 1/\cosh^2 t$ which is equal to $1-\tanh^2 t$, indeed.