Luboš Motl
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 May 19 answered How can there be multiple irreducible representations of a group each having distinct dimension? May 19 answered Sturm-Liouville Problem May 19 answered Additive group of rational numbers May 19 answered How to project the surface of a hypersphere into the full volume of a sphere? May 19 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. May 19 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. May 19 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? ;-) May 19 awarded Supporter May 19 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. May 19 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. May 19 comment How can I solve this non-linear differential equation? Dear @Ross, I don't think your comment is right. "Working in the reals" only means that you must filter the solutions at the very end to make sure that they're real if this is what you were asked about. However, all intermediate steps can and should use complex numbers, otherwise you're missing some solutions. This is, in fact, why complex numbers were first used. Some cubic equations have all 3 roots being real, but you still need complex numbers in intermediate steps (of the Cardan formula) to calculate these roots. This situation is completely analogous. May 19 comment How can I solve this non-linear differential equation? Yes, you may set $k$ to a negative number because $c$ can be complex. Any solution with any complex values of the integration constants is OK, and this particular choice even ends up with a real $y$ for real $t$. A related comment: the absolute value as a part of the logarithm's argument is counterproductive because it is not a holomorphic function. I think it's a good idea to avoid all such symbols that only work on the real axis - and create a wrong discontinuity when the argument of the absolute value vanishes. May 19 answered Trig equation help please May 19 comment How is the uniform boundedness principle compatible with this seemingly weak convergent sequence? Of course that I am allowed to use $x$ even if it is not in your arbitrarily chosen basis. The weak convergence is defined so that the inner products with all vectors in the Hilbert space converge to the right value, see en.wikipedia.org/wiki/Weak_convergence_(Hilbert_space) - it would be very counterproductive to make a definition of a vector's property (or the property of their sequence) that depends on the choice of basis. Important properties don't depend on the choice of bases. @Jonas, thanks for interpreting, you understood me well. May 19 revised Collecting coupons which arrive as a Poisson process added 398 characters in body; added 163 characters in body May 19 answered Collecting coupons which arrive as a Poisson process May 19 answered Is $\frac{m-1}{x}$ an unbiased estimator of $\theta$ for given pdf? May 19 comment How is the uniform boundedness principle compatible with this seemingly weak convergent sequence? If I understand you well: on the contrary: for the weak convergence it is enough to show the convergence of the inner product with any $\phi_i$ because that's how the weak convergence is defined. What I tried to argue is that you did not prove the convergence of the inner products to the "right" inner product and you could not because this convergence of inner products is not true. May 19 answered Complex Exponential Expansion May 19 revised How is the uniform boundedness principle compatible with this seemingly weak convergent sequence? added 7 characters in body; added 59 characters in body