# Luboš Motl

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bio website motls.blogspot.com location Czech Republic age 40 member for 2 years, 10 months seen Mar 12 at 7:48 profile views 2,421

Hi, I am a string theorist and a publicist.

# 259 Actions

 May19 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. May19 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. May19 answered Trig equation help please May19 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. May19 revised Collecting coupons which arrive as a Poisson process added 398 characters in body; added 163 characters in body May19 answered Collecting coupons which arrive as a Poisson process May19 answered Is $\frac{m-1}{x}$ an unbiased estimator of $\theta$ for given pdf? May19 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. May19 answered Complex Exponential Expansion May19 revised How is the uniform boundedness principle compatible with this seemingly weak convergent sequence? added 7 characters in body; added 59 characters in body May19 answered How is the uniform boundedness principle compatible with this seemingly weak convergent sequence? May19 answered How to prove the inequality $\Theta(x,y)\le \Theta(x,z)+\Theta(z,y)$? May18 answered Complex Exponents May18 answered Deduce plus and minus with Cross Product in 3th and 4th Maxwell equations May18 awarded Commentator May18 comment Global conformally flat coordinates in 2d spacetimes Dear @Dionigi, the Lorentzian-signature "disks" defined as any contractible manifolds can't be mapped to each other. Imagine that the shape is described by an equation involving $x^+,x^-$, the light-like coordinates. The conformal transformations in this case are separate reparametrizations of $x^+$ and of $x^-$. This is clearly not enough to relate all contractible shapes. For example, a null boundary of a "diamond" will always stay null under conformal transformations, and manifolds with piecewise spacelike and then timelike boundaries etc. will always have these pieces. May18 awarded Nice Answer May18 awarded Critic May18 comment Branch cut of the logarithm Thanks for your comment, but when you mention an "answer", what is the question you're trying to answer? May18 comment Global conformally flat coordinates in 2d spacetimes Apologies, I don't understand in what sense is $\Sigma \times {\mathbb R}$ a disk: isn't it a noncompact manifold?