Wednesday, February 6, 2013

cosmic conspiracy for rotation curves

what's the density distribution inside a dark matter halo that allows flat rotation curves? how does that correspond to the NFW-profile shape and what's the reason that the size of the galactic disk is roughly equal to the NFW scale radius?

bonus question: can you show using a power series that
\begin{equation}
\exp(A)\exp(B)=\exp(A+B)
\end{equation}
for two square, real-valued, commuting matrices $A$ and $B$, i.e. $\left[A,B\right]=0$. what changes if the two matrices are non-commuting?

1 comment:

  1. bonus questions: if the matrices are commuting, they can be treated like ordinary numbers - and one can substitute the exponential series into the Cauchy-product formula,
    \begin{equation}
    \exp(A)\exp(B)
    = \sum_i\frac{A^i}{i!}\times\sum_j\frac{B^j}{j!}
    = \sum_i\sum_j\frac{A^j}{j!}\frac{B^{i-j}}{(i-j)!}
    \end{equation}
    and isolate the exponential series by employing the binomial formula,
    \begin{equation}
    \ldots
    = \sum_i\frac{1}{i!}\sum_j{i\choose j}A^jB^{i-j}
    = \sum\frac{(A+B)^i}{i!}
    = \exp(A+B)
    \end{equation}
    in the case of non-commuting matrices, one arrives at the Baker-Hausdorff identities.

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