maths bonus question: where's the global maximum of $\sqrt[x]{x}$, $x>0$?

physics bonus question: in quantum mechnics, can a perfect sphere rotate?

imagine there hadn't been an inflationary epoch - how long would we need to wait until hydrogen atoms were formed and the cosmic microwave background was generated?

maths bonus question: where's the global maximum of $\sqrt[x]{x}$, $x>0$?

maths bonus question: where's the global maximum of $\sqrt[x]{x}$, $x>0$?

physics bonus question: in quantum mechnics, can a perfect sphere rotate?

does time pass slower in distant galaxies? if you could observe a planetary system at high redshift and observe the orbits of planets (or any other dynamical system driven by gravity), would the observation be consistent with a lower gravity instead of a slower passage of time? what about atomic processes like chemical reactions or radiative processes?

bonus question: can you show that for a real-valued matrix $A$ the matrix exponential $\exp(A)$ transforms according to

\begin{equation}

\exp(R^{-1}AR) = R^{-1}\exp(A)R

\end{equation}

with any invertible matrix $R$?

bonus question: can you show that for a real-valued matrix $A$ the matrix exponential $\exp(A)$ transforms according to

\begin{equation}

\exp(R^{-1}AR) = R^{-1}\exp(A)R

\end{equation}

with any invertible matrix $R$?

can you find an intuitive argument why the variance of density fluctuations seeded by inflation is proportional to the wave number? what would be the corresponding relation if we lived in a universe with 4 spatial dimensions instead of 3?

bonus question: can you show that if a square real-valued matrix $A$ is skew-symmetric, $A^t=-A$, then the matrix exponential $\exp(A)$ of that matrix is orthogonal?

bonus question: can you show that if a square real-valued matrix $A$ is skew-symmetric, $A^t=-A$, then the matrix exponential $\exp(A)$ of that matrix is orthogonal?

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?

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?

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